Transfer coefficients for evaporation of a system with a Lennard-Jones long-range spline potential

Transcription

1 PHYSICAL REVIEW E 75, Transfer coefficients for evaporation of a system with a Lennard-Jones ong-range spine potentia Jiain Ge, S. Kjestrup,* D. Bedeaux, and J. M. Simon Department of Chemistry, Norwegian University of Science and Technoogy, NO-749 Trondheim, Norway B. Rousseau Laboratoire de Chimie Physique, Bâtiment 349, CNRS UMR8, Université Paris-Sud, 945 Orsay Cedex, France Received 5 December 6; revised manuscript received 3 March 7; pubished June 7 Surface transfer coefficients are determined by nonequiibrium moecuar dynamics simuations for a Lennard-Jones fuid with a ong-range spine potentia. In earier work A. Røsjorde et a., J. Cooid Interface Sci. 4, 355 ; J.Xuet a., ibid. 99, 45 6, using a short-range Lennard-Jones spine potentia, it was found that the resistivity coefficients to heat and mass transfer agreed rather we with the vaues predicted by kinetic theory. For the ong-range Lennard-Jones spine potentia considered in this paper we find significant discrepancies from the vaues predicted by kinetic theory. In particuar the couping coefficient, and as a consequence the heat of transfer on the vapor side of the surface are much arger. Thermodynamic data for the iquid-vapor equiibrium confirmed the aw of corresponding states for the surface, when it is described as an autonomous system. The importance of these findings for modeing phase transitions is discussed. DOI:.3/PhysRevE PACS numbers: 68.3.g, 45.5.Jf I. INTRODUCTION Phase transitions are common in nature as we as in industry. So the study of the transport properties of the surface between two phases is very important. A phase transition invoves simutaneous transfer of heat and mass across a surface, and arge efforts have been devoted to mode this in a correct way 3. Many of these studies have assumed equiibrium between the phases at the phase boundary, i.e., continuity in the chemica potentias and the temperature. In such a case, the resistivities of the surface to heat and mass transfer pay no roe. We have questioned this assumption for iquid-vapor phase transitions 4,5,, not because we find very arge resistivities for this transition,6,, but because the couping of heat and mass at the surface is important. The couping coefficient measures how much of the enthapy of evaporation is taken from each of the adjacent phases 4. Therefore, the couping coefficient wi infuence the vaue of the heat fuxes into the homogeneous phases. So it is important to know its vaue and understand its origin. Linear kinetic theory KT 7 has since ong, predicted the transfer coefficients for the surface, aso the couping coefficient, and given expicit formuas for them. Kinetic theory appies to partices that are hard spheres, however. In ack of information on rea systems, the surface resistivities for heat and mass transfer have mosty been negected in the engineering description of phase transitions. Experiments have been done, which confirm the presence of a temperature difference across the iquid-vapor interface. In experiments the probem becomes at east two dimensiona. Phenomena ike convection may occur. We refer to Duan et a. 8 and references therein. For *Corresponding author. On eave from Institute Carnot de Bourgogne, UMR 59 CNRS-Universit de Bourgogne, F-78 Dijon, France. a detaied anaysis of the various theoretica descriptions and their appication to the experiments we refer to Bond and Struchtrup 9. aims to add to the understanding of the surface resistivities, and in particuar to give information about the coefficient that describes couping of heat and mass. We sha find a transfer coefficients by appying the nonequiibrium moecuar dynamics simuation technique NEMD to a system with a ong-range Lennard-Jones spine potentia, and compare the resuts to kinetic theory as we as to resuts obtained earier with a short-range Lennard-Jones spine potentia,. The ong-range potentia is a better approximation to the rea Lennard-Jones potentia than its short-range version. We want to know whether the good agreement with kinetic theory obtained earier is accidenta or not, and in genera to contribute to the systematic knowedge of these coefficients. In a study of transport properties of surfaces, one first must estabish the surface boundaries, and the equiibrium properties of the surface. This sha be done aso here. One needs to estabish the proper equation of state, before one can cacuate thermodynamic driving forces in the system. The surface and its equiibrium thermodynamic properties have been defined