In: Encylcopedia of Analytical Chemistry: Instrumentation and Applications, LIGHT SCATTERING, CLASSICAL: SIZE AND SIZE DISTRIBUTION CHARACTERIZATION

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1 In: Encylcopedia of Analytical Chemistry: Instrumentation and Applications,!!!! LIGHT SCATTERING, CLASSICAL: SIZE AND SIZE DISTRIBUTION CHARACTERIZATION!!!!!!! G. C. Berry Department of Chemistry Carnegie Mellon University Pittsburgh, PA, USA!!!! ABSTRACT!!! The use of classical, or time-averaged, light scattering methods to characterize the size and size distribution of macromolecules in dilute solutions or particles in dilute dispersions is discussed. The necessary scattering relations are presented systematically, starting with three cases at infinite dilution: the scattering extrapolated to zero angle, the scattering at small angle, and the scattering for arbitrary angle, including the inversion of the scattering data to estimate the size distribution. The relations needed to effect an extrapolation to infinite dilution from data on dilute solutions are also discussed. These sections are followed by remarks on light scattering methods, and concluding sections giving examples for several applications. The Rayleigh-Gans-Debye approximation is usually appropriate in the scattering from dilute polymer solutions, and is also adequate for the scattering from dilute dispersions of small particles. It is assumed when appropriate, but more complete theories are introduced where necessary, as in the use of the Mie-Lorentz theory for large spherical particles. Methods to suppress multiple scattering and non ergodic scattering behavior are discussed. berry 1 June 2014!

2 ! TABLE OF CONTENTS!!! 1. INTRODUCTION 1! 2. SCATTERING RELATIONS General Remarks Scattering at zero angle and infinite dilution Isotropic solute in the RGD regime Isotropic solute beyond the RGD regime Anisotropic solute Scattering at small angle and infinite dilution Isotropic solute in the RGD regime Isotropic solute beyond the RGD regime Anisotropic solute Scattering at arbitrary angle and infinite dilution Isotropic solute in the RGD regime Isotropic solute beyond the RGD regime Anisotropic solute The size distribution from scattering data at infinite dilution 21! 2.6 Extrapolation to infinite dilution EXPERIMENTAL METHODS Instrumentation Methods 28! 4. EXAMPLES Static scattering and size separation chromatography Light scattering from vesicles and stratified spheres Scattering from very large particles Intermolecular association Scattering with charged species Scattering from optically anisotropic solute Scattering from gels and dispersed particles The intramolecular structure factor for wormlike chains 63! 5. FREQUENTLY USED NOTATION 66! 6. REFERENCES 68! TABLES (3)! FIGURE CAPTIONS FIGURES (22) berry 1 June 2014!

3 1. INTRODUCTION 2. SCATTERING RELATIONS 2.1 General Remarks ϑ ϑ δ

4 ϑ, π λ ϑ λ = λ ο λ ο in vacuuo, ϑ ϑ ϑ ϑ ϑ R ϑ ϑ R ϑ R ϑ ϑ R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ ν ν ν

5 R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ R ϑ ϑ R ϑ R ϑ R ϑ R ϑ R ϑ ϑ R ϑ R ϑ R ϑ R ϑ R ϑ

6 R Vv ϑ R Hv ϑ µ µ Σ µ R Vv ϑ µ ϑ R Hv ϑ R ϑ R ϑ ψ ϑ ϑ ϑ ϑ ϑ ϑ ϑ π λ ψ ϑ ϑ ψ Π Π ρ ρ ϑ ϑ

7 ϑ ϑ ϑ Γ ϑ ϑ ϑ R ϑ ϑ ϑ Γ ϑ ψ Γ Γ Γ ϑ ϑ ϑ ϑ γ (δ/δ ο ϑ, (δ/δ ο ϑ γ

8 ψ δ ο ψ δ ο γ ϑ ϑ ϑ ϑ Γ Γ ϑ ϑ ϑ ϑ 2.2 Scattering at zero angle and infinite dilution Isotropic solute in the RGD regime. R R ψ ϑ ϑ R ψ R ψ Σ µ µ µ Σ ψ µ ψ ψ µ µ Σ µ Σψ µ ψ Σ µ ψ µ µ Σψ µ ψ µ µ Σ µ µ µ ψ µ ψ

9 ψ µ ψ Σ µ µ µ µ µ µ Π Π Π Π Π ψ ψ ψ ψ ψ Σ µ µ µ Σ µ µ µ µ µ µ ψ ψ ψ µ Isotropic solute beyond the RGD regime

10 R ψ µ Σψ µ µ µ ψ µ µ ψ µ µ λ µ λ µ λ ψ µ µ λ α α π λ α α α α πρν Α α ρ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ R Vv ϑ ν ϑ π λ ϑ ϑ α α R Vv ϑ ψ ψ α

11 α α α α α α α R Vv ν α α α τν τ α π ν α > 4 α α α α α α Σ µ µ µ λ µ λ µ α µ α µ α π λ

12 ε λ α µ µ λ α Anisotropic solute. R ψ δ ψ R ψ δ δ Σ µ µ µ δ µ δ R ψ R ψ δ 2.3 Scattering at small angle and infinite dilution Isotropic solute in the RGD regime. ϑ R R ϑ R ϑ ϑ ϑ

13 Σ µ µ µ ΣΣ ψ ψ µ ψ ψ µ µ ΣΣψ ψ µ Σ µ µ µ ΣΣ µ Σ µ µ µ µ µ µ µ µ Φ Φ µ µ µ µ ε ε+1 ε+1 α Σ µ µ µ α α α 1 α ε

14 ε ψ ψ ψ Isotropic solute beyond the RGD regime. Σ µ µ λ µ λ µ µ µ Σ µ µ λ µ µ µ λ µ λ µ

15 α λ α α π λ α α α α α α α α α α α α α α α α α α Σ µ µ µ α µ α µ µ Σ µ µ µ α µ λ µ α µ λ µ α µ λ α µ µ

16 λ α µ α µ Anisotropic solute. R ϑ R ϑ ϑ δ ϑ δ δ δ δ δ δ δ 2.4 Scattering at arbitrary angle and infinite dilution Isotropic solute in the RGD regime. ϑ R ϑ ψ ϑ Σ µ µ µ ΣΣ ψ ψ ψ µ

17 ϑ Σ µ µ µ ϑ µ ΣΣ ϑ µ µ µ µ ϑ µ µ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ µ ϑ ϑ Σ µ µ µ µ µ ϑ ϑ

