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Letter

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Nonzero Ideal Contribution to the of † ‡ ¶ § ∥ Marcello Sega,*, Balazś Fabiá n,́ , and Paĺ Jedlovszky , † University of Vienna, Boltzmangasse 5, A-1090 Vienna, Austria ‡ Institut UTINAM (CNRSUMR6213), UniversitéBourgogne Franche Comtè16 route de Gray, F-25030 Besancon,̧ France ¶ Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, Szt. Gellert́ teŕ 4, H-1111 Budapest, Hungary § Department of Chemistry, Eszterhazý Karolý University, Leanyká u. 6, H-3300 Eger, Hungary ∥ MTA-BME Research Group of Technical Analytical Chemistry, Szt. Gellert́ teŕ 4, H-1111 Budapest, Hungary

*S Supporting Information

ABSTRACT: Surface tension, the tendency of fluid interfaces to behave elastically and minimize their surface, is routinely calculated as the difference between the lateral and normal components of the or, invoking isotropy in space, of the virial tensor. Here we show that the anisotropy of the kinetic tensor close to a − vapor interface can be responsible for a large part of its surface tension (about 15% for water, independent from ).

he surface tension of a fluid can be obtained in several In water, however, the softest internal degree of freedom, the T ways from the microscopic variables describing the mode, has a frequency of about 1640 cm−1. This system: the so-called mechanical route links the surface tension corresponds, at room temperature, to an activation energy for fi of a planar interface to the imbalance between the normal (pN) the rst excited state of roughly 7.8kBT and a corresponding γp ∫ ∞ −3 and lateral (pT) components of the pressure tensor, = −∞pN average energy of about 3 × 10 k T. Excited stretching modes − B pT(z)dz. In a periodic system of size L, one can use the have even higher , and in this sense water volume average of the pressure tensor to write the surface are, for the sake of computing the surface tension, just rigid γp − tension as = L(pN pT)/2, where the factor 1/2 takes into bodies. In this case, the partition function is not separable any − account the presence of two interfaces. For a system of more into a kinetic and a configurational part,4 7 and the pointlike particles, the pressure tensor p can be accessed corresponding constraints acting on Cartesian coordinates and through the virial route,1,2 p =2(K − Ξ)/V, where V is the ∑ ⊗ velocities appear in the expression for the pressure tensor, as is system volume; K = 1/2 i mi vi vi is the kinetic energy 8 Ξ well-known for liquid crystals. Small molecular , on the tensor (corresponding to the ideal gas contribution); and is contrary, do not usually enjoy long-range order in the bulk, and the virial tensor, which, for pairwise-additive fij between ff Ξ − ∑ ⊗ the e ect of this coupling vanishes because of isotropy. The particle i and j, can be written as = 1/2 i>j rij fij.Ifno presence of an interface, introducing a preferential direction in constraints are present in the system, it is possible to invoke the 3 the system, can, however, change this picture substantially, so equipartition theorem, ⟨x∂H/∂x⟩ = k T (H being the Ξ B that in principle the equivalence between γp and γ is not Hamiltonian, k Boltzmann’s constant, and T the absolute B guaranteed any more, and a finite ideal gas contribution to the temperature), for the elements of the kinetic energy tensor and ρ − Ξ surface tension could appear. write the (average) pressure tensor as p = kBT1 2 /V, where ρ is the number of atoms and 1 is the unit tensor. In the course of extensive testing for the calculation of the surface tension of the SPC/E water model,9 we found that the This allows us to write an alternative expression for the surface Ξ γΞ − Ξ − Ξ difference between γp and γ amounts to about 15% at ambient tension, = L/V( N T), which is, on average, completely equivalent to the one involving the full pressure tensor, γp, but temperature. We were able to reproduce the same discrepancy ff has the advantage of not requiring the sampling of velocities. with di erent software packages, integrators (including The equivalence γp = γΞ, in other words, means that only the virial part of the pressure contributes to the surface tension, Received: April 26, 2017 whereas the ideal gas contribution is zero. This equivalence, Accepted: May 23, 2017 one should , is true only in the absence of constraints. Published: May 23, 2017

