Characterization of particles and particle systems

W. Pabst / E. Gegoová Chaacteization of paticles and paticle systems ICT Pague 007 Tyto studijní mateiály vznikly v ámci pojektu FRVŠ 674 / 007 F1 / b Tvoba předmětu Chaakteizace částic a částicových soustav.

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 CPPS 1. Intoduction Paticle size + equivalent diametes 1.0 Intoduction Paticle size is one of the most impotant paametes in mateials science and technology as well as many othe banches of science and technology, fom medicine, phamacology and biology to ecology, enegy technology and the geosciences. In this intoduction we give an oveview on the content of this lectue couse and define the most impotant measues of size (equivalent diametes). 1.1 A bief guide though the contents of this couse This couse concens the chaacteization of individual paticles (size, shape and suface) as well as many-paticle systems. The theoetical backbone is the statistics of small paticles. Except fo sieve classification (which has lost its significance fo paticle size analysis today, although it emains an impotant tool fo classification) the most impotant paticle size analysis methods ae teated in some detail, in paticula sedimentation methods, lase diffaction, micoscopic image analysis, as well as othe methods (dynamic light scatteing, electozone sensing, optical paticle counting, XRD line pofile analysis, adsoption techniques and mecuy intusion). Concening image analysis, the eade is efeed also to ou lectue couse Micostuctue and popeties of poous mateials at the ICT Pague, whee complementay infomation is given, which goes beyond the scope of the pesent lectue. The two final units concen timely pactical applications (aeosols and nanopaticles, suspension heology and nanofluids). Apat fom specific appendices to individual couse units, thee ae thee majo inte-unit appendices, which ae based on the knowledge of seveal couse units and concen in paticula isometic paticles (size chaacteization by lase diffaction and image analysis), oblate paticles (size and shape chaacteization, sedimentation and lase diffaction), polate paticles (size and shape chaacteization, image analysis + lase diffaction), as well as suspension heology. 1. Equivalent diametes Paticle size, in the sense commonly used, is a linea length measue, measued in SI unit [m]. In this sense it can be uniquely defined only fo sphees, whee it is the diamete (o adius). Fo all othe shapes, paticle size must be clealy defined via the measuing pocedue. Socalled deived diametes ae detemined by measuing a size-dependent popety of the paticle and elating it to a single linea dimension. The most widely used of these ae the equivalent diametes, in paticula the equivalent spheical diametes. 1

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 Impotant equivalent diametes ae: Volume-equivalent sphee diamete D volume = diamete of a sphee with the same volume as the paticle V paticle, i.e. 6 D volume V π = paticle e.g. fo a cube with edge length 1 µm (volume 1 µm ) we have D = 1. 4 µm. Suface-equivalent sphee diamete D suface = diamete of a sphee with the same suface as the paticle S paticle, i.e. 1 volume 6 D suface S π = paticle e.g. fo a cube with edge length 1 µm (suface 6 µm ) we have D = 1. 8 µm. Stokes diamete D S (= equivalent diamete coesponding to the diamete of a sphee with the same final settling velocity as the paticle undegoing lamina flow in a fluid of the same density and viscosity), defined via the Stokes elation 1 suface D S 18η v =, ( ρ ρ ) g S L whee η is the viscosity (of the pue liquid medium without paticles), ρ S the density of the solid paticles, ρ L the density of the pue liquid, g the gavitational acceleation and v the final settling velocity. Hydodynamic equivalent diamete D H (= diamete of a sphee with the same tanslational diffusion coefficient D tanslation as the paticle in the same fluid unde the same conditions), defined via the Stokes-Einstein elation D H = kt π η D, tanslation whee k is the Boltzmann constant, T the absolute tempeatue and η the viscosity of the liquid medium (the diffusion coefficient must be extapolated to zeo concentation). Sieve diamete sieve D (= equivalent diamete coesponding to the diamete of a sphee passing though a sieve of defined mesh size with squae o cicula apetues).

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 Lase diffaction equivalent diamete D L (= diamete of a sphee yielding on the same detecto geomety the same electonic esponse fom the optical signal, i.e. the diffaction patten, as the paticle); when the Faunhofe appoximation is valid, D L should coespond to the pojected aea diamete of the paticle in andom oientation. Pojected aea diamete D P (= equivalent diamete coesponding to the diamete of a sphee o cicle with the same pojected aea as the paticle); in geneal, D P, is oientation-dependent, paticulaly fo anisometic paticles; the equivalent aea diamete measued via micoscopic image analysis, D M, usually efes to pefeential (non-andom) stable oientation and thus is not the same as D P fo andom oientation; anothe equivalent aea diamete, conceptually analogous to the pojected aea diamete, is the andom section aea diamete, which can be measued fom andom cuts (plana sections, polished sections) via image analysis see CPPS-10. Volume-suface diamete D SV (Saute diamete) = atio of the cube of the volumeequivalent diamete to the squae of the suface-equivalent diamete, i.e. D D SV =. D V S This diamete is invesely popotional to the suface density (suface aea pe unit volume) S V, o the specific suface aea (suface aea pe unit mass), i.e. S M = S V ρ, whee ρ is the density. The elation between D SV and S V is (with values fo k SV given in Table 1.1) k SV S V =. DSV Table 1.1. Shape factos k SV fo sphees and the Platonic solids (egula polyheda). Shape k SV Sphee 6 Tetahedon 8.94 Octahedon 8.06 Cube 7.44 Dodecahedon (pentagonal faces) 6.59 Icosahedon (tiangula faces) 6.9 Othe equivalent diametes ae thinkable, but less fequently used, e.g. the peimeteequivalent diamete of a paticle outline etc. Apat fom the equivalent diametes thee ae othe size measues which can be used to quantify paticle size, mainly in micoscopic image analysis of D paticle outlines, among them the chod o intecept lengths (including the Matin diamete, i.e. the length of the chod dividing the pojected paticle aea into two equal halves) and the calipe o Feet diametes (including the maximum and minimum Feet diamete) see CPPS-9.

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems CPPS. Paticle shape and suface.0 Intoduction Paticle shape is a complex geometic chaacteistic. It involves the fom and habit of the paticle as well as featues like convexity and suface oughness. The liteatue on shape chaacteization is enomous and so is the numbe of possible definitions of shape factos. Hee we give only the minimum set of definitions which ae absolutely necessay fo undestanding (the liteatue on) paticle shape chaacteization. Since the distinction of shape and suface topology is moe o less a question of scale, we intoduce factal concepts as well..1 Shape chaacteization and measues of shape Paticle shape has at least two diffeent meanings: Shape (fom) in the sense of deviations fom spheical shape (e.g. egula polyheda), Shape (habit) in the sense of deviations fom isometic shape (e.g. spheoids). Apat fom these two meanings shape can denote the deviation fom oundness (ounded vesus angula) and deviations fom convexity (convex vesus concave shape). We define an isometic shape as a shape of an object (paticle) fo which, oughly speaking, the extension (paticle size) is appoximately the same in any diection. Moe pecisely, fo a paticle to be isometic, the atio of the maximum and minimum length of chods intesecting the cente of gavity of the convex hull of the paticle should not exceed the atio of the least isometic egula polyhedon, i.e. the tetahedon (simplex in D). Fo many pactical puposes, isometic paticles can appoximately be consideed (modeled) as spheical paticles. A size measue (e.g. an equivalent diamete) is often sufficient fo a desciption of isometic paticles. Note that the tem (an-) isometic efes to extenal shape of objects (paticles), while the tem (an-) isotopic efes to the intenal stuctue of media (mateials). Anisometic paticles have significantly diffeent extensions in diffeent diections. When the paticles (o thei convex hulls) ae centally symmetic (at least appoximately o in a statistical sense), i.e. possess a cente of symmety, they can be modeled as ellipsoids o ectangula paallelepipeds. In the geneal (tiaxial) case at least thee numbes ae needed to satisfactoily descibe the size and shape of such paticles (e.g. hydoxyapatite platelets in bones). Howeve, in pactice many anisometic paticles may be consideed as otationally symmetic, i.e. possessing an axis of otational symmety (e.g. disks / platelets and ods / fibes). In this case, only two numbes suffice fo a desciption of size and shape, e.g. the extension in the diection of the otational axis (maximum Feet diamete) and the maximum extension in the diection pependicula to it (minimum Feet diamete), o an equivalent diamete and an aspect atio. Although pismatic shapes fequently occu in pactice, the simplest and theefoe most popula model shapes fo otationally symmetic paticles ae Cylindes (with height H and diamete D ) and Spheoids, i.e. otay ellipsoids (with extension H in the diection of the otational axis and maximum extension D in the diection pependicula to the otational axis); they can be oblate (flattened, e.g. disks / platelets) o polate (elongated, e.g. ods / fibes). 4

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems In both cases an aspect atio can be defined as R = o vice vesa. All othe possible shape measues fo these model shapes can be educed to the aspect atio. In contast to cylindes, spheoids contain sphees as a special case ( R = 1). In pinciple, fo abitay paticle shapes in D (paticle outlines) the chod lengths intesecting the cente of gavity can be detemined in vaious diections; thus fo each paticle a plot of chod length (in m) vesus oientation angle (in adians) can be obtained, which may be evaluated via Fouie analysis: using pola coodinates, the shape of the paticle outline can be consideed to be a wave fom having a value of adius, fo values of θ lying between 0 and π. This wave fom can be expessed as a hamonic (Fouie) seies, i.e. H D ( ) = a + ( a cos nθ + b nθ ) n= 1 θ 0 n n sin. Fouie coefficients a n and b 0 descibing paticle shape. In pinciple, complete shape infomation is contained in the coefficients. A majo pactical difficulty, howeve, is to know the point at which the seies can be stopped (highe ode tems ae needed fo moe angula and iegula paticles). Moeove, the values of the coefficients depend on the choice of the oigin. Obviously, fo many-paticle systems this pocedue is usually not economical.. Factal geomety and suface oughness 1 The total length T of a line consisting of n identical units, each with length a, is T = na. Similaly, the total aea T of a squae of n units with aea a, is T = na, and the total volume T of a cube of n units with volume a, is T = na. Thus, in geneal, δ T = na, whee δ is an intege. In all the above cases the shape (hypevolume) can be consideed to be completely filled. Patial filling can be epesented by nonintege values of δ, with the degee of filling inceasing as the value of δ becomes geate. Thus an iegula paticle can be descibed by an exponent δ (non-euclidean o factal dimension), which contains infomation about the degee of volume filling, suface oughness o uggedness of the peimete of the D paticle outline (pojection o section). Iegula paticles with a ough suface o agglomeates can have factal dimensions between and. The factal dimension of the peimete of a D outline of an iegula paticle with a ough suface is between 1 and. That means, if the peimete (suface) is measued (tiled) with smalle and smalle pobes, then thei total length (aea) inceases the suface aea of a paticle (and similaly, the peimete of a D paticle outline) is not a uniquely defined value, but dependent on the size of the pobe used. The factal dimension δ is obtained fom the slope of the staight line fit in log-log-plots ( n vesus a ). The staight line fit in the log-log-plot (o, equivalently, the powe law fit in the lin-lin-plot) implies geometical similaity on diffeent length scales, i.e. diffeent degees of magnification (scale-invaiance, self-similaity), at least in a limited ange. Fo details on measuing techniques see CPPS-1.. 5

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems CPPS. Paticle packing, coodination numbes and factals.0 Intoduction The packing of paticles is of utmost pactical impotance in mateials science and technology as well as othe banches of science whee e.g. packed beds ae used (chemical engineeing, eacto technology), the poducts consist of paticula mateials (phamacology) o the systems involved ae intinsically ganula and poous (geosciences, petoleum engineeing). In paticula, when classical powe pocessing techniques ae used fo the poduction of ceamic o metal bodies a knowledge of paticle packing is essential to contol the subsequent high-tempeatue and / o high-pessue pocessing steps. The basic quantification of paticle packing involves the elative packing density (packing faction) and the coodination numbe. A moe detailed chaacteization of paticle systems, in paticula those exhibiting geometic self-similaity in a cetain ange of length scales, is possible via concepts of factal geomety..1 Packing faction and coodination numbes Fo monosized spheical paticles the densest packing is that with a packing faction (elative packing density = solids volume factions) of π 18 0. 74 (Keple s conjectue 1611, poved by Hales 1998; this appaently obvious esult gains its impotance fom the fact that in D space one can ceate suboptimal global packings with finite-sized clustes of sphees, e.g. tetahedal o icosahedal clustes, with local densities highe than the global maximum at the expense of having lage voids elsewhee, i.e. these high-density clustes cannot be spacefilling; e.g. identical non-ovelapping egula tetaheda cannot tile D space and the system is geometically fustated, meaning that local optimal packing ules ae inconsistent with global packing constaints). This maximum packing faction of 0.74 fo monosized sphees coesponds to hexagonal closest packing (hcp) o face-centeed cubic (fcc) and its stacking vaiants, all with a coodination numbe of 1 (i.e. a chosen paticles has 1 neaest neighbos in diect point contact). Simple cubic packing, on the othe hand, has a packing faction of 0.5 and a coodination numbe of 6. It is not known whethe stable packings of monosized sphees with lowe packing faction and coodination numbe exist in D space (diamond packing with a packing faction of 0.4 and a coodination numbe of 4 is unstable). Table.1 lists othe odeed packings of monosized sphees. Table.. Packing faction and coodination numbe of odeed packings of monosized sphees in D space. Packing type Packing faction Coodination numbe Closest packing 0.7405 1 (fcc / hcp) Tetagonal-sphenoidal 0.708 10 Body-centeed cubic 0.680 8 Othohombic 0.605 8 Simple cubic 0.54 6 Diamond 0.40 4 6

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Inspite of the fact that thee ae two ecognized packings with a coodination numbe of eight, thee have been attempts to appoximately coelate the packing faction φ S and the coodination numbe N C, e.g. by the elations N C π =, 1 φ S N C ( 1 φ ) 0. 8 = 14 10.4 S. The latte elation pedicts that the coodination numbe fo densely aanged paticles appoaches 14 when the packing faction appoaches 1, i.e. 100 %. Theefoe, the Kelvin teta-kai-decahedon (tuncated octahedon with 14 faces, i.e. 6 squaes and 8 hexagons) has become the pefeed basic model shape fo sinteed micostuctues see the couse Technology of Ceamics at the ICT Pague. When the packing is andom (i.e. not odeed), the packing faction fo monosized sphees is appox. 0.64, and the aveage coodination numbe is 7. Taditionally, this packing type has been called andom-close packing (cp stuctue). Although Toquato (000, 00) has shown that the cp stuctue is ill defined and has eplaced it by the concept of the maximally andom jammed state (mj stuctue), the best estimate fo the mj packing faction is still 0.64 in the case of monosized sphees. Highe packing factions can be achieved by polydispese paticle systems and nonspheical (e.g. polyhedal o anisometic) paticles, but eliable theoetical pedictions ae difficult in these cases. In pactice, empiical ules and expeience with eal systems ae invoked see the couse Technology of Ceamics at the ICT Pague.. Mass and suface factals When paticles aggegate, e.g. fom a paticulate sol o a macomolecula solution with polyfunctional monomes, they commonly fom factal stuctues. A mass factal (object) is distinguished fom a conventional Euclidean object by the fact that its mass M inceases with its size (equivalent adius) accoding to the elation d m M, whee d m is the mass factal dimension ( 0 d m ). Fo a Euclidean object M, but fo a factal d m <, that means the density of the object ( ρ M ) deceases as it gets bigge; a tee-like stuctue is an example of a mass factal. A suface factal (object) has a suface aea S inceasing moe steeply than popotional to, i.e. d s S, whee d s is the suface factal dimension ( d s ); a cumpled piece of pape is an example of a suface factal (it is not a mass factal, howeve, since its mass inceases as M ). 7

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Fo Euclidean objects (nonfactal with a smooth suface) d m = and d s =, fo mass factal objects d m = d s, fo suface factal objects the mass factal dimension equals the Euclidean dimension, i.e. d m =, and < d s <. The thee most popula techniques to detemine factal dimensions ae: Adsoption of gas o solute molecules (specific suface measuements) Pfeife-Avni appoach: m a σ d d m D, whee a is the amount of adsobate adsobed on the adsobent (e.g. numbe of adsobate molecules pe unit volume of adsobent o moles of adsobate pe unit mass of adsobent), σ is the equivalent aea o coss-section of the adsobate molecule (when a linea size measue, e.g. an equivalent diamete, is used the exponent is d m instead of d m ) and D is a linea measue of paticle size (e.g. an independently measued mean equivalent diamete); theoetically, eithe σ o D can be vaied (in pactice usually σ ). Altenative vaiants of the adsoption technique use a modified Fenkel-Halsey Hill equation o the Kiselev equation (Neimak-Kiselev appoach) see CPPS-1.. Mecuy intusion (volume-weighted poe-size distibution measuements): futhe details see CPPS-1.. dv d ( ) ds, Small-angle scatteing (Pood egion): Small-angle scatteing can use neutons (SANS), X-ays (SAXS), o visible light (static light scatteing o dynamic / quasielastic light scatteing QELS) length scales fom 0.1 nm to 1 µm. The scatteing cuve, i.e. the log-log plot of scatteed intensity as a function of the invese length measue 4π θ k = sin, λ whee λ is the wavelength and θ the scatteing angle, can be divided into thee egions: o Bagg egion at lage scatteing angles ( k β 1, whee β is the bond length), fom which infomation concening inteatomic spacings is obtained via Bagg s law (in amophous systems diffuse peaks adial distibution cuves). o Guinie egion at vey small scatteing angles ( k γ 1), fo which the scatteed intensity is exponentially elated to the adius of gyation γ, i.e. 8

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems I ( k) exp( k γ ) infomation on the mass o adius of macomolecules. o Pood egion at intemediate scatteing angles ( γ >> k 1 >> β ), fo which the scatteed intensity decays acccoding to a powe law, i.e. as P ( k) k I, whee P is the Pood slope, which can be intepeted in tems of factal dimensions as P = d s d m. Since fo mass factal objects d m = d s P = d m, i.e. the mass factal dimension is obtained diectly fom the slope. Fo suface factal objects d m = P = d s 6. Howeve, polydispesivity of poe sizes (intestitial voids in an aggegate / agglomeate of paticles) with a numbe-weighted poe size distibution coesponding to a powe law also yields a powe-law decay fo the scatteed intensity. That means, physically meaningful factal dimensions can be deived fom the Pood plot only when the type and degee of polydispesivity is known. Table. gives examples of Pood slopes fo vaious stuctues of paticles aggeagates / agglomeates. Table.. Pood slopes fo vaious stuctues of paticles aggeagates / agglomeates. Stuctue Pood slope Type of factal Linea polyme Mass (andom walk) Linea polyme 5/ 1.67 Mass swollen (self-avoiding walk) Banched polyme 16/7.9 Mass Banched polyme Mass swollen Diffusion-limited.5 Mass aggegate Multipaticle diffusionlimited 1.8 Mass aggegate Pecolation cluste.5 to.6 Mass Factally ough suface to 4 Suface Agglomeate of paticles o poous medium with smooth sufaces 4 Non-factal 9

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 CPPS 4. Small paticle statistics 4.0 Intoduction Small paticle statistics is teated in detail in Hedan s book (1960). In the pe-compute ea, analytical functions (and the coesponding special gaphical papes) wee used wheneve possible to epesent measued paticle size distibutions. These functions have the advantage that they can be chaacteized by a few fit paametes, fom which all statistical values can be detemined. Real paticle size distibutions, howeve, do usually not fit any analytical function exactly, and theefoe today a numeical (tabula o gaphical) epesentation is pefeed. In ode to educe the infomation contained in a complete distibution, statistics can be applied. 4.1 Gaphical epesentation of size distibutions Paticle size distibutions can be epesented as histogams (discete distibutions) o as continuous cuves, when the size classes ae sufficiently close (usually the bin width of a size class is chosen by dividing the oveall width of the distibution by the squae oot of the numbe of measued paticles). The size measue (usually an equivalent diamete x i, coesponding to the aveage in a size class i ) is given on the abscissa (x-axis), while the odinate (y-axis) shows the statistical weight of each size class. This statistical weight can be the numbe of paticles in a size class numbe-weighted distibution (with index 0), the total length of all paticles (= sum of all equivalent diametes) in the selected size class length-weighted distibution (with index 1), the total suface of all paticles (= sum of the suface aeas of equivalent sphees, as calculated fom the equivalent diametes) in the selected size class sufaceweighted distibution (with index ), the total volume of all paticles (= sum of the volumes of equivalent sphees, as calculated fom the equivalent diametes) in the selected size class volumeweighted distibution (with index ), the total mass of all paticles in the selected size class mass-weighted distibution (which is identical to the volume-weighted distibution when all paticles in a sample have the same density). Paticle size distibutions can be epesented eithe in diffeential fom as fequency cuves / histogams o, moe pecisely, pobability density distibutions (denoted q ), n x q, ( x ) i i i = ni xi whee n i is the numbe of paticles in the i -th size class with aveage size (equivalent diamete) x i, o in integal fom as cumulative cuves / histogams (denoted Q ), which can be undesize, 10

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 o ovesize, Q i n n = n= 1 x ( x ) = q ( x ) x q ( x)dx i Q ovesize x i min ( x ) = 1 Q ( x ) i (in this couse we always efe to the undesize distibution, if not explicitly stated othewise). Numbe-weighted paticle size distibutions ( q 0, Q 0 ) ae the pimay esults of counting methods such as micoscopic image analysis, while volume-weighted distibutions ( q, Q ) ae the pimay esults of ensemble methods such as lase diffaction. (The massweighted distibutions obtained using sedimentation methods ae identical to the volumeweighted distibutions if the density of all paticles is the same.) Length- and sufaceweighted distibutions ae athe uncommon in pactice. Note that numbe-weighted and volume-weighted distibutions cannot be diectly compaed. They can be compaed only afte one of them has been tansfomed into the othe (which equies eithe the assumption that the shape is size-invaiant o an independent measuement of the shape-size dependence). Although compaable afte applying this kind of tansfomation, ( q 0, Q 0 ) ( q, Q ) o ( q, Q ) ( q 0, Q 0 ), the esults cannot be expected to coincide in geneal, because diffeent methods measue diffeent equivalent diametes. Only fo spheical paticles (o appoximately fo isometic paticles) coincidence may be expected in pinciple ( standad efeence mateials fo calibation puposes). In pactice, the degee of coincidence can be limited by the diffeent measuing anges and othe method-specific eos. i 4. Statistical mean values In geneal, the mean values fo the diffeent types of distibutions ae: x k = x k i x x i i n n i i 1 k = x k + i x i n n i i 1 k, whee denotes the type of distibution ( = 0, 1,, fo numbe-weighted, lengthweighted, suface-weighted and volume-weighted, espectively) and k denotes the type of aveage (e.g. hamonic mean k = 1, geometic mean k = 0, aithmetic mean k = 1, quadatic mean k = etc.). Fo these aveages Cauchy s majoity elation holds:... xh xg x A xq... The geometic mean is calculated via the elation q ( xi ) log q ( x ) x ni log xi xi log x = i G =. x n i i i 11

