Phase diagram for an attractive square-well plus a linear tail potential within the van der Waals-like theory

The phase equilibrium behavior of a typical colloidal dispersion by the van der Waals-like theory was studied.

In this theory, the colloid–colloid potential is split into two parts: a repulsive part modeled by the hard-core potential and an attractive part modeled by the square-well potential corrected by a linear tail.

The theory is numerically elegant since the colloidal free energy is analytic and, physically, the model can be used to study the role played by the strength and range of interactions on the formation of coexisting phases.

The effect of the strength of the interaction can be accounted for by varying the width of the square-well potential (simulated by γ parameter) while the range of the interaction—by changing the “slope” of the linear potential (simulated by λ parameter).

In this work, we consider three separate cases.

First, we fixed the threshold values γth, being γLLth = 0.221 for liquid–liquid and γSSth = 0.0329 for solid–solid, and increased λ to examine the appearance of the liquid–liquid and solid–solid coexistence curves which were both calculated with respect to their liquid–solid counterparts.

Second, we fixed the threshold values λth, being λLLth = 0.325 for liquid–liquid and λSSth = 0.0456 for solid–solid, and increased γ (keeping the λ-slope unchanged) to bring in the strength of the interaction.

In these two cases, the calculations showed that a switching on of λ or γ has the consequence of inducing stable liquid–liquid or metastable solid–solid coexisting phases.

Finally, we maintained a given γ plus λ.

Here the calculated liquid–liquid and solid–solid phase diagrams can be analyzed in finer details to exhibit the combined influences of the strength and range of interactions.


A long-standing problem in modern condensed matter physics is the understanding of the mechanism of phase separation in both simple and complex liquids.

Recently, several novel experiments were reported to have observed the liquid–liquid phase separation in monatomic liquids,1,2 molecular liquids3 as well as in complex liquids.4–6

Stimulated by these laboratory findings, a number of computer-simulation experiments were performed7–10 and claimed to have successfully reproduced or captured the essential characteristics of some of the above liquid–liquid phase transitions.

Perhaps more interesting is that these simulation studies predict in addition an isostructural solid–solid transition for a few simple parametric “model” systems.10–12

Implicit in all of these empirical endeavors is the quest for understanding the link between the role of the range and strength (in particular, the attractive nature) of inter-particle interactions and the possible mechanisms underlying these complicated phenomena.13–15

In this respect, recent experimental and computer simulation results have had a great impact on the theoretical development of liquid–liquid, liquid–solid and solid–solid phase equilibria.

One of the earlier quantitative works of this kind was that by Victor and Hansen16 who, based on the idea of the Week, Chandler and Andersen,17 worked out a first-order thermodynamic perturbation theory for a charge-stabilized colloidal dispersion induced at high salt concentrations.

The theory emphasizes the attraction between charged colloids as arising from the long range van der Waals type and predicts a liquid–liquid phase transition showing features quite similar to those observed by Kotera et al18. for the polystyrene latex particles and by Hachisu for the gold sol system.19

The mechanism for the occurrence of the phase transition has been attributed to the existence of a second minimum in the colloid–colloid potential, a characteristic feature simulated in their theoretical studies by an excess salt condition.

Following a somewhat different perturbation theory of Gast et al.,20 Kaldasch et al21. subsequently extended the work of Victor and Hansen16 to second order and included in the phase diagram calculations the liquid–liquid phase transition boundary determined with respect to the liquid–solid coexistence curves.

It should be mentioned moreover that the same first-order theory as Victor and Hansen has been reported also by Lai et al22. who analyzed and compared in some details the phase separation results between the well-known DLVO model23 and a realistic model previously proposed by Belloni.24

A more quantitative second-order thermodynamic perturbation has very recently been reported by Lai and Wu.25

This latter work made a detailed comparison of the theory with flocculation scenario observed in charged colloids.

Parallel to these realistic calculations but emerging around 1994 are two theoretical approaches to the study of phase separation.

The first approach attempts to understand the phase separation employing the well-known classical density functional theory.

In this method, the free energy functional was constructed in terms of the one and two particle density distribution functions, both for the liquid and solid counterparts.

With the recent development of the inhomogeneous density functional theory for the hard-sphere system in solid state and in conjunction with the liquid state theory already available in the literature, several simple model systems have been investigated.26–29

These studies confirm some of the computer-simulation findings and predict generally the occurrence of the isostructural solid–solid phase transition.30

On theoretical basis, this approach is rigorous.

The second approach which relies more on simple models follows closely the original ideas of van der Waals to account for the vapor–liquid transition.

