Light Scattering. K c 2 A

Transcription

1 Light Scattering Kc ΔR( θ ) K c P( θ ) A ΔR( θ ) 1 P( θ ) 1 χ ρ V M 1 w optical constants + A c polymer concentration excess Raleigh ratio +... particle scattering factor, known for various particle geometries 1 nd virial coefficient R solution solvent Scattering arises from Light (Δα) polarizability fluctuations X-ray (Δρ) electron density variations Neutron (Δb) neutron scattering length variation R We will derive this. Note the nice set of variables that we would like to be able to determine ~ excess scattering intensity

2 Scattering arises from Density Fluctuations A dilute gas in vacuum Consider small particles: d p << λ radiation (situation:~ point scatterers) Particles d p Incident radiation Scattered Intensity at scattering angle θ to a detector r away from sample: I θ I 8π 4 1+ cos θ 0 ( ) α polarizability of molecule α λ 4 r I 0 incident beam intensity For N particles in total volume V (assume dilute, so no coherent scattering) λ I θ ' N V I N θ ε + 4π α V 1 ε dielectric constant ε n, ε(ω) frequency dependent

3 Fundamental relationship: index of refraction polarizability n 1+ 4 π N V α Can approximate n gas 1+ dn dc c dn refractive index increment dc c conc. of particles per unit volume So by analogy for a polymer-solvent solution: n n 0 + dn dn c n n 0 + n 0 c dc dc dn dn ngas 1 + c 1+ c +... dc dc Solving gives α 1 ( dn/dc) c the polarizability π N /V ( )

4 I θ ' simplifying Rayleigh and the Molecular Weight of Gases N V I θ ' 8π ( 1+ cos θ) (dn / dc) c λ 4 r π (N /V ) ( ) I π 1+ cos θ 0 λ 4 r (N /V ) dn dc c and since This expression contains several parameters dependent on scattering geometry, so we define Rayleigh Ratio, R as which equals R π dn dc 4 λ N AV R M c I 0 Where K is a lumped optical constant I θ ' ( 1+ cos θ)/r Or just R K M c Note, for polymer-solvent solution: dn π π dn n 0 dc K dc 4 K λ N AV λ 4 N AV Rayleigh measured the molecular weight of gas molecules using light scattering! N V M c / N AV

5 Scattering from Fluctuations II A dilute gas of polymer chains in solution polymer coil (solute) solvent Debye: Re-identify fluctuations as chains in a solvent and extend Rayleigh s idea to polymers in solution pure solvent Incident radiation λ coil/solvent d p V solution Now for polymer coils: λ ~ d p Recognize 4 features in a binary component system: 1. Each cell has on average, the same number of solvent molecules but there are variations. Fluctuations in solvent density will give rise to some (weak) scattering (subtract off pure solvent scattering).. Fluctuations in the number of solute molecules (chains) will give rise to significant scattering 3. Fluctuations in the concentration of solute create osmotic forces 4. Polymer chains are large and cannot treat them as point scatterers P(θ) 1

6 The Features of Excess Scattering Feature 1. Define ΔR R solution R pure solvent Excess Rayleigh Ratio Feature. Remaining scattering arises from fluctuations in solute concentration dn I π 1+ cos ΔR depends on and dc ( ) 0 ( θ) dn c δc λ 4 r (N /V ) dc Einstein-Smoluchowski mean squared concentration fluctuation per unit vol. I θ ' ( δc ) RTc ( ) δvn AV π / c Feature 3. A local osmotic pressure will arise due to local concentration differences, this effect acts to suppress solute concentration fluctuations. Note in the gas-vacuum system, such an effect is not present. 1 π RT 1 ( c M + χ) c +L V 1 ρ π c 1 RT M + A c + L

7 Feature 3 cont d + + λ π L AV 4 0 solvent solution c A M 1 N c dc dn n R R ΔR Kc 1 1 M + A c +L similar to the Rayleigh scattering for gases but with a new term depending A AV 4 N M c dc dn R λ π Gas in vacuum Polymer in solution ΔR