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 It can be shown that the hamonic mean of the volume-weighted distibution equals the aithmetic mean of the suface-weighted distibution (Hedan s theoem). Theefoe the specific suface (suface density) is invesely popotional to the hamonic mean of the volume-weighted distibution, also called natual mean o Saute mean (mean volume-tosuface paticle size). Its ecipocal, i.e. the atio between the second and thid moment (se below) is popotional to the specific suface of the powde, the popotionality facto being 6 fo sphees and geate than 6 fo othe paticle shapes. 4. Othe basic paametes chaacteizing size distibutions Othe paametes, which ae not statistical mean values (aveages), can be used to chaacteize paticle size distibutions. The most impotant of them ae: Quantiles: paticle sizes coesponding to a selected cumulative weight; the most impotant quantiles ae the lowe decile ( x 10 ), the median value ( x 50 ), and the uppe decile ( x 90 ) thei physical meaning is evident fom the cumulative (undesize) cuve (histogam): 10 % (with espect to numbe in Q 0, with espect to volume in Q etc.) ae smalle than x 10 etc. Median: the special quantile x 50, which divides the paticle population into two equal pats (with espect to numbe in Q 0, with espect to volume in Q etc.) Span: a measue of the width (beadth) of a distibution, defined as Span x x 90 10 =. x50 Mode: the most fequent value (with espect to numbe in Q 0, with espect to volume in Q etc.) in a distibution, coesponding to the maximum in the fequency cuve (o moe pecisely, pobability density distibution); distibutions and paticle systems with one mode ae called monomodal, with two bi- and with thee ti-modal (in geneal multimodal); paticle systems with one vey naow mode ae called monodispese, with two bidispese etc. (in contast to polydispese systems, which exhibit a boad distibution); in the exteme case of stictly monodispese sphees, the fequency cuve would be a Diac delta distibution and the coesponding cumulative cuve a Heaviside step function. Vaiance ( σ ): a measue of the width (beadth) of a distibution, defined as whee ( x ) ( x x ) q i i A σ =, N 1 N = ni fo numbe-weighted distibutions and N x n i in geneal. The = i standad deviation is the squae oot of the vaiance (σ ) and the coefficient of vaiation is the standad deviation divided by the aithmetic mean ( σ x ). A 1

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 Skewness: a measue of the distotion fom a symmetical distibution, defined as S ( N )( N ) ( x ) ( x x ) q ( x ) ( x x ) = N q i i A i i A 1 σ N σ. A symmetic distibution is defined as having zeo skewness. S is positive if the distibution is ight-skewed (steep ise on the left, long tail on the ight side, i.e. moe mateial in the coase size ange) and negative if left-skewed. Kutosis: a measue of the peakedness (shape) of the distibution, defined as K N ( N 1) ( N 1)( N )( N ) 4 ( xi ) ( xi x A ) ( N 1) 4 ( N )( N ) ( x )( x x ) q q i i = 4 σ N σ A 4 A nomal distibution (Gauss distibution) is defined as having zeo kutosis (being mesokutic). K is positive if the distibution is leptokutic (shape o naowe than the nomal distibution) and negative of platykutic (flattened maximum). Of couse, all these paametes ae diffeent fo each type of distibution (of the same sample), i.e. numbe-weighted, length-weighted, suface-weighted and volume-weighted. 4.4 The moment notation In the moment notation, mean values ae defined though the moments of diffeent types of distibutions. When the diffeential aea q ( x)dx below the fequency cuve (pobability density distibution) q ( x) (with = 0, 1,, denoting a numbe-weighted, length-weighted, suface-weighted and volume-weighted distibution, espectively) is multiplied by the leve k x ( k = -, -, -1, 0, 1,, ), the so-called moments esult. Complete geneal moment (of k-th ode) of the q ( x) distibution: M k, = xmax k x q k + 1 x i= 1 min N 1 k + ( ) = ( + 1 k 1 x dx q, i xi xi 1 ). This geneal moment is called complete, because the integation extends ove all paticle sizes. The coesponding incomplete geneal moment would be defined by the integal between two selected values x1 xmin and x xmax. Note that M 1. Complete cental moment (of k-th ode) of the q ( x) distibution: mk = x max k ( x x ) q ( x)dx, 1,, x min 0,0 = 1

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 whee x 1, is the aveage size (aithmetic mean diamete) defined as x 1, = M 1, = M M 1+,0,0 (aithmetic mean, i.e. the abscissa value of the cente of gavity of the ( x) a special case (fo k = 1) of the geneal mean q cuve), which is x k, M M k +,0 k + e, e = k M k, = k = k. M,0 M e, e Note that the nomalization condition fo geneal moments is M xmax 0, = q ( x) dx = 1. x min Impotant is the expectation value (k-th geneal moment of the numbe-weighted distibution) E xmax k k ( x ) = M k, 0 = x q0 ( x)dx and the vaiance (second cental moment of the numbe-weighted distibution) x min s = m, 0 = x max ( x x1,0 ) q0 ( x)dx x min. The k-th moment of a q ( x) distibution can be detemined fom a given ( x) via the equation q 0 distibution M k, M k +,0 =. M,0 This equation also allows a physical intepetation of the moments, e.g. M 1, = M,0 M, 0 coesponds to a suface-volume atio and M, = M 0,0 M,0 = 1 M, 0 to a ecipocal volume. Moe geneally, the k-th moment of a q ( x) distibution can be detemined fom a given q e ( x) distibution via the equation M k, M k + e, e =. M e, e Using two diffeent moments, any mean value of the distibution can be calculated. 14

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 4 4.5 The moment-atio notation In the moment-atio notation, mean values ae expessed as the atio between two moments of the numbe-weighted distibution of the size measue x (usually an equivalent diamete). The quantity D p, q is the mean size obtained fom summing discete individual x values to the powe of p (elationship between the signal and x ) and nomalizing by a sum of x values to the powe of q (elationship between the statistical weight of each paticle to its x value), i.e. 1 p q p x i D p, q fo p q q x i = p x i ln xi D = p, q exp fo p = q p xi In othe wods, D p, q (afte extaction of a pope oot) is the aithmetic mean of the q p q distibution obtained by plotting x against x. When a cetain equied mean value cannot be measued diectly, but two othe mean size values ae known, then the equied mean size can be calculated using the elation D p q p c ( D ) p q p, c q c ( D ) p q, = fo p q. q, c Fo example, a gaticule (gid) can be used to measue the total intecept length fom andom sections (cuts) of all paticles by optical o electon micoscopy; divided by the numbe of paticles this yields the mean intecept length D 1, 0 (aithmetic mean of the numbe-weighted distibution). If digital image analysis is used to measue the pojected aeas of all paticles and the total pojected aea is divided by the numbe of paticles this yields D, 0. Similaly, the Coulte pinciple measues D, 0 and lase diffaction, sedimentation and sieving D 4,. In dynamic light scatteing (DLS), also called photon coelation spectoscopy (PCS), the scatteing intensity is popotional to the volume squaed o the sixth powe of the paticle dimension, when the paticles ae smalle than the wavelength of light (Rayleigh-Debye-Gans theoy, fist-ode appoximation). Thus, the mean size obtained fom DLS (PCS) is 6 1 I i = Di D = D6,5 =. 5 I D i i Di 1 ( ) This mean size value ( D 6, 5 ) is always smalle than the weight aveage D 4,. 1 paticles, the DLS (PCS) mean size ( ) 1 D is smalle than D 6, 5.. Fo lage 15

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 5 CPPS 5. Sedimentation methods 5.0 Intoduction Apat fom sieve classification, which has lost its fome significance fo paticle sizing, sedimentation methods ae the most pominent taditional methods used fo paticle size analysis. Advantages ae thei conceptual claity and pactical simplicity, without the need of sophisticated equipment. Disadvantages ae that sedimentation methods ae elatively timeconsuming, the measuing ange is elatively naow and the esults ae vey sensitive to sample pepaation. In paticula it is essential to achieve optimal deagglomeation. Too lage paticles develop tubulent motion, too small paticles agglomeate and ae subject to Bownian motion ange 1 100 µm (centifugal sedimentation down to 0.1 µm). 5.1 Measuing pinciple, equipment and pocedue Pinciple of sedimentation methods: fom a polydispese paticle system suspended in a liquid medium lage paticles exhibit faste settling unde the influence of gavitation (and possibly centifugal foces) than small paticles. Common taditional equipment is the Andeasen pipette afte pepaing the suspension accoding to a standadized ecipe (deagglomeation by deflocculants, stiing, agitating, ultasonication, possibly boiling etc.) the suspension is allowed to settle. At pedetemined time intevals small volume (10 ml) samples ae taken by the pipette fom a fixed position in the sedimentation column (600 ml, moe than 0 cm high) to detemine the concentation of solids which ae still in suspension (afte the lage size factions have aleady settled out). Fo efficient measuements sampling time intevals should gow in a geometic seies, so that a complete measuement can last seveal days when submicon paticles ae pesent. Othe common equipment fo paticle sizing via sedimentation methods ae sedimentation balances, in which the mass incement of the sediment is continuously ecoded, o photo- and X-ay sedimentogaphs, in which the cuvette (sedimentation column) is scanned in ode to detemine the paticle concentation via attenuation of light o X-ays. With the latte, the measuement times can be educed to a few minutes. Necessay conditions fo eliable esults ae the absence of paticle-paticle inteactions ( dilute suspensions) and lamina flow ( Reynolds numbes below appox. 1; theefoe lage paticles have to be eliminated befoe measuement, usually by using 6 µm sieves; the sieve faction > 6 µm can be included in the final esult). 5. Standad data evaluation The standad evaluation of sedimentation measuements is pefomed by the classical Stokes fomula fo settling sphees Stokes diamete D (equivalent sphee diamete): D S 18η h =, ( ρ ρ ) g t S S L 16

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 5 whee η is the viscosity (of the pue liquid medium without paticles), ρ S the density of the solid paticles, ρ L the density of the (pue) liquid, g the gavitational acceleation, h the sedimentation path (height of the column above the sampling point) and t the sedimentation time (sampling time). Note that the velocity is v = h t only unde steady-state conditions, i.e. when the acceleation stage has been exceeded and the final settling velocity has been eached. This is usually the case afte a few seconds. The Stokes equation can be deived fom the foce equilibium F F + F = 0, B G whee F B is the lift foce (buoyancy foce) acting on the paticle in the (specifically lighte) liquid medium R F B 4 = π R ρ L g, F G the gavitational foce acting on the paticle F G 4 = π R ρ S g, and F R the esistance foce (fiction foce) exeted by the viscous liquid medium on the paticle F R = 6π η R v, with v being the (final) velocity of the paticle elative to the liquid medium and R = D S / the paticle adius (equivalent sphee adius). Apat fom seveal assumptions of physical chaacte (laminaity of flow, steady flow with final velocity), the validity of the Stokes equation is essentially based on the geometical assumption that the paticles ae spheical. Since this is usually not the case fo eal systems, the Stokes diametes D S coespond to equivalent diametes of hypothetical sphees with the same settling behavio as the iegula, anisometic paticles in question. The esults of sedimentation methods ae mass-weighted size distibutions. When all paticles have the same density, these esults can be consideed as identical to volumeweighted size distibutions, i.e. Q cuves. 5. Nonstandad data evaluation and shape detemination of oblate paticles The Stokes equation can be modified and adapted to flat cylindes and oblate spheoids. This modified Stokes equation can be used to eintepet the esults in the case of oblate paticles. Based on this eintepetation, paticle shape can be quantified when the sedimentation esults ae known and the size distibuition has been independently measued by image analysis o lase diffaction see CPPS-Appendix-oblate. 17

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 6 CPPS 6. Lase diffaction I Theoy 6.0 Intoduction The theoy of lase diffaction is a special banch of electomagnetic scatteing theoy. In its classical (i.e. non-quantum-mechanical) fom it is based on the Maxwell equations and its solutions. Mie theoy is the exact classical theoy of light scatteing with small paticles. It is elaboated fo sphees and numeical solutions ae available today, which can be implemented in compute algoithms. Altenatively, appoximate analytical solutions ae available fo paticles much smalle o lage than the wavelength of light (Rayleigh / Rayleigh-Debye- Gans scatteing and Faunhofe diffaction, espectively). The Faunhofe appoximation, which is close to geometical optics than the othe appoximations, is commonly used in lase diffaction instuments fo paticle sizing. 6.1 Inteaction between light and matte Light is electomagnetic adiation in the fequency ange (ν ) fom appox. 10 1 Hz (IR) to 10 17 Hz (UV), coesponding to the wavelength ange ( λ ) fom nm to 0 µm. The convesion between fequency and wavelength is via the speed of light c = λν (in vacuum 00 000 km/s). Visible light (i.e. the pat of the electomagnetic spectum to which the human eye is sensitive) anges fom appox. 400 nm (violet) to 750 nm (ed). The optical popeties of matte (paticles) ae descibed by the complex efactive index, N = n + iκ, whee the eal pat accounts fo efaction accoding to Snell s law and the imaginay pat is elated to the absoption coefficient a via the elation 4π κ a =. λ This absoption coefficient occus in the Lambet-Bee law descibing the exponential attenuation of light intensity (iadiance) I as the light wave taveses a medium of thickness z, i.e. I = I exp ( a z) 0, whee I 0 is the intensity of the incident light (magnitude of the Poynting vecto). Geneally, extinction of light in a medium occus by the combination of absoption and scatteing. The absobed adiation enegy can be tansfomed into heat o e-adiated as fluoescence o phosphoescence. Scatteing geneally occus in all diections and includes efaction and eflection as special cases. Light is chaacteized by the wave vecto k (diecting into the diection of popagation of the tansvese light wave), whose magnitude is the wave numbe k = π λ. A elative efactive index between two media can be defined as m = N 1 N. 18

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 6 6. Rayleigh scatteing, Rayleigh-Debye-Gans appoximation, and Mie theoy When the paticle is much smalle than the wavelength of light ( D << λ and D m << λ ), then each pat of the paticle expeiences the same homogeneous electic and magnetic field of incident light and the paticle behaves like a dipole adiating in all diections, iespective of its shape Rayleigh scatteing (with scatteing angle θ ): Thus, if the quantity ( )( ) ( 1+ θ ) 6 D m 1 I I 0 cos. 4 λ m + m 1 m + is independent of the wavelength (this is not always tue, because the complex efactive index geneally depends on fequency, mainly fo 4 metallic paticles), the scatteed intensity is invesely popotional to λ, as long as extinction is dominated by scatteing. When extinction is dominated by absoption, the intensity is invesely popotional to λ. In eithe case shote wavelengths ae extinguished moe than longe ones eddening of the spectum of light upon tansmission though hetegeneous media (aeosols, paticle suspensions, fluids with density fluctuations) blue sky duing daytime, ed sky at sunise / sunset, use of ed taffic lights in dust, fog / mist and haze. When the paticles ae too lage to be teated as single dipoles but still small enough to be teated as independent Rayleigh scattees, they can be teated in the Rayleigh-Debye-Gans appoximation if thei efactive index is close to that of the medium (i.e. m 1 << 1) and the condition D m 1 << λ is fulfilled (in pactice up to a few 100 nm). When the shape of the paticles is known (which implies knowledge of the shape-dependent scatteing facto), size infomation can be extacted by measuing the angula scatteing intensity (without knowledge of the efactive index of the paticle). Fo paticles of abitay size, Mie theoy can be applied to evaluate scatteing data (numeical solution). In ode to apply Mie theoy, the complex efactive index of the paticle (and the medium) must be known (fo the light wavelength used). With inceasing paticle size the scatteed intensity becomes pefeentially diected to the fowad diection. Note that Mie theoy has been deived fo optically isotopic paticles of spheical shape. 6. Faunhofe appoximation When the paticle size is much lage than the wavelength of light D >> λ, the paticle emoves an amount of light enegy coesponding to twice its coss-section aea (extinction paadox). One aeal coss-section is emoved by eflection, efaction and absoption, and one via diffaction. Diffaction by paticles is an edge effect (compaable to diffaction by an apetue), and fo lage paticles, intefeence aises mainly fom the paticle outline, i.e. only the pojected aea pependicula to the light popagation diection mattes, not the volume and the intenal stuctue (optical popeties) of the paticles Faunhofe appoximation. Moe pecisely, the Faunhofe diffaction patten is the Fouie tansfom of the paticle pojection. Analytical solutions ae known fo a vaiety of shapes. Fo sphees, which scatte as if they wee opaque disks, the Faunhofe diffaction equation is 19

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 6 J ( α sinθ ) 1 I I 0, α sinθ whee α π D λ J 1... the spheical Bessel function of fist kind. In pactice the Faunhofe appoximation applies to paticles lage than a few µm, o highly absoptive paticles (with absoption coefficients highe than 0.5), o paticles with significant diffeent efactive index contast elative to the medium ( m > 1.). Because fo lage paticles the scatteing intensity is concentated in the fowad diection, typically at angles smalle than 10, Faunhofe diffaction is also known as fowad scatteing o lowangle lase light scatteing (LALLS). In Faunhofe diffaction by a sphee, the angle of the fist minimum of scatteing intensity is simply elated to the paticle size via the elation = is a dimensionless size paamete and ( ) 1. λ sin θ ( fist minimum) =, D and most of the scatteing intensity is concentated close to the cente of the intefeence patten, see Table 6.1. Table 6.1. Intensity distibution of Faunhofe diffaction fom a sphee. Intensity ing Radial position Relative intensity Integal intensity in I I 0 the whole ing [%] Cental max. 0 1 8.8 Fist min. acsin ( 1. λ D) 0 0 Second max. acsin ( 1.64 λ D) 0.0175 7. Second min. acsin (. λ D) 0 0 Thid max. acsin (.68 λ D) 0.004.8 Thid min. acsin (.4 λ D) 0 0 Fouth max. acsin (.70 λ D) 0.0016 1.5 acsin 4.4 λ D 0 0 Fouth min. ( ) Table 6.. Common lase light souces. Lase type Powe [mw] Wavelength [nm] Remak He-Ne gas lase 1 50 54.5, 594.1, 61.0, 6.8 A ion lase 0 000 488, 514.5 Wate-cooling needed Diode lase 0.1-00 405, 450, 65, 650, 670, 685, 750, 780 Low cost 0

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 7 CPPS 7. Lase diffaction II Pactice 7.0 Intoduction Lase diffaction is the most widely used method fo paticle size analysis today. Although the undelying physical pinciples of scatteing and diffaction wee known fo moe than 100 yeas (Mie theoy, 1908), paticle sizes based on the diffaction pinciple could be developed only afte the invention of the lase (aound 1960), and the outine use of these instuments in pactice equied poweful computes (since the 1970es and 1980es). Commecial instuments today ae fast, flexible (fom laboatoy batch measuements to in-line poduction contol, fom suspensions to dy powdes, fom nanometes to millimetes) and yield highly epoducible esults. Theefoe they ae gadually eplacing othe paticle sizing methods, in paticula sedimentation methods, in most banches of industy. 7.1 Typical equipment and sample teatment A typical lase diffaction instument (paticle size) consists of a light souce (the lase), a sample chambe in the fom of a flow-though cell (e.g. a glass cuvette connected to a liquid esevoi, appox. 500 ml) and a photodetecto (e.g. a half-cicle, quate-cicle o wedge-shaped segmented detecto o a CCD-type detecto), which tansfoms the optical signal (intefeence patten, i.e. the light intensity in dependence of the scatteing angle) into an electic signal (fom the individual photodetecto segments), which is then tansfeed to the compute and used fo data geneation. The geomety of the photodetecto may become impotant when size measuement is to be coupled with shape measuement (based the deviation of the intefeence patten fom cicula symmety) o oientation measuement of anisometic paticles (fibes) cuent eseach. The distance between lase, sample chambe and photodetecto as well as the position and spatial esolution of the photodetecto (distance of detecto segments) detemine the measuing ange which can be achieved. Typically it is fom 0.1 µm to moe than 1 mm, but new instuments pincipally enable measuements in the nanosize ange as well. Fouie optics (with a Fouie lens between the sample chambe and the detecto) o evese Fouie optics (using a convegent lase beam with a Fouie lens between lase and sample chambe) is used to ensue that light scatteed at a specific angle will fall onto a paticula detecto element, egadless of the paticle s position in the beam. The liquid esevoi (which can be an extenal beake) contains the suspension (usually a powde sample dispesed in wate) and is mechanically agitated by ultasonics and possibly a stie. One of the advantages of lase diffaction, in contast to othe sizing methods, is the fact that ultasonication can be used even duing measuement (and not only as an auxiliay technique fo sample pepaation befoe measuement). Duing measuement the suspension is steadily pumped with a chosen flow velocity (adjustable accoding to the density of the paticles to avoid settling in the system) though the flow-though cell. Altenatively, a dy-dispesion unit can be used in some instuments, fom which the sample is conveyed though the glass cuvette by a ai steam as a dy powde. Sample pepaation has to be adapted to the chaacte of the paticles (type of mateial as well as paticle size), but is usually less demanding than fo sedimentation and othe sizing methods. Of couse, submicon and especially nanosized paticles tend to exhibit stong agglomeation effects, and poweful deflocculants o othe ticks may have to be used to achieve deagglomeation. 1

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 7 7. Measuing pinciple and data evaluation Lase diffaction is an ensemble method, i.e. a lage numbe of paticles is illuminated simultaneously and the diffaction patten ecoded by the photodetecto is assumed to be the supeposition of the intefeence pattens of the individual paticles. In ode to ensue that the latte is eally the case, the concentation of the paticle system (usually a suspension) has to be sufficiently low so that paticle ovelap and multiple scatteing is avoided. On the othe hand, the concentation must be high enough to achieve a easoanble signal-to-noise atio. The standad method fo data evaluation in lase diffaction is based on the Faunhofe appoximation. Fo a polydispese powde sample the usual evaluation pocedue consists in a deconvolution of the diffaction patten accoding to the integal equation I I 0 0 ( α sinθ ) J1 α sinθ f ( D)dD, whee the function ( D) f is the desied paticle size distibution (pobability density). This is a so-called invese poblem (in mathematical tems ill-posed and ill-conditioned), fo which questions of existence and uniqueness of the solutions geneally aise. In commecial equipment the solution is usually based on popietay algoithms. When the paticles ae not lage enough to justify the application of the Faunhofe appoximation (valid fo D >> λ ), the exact Mie theoy should be used fo data evaluation (highly ecommended fo paticles smalle than 1 µm), i.e. the complex efactive index of the mateials should be known. 7. Data intepetation The pimay esults of lase diffaction measuements ae volume-weighted size distibution cuves o histogams. These can be tansfomed into suface-, length- o numbe-weighted cuves (histogams), each with its own statistical values see Appendix-CPPS-7-A. Execise poblem Given the numeical values in Appendix-CPPS-7-A (alumina powde) tabulate the cumulative pecentage values of the Q distibution in steps of 0. µm, i.e. 0., 0.4, 0.6 etc. fom 0. µm to 6 µm. Based on these values calculate (assuming of sphecial shape wheeve necessay) 1. the pobability density distibution (fequency histogam) q,. the suface-, length- and numbe-weighted distibutions ( q, q1, q0 and Q, Q1, Q0 ). the hamonic, geometic, aithmetic, and quadatic mean fo each distibution, 4. the mode, median and span of each distibution, 5. the vaiances, standad deviations, coefficients of vaiation, skewnesses and kutoses, 6. the geneal moments M, 0, M, 0, M 1, 0, M 0, 0, M 1, 0, M 1, 1, M 1,, M 1,, M, 0, M, 0, M 4,0, as well as the cental moment m, 0, 7. the moment atios fom D 0, 0 to D 6, 6 (i.e. those with index pais 00, 10, 11, 0, 1,, 0, 1,,, 40, 41, 4, 4, 44, 50, 51, 5, 5, 54, 55, 60, 61, 6, 6, 64, 65, 66), and compae the esults with the compute pintouts given in Appendix-CPPS-7-A.