Baus and coworkers31 are the early main contributors to this approach.33

In this theory, directed to charged colloidal suspension, the total potential energy of particles is assumed to be composed of an idealized hard-sphere repulsion to which a model attractive part is added.

The Gibbs–Bogoliubov inequality is then employed to construct the Helmholtz free energy in terms of a reference system free energy, and a correction consisting of a difference in potential energies of the model and reference systems, with the distribution function of the reference system taken to account for the thermal ensemble average.

By taking the hard spheres as reference system, and in conjunction with the cell model for liquid and an analogue for solid, these authors succeeded in deriving theoretically self-consistent liquid and solid free-energy expressions in terms of the attractive part of the potential.

Several attractive model potentials were investigated and most show liquid–liquid as well as solid–solid phase transitions.

In view of the mathematical simplicity and the transparency of the underlying theory, we have adopted in the present study of phase separation a model system that has much relevance to real physical systems.

The calculation is both a supplement to the works of Baus and coworkers and an extension of our recent communication.32

The model system studied in this paper is the attractive square-well plus a linear tail potential schematically shown in Fig. 1.

The choice for this potential is twofold.

First, this potential can be compared unambiguously with the widely studied square-well potential (λ = 0 in Fig. 1) to account for the varied strength of the interaction, and by increasing λ, it can be used to examine the range of the interaction on the phase transition.

Most importantly, the present model potential mimics closely the embellished form of the colloid–colloid potential function customarily employed in more complicated thermodynamic perturbation theory seeking for an analytical hard sphere reference system (see Fig. 1 in each of refs. 16 and 22).

Second, thermodynamic quantities for this potential can be solved also in analytical forms thus permitting an accurate evaluation of the phase diagrams.

The paper is therefore organized as follows.

In Section 2, we describe the van der Waals theory and give the expressions for the thermodynamic functions needed in the phase diagram calculation.

Then, in Section 3 we present our numerical results for the three cases showing the role played by γ and λ.

A conclusion that summarizes our present work is given in Section 4.


In this section, main ideas of the van der Waals theory recently advanced by Baus et al31. were summarized.

We present details of our derived expressions for the attractive square-well plus a linear tail potential.

These equations will be used in the following numerical calculations.

Van der Waals theory: Gibbs–Bogoliubov inequality

The theoretical framework of the van der Waals theory of Baus et al. begins with the well-known Gibbs–Bogoliubov inequality which reads as follows.

Consider a thermodynamic state at temperature T and density ρ for a physical system whose Helmholtz free energy and Hamiltonian are denoted by Ft and Ht, where the subscript t refers to a true system.

Suppose at the same T and ρ, we can find a reference system (symbolized by a subscript r) whose Fr, Hr, and related quantities such as the structure, thermodynamic functions, etc. are known, the inequality gives Ft ≤ Fr + 〈Ht − Hrrwhich states that the true Ft of a physical system is bounded above by a reference Fr plus a correction term consisting of the difference in Hamiltonians of the true and reference systems.

The 〈…〉r in eqn. (1) means an ensemble average taken over the distribution function of the reference system.

Note that the perturbation in eqn. (1) is just the difference in potential energies Vt and Vr of the true and reference systems, respectively.

Next, we write Vt = VtR + VtA splitting the pair potential into a repulsive part and an attractive part and similarly for the reference system, namely, VrR and VrA.

In accord with the van der Waals theory, we take the reference system to be a system of hard spheres, VrR → VrHS, which thus implies VrA = 0 and approximate the VtR to be hard-sphere-like repulsive (R → HS) also.

Setting VtR → VtHS = VrHS, simplifies eqn. (1) to where ρHS(r) is the one-particle density associated with the reference system VrHS.

Note that in arriving at eqn. (2), we have neglected all two-particle correlations and set the lowest upper bound as a reasonable estimate.34

The FHS in eqn. (2) is thus the hard sphere free energy which we will utilize below separately for the liquid and solid phases.

Helmholtz free energies of a liquid and a solid

To emphasize the theoretical consistency in the construction of free energies for the liquid and solid, which is essential for an accurate calculation of the phase diagram, we shall present these equations in parallel, stressing the analogy between them.

Since eqn. (2) involves only the attraction, it is convenient to write VtA(r) = −εϕA(r) where ε ≥ 0 is an energy parameter measuring the strength of attraction.

Next, we define where the reduced distance x = r/σ, σ being the hard sphere diameter related to the volume fraction η by σ = (6ηρ)1/3 and φ(x) ≥ 0 describing the range of interaction.

For a liquid (solid) which is homogeneous (inhomogeneous), the one-particle density function is ρ → ρHS(r) = ,  = N/V being the uniform number density for the liquid (solid).

