Thermodynamic Properties

 The accurate expressions for $\Omega(E)$ developed in the previous section can be used to give a wide range of thermodynamic functions, in fact, all except those that depend on derivatives of N or V. The exceptions arise because the thermodynamic properties of small clusters are discontinuously dependent on N and because the volume of a cluster is hard to define[172]. The formulae for the functions illustrated in this section are given in the appendix. They have, for the most part, been derived in previous work within the harmonic approximation[153,133]; the extension to incorporate anharmonicity follows simply from the expressions given in the previous section. The results are given for the most accurate partition functions: the first-order corrected $\gamma$ formula for LJ₅₅ using sample A and the second-order corrected n^* formula for LJ₁₃ using sample D.

  

Figure 3.9: Comparison of harmonic and anharmonic thermodynamic functions of LJ₅₅: (a) the microcanonical (solid line) and canonical (dashed line) caloric curves, (b) the isopotential caloric curve, (c) C_v(T) and (d) A(T). In all except (a) the anharmonic curve is denoted by a solid line and the harmonic by a dashed line. Energies are measured with respect to the global minimum icosahedron.

Harmonic and anharmonic results for the caloric curves, the heat capacity C_v, the Helmholtz free energy A, the transition temperature $T_\half$ and the latent heat L_m are compared in Figures 3.9 and 3.10 and Table 3.8. We define $T_\half$ as the temperature for which the two states have an equal Landau free energy, A_L(E_c). L_m is the internal energy difference between the two states at $T_\half$and was obtained by extrapolating the caloric curves for each state to $T_\half$ using equation 3.29. For LJ₁₃, this procedure simply gives $L_m=U_{{\romannumeral2}}-U_{{\romannumeral1}}$, where U_i is the internal energy of region i. For LJ₅₅, we have used $L_m=U_{{\romannumeral5}{\rm -} {\romannumeral6}}-U_{{\romannumeral1}{\rm -} {\romannumeral4}}$, where $U_{{\romannumeral1}{\rm -} {\romannumeral4}}$ is the internal energy for the region of the PES formed from the combination of regions 1-4, and so L_m does not include the latent heat associated with the disordering of the surface that occurs before melting. Just integrating C_v over the transition region would overestimate L_m because it would also include the energy needed to raise the temperature of the cluster.

  

Figure 3.10: Comparison of harmonic and anharmonic thermodynamic functions of LJ₁₃: (a) the microcanonical (solid line) and canonical (dashed line) caloric curves, (b) the isopotential caloric curve, (c) C_v(T) and (d) A(T). In all except (a) the anharmonic curve is denoted by a solid line and the harmonic by a dashed line. Energies are measured with respect to the global minimum icosahedron.

  

Table 3.8: Transition temperatures and latent heats for LJ₁₃ and LJ₅₅.

The anharmonic caloric curves are displaced downward and away from their harmonic equivalents because of the increased densities of states associated with higher potential energy, lower temperature regions of the PES. The anharmonic heat capacity curve of LJ₅₅ is in very good agreement with results from the multi-histogram MC[135] and J-walking[173] simulations. The peak in the heat capacity curve is larger and sharper when anharmonicity is included. This change can be understood by considering a two level system, which is a reasonable model to describe the equilibrium between solid-like and liquid-like states of the cluster. The partition function can be written as $Z=\sum_i Z_i$, where the sum is over the two distinct regions of phase space. It follows that

$$U=-\left(\partial\ln Z\over\partial\beta\right)_{N,V} =-{1\... ...left(\partial Z_i\over\partial\beta\right)_{N,V}=\sum_ip_i U_i,$$ (3.40)

where p_i=Z_i/Z and $U_i=-(\partial\ln Z_i/\partial\beta)_{N,V}$. Hence,

 $$\begin{eqnarray} C_v&=&\left({\partial U\over\partial T}\right)_{N,V} =\sum_i\le... ...over\partial T}\right)_{N,V} \qquad\hbox{when}\quad p_1=p_2=\half,\end{eqnarray}$$

(3.41)


(3.42)

where $L_m=U_2(T_\half)-U_1(T_\half)$ and $C_{v,i}=(\partial U_i/\partial T)_{N,V}$.As

$$\left({\partial p_i\over\partial T}\right)_{N,V}={1\over kT^... ...\over\partial\beta}\right)_{N,V}\right)={p_i\over kT^2}(U_i-U),$$ (3.43)

$$\begin{eqnarray} \left({\partial p_2\over\partial T}\right)_{N,V}&=&{p_1p_2\over... ...\\ &=&{L_m\over 4kT_\half^2}\qquad \hbox{when}\quad p_1=p_2=\half.\end{eqnarray}$$

(3.44)


(3.45)

Substituting this result into equation 3.42 gives

$$C_v=\half(C_{v,1}+C_{v,2})+{L_m^2\over 4kT_\half^2}.$$ (3.46)

The greater anharmonicity of liquid-like minima causes L_m to be larger when anharmonicity is included. This change has two effects: it increases the area under the heat capacity peak, and causes the cluster to change between the two states more rapidly with temperature, decreasing the width of the peak.

