Subsections

• n-Particle Distribution Functions
• Correlation Functions
• Pair Correlation Function g(r) and Scattering
• g(r) and Thermodynamic Functions
• Integral Equations for g(r). Kirkwood Approximation
  • Potential of mean force:
  • Integral Equation for g⁽²⁾:
  • Conclusion:
• Closures
• Direct Correlation Function. Ornstein-Zernike Equation
• Percus-Yevick Approximation
  • Problems with PY equation

  Distribution Functions

  n-Particle Distribution Functions

  According to the Gibbs distribution, Probability, that n particular particles of the N-particle system will be found at the positions ${\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_n}$ is:

  $$\begin{split} \Prob^{(n)}({\mathbf{r}_1,\mathbf{r}_2,\dots,... ...-\beta U_N)\, d\mathbf{r}_{n+1}\dots d\mathbf{r}_N \end{split}$$ (1)

  The probability that any n particles are in the set of points: $\mathbf{r}_1$, $\mathbf{r}_2$,..., $\mathbf{r}_n$, irrespective of the position of the rest N-n particles is:

  $$\rho^{(n)}=\frac{N!}{(N-n)!}\Prob^{(n)}({\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_n})$$

  The factorials account for the number of ways to choose n particles from N.

  The simplest distribution function is the single particle distribution function $\rho^{(1)}(\mathbf{r}_1)$. This is just a probability to find a molecule at the point $\mathbf{r}_1$.

  Homogeneous fluid:

  $\rho^{(1)}(\mathbf{r}_1)=const=\rho$--the overall density in the system.

  Crystal:

  the periodic function of the coordinates, corresponding to the crystal lattice.

  Inhomogeneous systems:

  confined liquids, or adsorbed monolayers. It is not constant, and contains information about the interface structure.

  Correlation Functions

  .

  It is customary to define the n-particle correlation function  

  $$\rho^{(n)}({\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_n})=\rho^n g^{(n)}({\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_n})$$ (2)

  If all particles are independent, $\rho^{(n)}=\rho^n$ and thus g⁽ⁿ⁾ simply corrects for particle correlations.

  Obvious relationship between the correlation functions:  

  $$\begin{aligned} g^{(n)}({\mathbf{r}_1,\dots,\mathbf{r}_n})... ...^{(n+1)}({\mathbf{r}_1,\dots,\mathbf{r}_{n+1}}) \end{aligned}$$ (3)

  For the reasons that we'll see below, the most important is the pair correlation function $g^{(2)}(\mathbf{r}_1,\mathbf{r}_2)$.In a homogeneous liquid of spherically symmetric molecules it depends only on the distance $r=\left\vert\mathbf{r}_1-\mathbf{r}_2\right\rvert$ between two molecules. For this reason it is also called the radial distribution function

  $$\rho g^{(2)}(\mathbf{r}_1,\mathbf{r}_2)d\mathbf{r}= \rho g(r)d\mathbf{r}$$

  It is a ``probability'' to find a second molecule in $d\mathbf{r}$ given that there is a molecule at the origin of $\mathbf{r}$.

  It is not normalized by 1 (see problem in the Quiz)

  Due to the hard core repulsion of the molecules, and as the molecules in liquid are uncorrelated at large distances.

  $$g\to \begin{cases} 0,& r\to0 \\ 1,& r\to\infty , \end{cases}$$

  
  
  

  Pair Correlation Function g(r) and Scattering

  
  
  

  Consider the sample of gas, liquid, solid, glass, etc., containing N molecules, irradiated by X-rays, or neutrons, or light, or electrons, etc., with a wave vector of $\mathbf{k}_0$.We set detector to measure the radiation scattered at angle $\phi$.

