!berry 1 June 2014
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
!
TABLE OF CONTENTS !!!
1. INTRODUCTION 1 !
2. SCATTERING RELATIONS 1 2.1 General Remarks 1 2.2 Scattering at zero angle and infinite dilution 6
2.2.1 Isotropic solute in the RGD regime 6 2.2.2 Isotropic solute beyond the RGD regime 7 2.2.3 Anisotropic solute 9
2.3 Scattering at small angle and infinite dilution 10 2.3.1 Isotropic solute in the RGD regime 10 2.3.2 Isotropic solute beyond the RGD regime 12 2.3.3 Anisotropic solute 13
2.4 Scattering at arbitrary angle and infinite dilution 14 2.4.1 Isotropic solute in the RGD regime 14 2.4.2 Isotropic solute beyond the RGD regime 18 2.4.3 Anisotropic solute 20
2.5 The size distribution from scattering data at infinite dilution 21 2.6 Extrapolation to infinite dilution 24
!3. EXPERIMENTAL METHODS 27
3.1 Instrumentation 27 3.2 Methods 28
!4. EXAMPLES 31
4.1 Static scattering and size separation chromatography 31 4.2 Light scattering from vesicles and stratified spheres 33 4.3 Scattering from very large particles 35 4.4 Intermolecular association 38 4.5 Scattering with charged species 41 4.6 Scattering from optically anisotropic solute 43 4.7 Scattering from gels and dispersed particles 45 4.8 The intramolecular structure factor for wormlike chains 63
!5. FREQUENTLY USED NOTATION 66 !
6. REFERENCES 68 !
TABLES (3) !
FIGURE CAPTIONS FIGURES (22)
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45
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)E (q,τ) of the scattered intensities is given by the function
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)E (q,∞) obtained for τ large enough that
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)E (q,τ) is given by
g(2)
E (q,τ) = G(2)E (q,τ)/G(2)
E (q,∞) (81)
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)E (q,τ) is related to the more fundamental
ensemble-averaged electric field autocorrelation g(2)E (q,τ) by the expression
g(2)E (q,τ) = 1 + β[g(1)
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 β2). Equation �44� is the
expression for g(2)E (q,τ) normally applied with dilute solutions of polymers or suspensions of particles. The
46
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)E (q,τ). The arrangement used in the detector optics of many light
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)E (q,τ) is not
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)E (q,τ) = 1 + βX2
Fg(1)F;E(q,τ)2 + 2βnXF( 1– XF)g(1)
F;E(q,τ) (83)
where XF and g(1)F;E(q,τ) are, respectively, the ensemble-averaged parameters for the fraction of the total
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)
4.7.2 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)T (q,τ) given by
G(2)T (q,τ) = 〈I(q,0)I(q,τ)〉T (84)
47
Similar to the preceding, the corresponding function G(2)T (q,∞) obtained for τ large enough that the
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)T (q,τ) is given by
g(2)
T (q,τ) = G(2)T (q,τ)/G(2)
T (q,∞) (86)
Before proceeding to detailed analysis of g(2)T (q,τ) for samples exhibiting non ergodic behavior, it is useful
to consider qualitative examples of g(2)T (q,τ) and g(2)
E (q,τ) for a single speckle presented in Table 3 following
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)T (q,τ) and g(2)
E (q,τ) are constant, but differ in their values: for
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)〉2T and g(2)
T (q,τ) = 1 for all τ; however, if the scattering is averaged as the sample is moved (rotated or
translated), the statistics become those of a zero-mean Gaussian, so that 〈I(q,0)I(q,τ)〉E = 〈I2(q)〉E and
g(2)E (q,τ) = 2 for all τ. For ergodic media, such as a dilute solution or suspension, time and ensemble
averages are equivalent, and g(2)T (q,τ) = g(2)
E (q,τ) for all τ, decreasing from 2 for τ = 0 to 1 for very large τ.
Finally, the scattering for non ergodic media presents a mixture of the preceding cases such that the
fluctuations cause g(2)E (q,τ) to decrease from 2 for τ = 0 to approach a constant value g(2)
E (q,∞) representing a
rigid behavior with increasing τ, whereas these two effects cause g(2)T (q,τ) decreases from ≤ 2 for τ = 0
(fluctuating behavior) to 1 for very large τ (rigid behavior). In practice, as mentioned in the following, it is
possible for G(2)T (q,τ) to exhibit a very slowly relaxing component, leading to a plateau that would be
followed by further decrease in G(2)T (q,τ) for still larger τ, giving rise to an apparent non ergodic behavior if
not included.
<Table 3>
Three methods are of interest with samples that appear to exhibit non ergodic behavior:
48
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)〉T, using the expression for so-called heterodyne
behavior, i.e., the experimental arrangement with an external static scatterer used in electrophoretic
scattering. In this case data on G(2)T (q,τ) are interpreted to yield g(1)
F (q,τ) = g(1)E (q,τ) – g(1)
E (q,∞).
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)T (q,τ) to yield g(1)
E (q,τ).
3: Methods to permit averaging and measurement durations such that G(2)T (q,τ) becomes a reliable
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)T (q,τ) may
be analyzed with Equation (83) in the form (150-152)
g(2)T (q,τ) = 1 + βX2
Fg(1)F;E(q,τ)2 + 2βnXF( 1– XF)g(1)
F;E(q,τ) (87)
Here, g(1)F;E(q,τ) is related to the total ensemble-averaged electric field correlation function given by the
relation
g(1)F;E(q,τ) = g(1)
E (q,τ) – g(1)E (q,∞) (88)
and XF = 〈I(q)〉F,T/〈I(q)〉T. Since g(1)F (q,∞) = 0 and g(2)
T (q,∞) = 1, XF is given experimentally by the result
XF = 1 – [2 – g(2)T (q,0)]1/2 (89)
Solution of Equation (87) for g(1)F;E(q,τ) gives
g(1)F (q,τ) = 1 + (1/XF){– 1 + [1 + g(2)
T (q,τ) – g(2)T (q,0)]1/2} (90)
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)T (q,0)] by [1 + β – g(2)
T (q,0)]/β. (152) These data permit evaluation of 〈I(q)〉F,T =
XF〈I(q)〉T for use in the analysis of R(ϑ,c), as well as a diffusion constant D using g(1)F (q,τ) in Equation �44�.