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 8 CPPS 8. Othe methods Dynamic light scatteing, electozone sensing, optical paticle counting 8.0 Intoduction Paticle sizing methods can be divided into ensemble techniques and counting techniques. Ensemble techniques have typically low esolution and low sensitivity, but a boad dynamic size ange and high statistical accuacy. Examples ae sieving, sedimentation, lase diffaction and dynamic light scatteing (see below). On the othe hand, counting techniques ae typically high-esolution, high-sensitivity techniques with naow dynamic size ange and low statistical accuacy. Examples ae micoscopic image analysis, electozone sensing and optical paticle counting (see below). Counting techniques ae bette suited to detect a few small o lage paticles lying beyond selected size limits. 8.1 Dynamic light scatteing (photon coelation spectoscopy) Dynamic light scatteing (DLS, also called photon coelation spectoscopy PCS, a special case of quasi-elastic light scatteing - QELS) is the method of choice fo sizing submicon paticles (< 1 µm). Measuing pinciple: Fluctuations of the scatteed light intensity (i.e. tempoal vaiation in the µs to ms time scale) ae ecoded (at a given scatteing angle) and analyzed decay constant of the autocoelation function (ACF) diffusion coefficient size infomation. Lowe size limit (a few nm, depending on the elative efactive index) detemined by expeimental noise, uppe size limit (a few µm, depending on paticle density and fluid viscosity) by sedimentation (paticles to be analyzed must be stably suspended). No optical popeties of the paticle and no calibation needed. Instumental equipment and sample concentation: Light souce (e.g. He-Ne, A ion o diode lase) fo coheent and possibly linealy polaized light (note that coheence descibes light waves that ae in phase both in time and space tempoal and spatial coheence coheence length = coheence time speed of light), deliveing and collecting optics (e.g. fibe optics), sample module (e.g. glass cuvette), photodetecto system (photodiode o photomultiplie tubes), electonic system (amplifie and pulse disciminato) and coelato (hadwae o softwae to measue the ACF). The scatteing angle (ange) is chosen in ode maximize infomation and to incease the signal-to-noise atio. The masueing time should be long enough to poduce a smooth ACF. The paticle concentation should be low enough to avoid multiple scatteing and paticle-paticle inteactions ( mean distance between paticles should be at least 0 times thei diamete), but high enough to achieve a good signalto-noise atio ( difficult to achieve fo paticles lage than 1 µm). Data evaluation and intepetation: The ACF decays with time, e.g. fo monodispese paticles accoding to t C ( t ) = A exp + B, τ whee A = I () t I () t S S and B = I () t S and the chaacteistic decay time τ (fo polydispese systems a spectum of decay times, which has to be deconvoluted) is elated to

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 8 the diffusion coefficient D T (fo spheical paticles only tanslational diffusion due to Bownian motion; fo anisometic paticles otational diffusion can also be measued) via the elation 1 D T =, K τ whee K is the magnitude of the scatteed wave vecto 4π n θ K = sin, λ which is a constant, not containing infomation on the paticle s optical popeties ( n is the efactive index of the liquid medium). On the othe hand, the hydodynamic equivalent adius R H is elated to the diffusion coefficient via the Stokes-Einstein elation, e.g. fo spheical paticles D T = kt 6π η R, H whee k is the Boltzmann constant, T the absolute tempeatue and η the viscosity of the liquid medium (the diffusion coefficient must be extapolated to zeo concentation). Simila elations ae available fo non-spheical paticles. The pimaily intensity-weighted distibution can be tansfomed into a volumeweighted ot numbe-weighted distibution via appopiate weighting factos (fo Rayleigh scatteing, Rayleigh-Debye-Gans scatteing o Mie theoy, depending on paticle size). Additionally, in macomolecula solutions DLS can be used to detemine the volume- o numbe-aveaged molecula weight. 8. Electozone sensing (Coulte counte technique) The electical sensing zone method (Coulte counte) was invented in the ealy 1950es and since then it has become one of the most widely used paticle sizing techniques in medicine and phamaceutical technology. The instumental equipment is based on a tube with an oifice (apetue, sensing zone ) placed in an electolyte solution containing a low concentation of paticles. The device has two electodes, one inside and one outside the oifice and a cuent flows between them though the electolyte solution. When paticles pass though the oifice o sensing zone (via liquid flow diven by suction in the inne containe), a volume of electolyte equivalent to the immesed volume of the paticle is displaced, causing a shottem change in the conductivity acoss the oifice (i.e. the cuent between the two electodes deceases when the paticles ae electically insulating). This esistance change can be measued eithe as a voltage pulse o a cuent pulse. By measuing the numbe of pulses and thei amplitudes, one can obtain infomation about the numbe of paticles and the volume of each individual paticle (independent of paticle shape). numbe of pulses numbe of paticles, pulse amplitude popotional to the paticle volume: 4

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 8 U Vσ I, 4 R (U is the amplitude of the voltage pulse, V the paticle volume, σ the electolyte esistivity, I the apetue cuent, R the apetue adius). The electical esponse is independent of paticle shape. The lowe size limit (0.4 µm) is detemined by the signal-to-noise atio, the uppe size limit (seveal hunded µm) by the ability to suspend paticles unifomly in the beake (sedimentation). Fo measuements in a wide ange, two o moe apetues have to be used and the esults ovelapped to povide a complete size distibution. The paticle size can be channelized using a pulse height analyze and a numbeweighted paticle size distibution is obtained ( q 0, Q 0 ). The advantage of this counting method ae that it measues a paticle volume ( equivalent volume diamete D V ), unbiased by paticle shape. It counts and sizes with high esolution, sensitivity and epoducibility. The limitations (dawbacks) ae that electically conducting paticles (metals) cannot be measued, that an electolyte solution must be used (i.e. pue oganic solvents, e.g. ethanol, cannot be used). Oifice blocking by lage paticles may lead to infomation loss concening small paticles. Eos ae to be expected when the paticles ae poous, since open poes may o may not be filled with electolyte solution, so that the effectively displaced volume can be consideably smalle than the convex hull of the paticle. Although standad measuements can be in a few minutes, easonable statistics may equie long uns (moe than 0 min). Stay signals (electonic noise, e.g. fom an electon micoscope in the same oom) can cause high backgound counts. 8. Optical paticle counting (single-paticle optical sensing) Optical paticle counting (OPC, also called single paticle optical sensing - SPOS) is one of the main technologies fo envionmental monitoing (atmospheic aeosol monitoing, clean oom monitoing, clean wate contol) and industial quality contol (of liquid- o gas-bone paticle systems), due to its ability to make in-situ measuements (especially when simple monitos ae used). Simila to electozone sensing and image analysis it is a counting method (in contast to sedimentation, lase diffaction and DLS, which ae ensemble methods), yielding a numbe-weighted size distibution. Compaed to ensemble methods which have elatively low esolution, but a boad dynamical ange and high statistical accuacy, OPC is a high-esolution technique, but with elatively naow dynamical ange and low statistical accuacy. It is ideally suited to detect unwanted single paticles with a size lying outside specified limits, but the shape of the size distibution is not vey eliable. Measuing pinciple: Light scatteing technique (fo small paticles down to appox. 50 nm, mainly ai-bone powdes and aeosols) o light extinction technique (fo lage paticle fom appox. 0.5 µm to moe than 1 mm, mainly liquid-bone paticles); the scatteed intensity is dependent on the sixth powe of size fo small paticles (Rayleigh egime) and on the second powe of size fo lage o highly absoptive paticles (Faunhofe egime); in the Faunhofe egime (typically > 1-10 µm, depending on absoption) light extinction OPC (light blockage OPC) measues the pojected aea diamete. Each time the paticle taveses the beam, some pat of the beam is blocked (via scatteing o absoption by the paticle), the light flux detected by the photodetecto is educed and a negative signal pulse is poduced (pulse amplitude paticle size). Instumental equipment and sample concentation: Light souce (e.g. gas o diode lase), sensing zone (e.g. a glass cuvette), collecting optics and photodetecto (usually in 5

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 8 fowad diection; fo scatteing OPC sometimes at 90 ); volumetic designs illuminate the whole coss-section ( absolute fequencies), in-situ designs only a pat ( spectomete design fo the size distibution, monito design only to detect contamination single paticles). In OPC often the absolute concentation (counts pe unit volume) is of inteest volumetic meteing and flow contol. The electonic system convets light intensity pulses to electonic pulses, counts the individual pulses and sots them accoding to thei amplitude into pedefined channels (pulse-height analyze, multichannel analyze - MCA and / o chagecoupled device - CCD). Fo OPC measuements the paticle concentation must be vey low; extemely clean vehicle fluids must be used to pepae the highly dilute suspensions. Data evaluation and intepetation: Data evaluation is based on matix invesion schemes (e.g. Phillips-Twomey egulaization), but do not equie complicated mathematical models like lase diffaction. OPC intuments need calibation with paticles of the same optical popeties (othewise the esults ae only optically equivalent diametes, which ae not compaable even fo spheical paticles); if popetly calibated, good coelation of OPC and EZS esults can be expected. Fo non-spheical paticles the size measued closely appoaches the volume-equivalent diamete when the paticle size is smalle than the light wavelength and the pojected aea diamete in the Faunhofe egime. In geneal, the esults depend on paticle size, shape and oientation, as well as light wavelength, flow ate and elative efactive index. 6

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 9 CPPS 9. Image analysis I Fee paticles, quantitative size and shape detemination 9.0 Intoduction The aim of image analysis is the eduction of the complex visual infomation contained in images to easily intepetable quantitative infomation in the fom of simple gaphs (e.g. size distibutions) o even a few numbes (e.g. aveage o mean values). Thus, inevitably, some infomation is lost, and the use must ensue that the essential infomation is extacted. Pojection is the basic technique to obtain size and shape infomation fo paticles and paticle systems via micoscopic image analysis. Thus, in the case of anisometic paticles (disks / platelets o needles / fibes) it has to be taken into account that the oientation duing measuement is usually not andom, unless special methods of sample pepaation ae used. Theefoe, fo example, the thickness (height) of platelets is usually not accessible via image analysis. Apat fom this, sample pepaation has to ensue that agglomeation of paticles is avoided as fa as possible (difficult mainly fo fibes and nanopaticles). Since the esolution limit of optical micoscopy (light micoscopy) is of ode 1 µm, paticles of such a size and smalle should be chaacteized by SEM (down to appox. 10 nm), TEM (down to appox. 1 nm) o scanning pobe micoscopes (scanning tunneling micoscope STM, atomic foce micoscope AFM). Due to diffaction finges occuing fo small paticles, only dimensions lage than a few µm can be eliably measued by optical micoscopy. Image analysis is taditionally pefomed with static micogaphs, although dynamic eal-time o even in-line measuement systems ae available today. Image analysis can be done manually (by selecting and making each object by hand, i.e. via the use inteface, e.g. the compute) o automatically. Automatic image analysis geneally equies much highe image quality (e.g. contast) and usually also image pocessing (i.e. digital image modification to obtain a binay image accoding to the opeato s specifications) pio to image analysis pope. Automatic image analysis (and the image pocessing steps equied) is useful fo outine measuements (mainly in industy), but is beyond the scope of this couse. 9.1 Basic size and shape measues Calipe diamete (Feet diamete): Nomal distance between two paallel tangent planes touching the paticle suface (in D) o two paallel tangents touching the paticle outline (in D); these values ae dependent on paticle oientation, theefoe a single measuement has little significance eithe measuements in all diections (fo one single paticle) o measuement of a sufficient numbe of paticles in andom oientation (if all paticles ae of the same size and shape) o measuement of two mutually pependicula nomal values fo each paticle such that the aea of the ectangle enclosing the paticle outline becomes a minimum (detemined by the min minimum Feet diamete D F and the dimension pependicula to it, the so-called max maximum Feet diamete D F ) aspect atio R fo each paticle (o aspect atio distibution fo an ensemble of paticles), 7

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 9 D R =. D max F min F (Sometimes the invese definition is used, and many othe shape factos have been defined in the liteatue geneally the definition has to be given in each context.) Fo a completely convex paticle outline in D the aveage Feet diamete D F (aveaged ove all diections) is elated to the paticle peimete P via the elation P D F =. π The calipe o Feet diametes can be consideed as 1D pojections of D paticle outlines onto a line, i.e. the aveage Feet diamete coesponds to an aveage pojection (ensemble aveage fo a system of andomly oiented paticles). Note, howeve, that the eos in detemining peimetes fom digitalized pixel images ae usually lage and theefoe D F thus calculated is usually not vey eliable, especially fo small paticles (fo altenative methods to measue the peimete by image analysis see ou couse Micostuctue and popeties of poous mateials (ICT Pague). Chod length (intecept length): Secant length inside the paticle (dependent on diection and position); the mean chod length (in D o D) of a single paticle (o identical paticles in andom oientation and position) is pincipally defined as the aveage of all (in pactice, many) paallel chod lengths in a single diection, aveaged ove all (in pactice, a few) diections. It is elated in D to the aea-topeimete atio via the elation D C A = π P and in D to the volume-to-suface atio via the elation V D C = 4. S These elations ae valid fo each single paticle as well as fo systems of paticles (intepeted as ensemble aveages). Pojected aea diamete: Equivalent diamete D P of a cicle with the same aea as the D pojection of the paticle (pojected aea A P ): D P 4 AP =. π The pojected aea can be consideed as a D pojection of a D paticle and in the case of completely convex paticles its aveage value is elated to the paticle suface S via the Cauchy elation (Cauchy s steeological theoem, 1840): 8

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 9 S A P = 4 (ensemble aveage fo a system of andomly oiented paticles). In contast to peimetes the measuement of aeas fom digitalized pixel images gives quite eliable esults. Thee types of shape factos ae commonly used: Aspect atio (axial atio, Heywood s elongation atio, 1946): Ratio of maximum to minimum Feet diamete o vice vesa, i.e. Aspect atio = maximum Feet diamete minimum Feet diamete (values fom 1 to o fom 0 to 1, depending on the definition vaiant); a measue of elongation o flattening (anisomety) of the convex hull of paticles; note, howeve, that only fo polate spheoids (ods) the aspect atio detemined via image analysis in D is (close to) the tue D aspect atio. Ciculaity (oundness): Ratio of the peimete squaed to the pojected aea times 4 π, i.e. P Ciculaity = 4π A P (values 1, i.e. = 1 fo cicles and > 1 fo non-cicles); a combined measue of iegulaity (anisomety and non-smoothness) fo convex and non-convex paticles; an analogous shape facto in D in Waddell s spheicity facto (Waddell 19), defined as suface aea of sphee of same volume Spheicity =, suface aea of paticle which is simply the squaed atio of the volume-equivalent and suface- D. equivalent diamete, i.e. ( ) V D S Concavity (non-convexity): Ratio of the diamete of the smallest cicumscibed cicle (sphee) to the diamete of the lagest inscibed cicle (sphee) centeed in the cente-of-mass of the paticle, i.e. Concavity = diamete of cicumscibed cicle diamete of inscibed cicle ( sphee) ( sphee) (values fom fo sta-like objects to 1 fo cicles); only useful fo isometic paticles. 9

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 10 CPPS 10. Image analysis II Gains in polycystalline mateials, steeology 10.0 Intoduction Sectioning is the basic technique to obtain D images fom which infomation on D micostuctues can be extacted. Plana sections (polished sections) ae the simplest and most typical type of pobe available to investigate the micostuctue of polycystalline mateials. Othe pobes, which ae beyond the scope of this couse, ae thin sections and so-called disecto pobes and finally tomogaphic sectioning techniques. These ae teated in ou couse Micostuctue and popeties of poous mateials at the ICT Pague. Steeology can be consideed as a subdiscipline of stochastic geomety and aims at obtaining infomation on the D micostuctue fom D cuts (plana sections). Classical steeology is based on the assumption that thee exist mean values chaacteizing the micostuctue of the mateial, which ae invaiant unde affine tansfomations (i.e. non-distoting tanslations and otations). Theefoe, unless stated othewise, we assume that the micostuctue of the mateial unde investigation is isotopic, unifom and andom (IUR assumption). Moe geneal cases ae teated in ou couse Micostuctue and popeties of poous mateials. 10.1 Steeological teminology and the Delesse-Rosiwal law The basic symbols used in steeology and thei coesponding physical dimensions (units) ae P = numbe of points (e.g. pixels o intesection points in a measuing gid) N = numbe of objects (e.g. gains o poes) L = line o cuve length (of a pobe o a featue!) [m] A = aea of micostuctual featues in a plana section (always plane) [m ] S = suface o inteface aea of featues in D space (geneally cuved) [m ] V = volume of micostuctual featues in D space [m ] M = cuvatue (integal mean cuvatue) [m]. Note that N makes sense only when individual objects can be distinguished. This is not the case fo a bicontinuous micostuctue, fo example. All othe quantities ae applicable to abitay micostuctues, if the micostuctual featues of inteest ae clealy distinguishable (phases o gains). It is common pactice in steeology to wite atios as indexed quantities instead of factions. Hence, we have the following shothand notation fo steeological atios: P P = point faction = numbe of points hitting the featue (phase o gain) of inteest divided by the total numbe of points placed on the image, L L = line faction = cumulative length of lines hitting the featue (phase o gain) of inteest divided by the total length of lines placed on the image, A A = aea faction = cumulative aea of a featue (phase o gain) divided by the total aea of the measued image, V V = volume faction of a micostuctual featue in D space (when the featue of inteest ae poes, i.e. the void phase, V = φ is the poosity). V 0

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 10 In geneal, micostuctues can be anisotopic (e.g. tansvesally isotopic composites) and non-unifom (e.g. gadient mateials). In ode to obtain a eliable and coect quantitative desciption of abitay micostuctues in D, the sampling pocedue must ensue that the pobes intesect the micostuctue isotopically, unifomly and andomly (IUR equiement on the pobes). That means, pobing should take all oientations and positions into equal account and sampling should not be influenced by micostuctual systematicity (e.g. peiodicity). On the othe hand, if the micostuctue itself is isotopic, unifom and andom (IUR micostuctues), then any pobe is as good as any othe. Of couse, in both cases statistical accuacy equies additionally a sufficient numbe of measuements (and measued events ). The most fundamental steeological elationship is the Delesse-Rosiwal law, V = A = L = P. V A L P That means, the volume faction of a phase can be detemined by measuing the aea faction of that phase on a plana section o, equivalently, by using a supeimposed line gid o point gid to measue the cumulative line length o the numbe of points hitting the phase, espectively. As mentioned befoe, in the case of abitay micostuctues, pobing (sampling) must satisfy the IUR equiement, while fo IUR micostuctues, one plana section is in pinciple sufficient if only the numbe of measued events (hits) is lage enough to make the atio accuate fom the viewpoint of statistics. Note that, fo easons of simplicity, we omit angula backets with steeological quantities, but with the implicit undestanding that steeological theoems have to be intepeted as expected value theoems and the coesponding micostuctual quantities as estimatos moe details see ou couse Micostuctue and popeties of poous mateials. Othe fundamental atios used in steeology ae P L = numbe of intesection points with featue lines o cuves (e.g. gain section peimetes) pe unit length of a pobe line (e.g. a line in a supeimposed gid) [m 1 ], N L = numbe of objects pe unit length of a pobe line [m 1 ], N A = numbe of objects pe unit aea of a plana section [m ], N V = numbe of objects pe unit volume in D space [m ], L A = cumulative length of featue lines o cuves (e.g. gain section peimetes) pe unit aea in a plana section [m 1 ], L V = cumulative length of featue lines o cuves pe unit volume in D space [m ], sometimes misleadingly called specific line length (should be: line length density o shotly line density ), S V = cumulative suface o inteface aea of featues pe unit volume in D space [m 1 ], sometimes misleadingly called specific suface aea (should be: suface aea density o shotly suface density ) M = cuvatue density (integal mean cuvatue pe unit volume) [m ]. V Note that the topological chaacteistics N V (numbe density) and C V (connectivity density, not teated hee), povide basic infomation on abitay D micostuctues, iespective of the objects being convex o even well-defined inclusions o not Eule chaacteistic, measues of cuvatue (see ou couse Micostuctue and Popeties of Poous Mateials ) 1

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 10 10. Basic steeological elations Usually the quantities pe unit volume (i.e. N V, L V, S V, M V ) cannot be diectly measued diectly (except fo N V, when the disecto pobe is used see ou couse Micostuctue and popeties of poous mateials ) and it is one of the main tasks of steeology to detemine them indiectly via the othe, diectly measueable, quantities ( P L, N A, L A ). This is done using the following steeological elations: 4 Suface aea density (fom line intecept count o length pe aea): SV = PL = LA, π Line length density (fom the aea object o point count): LV = N A = PA, Cuvatue density (fom the aea tangent count): M V = π TA. The following aveages can be defined fo all micostuctual featues: Mean intecept length in D [m]: Mean coss section [m ]: A A = N A A D C 4V V = (in D: SV π V = M V V π mean pojected aea ) mean peimete Mean suface cuvatue (aveage cuvatue of the whole suface) [m 1 ]: M V H =. If the micostuctual featues ae individual objects (i.e. the micostuctue is an ensemble of objects), the following aveages can be defined additionally: Mean volume [m ]: V = V V NV Mean suface [m ]: S = S V NV Mean calipe diamete ( mean height, defined only fo convex bodies but of N A M V othewise abitay shape) [m]: D = = NV π NV Mean chod length [m]: Y = L L N L The numbe of objects pe unit volume can be eithe measued diectly fom tanspaent slices (via the so-called disecto count, which is a D equivalent of the D aea tangent count used to detemine the cuvatue density) o detemined fom the mean diamete (if known) S V N V N A π M V = =. D D In most cases, howeve, the size distibution (and thus the mean diamete) is not known a pioi, and the invese poblem must be solved (so-called unfolding of size distibutions).

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 10 10. Unfolding size distibutions Without shape assumptions it is not possible to infe the tue D size (distibution) of (convex) objects (inclusions, e.g. gains o poes, in a matix) fom measuements on D plane sections. Fom the mathematical viewpoint the poblem is ill-posed and ill-conditioned, i.e. small statistical vaiations in the D measuements lead to lage eos in the D esults. Even if it is assumed that the shape is the same fo all objects (inclusions) and size-invaiant, the solution is not tivial. When the objects (inclusions) ae sphees, one could in pinciple apply a simple gaphical unfolding pocedue (Lod-Willis pocedue, unfolding of linea intecepts), which is based on the fact that the fequency distibution of linea intecepts in a single sphee is a tiangula distibution ( see ou couse Micostuctue and popeties of poous mateials ), but the lage eos intoduced in the small-size faction devaluates the method in pactice. An altenative method is based on the simultaneous solution of a system of equations (Saltykov method). In pactice the lagest diamete obseved in the plana section is assumed to be the tue maximum sphee diamete and based on this value, the whole size ange is subdivided into n size classes (usually up to 10 15 classes of bin width δ ). Then the n - dimensional vecto N Ai (i.e. the N A values fo each size class, coesponding to the measued cicle diametes) is tansfomed into anothe n -dimensional vecto N Vj (i.e. the equied N V values fo each size class, coesponding to the desied tue sphee diametes) via a quadatic matix (Saltykov-matix, see below). Using the Einstein summation convention, this Saltykov-tansfomation can be witten biefly as N Vj 1 = α ij N δ Ai. Simila tansfomation matices ae available fo othe isometic shapes as well (e.g. polyheda and cylindes), but it has to be emphasized that the esults ae only easonable when the assumption of a unique, size-invaiant shape is at least appoximately fulfilled. In the case of otational ellipsoids (spheoids), which can be useful model shapes fo all kinds of anisometic inclusions a genealized vesion of the Saltykov tansfomation is available (DeHoff-tansfomation), N Vj ( R) K = α ij N δ Ai, whee K is a function of the aspect atio (axial atio) R (hee defined to be always < 1; the values of K ( R) ae > 1 fo polate and < 1 fo oblate spheoids). In ode to decide fom a D plana section whethe the shape is polate o oblate, one consides the most isometic (equiaxed) sections. If thei diamete coesponds to the length of the anisometic sections, the inclusions (e.g. gains o poes) ae oblate, othewise polate. It is in pinciple thinkable to extend this matix unfolding appoach to convex featues that do not all have the same shape (using a fouth-ode tansfomation matix). Statistical eos, howeve, devaluate this idea in pactice.

