Phase behaviour of binary mixtures of diamagnetic colloidal platelets in an external magnetic field
aa r X i v : . [ c ond - m a t . s o f t ] A p r Phase behaviour of binary mixtures of diamagnetic colloidal platelets in an externalmagnetic field
Jonathan Phillips
H.H. Wills Physics Laboratory, University of Bristol,Royal Fort, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
Matthias Schmidt
Theoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth,Universit¨atsstraße 30, D-95440 Bayreuth, Germany andH.H. Wills Physics Laboratory, University of Bristol,Royal Fort, Tyndall Avenue, Bristol BS8 1TL, United Kingdom (Dated: 23 December 2010, to appear in J. Phys.: Condensed Matter)Using fundamental measure density functional theory we investigate paranematic-nematic andnematic-nematic phase coexistence in binary mixtures of circular platelets with vanishing thick-nesses. An external magnetic field induces uniaxial alignment and acts on the platelets with astrength that is taken to scale with the platelet area. At particle diameter ratio λ = 1 . λ = 2, demixing into two nematic states with differ-ent compositions also occurs, between an upper critical point and a paranematic-nematic-nematictriple point. Increasing the field strength leads to shrinking of the coexistence regions. At highenough field strength a closed loop of immiscibility is induced and phase coexistence vanishes ata double critical point above which the system is homogeneously nematic. For λ = 2 .
5, besidesparanematic-nematic coexistence, there is nematic-nematic coexistence which persists and hencedoes not end in a critical point. The partial orientational order parameters along the binodals varystrongly with composition and connect smoothly for each species when closed loops of immiscibilityare present in the corresponding phase diagram.
I. INTRODUCTION
Dispersions of colloidal plateletlike particles, such asgibbsite [1, 2], montmorillonite [3, 4] or iron-rich beidel-lite [5, 6], are susceptible to the influence of magneticfields, since the particles possess nonvanishing diamag-netic anisotropy. When a magnetic field is applied to aninitially isotropic (I) platelet dispersion, the field inducesorientational order in the system, thus breaking the rota-tional symmetry; an orientationally ordered paranematic(P) phase results. The paranematic phase has interest-ing optical properties, similar to those of the nematic(N) phase. When observed through crossed polarisers,samples of gibbsite suspensions have been shown to ex-hibit field-induced birefringence [7]. Birefringence gradi-ents have also been theoretically modelled for a simplemodel system [8]. The effects of a magnetic field onmontmorillonite platelets were studied in Ref. [4] andon hematite platelets in Ref. [9]. Unlike the gibbsiteplatelets, hematite platelets are ferromagnetic and an I-Ntransition was not observed; rather the authors found aclustering effect whereby chains of particles were formeddue to the platelet-platelet interactions. Such clusteringhas also been observed in simulations [10]. Experimentalinvestigations of gibbsite platelets, whereby the suspen-sions were exposed to magnetic fields [7, 11], showed thata paranematic phase occurs in these systems.The phase behaviour of rods in external aligningfields is well-studied, see e.g. [12–17] for studies of one-component systems. In Ref. [12] the effect of externalfields on the phase behaviour of rigid rods, freely jointed rods and semiflexible rods was analysed with Onsagertheory [18]: the P-N transition was found to terminateat a critical point in all three cases at a high enoughfield strength. In Ref. [13] the theories of Landau andde Gennes [19] and of Maier and Saupe [20] were usedto analyse the effects of an applied field in the nematicphase. The magnetic field-induced birefringence in solu-tions of tobacco mosaic virus (TMV) particles was stud-ied in Ref. [14] both experimentally and theoretically (us-ing extensions of Onsager theory). In Ref. [15] the phasebehaviour of monodispese rods with varying aspect ra-tio was studied using the Parsons-Lee scaling [21, 22] ofthe Onsager functional. It was found that the bifurca-tion density decreases with increasing field strength. Thenematic order of model goethite nanorods in a magneticfield was investigated in Ref. [23], also using Parsons-Leetheory. The goethite rods were modelled as charged sphe-rocylinders with a permanent magnetic moment alongthe long axis of the rods. This encourages the rods toalign parallel to the field at low field strengths. However,goethite rods possess a negative diamagnetic susceptibil-ity