Gas-liquid coexistence in a system of dipolar soft spheres

Ran Jia, Heiko Braun, and Reinhard Hentschke*Fachbereich Mathematik und Naturwissenschaften, Bergische Universität, D-42097 Wuppertal, Germany

�Received 26 July 2010; revised manuscript received 9 November 2010; published 3 December 2010�

The existence of gas-liquid coexistence in dipolar fluids with no other contribution to attractive interactionthan dipole-dipole interaction is a basic and open question in the theory of fluids. Here we compute thegas-liquid critical point in a system of dipolar soft spheres subject to an external electric field using moleculardynamics computer simulation. Tracking the critical point as the field strength is approaching zero we find thefollowing limiting values: Tc=0.063 and �c=0.0033 �dipole moment �=1�. These values are confirmed byindependent simulation at zero field strength.

DOI: 10.1103/PhysRevE.82.062501 PACS number�s�: 64.70.F�, 05.50.�q, 61.20.Qg, 82.35.�x

I. INTRODUCTION

Somewhat over a decade ago a number of theoretical pa-pers predicted the absence of a gas-liquid �g-l� critical pointin systems consisting of particles with no other source ofattraction than dipole-dipole interaction �1–4�. This conclu-sion is in agreement with a number of computer simulationresults obtained for dipolar systems. Caillol �5� searchedwithout success for the g-l transition in systems of dipolarhard spheres �DHS� using NPT and Gibbs–Ensemble–MonteCarlo. This work was motivated by an early Monte Carlo�MC� study of a 32-particle DHS system claiming that thesystem undergoes g-l phase separation �6�. Van Leeuwen andSmit �7� studied a modification of the Stockmayer potential�Lennard-Jones �LJ� plus dipole-dipole interaction�, wherethe dispersion attraction is multiplied by a factor �. In thelimit �→0 the model reduces to the dipolar soft sphere�DSS� potential. For � less than a certain threshold van Leeu-wen and Smit conclude that g-l criticality is absent due tochain formation. At the same time Stevens and Grest �8�studied the DSS system in an applied field. Whereas for non-zero field strengths they do observe coexistence, their con-clusion in the zero field case is that coexistence most likelydoes not occur. Szalai et al. �9� use computer simulation toinvestigate thermodynamics and structural properties of thedipolar Yukawa hard sphere �DYHS� fluid. They find that athigh dipole moments the g-l coexistence disappears whilechain-like structures appear in the low density fluid phase.

On the other hand other researchers, in addition to theaforementioned study by Ng et al. �6�, have reached the op-posite conclusion. McGrother and Jackson �10� induce g-lcoexistence in a hard-core dipolar system by making themolecules nonspherical, i.e., they consider hard spherocylin-ders with central longitudinal point dipole moments. DHSare again studied via MC in Ref. �11� by Camp et al. Basedon their calculation of the equation of state and the free en-ergy the authors find evidence in favor of an isotropic fluid-to-isotropic fluid phase transition. Pshenichnikov andMekhonoshin �12� apply MC to simulate DHS using openboundaries. Applying an extra field confining the particles toa spherical region they do observe a gaslike distribution

within this region or a pronounced clustering depending onthe strength of dipolar interaction. They interpret this as in-dication for phase separation in the DHS bulk system. Gan-zenmüller and Camp �13� track the g-l coexistence whichthey find in systems of charged hard dumbbells as the dumb-bell length is decreased toward the DHS limit. Via extrapo-lation of their MC results obtained for finite dumbbell lengththey find a g-l critical point in the DHS limit. Comparableresults are obtained by two of the present authors using a softsphere version of this system �15�. Almarza et al. �14� useMC to investigate a mixture of hard spheres and DHS. Theyfind critical parameters for the g-l equilibrium extrapolatedfrom their mixture results in the limit of vanishing neutralhard sphere concentration in accord with the extrapolationresults in Ref. �13�, when the dumbbells approach the DHSlimit. Kalyuzhnyi et al. �16� use MC to study the g-l coex-istence in the DYHS fluid. Again the critical point may betracked as the DHS limit is approached by decreasing thestrength of the attractive Yukawa potential. These authorsfind a critical point for values of the control parameter rep-resenting the “distance” from the DHS limit, which are farlower than the limit set by the earlier study in Ref. �9�. Con-tinuation of this work in Ref. �17� however results in theconclusion that phase separation is not observable beyond acritical value of the aforementioned parameter.