  

Figure 3.11: Probabilities of LJ₅₅ being in regions I, II, III, IV and VI of the PES for (a) canonical, (b) microcanonical and (c) isopotential ensembles.

  

Figure 3.12: Plots for LJ₅₅ of (a) ${\cal P}(E)$ at T=0.2985, (b) A_L(E_c) at T=0.2985 (c) A_L(E_c) at a number of temperatures as labelled and (d) the free energy barrier heights (solid lines) and the free energy difference between solid-like and liquid-like states (dashed line) for A_L(E_c). In (a) and (b) the contributions of the different regions of the PES are also indicated. In (c) the results from our analytic partition function (solid line) are compared to simulation results (data points)[133].

From Figure 3.11 it can be seen that the probability of LJ₅₅ being in regions 2 and 3 is lowest in the canonical ensemble and highest in the isopotential ensemble. This result is due to the dependence of the partition function on the independent variables, T, E and E_c, of the three ensembles. In the canonical ensemble, Z is exponentially dependent on T, and is the most steeply varying of the three partition functions. In the microcanonical and isopotential ensembles, $\Omega$ and $\Omega_c$ are dependent on powers of E and E_c, respectively. As the exponent of E_c is lower than that for E, $\Omega_c$ is the slowest varying of the partition functions. For LJ₅₅ $n_{{\romannumeral1}}/n_{{\romannumeral2}- {\romannumeral3}} \gg n_{{\romannumeral2}-{\romannumeral3}}/ n_{{\romannumeral6}}$ (Table 3.9), where $n_{{\romannumeral1}}$, $n_{{\romannumeral2}-{\romannumeral3}}$ and $n_{{\romannumeral6}}$ are the number of minima in regions 1, 2-3 and 6, respectively. Consequently, as the partition function becomes more steeply varying, the defective icosahedra are seen for a narrower range of the independent variable, i.e. the contribution of the liquid-like states overtakes the defective icosahedra at an earlier stage in the generation of surface defects.

  

Figure 3.13: Plots for LJ₁₃ of (a) A_L(E_c) at T=0.2869, and (b) the free energy barrier heights (solid lines) and the free energy difference between solid-like and liquid-like states (dashed line) for A_L(E_c). In (a) the contributions of liquid-like and solid-like states are shown.

The Landau free energy, A_L(Q), is the free energy of a system for a particular value of an order parameter Q. It is defined by

$$A_L(Q)=A-kT\ln p_Q(Q),$$ (3.47)

where p_Q(Q) is the canonical probability distribution of the order parameter. The presence of two minima in A_L(Q) indicates that there are two thermodynamically stable states at this temperature. As A is independent of Q, knowledge of p_Q(Q) is sufficient to calculate the Landau free energy barrier. When E_c is used as an order parameter, simulations have shown that LJ₅₅ has two Landau free energy minima which correspond to the solid-like and liquid-like states[133]. Bimodality in the canonical energy distribution function, ${\cal P}(E)$, implies that there are two Landau free energy minima as a function of the order parameter, E. Figures 3.12 and 3.13 show that both LJ₁₃ and LJ₅₅ have a range of temperature for which two Landau free energy minima are observed. As can be seen from 3.12c the predicted free energy curves for LJ₅₅ are in very good agreement with those from simulation[133], confirming the accuracy of our analytic partition function. The states corresponding to the Mackay icosahedron and the defective icosahedra all contribute to the low potential energy minimum of A_L(E_c), but the free energies of regions IV and V are never low enough to affect the thermodynamics. Although the free energy barrier for LJ₁₃ is much smaller, it has been observed in some simulations[142,174].