18 ϑ ϑ ϑ ϑ β β ϑ ϑ ϑ ϑ ϕ ϕ ϕ ϕ ϕ ϕ ϕ Σ µ µ µ

19 ϑ µ µ µ 1/ε µ µ ε µ µ µ ε ϑ µ µ π π ϑ π R ϑ ε ϑ ϑ ϑ

20 ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ψ ψ ψ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ρ ρ ψ R R ϑ ϑ ψ ψ ϑ R ϑ Isotropic solute beyond the RGD regime. ϑ α ϑ α ϑ ϑ ϑ ϑ ϑ ϑ

21 ϑ ϑ α π λ λ λ α λ ϑ ϑ ϑ α α ϑ α + 4α α ϑ α ϑ α ϑ α ϑ λ α π λ

22 R ϑ Σ µ α λ µ µ α µ ϑ αµ ϑ µ µ R ϑ λ, ϑ αϑ αϑ) λ λ α >> 1 α ϑ ϑ α α ϑ

23 ϕ ϕ π ϑ ϕ ϑ ϑ ϑ ϑ π/2 ϑ π/2 ϑ ϑ π/4 ϑ π/4 ϑ Anisotropic solute. ϑ ϑ ϑ ϑ ϑ ϕ ϑ ϕ ϑ π/4 2.5 The size distribution from scattering data at infinite dilution ϑ ϑ ϑ

24 R ϑ ϑ R ϑ R ϑ ϑ, ϑ,0 ϑ ϑ ϑ R ϑ ϑ

25 τ τ Σ µ µ µ ϑ µ τ γ µ ϑ γ µ µ ϑ Σ µ µ µ ϑ µ τ ϑ µ µ µ µ µ µ µ ϑ µ ϑ ϑ ϑ ϑ

26 ϑ ϑ ϑ µ R ϑ λ 2.6 Extrapolation to infinite dilution R ϑ R ϑ R ϑ R ϑ ϑ ϑ Γ ϑ ϑ ψ R ϑ ϑ ϑ

27 ϑ R ϑ ϑ ϑ Γ R R α α α Π π π

28 η η η Γ δ Γ Γ ϑ ϑ R ϑ ϑ Γ ϑ R ϑ ϑ Γ ϑ ϑ ϑ Γ R ϑ ϑ

29 ϑ R ϑ ϑ R R R ϑ Γ ϑ Γ ϑ R ϑ Γ ϑ ϑ ϑ, ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ R ϑ ϑ ϑ ϑ ϑ ϑ ϑ 3. EXPERIMENTAL METHODS

30 3.1 Instrumentation ϑ

31 ϑ δϑ ϑ ϑ ϑ ϑ 3.2 Methods

32 ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ λ λ λ ο ϑ ϑ λ ϑ π ϑ R ϑ R ϑ R ϑ R ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ λ ϑ R R ϑ ϑ ϑ ϑ ϑ λ λ R ϑ ϑ R ϑ R ϑ ϑ R ϑ

33 ϑ ϑ ϑ ϑ 4. EXAMPLES 4.1 Static scattering and size separation chromatography Γ Σ R ϑ R ϑ ϑ Σ ϑ ϑ R ϑ ϑ ϑ ϑ

34 R ϑ ϑ ϑ Σ n n

35 ϑ R ϑ Γ µ R ϑ µ µ µ ϑ ν µ ν µ µ µ ϑ µ µ ϑ Σ µ ϑ µ µ ϑ µ ϑ µ µ µ µ µ µ µ µ 4.2 Light scattering from vesicles and stratified spheres

36 π π ϑ β β β π π β β π β β ϑ ϑ Γ

37 Γ µ ϑ, ψ ψ ψ ψ ψ ψ ρ ψ ρ ψ ρ

38 4.3 Scattering from very large particles α ϑ ϑ ϑ ϑ,ϑ ϑ ϑ ϑ ϑ αϑ αϑ αϑ ϑ ϑ ϑ ϑ ϑ Σ µ µ α µ α µ ϑ α µ ϑ α µ ϑ α µ ϑ µ µ µ ϑ

39 ϑ ϑ ϑ R ϑ ϑ R ϑ α ϑ α α α ( α π λ α λ ϑ ϑ α ϑ ϑ) ϑ α ϑ αϑ ϑ α α α α ϑ αϑ ϑ R ϑ ϑ ϑ ϑ ϑ µ

40 ϑ ϑ R ϑ R ϑ µ α ϑ ϑ ϑ ϑ ϑ 4.4 Intermolecular association

41 R Vv ϑ R Vv ϑ ϑ R Vv ϑ R Vv ϑ ε γ ε γ ϑ

42 Σ µ Σ ν µ µ ν ν µν µµ µ µν µµ νν ζ ζ ε ζ ζ ϑ α ϑ ζ ϑ ζ ζ γ/2 ζ ζ ζ ζ ϑ ε ϑ ε γ ζ R Vv ϑ ϑ R Vv ϑ Σ R µ Vv ϑ µ Σ µ ϑ µ ϑ

43 ϑ ϑ µ µ µ ϑ τ ϑ Σ µ µ ϑ τ γ µ ϑ µ ϑ R Vv ϑ µ R Vv ϑ τ ϑ µ µ µ 6πη γ µ ϑ µ µ µ µ µ µ µ 4.5 Scattering with charged species

