Spherocylinders, hard

The method has been extended to mixtures of hard spheres, to hard convex molecules and to hard spherocylinders that model a nematic liquid crystal. For mixtures m. subscript) of hard convex molecules of the same shape but different sizes. Gibbons [38] has shown that the pressure is given by... 

Stroobants A, Lekkerkerker FI N W and Frenkel D 1986 Evidence for smectic order in a fluid of hard parallel spherocylinders Phys.Rev.Lett 57 1452-5 Erratum 57 2331... 

Let us enter the world of liquid crystals built by the purely entropic forces present in hard body systems. The phase diagram of hard spherocylinders (HSC) shows a rich variety of liquid crystalline phases [71,72]. It includes the isotropic, nematic, smectic A, plastic, and solid phases [73]. In a plastic crystal the particle centers lie on lattice sites, but the orientations of the... 

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

The first of these was by Vieillard-Baron [5] who investigated a system of spherocylinders but failed to detect a liquid crystal phase primarily because the anisometry, L/D, of 2 was too small [37]. He also attempted to study a system of 2392 particles with the larger L/D of 5 but these simulations had to be abandoned because of their large computational cost. However, in view of the ellipsoidal shape of the Gay-Berne particles it is the behaviour of hard ellipsoids of revolution which is of primary relevance to us. 

Bolhuis, R Frenkel, D., Tracing the phase-boundaries of hard spherocylinders, J. Chem. Phys. 1997,106, 666-687... 

Scaled Particle Theory for Wormlike Hard Spherocylinders.93... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

The hard-core repulsion prevents spherocylinders from overlapping. This effect reduces the space available for the cylinders, and gives rise to a loss of their translational entropy ( —S ). Many statistical thermodynamic techniques were used to calculate it, as has been extensively reviewed by Vroege and Lekkerkerker [9]. 

The scaled particle approach is exact at the limit of infinite dilution and makes it possible to formulate static solution properties at finite dilutions in an approximate way. In this approach, we first calculate Sex for one hypothetical scaled particle with a size smaller or larger than the real particle and then find Sex of the real size by interpolation. For the wormlike spherocylinder, the scaled particle is assumed to have a cylinder length kL and a hard-core diameter Kd where X and k are scaling factors. The persistence length q of the scaled particle may be chosen rather arbitrarily. Here we do not scale q of the scaled particle but take it to be the same as the real q [17, 18],... 

The orientation dependent parameter p defined by Eq. (11) becomes unity in the isotropic state, and decreases as the polymers are uniaxially oriented. Therefore, it follows from Eqs. (9) and (10) that the wormlike hard spherocylinder system has a smaller translational entropy loss from the ideal solution in the liquid crystal state than in the isotropic state. This difference drives the system to form a liquid crystal phase. However, in order to determine the equilibrium orientation of the system, the orientation dependence of Sor has to be formulated, and this is done in Sect. 2.3. 

This leads to the following expressions for the osmotic pressure n of the solution and the chemical potential p of the wormlike hard spherocylinder ... 

Fig. 7. Comparison of experimental phase boundary concentrations between the isotropic and biphasic regions for various liquid-crystalline polymer solutions with the scaled particle theory for wormlike hard spherocylinders. ( ) schizophyllan water [65] (A) poly y-benzyl L-glutamate) (PBLG)-dimethylformamide (DMF) [66-69] (A) PBLG-m-cresoI [70] ( ) PBLG-dioxane [71] (O) PBLG-methylene chloride [71] (o) po y(n-hexyl isocyanate) (PHICH°Iuene at 10,25,30,40 °C [64] (O) PHIC-dichloromethane (DCM) at 20 °C [64] (5) a po y(yne)-platinum polymer (PYPt)-tuchIoroethane (TCE) [33] ( ) (hydroxypropyl)-cellulose (HPC)-water [34] ( ) HPC-dimethylacetamide (DMAc) [34] (N) (acetoxypropyl) cellulose (APC)-dibutylphthalate (DBP) [35] ( ) cellulose triacetate (CTA)-trifluoroacetic acid [72]...

Fig. 8. Comparison of experimental phase boundary concentrations between the biphasic and liquid crystal regions for various liquid crystalline polymer solutions with the scaled particle theory for hard wormlike spherocylinders. The symbols are the same as those in Fig. 7...

Fig. 10. Theoretical ternary phase diagram calculated from the scaled particle theory for worm-like hard spherocylinders with (Ni, N2) = (0.930,0.070), d = 1.52nm, q = 200nm, and ML = 2150 nm-1 [17]...

With the wormlike hard spherocylinder system with u = u0 taken as a reference system, and the electrostatic interaction we, regarded as a thermodynamic perturbation, we can derive by perturbation calculation [82]... 

In concluding this section, we should touch upon phase boundary concentration data for poly(p-benzamide) dimethylacetamide + 4% LiCl [89], poly(p-phenylene terephthalamide) (PPTA Kevlar)-sulfuric acid [90], and (hydroxy-propyl)cellulose-dichloroacetic acid solutions [91]. Although not included in Figs. 7 and 8, they show appreciable downward deviations from the prediction by the scaled particle theory for the wormlike hard spherocylinder. Arpin and Strazielle [30] found a negative concentration dependence of the reduced viscosity for PPTA in dilute Solution of sulfuric acid, as often reported on polyelectrolyte systems. Therefore, the deviation of the Ci data for PPTA in sulfuric acid from the scaled particle theory may be attributed to the electrostatic interaction. For the other two systems too, the low C] values may be due to the protonation of the polymer, because the solvents of these systems are very polar. 

Hard spherocylinders behave differently. Frenkel et al. [411] have studied systems consisting of these objects having length to diameter ratios of both 3 and 5 and have employed both the Monte Carlo method... 

Hard spherocylinders (cylinders with hemispherical end caps) were studied using computer simulations [118]. In addition to a nematic phase, such particles also display a smectic-A phase, in which the particles are arranged in liquid-like layers. To observe this transition, rather monodisperse particles are needed. The smectic-A phase was indeed observed in suspensions of TMV particles [17]. 

As a result, the Berthelot equation may be of interest for fluids of nonspherical molecules. However, it can apply only to molecules which are nearly spherical. For molecules which are not close to spherical, it would be better to follow Rigby (21) and use Equation 47 or 49 with F0 being the equation of state of some appropriate fluid of nonspherical molecules. Analogues of Equation 15 are known for a wide class of fluids of nonspherical molecules (22). For example, for a fluid of hard prolate spherocylinder... 

Cotter, M. A., Hard spherocylinders in an anisotropic mean field a simple model for a nematic liquid crystal, J. Chem. Phys., 66, 1098-1106 (1977a). 

Lagomarsino, M.C., Dogterom, M., Dijkstra, M. Isotropic nematic transition of long, thin, hard spherocylinders confined in a quasi-two-dimensional planar geometry. J. Phys. Chem. 119, 719-721 (2003)... 

Perera, A. and G. N. Patey. 1988. The solution of the hypernetted-chain and Percus-Yevick approximations for fluids of hard spherocylinders. Journal of Chemical Physics. 89, 5861. 

Consider a medium consisting of elongated, cylindricaUy symmetric hard-core molecules in the form of spherocylinders. For spherocylinders shown in Fig. 6.19b we may introduce parameter... 

Onsager used a low-density expansion, that is small packing factor r =pVm-After a cumbersome calculation procedure he has found the excluded volume Vexdi i j) that depends on orientation of the rods. Then, using (6.63) and several approximations concerning averaging, the free energy and the equation of state for hard spherocylinders have been found. 

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