In the present work we discuss the results of extensiveMD simulations of the DSS system with and without appliedfield. We find that the DSS system indeed undergoes a g-llike transition from a low density phase containing smallrings in a monomer dominated background to a high densitydisordered chain structure. The values of the critical param-eters are Tc�0.063 and �c�0.0033 �dipole moment �=1�.

II. MODEL AND SIMULATION METHODOLOGY

We use our previous model for the Stockmayer fluid inRef. �18� omitting the r−6-dispersion attraction, i.e.,

U = �i�j

4

rij12 −

1

2�� iTij�� j − �� i · E� i

ext −1

2g�� i · M� i �1�

is the total potential energy �cf. Eq. �1� in Ref. �18��. In ourunits the usual Lennard-Jones parameters � and � are equalto one. In addition 4�o=1 ��o: vacuum permittivity�. Thefirst term describes the soft core repulsion between DSS par-

*Author to whom correspondence should be addressed;hentschk@uni-wuppertal.de

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ticles i and j separated by the distance rij. Only particleswithin the cutoff distance rcut �here: rcut=5.5� interact explic-itly. Notice that tests with larger cutoffs up to 9.5 do notyield significantly different results. The particle number isN=900. Variation of N by a factor of two did not affect thepresent results. Soft-repulsive interactions from beyond rcutare included in standard fashion. The remaining terms, mak-ing use of the summation convention �i� j�, are due to theinteraction between point dipole moments, �� i��i=1�, locatedon every DSS site i. The quantity Tij is the dipole tensor �cf.Eq. �6� in Ref. �18��. E� i

ext is the constant external field as it isfelt at site i. We keep the index i as a reminder that therelation between a true external field E� ext, e.g., generated bycapacitor plates between the system is placed, and E� i

ext doesdepend on how long-range interactions are handled in thesimulation. Two scenarios are particular relevant: �i� fixedcharge density on the capacitor plates; �ii� fixed potential. �i�corresponds to E� ext, the electric field in a slit separating thedielectric from a capacitor plate, held constant. �ii� corre-sponds to E� ��, the average or Maxwell field inside the di-electric, held constant. We use the following relations,E� ext=�E� �� and E� i

ext= 32�+1E� ext. For nonvanishing electric field

the dielectric constant, �, is computed via �M� i� / �4rcut3 /3�

= ��−1�E� �� / �4�, where M� i is the total dipole momentinside the cutoff sphere surrounding particle i. For vanishingelectric field � is computed via Eq. �4� in Ref. �23�. Thelast term in Eq. �1� is a reaction field term correcting theneglect of explicit dipole-dipole interactions beyondrcut�g= 2��−1�

�2�+1�1

rcut3 �.

We carry out NVT-MD simulations. Forces, torques, andpressure are computed according to Eqs. �9�–�12� in Ref.�18�. The translational motion of the particles is governed byr�̈i=F� i, whereas the equation of motion governing the dipole

orientation follows via N� i=I��̈ i. Here I is the moment ofinertia with respect to the momentary axis of rotation. The

