---
abstract: |
    The objective of this paper is to correct a substantial, widespread error in parts of the quasielastic scattering literature. This error leads to entirely erroneous interpretations of quasielastic scattering spectra. The error, which is most prominent for interpreting spectra of dilute probe particles diffusing in complex fluids, arises from a valid calculation that is being invoked under circumstances in which its primary assumptions are incorrect.

    Quasielastic scattering from dilute probes yields the incoherent structure factor $g^{(1s)}(q,t) = \langle \exp(i q \Delta x(t)) \rangle$, with $q$ being the magnitude of the scattering vector ${\bf q}$ and $\Delta x(t)$ being the probe displacement parallel to ${\bf q}$ during a time interval $t$. The error is the claim that $g^{(1s)}(q,t)$ for dilute probe particles uniformly reduces to $\exp(- q^{2} \langle (\Delta x(t))^{2} \rangle/2 )$, regardless of the nature of the surrounding medium. If true, this claim would allow one to use quasielastic scattering to determine the time-dependent mean-square probe displacements in complex fluids. In reality, $g^{(1s)}(q,t)$ is determined by all even moments $\langle (\Delta x(t))^{2n} \rangle$, $n = 1, 2, 3,\ldots$, of the displacement distribution function $P(\Delta x,t)$. Only in the very special case of monodisperse probes in a simple Newtonian solvent is $g^{(1s)}(q,t)$ entirely determined by $\langle (\Delta x(t))^{2} \rangle$. Furthermore, the Langevin equation approach that ties $g^{(1s)}(q,t)$ to $\langle (\Delta x(t))^{2} \rangle$ *also requires with equal certainty* that $g^{(1s)}(q,t)$ relaxes as a simple exponential $\exp(- \Gamma t)$, $\Gamma$ being a time-independent constant. Contrariwise, *if the spectrum is not a simple exponential in time, $g^{(1s)}(q,t)$ is not determined by $\langle (\Delta x(t))^{2} \rangle$.* Several related subsidiary errors are discussed.
author:
- 'George D. J. Phillies'
title:  Interpretation of Quasielastic Scattering Spectra of Probe Species in Complex Fluids 
---

Introduction
============

Quasielastic scattering of light, x-rays, and neutrons has proven to be a powerful experimental tool for the study of complex fluids. For dilute macromolecule solutions, quasielastic scattering has extensive analytic applications for particle sizing[@phillies1990z]. With non-dilute solutions and more complex systems, quasielastic scattering reveals consequences of intermacromolecular and other interactions.

In a substantial enhancement of the classical method, the fluid of interest is doped with trace concentrations of monodisperse probe particles. Probes have ranged from small molecules to micron-size colloidal particles. The diffusion of probe particles through the complex fluid is then observed. Early successful studies of probe diffusion in complex fluids using quasielastic light scattering were by Hallett and co-workers[@gray1974a; @turner1976a], who in 1974 and 1976 observed the diffusion of polystyrene spheres through hyaluronic acid and dextran solutions, and compared probe diffusion with the rheological properties of their solutions. The subsequent four decades have seen an enormous extension of this approach[@phillies2011a], including studies of probes in highly viscous simple liquids[@phillies1981z], polymer melts[@lin1986a], chemically cross-linked gels[@schmidt1989b], surfactant solutions[@phillies1993b], protein solutions[@ullmann1985a], and the interior of living cells[@lubyphelps1987a]. More recently, quasielastic x-ray scattering has been used to extend the range of distance scales over which diffusion can be observed[@dierker1995a].

Probe diffusion has also been studied by a series of other physical techniques, each technique being sensitive to distinctive range of time and distance scales or other features of probe motion. For example, fluorescence correlation spectroscopy[@magde1972a], which by varying the probe concentration can measure both the self diffusion coefficient and the mutual diffusion coefficient of the labeled species[@phillies1975a; @scalettar1989a], has in recent years been extensively used to measure tracer diffusion. Recent work using probe diffusion is sometimes termed *microrheology*, the term microrheology referring to a particular model[@mason1996a] for interpreting quasielastic scattering spectra. In some studies, probe diffusion has been viewed as being of interest because it is a path to measuring the viscoelastic properties of the solution. In other studies, probe diffusion has been viewed as being of interest because it measures solution properties that are not the same as the viscoelastic properties of the solution.

A valuable complement to quasielastic scattering studies of probe diffusion is provided by measurements on probes subject to external driving forces. The overwhelming bulk of the literature on driven probe motion is provided by studies of capillary electrophoresis. The electrophoretic literature primarily emphasizes improving separations of different charged species. However, electrophoretic separations often use solutions of neutral polymers as the support medium. At the same time as these experiments are performing separations, these experiments are also giving information on the dynamics of the neutral polymers[@phillies2011a; @phillies2012e]. A substantial literature exists on buoyancy-driven probe motion in the ultracentrifuge[@laurent1963a; @ye1998a]. In a few experiments, magnetic[@hough1999a; @schmidt2000a] or optical[@amblard1996a] tweezers were used to examine oscillatory driven movements of probes. Tweezer experiments are particularly interesting because the experimenter can separately control two of the three: drive force, drive frequency, and particle displacement. An alternative complement to probe diffusion is provided by tracking probes in complex fluids in which the fluid itself is performing driven motion, e.g., shear[@tapadia2006a].

