Orientational Ordering in Spatially Disordered Dipolar Systems G. Ayton∗, M. J. P. Gingras†, and G. N. Patey∗ ∗Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 †TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3 (November 5, 2018) ple interpretation given above. More importantly, Zhang and Widom propose that, despite the strong frustration This letter addresses basic questions concerning ferroelec- present in randomly frozen systems, long-range ferroelec- tric order in positionally disordered dipolar materials. Three models distinguished by dipole vectors which have one, two or tric order is possible above a critical density. Their work three components are studied by computer simulation. Ran- implies that the spin-glass behavior observed in ferroflu- domly frozen and dynamically disordered media are consid- ids [6] results from the low concentration of magnetic par- ered. It is shown that ferroelectric order is possible in spatially ticles, whereas the ferroelectric liquid crystalline phase random systems, but that its existence is very sensitive to the found in computer simulations of dipolar fluids [4,5] arises dipole vector dimensionality and the motion of the medium. because of the high particle density considered. In the A physical analysis of our results provides significant insight present letter we examine the validity of this argument into the nature of ferroelectric transitions. and address the general question “Can long-range fer- roelectric order spontaneously arise in a system without PACS numbers: 64.70.Md, 77.80.-e, 82.20Wt fined-tuned positional correlations?” We investigate the behavior of dense spatially disor- Recently, spatially disordered dipolar materials have dered dipolar systems using constant temperature molec- attracted considerable attention. Diluted lattices [1–3], ular dynamics (MD) and Monte Carlo (MC) simulations. fluid phases [4,5] and amorphous frozen ferrofluids [6] Systems where the dipole vector has one, two and three have been examined experimentally [1,2,6], theoretically components are considered. The first two of these are [3,7–11] and with computer simulations [3–5]. It has been commonly referred to as the Ising and XY models and shown that in the absence of long-range positional order, for notational simplicity we shall refer to the three com- the strong spatial-orientational coupling intrinsic to dipo- ponent dipole as the XYZ model. In all cases the pair lar forces can lead to interesting new phase behavior. For potential, u(12), is of the generic form example, frozen ferrofluids containing magnetic particles 12 in a non-magnetic solvent exhibit magnetic irreversibil- u(12) = 4ε(σ/r) + uDD(12) , ities reminiscent of randomly frustrated magnetic spin where 4ε(σ/r)12 and glasses [12]. On the other hand, computer simulations of simple dipolar fluids clearly indicate the existence of a 5 3 uDD(12) = −3(µ1 · r)(µ2 · r)/r + µ1 · µ2/r ferroelectric liquid crystal phase [4,5]. A simple interpretation of these observations might be are the soft-sphere and dipole-dipole interactions. The as follows. In frozen ferrofluids, the quenched positional parameters ε and σ are the fundamental units of energy disorder creates random frustration and the system be- and length, µi is the dipole of particle i and r is the arXiv:cond-mat/9505150v1 30 May 1995 haves as a spin glass [12]. The fluid systems [4,5] differ intermolecular vector. The long-range dipolar interac- from the frozen case in that the strongly coupled transla- tions were taken into account using periodic boundary tional and rotational degrees of freedom are in full ther- conditions and the Ewald summation method assuming mal equilibrium. This allows the development of short- a perfectly conducting surrounding continuum [4,15–17]. range spatial correlations resembling those seen in the fer- The existence of a ferroelectric phase can be detected by roelectric tetragonal-I lattice [13] and, consequently, fer- calculating the average polarization P per particle de- roelectric order develops in the liquid phase [4]. In view N dˆ dˆ fined as P = 1/Nh| Pi=1 µˆ i · |i, where is a unit vec- of these observations, and recalling that perfect crystals tor in the direction of the total instantaneous moment, exhibit ferroelectric or antiferroelectric long-range order M N = Pi=1 µi, and N is the number of particles in the depending on the lattice symmetry [13,14], one might system. argue that specific spatial correlations are required for It is convenient to characterize dipolar soft-sphere sys- ferroelectric order. tems by the reduced density, ρ∗ = Nσ3/V , the reduced ∗ In recent papers Zhang and Widom [11] have put for- temperature, T = kBT/ε, and the reduced dipole mo- ward a mean field theory for spatially disordered dipolar ment, µ∗ = (µ2/εσ3)1/2, where V is the volume of the systems. They argue that