Configurational Temperatures and Interactions in Charge-Stabilized Colloid

§ III. Hyperconfigurational temperatures

Choosing \boldsymbol{B}(\boldsymbol{\Gamma})=\{ q_{1}^{s},\cdots,q_{{3N}}^{s},0,\cdots,0\} with s=1,2,3\cdots in Eq. (1) yields a hierarchy of so-called hypervirial temperatures (6), which reduce to Clausius' virial temperature for s=1. By the same token, we propose that \boldsymbol{B}(\boldsymbol{\Gamma})=\{ F_{1}^{s},\cdots,F_{{3N}}^{s},0,\cdots,0\} with s>0, yields the set of “hyperconfigurational temperatures” (5),

k_{B}T_{h}^{{(s)}}=\frac{\left<\sum _{{i=1}}^{{3N}}F_{i}^{{s+1}}\right>}{\left<-s\sum _{{i=1}}^{{3N}}F_{i}^{{s-1}}\partial _{i}F_{i}\right>}, (9)

of which T_{h}^{{(1)}} is equivalent to the standard configurational temperature. Here, F_{i} is the magnitude of the i-th element in the set of 3N components of the forces on the N particles. Because F_{i} is non-negative, T_{h}^{{(s)}} is well defined for any positive real value of s. Negative values of s would yield diverging temperatures because at least some of the F_{i} will be vanishingly small for any system substantially larger than the range of interactions.

A simple example motivates introducing this new hierarchy of expressions. If, for example, a system is characterized by Coulomb pair interactions, u(r)=1/r in d=3 dimensions, each term of the denominator, \nabla _{r}^{2}u(r)=r^{{1-d}}\,\partial _{r}(r^{{d-1}}\partial _{r}u(r)), of Eqs. (6) and (7) vanishes. Consequently, the associated configurational temperature expressions in Eqs. (6), (7) and (8) diverge unphysically. The hyperconfigurational temperatures, by contrast, are still well defined with \partial _{x}F_{x}=(1-3x^{2}/r^{2})/r^{3}, and F_{x}^{{s-1}}\partial _{x}F_{x}+F_{y}^{{s-1}}\partial _{y}F_{y}+F_{z}^{{s-1}}\partial _{z}F_{z}\ne 0 for s\ne 1. Consequently, the hyperconfigurational temperatures should apply to any system whose pair potential is continuous and differentiable. This suggests that they will be useful for studying systems whose interactions are not known a priori.

Additional useful results emerge for systems such as colloidal monolayers whose interactions are isotropic. In this case, the Cartesian coordinates may be analyzed independently

\displaystyle k_{B}T_{{\text{conF}}} \displaystyle=\frac{\left<\sum _{{j=1}}^{N}\boldsymbol{F}_{j}^{2}\right>}{\left<-\sum _{{j=1}}^{N}\nabla _{j}\cdot\boldsymbol{F}_{j}\right>}=\frac{\left<\sum _{{j=1}}^{N}(F_{{j,x}}^{2}+F_{{j,y}}^{2})\right>}{\left<-\sum _{{j=1}}^{N}(F^{\prime}_{{j,x}}+F^{\prime}_{{j,y}})\right>}
\displaystyle=\frac{\left<\sum _{{j=1}}^{N}F_{{j,x}}^{2}\right>}{\left<-\sum _{{j=1}}^{N}F^{\prime}_{{j,x}}\right>}=\frac{\left<\sum _{{j=1}}^{N}F_{{j,y}}^{2}\right>}{\left<-\sum _{{j=1}}^{N}F^{\prime}_{{j,y}}\right>}. (10)

Setting \boldsymbol{B}(\boldsymbol{\Gamma})=\partial _{x}V(\{ q_{i}\}) or \partial _{y}V(\{ q_{i}\}) in Eq. (1) leads to same results. We will refer to the two terms in Eq. (10), as well as analogous results for other temperature expressions, as the Cartesian components of the configurational temperature, T_{x} and T_{y}, respectively.