here foowing Gibbs. In the description of the transfer coefficients, we use nonequiibrium thermodynamics theory, as deveoped for surfaces 4. The basic hypothesis of nonequiibrium thermodynamics is the hypothesis of oca equiibrium. When the surface is defined as a separate system, as mentioned above, one must verify that the equation of state for the surface appies equay we in the presence as in the absence of gradients across the surface. This was done earier, working with Lennard-Jones spine partices, and with n-octane 6. In both cases the systems were exposed to a arge temperature gradient. We sha see that the assumption is true aso here, meaning that cassica nonequiibrium thermodynamics can be used /7/756/ The American Physica Society

2 GE et a. In a system with onger interacting potentias, we therefore want to answer the foowing main questions: Is the surface sti in oca equiibrium in a temperature gradient? How wi the transfer resistivities vary with the range of the potentia? Can kinetic theory predict the correct resuts in this case? Whie the answer to the first question is yes, as we sha see, the changes are such that the answer to the ast question is no. We furthermore sha show that the surface obeys the aw of corresponding states aso in the presence of a arge temperature gradient. II. NONEQUILIBRIUM THERMODYNAMICS FOR THE SURFACE The primary quantity of a nonequiibrium system is its entropy production. This quantity governs the equations of transport. For a surface, nonequiibrim thermodynamics gives the equations of transport from the excess entropy production. For a surface in a stationary state, the excess entropy production rate with heat and mass transport into and through the surface is s = J q g T s T g + J q T T s J T s g T s. Here J q g is the measurabe heat fux from the gas into the surface and J q is the measurabe heat fux out of the surface into the iquid. J is the mass fux across the surface, which is constant in a stationary state. According to the hypothesis of oca equiibrium, the surface is a separate thermodynamic system and has its own temperature T s. The temperature next to the surface on the iquid side is T and on the gas side it is T g 4,5. The chemica potentia next to the surface on the iquid side is and on the gas side it is g. The positive direction is defined as being from the vapor to the iquid. The notation foows de Groot and Mazur 3 and Bedeaux and Kjestrup 4. In a stationary state, the tota heat fux through the surface is constant, J q = J q + H J = J q g + H g J. It foows that J q = J q g + J vap H. 3 Here H is the moar enthapy and vap H=H g T g H T is the heat of evaporation. Using this reation to eiminate the term containing J q in the excess entropy production, and noticing the thermodynamic identity T T Ts T s = H s T 4 T both for the vapor and the iquid, we can reduce the excess entropy production to s = J q g T T g J T T g T = J q g X q + JX g. 5 In Eq. 4 the enthapy can be taken at T g, T s or T to inear order in the differences. If one uses the measurabe heat fux T s in the iquid as a variabe, we obtain the aternative expression s = J q X q + JX. 6 In the above equations, X q, X g, and X are the therma driving force and chemica driving forces, respectivey, X q = T = T T g, X g = T TT = T T g T, X PHYSICAL REVIEW E 75, = T g TT g = T g T g g T g. 7 According to cassica nonequiibrium thermodynamics, the thermodynamic forces are inear combinations of their conjugate fuxes X q = r s,g qq J q g + r s,g q J, X g = r s,g q J q g + r s,g J. 8 Here the coefficients r s,g qq and r s,g are the two main resistivities of the surface. They are for heat transfer and mass transfer, respectivey. The coefficients r s,g q and r s,g q are the couping resistivities. They describe the mass transport due to the temperature difference and the heat transport due to the mass fux, respectivey. According to the Onsager reciproca reations r s,g q =r s,g q. The above equations 8 are appropriate when the measurabe heat fux on the gas side is used. In the same way we obtain equations which use the measurabe heat fux on the iquid side: X q = r s, qq J q + r s, q J, X = r s, q J q + r s, J. 9 The two sets of equations are equivaent. The resistivities of the two sets are reated by r s, qq = r s,g qq, r s, q = r s, q = r s,g q vap Hr s,g qq, s, r = r s,g vap Hr s,g q + vap H r s,g qq. The interface fim resistivities from kinetic theory were discussed by Xu et a.. The authors proposed that the surface temperature be used in the expressions obtained from Cipoa et a. 7, rather than the iquid temperature. This gave a better fit to the simuation data. Aso it is more appropriate that a property of the surface depends on the temperature of the surface. The revised expressions, proposed by Xu et a., were r s,g qq T s.764 = M RT s c g T s 3RT s, 664-

