Theory Of Concentration Fluctuations Research Articles

Renormalized one-loop theory of correlations in disperse polymer blends.

Polymer blends are critical in many commercial products and industrial processes and their phase behavior is therefore of paramount importance. In most circumstances, such blends are formulated with samples of high dispersity, which have generally only been studied at the mean-field level. Here, we extend the renormalized one-loop theory of concentration fluctuations to account for blends of disperse polymers. Analyzing the short and long length-scale fluctuations in a consistent manner, various measures of polymer molecular weight and dispersity arise naturally in the free energy. Thermodynamic analysis in terms of moments of the molecular weight distribution(s) provides exact results for the inverse susceptibility and demonstrates that the theory is not formally renormalizable. However, physically motivated approximations allow for an "effective" renormalization, yielding (1) an effective interaction parameter, χe, which depends directly on the sample dispersities (i.e., Mw/Mn) and leaves the form of the mean-field spinodal unchanged, and (2) an apparent interaction parameter χa that depends on higher-order dispersity indices, for instance Mz/Mw, and characterizes the true limits of blend stability accounting for long-range off-critical fluctuations. We demonstrate the importance of dispersity on several example systems, including both "toy" models that may be realized in computer simulation and more realistic industrially relevant blends. We find that the effects of long-range fluctuations are particularly prominent in blends where the component dispersities are mismatched, especially when there is a small quantity of the high-dispersity species. This can be understood as a consequence of the shift in the critical concentration(s) from the monodisperse value(s).

Non-equilibrium fluctuations induced by the Soret effect in a ternary mixture.

We present, based on fluctuating hydrodynamics, the theory of concentration fluctuations in a ternary mixture subjected to a stationary temperature gradient, so that composition gradients are present due to thermal diffusion (Soret effect). We neglect gravity and confinement (boundary conditions) but consider a completely generic diffusion matrix, including cross-diffusion effects. We find, as in the case of binary mixtures, an important non-equilibrium enhancement of the concentration fluctuations, which is proportional to the square of the gradient and inversely proportional to the fourth power of the fluctuations wave number, q(-4). The results of this paper are expected to be asymptotically correct for fluctuations of large q, while for shorter q gravity and confinement effects need to be incorporated. Comparison with previous work in the topic is included.

Disorder-Order-Crystalline State Transitions of PEO-b-PPO-b-PEO Copolymers and their Blends: SAXS/WAXS/DSC Study

The structure in the symmetrical triblock copolymers PEO-b-PPO-b-PEO and their blend with PEO, studied by small angle x-ray scattering (SAXS), wide angle x-ray scattering (WAXS), and differential scanning calorimetry (DSC), pass from melt to the solid state on cooling. On subsequent heating back to the melt, they pass through disordered and ordered molten states, crystalline structure, and finally back to a disordered melt state. At high temperatures these systems are in the melt in the disordered state approximately described by the mean-field theory. The characteristic lengths of these systems, obtained from SAXS, are proportional to R g . At lower temperatures, their structure changes to a disordered state which can be described by the concentration fluctuation theory. During cooling, the disordered melt structure changes abruptly into the ordered melt structure. The crystallization destroys this melt structure, forming a new lamellar structure with different periodicity. During heating near the melting point, lamellar periodicity increases very steeply. After melting, the crystalline structure transforms directly to the disordered state.

The electronic origins of atomic short-range order in disordered fee Cu-Ni-Zn ternary alloys

Ternary alloys constitute an important and interesting laboratory for studying electronically driven ordering in alloys because (in contrast to binary alloys) the composition and electron-per-atom ratio (e/a) may be varied independently. Using their recently developed first-principles theory of atomic short-range order (ASRO) in alloys with an arbitrary number of components, the authors discuss ASRO in the ternary Cu-Ni-Zn system. Fermi-surface nesting is found to drive the ordering tendencies of the fcc portion of the phase diagram, yielding ASRO whose dominant wave vector varies with e/a in a predictable manner. The authors' concentration-fluctuation theory yields both the wave vector and polarization of the concentration wave, necessary to fully specify the ASRO in an alloy of more than two components.