  The intensity of the radiation scattered at angle $\phi$ with the wave vector $\mathbf{k}_s$, and $\mathbf{Q}=\mathbf{k}_s-\mathbf{k}_0$ is  

  $$\begin{split} I(\mathbf{Q}) \propto \left\langle \left\ver... ...xp(-i\mathbf{Q}\mathbf{r}_j) \right)\right\rangle \end{split}$$ (4)

  We introduce microscopic density  

  $$\rho(\mathbf{r}) = \sum_i^N \delta(\mathbf{r}-\mathbf{r}_i)$$ (5)

  It's Fourier transform can be defined as:

  $$\begin{aligned} \rho(\mathbf{k})&=\int d^3\mathbf{r}\rho(\ma... ...{k}\rho(\mathbf{k}) \exp( i\mathbf{k}\mathbf{r}). \end{aligned}$$

  From (5) and (4)  

  $$\begin{aligned} \rho(\mathbf{k})&=\sum_{i=1}^N \exp(-i\mat... ...\rho(\mathbf{Q}) \rho(-\mathbf{Q}) \right\rangle \end{aligned}$$ (6)

  The last equation defines the structure factor $S(\mathbf{Q})$, which is related to the density correlations and to the $g(\mathbf{r})$. 

  $$\begin{aligned} S(\mathbf{Q}) = & \frac 1N \left\langle \su... ...(\mathbf{r}_i-\mathbf{r}_j)\bigr) \right\rangle\\ \end{aligned}$$ (7)

  The first term just equals 1. The second term is an average of a sum over all pairs of molecules of a function $f(\mathbf{r}_i, \mathbf{r}_j) = \exp(i\mathbf{Q}\mathbf{r})$ of their separation $\mathbf{r}=\mathbf{r}_i-\mathbf{r}_j$. As any average, it can be rewritten as an integral of the $f(\mathbf{r})$ with the probability to observe $\mathbf{r}$. It is $\rho^{(2)}(\mathbf{r})=\rho^2 g(\mathbf{r})$, given there is a molecule at the origin.

  $$\begin{aligned} S(\mathbf{Q})=& 1 + \frac 1N \rho^2 \sum_i ... ...f{r}\, \exp(-i\mathbf{Q}\mathbf{r}) g(\mathbf{r}) \end{aligned}$$

  or just  

  $$S(\mathbf{Q}) = 1 + (2\pi)^3 \rho g(\mathbf{Q})$$ (8)

  with  

  $$\rho g(\mathbf{r}) = (2\pi)^{-3}\int d\mathbf{Q}(S(\mathbf{Q})-1) \exp(i\mathbf{Q}\mathbf{r})$$ (9)

  Note:

  In small angle scattering and fluctuation theory it is customary to subtract the forward scattering in the definition of $S(\mathbf{Q})$. Then  

  $$S(\mathbf{Q})=\frac 1N \left(\left\langle \rho(\mathbf{Q}) ... ...hbf{Q}) \right\rangle-\left\langle \rho\right\rangle^2\right)$$ (10)

  g(r) and Thermodynamic Functions

  Energy:

  In the case of pairwise interactions, the average potential energy is a sum over pairs:

  $$\begin{aligned} \left\langle U\right\rangle = & \left\langl... ...t d \mathbf{r}\, v(\mathbf{r}) g(\mathbf{r}) \end{aligned}$$

  The total energy is then given as;

  $$\frac{E}{NkT} =\frac 32 + \frac {\rho}{2kT} \int d \mathbf{r}\, v(\mathbf{r}) g(\mathbf{r},\rho,T)$$

  Pressure:

  $$p=kT\left( \frac{\partial \log Z}{\partial V}\right) _{N,T}$$

  where

  $$Z= \int_0^V d\mathbf{r}_1 \dots \int_0^V d\mathbf{r}_N\, \exp\bigl(-\beta U({\mathbf{r}_1\dots\mathbf{r}_N})\bigr)$$