Quasielastic scattering of light and other photons is most commonly studied via correlation methods, in which the quantity measured directly is the intensity-intensity time correlation function $$S(q,t) = \langle I(q,\tau) I(q, \tau+t) \rangle,
     \label{eq:Sqtdef}$$ Here $q$ is the magnitude of the scattering vector, $I(q,\tau)$ and $I(q, \tau+t)$ are the scattering intensities over short time intervals near $\tau$ and $\tau+t$, and the brackets $\langle \cdots \rangle$ represent an average. Scattering is said to be due to scatterers within the medium, scatterers generally being represented mathematically by points whose locations are known. Scattering from extended bodies, such as high-molecular-weight polymer chains, is often treated by representing the scattering body as a series of scattering points whose relative positions are partly fixed. If the volume being observed is much larger than the volumes over which particle positions and displacements are correlated, quasielastic scattering corresponds to the intermediate structure factor (or field correlation function) $g^{(1)}(q,t)$ via[@crosignani1975a] $$S(q,t) = A |g^{(1)}(q,t)|^{2} + B.
     \label{eq:Sqg1def}$$ In this equation $A$ and $B$ are constants determined by details of the experimental apparatus; these constants have no effect on the time dependence. Homodyne rather than heterodyne detection of the scattered light is assumed. The factorization of $S(q,t)$ into $g^{(1)}(q,t)$ is sometimes termed the “Gaussian approximation”. This Gaussian approximation is not related to the Gaussian approximation for the particle displacements as discussed below.

The intermediate structure factor is in turn determined by the time-dependent positions of the scattering particles via $$g^{(1)}(q,t) = \left\langle \sum_{i=1}^{N} \sum_{j=1}^{N} \exp(\imath {\bf q} \cdot ({\bf r}_{i}(t+\tau) -  {\bf r}_{j}(\tau) )) \right\rangle
    \label{eq:g1qgeneral}$$ In this equation, sums on $i$ and $j$ proceed separately over all $N$ particles in the system, while ${\bf r}_{i}(t+\tau)$ and ${\bf r}_{j}(\tau)$ are the locations of scatterers $i$ and $j$ at times $t+\tau$ and $\tau$, respectively.

In applying eq \[eq:g1qgeneral\], two particularly interesting experimental circumstances are described as mutual diffusion measurements and as probe diffusion measurements. In a measurement on a binary solvent: scatterer system, the scattering particles may be concentrated or dilute. Quasielastic scattering on such a system measures the mutual diffusion coefficient, which describes the diffusion of the scatterers down a concentration gradient[@phillies1974a; @phillies1974b]. Tracer diffusion experiments examine ternary solvent: matrix : probe systems. In these systems the matrix component may be dilute or concentrated, is substantially responsible for the system’s rheological and other interesting properties, but is nearly optically inert. Conversely, the probe (tracer) component is dilute, has virtually no effect on rheological and other properties of the solvent: matrix system, but dominates scattering by the ternary mixture. If matrix scattering is not entirely negligible, there are established, reliable ways to isolate the probe scattering, based on spectral subtraction at the level of the field correlation function.

Because probe particles very nearly do not interact with each other, the field correlation function for probe diffusion reduces (up to normalization constants) to the incoherent scattering function $$g^{(1s)}(q,t) =  \langle \exp(\imath q \Delta x(t))\rangle.
 \label{eq:g1sandr}$$ with $\Delta x(t)$ being the component parallel to ${\bf q}$ of $\mathbf{\Delta r}(t) = {\bf r}_{i}(t+\tau) -  {\bf r}_{i}(\tau)$. Probe motions perpendicular to $\mathbf{q}$ do not contribute to $g^{(1s)}(q,t)$. In moving from eq \[eq:g1qgeneral\] to eq \[eq:g1sandr\], terms of eq \[eq:g1qgeneral\] in which $i \neq j$ were taken to average to zero, because the relative positions of dilute probes are very nearly uncorrelated. An expression formally identical to eq \[eq:g1sandr\] describes diffusion measurements using pulsed-field-gradient nuclear magnetic resonance, though with this method $q$ has an entirely different meaning, namely in the simplest case $q = \gamma \delta g$, where $\gamma$ is the gyromagnetic ratio, $\delta$ is a pulse width, and $g = dB/dz$ is the field gradient.

The averages in eqs \[eq:g1qgeneral\] and \[eq:g1sandr\] may formally be phrased as averages over displacement distribution functions such as $P(\Delta x, t)$, which gives the time-dependent probability that a scattering particle will displace through $\Delta x$ during time $t$. Two previous papers[@phillies2005a; @phillies2012a] examined how $g^{(1s)}(q,t)$ and $g^{(1)}(q,t)$ are actually related to the displacement distribution functions. The two prior papers were primarily concerned with establishing formal relationships between dynamic structure factors and probabilities for scatterer displacement. The significance of these relationships for the interpretation of experimental measurements was at most a secondary consideration. This paper focuses on interpreting experimental measurements.

Section II of this paper presents the correct general relationship between $g^{(1s)}(q,t)$ and $P(\Delta x, t)$. Section III discusses the special case of probe particles in a purely Newtonian fluid. Section IV notes experimental findings bearing on the relative significance of Sections II and III. Section V considers paths for interpreting probe diffusion spectra. Section VI treats the determination of $P(\Delta x, t)$, relationships between $g^{(1s)}(q,t)$ and trapping/hopping behavior, and, closing on a positive note, cases in which quasielastic scattering from diffusing probes, correctly interpreted, has given valuable information about complex fluids and the objects diffusing in them.