Since no ensemble averaged estimate of g(1)E (q,∞) noted in Equation (88) the method does not yield an
ensemble average for g(1)E (q,∞), and hence unlike the methods discussed below, it cannot provide an
estimate for g(1)E (q,τ) = g(1)
F;E(q,τ) + g(1)E (q,∞).
An example of the use of Equations (87-90) to interpret data on g(2)T (q,τ) for a poly(N-
isopropylacrylamide) hydrogel is shown in Figure (16) for g(2)T (q,τ) determined at different positions. (150)
As may be seen, although the data at different positions display quite different values for g(2)T (q,τ) and XF,
the calculated g(1)F (q,τ) and 〈I(q)〉F,T are independent of the position in the gel, as expected with the
assumptions made in using this method.
<Figure 16>
A full evaluation of g(1)E (q,τ) from g(2)
T (q,τ) and an ensemble-averaged total intensity 〈I(q)〉E is provided by
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)T (q,τ). The analysis presumed that each particle is constrained to
movements from the average position, with a mean-square displacement 〈δ2〉 along wave vector q to give
g(2)T (q,τ) = 1 + X2
Eβ{g(1)E (q,τ)2 – g(1)
E (q,∞)2} + 2XE( 1– XE)βn{g(1)E (q,τ) – g(1)
E (q,∞)}(91)
for a suspension of identical spheres, where XE is the ratio XE = 〈I(q)〉E/〈I(q)〉T with 〈I(q)〉T the time-
averaged intensity during measurement of g(2)T (q,τ) and 〈I(q)〉E the ensemble-averaged intensity. Repeated
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)E (q,τ) from Equation (91) gives
g(1)E (q,τ) = 1 + (1/XE){ – 1 + [1 + g(2)
T (q,τ) – g(2)T (q,0)]1/2} (92)
for β = 1. Pusey and van Megen noted that if a location in the cell is located for which XE = 1, then
Equation (2) simplifies to read
g(1)E
(q,τ) = [1 + g(2)T
(q,τ) – g(2)T
(q,0)]1/2; for XE = 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 0.95.
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)E
(q,τ) by a well-established g(1)E
(q,∞).
A model with non-interacting, harmonically bound particles was presented as an example of a model of
constrained mobility, permitting evaluation of g(1)E
(q,τ) in terms of the diffusion coefficient D and the mean-
square displacement 〈δ2〉 along q: (149)
g(1)E
(q,τ) = exp{– q2〈δ2〉[1 – exp(±�Dτ/〈δ2〉)]} (94)
g(1)E
(q,τ) = 1 – Dq2τ + …. (95)
g(1)E
(q,∞) = exp{– q2〈δ2〉} (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 〈δ2〉1/2 per
Equation (96).
An example of the use of the direct determination of g(1)E
(q,∞) from measurement of g(2)E (q,τ) obtained
over a very long time is given in Figure (17), along with an evaluation of g(1)E
(q,∞) from of g(2)T
(q,τ) and XE
using Equation (92). (153) The direct determination of g(1)E
(q,∞) required 13 hours, whereas the one on
g(2)T
(q,τ) required 30 minutes; the sample comprised an aqueous polyacrylamide gel (2.5 wt% polymer)
containing 0.02 wt% of 82 nm polystyrene spheres, with most of the scattering arising from the spheres.
A substantial difference between g(2)T (q,τ) and of g(2)
E (q,τ) is shown in the upper part, along with the
agreement between the g(1)E (q,∞) determined directly from g(2)
E (q,τ) and that calculated from g(2)T (q,τ) using
Equation (92). (154) Additional examples of g(2)E (q,τ) calculated from g(2)
T (q,τ) using Equation (92) and a
detailed analysis of the results are available on gels of polystyrene particles in polyacrylamide gels. (155)
<Figure 17>
The use of Equation (92) requires an optical arrangement with β ≈ 1, and is based on an assumption that
g(2)T (q,τ) and the average intensity 〈I(q)〉F,T for the fluctuating component are invariant with position in the
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)E (q,∞). Consequently, methods were developed to provide the averaging needed
for a direct evaluation of G(2)E (q,τ). As mentioned in the preceding, the image of the scattered light on a
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)T (q,τ) over a long
enough time to permit evaluation of G(2)E (q,τ) by averaging those results. (156-158) The 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)T (q,τ) continuously. (156)
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 τmin = T on of the order 1s, with the maximum τ limited by the
available software and the patience of the experimenter (157-158) Averaging these data finally yields G(2)E (q,τ),
from which g(1)E (q,τ) may be determined, including the g(1)
E (q,∞) contribution provided that the data are
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)T (q,τ) for different positions by manual movement of the cell.
