Phase diagram of hard colloidal platelets: a theoretical account
We construct the complete liquid crystal phase diagram of hard plate-like cylinders for variable aspect ratio using Onsager’s second virial theory with the Parsons-Lee decoupling approximation to account for higher-body interactions in the isotropic and nematic fluid phases. The stability of the solid (columnar) state at high packing fraction is included by invoking a simple equation of state based on a Lennard-Jones-Devonshire (LJD) cell model which has proven to be quantitatively reliable over a large range of packing fractions. By employing an asymptotic analysis based on the Gaussian approximation we are able to show that the nematic-columnar transition is universal and independent of particle shape. The predicted phase diagram is in qualitative agreement with simulation results.
pacs:64.70.mf, 82.70.Dd, 05.20.-y
Many colloidal dispersions, such as natural clays, and macromolecular systems consist of oblate or disk-shaped mesogens whose intrinsic ability to form liquid crystalline order gives rise to unique rheological and optical properties. Despite their abundance in nature, the statistical mechanics of fluids containing non-anisometric particles in general (and oblate ones in particular) has received far less attention than that of their spherical counterparts. The possibility of a first order disorder-order transition from an isotropic to a discotic nematic fluid of platelets was first established theoretically by Onsager Onsager (1949) in the late 1940s. Although originally devised for rod-like particles in solution, his theory also makes qualitative predictions for plate-like particles based on the central idea that orientation-dependent steep repulsive interactions alone are responsible for stabilising nematic order.
The intrinsic difficulty with platelets, as pointed out by Onsager in his original paper, is that the contribution of third and higher body correlations can no longer be neglected like for thin rod-like species. Consequently, the original second-virial treatment is expected to give qualitative results at best Forsyth et al. (1977). In a pioneering simulation study, Eppenga and Frenkel Eppenga and Frenkel (1984) provided numerical evidence for an isotropic-nematic transition in systems of infinitely thin circular disks and found the transition densities to be much smaller and the first order nature of the transition to be much weaker than predicted by the Onsager theory. Owing to the simplicity of the model, the discrepancy can be attributed entirely to the neglect of third and higher body virial terms in the theory.
At high densities, the nematic phase of disk-shaped particles becomes unstable with respect to columnar order, characterised by a planar (2D) hexagonal arrangement of columns each with a liquid internal structure. Similar to the formation of the nematic phase, the stability of the columnar phase can be explained solely from entropic grounds Frenkel (1989); Veerman and Frenkel (1992). Although the system loses configurational entropy because of the partial crystallisation associated with columnar order, this loss is more than offset by a simultaneous increase in translational entropy, i.e. the average available space for each particle increases.
Attempts to improve Onsager’s second virial theory have met with variable success (see Ref. Harnau, 2008 for a recent overview). These approaches usually involve integral equation or geometric density functional methods whose applicability is often restricted to isotropic fluids Costa et al. (2005); Cheung et al. (2008), models with parallel or restricted orientations Harnau et al. (2002) or particles with vanishing thickness Esztermann et al. (2006); Harnau et al. (2001). A recent generalisation of the fundamental measure approach towards arbitrarily shaped hard convex bodies provides a potentially promising avenue to address more realistic models for liquid crystal ordering Hansen-Goos and Mecke (2009). The influence of higher-body correlations can only be assessed numerically via computer simulation Masters (2008). For cut spheres, the virial coefficients have been quantified up to the 8th order both in the fluid isotropic You et al. (2005) and nematic state Duncan et al. (2009). Despite the large number of virial terms, the convergence of the virial expansion of the free energy was found to be insufficient to provide an accurate description of dense nematic and columnar states. Alternatively, Scaled Particle Theory (SPT) can be used to incorporate higher virial terms in an indirect manner. Whilst SPT produces reasonable results for infinitely thin disks Savithramma and Madhusudana (1981), its extension to finite aspect ratios leads to poor predictions for the isotropic-nematic transition densities Wensink (2008).