which leads to alignment perpendicular to the fieldat higher field strengths. These competing effects werefound to yield rich phase diagrams including biaxial ar-rangements of the particles. The phase separation in sus-pensions of semiflexible fd -virus particles was studied inRef. [16], where a P-N phase transition was found. Ef-fects of an external field on the isotropic, nematic andsmectic-A phases of spherocylinders were compared withsimulations and theory in Ref. [17], with results fromboth approaches being in good agreement.Even when neglecting positionally ordered phases(such as columnar and crystal phases), the bulk phase be-haviour of binary mixtures of non-spherical colloidal par-ticles can be very rich, often including isotropic-isotropic(I-I), isotropic-nematic (I-N) and nematic-nematic (N-N) phase coexistence, depending on the value of the sizeasymmetry parameter of the two species. The asymme-try parameter may quantify the difference in thickness orlengths of the two species. An example are binary mix-tures of thick and thin hard rods in an external field [24–26]. A general feature of the phase behaviour of binarymixtures is a widening of the biphasic region on increas-ing the asymmetry parameter. In a certain range of thesize asymmetry there is typically an I-N-N and/or an I-I-N triple point. Coexistence between two nematic statesmay or may not end in a critical point depending on thesystem under study and the value of the asymmetry pa-rameter. A well-studied system is the Zwanzig model forbinary hard platelets, where the particles are restrictedto occupy three mutually perpendicular directions. Thiswas shown to exhibit rich bulk phase diagrams [27–29].We recently explored the phase behaviour of binary mix-tures of hard platelets with zero thickness and continu-ous orientations [30] using fundamental measure theory(FMT).Platelets can be characterized by a diamagnetic sus-ceptibility tensor that is diagonal in the platelet frameof reference, with components χ k in the platelet planeand χ ⊥ normal to it. The diamagnetic anisotropy ∆ χ ≡ χ k − χ ⊥ = 0, in general, and it may be positive or nega-tive depending on the properties of the platelet material.For Gibbsite platelets ∆ χ <
0, therefore the plateletstend to align with their normals perpendicular to the di-rection of the applied field. In order for the platelets toalign uniaxially in the presence of the field, the sampleswere placed on a central stage and rotated in a horizon-tally applied field. In Ref. [8], FMT was used to studythe effects of an external field on the phase behaviourof monodisperse platelets. It was found that above acritical field strength the P-N coexistence ceases to existand the system is homogeneously nematic. In Ref. [31],van den Pol et. al. have experimentally investigated thegeneral phase behaviour of the boardlike goethite col-loidal particles ( α -FeOOH) in the presence of an externalmagnetic field. The particles were found to align parallelto a small magnetic field and perpendicular to a largemagnetic field; this had already been known since theobservations of Lemaire et. al. [32]. This effect is dueto the particles having a permanent magnetic momentalong their long axis but the magnetic easy axis beingthe short axis. An exciting prospect is that suspensionsof beidellite platelets, which have a disk-like morphology,have recently been shown to undergo an I-N transition[5] and the nematic phase aligns strongly in the pres-ence of an externally applied magnetic or electric field[6]. These platelets possess a positive diamagnetic sus-ceptibility and, as such, a simpler experimental setupwould be required to investigate the P-N transition; the platelets are expected to align with their normals parallelto the magnetic field.Since Rosenfeld’s pioneering work [33–35] there hasbeen much interest in the development of FMT for non-spherical particles, see e.g. [36]. In the current investi-gation we use the FMT of Ref. [30], which is the mix-tures generalization of the theory proposed in Ref. [37],to study binary mixtures of diamagnetic platelets in amagnetic field. We consider three different size ratios rep-resentative of the different topologies of the bulk phasediagram and the full range of external field strengths.We investigate how the phase behaviour for each of thesethree size ratios changes on increasing the external fieldstrength, which we take to scale with the platelet areaand to induce uniaxial alignment.This paper is organised as follows. In Sec. II we out-line the density functional theory for the model system.The phase diagrams and results for order parameters arepresented in Sec. III and we conclude in Sec. IV. II. THEORYA. Pair Interactions, Model Parameters andExternal Orienting Field