© XXXX American Chemical Society 2608 DOI: 10.1021/acs.jpclett.7b01024 J. Phys. Chem. Lett. 2017, 8, 2608−2612 The Journal of Letters Letter quaternions to describe the rigid body motion), thermostats, with the molecular structure. For a symmetric top, correspond- and electrostatic treatments. In particular, we reproduced the ing to the case of a linear , such as O2, where I = same behavior also in the microcanonical ensemble, guarantee- diag(I, I, 0), the ideal gas surface tension contribution of the ith γid − θ θ ing conservation of the total energy to within at least 1 ppm, atom is i = kBTLP2(cos i)/V, where P2(cos ) = 3/2 2 θ − θ with no evident drift within one nanosecond of simulation. The cos ( ) 1/2 is the second-order Legendre polynomial and i asymmetry of the kinetic energy tensor does not show any identifies the angle between the molecular axis and the dependence on system size or time step, ruling out other macroscopic surface normal z.̂ For a flat, asymmetric top like − known effects that seemingly violate equipartition.10 14 water, initially laying in the xz plane with the dipole vector ff Although this e ect does not depend on the implementation oriented along the z axis, I = diag(Ix, Ix + Iz, Iz), and the ideal of the constraints, substituting them with harmonic springs gas contribution of the ith atom located, in the molecular frame, ′ ′ completely removes the asymmetry and allows the recovery of at (xi ,0,zi ), takes the form the equality γp = γΞ, confirming that it is the rigid arrangement Lk Tm of atoms in the molecules that is at the origin of this apparent γ id =+B i [fP (II ) (cosθδ ) gP ( ) (cos )] violation of equipartition. i 22 VIxyz I I (1) Because the molecules at the liquid− vapor interface differ from the bulk ones only in the first two or ′2 − ′2 ′2 2 ′2 δ 15,16 where f(I)=IxIz(xi zi ); g(I)=IxIyxi + miIzzi ; and , the three layers, it is reasonable to expect the kinetic energy angle between the molecular plane and the surface normal, is tensor to be anisotropic only in proximity to the interface. The related to the Euler angles through the expression cos(δ)= fi kinetic energy tensor is a well-de ned local quantity; therefore, cos(ψ) sin(θ). The derivation of eq 1 can be found in the it is possible to calculate, without the ambiguity that Supporting Information. fi 17 characterizes the con gurational part of the pressure, its The surface tension profile calculated using eq 1 as γ(z)= fi ⟨ ∑ ⊗ δ − pro le along the surface normal, K(z)= 1/2 imivi vi (z ∑ ⟨γ δ(z−z )⟩ reproduces strikingly well the ideal gas rotational ⟩ i i i zi) , where it is assumed that the center of mass of the liquid contribution obtained using the kinetic energy tensor, as is shifted at the origin of the coordinate system. The reported in Figure 1 with full circles and the line, ff di erence between the normal and the lateral components of respectively. The integrals of the two curves differ only by 0.5%. γid K(z) can be used to compute the ideal gas contribution (z)= This shows that, in fact, equipartition is not violated, as it has γp − γΞ (z) (z) to the surface tension, shown in Figure 1, which been used to derive eq 1, taking into account the correlations between kinetic and positional degrees of freedom, introduced by the presence of constraints. The explicit relation between the ideal gas contribution and the molecular ordering can be now used to consider from a different perspective some other results related to the surface tension. More precise structural information can be gained by identifying the molecules composing successive molecular layers below the interface18,19 and by calculating their contribution to the ideal surface tension γid. Such a decomposition, also shown in Figure 1, shows that the difference between the normal and lateral kinetic components of the stress tensor is indeed located mainly at the first molecular layer, with some minor deviations showing up in the second one, and with the third layer being already characterized by nearly zero-sum oscillations, which have a modest contribution to the total value of γid. We already Figure 1. Ideal gas rotational contribution γid to the surface tension observed this kind of oscillation in the full surface tension profile as computed from the kinetic energy tensor (solid line) and profile of water,16 but their physical significance was not clear from eq 1 (full circles); contribution of successive molecular layers because of the virial contribution to the pressure profile not calculated from the kinetic energy tensor (shaded ) and from eq 1 being a well-defined quantity.20 The kinetic part of the profile, (open circles). however, does not suffer from these interpretation problems, and the profile obtained by using eq 1 to compute the is indeed concentrated in proximity to the interfaces. The ideal contribution of different layers (Figure 1, open circles) shows gas contribution originates only from the rotational degrees of that the oscillations are the result of the correlation between freedom of the molecules, as the translational ones (that is, the molecular orientation and deviation from the mean layer molecular centers of mass positions) behave isotropically. position. The coupling of the kinetic degrees of freedom to the Not only the surface tension, but also the kinetic energy positional ones can be exploited to derive an expression for the density shows departure from the constant value one would ideal gas surface tension profile of rigid molecules, as a function expect. For water molecules, the relation between kinetic ′ ⟨ ⟩ of molecular orientations. Using the atomic positions ri and the energy E, number of atoms N, and temperature is kBT = E /N, ω τ angular velocity vector in the molecular comoving frame, the which can be written naively in a local form as kB (z)= ω ρ τ velocity of each atom in the lab frame can be written as vi = TrK(z)/ (z), with (z) representing the kinetic energy density × ϕ θ ψ ′ fi fi fi R( , , )ri ,whereR is the Euler rotation matrix pro le. The pro le so de ned departs from the constant value parametrized by the three Euler angles ϕ, θ, and ψ. With the T in the proximity of the interface (see Figure 2). This, again, is help of the equipartition theorem, quadratic terms in the not a violation of the equipartition theorem, which can be used components of ω appearing in the average can be expressed as to derive the correct expression for the average kinetic energy functions of the components of the inertia tensor I associated contribution of the ith atom in the water molecule

