Use of Parsons-Lee and Onsager theories to predict nematic and demixing behavior in binary mixtures of hard rods and hard spheres

Use of Parsons-Lee and Onsager theories to predict nematic and demixing behavior in binary mixtures of hard rods and hard spheres Alejandro Cuetos Soft Condensed Matter, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands Bruno Martínez-Haya*and Santiago Lago Departamento de Sistemas Físicos, Químicos y Naturales, Universidad Pablo de Olavide, 41013 Seville, Spain Luis F. Rull Departamento de Física Atómica, Molecular y Nuclear, Area de Física Teórica, Universidad de Sevilla, Apdo. 1065, 41080 Seville, Spain 共Received 21 September 2006; revised manuscript received 26 March 2007; published 4 June 2007兲 Parsons-Lee and Onsager theories are formulated for the isotropic-nematic transition in a binary mixture of hard rods and hard spheres. Results for the phase coexistence and for the equation of state in both phases for mixtures with different relative sizes and composition are presented. The two theories explain correctly the general behavior observed in experiments and computer simulations for these fluids. In particular, the theory accounts for the destabilization of the nematic phase when spherical or globular macromolecules are added to a system of rodlike colloids, and the entrance of the system into a demixed regime at high volume fractions of the spherical particles. Upon demixing a nematic state rich in rods coexists in equilibrium with an isotropic state much more diluted in the rodlike component. Onsager theory fails on quantitative grounds for aspect ratios of the rodlike molecules smaller than 100, and in the cases where the molar fractions of spheres becomes close to unity. On the contrary, the Parsons-Lee approximation remains accurate down to aspect ratios as small as 5. The spinodal analysis indicates that the isotropic-isotropic and nematic-nematic coexistences become feasible for sufficiently large spheres and long rods, respectively. The latter type of coexistence interferes partially with the isotropic-nematic coexistence regime of interest to the present work. Overall, the study serves to rationalize and control key aspects of the behavior of these binary nematogenic colloidal systems, which can be tuned with an appropriate choice of the relative size and molar fractions of the particles. DOI: 10.1103/PhysRevE.75.061701 PACS number共s兲: 64.70.Md, 61.30.Cz, 83.80.Xz, 87.15.Aa I. INTRODUCTION One of the aims of statistical mechanics is to establish a link between molecular interactions and phase behavior. This is a particularly difficult task in the field of liquid crystals, due the anisotropy of the molecular shapes and interactions, which are the essential features retained even in the most simple models. The major importance of molecular shape has nowadays become a consolidated aspect with interdisciplinary recognition. Properties such as the stability of colloidal suspensions or the conformational behavior of polymeric systems can be controlled with the addition of particles with the appropriate shape and molar fraction. In biological sciences, it has been established that excluded volume effects play an essential role in the equilibrium and dynamical properties of macromolecules in physiological environments 关1,2兴. Historically, Onsager 关3兴and Maier and Saupe 关4兴pio- neered the study of liquid crystals since the early 1940’s. The original works of Onsager have been revised by several authors 关5–10兴and have been applied to the study of fluid mixtures involving rodlike and platelike particles 关11–16兴. Onsager theory, despite its inherent limitations, presents one fundamental advantage with respect to its more accurate counterparts. The simple second virial approximation on which the theory relies, provides an intuitive framework for the understanding of the physical problem in terms of the steric effects associated to the particular molecular geometries considered. For finite size molecules, Onsager theory tends to become inaccurate at the high densities typical of liquid crystal transitions, and numerous more refined approaches have been developed 关17–19兴. Among these, Parsons-Lee 关20,21兴and scaled particle 关22兴theories have been of historical importance. One of the most extensive theoretical studies to date of the global phase diagram of polydispersed rodlike particles has been reported by Velasco, Mederos, and co-workers from Parsons-Lee and density functional theories 关23–25兴. These authors characterized in detail the phase diagram of mixtures of hard rods of different sizes but devoted, however, limited attention to the specific case of the rod/sphere mixture object of the present work. In the present article, we outline an application of Onsager and Parsons-Lee theories to the particular case of the binary mixture of spherical and rodlike colloids, modeled by hard spheres 共HS兲and hard spherocylinders 共HSC兲, respectively. This kind of mixtures have received considerable attention in recent years, mainly within the context of the stability of colloidal suspensions 关27–31兴. Although the liquid crystal properties of these binary mixtures have been more scarcely investigated, significant contributions have been made to this respect from the experimental and theoretical points of view. The wealth of the phase behavior of these systems has been extensively demonstrated experimentally *Email address: [email protected] PHYSICAL REVIEW E 75, 061701 共2007兲 1539-3755/2007/75共6兲/061701共12兲©2007 The American Physical Society061701-1 by Fraden and co-workers 关19,32–36兴, as well as by other groups 关37–39兴. Experiments