8 Scattering from Polymer Solutions Feature 4. Polymer chains are large and can not be assumed as point scatterers for visible light λ ~ 6,38Å He-Ne laser Å Coil size ~ typically Å depending on molecular weight of polymer Therefore need to consider self-interference of monomers in polymer coil on scattered intensity Therefore P(θ) term is important

9 Scattering of Polymer Solutions Introduce finite size chain scattering factor P(θ) 1 ΔR Kc P(θ) 1 + A c +L For a Gaussian coil P θ (Debye, 1939) () u M w u 1+ e u ( ) where u 4 π n 0 λ sin θ R g q scattering vector, u q R g In the limit of very small θ, P(θ > 0) 1 Useful approximation for small, nonzero θ P() θ 1 u 3 and 1 P() θ 1+ u 3 for u1/ <<1

10 Scattering in a Polymer-Solvent System ( θ), c 0, A 0 In general, P 1 therefore rewrite excess Rayleigh ratio as Kc 1 u + Ac ΔR M L 3 u q R g Kc ΔR 1 M + A c +L 1+ 16π n 0 θ sin R 3λ g Equation is valid for dilute solutions and scattering angles θ such that u 1 / << 1 Next: how to plot scattered intensity vs. c and versus q to extract M and A

11 Zimm Plot Analysis of Light Scattering Data Light Scattering Experiment q o ~ Sample Scattering Measurements: Detector ~ q Collect Scattered Intensity at typically 8 angles: θ 30, 37.5, 40, 60, 75, 90, 105, 10 Measure ΔR typically for 5 solute concentrations: c 1, c, c 3, c 4, c Pure (no dust!) solvent R solvent. Polymer solutions [c,i, I(θ i )] R polymer 3. Construct Zimm Plot ΔR R polymer -R solvent Kc Note: is a function of variables, c and θ. ΔR θ Zimm s cool idea was to plot y(p,q) vs (p+q) to separate out dependence on each variable: called Zimm plot

12 Construction of a Zimm Plot The Master Equation: Kc ΔR 1 M w θ 16π n 0 sin + A c λ R g do double extrapolation: θ 0 o A & c 0 <R g > & M w M w Kc ΔR( θ, i c i Mole/gram Units ) conc. dilute small angle θ fixed for a set of concentrations c fixed for a set of angles large angle sin (θ/) + k c dimensionless axis

13 Zimm Plot Sample Data Set Kc ΔR( θ, i c i x 10-3 ) θ 30 o 37.5 o 45 o 60 o 75 o 90 o 105 o 10 o 135 o 14.5 o 150 o x c g/cm 3 1 x x x K π dn n 0 dc λ 4 N AV For c 0.00, Kc θ 30 ΔR( θ, ) i c i 3.18 n dn dc cm 3 g 1 λ 5.461x10 5 cm N AV 6.0 x10 3 mole 1

14 Plotting the Data vertical data set θ30 c varies (decreases) Extrapolating to zero concentration (.67, 3.18) Kc ΔR θ Image removed due to copyright restrictions. Please see, for example, from horizontal data sets θ variable c x10-3 Plot typical data point : 3 3 θ 30,c 10 g / cm These points are determined by extrapolating the equations of each regression line x y-axis: Kc 3.18 ΔR θ x-axis: sin 30 + k( 10 3 ) ( 10 3 ) 0.67 Pick constant to spread the plot

15 Plotting the data cont d Extrapolating to zero scattering angle (.67, 3.18) 100c Image removed due to copyright restrictions. Please see, for example, from

16 Extrapolate the extrapolated data to obtain: M w, A R g A Kc Image removed due to copyright restrictions. ΔR θ Please see, for example, R from g

17 An example: A, R g and M w Note: units on y-axis are mole/g. The x-axis is dimensionless. Kc ( 1 + A ) R 0 M c + ( 0 1+ <s > ) z w -1 mole g θ 0 line Kc R 0 Kc R 0 The intercept occurs at a y-value ( ) of 1.36 x 10-6 mole/g. Hence, M w 7.35 x 10 5 g/mole. Now, consider the c 0 line. Where c 0, 1 M w 0 ( 1+ < s > z ) 0 KC / R x c 0 line c + Sin θ slope slope 1 M w Measure the slope of the c 0 line. y x 5.5 x 10-7 mole/g x 10-6 mole/g <s ( > z ) M M w From intercept, w 735,000 g/mol <s > z w 3( 1.6 x 10-6 mole/g ) (7.35 x 10 5 g/mole)(5.461 x 10-5 ) 16 (1.5014).33 x cm Figure by MIT OCW.

