Bethe-lattice calculations for the phase diagram of a two-state Janus gas
aa r X i v : . [ c ond - m a t . s o f t ] J a n Bethe-lattice calculations for the phase diagram of atwo-state Janus gas
Danilo B. Liarte
E-mail: [email protected]
Institute of Physics, Caixa Postal 66318, CEP 05314-970, S˜ao Paulo, SP, Brazil
Silvio R. Salinas
E-mail: [email protected]
Institute of Physics, Caixa Postal 66318, CEP 05314-970, S˜ao Paulo, SP, BrazilJanuary 2015
Abstract.
We use a simple lattice statistical model to analyze the effects ofdirectional interactions on the phase diagram of a fluid of two-state Janus particles.The problem is formulated in terms of nonlinear recursion relations along the branchesof a Cayley tree. Directional interactions are taken into account by the geometry of thisgraph. Physical solutions on the Bethe lattice (the deep interior of a Cayley tree) comefrom the analysis of the attractors of the recursion relations. We investigate a numberof situations, depending on the concentrations of the types of Janus particles andthe parameters of the potential, and make contact with results from recent numericalsimulations.
1. Introduction
The production and characterization of solutions of Janus particles, whose sphericalsurface is divided into hydrophobic and hydrophilic hemispheres, have attracted theattention of a number of authors [1, 2, 3, 4, 5, 6]. Studies of the behavior of thesecolloidal systems must take into account that Janus particles interact in a differentway depending on their relative orientations. A simple form of a pairwise directionalpotential has been proposed by Kern and Frenkel [7], who introduced a model of hardspheres with the addition of a square-well potential with directional attractive short-range interactions. The Kern-Frenkel model has been used in several analytical andnumerical investigations (see e.g. [8, 9, 4, 6]). In particular, a simplified two-stateversion of the Kern-Frenkel model, which is reminiscent of the Zwanzig approximationfor liquid-crystalline models, and can be experimentally realised by the application ofan electric field, has been extensively studied in [10, 11].These recent calculations provided the motivation to introduce a simple latticestatistical model to analyze the effects of directional interactions in a Janus gas of ethe-lattice calculations for the phase diagram of a two-state Janus gas a and b , with hydrophobic (hydrophilic) hemispheres in the upper (lower) half part oftheir respective spherical surfaces, which amounts to considering a two-state, Ising-like,representation of a Janus gas (see figure 1). In a grand canonical formulation, we adda chemical potential to control the density of each type of particle. Taking advantageof the geometrical structure of this tree, it becomes particularly simple to introducedirectional interactions between first-neighbor sites along the branches of the graph.We choose the parameters of the potential, and the concentration of the two types ofparticles, to make contact with the available numerical simulations. Generation j+1j j-1... (a) (b)
Figure 1.
Sketch of some generations of a Cayley tree of ramification r = 2. Thetwo-state system is formed by Janus particles of type a (depicted as circles with redheads up in this figure) and of type b (blue heads up). The trees on the left (a), and onthe right (b), illustrate macroscopic configurations with a high density of a -particles,and a modulated (“striped”) phase, respectively. This article is organized as follows. In Section I, we describe the lattice statisticalmodel on a Cayley tree. We show that the solutions on the Bethe lattice (deep inthe interior of the tree) can be obtained from the analysis of the recursion relationsassociated with a nonlinear discrete map. In sections II and III, we analyze someparticular cases, and make contact with results from simulations. In particular, weshow that attractive interactions of the Kern-Frenkel form lead to layered modulatedstructures, which have indeed been found in the simulations [11]. Although we donot make quantitative contacts with the simulations, we do claim to have used amuch simpler approach to obtain a number of analytical qualitatively results, for afull range of model parameters. In particular, we provide a unified view of the phasetransitions described by a “cubic diagram” of interaction parameters drawn by Fantoniand collaborators [11]. ethe-lattice calculations for the phase diagram of a two-state Janus gas