2609 DOI: 10.1021/acs.jpclett.7b01024 J. Phys. Chem. Lett. 2017, 8, 2608−2612 The Journal of Physical Chemistry Letters Letter

vanishing changes using a linear fit, resulting in a surface tension estimate of 59.5 ± 0.1 mN/m. This has to be compared with the mechanical route results of 59.9 ± 0.2 mN/m (full pressure tensor) and 51.2 ± 0.2 mN/m (virial contribution only). The test area method is therefore an appropriate way to obtain the surface tension with Monte Carlo methods when rigid molecules are present. Finally, we report a surprising result obtained from the analysis of the temperature dependence of the ideal gas contribution on the surface tension. Because the orientational preference of water molecules at the surface has to vanish when approaching the critical point, one can expect γid to decrease Figure 2. Upper panel: kinetic energy density profile τ(z) calculated Ξ τ ρ when the temperature increases. The values of γp and γ show, using the atomic expression kB (z)=TrK(z)/ (z) (blue squares), using eq 2 (red circles), and using the molecular expression (yellow in fact, a similar decreasing pattern, reaching convergence as the triangles), as described in the text. Lower panel: atomic density profile temperature approaches the critical value, as shown in Figure 3. of the whole system and of the first three layers, normalized to the density in the liquid region.

