Stress distribution and the fragility of supercooled melts arXiv:1001.2844v1 [cond-mat.dis-nn] 16 Jan 2010 Dmytro Bevzenko and Vassiliy Lubchenko Department of Chemistry, University of Houston, Houston, TX 77204-5003 (Dated: October 30, 2018) We formulate a minimal ansatz for local stress distribution in a solid that includes the possibility of strongly anharmonic short-length motions. We discover a broken-symmetry metastable phase that exhibits an aperiodic, frozen-in stress distribution. This aperiodic metastable phase is char- acterized by many distinct, nearly degenerate configurations. The activated transitions between the configurations are mapped onto the dynamics of a long range classical Heisenberg model with 6-component spins and anisotropic couplings. We argue the metastable phase corresponds to a deeply supercooled non-polymeric, non-metallic liquid, and further establish an order parameter for the glass-to-crystal transition. The spin model itself exhibits a continuous range of behaviors between two limits corresponding to frozen-in shear and uniform compression/dilation respectively. The two regimes are separated by a continuous transition controlled by the anisotropy in the spin- spin interaction, which is directly related to the Poisson ratio σ of the material. The latter ratio and the ultra-violet cutoff of the theory determine the liquid configurational entropy. Our results suggest that liquid’s fragility depends on the Poisson ratio in a non-monotonic way. The present ansatz provides a microscopic framework for computing the configurational entropy and relaxational spectrum of specific substances. If cooled sufficiently rapidly, a liquid may fail to crystal- lize, but will instead remain in a metastable, supercooled state. Upon further cooling, the relaxation times in a su- percooled liquid grow very rapidly as the mass transport becomes activated, in contrast with the mainly collisional transport near the fusion temperature. Because the lo- cal structures are much longer-lived than the vibrational equilibration times, the activated-transport regime repre- sents a state with a broken translational symmetry, even though the corresponding, aperiodic structure shows no obvious distinction from a snapshot of an ordinary, uni- form liquid. (It is said the heterogeneity is “dynamical.”) Since the symmetry is broken gradually with lowering the temperature - beginning with the highest frequency mo- tions - a transition into this “aperiodic-crystal” state is not sharp, but, instead, is a soft cross-over centered at a temperature T$_{cr}$ , 1,2 corresponding universally to vis- cosity 10 Ps or so. 2,3 The cross-over into the activated regime is a finite-dimensional analog of a mean-field ki- netic catastrophe of the mode-coupling theory (MCT), whereby the motional barriers would diverge at a tem- perature T$_{A}$ , even though the configurational entropy is still extensive. 4,5 figurations available to a region of a supercooled liquid containing N beads, i.e. e s$_{c}$$^{N/k$_{B}$}$. The bead usually con- tains two-three atoms, but could be bigger for molecular liquids containing large rigid units such as benzene. The other length scale emerging during the crossover is the so called Lindemann length d$_{L}$ , 1,6,7 which is the molec- ular displacement at the mechanical stability edge. This length is nearly universal: d$_{L}$ ≃ a/ 10, and characterizes bead displacements during transitions between distinct aperiodic packings in the metastable, aperiodic crystal phase. One may view the crossover into the activated liquid regime as a “localization” transition, whereby the emerging metastability of local structures is signaled by a discontinuous transition from a uniform liquid to a state with a non-zero force constant of the Einstein-oscillator. 1 The random first order transition (RFOT) theory utilizes this view to analyze the activated transport in chemically distinct (non-polymeric) fluids in a unified fashion. 8–10 (See Ref. 11 for a review.) The RFOT theory predicts that in a fully developed activated regime, the structural re- laxation time is determined solely by the configurational entropy per bead: 2,9 In additional contrast to the mean-field transition at T$_{A}$ , the cross-over at T$_{cr}$ exhibits two emerging length scales: One length scale is the molecular length a that signifies the volumetric size of a chemically rigid unit - of- ten called the “bead” - that is not significantly perturbed during activated transport. Conversely, the beads inter- act with each other weakly, comparably to the Lennard- Jones interaction. 2 The bead may be thought of as a coarse-graining length, beyond which activated motions are largely independent of chemical detail, but fully char- acterized by a single, bulk quantity. This bulk quantity is the excess liquid entropy relative to the corresponding crystal, usually called the configurational entropy. The magnitude of the configurational entropy per bead, s$_{c}$ , directly gives the number of alternative aperiodic con- where τ$_{vibr}$ ≃ 1 psec is the microscopic time scale char- acterizing vibrational relaxation. By Eq.(1), system- specific deviations from the pure Arrhenius temperature dependence of τ result from variations in the value of the heat capacity jump at the glass transition tempera- ture T$_{g}$ per bead: Δ c$_{p}$ ≡ T ( ∂s$_{c}$/∂T ) | T = $_{T$_{g}$}$. The so called fragility index m ≡ d (log τ ) /T d (1 /T ) | T = T$_{g}$ gives a quan- titative measure of that deviation. Small and large de- viations from the Arrhenius T -dependence (correspond- ing to small and large m respectively) are often called strong and fragile behaviors. 12,13 The RFOT theory pre- dicts m ≃ 34 . 7Δ c$_{p}$ , 2,9,14 in excellent agreement with ex- periment. Hall and Wolynes have put forth a simple specific τ = τ$_{vibr}$ exp(32 k$_{B}$/s$_{c}$ ) , (1)