Informational thermodynamic model for nanostructures

Abstract

Nanostructures may be fabricated from metal nanoclusters such as gold or platinum, which are of interest for catalytic and structural characteristics, or from nano forms of carbon allotropes. Here, informational thermodynamic properties such as free energy, enthalpy, and entropy are calculated using a graph network model at T = 298.15 K. We calculate the partition function using Euclidean adjacency matrices from the Hamiltonian and estimated bond energies. The summed atomic displacement from the Kirchhoff index has power law behavior, while the thermodynamic properties exhibit large $N$ logarithmic behavior: however, the data shows structurally related anomalies.

Appendix

We recently published some results describing power law behavior with the thermodynamic properties [48–50]. We now show that these power law relationships are equivalent to the logarithmic properties shown in this manuscript.

Suppose the thermodynamic property is $P$ and we consider

$$\begin{aligned} \frac{P}{N_B }=\frac{A+B\ln (N)}{cN} \end{aligned}$$

(14)

where $N_{B}$ is the number of bonds and $A$, $B$, and $c$ are constants with $N_{B}$ = cN. We then have

$$\begin{aligned} N=cN-(c-1)N=cN\left( {1-\frac{c-1}{c}} \right) \end{aligned}$$

(15)

and thus

$$\begin{aligned} \ln (N)=\ln (cN)+\ln \left( {1-\frac{c-1}{c}} \right) . \end{aligned}$$

(16)

If we use this expression in Eq. (14), we have

$$\begin{aligned} \frac{P}{N_B }=\frac{A}{cN}+B\frac{\ln (cN)}{cN}+B\frac{\ln \left( {1+\frac{c-1}{c}} \right) }{cN}=\frac{A+B\ln \left( {\frac{2c-1}{c}} \right) }{cN}+B\frac{\ln (cN)}{cN}.\qquad \end{aligned}$$

(17)

Now both terms on the right go to zero as $N$ $\rightarrow$ $\infty$, but the first one is faster than the second one. Hence for large N, the dominant term is BN $_{B}^{-1}$ ln ($N_{B})$. Because ln ($N_{B})$ $<$ ($N_{B})^{\alpha }$ for $N$ $\rightarrow$ $\infty$ and $\alpha$ $>$ 0, we shall have the result we are intending to prove

$$\begin{aligned} \frac{P}{N_B }\sim B(N_B )^{-1+\alpha },N\rightarrow \infty ,\alpha >0, \end{aligned}$$

(18)

and the original expression for $P$ (Eq. 14) shows the logarithmic behavior.