44 R ϑ R ϑ Π Π κ κ κ π Α Β µ µ µ µ µ Β ε ε πε ο ε ε Β ε κ π Α µ µ ε Β κ µ κ R κ κ κ κ

45 R ϑ Γ R ϑ ϑ Γ ϑ ϑ Γ ϑ ϑ ϑ ϑ ϑ κ ϑ κ κ κ κ κ ad hoc 4.6 Scattering from optically anisotropic solute

46 ψ ρ γ ρ ρ ψ δ ο γ ψ δ δ/δ ο δ/δ ο R R R δ δ ο R δ ο R R δ δ δο ϑ π/4

47 4.7 Scattering gels and suspensions of dispersed particles The characterization of gels and suspensions of dispersed particles can introduce a number of complicating factors, including non ergodic behavior, in which the observed light scattering intensity depends on the position in the sample from which the scattering arises, and multiple scattering, in which the initially scattered ray acts a source of scattering before exiting the sample; in the extreme, multiple scattering gives rise to a turbid appearance. Methods to suppress these effects to obtain the meaningful characterization of the sample are discussed in the following. These are based on the use of autocorrelation of the scattered intensity to augment the measurements of the total scattered intensity; the electric field autocorrelation function was introduced briefly in Section 2.5 in the context of dilute solutions. The ensemble-averaged autocorrelation G (2) (q,τ) of the scattered intensities is given by the function E G (2) E (q,τ) = I(q,0)I(q,τ) E (79) where I(q,τ) is the scattered intensity at time τ after the measurement of the intensity I(q,0), and subscript "E" indicates the ensemble-average. The corresponding function G (2) (q, ) obtained for τ large enough that E the intensities I(q,0) and I(q,τ) are no longer correlated is given by G (2) E (q, ) = I(q,0)I(q, ) E = I(q,0) E I(q, ) E = I(q) 2 E (80) Finally, the normalized function g (2) (q,τ) is given by E g (2) (q,τ) = G(2) E E (q,τ)/g(2) (q, ) (81) E If the scattering volume contains many uncorrelated regions then scattering sampled over the full ensemble is a zero-mean complex Gaussian variable, and g (2) (q,τ) is related to the more fundamental E ensemble-averaged electric field autocorrelation g (2) (q,τ) by the expression E g (2) (q,τ) = 1 + β[g(1) E E (q,τ)]2 (82) where β is a measure of the coherence of the scattering, with β = 1 for full coherence, decreasing monotonically with decreasing coherence (some authors designate this parameter as β 2 ). Equation 44 is the expression for g (2) (q,τ) normally applied with dilute solutions of polymers or suspensions of particles. The E 45

48 condition for coherence may be visualized by its appearance on a screen in the far field, it will appear as a field of speckles. For a scattering from an ergodic sample, the intensity from the speckles will wax and wane with time; the effects of non ergodicity are considered in the following paragraphs. The parameter β will be unity for the scattering confined to a single speckle, but will decrease if the scattering from more speckles is averaged to determine g (2) (q,τ). The arrangement used in the detector optics of many light E scattering photometers utilizes a pinhole to adjust the number of coherent areas from which the scattering is accepted, and hence the value of β, with β increasing with decreasing pinhole size. Since g (1) (q,τ) is not E altered with decreased β, an arrangement with β less than unity may be accepted as increasing β will be accompanied by reduced intensity, and decreasing signal to noise in the determination of g (1) E (q,τ). For reference in the following we note that some applications, such as electrophoretic light scattering, require scattering from a mixture of the scattering from a solution or suspension with that from a static source, such that g (2)(q,τ) = 1 + E βx2 F g(1) F;E (q,τ)2 + 2β n X F ( 1 X F )g (1) (q,τ) (83) F;E where X F and g (1) (q,τ) are, respectively, the ensemble-averaged parameters for the fraction of the total F;E intensity due to the solution or suspension and the electric field autocorrelation function for the solution or suspension. The exponent n on β has variously been given the values 1 or 1/2. (149) Non ergodic behavior in light scattering. The light scattering experiment involves averages over time at a fixed location. Non ergodic light scattering behavior is marked by results that depend on the volume element in the sample from which the scattering is obtained. This does not occur if the scattering components have free access to all diffuse throughout the sample on length scales probed by the scattering. Thus, non ergodic behavior is not expected with a dilute polymer solution or particle suspension. However, constraints to such motion may be imposed in gels or more concentrated suspensions. In that case, light scattering may not yield the ensemble-averaged quantities assumed throughout the preceding sections, but instead give time-averaged measurements representing the behavior a particular volume element in the sample, with the time-averaged autocorrelation function G (2) (q,τ) given by T G (2) T (q,τ) = I(q,0)I(q,τ) T (84) 46

49 Similar to the preceding, the corresponding function G (2) (q, ) obtained for τ large enough that the T intensities I(q,0) and I(q,τ) are no longer correlated is given by G (2) T (q, ) = I(q,0)I(q, ) T = I(q,0) T I(q, ) T = I(q) 2 T (85) Finally, the normalized function g (2) (q,τ) is given by T g (2) (q,τ) = G(2) T T (q,τ)/g(2) (q, ) (86) T Before proceeding to detailed analysis of g (2) (q,τ) for samples exhibiting non ergodic behavior, it is useful T to consider qualitative examples of g (2) (q,τ) and g(2) (q,τ) for a single speckle presented in Table 3 following T E the discussion of these in reference [149] (since a single speckle provides fully coherent scattering, β = 1 for the examples in Table 3). In the first example for rigid media, e.g., for the scattering from a rigid material or a ground glass screen, both g (2) (q,τ) and g(2) (q,τ) are constant, but differ in their values: for T E g (2) T (q,τ) the intensity is invariant at any spot in the medium, and thus I(q,0)I(q,τ) T = I(q,0) T I(q,τ) T = I(q) 2 T and g(2) (q,τ) = 1 for all τ; however, if the scattering is averaged as the sample is moved (rotated or T translated), the statistics become those of a zero-mean Gaussian, so that I(q,0)I(q,τ) E = I 2 (q) E and g (2) (q,τ) = 2 for all τ. For ergodic media, such as a dilute solution or suspension, time and ensemble E averages are equivalent, and g (2) (q,τ) = g(2) (q,τ) for all τ, decreasing from 2 for τ = 0 to 1 for very large τ. T E Finally, the scattering for non ergodic media presents a mixture of the preceding cases such that the fluctuations cause g (2) (q,τ) to decrease from 2 for τ = 0 to approach a constant value g(2) (q, ) representing a E E rigid behavior with increasing τ, whereas these two effects cause g (2) (q,τ) decreases from 2 for τ = 0 T (fluctuating behavior) to 1 for very large τ (rigid behavior). In practice, as mentioned in the following, it is possible for G (2) (q,τ) to exhibit a very slowly relaxing component, leading to a plateau that would be T followed by further decrease in G (2) (q,τ) for still larger τ, giving rise to an apparent non ergodic behavior if T not included. <Table 3> Three methods are of interest with samples that appear to exhibit non ergodic behavior: 47

