IMPROVED METHOD FOR CALCULATING SURFACE TENSION AND APPLICATION TO WATER 1
2
Hong Peng , Anh V Nguyen and Greg R Birkett*
School of Chemical Engineering, The University of Queensland Brisbane, Queensland 4072, Australia 1 2
Author’s Email:
[email protected]
Author’s Email:
[email protected]
*Corresponding author:
[email protected] ABSTRACT We have proposed an improved method for calculating surface tension of liquid-vapour interface based on the thermodynamic route. This method combines the central difference scheme used in the test-area method and the Bennett weighting factor. This improved method shows faster convergence than the test area method as the weighting factor reduces the asymmetric bias of two perturbations. In addition, this method is used to calculate the surface tension of water-methanol mixtures. The values obtained agree with experimental data and other simulation data. This method provides the new surface tension calculation procedure which is independent with on evaluation of complex pressure tensor when using the classical Kirkwood and Buff method. INTRODUCTION Surface tension is central to many daily and industrial activities, ranging from washing with soap, foaming in kitchen sinks to bubble coalescence in chemical and biological reactors. The measurement of surface tension is now well established. However, the determination of surface tension by molecular simulation still remains a challenge to scientists. There are two general routes employed to determine surface tension by molecular simulation. The first and most widely used method is based on the mechanical definition of surface tension. The mechanical route evaluates surface tension, γ, using the normal, PN, and tangential, PT, pressure tensors and is given in equation(1).
γ =∫
∞
−∞
( PN − PT ) dz
(1)
This equation was first introduced by Bakker in 1928 (Bakker, 1928) but is often referred to as the Kirkwood and Buff method (Kirkwood and Buff, 1949). The mechanical route has proven useful for a number of applications but the evaluation of the pressure tensors can be cumbersome and is a large overhead in Monte Carlo simulations. The second route is to use the thermodynamic definition of surface tension. In the canonical ensemble (where the temperature, T, volume, V, and number of particles, N, are held constant), the thermodynamic definition is given in equation (2) along with the finite difference approximation.
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H. Peng, AV. Nguyen, GR. Birkett
∂F
∆F
γ = = lim ∂A NVT ∆A→0 ∆A NVT
(2)
where F is the Helmholtz free energy and A is the interfacial area of the system. The accuracy of the finite difference approximation in equation (2) comes down to two factors: 1. the accuracy of the finite difference approximation for a given system and step size and 2. The accuracy of the method used to evaluate the free energy difference between the states (which differ in this case by their interfacial surface area). The difference in the free energy between any two systems can be calculated using the partition functions, Q0 and Q1, for systems 0 and 1, respectively, as per equation(3).
∆F0→1 = F1 − F0 =
Q ln 0 β Q1 1
(3)
where β is given by 1/ ( kBT ) . The canonical partition function is dependent on the configurational energy, U, and the configurational space, rN, and is described as Q=
1 exp ( − β U ) dr N N !Λ 3 N ∫
(4)
where Λ is the Thermal de Broglie wavelength. Various methods can be used for calculating the free energy difference in equation(4). For example, the Bennett method (BM) (Bennett, 1976) the free-energy difference is obtained by considering two equilibrated systems, denoted as 0 and 1, with the same values for N, V and T, but different interfacial areas, A0 and A1, respectively. A perturbation in interfacial area for system 0 from A0 to A1, whilst maintaining all values for N, V and T, is denoted as 0 → 1’ and referred to as a forward perturbation. And for system 1, a change from A1 to A0, is denoted as 1 → 0’ and referred to as the reverse perturbation. The difference in configurational energy between system 0, U0, and the perturbed system 1’, U1’, is the change in the potential energy due to the perturbation, ∆U0→1’. In a canonical ensemble, Bennett (Bennett, 1976) showed that the ratio of the partition functions can be calculated using equation (5). W ( r N ) exp ( − β U 0' ) Q0 = Q1 W ( r N ) exp ( − β U1' )
1
(5)
0
where the angle brackets with the subscripts 0 and 1 describe the canonical ensemble averages of the perturbation steps for equilibrated systems 0 and 1, respectively, and W r N is a weighting function of some description. A prudent choice in the weighting
( )
function can improve the accuracy of the calculation in equation (5) by maximizing the weighting of the calculation in the area of configuration space where the forward and reverse perturbations overlap. Bennett investigated what form of the weighting function was optimal and found that using the Fermi-Dirac function, f(x) = (1+exp(x))-1 together
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H. Peng, AV. Nguyen, GR. Birkett with a shifted potential was optimal. Implementing this in equation(5) , gives equation(6) which is the basis of the Bennett method.
