Hard rods

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

This is an example of a classical non-ideal system for which the PF can be deduced exactly [13]. Consider N hard rods of length [Pg.459]

Frenkel D 1988 Thermodynamic stability of a smectic phase in a system of hard rods Nature 332 822-3... 

For one-dimensional eonfined hard-rod [49,94,95] and Tonks-Takahashi fluids [49,96,97] the elose relationship between stratifieation and the oseillatory deeay of has been demonstrated analytieally. On the basis of a... 

Z. Tang, L. E. Scriven, H. T. Davis. Size selectivity in adsorptions of poly-disperse hard-rod fluids in micropores. J Chem Phys 97 5732-5737, 1992. 

A very simple model that predicts lyotropic phase transitions is the hard-rod model proposed by Onsager (Friberg, 1976). This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder where the... 

Comparison with Other Theories for Hard Rods.100... 

Table 4 compares different theoretical approaches with respect to the equations of state and the second and third virial coefficients (B2, B3) for a hard rod solution in the isotropic state B2 and B3 are the parameters appearing in the expansion... 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

Unlike for hard rods, no analytical solution exists for one-dimensional Lennard-Jones (LJ) rods. Molecular dynamics simulations have revealed a 1/f3 behavior in this system also [184, 185]. 

The important point now to note is that in one-dimensional hard rods, F(q, t) decays mostly as a Gaussian function of time. At small q decay, F(q, t) can be given as... 

In a hard-rod system, at sufficiently high volume fraction a transition is usually expected from the nematic to the smectic A phase [37], a lamellar phase with layers perpendicular to the nematic director. However, as elegantly demonstrated by Livolant [29], in DNA the smectic phase is replaced by columnar ordering this behavior can easily be explained on the basis of strand flexibility [38] or length polydispersity [39], both favoring the COL phase over smectic. 

Glass plates, 6x6 in. tubing, soft, tubing, hard, rod. 

Fig. 2. The partition function Z for athermal solutions of hard rods having an axial ratio x = 100 shown as a function of the disorder index y for the volume fractions indicated. Calculations were carried out according to Eq. (C-1) of the Appendix with x = 0. The logarithm of the partition function relative to the state of perfect alignment (y = 1) is plotted on the ordinate...

Table 1. Resume of Calculations of Biphasic Equilibria in Systems of Hard Rods...

This conclusion was reached, tentatively, by Frenkel, Shaltyko and Elyashevich A phenomenological analysis presented by Pincus and de Gennes predicted a first-order phase transition even in the absence of cooperativity in the conformational transition. These authors relied on the Maier-Saupe theory for representation of the interactions between rodlike particles. Orientation-dependent interactions of this type are attenuated by dilution in lyotropic systems generally. In the case of a-helical polypeptides they should be negligible owing to the small anisotropy of the polarizability of the peptide unit (cf. seq.). Moreover, the universally important steric interactions between the helices, regarded as hard rods, are not included in the Maier-... 

A theoretical treatment has recently been carried out by the author in collaboration with Matheson along the lines discussed above with appeal only to the spatial requirements of hard rods as represented in the lattice model, orientation-dependent interactions being appropriately ignored. The two transitions, one conformational and the other a cooperative intermolecular transition, are found to be mutually affected each promotes the other as expected. The coil-helix conformational transition is markedly sharpened so that it becomes virtually discrete, and hence may be represented as a transition of first-order. These deductions follow from the steric interactions of hard rods alone intermolecular attractive forces, either orientation-dependent or isotropic, are not required. 

A. Lattice Theory for Hard Rods with Exact Treatment of the Orientation Distribution... 

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

Figure 11.21 Predictions of the dimensionless Frank constants for hard rods as functions of reduced concentration nL"dv/A, using a numerical solution of the Onsager equation. At the concentration v = v, the abscissa has the value n L Ju/4 = 4. (From Lee and Meyer, reprinted with permission from J. Chem. Phys. 84 3443, Copyright 1986, American Institute of Physics.)...

---