Note that ρHS(r) = ∑Ni = 1ψ(r − Ri) where ψ(r − Ri), Ri being Bravais lattice sites in a perfect crystal, is the density profile satisfying the normalized condition ∫drψ(r) = 1.

Subject to these choices, ΔF in eqn. (2) can be written in reduced energy units as Similarly, the hard-sphere reduced free energy fHS = FHS/(Nε) can be cast in the form within the spirit of the free-volume approximation.

Here δ = /ρ0 (ρ/ρcp) for liquid (solid), T* = kBT/ε is the reduced temperature, ρ0 (ρcp = √2/σ3) is the maximum density of liquid (solid at compact crystal structure) for which a liquid (solid) is stable and Λ = h/(2πmkBT)1/2 is the thermal de Broglie wavelength of an hard sphere.

We should stress at this point that, to ensure theoretical consistency in the construction of the free energy f = fHS + Δf, the cell model has been applied on ‘equal footing’ to both the liquid and solid phases.

For example, we introduce, as is done in eqn. (5), the free volume (free edge) parameter α which is the fraction of the total volume V that is freely available to hard spheres in a liquid (solid) phase as In this work, we have set ρ0 = (6η0σ3)1/3 with an estimated η0 following the quantitative analysis of Baus et al.31

Also, for the case of solid, δ = 1/x31, x1 = r1/σ being the reduced nearest-neighbor distance for a given crystal structure.

We are now in a position to present our derived expressions for the linear attractive potential shown schematically in Fig. 1.

First, Γ for the liquid phase in eqn. (4) can be written as which reduces to the Γ of square-well and linear potentials for λ = 0 and γ = 0, respectively.

For the solid phase, we have where in which In eqn. (10), n1 is the number of nearest neighbors and ζσ2 is the inverse width of the Gaussian density profile G(a,x) = ∫x0dta−1et is the incomplete gamma function and erf(x) = (2/√π)∫x0dtet2 is the error function.

Given eqns. (5), (8) and (9) for f, it is straightforward to calculate the pressure p = ερ2(∂f/∂ρ) and chemical potential μ = ε(∂/∂ρ)(ρf) from which the corresponding critical points (ηc,T*c) can be determined by ∂p(ρc,T*c)/∂ρ = 0 and ∂2p(ρc,T*c)/∂ρ2 = 0 yielding the critical parameters for the liquid–liquid transition.

These predictions for the critical points are reminiscent of the celebrated van der Waals theory for the gas–liquid transition; the parameter Γ now plays the same role as the attraction parameter while η0 simulates the excluded volume factor (see, for example, eqn. (3.49) in .ref. 35

In fact one can write down the same van der Waals equation of state if the parameters a and b appearing in eqn. (3.49) of ref. 35 are replaced by εΓ/ρ0 and 1/ρ0, respectively).

The critical points for the solid–solid phase transition, however, have to be determined numerically.

Numerical results and discussion

As pointed out above, the present van der Waals theory needs ρ0 and ρcp as sole input to the calculation.

For these estimates, we follow Coussaert and Baus31 fixing πσ3ρ0/6 = 0.4952 and ρcp = √2/σ3; these choices are based on: (a) the exact virial expansion of the hard-sphere free energy up to the second virial coefficient; and (b) the criterion of maximum density for a stable liquid or crystalline phase to exist.

In the following we assume the face-centered cubic (fcc) crystalline structure for the solid phase.

Liquid–liquid versus liquid–solid phase transitions: fixed λ or γ

We have applied eqns. (5), (8) and (9) to study the occurrence of the liquid–liquid and solid–solid coexisting phases calculated first at their threshold values γLLth = 0.221 and γSSth = 0.0329, respectively.32

By increasing the range of the interaction λ from 0 (dashed curves), which corresponds to the threshold conditions for the square-well potential, to 0.01 (dotted curves) and then 0.05 (solid curves), we see from Fig. 2 that the stable liquid–liquid phases progressively evolve with the critical T*c increases from the threshold T*th = 0.481 to T*c = 0.495 and then to 0.549.

These results are in reverse order for the solid–solid transition, shown in Fig. 3.

Here, γSSth = 0.0329 (and λ = 0) is the threshold value marking a stable solid–solid transition just about to appear.

When λ is increased, which implies extending a wider range of the interaction, the coexistence of solid–solid phases becomes metastable with respect to their stable liquid–solid counterparts.

It is interesting to compare these results with cases in which we fix the range of the interaction at the threshold values λLLth = 0.325 and λSSth = 0..045632

We again start with the threshold range of the interaction (the triangular potential, γ = 0) at which values the stable liquid–liquid and solid–solid branches are just about to develop.