We consider a binary mixture of hard circular plateletswith vanishing thickness and continuous positional andorientational degrees of freedom. Particles of species 1and 2 possess radii R and R , respectively, and we take R > R . The pair potential u ij between two particlesof species i and j , where i, j = 1 ,
2, models hard coreexclusion and is hence given by u ij ( r − r ′ , ω , ω ′ ) = ( ∞ if particles overlap0 otherwise, (1)where r and r ′ are the positions of the particle centresand ω and ω ′ are unit vectors indicating the particleorientations (normal to the particle surface). As a controlparameter that characterises the radial bidispersity weuse the size ratio λ = R R > . (2)The effect of a magnetic field on the diamagnetic plateletsis described by an external potential for each species, V ( i )ext ( θ ) = β − W i sin θ, i = 1 , , (3)where β = 1 / ( k B T ), with k B being the Boltzmann con-stant and T absolute temperature; θ is the angle betweenthe platelet orientation ω and the direction of the exter-nal field. The strength of the external potential of species i is related to the material and field properties via W i = − β B ∆ χ i (4)where B is the magnetic flux density (measured in T)and ∆ χ i = χ ( i ) k − χ ( i ) ⊥ is the diamagnetic susceptibilityanisotropy (with units of JT − ) of species i , with χ ( i ) k and χ ( i ) ⊥ being the susceptibilities perpendicular and parallelto the field, respectively [45]. In general both W and W constitute further control parameters. We restrict our-selves in the following to special cases and assume that W i scales with the platelet area, i.e. W i ∼ R i . This im-plies the relationship W = λ W , and we hence take W to be our second control parameter, besides the size ratio λ itself. Scaling with the platelet area is motivated by theassumption that the platelets interact with an externalfield in a manner proportional to their mass (neglectingany effects of thickness). We could well envisage thatscaling the strength of the potential e.g. with the radiuswould be another, different yet sensible, choice. We ne-glect platelet-platelet interactions due to induced dipolesbecause of their small magnitude, see e.g. the discussionin Ref. [8].The thermodynamic state is characterised by two di-mensionless densities c = ρ R and c = ρ R , where ρ and ρ are the number densities of the two species, ρ i = N i /V , where N i is the number of particles ofspecies i = 1 , V is the system volume. Thecomposition (mole fraction) of the (larger) species 2 is x = ρ / ( ρ + ρ ) and the total dimensionless concentra-tion is c = ( ρ + ρ ) R = c + c . B. Density Functional Theory
Density functional theory (DFT) is formulated on thelevel of the one-body density distributions ρ i ( r , ω ) of eachspecies i . The variational principle [38] asserts that min-imising the grand potential functional Ω yields the trueequilibrium density profile, δ Ω([ ρ , ρ ] , µ , µ , V, T ) δρ i ( r , ω ) = 0 , i = 1 , µ i is the chemical potential of species i . The grandpotential functional is given byΩ([ ρ ,ρ ] , µ , µ , V, T ) = F id ([ ρ , ρ ] , V, T )+ F exc ([ ρ , ρ ] , V, T ) + X i =1 Z d r Z d ω ρ i ( V ( i )ext ( r , ω ) − µ i ) , (6)where the spatial integral (over r ) is over the system vol-ume V and the angular integral (over ω ) is over the unitsphere. The inter-particle interactions are described bythe excess (over ideal gas) contribution to the Helmholtzfree energy, F exc ([ ρ , ρ ] , V, T ). We skip the explicit defi-nition of the FMT approximation here; this can be foundin Ref. [30]. The free energy functional for a binary ideal gas of uniaxial rotators is given by βF id ([ ρ , ρ ] , V, T ) = X i =1 Z d r Z d ω ρ i ( r , ω ) × [ln( ρ i ( r , ω )Λ i ) − , (7)where Λ i is the (irrelevant) thermal wavelength of species i . For bulk fluid states (i.e. with the density distributionnot depending on r ) the orientational distribution func-tions (ODFs), Ψ i ( θ ), i = 1 ,