General Case\[sectiongeneralcase\]
==================================

This section summarizes what $g^{(1s)}(q,t)$ and $g^{(1)}(q,t)$ actually reveal about particle displacements. Extended derivations have appeared previously in two earlier papers, refs and , and are not repeated here. For probe diffusion, the intermediate structure factor is always determined by the displacement distribution function, namely the average in eq \[eq:g1sandr\] can be written as $$g^{(1s)}(q,t) = \int_{-\infty}^{\infty}  d(\Delta x)  \exp(i q \Delta x) P(\Delta x, t).
        \label{eq:g1sPDelta}$$ $P(\Delta x, t)$ is taken to be properly normalized.

On taking a Taylor series expansion of the exponential in powers of $q$, reflection symmetry, namely $P(\Delta x, t) = P(-\Delta x, t)$, eliminates all terms odd in $q$. As a result, $g^{(1s)}(q,t)$ and its logarithm are necessarily power series in $q^{2}$. The coefficients of the $q^{2n}$ are generated by the even moments $\langle(\delta x)^{2n}\rangle$ of $P(\Delta x, t)$. As shown previously[@phillies2005a], the lead terms of an expansion for $g^{(1s)}(q,t)$ are $$g^{(1s)}(q,t) = N \exp\left( - \frac{1}{2} q^2 \langle \Delta x(t)^{2} \rangle + \frac{1}{24} q^{4}( \langle \Delta x(t)^{4} \rangle - 3\langle \Delta x(t)^{2} \rangle^{2}) - {\cal O}(q^{6})\right).
      \label{eq:g1sanddisplacements}$$ All even moments $\langle(\delta x)^{2n}\rangle$ are required for the complete expansion.

It was early shown that quasielastic scattering from a binary solvent: macromolecule system determines the mutual diffusion coefficient, not the self diffusion coefficient[@phillies1974a; @phillies1974b]. Theoretical approaches to computing $g^{(1)}(q,t)$ and the mutual diffusion coefficient of non-dilute colloid solutions have historically followed routes very different from the routes that are based on the displacement distribution function $P(\Delta x, t)$. Only very recently[@phillies2012a] was a solution for $g^{(1)}(q,t)$ in terms of displacement distribution functions obtained. In this solution, the expansion of eq \[eq:g1qgeneral\] was shown to require averages over two different displacement distribution functions, namely $P(\Delta x, t)$ and a new distribution function $P_{2}(\Delta x, t, \mathbf{R}_{12})$. $P_{2}$ is a two-particle conditional displacement distribution function, in which $\Delta x$ is the displacement of particle $1$ during $(0, t)$ given that the vector $\mathbf{R}_{12}$ from particle $1$ to some particle $2$ at time $0$ has a given value.

Special Case: Probes in Simple Newtonian Liquids \[sectionsimple\] 
===================================================================

The earliest quasielastic scattering experiments were performed on dilute suspensions of monodisperse scattering particles in simple Newtonian solvents. Cummins, et al.’s results on polystyrene spheres in water[@cummins1964a] are the archetype. The resulting spectra were interpreted by invoking a mechanical model for the motions of diffusing particles. The mechanical model was provided by the Langevin equation, which in one dimension is $$m\frac{d^{2} x(t) }{dt^{2}} = - f \frac{dx(t)}{dt} +{\cal F}_{x}(t).
     \label{eq:Langevin}$$ Here $x(t)$ is a coordinate of the diffusing particle, $m$ is the particle mass, $f$ is the particle’s drag coefficient, and ${\cal F}_{x}(t)$ is the random force, called *random* because in the Langevin model the values of ${\cal F}_{x}(t)$ at different instants in time are uncorrelated. Within the model, ${\cal F}_{x}$ cannot be predicted beyond stating that ${\cal F}_{x}$ has certain statistical properties.

The canonical literature treatment of the Langevin model as applied to quasielastic light scattering is the volume by Berne and Pecora[@berne1976a], notably their Section 5.9. Berne and Pecora show that the Langevin model is appropriate for polystyrene spheres in water, on the time and distances scales observed by quasielastic light scattering. From the Langevin model and the requirement that the system remains in thermal equilibrium, a series of conclusions about the statistical properties of the particle motion follow. In particular:

(i)

:   The mean-square average value of ${\cal F}_{x}(t)$ must be consistent – the fluctuation-dissipation theorem – with the drag coefficient $f$ and the thermal energy $k_{B}T$.

(ii)

:   The distribution $P(\Delta x)$ of particle displacements $\Delta x$ during a time interval $\Delta t$ is the same for all time intervals $(t, t+\Delta t)$.

(iii)

:   Velocity correlations are evanescent. For time steps appreciably longer than $m/f$, which for Brownian particles is actually a quite short time, particle displacements in a series of time steps are very nearly independent from each other.

Conclusion (ii) corresponds to the statement that $x(t)$ is the sum of a series of identically-distributed random variables. Conclusion (iii) corresponds to the independent statement that the time evolution of $x(t)$ is described by a Markoff process. In this very special case, the distribution of particle displacements is described by Doob’s Theorem[@doob1942a]. Doob’s theorem is closely related to the central limit theorem. Doob’s theorem treats random processes such as $\Delta x(t)$, while the central limit theorem treats random variables. For the Langevin model, Doob’s Theorem shows that the distribution of particle displacements is a Gaussian $$P(\Delta x) =   \left(2\pi  \langle \Delta x^{2} \rangle \right)^{-1/2} \exp(- (\Delta x(t))^{2}/ 2 \langle \Delta x^{2} \rangle ).
    \label{eq:gaussianform}$$ For this special case, the incoherent scattering function reduces to $$g^{(1s)}(q,t) = \exp(- q^{2} \langle (\Delta x(t))^{2} \rangle/2).
 \label{eq:g1swrong}$$ Equation \[eq:g1swrong\] is quite accurate for the systems considered in Ref. [@berne1976a], namely highly dilute solutions of monodisperse objects in simple Newtonian solvents.

