Self-motion response and incoherent scattering function in classical liquids

Abstract

The self-motion response function and incoherent scattering function S_s(k, ω) for simple classical liquids is studied using an exact representation presented in a previous paper. The latter can be termed a generalized mean field representation to distinguish it from the generalized hydrodynamic representation introduced elsewhere. It is shown that the present formalism offers a natural and convenient way of relating the experimentally determined S_s(k, ω) to some basic quantities involving only the interaction. Using a small part of the recent experimental data on incoherent neutron scattering in liquid argon, we are able to calculate S_s(k, ω) and other quantities of interest and to compare with the rest of the data     

A master equation for S(q,ω) leading to the hydrodynamic regime for simple fluids

A master equation relating the coherent and incoherent dynamical structure factors S(q, ω) and S_S(q, ω) is introduced on the basis of physical arguments suggested by previous theories. It is shown that, once S_S(q, ω) and the first six moments of S(q, ω) are known, the resulting expression for S(q, ω) contains the correct hydrodynamic regime as well as the free particle behaviour. The relationships thus introduced between the transport coefficients and the molecular quantities are compared to the experimental data for liquid argon. Molecular dynamics data are also well reproduced.     

Evidence for hydrophobic attraction between latex particles in aqueous systems as investigated by turbidimetry

We report the evidence for attractive interaction of latex particles which are covered by poly(ethylene oxide) chains. These particles are suspended in aqueous solutions of ammonium sulfate. The interaction is probed by measurements of the turbidity of the suspensions up to 70 g/l. Turbidity is insensitive to multiple scattering and allows the static structure factor, S(q) [q=(4πn ₀/λ₀)sin(θ/2), where θ is the scattering angle, n₀ is the refractive index of the medium and λ₀ is the wavelength in vacuo], to be determined at small q values. The analysis of S(q) at small q values yields information about possible attraction of the particles. The analysis of the turbidity data furthermore shows that no aggregation took place in these systems. A weak but long-range attractive interaction was found at ammonium sulfate concentrations of 0.01 and 0.1 M. The relation of this attractive force to hydrophobic forces is discussed. Received: 9 March 2000/Accepted: 28 June 2000     

Quasielastic scattering of neutrons from freely jointed polymer chains in dilute solutions

Starting with Kirkwood's Fokker–Planck equation for the polymer configuration-space distribution function and using the Zwanzig–Mori projection operator technique we have calculated the scattering law S_(q,w) for a freely jointed model polymer chain in a dilute solution. When memory effects are neglected, the theory predicts a Lorentzian for S_(q,w) with a halfwidth Ω_(q), which we have determined as a function of the momentum transfer q for all values of q. The results are compared with recent neutron scattering experiments on deuterated polytetrahydrofuran and polystyrene in dilute solution in CS₂. It is found that the observed q dependence of Ω(q) is represented satisfactorily by the present theory with a bond length b of about 6.3 Å for polystyrene and 3.8 Å for polytetrahydrofuran, and a friction coefficient ζ = 4πη0b where η₀ is the viscosity of the solvent.     

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Characteristic lengths and the structure of salt free polyelectrolyte solutions. A small angle neutron scattering study

We have studied salt free semi dilute polyelectrolyte solutions by small angle neutron scattering. Specific labelling associated with an extrapolation method has allowed the separation of the form factor of a single polyelectrolyte chainS ₁(q) and the structure factorS ₂(q). Two lengths are deduced from these two factors: the persistence lengthb _t which characterizes the electrostatic interactions along the chain by a fitting ofS ₁(q) with calculation of the scattering function for a wormlike chain, and fromS ₂(q),q _m ^–1 which characterizes the interactions between chains. These two lengths vary in the same way with the concentration of polyions (b _t C _p ^–1/2 ,q _m ^–1 C _p ^–1/2 ) and a constant relation exists between them: only one length is then necessary to describe the structure of polyelectrolyte soltuion on this semidilute concentration range.Laboratoire Commun CEA-CNRS.     