Dynamical theory of concentration fluctuations in polymer solutions under shear.

Dynamical theory of concentration fluctuations in polymer solutions under shear.

Simulation study of reaction fronts.

Using random walkers on a square lattice we have simulated a diffusion-reaction process A+B\ensuremath{\rightarrow}C; initially, the reactants are segregated on either side of a straight boundary. Quantities studied include the position and width of the reaction front and the position-dependent reaction rates and densities of the reactants. Our results agree with the scaling theory of G\'alfi and R\'acz for the long-time behavior of the system indicating that the scaling assumption is valid and that the neglect in the scaling theory of concentration fluctuations and of the discrete (particle) nature of the reactants is not important. Furthermore, although the scaling theory is not explicitly constructed for the case of unequal diffusion coefficients of the two initial species, our results indicate that it is correct in this regime as well.

Ultrasonic Viscoelasticity and Excess Absorption in Polymer Solution

Shear moduli are measured by the reflection method for many polymer-toluene solutions. The results have good agreement with theoretical values by Rouse and Lamb. Longitudinal velocity and absorption are measured for poly(vinylisobuthylether) solution in toluene by the interferometric method. Comparing both moduli, it is found that the ratio of the dynamic volume viscosity to the dynamic shear one is 20–40. The phenomena are discussed on the basis of several theories and found to be well described by the concentration fluctuation theory by V. P. Romanov and V. A. Solove’ev.

The statistical mechanical theory of concentration fluctuations in mixtures

A general theory of concentration fluctuations in mixtures is derived. The resulting expressions, which include the Kirkwood–Buff theory as a particular case, relate different partial molar properties to a single combination of molecule number fluctuations within differently defined subsystems. The formalism, therefore, gives rise to a common definition for partial molar energies, volumes, enthalpies, and entropies in terms of molecule number fluctuations exclusively. Likewise, a second function of molecule number fluctuations describes both the mixture’s isobaric specific heat as well as its isothermal compressibility. The partial molar energy and enthalpy definitions, moreover, suggest new methods for the study of mixture properties via computer simulations.

Interaction between magnetic and compositional order in Ni-rich NicFe1−c alloys (invited)

We have developed a first-principles electronic theory of concentration fluctuations in spin polarized binary alloys. It is a mean field theory of the state of compositional order and it is based on the local spin density (LSD) approximation for describing the electrons. The usual averages over the statistical mechanical ensemble are carried out with the aid of the self-consistent Korringe–Kohn–Rostoker coherent-potential approximation (SCF-KKR-CPA). To illustrate the main consequences of the theory we study the compositional short-range order in the NicFe1−c alloy system. We find that the ordering energy is almost entirely of magnetic origin.

Comments on paper ‘the effect of gaussian particle‐pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence’ by B. L. Sawford (Q.J. April 1983, 109, 339–353)

AbstractIn this brief note we examine the necessity of distinguishing between the use of forward and reversed transition densities in the computation of second moments of concentration fields advocated by Sawford. In contrast to Sawford, we show that the forward densities are always correct and that for incompressible flow the forward and reversed formulations are equivalent. This equivalence implies a simple constraint on particle‐pair separation densities for incompressible flow.

Comments on paper 'The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulance'

Comments on paper 'The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulance'

The effect of Gaussian particle‐pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence

AbstractThe consequences for the theory of concentration fluctuations of different proposals by L. F. Richardson and G. K. Batchelor about the rate of separation of pairs of marked particles (giving rise to non‐Gaussian and Gaussian particle separation probability density functions respectively) are explored via two Lagrangian Monte Carlo models.The models are applied to an instantaneous one‐dimensional cloud source (which is approximately equivalent to a continuous line source) and in many respects give similar results for concentration fluctuations. The crucial difference is that whereas the non‐Gaussian model predicts, in agreement with observation, that at large time the fluctuations remain the same order of magnitude as the mean field (the ratio depending only on the source size), the Gaussian model incorrectly predicts that fluctuations ultimately vanish compared with the mean field.The reason for the failure of the Gaussian model is explored by partitioning the total fluctuations into contributions due to the variation of the distribution of material within the cloud (i.e. in coordinates relative to the centre‐of‐mass) and due to motions of the cloud as a whole (meandering). It is shown that at large time the Gaussian model accounts only for the meandering contribution to fluctuations, and by smoothing out all internal structure of the doud eliminates relative fluctuations.

