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We employ a generalized van der Waals-Onsager perturbation theory to construct a free energy functional capable of describing the thermodynamic properties and orientational order of the isotropic and nematic phases of attractive disc particles. The model mesogen is a hard (purely repulsive) cylindrical disc particle decorated with an anisotropic square-well attractive potential placed at the centre of mass. Even for isotropic attractive interactions, the resulting overall inter-particle potential is anisotropic, due to the orientation-dependent excluded volume of the underlying hard core. An algebraic equation of state for attractive disc particles is developed by adopting the Onsager trial function to characterize the orientational order in the nematic phase. The theory is then used to represent the fluid-phase behaviour (vapour-liquid, isotropic-nematic, and nematic-nematic) of the oblate attractive particles for varying values of the molecular aspect ratio and parameters of the attractive potential. When compared to the phase diagram of their athermal analogues, it is seen that the addition of an attractive interaction facilitates the formation of orientationally-ordered phases. Most interestingly, for certain aspect ratios, a coexistence between two anisotropic nematic phases is exhibited by the attractive disc-like fluids.

Liquid crystals [

Discotic LCs tend to comprise poly-aromatic cores of oblate geometry, e.g., hexa-alkanoyloxy benzenes, or metal organic complexes of phenyl pyridines, porphyrazines, phthalocyanines, triphenylenes,

In computer simulation studies [

The theoretical treatment of hard non-spherical fluids dates back to the 1940s [

For a proper description of the temperature dependence of the thermodynamic properties of a system, both repulsive and attractive inter-molecular interactions have to be considered on an equal footing. This is in line with a van der Waals description of fluids, where the hard core is treated as the reference in determining the fluid structure [

As has already been mentioned, purely repulsive discotic particles have been shown to exhibit columnar phases in the higher density region. This was first demonstrated by Veerman and Frenkel [

Of particular importance here is the closed-form algebraic equation of state (EOS) developed for attractive hard spherocylinders [

In this section, we describe the perturbation theory [

Here, ^{ref}, is taken from an expression for the hard-core particles. Following the Parsons-Lee approach (PL) [

where

where _{m} is the packing fraction, _{m} is the molecular volume. Alternatively, a improved generic equation of state for hard disc-like particles developed in [

In the high-temperature limit, the attractive free energy can be approximated at first order [

where _{ref} represents an ensemble average over all configurations of the reference system. In the canonical ensemble, the mean-attractive energy is formally written as:

with the configurational integral defined as:

which is a function of the positions, ^{N}^{N}_{ij}_{i}_{j}^{ref} (_{12}, _{1}, _{2}), defined as:

The attractive term can thus be re-written as a sum of pair contributions. When considering a nematic phase where the distribution of particle positions is uniform, the single-particle density of the system can be factorised as ^{ref}(_{i}_{i}_{i}

Since the particles of interest are non-spherical, both the attractive interaction and the excluded volume are complicated functions of the relative positions and orientations of particles [

where _{2}(cos _{1} · _{2}), between the principle axes of the two discs. In order to make the approach tractable, we use the low-density limit to approximate the pair distribution function of the reference system:

where ^{ref} (_{12}, _{1}, _{2}) is the pair interaction of the reference system. Since the reference system corresponds to the hard-core particles, ^{ref} (_{12}, _{1}, _{2}) is a purely repulsive interaction, which is infinity when hard cores overlap, and zero otherwise. Introducing the ASW attractive potential in

where the double orientational average of a quantity _{1}, _{2}) is defined as:

Combining the separate contributions, the free energy of the full system is obtained in terms of angle averages of the configurational contributions:

In this expression for the free energy, one can recognize the coupling between the repulsive and attractive contributions of the pair potential. The excluded volume is seen in both the terms corresponding to the isotropic and the anisotropic attractive contributions; the last term in 〈_{exc}(_{2}(cos _{ω⃗}_{1,}_{ω⃗}_{2} also constitutes a direct coupling of the two contributions. The single-particle orientational distribution function in the isotropic state is constant, _{iso}(

We start by examining the angle averages of the purely repulsive contributions, _{exc}(

where the coefficient, _{i}^{4}

where the coefficients are given by:

and the molecular volume of a cylindrical disc is _{m} = ^{2}/4. Using the notation of