Frequency spectra of disordered metals: especially liquid Rb

The phonon frequency spectrum g(ω) of a crystal, such as body centred cubic (bcc) Rb, is known to be characterized by the Van Hove singularities at ω?≠?0. However, for a liquid metal like Rb, g(ω) has a single, hydrodynamic-like singularity, namely a cusp ∝ ω ^(1/2), at ω?=?0. Here, we note first that computer simulation on liquid Rb near freezing has revealed a rather well-defined Debye frequency ω_D. Therefore, we propose here a zeroth-order model g ₀ (ω ) of g(ω) for Rb, which combines the Debye model with the ‘hydrodynamic’ ω ^(1/2) cusp. The corresponding velocity autocorrelation function 〈 v (t)·v (0)〉 has correctly a long-time tail ∝ t ^-(3/2). The terms from g ₀ (ω ) involving ω_D are then damped by weak exponential factors exp (-α _i t), and the resulting first-order approximation, g ₁ (ω ) say, to the frequency spectrum is found to have features in common with the molecular dynamics (MD) simulation form. Thus ω_D is fixed, as well as transport coefficients for the known thermodynamic state. The article concludes with a more qualitative discussion on supercooled liquids, and on metallic glasses such as Fe, for which MD simulations exist.     

Some properties of the structure factor S(q) in two-dimensional classical liquids near freezing

We explore general properties of the main peak of the structure factor S(q) near the melting temperature T _melt in liquids confined in two dimensions, especially for the one component plasma model and for monatomic liquids interacting through inverse twelfth-power potentials. Those properties are the height of the peak, S(q _m), where q _m is the position of maximum in the peak, and the ratio between S(q _m) and q _m/Δq, where 2Δq is the width of the peak. The results obtained are then compared with those for similar systems in three dimensions. Other magnitude that we use to compare two-dimensional and three-dimensional simple liquids is r _m/Δr, where r _m is the position of the main peak in the pair distribution function g(r) and 2Δr is the width of that peak.     

Length scale hierarchy in sol,gel, and coacervate phases of gelatin

Length scale hierarchy in gelatin sol, gel, and coacervate (induced by ethanol) phases, having same concentration of gelatin in aqueous medium (13% w/v), has been investigated through small angle neutron scattering and rheology measurements. The static structure factor profile, I(q) versus wave vector q, was found to be remarkably similar for all these samples. This data could be split into three distinct q‐regimes: the low‐q regime, I_ex(q) = I_ex(0)/(1+q²ζ²)² valid for q < 3R_g^?1; the intermediate q‐regime, I(q) = I(0)/(1+q²ξ²) for 3R_g^?1 < q < ξ^?1; and the asymptotic regime, I(q) = (c/q) exp(?R_c²q²/2) for q > ξ^?1. Consequently, three distinct length scales could be deduced from structure factor data: (a) inhomogeneity of size, ζ = 20 ± 1 nm for all the three phases; (b) average mesh size, ξ₀ = 2.6 ± 0.2 nm for sol and gel, and smaller mesh size, ξ_os = 1.2 ± 0.2 nm for coacervate; and (c) cross section of gelatin chains, R_c = 0.35 ± 0.04 nm. In addition, the structure factor data obtained from coacervating solution analyzed in the Guinier region, I(q) = exp(?q²R_g²/3), yielded value of typical radius of gyration of clusters, R_g ≈ 69 nm that indicated existence of triple‐helices of length, L ≈ 239 nm; (d) Frequency and temperature sweep measurements conducted on coacervate samples revealed two other length scales: (e) viscoelastic length, ξ_ve = 14 ± 2 nm and (f) correlation length at melting, ξ_T = 500 ± 70 nm. Thus, existence of six distinct length scales, (a–f), ranging from 1.2 to 500 nm has been established in the coacervate phase of gelatin–ethanol–water system. Results are discussed within the framework of Landau‐Ginzburg treatment of dynamically asymmetric systems (Prog Theor Phys 1977, 57, 826; Phys Rev A 1991, 44, R817; J Phys II (France) 1992, 2, 1631). © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1653–1667, 2006     

Solute alignment in liquid crystal solvents The Saupe ordering matrix for anthracene dissolved in uniaxial liquid crystals

Abstract

We have described a theory for U, the potential of mean torque of rigid solutes at infinite dilution in a uniaxial liquid crystal phase; this may be used to calculate (S _xx - S _yy) and S _zz, the principal elements of the Saupe ordering matrix. In its simplest form U(ω) contains only second-rank terms and the dependence of the biaxiality (S _xx - S _yy) is determined by ω, a parameter which describes the departure of the potential of mean torque from cylindrical symmetry, and is predicted to be temperature independent. If dispersion forces are responsible for the magnitude of the orientational order parameter then ω should be independent of the solvent and depend only on the anisotropy in the electric polarizability of the solute. Indeed, this independence should result for any pair potential which can be factorized into a product of solute and solvent properties. These predictions are tested here by determining values of S _zz and (S _xx - S _yy) for anthracene-d ₁₀ as a solute in several liquid crystal solvents, from the quadrupolar splittings obtained from the deuteron N.M.R. spectra. It is found that ω has a strong dependence on the nature of the solvent, which demonstrates that the solute ordering cannot be determined primarily by dispersion forces, or by a factorizable potential. There is also a weaker temperature dependence of λ observed for each binary mixture, and we show how this might be caused by a dependence of ω on solvent ordering, or by the inclusion of a fourth-rank term in U(ω).     

