Characterization of particles and particle systems

Transcription

1 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.

2 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

3 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).

4 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.

5 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

6 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

7 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 (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 (fcc / hcp) Tetagonal-sphenoidal Body-centeed cubic Othohombic Simple cubic Diamond

8 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 = 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

9 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

10 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

11 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

12 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

13 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 =. 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

14 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

15 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

16 PABST & GREGOROVÁ (ICT Pague) Chaacteization of paticles and paticle systems 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

17 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 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

18 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

19 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 Hz (IR) to 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 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

20 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

21 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 Fist min. acsin ( 1. λ D) 0 0 Second max. acsin ( 1.64 λ D) Second min. acsin (. λ D) 0 0 Thid max. acsin (.68 λ D) Thid min. acsin (.4 λ D) 0 0 Fouth max. acsin (.70 λ D) 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 , 594.1, 61.0, 6.8 A ion lase , Wate-cooling needed Diode lase , 450, 65, 650, 670, 685, 750, 780 Low cost 0

22 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

23 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.