2. Formulation of the problem on the Bethe Lattice
We consider Janus particles restricted to two orientations. These particles arerepresented by a set variables { t i } , on the sites of a tree, so that t i = 1 is associated witha particle of type a on site i , and t i = 0 represents a Janus particle of type b on site i .The pair interaction energy between nearest-neighbor sites i and j along the branchesof the tree is given by E ij = − ( ǫ aa + ǫ bb − ǫ ab − ǫ ba ) t i t j − ( ǫ ab − ǫ bb ) t i − ( ǫ ba − ǫ bb ) t j − ǫ bb . (1)Note that aa and bb particles interact with energies − ǫ aa and − ǫ bb , respectively. Also,note that ab and ba particles interact with (in general) different energies, − ǫ ab = − ǫ ba .Of course, if ǫ ab = ǫ ba we regain the results for a usual lattice gas representation of abinary liquid mixture [12].In the appendix, we use standard treatments for a Cayley tree [14], in order toobtain the recursion relationsΞ aj +1 = z h e K aa Ξ aj + e K ab Ξ bj i r (2)and Ξ bj +1 = h e K ba Ξ aj + e K bb Ξ bj i r , (3)for the partial grand partition function Ξ aj (Ξ bj ) that is associated with the sub-treegenerated by a site at generation j which is occupied by a particle of type a ( b ). K kl = βǫ kl , for k, l = a, b , β = 1 /k B T is the inverse temperature, z = exp( βµ ) isthe fugacity, and µ is the chemical potential (associated with particles of type a ). It isnow convenient to define the density of particles of type a in generation j , ρ j = Ξ aj Ξ aj + Ξ bj , (4)so that Eqs. (A.3) and (A.4) may be written as a single recursion relation ρ j +1 = f ( ρ j ) , (5)with f ( x ) = ( z " e K ba x + e K bb (1 − x ) e K aa x + e K ab (1 − x ) r ) − , (6)where 0 ≤ x ≤
1. This is the central result of this formulation. At this point theproblem is reduced to analyzing the general map given by Eq. (5), from which weobtain the main features of the phase diagrams in terms of temperature, T = 1 / ( k B β ),and chemical potential µ (and with different choices of the energy parameters).It is interesting to investigate the limit of infinite coordination of the tree, r → ∞ ,with fixed values rǫ aa , rǫ bb , rǫ ab , and rǫ ba . In this limit, it is easy to show that f ( x ) → f ∞ ( x ) = (cid:16) e K + K x (cid:17) − , (7)where K = β ( µ + δ ) , K = β ∆ , (8) ethe-lattice calculations for the phase diagram of a two-state Janus gas r ( ǫ ab + ǫ ba − ǫ aa − ǫ bb ) , δ = r ( ǫ bb − ǫ ab ) . (9)This limit is known to lead to the solutions for an analogous fully-connected, mean-fieldmodel [13, 14]. It is important to remark that the phase diagrams depend on just twoparameters, K and K , and that the parameter ∆ plays a quite special role.At high temperatures, both f ( x ) and f ∞ ( x ) tend to 1 /
2, so that ρ → / T → ∞ . At low temperatures, there are two possible scenarios depending on the signof ∆. If ∆ <
0, there is a discontinuous (first-order) line of transitions in a diagramin terms of chemical potential and temperature. This border, which separates stateswith low- and high-density of particles of type a , ends at a critical point. In the secondscenario, for ∆ >
0, the analogous phase diagram displays a critical line enclosing acycle-2 periodic phase. This approach provides a unified view of the phase transitionsdescribed by the choice of interaction parameters according to the cubic diagram ofFantoni and collaborators (see table 1). Except for the HS and SW models, whichare non-interacting models in our lattice approach, all of the cases in this diagram areshown to fit into one of these two scenarios. Also, although the finite-coordination mapis more respectable than the mean-field limit, we find no qualitative changes for treeswith ramification r > r = 1 corresponds to a one-dimensional model, which has nophase transition at finite temperature). We will discuss these points in the followingsections. Model ǫ aa ǫ ab ǫ ba ǫ bb ∆HS 0 0 0 0 0A0 0 ǫ r ǫ I0 ǫ ǫ − r ǫ J0 0 ǫ ǫ r ǫ B0 ǫ ǫ ǫ − r ǫ SW ǫ ǫ ǫ ǫ Table 1.
Definition of the models and corresponding values of ∆ according tonomenclature defined in [11].