Reentrant behavior of poly(vinyl alcohol)–borax semidilute aqueous solutions

 The reentrant behavior of Poly(vinyl alcohol) (PVA)–borax aqueous semidilute solutions with a PVA concentration of 20 g/l and borax concentrations varies from 0.0 to 0.20 M was investigated using dynamic light scattering (DLS) and dynamic viscoelastic measurements. Two (fast and slow modes) and three (fast, middle, and slow) relaxation modes of PVA semidilute aqueous solutions without and with the presence of borax, respectively, were observed from DLS measurements. The fast and middle relaxation modes were q ²-dependent (q is the scattering vector) characteristic of diffusive behavior; however, the slow modes were q ³-dependent, characteristic of intraparticle dynamics. The experimental results showed that the slow relaxation mode dominates the DLS relaxation. The DLS slow mode relaxation time, τ_s, and the viscoelastic modulus G′(ω) and G′′(ω) data had a similar trend and demonstrated reentrant behavior as the borax concentration was increased from 0.0 to 0.20 M, i.e. τ_s, G′(ω), and G′′(ω) fluctuated with increasing borax concentration. The excluded-volume effect of polymers, charge repulsion among borate ions bound on PVA molecules, and intermolecular cross-linking didiol–borate complexation caused an expansion of the polymer chain; however, the screening effect of free Na⁺ ions on the negative charge of the borate ions bound on PVA and intramolecular cross-linking didiol–borate complexation led to a shrinkage of the polymer chain. The reentrant behavior was the consequence of the balance between expansion and shrinkage of the PVA–borate complex. Received: 26 March 1999/Accepted in revised form: 3 September 1999     

Quasielastic light scattering study of chemically crosslinked gelatin solutions and gels

Quasielastic light scattering measurements are reported for experiments performed on mixtures of gelatin and glutaraldehyde (GA) in the aqueous phase, where the gelatin concentration was fixed at 5 (w/v) and the GA concentration was varied from 1×10⁻⁵ to 1×10⁻³ (w/v). The dynamic structure factor, S(q,t), was deduced from the measured intensity autocorrelation function, g ₂(τ), with appropriate allowance for heterodyning detection in the gel phase. The S(q,t) data could be fitted to S(q,t)=Aexp(−D _f q ² t)+Bexp(−t/τ_c)^β, both in the sol (50 and 60 ^∘C) and gel states (25 and 40 ^∘C). The fast-mode diffusion coefficient, D _f showed almost negligible dependence on the concentration of the crosslinker GA; however, the resultant mesh size, ξ, of the crosslinked network exhibited strong temperature dependence, ξ∼(0.5−χ)^1/5exp(−A/RT) implying shrinkage of the network as the gel phase was approached. The slow-mode relaxation was characterized by the stretched exponential factor exp(−t/τ_c)^β. β was found to be independent of GA concentration but strongly dependent on the temperature as β=β₀+β₁ T+β₂ T ². The slow-mode relaxation time, τ_c, exhibited a maximum GA concentration dependence in the gel phase and at a given temperature we found τ_c(c)=τ₀+τ₁ c+τ₂ c ². Our results agree with the predictions of the Zimm model in the gel case but differ significantly for the sol state. Received: 25 May 1999 /Accepted in revised form: 27 July 1999     

Interaction of blockcopolymer micelles as probed by small-angle neutron and by small-angle X-ray scattering

 The analysis of the interaction of micelles formed by a blockcopolymer is given by means of small-angle X-ray (SAXS) and small-angle neutron scattering (SANS). The blockcopolymer consists of poly(styrene) and poly(ethylene oxide) (molecular weight of each block: 1000 g/mol) and forms well-defined micelles (weight-association number: 400, weight-average diameter: 15.4 nm) in water. The internal structure has been studied previously (Macromolecules 29:4006 (1996)) by SAXS. There it has been shown that the micelles are spherical objects. The structure factor S(q) as a function of the scattering vector q (q=(4π/λ) sin (θ/2); λ: wavelength of the radiation in the medium; θ: scattering angle) can be extracted from both sets of small-angle scattering data (SANS: q≤0.4 nm^-1; SAXS: q≤0.6 nm^-1). It is shown that particle interaction in the present system can be described by assuming soft interaction which is modeled by a square-step potential. Received: 12 May 1997 Accepted: 9 July 1997     