∆F0→1 =
f ( β∆U1→0' + C ) 1 C Q 1 ln 0 = ln + β Q1 β f ( β ∆U 0→1' − C ) 0 β 1
(6)
where the and C is an adjustable parameter used to maximize the overlap of the forward and revere perturbations. The optimal value for C requires knowledge of the overlap between the two simulations and is given by equation(7). Q C = ln 0 (7) Q1 We cannot know the result of equation (7) before we start a simulation, however, C can be estimated with increasing accuracy during the simulation using equation(7). It may seem inconsistent to use an equation that requires C, to calculate C, it must be remembered that a property of equation(7) is that it will converge for any value of C. A suitable value for C decreases convergence time and better estimates of C will be calculated as the simulation progresses. The Bennett method has been rarely used to evaluate the surface tension due to the need of two more simulations (Errington and Kofke, 2007) although this procedure has been widely used to calculate other properties in simulations (Lu et al., 2003, Harismiadis et al., 1996, Lu and Kofke, 2001). Another method for estimating the free energy difference is the Zwanzig theory for high-temperature perturbation, as given in equation(8) (Zwanzig, 1954). This equation is for a single system 0 which is perturbed to system 1. 1 ∆F0→1 = − ln exp ( − β∆U ) 0 (8)
β
The test-area method (Gloor et al., 2005), uses equation (8) to calculate the free energy difference for perturbations that increase (1) and decrease (-1) the interfacial area from a specific reference system 0 by ∆A and −∆A , respectively. The resulting free-energy differences, ∆F0→1and∆F0→-1, are averaged and subbed into 2 as the estimate for ∆F to give equation(9).
γ=
∆F0→1 − ∆F0→−1 2∆A
(9)
The test-area method has been used to calculate the surface tension of simple Lenard-Jones (LJ) fluids (Gloor et al., 2005, Errington and Kofke, 2007), water (Vega and Miguel, 2007, Ghoufi et al., 2008) and related mixtures (Bahadur et al., 2007). It is noted that the test-area scheme only requires a single simulation to determine surface tension. However, under some special conditions the test-area method can provide less accurate surface tension than the Bennett method. Examples include the cases where asymmetric bias is inherent in the simulation of free-energy differences (e.g., the square-well fluid) (Errington and Kofke, 2007). This problem with averaging forward and reverse free energy differences has also been discussed by Lu et al (Errington and Kofke, 2007, Lu et al., 2003) who recommend that this method should be avoided due to the unpredictable consequences of combing the free energy changes this way. In this paper, we propose an improved method by using the advantages of the Bennett method (improved accuracy and precision) and the Test Area method (requires Page 3
H. Peng, AV. Nguyen, GR. Birkett a single simulation). The proposed scheme is shown schematically in Table 1. The key assumption to the proposed method is that ∆F1→0’ = ∆F0→-1 such that the two perturbations have equivalent weights in the Bennett method. Substituting this into equation (6) and using the slightly neater (but equivalent) weighting factor used by Lu et al. (Lu et al., 2003), gives equation (10), which is used in this study to calculate surface tension. Table 1 The Bennett method, the test-area method and the improved method proposed in this paper. The state (-1) presents the negative (compression) perturbation, while the state (1) is the positive (expansion) perturbation, all relative to the reference state (0). Dash line square reflects the virtual perturbation state. W represents the weighting function.