angle of rotation vector �� i can be replaced by �� i using ��̇ i

��� i=��̇ i and ��̇ i ·�� i=0. The resulting equation of motion for�� i can be found in section 8.2 of Ref. �20� or in the appendixof Ref. �19�. The equations of motion are integrated using thevelocity Verlet algorithm. Temperature is controlled via theweak coupling method of Berendsen et al. �21�. Notice thatthe rotational temperature is given by �2�2�−1���̇ 2�=T. Atequilibrium the rotational temperature must be equal to thetranslational temperature of course. Notice also that here weset the moments of inertia with respect to the major axesequal to one in LJ units. G-L phase coexistence curves areobtained by the same method as introduced previously�22,23�, i.e., phase coexistence is established using the Max-well construction method applied to simulation isotherms atdifferent temperatures. The toughest challenge every meth-ods applied to dipolar systems faces is the formation of ex-tended chains and related structures �loops, etc.� �17�. Theformer method of choice, Gibbs-Ensemble MC, and also thenow more frequently used grand-canonical MC �in theframework of histogram reweighing techniques and mixed-field finite size scaling� run into problems due to the lowprobabilities of inserting or deleting particles from extended

clusters. The problem is reduced via biased particle orienta-tion. This however is not straightforward because no system-atic biasing scheme is available. Here we rely on MD, be-cause it avoids the above sampling problem. In addition allisotherms are sampled along both directions in order to de-tect possible deviations. Of course, there is no ultimate proofthat long relaxation times associated with extended aggre-gates do not affect the result. The critical parameters areobtained via application of the following scaling relations�24� using Ising critical exponents � �0.110, ��0.326,��0.5 �25��: �F−�G�A0t�+A1t�+�, ��F+�G� /2��c+D0t1− +D1t, and P− Pc� P0t+ P1t2− + P2t2�t= �T−Tc� /Tc�. The Ising universality class is assumed in thiscase. This is in line with previous work aimed at the criticalparameters of near DSS systems �17�, where the mapping onto the ordering parameter distribution of the 3D Ising univer-sality class is part of the mixed-field finite size scaling em-ployed. In general good matching of the data to the limitingIsing-class distribution is taken as consistency check forIsing-type criticality.

III. RESULTS

Figure 1 is a graphical summary of past attempts to locatethe g-l critical point in systems of dipolar spheres includingour present results. The horizontal lines represent isothermsalong which no g-l phase separation was observed. Filledsymbols refer to temperature-density-coordinates in caseswhere a DSS/DHS critical point is found or it is claimed thata critical point is found. Open symbols also indicate criticalpoints, but the respective systems are not �quite� DSS orDHS. The attendant models do contain a control parameterand only for a certain limiting value of this control parameterthe respective models are DSS or DHS �cf. the introduction�.In these cases Fig. 1 shows the critical points obtained di-rectly from simulation corresponding to control parametervalues closest to the DHS or DSS limit, even if extrapolationto the respective DSS or DHS limit does not yield finitecritical parameters. The solid-diamond-result probablyshould be excluded, because of the small system size �32particle�. The results of Camp et al. �solid square and dottedcircle� are obtained with different models. While in the firstcase �charged hard dumbbells� extrapolation to the DHSlimit was possible, the same was not possible in the second

����

��

��

��

��

����

0.0 0.1 0.2 0.3 0.4Ρ0.0

0.1

0.2

0.3

0.4

0.5T

FIG. 1. Graphical summary of past attempts to locate the gas-liquid critical point in systems of dipolar spheres including the re-sults of this work. Horizontal lines �5�; open circle �7�; open square�8�; open triangle �9�; dotted circle �17�; solid diamond �6�; solidsquare �13�; solid circles �15�: �higher density� and this work �lowerdensity��.