3 TRANSFER COEFFICIENTS FOR EVAPORATION OF A PHYSICAL REVIEW E 75, TABLE I. Reduced units. Variabe Reduction formua FIG.. A snapshot of the MD box with 496 partices. The axes are not drawn with the same scae, see text for aspect ratio. r s,g q T s = r s,g q T s =.5475 M T s c g T s 3RT s, r s,g T s = 4.346R c T s M c g T s 3RT s. Here c g T s is the density of a vapor in equiibrium with a iquid at the temperature of the surface, M is the moar mass, and R is the gas constant. The condensation coefficient, c, gives the fraction of partices that condense at the surface after coision with the iquid. Vaues between. to. have been reported 4 for this coefficient. III. MOLECULAR SIMULATION METHOD A. Simuation system The nonequiibrium moecuar dynamics simuation NEMD method was described in more detai esewhere. Here we briefy describe its main aspects. The simuation box, is a noncubic, rectanguar one, with ength ratios L x /L y =L x /L z =6. Here L i is the ength of the box in the ith direction. We simuated 496 argonike partices. Periodic boundary conditions were used. Normay the size of the box was about moecuar diameters in the direction aong the surface and about 6 moecuar diameters 6 in the direction norma to the surface, the x direction. The NEMD simuations were done with a constant number of partices N and voume V. In equiibrium the thermostatted regions were thermostatted to the same temperature. This produced a canonica ensembe. Different overa densities of the mode system were obtained by varying the size of the box. The densities were chosen such that the mean free path in the vapor was smaer than the thickness of the vapor ayer in the direction norma to the surface. The mean free path was cacuated using the standard formua =/ cn A. For two out of three simuations the mean free path was arger than up to times the ength of the box in the direction parae to the surface. In view of the fact that we cacuate resistivities for transport norma to the surface this is not expected to be a probem. No behavior of the data indicated such an effect. The simuation box was divided in the x direction into 8 equa panar ayers parae to the surface. A snapshot of the system, which ceary shows the vapor and the iquid phases, is given in Fig.. In order to create a temperature gradient, the ayers at the end of the box, ayers and 7 8, which we caed the hot zones, were thermostatted to a high temperature T H, and Mass m * =m/m Distance r * =r/ Energy U * =U/ Time t * =t/ /m Temperature T * =kt/ Density c * =c 3 N A Pressure p * = p 3 / Veocity v * =v m / Surface tension * = / the ayers in the center of the box, ayers 63 66, caed the cod zone, were thermostatted to a ow temperature T L. This is done by the heat exchange HEX agorithm 5, which adds or withdraws energy in these ayers. In this way, a heat fux from the ends of the box to the center was created. We simuated a mass fow using the mass exchange MEX agorithm 5, which moves partices from the cod zone to the hot zone. This is done by randomy seecting a partice in the cod zone and changing its ocation up or down by haf a box ength. It was then verified that the partice at its new ocation had no overap with any of the partices aready present. In the case of overap the move was rejected. To cacuate the driving forces, the equiibrium properties were necessary. The equiibrium properties of the system, were studied by equiibrium moecuar dynamics EMD simuations and some Gibbs ensembe Monte Caro GEMC simuations 6. To obtain the equiibrium phase diagram by moecuar dynamics, we performed simuations on the above system. We first run the code for miion time steps at nonequiibrium states with the cod zone thermostatted at a ower temperature and the hot zone at a higher temperature. Every time step was.5 in reduced units see Tabe I. This created an interface in the ce. Then we set the high temperature equa to the ow temperature and run the code for 5 miion time steps at the appropriate equiibrium state. This gave the density and pressure of vapor and iquid in coexistence. GEMC simuations at constant voume, temperature, and tota number of partices aow for direct simuation of phase equiibria in a pure component without an expicit interface. Two cubic simuation boxes were simuated and therma, mechanica, and chemica equiibrium between both phases were obtained by means of Monte Caro moves, incuding partice transation, voume changes at constant tota voume, and partice transfer between phases. Starting from an initia configuration, equiibrium was reached and the density of both phases and the vapor pressure were obtained. B. Potentia A Lennard-Jones spine potentia was used to describe the partice interaction. The potentia is expressed in terms of the interpartice distance r ij between any pair of partices i and j, 664-3