⎧ 2222⎫ kT⎪⎪3 x′ x′ z′ z′ =+B ⎨ i + i + i + i ⎬ emii⎪⎪ 2 ⎩ M Iy Iz Ix Iy ⎭ (2) where M is the mass of the molecule and the term 3/M is the molecular center of mass velocity (translational) contribution to the atom’s kinetic energy; the remaining terms are the rotational contributions (interestingly, for the symmetric top, the rotational contribution of each atom is simply kBT/2). The atomic kinetic energy density profile can thus be written using τ ∑ ⟨ δ − ⟩ ρ only atomic positions as kB (z)= i ei (z zi) / (z) and is γp γΞ fi Figure 3. Upper panel: surface tension and virial contribution as shown in Figure 2 to reproduce very well the kinetic de nition. a function of temperature. Lower panel: ratio γid/γp =1− γΞ/γp. Error Obviously, this quantity does not correspond to the usual bars are always smaller than the symbols in the upper panel and are for thermodynamic temperature, which is expected to be constant all of the order of 0.1−0.2 mN/m. across the interface. The temperature so defined loses its meaning once correlations between momenta and atomic What is quite remarkable, however, is that the relative positions occur. However, by using a molecular-based definition contribution γid/γp is, to a good approximation, independent of the kinetic energy, in which the translational and rotational of the temperature and oscillates within a few fractions of a contributions are concentrated at the center of mass of the percent around 15%. We tested different rigid water models molecule, one obtains the expected constant profile, as shown (SPC,23 TIP3P,24TIP4P,24 TIP4P-2005,25 TIP5P,26 and also in Figure 2. TIP5P/E27), obtaining in all cases a temperature-independent As it is clear that the kinetic part of the pressure tensor is ratio of γid/γp, although the value itself is model-dependent, needed to compute the correct value of the surface tension if ranging from about 9 to 15%, as shown in Table 1. There is, rigid molecules are present in the system, one might wonder if therefore, a direct proportionality between the orientational this can create problems for methods like Monte Carlo, which order of molecules, of which γid is representative, and the do not provide explicit access to momenta. In fact, if one wants surface tension of the system, γp ∝ γid. to compute the surface tension through explicit calculation of the pressure tensor elements, there is no other way but to ■ METHODS include the kinetic contribution through formulas like eq 1. Simulations have been performed using the GROMACS 5.1,28 However, this is not the only possible route to the surface LAMMPS,29 and ESPResSo30 molecular simulation packages 21 tension: the test-area method (of which also local variants using either single or double precision. Water molecules have 22 exist ), for example, follows a thermodynamic route to been kept rigid by either solving the constrained equation of compute the surface tension as the limit toward vanishing motion using the SHAKE31 or SETTLE32 algorithms or, in the cross-sectional surface area perturbations ΔA (at constant case of ESPResSo, by solving the rigid-body dynamics using volume) of the associated changes in Helmholtz free energy F, quaternions. The difference between γΞ and γp has been shown γ Δ Δ − ⟨ −Δ ⟩ so that = limΔA→0 F/ A = kBT ln exp( U/kBT) , where to persist independently from short-range forces at the cutoff ΔU is the change in potential energy between the perturbed distance ( truncation vs potential shift), mesh size and and not perturbed state, and the ensemble average is performed accuracy in the smooth particle mesh Ewald33 (sPME) or over configurations sampled from the unperturbed state. particle−particle particle−mesh Ewald34 method, number of Because this is (in the limit of small perturbations) the reciprocal vectors and β parameter in plain Ewald method, thermodynamic definition of the surface tension, one might integration time step from 0.1 to 1 fs, simulation box size, type expect it to yield the total surface tension, including the kinetic of thermostat (Berendsen35 vs Nose−́Hoover36,37)and contributions. To test this, we computed the surface tension for ensemble (microcanonical vs canonical). The actual value of water at T = 300 K using the test area method with area the surface tension depends on the short-range interactions changes of 0.1, 0.05, and 0.01% and extrapolated the results to cutoff value as well as on the parameters used to compute the

2610 DOI: 10.1021/acs.jpclett.7b01024 J. Phys. Chem. Lett. 2017, 8, 2608−2612 The Journal of Physical Chemistry Letters Letter

