Solvation Free Energy

The goal of this section is to show how to compute the free energy of solvation of structureless solute particles $A$ dissolved in a solvent $S$. The solvation free energy is computed according to the coupling parameter method in terms of the pair correlation function $g(r_A, r_S)$, where $r_A$ and $r_S$ are the coordinates of the solute and solvent molecules, respectively. Consider a solute-solvent mixture where solute particles with no internal structure interact with solvent molecules according to a pairwise additive potential $V_{AS}$. Assume that the solute concentration is so low that the interactions between solute particles can be neglected. The canonical partition function of the system is

$$Z_{\lambda} = \frac{Z_A^{(id)} Z_S^{(id)}}{V^{N_A+N_S}} \int dr^{... ... \int dr^{N_S} e^{-\beta V_S (r^{N_S}) -\beta V_{SA} (r^{N_S},r^{N_A})\lambda},$$ (409)

where the coupling parameter $\lambda=1$ and where

$$Z^{(id)}=\frac{1}{h^{3N} N!} \int d r^{3N} \int d {p}^{3N} e^{-\b... ...\frac{p^2}{2m}} =\frac{V^N}{N! h^{3N}} \int d p^{3N} e^{-\beta \frac{P^2}{2m}}.$$ (410)

In the absence of solute-solvent interactions (i.e., $\lambda=0$), the energy of the system becomes factorizable into the solute and the solvent contributions to the total energy and the free energy of the system is $A^{(id)}(N_S, N_A, V, T)=-\beta$   ln$Z_{\lambda=0}$. The change in free energy due to a differential change in $\lambda$ is

$$-k T \frac{d \text{ln} Z_{\lambda}}{d \lambda} = -k T \frac{\int ... ...nt d r^{N_S} e^{-\beta V_S(r^{N_S}) -\beta V_{SA}(r^{N_S}, r^{N_A})\lambda } }.$$ (411)

and assuming that

$$V_{SA}(r^{N_S}, r^{N_A}) =\sum_{j=1}^{N_A}\sum_{k=1}^{N_S} u_{AS}(\vert r_j-r_k\vert),$$ (412)

we obtain

$$-k T \frac{d \text{ln} Z_{\lambda}}{d \lambda} = N_A N_S \frac{\i... ... -\beta V_{SA}} }{\int d r^{N_A} \int d r^{N_S} e^{-\beta V_S -\beta V_{SA}} }.$$ (413)

Introducing the the pair correlation function $g_{AS}= \rho_{AS}/\rho_A \rho_B$, where $\rho_{AS}$ is the joint probability of finding a solute particle at $r_{A_1}$ and a solvent molecule at $r_{S_1}$,

$$\rho_{AS} = N_A N_S \int d r^{N_A-1} \int d r^{N_S-1} \frac{e^{-\... ... -\beta V_{SA}} }{\int d r^{N_A} \int d r^{N_S} e^{-\beta V_S -\beta V_{SA}} },$$ (414)

we obtain

$$- k T \frac{d \text{ln} Z_{\lambda}}{d \lambda}=\int dr_{A_1} \int dr_{S_1} u(\vert r_{A_1}-r_{S_1}\vert) \rho_A \rho_S g_{AS},$$ (415)

or

$$- k T \frac{d \text{ln} Z_{\lambda}}{d \lambda} = \rho_A \rho_S V \int d r u_{AS} (r) g_{AS} (r).$$ (416)

Therefore, the change in free energy due to a differential change in the coupling parameter $\lambda$ is

$$- k T \frac{d \text{ln} Z_{\lambda}}{d \lambda}=N_A \int d r u_{AS} (r) \rho_S g_{AS} (r),$$ (417)

and the total free energy of the system is

$$A(N_S, N_A, V, T) = A^{(id)}(N_S, N_A, V, T) + (\Delta A)_{solv},$$ (418)

where the free energy of solvation $(\Delta A)_{solv}$ is,

$$(\Delta A)_{solv} = N_A \int_0^1 d \lambda \int d r u_{AS} (r) \rho_S g_{AS}^{(\lambda)} (r).$$ (419)

The approach implemented in this section, where a coupling parameter is introduced to investigate the contributions of specific interactions to an ensemble average, is called coupling parameter method.