The measurement time required to obtain a satisfactory evaluation of g(1)E (q,∞) by the methods described
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)
E (q, τ) g(1)ROT(q,τ)]2 (97)
with
g(1)OBS(q,τ) = g(1)
E (q,τ) g(1)ROT(q,τ) (98)
where g(1)OBS(q,τ) and g(1)
ROT(q,τ), respectively, are normalized electric field correlation functions as observed
and accounting for the effects of the cell rotation. The function g(1)ROT(q,τ) may be determined
experimentally by evaluation of g(1)OBS(q,τ) for a rigid sample, for which g(1)
E (q,τ) = 1. For rotation of a
cylindrical cell about its axis,
g(1)ROT(q,τ) =
2J1(qRστ)
qRστ (99)
where στ = 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)OBS(q,τ) calculated from the stream of data collected will appear as
a series of peaks (echos) for τ = nT, with the shaped specified by g(1)ROT(q,τ), but with a peak value
modulated by g(1)E (q,τ = nT). Although the peaks will appear as a linearly progressing sequence, a procedure
is available to increase the separation between the longer to reduce the time on calculations that do
53
not add usefully to the estimate for g(1)E (q,τ). (159) As with the interleaved method mentioned above (157-158)
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)
4.7.2 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 SKRWRQ 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 cross-
correlation 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 auto-
correlation 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 thLV section was
introduced by Phillies and implemented in an instrument restricted to the scattering at 90°. (161-162) Since
that time, both the theory and experimental methods have been refined. (163-167)
The theoretical treatment of cross-correlation involves an ensemble-averaged autocorrelation G(2)12(q,τ) of
the scattered intensities from each of the two scattering detectors, given by the function
G(2)12(q,τ) = 〈I1(q,0)I2(q,τ)〉E (100)
54
where the subscript "E" has been suppressed on G(2)12(q,τ) for simplicity. Division by the two average
intensities gives the intensity cross-correlation function g(2)12(q,τ), relate to the results in an expression for
the field cross-correlation g(1)12(q,τ) by the expression
g(2)12(q,τ) = 1 + β12[g(1)
12(q,τ)]2 (101)
(again suppressing notation to indicate the ensemble average), where β12 involves the coherence factors β1
and β2 from each of the detectors, a factor βV accounting for the incomplete overlap of the scattering
volume viewed by the two detectors, a factor βS accounting for incomplete separation of the scattering
detected by the two detectors, discussed in more detail below, and βMS a measure of the single to multiple
scattering:
β12 = [β1β2]1/2βVβSβMS (102)
βMS = 〈I
ss1 (q)〉〈I
ss2 (q)〉
〈I1(q)〉〈I2(q)〉 (103)
where 〈I
ss1(q)〉 and 〈I
ss2(q)〉 are the single scattered components of the total intensities 〈I1(q)〉 and 〈I2(q)〉,
respectively. As discussed further in the following, βS varies from 1 to 0.25, depending on the arrangement
used in the cross-correlation analysis (168) Although β1, β2 and βV may all be controlled, it is best to
determine [β1β2]1/2βVβS as the value of (β12)single, which may depend on q, from measurements of β12 on a
system with no multiple scattering, such as a dilute polymer solution or particle suspension, so that βMS = 1.
Such an evaluation will subsequently permit use of β12/{[β1β2]1/2βVβS} = β12/(β12)single to compute βMS 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 βS = 1 by placing a laser line pass filter in front of each of the
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 βV 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 βS = 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 βS 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 βS 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 βS as mentioned above (the 3D LS Photometer, LS
Instruments; http://www.lsinstruments.ch/).
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 auto-
correlation and β12 ≈ 0.45 for a dilute solution in Figure (19) without multiple scattering suggests that in βS
= 0.5, indicating one of the difficult alignment issues. The data shown in Figure (19) were analyzed to give
a hydrodynamic radius Rh determined in by the usual expression for dilute solutions for a monodisperse
scatterer (17):
g(1)12(q,τ) = exp[– Dq2τ] (104)
with D the diffusion constant in dilute solutions or suspensions, and Rh = kT/6πηD, with η the viscosity of
the media. The values of Rh 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
substantial effect of multiple scattering on R(ϑ,c) determined from 〈Iss
(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>
4.7.3 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. (170-172) 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)E (ϑ,τ); here, for reasons explained in the following, the notation indicates a
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)E (ϑ,τ) requires a value for l* for transmission, but not in
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)E (ϑ,τ) with the non ergodic behavior of the gel. Two
correlators analyze the scattering received via an optical fiber, with the light divided into two optical fibers
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)2Dτ with D the diffusion constant for diffusive (Brownian)
motion of the particles or by replacing Dτ by 〈Δr2(τ)〉/6 for non-diffusive particle motion where 〈Δr2(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)E (ϑ,τ) = 1 + β[g(1)
E (ϑ,τ)]2 (105a)
g(1)E (ϑ,τ) = ∫
∞0 ds P(s)exp[–(s/l*)(2τ/τ0)] (105b)
where β ≈ 1 in optical arrangements relevant to DWS, and P(s) depends on the geometric nature of the
experimental optical arrangement and τo is a time constant characteristic of the process, see below. (170-
171,174) 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* = 〈zo〉/l*, where 〈zo〉 ≈ 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 + (4x2/9)]sinh[ãx] + (4x/3) cosh[ãx]} (106a)
g(1)E (0,τ) ≈
[ã+ (4/3)]x[1 + (4x2/9)]sinh[ãx] + (4x/3) cosh[ãx] ; for x << 1 (106b)
58
g(1)E (π,τ) =
sinh[(ã – a*)x] + (2x/3) cosh[(ã – a*)x][1 + (4x2/9)]sinh[ãx] + (4x/3) cosh[ãx] (107a)
, g(1)E (π,τ) ≈
exp(–a*x)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 (106-107). (171,175) A method has been given to correct g(1)E (0,τ) for the
effects of reflection at the air-glass interfaces. (176) The parameter ã may be determined from the static
optical transmission Tscat as affected by the scattering (i.e., without absorption at the scattering wavelength,
which if present requires a known correction) by the expression (176)
Tscat = 53ã/01
2341 +
43Nã
-1
(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)〈Δr2(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��DUH�XVHG to
determine either D or 〈Δr2(τ)〉. Examples of typical behavior for 〈Δr2(τ)〉 DUH given in Figure (20), with the
tangent to 〈Δr2(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), (177-180) the DSSOLFDWLRQ of WKH
method in its application to the use of 〈Δr2(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 〈Δr2(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), (175,181-183)
mv̇(t) = frand(t) – ∫ t
0 dτ ζ(t - τ) v(τ) (109)
where ζ(t) is a generalized time-dependent memory function, frand(t) represents random forces acting on the
particle, v(t) is the particle velocity, v̇(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 < 10-6s), the
result relates the Laplace transform 〈 ~Δr2(s)〉 of the mean-square displacement in terms to η'(s)
η'(s) = ~G(s)/s = kT
πRs2〈 ~Δr2(s)〉
(111)
where ~G(s) = s~GR(s), with ~GR(s) the Laplace transform of the shear stress relaxation modulus GR(t). For
freely diffusing particles, s2〈 ~Δr2(s)〉 = 6D, and Equation (111) is seen to be a frequency-dependent form of
the usual Stokes-Einstein expression ηo = kT/6πRD. (183) Alternatively, since the Laplace transforms of
GR(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)〈 ~Δr2(s)〉, or after inverse Laplace transformation,
(185)
〈Δr2(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∞ – R0]ρ( t) (113)
where ρ(t) decreases from unity to zero with increasing t, R0 ≈ 10-10 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 Je for a solid. The retardation
function ρ(t) is often represented to within experimental uncertainty as a sum of weighted exponential
terms exp(-t/λi), e.g., by methods used to represent g(1)E (q,τ) as a weighted sum of exponential terms
exp(-q2Diτ) 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 JN 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 Je < JN. (186) These attributes are seen in the
experimentally determined 〈Δr2(t)〉, e.g., 〈Δr2(t)〉 ∝ t, perhaps for the entire accessible range of t for a low
viscosity fluid, and 〈Δr2(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 〈Δr2(t)〉 determined from g(1)E (0,τ) for a polymer solution and
gels prepared therefrom. (186-187) The similarity between 〈Δr2(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
〈Δr2(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 〈Δr2(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),
J'(ω) = R∞ – ω[R∞ – R0] ∫ ∞
0 dτ ρ( τ) sin(ωτ) (114a)
J"(ω) = (1/ωη) + ω[R(∞ – R0] ∫ ∞
0 dτ ρ( τ) cos(ωτ) (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 〈Δr2(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)]0.8R(t)}ωτ = 1 (115a)
J'(ω) = (1/ωη) + {[1 – m(2t/3)]0.8R(t)}ωτ = 1 (115b)
m(t) = ∂ln R(t)/∂ln t (115c)
where R(t) and η would be derived from 〈Δr2(t)〉 using Equation (112) in opto-microrheology.
