arXiv:cond-mat/9505150v1 30 May 1995 re,adta hsi o xlctyicue ntesim- the ferroelectric in included yielding explicitly in not is role the this of key that and nature a order, long-range plays the forces that dipolar dipolar argue disordered They spatially for theory systems. field mean a ward might one order. [13,14], ferroelectric symmetry that lattice argue the order on long-range crystals depending antiferroelectric perfect view or that In ferroelectric recalling [4]. exhibit and phase observations, liquid these the of in fer- develops consequently, order and, roelectric [13] lattice fer- short- tetragonal-I the in of roelectric seen development those resembling ther- the correlations full spatial allows range in This are freedom equilibrium. differ of transla- mal [4,5] degrees coupled systems strongly rotational the fluid and that The tional in case [12]. frozen be- the glass from system spin the a and as positional haves frustration quenched random the creates ferrofluids, disorder frozen In follows. as a of existence [4,5]. phase the crystal indicate liquid of clearly ferroelectric simulations fluids computer hand, dipolar spin other simple magnetic the On frustrated [12]. randomly glasses irreversibil- of magnetic reminiscent exhibit solvent ities non-magnetic a particles For magnetic in containing behavior. ferrofluids phase new frozen interesting example, to lead can forces dipo- lar order, to intrinsic positional coupling long-range spatial-orientational of strong the absence been the has in It that [3–5]. shown [6] simulations computer ferrofluids with theoretically and [1,2,6], frozen [3,7–11] experimentally amorphous examined and been have [1–3], [4,5] lattices phases Diluted fluid attention. considerable attracted 82.20Wt 77.80.-e, 64.70.Md, numbers: PACS transitions. insi ferroelectric significant of provides nature results medium. the our the into of of analysis motion t physical the to A and sensitive very dimensionality is vector existence spa dipole its in that possible is but consid- order systems, ferroelectric are random that media shown is disordered It Ran- dynamically ered. simulation. and computer frozen by domly studied o two are one, have components T which three vectors materials. dipole by dipolar distinguished disordered models positionally in order tric hslte drse ai usin ocrigferroele concerning questions basic addresses letter This nrcn aesZagadWdm[1 aeptfor- put have [11] Widom and Zhang papers recent In be might observations these of interpretation simple A have materials dipolar disordered spatially Recently, ∗ eateto hmsr,Uiest fBiihClmi,V Columbia, British of University Chemistry, of Department specific rettoa reigi ptal iodrdDplrSy Dipolar Disordered Spatially in Ordering Orientational pta orltosaerqie for required are correlations spatial † RUF 04Wsro al acue,BiihClmi,C Columbia, British Vancouver, Mall, Wesbrook 4004 TRIUMF, .Ayton G. ∗ .J .Gingras P. J. M. , Nvme ,2018) 5, (November tially hree ght he c- r 1 oetdpl steXZmdl nalcsstepair the cases all In model. com- XYZ three and the potential, the models as to XY refer dipole are and shall ponent we these Ising simplicity of the notational two as for first to three The referred and two commonly one, considered. has are vector components dipole the simulations. (MC) where Carlo Systems Monte and (MD) molec- dynamics temperature ular constant using systems dipolar dered without system correlations?” a positional in fined-tuned arise argument question spontaneously this order general the roelectric of the In validity address the considered. and examine density we particle letter high present the arises of [4,5] phase fluids because dipolar crystalline of simulations liquid computer in ferroelectric found the par- magnetic whereas of concentration ticles, low ferroflu- the in from results observed work [6] behavior Their ids spin-glass density. critical the a that above implies possible is ferroelec- order long-range frustration tric systems, strong frozen the randomly despite in present that, propose Zhang Widom importantly, More and above. given interpretation ple hr 4 where n length, and r h otshr n ioedpl neatos The interactions. dipole-dipole and parameters soft-sphere the are h xsec fafrolcrcpaecnb