The turning points in A_L(E_c) and ${\cal P}(E)$ correspond to points on the isopotential and the microcanonical caloric curves, respectively. This can be demonstrated for A_L(E_c) by solving the equation $(\partial A_L/ \partial E_c)_{N,V}=0$.The solution is $1/kT=(\partial \ln \Omega_c/\partial E_c)_{N,V}$, which is simply the definition of the isopotential temperature. The maxima in A_L(E_c) correspond to the segment of the caloric curve with negative slope, and the minima to segments with positive slope. Therefore the temperature range for which A_L(E_c) has two minima is the same as the depth of the Van der Waals loop in the isopotential caloric curve, as can be seen by comparing Figures 3.9b and 3.12d, and 3.10b and 3.13b.

Comparing the results for LJ₁₃ and LJ₅₅ the effects of size are apparent. For LJ₅₅ the melting transition is much more pronounced: it has a sharper peak in C_v, more pronounced Van der Waals loops in the microcanonical and isopotential caloric curves, a larger Landau free energy barrier, a larger temperature range for which solid-like and liquid-like clusters coexist, a larger latent heat per atom and a higher melting temperature. This behaviour is closer to the first-order phase transition of bulk matter.

  

Figure 3.14: Heat capacity of LJ₅₅ when only regions V and VI are included in the partition function.

One feature of the superposition method is that it allows one to restrict the regions of configuration space considered, and hence to look at thermodynamic properties which would be difficult to obtain by other methods. For example, we can study the thermodynamic effects of structural relaxation in the absence of a transition to the solid-like state, thus enabling us to study supercooled liquid-like and perhaps even glass-like clusters. This could not easily be done by simulation since for LJ₅₅ the cluster readily passes into the solid-like region of configuration space on cooling. Shown in Figure 3.14 is $C_{v,{\romannumeral5}-{\romannumeral6}}$, the heat capacity when only regions 5 and 6 are included in the partition function. It shows a clear peak at T=0.252 (0.84$T_\half$), which has a height above the background C_v that is about a seventh of that for melting. It resembles the heat capacity peak that is often seen at the bulk glass transition. Such an interpretation should be caveated by the fact that the peak is dependent on where one decides the low energy tail of the liquid-like minima finishes.

  

Figure 3.15: Plot of G_m(E_c) for LJ₅₅ as a function of the potential energy calculated using sample B.

Estimates of the number of minima in different regions of the PES can be obtained from the quench frequencies. Using equations 3.6, 3.7 and 3.9 an expression for g_s can be derived:  

$$g_s={\gamma(E')_s h_s\prod^{\kappa}_{j=1} \nu^s_j\over \gamm... ...{C^\kappa_l a_s^l(E'-E^0_s)^{\kappa+l-1}\over\Gamma(\kappa+l)}.$$ (3.48)

From g_s, the sum of minima, G_m(E_c), can easily be calculated

$$G_m(E_c)=\sum_{E^0_s<E_c}g_s.$$ (3.49)

This approach has been applied to LJ₅₅. From Figure 3.15 it can be seen that above $-270\epsilon$ the number of minima rises exponentially with the energy. The total number of minima in the energy range probed by the MD quenching (potential energies up to $-259\epsilon$) is $8.3\times10^{11}$.This value is much less than the total number of minima predicted by extrapolating Tsai and Jordan's results for small clusters and so suggests that the number of minima will continue to rise exponentially above $-259\epsilon$.The present method has also been used to estimate the number of minima in the energy ranges I-VI (Table 3.9). The number of minima in range II is known[108] to be 11. The results from the quench frequencies agree well with this figure.

 

Region number of minima

anharmonic harmonic

I 1 1

II 11.3 6.3

III 994.6 457.9

IV 81.5 2871.2

$$1.26\times10^5 1.22\times10^6$$ V

$$8.30\times10^{11} 3.11\times10^{12}$$ VI

The methods developed here should be applicable to other types of clusters, although, as in our examples, the most appropriate form for the anharmonic terms may depend on the type and size of cluster considered. The advantages of the superposition method are that once the partition function is obtained it can be used to calculate a whole gamut of thermodynamic properties analytically, and that it allows one to consider the roles of different regions of configuration space and so gain more physical insight into the system, particularly the relationship between structure and thermodynamics. (If one simply wants $\Omega(E)$ without these extras, one should use a simulation method, such as multi-histogram MC[141].) However, the method currently relies upon there being an energy at which all relevant regions of phase space are sampled. This condition does not hold for larger clusters, such as LJ₁₄₇. To extend the method to such systems an analogue of the multi-histogram technique would need to be developed, in which the information from quenching at different energies could be combined. The expressions for the density of states could also be used to calculate accurate rate constants using RRKM (Rice-Ramsperger-Kassel-Marcus) theory[166,175], and so aid quantitative elucidation of dynamic properties from a knowledge of the transition states on the PES.