5.3. Gibbs Ensemble and Phase Coexistence#
5.3.1. 1. Phase Coexistence#
Total system is closed, sum of two phases:
( E = E_A + E_B )
( S = S_A + S_B )
( V = V_A + V_B )
( N = N_A + N_B )
Since the total system is closed, ( E, V, N ) are constant:
( dE = 0 = dE_A + dE_B \Rightarrow dE_A = -dE_B )
( dV = 0 = dV_A + dV_B \Rightarrow dV_A = -dV_B )
( dN = 0 = dN_A + dN_B \Rightarrow dN_A = -dN_B )
Entropy differential: [ dS = dS_A + dS_B = \left( \frac{1}{T_A} - \frac{1}{T_B} \right) dE_A + \left( \frac{P_A}{T_A} - \frac{P_B}{T_B} \right) dV_A - \left( \frac{\mu_A}{T_A} - \frac{\mu_B}{T_B} \right) dN_A ]
At equilibrium, entropy is maximized ( \Rightarrow dS = 0 ), so:
( T_A = T_B )
( P_A = P_B )
( \mu_A = \mu_B )
5.3.2. 2. Simulating Phase Coexistence#
Direct (interfacial) coexistence: two phases in one box
GCMC + NPT: simulate ( S(P), \mu(P) ); find coexistence where ( P ) and ( \mu ) match
GCMC with histogram reweighting: accurate but expensive
5.3.3. 3. Gibbs Ensemble#
Idea: couple two systems to generate equal ( P ) and ( \mu )
Boxes represent bulk phases, no true interface
Probability of observing both systems in a state: [ \propto \frac{V_A^{N_A} V_B^{N_B}}{N_A! N_B!} \exp[-\beta(U_A + U_B)] ]
5.3.3.1. Move Set#
Displacement (NVT move): [ P_{\text{accept}} = \min\left(1, e^{-\beta \Delta U}\right) ]
Volume change (move ( \Delta V ) from box B to A): [ P_{\text{accept}} = \min\left(1, \left(\frac{V_A + \Delta V}{V_A}\right)^{N_A} \left(\frac{V_B - \Delta V}{V_B}\right)^{N_B} e^{-\beta \Delta U}\right) ]
Particle transfer (move particle from B to A): [ P_{\text{accept}} = \min\left(1, \frac{V_A}{V_B} \frac{N_B}{N_A + 1} e^{-\beta \Delta U}\right) ]
Typical move set: 100 displacements : 1 volume change : 200 transfers
5.3.4. 4. Gibbs–Duhem Integration#
If one coexistence point is known, use thermodynamics to integrate: [ -S_A dT + V_A dP - N_A d\mu_A = 0 ] [ -S_B dT + V_B dP - N_B d\mu_B = 0 ]
Subtracting: [ d(\mu_B - \mu_A) = -(S_B - S_A) dT + (V_B - V_A) dP ]
At coexistence: [ \frac{dP}{dT} = \frac{S_B - S_A}{V_B - V_A} = \frac{\Delta H}{T \Delta V} ]
This is the Clapeyron equation (or Clausius–Clapeyron if ideal gas)
Most useful for solids where other techniques are harder or fail