A simpler strategy to account for higher-body particle correlations in the isotropic and nematic fluid state is provided by the so-called Parsons-Lee decoupling approximation Parsons (1979); Lee (1987, 1989). The basic assumption of this approach is that the pair correlation function of a fluid of hard anisometric bodies, which depends rather intractably on the centre-of-mass distance vector and orientational unit vectors and , can be mapped onto that of a hard sphere fluid with the same packing fraction via:
with some reference distance (e.g. particle diameter) and the distance of closest approach of a pair of hard anisometric bodies at a given set of orientation unit vectors. In case of hard spheres the distance of closest approach is simply the hard sphere diameter . Eq. (1) provides a natural route of decoupling the translational and orientational degrees of freedom. Starting from the generalised virial equation it is possible to derive an expression for the excess free energy which is similar to the one from Onsager with the particle density replaced by a rescaled density involving the hard sphere excess free energy. Whilst the decoupling approximation is known to work well for short hard spherocylinders McGrother et al. (1996), its merits for plate-like cylinders have not been investigated so far. This we intend to do in the present paper.
As for the columnar state, the high degree of positional and orientational order can be exploited to devise simple free-volume approaches inspired by the Lennard-Jones-Devonshire (LJD) cell model Lennard-Jones and Devonshire (1937); Wood (1952); Salsburg and Wood (1962). This was first done by Taylor and Hentschke Taylor et al. (1989); Hentschke et al. (1989) for the high-density liquid crystal states of parallel cylinders which do not exhibit an isotropic phase. The approach was further developed and modified in Ref. Wensink, 2004 showing that a quantitatively reliable equation of state for the columnar phase can be obtained by accounting for the orientational entropy of the particles, neglected in the original version. In this paper, we will combine the Onsager-Parsons approach for the isotropic and nematic fluid state with the modified LJD cell theory for the columnar phase to trace the complete phase diagram for freely rotating hard cylinders as a function of thickness-to-diameter ratio. The theoretical predictions will be tested against simulation results for hard cut-spheres. In view of the inherent difficulty of capturing multi-particle correlations in dense plate fluids, the overall performance of the present theory must be deemed satisfactory. Although quantitative agreement with simulation data is generally lacking, the theory does manage to reproduce the generic features of the phase diagram and provides a simple theoretical underpinning for the relative stability of nematic and columnar order as a function of the plate aspect ratio.
This paper is constructed as follows. Section II and III are devoted to a detailed exposition of the Onsager-Parsons and modified LJD theories, respectively. The phase diagram emerging from the present theory will be presented and discussed in Section IV. Next, algebraic forms of the nematic and columnar free energy are given which allow us to obtain universal scaling results for the nematic-columnar transition. Finally, some concluding remarks are formulated in Section VI.
Ii Onsager-Parsons theory for the isotropic-nematic transition
Let us consider a system of hard cylinders with length and diameter in a macroscopic volume . For disk-like cylinders we consider here the aspect ratio is much smaller than unity. The particle concentration is expressed in dimensionless form via . Following Ref. Wensink et al., 2001, the Helmholtz free energy within the Onsager-Parsons-Lee approach takes the following form:
with the thermal energy ( represents Boltzmann’s constant and temperature) and the dimensionless thermal volume of a platelet including contributions from the rotational momenta. The brackets , denote single and double orientational averages involving some unknown distribution of the orientation unit vector of the plate normal which is normalised according to . Several entropic contributions can be distinguished in Eq. (2). The first two are exact and denote the ideal translational and orientational entropy, respectively. The last term represents the excess translational or packing entropy which accounts for the particle-particle interactions on the approximate level of pair-interactions. The key quantity here is the excluded volume between two plate-like cylinders at fixed inter-particle angle with . This quantity has been calculated in closed form by Onsager Onsager (1949) and reads:
with the complete elliptic integral of the second kind. Although the structure of Eq. (2) is similar to the classic Onsager second-virial free energy, the effect of higher order virial terms are incorporated via the scaling factor
which depends on the total plate volume fraction . The rescaled density stems from the Parsons-Lee method Parsons (1979); Lee (1987, 1989); Wensink et al. (2001) which involves a mapping of the plate pair distribution function onto that of a hard sphere system via the virial equation. The free energy can ultimately be linked to the Carnahan-Starling expression for hard spheres, which provides a simple strategy to account for the effect of higher-body particle interactions, albeit in an implicit and approximate manner. As required, approaches unity in the limit in which case the original second-virial theory is recovered.
Let us now specify the orientational averaging. By definition, all orientations are equally probable in the isotropic (I) phase and . The orientational entropy then vanishes:
If we use the random isotropic averages , and the excluded volume entropy reduces to:
With this, the free energy of the isotropic phase is fully specified.