the long-range nature of the sample, T is the absolute temperature and kB is the dipolar forces plays a key role in yielding ferroelectric Boltzmann constant. All results explicitly presented are order, and that this is not explicitly included in the sim- for µ∗ = 4 and ρ∗ =0.8. This density is well within the 1 range for which Zhang and Widom predict a ferroelec- tion (i.e., ρ∗ = 1.05 [18], T ∗ = 10.5) also showed no tric phase [18]. For µ∗ = 4 and ρ∗ = 0.8, Zhang and long-range ferroelectric order. In brief, for the randomly Widom predict a ferroelectric phase for the Ising case if frozen XY and XYZ models we find no evidence of a fer- T ∗ ≤ 35.2 and for the XYZ model if T ∗ ≤ 4.8. roelectric state in the thermodynamic limit. This clearly We first consider frozen systems. Suitable spatially disagrees with the calculations of Zhang and Widom disordered configurations were generated by carrying out which for the XYZ model predict a stable ferroelectric an MD simulation of a soft-sphere fluid at T ∗ = 10.5 phase well within the temperature-density range consid- and ρ∗ = 0.8. Fluid-like configurations were then se- ered here. lected at random for dipolar rotational MD simulations. In order to gain further insight into the nature of fer- Following this approach we could obtain a frozen state roelectric order (or the lack thereof) in spatially random at a much higher density than is possible from a ran- systems, we consider a “toy model” where the transla- dom parking algorithm. Unfortunately, it is impossible tional motion is completely decoupled from the dipolar to have a truly “random” and structureless spatial con- interactions. The soft-sphere fluid acts as a “substrate” figuration (i.e., with the radial distribution function g(r) which moves at a fixed translational temperature inde- equal to one for r ≥ σ [11]) at this density. However, at pendent of the embedded dipoles. The dipoles themselves T ∗ = 10.5 the local structure in the soft-sphere fluid is interact and are equilibrated at a different rotational tem- weak and very short-ranged. perature. Of course, the “equilibrium” state achieved by Polarization results for randomly frozen systems are the dipoles will depend on the underlying motion of the shown in Fig. 1 [19]. The XY and XYZ values were ob- substrate. This model is similar in spirit to those used tained with MD simulations. The discrete nature of the in recent studies of non-equilibrium phase transitions in Ising model renders it inappropriate for MD so MC cal- magnetic systems subject to Levy flights [21]. It must be culations were used. The average polarization obtained emphasized that this technique is presented only as a use- at the lowest temperature where equilibrium could be ful simulation tool and we do not imply any real physical ∗ ∗ achieved, Tmin, is plotted vs 1/N. The values of Tmin mechanism for the decoupling. The moving substrate is a are 10.0, 4.0, and 3.5 for the Ising, XY and XYZ models, means of simulating dipolar systems in a dynamically ran- respectively. Below these temperatures MD or MC runs dom medium that lacks any specific spatial correlations for the same configuration started from perfectly ordered which may favor ferroelectric ordering. The translational and disordered states (i.e., two replicas) did not converge diffusion rate of the substrate can be controlled by the to the same result within a reasonable computation time particle mass. Extrapolation to infinite mass should pro- (i.e., about a week). Possibly with greater computational vide information about the randomly frozen system. ∗ ∗ effort Tmin could be pushed a little lower, but the val- In Fig. 2, we have plotted P vs T (rotational) for ues given above are well within the range where Zhang the XYZ model. Here, the decoupled substrate is a soft- and Widom predict a ferroelectric phase. For the Ising sphere fluid again at ρ∗ = 0.8 and T ∗(translational) = model the equilibrated system at T ∗ = 10.0 is nearly 10.5. It is convenient to introduce the reduced mass m∗ = 100% polarized. Moreover, for the Ising case the po- m/m′, where m′ is a reference mass defined such that ∗ ∗ ′ 2 1/2 larization at Tmin exhibits very little number dependence the reduced simulation timestep ∆t ≡ (ǫ/m σ ) ∆t = and certainly does not appear to vanish in the thermody- 0.00125. Figure 2 includes results for m∗ = 1, 5 and 10. namic limit. This, together with the plot of P vs T ∗ and Spontaneous polarization develops for all systems and the heat capacity calculations (see Fig. 3 below), strongly temperature at which P begins to grow decreases with suggests that ferroelectric order develops spontaneously increasing mass. For example, from the P vs T ∗ plot in the spatially disordered Ising system with the transi- there appears to be a ferroelectric transition at T ∗ ≈ 2 tion occurring at T ∗ ≈ 25 for ρ∗ = 0.8.