3. Discontinuous transition ( ∆ < ) We first consider a simple case, ǫ aa = ǫ bb = ǫ/r > ǫ ab = ǫ ba = 0, which correspondsto I in the cubic representation of Fantoni and collaborators [11]. The attractors ofthe map can be visualized if we draw graphs of f ( ρ ), given by Eq. (6) in terms of ρ ,as shown in figure 2, for a fixed temperature ( k B T /ǫ = 0 . r = 5), and for several values of the chemical potential ( µ/ǫ = − . − . .
17 and 0 . f ( ρ ) = ρ , so that we plot ρ as the black dotted line in this figure. These plots cover all the three qualitatively ethe-lattice calculations for the phase diagram of a two-state Janus gas <
0. There can be a single stable fixed point (blackcurves), a stable and a marginally stable fixed point (blue curves), and two stable andone unstable fixed points (red curve). The plots suggest a first-order transition from ahigh to a low density phase of a -particles. The blue curves indicate the emergence (orvanishing) of two fixed points, as well as their stability threshold. In a phase diagram interms of temperature and chemical potential, the blue curves describe the behavior ofthe map along the spinodal lines. The red curve shows the behavior of the map at thetransition (in the next paragraph, we show that, below a certain critical temperature,there is a first-order phase transition at µ = | ∆ | / − δ , with µ = 0 in the I case ofthe cubic diagram). A numerical inspection of Eqs. (5) and (6) leads to no additionalcharacteristic structures of the map. Figure 2.
Plots of f ( ρ ) for ∆ <
0. We assume temperature k B T /ǫ = 0 .
3, ramification r = 5, and several values of the chemical potential, µ/ǫ = − . − .
17; 0; 0 .
17; and 0 . µ ). The black dottedline corresponds to ρ . In figure 3, we draw some phase diagrams in terms of ( µ + δ ) / | ∆ | and k B T / | ∆ | for a tree of finite coordination ( r = 5) and at the infinite coordination limit. There isa critical point at ( µ c + δ ) / | ∆ | = 1 /
2, and k B T c / | ∆ | = 1 /
4. The black solid line, for( µ + δ ) / | ∆ | = 1 /
2, and k B T / | ∆ | < /
4, is a first-order boundary. The spinodal limits,which are represented by the dashed lines, are given by the equations x = f ( x ) , | f ( x ) | = 1 . (10)In the infinite-coordination limit, we have the set of parametric equations k B T | ∆ | = ρ (1 − ρ ) , (11)and ( µ + δ ) | ∆ | = ρ (1 − ρ ) ln 1 − ρρ + ρ, (12) ethe-lattice calculations for the phase diagram of a two-state Janus gas Figure 3.
Phase diagram (temperature versus chemical potential) for ∆ <
0. Thesolid black line of first-order transitions ends at a critical point. We also show spinodallines (dashed) for a tree of finite coordination (red lines) and in the mean-field limit(black lines). with 0 ≤ ρ ≤
1. In order to describe the first-order boundary, we note that, at a fixedpoint of f ∞ , we have( µ + δ ) / | ∆ | = ρ + k B T | ∆ | ln 1 − ρρ . (13)Therefore, ( µ + δ ) / | ∆ | − / ρ − /
2, so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z / ρ [( µ + δ ) / | ∆ | − / dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z − ρ / [( µ + δ ) / | ∆ | − / dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14)where ρ is the smallest solution of ( µ + δ ) / | ∆ | = 1 /
2. If we resort to a Maxwellconstruction [14], this leads to a first-order boundary, given by ( µ + δ ) / | ∆ | = 1 / k B T | ∆ | = 12 1 − ρ ln 1 − ρρ , (15)so that the two coexistent densities are the solutions ( ρ, − ρ ) of Eq. (15), with0 ≤ ρ ≤
1, and converge to ρ → / T c = | ∆ | / k B . In figure 4, we show thecoexistence curve (black solid curve) and a few tie lines (in blue) in the mean-fieldlimit. We also show isotherms of the chemical potential (red dotted curves), whichare adequately scaled with temperature so that the corrected solutions, according toMaxwell’s construction, correspond to the coexistence tie lines. Similar graphs andresults can be numerically obtained for trees of finite coordination (with q ≥ I and B in the cubic diagram of parameters are fully understood according tothe general behavior displayed in figures 3 and 4. ethe-lattice calculations for the phase diagram of a two-state Janus gas Figure 4.