Method Test area method
Perturbation Scheme
-1
0
0
Bennett method
0 Improved method
w
-1
w w
0
1
1 1 w
1
A single simulation is performed in which a reference system 0 equilibrates and then samples configurations relevant to an interfacial area A0. An area change, ∆A, is then performed to generate a virtual perturbed state (System 1, expansion of interfacial area: A1 = A0 (1+ ∆A), or System -1, compression of interfacial area: A-1 = A0 (1-∆A). The perturbation in area is made along x and y axis by equal ratio, L( x , y )(1,−1) = L( x, y )( 0) 1 ± 0.5∆A . To keep the volume constant, the corresponding change
along z axis is, L( z )( −1,1) = L( z )( 0 ) (1 ± 0.5∆A ) .The corresponding energy differences ∆U0→1
and ∆U0→-1 are obtained by perturbation steps are repeated many times through the simulation to collect the ensemble averages in equation (10) to calculated the surface tension.
W ( ∆U ) exp ( − β∆U / 2 ) 1 ln β∆A W ( ∆U ) exp ( − β∆U / 2 )
→
1 -
0
(10) → The weighting function W (∆U) in the equation (10) is described by a Gaussianlike hyperbolic secant function, i.e., 1 0
γ=
W ( ∆U ) = cosh ( β∆U − C ) 2
(11)
Applying the Bennett equation (6) for determining the free-energy differences of the forward (expansion) and reverse (compression) perturbations, ∆F0→1 and ∆F0→−1 , and employing the central difference equation, i.e. equation (9) of the test-area method give. Page 4
H. Peng, AV. Nguyen, GR. Birkett The required Bennett factor can be determined at the crossing point of the probability distribution functions of free-energy perturbations for the forward and reverse directions in the Bennett method. The crossing point is selected to gain the greatest overlap of samplings and maximize the weighting on the configurational energy differences, giving
→
1 -
0
(12)
1 0
W ( ∆U ) exp ( − β∆U / 2 ) C = ln W ( ∆U ) exp ( − β∆U / 2 )
→
Equation (10) in conjunction with equations (11) and (12) was used to calculate surface tension. The improved method and the test-area method are used to evaluate the surface tension of water-vapour and water-methanol mixtures using an in-house Monte Carlo (MC) simulation code.
METHODS Interaction Energy Models Isotropic site-site pair-wise potentials were used in this study. The interaction potential between two molecules is: Α
Β
φij = ∑∑ a =1 b =1
qia q bj 4πε 0 rijab
C
D
+ ∑∑ 4ε c =1 d =1
cd ij
σ cd ij cd r ij
12
σ ijcd − cd rij
6
(13)
where ϕij is the interaction energy between fluid molecules i and j, Α and Β are the number of charges on the molecules i and j, respectively, C and D are the number of LJ sites on the molecules i and j, respectively, ε0 is the permittivity of a vacuum, rijab is the separation between the charge a on molecule i and the charge b on molecule j having charges qia and qjb, respectively and rijcd is the separation between the LJ site c on molecule i and the LJ site d on molecule j with combined LJ well depth of εijcd and a combined LJ collision diameter of σijcd for the two sites. The combined LJ well-depth and collision diameter of two sites is calculated based on the Lorentz-Berthelot mixing rules (Potoff et al., 1999). For water models, the TIP4P/2005 (Abascal and Vega, 2005) model was mainly used. It consists of one LJ site located at the oxygen and three coulomb sites; two positive charges located at the two hydrogen sites and a negative charge placed along the bisector of the H-O-H angle (the ‘M’ site). The distance of OH bond is 0.9572Å and the angle of H-O-H is 104.52°. Both LJ and electrostatic interactions were truncated at 1.3 nm. No long-range corrections were used for either LJ or electrostatic interactions. The OPLS model described by Jorgensen (Jorgensen et al., 1993) is used to simulate the methanol molecules. The mole of fraction of methanol in the liquid water varied from 0% to 100% with interval value of 25%. The ratio of number of molecules is listed in Table 2 (Chang and Dang, 2005). The temperature used in simulation is 300K.
