Vincent Krakoviack 02/2020
Low-wavenumber Raman scattering of molecular glass-forming liquids
In connection with the liquid to glass transition, much attention has been given to the relaxation dynamics of dense fluids; in particular, Raman spectroscopy has been employed to study fast dynamic processes. Depolarized light scattering spectra are presented of several glass-forming molecular liquids, a series of disubstituted benzenes and glycerol, in their supercooled regime (at temperatures below the melting point) in the wavenumber range from 5 to 200 cm-1. Some common features are encountered which combine two contributions, vibrational and relaxational; in the framework of the dipole-induced-dipole formalism, schematic mode coupling equations are adjusted to the experimental susceptibilities, and the shape of the spectra, including the so-called boson peak, is analysed in terms of damped oscillators.
Study of the depolarized light scattering spectra of supercooled liquids by a simple mode-coupling model
By using simple mode coupling equations, we investigate the depolarized light scattering spectra of two so-called "fragile" glassforming liquids, salol (phenylsalicylate) and CKN (Ca0.4K0.6(NO3)1.4), measured by Cummins and co-workers. Nonlinear integrodifferential equations for the time evolution of the density-fluctuations autocorrelation functions are the basic input of the mode coupling theory. Restricting ourselves to a small set of such equations, we fit the numerical solution to the experimental spectra. It leads to a good agreement between model and experiment, which allows us to determine how a real system explores the parameter space of the model, but it also leads to unrealistic effective vertices in a temperature range where the theory makes critical asymptotic predictions. We finally discuss the relevance and the range of validity of these universal asymptotic predictions when applied to experimental data on supercooled liquids.
Quantitative investigation of the mean-field scenario for the structural glass transition from a schematic mode-coupling analysis of experimental data
A quantitative application to real supercooled liquids of the mean-field scenario for the glass transition (Tg) is proposed. This scenario, based on an analogy with spin-glass models, suggests a unified picture of the mode-coupling dynamical singularity (Tc) and of the entropy crisis at the Kauzmann temperature (TK), with Tc > Tg > TK. Fitting a simple set of mode-coupling equations to experimental light-scattering spectra of two fragile liquids and deriving the equivalent spin-glass model, we can estimate not only Tc, but also the static transition temperature Ts corresponding supposedly to TK. For the models and systems considered here, Ts is always found above Tg, in the fluid phase. A comparison with recent theoretical calculations shows that this overestimation of the ability of a liquid to form a glass seems to be a generic feature of the mean-field approach.
Adsorption of a fluid in an aerogel: Integral equation approach
We present a theoretical study of the phase diagram and the structure of a fluid adsorbed in high-porosity aerogels by means of an integral-equation approach combined with the replica formalism. To simulate a realistic gel environment, we use an aerogel structure factor obtained from an off-lattice diffusion-limited cluster-cluster aggregation process. The predictions of the theory are in qualitative agreement with the experimental results, showing a substantial narrowing of the gas-liquid coexistence curve (compared to that of the bulk fluid), associated with weak changes in the critical density and temperature. The influence of the aerogel structure (nontrivial short-range correlations due to connectedness, long-range fractal behavior of the silica strands) is shown to be important at low fluid densities.
What can be learned from the schematic mode-coupling approach to experimental data ?
We propose a detailed investigation of the schematic mode-coupling approach to experimental data, a method based on the use of simple mode-coupling equations to analyze the dynamics of supercooled liquids. Our aim here is to clarify different aspects of this approach that appeared so far uncontrolled or arbitrary, and to validate the results obtained from previous works. Analyzing the theoretical foundations of the approach, we first identify the parameters of the theory playing a key role and obtain simple requirements to be met by a schematic model for its use in this context. Then we compare the results obtained from the schematic analysis of a given set of experimental data with a variety of models and show that they are all perfectly consistent. A number of potential biases in the method are identified and ruled out by the choice of appropriate models. Finally, reference spectra computed from the mode-coupling theory for a model simple liquid are analyzed along the same lines as experimental data, allowing us to show that, despite the strong simplification in the description of the dynamics it involves, the method is free from spurious artifacts and provides accurate estimates of important parameters of the theory. The only exception is the exponent parameter, the evaluation of which is hindered, as for other methods, by corrections to the asymptotic laws of the theory present when the dynamics is known only in a limited time or frequency range.