4 GE et a. u/ε L-J L-J spine (ong-range) L-J spine (short-range) r/σ FIG.. The Lennard-Jones potentia, the short- and the ongrange Lennard-Jones spine potentias. ur ij = 4/r ij /r ij 6, r ij r s, ar ij r c + br ij r c 3, r s r ij r c,, r c r ij, where r c =.5 is the truncated distance and r s =48/67r c. The constants a and b were chosen so that the potentia and its derivative are continuous at r s.inr c the potentia and its derivative are aso continuous. In the simuations, we used argonike partices with mass m = kg, diameter =3.4 m, and the potentia depth /k =4 K. Here k is Botzmann s constant. This resuted in a = J/m and b= J/m 3. Pots of the PHYSICAL REVIEW E 75, Lennard-Jones potentia and the short- and the ong-range Lennard-Jones spine potentias are given in Fig.. Reduced units in terms of potentia parameters were used for a variabes in the computer codes, see Tabe I for definitions. C. Case studies GEMC simuations were performed in order to obtain the critica coordinates of the ong-range Lennard-Jones spine potentia. A set of 4 simuations at reduced temperatures ranging from.9 to.3 were reaized. For each simuation, we used a tota number of 5 partices. Probabiity attempts for the different moves were.88 for transation,. for transfer, and. for voume change. A simuations ran for miion Monte Caro steps and average properties were computed from the ast 8 miion steps. We performed nonequiibrium moecuar dynamics simuations in the study of the transport properties of the surface, see Tabe II. Every simuation ran miion time steps with a time step of 5 4 in reduced units, which is equivaent to about 5 s in rea time. The first 3 miion time steps were abandoned to avoid transient effects. A the properties were averaged over 7 miion time steps. Because the system is symmetric about the centra pane in the midde of the box in a stationary state, in the x direction, the mean of the properties in each haf-box was furthermore cacuated to obtain better statistics. The density in the vapor phase is very ow. Accuracy of the vaues cacuated for the vapor phase is obtained by running the stationary state simuations ong enough. TABLE II. NEMD simuation conditions, in reduced and rea units. Simuation c * T H * T N * J * c mo/m 3 T H K T L K J mo/m s

5 TRANSFER COEFFICIENTS FOR EVAPORATION OF A Among the NEMD simuations for the surface transfer coefficients, had a mass fux, and the other did not see Tabe II. We chose a suitaby high temperature, ow temperature, and overa density so that two vapor-iquid surfaces appeared in the box. The temperature gradients went beyond usua aboratory vaues. They were of the order of 9 K/m in the vapor and 8 K/m in the iquid. D. Defining the surface The extension of the surface is an issue of discussion. On the gas side, a good equation of state is avaiabe. This is the Soave-Redich-Kwong SRK equation 7. p = RT v b a vv + b, 3 where the coefficients were found to be given by, a = R T c p c T T c, b = RT c p c. 4 From knowedge of the critica coordinates T c =38.9±. K and p c =4.5 6 Pa obtained by GEMC the gas phase moar voume v and moar density =/v can be computed as a function of pressure and temperature by soving the SRK equation. In each ayer this density was compared with the one obtained from NEMD. For the pressure in this cacuation it is appropriate to use the voume average of the pressure norma to the surface, see beow. A ayer is defined as a interfacia ayer when these densities differ seriousy. On the iquid side, we do not have an equation of state to hep us determine the first ayer of the surface. But the transition from iquid to the surface is rather abrupt. From the iquid ayer to the surface ayer, the density changes markedy. As the first surface ayer we took the ayer next to the ast ayer on the iquid side with a density that is sti equa to the iquid density. The interfaces were found to be roughy ayers thick. Given this thickness, there was no indication that the interface was not perfecty fat. The resuts reported refer to this choice. We have earier found that the excusion and/or incusion of one extra ayer does not ater the resuts significanty. E. Cacuations The fuxes and thermodynamic properties of each ayer were cacuated as time averages