2, are related to the one-bodydensity distributions by ρ i ( r , ω ) = ρ i Ψ i ( θ ). There is nodependence of the ODF on the azimuthal angle φ sincethe platelets are uniaxial rotators, and we assume thatonly uniaxial states are formed. A powerful feature ofDFT is that V ( i )ext ( r , ω ) (3) appears explicitly in the grandpotential and therefore enters straightforwardly into theminimisation procedure (5); see the appendix for the ex-plicit form of the corresponding Euler-Lagrange equa-tions that we solve numerically.The requirements for phase coexistence between twophases A and B are the mechanical and chemical equi-libria between the two phases and the equality of tem-perature in the two coexisting phases (which is trivialin hard-body systems). Hence we have the non-trivialconditions: the equality of pressure p A = p B and theequality of chemical potentials µ Ai = µ Bi , where i = 1 , A, B labels the phase. Wecalculate the total Helmholtz free energy F = F id + F exc numerically by inserting Ψ i ( θ ) into the free energy func-tional. Likewise, the pressure can be obtained numeri-cally as p = − F/V + P i =1 ρ i ∂ ( F/V ) /∂ρ i and the chem-ical potentials as µ i = ∂ ( F/V ) /∂ρ i . We define a reducedpressure p ∗ = βpR and reduced chemical potentials µ ∗ i = βµ i . The equations for phase coexistence are threeequations for four unknowns (two statepoints each char-acterised by two densities) hence regions of two-phasecoexistence depend parametrically on one free parame-ter (which can be chosen arbitrarily, e.g. as the value ofcomposition x in one of the phases) and are solved nu-merically with a Newton-Raphson procedure [39]. Theresulting set of solutions yields the binodal. P-N-N triplepoints are located where the P-N and N-N coexistencecurves cross. In the fieldless case there is, of course, nota paranematic phase, but an isotropic phase.We characterize orientationally ordered phases (P andN) of the binary mixture by two partial order parameters, S and S , defined by S i = 4 π Z π/ dθ sin( θ )Ψ i ( θ ) P (cos θ ) , (8)where P (cos θ ) = (3 cos θ − / θ . III. RESULTS
We first review the behaviour of the pure system un-der the influence of an aligning field [8]. The inset ofFig. 1(a) shows the phase diagram for a system com-posed of particles of species 1 only. Upon increasing thefield strength W , the coexisting concentrations c ini-tially shift to lower values. The biphasic density gap de-creases slightly as the strength of the external potentialis increased. At approximately W = 0 .
02, the parane-matic coexistence concentration starts to increase, whilethe nematic coexistence concentration continues to de-crease. The two branches of the binodal meet at a criticalpoint at c crit1 = 0 .
42. For W > W crit1 = 0 . λ = 1 in main plotof Fig. 1(a). However, due to the definition of c (recallthat c = ρ R , using the radius of species 1 in order toobtain a dimensionless quantity) and the scaling of W with the square of the size ratio ( W = λ W ), the phasediagram of the pure system of species 2 displays strongvariation with size ratio λ , as shown in Fig. 1(a). A shiftto both smaller values of c and of W occurs upon in-creasing the value of λ . However, this effect is entirelydue to the choice of coordinates, which for both puresystems are related via c = c /λ and W = W /λ .Numerical values for the location of the critical point aresummarised in Tab. I. λ λ λ c crit2 W crit1 c crit2 = λ − c crit and criticalfield strength W crit1 = λ − W crit for a range of size ratios λ ,where c crit and W crit are the critical concentration and fieldstrength in the pure system (without species index). The variation of the order parameter S for monodis-perse platelets with field strength is displayed inFig. 1(b). In the field-free case, W = 0, the nematicphase at coexistence possesses an unusually small orderparameter, see e.g. the discussion in Ref. [40]. For allsize ratios, as W is increased, the coexistence value of S in the paranematic phase increases monotonically, andthe value of S along the nematic branch of the binodaldecreases with increasing field strength. This is consis-tent with the fact that the coexistence density decreasesas the field strength increases, overcompensating for theordering effect caused by the applied field. At the criti-cal point the nematic order parameter takes on the value S = 0 .
27. For increasing values of λ = 1 , . , . ≤ x ≤
1. In Fig. 2 the phase diagram for λ = 1 . W = 0 .
15 (which corresponds to W = λ W =0 . W = 0, there is I-N phasecoexistence over the entire range of compositions x . Wedisplay this phase diagram (and subsequent ones) bothin the ( c , c ) representation [Fig. 2(a)] as well as in the( x, p ∗ ) representation [Fig. 2(b)]. Tie-lines are omittedfor clarity; in the ( c , c ) representation these connectthe lower branch of the binodal to the upper branch insuch a way that the isotropic (or paranematic) phase isrich in (the smaller) species 1 and the nematic phase isrich in (the larger) species 2. In the ( x, p ∗ ) representation[Fig. 2(b)] the tie lines are (trivially) horizontal due tothe condition of equal pressures in the coexisting phases.For W = 0 .