typically employ mixtures of spherical colloids or globular polymers with rodlike particles, such as the Tobacco Mosaic virus 关19,33兴, the bacteriophage fd virus 关32兴, nanorods of cellulose 关37兴, and boehmite 关38兴or organic liquid crystals 关39–42兴. The fd virus, in particular, depending on environmental parameters such as pH or ionic strength, can have effective length-to-breadth ratios ranging from a few tens to up to around 100. This latter value has been established as a rough limit for the accurate applicability of Onsager theory for pure fluids of prolate molecules 关19兴. Interestingly, much of the rich phenomenology observed experimentally remains to a large extent unexplained, although some relevant effects have been reproduced in theoretical and computer simulation studies 关23–26,30,43–46兴. The main motivation of the current contribution is the elucidation of the capability of Onsager and Parsons-Lee theories 共presented in Sec. II兲for predicting, on qualitative and quantitative grounds, nematic ordering and demixing transitions in rod-sphere binary mixtures. Perhaps surprisingly, a systematic study of this nature has not been performed up to date for rod-sphere mixtures. II. THEORY The Onsager theory for the HSC-HS binary mixture here outlined represents an extension of the original approach to the free energy for the pure rod system 关3兴. It complements similar approaches applied previously to mixtures of rodlike molecules without specific attention to rod-sphere mixtures 关5,13兴. Onsager treated the pure rod system formally as a polydispersed mixture by considering particles with different orientations in space as belonging to different molecular species 关3兴. It was assumed that the probability distribution for the orientation of the rodlike particles with respect to a fixed frame is determined by a single-particle distribution function f共⍀兲. In the isotropic phase all orientations are equally probable so that f共⍀兲=共4 ␲ 兲−1, whereas in the nematic phase, due the axial symmetry of the particles, f共⍀兲=共2 ␲ 兲−1h共 ␪ 兲, ␪ being the angle between the molecule and the nematic director vector. Onsager’s approach provides a natural means of introducing the spherical particles in the theory, namely, as one additional species of the formal multicomponent mixture constituted by the HSC particles with different orientations. With these premises, a mimic of the procedure followed by Onsager leads to the following expression for the free energy of the binary mixture of Ncfreely rotating hard spherocylinders and Nsspheres: ␤ F N=fiso id +f1+f2,共1兲 f1=−1+ln ␳ +xc ␴ 关f兴+xcln xc+xsln xs,共2兲 f2=兺 ij xixj ␳ 2 冕冕 d⍀id⍀jf共⍀i兲f共⍀j兲vexc;ij共⍀i,⍀j兲共3兲 with ␴ 关f兴⬅ 冕 f共⍀兲ln关4 ␲ f共⍀兲兴d⍀.共4兲 In the above equation, Nis the total number of particles, ␳ the particle density, xcand xsdenote the molar fractions of HSC and HS particles, and fiso id is the concentrationindependent part of the free energy per particle in units of thermal energy ␬ T= ␤ −1. The subscripts i,jrun on the types of particles in the mixture, i.e., spherocylinders and spheres. The functions vexc;ij共⍀i,⍀j兲denote the physical volume of space mutually excluded between pairs of HSC-HSC, HSCHS, and HS-HS particles, which will be henceforth denoted by vcc,vcs, and vss, respectively: vcc共 ␥ 兲=2L2 ␴ 兩sin ␥ 兩+2 ␲␴ 2L+共4/3兲 ␲␴ 3⬅8 ␲ a兩sin ␥ 兩+vca, 共5兲 vcs = ␲ 冋 共D+ ␴ 兲3 6+共D+ ␴ 兲2 4L 册 ,共6兲 vss =4 3 ␲ D3.共7兲 Here, Land ␴ stand for the length and diameter of the HSC particles, respectively, Dis the diameter of the HS particles, and ␥ 共⍀i,⍀j兲is the angle between the axes of the HSC particles. We introduce at this point the Parsons-Lee formulation for the rod-sphere mixture, in which a scaling approximation is considered that takes effectively into account higher virial coefficients than Onsager theory 关20,21,47兴. Within the Parsons-Lee approximation, the integration of the virial equation for the pressure leads to an expression for the free energy of the HSC-HS binary mixture similar to Eq. 共1兲with f2=−1 6兺 ij xixj 冕 d ␳ 冕 dr 冕冕 d⍀id⍀jf共⍀i兲f共⍀f兲gij共r,⍀i,⍀j兲关 ␤ ⵜrVij共r,⍀i,⍀j兲·r兴共8兲 ⬇−1 2 冕 d ␳ gHS;vm兺 ij xixj 冕冕 d⍀id⍀jf共⍀i兲f共⍀j兲vexc;ij共⍀i,⍀j兲,共9兲 CUETOS et al. PHYSICAL REVIEW E 75, 061701 共2007兲 061701-2 where Vij is the intermolecular potential between a pair of particles iand j. For hard core interactions, such as the ones presently considered, an additional approximation must be assumed in order to obtain an expression for the radial distribution functions at contact between the different molecular components gij共 ␴ ij兲共with ␴ ij being the pair contact distance兲. Such contact values arise from the integration of the ␦ 共r − ␴ ij兲functions given by the gradient of the discontinuous Vij potential. One of the simplest choices consists of assuming gij共 ␴ ij兲=gHS;vmfor all relative pair orientations. Here, gHS;vm stands for the contact value of a pair of hard spheres with effective diameters corresponding to the same average molecular volume vmas the mixture under study. For the present case, vm=xsvs+xcvc, where vcand vsdenote the molecular volumes of the HSC and HS particles, respectively. The introduction of this approximation into Eq. 共8兲leads to the above Eq. 共9兲. Even though several more complex functionalities for gij共 ␴ ij兲were tested within the present work, none of them yielded a better agreement between theory and simulation. The integral on the density in Eq. 