The equilibrium state corresponds to the minimum of the total free energy,

where _{L} is a Lagrange undetermined multiplier, which ensures that ∫ _{OTF}(

and is seen to depend on the parameter _{0}), where _{0} is the director of the nematic phase. One should note that when _{OTF}(_{L}, in

Using _{2}(sin _{2}

Collecting all of the terms, one obtains the Helmholtz free energy for the system of attractive cylinder disc (ACD) in terms of the orientational parameter

Using the OTF to represent the orientational order of the nematic phase, the free energy is expressed in an explicit algebraic form. As a result, the functional variation of

where the coefficients are given by:

The equilibrium value of

where _{j}_{eq} = _{0}.

The expressions for the chemical potential, ^{nem}, and the compressibility factor (equation of state), ^{nem} = _{V}_{,}_{T}_{N}_{,}_{T}

and

where

An important quantity that is used to characterize the degree of orientational order of the nematic phase is the nematic order parameter, _{2}, which is commonly defined as the orientational average of the second Legendre polynomial, _{2}(cos(

Because the OTF is used to represent the orientational distribution in nematic phase, _{OTF}(_{2} can be expressed as a function of the orientational parameter

In the case of the isotropic phase, where no orientational order is exhibited, the orientational averages of

The orientational averages in the isotropic phase can be evaluated as:

From the expressions for free energy of the isotropic phase (cf.,

and

Before proceeding, we should note that the PL approach for the isotropic-nematic phase of hard disc-like particles has been compared with a generic equation of state, which accounts for both negative and positive contributions of the higher-body virial coefficients [

The fluid-phase diagrams of attractive cylindrical discs of various aspect ratios with square-well attractive interaction are calculated by equating the pressure and chemical potential of the coexisting phases at a given temperature. For the ACD model, the range of the attractive potential is characterized by the parameter _{m}/_{0}, and packing fraction, _{m}. Two dimensionless scales for the temperature have been used in our study: the dimensionless temperature, _{0}, in terms of the isotropic square-well depth, _{0}; and the reduced van der Waals-like temperature,

First, we focus on discs with a spherically symmetric SW potential, so that the orientation-dependent attractive contribution is inactive, _{2} = 0. Although the attractive interaction is isotropic, the orientation-dependent excluded volume gives rise to an overall interaction between the discs, which is anisotropic. The fluid-phase behaviour of ACDs of aspect ratio _{c} = 0.10362 and

The model developed in our current work allows one to assess the effect of the anisotropy of the underlying hard disc on the phase behaviour of the ACD system. The phase diagram of hard discs of aspect ratio

When the aspect ratio is increased to _{1}-N_{2} coexistence is bounded by the I-N_{1}-N_{2} triple line and nematic-nematic critical point (
_{2}) decreases slightly, while that of the lowdensity nematic state (N_{1}), though still highly ordered, behaves as a monotonically increasing function of temperature up to

As the aspect ratio of the discs is made larger (corresponding to thinner particles), the isotropic-nematic moves to lower densities. For particles that are moderately thin (_{1}-N_{2} phase behaviour with increasing aspect ratio has been observed for attractive rod-like LC molecules [

Also included in

By varying the parameter

The fluid-phase behaviour of discs characterized by an aspect ratio _{1}) becomes lower as

These examples indicate that the long-ranged isotropic SW attraction slightly destabilizes the orientationally-ordered phases. In the case of the thicker discs (_{2} = 0), while attractive discs with short-ranged attractive interactions (

In real systems, the dispersion forces are associated with the functional groups distributed at various points in the molecule. The assignment of a central isotropic attraction is but a simplistic first-order approximation. Recognizing the morphological anisotropy of discotic LCs, one can postulate the existence of additional orientation-dependent (anisotropic) attractions. The effect of including anisotropic attractions on the fluid-phase diagram of discs with _{2} = 0.3_{0}, the VLE becomes metastable with respect to the isotropic-nematic coexistence. A similar phenomena was also reported for the systems of attractive spherical and rod-like particles with anisotropic attractions [