50 1: The scattering may be treated as one with a fluctuating component with time-averaged intensity I(q) F,T and a time-averaged total intensity I(q), using the expression for so-called heterodyne T behavior, i.e., the experimental arrangement with an external static scatterer used in electrophoretic scattering. In this case data on G (2) (q,τ) are interpreted to yield g(1) (q,τ) = g(1) (q,τ) g(1) (q, ). T F E E 2: Use of the ensemble-averaged total intensity I(q) E determined by averaging the total scattering obtained at different locations in the sample, e.g., by translation or rotation of the sample cell, and a theoretical evaluation of G (2) (q,τ) to yield g(1) (q,τ). T E 3: Methods to permit averaging and measurement durations such that G (2) (q,τ) becomes a reliable T estimate of G (2) E (q,τ). In the first method, it is assumed that the non ergodic behavior in the scattering from a gel or concentrated suspension is caused by clusters of some kind that are either completely stationary or move so slowly that they can be assumed to act as a static source in a heterodyne mode in mixing with the fluctuating source from the scattering from the solution or suspension, such that the observed g (2) (q,τ) may T be analyzed with Equation (83) in the form ( ) g (2)(q,τ) = 1 + T βx2 F g(1) F;E (q,τ)2 + 2β n X F ( 1 X F )g (1) (q,τ) (87) F;E Here, g (1) (q,τ) is related to the total ensemble-averaged electric field correlation function given by the F;E relation g (1) F;E (q,τ) = g(1) (q,τ) g(1) (q, ) (88) E E and X F = I(q) F,T / I(q) T. Since g (1) (q, ) = 0 and g(2) (q, ) = 1, X F T F is given experimentally by the result X F = 1 [2 g (2) T (q,0)]1/2 (89) Solution of Equation (87) for g (1) (q,τ) gives F;E (q,τ) = 1 + (1/X F F){ 1 + [1 + g (2) (q,τ) g(2) T T (q,0)]1/2 } (90) g (1) 48

51 where β has been taken to be unity. Otherwise, if β < 1 and n = 1 in Equation (87), the result would be modified by replacing [2 g (2) (q,0)] by [1 + β g(2) (q,0)]/β. (152) These data permit evaluation of I(q) T T F,T = X F I(q) T for use in the analysis of R(ϑ,c), as well as a diffusion constant D using g (1) (q,τ) in Equation 44. F Since no ensemble averaged estimate of g (1) (q, ) noted in Equation (88) the method does not yield an E ensemble average for g (1) (q, ), and hence unlike the methods discussed below, it cannot provide an E estimate for g (1) (q,τ) = g(1) (q,τ) + g(1) (q, ). E F;E E An example of the use of Equations (87-90) to interpret data on g (2) (q,τ) for a poly(n- T isopropylacrylamide) hydrogel is shown in Figure (16) for g (2) (150) (q,τ) determined at different positions. T As may be seen, although the data at different positions display quite different values for g (2) T (q,τ) and X F, the calculated g (1)(q,τ) and I(q) are independent of the position in the gel, as expected with the F F,T assumptions made in using this method. <Figure 16> A full evaluation of g (1) (q,τ) from g(2) (q,τ) and an ensemble-averaged total intensity I(q) is provided by E T E an alternative procedure, based on a theoretical treatment of the scattering from moderately concentrated solutions of spherical particles in a suspension developed by Pusey and van Megen. (149) In this treatment it is assumed that the (apparent) non ergodic behavior observed for an aqueous suspension of polystyrene spheres is due to constrained diffusion of the particles about their unchanging mean position in the suspension during the measurement of g (2) (q,τ). The analysis presumed that each particle is constrained to T movements from the average position, with a mean-square displacement δ 2 along wave vector q to give g (2)(q,τ) = 1 + T X2 E β{g(1) E (q,τ)2 g (1) E (q, )2 } + 2X E ( 1 X E )β n {g (1) (q,τ) g(1) (q, )}(91) E E for a suspension of identical spheres, where X E is the ratio X E = I(q) E / I(q) T with I(q) T the timeaveraged intensity during measurement of g (2)(q,τ) and I(q) the ensemble-averaged intensity. Repeated T E measurements of the total intensity at various positions in the sample are used as the measure of I(q) E, e.g., by translation or rotation of the sample cell, along its vertical axis in the original studies. Evaluation of g (1) (q,τ) from Equation (91) gives E g (1) (q,τ) = 1 + (1/X E E){ 1 + [1 + g (2) (q,τ) g(2) T T (q,0)]1/2 } (92)

52 for β = 1. Pusey and van Megen noted that if a location in the cell is located for which X E = 1, then Equation (2) simplifies to read g (1) E (q,τ) = [1 + g(2) (q,τ) g(2) T T (q,0)]1/2 ; for X E = 1 (93) They also concluded that evaluation of the exponent n in Equation (91) is complex, and recommended that the best approximation is to put n = 1, and to work with detector optics that makes β greater than Finally, they suggest that the results may be used for systems for which the non ergodic behavior arises from causes other than the restricted mobility of the spheres primarily of concern in their paper, including, for example, polymeric gels or suspensions of particles, where the non ergodic behavior may be caused by the presence of clusters, such as polymeric aggregates in gels, or clusters of the particles, perhaps via van der Waals attractive interactions in suspensions. The method, which is applied to a single speckle requires β = 1 and scattering centers characterized by g (1) (q,τ) by a well-established g(1) (q, ). E E A model with non-interacting, harmonically bound particles was presented as an example of a model of constrained mobility, permitting evaluation of g (1) (q,τ) in terms of the diffusion coefficient D and the mean- E square displacement δ 2 along q: (149) g (1) E (q,τ) = exp{ q2 δ 2 [1 exp( Dτ/ δ 2 )]} (94) g (1) E (q,τ) = 1 Dq2 τ +. (95) g (1) E (q, ) = exp{ q2 δ 2 } (96) Consequently, according to Equation (95), the particles diffuse for short times as if in a dilute suspension, with the constraint to that motion realized as their root-mean-square displacement approaches δ 2 1/2 per Equation (96). An example of the use of the direct determination of g (1) (q, ) from measurement of g(2) (q,τ) obtained E E over a very long time is given in Figure (17), along with an evaluation of g (1) (q, ) from of g(2) (q,τ) and X E T E using Equation (92). (153) The direct determination of g (1) (q, ) required 13 hours, whereas the one on E g (2) (q,τ) required 30 minutes; the sample comprised an aqueous polyacrylamide gel (2.5 wt% polymer) T