Light scattering from spherulitic materials

The azimuthal angular dependence of the depolarized component of the light scattered from spherulitic materials is derived by an algebraic method that avoids the difficult angular integrations of the usual approach. The result appears as a sum of products of two factors, a molecular factor, that depends only on the structure and the scattering angle θ, and a geometrical factor that depends only on the azimuthal angle ? and the scattering angle θ. The molecular factors are evaluated for models of spherulitic structure that assume a constant tilt of the optical polarizability tensor. The radial distribution, in principle, is arbitrary, and an evaluation for the layered spherulite is made. If the tilt angle is ω when the azimuthal patterns depend only on a linear combination of P₂(cos ω) and P₄(cos ω), where P_n(x) is the Legendre polynomial of order n. In our theory the V_H scattering pattern is a four-leaf clover whose axes are restricted by the theory to be at either 0 or 45° to the polarization directions.     

Relations Between Atomic Transport Coefficients in Simple Liquids and Moments of Neutron Scattering Functions

Abstract

A possible alternative to Kubo theory is discussed for relating atomic transport coefficients in simple liquids. The arguments used result in a new expansion, in which the density-density response function x is expanded in a power series in the self response function X_s. The coefficients of this expansion can be determined, in principle, to all orders, from the moments of the neutron scattering functions S(qω) and S _s(qω). It is then proposed, by appeal to the hydrodynamic equations, that the radius of convergence of this series can be used to relate diffusion to the sound wave attenuation coefficient. Finally, this x-x_s expansion allows a direct comparison of the exact theory presented in this paper with earlier approximate theories relating incoherent and coherent neutron scattering from liquids.     

Calculation of X‐ray scattering intensities by means of the coupled cluster singles and doubles model

Total X‐ray scattering intensity σ_ee(q) is very sensitive to electron correlation effects. In this study σ_ee(q) of N₂, CO, and N₂O have been computed by the coupled cluster singles and doubles (CCSD) method and compared with configuration interaction singles and doubles (CISD) calculations as well as experimental observations. σ_ee(q) curves by CCSD calculations are rather close to those by CISD, but although small, there still exist some discrepancies between calculated and observed values. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1315–1320, 2001     

Synthesis and Characterization of Polymethacrylates,Polyacrylates, and Poly(Methylsiloxane)S Containing 4-[S(-)-2-Methyl-1-Butoxy]-4′-(ω-Alkanyl-1-OXY)-α-Methylstilbene Side Groups

Abstract

The synthesis and characterization of polymethacrylates, polyacrylates, and poly(methylsiloxane)s containing 4-[S(-)-2-methyl-1-butoxy]-4′-(ω-alkanyl-1-oxy)-α-methylstilbene side groups with ω-alkanyl from 11-undecanyl to 2-ethyl are presented. According to both differential scanning calorimetry and thermal optical polarized microscopy analyses, the poly(methylsiloxane)s containing 1-octyl and 1-hexyl as ω-alkanyl groups exhibit enantiotropic S _A and S _C* mesophases. All other polymers display only an enantiotropic S _A mesophase.     

Unusual finite size effects in the Monte Carlo simulation of microphase formation of block copolymer melts

Extensive Monte Carlo simulations are presented for the Fried-Binder model of block copolymer melts, where polymer chains are represented as self and mutually avoiding walks on a simple cubic lattice, and monomer units of different kind (A, B) repel each other if they are nearest neighbors (ε_AB > 0). Choosing a chain length N = 20, vacancy concentration Φ_v = 0,2, composition ƒ = 3/4, and a L × L × L geometry with periodic boundary conditions and 8 ≤ L ≤ 32, finite size effects on the collective structure factor S(q) and the gyration radii are investigated. It is shown that already above the microphase separation transition, namely when the correlation length ξ(T) of concentration fluctuations becomes comparable with L, a nonmonotonic variation of both S(q) and the radii with L sets in. This variation is due to the fact that the wavelength λ^*(T) of the ordering (defined from the wavenumber q^* where S(q) is maximal at λ^* = 2 π/q^*) in general is incommensurable with the box. The competition of two nontrivial lengths ξ(T), λ^* (T) with L makes the straigthforward application of finite size scaling techniques impossible, unlike the case of polymer blends. Since also the specific heat is found to have a broad rounded peak near the transition only, locating the transition accurately from Monte Carlo simulations remains an unsolved problem.