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 a recent paper,¹ Gradenigo et al. have reported on a computer simulation study of a glass-forming liquid constrained by amorphous boundary conditions representative of its equilibrium bulk configurations. This setup has been put forward a few years ago as a possible tool to probe the existence of nontrivial static correlations in bulk glassy systems, through the investigation of point-to-set correlation functions such as configurational overlaps.^2,3 Accordingly, quantities of this type are reported in Ref. 1 for two related geometries, a semi-infinite fluid in contact with a single wall (wall geometry) and a fluid slab sandwiched between two parallel walls (slit geometry). These data are then analyzed in terms of the interplay between the boundary conditions and the complex coarse-grained free-energy landscape postulated by the random first-order transition (RFOT) theory for bulk glassy liquids.^2,4,5

However, it was recently observed⁶ that, outside the realm of glassy physics, constrained systems such as those of Ref. 1 are just special instances of a generic model for fluids in contact with random substrates previously studied with standard tools of the theory of simple liquids.⁷ In this framework, it is customary to quantify the direct influence of the quenched-disordered solid boundary on the microscopic fluid configurations via the so-called blocking or disconnected two-point density correlation function, i.e., the covariance of the random density profile established in the presence of the amorphous surface. This correlation function can be straightforwardly turned into configurational overlaps that are analogues of those measured in Ref. 1 and not necessarily bound to be featureless objects, even in the absence of specific glassy features.^6,8

From this observation, the question naturally arises, whether one really needs to appeal to any particular glassy phenomenology to interpret the behavior of the overlap functions reported in Ref. 1. This is the point addressed in this comment, through a direct comparison between the wall and slit geometries. For reference, it should be recalled that, in a RFOT-inspired analysis, the two geometries are expected to be ruled by different physics, resulting in well distinct characteristic lengthscales,^4,5 and this is how the data are described in Ref. 1.

The present discussion is guided by an asymptotic result derived in Ref. 6 for the disconnected total correlation function of a nonglassy liquid, h_dis(x, y), when at least one of the points x or y is far enough from any amorphous boundary. Indeed, one then gets

with ρ the number density of the fluid, h(r) its total correlation function in the bulk, and χ(r) the indicator function of the domain from which the fluid particles are excluded by the amorphous boundaries.

The linearity of this equation with respect to χ(r) suggests to investigate the range of validity of a simple superposition approximation, in which, for the geometries considered here, the effect on the fluid of the two walls in the slit geometry would merely be the sum of the effects of the two walls taken individually. Note that such a linear regime can be expected on general grounds and that Eq. (1), which represents the leading asymptotic contribution to it, only plays the role of a formal proof of existence. In terms of the configurational overlaps reported in Ref. 1, such a superposition approximation leads to the compact relation

with q_c(d) the overlap at the central plane of a slit of width 2d, q(d) the overlap at a distance d from a single wall, and q₀ the trivial ideal-gas contribution to these functions.

Equation (2) is tested at four temperatures in Fig. 1, where the data of Ref. 1 are plotted accordingly. It appears quite reasonable at all temperatures for d ≳ 1 and, in this domain, 2[q(d) − q₀] and q_c(d) − q₀ can be both described by the same exponential decay. This suggests that, in this regime in both geometries, the behavior of the system is ruled by rather simple amorphous-boundary effects.

Gradenigo et al. have shown that 2[q(d) − q₀] keeps an exponential behavior down to d = 0 at all studied temperatures, while q_c(d) − q₀ remains exponential down to d = 0.75 (the smallest value considered for the slit geometry) at the two highest temperatures and tends to flatten for d ≲ 1 at the two lowest. Such a bending is a foreseeable consequence of the presence of two nearby facing boundaries, i.e., of confinement: For narrow slits, the combined action of the two amorphous walls is indeed expected to restrain the decay of the correlations of the disorder-induced fluid density profile when d increases more strongly than a mere linear superposition of independent boundary effects. In fact, such a leveling-off disrupting at short distances a medium-to-long-range exponential decay occurs in models as simple as the one-dimensional Ising model, as shown in the supplementary material.⁹ If no bending is seen at the highest temperatures, this might only mean that the breadth of the raw confinement effect, expected to decrease as temperature increases, is too small at these temperatures for it to appear in the probed slit-width window.

Therefore, by looking at the data of Ref. 1 from the angle of the physics of simple liquids in contact with disordered substrates, i.e., of raw boundary and confinement effects,^6,7 we do not see any compelling evidence that any specific glassy phenomenology has to be appealed to in order to give an account of the observed behaviors. In particular, in the studied temperature range, there is no obvious sign of distinct and complex physics in the wall and slit geometries, both described by the same simple exponential decay law starting at quite small distances already, at variance with expectations from the RFOT scenario, for instance.^1,4,5

The contrasting conclusions of Ref. 1 and of this comment clearly point towards difficulties in the interpretation of measured point-to-set correlation functions. They actually are complex objects, possibly blending ingredients from the physics of normal and glassy liquids that are not easily sorted out.⁶ In fact, these difficulties were already acknowledged in Ref. 1, where warnings were raised, based on the experience with an alternative setup, the so-called random pinning geometry.^8,10 However, they could not be made concrete, due to the lack of simple results such as Eq. (1) that appeared more recently.

Finally, it should be mentioned that the predictions of the RFOT theory for the point-to-set correlations are one aspect of an elaborate scenario, also involving features such as a bimodal distribution of overlaps recently observed in computer simulations of spherical cavities.¹¹ It remains a challenge for the future to find out whether the simple picture put forward in this comment could also account for these additional aspects.

The authors of Ref. 1 are warmly thanked for sharing their data with us and thus making the present analysis possible.