共9兲is solved by means of the Carnahan-Starling equation for the compressibility of the HS fluid ZHS 关48兴: f ˆ共 ␩ ;vm兲⬅ 冕 d ␳ gHS;vm= 冕 d ␳ ZHS −1 4 ␩ =1 vm 4 ␩ −3 ␩ 2 共1− ␩ 兲2, 共10兲 where ␩ = ␳ vmdenotes the packing fraction of the system. The combination of the different expressions outlined above leads to the following compact formulation for the free energy of the HSC-HS mixture within the Onsager and Parsons-Lee approximations ␤ F N=fiso id −兵1−ln ␳ −共xcln xc+xc ␴ 关f兴兲 −xsln xs −⌳„xc 2共2ag关f兴+vca兲+2xcxsvcs +xs 2vss…其共11兲 with g关f兴⬅ 4 ␲ 冕 兩sin ␥ 兩f共⍀兲f共⍀⬘兲d⍀⬘ and ⌳= ␳ 2共Onsager兲or ⌳=f ˆ共 ␩ ;vm兲 8共Parsons-Lee兲.共12兲 In the limit L*Ⰷ1, L*ⰇD*共where L*=L/ ␴ and D*=D/ ␴ 兲, the Parsons-Lee expression for ⌳converges to the Onsager form and, hence, becomes exact. One of the most crucial aspect in both Onsager and Parsons-Lee theories is the determination of the orientation distribution function f共⍀兲关or h共 ␪ 兲for axial-symmetric systems兴, which minimizes the free energy of the system. Several functional forms and variational parameters have been proposed for this purpose 关3,8,49,50兴. In this work, we have followed the alternative approach of minimizing the free energy with respect to variations in f共⍀兲关10兴, which leads to the integral equation ln关4 ␲ f共⍀兲兴 =␭−⌳16axc ␲ 冕 兩sin ␥ 共⍀,⍀⬘兲兩f共⍀兲d⍀. 共13兲 Here, ␭is the Lagrange multiplier, which is determined by the normalization condition. Equation 共13兲has always the isotropic phase solution f共⍀兲=共4 ␲ 兲−1. For the calculation of the nematic phase solution, we have applied a well-known self-consistent methodology 关5,51兴. The method is based on the even Legendre expansions of h共 ␪ 兲and of 兩sin ␥ 兩, which by making use of the addition theorem of the Legendre polynomials, and integrating Eq. 共13兲on the azimuthal angle, leads to the following expression for the distribution function: h共 ␪ 兲=兺 n=0 ⬁ a2nP2n共cos ␪ 兲 =Kexp 冋 −32axc⌳兺 n=0 ⬁2 2共2n兲+1a2nd2nP2n共cos ␪ 兲 册 , 共14兲 where Kis a normalization constant, a2nare the Legendre expansion coefficients for h共 ␪ 兲, and d2nare the 共analytical兲 Legendre coefficients for 兩sin ␥ 兩关52兴. In order to determine the function h共 ␪ 兲of minimum free energy for a given density and molar fraction, Eq. 共14兲is solved iteratively until self-consistent convergence is obtained in the anexpansion coefficients. A convergence test revealed that terms up to 2n=20 had to be included. Once a converged solution for h共 ␪ 兲has been obtained, the different thermodynamic quantities for the HSC-HS fluid can be calculated from the corresponding partial derivatives of the free energy. In particular, it is worthwhile to notice that the equation of state is given by the following expression: ␤ p= ␳ + ␳ 2d⌳ d ␳ 兵xc 2共2ag关f兴+vca兲+2xcxsvcs +xs 2vss其 共15兲 which becomes fully analytical in the isotropic phase, where g关f兴=1. The coexistence between the isotropic and nematic phases can be characterized by solving the system of equations pI =pN, ␮ c I= ␮ c N, and ␮ s I= ␮ s N, where the superscripts Iand N indicate the isotropic and nematic phases, respectively. In this system of equations, the unknown quantities are the packing fractions of the two phases and the molar fraction of spherocylinders in the nematic phase xc N. The choice of xc I instead of xc Nled to equivalent results but with a slower convergence of the iterative method. The numerical solution to the coexistence equations was carried out following the procedure outlined in Ref. 关51兴. Only for the fluids of shorter particles 共L*艋20兲the present work employs the more efficient genetic algorithm described in the appendix below. We finally describe an extension of the methodology outlined above in order to incorporate an evaluation of the range of stability of the I-Iand N-Nphase coexistences in the USE OF PARSONS-LEE AND ONSAGER THEORIES TO…PHYSICAL REVIEW E 75, 061701 共2007兲 061701-3 HSC-HS fluid. In a pioneering study based on scaled particle theory, Lekkerkerker and Stroobants characterized these types of coexistence for binary mixtures of hard rods and polymers 关54兴. It will be shown that within the HSC-HS geometries scoped in the present work, the I-Icoexistence is not relevant and the N-Ncoexistence becomes stable for a well delimited range of particle geometries. Otherwise, the phase diagram will be dominated by I-Ncoexistence, which will therefore constitute the main concern of the investigation. Spinodal curves delimiting the region of possible occurrence of I-Iand N-Ncoexistence were calculated extending, to the frame of Parsons-Lee theory, the formalism proposed by van Roij and Mulder within Onsager approximation 关12,55兴, which relies on the stability conditions of a mixture in a thermodynamic phase expressed as 冉 ␦ 2 ␤ F/N ␦ v2 冊 x ⬎0, 冉 ␦ 2 ␤ F/N ␦ x2 冊 v ⬎0, 冉 ␦ 2 ␤ F/N ␦ v2 冊 x 冉 ␦ 2 ␤ F/N ␦ x2 冊 v − 冉 ␦ 2 ␤ F/N ␦ x ␦ v 冊 2 ⬎0, 共16兲 where xis the molar fraction of one of the components and v=1/ ␳ . The application of the above inequalities to the Parsons-Lee expression for the free energy given by Eq. 共11兲, with ⌳=f ˆ共 ␩ ;vm兲/8, leads to lengthy expressions which are, however, straightforward to obtain. The explicit dependence of ⌳on the composition of the mixture through vmmust be taken into account. This is in contrast to Onsager theory where ⌳is only dependent on ␳ . If any one of the three inequalities of Eq. 共16兲becomes negative the homogeneous mixing of the binary system becomes unstable and, consequently, a demixing transition is produced. Hence, the spinodal curve in a given binary mixture separates the region of the phase diagram where the inequalities of Eq. 