Langevin dynamics of the Coulomb frustrated ferromagnet: A mode-coupling analysis
We study the Langevin dynamics of the soft-spin, continuum version of the Coulomb-frustrated Ising ferromagnet. By using the dynamical mode-coupling approximation, supplemented by reasonable approximations for describing the equilibrium static correlation function, and the somewhat improved dynamical self-consistent screening approximation, we find that the system displays a transition from an ergodic to a nonergodic behavior. This transition is similar to that obtained in the idealized mode-coupling theory of glass-forming liquids and in the mean-field generalized spin glasses with one-step replica symmetry breaking. The significance of this result and the relation to the appearance of a complex free-energy landscape are also discussed.
Relating monomer to centre-of-mass distribution functions in polymer solutions
A relationship between the measurable monomer-monomer structure factor, and the centre-of-mass (CM) structure factor of dilute or semi-dilute polymer solutions is derived from Ornstein-Zernike relations within the "polymer reference interaction site model" (PRISM) formalism, by considering the CM of each polymer as an auxiliary site and neglecting direct correlations between the latter and the CM and monomers of neighbouring polymers. The predictions agree well with Monte Carlo data for self-avoiding walk polymers, and are considerably more accurate than the predictions of simple factorization approximations.
Site-averaging in the integral equation theory of interaction site models of macromolecular fluids: An exact approach
A simple "trick" is proposed, which allows us to perform exactly the site-averaging procedure required when developing integral equation theories of interaction site models of macromolecular fluids. It shows that no approximation is involved when the number of Ornstein-Zernike equations coupling the site-site correlation functions is reduced to one. Its potential practical interest for future theoretical developments is illustrated with a rederivation of the so-called molecular closures.
Influence of solvent quality on effective pair potentials between polymers in solution
Solutions of interacting linear polymers are mapped onto a system of "soft" spherical particles interacting via an effective pair potential. This coarse-graining reduces the individual monomer-level description to a problem involving only the center of mass (c.m.) of the polymer coils. The effective pair potentials are derived by inverting the c.m. pair distribution function, generated in Monte Carlo simulations, using the hypernetted chain closure. The method, previously devised for the self-avoiding walk model of polymers in good solvent, is extended to the case of polymers in solvents of variable quality by adding a finite nearest-neighbor monomer-monomer attraction to the previous model and varying the temperature. The resulting effective pair potential is found to depend strongly on temperature and polymer concentration. At low concentration the effective interaction becomes increasingly attractive as the temperature decreases, eventually violating thermodynamic stability criteria. However, as polymer concentration is increased at fixed temperature, the effective interaction reverts to mostly repulsive behavior. These issues help to illustrate some fundamental difficulties encountered when coarse-graining complex systems via effective pair potentials.
An integral equation approach to effective interactions between polymers in solution
We use the thread model for linear chains of interacting monomers, and the "polymer reference interaction site model" (PRISM) formalism to determine the monomer-monomer pair correlation function hmm(r) for dilute and semi-dilute polymer solutions, over a range of temperatures from very high (where the chains behave as self-avoiding walks) to below the theta temperature, where phase separation sets in. An inversion procedure, based on the HNC integral equation, is used to extract the effective pair potential between ``average'' monomers on different chains. An accurate relation between hmm(r), hcc(r) [the pair correlation function between the polymer centers of mass (c.m.)], and the intramolecular form factors is then used to determine hcc(r), and subsequently extract the effective c.m.-c.m. pair potential vcc(r) by a similar inversion procedure. vcc(r) depends on temperature and polymer concentration, and the predicted variations are in reasonable agreement with recent simulation data, except at very high temperatures, and below the theta temperature.
Liquid-glass transition of a fluid confined in a disordered porous matrix: A mode-coupling theory
We derive an extension of the mode-coupling theory for the liquid-glass transition to a class of models of confined fluids, where the fluid particles evolve in a disordered array of interaction sites. We find that the corresponding equations are similar to those describing the bulk, implying that the methods of investigation which were developed there are directly transferable to this new domain of application. We then compute the dynamical phase diagram of a simple model system and show that new and nontrivial transition scenarios, including reentrant glass transitions and higher-order singularities, can be predicted from the proposed theory.
Coarse-graining diblock copolymer solutions: a macromolecular version of the Widom-Rowlinson model
We propose a systematic coarse-grained representation of block copolymers, whereby each block is reduced to a single ``soft blob'' and effective intra- as well as intermolecular interactions act between centres of mass of the blocks. The coarse-graining approach is applied to simple athermal lattice models of symmetric AB diblock copolymers, in particular to a Widom-Rowlinson-like model where blocks of the same species behave as ideal polymers (i.e. freely interpenetrate), while blocks of opposite species are mutually avoiding walks. This incompatibility drives microphase separation for copolymer solutions in the semi-dilute regime. An appropriate, consistent inversion procedure is used to extract effective inter- and intramolecular potentials from Monte Carlo results for the pair distribution functions of the block centres of mass in the infinite dilution limit.
