Lennard-Jones fluid simulation website
ENSEMBLE:: | Micocanonical (NVE) |
INTEGRATOR: | Velocity-Verlet |
EQUILIBRATION: | Long enough to stabilize interface |
PRODUCTION: | 1000 t* |
TIME STEP: | 0.01 t* |
N: | 1600 |
( L_{x}, L_{y}, L_{z} ) | ( 12.1σ, 12.1σ, 40σ ) |
Δz | 0.1σ |
T* | r_{c}/σ | γ* | +/- | |
0.700 | 2.5 | 1.1356 | 0.022 | |
0.700 | 4.0 | 1.1367 | 0.006 | |
0.800 | 2.5 | 0.9043 | 0.002 | |
0.800 | 4.0 | 0.9057 | 0.003 | |
0.816 | 6.0 | 0.8720 | 0.014 | |
0.835 | 6.0 | 0.8260 | 0.013 | |
0.850 | 2.5 | 0.8175 | 0.019 | |
0.850 | 4.0 | 0.8130 | 0.028 | |
0.898 | 6.0 | 0.7026 | 0.013 | |
0.963 | 6.0 | 0.5743 | 0.011 | |
1.000 | 2.5 | 0.5037 | 0.012 | |
1.000 | 4.0 | 0.4993 | 0.012 | |
1.030 | 6.0 | 0.4554 | 0.011 | |
1.100 | 2.5 | 0.3107 | 0.018 | |
1.100 | 4.0 | 0.2973 | 0.013 | |
1.100 | 6.0 | 0.3050 | 0.006 |
6b. Finite-Size Scaling and Transition-Matrix Monte Carlo
An alternative route to interfacial tension is the combination of finite-size scaling and transition-matrix Monte Carlo. [1] This methodology uses grand-canonical transition-matrix Monte Carlo (GC-TMMC) [6-10] to calculate apparent surface tensions at various system sizes and then employs finite-size scaling arguments to extrapolate the data to the thermodynamic limit. As noted on a previous page, GC-TMMC provides the particle number probability distribution, that is the the probability of observing N particles in a simulation box of volume V at temperature T and chemical potential µ. The distribution of interest here is the one at which the chemical potential corresponds to liquid-vapor phase coexistence µ_{sat}. From this distribution, an apparent surface tension L can be calculated
where F_{L} is the height of the free energy barrier separating the two coexisting phases. Since cubic simulation boxes were used here, V = L^{3}. Because F_{L} is system-size-dependent, so too is the apparent surface tension. The thermodynamic surface tension is obtained by applying the following scaling proposed by Binder [3]
where c_{1} and c_{2} are unknown constants, β is 1/(k_{B}T), and γ_{∞} is the thermodynamic surface tension. For large L, the apparent surface tension is a linear function of the scaling variable (ln L)/L^{2}, where the y-intercept corresponds to the thermodynamic surface tension. For the Lennard-Jones fluid, the linear scaling regime appears (empirically) to be applicable to for L ≥ 9σ.
The surface tension of the Lennard-Jones fluid was calculated using the finite-size scaling and grand-canonical transition-matrix Monte Carlo (FSS/GC-TMMC) methodology. Run parameters are summarized in the table below:
METHOD: | GC-TMMC |
k_{B}T/ε | 0.70, 0.85, and 1.10 |
Prob. of Disp. Move: | 0.70 |
Prob, of Ins/Del. Move: | 0.30 |
Biasing Function Update Frequency |
1 million trials |
Simulation Length: | 1 - 20 billion MC trials |
Effect of Truncation Using Standard Long Range Corrections:
At a reduced temperature of 0.85, standard long range corrections (sLRC's) were used, and three cutoff distances were compared, r_{c} = 3σ, 4σ, and L/2. Phase coexistence properties obtained for each cutoff were in agreement with data presented elsewhere. Apparent and thermodynamic (bold) surface tension data are given below, along with uncertainties which represent the standard deviation from at least 3 separate simulations per point.