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 10 Figue 10.1. Saltykov matix. 4

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 CPPS 11. Cystallite size fom XRD line pofile analysis 11.0 Intoduction The shape and boadening (beadth) of XRD line pofiles povides micostuctual infomation, including the aveage size, size distibution and shape of cystallites in the ange 5 100 nm (i.e. cystalline nanopaticles and nanocystalline mateials), lattice faulting / twinning and density and spatial aangement of dislocations. This micostuctual infomation is geneally convoluted togethe and can be gouped into size-boadening and stain- / distotion-boadening contibutions. Line pofile analysis attempts to extact micostuctual infomation fom the obseved (ecoded) line pofiles. This commonly involves the solution of an invese poblem (integal equation), fo which questions of existence and uniqueness may aise. Theefoe, cuently, line pofile analysis still is a field of intense eseach activities. Concomitantly with the pogess in nanotechnology and the need fo a staightfowad outine size and shape analysis in the nanosize ange, thee ae many attempts to develop and select suitable standad mateials, and to compae the size distibutions and shape infomation obtained by XRD methods with the esults of image analysis using SEM, TEM and possibly, scanning pobe micoscopes such as the scanning tunneling micoscope (STM) and the atomic foce micoscope (AFM). XRD line pofile analysis seems to be a pomising tool to be applied in fast and convenient outine measuements (as soon as eliable evaluation pocedues ae established and the physical intepetation of the esults is claified). 11.1 Schee fomula and Williamson-Hall plot When the cystallite size is smalle than appox. 100 nm XRD line pofile boadening becomes a measuable effect. The classic elation used fo the calculation of cystallite size is the Schee fomula: K λ D Schee =, β cosθ whee D Schee is the volume-weighted appaent cystallite size, β size the integal beadth of the line pofile (XRD peak) caused by small cystallite size, θ the diffaction angle, λ the X-ay wavelength (e.g. 0.154 nm fo CuK α1 adiation) and K the (shape-dependent) Schee constant (e.g. 0.94 fo cubic cystallites). Howeve, apat fom geneic instumental boadening, anothe impotant cause fo boad XRD peaks is stain boadening (distotion boadening caused by dislocations, stacking faults and twins). This boadening is elated to the micostain ε = d d (with d being the inteplana distance between (hkl) planes) via the elation β ε size = 4ε tanθ. Due to the diffeent angula dependence of the two boadening effects, they can in pinciple be distinguished. Taditionally this is done using the Williamson-Hall plot, based on the elation 5

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 * β β cosθ = = λ K D 4ε + sinθ, λ * whee β is the obseved integal beadth. When β values ae plotted against sin θ and fitted by linea egession, a line is obtained with a slope popotional to micodefomations and an intecept (extapolated to θ = 0 ) invesely popotional to the cystallite size D. Schee 11. Theoy of nanocystallite size boadening of X-ay line pofiles Constuctive (i.e. non-destuctive) intefeence of the scatteed (diffacted) X-ay beam in a cystal occus only when the Bagg condition is met: s f s = g, i whee the wave vectos s f and s i efe to the diffacted and incident wave, espectively (with s f = s i = 1 λ, whee λ is the X-ay wavelength), and g is the diffaction vecto with g = 1 d and d being the inteplana spacing (inteatomic spacing fo a set of ( hkl ) planes). Taking into account the angle between s f and s i the Bagg condition can be witten as d sinθ 0 = λ whee θ 0 is the Bagg angle. The esulting diffaction spot (in ecipocal space) is and intensity distibution (in D) which is infinitesimally naow and of infinite intensity. Diffaction of monochomatic X-ays fom an ideal cystal of infinite size ( D ): I ( s) Aδ ( s) =, whee δ (...) denotes the Diac delta distibution; this elation follows fom I () s ( πds) sin A πds, whee the ecipocal space unit is defined as: s = ( sinθ sinθ 0 ). λ Fo spheical cystallites with finite size (diamete D < 100 nm) the size-boadened line pofile (Fouie tansfomation of the cystallite) is + τ ( s D) = V ( t, D) I, cos πst dt, τ 6

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 whee t defines the displacement of a cystallite shifted paallel to the diffaction vecto, V ( t, D) is the so-called common-volume o ghost function of the cystallite and the dimension τ defines the maximum thickness of the cystallite paallel to the diffaction vecto g (fo anisometic cystallites a function of the cystallite size D in the espective diection; thus, pincipally, shape infomation can be extacted fom consideing the line pofile diffeences at diffeent angles). The diffaction spots and line pofiles ae Fouie tansfoms of the cystallites (as a diect consequence of the fa-field appoximation). Theefoe small cystallites coespond to boad diffaction spots (line pofiles). The Fouie coefficients A () t ae whee ( 0) ( t,0) ( 0) V A () t =, V V is the volume of the cystallite. Based on these Fouie coefficients, an aeaweighted aveage cystallite size can be defined as ( ) 1 d A t t A =. d t Altenatively, the integal beadth β (a diect consequence of the additivity of I ) can be defined in ecipocal space units as t= 0 1 + τ 1 β = I( s, D) ds = A( t) dt, I mex 0 and, based on this integal beadth, anothe size measue, the volume-weighted aveage size can be defined as 1 t V = β. Both t A (column length) and t V (domain size) coespond to the appaent size (thickness) of the cystallite in the diection of the diffaction vecto g. A shape assumption is equied to intepet them in tems of physical cystallite dimensions. This geneally equies detemining the Schee constant in ode to elate them to the actual thickness τ of the cystallite, paallel to g. Given a diffaction patten consisting only of size-boadened line pofiles, each (hkl) defines the thickness of the cystallite in a paticula diection. Using this infomation, an aveage cystallite shape can be obtained. In a nanocystalline mateial with cystallites of the same shape, but with a distibution D s I s, D, of sizes P ( ), the size boadening, f ( ), consists of a supeposition of line pofiles, ( ) weighted by P ( D), i.e. 0 () s = I( s D) P( D) f, dd. 7

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 The modified common-volume function now involves the contibution of cystallites in the size ange between D and D + dd, i.e. ~ V and the Fouie coefficients ae given as t () t = V ( t, D) P( D)dD ~ V A () t = ~. V The difficulty in detemining P ( D) fom the obseved line pofile, g () s, is that thee ae othe contibution to line boadening which must be emoved (instumental boadening, backgound and noise). The aea- and volume-weighted aveage sizes (appaent theicknesses) can be elated to a spheical nanocystallite model aea- and volume-weighted aveage diametes: ( t) ( 0) D A = t A, 4 D V = t V. The atio of the two aveage sizes also gives a qualitative indication as to the pesence of a cystallite diamete distibution: if t A tv = DA DV = 8 9 0. 89 fo spheical cystallites, then the size distibution is a Diac delta-distibution about D 0, i.e. all cystallites gave the same size. Othe values of this atio ae indicative of a size distibution with finite beadth. When a log-nomal size distibution is assumed fo spheical cystallites, i.e., ( D D ) 1 1 ln 0 P ( D) = exp ln, π D ln σ σ 0 0 D A and D V can be elated to the median diamete D 0 and the log-nomal vaiance σ 0 [Kill & Biinge 1998]. D V coesponds to the cystallite size detemined by the Schee fomula. A new method of cystallite size detemination, poposed by [Kužel et al. 004] and usable fo thin films (foils), is based on the measuement of the small-angle diffuse scatteing of the tansmitted wave. Standad efeence mateials fo XRD line pofile analysis: NIST SRM 660 a LaB 6 powde fo coection of instumental boadening NIST SRM CeO powde (cubic, isometic-spheical) in the size ange 10-0 nm with aveage cystallite diamete 17 nm (in pepaation) NIST SRM ZnO powde (hexagonal, pismatic-cylindical) in the size ange 40-60 nm (in pepaation). 8

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 Line-boadening analysis (LBA) is the method to extact cystallite size and inhomogeneous stain (associated with local lattice defomations due to defects) fom diffaction data two main phenomenological appoaches elated to two diffeent cystallite-size dimensions: One based on the integal beadth of the diffaction lines volume-aveaged appaent dimension in the diection nomal to the eflecting planes (domain size) D V [Klug & Alexande 1974], One based on the Fouie analysis of the line pofile aea-aveaged appaent dimension in the diection nomal to the eflecting planes (column length) D A [Betaut 1949]. The second deivative of the Fouie coefficient is elated to the column-length distibution function, while detemination of the eal cystallite size distibution includes the thid Fouie coefficients lage eos and uneliable because of the appoximations inheent in sizestain sepaation appoaches. An unbiased detemination of cystallite size and stain can be undetaken only if the diffaction lines do not ovelap. Othewise, patten fitting and decomposition must be made befoe LBA. In the simplest case line pofiles can be consideed as a convolution of two types of analytical functions: Gauss functions, loosely associated with stain boadening, Loentz functions (Cauchy functions), loosely associated with size boadening. It is widely accepted that the best analytical fucntion fo pofile fitting is a convolution of the two, i.e. a Voigt function. Peason VII and pseudo-voigt functions wee then intoduced as satisfactoy appoximations to the Voigt function, but faste to evaluate advantage in Rietveld efinement. One of the most fequently used functions fo pofile fitting is the Thompson-Cox-Hastings pseudo-voigt function, whee the full widths at half maximum (FWHM) of the Gauss and Loentz components ae constained to be the same and equal to the width of the pseudo-voigt function itself. Note, howeve, thee is no a pioi eason to believe that a simple Voigt function can sucessfully descibe all size- and stain-boadening effects. Fo example, supe-loentzian line pofiles (with tails falling off moe slowly than the Loentzian function) can be moe adequately modeled by a pioi assuming a size distibution, e.g. multimodal o boad log-nomal. The log-nomal size distibution fo spheical cystallites is chaacteized by two paametes, the aveage adius R and the dispesion σ R o, altenatively, the dimensionless atio c = σ R : R f () = 1 π ln 1 ( + c) Volume-aveaged and aea-aveaged diametes: ln exp ln ( 1 c) R + D V = ( 1 c) 4R + D A = 1 ( R 1+ c ) ( 1+ c). 9

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 11 In pinciple, the values D V and D A can be used to calculate the distibution paametes R and c fo the log-nomal distibution. Howeve, except fo mateials with highest symmety the line pofiles commonly ovelap and have to be econstucted (by fitting with simple analytical functions such as the Voigt function o its appoximations o by whole-patten fitting ), befoe Fouie analysis of the line pofiles can be pefomed. When a Voigt function is used the following inequality holds between D and V D : 1 DV π e efc 1.1 <. DA The lowe limit of this atio (which ensues that the column-length distibution function is always positive) constaints the dispesion paamete c 0. 164, i.e. the Voigt function is not appopiate fo vey naow distibutions (this lowe limit is not necessay fo the pseudo- Voigt function, which ensues positivity of D A automatically). Both the Voigt and the pseudo-voigt function ae inadequate fo vey boad distibutions ( c > 0. 4 ). Assuming a pioi a log-nomal size distibution leads to easonable line pofile fits. A Execise poblems 1. Fo a given XRD patten detemine the cystallite size and micostain using the Williamson-Hall plot (neglecting instumental boadening).. Calculate the aveage cystallite adius R and the dispesion paamete c fo cubic ceia (CeO ) when the domain size D V is. nm and the column length D A is 16.8 nm and a log-nomal size distibution is assumed (solution: 9.0 nm and 0.181).. Calculate the domain size D V and the column length D A fo cubic ceia (CeO ) when the cystallite size distibution is log-nomal with and aveage cystallite adius R of 16.8 nm and a dispesion paamete c of.8 (solution: 140.8 nm and.8 nm). 40

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 CPPS 1. Adsoption methods and mecuy intusion 1.0 Intoduction The adsoption of gas molecules in a gaseous medium (o solute molecules in a liquid medium, the solvent) can be used to chaacteize the suface of poous and / o paticulate mateials. Also mecuy intusion, which is commonly used to detemine poosity and poesize distibutions, can unde cetain cicumstances be used to chaacteize paticle size. Specific suface aea measuement by adsoption methods is most fequently based on the BET model (198), while paticle sizing by mecuy intusion is usually based on the socalled Maye-Stowe theoy (1965). 1.1 Pinciples of adsoption, classification of isothems, BET method The specific suface aea [m /g] of a powde o a poous mateial can be tansfomed into suface density [m /cm ] when the helium-pycnometic density [g/cm ] is known. Since the helium-pycnometic density diffes fom the tue density (e.g. the theoetical X-ay density of cystalline substances) when closed (isolated) poes ae pesent, all these chaacteistics usually efe only to the open void space accessible fom outside. When a solid suface is exposed to a gaseous atmosphee, gas molecules impinge on the suface and a cetain pecentage (depending on the patial pessue of the gas) of them sticks to the suface, i.e. is adsobed. The equilibium amount a of adsobed gas in [moles/g], i.e. in moles of adsobate on 1 g of adsobent, o in [cm at STP/g], i.e. in volume of adsobate at standad tempeatue (0 C = 7 K) and pessue (1 atm = 101. kpa) on 1 g of adsobent, as a function of elative pessue x = p p0 (whee p 0 is the satuation vapo pessue, e.g. 101. kpa fo nitogen at 77 K) at a cetain constant tempeatue, is called an adsoption isothem. Thus, the adsoption isothem epesents a dynamical equilibium situation between condensation on and evapoation fom the solid suface. Accoding to the IUPAC classification, poe sizes can be divided into micopoes (< nm), mesopoes (-50 nm) and macopoes (> 50 nm). In mico- and mesopoous mateials the adsobed gas can attain a liquid-like state (capillay condensation) and thus the amount of gas molecules appaently adsobed geatly exceeds that needed fo monolaye (o even multilaye) adsoption unealistically high suface aea values. Six types of adsoption isothems ae commonly distinguished, the most impotant of which ae: Type I (Langmui): fo micopoous mateials; monolaye adsoption, with a nealy hoizontal pat, fom which the amount of absobate in the monolaye (monolaye capacity) a m can be ead off and the specific micopoe volume [cm /g] can be estimated as V mico = a v, m whee v is the mola volume of the liquid adsobate (e.g. 4.6 cm /mole fo liquid nitogen at 77 K). 41

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 Type II (Bunaue-Emmett-Telle, BET): fo nonpoous o macopoous mateials; multilaye adsoption, which can be descibed in the elative pessue ange 0.05 < x < 0. via the equation a a = Cx ( 1 x)( 1 x Cx) m + whee C is a constant elated to the diffeence between heat of adsoption and heat of condensation. This BET equation can be ewitten in the altenative fom, a x ( 1 x) = 1 a C m + C 1 x. a C m When the l.h.s. of this equation is plotted against x and fitted by linea egession, the values of the monolaye capacity a m and the BET constant C can be detemined fom the slope s and the intecept i ( a m = 1 ( s + i) and C = 1 + s i ). As soon as a m is known, the specific BET suface can be calculated as S a N σ, BET = m A whee N A is Avogado s numbe (6.0 10 molecules pe mole) and σ the aveage coss-sectional aea of the adsobate gas molecule (e.g. fo nitogen 0.16 nm pe molecule). Note that the S BET value calculated fom type II (and IV) isothems coesponds to a ealistic specific suface value only when micopoes ae absent. High values of the BET constant C (i.e. shap knees in the isothem) usually indicate the pesence of micopoes and devaluate the esults. On the othe hand, values C < esult, then the isothems ae of type III o V (i.e. thee is no knee in the isothem) and indicate that the inteaction between adsobate and adsobent is weak. Also in this case the esults have to be discaded. Specific suface deteminations via so-called single-point methods (e.g. using the chomatogaphic Nelssen-Eggetsen technique, which does not equie vacuum equipment) ae based on the simplifying assumption that C. In othe wods, the intecept in the BET gaph is neglected and a staight line is dawn fom the point in question to the oigin; the monolaye capacity is then calculated fom the slope of this line as a m = 1 ( s + i) 1 s. The pinciple of this technique is that fom a gas mixtue containing nitogen and hydogen o helium, nitogen is pefeentially adsobed when the dy sample is cooled down by inseting the sample flask into liquid nitogen. A themal conductivity detecto is used to detemine the concentation change (patial pessue change) in the gas mixtue and thus the patial pessue of the adsobate. In pactice this technique is used mostly as a compaative technique, i.e. using a efeence standad of known specific suface S ef. The sample specific suface S is then calculated fom the peak aeas (signal esponses) of the efeence A ef and sample A (with weights W ef and W, espectively) as ef ( A W ) S =. S ( A W ) ef ef 4

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 Type IV isothems exhibiting hysteesis ae chaacteistic of mesopoous mateials exhibiting capillay condensation at sufficiently high elative pessues (> 0.4) and allow the detemination of the specific suface S BET as well as the poe size distibution (of mesopoes; note that mecuy poosimety yields the size distibution of macopoes). The basic equation fo poe size detemination is the modified Kelvin equation γ v cosθ = τ ( x), RT ln x whee is the (equivalent) poe adius, γ and v the suface tension and the mola volume of the liquid adsobate (fo liquid nitogen γ = 8. 88 10 - N/m at 77 K) and τ ( x) the equilibium thickness of the adsobed film (befoe capillay condensation o duing o afte evapoation), fo which empiical appoximations ae available, e.g. when x > 0. 65 fo nitogen (with τ in nm) the Halsey elation, 1 5 τ ( x) = 0.54, ln x o simila elations. Since duing nitogen adsoption at 77 K nitogen is pefectly wetting, θ = 0 and thus cos θ = 1. By calculating the values fo diffeent elative pessues (usually fom the desoption isothem) yields a poe size distibution. Note that, simila to mecuy poosimety, this poe size detemination is based on the idealized model of staight, non-intesecting, cylindical poes (capillaies) with cicula coss-section. That means, fo a eal system (poous mateial) the esults have to be intepeted in tems of equivalent diametes. Note that the Kelvin equation is valid only as long as the continuum appoach is justified, i.e. at best down to adii of appox. 1 nm (i.e. not fo micopoes). Isothems of type III (fo nonpoous mateials) and type V (fo mesopoous mateials) may occu when the inteaction between adsobate and adsobent is weak. They ae less common in pactice (simila to the stepped type VI isothem) and no standad evaluation pocedues ae available to extact micostuctual infomation fom them. 1. Pinciples of mecuy intusion and Maye-Stowe theoy Mecuy intusion poosimety is based on the Washbun equation, γ cosθ =, p whee γ is the suface tension of mecuy (usually conventionally taken as 0.480 N/m, although moden eseach favos 0.485 N/m as the best values fo highly pue mecuy at oom tempeatue in vacuum), θ the mecuy contact angle at the mateial suface (usually 4

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 taken as 140, although it is stongly dependent on the substate and tempeatue and vaies at least between 10 and 160 ) and p the hydostatic pessue. This equation is deived fom the foce balance π p = π γ cosθ (in wods, aea times pessue equals peimete times effective suface tension) and can fo pactical puposes be witten as = 75 p, with in nm and p in MPa. Since mecuy is a non-wetting liquid, high pessues ae needed to fill small poes (the highe the pessue the smalle ae the poes intuded by mecuy). Note that at 0.1 MPa (i.e. appoximately atmospheic pessue, since 1 atm = 101.5 kpa) poes lage than 7 µm ae aleady filled, while pessues highe than 100 MPa (appox. 1000 atm) ae needed to fill poes (o bette poe thoats, due to the idealized model of equivalent cylindical poe channels) smalle than 7 nm. Fo poes (poe thoats) lage than 7 µm a lowpessue unit with vacuum has to be used. The volume of mecuy pessed into the sample is ecoded electically with a capillay dilatomete and the esulting intusion cuve can be diectly conveted into a cumulative poe size distibution, fom which a fequency distibution can be obtained by deivation. When powde beds ae analyzed by mecuy intusion, the peak at low pessues coesponds to the size (usually in the micomete ange) of the intestitial voids between powde paticles and has nothing to do with the intenal poosity of the paticles itself. Accoding to a simple theoy by Maye and Stowe (1965) the beakthough pessue p (usually detemined fom a e-intusion cuve, afte the agglomeates have been destoyed by the fist intusion) equied to foce mecuy into the void spaces between egulaly o nonegulaly packed sphees of diamete D is given by the elation γ p = κ, D whee γ is the suface tension of mecuy (hee assumed to be 0.485 N/m) and the (dimensionless) Maye-Stowe constant κ depends on the mecuy contact angle at the paticle suface and on the intestitial void space (intepaticle poosity). Maye and Stowe consideed egula packings fom simple cubic (void faction 0.48) to close packed (fcc o hcp, void faction 0.6), fo which κ vaies in the wide ange fom appox. 5 to 17 (assuming a contact angle of 10 ). Fo andom packing (void faction appox. 0.6) the κ value is appox. 10. Expeimental measuements by Pospěch & Schneide (1989) have indeed confimed this value as a easonable estimate e.g. fo compaison with mean paticle sizes ( integal mean size and mode) detemined fom image analysis. Fo compaison with sedimentation data the same authos ecommend a κ value lowe by appox. 40 % (i.e aound 7). Howeve, expeimental scatte is geneally high (coesponding to κ values of 8-1 fo compaison with image analysis and 6-10 fo compaison with sedimentation), and thus the aveage paticle size calculated via Maye-Stowe theoy should be consideed only as a ough estimate. 1. Detemination of factal dimensions Concepts of factal geomety, elaboated by Mandelbot fom the late 1960es onwads, have been successfully applied in the study of solid sufaces. Factal objects ae self-simila, i.e. they look simila at all levels of magnification (at least in a cetain ange of length scales). The degee of oughness (topogaphy) of iegula sufaces can be chaacteized by the factal dimension D, which may diffe fom the Euclidean dimension of the suface d =. The factal dimension of an iegulaly shaped solid can vay between and, depending on 44

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 suface oughness and / o poosity, i.e. the suface of such solids can in some sense be almost volume-filling. By analogy, a factal cuve has 1 < D <, i.e. can almost fill an aea. Thee exist seveal expeimental methods to detemine the factal dimension, e.g. small-angle X-ay scatteing (SAXS), small-angle neuton scatteing (SANS), adsoption techniques and mecuy poosimety. Real solids have suface aeas which ae popotional to D, whee D is the factal dimension anging fom (fo pefectly flat sufaces) to (fo extemely ough sufaces). Well-defined mathematical examples of factal objects ae the 1D Koch cuve (with D = log 4 log = 1.6... ), the Koch suface (tanslated Koch cuve) in D (with D = 1 + log 4 log =.6... ), the D Menge sponge (with D = log 0 log =.7... ) and the D Menge sieve (with D = log 8 log = 1.89... ). Note that the tanslated Menge sieve in D has a factal dimension of D = 1 + log8 log =.89..., which is highe than that of the D Menge sponge, coesponding to the fact that its solid volume is lage. The Koch suface is an example of a low-d suface (vey ough, but not volume filling), the Menge sponge of a high-d suface (almost volume-filling). Cuves, sufaces and volumes can be measued by isometic, egula 1D, D o D yadsticks (usually hypecubes o hypesphees) chaacteized by a linea size. The size of the object N () is then chaacteized by the scaling law of the type N D ( ) C =, (powe law), letting 0, whee D is the factal dimension and C is a constant, which 1 1 π equals L, A and V in 1D, D and D, espectively. This elation can be ewitten to π 4 define the factal dimension as ( ) log N D = lim. 0 log In pactice, (specific) suface aeas (o suface aea densities) ae usually measued via the monolaye capacity n, i.e. the numbe of moles n of adsobate coesponding to the fomation of a monolaye on the adsobent. Theefoe the factal dimension can in pinciple be detemined by using diffeent (spheical) molecules, accoding to the elation n D. When the adsobate molecules of a seies ae not spheical (but geometically simila), it is moe useful to use the effective coss-section σ, i.e. n D σ. Factality of the suface implies a staight line of the log n - log - (o log n - logσ -) plot in the scale ange of self-similaity. In this ange the factal dimension can easily be detemined fom the slope of the line log n = ( D ) logσ + const, which elates the numbe of adsobate molecules n (in moles) in the adsobed monolaye to the effective adsobate coss-section σ (which has to be appopiately estimated). 45