共16兲are fulfilled from the region where any of the three inequalities is not satisfied. For the present study, the spinodal was determined in an overwhelming majority of cases by the last inequality, which combines the mixing and mechanical stabilities of the system. Only occasionally the second inequality 共mixing equilibrium兲was found to be limiting. No evidence for breaking of the mechanical stability condition expressed by the first inequality was detected. III. RESULTS AND DISCUSSION Figure 1illustrates typical isotropic-nematic coexistence diagrams predicted by Parsons-Lee theory for the HSC-HS mixture. The graphs depict the packing fractions, ␩ = ␳ vm,of the coexisting isotropic and nematic states plotted as a function of the volumetric fraction of spheres xv⬅xsvs/vmin each of the phases. The particular cases of HSC with L* =300, 20, and 5 and HS with D*=10, 2, and 1, respectively, are chosen to illustrate the behavior of different molecular geometries. In each of the ␩ -xvdiagrams tie lines joining some of the coexisting states are drawn for reference. A first inspection of Fig. 1already reveals the occurrence of two different coexistence regimes for any given molecular geometry. In the low xvregion, the isotropic and nematic branches evolve close to each other and present only a slight composition asymmetry. On the other hand, as the concentration of spheres is sufficiently increased, a turning point is observed, where the nematic branch attains a maximum xv value. Beyond the turning point, the binary fluid enters a demixed coexistence regime with a very asymmetric molar fraction of the two components in the isotropic and nematic states. This leads to a phase diagram with a reentrant nature; at 共fixed兲low volume fraction of spheres, upon increasing ␩ the system evolves from the isotropic phase to a I-Ncoex- istence region, then to a nematic region and again to a I-N coexistence region. For the HSC-HS systems of Fig. 1with L*=300, 20, and 5, the nematic coexistence branch reaches maximum values of xv⬇0.43, 0.16, and 0.074, respectively. In fact, as the isotropic branch evolves to the limit xv→1, the volumetric fraction of spheres in the corresponding nematic 0.0 0.2 0.4 0.6 0.8 1. 0 0.4 0.5 0.6 0.7 η N I 0.2 0.3 0.4 0.5 η N I 0.01 0.1 HS volume fraction, xV L*=5 D*=1 L*=20 D*=2 L*=300 D*=10 η NI 0.00 0.02 0.04 0.06 0.40 0.44 N I FIG. 1. Typical isotropic-nematic coexistence diagrams predicted by Parsons-Lee theory for the HSC-HS fluid. The packing fractions are represented versus the volumetric fraction of spheres xv=共xsvs兲/vmin the coexisting isotropic and nematic states. Tie lines joining some of the coexisting states are included for illustration. The results shown correspond to binary mixtures with the values for L*=L/ ␴ and D*=D/ ␴ indicated in each panel. The inset in the bottom panel compares the Parsons-Lee coexistence curves with the isotropic 共solid symbols兲and nematic 共open symbols兲boundary states reported in Ref. 关43兴from Monte Carlo 共NPT-MC兲simula- tions at constant composition. CUETOS et al. PHYSICAL REVIEW E 75, 061701 共2007兲 061701-4 state of coexistence becomes progressively smaller and tends to vanish. In the demixed regime, the relative magnitude of the packing fractions of the coexisting states depends on the relative sizes of the HSC and HS particles. This is noteworthy since in the mixed regime the I-Ncoexistence implies systematically a greater packing fraction in the nematic phase. Figure 1shows that for L*=300, D*=10 the sphererich isotropic branch eventually becomes more compact than its nematic counterpart. For the other two cases L*=20, D* =2 and L*=5, D*=1, the demixed nematic states are always denser than the isotropic ones. Furthermore, the coexistence diagram for L*=5, D*=1 displays a nonmonotonous behavior of the packing fraction in the isotropic branch. It will be shown below that this feature is associated to the demixing regime mixtures with small D/Lratios, where the isotropic branch becomes less dense as it expels HSC particles. The appearance of highly partitioned isotropic and nematic phases in rod-sphere mixtures was originally predicted theoretically by Flory 关53兴. In fact, coexistence diagrams in qualitative agreement with the ones shown in Fig. 1have been observed experimentally for different systems 关19,32,33,35–41兴. Lekkerkerker and Stroobants 关54兴de- scribed in detail this qualitative behavior for mixtures of colloidal rods and flexible polymers, guided by scaled particle theory. It is therefore gratifying to confirm that Parsons-Lee theory 共and also Onsager theory as shown below兲captures, at least on qualitative grounds, the behavior observed in real systems on the basis of depletion interaction effects. Figure 1also shows that at low values of xvwithin the mixed regime, the nematic phase is perturbed by the introduction of spherical particles on the dominant bulk of rodlike particles. Any increment in the fraction of spheres results in a systematic displacement of the I-Ntransition toward higher packing fractions of both coexisting states. This type of retardation of the nematic phase has been found in a recent simulation study of the HSC-HS mixture with L*=5 and D*=1 关43兴, and appears to be a general effect for small and moderate values of xv. Figure 1共bottom panel兲illustrates the fair agreement existing between the Parsons-Lee coexistence diagram and the NPT-MC simulation data of Ref. 