A similar use of the tangent α(τ) = ∂ln 〈Δr2(τ)〉/∂ln τ has been incorporated into an approximation used
for the Laplace inversion and the subsequent Fourier transformation to frequency space to represent
〈Δr2(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〈Δr2(τ)〉)]Γ[ 1 + α(τ)]}τω = 1 (116a)
G'(ω) = |G*(ω)|cos(πα(ω)/2) (116b)
G"(ω) = |G*(ω)|sin(πα(ω)/2) (116c)
where Γ[…] is the gamma function; the behavior of 〈Δr2(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/R0 and G"(ω) tends to zero; whereas, for small ω, G'(ω) ∝ ω2 for a fluid or becomes
independent of ω and equal to the equilibrium modulus Ge = 1/Je for a solid, such as a gel. The data on
G'(ω) and G"(ω) given in Figure (21), determined using the 〈Δr2(τ)〉 given in that figure for a colloidal
dispersion, (175) show that in that case the data on 〈Δr2(τ)〉 extend to large enough τ (small enough ω) so
that G'(ω) may have nearly reached the limiting low-ω behavior with G'(ω) = Ge 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; http://www.lsinstruments.ch/).
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)E (0,τ) for non ergodic samples, as
would be encountered in the study of gels or other solid materials, and the use of software to compute and
present results on 〈Δr2(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)E (0,τ) for a non ergodic sample, or failure of the optical arrangement to
conform to the analytical expression given for g(1)E (0,τ) (the expressions given in Equations (106-107) are
for a particular optical arrangement, with different expressions needed for other arrangements (171));
additional issues are discussed in detail in the literature. (175-176,184) 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.
4.8 The intramolecular structure factor for wormlike chains The increasing interest in the scattering from wormlike macromolecules and micelles warranted an
improved version of the PVv(ϑ,0) for the behavior at large q (as only the Vv scattering will be of interest in
this section, the subscript "Vv" will be suppressed in the following). The limitations of the versions for
PRF(ϑ,0) for the random flight and the persistent linear chains for âq greater than about 2 are well known,
where â is the persistence length of the chain. For example, as shown by Equation (38) in Section 2.4.1,
the random flight chain model does not provide the relevant properties for âq > 2, as it misses the rodlike
character of any real chain in such a range of q, for which the rodlike expression in Table 1 is more
appropriate for a chain molecule, albeit requiring some modification for "thick" micelle rodlike structures
(see below). Improved treatments presented in numerical formats addressing these deficiencies are
discussed in the following, providing a crossover from PRF(ϑ,0) for R2Gq2 << 1 to P(ϑ,0) for the rodlike
chain for âq >> 1. The results for P(ϑ,0) are presented as a fairly complicated expression involving a
number of numerical parameters chosen to fit the numerical results of the calculation of P(ϑ,0). They are
well represented by PW-RF(ϑ,0) obtained from PRF(ϑ,0) in the range 0.1 ≤ L/â ≤ 20,000 with R2G =
(âL/3)S(â/L) for the wormlike chain used in place of the value âL/3 for the random-flight chain, see Table 1.
A set of curves for the functions (âLq2/3)P(ϑ,0) and (Lq/π)P(ϑ,0) as functions of âq for the Kratky-
Porod wormlike chain model developed by Yamakawa and Yoshizaki (2,189) as presented elsewhere is
reproduced in Figure (22). (190-191) The curves were obtained using a set of data given as bilogarithmic
plots of (L/2â)P(ϑ,0) vs 2âq for twelve values of L/â, ranging from 0.6 to 1,280. (192) Additional
calculations have produced similar results. (193-195)
<Figure 22>
Although the computations and various numerical presentations are complex one might anticipate that a
Padé approximation for P(ϑ,0) calculated for persistent chain models could provide a useful approximation
in many cases, using the limiting forms for the random flight and rodlike chains for large and small L/â,
respectively, e.g., see the discussion of these limits in Section 2.4.1. It is found that the following
expression provides a satisfactory representation of the data in Figure (22) for L/â >5: (190-191)
P(ϑ,0) = #$%
&'(
PmW-RF
(ϑ,0) + #$%
&'(1 – exp[–(âq)2]
1 + Lq/π
m 1/m
(11�)
with m = 3 provides a good representation of the more complex crossover expression for L/â > 5; the
second term in the brackets is devised to go to zero as q tends to zero, and to give the correct asymptotic
behavior for a rodlike chain for larger q. (Unfortunately, the prior rendition of Equation (11�) in
reference [191] included an error in which the term Lq/π was entered as Lq2/π�. The deviation
of the Padé approximation with m = 3 from the numerical P(q,0) for smaller L/â reflects the sharpening
character of the crossover noted above, and accordingly, may be minimized by permitting m to increase
with decreasing L/â, e.g., m equal to 6 for L/â of 2.5 to 0.6.