01, the binodal still connects to the axes(which correspond to the pure systems). Recall that theP-N transition still occurs in the pure systems at this fieldstrength, cf.
Fig. 1(a). However, the isotropic phase hasnow become a weakly-ordered paranematic phase. Hencethere is P-N phase coexistence over the entire range ofcompositions. The isotropic phase has been replaced bya paranematic phase, because the order parameter alongthe lower branch of the binodal is non-zero, see Fig. 2(c)and (d), where the partial nematic order parameters areshown for species 1 and 2, respectively. On increasing W to 0 .
03, the upper and lower branches of the binodalstill persist to the pure system of smaller platelets, con-sistent with the findings of Ref.[8]. However, the binodaldoes not touch the c -axis, indicating that there is nolonger a phase transition in the pure system of species2 (we found the critical field strength for the monodis-perse system at λ = 1 . . .
03, Fig. (1a)). Hence the two branches of the binodalconnect at a (lower, in pressure) critical point. There-fore the state of the system changes continuously fromparanematic to nematic for compositions greater thanabout 0.7 by increasing the pressure. For compositionsless than this value, increasing the pressure from belowthe lower branch of the binodal to the upper branch of thebinodal, the system, as before, passes through a bipha-sic region. For W = 0 .
05 the departure of the binodalfrom c = 0 (and x = 0) occurs as is consistent withthe critical field strength being W crit1 = 0 .
045 in the puresystem. The result is a phase diagram in which the twobranches of the binodal have joined to form a closed loopof immiscibility. There is a larger range of compositionstowards the x = 1 side of the phase diagram (approx-imately x > .
45) than towards the x = 0 side of thephase diagram (approximately x < . W = 0 , . c W . . . . . . . . . . . . . . c W . .
45 0 . . . . . . (a)2.5 1.52 1 x = 0NP S W . . . . . . . . . . . . . .
05 0 .
3P N (b)1.522.5 λ = 1 FIG. 1: Paranematic-nematic phase diagram of the one-component system(s). (a) Behaviour of the pure system ofspecies 2 (i.e. x = 1). The variation of the paranematic (P)and nematic (N) coexistence concentrations, c (horizontalaxis) with the strength of the aligning field W (vertical axis)is shown for λ = 1 , . , . x = 0). This is equivalent to the case λ = 1 in the main plot when identifying the horizontal axes.The critical point is depicted as a filled circle. (b) Variationof the orientational order parameter S = S along the parane-matic and nematic branches of the binodal (horizontal axis)of the pure system of species 2, with increasing field strength W (vertical axis) for the same size ratios as in (a). Criticalpoints are depicted as filled circles. c c . . . . . . . . . . . . . .
18 P N (a) x p ∗ . . . . . . . . . . . .
82 P N (b) x S . . . . . . . . . . . . .
91 (c) x S . . . . . . . . . . . . .
91 (d)
FIG. 2: Phase behaviour of binary platelet mixtures withsize ratio λ = 1 . W =0 , . , . , . , . , . , .
12 and 0 .
15 (with W = λ W )from outside to inside. Shown are phase diagrams in (a) the( c , c ) representation and (b) in the ( x, p ∗ ) representation.The partial order parameters S i along the binodal are shownin (c) for species 1 and in (d) for species 2. and 0 .
03 the paranematic and nematic branches of thebinodal do not connect on the low-composition side ofthe order parameter graph since these values of W areless than W crit1 = 0 . . ≤ W ≤ .
15 the twobranches of the binodal connect on the low-compositionside of the graph. For W = 0 and 0 .
01 the two branchesof the binodal do not connect on the high-compositionside of the graph since these values of W are smallerthan W crit2 = λ W crit1 = 0 . W ≥ .