18 (i) (ii) getting A, R g and M w r < r6 s> 6< R g > r 6 (.33 x cm ) 1.40 x cm 1/ 1.18x10-5 cm 1,18 A Now consider the θ 0 line. At θ 0 (neglecting higher order c terms) Kc ΔR 1 + A c Mw θ slope A The density of polystyrene (ρ ) is 1.05g/cm 3. The molar volume of benzene V 1 MW/ρ 1. The molecular weight of benzene is g/mole and its density is g/cm 3 (from CRC Handbook of Chemistry and Physics). from the Zimm plot slope 1.5 x10 6 mole/g x10 g/cm x10 4 mole cm 3 g A A 4.55x10 4 mole cm g 3 Pretty big molecules!

19 A, R g and M w (ii) The relationship between A and χ is given by: χ 1 χ 1 mole cm x10 4 g A V 1 ρ Substituting the values, 78.11g/mole g /cm g/cm 3 ( ) χ (So, of course, the polymer solution used for light scattering will be a single phase since χ < 1/ for miscibility of solvent and polymer).

20 Noncrystalline Materials The structure of noncrystalline materials (i.e. polymer glasses, amorphous polymer melts) is characterized by short range order (SRO) SRO develops due to excluded volume and locally dense packing (glasses ~ are only 10% less dense than crystals) Pair distribution function g(r) is a dimensionless function used to quantify SRO. In polymers SRO is primarily due to covalent intra-molecular bonds and neighboring intermolecular packing. Properties of noncrystalline polymers are heavily influenced by τ*, the characteristic relaxation time relative to an experimental observation time, t. liquid (melt) t» τ* rubbery t τ* τ*(t)? glassy t «τ*

21 φ fixed θ Structural Features of Noncrystalline Polymers SRO in Polymers C C 3 θ Due to covalent intra-molecular bonds C 1 C C 1 C 3 C 1 C 4 constant constant varies Due to chain-chain inter-chain distances ~ 5Å gives rise to peaks at large q C 1 Inter-chain C 1 C 1 ~ 5Å c 1 C 1, Typical X-ray signature of noncrystalline materials - broad overlapping peaks from multiple distances: I(q) inter- intra- q

22 Pair Distribution Function g(r) Å Figure by MIT OCW. The Pair Distribution Function g(r) addresses the distances between the centers of mass of pairs of units. Since glasses and liquids are isotropic, the magnitude of the inter-unit distance is of interest. The scalar distance r ij between molecule i and molecule j is: r ij r i - r j

23 r ij r i - r j g(r) cont d Characterize the set of distances {r ij } from an average unit i to every other unit j 1.. N. g(r) counts the number of units dn in a small spherical shell sampling volume element of size dv at each distance r from a reference unit, dv 4πr dr The statistical average of these numbers for many units chosen as the reference is divided by the average unit density ρ Figure showing the pair-distribution functions for gas, liquid or glass, and monatomic crystal removed due to copyright restrictions. See Figure.5 in Allen, S. M., and E.L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, g( r) 1 ρ dn( r, dv ( r, r r + + dr) dr)

24 Features of g(r) in Glasses & Liquids g(r) 1.0 At several unit diameters, the average number of units/vol. becomes constant ξ 0 R o r Figure by MIT OpenCourseWare. Due to excluded volume, g(r) 0 for distances less than R o Liquids and glasses are strongly correlated at the shortest distance between units, the maximum occurs at slightly > R o, this largest peak is the average distance to the first shell to the nearest-neighbor unit. C 1 C g(r) intra C 1 C C3 g(r) inter Superposed correlations