The behavior displayed in Figure (22) is usually considered in three regimes: Regimes I for which R2Gq2 is
small enough so that P(ϑ,0) is fitted by PI(ϑ,0) ≈ PW-RF(ϑ,0), a crossover in Regime II, with P(ϑ,0) ≈
PII(ϑ,0), and a rodlike behavior in Regime III for large âq for which P(ϑ,0) ≈ PIII(ϑ,0) with
PIII(ϑ,0) = (π/Lq)Psection(ϑ,0) (11�)
where Psection(ϑ,0) ≈ 1 for a scatterHU for which the effective radius Rc is small enough that Rcq << 1, despite
the large âq. For cylindrical symmetry of the scatterHU about its long axis,
Psection(ϑ,0) = #$%
&'(2J1(Rcq)
Rcq
2
≈ exp[– (Rcq/2)2] (11�)
where J1(…) is the first-order Bessel function of the first kind, and the exponential function is within 10%
of the Bessel function fo Rcq < 2, deviating rapidly with increasing Rcq. Regime III is rarely reached with
light scattering, but can be dominant for neutron or x-ray scattering, i.e., since ϑ < π, â/λ must be larger
than about (3/2π2)(1 + 4â/L) ≈ 0.24 if the transition to Regime III is to be observed�
Analysis of (âLq2/3)P(ϑ,0)PII(ϑ,0) and âLq2/3)P(ϑ,0)PIII(ϑ,0) shows that the intersection between these
two functions occurs for a q = q* given by
âq* ≈ (6/π)S(â/L)-1 ≈ (6/π)(1 + 4â/L) �����������(1��)
providing a means to estimate â if Regime III is to be observed, indicating that it will usually require the
smaller wavelenths of neutrons or x-rays for the analysis. If thDW regime is reached, e.g., in neutron or x-ray
scattering��(TXDWLRQ������ provides a means to determine â. If not, methods discussed in Section 2.4.1, �VXFK�DV
evaulation of the dependence of R2G on the chain coutour length L obtained for R2
Gq2 << 1, using the
expression for R2G = (âL/3)S(â/L) = (âL/3)S(â/L) given in Table 1, or the comparison of P(ϑ,0)) over the
available range of q with the improved versions mentioned above, displayed in Figure (22). (192-195)
EHUU\�������
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crosslinked polymer gels by diffusing-wave spectroscopy. Macromol Symp 207, 17-30 (2004).
188. D. J. Plazek, N. Raghupathi, S. J. Orbon, 'Determination of dynamic storage and loss compliances
from creep data', J Rheol, 23, 477-88 (1979).
189. T. Yoshizaki, H. Yamakawa, 'Scattering functions of wormlike and helical wormlike chains',
Macromolecules, 13, 1518-25 (1980).
190. G. C. Berry 'Total Intensity Light Scattering from Solutions Macromolecule',
������in Soft-Matter Characterization, eds. R. Borsali, R. Pecora, New York, Springer, 41-132. 2008.
191. G. C. Berry, 'Light Scattering', in Monitoring Polymerization Reactions, eds. W. F. Reed,
A . M. Alb, Johh Wiley & Sons, Hoboken NJ, 150-70 2014.
192. J. S. Pedersen, P. Schurtenberger, 'Scattering Functions of Semiflexible Polymers with and without
Excluded Volume Effects', Macromolecules, 29, 7602-12 (1996).
193. H. U. Ter Meer, W. Burchard, 'Determination of chain flexibility by light scattering', Polym Commun,
26, 273-5 (1985).
��
194. A. L. Kholodenko, 'Analytical calculation of the scattering function for polymers of arbitrary
flexibility using the Dirac propagator', Macromolecules, 26, 4179-83 (1993).
195. D. Potschke, P. Hickl, M. Ballauff, P.-O. Astrand, J. S. Pedersen, 'Analysis of the conformation of
worm-like chains by small-angle scattering: Monte Carlo simulations in comparison to analytical
theory', Macromol Theory Simul, 9, 345-53 (2000).
berry1/6/99
55
Table 1Intramolecular Structure Factor for Several Models (a)
(Rayleigh-Gans-Debye Approximation)
Model R 2 G
(b) PVv(ϑ,0)
random-flight linear coil âL/3 u = âLq2/3 pc (u) = (2/u2)[u – 1 + exp(–u)] persistent (wormlike) linear chain(c) (âL/3)S(â/L) v = uâ/L exp(v)
i=0Σ(–v/i!)−1pc[(v + i)L/â]
disk ("infinitely thin") (d) R2/2 y = Rq (2/y2)[1 – J1(2y)/y] sphere 3R2/5 y = Rq (9/y6)[sin(y) – ycos(y)]2
shell ("infinitely thin") R y = Rq [sin(y)/y]2
rod ("infinitely thin") (e) L2/12 x = Lq p1(x) = (2/x2)[xSi(x) – 1 + cos(x)]
5 With one exception, the original citations for the entries for PVv(ϑ ,0), along withexpressions for a number of additional models may be found in reference 21h. Allfunctions are for full orientational averaging.
6 â is the persistence length; L = M/ML is the chain contour length (mass per unit lengthML); R is the radius for a disk or shell.
7 S(Z) = 1 – 3Z + 6Z2 – 6Z3[1 – exp(–Z−1)] ≈ (1 + 4Z)−1; the expression for PVv(ϑ,0)is limited to â/L < 0.1; a more complete representation may be found in reference 52.
8 J1(…) is the Bessel function of 1st order and kind.
9 Si(x) = ∫ 0 x ds{sin(s)/s} is the sine integralFor rods with optically anisotropic
elements:(10, 18. 41)
(1 + 4δ2/5)PVv(ϑ ,0) = p1(x) + δ2{(4/5) p3(x) – (2 – δ−1)m1(x) + (9/8)m2(x) + m3(x)}PHv(ϑ ,0) = p3(x) + (5/8)sin2(ϑ/2)m2(x)
p2(x) = (6/x3)[x – sin(x)]p3(x) = (10/x5)[x3 + 3xcos(x) – 3sin(x)]
m1 = p1 – p2
m2 = 3p1 – p2 – 2p3
m3 = p3 – p2
berry1/6/99
56
Table 2
Light Scattering AverageMean-Square Radius of Gyration and Hydrodynamic Radius
(Rayleigh-Gans-Debye Approximation)
(R 2 G)LS (RΗ)LS
Exact Relation(a)(1/Mw)ΣwµMµ (R2
G)µ Mw
/Σ wµMµ (RΗ−1)µ
Approximation for(b)
RΗ ∝ R G ∝ Mε/2(R2
G/Mε) M(ε+1)ε+1
/Mw (RΗ/Mε/2) Mw
/M(1–ε/2)1–ε/2
Random-flight coil(c);ε = 1
(R2G/M) Mz (RΗ/M1/2) M
w/M(1/2)
1/2
≈ (RΗ/M1/2) Mw.