关43兴in the small xvregime 共xv⬍0.07兲for the particular case of the L* =5, D*=1 mixture. A similar level of quantitative agreement is found for the coexistence pressures 共not shown兲. Figure 2 shows snapshots corresponding to NPT-MC simulations of nematic states for the L*=5, D*=1 HSC-HS fluid with xv values in the mixed and demixed regimes. The simulations corroborate the theoretical predictions by showing that the mixed case maintains the bulk structure of the nematic phase of the pure HSC fluid, whereas a clear segregation of the rodlike and spherical molecules takes place at high xv. A question that emerges at this point is whether the demixing transitions predicted in the present theoretical treatment are stable with respect to liquid crystalline transitions other than the nematic one. The most plausible situation to this respect would be the entrance of the fluid into a lamellar phase in which the HSC and HS particles arrange in alternate layers. Phases of this kind have been investigated previously 关23–25,43,44兴. For the L*=5, D*=1 HSC-HS fluid it has been shown that the lamellar phase becomes stable at packing fractions in the range ␩ =0.45–0.50, which overlaps with the demixing regime found in the present study for the I-N coexistence of that particular mixture. In fact, in the snapshot for the demixed state shown in Fig. 2it can be appreciated that the segregated HSC particles are more tightly packed than a typical nematic phase and display an arrangement more related to a smectic or solidlike phase. Hence, the application of the present theory, in which the demixing transitions are treated within the context of nematic ordering is limited to HSC elongations greater than L*=5. Fortunately, as L*is increased the transition to the lamellar phase is also delayed to greater packing fractions and, in addition, the demixed nematic regime takes place at smaller packing fractions 共see Fig. 1兲. In this way, for the particular case of L* =20, the smectic A phase becomes stable at ␩ ⬇0.5 关58兴,a value well above the range where the I-Ndemixing occurs 共see Fig. 1兲. It follows that the conclusions related to the demixing behavior of the HSC-HS fluid drawn from the present theoretical treatment are expected to be robust, within the framework of the Parsons-Lee and Onsager approximations, for mixtures of HSC particles with elongations of L*⬇10–20 or greater. We will now focus the discussion in more detail on the theoretical predictions for the dependence of the I-Ncoexist- a ) b) FIG. 2. Snapshots from Monte Carlo simulations of the HSC-HS fluid with L*=5, D*=1 in 共a兲a mixed nematic state with xv=0.065 and ␩ =0.442 and 共b兲a demixed nematic state with xv =0.536 and ␩ =0.412. The computations were performed as described in Ref. 关43兴. USE OF PARSONS-LEE AND ONSAGER THEORIES TO…PHYSICAL REVIEW E 75, 061701 共2007兲 061701-5 ence diagram on the relative sizes of the HSC and HS particles. Figure 3compares the I-Ncoexistence branches for HSC-HS fluids with different HSC elongations ranging L* =5–300. As can be seen, the main features of the coexistence diagrams are similar for the different L*values and reproduce the qualitative behavior observed in Fig. 1.Inall cases, the isotropic states at low volume fractions of spheres coexist with nematic states with relatively small differences in xvand ␩ , although the nematic state is always appreciably poorer in spheres and displays systematically greater packing fractions. As the value of xvgrows in the isotropic branch, the differences in both xvand ␩ with the nematic branch vary progressively. Eventually, the system enters the demixing regime described above, in which nematic states rich in spherocylinders coexist with isotropic states rich in spheres. It is interesting to observe that the maximum value of xv attained in the nematic branch before demixing occurs, shifts toward greater values of xvas L*grows for fixed D*. This implies that the homogeneity of the isotropic and nematic phases, and hence the “resistance” of the system against demixing, in the HSC-HS fluid is enhanced with growing L*. Such trend is a natural consequence of the dominance of the HSC-HSC pair interactions 关e.g., the vcc共 ␥ 兲term of Eq. 共5兲兴 in the high L*limit, which tends to overrun the depletion effects that drive the demixing behavior. The greater weight of the pair HSC interactions is also responsible for the shift of the I-Ncoexisting branches globally toward smaller packing fractions with growing L*. Figure 3shows that the coexistence diagram resulting from Onsager theory agrees well with the Parsons-Lee one only for the two cases of greater elongation L*=100 and 300. On the contrary, the quantitative accuracy of the Onsager approximation deteriorates for smaller elongations and in particular for the three depicted fluids with L*=50, 20, and 5. The bottom panel of Fig. 3compares the coexistence pressure branches from Parsons-Lee and Onsager theory for two illustrative cases. Note that in this diagram the recognition of the coexisting states, both having the same pressure, is straightforward. The agreement between both theoretical approaches is better in the pressure diagram than in the packing fraction diagram, in particular for the L*=20, D*=2 case. Hence, Onsager estimation of coexisting pressures appears to be more robust than that of packing fractions. Overall, we found that Onsager results for the isotropic-nematic coexistence of the HSC-HS fluid diverged appreciably from their Parsons-Lee counterparts in general for fluids with L*⬍100. For elongations of the HSC particles greater than L*=100, Onsager theory provided reliable results except in regions of the phase diagram involving volume fractions of spheres too close to unity, as further shown below. Figure 4depicts the dependence of the I-Ncoexistence diagram with the diameter of the spherical particles for two fixed HSC elongations, namely, L*=300 and 20. For the fluids with L*=300 the coexistence curves from both Onsager and Parsons-Lee theories are shown for comparison. For the L*=20 fluids only the Parsons-Lee 共PL兲theory is considered. Irrespectively of whether the Onsager or the PL approach was employed, the computations for L*=300, D*艋3 could be calculated up to neatly beyond the reversal of the nematic branch toward the demixed regime. However, a point beyond FIG. 3. Isotropic-nematic coexistence diagram predicted by Parsons-Lee 共solid curves兲and Onsager 共dashed curves兲theories for HSC-HS fluids with different elongations of the HSC particles L* and diameters of the HS particles D*as indicated. The dimensionless pressure of the mixture is defined as bP*=b ␤ P ␴ 3, with b =共 ␲ /4兲L*2D*. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. 0 0.2 0.3 0.4 η2 2 4 0.7 1.5 1 x v 3 D*=4 3 L*=20 N I 0.01 0.1 η 31 15 10 1 D*=5 L*=300 NI FIG. 4. Parsons-Lee 共solid curves兲and Onsager 共dashed curves, top panel only兲isotropic-nematic coexistence diagrams for HSC-HS fluids with fixed rod length of the HSC particles 共L*兲and different diameter of the HS particles 共D*兲: Top panel: L*=300 and D*=1–15, bottom panel: L*=20 and D*=0.7–4. CUETOS et al. PHYSICAL REVIEW E 75, 061701 共2007兲 061701-6 which no converged results could be obtained was eventually reached. It will be discussed below that the observed lack of convergence in the demixing region of the I-Ncurves may be attributed to the entrance of the system into a N-Ncoexist- ence region. One of the main conclusions that can be drawn from the results of Fig. 4is that the stability of the mixed I-Ncoex- istence is reduced with the increase in size of the spherical component in the binary mixture. According to the PL calculation, for L*=300 and increasing D*=1, 3, 5, and 15, I-N states coexist under mixed conditions up to progressively smaller HS volume fractions, of xv=0.71, 0.66, 0.60, and 0.28, respectively. For L*=20, a similar trend is found, although in this case the transitions to the demixing regime takes place at roughly the same xv⬇0.165 within D* =0.7–1.5, with a shallow maximum of xv⬇0.167 for D* =1.25. This same behavior was found for L*=5 共not shown兲 where the turning point into the demixing regime is delayed to xv⬇0.83 for D*=1.5, in comparison to xv⬇0.72 for both D*=1 and 2. Such weak dependence of the demixing transition on the size of the spherical articles in the region of small D*values, even featuring a non-monotonous character, appears to be reminiscent of the excluded volume contributions contained in the second order virial terms of the free energy. In fact, Onsager theory predicts a much more pronounced nonmonotonous dependence on D*of the stability of the mixed regime. Within the Onsager approximation, for the L*=300 fluid the demixing regime enters, for instance, at xv=0.49, 0.64, 0.60, and 0.27 for D*=1, 2, 3, and 15, with a detailed computation showing that the range of stability of the mixed I-Ncoexistence becomes indeed maximum for D*⬇3. This is the reason for the strong discrepancy between the Onsager and Parsons-Lee coexistence curves for the L* =300, D*=1 fluid that can be appreciated in Fig. 4. A further remarkable aspect illustrated in Fig. 4is the appreciable maximum in ␩ displayed by the isotropic branch of the L*=20 fluid in the small D*limit. Such trend is observed for D*艋1 and indicates that the expel of HSC particles from the isotropic phase as the system enters the demixing regime leads to a less efficient packing of the fluid in that phase. A similar behavior of the isotropic branch for small D*was observed for L*⬍20, as illustrated in Fig. 1for L*=5. As the volume of the HS particles becomes comparable or greater than that of the HSC particles, this effect vanishes and the packing fraction isotropic branch increases monotonously over the whole xvrange. In fact, for the mixtures with L*=20 and D*⬎2, the packing fraction of the isotropic branch for xvclose to unity reaches values typical of the solid phase of the bulk HS system. Hence, these latter cases correspond to a situation in which the HS particles freeze upon segregation from the HSC particles. The pertinence of this remark is related to the experimental observation of a similar type of behavior reported by Fraden and co-workers for mixtures of rodlike viruses and spherical colloids 关33兴. In these experiments the colloids were found to form arrangements of solidlike filaments segregated from the bulk of nematic viruses. This phenomenology, still without a clear explanation, presents similarities with the nematic HSC-solid-HS phase separation described here. Figure 5illustrates the stability of the isotropic and nematic phases against I-Iand N-Ndemixing transitions. This is done by means of the Parsons-Lee spinodal curves resulting from the mapping on the xv- ␩ plane of the conditions defined in Eq. 共16兲. Demixing could take place if the packing fractions associated to the corresponding spinodal become smaller than the packing fractions obtained for I-Ncoexist- ence. Spinodals for I-Idemixing were computed for HSC aspect ratios ranging L*=5–1000 and HS diameters up to D*=1000. Figure 5shows only some illustrative examples. The detailed analysis of these spinodals indicates that I-I coexistence becomes feasible for sufficiently large D*values, namely, e.g., D*⬎4, 6, 10, 20, 30, and 90 for L*=5, 10, 20, 100, 300, and 1000, respectively. It can be noted that these HS diameters are significantly greater than the ones considered above for the I-Ncoexistence and therefore do not interfere with the diagrams of Figs. 2–4. This conclusion is supported by the fact that our algorithm never converged to a I-Ncoexistence configuration of low orientational order in the a priori nematic phase. This type of convergence would have been plausible, since the h共 ␪ 兲distribution was free to adjust through its Legendre coefficients to the configuration of minimum free energy. This result is also consistent with the prediction outlined in Ref. 