r_{c} | L/σ | γ_{L}* | +/- |
3.0σ | 7 | 0.712 | 0.0003 |
3.0σ | 8 | 0.755 | 0.0005 |
3.0σ | 9 | 0.785 | 0.0009 |
3.0σ | 10 | 0.812 | 0.0004 |
3.0σ | 12 | 0.859 | 0.0007 |
3.0σ | ∞ | 0.986 | 0.0081 |
4.0σ | 8 | 0.747 | 0.0001 |
4.0σ | 9 | 0.771 | 0.0006 |
4.0σ | 10 | 0.786 | 0.0009 |
4.0σ | 12 | 0.808 | 0.0008 |
4.0σ | ∞ | 0.873 | 0.0072 |
L/2 | 7 | 0.712 | 0.0025 |
L/2 | 8 | 0.750 | 0.0009 |
L/2 | 9 | 0.771 | 0.0008 |
L/2 | 10 | 0.783 | 0.0016 |
L/2 | 12 | 0.795 | 0.0023 |
L/2 | 14 | 0.804 | 0.0013 |
L/2 | ∞ | 0.837 | 0.0020 |
An alternative treatment of long-range correction is the lattice summation. [4,5] While this is a more robust treatment, it is computationally expensive. The dominant contribution to the Lennard-Jones pair potential at large separations is the dispersive (inverse sixth power) term , and therefore, one focuses on the long range corrections to this contribution. Applying the lattice summation to the dispersive term of the potential, the total dispersion interaction in a system U_{6} can be written as the sum a real-space term and reciprocal space term
.
The real-space contribution to the total dispersion energy is
where B_{jk} = -B_{j}B_{k} and
In the above expressions, r_{i} is the position of particle i, R_{L} is a lattice translation vector, and ν is an adjustable parameter controlling the range of the real-space interaction. In the literature, ν is commonly expressed as α/L where α is a dimensionless parameter and L is the length of the simulation box. The prime in the summation indicates that terms for which j=k when L=0 are excluded. While the summation is formally an infinite sum, when evaluated in practice, it is truncated at L=0 and only to those pairs of particles separated by a distance less than a cutoff r_{c}.
The reciprocal space contribution to the dispersion interaction is
where S(h) is the structure factor
and the function J(h) is
where h = (2π/L)(k_{x},k_{y},k_{z}) is a reciprocal lattice vector of magnitude h, and V is the system volume. The summation over h is truncated at some maximum lattice vector h_{max}. In practice evaluation of the lattice summation requires specification of three parameters, r_{c}, ν, and h_{max }= (2π/L)k_{max}.
At a reduced temperature of 0.85, the surface tension of the Lennard-Jones fluid has evaluated using the lattice summation within the FSS/GC-TMMC methodology. Results and lattice sum parameters are summarized below. Other GC-TMMC run parameters are identical to those listed above.
r_{c} | L/σ | α | k_{max} | γ_{L}* | +/- | |
3.0σ | 7 | 5.6 | 6 | 0.7134 | 0.00017 | |
3.0σ | 8 | 6.4 | 7 | 0.7598 | 0.00032 | |
3.0σ | 9 | 7.2 | 8 | 0.7697 | 0.00031 | |
3.0σ | 10 | 8.1 | 9 | 0.7825 | 0.00026 | |
3.0σ | 12 | 9.8 | 11 | 0.7948 | 0.00034 | |
3.0σ | 14 | 12 | 14 | 0.8039 | 0.00040 | |
3.0σ | ∞ | → | 0.8378 | 0.00094 | ||
L/2 | 7 | 5.0 | 5 | 0.7142 | 0.00017 | |
L/2 | 8 | 4.4 | 5 | 0.7506 | 0.00032 | |
L/2 | 9 | 4.2 | 4 | 0.7701 | 0.00031 | |
L/2 | 10 | 4.2 | 4 | 0.7830 | 0.00028 | |
L/2 | 12 | 4.2 | 4 | 0.7970 | 0.00032 | |
L/2 | 14 | 4.2 | 4 | 0.8043 | 0.00034 | |
L/2 | ∞ | → | 0.8394 | 0.00084 |
Below are results at reduced temperatures of 1.10 and 0.70 using standard
long range corrections (sLRC) and lattice sums (LS).