1/2 (M
w/M
n)0.10
Rodlike chain(c);ε ≈ 2
(R2G/M2) MzMz+1 (RΗ/M) M
w
Sphere(c);ε = 2/3
(R2G/M2/3) M(5/3)
5/3/Mw
≈ (R2G/M2/3) M
z
2/3(M
w/M
z)
0.10
(RΗ/M1/3) Mw
/M(2/3)2/3
≈ (RΗ/M1/3) Mw.
1/3 (M
w/M
n)0.10
(a) For optically isotropic solute, and with ∂n/∂c the same for all scattering elements
(b) M(α) = ∑wµMµα
1/α and Mn = 1/Σ
µwµMµ
−1; Mw = Σ
µwµMµ; Mz = Σ
µwµMµ
2/Σ
µwµMµ
(c) Approximations are for a solute with a Schulz-Zimm (two-parameter exponential)distribution of M, for which M(α) ≈ Mw{Γ(1+h+α)/Γ(1+h)}1/α/(1+h), see reference 42.
66
Table 3. Media type dependent examples of g(2)T (q,τ) and g(2)
E (q,τ) for a single speckle
Media Type τ; correlation time g(2)T (q,τ)
Time-averaged
g(2)E (q,τ)
Ensemble-averaged
Rigid media:
Non fluctuating scatters
τ = 0
τ ⇒ ∞
1
1
2
2
Ergodic media:
Fully fluctuating scatters
τ = 0
τ ⇒ ∞
2
1
2
1
Non ergodic media: τ = 0
τ ⇒ ∞
≥ 1; ≤ 2
1
2
≥ 1; ≤ 2
berry
Figure Captions
Figure 1 The ratio MLS/M (= msph(ñ,α)2) of the light scattering averaged molecular weight MLS formonodisperse spheres of radius R to the molecular weight M as a function of the relativerefractive index ñ for the indicated values of the size parameter α = 2πR/λ. The dashed linesgive the limiting behavior for small ñ – 1 (see Equation (15)).
Figure 2 The ratio (R2G)LS/(3R2/5) (= ysph(ñ,α)) of the light scattering averaged mean square radius of
gyration (R2G)LS for monodisperse spheres of radius R to the geometric square radius of
gyration as a function of (a) the size parameter α = 2πR/λ for the indicated values of therelative refractive index ñ, and (b) the relative refractive index ñ for the indicated values of thesize parameter α = 2πR/λ.
Figure 3 Examples of PVv(ϑ ,0)-1 versus q2R2G
for monodisperse model structures (see Table 1):(a) rodlike chains (R), random-flight linear chains (C), disks (D), spheres (S), shells (Sh) and
the exponential function exp(–q2R2G/3) (E), included for comparison;
(b) the persistent or wormlike chain, for the indicated values of the ratio of the persistencelength â to the contour length L.
Figure 4 Examples of PVv(ϑ ,0)-1 versus q2(R2G)LS
for model structures with a weight distribution givenby the two-parameter Zimm-Schulz distribution function, with h = 1/[(Mw/Mn) – 1] per thediscussion in the text, for random-flight linear chains (a) and spheres (b), for the indicatedvalues of h, with h decreasing from top to bottom in each set of curves (the uppermost curvefor the monodisperse case is bold in each set). The dashed line in (b) is the initial tangent.
Figure 5 Examples of [PVv(ϑ ,0)]BR vs. q2R2G for comb-shaped branched chain polymers divided by
[PVv(ϑ ,0)]LIN for linear chains with the same R2G (not the same molecular weight). The number
of branches is indicated, along with the fraction ϕ of mass in the backbone of the branchedchain. From reference 53.
Figure 6 Examples of PVv(ϑ ,0) vs. q2(R2G)LS for spheres with size parameter α = 2πR/λ = 4 for the
indicated values of the relative refractive index ñ. The angular range is 0 to 180 degrees in allcases except for ñ = 2. The RGD limiting case for very small ñ – 1 is given by the dashedcurve. The dashed line gives the initial tangent. Values of (R2
G)LS/(3R2/5) may be seen for thisα in Figure 2.
berry
Figure 7 An example of the use of multi-angle light scattering as an SEC detector in the analysis ofpoly(di-n-hexylsilane), PDHS, and poly(phenyl-n--hexylsilane), PPHS.(a) Mw and (R2
G)LS resulting from analysis of the multi-angle scattering data;(b) The response from the differential refractive index, normalized to give the same peak
response.From reference 1.
Figure 8 An example of the use of multi-angle light scattering as an SEC detector in the analysis ofrandomly branched poly(methyl methacrylate), PMMA.(a) Mw and (R2
G)LS/Mw for the eluent for heterodisperse linear (•) and branched polymers (°)resulting from analysis of the multi-angle scattering data, with the dashed line and solidcurves giving power law and polynomial extrapolations, as discussed in the text;
(b) The ratio [(R2G)LS]BR/[(R2
G)LS]LIN from the data in part (a), using the powerlaw (°) andpolynomial (•) extrapolations to estimate [(R2
G)LS]LIN.Adapted from figures in reference 132.
Figure 9 An example of the scattering from an aqueous dispersion of heterodisperse hollowpolystyrene spheres.(a) the scattering function and fits thereto using the RGD approximation and the Mie theory;(b) the number fraction distribution of particles with a give size deduced from the inversion of
the scattering function using the RGD approximation and the Mie theory; an estimatedetermined from analysis of the dynamic light scattering is included for comparison.
From figures in reference 107.
