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CUETOS et al.PHYSICAL REVIEW E 75, 061701 2007rods, I-I demixing becomes only stable for comparably largesphere sizes.The N-N Parsons-Lee spinodals for different HSC-HSmolecular sizes are shown in the lower panels of Fig. 5. Thecorresponding nematic branches of the I-N coexistencecurves are also shown for comparison. In this case, the computationof the spinodals from the conditions of Eq. 16involves the iterative optimization of h for each HS volumefraction and packing fraction of the mixture throughEqs. 13 and 14. For the mixtures involving HSC particleswith L * =20 middle panel of Fig. 5, the comparison withthe I-N coexistence curves indicates that for D * 1, N-Ndemixing requires too large packing fractions and is thereforeunstable with respect to I-N demixing within theParsons-Lee approximation. Only for D * 1 would the N-Nphase separation be allowed, so that the present results forI-N equilibrium for this range of geometries should be consideredwith caution. Interesting, the nematic branch of theI-N coexistence curve for L * =20, D * =0.7 displays a discontinuityafter the turning point i.e., already within the domainof demixing where a small change in the composition of theisotropic branch induces an abrupt change in both the compositionand packing fraction of the coexisting nematic state.It is interesting to notice that the appearance of such discontinuityis coincident with the evolution of the spinodal topacking fraction values neatly below those of the I-N curve.It would therefore be plausible that the range of thecomposition-packing fraction diagram not covered by theI-N curve corresponds to the domain of stability of N-N coexistence.This expectation is consistent with the descriptionof the overall phase diagram given by Lekkerkerker andStroobants 54.For the case of HSC particles with L * =300 bottom panelof Fig. 5, I-N coexistence is favored in all cases with D *5. This is illustrated by the comparison for D * =10 showingthat too large values are required for N-N equilibrium.For smaller sizes of the HS particles, represented by D * =1 inFig. 5, a composition domain appears in which N-N separationcompetes favorably, according to the smaller of thespinodal with respect to the nematic branch of the I-N coexistence.Such a situation arises only for sufficiently high valuesof x v i.e., for large molar fractions of the HS particles,whereas for low values of x v the I-N separation remainsstable. In this case, the entrance of the possible N-N demixingtakes place in a region where the algorithm to calculatethe I-N coexistence did not converge to any solution seeFig. 4. Such lack of convergence is likely to be related to adiscontinuous gap in the nematic branch similar to the onedescribed above for the case L * =20, D * =0.7. In this case,however, the boundary nematic states are too different fromeach other to be linked by the genetic algorithm employed inthe present work. This interpretation then suggests the entranceof the fluids L * =300, D * 3 in the N-N coexistenceregion at some point after the turning point of the I-Nbranch.A modification of the present methodology consideringindependent h distributions for the coexisting phases i.e.,allowing N-N equilibrium led to the same I-N coexistingcurves within the range of convergence of the original algorithmone of the orientational distributions became isotropic.However, the algorithm did not reach convergence forthe pressures and chemical potentials within the expecteddomain of N-N stability. In conclusion, in order to characterizein a more complete way the phase diagram of HSC-HSmixtures, an extended investigation would be required. Suchinvestigation, in addition to implementing an efficient algorithmto characterize in detail the N-N coexistence binodalsshould incorporate the lamellar phase into the treatment. Thislies beyond the aim of the present work, where the domain ofstability of the I-N coexistence diagram has been determinedfor particle sizes where the lamellar phase is not expected tointerfere. For these cases, the region where N-N separationmay become stable within the Parsons-Lee approximationhave been established e.g., L * =20, D * 1orL * =300, D *3 for sufficiently high x v values.Figure 6 represents the Parsons-Lee equations of state atconstant mixture composition isopleths for rod-sphere mixtureswith L * =20 and D * =1,2,4 and with L * =300 and D *=2,5,7. The range of volume fractions of spheres consideredin each case corresponds to the region of relativelysmall composition asymmetry in the coexisting I-N diagramsee Fig. 3. Such region lies before the entrance of the systeminto the demixed regime and sufficiently anticipatedwith respect to the region where N-N separation may becomerelevant. The different isopleths show the expected positivecorrelation between pressure and packing fraction. The displacementof the I-N transition toward greater values withgrowing x v i.e., destabilization of the nematic phase by thedepletion effects induced by the spherical component canalso be appreciated, as already noticed above in the coexistencediagrams. For the smaller HS diameters considered inFig. 