05 the par-tial order parameters measured along the binodal formclosed loops. These islands become smaller with increas-ing field strength and eventually coalesce to a point whenthe double critical point is reached. The partial orderparameters of species 2 [Fig. (2d)] follow a similar pat-tern except that the order at a given statepoint is higherthan that for species 1, as one could expect, given thatspecies 2 is of the larger size.In Fig. 3 we show results for λ = 2. Increasing thesize ratio to this value leads to an increase of the size ofthe I-N biphasic region [30]. The fieldless case possessesa reentrant phenomenon whereby the system undergoesthe following change of state when increasing the pres-sure at fixed mole fraction at around x = 0 . → I-N → N → I-N → N. In addition,there is N-N coexistence between a nematic phase richin species 1 (N ) and a nematic phase rich in species 2(N ) ending in an upper critical point and an I-N-N triplepoint. Applying a small field strength of W = 0 .
02, the c c . . . . . . . . . . . . x p ∗ . . . . . . .
53 NP (b) x S . . . . . . . . . . . . .
91 (c) x S . . . . . . . . . . . . .
91 (d)
FIG. 3: Same as Fig. 2 except for size ratio λ = 2 and forexternal field strengths W = 0 , . , . , . binodal no longer reaches the pure system of species 2,as W = 0 . > W crit1 = 0 . λ = 2. An effectof this is that the reentrant part of the phase diagramalters: the range of compositions for which the systemundergoes P → P-N → N → P-N → N at just over x = 0 . x = 0 .
6. Aside fromthese differences, the rest of the phase boundaries fol-low closely those of the fieldless case, though remainingslightly inside those of the latter. There is N-N coexis-tence ending in an upper critical point. The triple pointis retained as a P-N-N line in the ( x, p ∗ ) representationand a triangle in the ( c , c ) representation although wedo not show these features in the plots for clarity. Uponincreasing the field, triple phase coexistence vanishes, i.e.the triple point collapses onto two-phase coexistence. Wehave not calculated the precise value of the external fieldwhere this happens. We expect this value to be differentfrom the values where the binodal detaches from either ofthe density axes (i.e. differ from the critical field strengthsin the pure systems). Applying a field strength W = 0 . W crit1 = 0 . ) and the paranematic phase have merged.In Fig. 4 we present results for λ = 2 .
5. The field-less case displays I-N coexistence, an I-N-N triple pointand coexistence between two nematic states, which doesnot end in a critical point. Applying just a small field W = 0 .
01 has a considerable effect on the phase be-haviour: the pure system of species 2 looses the P-Ntransition as W = 0 . > W crit1 = 0 . c c . . . . . . . . . . . . . . x p ∗ . . . . . . . . . x S . . . . . . . . . . . . .
91 (c) x S . . . . . . . . . . . . .
91 (d)
FIG. 4: Same as Fig. 2 except for λ = 2 . W = 0 , . , , . , W = 0 . , , . S i in (c)and (d) are due to numerical artifacts. the transition persists in the pure system of species 1.Hence, there is still P-N coexistence and indeed a tailbetween about x = 0 . x = 0 . W = 1, there is no P-N tran-sition for either of the pure limits. There is, however,a large immiscibitity gap between two distinct nematicphases, N and N . Increasing the field strength raisesthe phase coexistence to higher pressures and narrowsthe phase coexistence region. However, large steps infield strength are required to have a significant effect onthe system. W = 7 (which corresponds to W = 49)is approximately 45 times stronger than the field reqiredto homogenise the system at λ = 1 . W . This formsa limit to the densities at which we may probe at λ = 2 . θ -grids. IV. CONCLUSIONS AND OUTLOOK
We have investigated the effects of an external align-ing (magnetic) field on the phase behaviour of binarymixtures of circular hard core platelets with zero thick-ness. Using the FMT of Refs. [30, 37], we have tracedparanematic and nematic phase boundaries and have ex-amined the partial nematic order parameters at coexis-tence. Three different representative values for the ra-dial bidispersity ( λ = 1 . , , .
5) have been studied. Thetopologies for each of these values are different from eachother in the fieldless case. For the smallest size ratio con-sidered, λ = 1 .
5, the fieldless case shows only I-N phasecoexistence over the entire range of compositions. For λ = 2, besides I-N coexistence there is also N-N coex-istence ending in an upper critical point and an I-N-Ntriple point occurs. For λ = 2 . λ = 1 . λ = 2 .
