Nucleation in a Sheared Ising Model: Effects of External Field

Abstract

Simulations using the Forward Flux Sampling method have shown a nonmonotonic dependence of the homogeneous nucleation rate on the shear rate for a sheared two dimensional Ising model [R. J. Allen et al, arXiv:cond-mat/0805.3029]. For quasi-equilibrium systems (i.e. in the absence of shear), Classical Nucleation Theory (CNT) predicts the dependence of the critical cluster size and the nucleation rate on the external magnetic field. We investigate the behaviour of the sheared Ising model as a function of the external field. At low external field strength, the same nonmonotonic behaviour holds and the peak in the nucleation rate is remarkably insensitive to the field strength. This suggests that the same external field-dependence holds for the enhancement of nucleation by shear at low shear rates and the suppression of shear at high shear rates. At high field strength, the nucleation behaviour is qualitatively different. We also analyse the size and shape of the largest cluster in the transition state configurations, as a function of the external field. In the sheared system, the transition state cluster becomes larger and more elongated as the field strength decreases. We compare our results for the sheared system to the predictions of the CNT for the quasiequilibrium case, and find that the CNT cannot easily be used to describe nucleation in the system under shear.Nucleation in driven systems is a widespread and important phenomenon that remains poorly understood. For "quasi-equilibrium" systems, in the absence of external driving, Classical Nucleation Theory (CNT) predicts the dependence of the nucleation rate and the critical cluster size on the degree of supersaturation and the interfacial tension of the growing cluster, for moderate values of the supersaturation. No such simple theoretical description is available for nucleation in driven systems. In this paper, we consider nucleation in a system under shear. In experimental sheared systems, nucleation is likely to be affected by a wide range of factors including transport of particles to the growing cluster, changes in cluster shape, shear-induced ordering of the cluster, polydispersity and shear-induced cluster breakup. Intuition suggests that nucleation will be suppressed for high enough shear rates. At low shear rates, some simulations and experiments have observed suppression of nucleation, 1)-3) while others have suggested that nucleation may be enhanced by the shear. 4)-7)In a recent paper, we studied nucleation in a sheared two-dimensional Ising model, for a fixed value of the external magnetic field. 8) Although this is a highly simplified model system, it nevertheless shows a nonmonotonic dependence of the nucleation rate as a function of the shear rate. The nucleation rate increases, apparently linearly, with the shear rate for weak shear, achieves a peak for shear ratė γ max , and decreases nonlinearly as the shear rate is further increased. These results were obtained using the Forward Flux Sampling rare event simulation method. 9)-11) typeset us...

Studying rare events using forward-flux sampling: Recent breakthroughs and future outlook

“…Here, P A (λ) is called the splitting probability, and is the likelihood that a configuration with order parameter value λ falls back to A. In general, P A (λ) can be estimated from: (38) with f (λ) the attachment rate. In general, Eq.…”

“…In a lattice gas, each vertex will be either occupied (n i = 1) or vacant (n i = 0). In recent years, FFS has been extensively utilized for studying the sensitivity of the nucleation kinetics in the Ising and lattice gas models to the existence and energetic properties of impurities, 34,36,41 mechanical deformations, 37,38 internal and external interfaces, 35,36,[41][42][43] and the topological properties of the underlying network. 39,40,44,45 Since nucleation will only occur if the underlying transition is first-order, all these simulations were conducted in the presence of an external magnetic field or chemical potential, and the starting basin A is always set to the respective metastable phase in each system.…”

“…By doing so, the nucleation rate increased monotonically with shear rate, as the stochasticity in shear direction suppressed cluster breakup. In a follow-up study, 38 the same authors used FFS to probe the effect of external magnetic field on nucleation under shear, and found the non-monotonicity to disappear in the presence of sufficiently strong external fields.…”

“…Therefore, developing more robust and efficient variants and implementations of FFS has become an intense focus of research, with earlier methodological efforts covered in several comprehensive reviews published shortly after its development. 32,33 In recent years, FFS has been extensively utilized for studying a wide variety of rare events, such as phase transitions in the Ising [34][35][36][37][38][39][40][41][42][43][44][45] and Potts 46-49 models, crystal nucleation, 50-80 evaporation, [81][82][83][84][85][86] phase separation, 87,88 coalescence, 89 wetting, 90,91 protein folding, rupture and aggregation, [92][93][94][95][96][97][98] DNA hybridization, [99][100][101][102][103][104][105] polymer relaxation and translocation, [106][107][108][109][110][111] ion transport 112 and genetic [113]…”

Incorporating friction and collective shear moves into a lattice gas

“…Cirillo et al ͓24͔ performed a much more extensive numerical analysis of this sheared Ising model using the same dynamics. The sheared Ising model was also used by Allen et al ͓14,25͔ to study nucleation and growth processes using Forward Flux Sampling ͓26-28͔. In the latter work it was found that shear introduces competing processes ͑shear-mediated breakup at large shear rates, and shear-induced coalescence at small shear rates͒ that produce a nonmonotonic dependence of the cluster growth rate on the rate of shear.…”