Liquid-glass transition of confined fluids: Insights from a mode-coupling theory
The dynamics of confined glassforming liquids is discussed on the basis of the recent extension of the mode coupling theory for the liquid-glass transition to the model of the quenched-annealed binary mixture. It is in particular shown that, in confinement, the collective density correlation functions always decay to a non-zero infinite time value, even in the fluid state, and some clarification is given about the question of the relation between structure and dynamics in confined fluids.
Multiscale coarse graining of diblock copolymer self-assembly: from monomers to ordered micelles
Starting from a microscopic lattice model, we investigate clustering, micellization and micelle ordering in semi-dilute solutions of AB diblock copolymers in a selective solvent. To bridge the gap in length scales, from monomers to ordered micellar structures, we implement a two-step coarse graining strategy, whereby the AB copolymers are mapped onto ``ultrasoft'' dumbells with monomer-averaged effective interactions between the centres of mass of the blocks. Monte Carlo simulations of this coarse-grained model yield clear-cut evidence for self-assembly into micelles with a mean aggregation number n of roughly 100 beyond a critical concentration. At a slightly higher concentration the micelles spontaneously undergo a disorder-order transition to a cubic phase. We determine the effective potential between these micelles from first principles.
Heterolytic Splitting of H2 and CH4 on γ-Alumina as a Structural Probe for Defect Sites
A combined use of DFT periodic calculations and spectroscopic studies (IR and solid-state NMR) shows that a γ-alumina treated at 500 ℃ under high vacuum contains surface defects, which are very reactive toward H2 or CH4. The reaction of H2 on defect sites occurs at low temperature (ca. 25 ℃) on two types of Al atoms of low coordination numbers, AlIII or AlIV, to give AlIV-H and AlV-H, respectively. The amount of defects as titrated by H2 at 25 and 150 ℃ is 0.043 and 0.069 site/nm2, respectively, in comparison with 4 OH/nm2). In contrast, CH4 reacts selectively at 100-150 ℃ on the most reactive AlIII sites to form the corresponding AlIV-CH3 (0.030 site/nm2). The difference of reactivity of H2 and CH4 is fully consistent with calculations (reaction and activitation energy, ΔE and ΔE‡).
Mode-coupling theory for the slow collective dynamics of fluids adsorbed in disordered porous media
We derive a mode-coupling theory for the slow dynamics of fluids confined in disordered porous media represented by spherical particles randomly placed in space. Its equations display the usual nonlinear structure met in this theoretical framework, except for a linear contribution to the memory kernel which adds to the usual quadratic term. The coupling coefficients involve structural quantities which are specific of fluids evolving in random environments and have expressions which are consistent with those found in related problems. Numerical solutions for two simple models with pure hard core interactions lead to the prediction of a variety of glass transition scenarios, which are either continuous or discontinuous and include the possibility of higher-order singularities and glass-glass transitions. The main features of the dynamics in the two most generic cases are reviewed and illustrated with detailed computations. Moreover, a reentry phenomenon is predicted in the low fluid-high matrix density regime and is interpreted as the signature of a decorrelation mechanism by fluid-fluid collisions competing with the localization effect of the solid matrix.
Comment on "Spherical 2+p spin-glass model: An analytically solvable model with a glass-to-glass transition"
Guided by old results on simple mode-coupling models displaying glass-glass transitions, we demonstrate, through a crude analysis of the solution with one step of replica symmetry breaking (1RSB) derived by Crisanti and Leuzzi for the spherical s+p mean-field spin glass [Phys. Rev. B 73, 014412 (2006)], that the phase behavior of these systems is not yet fully understood when s and p are well separated. First, there seems to be a possibility of glass-glass transition scenarios in these systems. Second, we find clear indications that the 1RSB solution cannot be correct in the full glassy phase. Therefore, while the proposed analysis is clearly naive and probably inexact, it definitely calls for a reassessment of the physics of these systems, with the promise of potentially interesting developments in the theory of disordered and complex systems.