eetdby polarization detected be average can the phase ferroelectric calculating a of [4,15–17]. existence continuum The assuming surrounding method boundary conducting summation perfectly periodic a Ewald using the account and interac- into conditions dipolar taken long-range were The tions vector. intermolecular ndas fined o ntedrcino h oa ntnaeu moment, instantaneous total the of direction M the in tor esb h eue density, reduced the by tems system. sample, temperature, ment, otmn osat l eut xlctypeetdare presented explicitly for results All constant. Boltzmann eivsiaetebhvo fdnesailydisor- spatially dense of behavior the investigate We ti ovnett hrceiedplrsf-peesys- soft-sphere dipolar characterize to convenient is It † = µ n .N Patey N. G. and , ∗ novr rts ouba aaaVT1Z1 V6T Canada Columbia, British ancouver, u P µ and 4 = DD ε ∗ i N P T ( =1 σ/r ( = 1)= (12) u 1 = 1) so h eei form generic the of is (12), steaslt eprtr and temperature absolute the is ε µ u µ ) i µ and T 1)=4 = (12) 12 and , i /N 2 ∗ ρ /εσ stedpl fparticle of dipole the is ∗ and − = h| σ 0 = 3( 3 P k r h udmna nt fenergy of units fundamental the are N ) nd 6 2A3 V6T anada µ B ∗ 1 i N . ε / 1 T/ε =1 .Ti est swl ihnthe within well is density This 8. stenme fprilsi the in particles of number the is 2 ( · σ/r where , r µ ˆ )( n h eue ioemo- dipole reduced the and , i ) µ · 12 d 2 ˆ ρ ∗ |i + · r where , V = stems u ) /r DD Cnln-ag fer- long-range “Can Nσ stevlm fthe of volume the is 5 P (12) + 3 /V e atcede- particle per µ d ˆ i 1 , h reduced the , saui vec- unit a is and · µ 2 k /r B r 3 sthe is sthe is 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. We note that for m∗ = 1. To verify that this is a real thermody- this transition temperature is much lower than the value namic transition, we have calculated the Binder ratio ∗ 4 2 2 (T = 35.2) predicted by Zhang and Widom. [12], gBinder ≡ (5/2) − (3/2)h|M| i/h|M| i , for systems ∗ The situation for the XY and XYZ models is very dif- with 64, 108 and 256 particles. A plot of gBinder vs T ferent. Although significant polarization was observed (see Fig. 2, inset) shows a clear crossing, and hence a at finite temperatures, P decreases monotonously with transition, at T ∗ = 1.9. Simulations were also carried increasing system size and appears to approach zero for out with m∗ = 20 but no significant orientational order an infinite system. Furthermore, the behavior of various was found above T ∗ = 0.1. Very slow convergence pre- spin-glass correlation functions and susceptibilities [12] vented calculations at lower temperatures. suggests that both systems are entering an anisotropic We have also investigated dynamically disordered XY spin-glass phase at nonzero temperature [20], and that and Ising systems. The XY model behaves much like the observed polarization for the XY and XYZ models the XYZ system described above. For the Ising case, ro- is due to a combination of short-range ferroelectric cor- tational dynamics cannot be used and a suitable Monte relations and finite-size effects. Test calculations for the Carlo scheme which allowed the substrate to move in- XYZ model using a denser frozen soft-sphere configura- dependent of the Ising dipoles was employed. P vs T ∗

2 results for m∗ = 1, m∗ = 5, and the randomly frozen and a ferroelectric phase is possible. As the mass of a system (m∗ = ∞) are shown in Fig. 3. We see that for substrate particle is decreased, the translational motion the Ising model the ordering behavior is essentially inde- becomes faster and the above condition is met at higher pendent of mass and that the results for the dynamically and higher rotational temperatures. Thus the observed disordered system with m∗ = 5 lie very close the those transition temperatures increase with decreasing mass. for the randomly frozen case. Heat capacities obtained As the substrate particles become sufficiently light, the by differentiating the average dipolar energy with respect transition temperature is determined only by the reac- to the rotational temperature are also shown in Fig. 3. tion field and hence becomes independent of mass. In The randomly frozen and m∗ = 5 results are very similar fact, m∗ = 1 gives essentially this limiting behavior and and indicate a phase