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case �dipolar Yukawa-fluid�. Here the symbol marks thesimulation result for the g-l critical point closest to the DHSlimit. Notice that all DSS and DSS-like results are mappedonto the DHS system using the following conversion: wecompute an effective hard core diameter, �ef f�T�, via B2�T��B2�HS�=4� /6��ef f

3 , where B2�HS� is the second virial co-efficient of hard spheres and B2�T� is the second virial coef-ficient of soft spheres at temperature T. The result is�ef f�T�= ���3 /4��1/3�4 /T�1/12 where ���3 /4��1/3�1.07011.Note that the corresponding result using the Barker-Henderson formula �26� is �ef f�T��1.0555�4 /T�1/12. The as-sumed conversion is Tc

DHS��ef f3 �Tc

DSS�TcDSS and �c

DHS

��ef f3 �Tc

DSS��cDSS.

Figures 2 shows the critical temperature, Tc, and the criti-cal density, �c, as function of field strength, E, for E=E��

and E=Eext �Tc only�. In the former case we obtain a mo-notonous decrease of Tc with decreasing E��. This is quali-tatively identical to the corresponding result for the Stock-mayer system �cf. Fig. 7 in Ref. �18��. The difference is thathere Tc is much lower, due to the absence of dispersion at-traction, and the initial quadratic increase of Tc crosses overto saturation at much smaller field strength. The initial in-crease of �Tc=Tc−Tc,E=0 is proportional to given by �E���2

as can be shown on quite general thermodynamic grounds�27� �paragraph 18; for a more detailed discussion see Ref.�28��. An analogous thermodynamic perturbation expansionalso yields ��c� �E���2. The inserts in Fig. 2 include qua-dratic fits �solid lines�, i.e., we use Tc=Tc,E=0−AT�E���2 and�c=�c,E=0−A��E���2, where Tc,E=0, �c,E=0, AT, and A� are fitparameters, to Tc and �c at the lowest field values. For theamplitudes we obtain AT�12.3 and A��2.5, respectively. Inthe case of the Stockmayer system �18�, where the conditionsare chosen to suppress aggregate formation, the correspond-ing amplitudes may be obtained in good agreement with thesimulation via a mean field theory, based on the van der

Waals equation combined with Onsager’s theory of a singledipole embedded in an infinite dielectric �18�. Unfortunately,this mean field theory does not apply here. In comparison tothe Stockmayer system, whose near critical compressibilityfactor is close to 0.35, a value indicating simple liquid be-havior, the same quantity now is much smaller ��0.04�. Thiscan be understood on the basis of Flory’s equation of statefor linear polymers �29� �reversible dipole chains in thepresent case�, which yields a compressibility factor propor-tional to the inverse polymer length, n, for large n. However,Onsager’s theory does not apply to dipole chains, and thecomputation of the aforementioned amplitudes is not in ac-cord with the above values.

Figure 2 includes critical temperatures obtained at con-stant Eext. It was shown in previous work on the Stockmayerfluid that a nonzero Eext lowers the critical temperature �18�.Here this can be used in principle to bracket the DSS criticaltemperature. However, it is numerically impossible to obtainvan der Waals loops which can be used for determining boththe critical temperature and the critical density. Neverthelessit is possible to simulate near critical isotherms to an extendthat makes the observation of a pressure decrease possibleand thus allows the estimation of at least Tc. The valuesobtained in this work are shown as crosses in Fig. 2.

Also included in Fig. 2 are the results obtained in Ref. �8�on the DSS system in an external field. Note that these au-thors use a dipole moment �=2.5 instead of the value �=1.0 used in this work. We convert their critical parametersto the case �=1 requiring the invariance of theNVT-Boltzmann factor. The resulting mapping is T=�−8/3T�,E��=�−5/3H�, and �=�−2/3��, where �=�� /��=2.5�. The in-dex � here refers to the reduced quantities indicated by thesame index in Ref. �8�. The external field dependence of theconverted critical temperature follows our results in reason-able accord. However, in the case of the critical density thereare significant deviations as the field strength becomes small.This may be caused by sampling problems of the Gibbs-Ensemble MC algorithm, used here without additional bias-ing scheme as discussed above