53 containing 0.02 wt% of 82 nm polystyrene spheres, with most of the scattering arising from the spheres. A substantial difference between g (2) (q,τ) and of g(2) (q,τ) is shown in the upper part, along with the T E agreement between the g (1) (q, ) determined directly from g(2) (q,τ) and that calculated from g(2) (q,τ) using E E T Equation (92). (154) Additional examples of g (2) (q,τ) calculated from g(2) (q,τ) using Equation (92) and a E detailed analysis of the results are available on gels of polystyrene particles in polyacrylamide gels. (155) <Figure 17> T The use of Equation (92) requires an optical arrangement with β 1, and is based on an assumption that g (2)(q,τ) and the average intensity I(q) for the fluctuating component are invariant with position in the T F,T cell, even though the total intensity I(q) T may vary with position, giving rise to the non ergodic behavior. Although these constraint on β is relaxed with the use the use of Equations (89) (90), that method does not provide an evaluation of g (1) (q, ). Consequently, methods were developed to provide the averaging needed E for a direct evaluation of G (2) (q,τ). As mentioned in the preceding, the image of the scattered light on a E screen in the far-field reveals a field of speckles, the intensity of which fluctuates in time unless the scattering centers are stationary. As noted above, the variation of this field with rotation or translation of the sample has been used to determine I(q) E for use in Equation (90). (149,151,155) With improved computational assets, it became possible to collect data on many speckles and compute G (2) (q,τ) over a long T enough time to permit evaluation of G (2) ( ) (q,τ) by averaging those results. The E methods are implemented by rotating the light scattering cell so that the field of speckles changes, to return to the original configuration after one full rotation, with slightly altered intensities of the speckles in the field, unless the scattering entities are static. The rotation will direct many speckles into the detector during one full rotation (of the order of 1,000). An early example of this method utilized a relatively slow rotation to calculate G (2) (156) (q,τ) continuously. T The rotation translates spatial fluctuations into temporal fluctuations, resulting in the desired averaging modulated by a cutoff as a particular speckle leaves the field of view, limiting the result to short τ, of the order τ < 1 s. (156) Interleave methods at faster rotation to compute G (2) T (q,τ) for each of these speckles, with τ = nt, where n is the number of rotations with period T starting with the onset of the calculation of G (2) T (q,τ); the shortest correlation time is then τ min = T on of the order 1s, with the maximum τ limited by the available software and the patience of the experimenter ( ) Averaging these data finally yields G (2) E (q,τ), from which g (1) (q,τ) may be determined, including the g(1) (q, ) contribution provided that the data are E E

54 collected and analyzed for a time long enough for τ to reach the limiting value behavior. The time for such an analysis may be several orders of magnitude shorter than would be required for a comparable analysis with sequential measurements of G (2) (q,τ) for different positions by manual movement of the cell. T The measurement time required to obtain a satisfactory evaluation of g (1) (q, ) by the methods described E above may be reduced by the so-called "echo method", in which the photon counts are collected as a single stream of data as the sample is rotated. (159) In a method utilizing a vertical cylindrical cell, rotated about the cell axis, data were analyzed using the expression G (2) OBS (q,τ) = 1 + β[g(1) (q, τ) g(1) E ROT (q,τ)]2 (97) with g (1) OBS (q,τ) = g(1) (q,τ) g(1) E ROT (q,τ) (98) where g (1) (q,τ) and g(1) (q,τ), respectively, are normalized electric field correlation functions as observed OBS ROT and accounting for the effects of the cell rotation. The function g (1) (q,τ) may be determined ROT experimentally by evaluation of g (1) (q,τ) for a rigid sample, for which g(1) (q,τ) = 1. For rotation of a OBS E cylindrical cell about its axis, g (1) ROT (q,τ) = 2J 1(qRσ τ ) qrσ τ (99) where σ τ = 2sin(ωτ/2) and R is the radius of the scattering volume; ω is the angular velocity of the rotation. With this function, [g (1) ROT (q,τ)]2 appearing in Equation (99) is periodic with period T = 2π/ω, with main maxima, or echos, of amplitudes that are unity for τ = nt, where n = 0, 1, 2, 3,, and is close to zero otherwise. Thus, following n rotations g (1) (q,τ) calculated from the stream of data collected will appear as OBS a series of peaks (echos) for τ = nt, with the shaped specified by g (1) (q,τ), but with a peak value ROT modulated by g (1) (q,τ = nt). Although the peaks will appear as a linearly progressing sequence, a procedure E is available to increase the separation between the longer to reduce the time on calculations that do

55 not add usefully to the estimate for g (1) E (q,τ). (159) As with the interleaved method mentioned above ( ) the smallest value of τ that may be determined by the echo method is limited by the rotational period T. The example in Figure (18) shows results for an ergodic scattering system obtained by (156), displaying the agreement between these for the range of τ for which the overlap, and reduced scatter of the echo derived data at the longest τ. <Figure 18> A light scattering cell has been described with the sample confined in a slab-shaped cylindrical cell, permitting use of the echo method for scattering angles down to 30, and also applicable for the methods described in the next section re multiple scattering. (160) Cross-correlation to suppress weak multiple scattering effects. As the concentration of solute in a solution or particles in a suspension increases, one will usually encounter the effects of multiple scattering, in which a scattered in the light scattering cell suffers more than a single scattering event before leaving the cell. Attempts to determine the static scattering behavior, e.g., for analysis of thermodynamic and conformational properties by studies on R(ϑ,c) described in preceding sections, require the suppression of multiple scattering, e.g., by using thin cells, or the suppression of the effects of multiple scattering in some way. Effects experienced with more concentrated, turbid systems are discussed in the next section. Multiple scattering destroys the coherence required for auto-correlation in the scattered intensity, even though it adds to the total intensity. As a consequence, analysis of a properly defined crosscorrelation intensity function permits evaluation of properties of the scattering that does not experience rescattering before exiting the scattering cell. Cross-correlation requires two distinct, simultaneous autocorrelation experiments, each with its own incident laser source (mutually incoherent) and detector arrangement, on the same scattering volume and with different angles providing the same q. The use of cross-correlation methods to suppress the effects of multiple scattering discussed in th section was introduced by Phillies and implemented in an instrument restricted to the scattering at 90. ( ) Since that time, both the theory and experimental methods have been refined. ( ) The theoretical treatment of cross-correlation involves an ensemble-averaged autocorrelation G (2) (q,τ) of 12 the scattered intensities from each of the two scattering detectors, given by the function G (2) 12 (q,τ) = I 1(q,0)I 2 (q,τ) E (100) 53

