# 6. Surface Tension

The interfacial tension between bulk liquid and vapor phases, that is the surface tension, was determined using two different methodologies, molecular dynamics simulation of the liquid-vapor interface and the combination of finite-size scaling and grand-canonical Monte Carlo. [1] Interfacial properties, unlike bulk fluid properties, are sensitive to the details of the actual potential used in the simulation due to the inhomogeneous nature of the interface. It is therefore crucial that long range corrections be treated correctly.

6a. Explicit interfacial simulations:

Molecular dynamics simulations of the liquid-vapor interface were performed in a simulation cell of dimensions Lx × Ly × Lz (where Lx = Ly < Lz).  Long range corrections to the force, energy, and pressure tensor were applied at every time step using the density-profile-based scheme introduced by Janecek. [2] Those who are interested are referred to the Ref. [2] for the mathematical expressions and their derivations. The surface tension g was calculated by taking the difference in the diagonal components of the pressure tensor

g = [ 2Pzz - ( Pxx + Pyy) ] / (2A)

where Pkk is the pressure tensor component and A is the total interfacial area (A=2LxLy).  The surface tension was determined over the reduced temperature range of 0.70 to 1.10 using cutoff distances of rc/σ = 2.5, 4.0, and 6.0. Uncertainties represent the standard deviation of at least three separate simulations. In all cases, the properties of the coexisting fluid phases were in excellent agreement with those reported in other pages on this website. Important details of the simulations are given in the table below:

 ENSEMBLE:: Micocanonical (NVE) INTEGRATOR: Velocity-Verlet EQUILIBRATION: Long enough to stabilize interface PRODUCTION: 1000 t* TIME STEP: 0.01 t* N: 1600 ( Lx, Ly, Lz ) ( 12.1σ, 12.1σ, 40σ ) Δz 0.1σ

Several simulations with N=2500 and box dimensions 11σ × 11σ × 66σ were performed to investigate system size effects. Within the uncertainties, none were found. An initial configuration was generated by placing a liquid slab in the middle of an empty tetragaonal simulation box. As long as the overall density of the system is between that of the saturated vapor and liquid, then two liquid-vapor interfaces will form. Of crucial importance in an interfacial simulation is the formation and stabilization of the interface itself. For this reason, a predetermined equilibration period throughout all the runs was not used. Rather, the stability of the interface was explicitly checked for each run. In the table below, the mean surface tension and standard deviation from at least three molecular dynamics simulations are reported. In all cases, the densities of the coexisting phases were in agreement with saturation data provided elsewhere on this website.

 T* rc/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 µsat. From this distribution, an apparent surface tension L can be calculated

where FL is the height of the free energy barrier separating the two coexisting phases. Since cubic simulation boxes were used here, V = L3. Because FL 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 c1 and c2 are unknown constants, β is 1/(kBT), and γ is the thermodynamic surface tension. For large L, the apparent surface tension is a linear function of the scaling variable (ln L)/L2, 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 L9σ.

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 kBT/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, rc = 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.

 rc L/σ γL* +/- 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 U6 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 Bjk = -BjBk and

In the above expressions, ri is the position of particle i, RL 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 rc.

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)(kx,ky,kz) 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 hmax. In practice evaluation of the lattice summation requires specification of three parameters, rc ν, and hmax = (2π/L)kmax.

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.

 rc L/σ α kmax γL* +/- 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:

 rc L/σ LRC Scheme α kmax γL* +/- 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:

 rc L/σ LRC Scheme α kmax γL* +/- 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].

## References

[1] J. Errington, Phys. Rev. E 67, 012102 (2003).

[2] J. Janecek, J. Chem. Phys B 110, 6264 (2006).

[3] K. Binder, Phys. Rev. A 25, 1699 (1982).

[4] N. Karasawa and W. A. Goddard III, J. Phys. Chem. 93, 7320 (1989).

[5] J. Lopez-Lemus and J. Alejandre, Mol. Phys. 100, 2983 (2002).

[6] V. K. Shen and D. W. Siderius, J. Chem. Phys., 140, 244106, (2014).

[7] V. K. Shen and J. R. Errington, J. Phys. Chem. B 108, 19595, (2004).

[8] V. K. Shen and J. R. Errington, J. Chem. Phys. 122, 064508, (2005).

[9] V. K. Shen, R. D. Mountain, and J. R. Errington, J. Phys. Chem. B 111, 6198, (2007).

[10] D. W. Siderius and V. K. Shen, J. Phys. Chem. 117, 5681, (2013).