# Equation of state for hard rods

## Some context:

For ideal gases, the thermodynamic equation of state is the well-known: $$pV = nRT \tag{1}$$ where $n$ is the amount of substance, $R$ the universal gas constant and $P,V,T$ are pressure, volume and temperature respectively.

On a more realistic level, we have the van der Waals correction of $(1)$ where he accounts both for the finite volume of the gas molecules and their interactions.

## Hard systems:

The equation of state for a system of hard[*] spheres has also been solved (upto a very good approximation), given by the Carnahan-Starling equation of state (and there have been further corrections to this):

$$Z = \frac{pV}{Nk_B T} = \frac{1+\eta + \eta^2 -\eta^3}{(1-\eta)^3} \tag{2}$$ where $Z$ is the compressibility factor, $k_B$ the Boltzmann constant and $\eta$ the packing fraction of spheres, i.e., $\eta = \pi N \sigma^3/ (6V),$ with $\sigma$ the diameter of a sphere.

One importance of having an analytic expression for the equation of state of such system, is that if we know e.g., from simulations/experiments, at what packing fraction the system undergoes a certain phase transition, we can calculate the corresponding values for all other macroscopic properties of the system, such as the pressure at which the transition occurs.

[*]: hard here means that the only interaction between the spheres is the excluded-volume interaction, where if there's any overlap between spheres, the energy is infinite and $0$ otherwise. This also means that all hard systems are athermal (their behaviour is temperature independent). So in other words, hard systems and all the problems revolving around them, such as their phase transition behaviours, are essentially geometric problems, tightly related to e.g., sphere packing problems.

Now the question concerns systems of hard spherocylinders (rods):

• Similar to Eq. (2), is there a known approximation for the equation of state of hard rods?
• Again the aim is to calculate the corresponding pressure value $p$ at a phase transition for a system of hard rods, knowing the packing fraction of the system at that point. To clarify, by a point of interest, I simply mean for instance the packing fraction $\eta_j$ at which the system jams, or the fraction at which $\eta_c$ the rods have connectivity percolation.
• I didn't think there was a rigorous justification of the Carnahan-Starling equation of state (even in terms of any kind of perturbative / asymptotic series) so I wouldn't call the hard sphere case "solved" (at least not on a site geared towards mathematicians). – j.c. Oct 11 '17 at 15:27

I don't know if there are any expressions that take such a simple form as the C-S equation of state. Note that spherocylinders have an additional geometrical parameter $L/D$ relating the length $L$ to the diameter $D$ of the "caps" and there's also additional complexity in that they can undergo multiple phase transitions, e.g. from an isotropic liquid to a nematic liquid to a smectic crystal to a 3D crystal. One would expect of course very different "equations of state" in each of the phases.
Density functional theory for hard spherocylinders: phase transitions in the bulk and in the presence of external fields by Hartmut Graf and Hartmut Löwen, J. Phys.: Condens. Matter 11, 1435 (1999). Using density functional theory arguments, this paper describes approximations to the free energy of the spherocylinder system in various phases and also derives some phase diagrams. Note that "p" is not pressure in this paper but the geometric ratio $L/D$.