T*=1.10:
r_{c} | L/σ | LRC Scheme | α | k_{max} | γ_{L}* | +/- |
L/2 | 7 | sLRC | 0.2027 | 0.00117 | ||
L/2 | 8 | sLRC | 0.2186 | 0.00051 | ||
L/2 | 9 | sLRC | 0.2337 | 0.00080 | ||
L/2 | 10 | sLRC | 0.2483 | 0.00062 | ||
L/2 | 12 | sLRC | 0.2732 | 0.00078 | ||
L/2 | 14 | sLRC | 0.2888 | 0.00099 | ||
L/2 | ∞ | → | 0.343 | 0.0021 | ||
3σ | 7 | LS | 6.0 | 7 | 0.2023 | 0.00055 |
3σ | 8 | LS | 6.5 | 7 | 0.2175 | 0.00042 |
3σ | 9 | LS | 7.5 | 9 | 0.2334 | 0.00087 |
3σ | 10 | LS | 8.5 | 10 | 0.2481 | 0.00047 |
3σ | 12 | LS | 10 | 12 | 0.2721 | 0.00074 |
3σ | 14 | LS | 12 | 14 | 0.2877 | 0.00096 |
3σ | ∞ | → | 0.340 | 0.0025 | ||
L/2 | 7 | LS | 5.5 | 5 | 0.2025 | 0.00025 |
L/2 | 8 | LS | 4.5 | 5 | 0.2175 | 0.00011 |
L/2 | 9 | LS | 4.5 | 5 | 0.2337 | 0.00014 |
L/2 | 10 | LS | 4.5 | 5 | 0.2483 | 0.00048 |
L/2 | 12 | LS | 4.5 | 5 | 0.2728 | 0.00066 |
L/2 | 14 | LS | 4.5 | 5 | 0.2886 | 0.00110 |
L/2 | ∞ | → | 0.340 | 0.0013 |
T*=0.70:
r_{c} | L/σ | LRC Scheme | α | k_{max} | γ_{L}* | +/- |
L/2 | 7 | sLRC | 1.072 | 0.001 | ||
L/2 | 8 | sLRC | 1.100 | 0.003 | ||
L/2 | 9 | sLRC | 1.111 | 0.002 | ||
L/2 | 10 | sLRC | 1.121 | 0.004 | ||
L/2 | 12 | sLRC | 1.138 | 0.003 | ||
L/2 | ∞ | → | 1.181 | 0.010 | ||
3σ | 7 | LS | 6.0 | 7 | 1.077 | 0.001 |
3σ | 8 | LS | 6.8 | 8 | 1.100 | 0.001 |
3σ | 9 | LS | 7.7 | 9 | 1.115 | 0.001 |
3σ | 10 | LS | 8.65 | 10 | 1.125 | 0.002 |
3σ | 12 | LS | 10.3 | 12 | 1.135 | 0.002 |
3σ | ∞ | → | 1.181 | 0.001 | ||
L/2 | 7 | LS | 5.0 | 6 | 1.077 | 0.001 |
L/2 | 8 | LS | 4.75 | 5 | 1.100 | 0.001 |
L/2 | 9 | LS | 4.65 | 5 | 1.114 | 0.002 |
L/2 | 10 | LS | 4.50 | 5 | 1.124 | 0.001 |
L/2 | 12 | LS | 4.50 | 5 | 1.135 | 0.003 |
L/2 | ∞ | → | 1.175 | 0.008 |
Lattice sum parameters were chosen such that the estimated error in the dimensionless dispersion energy per particle was no more than 0.005. Expressions for this type of estimate can be found in Ref. [4].
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