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H. Peng, AV. Nguyen, GR. Birkett Table 2. Simulation Mole fraction of water-methanol mixture Fraction (%)
0
25
50
75
100
Nwater
1000
600
350
150
0
Nmethanol
0
200
350
450
500
Simulation Procedure The simulations were performed in the canonical NVT ensemble by using an in-house Monte Carlo code. For each simulation, a two-stage run was used to obtain the properties of the system. Firstly, a cubic box of the bulk liquid was run and equilibrated by at least 200,000 cycles (each cycle consists of number of molecules moves). The second stage was specially designed to determine the interface properties; surface tension and the local density distribution. The equilibrium bulk box of liquid from the last run of the first stage was used for the initial configuration. To make the interface of vapor-liquid, two empty boxes with the same geometry, were added on each side of bulk box of liquid along z-axis (Figure 1). This new system with two interfaces is equilibrated for 200,000 cycles following by 400,000 cycles to produce constant (reliable) values for surface tension. The last 100,000 cycles are divided 5 blocks to obtain the mean values and standard deviation for each simulation condition.
x z y
L
L
L
Figure 1. A schematic of the simulation box at the second computation stage with the liquid cubic slab sandwiched between two vapor phase cubic boxes.
RESULTS Convergence of thermodynamic route method There is the issue in simulations of inhomogeneous system is whether the runs have been performed over a sufficiently long period time to obtain reliable value of surface tension. From our simulation in Figure 2, it is shown that the value at the first 100,000 cycles fluctuated followed a slow convergence trend which ranged from 200,000 cycles to 400,000 cycles. The surface tension profile by improved method proposed shows the much earlier convergence just at near 200,000 cycles than the value by test-area method which converged near the end of simulation around 400,000 cycles.
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H. Peng, AV. Nguyen, GR. Birkett 80 Improved method Test area
Surface tension[nN/ m]
60
40
20
0 0
10
20
30
40 x 104
Monte Carlo Cycles
Figure 2. Surface tension versus Monte Carlo simulation cycles as obtained by the testarea method and the improved method developed in this paper.
Effect of box size on surface tension In the molecular simulation, box size is the critical parameter on the accuracy of surface tension. We vary the simulation box size from 2.7 nm to 3.5 nm with the number of particles ranging from 656 to 1428 at 300K. As shown in Figure 3, for the box size more than 3.0nm, the average values of surface tension are similar and close with the reported value (Vega and Miguel, 2007). When box size is larger than 30nm, the effect of system size is limited.
75
Surface tension [mN/ m]
70 65 60 55 50 45 2.50
2.75
3.00
3.25
3.50
3.75
Box size L[nm] Figure 3. Surface tension versus box sizes as obtained by the improved method developed in this paper. Page 7
H. Peng, AV. Nguyen, GR. Birkett
Surface tension of water-vapour phase To verify our approach could be applied for water, improved method is used to calculate its surface tension at two temperatures. The simulated values of surface tension are summarized in Table 3 in comparison with test-area method and the reported values (Vega and Miguel, 2007) . In our simulation, no long range correction for both LJ and CL interactions is applied but the simulation data is reasonable consistent with the values with long range correction. Maybe, the large cutoff distance in the simulation ensures that the effect of tail correction is small. The relative errors of surface tension by improved method for both temperatures are smaller than the value from test area method. This could be attributed to the use of weight function optimizing the two perturbation ways. Table 3. Simulation value of surface tension Surface Tension (mN/m) T (K)
Improved method
Test-area method
Experiments
300
64.70±5.43
60.22±6.82
71.73
450
37.2±0.85
36.1±1.75
42.88
Surface tension of water-methanol mixtures We also use the improved method to calculate the surface tension of water methanol mixtures. The mole of fraction of methanol in the liquid water varied from 0% to 100% with interval value of 25%. The calculated surface tension is shown in Figure 4 compared with experimental value (Vazquez et al., 1995) and the simulation values by Chang and Dang (Chang and Dang, 2005). The calculated value is expected higher than experimental value and close to the MD model except that pure water value which is lower than both experimental and MD simulation data. As we discussed before, the rigid non-polarization water model only could provide the value lower than experimental data. In general, the overall dependence of the surface tension on mixture composition agrees well with the experimental trend. When Liquid water is mixed with the amount of methanol molecular, the surface tension of mixture decreased with the elevated the mole fraction of methanol. It may be caused by the strong tendency for methanol to concentrate at the interface.
Surface tension [mN/ m]
100 Experiment
80
This study 60
MD
40 20 0 0
20 40 60 80 Methanol concentration in water [%]
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100
H. Peng, AV. Nguyen, GR. Birkett Figure 4. Surface tension of water-methanol mixtures at 300 K. Experimental data are shown with continuous line (Vazquez et al., 1995) . MD results are taken from (Chang and Dang, 2005).
