Binary mixture bond probability calculator

Thermodynamic parameters

Solute fraction (x_B) 0.50

x_A = 0.99 x_B = 0.50 x_B = 0.99

Temperature (T) 1000 K

200 K 1000 K 2500 K

Enthalpy magnitude (|Ω|) 12.0 kJ/mol

1 kJ/mol 12 kJ/mol 40 kJ/mol

Coordination number (z) 12

Critical behaviour (equiatomic x = 0.5)

Quasi-chemical T_c^QCT: -- K

Bragg-Williams T_c^BW: -- K

Phase state (clustering): Homogeneous

Calculated bond probabilities across three thermodynamic regimes P_AA + P_BB + P_AB = 1

Regime (i)

Random mixture

Ideal spatial distribution with zero enthalpy of mixing (Ω = 0).

P(A-A) 0.2500

P(B-B) 0.2500

P(A-B) 0.5000

Warren-Cowley α_1: 0.0000

Regime (ii)

Positive enthalpy

Clustering and demixing preference (+Ω = like bonds favoured).

P(A-A) 0.2850

P(B-B) 0.2850

P(A-B) 0.4300

Warren-Cowley α_1: +0.1400

Regime (iii)

Negative enthalpy

Short-range ordering (-Ω = unlike A-B bonds favoured).

P(A-A) 0.2050

P(B-B) 0.2050

P(A-B) 0.5900

Warren-Cowley α_1: -0.1800

Theoretical formulation and equations

Quasi-chemical equilibrium

The nearest-neighbour exchange reaction between pairs satisfies the equilibrium mass action relation:

(A-A) + (B-B) ⇌ 2(A-B)

P_AB^2 / (P_AA · P_BB) = 4 · exp(-2Ω / (z R T))

Where Ω is the regular solution enthalpy of mixing parameter, z is coordination number, and R is the molar gas constant.

Pair probabilities solution

Subject to boundary constraints 2P_AA + P_AB = 2x_A and 2P_BB + P_AB = 2x_B, the exact solution yields:

β = √[ 1 + 4 x_A x_B (exp(2Ω / (z R T)) - 1) ]

P_AB = 4 x_A x_B / (β + 1)

P_AA = x_A - 0.5 P_AB, P_BB = x_B - 0.5 P_AB

Short-range order and critical limits

The Warren-Cowley first-shell short-range order parameter α_1 and critical demixing temperatures are:

α_1 = 1 - P_AB / (2 x_A x_B) = (β - 1) / (β + 1)

T_c^BW = Ω / (2 R)

T_c^QCT = Ω / [ z R ln( z / (z - 2) ) ]