Table 1. Temperature-Independent Ratio γid/γp (%) for Different Water Models

SPC SPC/E TIP3P TIP4P TIP4P-2005 TIP5P TIP5P/E 14.3(2) 15.0(1) 15.4(2) 11.5(1) 11.4(1) 9.2(5) 8.7(4) electrostatic interaction. The data reported in Figures 1 and 2 (2) Hünenberger, P. H. Calculation of the group-based pressure in are obtained by simulating 1488 water molecules in a 3.6 × 3.6 molecular simulations. I. A general formulation including Ewald and × 9.6 nm3 simulation box with the velocity-Verlet algorithm, particle-particle-particle-mesh electrostatics. J. Chem. Phys. 2002, 116, smoothly switching the short-range forces to zero in the 6880−6897. interval between 1.55 and 1.6 nm, using sPME with an accuracy (3) Tolman, R. C. A general theory of energy partition with −8 applications to quantum theory. Phys. Rev. 1918, 11, 261−275. of 10 for the real part of the screened electrostatic potential at ’ 1.55 nm and a reciprocal space grid of 128 × 128 × 256, the (4) Ryckaert, J. P.; Ciccotti, G. 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Periodic boundary determining the layer structure using a cutoff (0.35 nm) condition induced breakdown of the equipartition principle and other based cluster search that associates the liquid phase to the kinetic effects of finite sample size in classical hard-sphere molecular largest cluster in the system.39 The low inaccuracies for the dynamics simulation. J. Chem. Phys. 2006, 125, 164102. (11) Uline, M. J.; Siderius, D. W.; Corti, D. S. On the generalized values of the surface tension as well as smooth profiles were equipartition theorem in molecular dynamics ensembles and the obtained by analyzing 30 000 samples over 3 ns for the data microcanonical of small systems. J. Chem. Phys. 2008, reported in Figures 1 and 2 and 50 000 samples over 50 ns for 128, 124301. each system and temperature for the data reported in Figure 3 (12) Eastwood, M. P.; Stafford, K. A.; Lippert, R. A.; Jensen, M. Ø; and Table 1. Maragakis, P.; Predescu, C.; Dror, R. O.; Shaw, D. E. Equipartition and the calculation of temperature in biomolecular simulations. J. Chem. ■ ASSOCIATED CONTENT Theory Comput. 2010, 6, 2045−2058. *S Supporting Information (13) Pastor, R. W.; Brooks, B. R.; Szabo, A. An analysis of the The Supporting Information is available free of charge on the accuracy of Langevin and molecular dynamics algorithms. Mol. Phys. ACS Publications website at DOI: 10.1021/acs.jpclett.7b01024. 1988, 65, 1409−1419. Derivation of the equations for the ideal gas surface (14) Hairer, E.; Wanner, G.; Lubich, C. Geometric Numerical Integration; Springer-Verlag: Berlin/Heidelberg, 2006; pp 179−236. tension contribution (PDF) (15) Sega, M.; Fabiá n,́ B.; Horvai, G.; Jedlovszky, P. How is the surface tension of various liquids distributed along the interface ■ AUTHOR INFORMATION normal. J. Phys. Chem. C 2016, 120, 27468−27477. Corresponding Author (16) Sega, M.; Fabiá n,́ B.; Jedlovszky, P. Pressure profile calculation *E-mail: [email protected]. with mesh Ewald methods. J. Chem. Theory Comput. 2016, 12, 4509− ORCID 4515. (17) Schofield, P.; Henderson, J. of inhomoge- Marcello Sega: 0000-0002-0031-905X neous fluids. Proc. R. Soc. London, Ser. A 1982, 379, 231−246. Paĺ Jedlovszky: 0000-0001-9304-435X (18) Partay,́ L. B.; Hantal, G.; Jedlovszky, P.; Vincze, Á.; Horvai, G. A Notes new method for determining the interfacial molecules and character- The authors declare no competing financial interest. izing the surface roughness in computer simulations. Application to the liquid-vapor interface of water. J. Comput. Chem. 2008, 29, 945− ■ ACKNOWLEDGMENTS 956. ́ ́ The authors acknowledge useful discussions with Giovanni (19) Sega, M.; Fabian, B.; Jedlovszky, P. Layer-by-layer and intrinsic analysis of molecular and thermodynamic properties across soft Ciccotti, Simone Melchionna, and Francesco Sciortino. This ł interfaces. J. Chem. 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2611 DOI: 10.1021/acs.jpclett.7b01024 J. Phys. Chem. Lett. 2017, 8, 2608−2612 The Journal of Physical Chemistry Letters Letter

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2612 DOI: 10.1021/acs.jpclett.7b01024 J. Phys. Chem. Lett. 2017, 8, 2608−2612