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 The afoementioned method uses yadsticks of diffeent size. Fo powdes and powde compacts consisting of diffeently sized paticles, a complementay appoach can be chosen: the suface aea of systems with diffeently sized paticles (each chaacteized by the mean diamete d ) is measued with the same yadstick (adsobate molecule). In this case the monolaye capacity (and thus the measued suface aea) has the dependence D n S d, and the factal dimension can be detemined fom log S = ( D ) log d + const, when d has been independently measued, e.g. by electon micoscopy. Methods based on the afoementioned elations ae called molecula tiling (Pfeife-Avni appoach). To apply this methods in pactice, the monolaye capacity has to be detemined, usually via the BET appoach. Othe methods ae based on a factal Fenkel-Halsey-Hill equation (FHH appoach) N ln o on the Kiselev equation fo the adsobate vapo intefacial aea capillay condensation (themodynamic o Neimak-Kiselev appoach), p p 0 D S LV in the egion of S LV RT = γ N max N ln p p 0 dn. Based on this equation the suface aea of the adsobed film can be intepeted as that of the adsobent that would be measued by sphees with adius m, i.e. fo a factal suface we have D S LV m, and the factal dimension can be obtained fom a plot of ln S LV vesus ln m. In pinciple a simila method can be applied to the intusion (and extusion) of non-wetting fluids. The poe size distibution measued by mecuy poosimety can be witten as dv d ( ) p dv ( p) D = dp, o equivalently, using the invese popotionality 1 p (Washbun equation), dv dp ( p) 4 Hence, the factal dimension D can be detemined diectly fom the slope of log ( dv dp) vesus log p. p D. 46

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 CPPS 1. Aeosols and nanopaticles 1.0 Intoduction Aeosols, i.e. dispesions of tiny paticles (solid paticles, liquid dops, o composite paticles with a solid coe coveed by liquid shell) in gaseous media, e.g. ambient ai, ae of impotance in envionmental and health science and technology as well as many othe fields such as meteoeology, enegy technology, powde technology, and mateials technology. They include dust, fumes, smoke, fog / mist / clouds, haze and smog. In mateials technology aeosols play a majo today in connection with gas-phase synthesis and elated synthesis outes (e.g. chemical vapo synthesis) of nanopowdes. Ubiquitous phenomena connected with aeosols (and with many suspensions) ae coagulation, i.e. the fomation of lage seconday paticles ( soft o had agglomeates, the latte sometimes called aggegates) fom small pimay paticles, as well as coalescence and sinteing. Simila to othe colloidal systems, aeosols exhibit inteesting and unique optical popeties. Nanopaticles have been used in vaious poducts fo seveal decades, but the expected incease of poduction and use of newly developed mateials makes the question of thei safety to life and the envionment inceasingly impotant. Howeve, fundamental knowledge concening the toxicity of these mateials is still missing o contovesial. 1.1 Aeosols Classification Aeosol a dispesion of solid paticle, liquid dops, o composite paticles (solid coe with liquid shell) in a gas, usually ai taditional classification of aeosols: Dust: Solid paticles fomed by disintegation pocesses (e.g. cushing, ginding, dilling), can be classified by sceening (sieving), Fumes: Solid paticles poduced by physicochemical eactions (e.g. combustion, sublimation) fom metals, typically smalle than 1 µm (too small to be sceened), Smoke: Solid paticles poduced by buning (oxidation) of oganic matte (e.g. coal, wood, oil), typically smalle than 1 µm, cannot be sized on sceens, Mist, fog, clouds: Aeosols poduced by the disintegation of liquid o the condensation of vapo spheical liquid doplets, small enough to float in modeate ai cuents (when > 100 µm dizzle o ain dops), Haze: Solid paticles in the atmosphee (in the pe-condensation state of ai) with adii < 0.1 µm (Aitken coes / nuclei) gown in the pesence of atmospheic moistue; foggy haze = equilibium state achieved by condensation of moistue on lage (0.1 µm < R < 1 µm) and giant (> 1 µm) paticles (condensation nuclei) of soluble salts, Smog: Combination of smoke and fog, esulting fom impuities in the atmosphee (poducts fom photochemical eactions o volcanic activity, smoke fom wood fies etc.) and thei combination with wate vapo, typically smalle than 1 µm. In atmospheic physics and meteoology the tem atmospheic aeosol is fequently used synonymously with atmospheic haze subdivided into continental aeosols ( haze M, consisting of insoluble soil components such as quatz o clay mineals o hygoscopic sulfates), sea aeosols ( haze L, consisting of soluble sea salt condensation coes) and highaltitude statospheic aeosols, including submicon paticles in dust layes ( haze H ). 47

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 1. Condensation and evapoation phenomena in aeosols Ealy investigations: Coulie 1875, Aitken 1880, Wilson 1897. Types of nucleation: homogeneous (condensation of vapo on clustes of simila vapo molecules, supecooling is common, fo wate e.g. down to 40 C) heteogeneous (condensation on nuclei of dissimila mateial, e.g. ions o ionic clustes o small solid paticles acting as condensation coes / nuclei). Homogeneous nucleation is a thee-step pocess: Vapo must become sufficiently supesatuated Small clustes of molecules (embyos) must fom Vapo must condense on these embyos nucleus doplet In heteogeneous nucleation the second step is omitted (nuclei can be soluble o insoluble). The theoy developed fo condensation and evapoation of liquid aeosols can also be applied to the fomation of solid aeosols (nanopaticles) by gas-phase eactions. 1. Optical popeties of aeosols Extinction: Combination of scatteing and absoption. Unde the assumptions of quasi-elastic independent, single scatteing the extinction of light is given by Bougue s law (1760), also called Lambet-Bee s law: I = I exp( a ), 0 z whee I is the light intensity (o iadiance o luminous flux), z the path fom souce to ecepto and a the extinction coefficient (also called attenuation coefficient o tubidity), which is invesely popotional to the paticle size as long as the paticle size is lage enough fo the extinction efficiency facto Q ext to be constant (asymptotically appoaching with inceasing paticle size d ). Theefoe e.g. small paticles poduce moe haze in the atmosphee than lage paticles. Fo visible light, aeosols ae most optically active in the 0.1 1 µm diamete ange whee the extinction efficiency is highest. The extinction efficiency facto is defined as the atio of enegy flux extinguished by a single paticle to that incident on this paticle. Fo vey small paticles with even a little bit of absoption Qext Qabs, while fo pue scattees Q = Q is given by ext scat whee Q 8 4 N 1 = = ext Qscat α, N + 48

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 π D α =, λ with D being the paticle diamete. Fo α << 1 this special case of light scatteing is called Rayleigh scatteing (scatteing of light by molecules making up the atmosphee; blue light is scatteed about 9 times as efficiently as ed light blue sky at daytime, ed sky duing sunise and sunset). Fo lage paticles the elationship of Q ext on D is moe complex, fist ising fom zeo accoding to the fouth-powe elation, then oscillating (mainly fo pue scattees; absobing paticles exhibit essentially no oscillations, only a slight oveshoot to values above ) and finally attaining the asymptotic value of. The asymptotic value of implies that a paticle can emove light fom an aea equal to twice its coss-section ( extinction paadox ). In the case of a polydispese aeosol we have fo the extinction coefficient π a = D f ( D) QextdD 4 and fo the paticle mass concentation (mass pe unit volume) π C = D f ( D) dd 6 ρ, whee ρ is the paticle density (typically.4.7 g/cm ) and f ( D) dd the numbe of paticles pe unit volume having diametes between D and D + dd, i.e. a pobability density function with the nomalization condition (total numbe of paticles pe unit volume) f ( D) dd = f. Theefoe, in the case of lage paticles ( Q ext = ) the paticle mass concentation and the extinction coefficient ae elated via the suface mean diamete (Saute mean diamete) ( D) ( D) ρ D f dd ρ C = γ = DSautea D f dd. If the polydispese aeosol can be descibed by a lognomal size distibution with geometic mean diamete D and geometic standad deviation σ, then G G (.5ln σ ) ρ a D G exp C = G. Note, howeve, that the aveage size of atmospheic aeosol paticles can vay makedly, depending on the humidity of ai (and thus the moistue content of the paticles). Fo soluble nuclei this can be even moe complicated due to the hysteesis effect, which leads to diffeent diametes fo ising and falling humidity, especially when the ai is moving. Theefoe, mass concentation measuements deived fom extinction measuement should be consideed eliable only fo cases whee the elative humidity is less than 40 %. 49

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 The fundamental optical paamete fo aeosols is the efactive index, defined as the atio of the speed of light in vacuum to the speed of light in the mateial. When thee is appeciable absoption of adiation in the aeosol paticle in addition to scatteing it is necessay to conside the complex efactive index: N = n in = n iκ, whee in the case of visible light the eal pat n is fo aibone paticles mostly between 1. and 1.7 (1.0 fo vacuum and ai, 1. fo wate, 1. fo ice, 1.7 fo alumina,.4 fo diamond) and the imaginay pat between 0 and 1 (typically of ode 0.1). Cabon paticles have a complex efactive index of appox. N( cabon) = i. Note that classical Mie theoy (1908) consides ccatteing by homogeneous spheical paticles [Keke 1969], but aeosol paticles ae inhomogeneous, consisting of at least two layes (coe + shell) [Babenko et al 00]. The eal and imaginay pats of the complex efactive index of the composite aeosol paticles (coe paticles coveed by with wate films) can be calculated as follows: whee ( n )( ) 1 n = n γ, w + 0 n w 1 ( κ κ )( ) 1 κ = κ γ w + 0 w 1, ρ 0mw γ = ρ m w 0 ρ0 = µ ρ w ( ϕ) ϕ, 1 ϕ with ϕ the elative humidity and ( ϕ) µ the coefficient of mass incease due to the wate film, which is given in Table 1 [Hanel 1976, Babenko et al. 00]. ϕ µ ( ϕ) 0.1 0.050 0. 0.065 0. 0.080 0.4 0.095 0.5 0.110 0.6 0.17 0.7 0.140 0.8 0.146 0.9 0.150 0.95 0.156 0.975 0.160 Altenative expessions use the knowledge of the size change of aeosol paticles in compaison to thei dy coes (with D 0 being the dy paticle diamete and D ( ϕ) the aeosol paticle diamete at elative humidity ϕ, which can be descibed e.g. via a elation of the type 50

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 D ( ϕ) D 0 ( ϕ) ε = 1 whee ε is appox. 0.5 ± 0.08 (0.17 fo continental aeosols, 0. fo sea aeosols) [Kasten 1969, Babenko et al. 00]. Then we have fo the composite density [Hanel 1976] ( ϕ) = ρ + ( ρ ρ )( D( ϕ) ) ρ = ρ D, and fo the eal and imaginay pat of the complex efactoy index, espectively: w 0 w, 0 n ( ) = n + ( n n )( D( ϕ) ) = n ϕ D, w 0 w 0 ( ϕ) = κ + ( κ κ )( D( ϕ) ) κ = κ D. w Popula size distibution models fo aeosols ae the (Junge-type) powe distibution, f 0 w b ( R) a R =, the genealized (modified) fou-paametic gamma distibution, and the lognomal distibution, f α β ( R) a R ( b R ) = exp, 0 ( R R ) c 1 ln 0 f ( R) = exp, π R lnσ lnσ whee a, b, c, α, β ae fit paametes, R 0 is the median adius and ln σ the standad deviation. 1.4 Coagulation phenomena in aeosols and nanopaticle systems Due to thei small size and lage specific suface aea the shot-ange suface foces such as electostatic (Coulombic) o van de Waals foces can ovecompensate long-ange volume foces such as gavitation and thus aeosol paticles in the atmosphee (and similaly suspensions of nanopaticles dispesed in liquids) exhibit a geneal tendency to inteact with each othe and to coagulate. When the elative motion among paticles is caused by Bownian motion (Bownian diffusion), the pocess is called Bownian coagulation. It is always pesent, but usually pedominates (ove othe causes of coagulation) only when the paticle size is vey small (< 10 nm). When the elative motion aises fom extenal foces such as gavity, electical foces, aeodynamic foces and ultasound the pocess is called kinematic coagulation (gavitational, electostatic, tubulent and sonic coagulation o agglomeation, espectively). The ultimate goal of coagulation theoy is to descibe how paticle (numbe) concentations, paticle size (distibutions) and coagulation ates change with time. The simplest theoy, going back to Smoluchowski [Smoluchowski 1911, 1917], who deived it 51

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 oiginally fo coagulation in dilute electolytes, concens the Bownian coagulation of a monodispese aeosol of spheical paticles [Whytlaw-Gay & Patteson 19]. The usual fom of the coagulation equation is dn dt = K B N, whee dn dt is the change in paticle numbe concentation ( N ) with time and K B is the Bownian coagulation coefficient (in the continuum egime), K kt = πd P DB, µ B = (with paticle diamete d P and gas viscosity µ ), which is elated to the Bownian diffusion coefficient D via the Stokes-Einstein elation, B D B kt =. πµ d P Invoking the initial condition that N = N 0 at t = 0, the solution of the coagulation equation is N N 1 =. 1+ K N t 0 Seveal simplifying assumptions ae implict in the Smoluchowski model (paticles adhee at evey collision, so that only the diffusional flux towad a single cental paticle acting as a sink has to be consideed, and only the fist few paticle collisions ae consideed, so that paticle size changes so slowly that it can be neglected), but expeimental data fom both monodispese and polydispese aeosols follow this geneal fom of the equation often supising well. Howeve, the coagulation constant may be appeciably lage due to othe than Bownian foces (see below) and due to polydispesivity. Redefining K = 4 K B the Smoluchowksi coagulation equation eads dn dt B 0 1 KN =. Fo the coagulation kinetics of polydispese aeosols no explicit solution exists. Howeve, fo an aeosol with discete size classes a system of diffeential equations can be set up, taking into account the incease in paticles due to combination fom smalle paticles and the coesponding loss of paticles in the smalle size class, i.e. k 1 dn k 1 = dt K j( k = j) N j N k = j j= 1 The total numbe of paticles of all sizes (pe unit volume) is equal to the sum of the numbes of paticles in the individual size classes (pe unit volume) and can be witten as all j K kj N k N j. 5

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 N =. N k all k Theefoe the total change in paticle numbe concentation is dn dt 1 = all k all j K kj N k N j. Note that no souces o sinks have been assumed fo the coagulating paticles. Fo an aeosol with a continuous size distibution the coesponding population balance is given by the following nonlinea intego-diffeential equation (thus cicumventing the poblem of many equations), dn dt ( v, t) = 1 v (, v ϕ) N( ϕ, t) N( v ϕ, t) dϕ K( ϕ, v) N( ϕ, t) N( v, t) K ϕ dϕ, 0 0 whee v and the integation vaiable ϕ denote paticle volumes (used athe than diametes, because volumes ae additive), N ( v, t) is the paticle size distibution function at time t and K ( ϕ,v) is the so-called collision kenel fo two paticles of volume ϕ and v. The collision kenel is dependent on the coagulation mechanism. To detemine the total numbe of paticles pe unit volume it is necessay to integate ove all paticle volumes, i.e. 0 () t = N( v t) N = N, dv. Then the change in the total numbe concentation with time becomes dn dt () t 1 = 0 0 K ( ϕ, v) N( ϕ, t) N( v, t) dv dϕ. As mentioned above, the collision kenel depends ion the coagulation mechanism. Fo Bownian coagulation the collision kenel can be deived eithe by the kinetic theoy of gases o by continuum diffusion theoy, dependeing on paticle size. Paticles much smalle than the mean fee path λ of the suounding gas molecules behave like molecules, and the kinetic theoy of gases must be used to deive the collision kenel, wheeas fo paticles much lage, the continuum diffusion theoy should be used. Geneally the Knudsen numbe λ Kn = is used to define the paticle size egime: in the fee molecule egime (fm) Kn > 50, in the continuum egime (co) Kn < 1. The coesponding collision kenels ae 5

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 whee ( 4 ) 1 6 ( 6 ) 1 K K 1 1 1 (, ) ( 1 ) 1 fm fm v ϕ = K B v + ϕ +, v C v 1 v 1 1 ( ) ( ) ( ) ( ) v, ϕ K v + ϕ + ϕ C ϕ 1 ϕ = co B, K fm B = π kt ρ is the Bownian coagulation coefficient fom the fee molecule egime (with ρ being the paticle density) and C (...) = 1+ Kn[ 1.14 + 0.558 exp( 0.999 Kn) ] is the gas slip coection facto. A coection f Kn fo the tansition egime (1 < Kn < 50) has been given e.g. by Dahneke [Dahneke 198]: function ( ) D f 1+ Kn =, 1+ Kn + Kn D ( Kn D ) whee D Kn D = K co v, ϕ K fm v,ϕ. The coesponding geneal collision kenel, vaild also in the tansition egime is then Kn is given by the atio ( ) ( ) K geneal B D D ( ) 1 1 C ( ) ( ) ( v) C( ϕ) v = K v + ϕ + ϕ f ( Kn D ). v ϕ, B 1 1 The collision kenels of kinematic coagulation mechanisms (gavitational, electostatic, tubulent and sonic) ae moe complicated. Depending on the conditions, gavitational coagulation (caused by the fact that the elative motion duing gavitational settling caused by diffeing paticle size leads to additional collisions) and electostatic coagulation (caused by the suface chage of paticles, leading to attaction o epulsion) may enhance o decease the coagulation ate, while tubulences always lead to additional collisions and thus tubulent coagulation (elated to the inetia of aeosol paticles) can be teated in a manne simila to Bownian coagulation except that the diffusion coefficients ae much lage (so-called tubulent o eddy diffusion coefficient). Fo paticles smalle than 10 nm Bownian diffusion usually dominates. Sonic coagulation is not well undestood but it is an empiically known fact that unde cetain cicumstances the action of ultasound can lead to agglomeation. One of the insteesting featues of coagulation is that afte sufficiently long time all coagulating aeosols ae pedicted to attain the same (quasi-) steady-state size distibution, egadless of the aeosol s initial size distibution [Fiedlande 1965, Fiedlande & Wang 1966, Wand & Fiedlande 1967]. When this so-called self-peseving size distibution is eached, gains by coagulation in the numbe of paticles of a cetain size class ae compensated by losses fom that size eithe by coagulation o by sedimentation (note that the self-peseving paticle size distibution is only quasi-steady-state, because without a paticle souce the system would eventually un out of paticles and exhibit a null size distibution function). 54

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 1 1.5 Nanopaticles and thei safety aspects Nanopaticles ae in many aspects diffeent fom thei lage-sized countepats and the coesponding bulk mateials. They ae on the same size scale as most elements of living cells, including poteins, nucleic acids, lipids and cell oganelles. Thus, nanopaticles (and nanomateials) can inteact with biological systems in an unfoseen way. On the one hand, nanosystems may be specifically engineeed to inteact with biological systems (e.g. fo paticula medical applications such as dug delivey systems). On the othe hand, the poduction of nanopaticles o thei occuence as a by-poduct of combustion pocesses may advesely affect a wide ange of oganisms thoughout the envionment. Moeove, as a esults of thei inceased eactivity (due to the lage specific suface) aeosols with nanopaticles can be highly explosive, although thei lage-size countepats ae not. The inceasing poduction, mainly of metal oxide nanopaticles (of commeical inteest ae e.g. SiO, TiO, Al O, ZO, ZnO and ion oxides) and new cabon mateials such as nanotubes (single-walled SWNT - and multi-walled - MWNT) and fulleenes (e.g. C 60 ), will enhance the possible exposue at wok places. Similaly, the nanopaticles unintentionally poduced by combustion pocesses (e.g. in diesel engines o oil bunes), ae eleased into the envionment and affect the whole population. Ambient aeosols may typically contain oganic and elemental cabon, metals and thei oxides as well as chloides, nitates and sulfates. Although man-made nanopaticles have been occuing in the envionment at least since the industial evolution and have been used in vaious poducts fo seveal decades, the expected incease of poduction and use of newly developed mateials makes the question of thei safety to life and the envionment inceasingly impotant. Howeve, although advese health effects based essentially on the size and shape of paticles have been known fo a long time fom expeience with wokes in the silicate industy (e.g. silicose fom fine quatz dust and clay mineals) as well as in the inoganic fibe industy (e.g. lung cance fom asbestos fibes), fundamental knowledge concening the toxicity of nanopaticles and nanomateials is still missing o contovesial. Without detailed knowledge of possible advese effects, nanopaticle exposue should be avoided at wok places as well as in the population and the envionment. Multiple studies (in vito and in vivo) ae necessay to claify the biological effects of nanopaticles. 55

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 14 CPPS 14. Suspensions and nanofluids 14.0 Intoduction Suspensions, i.e. dispesions of solid paticles in liquid media, ae ubiquitous in mateials technology and many othe technological and non-technological fields, including geosciences, medicine and ecology. In mateials technology casting o injection molding of suspensions ae popula shaping techniques, e.g. to fabicate ceamics o paticulate-, fibe- o plateleteinfoced plastic pats. Suspensions used in mateials technology contain paticles with a size anging fom submicon (i.e. hundeds of nm) to tens of micometes, but in pinciple the paticles can be much lage (e.g. in mud steams occuing in geosciences). Nanofluids, on the othe hand, ae suspensions with nanopaticles, i.e. paticles which ae smalle than 100 nm in at least one dimension. Due to thei small paticle size (colloidal ange), nanofluids ae usually moe stable against settling. Thei pepaation, howeve, is moe involved and agglomeation can be a majo poblem. The main field of potential application of nanofluids is as heat tansfe media. Theefoe, apat fom heology, the effective themal conductivity of nanofluids is of paticula inteest. 14.1 Suspension heology Fo an intoduction to suspension heology see Appendix-CPPS-14-A. The key point of suspension heology is the pediction of the effective viscosity of a suspension based on the knowledge of the volume faction of paticles and, possibly, paticle shape. The basic equation of suspension heology is the Einstein equation fo dilute suspensions of igid spheical paticles: η = 1+. 5 φ, whee φ is the solids volume faction and η the elative suspension viscosity, defined as η vis cosity of suspension η = =. η vis cosity of pue liquid Popula extensions of the Einstein elation to non-dilute systems ae: 1 Eiles elation: 1 1.5 1 η = + φ φ (contains the Einstein elation in the dilute limit), 1 Mooney elation: η = exp Cφ 1 φ (educes to the Einstein elation if C is chosen to be. 5 ), 0 56

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 14 1 Kiege elation: η = 1 φ (educes to the Einstein elation if N is chosen to be N.5 φ ), max whee φ is the educed volume faction, defined as φ volume faction φ = =. maximum volume faction φ max Fo anisometic paticles the Einstein elation can be genealized to the Jeffey elation, η = 1+ [ η]φ, whee the so-called intinsic viscosity [ η ] is a function of paticle shape. This complicated poblem has been solved fo dilute systems of otational ellipsoids (i.e. spheoids, polate and oblate) with Bownian motion by Jeffey (19) and fo non-dilute systems of polate paticles (fibes) with o without Bownian motion by Benne (1974) see Appendix- CPPS-14-B. Today, fibe suspension ae much bette investigated than platelet suspensions. Nevetheless, even fo fibes a pediction of suspension viscosity fo non-dilute systems is usually extemely difficult, and in pactice the Kiege elation is mostly used with N = and the maximum volume faction being linealy dependent on the aspect atio R, i.e. φ = a b R. max 14. Rheology and themal conductivity of nanofluids Pedictive elations fo the effective viscosity of nanofluids ae in pinciple analogous to those fo odinay suspensions see Appendix-CPPS-14-C. The fact (empiical finding) that the Einstein elation always undeestimates the actual viscosity incease with solid volume faction, can be accounted fo eithe by using a nonlinea elation o, sometimes, by eintepeting the volume faction in tems of an effective, equivalent, o appaent volume faction. In nanofluids, and to a cetain extent in odinay suspensions as well, the physical intepetation of this appaently enhanced volume faction may be agglomeation. Fo nanofluids, additionally, the fluid suface laye on the dispesed nanopaticles, which is known to exhibit stuctue and popeties diffeent fom the bulk fluid, may be volumetically significant. One the most inteesting models fo pactical use is the Chen model [Chen et al. 007] see Appendix-CPPS-14-C. Nanofluids usually exhibit enhanced themal conductivity in compaison to the base fluid and it is commonly ageed that thee is measuable enhancement of themal conductivity even fo vey low volume factions of solids (< 1 %). Pincipally this enhancement is simply a plausible consequence of the fact that most solids have highe sometimes consideably highe themal conductivity than the base fluid, cf. Table 14-C-1. Popula models fo the effective themal cobnductivity of nanofluids ae the Maxwell-Eucken model, the Buggeman-Landaue model and the Hamilton-Cosse model see Appendix-CPPS-14-C. 57