关54兴that for sufficiently long 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 η 00.1 0.2 0.2 0.4 η D*=4 D*=0.7 D*=1, 0.7 D*=4 D*=1 L*=20 0 0.2 0.4 0.6 x v 0 0.1 η D*=1 D*=10 D*=1 D*=4 D*=10 D*=10 D*=1 D*=1 L*=20 L*=300 L*=300 FIG. 5. Top panel: isotropic-isotropic spinodal demixing curves for HSC-HS fluids with L*=300, D*=1 and 10 共solid curves兲and L*=20, D*=1 and 4 共dashed curves兲. Middle panel: Nematicnematic spinodal demixing curves 共dashed lines兲for HSC-HS fluids with L*=20, D*=0.7, 1, and 4. The corresponding nematic branches of the I-Ncoexistence are also included for direct comparison. Note the discontinuity in the I-Ncurve for D*=0.7 共symbols兲not present in the closely overlapping curve for D*=1 共solid line兲. Bottom panel: Same as middle panel but for L*=300, D*=1 and 10. The I-I or N-Ndemixing transitions can take place at packing fractions above the corresponding spinodals. USE OF PARSONS-LEE AND ONSAGER THEORIES TO…PHYSICAL REVIEW E 75, 061701 共2007兲 061701-7 rods, I-Idemixing becomes only stable for comparably large sphere sizes. The N-NParsons-Lee spinodals for different HSC-HS molecular sizes are shown in the lower panels of Fig. 5. The corresponding nematic branches of the I-Ncoexistence curves are also shown for comparison. In this case, the computation of the spinodals from the conditions of Eq. 共16兲 involves the iterative optimization of h共 ␪ 兲for each HS volume fraction and packing fraction of the mixture through Eqs. 共13兲and 共14兲. For the mixtures involving HSC particles with L*=20 共middle panel of Fig. 5兲, the comparison with the I-Ncoexistence curves indicates that for D*艌1, N-N demixing requires too large packing fractions and is therefore unstable with respect to I-Ndemixing within the Parsons-Lee approximation. Only for D*⬍1 would the N-N phase separation be allowed, so that the present results for I-Nequilibrium for this range of geometries should be considered with caution. Interesting, the nematic branch of the I-Ncoexistence curve for L*=20, D*=0.7 displays a discontinuity after the turning point 共i.e., already within the domain of demixing兲where a small change in the composition of the isotropic branch induces an abrupt change in both the composition and packing fraction of the coexisting nematic state. It is interesting to notice that the appearance of such discontinuity is coincident with the evolution of the spinodal to packing fraction values neatly below those of the I-Ncurve. It would therefore be plausible that the range of the composition-packing fraction diagram not covered by the I-Ncurve corresponds to the domain of stability of N-Nco- existence. This expectation is consistent with the description of the overall phase diagram given by Lekkerkerker and Stroobants 关54兴. For the case of HSC particles with L*=300 共bottom panel of Fig. 5兲,I-Ncoexistence is favored in all cases with D* 艌5. This is illustrated by the comparison for D*=10 showing that too large ␩ values are required for N-Nequilibrium. For smaller sizes of the HS particles, represented by D*=1 in Fig. 5, a composition domain appears in which N-Nsepara- tion competes favorably, according to the smaller ␩ of the spinodal with respect to the nematic branch of the I-Ncoex- istence. Such a situation arises only for sufficiently high values of xv共i.e., for large molar fractions of the HS particles兲, whereas for low values of xvthe I-Nseparation remains stable. In this case, the entrance of the possible N-Ndemix- ing takes place in a region where the algorithm to calculate the I-Ncoexistence did not converge to any solution 共see Fig. 4兲. Such lack of convergence is likely to be related to a discontinuous gap in the nematic branch similar to the one described above for the case L*=20, D*=0.7. In this case, however, the boundary nematic states are too different from each other to be linked by the genetic algorithm employed in the present work. This interpretation then suggests the entrance of the fluids L*=300, D*艋3 in the N-Ncoexistence region at some point after the turning point of the I-N branch. A modification of the present methodology considering independent h共 ␪ 兲distributions for the coexisting phases 共i.e., allowing N-Nequilibrium兲led to the same I-Ncoexisting curves within the range of convergence of the original algorithm 共one of the orientational distributions became isotropic兲. However, the algorithm did not reach convergence for the pressures and chemical potentials within the expected domain of N-Nstability. In conclusion, in order to characterize in a more complete way the phase diagram of HSC-HS mixtures, an extended investigation would be required. Such investigation, in addition to implementing an efficient algorithm to characterize in detail the N-Ncoexistence binodals should incorporate the lamellar phase into the treatment. This lies beyond the aim of the present work, where the domain of stability of the I-Ncoexistence diagram has been determined for particle sizes where the lamellar phase is not expected to interfere. For these cases, the region where N-Nseparation may become stable within the Parsons-Lee approximation have been established 共e.g., L*=20, D*⬍1orL*=300, D* 艋3 for sufficiently high xvvalues兲. Figure 6represents the Parsons-Lee equations of state at constant mixture composition 共isopleths兲for rod-sphere mixtures with L*=20 and D*=1,2,4 and with L*=300 and D* =2,5,7. The range of volume fractions of spheres considered in each case corresponds to the region of relatively small composition asymmetry in the coexisting I-Ndiagram 共see Fig. 3兲. Such region lies before the