Figure 3 consists of a series of simulation snapshotsshowing the low and the high density phase at coexistencenear Tc. The bottom pair of snapshots is for E��=0.5. Thestructures are similar to the corresponding ones by Stevensand Grest �8� obtained for H�=1.0 �or �0.22 if converted asdescribed above�. As the field strength is reduced, one ob-serves the increase of orientation disorder. On the low den-sity side of coexistence smaller chains or linear aggregatesappear. Very close to the DSS limit there are small loopsdispersed in a monomer background. Because the coexist-ence curve is very flat close to Tc, we note that at a slightlyhigher T one observes more and larger loops still dispersedin a monomer dominated background. Note that loops ratherthan linear aggregates occur at small field strength and lowaggregation numbers because the energetic gain of an extrareversible bond outweighs the entropic loss �cf. Ref. �23��.

Figure 4 shows the number average aggregate size, �L�,setting the density, �, and the temperature, T, equal to thecritical values for different strengths of the external fieldE��. There is a pronounced and expected increase of �L�when E�� becomes small. The average aggregate size in-

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� ��

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�

0.0 0.2 0.4 0.6 0.8 1.0 1.2 E

0.02

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0.14

Tc

E���

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0.010

0.015

0.020

0.025

0.030Ρc

���������

0.01 0.1 1 E���0.00010.0010.01

�Ρc

FIG. 2. Top: gas-liquid critical temperature, Tc, vs field strength,E. Circles: E=E��; crosses: E=Eext. Inset: double logarithmic plotof �Tc=Tc−Tc,o vs E�� including a quadratic fit to low E data.;bottom: gas-liquid critical density, �c, vs. field strength, E��. Inset:double logarithmic plot of ��c=�c−�c,o vs E��. Open squares inboth panels are results computed on the basis of Ref. �8� using theconversion relations explained in the text.

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creases but apparently does not diverge in the limit E��

→0, i.e., in the DSS limit. We determine whether or not twoparticles are neighbors in the same aggregate via a distancecriterion. Two dipoles are considered neighbors if they arecloser than ro. Even though the value of ro affects the result,as demonstrated by the example shown as inset in Fig. 4, thegeneral conclusions do not depend on ro �within a reasonablerange�. If ro is too small, no neighboring particle is found.For too large ro, the distinction between aggregates is lost�typically for ro greater are close to �−1/3�. Here we use ro=2.0. Note that �L� includes monomers, i.e., �L�=1 meansthat only monomers are present. We observe that close toE��=0 the number of rings and the number of chains arecomparable.

It is also interesting to map the critical temperature anddensity obtained in this work onto the DHS system �cf. Fig.1�. Multiplication of the critical temperature Tc�0.063, with�ef f�Tc�3 yields 0.22. Similarly we may map the critical den-

sity �c�0.0033 to the hard sphere case via �ef f3 �c which

yields 0.011. These values correspond to the left of the twosolid circles in Fig. 1. The closeness of this and the dumbbellresult presents additional evidence for the existence of a g-llike critical point in a purely dipolar system.

In this context it is also interesting to relate the dumbbellsystem to the so called restricted primitive model �RPM�,i.e., charged hard spheres, of ionic liquids. A rather goodmean field description of g-l criticality in the RPM is basedon the idea of ion clusters as basic entities, where ion pairs�dumbbells� appear to be of dominant importance �30�. Anatural dumbbell length for ion pairs is d=1. In an extensionof previous work �15�, where the focus is on the limit d→0, we have computed the critical parameters for chargedsoft dumbbells with d=1.0. The result is Tc

CSD=0.017 and�c

CSD=0.011. Using a similar conversion as in the previousparagraph �with �ef f instead of �ef f

3 in the case of Tc; cf. Eq.�1� in Ref. �31�� we obtain for charged hard dumbbellsTc

CHD�0.03 and �CHD�0.1 �here the number of ions pervolume�. This is reasonably close to the RPM critical param-eters of Panagiotopoulos, i.e Tc

RPM =0.0489 and �cRPM

=0.076 �31�. Note that the value of the charge used in Ref.�31� in our units is one corresponding to our �=1. The close-ness of these two sets of critical parameters together with theresults that the dumbbell critical point approaches smoothlythe DSS value with decreasing dumbbell length presents evi-dence that the observed DSS critical point indeed is a gas-liquid critical point.