However, Berne and Pecora[@berne1976a], especially their Appendix 5.A and Section 5.9 leading to their eq 5.9.6, also prove the other important consequence of the Langevin model and Doob’s theorem, namely that the Langevin model determines the exact value of $\langle (\Delta x(t))^{2} \rangle$. On the time and distance scales accessible to quasielastic scattering, the Langevin model requires $$\langle (\Delta x(t))^{2} \rangle = 2 D t.
     \label{eq:meansquare}$$ Here $k_{B}$ is Boltzmann’s constant and $T$ is the absolute temperature. $D = k_{B}T/f$ is the diffusion constant, a quantity *that does not depend on time*. Time independence of $D$ is required by the calculation, because $D$ results from a time integral over ($0 \leq t \leq \infty$).

Equations \[eq:g1swrong\] and \[eq:meansquare\] come as a package; they are equally consequences of the Langevin model. Correspondingly, Berne and Pecora show for diffusing monodisperse Brownian particles that the Langevin model requires that the field correlation function is a simple exponential $$g^{(1s)}(q,t) = \exp(- q^{2} D t).
      \label{eq:g1ssimple}$$ For unclear reasons – the literature error noted in the introduction – Berne and Pecora’s entirely correct Chapter 5 is being misread as proving that eq \[eq:g1swrong\] is always correct, even when the time relaxation of $g^{(1s)}(q,t)$ is not a simple exponential. Berne and Pecora in fact prove exactly the opposite, namely the contrapositive of their result is that if the relaxation is not a single exponential, then the Langevin model must not be applicable to the system, and therefore invocation of the the Langevin Model prediction eq \[eq:g1swrong\] is incorrect.

Berne and Pecora’s discussion refers purely and exclusively to the special case of particle motion that is described by the Langevin equation, in which case eqs \[eq:gaussianform\]-\[eq:g1ssimple\] are all correct. This special case corresponds to most of the experiments that were of interest at the time that ref was written, namely applications of quasielastic scattering for particle sizing[@dahneke1983]. For diffusing dilute colloids, the $t$ and $q$ dependences of $g^{(1s)}(q,t)$ are precisely as predicted by the Langevin model, in particular $g^{(1s)}(q,t) \sim  \exp(- \Gamma t)$ with $\Gamma \propto q^{2}$. The quasielastic scattering spectrum only leads to the time-dependent mean-square particle displacement if the spectrum is a pure exponential in $q^{2} t$. In this special case, the mean-square displacement increases linearly with $t$, so that the short-time and long-time behaviors of $g^{(1s)}(q,t)$ are one and the same.

If the decay of the field correlation function is not a simple exponential in $t q^{2}$, then eq \[eq:gaussianform\] and the Langevin model cannot not possibly describe how the scattering particles move. If eq \[eq:gaussianform\] described the particle motions, then the spectrum would be a simple exponential. In systems in which the spectrum is more complex than a simple exponential, eq \[eq:g1swrong\] is invalid. That is, if $g^{(1s)}(q,t)$ is not a simple exponential in $t$, $\log(g^{(1s)}(q,t))$ does not reveal the mean-square displacement of the particles.

Why does eq \[eq:g1sanddisplacements\] ever reduce to eq \[eq:g1swrong\]? If and only if $P(\Delta x, t)$ is a Gaussian in $\Delta x$, $P(\Delta x, t)$ is entirely characterized by its second moment $\langle \Delta x^{2} \rangle$. For a Gaussian displacement distribution function, the higher moments of $P(\Delta x, t)$ have values such that the coefficients of the higher-order terms ($q^{2n}$ for $n \geq 2$) of eq \[eq:g1sanddisplacements\] all vanish. For a Gaussian $P(\Delta x, t)$, the only non-zero part of eq \[eq:g1sanddisplacements\] is eq \[eq:g1swrong\]. This disappearance of the higher-order terms is unique to a Gaussian $P(\Delta x, t)$. For any other $P(\Delta x, t)$, the higher-order terms of eq \[eq:g1sanddisplacements\] do not vanish.

Experimental Findings \[sectionexperiment\]
===========================================

What do experiments say about $P(\Delta x, t)$ and $g^{(1s)}(q,t)$? There are systems in which the Langevin model is adequate, namely dilute monodisperse particles suspended in simple Newtonian fluids. The Langevin model provides the solid foundation for particle sizing via quasielastic scattering[@dahneke1983]. For probe diffusion in complex fluids, experiment provides a far more complex picture. Consider a few representative experiments that have determined $P\Delta x, t)$ or $g^{(1s)}(q,t)$ for probe particles in complex fluids.

On relatively long time scales, $P(\Delta x,t)$ is accessible via particle tracking methods. As examples, note experimental studies by Apgar, et al.[@apgar2000a], Tseng and Wirtz[@tseng2001a], and Xu, et al.[@xu2002a] on probe diffusion in glycerol, actin solutions and gels, and gliadin solutions. These authors used video recording and computer image analysis to track simultaneously the motions of large numbers of particles. They report $P(\Delta x, t)$ and $\langle \Delta x(t)^{2} \rangle$ in their systems. Probes in glycerol follow eqs \[eq:gaussianform\] and \[eq:meansquare\]. For probes in various complex fluids, $P(\Delta x, t)$ has decidedly non-Gaussian forms. In these systems, the mean-square displacement does not increase linearly in time. By direct measurement, eq \[eq:gaussianform\] and \[eq:meansquare\] are not uniformly correct for probes in polymer solutions.