Fluids Confined in Porous Media: A Soft-Sponge Model
The morphology of many porous materials is sponge-like. For describing fluid adsorption in such materials, we propose here a quite general sponge model which is built by digging spherical cavities in a continuum. In contrast to the hard-sponge model proposed recently by Zhao et al. [Zhao, S. L.; Dong, W.; Liu, Q. H. J. Chem. Phys. 2006, 125, 244703], the continuum in the present soft-sponge model is permeable to fluid particles. Although the general expression of the fluid-matrix interaction potential is not pair additive, we were able to extend some statistical-mechanics formalism of liquid state to deal with this model. We derived the diagrammatic expansions of various correlation functions and Ornstein-Zernike equations. Usually, one would not expect that the thermodynamic quantities (e.g., internal energy) of a system with a nonpair-additive interaction can be completely determined from the structural information at the two-body level. We found a remarkable result that for the soft-sponge model considered here, the internal potential energy can be determined from only two-body correlation functions. In the particular case of a soft-sponge model with nonoverlapping cavities, we show that the fluid-matrix interaction can be also described by a pair-additive potential. In this case, the Madden-Glandt formalism applies. The Ornstein-Zernike equations obtained by using the pair-additive fluid-matrix interaction potential look to be quite different from those obtained by starting with the nonpair-additive potential. We found the relationship between the two descriptions and show how the two sets of Ornstein-Zernike equations can be transformed from each other for a soft-sponge model with nonoverlapping cavities.
Entropic self-assembly of diblock copolymers into disordered and ordered micellar phases
We investigate the self-assembly of an athermal model of AB diblock copolymers into disordered and ordered micellar microphases. The original microscopic lattice model with ideal A strands and self-avoiding B strands is mapped onto a system of ultrasoft dumbbells, with monomer-averaged effective interactions between the centers of mass (CMs) of the two blocks. Extensive Monte Carlo simulations of this coarse-grained model are reported for several length ratios f = LA/(LA + LB) of the two strands of lengths LA and LB. Clear-cut evidence is found for clustering and self-assembly into micelles with a mean aggregation number of n̅ ≃ 100 beyond a critical micellar concentration (cmc) in the semidilute regime. The cmc is found to decrease with increasing f, as predicted by an analytic calculation based on the random phase approximation. The initially disordered dispersion of polydisperse spherical micelles undergoes a disorder−order transition to a micellar crystal phase at higher copolymer concentrations. The effective pair potential between the micellar CMs is determined by inverting the measured CM−CM pair distribution function and is found to become steeper with increasing density.
Tagged-particle dynamics in a fluid adsorbed in a disordered porous solid: Interplay between the diffusion-localization and liquid-glass transitions
A mode-coupling theory for the slow single-particle dynamics in fluids adsorbed in disordered porous media is derived, which complements previous work on the collective dynamics [V. Krakoviack, Phys. Rev. E 75, 031503 (2007)]. Its equations, such as the previous ones, reflect the interplay between confinement-induced relaxation phenomena and glassy dynamics through the presence of two contributions in the slow part of the memory kernel, which are linear and quadratic in the density correlation functions, respectively. From numerical solutions for two simple models with pure hard-core interactions, it is shown that two different scenarios result for the diffusion-localization transition depending on the strength of the confinement. For weak confinement, this transition is discontinuous and coincides with the ideal glass transition, such as in one-component bulk systems, while, for strong confinement, it is continuous and occurs before the collective dynamics gets nonergodic. In the latter case, the glass transition manifests itself as a secondary transition, which can be either continuous or discontinuous, in the already arrested single-particle dynamics. The main features of the anomalous dynamics found in the vicinity of all these transitions are reviewed and illustrated with detailed computations.
Statistical mechanics of homogeneous partly pinned fluid systems
The homogeneous partly pinned fluid systems are simple models of a fluid confined in a disordered porous matrix obtained by arresting randomly chosen particles in a one-component bulk fluid or one of the two components of a binary mixture. In this paper, their configurational properties are investigated. It is shown that a peculiar complementarity exists between the mobile and immobile phases, which originates from the fact that the solid is prepared in presence of and in equilibrium with the adsorbed fluid. Simple identities follow, which connect different types of configurational averages, either relative to the fluid-matrix system or to the bulk fluid from which it is prepared. Crucial simplifications result for the computation of important structural quantities, both in computer simulations and in theoretical approaches. Finally, possible applications of the model in the field of dynamics in confinement or in strongly asymmetric mixtures are suggested.