56 where the subscript "E" has been suppressed on G (2) (q,τ) for simplicity. Division by the two average 12 intensities gives the intensity cross-correlation function g (2) (q,τ), relate to the results in an expression for 12 the field cross-correlation g (1) (q,τ) by the expression 12 g (2) 12 (q,τ) = 1 + β 12[g (1) 12 (q,τ)]2 (101) (again suppressing notation to indicate the ensemble average), where β 12 involves the coherence factors β 1 and β 2 from each of the detectors, a factor β V accounting for the incomplete overlap of the scattering volume viewed by the two detectors, a factor β S accounting for incomplete separation of the scattering detected by the two detectors, discussed in more detail below, and β MS a measure of the single to multiple scattering: β 12 = [β 1 β 2 ] 1/2 β V β S β MS (102) β MS = Iss 1(q) I ss 2(q) I 1 (q) I 2 (q) (103) where I ss 1(q) and I ss 2(q) are the single scattered components of the total intensities I 1 (q) and I 2 (q), respectively. As discussed further in the following, β S varies from 1 to 0.25, depending on the arrangement used in the cross-correlation analysis (168) Although β 1, β 2 and β V may all be controlled, it is best to determine [β 1 β 2 ] 1/2 β V β S as the value of (β 12 ) single, which may depend on q, from measurements of β 12 on a system with no multiple scattering, such as a dilute polymer solution or particle suspension, so that β MS = 1. Such an evaluation will subsequently permit use of β 12 /{[β 1 β 2 ] 1/2 β V β S } = β 12 /(β 12 ) single to compute β MS for systems with multiple scattering, and hence evaluation of the single scattered intensity, permitting computation of R(ϑ,c) for use in analysis of the static scattering. Two methods have been in the forefront of cross-correlation technology: (1) scattering with incident laser light of two different wavelengths, with the incident light and the detectors all in the scattering plane, and (2) a so-called 3-D arrangement, with scattering with a single wavelength, but with the two incident beams lying in the same plane orthogonal to the scattering plane, with one above and the other below that plane by some angle, with the two detectors similarly positioned above and below the scattering plane by corresponding angles. Each of these methods offers advantages and disadvantages. For example, with the two color arrangement, one can ensure that β S = 1 by placing a laser line pass filter in front of each of the 54

57 relevant detectors, but great care must be given to accurate adjustment of the angle between the two detectors to ensure that account is taken of the different wavelengths of each required to give the q in common for the detected scattering, but β V may depend on the scattering angle, especially at large scattering angle. An elegant photometer has been custom designed and constructed using a 4-arm goniometer to permit accurate setting of the angles of incident beams and detectors, single-mode fiber optics to guide the incident and scattered light, and laser line filters to suppress contamination of the detectors by light of the wrong wavelength. (169) Examples of data taken with this instrument will be discussed in the following. On the other hand, although the setup is easier, the optical arrangement for the detectors in the 3-D method permits their contamination by light scattered from both incident beams. If not suppressed, that contamination will result in β S = 0.25, reducing the sensitivity of the cross-correlation result. The use of oppositely polarized incident beams of the incident light and appropriately oriented polars in front of the detectors may suppress the cross-talk over a reasonable range of scattering angle if a flat cell is utilized, with a result that did about double β S in one arrangement. (160) The use of a flat cell also facilitates the use of the echo method to obtain an ensemble average if the scattering system demonstrates non ergodic behavior. A method to suppress the effects of this contamination by modulating the light beam intensity and gating the detector outputs at a frequency much greater than any of interest in the system dynamics to temporally separate the detectors, giving the desired increase in β S to close to unity. (168) A commercial photometer is available for such measurements, including the beam modulation and gating of the detector outputs needed to enhance the value of β S as mentioned above (the 3D LS Photometer, LS Instruments; Some experimental results obtained by two color cross-correlation on aqueous dispersions of polystyrene spheres are displayed in Figure (19). (169) Based on the data seen for β 0.9 for normal autocorrelation and β for a dilute solution in Figure (19) without multiple scattering suggests that in β S = 0.5, indicating one of the difficult alignment issues. The data shown in Figure (19) were analyzed to give a hydrodynamic radius R h determined in by the usual expression for dilute solutions for a monodisperse scatterer (17) : g (1) 12 (q,τ) = exp[ Dq2 τ] (104) with D the diffusion constant in dilute solutions or suspensions, and R h = kt/6πηd, with η the viscosity of the media. The values of R h determined via the cross-correlation were found to be independent of the transmission as the concentration of the spheres increased, whereas the similar value determined from the auto-correlation function g (1) E (q,τ) decreases rapidly owing to the effects of multiple scattering. (169) The

58 substantial effect of multiple scattering on R(ϑ,c) determined from I ss (q) for the scattering from spheres in the Mie scattering regime are illustrated in Figure (19), showing the smearing of the minima characteristic of multiple scattering. <Figure 19> Diffusing wave spectroscopy in turbid media. With increasing concentration dispersions the multiple scattering can become so pervasive that the suspension becomes too turbid for use of the methods described in the previous paragraphs. In the diffusing wave spectroscopy (DWS) method described in this section the scattering is evaluated in the forward and back directions for a turbid suspension. ( ) This method, which utilizes the scattering from a single speckle or coherence area, is based on the notion that in the highly multiple scattering environment, the direction of a photon is randomized by the very large number of multiple scattering events, with a resultant change in the phase of its wavevector giving rise to effects that may be approximated by the contribution on an averaged event to compute the effect on the field autocorrelation g (2) (ϑ,τ); here, for reasons explained in the following, the notation indicates a E scattering angle ϑ instead of the usual magnitude q of the scattering wavevector. The transport mean free path l* a photon must travel before its direction is completely randomized is an important parameter in the model. Usually, l* is much larger than the scattering mean free path l that a photon must travel to undergo a scattering event, i.e., l* > l. Owing to the randomization of the scattering wavevector, the scattering angle is not important, and either the transmitted or the backscattering is used in the measurement of g (2) E (ϑ,τ), i.e., ϑ either 0 or (essentially) π radians, respectively; since q does not enter in the final analysis, these angles should be taken as nominal values. As developed in the following, the transmission and backscattering differ in that the interpretation of g (2) (ϑ,τ) requires a value for l* for transmission, but not in E backscattering. Although the backscattering mode may be the only option if the suspension is very turbid, some of the assumptions made in the model may not be valid, resulting in inaccurate analysis in the backscattering mode, in particular whether the photon scattering wavevector is randomized in the penetration length for the light. (171) A schematic diagram indicating the length l* in comparison to the distance l between scattering events, and a flat cell arrangement used to study the transmitted scattering from a gel, with non ergodic effects in the scattering is given in Figure 1 of reference [173]. The scattering from the gel is passed through a second cell containing a slightly turbid suspension with ergodic behavior to assist in the averaging needed to obtain g (2) (ϑ,τ) with the non ergodic behavior of the gel. Two E correlators analyze the scattering received via an optical fiber, with the light divided into two optical fibers