Quasielastic scattering spectra of probes in polymer solutions are often markedly non-exponential. For polystyrene latex sphere probes in hydroxypropylcellulose: water, this author and Lacroix[@lacroix1997a] found stretched exponentials in time $$g^{(1s)}(q,t) =  a \exp(- \theta t^{\beta}) \equiv a \exp(- (t/\tau)^{\beta}).
     \label{eq:gisqeext}$$ Here $\beta$ is a scaling exponent while $\theta$ and $\tau$ are prefactors. A series of papers by Streletzky and collaborators on the same chemical system (most recently, ref. ) established by viewing a much wider range of delay times that $g^{(1s)}(q,t)$ is in fact a sum of two stretched exponentials in time.

Finally, I note a very simple model system in which eqs \[eq:gaussianform\] and \[eq:g1swrong\] fail. The system is a dilute aqueous dispersion of polystyrene spheres, in which the spheres are of two different sizes. There are no sphere-sphere interactions. Each sphere individually performs Brownian motion as described by the Langevin equation. Therefore, for each sphere in the mixture, $P(\Delta x, t)$ is a Gaussian in $\Delta x$ and $\langle (\Delta x)^{2} \rangle$ increases linearly in time. For the mixture as a whole, the mean-square displacement averaged over all the particles must therefore also increase linearly with time, at a rate determined by a weighted average of the diffusion coefficients of the two sphere species.

However, the mixture’s field correlation function is a sum of two exponentials $$g^{(1s)}(q,t) = A_{1} \exp(-D_{1} q^{2} t) + A_{2} \exp(-D_{2} q^{2} t)
      \label{eq:doubleexponential}$$ where $A_{i}$ is the scattering amplitude of species $i$ and $D_{i}$ is the diffusion coefficient of species $i$. Correspondingly, the displacement distribution function for all the particles in solution is a weighted sum of two Gaussians, one Gaussian for each sphere size.

Suppose one were used eq \[eq:g1swrong\] to determine $\langle (\Delta x)^{2} \rangle$ of the sphere mixture. According to eq \[eq:g1swrong\] $$\langle (\Delta x)^{2} \rangle = - \ln( g^{(1s)}(q,t))/q^{2}.
      \label{eq:meansquare2}$$ At short times, $g^{(1s)}(q,t)$ contains contributions from both species, and the $\langle (\Delta x)^{2} \rangle$ computed from eq \[eq:meansquare2\] is the weighted average of the mean-square displacements of the two species. At long times, in eq \[eq:doubleexponential\] the exponential corresponding to the more rapidly-moving species has decayed to zero, so the nominal mean-square displacement from eq \[eq:meansquare2\] is determined by scattering from the larger, more slowly-moving spheres. For this simple bidisperse sphere suspension, eq \[eq:meansquare2\] asserts that the slope of $\langle (\Delta x)^{2}\rangle$ depends on time, $\langle (\Delta x)^{2}\rangle$ increasing more rapidly at short times and more slowly at long times. The assertion is incorrect. In this system $\langle (\Delta x)^{2}\rangle$ increases linearly with $t$. At large times the smaller spheres continue to move, but they stop contributing to $g^{(1s)}(q,t)$ or to the nominal $\langle (\Delta x)^{2}\rangle$ from eq \[eq:meansquare2\].

Experiment thus demonstrates that neither eq \[eq:gaussianform\] nor eq \[eq:g1ssimple\] is generally valid for probe diffusion in complex fluids. Even in a Newtonian fluid, a model system in which $g^{(1s)}(q,t)$ does not decay exponentially in time does not follow eqs \[eq:g1swrong\] and \[eq:meansquare2\], a result that is exactly as required by Doob’s theorem. Interpretations of quasielastic scattering spectra for probes in complex fluids, based on the Gaussian approximation of eq \[eq:g1swrong\], are therefore incorrect. Interpretations of spectra of probes in complex fluids in terms of particle displacements are properly based on eq \[eq:g1sanddisplacements\], which correctly reflects the non-Gaussian displacement distribution function of real probes.

Interpretations of Quasielastic Scattering Spectra \[sectioninterpretations\]
=============================================================================

First, every single physical $g^{(1s)}$ viewed only as a function of $t$ corresponds to a system in which the mean-square particle displacement increases linearly with time. However, the correspondence is not unique. The same $g^{(1s)}(q,t)$ may also correspond to systems in which particle thermal motions are more complex. In consequence, from a $g^{(1s)}(q,t)$ measured over a full range of times and a single $q$ one cannot infer how the particle displacement depends on time.

This result has a purely mathematical basis, namely that the field correlation function can always be represented via a Laplace transform as $$g^{(1s)}(q,t) = \int_{0}^{\infty} A(\Gamma) \exp(- \Gamma t).
    \label{eq:laplace}$$ Here $\Gamma$ is a relaxation rate and $A(\Gamma)$ is the contribution of relaxations having decay rate $\Gamma$ to $g^{(1s)}(q,t)$. So long as the system does not have relaxational modes that have negative amplitudes, $A(\Gamma)$ is everywhere positive or zero. In this case, there is always a system having the same $g^{(1s)}(q,t)$ as the system of interest, and in which $ \langle (\Delta x(t) )^{2} \rangle$ increases linearly in time. The system can be physically constructed as a solution of of polystyrene spheres of all different sizes. The composition of the mixture is determined by $A(\Gamma)$: One adds to the mixture just enough polystyrene spheres having diffusion coefficient $\Gamma/q^{2}$ so that their contribution to the scattering spectrum is $A(\Gamma)$. For each sphere, $\langle (\Delta x(t) )^{2} \rangle$ increases linearly in time, so therefore $\langle (\Delta x(t) )^{2} \rangle$ of the mixture also increases linearly in time. Thus, an arbitrary physically-acceptable (i.e., $A(\Gamma) > 0 \ \forall \ \Gamma$) form for $A(\Gamma)$ corresponds as one non-unique possibility to a system in which $\langle (\Delta x(t) )^{2} \rangle$ increases linearly in time.