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Appendix-CPPS-4 CPPS-4-A. Analytical distibution functions In unit CPPS-4 we have focused on those pats of statistics which ae specific to systems of small paticles and theefoe usually not included in standad textbooks of geneal statistics. Theefoe we do not give a complete account of analytical functions that can be used to appoximate size distibutions and to pefom statistical data analysis see standad textbooks of mathematics and statistics. A few popula analytical distibution functions ae: 1. Nomal distibution (Gauss-Laplace) fo < x < F f ( x) 1 1 x = exp σ π σ 1 x 1 t x σ x A A ( x) = exp d t σ π x A = aithmetic mean (= median = mode) σ = (aithmetic) standad deviation ( σ = vaiance) The inflection points in the pobability density distibutions (fequency cuves) coespond to F ( x) = 15.9 % and F ( x) = 84.1 % in the cumulative cuve. Using pobability pape, the two coesponding x values can be ead off and the standad deviation can be calculated diectly using the elation σ = x 84.1 x15.9. Log-nomal distibution: f ( x) 1 = log σ G 1 log x log x exp π logσ G G x G = geometic mean σ = (geometic) standad deviation G. Gaudin-Schumann distibution (Závesky-Špaček, powe-law) fo 0 x xmax : F ( x) = x x max n, 58

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices n n 1 f ( x) = x. x n max The cumulative cuve of this distibution does not exhibit inflection points, and the fequency cuve does exhibit a maximum (mode). The fit paametes x max ( theoetical maximum gain size ) and n ( gain size exponent ) can be detemined gaphically by linea egession on log-log pape accoding to the elation log F (slope = n, intecept at ( x) = 100 % ( x) nlog x nlog xmax =. F gives x max ). 4. Weibull distibution (o RRSB Rosin-Rammle-Speling-Bennett distibution, a combination of exponential and powe law) fo x > 0: F ( x) m x = 1 exp, x W f ( x) = m x W x x W m 1 exp x x W m. The fit paametes (position paamete x W and Weibull modulus m ) can be detemined using double-log gaphs. This distibution can descibe sigmoidal cumulative cuves with inflection points. CPPS-4-B. Compaison of moment notation and moment-atio notation Distibution type and mean value Numbe-weighted, geometic Numbe-weighted, aithmetic (mean length) Length-weighted, geometic Length-weighted, aithmetic 1, 1 Mean suface, 0 Suface-weighted, aithmetic (Saute mean) 1, Mean volume, 0 Volume-weighted, geometic Volume-weighted, aithmetic (De Bouckee mean) Moment notation Moment-atio notation D 0,0 x 1,0 D 1, 0 D 1,1 x D, 1 x D, 0 x D, x D, 0 D, x 1, D 4, 59

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Appendix-CPPS-7 Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 60

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Intepolation Values... C:\Pogam Files\a \fitsch\equal-size-intevals.fps ****** % <= 0.100 µm 0.1 % <= 0.150 µm 0. % <= 0.00 µm 1. % <= 0.50 µm.9 % <= 0.00 µm 5.4 % <= 0.50 µm 8.4 % <= 0.400 µm 1.0 % <= 0.450 µm 15.7 % <= 0.500 µm 19.5 % <= 0.550 µm. % <= 0.600 µm 7.1 % <= 0.650 µm 0.8 % <= 0.700 µm 4.4 % <= 0.750 µm 7.8 % <= 0.800 µm 41.1 % <= 0.850 µm 44.4 % <= 0.900 µm 47.5 % <= 0.950 µm 50.4 % <= 1.000 µm 5. % <= 1.050 µm 55.8 % <= 1.100 µm 58. % <= 1.150 µm 60.6 % <= 1.00 µm 6.8 % <= 1.50 µm 64.9 % <= 1.00 µm 67.0 % <= 1.50 µm 68.8 % <= 1.400 µm 70.6 % <= 1.450 µm 7. % <= 1.500 µm 7.9 % <= 1.550 µm 75. % <= 1.600 µm 76.7 % <= 1.650 µm 78.1 % <= 1.700 µm 79. % <= 1.750 µm 80.5 % <= 1.800 µm 81.7 % <= 1.850 µm 8.7 % <= 1.900 µm 8.7 % <= 1.950 µm 84.7 % <=.000 µm 85.5 % <=.050 µm 86.4 % <=.100 µm 87. % <=.150 µm 87.9 % <=.00 µm 88.5 % <=.50 µm 89. % <=.00 µm 89.9 % <=.50 µm 90.5 % <=.400 µm 91.1 % <=.450 µm 91.7 % <=.500 µm 9.6 % <=.600 µm 9.5 % <=.700 µm 94. % <=.800 µm 95.0 % <=.900 µm 95.5 % <=.000 µm 96.5 % <=.00 µm 97. % <=.400 µm 97.7 % <=.500 µm 97.9 % <=.600 µm 98.4 % <=.800 µm 98.8 % <= 4.000 µm 99.1 % <= 4.00 µm 99. % <= 4.400 µm 99.4 % <= 4.500 µm 99.5 % <= 4.600 µm 99.6 % <= 4.800 µm 99.7 % <= 5.000 µm 99.8 % <= 5.00 µm 99.9 % <= 5.400 µm 99.9 % <= 5.500 µm 99.9 % <= 5.600 µm 99.9 % <= 5.800 µm 100.0 % <= 6.000 µm 100.0 % <= 6.00 µm 100.0 % <= 6.400 µm 100.0 % <= 6.500 µm 100.0 % <= 6.600 µm 100.0 % <= 6.800 µm 100.0 % <= 7.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 8.500 µm 100.0 % <= 9.000 µm 100.0 % <= 9.500 µm 100.0 % <= 10.000 µm 100.0 % <= 1.000 µm 100.0 % <= 14.000 µm 100.0 % <= 16.000 µm 100.0 % <= 18.000 µm 100.0 % <= 0.000 µm Intepolation Values... C:\Pogam Files\a \fitsch\five-pecent-steps.fpv 1.0 % <= 0.44 µm.0 % <= 0.78 µm 5.0 % <= 0.4 µm 10.0 % <= 0.4 µm 15.0 % <= 0.491 µm 0.0 % <= 0.557 µm 5.0 % <= 0.6 µm 0.0 % <= 0.690 µm 5.0 % <= 0.759 µm 40.0 % <= 0.8 µm 45.0 % <= 0.910 µm 50.0 % <= 0.99 µm 55.0 % <= 1.085 µm 60.0 % <= 1.187 µm 65.0 % <= 1.0 µm 70.0 % <= 1.4 µm 75.0 % <= 1.589 µm 80.0 % <= 1.779 µm 85.0 % <=.019 µm 90.0 % <=.61 µm 95.0 % <=.907 µm 98.0 % <=.619 µm 99.0 % <= 4.10 µm 100.0 % <= 10.490 µm 61

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.807 µm Vaiance.4 µm Geometic Mean Diamete.658 µm Mean Sque Deviation.584 µm Quadatic Squae Mean Diamete.995 µm Aveage Deviation.415 µm Hamonic Mean Diamete.544 µm Coefficiant of Vaiation 7.49 % Statistical Modes... Skewness.169 Mode.59 µm Cutosis 7.09 Median.64 µm Span 1.98 Mean/Median Ratio 1.7 Unifomity.61 Specific Suface Aea Density Fom Facto 11078.55 cm/cm 6

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Intepolation Values... C:\Pogam Files\a \fitsch\equal-size-intevals.fps ****** % <= 0.100 µm 0.8 % <= 0.150 µm 1.6 % <= 0.00 µm 4.6 % <= 0.50 µm 9.7 % <= 0.00 µm 15.8 % <= 0.50 µm.4 % <= 0.400 µm 9.1 % <= 0.450 µm 5.4 % <= 0.500 µm 41. % <= 0.550 µm 46.6 % <= 0.600 µm 51.5 % <= 0.650 µm 55.9 % <= 0.700 µm 59.9 % <= 0.750 µm 6.5 % <= 0.800 µm 66.8 % <= 0.850 µm 69.8 % <= 0.900 µm 7.4 % <= 0.950 µm 74.9 % <= 1.000 µm 77.1 % <= 1.050 µm 79.0 % <= 1.100 µm 80.8 % <= 1.150 µm 8.4 % <= 1.00 µm 8.9 % <= 1.50 µm 85. % <= 1.00 µm 86.5 % <= 1.50 µm 87.5 % <= 1.400 µm 88.6 % <= 1.450 µm 89.4 % <= 1.500 µm 90. % <= 1.550 µm 91.1 % <= 1.600 µm 91.8 % <= 1.650 µm 9.4 % <= 1.700 µm 9.0 % <= 1.750 µm 9.5 % <= 1.800 µm 94.0 % <= 1.850 µm 94.5 % <= 1.900 µm 94.9 % <= 1.950 µm 95. % <=.000 µm 95.6 % <=.050 µm 96.0 % <=.100 µm 96. % <=.150 µm 96.5 % <=.00 µm 96.8 % <=.50 µm 97.0 % <=.00 µm 97. % <=.50 µm 97.5 % <=.400 µm 97.7 % <=.450 µm 97.9 % <=.500 µm 98.1 % <=.600 µm 98.4 % <=.700 µm 98.6 % <=.800 µm 98.8 % <=.900 µm 99.0 % <=.000 µm 99. % <=.00 µm 99.5 % <=.400 µm 99.5 % <=.500 µm 99.6 % <=.600 µm 99.7 % <=.800 µm 99.8 % <= 4.000 µm 99.8 % <= 4.00 µm 99.9 % <= 4.400 µm 99.9 % <= 4.500 µm 99.9 % <= 4.600 µm 99.9 % <= 4.800 µm 100.0 % <= 5.000 µm 100.0 % <= 5.00 µm 100.0 % <= 5.400 µm 100.0 % <= 5.500 µm 100.0 % <= 5.600 µm 100.0 % <= 5.800 µm 100.0 % <= 6.000 µm 100.0 % <= 6.00 µm 100.0 % <= 6.400 µm 100.0 % <= 6.500 µm 100.0 % <= 6.600 µm 100.0 % <= 6.800 µm 100.0 % <= 7.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 8.500 µm 100.0 % <= 9.000 µm 100.0 % <= 9.500 µm 100.0 % <= 10.000 µm 100.0 % <= 1.000 µm 100.0 % <= 14.000 µm 100.0 % <= 16.000 µm 100.0 % <= 18.000 µm 100.0 % <= 0.000 µm Intepolation Values... C:\Pogam Files\a \fitsch\five-pecent-steps.fpv 1.0 % <= 0.179 µm.0 % <= 0.11 µm 5.0 % <= 0.54 µm 10.0 % <= 0.0 µm 15.0 % <= 0.4 µm 0.0 % <= 0.81 µm 5.0 % <= 0.40 µm 0.0 % <= 0.457 µm 5.0 % <= 0.497 µm 40.0 % <= 0.59 µm 45.0 % <= 0.585 µm 50.0 % <= 0.64 µm 55.0 % <= 0.690 µm 60.0 % <= 0.751 µm 65.0 % <= 0.8 µm 70.0 % <= 0.904 µm 75.0 % <= 1.00 µm 80.0 % <= 1.16 µm 85.0 % <= 1.9 µm 90.0 % <= 1.5 µm 95.0 % <= 1.959 µm 98.0 % <=.549 µm 99.0 % <=.999 µm 100.0 % <= 8.1 µm 6

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.544 µm Vaiance.145 µm Geometic Mean Diamete.454 µm Mean Sque Deviation.8 µm Quadatic Squae Mean Diamete.66 µm Aveage Deviation.59 µm Hamonic Mean Diamete.8 µm Coefficiant of Vaiation 69.84 % Statistical Modes... Skewness.615 Mode.4 µm Cutosis 11.899 Median.49 µm Span 1.707 Mean/Median Ratio 1.4 Unifomity.55 Specific Suface Aea Density Fom Facto 157146.9 cm/cm 64

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Intepolation Values... C:\Pogam Files\a \fitsch\equal-size-intevals.fps ****** % <= 0.100 µm.7 % <= 0.150 µm 6.0 % <= 0.00 µm 1. % <= 0.50 µm. % <= 0.00 µm.6 % <= 0.50 µm 4. % <= 0.400 µm 51.8 % <= 0.450 µm 58.9 % <= 0.500 µm 65.0 % <= 0.550 µm 70.1 % <= 0.600 µm 74.4 % <= 0.650 µm 77.9 % <= 0.700 µm 81.0 % <= 0.750 µm 8.5 % <= 0.800 µm 85.6 % <= 0.850 µm 87.5 % <= 0.900 µm 89.1 % <= 0.950 µm 90.5 % <= 1.000 µm 91.6 % <= 1.050 µm 9.6 % <= 1.100 µm 9.5 % <= 1.150 µm 94. % <= 1.00 µm 94.9 % <= 1.50 µm 95.4 % <= 1.00 µm 96.0 % <= 1.50 µm 96.4 % <= 1.400 µm 96.8 % <= 1.450 µm 97.1 % <= 1.500 µm 97.4 % <= 1.550 µm 97.7 % <= 1.600 µm 97.9 % <= 1.650 µm 98.1 % <= 1.700 µm 98. % <= 1.750 µm 98.5 % <= 1.800 µm 98.6 % <= 1.850 µm 98.8 % <= 1.900 µm 98.9 % <= 1.950 µm 99.0 % <=.000 µm 99.1 % <=.050 µm 99. % <=.100 µm 99. % <=.150 µm 99. % <=.00 µm 99.4 % <=.50 µm 99.4 % <=.00 µm 99.5 % <=.50 µm 99.5 % <=.400 µm 99.6 % <=.450 µm 99.6 % <=.500 µm 99.7 % <=.600 µm 99.7 % <=.700 µm 99.8 % <=.800 µm 99.8 % <=.900 µm 99.8 % <=.000 µm 99.9 % <=.00 µm 99.9 % <=.400 µm 99.9 % <=.500 µm 99.9 % <=.600 µm 100.0 % <=.800 µm 100.0 % <= 4.000 µm 100.0 % <= 4.00 µm 100.0 % <= 4.400 µm 100.0 % <= 4.500 µm 100.0 % <= 4.600 µm 100.0 % <= 4.800 µm 100.0 % <= 5.000 µm 100.0 % <= 5.00 µm 100.0 % <= 5.400 µm 100.0 % <= 5.500 µm 100.0 % <= 5.600 µm 100.0 % <= 5.800 µm 100.0 % <= 6.000 µm 100.0 % <= 6.00 µm 100.0 % <= 6.400 µm 100.0 % <= 6.500 µm 100.0 % <= 6.600 µm 100.0 % <= 6.800 µm 100.0 % <= 7.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 8.500 µm 100.0 % <= 9.000 µm 100.0 % <= 9.500 µm 100.0 % <= 10.000 µm 100.0 % <= 1.000 µm 100.0 % <= 14.000 µm 100.0 % <= 16.000 µm 100.0 % <= 18.000 µm 100.0 % <= 0.000 µm Intepolation Values... C:\Pogam Files\a \fitsch\five-pecent-steps.fpv 1.0 % <= ********* µm.0 % <= 0.110 µm 5.0 % <= 0.189 µm 10.0 % <= 0. µm 15.0 % <= 0.59 µm 0.0 % <= 0.84 µm 5.0 % <= 0.08 µm 0.0 % <= 0. µm 5.0 % <= 0.57 µm 40.0 % <= 0.8 µm 45.0 % <= 0.410 µm 50.0 % <= 0.49 µm 55.0 % <= 0.471 µm 60.0 % <= 0.508 µm 65.0 % <= 0.550 µm 70.0 % <= 0.599 µm 75.0 % <= 0.657 µm 80.0 % <= 0.7 µm 85.0 % <= 0.8 µm 90.0 % <= 0.981 µm 95.0 % <= 1.60 µm 98.0 % <= 1.669 µm 99.0 % <=.006 µm 100.0 % <= 59.400 µm 65

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 66

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Intepolation Values... C:\Pogam Files\a \fitsch\equal-size-intevals.fps ****** % <= 0.100 µm 1.5 % <= 0.150 µm 17.4 % <= 0.00 µm 9.5 % <= 0.50 µm 4.5 % <= 0.00 µm 55.6 % <= 0.50 µm 65.4 % <= 0.400 µm 7. % <= 0.450 µm 78.9 % <= 0.500 µm 8. % <= 0.550 µm 86.8 % <= 0.600 µm 89.4 % <= 0.650 µm 91.4 % <= 0.700 µm 9.0 % <= 0.750 µm 94. % <= 0.800 µm 95. % <= 0.850 µm 96.1 % <= 0.900 µm 96.7 % <= 0.950 µm 97. % <= 1.000 µm 97.7 % <= 1.050 µm 98.0 % <= 1.100 µm 98. % <= 1.150 µm 98.6 % <= 1.00 µm 98.8 % <= 1.50 µm 99.0 % <= 1.00 µm 99.1 % <= 1.50 µm 99. % <= 1.400 µm 99. % <= 1.450 µm 99.4 % <= 1.500 µm 99.5 % <= 1.550 µm 99.6 % <= 1.600 µm 99.6 % <= 1.650 µm 99.7 % <= 1.700 µm 99.7 % <= 1.750 µm 99.7 % <= 1.800 µm 99.8 % <= 1.850 µm 99.8 % <= 1.900 µm 99.8 % <= 1.950 µm 99.8 % <=.000 µm 99.9 % <=.050 µm 99.9 % <=.100 µm 99.9 % <=.150 µm 99.9 % <=.00 µm 99.9 % <=.50 µm 99.9 % <=.00 µm 99.9 % <=.50 µm 99.9 % <=.400 µm 99.9 % <=.450 µm 100.0 % <=.500 µm 100.0 % <=.600 µm 100.0 % <=.700 µm 100.0 % <=.800 µm 100.0 % <=.900 µm 100.0 % <=.000 µm 100.0 % <=.00 µm 100.0 % <=.400 µm 100.0 % <=.500 µm 100.0 % <=.600 µm 100.0 % <=.800 µm 100.0 % <= 4.000 µm 100.0 % <= 4.00 µm 100.0 % <= 4.400 µm 100.0 % <= 4.500 µm 100.0 % <= 4.600 µm 100.0 % <= 4.800 µm 100.0 % <= 5.000 µm 100.0 % <= 5.00 µm 100.0 % <= 5.400 µm 100.0 % <= 5.500 µm 100.0 % <= 5.600 µm 100.0 % <= 5.800 µm 100.0 % <= 6.000 µm 100.0 % <= 6.00 µm 100.0 % <= 6.400 µm 100.0 % <= 6.500 µm 100.0 % <= 6.600 µm 100.0 % <= 6.800 µm 100.0 % <= 7.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 7.00 µm 100.0 % <= 7.400 µm 100.0 % <= 7.500 µm 100.0 % <= 7.600 µm 100.0 % <= 7.800 µm 100.0 % <= 8.000 µm 100.0 % <= 8.500 µm 100.0 % <= 9.000 µm 100.0 % <= 9.500 µm 100.0 % <= 10.000 µm 100.0 % <= 1.000 µm 100.0 % <= 14.000 µm 100.0 % <= 16.000 µm 100.0 % <= 18.000 µm 100.0 % <= 0.000 µm Intepolation Values... C:\Pogam Files\a \fitsch\five-pecent-steps.fpv 1.0 % <= ********* µm.0 % <= ********* µm 5.0 % <= ********* µm 10.0 % <= 0.118 µm 15.0 % <= 0.187 µm 0.0 % <= 0.1 µm 5.0 % <= 0.4 µm 0.0 % <= 0.5 µm 5.0 % <= 0.70 µm 40.0 % <= 0.88 µm 45.0 % <= 0.06 µm 50.0 % <= 0.5 µm 55.0 % <= 0.47 µm 60.0 % <= 0.71 µm 65.0 % <= 0.98 µm 70.0 % <= 0.49 µm 75.0 % <= 0.465 µm 80.0 % <= 0.511 µm 85.0 % <= 0.57 µm 90.0 % <= 0.664 µm 95.0 % <= 0.85 µm 98.0 % <= 1.094 µm 99.0 % <= 1.1 µm 100.0 % <= 5.045 µm 67

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 68

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 69

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 70

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 71

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 7

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.807 µm Vaiance.4 µm Geometic Mean Diamete.658 µm Mean Sque Deviation.584 µm Quadatic Squae Mean Diamete.995 µm Aveage Deviation.415 µm Hamonic Mean Diamete.544 µm Coefficiant of Vaiation 7.49 % Statistical Modes... Skewness.169 Mode.59 µm Cutosis 7.09 Median.64 µm Span 1.98 Mean/Median Ratio 1.7 Unifomity.61 Specific Suface Aea Density Fom Facto 11078.55 cm/cm 7

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.807 µm Vaiance.4 µm Geometic Mean Diamete.658 µm Mean Sque Deviation.584 µm Quadatic Squae Mean Diamete.995 µm Aveage Deviation.415 µm Hamonic Mean Diamete.544 µm Coefficiant of Vaiation 7.49 % Statistical Modes... Skewness.169 Mode.59 µm Cutosis 7.09 Median.64 µm Span 1.98 Mean/Median Ratio 1.7 Unifomity.61 Specific Suface Aea Density Fom Facto 11078.55 cm/cm 74

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.807 µm Vaiance.4 µm Geometic Mean Diamete.658 µm Mean Sque Deviation.584 µm Quadatic Squae Mean Diamete.995 µm Aveage Deviation.415 µm Hamonic Mean Diamete.544 µm Coefficiant of Vaiation 7.49 % Statistical Modes... Skewness.169 Mode.59 µm Cutosis 7.09 Median.64 µm Span 1.98 Mean/Median Ratio 1.7 Unifomity.61 Specific Suface Aea Density Fom Facto 11078.55 cm/cm 75

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.807 µm Vaiance.4 µm Geometic Mean Diamete.658 µm Mean Sque Deviation.584 µm Quadatic Squae Mean Diamete.995 µm Aveage Deviation.415 µm Hamonic Mean Diamete.544 µm Coefficiant of Vaiation 7.49 % Statistical Modes... Skewness.169 Mode.59 µm Cutosis 7.09 Median.64 µm Span 1.98 Mean/Median Ratio 1.7 Unifomity.61 Specific Suface Aea Density Fom Facto 11078.55 cm/cm 76

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.544 µm Vaiance.145 µm Geometic Mean Diamete.454 µm Mean Sque Deviation.8 µm Quadatic Squae Mean Diamete.66 µm Aveage Deviation.59 µm Hamonic Mean Diamete.8 µm Coefficiant of Vaiation 69.84 % Statistical Modes... Skewness.615 Mode.4 µm Cutosis 11.899 Median.49 µm Span 1.707 Mean/Median Ratio 1.4 Unifomity.55 Specific Suface Aea Density Fom Facto 157146.9 cm/cm 77

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.544 µm Vaiance.145 µm Geometic Mean Diamete.454 µm Mean Sque Deviation.8 µm Quadatic Squae Mean Diamete.66 µm Aveage Deviation.59 µm Hamonic Mean Diamete.8 µm Coefficiant of Vaiation 69.84 % Statistical Modes... Skewness.615 Mode.4 µm Cutosis 11.899 Median.49 µm Span 1.707 Mean/Median Ratio 1.4 Unifomity.55 Specific Suface Aea Density Fom Facto 157146.9 cm/cm 78

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.544 µm Vaiance.145 µm Geometic Mean Diamete.454 µm Mean Sque Deviation.8 µm Quadatic Squae Mean Diamete.66 µm Aveage Deviation.59 µm Hamonic Mean Diamete.8 µm Coefficiant of Vaiation 69.84 % Statistical Modes... Skewness.615 Mode.4 µm Cutosis 11.899 Median.49 µm Span 1.707 Mean/Median Ratio 1.4 Unifomity.55 Specific Suface Aea Density Fom Facto 157146.9 cm/cm 79