Mode-coupling theory predictions for the dynamical transitions of partly pinned fluid systems
The predictions of the mode-coupling theory (MCT) for the dynamical arrest scenarios in a partly pinned (PP) fluid system are reported. The corresponding dynamical phase diagram is found to be very similar to that of a related quenched-annealed (QA) system. The only significant qualitative difference lies in the shape of the diffusion-localization lines at high matrix densities, with a reentry phenomenon for the PP system but not for the QA model, in full agreement with recent computer simulation results. This finding clearly lends support to the predictive power of the MCT for fluid-matrix systems. In addition, the predictions of the MCT are shown to be in stark contrast with those of the random first-order transition theory. The PP systems are thus confirmed as very promising models for differentiating tests of theories of the glass transition.
Mode-coupling theory of the glass transition for confined fluids
We present a detailed derivation of a microscopic theory for the glass transition of a liquid enclosed between two parallel walls relying on a mode-coupling approximation. This geometry lacks translational invariance perpendicular to the walls, which implies that the density profile and the density-density correlation function depends explicitly on the distances to the walls. We discuss the residual symmetry properties in slab geometry and introduce a symmetry adapted complete set of two-point correlation functions. Since the currents naturally split into components parallel and perpendicular to the walls the mathematical structure of the theory differs from the established mode-coupling equations in bulk. We prove that the equations for the nonergodicity parameters still display a covariance property similar to bulk liquids.
Simple physics of the partly pinned fluid systems
In this paper, we consider some aspects of the physics of the partly pinned (PP) systems obtained by freezing in place particles in equilibrium bulk fluid configurations in the normal (nonglassy) state. We first discuss the configurational overlap and the disconnected density correlation functions, both in the homogeneous and heterogeneous cases, using the tools of the theory of adsorption in disordered porous solids. The relevant Ornstein-Zernike equations are derived, and asymptotic results valid in the regime where the perturbation due to the pinning process is small are obtained. Second, we consider the homogeneous PP lattice gas as a means to make contact between pinning processes in particle and spin systems and show that it can be straightforwardly mapped onto a random field Ising model with a strongly asymmetric bimodal distribution of the field. Possible implications of these results for studies of the glass transition based on PP systems are also discussed.
Comment on "Static correlations functions and domain walls in glass-forming liquids: The case of a sandwich geometry" [J. Chem. Phys. 138, 12A509 (2013)]
In this Comment, we argue that the behavior of the overlap functions reported in the commented paper can be fully understood in terms of the physics of simple liquids in contact with disordered substrates, without appealing to any particular glassy phenomenology. This suggestion is further supported by an analytic study of the one-dimensional Ising model provided as Supplementary Material.
Dynamics of fluids in quenched-random potential energy landscapes: A mode-coupling theory approach
Motivated by a number of recent experimental and computational studies of the dynamics of fluids plunged in quenched-disordered external fields, we report on a theoretical investigation of this topic within the framework of the mode-coupling theory, based on the simple model of the hard-sphere fluid in a Gaussian random field. The possible dynamical arrest scenarios driven by an increase of the disorder strength and/or of the fluid density are mapped, and the corresponding evolutions of time-dependent quantities typically used for the characterization of anomalous self-diffusion are illustrated with detailed computations. Overall, a fairly reasonable picture of the dynamics of the system at hand is outlined, which in particular involves a non-monotonicity of the self-diffusion coefficient with fluid density at fixed disorder strength, in agreement with experiments. The disorder correlation length is shown to have a strong influence on the latter feature.
Dynamics of a noninteracting colloidal fluid in a quenched Gaussian random potential: A time-reversal-symmetry-preserving field-theoretic approach
We develop a field-theoretic perturbation method preserving the fluctuation-dissipation relation (FDR) for the dynamics of the density fluctuations of a noninteracting colloidal gas plunged in a quenched Gaussian random field. It is based on an expansion about the Brownian noninteracting gas and can be considered and justified as a low-disorder or high-temperature expansion. The first-order bare theory yields the same memory integral as the mode-coupling theory (MCT) developed for (ideal) fluids in random environments, apart from the bare nature of the correlation functions involved. It predicts an ergodic dynamical behavior for the relaxation of the density fluctuations, in which the memory kernels and correlation functions develop long-time algebraic tails. A FDR-consistent renormalized theory is also constructed from the bare theory. It is shown to display a dynamic ergodic-nonergodic transition similar to the one predicted by the MCT at the level of the density fluctuations, but, at variance with the MCT, the transition does not fully carry over to the self-diffusion, which always reaches normal diffusive behavior at long time, in agreement with known rigorous results.
Vincent Krakoviack 02/2020