In the nematic (N) phase, the particles on average point along a common nematic director and the orientation distribution is no longer a trivial constant. For a uniaxial nematic phase, involving the polar angle between the plate normal and the director, with being a peaked function around and .
The equilibrium form follows from the minimum condition of the free energy:
where the Lagrange multiplier ensures the normalisation of . Applying the condition to Eq. (2) leads to a self-consistency equation for :
which needs to be solved numerically for a given particle concentration Herzfeld et al. (1984). Note that the isotropic distribution is a trivial solution of Eq. (8), irrespective of . At higher densities, nematic solutions of Eq. (8) will appear which give rise to a lower free energy than the isotropic one. The nematic order parameter , defined as:
[where ] is used to distinguish the isotropic state () from the nematic (). Once the equilibrium orientational distribution function is known, the pressure and chemical potential can be specified to establish phase equilibria between isotropic and nematic states.
Iii LJD cell theory for the columnar phase
To describe the thermodynamical properties of a columnar phase we use an extended cell theory as proposed in Ref. Wensink, 2004. In this approach, the structure of a columnar phase is envisioned in terms of columns ordered along a perfect lattice in two lateral dimensions with a strictly one-dimensional fluid behaviour of the constituents in the remaining direction along the columns. As for the latter, the configurational integral of a system of parallel platelets with thicknesses and diameter with their centre-of-mass moving along the plate normal on a line of length is formally given by Tonks (1936):
with the thermal de Broglie wavelength. The columns are assumed to be strictly linear and rigid so that fluctuations associated with bending of the columns can be ignored. Next, we allow the platelets to rotate slightly about their centre-of-mass. At high packing fractions, the rotational freedom of each platelet is assumed to be asymptotically small and the configurational integral above may be approximated as follows:
where represents the total 1D thermal volume including contributions arising from the 3D rotational momenta of the platelet. Furthermore, is an orientational partition integral depending on the orientational probability distribution . In the mean-field description implied by Eq. (11) there is no coupling between the orientational degrees of freedom of the platelets. The rotational freedom of the platelets is expressed in an effective entropic thickness, defined as
up to leading order in the polar angle which measures the deviation of the plate normal from the direction of the column. A prefactor of ‘’ in Eq. (12) has been included to correct in part for the azimuthal rotational freedom and captures the effect that the excluded length between two platelets at fixed polar angles becomes minimal when the azimuthal orientations are the same. The free energy of the 1D fluid then follows from :
in terms of , the reduced linear density and effective thickness .
Similar to the nematic case in the previous Section, the equilibrium form is found by a formal minimisation of the free energy under the normalisation constraint. The corresponding stationarity condition is given by Eq. (7). Since the free energy Eq. (13) depends only on one-particle orientational averages, the equilibrium ODF can be obtained in closed form and turns out be of a simple exponential form:
The orientational averages are now easily carried out and the leading order expressions for the orientational entropy and entropic thickness are given by:
As a measure for the orientational order along the column, we can define a nematic order parameter [cf. Eq. (9)] related to via:
Let us now turn to the free energy associated with the positional order along the lateral directions of the columnar liquid crystal. A formal way to proceed is to map the system onto an ensemble of disks ordered into a 2D lattice. Near the close packing density, the configurational integral of the system is provided in good approximation by the LJD cell theory Lennard-Jones and Devonshire (1937); Wood (1952); Salsburg and Wood (1962); Kirkwood (1950). Within the framework of the cell model, particles are considered to be localised in ‘cells’ centred on the sites of a fully occupied lattice (of some prescribed symmetry). Each particle experiences a potential energy generated by its nearest neighbours. In the simplest version, the theory presupposes each cell to contain one particle moving independently from its neighbours. The -particle canonical partition function can then be factorised as follows:
For hard interactions, the second phase space integral is simply the cell free area available to each particle. If we assume the nearest neighbours to form a perfect hexagonal cage, the free area is given by with the nearest neighbour distance. The configurational integral then becomes
where the (lateral) spacing is a measure for the translational freedom each particle experiences within the cage. The free energy associated with the LJD cell theory is given by:
The LJD equation of state associated with Eq. (20) provides a very accurate description of a 2D solid at densities near close-packing Alder et al. (1968). If we now apply the condition of single-occupancy (i.e. one array of platelets per column) we can use to relate the plate volume fraction (with the particle volume) to the reduced linear density via:
in terms of the reduced packing fraction with the value at close packing. The total free energy of the columnar state is now obtained by adding the fluid and LJD contributions:
where the ideal contribution is identical to that of Eq. (2). The final step is to minimise the total free energy with respect to . The stationarity condition yields a third-order polynomial whose physical solution reads:
With this, the free energy for the columnar state is fully specified. Unlike the nematic free energy, the columnar free energy is entirely algebraic and does not involve any implicit minimisation condition to be solved (cf. Eq. (7)). The pressure and chemical potential can be found in the usual way by taking the appropriate derivative of Eq. (22). In Sec. V we will show that the nematic free energy can also be recast in closed algebraic form using a simple variational form for the ODF, similar to Eq. (14).