It has repeatedly been found that $g^{(1s)}(q,t)$ decays in time as the stretched exponential of eq \[eq:gisqeext\]. If one interpreted this time dependence by applying eq \[eq:g1swrong\], one would conclude $$\langle (\Delta x(t) )^{2} \rangle = \theta t^{\beta}.
         \label{eq:subdiffusive}$$ In the common case $\beta < 1$, from eq \[eq:subdiffusive\] one would infer that the mean-square particle displacement increases less rapidly at large times than would be the case for simple diffusion, a behavior that has been termed ’subdiffusive’ motion. The inference is incorrect. A more reasonable interpretation for $\beta <1$ is that diffusion in the complex fluid is created by modes having a range of relaxation times, some longer than others, the contribution of the slower modes to the spectrum becoming more important at longer times.

It is not suggested that there cannot exist subdiffusive motion. Such motion has unambiguously been observed experimentally. Amblard, et al.,[@amblard1996a] studied probe motion in f-actin solutions, using magnetic tweezers and video microscopy. Small-bead motion was diffusive; larger-bead diffusion was subdiffusive with $\beta \approx 3/4$. The non-classical drag forces for diffusive motion and for driven motion are the same, in the sense that the displacement under an applied force and the mean-square displacement for diffusion have the same linear or sublinear time dependences, depending on the probe size.

There are sometimes suggestions that probe behavior should approach Stokes’ Law and the Stokes Einstein-equation as the size of the probes is increased. Amblard, et al.’s experiments show precisely the opposite behavior. Deviations from simple diffusion are larger for the larger particles. Whenever particle motion is subdiffusive, the light scattering spectrum will not be a simple exponential. The scattering spectrum will follow eq \[eq:g1sanddisplacements\], showing that the relationship between $g^{(1s)}(q,t)$ and $\langle (\Delta x(t) )^{2} \rangle$ is a neither-way street. Just as one cannot in general calculate $\langle (\Delta x(t) )^{2} \rangle$ from $g^{(1s)}(q,t)$, so also one cannot in general calculate $g^{(1s)}(q,t)$ from $\langle (\Delta x(t) )^{2} \rangle$, because all higher moments of $P(\Delta x)$ are needed in order to calculate $g^{(1s)}(q,t)$.

Discussion\[sectiondiscussion\]
===============================

The primary objective of this paper was to correct a widespread literature error, namely the false belief that that $g^{(1s)}(q,t)$ can in general be used to determine the mean-square displacement of probes in complex fluids. The belief appears to have arisen from a misreading of Berne and Pecora’s excellent monograph, Chapter 5, in which Berne and Pecora discuss the motions of monodisperse probe particles in simple Newtonian fluids by using the Langevin model. The spectra of monodisperse Langevin-model particles are without exception single exponentials. The calculation is correct, but does not refer to non-Newtonian fluids or polydisperse probe systems. Having corrected this error, I turn to several subsidiary points of misinterpretation.

The functional form of $P(\Delta x, t)$ can be inferred, at least approximately, from the angular dependence of $g^{(1s)}(q,t)$, namely as seen in eq \[eq:g1sPDelta\] the correlation function $g^{(1s)}(q,t)$ at fixed $t$ is the spatial Fourier transform of $P(\Delta x, t)$. If $g^{(1s)}(q,t)$ is determined sufficiently accurately over an adequate range of $q,$ an inverse spatial Fourier transform can take the experimenter back from $g^{(1s)}(q,t)$ to $P(\Delta x, t)$. To the author’s knowledge, this inversion has only been done for the ubiquitous polystyrene spheres in distilled water, for which $g^{(1s)}$ is a simple exponential $\exp(- \Gamma t)$, while $\Gamma$ is found to be accurately linear in $q^{2}$, showing that $P(\Delta x, t)$ in this system has the Gaussian form of eq \[eq:gaussianform\]. Measurements of the $q$-dependence of $g^{(1s)}$ for probes in complex fluids are less common, though note, e. g., Streletzky and Phillies[@streletzky1998q]. These authors found spectra having multiple relaxational modes, some of which had relaxation rates that did not scale linearly in $q^{2}$, proving that the modes did not correspond to Gaussian displacement distribution functions.

Particle tracking is sometimes used to generate the simplified $\langle \Delta x(t)^{2} \rangle$, rather than the full $P(\Delta x, t)$. The simplification is potentially hazardous, because if one has not determined $P(\Delta x, t)$ one does not know if the particle motion process corresponds to simple diffusion. Any physically reasonable $P(\Delta x, t)$ has some second moment $\langle \Delta x(t)^{2} \rangle$, but if the form of $P(\Delta x, t)$ is unknown, one cannot tell if it is meaningful to characterize $P(\Delta x, t)$ by its second moment or by the corresponding nominal diffusion coefficient $$D(t) = \langle \Delta x(t)^{2} \rangle/ 2 t
      \label{eq:timedependentD}$$ The notion that a diffusive process can be characterized by $\langle \Delta x(t)^{2} \rangle$ corresponds to particle diffusion in simple Newtonian liquids, in which $D(t)$ as defined here is a constant independent of time. In a complex fluid, characterization of probe motion via measurement of the second moment $\langle \Delta x(t)^{2} \rangle$ must be expected to be inadequate.