entrance of the system into the demixed regime and sufficiently anticipated with respect to the region where N-Nseparation may become relevant. The different isopleths show the expected positive correlation between pressure and packing fraction. The displacement of the I-Ntransition toward greater ␩ values with growing xv共i.e., destabilization of the nematic phase by the depletion effects induced by the spherical component兲can also be appreciated, as already noticed above in the coexistence diagrams. For the smaller HS diameters considered in Fig. 6it is observed that the system pressure increases appreciably with growing xvin both the Iand Nphases 共at fixed ␩ 兲. This behavior, also found in Monte Carlo simulation studies 关43兴, occurs as a natural consequence of the multiplication of the number density of the system with any increase of xv. This in turn follows from the much smaller volume of the spherical particles in comparison to the rods. The opposite effect is observed for sufficiently large HS diameters once the HS molecular volume becomes greater than the HSC one. In this case, any increase in the volume fraction of spheres, for fixed ␩ , leads to a decrement in the absolute number of particles per unit volume and, hence, to a decrease in the system pressure. Interestingly, for intermediate D*values, where the HS and HSC molecular volumes become comparable, a situation arises where the composition has a weak effect on the pressure-packing fraction representation of the isopleth. This is a consequence of the effective compensation of the excluded volume contributions that drive the system pressure. Moreover, Fig. 6shows that such steric effects vary significantly from the isotropic to the nematic phase so that it is possible to find roughly composition-independent isopleths in one phase and appreciable composition effects on the other phase. In fact, for the mixtures L*=20, D*=2 and L* =300, D*=5, whereas a close overlap is found for the isopleths with the different xvvalues in the isotropic phase, a significant shift of pressure with xvis still observed in the nematic phase. The complementary situation of an almost negligible dependence on xvof the isopleths in the nematic CUETOS et al. PHYSICAL REVIEW E 75, 061701 共2007兲 061701-8 phase is found for L*=20, D*=4 and L*=300, D*=7.Itis worthwhile noticing that the weakest composition dependence of the isopleths in the isotropic phase is found, in both the L*=20 and L*=300 mixtures, for molecular volume ratios between the HS and HSC particles of vs/vc⬇0.25. On the other hand, for the nematic phase, this situation is found for vs/vc⬇2 for L*=20 and vs/vc⬇0.75 for L*=300. These values are in contrast to the common choice of vs=vc共i.e., D*=3.1 for L*=20 and D*=7.7 for L*=300兲, often employed when the HS fluid is taken as reference system for the HSC fluid in approximate theories. In any case, only for mixtures involving relatively short rods it seems possible to find an effective value of D*for which the isopleth of the mixture becomes insensitive to the composition of the mixture simultaneously in the isotropic and the nematic phases. Simulation or experimental studies of the equation of state 共EOS兲of HSC-HS mixtures, that could provide a reference for the evaluation of the present theoretical results, are scarce. Isopleths for the isotropic and nematic phases have been reported in previous works from NPT-MC simulations for the L*=5, D*=1 fluid 关43兴. For the present paper, we have extended those simulations to include the case L*=5, D*=3 employing the same methodology described in Ref. 关43兴. Figure 7illustrates the level of agreement observed between the Parsons-Lee isopleths in the isotropic and nematic phases and the results from NPT-MC simulations for these particular cases. In addition, alternative EOS for the isotropic phase of the HSC-HS system from scaled particle theory proposed by other authors 关56,57兴are also shown. It can be appreciated that these latter EOSs, originally developed for shorter HSC molecules, remain fairly accurate at the L*=5 elongation considered. Nevertheless, the EOS from Parsons-Lee theory compares better with the simulation data than any of the curves from scaled particle theory. Similar or better level of accuracy for the Parsons- Lee EOS can be expected for mixtures with greater L*val- ues. In the immediacy of the isotropic-nematic transition and in the nematic phase itself, although Parsons-Lee theory compares fairly well with simulation, the agreement is worse than in the isotropic phase. This can be attributed to the intrinsic limitations of the Parsons-Lee approximation to manage orientational order. In fact, the Parsons-Lee results presently obtained for the nematic phase of HSC-HS mixtures, show a similar accuracy when compared to MC data as found for the pure HSC fluid 关48兴. 0 20 40 60 80 0 10 0 20 0 30 0 40 0 50 0 0 20 40 60 80 0 50 10 0 15 0 0.0 0.1 0.2 0 20 40 60 80 100 0.00 0.01 0.02 0.03 0 50 10 0 15 0 0.15 0.15 0.01 0.05  xV=0.15 0.10 bP* L*=20 D*=1 L*=20 D*=2 L*=20 D*=4  bP* 0.3 0.2 0.1 xV=0.4 L*=300 D*=7 L*=300 D*=2 L*=300 D*=5 0.01 0.4 0.1 0.1 xV=0.4 0.01 b P* 0.1 0.4 0.4 0.1 FIG. 6. Equations of state at constant composition 共isopleths兲predicted by Parsons-Lee theory for the isotropic and nematic phases of binary HSC-HS mixtures with fixed L*=20 and 300 and different HS diameters D*. The indicated values for the volume fraction of spheres xvcorrespond to the mixed regime of the two phases 共see Fig. 3 for reference兲. The dimensionless pressure bP*is defined as in Fig. 3. USE OF PARSONS-LEE AND ONSAGER THEORIES TO…PHYSICAL REVIEW E 75, 061701 共2007兲 061701-9