INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 13 (2001) 9197–9206 PII: S0953-8984(01)27593-6 Freezing and melting criteria in non-equilibrium G P Hoffmann and H Lowen¨ Institut fur¨ Theoretische Physik II, Heinrich-Heine-Universitat¨ Dusseldorf,¨ Universitatsstraße¨ 1, D-40225 Dusseldorf,¨ Germany Received 3 August 2001, in final form 28 August 2001 Published 28 September 2001 Online at stacks.iop.org/JPhysCM/13/9197 Abstract Using non-equilibrium computer simulations, it is shown that various phenom- enological criteria for melting and freezing hold not only in equilibrium but in steady-state non-equilibrium as well. In particular, we study the steady state of charge-polydisperse Brownian particles shaken by a time-dependent oscillatory electric field. Among these criteria are the Lindemann melting rule, the Hansen–Verlet freezing rule and the dynamical freezing criterion proposed for colloidal fluids by Lowen,¨ Palberg and Simon. 1. Introduction The freezing and melting transition is far from being understood from first principles, even in thermodynamic equilibrium and for relatively simple systems interacting via a radially symmetric pair potential [1]. The essential difficulty arises from the collective many-body effects resulting in a broken translational symmetry across freezing. Recent microscopic descriptions [2,3] are based either on the liquid phase (such as density functional approaches) or on the crystal phase (such as solid–defect theories) but it is fair to say that a fully predictive theory with the interactions between the particles as the only input is not available. Therefore phenomenological criteria based on empirical observations in experiments or computer simulations have been very important for estimating the location of the freezing and melting lines. In general, these criteria work if the freezing occurs at high densities, in which case the relative density jump across freezing is small. The most traditional criterion is the Lindemann melting rule [4]: it states that a solid melts if the root mean square displacement of particles around their ideal lattice positions is roughly ten per cent of their nearest-neighbour distance. This rule has been tested for different materials [5, 6] and model fluids [7, 8] and indeed the relative displacement L was found to be within five to twenty per cent at melting. The Lindemann criterion cannot be used to describe two-dimensional melting because the root mean square displacement diverges there [9]. It can, however, be generalized by considering relative displacements between neighbouring particles only [10]. Furthermore the Lindemann rule was found recently to be valid in higher spatial dimensions [11], for re-entrant melting [12] and for interfacial freezing transitions [13]. 0953-8984/01/419197+10$30.00 © 2001 IOP Publishing Ltd Printed in the UK 9197 9198 G P Hoffmann and H Lowen¨ A second criterion is the Hansen–Verlet freezing rule [14]. It states that a (three- dimensional) liquid freezes if the amplitude Sm of the first maximum in the structure factor S(k) exceeds 2.85. For two-dimensional freezing, however, the actual value of the amplitude is getting much larger at freezing and is about 5.5 [15]. Third, a dynamical freezing rule for Brownian colloidal fluids was proposed by Lowen,¨ Palberg and Simon (LPS) [8]. This criterion is based on the ratio DL/D0 between the short- time and long-time diffusion coefficients. A fluid freezes if this ratio becomes smaller than ten per cent. This criterion was shown to be also valid in two dimensions [16] and was recently related to the Hansen–Verlet freezing rule via mode-coupling theory [17,18]. It is particularly useful for the crystallization transition of colloidal suspensions [19]. All these criteria have been developed and applied so far in equilibrium. In this paper we investigate whether these criteria are also valid in non-equilibrium situations [20]. Freezing and melting in non-equilibrium is, of course, much more complicated, as the traditional thermo- statistical concepts break down. In many non-equilibrium cases, however, the systems run into a steady-state situation where structural and dynamical correlations as well as phase transitions can be defined [21–24]. Consequently, the Lindemann parameter L, the liquid structure factor S(k) and the diffusion coefficients are still well-defined quantities there. Therefore it is tempting to check whether the phenomenological criteria are fulfilled in non-equilibrium. In fact, in experiments on two-dimensional colloids by Grier and Murray [25], the LPS criterion for freezing was used in non-equilibrium and was confirmed there. In this work we consider a particular steady-state non-equilibrium situation: a slightly polydisperse Brownian Yukawa fluid is driven by a