6 it is observed that the system pressure increases appreciablywith growing x v in both the I and N phases atfixed . This behavior, also found in Monte Carlo simulationstudies 43, occurs as a natural consequence of the multiplicationof the number density of the system with anyincrease of x v . This in turn follows from the much smallervolume of the spherical particles in comparison to the rods.The opposite effect is observed for sufficiently large HS diametersonce the HS molecular volume becomes greater thanthe HSC one. In this case, any increase in the volume fractionof spheres, for fixed , leads to a decrement in theabsolute number of particles per unit volume and, hence, to adecrease in the system pressure.Interestingly, for intermediate D * values, where the HSand HSC molecular volumes become comparable, a situationarises where the composition has a weak effect on thepressure-packing fraction representation of the isopleth. Thisis a consequence of the effective compensation of the excludedvolume contributions that drive the system pressure.Moreover, Fig. 6 shows that such steric effects vary significantlyfrom the isotropic to the nematic phase so that it ispossible to find roughly composition-independent isoplethsin one phase and appreciable composition effects on theother phase. In fact, for the mixtures L * =20, D * =2 and L *=300, D * =5, whereas a close overlap is found for the isoplethswith the different x v values in the isotropic phase, asignificant shift of pressure with x v is still observed in thenematic phase. The complementary situation of an almostnegligible dependence on x v of the isopleths in the nematic061701-8
USE OF PARSONS-LEE AND ONSAGER THEORIES TO…PHYSICAL REVIEW E 75, 061701 2007bP*80604020L*=20D*=1x V=0.150.100.050.01L*=300D*=2x V=0.40.30.20.1500400300200100bP*080604020010080L*=20D*=2L*=20D*=40.150.01L*=300D*=50.1L*=300D*=70.4x V=0.40.10.10.40150100500150FIG. 6. Equations of state at constant compositionisopleths predicted by Parsons-Lee theoryfor the isotropic and nematic phases of binaryHSC-HS mixtures with fixed L * =20 and 300 anddifferent HS diameters D * . The indicated valuesfor the volume fraction of spheres x v correspondto the mixed regime of the two phases see Fig. 3for reference. The dimensionless pressure bP * isdefined as in Fig. 3.bP*6040200.010.1500.0 0.1 0.21000.10.4500.00 0.01 0.02 0.03 0phase is found for L * =20, D * =4 and L * =300, D * =7.Itisworthwhile noticing that the weakest composition dependenceof the isopleths in the isotropic phase is found, in boththe L * =20 and L * =300 mixtures, for molecular volume ratiosbetween the HS and HSC particles of v s /v c 0.25. Onthe other hand, for the nematic phase, this situation is foundfor v s /v c 2 for L * =20 and v s /v c 0.75 for L * =300. Thesevalues are in contrast to the common choice of v s =v c i.e.,D * =3.1 for L * =20 and D * =7.7 for L * =300, often employedwhen the HS fluid is taken as reference system for the HSCfluid in approximate theories. In any case, only for mixturesinvolving relatively short rods it seems possible to find aneffective value of D * for which the isopleth of the mixturebecomes insensitive to the composition of the mixture simultaneouslyin the isotropic and the nematic phases.Simulation or experimental studies of the equation of stateEOS of HSC-HS mixtures, that could provide a referencefor the evaluation of the present theoretical results, arescarce. Isopleths for the isotropic and nematic phases havebeen reported in previous works from NPT-MC simulationsfor the L * =5, D * =1 fluid 43. For the present paper,we have extended those simulations to include the caseL * =5, D * =3 employing the same methodology described inRef. 43. Figure 7 illustrates the level of agreement observedbetween the Parsons-Lee isopleths in the isotropicand nematic phases and the results from NPT-MC simulationsfor these particular cases. In addition, alternative EOSfor the isotropic phase of the HSC-HS system from scaledparticle theory proposed by other authors 56,57 are alsoshown. It can be appreciated that these latter EOSs, originallydeveloped for shorter HSC molecules, remain fairlyaccurate at the L * =5 elongation considered. Nevertheless,the EOS from Parsons-Lee theory compares better with thesimulation data than any of the curves from scaled particletheory. Similar or better level of accuracy for the Parsons-Lee EOS can be expected for mixtures with greater L * values.In the immediacy of the isotropic-nematic transition andin the nematic phase itself, although Parsons-Lee theorycompares fairly well with simulation, the agreement is worsethan in the isotropic phase. This can be attributed to theintrinsic limitations of the Parsons-Lee approximation tomanage orientational order. In fact, the Parsons-Lee resultspresently obtained for the nematic phase of HSC-HS mixtures,show a similar accuracy when compared to MC data asfound for the pure HSC fluid 48.061701-9
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