Non-invertibility of Multiple-Scattered QELSS Spectra

We consider the spectrum S(q,t) and field correlation function f(q,t) of light quasielastically scattered from diffusing optical probes in complex viscoelastic fluids. Relationships between the single-scattering f_1(q,t) and the multiple-scattering f_m(t) are examined. We show that it is fundamentally impossibly to invert f_m(t) to obtain f_1(q,t) or particle displacement moments , except with assumptions that are certainly not correct in complex, viscoelastic fluids. For diffusing dilute probes in viscoelastic fluids, f_1(q,t) is determined by all even moments , n > 0$, of the particle displacement X; this information is lost in f_m(t). In the special case of monodisperse probes in a true simple fluid, f_1(q,t) can be obtained from f_m(t), but only because the functional form of f_1(q,t) is already known.Comment: 7 pages and 1 figur

The Kirkwood-Riseman Polymer Model of Polymer Dynamics is Qualitatively Correct

We use Brownian dynamics to show: For an isolated polymer coil, the Kirkwood-Riseman model for chain motion is qualitatively correct. The Rouse model for chain motion is qualitatively incorrect. The models are qualitatively different. Kirkwood and Riseman say polymer coils in a shear field perform whole-body rotation; in the Rouse model rotation does not occur. Our simulations demonstrate that in shear flow: Polymer coils rotate. Rouse modes are cross-correlated. The amplitudes and relaxation rates of Rouse modes depend on the shear rate. Rouse's calculation only refers to a polymer coil in a quiescent fluid, its application to a polymer coil undergoing shear being invalid.Comment: 23 pages, 10 figure

Falsity of the Rouse Mode Solution of the Rouse and Zimm Models

The Rouse (J. Chem. Phys. 21, 1272 (1953)) and Zimm (J. Chem. Phys. 24}, 269 (1956)) treatments of the dynamics of a polymer chain are shown to contain a fundamental mathematical error, which causes them to lose the zero-relaxation-rate, whole-body rotation motions dominant in viscosity and dielectric relaxation. As a result, the oft-cited mode solutions for these models are qualitatively incorrect. Comparison with the Wilson-Decius-Cross treatment of vibrational modes of polyatomic molecules reveals qualitatively the correct form for the solutions to the Rouse model.Comment: One text fil

Interpretation of Diffusing Wave Spectra in Nontrivial Systems

Mathematical methods previously used (Phillies, J. Chem. Phys., 122 224905 (2005)) to interpret quasielastic light scattering spectroscopy (QELSS) spectra are here applied to relate diffusing wave spectroscopy (DWS) spectra to the moments \bar{X^{2n}} of particle displacements in the solution under study. DWS spectra of optical probes are like QELSS spectra in that in general they are not determined solely by the second moment \bar{X^{2}}. In each case, the relationship between the spectrum and the particle motions arises from the field correlation function g^{(1)}_{s}(t) for a single quasi-elastic scattering event. In most physically interesting cases, g^{(1)}_{s}(t) receives except at the shortest times large contributions from higher moments \bar{X(t)^{2n}}, n >1. As has long been known, the idealized form g^{(1)}_{s}(t) =\exp(-2 q^{2} \bar{X(t)^{2}}), sometimes invoked to interpret DWS and QELSS spectra, only refers to (adequately) monodisperse, noninteracting, probes in purely Newtonian liquids and is erroneous for polydisperse particles, interacting particles, or particles in viscoelastic complex fluids. Furthermore, in DWS experiments fluctuations (for multiple scattering paths of fixed length) in the number of scattering events and the total-square scattering vector significantly modify the spectrum.Comment: 17 pages revision 31 pages incorporating responses to referee

Interpretation of Light Scattering Spectra in Terms of Particle Displacements

Quasi-elastic light scattering spectroscopy of dilute solutions of diffusing mesoscopic probe particles is regularly used to examine the dynamics of the fluid through which the probe particles are moving. For probes in a simple liquid, the light scattering spectrum is a simple exponential; the field correlation function $g^{(1)}_{P}(q,\tau)$ of the scattering particles is related to their mean-square displacements $\bar{X^{2}} \equiv < (\Delta x(\tau))^{2}>$ during $\tau$ via $g^{(1)}(q,\tau) = \exp(- {1/2} q^{2} \bar{X^{2}})$. However, historical demonstrations of this expression refer only to ideal Brownian particles in simple liquids, and show that if the form is correct then it is also true that $g^{(1)}(q,\tau) = \exp(- \Gamma \tau)$, a pure exponential in $\tau$. In general, $g^{(1)}_{P}(q,\tau)$ is not a single exponential in time. $g^{(1)}_{P}(q,\tau)$ reflects not only the mean-square particle displacements but also all higher-order mean displacement moments $\bar{X^{2n}}$. A correct general form for $g^{(1)}(q,\tau)$, replacing the generally-incorrect $\exp(- {1/2} q^{2} \bar{X^{2}})$, is obtained. A simple experimental diagnostic determining when the field correlation function gives the mean-square displacements is identified, namely $g^{(1)}(q,\tau)$ reveals $\bar{X^{2}}$ if $g^{(1)}(q,\tau)$ is exponential in $\tau$.Comment: Now 19 pages, no figures, accepted by Journal of Chemical Physics Revision: Add New Section V Spectrum of a Bidisperse System giving a concrete example of issues raised, and did minor rewrite