The black solid line is a coexistence curve (temperature versus concentrationof particles of type a ). We also show tie lines (blue), and isothems (red dotted lines)for the chemical potential in the mean-field limit.
4. Continuous transition ( ∆ > ) For ∆ >
0, at low temperatures, instead of a first-order boundary, there is a criticalline from a disordered to a cycle-2 periodic phase. We first sketch the properties of themap in the simple A case of the cubic diagram of interactions, with ǫ ab = ǫ/r > ǫ aa = ǫ bb = ǫ ba = 0. Now there are only two scenarios. The map has either asingle stable fixed point or one unstable fixed point and a stable cycle of period 2. Infigure 5a, we draw ρ (dotted line), f ( ρ ) (dashed lines), and f ◦ f ( ρ ) (solid lines), forfixed chemical potential, µ/ǫ = 0 .
5, ramification r = 7, and two values of temperature, k B T /ǫ = 0 .
25 (black) and 0 .
15 (red). A cycle-2 orbit can be graphically found as thesolution of ρ = f ◦ f ( ρ ) with ρ = f ( ρ ). This orbit represents modulated phases, withdensity oscillations along the generations of the tree (as illustrated in figure 1b). Theorbit of period 2 can also be visualized by means of a cob-web plot, which we show asthe blue dotted line in figure 5b. Note that the instability threshold occurs at the samepoint as the emergence of the stable cycle-2 phase, and that any fixed point locatedbetween the solutions of a stable cycle-2 orbit will be unstable. A numerical inspectionof f and f ∞ leads to no additional structures of the map beyond these two scenarios.In figure 6a we plot phase diagrams in terms of ( µ + δ ) / | ∆ | and k B T / | ∆ | , with ∆ > A in the cubic diagram of Fantoni and collaborators [11].Since there is a coincidence between stability and transition thresholds, we can use Eq.(10) to derive parametric equations for the critical line of the mean-field map, k B T | ∆ | = ρ (1 − ρ ) , (16)and µ + δ | ∆ | = ρ (1 − ρ ) ln 1 − ρρ − ρ. (17) ethe-lattice calculations for the phase diagram of a two-state Janus gas Figure 5. (a) Plots of ρ (dotted line), f ( ρ ) (dashed lines), and f ◦ f ( ρ ) (solidlines), for fixed chemical potential, µ/ǫ = 0 .
5, ramification r = 7, and two values oftemperature, k B T /ǫ = 0 .
25 (black) and 0 .
15 (red). A cycle-2 orbit can be graphicallyfound as the solution of ρ = f ◦ f ( ρ ) with ρ = f ( ρ ). (b) Cobweb plot to illustrate theperiod-2 cycle. Note that the minus sign in the second term on the r.h.s. of this last equation isresponsible for both the “soft” behavior near k B T / | ∆ | = 0 .
25, and the reentrantbehavior at low temperatures. For trees of finite coordination, we resort to a simplenumerical calculation to find the analogous critical line. In figure 6b we plot thecorresponding phase diagram in terms of ρ and k B T / | ∆ | . As we mentioned above,there is no indication of an alternative critical behavior in the region enclosed by thecritical line. (a) (b) Figure 6. (a) Critical lines in the phase diagram in terms of temperature andchemical potential (with ∆ > ρ × T plane. ethe-lattice calculations for the phase diagram of a two-state Janus gas
5. Duality and low coordination limit
We have found an interesting (dual) relation between the solutions for a cycle-2 orbitalong a symmetry line and the phase-separated densities along the coexistence curve.Consider the mean-field map with ∆ >
0, so that at ( µ + δ ) / | ∆ | = − /
2, and k B T / | ∆ | < /
4, there is a periodic solution of the form ( ρ, − ρ ). Thus, we canwrite 1 − ρ = f ∞ ( ρ ) = h e − (1 − ρ ) / T i − , (18)from which we have ρ = h e (1 − ρ ) / T i , (19)where ˜ T is a short-hand notation for k B T / | ∆ | . Hence, the solutions for the densityalong the coexistence curve for ∆ < µ + δ ) / | ∆ | = − / > I and A , for ∆ < >
0, respectively.In the I case, for ( µ + δ ) / | ∆ | = 1 / ρ = 1 /
2, we have f ′ ( ρ ) | ρ =1 / = r tanh | ∆ | r k B T ! , (20)so that, at the critical point,1 r = tanh | ∆ | r k B T c ! . (21)As it should be anticipated, T c = | ∆ | / k B is a trivial solution for the map in the mean-field limit ( r → ∞ ). Also, Eq. (21) has no solutions for r <
1, so T c → r → r = 1, since no transition is expected in one-dimensional systems with short-rangeinteractions. Similar results can be obtained in the A case (with ( µ + δ ) / | ∆ | = − / r isdecreased, and the solution ρ = 1 / r ifwe fix ( µ + δ ) / | ∆ | = − /
