6.1.1 Fluids and solids in a nutshell

Fluids:

Made up of classical mass points or rigid bodies, interacting by pair forces (and possibly torques).

Table 6.1: Isotropic model potentials in statistical-mechanical simulation: $u=u(r)$

Hard spheres	$u(r)= \infty \;\;{\rm if}\, r < r_{0}$ $= 0 \;\;\; {\rm if}\, r \geq r_{0}$	First approximation in many applications
Lennard-Jones	$$u(r)= 4 \,\epsilon \left[ \left(\frac{r}{\sigma}\right)^{-12} - \left(\frac{r}{\sigma}\right)^{-6} \right]$$	Noble gas atoms; nearly spherical molecules
Isotropic Kihara	${ u(r)=}$ ${ 4 \, \epsilon \left[ \left( \frac{\textstyle{r-a}}{\textstyle{\sig... ...2} - \left(\frac{\textstyle{r-a}}{\textstyle{\sigma-a}}\right)^{-6} \right] }$	Noble gas atoms; nearly spherical molecules ( $a=0.05-0.1 \, \sigma$)
Harmonic	$u(r)= A \, \left( r-r_{0}\right)^{2}$	Intramolecular bonds, if $kT$ is small compared to the bond energy
Morse	$u(r)= A\,\left[ e^{\textstyle{-2b(r\!-\!r_{0})}}- 2 e^{\textstyle{-b(r\!-\!r_{0})}} \right]$	Intramolecular bonds, if $kT$ is comparable to the bond energy
Born-Huggins-Mayer	$u(r)= \frac{\textstyle{q_{1}q_{2}}}{\textstyle{4\pi \epsilon_{0}r}} + B e^{\te... ...rac{\textstyle{C}}{\textstyle{r^{6}}}- \frac{\textstyle{D}}{\textstyle{r^{8}}}$	Ionic melts; $q_{i}$ are the ion charges

Table: Anisotropic model potentials in statistical-mechanical simulation: $u(12)=u(\mbox{$\bf r$}_{12},\mbox{$\bf e$}_{1},\mbox{$\bf e$}_{2})$

Hard dumbbells, hard spherocylinders, etc.	$u(12)= \infty \;\;$ if overlap $= 0 \;\;\;$ otherwise	First approximation to rigid molecules
Interaction site models, rigid	$u(12)=$ sum of isotropic pair energies ${u(r_{i(1),j(2)})}$, where several interaction sites $i$ and $j$ are in fixed positions on molecules $1$ and $2$, respectively	Rigid molecules
Interaction site models with non-rigid bonds	$u(12)=$ sum of isotropic pair energies, both intra- and intermolecular	Non-rigid molecules
Kramers-type	$u(12)=$ sum of isotropic pair energies, exclusively between sites on different molecules; certain intramolecular distances (bonds) and/or angles are fixed	Flexible molecules, from ethane to biopolymers
Stockmayer	$u(12)=$ Lennard-Jones + point dipoles	First approximation to small polar molecules
Anisotropic Kihara	$u(12)= 4 \, \epsilon \left[ \left( \frac{\textstyle{\rho_{12}}}{\textstyle{\si... ... - \left( \frac{\textstyle{\rho_{12}}}{\textstyle{\sigma}}\right)^{-6} \right]$ where $\rho_{12}$ is the shortest distance between two linear rods	Rigid linear molecules with distributed Lennard-Jones interaction
Gay-Berne	$u(12)=$ $4 \,\epsilon(12) \left[ \left( \frac{\textstyle{r_{12}-\sigma(12)+\sigma_{0}}}{\textstyle{\sigma_{0}}}\right)^{-12} \right.$ $- \left. \left( \frac{\textstyle{r_{12}-\sigma(12)+\sigma_{0}}}{\textstyle{\sigma_{0}}}\right)^{-6} \right]$ where $\sigma(12)$ and $\epsilon(12)$ depend on $\mbox{$\bf r$}_{12},\mbox{$\bf e$}_{1},\mbox{$\bf e$}_{2}$ and certain substance-specific shape parameters	Liquid crystal molecules of ellipsoidal shape, with smoothly distributed Lennard-Jones sites

At time $t$, combine the position vectors of the $N$ atoms into a vector $\mbox{$\bf\Gamma$}_{c}\equiv$ $\{\mbox{$\bf r$}_{1} \dots$ $\mbox{$\bf r$}_{N}\}$. The set of all possible such vectors spans the $3N$-dimensional ``configuration space'' $\Gamma_{c}$.

Let $a(\mbox{$\bf\Gamma$}_{c})$ be a property of the $N$-body system, depending on the positions of all particles (i.e. of the microstate $\mbox{$\bf\Gamma$}_{c}$). The thermodynamic average of $a$ is given by

$$\langle a \rangle = \int_{\Gamma_{c}} a(\mbox{$\bf\Gamma$}_{c}) \,p(\mbox{$\bf\Gamma$}_{c})\, d\mbox{$\bf\Gamma$}_{c}$$

where $p(\mbox{$\bf\Gamma$}_{c})$ is the probability density at the configuration space point $\mbox{$\bf\Gamma$}_{c}$.

Examples:

• Internal energy:
  

  $$U=NkT+\frac{1}{2} \langle \, \sum_{i} \sum_{j \neq i} u(r_{ij}) \, \rangle$$ (6.1)

  
  
  
  
  
• Pressure:
  

  $$p=\frac{NkT}{V} - \frac{1}{6V} \langle \, \sum_{i} \sum_{j \... ...i} r_{ij} \left. \frac{du}{dr} \right\vert _{r_{ij}} \rangle$$ (6.2)

  
  

Problem: the probability density $p(\mbox{$\bf\Gamma$}_{c})$ is known only up to an indetermined normalizing factor. Thus, $p_{can}(\mbox{$\bf\Gamma$}_{c})$ $\propto exp[-E(\mbox{$\bf\Gamma$}_{c})/kT]$, but the normalizing denominator $Q$ (the Configurational Partition Function) in

$$1 = \int p(\mbox{$\bf\Gamma$}_{c})\, d\mbox{$\bf\Gamma$}_{c}... ...tyle -E(\mbox{$\bf\Gamma$}_{c})/kT} \, d\mbox{$\bf\Gamma$}_{c}$$

is not known.

Spin systems:

Basic model of ferromagnetic solids: atoms are fixed on the vertices of a lattice. Their dipole vectors (spins) may have varying directions: either up/down (Ising model) or any direction (Heisenberg model).

Microscopic configuration $\mbox{$\bf\Gamma$}_{c}$: given by the $N$ spins on the lattice.

Example: Two-dimensional square Ising lattice; only the four nearest spins contribute to the energy of spin $\sigma_{i}$ ($= \pm 1$). (Three dimensions: six nearest neighbors). The total energy is

$$E=-\frac{A}{2}\sum_{i=1}^{N} \sum_{j(i)=1}^{4\; or \;6} \sigma_{i} \sigma_{j(i)}$$

($A$ ... coupling constant).

$\Longrightarrow$Magnetic polarization

$$M \equiv \sum_{i=1}^{N} \sigma_{i}$$

may be determined as a function of temperature. If a magnetic field $H$ is applied, the additional potential energy is $E_{H}=-H \sum_{i} \sigma_{i}$.