The Hydrodynamic Scaling Model for the Dynamics of Non-Dilute Polymer Solutions: A Comprehensive Review

This article presents a comprehensive review of the Hydrodynamic Scaling Model for the dynamics of polymers in dilute and nondilute solutions. The Hydrodynamic Scaling Model differs from some other treatments of non-dilute polymer solutions in that it takes polymer dynamics up to high concentrations to be dominated by solvent-mediated hydrodynamic interactions, with chain crossing constraints presumed to create at most secondary corrections. Many other models take the contrary stand, namely that chain crossing constraints dominate the dynamics of nondilute polymer solutions, while hydrodynamic interactions only create secondary corrections. This article begins with a historical review. We then consider single-chain behavior, in particular the Kirkwood-Riseman model; contradictions between the Kirkwood-Riseman and more familiar Rouse-Zimm models are emphasized. An extended Kirkwood-Riseman model that gives interchain hydrodynamic interactions is developed and applied to generate pseudovirial series for the self-diffusion coefficient and the low-shear viscosity. To extrapolate to large concentrations, rationales based on self-similarity and on the Altenberger-Dahler Positive-Function Renormalization Group are developed and applied to the pseudovirial series for $D_{s}$ and $\eta$. Based on the renormalization group method, a two-parameter temporal scaling ansatz is invoked as a path to determining the frequency dependences of the storage and loss moduli. A short description is given for each of the individual papers that developed the Hydrodynamic Scaling Model. Phenomenological evidence supporting aspects of the model is noted. Finally, directions for future development of the Hydrodynamic Scaling Model are presented.Comment: 58 pages, 167 equations, 116 reference

Capillary Electrophoresis as a Fundamental Probe of Polymer Dynamics

Capillary electrophoresis has long been been recognized as a powerful analytic tool. Here it is demonstrated that the same capillary electrophoretic experiments also reveal dynamic properties of the polymer solutions being used as the support medium. The dependence of the electrophoretic mobility on the size of the probe and the properties of the matrix polymers shows a unity of behavior between electrophoresis and other methods of studying polymer properties.Comment: 14 pages, 7 Figure

The VW transformation -- A Simple Alternative to the Wilson GF Method

An alternative, the VW transformation, is proposed to replace the Wilson GF method for calculating molecular vibration frequencies and normal modes. The VW transformation yields precisely the same eigenmodes and and eigenfrequencies that are found with the GF method. The transformation proceeds entirely in the mass-normalized Cartesian coordinates of the individual atoms, with no transformations to internal coordinates. The VW transformation thus offers an enormous computational simplification over the GF method, namely the mathematical apparatus needed to transform to internal coordinates is eliminated. All need for new researchers to understand the complex matrix transformations underlying the GF method is thus also removed. The VW transformation is not a projection method; the internal vibrations remain dispersed over all 3N atomic coordinates. In the VW method, the 3N x 3N force constant matrix V is replaced with a new 3N x 3N matrix W. The replacement is physically transparent. The matrix W has the same internal normal modes and vibration frequencies that V does. However, unlike V, W is not singular, so the normal modes and normal mode frequencies can be obtained using conventional matrix techniques.Comment: 3 pages, no figure

Readings and Misreadings of J. Willard Gibbs Elementary Principles in Statistical Mechanics

J. Willard Gibbs' Elementary Principles in Statistical Mechanics was the definitive work of one of America's greatest physicists. Gibbs' book on statistical mechanics establishes the basic principles and fundamental results that have flowered into the modern field of statistical mechanics. However, at a number of points, Gibbs' teachings on statistical mechanics diverge from positions on the canonical ensemble found in more recent works, at points where seemingly there should be agreement. The objective of this paper is to note some of these points, so that Gibbs' actual positions are not misrepresented to future generations of students.Comment: 11 pages, no figures. This is a major revision of the prior version. For the prior version, I received from a particular journal a truly magnificent four-page referee report correctly raising all sorts of issues. I by and by did the requested revisions, as seen her

FCS and RICS Spectra of Probes in Complex Fluids

The Fluorescence Correlation Spectroscopy (FCS) spectrum G(t) and Raster Image Correlation Spectroscopy (RICS) spectrum R(t) of dilute diffusing particles are determined by the displacement distribution function P(x,t) of the particles and by the experimental parameters of the associated optical trains. This letter obtains the general relationships between P(x,t) and these spectra. For dilute diffusing molecules in simple liquids, P(x,t) is a Gaussian in the displacement x; the corresponding G(t) is a Lorentzian in ()^(1/2). In complex fluids such as polymer solutions, colloid and protein solutions, and the interior of living cells, P(x,t) may have a non-Gaussian dependence on x, for example an exponential in |x|. We compare theoretical forms for FCS and for RICS spectra of two systems in which P(x,t) is a Gaussian or an exponential in x, but in which the mean-square displacements are precisely equal at all times. If the G(t) and R(t) arising from an exponential P(x,t) are interpreted by using the forms for G(t) and R(t) that are appropriate for a Gaussian P(x,t), the inferred diffusion coefficient may be substantially in error.Comment: 13 pages, two tables, no figure