CONCLUSION Here, we proposed an improved method on evaluation of surface tension of liquidvapour interface based on thermodynamic route. This method combined the central difference scheme used the test-area method and the weighting factor used in the Bennett technique. This improved method shows faster convergence than the test area method as the weighting factor reduces the asymmetric bias of two perturbations. The value of surface tension of water-vapour obtained by this improved method agrees with the expected values within the small uncertainty. In addition, this method is also used to evaluate the surface tension of water-methanol mixtures and obtained the close value with experimental data and other simulation data.
REFERENCES ABASCAL, J. L. F. & VEGA, C. (2005) A general purpose model for the condensed phases of water: TIP4P/2005. The Journal of Chemical Physics, 123, 234505. BAHADUR, R., RUSSELL, L. M. & ALAVI, S. (2007) Surface Tensions in NaClWater-Air Systems from MD Simulations. The Journal of Physical Chemistry B, 111, 11989-11996. BAKKER, G. (1928) Kapillarität und Oberflächenspannung, Leipzig, Akad. Verlagsgesellschaft. BENNETT, C. H. (1976) Efficient estimation of free-energy differences from montecarlo data. Journal of Computational Physics, 22, 245-268. CHANG, T.-M. & DANG, L. X. (2005) Liquid-Vapor Interface of Methanol-Water Mixtures: A Molecular Dynamics Study. The Journal of Physical Chemistry B, 109, 5759-5765. ERRINGTON, J. R. & KOFKE, D. A. (2007) Calculation of surface tension via area sampling. The Journal of Chemical Physics, 127, 174709. GHOUFI, A., GOUJON, F., LACHET, V. & MALFREYT, P. (2008) Surface tension of water and acid gases from Monte Carlo simulations. The Journal of Chemical Physics, 128, 154716. GLOOR, G. J., JACKSON, G., BLAS, F. J. & MIGUEL, E. D. (2005) Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials. The Journal of Chemical Physics, 123, 134703. HARISMIADIS, V., VORHOLZ, J. & PANAGIOTOPOULOS, A. (1996) Efficient pressure estimation in molecular simulations without evaluating the virial. Journal of Chemical Physics, 105, 8469-8470. JORGENSEN, W., MAXWELL, D. & TIRADO-RIVES, J. (1993) Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. Q. ReV. Biophys, 26, 49. KIRKWOOD, J. G. & BUFF, F. P. (1949) The Statistical Mechanical Theory of Surface Tension. The Journal of Chemical Physics, 17, 338-343. LU, N. & KOFKE, D. A. (2001) Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling. Journal of Chemical Physics, 114, 73037311.
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H. Peng, AV. Nguyen, GR. Birkett LU, N., SINGH, J. & KOFKE, D. (2003) Appropriate methods to combine forward and reverse free-energy perturbation averages. The Journal of Chemical Physics, 118, 2977. POTOFF, J. J., ERRINGTON, J. R. & PANAGIOTOPOULOS, A. Z. (1999) MOLECULAR PHYSICS LECTURE Molecular simulation of phase equilibria for mixtures of polar and non-polar components. Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 97, 1073 - 1083. VAZQUEZ, G., ALVAREZ, E. & NAVAZA, J. M. (1995) Surface Tension of Alcohol Water + Water from 20 to 50 .degree.C. Journal of Chemical & Engineering Data, 40, 611-614. VEGA, C. & MIGUEL, E. D. (2007) Surface tension of the most popular models of water by using the test-area simulation method. The Journal of Chemical Physics, 126, 154707. ZWANZIG, R. W. (1954) High-Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. The Journal of Chemical Physics, 22, 1420-1426.
Brief Biography of Presenter Hong Peng, is the Ph.D candidate in school of chemical Engineering, The university of Queensland. His main project is about molecular simulation on interface phenomena of water and its mixtures with ions and surfactants cross liquid-vapour-solid phases. Before taken this Ph.D scholarship in Uni, he has worked in BHPBilliton technology and Olympic Dam for three years on various hydrometallurgy process development of titanium, nickel and uranium immediately after obtained his master degree in bioleaching area.
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