  We can assume, that our liquid is in a cubic container and make the coordinate change

  $$x'=\chi x, \quad y'= \chi y, \quad z' = \chi z, \quad \chi=V^{-1/3}$$

  Then the distance between the particles:

  r_ij=V^1/3 r_ij'

  and  

  $$\begin{aligned} \left( \frac{\partial Z}{\partial V} \righ... ...ft(-\beta U\right) \frac{\partial U}{\partial V} \end{aligned}$$ (11)

  with

  $$\frac{\partial U}{\partial V}=\sum_{i\neq j} \frac{dv(r_{ij})}{dr_{ij}}\frac{dr_{ij}}{dV}$$

  Now after going back to the initial coordinates and substituting the $\partial\log Z_N/\partial V=Z_N^{-1}\partial Z_N/\partial V$, and again substituting the average of the sum over pairs by the integral with g(r), as before, we finally obtain for the spherically isotropic case:  

  $$\frac{p}{kT} = \rho - \frac{\rho^2}{6kT}\int_0^\infty r\frac{dv}{dr}g(r)4\pi r^2 \, dr$$ (12)

  This equation is called pressure equation. This is essentially an equation of state in terms of the g(r).

  Virial Coefficients:

  If we expand:  

  $$g(r,\rho,T)=g_0(r,T)+\rho g_1(r,T) + \rho^2 g_2(r,T) \dots$$ (13)

  and substitute it into (12) to get:

  $$\frac{p}{kT} = \rho - \frac{\rho^2}{6kT} \sum_{j=0}^{\infty} \rho^j \int_0^\infty r\frac{dv}{dr}g_j(r,T)4\pi r^2 \, dr,$$

  and compare the coefficients of the equal power of the density with the virial expansion from the previous lecture, we obtain for the virial coefficients:

  $$B_{j+2}(T)=-\frac{1}{6kT}\int_0^{\infty}r\frac{dv}{dr}g_j(r,T)4\pi r^2 \,dr$$

  Chemical Potential:

  So far we have established relationships of the g(r) to the mechanical thermodynamic properties. To have a complete description of the liquid we need a non-mechanical thermodynamic function. (Why?) Let it be the chemical potential. Let us assume that the system's energy is described by an additional coupling parameter $\kappa$: 

  $$\begin{split} U(\mathbf{r}_1, \dots, \mathbf{r}_N, \kappa)... ...pa v(r_{1j}) + \sum_{2\leq j < i \leq N} v(r_{ij}) \end{split}$$ (14)

  This is equivalent to changing all interactions with the particle number 1. All the other interactions remain unchanged. Changing $\kappa$ from to 1 switches between the system of particles 2,..., N, to the complete system of particles 1,..., N. We will denote their configurational integrals as Z_N-1 and Z_N respectively. By definition (with $\tilde\mathbf{r}=(\mathbf{r}_1,\dots,\mathbf{r}_N)$) 

  $$\begin{aligned} Z_N(\kappa)=&\int d\tilde\mathbf{r}\exp\bigl... ...e\mathbf{r}\sum_{j=2}^{N}v(r_{1j}) \exp(-\beta U) \end{aligned}$$ (15)

  Dividing by Z_N, and again replacing the average of pair contributions by the integral with $g(r,\kappa)$: 

  $$\frac{\partial \log Z_N}{\partial \kappa} = -\frac{\rho}{kT} \int_0^{\infty} v(r) g(r,\kappa) 4\pi r^2 \, dr$$ (16)

  Then

  $$VZ_{N-1}=Z_{N}(\kappa=0), \quad Z_{N}=Z_{N}(\kappa=1)$$

  For very large N

  $$\begin{aligned} \mu =&\left( \frac{\partial A}{\partial N} \... ...a \bigl(\frac{\partial Z_N}{\partial \kappa}\bigr)\end{aligned}$$

  Putting this together with (16) gives finally:

  $$-\frac{\mu}{kT} = \log \rho \Lambda^3 + \frac{\rho}{kT}\int_0^1 d\kappa \int_0^{\infty} v(r) g(r,\kappa) 4\pi r^2\, dr$$

