^{1,2,a)}, Roberto Trozzo

^{1}, Andrea Cavagna

^{1,2}, Tomás S. Grigera

^{3,4}and Paolo Verrocchio

^{5,6,7}

### Abstract

The problem of measuring nontrivial static correlations in deeply supercooled liquids made recently some progress thanks to the introduction of amorphous boundary conditions, in which a set of free particles is subject to the effect of a different set of particles frozen into their (low temperature) equilibrium positions. In this way, one can study the crossover from nonergodic to ergodic phase, as the size of the free region grows and the effect of the confinement fades. Such crossover defines the so-called point-to-set correlation length, which has been measured in a spherical geometry, or cavity. Here, we make further progress in the study of correlations under amorphous boundary conditions by analyzing the equilibrium properties of a glass-forming liquid, confined in a planar (“sandwich”) geometry. The mobile particles are subject to amorphous boundary conditions with the particles in the surrounding walls frozen into their low temperature equilibrium configurations. Compared to the cavity, the sandwich geometry has three main advantages: (i) the width of the sandwich is decoupled from its longitudinal size, making the thermodynamic limit possible; (ii) for very large width, the behaviour off a single wall can be studied; (iii) we can use “anti-parallel” boundary conditions to force a domain wall and measure its excess energy. Our results confirm that amorphous boundary conditions are indeed a very useful new tool in the study of static properties of glass-forming liquids, but also raise some warning about the fact that not all correlation functions that can be calculated in this framework give the same qualitative results.

We warmly thank Giorgio Parisi for making us familiar with the aspect ratio scaling technique. We also thank Giulio Biroli and Chiara Cammarota for several important discussions. T.S.G. thanks the Dipartimento di Fisica of the Sapienza Universitá di Roma and ISC (CNR, Rome) for hospitality. The work of G.G. is supported by the “Granular-Chaos” project, funded by Italian MIUR under the Grant No. RBID08Z9JE. PV was partly supported by MICINN (Spain) through Research Contract Nos. FIS2009-12648-C03-01 and FIS2008-01323 (PV). T.S.G. was partially supported by ANPCyT (Argentina).

I. INTRODUCTION

II. MODEL AND SIMULATION DETAILS

III. DIFFERENT STATIC LENGTHSCALES

A. Point-to-set correlation length

B. Penetration length

IV. MOSAIC IN THE SANDWICH

A. Naive argument

B. Sharpening in the thermodynamic limit?

V. “ANTI-PARALLEL” BOUNDARY CONDITIONS

A. Interface energy

B. Aspect ratio scaling

VI. COMPARISON OF THE DIFFERENT LENGTHSCALES AND THE ISING CASE

VII. CONCLUSIONS

### Key Topics

- Boundary value problems
- 16.0
- Correlation functions
- 14.0
- Surface tension
- 12.0
- Domain walls
- 6.0
- Ising model
- 4.0

##### B32B

## Figures

Cartoon of the sandwich geometry. In the ‘parallel” (or αα) setup (top) both frozen walls are taken from the same equilibrium configuration, while in the “anti-parallel” (αβ) case, they come from different configurations (bottom).

Cartoon of the sandwich geometry. In the ‘parallel” (or αα) setup (top) both frozen walls are taken from the same equilibrium configuration, while in the “anti-parallel” (αβ) case, they come from different configurations (bottom).

Overlap at the center of the sandwich vs. sandwich half-width *d* in the parallel setup for (from left to right) *T* = 0.482, 0.350, 0.246, 0.202. Lines are exponential or compressed-exponential fits (see text). (Inset) Same data in semilog plot.

Overlap at the center of the sandwich vs. sandwich half-width *d* in the parallel setup for (from left to right) *T* = 0.482, 0.350, 0.246, 0.202. Lines are exponential or compressed-exponential fits (see text). (Inset) Same data in semilog plot.

Overlap in the parallel setup as a function of *z* for a sandwich of half-width *d* = 4 at several temperatures. The free region is wide enough that the overlap can reach its bulk value *q* _{0} near the center.