transition at T ∗ ≈ 25. reducing the mass further has little effect on the transi- The dependence of the ferroelectric transition temper- tion temperature. ature on particle mass is shown in Fig. 4. The transi- In conclusion, the answer to the question raised at the tion temperatures were estimated from the heat capaci- outset is a subtle one. Our results for the frozen Ising ties and results are included for the XY and XYZ models. system and for the dynamically disordered XY and XYZ Results for the Ising system are not plotted because the models clearly demonstrate that it is possible to have transition temperature is essentially independent of the ferroelectric states without fine-tuned positional corre- mass. We see that as the mass increases the transition lations. However, they also show that if a ferroelec- temperature drops for both the XY and XYZ models. tric phase is to exist in a positionally random system, As noted earlier, for large masses and low temperatures the long-range spatially-independent correlations arising convergence becomes prohibitively slow, but it seems rea- through the reaction field must dominate the shorter- sonable to assume that the graph would simply continue ranged position-sensitive correlations which generally act with the transition temperature approaching zero in the to frustrate ferroelectric order. For the Ising system infinite mass limit. This is consistent with the fact that the discrete nature of the interaction (i.e., there is just we did not find a ferroelectric phase for randomly frozen not much opportunity for favorable interactions among systems at finite temperatures. neighbors) limits the build up of short-range disordering The present and previous results can be interpreted as correlations and there is a clear ferroelectric transition. follows. It is useful to divide the local field, Elocal, ex- For the dynamically random systems the orientational perienced by a particle in an infinite medium into two correlations opposing ferroelectric order are reduced by parts such that, Elocal = R + E, where R is a reac- the motion of the substrate resulting in a ferroelectric tion field contribution [9,15] and E is everything else. phase. On the other hand, for the randomly frozen XY The reaction field arises from the long-range nature of and XYZ models we find no indication of a ferroelec- the dipolar interactions; it is independent of the spatial tric phase at finite temperature. Rather, the disordering correlations and favors ferroelectric order. The remain- fields dominate and these systems appear to form nonfer- ing contribution, E, is sensitive to positional correlations roelectric spin glasses at low temperature. Evidence for and may or may not favor ferroelectric order. Thus if this is provided both by our direct simulations of frozen R dominates a ferroelectric phase is to be expected, but systems and by the extrapolation of the dynamically dis- if E is important the existence or nonexistence of a fer- ordered results to infinite mass. This conclusion must roelectric phase will depend on the details of the spatial remain a little tentative because direct MD simulations correlations. This simple picture allows us to rationalize at very low temperatures are not practical. Nevertheless, the various systems considered. For fully coupled dipo- a ferroelectric phase in the randomly frozen XY and XYZ lar fluids [4,5] the short-range spatial correlations (and models seems highly unlikely. hence E) can adjust (i.e., become tetragonal-I-like) in order to favor (or at least allow) a ferroelectric state. On the other hand, for randomly frozen systems the spatial ACKNOWLEDGMENTS correlations cannot adjust and at equilibrium E domi- nates R frustrating the formation of a ferroelectric phase We thank Z. R´acz, M. Widom and H. Zhang for use- except for the Ising case. Apparently, the discrete nature ful discussions. The financial support of the National of the Ising model makes it much less susceptible to the Science and Engineering Research Council of Canada is development of disordering fields than are the XY and gratefully acknowledged. XYZ systems. The reaction field dominates in the Ising system and gives rise to a ferroelectric state. The behavior of the dynamically disordered systems can also be understood. If the translational diffusion of the substrate is sufficiently fast compared to the dipolar reorientational time, the extent of spatially dependent frustrating correlations is limited, R can prevail over E,