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =. µm Statistical Means... Aithmetic Mean Diamete.544 µm Vaiance.145 µm Geometic Mean Diamete.454 µm Mean Sque Deviation.8 µm Quadatic Squae Mean Diamete.66 µm Aveage Deviation.59 µm Hamonic Mean Diamete.8 µm Coefficiant of Vaiation 69.84 % Statistical Modes... Skewness.615 Mode.4 µm Cutosis 11.899 Median.49 µm Span 1.707 Mean/Median Ratio 1.4 Unifomity.55 Specific Suface Aea Density Fom Facto 157146.9 cm/cm 80

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 81

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 8

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 8

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 84

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 85

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 86

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 87

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 88

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.1 µm Statistical Means... Aithmetic Mean Diamete.8 µm Vaiance.06 µm Geometic Mean Diamete. µm Mean Sque Deviation.5 µm Quadatic Squae Mean Diamete.456 µm Aveage Deviation.171 µm Hamonic Mean Diamete.71 µm Coefficiant of Vaiation 65.58 % Statistical Modes... Skewness.601 Mode.106 µm Cutosis 1.851 Median.5 µm Span 1.68 Mean/Median Ratio 1.177 Unifomity.5 Specific Suface Aea Density Fom Facto 1185.97 cm/cm 89

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 90

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 91

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 9

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 9

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Meas.No. 00 Date 1..007 Time 10:5 Opeato ID Seial No. 778 AlO, D50 = 1 micon Measuing Range 0.1 [µm] - 100.9 [µm] Pump 50 [%] Resolution 10 Channels (0 mm / 8 mm ) Absoption 10.00 [%] Ultasonic On Measuement Duation 100 [Scans] Regulaization / Modell 579.601 Faunhofe Calculation selected. Mean Values... D4 = 1. µm D4 =.99 µm D41 =.81 µm D40 =.67 µm D =.81 µm D1 =.66 µm D0 =.55 µm D1 =.54 µm D0 =.46 µm D10 =.4 µm Statistical Means... Aithmetic Mean Diamete 1.6 µm Vaiance.706 µm Geometic Mean Diamete.995 µm Mean Sque Deviation.84 µm Quadatic Squae Mean Diamete 1.484 µm Aveage Deviation.61 µm Hamonic Mean Diamete.807 µm Coefficiant of Vaiation 68.579 % Statistical Modes... Skewness 1.591 Mode.9 µm Cutosis.07 Median.99 µm Span 1.95 Mean/Median Ratio 1.5 Unifomity.61 Specific Suface Aea Density Fom Facto 7488.1 cm/cm 94

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Appendix-CPPS-9 Table 9-A. Mean calipe diamete, suface and volume of standad shapes. Shape (and coesponding linea size measue) Sphee with adius Hemisphee with adius Oblate spheoid with semiaxes a = b and c ( c < a ) whee 1 c ε = 1 a Polate spheoid with semiaxes a = b and c ( c > a ), whee 1 Mean calipe diamete Suface Volume 4π 1.6 4 1 π + 4 ε c + a acsinε π a + c 1+ ε π ln ε 1 ε ε c + a acsinhε π 4. π 4 π a c π π a + c π acsin ε aε c 4 π a c c ε = 1 a Cylinde of length ( ) l and adius l + π π l + π l Thin od (needle), l 0 0 i.e. 0 Thin disc π π 0 (platelet), i.e. l 0 Tetahedon of 1 1 a 1.7a edge length a a accos 0. 91a a 0.1 a π 1 Cube of edge length a Octahedon of edge length a a 1 6 a accos 1. 18a π 6a a a a.46 1 9 a 0.47 a 95

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Dodecahedon of edge length a Tetakaidecahedo n of edge length a Icosahedon of edge length a Paallelepiped of edge lengths a, b and c 15 1 a accos + π.64 a a 5 10 1 [ 5( 5 + 5) ] 0.6 a 6( 1+ ) 15 1+ a accos π + 1.74a a b + c ( 5 5) 5 10 1 6.8a a 1 a 15 + 7 4 7.66 a 5 a 8 a 11.a a 8.66 5( + 5) 5 a + ( ab + bc + ca) 1.18a a bc a 96

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Appendix-CPPS-14 14-A. Intoduction to suspension heology 14-A-1. States of matte (basic heological classification) gas (compessible fluid) liquid (incompessible fluid) solid Deboah numbe: t De = t mateial pocess chaacteistic mateial time ( elaxation time) = chaacteistic pocess time ( defomation time) fluids ( De << 1, fo puely viscous fluids De 0 ) visco-elastic and elastico-viscous mateials ( De 1, Maxwell fluid, Kelvin solid) solids ( De >> 1, fo puely elastic solids De ) Ode of magnitude of viscosities fo diffeent classes of mateials (only oientational values fo typical mateials of the espective class!) Mateial Viscosity Remak Gases 10 µpas at oom tempeatue Wate, ethanol, mecuy 1 mpas at oom tempeatue Metal melts 1 mpas at high tempeatue Ceamic suspensions 10 mpas 1000 mpas appaent viscosity Oil 1 Pas appaent viscosity Ceamic pastes 10 Pas 1000 Pas appaent viscosity Glass melts 10 Pas at high tempeatue Solid glass 10 18 Pas Extapolated value 14-A-. Basic 1-D heological models Model Hooke Newton Maxwell Constitutive equation (1-D) Mechanical analogue Rheological chaacteistic Response τ = G γ sping ideally viscous on de-loading: defomation and stesses estoe instantaneously τ = µ & γ dashpot pefectly elastic on de-loading: defomation emains, stesses estoe instantaneously & τ τ sping and stess on de-loading: & γ = + G µ dashpot elaxation stesses elax slowly in seies 97

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices with µ t elax G t = telax τ = τ e 0 Kelvin τ = G γ + µ & γ sping and dashpot paallel Delayed esponse µ with t delay = G on de-loading: defomation elaxes slowly γ = γ 0 e t t delay on loading: γ = γ final 1 e t t delay (only hee γ is the shea defomation, γ& the shea ate!)!!! In the following γ is the shea ate!!! Definition of the 1-D shea stess τ and shea ate γ fo impotant geometies and flow types: In (ectangula) Catesian coodinates with flow in diection x (plane Couette flow): Shea stess τ τ xy ( = x-y-component of the stess tenso), dv shea ate γ x ( = y-diection gadient of the x-component of the velocity vecto dy = x-y-component of the ate of defomation tenso) In cylindical (semi-pola) coodinates with flow in axial diection (cylindical Poiseuille flow): Shea stess τ τ z ( = -z-component of the stess tenso), dv shea ate γ z ( = adial gadient of the axial velocity component = -zcomponent of the ate of defomation d tenso) In cylindical (semi-pola) coodinates with flow in tangential diection (cylindical Couette flow): Shea stess τ τ θ ( = -θ-component of the stess tenso), dv shea ate γ θ = ω ( = adial gadient of the angula velocity = -θ-component of d the ate of defomation tenso) 98

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 14-A-. Pinciple of heology fluid flow along solid sufaces Citical assumption (bounday condition): Fluid sticks on the solid suface (absence of wall slip) Note: All "nomal" monophase liquids (not gases! not suspensions!) ae assumed to stick on solid sufaces duing flow, iespective of thei physico-chemical adheence o wetting behavio with espect to these sufaces! Even mecuy sticks on the glass suface duing flow though capillaies! (Coleman, Makovitz & Noll, 1966) 14-A-4. Simple viscometes (basic fomulae) Falling ball viscomete: Capillay viscomete: D ( ρ s ρl ) g η = 18 v (Stokes equation) D p τ = 4L ( p = pessue diffeence ) 8 v γ = z D ( v z = mean velocity ) Rotational viscomete: M τ = ( M = toque ) π R1 h ω γ = ( ω = angula velocity ) 1 R1 R 14-A-5. Rational theoy of viscomety T = T (D) Linea case Newtonian fluids (Navie-Stokes fluids = -D "Newtonian fluids", compessible o incompessible): T = ξ ( t D) 1 + µ D Non-linea case non-newtonian fluids (puely viscous, Reine-Rivlin fluids, compessible o incompessible) whee the φ i ae scala functions of III D. T = φ (...) + D 0 1 + φ1(...) D φ (...) ρ, T and the thee pincipal invaiants I D, II D and 99

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Fo incompessible fluids (= liquids) div v = t D = I D = 0 and fo viscometic flows (equivalent to simple shea flows) det D = III D = 0. Futhemoe fo incompessible fluids the pessue is hydostatic and theefoe an abitay scala, into which the fist above.h.s. tem can be absobed. Fo viscometic flows the thid.h.s. tem detemines nomal stesses only. When these ae without concen we have fo the stess tenso [ ] Since 1 ( t D) t ( D ) T = p1 + φ ( II ) D 1 D II D = and t D = 0 we have fo the shea stess tenso ( ( D ) D τ = φ 1 t ) ("genealized Newtonian fluids", only incompessible!) Specialization to 1-D shea stesses Linea case Newtonian fluids τ = µ γ whee µ = viscosity (coefficient of dynamic shea viscosity). Nonlinea case Genealized Newtonian fluids τ = η( γ ) γ whee η ( γ ) = appaent viscosity. Special models fo non-newtonian liquids in 1-D: Powe law (Ostwald - De Waele model): Bingham model: Heschel-Bulkley model (genealized Bingham model): n τ = Kγ τ τ + Kγ = 0 n τ τ + Kγ Bid-Caeau model, (modified) Coss model, Pandtl-Powell-Eying model = 0 14-A-6. Tempeatue dependence of viscosity B T AFE - type equations: η = e (Ahenius, Andade, Fenkel, Eying) (theoy of eaction ates) WLF - type equations: (fee volume theoy) 0 η 0 A ( T T0 ) B+ ( T T ) 0 η = η e (William, Landel, Fey) T0 VFT - type equations: η = η e (Vogel, Fulche, Tamann) (modified fee volume theoy) 0 A T Puely empiical fit equations e.g. η = η e 0 A ( T T 0 ) 100

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 14-A-7. Concentation dependence of viscosity Fo dilute systems: Definition: Relative viscosity η vis cosity of η = = η vis cosity of 0 suspension pue liquid Einstein fomula: η = 1+. 5 φ (φ = volume faction of solids) (fo pefectly igid non-inteacting sphees) Second-ode fomulae fo igid sphees with fist-ode inteaction effects: η 1.5 = + φ + B φ +... ( B = 4.4 14. 1) Fo concentated systems: Definition: Reduced volume faction φ φ = = φ max volume faction maximum volume faction Note: The highest possible φ max fo a system of monosized sphees coesponds to closest packing, i.e. 74 %. Fo eal systems the value φ max is often close to the value of andom close packing of monosized sphees, i.e. 64 %. Shell model fomulae fo monodispese systems η = C φ φ 1 (whee C is a model-dependent paamete anging fom 0.589 to 1.15) Eiles fomula: 1 1 1.5 1 η = + φ φ 1 Mooney fomula: η = exp Cφ 1 φ (educes to Einstein if C is chosen to be. 5 ) 1 Kiege fomula: η = 1 φ (educes to Einstein if N is chosen to be.5 φ ) N max 101

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 14-B. Rheology of suspensions with anisometic paticles 14-B-1. Effective, elative and intinsic viscosity Effective popeties ae the macoscopic (i.e. oveall o lage-scale), popeties of multiphase mateials. In geneal they ae dependent on the constituent (i.e. phase) popeties and the micostuctue of the mateial. Fo two-phase solid-liquid mixtues with matix-inclusion type micostuctue (suspensions) the effective shea viscosity η (simply called "effective viscosity" in the sequel) can be assumed to be a function of the solids volume faction φ. Note, howeve, that the assumption of a dependence exclusively on φ is only justifiable on pagmatic gounds, i.e. when highe-ode micostuctual infomation is lacking. Note futhe, that in assuming the existence of a unique (i.e. not shea-ate dependent) shea viscosity, one implicitly assumes puely viscous behavio (i.e. no viscoelastic effects) and Newtonian (linea) behavio of the whole suspension (and not only fo the suspending medium). In the dilute limit, i.e. fo volume factions φ 0, the effective viscosity η of suspensions with igid, spheical paticles is given by the Einstein elation η = η 0 (1 +.5 φ). In this equation, φ is the solids volume faction, η denotes the effective suspension viscosity and η 0 the viscosity of the suspending medium (pue liquid). In ode to simplify notation in the following text, we intoduce the elative viscosity η, and the so-called intinsic viscosity [ η ], [ η] η η, η 0 η 1 lim. φ 0 φ Using intinsic viscosity, the Einstein elation can be fomally genealized to suspensions of anisometic paticles, i.e. η = 1+ [ η]φ. Jeffey, in a igoous teatment of the motion of a igid ellipsoids and spheoids of a cetain aspect atio, was the fist to calculate definite values fo [ η ] as a function of the paticle aspect atio. Theefoe this can be called Jeffey-Einstein elation. 14-B-. Intinsic viscosity as a function of the paticle aspect atio Jeffey (19) calculated the motion of a single ellipsoidal paticle immesed in a viscous liquid. He solved the equations of motion fo the case of slow lamina (ceeping) shea flow and showed that fo a otationally symmetic ellipsoid (spheoids) this motion is in geneal peiodic, with its axis of evolution descibing a cone about the pependicula to the plane of 10

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices the undistubed motion of the liquid (Jeffey's obit). Jeffey's investigation eveals no tendency of the spheoid to set its axis in any paticula diection with egad to the undistubed motion of the liquid. This finding is in ageement with moden eseach on fibe suspensions. Note, howeve, that the esult has been deived fo dilute suspensions (i.e. suspensions with non-inteacting fibes o platelets) in shea flow, and cannot be assumed to be valid fo eithe non-dilute suspensions (i.e. suspensions with inteacting fibes o platelets) o othe flow types, i.e. elongational flow. Accoding to Jeffey, only lowe and uppe bounds can be given fo the intinsic viscosity of spheoid suspensions. The lowe bound (i.e. the minimum value of [ η ]) coesponds to the smallest dissipation of enegy, the uppe bound (i.e. the maximum value of [ η ]) to the lagest. In moden notation, these bounds ae given by the following fomulae: 1. Minimum intinsic viscosity. Maximum intinsic viscosity 1 η, Ahd [ ] min = 4 1 h d hd [ η ] max = + +. hd ( h + d ) A B C In these equations, h is the length of the long axis ("height") and d the length of the shot axis ("diamete"), and the coefficients A, B, C ae A = 4 h 5 1 sin 4 θ 1 1 θ π ( 5 cos θ ) + ln tan + 4 cos θ sinθ 4 B = h 5 1 sin 4 + θ 1 1 θ π ln tan + cos θ sinθ 4 1 θ π ( + cos θ ) ln tan + 1 C = 4 h sin θ sinθ 4 fo polate spheoids, fo which θ accos h d, and A = 4 5 d 4 sin θ θ ( 5 cos θ ) cosθ + 1 sinθ B = 5 d 4 sin θ 1 θ ( + cos θ ) 1 cosθ sinθ θ ( 1+ cos θ ) cos 1 C = θ 4 d sin θ sinθ 10

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices d and fo oblate spheoids, fo which θ accos. Numeical values of the minimum and h maximum intinsic viscosity of spheoid suspensions ae listed in Table 14-B-1 as a function of the aspect atio R h d. Table 14-B-1. Minimum and maximum intinsic viscosity values of a suspension with oblate and polate spheoids as a function of the paticle aspect atio (calculated accoding to Jeffey). Aspect atio R Minimum of [ η ] Maximum of [ η ] 0.061 1/100.067 86. 1/50.07 4.9 1/40.075 5.4 1/0.080 6.9 1/0.089 18.4 1/10.116 9.96 1/9.1 9.11 1/8.10 8.7 1/7.19 7.4 1/6.151 6.58 1/5.168 5.74 1/4.19 4.91 1/. 4.08 1/.06.7 1.5.5.174.819.088.088 4.05.7 5.05.548 6.05.75 7.019.948 8.015 4.14 9.01 4.1 10.010 4.485 0.00 5.999 0.001 7.16 40.001 8.59 50.000 9.677 100.000 14.9.000 104

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Accoding to Benne (1974), in simple shea flow the intinsic viscosity of a suspension with axisymmetic paticles (possessing foe-aft symmety) is given by the expession 15 4 5 4B 15 BP [ η] = 5Q Q sin θ ( Q + 4Q ) sin θ cos φ + ( Q + 4 ) sin θ sin φ In this expession 1 4 Q is a dimensionless paamete and 5N B = K P = γ D the otay Péclet numbe, with γ being the shea ate and the otay Bownian diffusion coefficient D given by the Stokes-Einstein equation D kt =, 6 Vη 0 K whee k is the Boltzmann constant, T the absolute tempeatue, V the paticle volume and η 0 the viscosity of the suspending medium. The mateial constants N and K (connected to the otation of the axisymmetic paticle about a tansvese axis) ae dependent on the model shape chosen and the aspect atio (tue axis atio, paticle axis atio) R = a b, whee a is the pola adius (fo polate shapes half of the length, fo oblate shapes half of the thickness) and b the equatoial adius (fo polate shapes half of the thickness, fo oblate shapes half of the diamete). Fo spheoids in geneal K = ( R + 1) ( R α +α ) 1 and ( R 1) ( R α +α ) N =. 5 1 Moeove, Q 1 = 1 5α Q α 5 15α α = 1 6 105

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Q ( α + α ) 1 R 1 α = 1 5α R α1 + α α 4 and Q 4 1 = 1 5α 4 Rα ( R + 1) α In these expessions the α s ae defined as whee α = R R 1 ( β 1) 1 α = R ( β ) 1 1 R ( R 1) ( β + 5) R α = R 4 α R ( R 1) ( R + βr ) 4 = ( R 1) ( R + 1 ( 4 ) ) R α 5 = R 1 β 4 α R ( R 1) [( R + 1) ] 6 = β β = cosh R R 1 R 1 fo polate spheoids ( R > 1) and β = cos 1 R R 1 R fo polate spheoids ( R < 1). As a consequence R B = R 1 + 1 106

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices fo all spheoids. The volume of a spheoid is V 4π ab =. In the limiting case R >> 1 (long thin polate spheoid) K = R ( ln R 0.5) and B = 1 R Moeove, fo long thin polate spheoids, Q 1 = 6ln R 5 5R Q + = 15 R ( ln R 1.5) 5 Q = 10 R ( ln R 0.5) 6ln R Q 4 =. 5R Fo long slende ( R >> 1, but finite and blunt-ended) cylindical ods (with L 5. 45 ) K = R ln 1 L 1+ + 9 ln R ln R 8π and Moeove, fo long slende cylindical ods, L ln R B = 1. 4πR Q 1 = 5 107

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Q = 45 R ( ln R + ln 17 6) Q 1 L ln R R ln 1 L = 1 1+ 15 4πR ln R ln R 8π Q = 0 4 The volume of a cicula cylinde is V = πab. The mateial constants in the limiting case R << 1 (infinitesimally thin cicula disk of adius b ) can be obtained fom the geneal esults fo an oblate spheoid, by letting the pola adius a tend to zeo. Since, howeve, the volume V of such a disk and accodingly also its volume faction φ is zeo, the esults must be pesented in a diffeent fom, using the numbe density n (numbe of disks pe unit volume) as a concentation measue. Accoding to Benne (1974) and 6VK b B = 1 (coesponding to R = 0 ). Moeove, the following eplacements have to be made: φ Q 1 n b 45 16 φ Q n b 45 8 φ Q n b 45 φ Q 4 n b. 45 In the special case of sphees ( R = 1) the α integals educe to α 1 = α =, 4 and α 5 = α 6 = and thus the mateial constants ae 15 α = α 4 = 5 K = 1, 108

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices B = 0, 1 Q 1 =, Q = = Q Q4 The volume of a sphee (with adius b ) is V = πb. 6 Fo all axisymmetic paticles with foe-aft symmety the mateial paametes Q 1 though Q 4, as well as K, N, and B completely detemine the behavio of the suspension in abitay flow pocesses. Only five of these paametes ae independent. Note that fo all model shapes descibed above B 1. The invese case B 1 occus only fo elatively uncommon model shapes (e.g. cetain peanut-shaped bodies). The paamete Q 4 is deived fom Q 1 = Q BN. 4 All mateial paametes ae uniquely defined by the aspect atio R. Based on the knowledge of K and paticle size (volume V ) 1 the otay Péclet numbe can be estimated in ode to assess the influence of Bownian motion. It is common pactice to distinguish thee egimes: Dominant Bownian motion: P << 1 In this case ( zeo shea ate limit ) the intinsic viscosity is maximal. The uppe bound of the intinsic viscosity is [ ] = 5Q 1 Q + Q η. 0 Accoding to Benne (1974) this esult holds not only fo simple shea flows (as stated in ealie deivations), but fo any (homogeneous) shea flow, including e.g. uniaxial extension. Fo long thin polate spheoids ( R >> 1) this equation educes to the well-known appoximate esult of Kuhn and Kuhn (1945) 1 Apat fom diluteness of the suspension, the paticle system is assumed to be monodispese, with a constant aspect atio. This has to be kept in mind when a compaison with eal suspensions is intended. The extapolation of viscosity measuement esults in the non-dilute egion to the dilute egion may o may not be justified. Due to inteactions effective in the non-dilute egion, such an extapolation may lead to highe intinsic viscosity values. Similaly, the effect of polydispesity of eal systems on the esults is difficult to assess, especially when additionally the aspect atio is not sizeinvaiant o has a distibution with finite width. 109

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices R 1 8 [ η ] 0 = 15 + + ln 0.5 ln 1.5. R R 5 Table 14-B- lists the uppe bound of the intinsic viscosity [ η ] 0 (i.e. in the case P = 0 ) fo dilute suspensions of polate and oblate spheoids in dependence of the aspect atio, accoding to the exact numeical calculation of Scheaga, cf. Benne (1974). Table 14-B-. Uppe bound of the intinsic viscosity [ η ] 0 (i.e. in the case P = 0 ) fo dilute suspensions of polate and oblate spheoids in dependence of the aspect atio. R o 1 R Polate Oblate 1.5.5.908.854 4 4.66 4.059 5 5.806 4.708 10 1.6 8.04 16 7.18 1.10 5 55.19 18.19 50 176.8 5.16 Intemediate Bownian motion: R e + R e >> P >> 1 In this case (which necessaily equies R >> 1 fo polate and R << 1 fo oblate paticles) the intinsic viscosity is Weak Bownian motion: [ η ] = ( Q + Q ) 5 1 4 + 5 4 P >> 1 and P >> R e + Re whee R e is an equivalent aspect atio (emeging in a natual way in hydodynamic theoy), which is equal to the tue aspect atio (i.e. R e = R ) in the case of spheoids and R e = 8π L R ln R in the case of long slende cylindical ods. Table 14-B- lists equivalent aspect atios of slende ods in dependence of the tue aspect atio. 110

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices Table 14-B-. Equivalent aspect atios of slende ods in dependence of the tue aspect atio. R R e 1 -.98 4 4.1 5 4.89 10 8.17 16 11.9 5 17. 50 1. 100 57.8 1000 47 In the case of weak Bownian motion the goniometic factos ae given in Table 14-B- 4 in dependence of the equivalent aspect atio, accoding to the asymptotic esults of Hinch & Leal, cf. Benne (1974). Table 14-B-4. Goniometic factos in dependence of the equivalent aspect atio. R e sin θ P sin θ sin φ sin θ cos φ 1 0.667 0 0 0.690.165-0.716 4 0.758 8.6551-0.516 5 0.784 1.0-0.5810 10 0.86 51.090-0.750 16 0.905 18.9-0.866 5 0.96 1.55-0.8918 50 0.968 144.68-0.988 100 0.986 4998.81-0.9578 1 R -1 The values in this table efe to the case 1 (polate paticles). In ode to obtain the values of the goniometic factos in the ange 0 R e 1 (oblate paticles) the following eplacements have to be made: Re 1 R e R e sin θ sin θ sin θ sin φ sin θ sin φ sin θ cos φ sin θ cos φ. 111

