A phase-separation perspective on dynamic heterogeneities in glass-forming liquids C. Cammarota $^{∗}$, 1, 2 A. Cavagna, 2, 3 I. Giardina, 2, 3 G. Gradenigo $^{†}$, 4, 5 T. S. Grigera, 6 G. Parisi, 1, 2 and P. Verrocchio 4, 5, 7 $^{1}$Dipartimento di Fisica, Universit` a di Roma “Sapienza”, p.le Aldo Moro 5, 00185, Roma, Italy. $^{2}$Centre for Statistical Mechanics and Complexity (SMC), CNR-INFM. $^{3}$Istituto Sistemi Complessi (ISC), CNR, Via dei Taurini 19, 00185 Roma, Italy. $^{4}$Dipartimento di Fisica, Universit` a di Trento, via Sommarive 14, 38050 Povo, Trento, Italy. $^{5}$INFM CRS-SOFT, c/o Universit` a di Roma “Sapienza”, 00185, Roma, Italy. $^{6}$Instituto de Investigaciones Fisicoqu´ımicas Te´ oricas y Aplicadas (INIFTA) and Departamento de F´ısica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, and CCT La Plata, Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, c.c. 16, suc. 4, 1900 La Plata, Argentina. $^{7}$Instituto de Biocomputaci´ on y F´ısica de Sistemas Complejos (BIFI), Spain. We study dynamic heterogeneities in a model glass-former whose overlap with a reference config- uration is constrained to a fixed value. The system phase-separates into regions of small and large overlap, so that dynamical correlations remain strong even for asymptotic times. We calculate an appropriate thermodynamic potential and find evidence of a Maxwell’s construction consistent with a spinodal decomposition of two phases. Our results suggest that dynamic heterogeneities are the expression of an ephemeral phase-separating regime ruled by a finite surface tension. The conspicuous lack of a growing correlation length, contrasting with the very steep increase of the relaxation time, has been a puzzle in the physics of structural glasses for quite a long time. Arguably, the first breakthrough has been the discovery of dynamic heterogeneities [1], and the detection of a growing dynamical correlation length, ξ$_{d}$ [2, 3]. If we take two snapshots of the system sepa- rated by a time lag comparable to the α relaxation time, τ$_{α}$ , the particle displacements vary enormously across the system, and the typical size ξ$_{d}$ of the mobility-correlated regions increases on lowering the temperature. More recently, by studying the thermodynamics of sys- tems subject to amorphous boundary conditions [4, 5], an entirely different, fully static, correlation length ξ$_{s}$ has been discovered [6, 7]. ξ$_{s}$ also grows upon cooling, even though its surge occurs at lower temperatures than ξ$_{d}$ . The static correlation length has a natural interpreta- tion as the size of the cooperatively rearranging regions [8], and within the random first-order theory [9] it is de- termined by the balance between a surface tension cost and a configurational entropy gain of a rearrangement. Clearly, it would be desirable to unify the dynamic and the thermodynamic frameworks, so as to understand the interplay between the two correlation lengths. Although the static-dynamic connection is clear in mean-field sys- tems [10] and some progresses have been made in more realistic systems [11], we are quite far from a unifying picture in real glass-formers. Here we show that surface tension, which is a crucial ingredient of the thermody- namic framework, also plays a key role in the formation of dynamic heterogeneities. In so doing, we establish a much-needed further link between dynamic and thermo- dynamic relaxation in glass-forming liquids. Let us start with the standard measurement of the dynamic correlation length ξ$_{d}$ . Our glass-former is the well-known soft-sphere model in 3- d [26]. A useful tool to measure ξ$_{d}$ is the overlap, which quantifies how much a configuration at time t is similar to the reference con- figuration at t = 0. If we partition the system in small cubic boxes and let n$_{i}$ be the number of particles in box i , the local overlap is defined as q ( r $_{i}$, t ) ≡ n$_{i}$ ( t ) n$_{i}$ (0), where r i refers to the centre of cell i [27]. The spatial map of the local overlap tells us how much different regions of the system have decorrelated (respect to the initial con- figuration) over a time t . In Fig. 1 (top) we show two snapshots of the overlap field. We see that at t = τ$_{α}$ there are large heterogeneous regions, which eventually fade away for longer times. To quantify their size we must compute the overlap correlation function, or its Fourier transform S ( k, t ) (Fig. 2, left). In general, given a correlation function in Fourier space, it is well- established practice [12] to extract the correlation length ξ from the small- k linear interpolation of S − 1 vs. k $^{2}$, from which the correlation length is obtained as ξ ( t ) 2 = B/A . The validity of Eq. 2 is shown in the inset of Fig. 2, right. [28]. The time-dependent correlation lenght ξ ( t ) represents the size of the dynamical heterogeneities at time t . This lengthscale grows as the time approaches τ$_{α}$ and the heterogeneities become more extended (inset of Fig. 2, left). The largest value of ξ ( t ) (reached at the τ$_{α}$ ) defines the so-called dynamical correlation length, ξ$_{d}$ ≡ ξ ( τ$_{α}$ ) [2, 3, 13]. G ( r , t ) ≡ 〈 q (0 , t ) q ( r , t ) 〉 − 〈 q (0 , t ) 〉 〈 q ( r , t ) 〉 , (1) S ( k, t ) − 1 = A + Bk $^{2}$, (2) arXiv:1001.2539v1 [cond-mat.dis-nn] 14 Jan 2010 $^{∗}$Present address: CEA, Institut de Physique Theorique, Saclay, F-91191 Gif-sur-Yvette, France $^{†}$Present address: SMC-INFM and Dipartimento di Fisica, Uni- vesit´ a di Roma “Sapienza”, p.le Aldo Moro 5, 00185, Roma, Italy.