59 delivered to two, independent detectors. The signals from those detectors are cross-correlated to remove any effect of random after-pulsing that might affect measurements at very small τ. In addition to l* and l introduced in the preceding paragraph, parameters in the interpretation of g (2) E (ϑ,τ) in terms of the DWS model include the (spherical) particle radius R, the sample thickness L for transmission and a reduced time given by (2π/λ) 2 Dτ with D the diffusion constant for diffusive (Brownian) motion of the particles or by replacing Dτ by Δr 2 (τ) /6 for non-diffusive particle motion where Δr 2 (τ) is the mean-square particle displacement. To be most effective, the particles should have R close to the wavelength λ in the medium, so that the particle scattering factor will involve Mie scattering, and be strongly peaked in the forward directions for each scattering event. With the DWS model, it is assumed that owing to the randomization, a transmitted photon will experience (L/l*) 2 random walk steps on leaving the sample, with l*/l scattering events per step, or an average n = (L/l*) 2 (l*/l) number of scattering events. The field auto-correlation scattering events are averaged over q for each step, using the relevant particle scattering function for the particle. The calculation is sensitive to the experimental conditions, e.g., transmission or backscattering, point source or extended source to illuminate a wide area on the sample. The necessary averaging is represented in Equation (105) g (2) (ϑ,τ) = 1 + β[g(1) E E (ϑ,τ)]2 (105a) g (1) E (ϑ,τ) = 0 ds P(s)exp[ (s/l*)(2τ/τ 0)] (105b) where β 1 in optical arrangements relevant to DWS, and P(s) depends on the geometric nature of the experimental optical arrangement and τ o is a time constant characteristic of the process, see below. ( ,174) For example, for the use of an extended source, and scattering collected from a small area near the center defined by the illuminated area, g (1) E (0,τ) in transmission and g(1) E (π,τ) in backscattering are given by Equations (106) and (107), respectively, with ã = L/l* and a* = z o /l*, where z o l* is a distance from the illuminated face for which the scattering wavevector has become randomized: g (1) E (0,τ) = [ã+ (4/3)]{sinh[a*x] + (2x/3) cosh[a*x]} [a* + (2/3)]{[1 + (4x 2 /9)]sinh[ãx] + (4x/3) cosh[ãx]} (106a) g (1) E (0,τ) [ã+ (4/3)]x [1 + (4x 2 ; for x << 1 (106b) /9)]sinh[ãx] + (4x/3) cosh[ãx]

60 sinh[(ã a*)x] + (2x/3) cosh[(ã a*)x] g (1) (π,τ) = E [1 + (4x 2 /9)]sinh[ãx] + (4x/3) cosh[ãx] (107a) exp( a*x), g (1) (π,τ) E 1 + 2x/3 exp( γx) for ã >>1 and x << 1 (107b) with γ = a* + 2/3. It should be emphasized that other expressions must be used for optical arrangments not used in the calculation of Equations ( ). (171,175) A method has been given to correct g (1) (0,τ) for the E effects of reflection at the air-glass interfaces. (176) The parameter ã may be determined from the static optical transmission T scat as affected by the scattering (i.e., without absorption at the scattering wavelength, which if present requires a known correction) by the expression (176) T scat = 5 3ã/ Nã (108) For a diffusive process, x = (2π/λ)(6Dτ) 1/2 in either Equations (106) or (107). For a non diffusive process x = (2π/λ) Δr 2 (τ) 1/2 in Equation (106) for transmission, but should not be applied with Equation (107) in backscattering because the diffusion approximation and central limit theorem used in arriving at this result are valid only for long paths, and break down for short paths characteristic of backscattering in turbid media. (171) In practice, these expressions (or others, relating to alternative optical arrangements to determine either D or Δr 2 (τ). Examples of typical behavior for Δr 2 (τ) given in Figure (20), with the tangent to Δr 2 (τ) tending to unity with increasing τ for the solutions, but decreasing gradually with increasing τ until it decreases rapidly tending to zero over a short range in large τ for a densely crosslinked sample, these attributes are discussed further in the next paragraphs. (175) <Figure 20> Although DWS is used to characterize media for which the particle diffusion is diffusive, e.g., to evaluate properties changing with some processing time, e.g., as in Figure (20), ( ) the of method in its application to the use of Δr 2 (τ) in microrheology is of more interest here. For example, the scattering from polymeric solutions or gels serving as a matrix in a suspension containing a sufficient number of spherical particles to dominate the scattering and the resultant turbidity, but not so concentrated that the rheological properties of the matrix are affected. The calculation of the linear viscoelastic properties from Δr 2 (τ) based on its connection to a diffusion process begins with a version of a generalized Langevin equation incorporating a time-dependent memory function ζ(t), (175, ) 58

61 m v(t) = f rand (t) t dτ ζ(t - τ) v(τ) (109) 0 where ζ(t) is a generalized time-dependent memory function, f rand (t) represents random forces acting on the particle, v(t) is the particle velocity, v(t) its acceleration and m its mass. The merits and potential problems with approximations used in this calculation, and especially the use of the generalized Stokes-Einstein relation have been considered in detail. (184) The unilateral Laplace transform of Equation (108) involves the approximation that a generalized time-dependent Stokes-Einstein holds with proportionality of ζ(τ) and the real component η'(ω) dynamic viscosity η*(ω), so that after the Laplace transform (indicated by the "~"), ~ ζ(s) = 6πR ~ η'(s) (110) After rearrangement, and with neglect of an inertial term (which could cause inaccuracies for τ < 10-6 s), the result relates the Laplace transform ~ Δr 2 (s) of the mean-square displacement in terms to η'(s) η'(s) = ~ G(s)/s = kt πrs 2 ~ Δr 2 (s) (111) where G(s) ~ = s ~ G R (s), with ~ G R (s) the Laplace transform of the shear stress relaxation modulus G R (t). For freely diffusing particles, s 2 Δr ~ 2 (s) = 6D, and Equation (111) is seen to be a frequency-dependent form of the usual Stokes-Einstein expression η o = kt/6πrd. (183) Alternatively, since the Laplace transforms of G R (t) and the shear creep compliance J(t) are related by s G(s) ~ ~ J(s) = 1 for a linear viscoelastic material, Equation (111) may be rearranged to give ~ J(s) = (kt/πr) Δr ~ 2 (s), or after inverse Laplace transformation, (185) Δr 2 (t) = (kt/πr)j(t) = (kt/πr)[r(t) + t/η] (112) with η the viscosity and R(t) the recoverable creep compliance. For a linear viscoelastic material, it is useful to express R(t) in the form R(t) = R [R R 0 ]ρ( t) (113)