For the liquid–liquid case, increasing γ, which corresponds to adding to the system the strength of the interaction but at the same time extending the range (since the “slope” λ is kept unchanged), favors the occurrence of a stable liquid–liquid.

This trend differs from the solid–solid case where the metastable coexisting phases are induced due to weakening of the interaction between particles brought about by the extended range λ.

These features are clearly seen in Figs. 4 and 5.

We should comment further on the physical implication of the scenarios discussed above.

Let us first depict in Fig. 6 the typical two-body colloid–colloid potential.16

In our previous thermodynamic perturbation theory, the two-body colloid–colloid potential is split into a repulsive and an attractive part.

Since the repulsive interaction between colloidal particles is generally strong due to the huge amount of charge carried by the particles, the repulsive part of the potential is quite often modeled by a hard-sphere potential.

An effective hard-sphere diameter at x = S, denoted by the dotted curve, is substituted for the steep rising repulsion.

Next, the “plateau”, from S to the second minimum, and the tail are treated as an attractive perturbation.

Apart from the tail, the structure of the perturbation closely resembles that shown in Fig. 1.

Now it is seen that the increase or decrease of the basin is a reflection of the softness of the repulsion; a softer repulsive part will yield a larger basin and hence a larger γ, and vice versa.

The behavior of the liquid–liquid transition as presented here is in qualitative agreement with the general trend obtained here, namely, an increase of γ enhances the occurrence of a stable liquid–liquid transition.

As discussed in our preceding work,25 the mechanism is attributed to the appearance of a second minimum.

Liquid–liquid versus liquid–solid phase transitions: fixed λ plus γ

We turn to examining the phase diagrams for cases in which the strength and range of interactions are simultaneously operating but their sum is kept unchanged at a prescribed value ξ = λ + γ.

To this end, we first recall Fig. 3 for the solid–solid threshold value γSSth = 0.0329 that characterizes the square-well potential (λ = 0).

In Fig. 7, we fix ξ = 0.0329 and consider the combination γ = 0.01 and λ = 0.0229.

The situation describes an extremely narrow region with the same attractive strength of interaction and is corrected by a rapidly decaying linear tail.

Compared with our recent work for the square-well potential (Fig. 3 in ref. 32) at γ = 0.01, there we see a stable lower density fcc solid coexisting with a stable higher density fcc solid; the solid–solid transition occurs at a lower critical temperature T*c.

The extended range of the interaction λ ≠ 0 has resulted in lowering T*c for the appearance of stable solid–solid phases.

On the other hand, Fig. 5 shows that, at the same γ = 0.01 (dotted curves), when the range of the interaction is longer, i.e.λ = 0.0456, the solid–solid separation is driven to metastable phases.

It thus implies that there exists for the given γ = 0.01 a threshold λth at which value the solid–solid transition is just about to occur.

By varying λ between 0.0229 and 0.0456, we find that this threshold value is λSSth = 0.0346 for γ = 0.01.

Similarly, we determine the threshold λSSth = 0.0216 for γ = 0.02.

These latter results are depicted in Fig. 8.

The set of threshold values γ and λ, together with those of the threshold values γSSth and λSSth for the square-well and linear potentials, respectively, define a phase boundary demarcating the metastable solid–solid transition from the stable solid–solid counterpart.

This scenario is shown in Fig. 9.


The classic theory of van der Waals previously applied to understanding the gas–liquid separation has been extended to study the phase diagrams of the liquid–liquid and solid–solid coexisting phases calculated with respect to the liquid–solid counterparts.

By appealing to the cell model approximation aimed at treating the liquid and the solid on an equal footing, we follow Baus et al. and construct theoretically the self-consistent Helmholtz free energies separately for a liquid and a solid.

Analytical expressions for the pressures and chemical potentials of the latter phases then permit a straightforward calculation of the coexistence curves.

Employing this extended van der Waals theory, we investigate the square-well plus a linear tail attractive potential added to a repulsive hard-sphere system.

Our studies of the phase diagrams suggest the following:

(a) Given the threshold strength of interactions γLLth and γSSth, an increase in λ results in progressive development of the stable liquid–liquid phases whereas for the solid–solid coexisting phases, metastability results.

(b) In analogy with (a), we fixed the threshold range of interactions λLLth and λSSth and increased the interaction strength, γ; the same trend is observed, namely, the stable liquid–liquid phases evolve progressively whereas for the solid–solid phases, metastability results.

(c) The range of interaction, λ, plays a non-negligible role in lowering the critical T*c for the stable coexisting solid-solid phases.