The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulance

The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulance

Ultrasonic absorption in dioxane‐water mixtures

Dioxane‐water mixtures are a very common solvent system. However, their thermodynamic properties show marked deviations from ideal behavior. The mixtures also show ultrasonic relaxation phenomena in the 10–80‐MHz region. In 1966, Hammes and Knoche examined this system in the 10–160‐MHz frequency range. They interpreted their data in terms of two chemical relaxations caused by the H‐bonding processes: 2 W + 2 D ⇌ D + Q ⇌ Z. We have reexamined the system over a much wider frequency range of 0.2–610 MHz and conclude that the data are more sensibly explained by the concentration fluctuation theory of Romanov and Solovyev. The three parameters of this theory (Q, a thermodynamic parameter; D1, a mutual diffusion coefficient; l, an interaction distance) are examined in terms of the known thermodynamic and transport properties of the system. Finally, an attempt is made to relate this theoretical approach to a molecular model for such water mixtures.

Concentration fluctuations in chemical reactions

A theory of microscopic fluctuations, which is based on generalized fluctuation–dissipation assumptions, is used to describe concentration fluctuations in a uniform system undergoing chemical reactions. The theory is shown to agree with the linear Langevin-type theory of concentration fluctuations near equilibrium, and provides a method for calculating the time evolution of an arbitrary initial probability distribution for a system far from equilibrium. These results are easily compared to the birth and death (master equation) theory of chemical reactions, and recent work on the asymptotic solutions of these master equations indicates that in the limit of a macroscopic system, the two approaches are identical.

Rayleigh-Brillouin intensity ratios in aqueous electrolyte solutions

Rayleigh-Brillouin intensity ratios have been measured for a number of aqueous electrolytic solutions as a function of concentration. The calculated and experimental intensity ratios are in agreement within the experimental precision for solutions of alkali-metal and alkali-earth halides and nitrates at concentrations up to 3M, thus confirming applicability of simple concentration fluctuation theory for such systems. In solutions of 2–2 sulfates, the observed intensity ratios are significantly higher than the calculated values; the discrepancy remains to be explained although pronounced ionic association may be a factor.

Brillouin and Rayleigh Scattering Studies of Electrolyte Solutions

Rayleigh and Brillouin scattering studies have been carried out in a number of electrolyte solutions at various concentrations up to 4M. The apparatus consisted of an argon laser operating in a single mode, a thermostated octagonal light scattering cell with provisions for scattering at precisely known angles, a piezoelectric scanning Fabry-Perot interferometer, and a photomultiplier-photon-counting system. Three types of results have been obtained: (1) hypersonic velocity data from the frequency shift of the Brillouin components; (2) hypersonic absorption coefficients from the line widths of the Brillouin components; and (3) Rayleigh-Brillouin intensity ratios. The first and second types of results have been found to compare very favorably with those from ultrasonic velocity and absorption data at lower frequencies. The theoretical Rayleigh-Brillouin intensity ratios have been calculated from concentration fluctuation theory and found to be in good agreement with experiment at concentrations up to several molal except for 2-2 electrolytes such as MnSO4 and MgSO4. [Research supported by the Office of Naval Research and a NATO grant.]

The correlation theory of concentration fluctuations in dispersed systems

The fluctuations in the volume concentration of the dispersed phase are considered via the correlation theory of stationary random processes; the formula is derived for the rms perturbation in the concentration, which corresponds to a two point two-time correlation function, which is expressed as a simple integral.