3 ∗ [1] B.E. Vugmeister and M.D. Glinchuk, Rev. Mod. Phys. FIG. 1. The polarization P at Tmin vs 100/N for the 62, 4 (1990). randomly frozen Ising (circles), XY (squares) and XYZ (tri- [2] U.T. H¨ochli and M. Maglione, J. Phys. Condens. Matter angles). Results are included for 108 (XYZ only), 256, 392 1, 2241 (1989). (XYZ only), 500 and 864 particles. [3] H-J Xu, B. Bergersen, and Z. Racz, J. Phys. Condens. Matter 4, 2035 (1992); H-J Xu, Ph.D. thesis, University ∗ of British Columbia (1992). FIG. 2. P vs T (rotational) for dynamically random [4] D. Wei and G.N. Patey, Phys. Rev. Lett. 68, 2043 (1992); XY Z systems. The squares, triangles and circles are for ∗ Phys. Rev. A. 46, 7783 (1992). m = 1, 5 and 10, respectively. The error bars represent ∗ [5] J.J. Weis, D. Levesque, and G.J. Zarragoicoechea, Phys. one estimated standard deviation. gBinder vs T (rotational) Rev. Lett. 69, 913 (1992). is shown in the inset for N = 64 (squares), 108 (triangles) [6] W. Luo, S.R. Nagel, T.F. Rosenbaum, and R. E. Rosen- and 256 (circles) particles. weig, Phys. Rev. Lett. 67, 2721 (1991). [7] M.A. Zaluska-Kotur and M. Cieplak, Europhys. Lett. 23, ∗ 85 (1993). FIG. 3. P vs T (rotational) for the Ising model. Re- ∗ [8] B. Mertens, K. Levin, and G.S. Grest, Phys. Rev. B 49, sults are shown for dynamically random systems with m = 1 ∗ 15374 (1994). (squares) and m = 5 (triangles) and for the randomly frozen [9] D. Wei, G.N. Patey, and A. Perera, Phys. Rev. E 47, 506 case (circles). The heat capacities per particle, Cv/N, are ∗ (1993). plotted vs T (rotational) in the inset. [10] B. Groh and S. Dietrich, Phys. Rev. E 50, 3814 (1994); Phys. Rev. Lett. 72, 2422 (1994). [11] H. Zhang and M. Widom, J. Magn. Magn. Mater. 122, FIG. 4. The mass dependence of the disordered to ferro- ∗ 119 (1993); Phys. Rev. B 51, 8951 (1995). electric transition temperature TF (rotational). The squares [12] K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801 and triangles are results for the XY and XYZ models, re- ∗ (1986); K.H. Fischer and J.A. Hertz, Spin Glasses (Cam- spectively. The values of TF were obtained from plots of the ∗ bridge University Press, Cambridge, 1991). heat capacity, Cv/N, vs T (rotational) and a typical exam- [13] R. Tao and J.M. Sun, Phys. Rev. Lett. 67, 398 (1991). ple (XYZ model, m∗ = 1) is shown in the inset. The error [14] J.M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946). bars represent estimated uncertainties in the peak position. [15] S.W. de Leeuw, J.W. Perram, and E.R. Smith, Ann. Rev. Phys. Chem. 37, 245 (1986); Proc. R. Soc. A373, 27 (1980); A388 (1983). [16] P.G. Kusalik, J. Chem. Phys. 93, 3520 (1990). [17] For the present dense strongly dipolar systems the dielec- tric constants are sufficiently large that the reaction field is essentially independent of the exact value [15]. There- fore, conducting boundary conditions are an appropriate choice. [18] The critical volume fractions are 0.157 for the Ising sys- tem and 0.295 for the XYZ model [11]. Using the effec- tive hard sphere diameter 0.967σ [see J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, 2nd ed. (Aca- demic, London, 1986), chap. 6] the soft-sphere densities ρ∗ = 0.8 and 1.05 correspond to volume fractions of 0.379 and 0.497, respectively. [19] The results given in Fig. 1 are for typical frozen con- figurations. Tests show that the actual values of P vary a little from configuration to configuration but that the same general trend is found. Ideally, accurate values of P could be obtained by averaging over many frozen configu- rations for each N. However, this would be very expensive computationally and is not necessary here; we are only interested in the limiting value of P as N becomes large and not in its detailed N dependence. [20] M.J.P. Gingras, Phys. Rev. Lett. 71, 1637 (1993). [21] B. Bergersen and Z. R´acz, Phys. Rev. Lett. 67, 3047 (1991); H-J Xu, B. Bergersen, and Z. R´acz, Phys. Rev. E 47, 1520 (1993)

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