Iv Phase diagram
Fig. 1 presents an overview of the phase behaviour of a hard cylindrical platelets based on the theoretical approach described above, along with various simulation data for hard cut spheres available in literature. From Fig. 1a it is evident that the packing fractions associated with the isotropic-nematic coexistence increase for larger aspect ratio whereas the nematic-columnar transition remains virtually unaffected by the shape of the platelet. This observation is in line with the tentative phase diagram constructed by Veerman and Frenkel Veerman and Frenkel (1992). The trends can be understood qualitatively by noting that the onset of nematic order occurs if the fraction of excluded volume exceeds a certain universal value of about (as reflected in Fig. 1b) whereas columnar order only becomes stable beyond a critical packing fraction, typically . Whence:
which implies the presence of a triple aspect ratio, fixed by the intersection of both nematic binodals. Although at this particular value an isotropic-nematic-columnar triphasic coexistence occurs above a certain packing fraction, the system volume occupied by the nematic phase is always infinitesimally small and the situation thus differs from a regular tri-phasic coexistence occurring in e.g. binary mixtures at a given thermodynamic state point. Equating both expression we estimate the triple aspect ratio to be , which is very close to the value obtained from extrapolating the simulation binodals from Refs. Zhang et al., 2002; van der Beek et al., 2004. Beyond the triple aspect ratio, the platelets are no longer sufficiently anisometric to guarantee a stable nematic phase and direct transitions from the isotropic fluid to the columnar solid occur. We should note that our theory does not take into account the theoretically disputed cubatic phase, as an intermediate state between the isotropic and columnar phases. The issue of the stability of cubatic order with respect to columnar order is discussed in a recent simulation study by Duncan et al. Duncan et al. (2009). The transition densities from Veerman Veerman and Frenkel (1992) and Bates Bates and Frenkel (1998) are systematically larger than those reported by Zhang Zhang et al. (2002) and van der Beek van der Beek et al. (2004) and therefore give rise to a slightly higher estimate of the triple value ().
The theoretical value deviates considerably from the ones predicted from simulations, mainly because the predicted packing fractions of the coexisting nematic and columnar phases are too large. The equations of state presented in Fig. 2 demonstrate that the main source of error must be the chemical potential of either the nematic or columnar phase nematic branch, rather than the pressure. For , the predicted pressure in the nematic and columnar states are fairly close to the simulation results with discrepancies less than a few percent in both branches.
For larger aspect ratios () the agreement between theory and simulation is quite satisfactory, despite an increased shape difference between the cylinder and the cut sphere. For , the occurrence of cubatic order has been reported in simulation Veerman and Frenkel (1992) which is not taken into account in the present model. The isotropic binodal point in Fig. 1a at this value is taken to be the mean value between the onset of cubatic order and the transition to the columnar state, with the error bar indicating the boundaries of cubatic order. For the thickest species , the coexisting high density phase was found to be a solid rather than a columnar. Similarly, for smaller aspect ratios a continuous columnar-solid transition line could be located beyond the nematic-columnar moving toward higher packing fraction upon decreasing . In our simple cell-fluid model there is, however, no distinction between the columnar and solid states due to the absence of a freezing transition within the strictly 1D line fluid representing the structure along the column direction.