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 1 Fo long slende bodies ( R e >> 1, i.e. B 1), i.e. when P >> R e >> 1, the following asymptotic appoximations can be used, cf. Benne (1974): 1.79 sin θ = 1 R e sin θ sin φ = R e P.054 sin θ cos φ = 1+. R e The coesponding intinsic viscosity is R [ η ] = 0.15. ln R In the total absence of Bownian motion (i.e. absence of otay diffusion, coesponding to the limit P ) bodies with B 1 (common case) undego in simple shea flow a timepeiodic otation of the type fist descibed by Jeffey (19). The equations govening otation of an axisymmetic body suspended in a liquid medium exposed to simple shea flow ae 1 θ = γ B sin θ sin φ 4 & 1. & and φ = γ ( 1+ B cos φ ) Inspection of these equations shows, howeve, that bodies with B 1 undego an apeiodic motion in simple shea flow, ultimately adopting a stable teminal oientation. In this case the teminal oientations ae θ = π, tan φ = R and θ = π, tan φ = R fo B 1 and B 1, espectively. As a consequence, in the absence of otay Bownian motion the oientational distibution function is the Diac delta function (distibution) and the following goniometic factos may be employed to calculate the intinsic viscosity in simple shea flow sin θ = 1 sin θ sin φ = B 1 B 1 sin θ cos φ =. B In concluding we note that the intinsic viscosity is in geneal bounded fom below by the inequality [ η ] > 1. 11

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 14-C. Rheology and themal conductivity of nanofluids 14-C-1. Concentation measues and mixtue ules Tansfomation of mass factions (weight factions) w into volume factions φ and vice vesa: φ = w = ρ ρ s ρ w f ( 1 w) + ρ w f ρ φ s ( 1 φ) + ρ φ In these elations ρ s is the density of the solid paticles and ρ f the density of the base fluid, i.e. the liquid medium; φ and w (fo convenience without subscipt) efe to the dispesed phase, i.e. the solid paticles. Accodingly, the effective density of the suspension is s f s ( φ) ρ f ρ = φ ρ + 1. The pimay aim of suspension heology is the desciption of the effective viscosity of a suspension η at constant tempeatue in dependence of the volume faction of solids φ. In the classical appoach to nanofluids, volume factions, mass factions and densities ae elated by the same elations as fo odinay suspensions (see above). Moeove, if the nanofluid is in themal equilibium, the effective volumetic heat capacity C p = ρ c p (effective specific heat at constant pessue, efeed to unit volume) is usually assumed to be C P Ps ( φ) CPf = φ C + 1. Although fom a theoetical point of view this simple additivity of the volumetic heat capacities with espect to volume factions is questionable (an exact teatment has to take compessibility and themal expansion into account), this elation can be expected to be a easonable appoximation to eality. The exact elation is based on the additivity of the specific heats (i.e. heat capacities efeed to unit mass) with espect to mass factions, i.e. c P Ps ( w) cpf = w c + 1. Note that fo the calculation of the effective (volumetic) themal expansion coefficient β of suspensions, including nanofluids, the themal expansion of the solid phase paticles can usually be neglected, i.e. ( ) β f β = 1 φ, whee β f is the volumetic themal expansion coefficient of the base fluid. 11

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 14-C-. Effective viscosity and themal conductivity Pedictive elations fo the effective viscosity of nanofluids ae in pinciple analogous to those fo odinay suspensions (see above), i.e. the Einstein and Binkman elation. Additionally, seveal othe extensions o modifications of the Einstein elation as well as seveal elations of empiical oigin (obtained fom fitting expeimental data), have been poposed, e.g. η 1+ 10 φ ( 10.6 ) η = 1+ 10.6 φ + φ η 4 (.5 φ) + (.5 φ) + (.5 ) +... = 1+.5 φ + φ η = 1+ 9.1φ + 54 φ The fact (empiical finding) that the linea Einstein elation always undeestimates the actual viscosity incease with solid volume faction, can be accounted fo eithe by using a nonlinea elation o, sometimes, by eintepeting the volume faction in tems of an effective, equivalent, o appaent volume faction. In nanofluids, and to a cetain extent in odinay suspensions as well, the physical intepetation of this appaently enhanced volume faction may be agglomeation (since poous agglomeates fomed by clusteing of pimay paticles can act as seconday paticles duing flow pocesses). Fo nanofluids, additionally, the fluid suface laye on the dispesed nanopaticles, which is known to exhibit stuctue and popeties diffeent fom the bulk fluid, may be volumetically significant. One the most inteesting models fo pactical use is the Chen model [Chen et al. 007]. Unde easonable assumptions, this model contains one adjustable paamete which can be detemined fom expeimentally measued data and intepeted in physical tems, viz. as an effectively enhanced volume faction due to agglomeation. The Chen model is based on a Kiege-type elation, modified fo agglomeated suspensions, i.e. η [ η ] φ max φ = 1 a φ, max whee φ max is the maximum concenation (solids volume faction) at which flow can occu (appox. 0.605 fo high shea ate flows), [ η ] is the intinsic viscosity (appox..5 fo isometic paticles) and φ a is the appaent volume faction of the agglomeated solid paticles, which is elated to the tue volume faction φ via the factal dimension D, D a φ a = φ, whee and a ae the adii of the pimay paticles and the seconday paticles (agglomeates), espectively. Assuming that agglomeation is a diffusion limited aggegation pocess (DLA pocess) the factal dimension D is appox. 1.8 fo nanofluids. The esulting elation 114

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices φ η = 1 a 0.605 1. 1.515 can thus be used to extact an effective agglomeate size (adius a ) fom the measued concentation dependence of the viscosity ( η vesus φ cuve) when the pimay paticle size is known. Note that the above Kiege-type expession can be expanded in polynomial fom, [ η] φ + ([ η] φ ) + ([ η] φ ) +... η = 1+ a a a In othe wods, when the appaent volume faction of agglomeated paticles is taken instead of the tue volume faction, Einstein-type equations and its extensions ae obtained (see above). Indeed, expeimentally it is often found, that the atio a is between and 4, so that the intinsic viscosity emains at a value close to.5, as expected fo appoximately isometic agglomeates. Note futhe, that the tempeatue dependence of the effective viscosity of nanofluids, and suspensions in geneal, is almost entiely detemined by that of the base fluid (liquid medium) and can be descibed e.g. by the Vogel-Fulche-Tamann elation. In concluding this subsection we would like to emphasize that the heology of nanofluids is a hot topic of cuent eseach and many open questions emain. Fo stongly anisometic nanopaticles (e.g. cabon nanotubes) the viscosity incease with solids volume faction can be expected to be much stonge than pedicted by any of the above models. Nanofluids usually exhibit enhanced themal conductivity in compaison to the base fluid. Thee is some contovesy on whethe this enhancement is even lage than pedicted by classical models (ecent citical wok [Zhang et al. 007] denies this and claims that systematic measuement eos ae esponsible fo eoneously high themal conductivity values beyond the classical pedictions), but it is commonly ageed that thee is measuable enhancement of themal conductivity even fo vey low volume factions of solids (< 1 %). Pincipally this enhancement is simply a plausible consequence of the fact that most solids have highe sometimes consideably highe themal conductivity than the base fluid, cf. Table 14-C-1. Table 14-C-1. Themal conductivity values of mateials of inteest fo nanofluids. Mateial Themal conductivity [W/mK] at R.T. Base fluids Wate 0.61 Ethylene glycol 0.5 Oil 0.11 Nanopaticles MW-CNT (cabon nanotubes) 000 CuO 76.5 Al O (up to 4-46?) SiO 1.4 Fulleene 0.4 115

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices The most popula models fo themal conductivity pediction ae the Maxwell (o Maxwell-Eucken) model fo spheical paticles [Maxwell 189, Eucken 19], sometimes eoneously attibuted to Wasp [Wasp 1977], k = k k f s f ( k k ) k s + k f + s f φ =, k + k ( k k )φ the Buggeman-Landaue mean field model [Buggeman 195, Landaue 195], 1 k k = + λ + 8 4 k f s s λ with λ = ( φ 1) ( φ ) and the Hamilton-Cosse model fo non-spheical paticles [Hamilton & Cosse 196], k = k k f k = s + ( n 1) k + ( n 1) ( k k ) k s + f ( n 1) k ( k k )φ whee n is a shape facto, which is n = Ψ, whee Ψ is the spheicity, defined as the atio of the suface aea of a volume-equivalent sphee to that of the paticle in question. Note that fo sphees n =, i.e. the Hamilton-Cosse model contains the Maxwell model as a special case. High aspect atio paticles have high values of n and theefoe geneally moe potential fo themal conductivity enhancement than isometic nanopaticles of the same mateial (howeve, also the viscosity is extemely enhanced, which sets cetain limitations to the exploitation of this potential in pactical applications the volume faction has to be kept low enough to ensue sufficient fluidity). In the case of cabon nanotubes, which ae intinsically high-aspect atio mateials, the extemely high themal conductivity clealy outweighs the equiement of extemely low concentations to ensue fluidity. Specific models fo anisometic paticles, in paticula high aspect atio fibes and cabon nanotubes, ae the Yamada-Ota unit-cell model [Yamada & Ota 1980], k f s s ( k k ) k s + C k f + C s f φ =, k + C k ( k k )φ s f which can also be consideed as a modification of the Maxwell model (whee C is a shape 0. facto given fo cylindical paticles by C = φ ( l P d P ), whee l d P P is the aspect atio, i.e. the atio of fibe length and diamete) and the Jang-Choi model [Jang & Choi 004], s f f s f f φ, k k f, k k s fm φ θ ζ φ φ k s fm = 1 + cos + C Re P = 1+ cos θ ζ + Re P 1 1 p C φ 1 p, k f sp k f sp whee ζ is the Kapitza esistance (often teated as an adjustable paamete), fm is the adius of the molecules of the base fluid, sp the adius of the solid paticles, Re p the (nano-)paticle Reynolds numbe, P the Pandtl numbe and the cos-tem detemines the fibe oientation (via θ ): cos θ = 1 fo well aligned fibes and cos θ = 1 fo completely andom 116

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices oientation. This model pedicts an additional themal conductivity contibution via the paticle intefaces and a invese dependence on paticle size (last tem). Similaly, the model by Kuma et al. [Kuma 004] Cv P fm φ k = 1+, k 1 φ whee C is a constant and v P is given by the Stokes-Einstein elation f sp v P 8k T =, πη B P pedicts that the themal conductivity enhancement inceases with paticles size as 1 P (howeve, it also pedicts a tempeatue dependence in addition to that of the base fluid; which has not been expeimentally confimed so fa). Note that in the absence of micoconvection (caused by paticle motion), the Jang-Choi model simplifies to k k s = 1 φ + cos θ ζ φ. k f In ode to account fo themal conductivity inceases beyond the pedictions, the bulk themal conductivity of the solid paticles can be eplaced by an effective (appaent, equivalent) themal conductivity, e.g. that of agglomeates (aggegates), cf. [Chen et al. 007] fo a combined Maxwell-and-Buggeman model, o the nanopaticles with its adsobed patially odeed liquid laye (which exhibits stuctue and popeties significantly diffeent fom the bulk fluid and appoaching the coesponding solid phase stuctue and popeties) can be modeled as composite shells, i.e. the nanopaticles ae modeled as non-ovelapping equivalent paticles of geate size and the odeed liquid laye is assumed to have highe themal conductivity than the bulk themal conductivity of the base fluid. The Maxwell elation as modified by the Yu-Choi model is ( k k ) ( 1+ ε ) k pe + k f + pe f φ k =, k + k pe f ( k k )( 1+ ε ) φ whee ε is the atio of the odeed laye thickness to the nanopaticle adius and k pe is the effective themal conductivity of the composite-shell paticles given by k = [ ( 1 γ ) + ( 1+ ε ) ( 1+ γ )] ( 1 γ ) + ( 1+ ε ) ( 1+ γ ) pe k s whee γ is the atio of the themal conductivity of the odeed laye to that of the solid paticle coe. Denoting the atio between the themal conductivity of the solid paticles and the fluid ( phase contast ) as κ = k s k f, a shot-hand notation of the above elations can be given. In this shot-hand notation futhe models occuing in the liteatue can be given as follows: pe f γ, 117

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Appendices 1. Jeffey model [Jeffey 197]: k = ϑ 4 9ϑ 16 κ + κ + 1+ + + + +... φ ϑ φ ϑ. Davis model [Davis 1986]: ( κ 1) ( κ + ) ( κ 1). Lu-Lin elation [Lu & Lin 1996]: [ φ + f ( κ ) φ O( )] k = 1+ + φ φ k = 1+ κ φ + b φ 118

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Refeences CPPS-Refeences. Refeences to the Lectue Couse Chaacteization of Paticles and Paticle Systems 1. Adamczyk Z.: Paticles at Intefaces Inteactions, Deposition, Stuctue. Elsevie, Amstedam 006.. Allen T.: Paticle Size Measuement, Volume 1 Powde Sampling and Paticle Size Measuement, Volume Suface Aea Measuement (fifth edition). Chapman and Hall, London 1997.. Aste T., Weaie D.: The Pusuit of Pefect Packing. Institute of Physics Publishing. Bistol and Philadelphia 000. 4. Begaya F., Theng B. K. G., Lagaly G. (eds.): Handbook of Clay Science. Elsevie, Amstedam 006. 5. Benhadt C.: Ganulometie Klassie- und Sedimentationsmethoden. Deutsche Velag fü Gundstoffindustie, Leipzig 1990. 6. Benhadt C.: Paticle Size Analysis Classification and Sedimentation Methods. Chapman & Hall, London 1994. 7. Boccaa N., Daoud M. (eds.): Physics of Finely Divided Matte. Spinge, Belin 1985. 8. Bohen C. F., Huffman D. R.: Absoption and Scatteing of Light by Small Paticles. Wiley-VCH, Weinheim 004. 9. Bon M., Wolf E.: Pinciples of Optics. Thid edition. Pegamon Pess, Oxfod 1965. 10. Champeney D. C.: Fouie Tansfoms and Thei Physical Applications. Academic Pess, New Yok 197. 11. Choi. S. U. S., Zhang Z. G., Keblinski P.: Nanofluids, pp. 757-77 in Nalwa H. S. (ed.): Encyclopedia of Nanoscience and Nanotechnology Vol. 6. Ameican Scientific Publishes, Valencia / CA 004. 1. Dettmann J.: Fulleene. Bikhause, Basel 1994. 1. Dinge D. R.: Paticle Calculations fo Ceamists. Dinge Ceamic Consulting Sevices, Clemson 001. 14. Fejes Tóth L.: Reguläe Figuen. Teubne, Leipzig 1965. 15. Fiedich J.: Fomanalyse von Patikelkollektiven (Fotschittsbeichte VDI, Reihe Vefahenstechnik, N. 68). VDI Velag, Düsseldof 199. 16. Geman R. M.: Paticle Packing Chaacteistics. Metal Powde Industies Fedeation, Pinceton 1989. 17. Happel J., Benne H.: Low Reynolds Numbe Hydodynamics. Matinus Nijhoff / Kluwe, The Hague 198. 18. Heffels C.: On-line Paticle Size and Shape Chaacteization by Naow Angle Light Scatteing. Ph.D. Thesis, Technical Univesity Delft, Delft 1996. 19. Hedan G.: Small Paticle Statistics. Buttewoths, London 1960. 0. Hilbet D., Cohn-Vossen S.: Anschauliche Geometie. Spinge, Belin 19. 1. Hilliad J. E., Lawson L. R.: Steeology and Stochastic Geomety. Kluwe Academic Publishes, Dodecht 00.. Hulst, H. C. van de: Light Scatteing by Small Paticles. Wiley, New Yok 1957. 119

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Refeences. Jackson J. D.: Classical Electodynamics. Second edition. Wiley, New Yok 1975. 4. Jasmund K., Lagaly G. (eds.): Tonmineale und Tone Stuktu, Eigenschaften, Anwendungen und Einsatz in Industie und Umwelt. Steinkopff Velag, Damstadt 199. 5. Jelínek Z. K.: Paticle Size Analysis. John Wiley and Sons, New Yok 1974. 6. Jillavenkatesa A., Dapkunas S.J., Lum L.-S. H.: Paticle Size Chaacteization (National Institute of Standads and Technology Special Publication NIST 960-1, downloadable fom http://www.nist.gov/pacticeguide). US Govenment Pinting Office, Washington 001. 7. Johnson C. S., Gabiel D. A.: Lase Light Scatteing. Dove, New Yok 1994. 8. Jonasz M., Founie G. R.: Light Scatteing by Paticles in Wate Theoetical and Expeimental Foundations. Elsevie, Amstedam 007. 9. Jones D. S.: The Theoy of Electomagnetism. Pegamon, Oxfod 1964. 0. Kaye B. H.: A Random Walk Though Factal Dimensions (second edition). VCH, Weinheim 1994. 1. Kendall D. G., Baden D., Cane T. K., Le H.: Shape and Shape Theoy. John Wiley & Sons, Chicheste 1999.. Keke M.: The Scatteing of Light and Othe Electomagnetic Radiation. Academic Pess, New Yok 1969.. Klug H. P., Alexande L. E.: X-ay Diffaction Pocedues fo Polycystalline and Amophous Mateials. Second edition. John Wiley & Sons, New Yok 1974. 4. Kopp H.: Bildveabeitung inteaktiv. Teubne, Stuttgat 1997. 5. Köste E.: Ganulometische und mophometische Messmethoden an Minealkönen, Steinen und sonstigen Stoffen. Enke, Stuttgat 1964. 6. Kuma C. (ed.): Nanomateials Toxicity, Health and Envionmental Issues. Wiley- VCH, Weinheim 006. 7. Landau L. D., Lifshitz E. M.: Electodynamics of Continuous Media. Pegamon Pess, Oxfod 1960. 8. Lee K. W., Kwon S.-B.: Aeosol Nanopaticles Theoy of Coagulation, pp. 5 44 in Schwatz J. A., Contescu C. I., Putyea K. (eds.): Dekke Encyclopedia of Nanoscience and Nanotechnology. Macel Dekke, New Yok 004. 9. Leschonski K.: Patikelmeßtechnik 40. Lowell S., Shields J. E., Thomas M. A., Thommes M.: Chaacteization of Poous Solids and Powdes Suface Aea, Poe Size and Density. Kluwe Academic Publishes, Dodecht 004. 41. Mandelbot B. B.: Factals Fom, Chance and Dimension. Feeman, San Fancisco 1977 (Czech tanslation ). 4. Mandelbot B. B.: The Factal Geomety of Natue. Feeman, San Fancisco 198 (Geman tanslation ). 4. Maxwell J. C.: A Teatise on Electicity and Magnetism Vol. 1. Thid edition. Dove, New Yok 1954. 44. McCone W., Daftz R. G., Delly J. G.: The Paticle Atlas. Ann Abo Science Publishes, Ann Abo, 1967. 45. Mecke K., Stoyan D. (eds.): Mophology of Condensed Matte Physics and Geomety of Spatially Complex Systems. Spinge, Belin 00. 10

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Refeences 46. Mecke K., Stoyan D. (eds.): Statistical Physics and Spatial Statistics. Spinge, Belin 000. 47. Michaelides E. E.: Paticles, Bubbles and Dops Thei Motion, Heat and Mass Tansfe. Wold Scientific, New Jesey 006. 48. Mie G.: Beitage zu Optik tübe Medien, speziell kolloidale Metallösungen, Ann. Phys. 5, 77-445 (1908). 49. Milling A. J. (ed.): Suface Chaacteization Methods Pinciples, Techniques, and Applications. Macel Dekke, New Yok 1999. 50. Mittemeije E. J., Scadi P. (eds.): Diffaction Analysis of the Micostuctue of Mateials. Spinge, Belin 004. 51. Moison I. D., Ross S.: Colloidal Dispesions Suspensions, Emulsions and Foams. Wiley Intescience, New Yok 00. 5. Mühlenweg H.: Investigation of the Influence of Paticle Shape on Diffaction Spectoscopy. DFG Scholaship Repot 1997. 5. Mülle G.: Methoden de Sediment-Untesuchung (= Volume I of von Engelhadt W., Füchtbaue H., Mülle G. (eds.): Sediment-Petologie). Schweizebat, Stuttgat 1964. 54. Munshi P. (ed.): Computeized Tomogaphy fo Scientists and Enginees. CRC Pess (Taylo & Fancis), Boca Raton 007. 55. Nalwa H. S. (ed.): Encyclopedia of Nanoscience and Nanotechnology (10 volumes). Ameican Scientific Publishes, Valencia 004. 56. Ohse J., Mücklich F.: Statistical Analysis of Micostuctues in Mateials Science. John Wiley, Chicheste 000. 57. Piee A. C.: Intoduction to Sol-Gel Pocessing. Kluwe Academic Publishes, Boston 1998. 58. Rayleigh, Lod: On the light fom the sky, its polaization and colou, Philos. Mag. 41, 107-10, 74-79 (1871). 59. Reist P. C.: Aeosol Science and Technology (second edition). McGaw-Hill, New Yok 199. 60. Ruck B.: Lase-Dopple-Anemometie. AT Fachvelag, Stuttgat 1987. 61. Russ J. C., Dehoff R. T.: Pactical Steeology. Second edition. Kluwe Academic / Plenum Publishes, New Yok 000. 6. Russ J. C.: The Image Pocessing Handbook. Fifth edition. CRC Pess (Taylo & Fancis), Boca Raton 007. 6. Russel W. B., Saville D. A., Schowalte W. R.: Colloidal Dispesions. Cambidge Univesity Pess, Cambidge 1989. 64. Sahimi M.: Applications of Pecolation Theoy. Taylo & Fancis, London 1994. 65. Saltykov S. A.: Steeometische Metallogaphie. VEB Deutsche Velag fü Gundstoffindustie, Leipzig 1974. 66. Saxl I.: Steeology of Objects with Intenal Stuctue. Academia, Pague 1989. 67. Scalett B.: Chaacteization of paticles and powdes, pp. 99 15 in Book (volume ed.): Pocessing of Ceamics Pat 1 (= Vol 17 A of Cahn R. W., Haasen P. and Kame E. J. (seies eds.): Mateials Science and Technology A Compehensive Teatment, softcove edition). Wiley VCH, Weinheim 005. 68. Sinton C. W.: Raw Mateials fo Glass and Ceamics Souces, Pocesses and Quality Contol. John Wiley & Sons, Hoboken / NJ 006. 11

PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems Refeences 69. Stauffe D., Ahaony A.: Pekolationstheoie Eine Einfühung. VCH, Weinheim 1995. 70. Stauffe D., Ahaony A.: Intoduction to Pecolation Theoy. Revised second edition. Taylo & Fancis, London 1994. 71. Stoyan D., Mecke J.: Stochastische Geometie. Akademie-Velag, Belin 198. 7. Stoyan D., Kendall W. S., Mecke J.: Stochastic Geomety and its Applications. Second edition. John Wiley & Sons, Chicheste 1995. 7. Statton J. A.: Electomagnetic Theoy. McGa-Hill, New Yok 1941. 74. Sugimoto T.: Monodispesed Paticles. Elsevie, Amstedam 001. 75. Sunagawa I. (ed.): Mophology of Cystals Pat A. Tea Scientific and Reidel Publishing Company, Tokyo and Dodecht 1987. 76. Syvitski J. P. M. (ed.): Pinciples, Methods, and Application of Paticle Size Analysis. Cambidge Univesity Pess, Cambidge 1991. 77. Valvoda V., Polcaová M., Lukáč P.: Základy stuktuní analýzy. Univezita Kalova, Pague 199. 78. Waen B. E.: X-ay Diffaction. Addison-Wesley Publishing Company, Reading 1969. 79. Wasp F. J.: Solid-Liquid Sluy Pipeline Tanspotation. Tans Tech Publications, Belin 1977. 80. Williams C. S., Becklund O. A.: Optics a Shot Couse fo Enginees and Scientists. Wiley-Intescience, New Yok 197. 81. Wojna L.: Image Analysis Applications in Mateials Engineeing. CRC Pess, Baco Raton 1999. 8. Xu R.: Paticle Chaacteization Light Scatteing Methods. Kluwe Academic Publishes, Dodecht 000. 8. Zong C.: Sphee Packings. Spinge, New Yok 1999. 1