�1� R. P. Sear, Phys. Rev. Lett. 76, 2310 �1996�.�2� R. van Roij, Phys. Rev. Lett. 76, 3348 �1996�.�3� J. M. Tavares et al., Phys. Rev. E 56, R6252 �1997�.�4� Y. Levin, Phys. Rev. Lett. 83, 1159 �1999�.�5� J.-M. Caillol, J. Chem. Phys. 98, 9835 �1993�.�6� K.-C. Ng et al., Mol. Phys. 38, 781 �1979�.�7� M. E. van Leeuwen et al., Phys. Rev. Lett. 71, 3991 �1993�.�8� M. J. Stevens et al., Phys. Rev. Lett. 72, 3686 �1994�.�9� I. Szalai et al., J. Chem. Phys. 111, 337 �1999�.

�10� S. C. McGrother et al., Phys. Rev. Lett. 76, 4183 �1996�.�11� P. J. Camp et al., Phys. Rev. Lett. 84, 115 �2000�.�12� A. F. Pshenichnikov et al., Eur. Phys. J. E 6, 399 �2001�.�13� G. Ganzenmüller et al., J. Chem. Phys. 126, 191104 �2007�.�14� N. G. Almarza et al., J. Chem. Phys. 129, 234504 �2008�.�15� H. Braun et al., Phys. Rev. E 80, 041501 �2009�.�16� Yu. V. Kalyuzhnyi et al., EPL 84, 26001 �2008�.�17� G. Ganzenmüller et al., Mol. Phys. 107, 403 �2009�.

�18� R. Jia et al., Phys. Rev. E 80, 051502 �2009�.�19� W.-Z. Ouyang et al., J. Chem. Phys. 127, 164501 �2007�.�20� D. C. Rapaport, The Art of Molecular Dynamics Simulation

�Cambridge University Press, Cambridge, England, 1995�.�21� H. J. C. Berendsen et al., J. Chem. Phys. 81, 3684 �1984�.�22� R. Hentschke et al., Phys. Rev. E 75, 011506 �2007�.�23� J. Bartke et al., Phys. Rev. E 75, 061503 �2007�.�24� M. Ley-Koo et al., Phys. Rev. A 23, 2650 �1981�.�25� A. Pelissetto et al., Phys. Rep. 368, 549 �2002�.�26� J. A. Barker et al., J. Chem. Phys. 47, 4714 �1967�.�27� L. D. Landau et al., Electrodynamics of Continuous Media

�Pergamon Press, New York, 1984�, Vol. 8.�28� A. Onuki, EPL 29, 611 �1995�.�29� P. J. Flory, Principles of Polymer Chemistry �Cornell Univer-

sity Press, Ithaca, 1953�.�30� M. E. Fisher et al., Phys. Rev. Lett. 71, 3826 �1993�.�31� A. Z. Panagiotopoulos, J. Chem. Phys. 116, 3007 �2002�.

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30.0

15.0

7.0

�L�

���������������

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2.0 2.5 3.0 3.5ro

23456�L�

FIG. 4. Average aggregate size, �L�, vs E�� at criticality. Inset:�L� vs ro at E��=0.1.

FIG. 3. Simulation snapshots along the coexistence curve nearthe respective critical temperature. Top: E��=0.005, T=0.064, �=0.000 121 9 �left�, �=0.012 18 �right�; middle: E��=0.05, T=0.076, �=0.001 57 �left�, �=0.009 54 �right�; bottom: E��=0.5,T=0.114, �=0.0129 �left�, �=0.0202 �right�.

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