It is clear that a positive anisotropic attractive interaction stabilizes the nematic phase and enhances the propensity of the system to form orientationally-ordered states. For a system of discs with a large aspect ratio, e.g., _{2} is increased, while the system converges into the hard-disc limit in the high-temperature limit. For highly anisometric systems, the anisotropic shape and orientation-dependent attractions both contribute to the stabilization of the nematic phases. Hence, for an anisotropic strength of _{2} = 0.7_{0}, which is comparable to the isotropic attraction, the

In addition to attractive interactions with a positive anisotropy, for some molecules one would expect interactions with a negative anisotropy favouring a perpendicular configuration: e.g., the quadrupolar interactions between aromatic moieties give rise to both parallel (side-by-side) and perpendicular (Tshaped) relative orientations of the cores. These kinds of interactions can be represented in our ACD models using negative values for the parameter _{2}. In _{2} = −0.3_{0}. For discs with a larger aspect ratio of _{2} = −0.5_{0}, the system no longer exhibits the region of nematic-nematic coexistence seen for the equivalent system with less negative anisotropic interaction, _{2}

In this work, we present a closed-form equation of state for the description of the thermodynamic properties and orientational ordering of attractive hard-core cylindrical-disc fluids, which serves as a basic model for thermotropic discotic liquid crystals. With the aid of the Onsager trial function to represent the single-particle orientational distribution function in the nematic phase, the free energy is expressed in algebraic form and the functional variation of free energy reduced to a simple derivative with respect to the Onsager orientational parameter _{eq} is then obtained when higher-order terms in the expansion are neglected and ordered states (_{2} (or its ratio _{2}_{0} relative to the isotropic term), is key to the stabilization/destabilization of orientationally-ordered phases. The attractive disc model studied in the current work is a simple prototypical coarse-grained representation of real molecular interactions in discotic liquid crystals: neither will the repulsive interactions be purely repulsive in real systems nor will the attractive interactions be of the simple van der Waals square-well form. This having been said, many common discotic thermotropic particles comprise large fused aromatic cores, which will have lower energetic overlap volumes at the edges of the particles than in the central region, which is captured volumetrically at least with our square-well model. We are therefore confident that, qualitatively, at least, our model will describe the isotropic and nematic ordering behaviour of discotic thermotropic mesogens. A simple square-well hard-spherocylinder model of this generic form has been used to successfully represent the ordering behaviour of solutions of rod-like polypeptide (poly

It is important to point out that in liquid state theory [

The methodology developed here allows other interactions to be incorporated within the attractive hard-disc to provide a more realistic free energy functional. The molecular-based statistical associating fluid theory (SAFT) [

L.W. thanks the Department for Business Innovation and Skills of the UK, and the China Scholarship Council for funding a PhD studentship. Funding to the Molecular Systems Engineering Group from the Engineering and Physical Sciences Research Council (EPSRC) of the UK (grants GR/T17595, GR/N35991, EP/E016340 and EP/J014958), the Joint Research Equipment Initiative (JREI) (GR/M94426) and the Royal Society-Wolfson Foundation refurbishment scheme is also gratefully acknowledged.

The authors declare no conflict of interest.

The attractive cylindrical disc (ACD) model. The model is characterized by the thickness,

(_{0} + _{2}_{2}(cos _{0} ≠ 0 and _{2} = 0. The dimensionless properties are defined in terms of _{0}, as _{0} for the temperature, ^{3}_{0} for the pressure and _{m}_{m}

Temperature-density representation of the fluid-phase equilibria for attractive cylindrical discs (ACDs) with an aspect ratio of

(_{1}) and high-density nematic (N_{2}), and
_{1}-N_{2} three-phase coexistence line. See the caption of

The temperature dependence of the nematic order parameter, _{2}, of the coexisting nematic phases for attractive cylindrical discs (ACDs) of aspect ratio

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio _{1}-N_{2} three-phase coexistence lines in each case. See the caption of

The temperature dependence of the nematic order parameter, _{2}, of the coexisting nematic phases for attractive cylindrical discs (ACDs) of aspect ratio

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio of _{2}

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio of _{2}

(_{2}. The dot-dashed line represents th I-N_{1}-N_{2} three-phase coexistence line; (_{2} = 0.7_{0}. See the caption of

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio of _{2}

Temperature-density representation of the fluid-phase equilibria of attractive cylindrical discs (ACDs) with an aspect ratio of _{2}_{1}-N_{2} three-phase coexistence lines in each case. See the caption of