The assertion that the central limit theorem guarantees that $P(\Delta x, t)$ is a Gaussian in $\Delta x$ is sometimes describes as the “Gaussian Approximation”. Experiments such as those summarized above prove that this assertion is incorrect. The central limit theorem (for random variables) and Doob’s theorem (for random processes) are well-known. Where do their invocations go wrong? The central limit theorem and Doob’s theorem are statements about the sum of a large number of identically distributed, independent random processes. As applied to the diffusion problem, the displacement $\Delta x$ during experimentally accessible times can be expressed as the sum of a large number of far smaller steps $\delta x$, each taken during a far smaller time interval $\delta t$. If a random variable $\Delta x$ is the sum of a large number of identically distributed *independent* variables, i. e., if $\delta x(t)$ is a Markoff process, it must in general be the case that $P(\Delta x, t)$ is a Gaussian in $\Delta x$. This rationale fails because it refers to a sum of *independent* random processes. The process that generates $\delta x$ for probes in a viscoelastic fluid is not a Markoff process, because the system has memory. The central limit theorem and Doob’s theorem are therefore not applicable. For probes in complex fluids, the processes generating the steps $\delta x$ are highly correlated, because the “random” forces that determine the $\delta x(t)$ are controlled by the shear modulus $G(t)$, which in a complex fluid has a long correlation time. Correspondingly, the time correlation function of the random force $\langle {\cal F}_{x}(0) {\cal F}_{x}(t) \rangle$ is long-lived, not $\sim \delta(t)$, and in the Langevin equation the friction force $f \dot{x}(t)$ is replaced with a memory function $\sim \int ds \langle {\cal F}_{x}(0) {\cal F}_{x}(s) \rangle \dot{x}(t-s)$.

An alternative to the central limit theorem is the *small-$q$ approximation*. The nominal idea in the small-q approximation is that the rhs of eq \[eq:g1sanddisplacements\] is a power series in $q$. If one went to sufficiently small $q$, one might hope that the $q^{2}$ term in the exponential would become dominant, so that eq \[eq:g1swrong\] would approach being valid. This hope is not met. For the simplest case of a mixture of diffusing particles, $g^{(1s)}(q,t)$ is in fact a power series in $q^{2} t$. If one goes to smaller $q$, in order to keep fitting spectra equally accurately one needs to observe the same fractional degree of decay of $g^{(1s)}(q,t)$; one must therefore go out to longer times. At those longer times, the ($q^{4}$) terms are as significant as they were at larger $q$, smaller $t$, and the same $q^{2} t$. Said differently, the coefficients of the correct Taylor series (in $q$) expansion of $g^{(1s)}(q,t)$ are time-dependent. In order for the lead term of the expansion to be dominant, the expansion must be limited not only to small $q$ but also to to small $t$. If $t$ is large, no matter how small $q$ has been made, the higher-order in $q$ terms are as important as the lower-order terms. Only at small $q$ and small $t$ is a single-term small-$q$ expansion valid. The valid small-$q$ expansion is $1-q^{2} \langle \Delta x(t)^{2} \rangle$, which only describes the leading slope of $g^{(1s)}(q,t)$ at small times.

Consider spectra described by eq \[eq:gisqeext\]. The exponential in eq \[eq:g1sanddisplacements\] scales as $q^{2}$ or is a power series in $q^{q}$, so therefore $\theta$ should also be a power series in $q^{2}$, perhaps simply by being linear in $q^{2}$. Indeed, for probes in in some but not other water: hydroxypropylcellulose solutions, Streletzky[@streletzky1998q] confirmed experimentally $\theta \sim q^{2}$ over a wide range of $q$. If $\theta$ were replaced with $\tau^{-\beta}$, one would have $$\tau \sim q^{-2/\beta}.
        \label{eq:tauq}$$ $\beta$ is often in the range 0.5-1.0, so $\tau$ often depends on $q$ as $q^{-3\pm 1}$. If one interpreted $\tau$ to be a relaxation time, the $q$-dependence from eq \[eq:tauq\] would be strange indeed: The relaxation would occur more rapidly over large distances (small $q$) than over short distances (large $q$). This strange $q$-dependence is simply an artifact of the choice of parameterization of $g^{(1s)}(q,t)$, and the identification of $\tau$ as a relaxation time. In terms of eq \[eq:gisqeext\], $(\theta, \beta)$ provides a natural parameterization while $(\tau,\beta)$ is less transparent. If mean relaxation times are inferred from the spectral time moments $$\langle T_{n} \rangle = \int_{0}^{\infty} t^{n} g^{(1s)}(q,t) dt
      \label{eq:Tmoments}$$ of $g^{(1s)}(q,t)$, the choice of parameterizations in eq \[eq:gisqeext\] has no consequences. The two paramaterizations of $g^{(1s)}(q,t)$ must lead to the same $\langle T_{n} \rangle$.

Spectra of diffusing probes showing two relaxations on very different time scales are sometimes interpreted in terms of caging and hopping relaxations. The notion is that the medium supplies regions of low potential energy within which probes are free to move (“caging”). The regions are separated by barriers of high potential energy, across which probes only pass on rare occasion (“hopping”). The short time-scale relaxation is said to correspond to caging, while the long time-scale relaxation is said to correspond to hopping.