2. For finite ramification, we have to numerically calculate thevalue of ρ at the critical temperature.
6. Conclusions
We have considered a lattice gas of two-state Janus particles on the sites of a Cayley tree.Taking advantage of the geometrical structure of this graph, it is particularly simple tointroduce directional interactions between first-neighbor sites along the branches of thetree. The problem is formulated in terms of a set recursion relations, whose attractorscorrespond to physical solutions on the Bethe lattice (the deep interior of the Cayleytree). With relatively easy calculations, we can draw a number of phase diagrams ethe-lattice calculations for the phase diagram of a two-state Janus gas
Acknowledgement
We acknowledge the financial support provided by the Brazilian agencies CNPq andFapesp.
Appendix A. Derivation of recursion relations
In figure A1, we draw the configurations that are associated with two generations of aCayley tree of ramification r = 2 (corresponding to coordination q = r + 1 = 3). Thistree is constructed along a direction that leads to a consistent and unambiguous choiceof the interaction parameters. We assume that a and b particles interact with energy − ǫ ab if a particle of type a is on a site belonging to a certain generation and a particleof type b is on a nearest-neighbor site belonging to the next generation (and vice versa).There is a Boltzmann factor associated with each interaction between j and j + 1. Thereis also a fugacity term related to particles of type a ( t j +1 = 1). For this 3-coordinatedtree, let Ξ aj (Ξ bj ) be the partial grand partition function associated with the sub-treegenerated by a site at generation j which is occupied by a particle of type a ( b ). Thus,the partial partition functions at generations j and j + 1 obey the set of relationsΞ aj +1 = z (cid:20) e K aa (cid:16) Ξ aj (cid:17) + 2 e K aa + K ab (cid:16) Ξ aj Ξ bj (cid:17) + e K ab (cid:16) Ξ bj (cid:17) (cid:21) (A.1)and Ξ bj +1 = e K ba (cid:16) Ξ aj (cid:17) + 2 e K ba + K bb (cid:16) Ξ aj Ξ bj (cid:17) + e K bb (cid:16) Ξ bj (cid:17) , (A.2)where K kl = βǫ kl , for k, l = a, b , β = 1 /k B T is the inverse temperature, z = exp( βµ ) isthe fugacity, and µ is the chemical potential (associated with particles of type a ). Fora tree with a general ramification r , we have the more general equationsΞ aj +1 = z h e K aa Ξ aj + e K ab Ξ bj i r (A.3)and Ξ bj +1 = h e K ba Ξ aj + e K bb Ξ bj i r . (A.4) ethe-lattice calculations for the phase diagram of a two-state Janus gas exp(2 K aa )exp(2 K ba ) exp(2 K bb )exp(2 K ab )exp( K aa +K ab ) exp( K ab +K aa )exp( K ba +K bb ) exp( K bb +K ba ) t j+1 = t j+1 = 0 t j t j z zz z Figure A1.
Illustrations of the possible configurations at the j − th generation with t j +1 = 1 (top) and t j +1 = 0 (bottom). Note that we write the Boltzmann factorassociated with each configuration. A fugacity term z is present at the recursionrelation if t j +1 = 1. References [1] S. Jiang and S. Granick.
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