Figure 10 An example of the scattering from an heterodisperse aerosol.(a) the scattering function (solid curve) and fits thereto using the Fraunhofer diffraction
approximation and an assumed size distribution function;(b) the weight fraction distribution of particles with a give size deduced from the inversion of
the scattering function using the Fraunhofer diffraction approximation (solid curve) andthe function used in part (a) (dashed curve)Adapted from figures in reference 140.
berry
Figure 11 An example of the scattering from heterodisperse glass particles dispersed in differentsolvents to span a range of ñ.(a) the scattering function (points and solid curve) obtained using a lens with a 100 mm focal
length and solvents to give the indicated |ñ – 1|, and fits thereto using the Mie theory andthe known size distribution function (dashed curves);
(b) the cumulative weight distribution of particles with a give size deduced by severalmethods: filled squares, from microscopic measurements (the "known" distributionfunction), as reported by NIST (see text), unfilled squares and diamonds, from inversionof inversion of the scattering function using the Mie theory for |ñ – 1| ≈ 0.15 and twolenses, with focal lengths 300 mm (squares) and 100 mm (diamonds), and circles, forsolvents to give |ñ – 1| ≈ 0.02 calculated with the assumption of Fraunhofer diffraction(filled and dashed line) or anomalous diffraction (unfilled and solid line)Adapted fromfigures in reference 141.
Figure 12 Scattering functions for an illustrative example of a flexible chain polymer undergoing end-to-end dimerization.(a) dependence on angle, calculated as discussed in the text for a reduced equilibrium
constant ~Keq = 0.1 and the indicated values of (A2)MMc, with the constant equal to zero or0.2 for the solid and dashed curves, respectively;
(b) scattering extrapolated to zero angle as a function of (A2)MMc, for the indicated values of~Keq.
Figure 13 Scattering from a semi-flexible polyelectrolyte chain in solvents with low and high ionicstrengths (unfilled and filled circles, respectively).Adapted from figures in reference 150.
Figure 14 The dependence of the structure factor on qR for polystyrene spheres (R = 45 nm) immersedin deionized water, with the number concentration ν/particles·µm-3 = 2.53, 5.06, 7.59 and10.12 for the circles with increasing depth of the shading, respectively.Adapted from figures in reference 152.
Figure 15 The dependence of (R2G)LS and (δ/δο) 2
LS on the ratio of the weight average contour length to thepersistence length for rodlike molecules with a distribution of contour lengths.Adapted fromfigures in reference 10.
Figure 16. The intensity correlation function g(2)T (q,τ) for aqueous poly(N-isopropylacrylamide) hydrogels
analyzed via Equations (88-90) for several positions in the light scattering cell, each with a
different total intensity 〈I(q)〉T and factor XF = 〈I(q)〉F,T/〈I(q)〉T. The figure displays the
g(2)T (q,τ) for each position, along with the value of XF for that position, with the uppermost
curve giving g(1)F;E(q,τ) obtained from g(2)
T (q,τ) using Equation (90). The insert shows 〈I(q)〉F,T
obtained at the various positions, showing it to be nearly independent of position, despite the
wide variation in 〈I(q)〉F,T with position. (150), Copyright 2003. Reproduced with permission
from the American Chemical Society.
Figure 17. Autocorrelations functions for a polyacrylamide hydrogel containing 2.5 and 0.02 wt%
polymer and polystyrene spheres (diameter 82 nm), respectively.
Upper: The intensity correlation functions g(2)E (q,τ) measured by extensive averaging, and the
normal time average g(2)T (q,τ) obtained with much shorter averaging duration.
Lower: The ensemble averaged g(1)E (q,τ) determined directly from g(2)
E (q,τ), and that determined
from g(2)T (q,τ) with the use of Equation (92) and supporting measurements of XE, showing that
the two estimates are essentially equivalent. (153), Copyright 1993. Reproduced with permission
from Springer-Verlag.
Figure 18. Examples of g(1)E (q,τ) determined on a slowly relaxing colloidal suspension by two methods:
g(1)E (q,τ) from averages of 100 measurements of G(2)
T (q,τ), (+), and the echo technique, (o). The
data from the echo method are normalized to agree with the data from the averaged
measurements in the overlap region. The echo method is seen give better results at long τ than
the average of the 100 measurements, despite the much shorter experimental time needed for
the echo measurement than for the 100 averges. (159), Copyright 2004. Reproduced with
permission from the American Institute of Physics.
Figure 19. Upper: The function g(2)T (q,τ) – 1 determined at 90° scattering angle, either by a normal single
correlator or cross-correlation (unfilled and filled symbols, respectively, for aqueous
suspensions of spheres (48.5 nm radius) at two different transmission levels, showing the
suppression of multiple scattering by the cross-correlation the cross-correlation method.
Lower: The scattered intensity as a function of q for a suspension of spheres (322 nm radius)
determined either by a normal single correlator or cross-correlation (filled circles or stars,
respectively. The solid line represents the form factor given by Mie scattering, including the
effects of particle size distribution determined by electron microscopy. (169), Copyright 2012.
Reproduced with permission from the American Physical Society.
Figure 20. Upper: Examples of the linear viscoelastic recoverable compliance R(t) = J(t) – t/η for a linear,
high molecular weight polymer, with J(t) and η the creep compliance and the viscosity,
respectively. The mean-square-displacement 〈Δr2(t)〉 determined in DWS is proportional to J(t)
in Equation (112). Also shown is the shear relaxation modulus GR(t), denoted simply as G(t)
in the figure); the Laplace transform of GR(t) appears in the theory leading Equation (112);
comparisons of the dynamic compliance J'(ω) with J(t) and the dynamic modulus G(ω) with
GR(t) are also shown in the figure. (186)
Lower: (a) Examples of g(1)E (0,τ) for samples of poly(vinyl alcohol) chemically crosslinked gels
and a gel-precursor solution in water, obtained by transmission DWS with samples containing
≈ 5% polymer and 1% polystyrene spheres (535 nm radius); the ratio Rc of the crosslinker to
the polymer repeat units is given in the figure for the gels
(b) The mean-square-displacement 〈Δr2(t)〉 obtained from g(1)E (0,τ) in panel (a) using Equation
(106a). For the solution (and water dispersion) 〈Δr2(t)〉 ∝ t expected with Equation (112) when
the contribution from the recoverable compliance R(t) << t/η; the 〈Δr2(t)〉 for the gels are in
accord with R(t) expected for a gel, leading to the compliance Je of the gel for large t for the
sample with the larger crosslink density Rc, but showing that the data do not extend to large
enough t to obtain Je for the sample with smaller Rc. (187), Copyright 2004. Reproduced with
permission from John Wiley & Sons Inc.