The predictive power of the Onsager-Parsons theory for platelets is perhaps better highlighted in Fig. 1b, where the isotropic-nematic binodals are plotted in terms of the reduced number concentration . The agreement is reasonable for large aspect ratio but rather poor for thin platelets. In the limit of infinitely thin disks () the coexistence concentrations are identical to those obtained from Onsager’s second-virial theory viz. and Lekkerkerker et al. (1984). This is easily understood from the fact that the packing fraction of infinitely thin disks at a given finite number concentration is zero. Consequently, reduces to unity and the Parsons decoupling approximation involving the hard sphere excess free energy becomes ineffective. A similar reduction to the level takes place for infinitely thin rods (). However, contrary to rods, the effect of the third virial coefficient is finite for disks with vanishing thickness. In general, for isotropic systems of hard cylinders we have:
where the latter value is taken from Ref. Eppenga and Frenkel, 1984. Likewise, higher order virial contributions will also be non-zero. Simulation studies of the virial terms up to Eppenga and Frenkel (1984); You et al. (2005) reveal that higher order virial terms involve alternating positive and negative contributions of comparable magnitude, indicating just how complicated virial expansions are for dense fluids of platelets. The virial terms generated by the Onsager-Parsons free energy can be obtained from the virial expression for the excess free energy . Applying this to Eq. (2) gives:
It is clear that all contributions beyond are zero for thus leading back to the original Onsager result. For the reduced third, fourth and fifth virial coefficients in the isotropic phase are 0.170, 0.010 and 3.67 . Comparing these with the numerically exact values 0.508, 0.111 and -0.10 for cut spheres You et al. (2005) shows that higher-order correlations are systematically under-weighted by the Parsons method.
V Asymptotic results for the N-C transition
A simple rationale for the apparent independence of the NC transition with respect to particle shape can be obtained by comparing the free energy of the two states and exploiting the fact that the nematic order at densities close to the transition is very strong. In that case the average excluded volume Eq. (3) between the particles in the nematic phase can be approximated by retaining the leading order contribution for small inter-rod angles :
The orientational averages indicated by the brackets can be estimated using a Gaussian Ansatz for the ODF Odijk and Lekkerkerker (1985):
in terms of the following one-parameter Gaussian variational function:
where the normalisation factor follows from . The variational parameter is required to be much larger than unity so that is sharply peaked around . In that case, the integration over the polar angle in Eq. (29) can be safely extended to and . In the asymptotic limit, the normalization constant is given by and the double orientational average over the angle in the nematic phase is found to be Odijk and Lekkerkerker (1985):
to leading order in . Similarly, the orientational entropy can be approximated by:
The nematic order parameter [cf. Eq. (17)] follows from . These algebraic results allow the minimisation of the free energy with respect to to be carried out analytically and leads to a closed expression for the nematic free energy.
Using the asymptotic expressions above, and introducing the volume fraction as a density variable the following algebraic form for the Onsager-Parsons nematic free energy can be produced:
which is a sum of the ideal, orientational and excess parts, respectively. Similarly, one may derive for the columnar free energy:
with and combines the ideal, orientational, 1D fluid and cell contributions, respectively. The only explicit shape dependency is the contribution in the orientational part which is identical in both expressions and therefore does not affect the NC coexistence properties. Solving the coexistence conditions gives the universal coexistence values and and pressure . Furthermore, the normalised lateral columnar spacing is and the equilibrium variational parameters pertaining to the nematic order in the nematic phase and columnar phases are given by and , respectively.
We have combined the Onsager-Parsons theory with a simple LJD cell model to address the phase behaviour of hard cylindrical platelets with variable aspect ratio. The theoretical framework provides a simple, yet qualitative underpinning for the competitive stability of the isotropic, nematic and columnar states, observed in Monte Carlo computer simulations. Upon increasing the aspect ratio, the window of stability for the nematic phase decreases systematically up to a critical value, identified as a triple equilibrium. Beyond this value the anisometry of the plates is too small to warrant a stable nematic phase and direct transitions from an isotropic fluid to a columnar solid occur. It would be intriguing to verify whether the stability of cubatic order, suggested by the simulations, can be captured with a similar free volume concept. Rather than forming a closed packed assembly of columns, the cubatic phase must be envisioned in terms of interacting finite-sized stacks with random orientations. This will be explored in a future study.
Acknowledgements.We are grateful to George Jackson and Jeroen van Duijneveldt for fruitful discussions. HHW acknowledges the Ramsay Memorial Fellowship Trust for financial support.
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