62 where ρ(t) decreases from unity to zero with increasing t, R Pa -1 in the normally accessible experimental range used to determine R(t) (decreasing still further toward zero with still smaller times), with R(t) increasing to reach a value R with increasing t for large t. Here, R is the steady-state recoverable creep compliance for a fluid or the equilibrium compliance J e for a solid. The retardation function ρ(t) is often represented to within experimental uncertainty as a sum of weighted exponential terms exp(-t/λ i ), e.g., by methods used to represent g (1) (q,τ) as a weighted sum of exponential terms E exp(-q 2 D i τ) in dynamic light scattering on dilute solutions to investigate the distribution of molecular weight for samples. (186) For either a fluid or a solid, R(t) may exhibit a plateau with value J N over an intermediate range of t for a polymer or its solution reflecting a pseudo-network in the entanglement regime, although this property will be suppressed if J e < J N. (186) These attributes are seen in the experimentally determined Δr 2 (t), e.g., Δr 2 (t) t, perhaps for the entire accessible range of t for a low viscosity fluid, and Δr 2 (t) increasing gradually approaching a limiting value at large t for a solid. Examples for a schematic representations of R(t) and other linear viscoelastic functions are shown in Figure (20), along with experimental data on a Δr 2 (t) determined from g (1) (0,τ) for a polymer solution and E gels prepared therefrom. ( ) The similarity between Δr 2 (t) and J(t) is evident (the contribution of the term t/η to J(t) being dominant for the solution, but suppressed for the gel). Nevertheless, although J(t) may be determined with commercially available instrumentation, there does not seem to be any direct evaluation of the accuracy of the Equation (112) in the literature, even though such is certainly feasible using commercially available instrumentation for the range of t encompassed by the measurements of both Δr 2 (t) and J(t). Rather, most, if not all, of the examples in the literature utilize qualitative comparisons of the storage and loss components G'(ω) and G"(ω), respectively, derived from an analysis of Δr 2 (t=1/ω) in terms of the complex modulus G*(ω) along with the dynamic modulus G*(ω) given by: G*(ω) = {[G'(ω)] 2 + [G"(ω)] 2 } 1/2 (113) Both G'(ω) and G"(ω) may be computed from J(t), either by available exact or quite good approximate methods. (186) For example, G'(ω) = J'(ω)/ J*(ω) 2 and G"(ω) = J'(ω)/ J*(ω) 2, where J'(ω) and J"(ω) are the storage and loss components of the complex compliance J*(ω) = 1/G*(ω),

63 J'(ω) = R ω[r R 0 ] dτ ρ( τ) sin(ωτ) 0 (114a) J"(ω) = (1/ωη) + ω[r( R 0 ] dτ ρ( τ) cos(ωτ) 0 (114b) J*(ω) = {[J'(ω)] 2 + [J"(ω)] 2 } 1/2 (114c) The required range of the integration may usually be problematic given the limited (albeit large) range of t for which J(t) is known from Δr 2 (t) via Equation (112). Alternative approximate relations that provide close approximations to the dynamic compliances from R(t) are available, for example, (186,188) J'(ω) = {[1 m(2t)] 0.8 R(t)} ωτ = 1 (115a) J'(ω) = (1/ωη) + {[1 m(2t/3)] 0.8 R(t)} ωτ = 1 m(t) = ln R(t)/ ln t (115b) (115c) where R(t) and η would be derived from Δr 2 (t) using Equation (112) in opto-microrheology. A similar use of the tangent α(τ) = ln Δr 2 (τ) / ln τ has been incorporated into an approximation used for the Laplace inversion and the subsequent Fourier transformation to frequency space to represent Δr 2 (τ) -1 in terms of the dynamic moduli. (175) It may be noted that α(τ) = [R(τ)/J(τ)]m(τ) + t/ηj(t), so that α(τ) m(τ) for small τ, but that these differ for large τ, with α(τ) tending to unity and m(τ) tending to zero. The approximations lead to the result G*(ω) = {[kt/(πr Δr 2 (τ) )]Γ[ 1 + α(τ)]} τω = 1 G'(ω) = G*(ω) cos(πα(ω)/2) G"(ω) = G*(ω) sin(πα(ω)/2) (116a) (116b) (116c) where Γ[ ] is the gamma function; the behavior of Δr 2 (τ) at small and large τ are reflected in the properties of the moduli. Well-known limits exist for the dependence of G'(ω) and G"(ω) in the extremes of small and large ω: for both linear viscoelastic fluids and solids G"(ω) ω for small ω, and for large ω, G'(ω) tends to 1/R 0 and G"(ω) tends to zero; whereas, for small ω, G'(ω) ω 2 for a fluid or becomes independent of ω and equal to the equilibrium modulus G e = 1/J e for a solid, such as a gel. The data on G'(ω) and G"(ω) given in Figure (21), determined using the Δr 2 (τ) given in that figure for a colloidal dispersion, (175) show that in that case the data on Δr 2 (τ) extend to large enough τ (small enough ω) so

64 that G'(ω) may have nearly reached the limiting low-ω behavior with G'(ω) = G e for the gel, but that cannot be certain since the corresponding low-ω behavior with G"(ω) ω for small ω is not seen. <Figure 21> The opto-microrheology described above affords a useful method to obtain linear viscoelastic data, and a light scattering apparatus is available (DWS RheoLab II, LS Instruments; The commercial apparatus incorporates a flat cell with arrangements to use the two-cell measurements described above, with echo technology to assist determination of g (1) (0,τ) for non ergodic samples, as E would be encountered in the study of gels or other solid materials, and the use of software to compute and present results on Δr 2 (τ), G'(ω) and G"(ω). Examples in the literature from different laboratories, on differently designed equipment and methods of analysis, give inconsistent results on the comparison of viscoelastic results, almost always in the form of G'(ω) and G"(ω), from opto-microrheology with those from the traditional use of rheometers. Deviations by a factor of two are not unusual, e.g., see the example in Figure (21), even though some reports give close correspondence over the range of ω studied. The reported deviations could reflect some assortment of potential errors, including failure of the generalized Stokes-Einstein relation in the calculation of Equation (111) for the particular set of viscoelastic properties of interest, unwanted effects of the filler particles on the viscoelastic properties of the matrix at the concentration of particles needed to obtain the strong multiple scattering necessary for the theory to apply, failure of the "stick" boundary conditions assumed between the particles and the matrix, especially in a gel failure to obtain a full measure of g (1) (0,τ) for a non ergodic sample, or failure of the optical arrangement to E conform to the analytical expression given for g (1) (0,τ) (the expressions given in Equations ( ) are E for a particular optical arrangement, with different expressions needed for other arrangements (171) ); additional issues are discussed in detail in the literature. ( ,184) Despite these potential sources of error, with the appropriate equipment, opto-microrheology can at the least provide a method to discriminate between the viscoelastic properties of a range of samples of interest, at the best may provide useful viscolastic data on materials in the relevant viscosity range, approximately from > 0.1 mpa s (provided suspended particles do not settle from the suspension) to < 1 kpa s.