  From $\mu$, p, and E, we can get any other thermodynamic properties, if we knew $g(r,\kappa)$

  Compressibility:

  We know that

  $$\left\langle (\Delta V)^2\right\rangle = kTV\alpha_T$$

  or, since $V=N/\rho$, we obtain for grand canonical ensemble $\Delta N=\Delta V/\rho$ and

  $$\frac{\left\langle (\Delta N)^2\right\rangle}{N} = kT\rho\alpha_T$$

  On the other hand
  $$\begin{multline*} S(0) = \frac {1}{\left\langle N\right\rangle} \left\langle \su... ...gle}{\left\langle N\right\rangle} + \left\langle N\right\rangle \end{multline*}$$
  and

  $$S(0) = 1+ \rho\int g(\mathbf{r})\,d\mathbf{r}$$

  We obtained:

  $$\frac{(\Delta N)^2}{\left\langle N\right\rangle} + \left\langle N\right\rangle = 1+ \rho\int g(\mathbf{r})\,d\mathbf{r}$$

  or, since

  $$\left\langle N\right\rangle = \rho\int d\mathbf{r}$$

   

  $$kT\rho\alpha_T = 1 + \rho\int\bigl(g(\mathbf{r})-1\bigr)\,d\mathbf{r}$$ (17)

  This is called compressibility equation. If we know g(r), we can obtain compressibility $\alpha_T$

  Integral Equations for g(r). Kirkwood Approximation

  Potential of mean force:

  Describes the interaction between two particles, immersed into the medium. Very important for description of colloids, ions in solution, etc. Given that two particles are separated by $\mathbf{r}$, what is their interaction?

  Direct:

  via the intermolecular interaction potential

  Indirect:

  due to the presence of the other particles (medium)

  
  
  This potential should, of course, be a function of $\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1$, and related to the probability for $\mathbf{r}$ to occur, or $g^{(2)}(\mathbf{r})$. Indeed it is defined as:  

  $$g^{(2)} = \exp(-\beta \Psi(\mathbf{r}))$$ (18)

  It is easy to show (see the homework problem (5), that $\Psi(\mathbf{r})$ is in fact free energy of the whole system given that particles 1 and 2 are $\mathbf{r}$ apart, and all the rest of the system is averaged over. PMF is convenient to describe the interaction of the various solutes (colloid particles, ions or macromolecules in solution, etc.) It can also be used to describe the ``effective interaction'' of two molecules of the liquid, given their relative position, and averaged over the possible configurations of the rest.

  Integral Equation for g⁽²⁾:

  Consider triplet of particles: 1, 2, and 3: The net force, acting on particle 1 is composed of
  • Direct interaction between 1 and 2, $v(\mathbf{r}_{12})$, and
  • Mean force due to 3, averaged over all it's positions with the proper probability taking into account fixed positions of 1 and 2
   

  $$\begin{aligned} -g^{(2)}(\mathbf{r}_1, \mathbf{r}_2) &\nab... ...ho g^{(3)}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)\end{aligned}$$ (19)

  This is the first equation of the BBGKY (Bogolyubov, Born, Green, Kirkwood, Yvon) hierarchy. It relates the correlation functions of two adjacent orders, and the inter-molecular interactions potential v(r). Following the same line of argument we could derive similar equations relating g⁽ⁿ⁾, g⁽ⁿ⁺¹⁾, and v(r), for any n.

  Conclusion:

  If we know the high-order correlation functions, we could calculate, the lower- order ones, and then all the thermodynamic properties.

  Closures

  Still BBGKY is an infinite hierarchy of integral equations. The simple way to close it at the second order is due to Kirkwood. Assumption:  

  $$g^{(3)}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3) = g^{(2)}(\m... ...\mathbf{r}_1, \mathbf{r}_3) g^{(2)}(\mathbf{r}_2, \mathbf{r}_3)$$ (20)

  This is called superposition approximation. (SPA) Using this equation we can obtain the integral equation for g⁽²⁾ and try to solve it. Usually it can only be done numerically.