space-independent but time-dependent oscillatory external field that couples differently to the different particles according to their polydispersity. We have shown recently [26] that such a field shifts the freezing transition with respect to equilibrium freezing. The shift depends strongly on the field amplitude, scaling with the square of the relative polydispersity, and vanishes for very large field frequencies. In this paper we use extensive computer simulations to test the validity of the three different freezing and melting criteria mentioned above. We find that all three rules are indeed fulfilled provided the polydispersity is not too large which means that the system is rather close to equilibrium (see our discussion in section 3). Our results suggest that the freezing criteria are robust when turning from equilibrium to non-equilibrium systems, provided the system is not too far away from equilibrium. Thus our findings will be helpful in interpreting experiments which are typically carried out in non-equilibrium [25]. As a by-product, we also confirm that the freezing criteria are valid in equilibrium for small charge polydispersities. In this case, similar tests for the Hansen–Verlet rule and the Lindemann criterion have already been made by Tata and Arora [27, 28]. The paper is organized as follow. In section 2, we define our model. In section 3 we describe our simulation method. The different freezing criteria are tested and further results for the inherent system anisotropy are presented in section 4. We conclude in section 5. 2. Model Our model for non-equilibrium Brownian dynamics of charge-polydisperse colloids is defined as follows [26]: N colloidal particles confined to a volume (with a fixed number density ρ = N/ defining a mean interparticle spacing of a = ρ−1/3) are held at fixed temperature T . Two colloidal particles i and j are interacting via an effective screened Coulomb pair potential [29, 30] (−λ(r − σ )/σ ) V (r) = Z Z U σ exp ij i j 0 r (1) Freezing and melting criteria in non-equilibrium 9199 where r is the interparticle distance, U0 sets an energy and σ a length scale. The quantity λ is a dimensionless screening parameter. Zi > 0 denotes the reduced colloidal charge of the ith colloidal particle. The charges Zi are drawn from a given distribution p(Z) around a mean value Z which can be set equal to unity without loss of generality. The width of the distribution is governed by the prescribed relative polydispersity pZ = Z2 − 1. In our simulations we took either a bimodal distribution p(Z) = [δ(Z − 1+pZ) + δ(Z − 1 − pZ)]/2 (2) or a rectangular distribution √ √ 1/( 12pZ) for |Z − 1| < 3pZ p(Z) = (3) 0 otherwise. Apart from the direct interaction (1), there is an external oscillatory force Fi (t) resulting from an ac electric field which acts on the ith colloidal particle in the z-direction: Fi (t) =ezZi E0 sin(ωt) (4) where ω is the field frequency, E0 the field amplitude and ez the unit vector along the z- direction. Neglecting hydrodynamic interactions, the stochastic Langevin equations for the colloidal trajectories ri (t) = (xi (t), yi (t), zi (t)) (i = 1,...,N)read as k T r B d i (R) =−∇r Vij (|ri −rj |) + Fi (t) + F (t) (5) D t i i 0 d j=i where kB T is the thermal energy, D0 is the short-time diffusion coefficient and the random (R) forces Fi describe the kicks of the solvent molecules acting on the ith colloidal particle. These random forces are Gaussian random numbers with zero mean: (R) Fi = 0 and variance (k T)2 (R) (R) 2 B (Fi )α(t)(Fj )β (t ) = δαβ δij δ(t − t ) (6) D0 guaranteeing that 2 D0 = lim (ri (t) −ri (0)) /6t. (7) t→0 The subscripts α and β stand for the three Cartesian components. As has been shown earlier, the external field has no influence for a monodisperse system apart from adding a trivial overall dynamical mode [26]. Hence a finite polydispersity pZ > 0 is essential to drive the system into real non-equilibrium and the polydispersity itself measures the ‘distance’ from equilibrium. 3. Simulation methods and diagnostics for the freezing and melting transition We solved the Langevin equations (5) by means of a finite-time step Brownian dynamics simulation [31] using N = 864 particles in a cubic box with periodic boundary conditions. 2 The Brownian timescale is set by τ = kB Ta /U0D0 and a typical size of the time step was )t = 0.003τ. The system was carefully ‘equilibrated’ for a time of typically 100τ until a steady state was reached. In order to reduce the multi-dimensional parameter space we kept the density, the field amplitude and the dimensionless screening parameter fixed to ρ = 1/σ 3, 9200 G P Hoffmann and H Lowen¨ E0 = 10.5U0/a, λ = 5.8, such that the crystalline structure is either face-centred cubic (fcc) or hexagonal close packed (hcp) with random occupancy for small pZ.