Figure 21. The dynamic moduli G'(ω) and G"(ω) calculated from 〈Δr2(t)〉 shown in the insert; the data are
for an aqueous suspension volume fraction 0.56 of polystyrene spheres. The mean-square-
displacement 〈Δr2(t)〉 was obtained from g(1)E (0,τ) determined in transmission DWS using
Equation (106a). The dynamic moduli were obtained by analysis of ~G(s) = s~GR(s) given in
Equation (111), rather than by the evaluation from J(t) using Equation (112), with subsequent
calculation of G'(ω) and G"(ω) from J(t) by a method described in the text. (183), Copyright
1995. Reproduced with permission from the American Physical Society.
Figure 22. The functions (âLq2/3)P(ϑ,0) (upper) and (Lq/π)P(ϑ,0) (lower) vs âq for the Kratky–Porod
wormlike chain model (2,189) for chains of contour length L and persistence length â as
presented previously. (190) Copyright 2008. Reproduced with permission from Springer-Verlag.
For convenience of comparison, the values of L/â used are among those in an alternative
bilogarithmic representation of (L/2â)P(ϑ,0) vs 2âq presented in the literature (192): 640, 160, 80,
40, 20, 10, 5 for the curves from top to bottom in the lower panel, and all of these except 160 in
the upper panel for the curves from left to right.
berry
2
1.8
0 1
n - 1~
0.2 0.4 0.6 0.8
4.0
2.0
6.0
1.5
(b)
3
4
1
2
3
4
1
2
0 2 4 6 8
α = 2πR/ λ
(a)
1.2
1.4
2 G(R
)
/(
3R /
5)
2LS
berry
Sh
0 2 4 6 8 10
[P
(ϑ, 0
)]-1
Vv
1
2
3
4
5â/L∞
0.0050.010.030.050.08
0
(b) Wormlike chain
2Gq R 2
SD
C
R
(a) Various models
E
1
2
3
4
5
6
berry
1 2 3 4 5 60
h = ∞201051
[P
(ϑ, 0
)]-1
Vv
0 2 4 6 8 10
1
2
3
4
5
6
(a) Random-flight flexible chain
(b) Spheres
2Gq (R )2
LS
1
2
3
4
5
6
berry
u = (qR )G2
1.00
0.95
0.90
1.00
0.95
0.90
1.00
0.95
0.900 2 4 6 8 10 12
2
20
9
4
20
2
94
20
9
42
ϕ = 0.1
ϕ = 0.3
ϕ = 0.7
[P (
ϑ,0
)]
/[P
(ϑ,
0)]
BR
LIN
Vv
Vv
berry
0 20 40 60
2.0
1.8
1.6
1.4
RGD
2.0
1.8
1.6
1.4
1.2
1.05
1.1
α = 4
0
-1
-2
-3
-4
-5
80
2Gq (R ) 2
LS
Log
[P
(ϑ,0
)]V
v
berry
PPHS
PPHS
PDHS
PDHS
Elution Volume 18 20 22 24 26
10
100
1000
Ref
ract
ive
Inde
x
°
R
/n
m;
M/1
0 ;
4
G
•
berry
-1.4
-1.3
-1.2
0.2
0.4
0.6
0.8
1.0
5.0 6.0 7.0
(a)
(b)
Log M w
0.5
log
(R
/M ) w
2 LS/(
R
)2 LS
LIN
(R
)2 LS
BR
berry
Radius/µ m
Num
ber
Fra
ctio
n
q /nm-1
R
(ϑ, c
)/ar
bitr
ary
units
Vv
0.1 0.2 0.3 0.4
RGD
MIE
DYN
0.2
0.3
0.1
0
0.01
0.10
1.00
0.01 0.02 0.03 0.040
RGD
MIE
(a)
(b)
berry
Ring-photodiode Number
(a)
0 1 2 3
log(R/ µm)
0
0.04
0.08
0.12
Wei
ght F
ract
ion
R
(ϑ,0
)/ar
bitr
ary
units
Vv
0
1
2
0 10 20 30
(b)
berry
|n - 1|≈ 0.15~
|n -1| ≈ 0.02~
|n - 1|≈ 0.08~
|n -1| ≈ 0.02~
Cum
ulat
ive
Wei
ght F
ract
ion
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
Ring-photodiode Number
2 4 6 8N
orm
aliz
ed E
nerg
y
Size/µm
0
10 12 14
0 20 40 60 80 100 120 140 160
(a)
(b)
berry
1.0
0.8
1.2
1.4
1.6
1.8
0.0001
0.03
0.01
1.0
0.6
1.2
1.4
0.8
0 0.2 0.4 0.6 0.8 1
2Gq (R )2
monomer
0 0.2 0.40.1 0.3
2(A Mc) 2 monomer
(a)
(b)
cst.
+ K
cM
/R
(ϑ
,c)
mon
omer
V
v
0.03
0.1
0.3
0.2K
cM
/R
(0,
c)m
onom
er
Vv
0.0001
0.01
0.001
0.1
110
berry
10 √
[Kc/
RV
v(0,
c)]
310
Kc/
RV
v(ϑ,
0)6
Concentration (g/L)
0 0.2 0.4 0.6 0.8 1
sin (ϑ /2)2
10
20
30
40
0 0.2 0.4 0.616
12
8
4
(a)
(b)
berry
2
0 0.2 0.4 0.6 0.8 1 1.2
qR
2
1
0
S
(q,c
)V
v
berry
0
–1
Log [L /â ] w
–2 –1 0 1 2
–2
–1
0
oLo
g [(
δ/δ
)
]2 LS
(a)
(b)
Log
12(R
)
2 GLS
L L
z +
1z
Figure (16)
Figure (17)
Figure (18)
Figure (19)
Figure (20)
Figure (21)
105
7. The functions (âLq2/3)P(q,0) (upper) and (Lq/!)P(q,0) (lower) vs âq for the Kratky–Porod
wormlike chain model [5, 91], for chains of contour length L and persistence length â. For
convenience of comparison, the values of L/â used are the same are among those in an
alternative bilogarithmic representation (L/2â)P(q,0) vs 2âq presented in the literature [102]:
640, 160, 80, 40, 20, 10, 5 for the curves from top to bottom in the lower panel, and all of
these except 160 in the upper panel for the curves from left to right.