Liquid-glass transition of confined fluids: Some 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 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. PACS numbers: 64.70.Pf, 61.20.Lc, 61.25.-f In the past few years, there has been a strong interest for the dynamics of liquids under confinement and more specifically for their structural glass transition, which has been investigated by a variety of experimental techniques and by computer simulation [1, 2]. One of the main goals of these studies was to improve our understanding of the concept of cooperativity, a key ingredient of many glass transition theories [3]. Indeed, there are now many evidences that, in the bulk, the dynamics of deeply supercooled liquids is strongly inhomogeneous and that correlated clusters of molecules, the so-called dynamical heterogeneities, play a crucial role in the slowing-down of the dynamics when the temperature is decreased. But, up to now, many aspects of the characterization of these dynamical heterogeneities have remained elusive. For instance, there is no clear consensus on their shape, their size and their evolutions with temperature. This is where the interest for confined glassforming systems came in. Indeed, confinement is a means to introduce geometrical constraints and new characteristic lengthscales (pore size, film thickness. . . ) in the system under study. Thus, by looking at the way the dynamics is modified under confinement compared to the bulk, one can hope to gain some insight on the properties of the dynamical heterogeneities. For instance, in the simplest scenarios, one expects from finite size effects a cutoff on the slowing-down of the dynamics as temperature is varied, when the typical size of the heterogeneities in the bulk would become larger than the characteristic lengthscale of the confining medium. Confinement would thus provide an indirect probe of the properties of the dynamical heterogeneities. Glass transition of confined fluids: Mode-coupling theory 2 It turns out that the situation is more complex. Indeed, the previous line of reasoning requires that the physical phenomena which are specific of confined systems have a negligible impact on the dynamics of the imbibed fluid or at least that their influence is sufficiently well known that it can subtracted from the results. This is usually not the case and strong confinement effects are observed, in particular at the fluid-solid interface where structured layers of almost immobile molecules are often formed. So this is in fact the problem of dynamics under confinement as a whole which has to be addressed and not only at the level of a simple modification of the bulk dynamics. Theories dealing with the dynamics in confinement are far less developed than for the bulk and they usually are of a phenomenological nature [2]. There is yet a strong need for elaborate microscopic theories in this field. Indeed, the variety of systems to consider is immense. Porous media can differ in the shape, size, size distribution, connectivity, etc, of their pores. They can be made of various materials, leading to a wide range of fluid-solid interactions which adds to the already great variability of intermolecular interactions met with usual glassformers. A reasonable theoretical approach, able to catch many of these subtleties, would thus be of a great help. Applied to various models, it would allow to explore thoroughly the phenomenology of confined glassforming systems and maybe to disentangle the different physical effects which interplay in these systems. Recently, a step in this direction has been made with the extension of the mode coupling theory (MCT) for the liquid-glass transition [4, 5, 6, 7] to a particular class of confined systems, the so-called “quenched-annealed” (QA) binary mixtures [8]. In these systems, first introduced by Madden and Glandt [9], the fluid molecules equilibrate in a matrix of particles frozen in a disordered configuration sampled from a given probability distribution. This class of models, which describe situations of statistically homogeneous and isotropic confinement, is thought to be able to reproduce most of the physics of fluids confined in materials like Vycor, controlled porous glasses or aerogels, and, in fact, the fluid dynamics and glass transition in some of its instances have been the subject of recent studies by molecular dynamics simulations [10, 11, 12]. In this paper, we will give a short presentation of the proposed extension of the MCT to QA systems (a more detailed account is given in reference [8]) and, on the basis of this approach, we will discuss a few aspects of the physics of confined fluids which are of relevance for the interpretation of experimental and computer simulation results. Before dealing with dynamics, one first has to consider some peculiarities of the statics of QA systems. QA mixtures are systems with quenched disorder, so that their theoretical description requires two types of averages, a thermal average denoted by 〈· · ·〉, taken for a given realization of the matrix, and a disorder average over the matrix realizations, denoted by · · ·, to be taken after the thermal average. Like in the bulk, one is interested in the Fourier components of the microscopic fluid density, or, in short, density fluctuations, defined as ρfq(t) = Nf j=1 e j, where q denotes the wavevector, Nf is the fluid particle number and rj(t) is the position of fluid particle j at time t. A significant difference with the bulk is that, for a given matrix Glass transition of confined fluids: Mode-coupling theory 3 realization, the translational invariance of the system is broken by the presence of the quenched component. This results in non-zero average density fluctuations at equilibrium, i.e., 〈ρq〉 6 = 0. It is only after the disorder average that the symmetry is restored, leading to 〈ρq〉 = 0, hence the description of the model as statistically homogeneous and isotropic. This property has a well known impact on the equations describing the structural correlations in such systems, for instance the so-called replica Ornstein-Zernike (OZ) equations [13, 14], where it leads to the splitting of the total and direct correlation functions of the fluid, h (r) and c (r), respectively, into two contributions, connected [h(r) and c(r)] and blocked or disconnected [h(r) and c(r)]. The separation of h (r) into two terms leads to a similar property of the fluid structure factor S q = 〈ρfqρf−q〉/Nf = 1 + nf ĥ q , where nf is the fluid number density, which can be expressed as S q = S c q +S b q with S c q = 〈(ρq − 〈ρfq〉)(ρf−q − 〈ρf−q〉)〉/Nf = 1+nf ĥcq and S q = 〈ρfq〉〈ρf−q〉/Nf = nf ĥq, where f̂q denotes the Fourier transform of f(r). To fix all the notations, we define here the matrix-matrix and fluid-matrix structure factors and total correlation functions as well, which are given by S q = 〈ρq ρ−q〉/Nm = 1+nmĥ q and S q = 〈ρfqρ−q〉/ √ NfNm = √ nfnmĥ fm q , where Nm is the matrix particle number, nm is the matrix number density, and ρ m q = ∑Nm j=1 e iqsj , where sj is the fixed position of matrix particle j, is the q Fourier component of the quenched microscopic matrix density. The identification of two types of static fluid correlations has significant implications for the dynamics. Indeed, if one forgets for a moment the possibility of a dynamical ergodicity breaking, one expects, using standard arguments, that limt→∞ 〈ρfq(t)ρf−q〉 = 〈ρfq〉〈ρf−q〉, i.e., the normalized total density fluctuation autocorrelation function φq (t) = 〈ρfq(t)ρf−q〉/(NfS q ) does not decay to zero at long times, but rather lim t→∞ φTq (t) = S q