5, the coexistingnematic states become so well-ordered that the systemdoes not become homogeneously nematic up to the fieldstrengths we have applied.Results from computer simulation studies for this (or asimilar) model mixture are highly desirable, as are exper-imental studies. Colloidal platelets are often significantlypolydisperse in both radius and thickness, see for exam-ple Ref. [41], so effects due to polydispersity will play arole in experimental systems, which are not accounted for in the present theory. Recently, the P-N interface insuspensions of boardlike goethite particles has been in-vestigated experimentally [42]. Anticipating that similarstudies could be made in systems of colloidal platelets, afurther exciting avenue would be to investigate the prop-erties of the P-N interface using FMT, which has alreadybeen shown to compare well with experimental and sim-ulation results for the I-N interface [43]. It would be in-teresting to consider in theoretical work the effects thatare induced by finite thickness of the platelets. In thepresent study we restricted ourselves to mixtures withmoderate size asymmetry, as we expect the theory to de-scribe these accurately. Investigating highly asymmetricmixtures, possibly based on the depletion picture, is aninteresting issue for future work.
Appendix A: Self-Consistency Equations for theOrientational Distribution Functions
We give a summary of the equations that are necessaryto find the ODFs at a given composition, x and conce-tration, c . The excess free energy from FMT is the sumof the right hand sides of Eqs. (13) and (27) of Ref. [30].Inserting this, together with the ideal free energy (7) andthe external potential (3), into the grand potential func-tional (6), and employing the minimisation principle (5)leads to two coupled Euler-Lagrange equations for theODFs:Ψ ( θ ) = 1 Z exp (cid:20) − πc Z π/ dθ ′ sin θ ′ K ( θ, θ ′ )[(1 − x )Ψ ( θ ′ ) + 12 x ( λ + λ )Ψ ( θ ′ )] − πc Z π/ dθ ′ sin θ ′ Z π/ dθ ′′ sin θ ′′ L ( θ, θ ′ , θ ′′ )[(1 − x ) Ψ ( θ ′ )Ψ ( θ ′′ ) + 2 x (1 − x ) λ Ψ ( θ ′ )Ψ ( θ ′′ ) + x λ Ψ ( θ ′ )Ψ ( θ ′′ )] + W sin θ (cid:21) , (A1)Ψ ( θ ) = 1 Z exp (cid:20) − πc Z π/ dθ ′ sin θ ′ K ( θ, θ ′ )[ xλ Ψ ( θ ′ ) + 12 (1 − x )( λ + λ )Ψ ( θ ′ ) − πc Z π/ dθ ′ sin θ ′ Z π/ dθ ′′ sin θ ′′ L ( θ, θ ′ , θ ′′ )[ x λ Ψ ( θ ′ )Ψ ( θ ′′ ) + 2 x (1 − x ) λ Ψ ( θ ′ )Ψ ( θ ′′ ) + (1 − x ) λ Ψ ( θ ′ )Ψ ( θ ′′ )] + W sin θ (cid:21) , (A2)where the constants Z and Z are such that the nor-malisation R d ω Ψ i ( ω ) = 1, for i = 1 ,
2. The numericalprocedure is the same as that described in Ref. [30], whichis an extension of the procedure introduced in Ref. [44]. The integral kernel K ( θ, θ ′ ) is K ( θ, θ ′ ) = Z π dφ sin γ = Z π dφ p − ( ω · ω ′ ) = Z π dφ p − (cos θ cos θ ′ + sin θ sin θ ′ cos φ ) , (A3)where φ is the difference between the azimuthal angles ofthe two platelets and the kernel L ( θ, θ ′ , θ ′′ ) is L ( θ, θ ′ , θ ′′ ) = Z π Z π dφ ′ dφ ′′ | ω · ( ω ′ × ω ′′ ) | = Z π Z π dφ ′ dφ ′′ | sin θ (sin φ ′ sin θ ′ cos θ ′′ + cos θ ′ sin φ ′′ sin θ ′′ ) + cos θ (cos φ ′ sin θ ′ sin φ ′′ sin θ ′′ − sin φ ′ sin θ ′ cos φ ′′ sin θ ′′ ) | . (A4)The solutions of Eqs. (A1) and (A2), Ψ ( θ ) and Ψ ( θ ),are then inserted into the equations p A = p B and µ A i = µ B i , i = 1 , Acknowledgments
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