  SPA assumes, that the correlations between any two molecules are unaffected by the presence of the third one.

  Equation (20) has been tested directly by computer simulations. Although it is accurate only within 10% for hard spheres, it actually describes the (P,V,T) properties quite well. It also starts losing solutions at higher densities, which is believed to be an offset of solidification.

  Different closures, trying to capture three-body effects, and/or considering higher-order correlations, have been considered in the literature.

  Direct Correlation Function. Ornstein-Zernike Equation

  Instead of splitting the potential to direct an indirect, we can split pair correlation function into direct and indirect parts.

  Direct part:

  correlations due to interparticle interactions between 1 and 2.

  Indirect part:

  Due to the presence of other particles.

  Ornstein-Zernike approach:

  At small $\rho$ there are no correlations: $g(r)\approx1$ at $r\gt\sigma$. We introduce  

  $$h(r)=g(r)-1\to0\quad \text{small $\rho$, $r\gt\sigma$}$$ (21)

  Higher densities: $h(r)\ne0$. We introduce c(r) describing only binary collisions between 1 and 2 and

  $$h(r)\approx c(r)$$

  If potential is short-range, c(r) is also short range. At $\rho\to0$ PMF tends to interparticle potential, and

  $$c(r)\to f(r) = \exp(-\beta v(r))-1,\quad\text{small $\rho$}$$

  Even higher densities: both binary and ternary collisions:

  $$h(\mathbf{r}_{12}) = c(\mathbf{r}_{12}) + \rho\int c(\mathbf{r}_{13})c(\mathbf{r}_{23})\,d\mathbf{r}_3$$

  Then we add next terms...
  $$\begin{multline*} h(\mathbf{r}_{12}) = c(\mathbf{r}_{12}) + \rho\int c(\mathbf{... ...{r}_{34})c(\mathbf{r}_{42})\,d\mathbf{r}_3\,d\mathbf{r}_4+\dots \end{multline*}$$
  This could be represented as:  

  $$h(\mathbf{r}_{12}) = c(\mathbf{r}_{12}) + \rho\int c(\mathbf{r}_{13})h(\mathbf{r}_{23})\,d\mathbf{r}_3$$ (22)

  This is an exact equation. It is a definition of direct correlation function c(r).

  Percus-Yevick Approximation

  Model:

  Hard spheres.

  $$v(r) = \begin{cases} \infty,& r<\sigma\\ 0,& r\gt\sigma \end{cases}$$

  Pair correlation function:

  inside spheres g(r)=0 at $r<\sigma$

  Direct correlation function:

  small at $r\gt\sigma$. Percus-Yevick assumption: c(r)=0 at $r\gt\sigma$

  This allows analytical solution of equation (22) for hard spheres. It gives us correlation function g(r) (c(r) and h(r)).

  From pressure equation of state (12)  

  $$\frac{PV}{NkT} = \frac{1+2\eta+3\eta^2}{(1-\eta)^2},\quad \eta= \frac15\pi\sigma^3\rho$$ (23)

  From compressibility equation (17)  

  $$\frac{PV}{NkT} = \frac{1+\eta+\eta^2}{(1-\eta)^3}$$ (24)

  Problems with PY equation

  1.

  Equations (23) and (24) give different results! The theory is thermodynamically inconsistent.

  Reason:

  not any function g(r) gives thermodynamically consistent theory--only the correct one. We made assumption for c(r)--and must pay for it.

  Carhanan-Starling approximation:

  add equations (23) and (24) with weights (1/3 and 2/3)

  Even better idea:

  adjust the weights.

  2.

  Pressure diverges at $\eta\to1$ instead of $\eta\to0.634$.