Computer simulation studies by Luo and Phillies and Luo, et al., test the caging-hopping interpretation[@luo1995a; @luo1996a]. These simulations represented Brownian particles moving through a square lattice or a random glass of Lennard-Jones force centers. The force centers were immobile. Probe motions were generated via the Metropolis algorithm. These studies differed from some earlier work in that they determined not only time dependent mean-square displacements and effective diffusion coefficients but also obtained $P(\Delta r,t)$ and $g^{(1s)}(q,t)$. By varying the nominal temperature, trapping, hopping, and hindered diffusion behaviors were obtained. At low temperatures, probe particles explored the volume of their traps; after a certain relaxation time $\langle r^{2}(t) \rangle$ ceased to increase. At high temperatures, $P(\Delta r, t)$ was nearly Gaussian, with $\langle r^{2}(t) \rangle$ increasing linearly in time even at short times.

Luo, et al., evaluated $g^{(1s)}(q,t)$ for $q^{-1}$ extending from a small fraction of the size of a single potential energy minimum out to distances substantially larger than a typical distance between force centers. At low and high temperatures, $g^{(1s)}(q,t)$ showed nearly exponential relaxations, though at small $T$ and small $q$ the relaxation fell to a non-zero baseline. The baseline was non-zero because the particles were permanently trapped in small isolated volumes of the system. At intermediate temperatures, relaxations were single-exponential at large $q$ but double-exponential at small $q$. At the same intermediate temperatures, $P(\Delta r, t)$ was radically non-Gaussian, with local maxima and minima created by local potential energy minima, potential energy saddle points, and times required to traverse local energy maxima.

Other, physically different, systems also give bimodal spectra. In contrast to Luo, et al.’s probes moving though a fixed matrix, in which relaxations are only bidisperse for some values of $q$, relaxations of dilute bidisperse suspensions are double-exponential at all $q$. An alternative model system in which monodisperse particles show several very different classes of relaxation behavior is shown by Glotzer, et al.’s[@glotzer2000a] computer simulations of three-dimensional glasses, in which one finds distinct long-lived populations of slow and fast-moving particles, with the immobile particles in clumps and the rapidly moving particles lying in thin ribbons.

Thus, in order to distinguish between systems containing species with two different dynamic behaviors, and systems in which there is local trapping with escapes from the traps at longer times, it is experimentally necessary to study $g^{(1s)}(q,t)$ over a wide range of $q$. Observations at fixed $q$ of double-exponential relaxations do not reveal whether one is seeing trapping with hopping, or whether the system is in some sense dynamically bidisperse. Furthermore, in the cases in which $g^{(1s)}(q,t)$ was observed by Luo, et al., to be very nearly the sum of two exponentials, $P(\Delta r,t)$ on interesting distance scales had an elaborate dependence on $r$ with multiple maxima and deep minima. The interpretation that a biexponential $g^{(1s)}(q,t)$ must correspond to a $P(\Delta r,t)$ that is a sum of two Gaussians, each with a mean-square width increasing linearly in time, is categorically disproved by Luo, et al.’s measured forms for $P(\Delta r,t)$.

Finally, the observation that quasielastic scattering does not determine the mean-square probe displacement certainly does not mean that probe diffusion is ineffective as an experimental technique. Probe diffusion measurements can certainly be used to obtain novel information about complex fluids. The richness of the revealed information corresponds to the depth with which models for probe motion are constructed. As a positive conclusion, two successful applications of probe diffusion are noted:

\(i) A long-time question in the study of surfactant solutions is the determination of the aggregation number $n$ of surfactant molecules in micelles. One of many approaches to this question has been to use quasielastic scattering to determine an effective hydrodynamic radius of the micelles. Perhaps after some hydrodynamic modeling to account for micelle shape, spherical micelles being the simplest case, the measured diffusion coefficient can be transformed to an apparent hydrodynamic radius $r_{H}$, to a hydrodynamic volume $V_{h}$, and (taking into account the surfactant density and molecular weight) finally to a nominal aggregation number. This procedure was criticized by Kratohvil[@kratohvil1980a], who noted that the hydrodynamic volume of the micelle might well include solvent molecules rather than being composed of pure surfactant. Probe diffusion experiments prove that Kratovil was correct. The diffusion of probe particles through micellar solutions is retarded by hydrodynamic and direct interactions between the micelles and the probe particles. The degree of retardation is determined by the volume fraction of micelles in the solution. By combining quasielastic scattering measurements on surfactant solutions and on surfactant-probe mixtures, quasielastic scattering has been used to determine the size, volume fraction, and thus number density of micelles in solution, leading to determinations of the micellar aggregation number and, independently, the (substantial) degree of hydration of micelles, as seen in studies by Phillies and collaborators[@phillies1993a].

\(ii) Diffusion of mesoscopic probe particles in polymer solutions is not Stokes-Einsteinian. $D$ is not determined by the macroscopic viscosity $\eta$. Therefore, one cannot use the Stokes-Einstein equation for sphere diffusion $$D = \frac{k_{B}T}{6 \pi \eta R}
     \label{eq:SEeq}$$ (where $k_{B}$ is Boltzmann’s constant, $T$ is the absolute temperature, $\eta$ is the solution viscosity, and $R$ is the sphere radius) to determine the size of probe particles in polymer solutions. However, by using probes of known size, Ullmann and Phillies[@ullmann1983a] were able to quantitate the degree of failure of the Stokes-Einstein equation for their polymer solutions, allowing them to measure the size of unknown probe particles in the same solutions. This approach permitted a quantitative study of the extent of polymer adsorption onto particles chosen for their ability to bind polymers in solution.

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