Overlap in the parallel setup as a function of *z* for a sandwich of half-width *d* = 4 at several temperatures. The free region is wide enough that the overlap can reach its bulk value *q* _{0} near the center.

Overlap vs. *d* − *z* (distance from wall) at fixed *d* (same data as Fig. 3 ) with pure exponential fits. (Inset) Same data in semilog scale.

Overlap vs. *d* − *z* (distance from wall) at fixed *d* (same data as Fig. 3 ) with pure exponential fits. (Inset) Same data in semilog scale.

Overlap at center vs. sandwich half-width *d* at *T* = 0.203 and two values of *L*.

Overlap at center vs. sandwich half-width *d* at *T* = 0.203 and two values of *L*.

Excess energy per mobile particle at *T* = 0.246 and several values of *d*, together with the bulk (PBC) average value.

Excess energy per mobile particle at *T* = 0.246 and several values of *d*, together with the bulk (PBC) average value.

Excess energy vs. *d* at several temperatures and *L* = 16. Lines are exponential fits. (Inset) Same data in semilog scale.

Excess energy vs. *d* at several temperatures and *L* = 16. Lines are exponential fits. (Inset) Same data in semilog scale.

Penetration length λ vs. point-to-set correlation length ξ.

Penetration length λ vs. point-to-set correlation length ξ.

Energy decay length *l* vs. point-to-set correlation length ξ.

Energy decay length *l* vs. point-to-set correlation length ξ.

Energy decay length vs. two-spin correlation length for the Ising model in the square lattice. Data are from Monte Carlo simulations on a 100 × 100 lattice with single-flip Metropolis dynamics performed above the critical point, at temperatures *T* = 2.5*J*, *T* = 2.4*J*, *T* = 2.35*J* and *T* = 2.32*J*, where *J* is the Ising coupling constant (the critical point is *T* _{ c } ≈ 2.269*J*). The correlation length was obtained from a fit of the spin-spin space correlation function *C*(*r*) = ⟨*S*(0)*S*(*r*)⟩. To determine the length *l*, sandwich configurations were prepared as explained for the liquid case, measuring the excess energy Δ*E*(*d*) = *E* _{αβ} − *E* _{αα} for *d* = 1, 2, 3, 4, 5, 7.5, 10, 15, 20, 25 and 45 lattice spacings, and fitting to an exponential decay.

Energy decay length vs. two-spin correlation length for the Ising model in the square lattice. Data are from Monte Carlo simulations on a 100 × 100 lattice with single-flip Metropolis dynamics performed above the critical point, at temperatures *T* = 2.5*J*, *T* = 2.4*J*, *T* = 2.35*J* and *T* = 2.32*J*, where *J* is the Ising coupling constant (the critical point is *T* _{ c } ≈ 2.269*J*). The correlation length was obtained from a fit of the spin-spin space correlation function *C*(*r*) = ⟨*S*(0)*S*(*r*)⟩. To determine the length *l*, sandwich configurations were prepared as explained for the liquid case, measuring the excess energy Δ*E*(*d*) = *E* _{αβ} − *E* _{αα} for *d* = 1, 2, 3, 4, 5, 7.5, 10, 15, 20, 25 and 45 lattice spacings, and fitting to an exponential decay.

## Tables

Point-to-set correlation length ξ, and anomaly exponent ζ, from a fit of Eq. (19) ; penetration length λ, from a fit of Eq. (20) ; and excess energy decay lengthscale *l*, from a fit of Eq. (21) . At the highest temperature the value of *l* has large uncertainty as we have very few nonzero values of Δ*E*.

Point-to-set correlation length ξ, and anomaly exponent ζ, from a fit of Eq. (19) ; penetration length λ, from a fit of Eq. (20) ; and excess energy decay lengthscale *l*, from a fit of Eq. (21) . At the highest temperature the value of *l* has large uncertainty as we have very few nonzero values of Δ*E*.

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