using instantaneous veocities, kinetic energies, potentia energies, and the number of partices. The moar density in ayer =,,...,8 was given by c = 8N 5 VN A here N is the number of partices in ayer and V=L x L y L z is the voume of the system. The moar fux in ayer was Reduced Pressures Layer Number J = c v = c N v i, 6 i where v is the veocity of the center of mass in ayer. In the stationary state considered ony the x component of the moar fux and the barycentric veocity, J and v, are unequa to zero. The temperature in each ayer was found from the partices kinetic energy 3 N kt = mv i v. 7 i The temperatures T g and T were chosen to be equa to the temperatures of the ast ayers of the vapor and the iquid next to the surface ayers. These then give the therma force X q. For the temperature of the surface we used the vaue which is found from the kinetic energy of the surface ayers. In terms of the temperatures of the ayers that are part of the surface this gives T s = PHYSICAL REVIEW E 75, surface N T N. surface 8 The oca pressure tensor was cacuated by time averaging the microscopic pressure tensor p = 8 i, v i, + V imv F ij, r ij,, ji P xx P yy P zz FIG. 3. The reduced diagona pressures for simuation 7 as a function of ayer number. 9 where v i, is the veocity of partice i in the direction, F ij, is the force exerted on partice i by partice j in the direction, and r ij, is the component of the vector from partice j to partice i in the direction. The pressure tensor was found to be diagona. Away from the surface the diagona eements in the x, y, and z directions are the same. In the neighborhood of the surface it is found that the diagona eements aong the surface, p = p yy = p zz, and norma to the surface, p = p xx, are different. In Fig. 3 we pot the diagona eements of the reduced pressure tensor for simuation 7. In the work by Todd et a. 8,9 it is shown that the method of panes is a more efficient procedure to cacuate the pressure tensor when the system is inhomogeneous. They consider in particuar fuids 664-5

6 GE et a. under shear conditions and with arge or varying density gradients. In this paper there is no shear, the density gradient is smaer and smooth. Away from the interface we find the expressions given to be appropriate. In the interfacia region the method of panes eads to a norma pressure which is coser to being constant as it shoud be 6. As we did not further need this information, we did not pursue this further. The surface tension was computed from = r ij 3x A ij /r ij ur ij, ij where A=L y L z is the surface area of the cross section in the y and z directions. The x direction is the direction norma to the surface, and x ij =x i x j, whie ur ij is the derivative of the pair potentia with respect to interpartice distance, r ij. Equation gives the surface tension of one of the two surfaces by restricting the partices to be either in ayers 3 6 or in If the sum is not restricted in this manner one must divide by A. The tota heat fux J q is constant whie the measurabe heat fux J q varies due to the temperature dependence of the enthapy per moe H. The tota heat fux is J q = J q + HJ. The tota heat fux in ayer is cacuated from J q, = 8 V iayer i v mv i + i + v i mv i v i + F ij r ji ij = 8 V iayer i v 3 mv i + i + v i F ij r ji ij, where the potentia energy of partice i is i ur ij. 3 j In the stationary state considered ony the x component of the tota heat fux, J q,, is unequa to zero. In Figure 4 we pot the tota and the measurabe heat fux for simuation 7. In the work by Todd et a. 8,9 it is shown that the method of panes is aso a more efficient procedure to cacuate the heat fux when the system is inhomogeneous. They consider in particuar fuids under shear conditions. In the work of this paper there are no shear gradients and we find the expressions given to be appropriate aso through the interfacia region. The enthapy per moe in ayer is found from H = N A 5 k BT N + i + iayer F ij r 6 j ij. 4 To cacuate the chemica driving forces, we use the foowing formuas : Reduced Heat Fuxes X g = R n X Layer Number p p * T + R n T,p + c p g T g Tg = R n * T,p + v* T T p * T p T p p * T g + R n Tg,p + c p T T g T, 5 * T g,p + v* T g T g p * T g p. 6 These formuas are vaid to second order in the temperature difference. The input quantities are the system pressure, p, saturation pressure, p *, heat capacities at constant pressure, c p g and c p, the iquid moar voume at saturation, v *. The fugacity coefficient was obtained using the foowing formua: RT n T,v = RT b v b +n PHYSICAL REVIEW E 75, FIG. 4. The reduced heat fuxes for simuation 7 as a function of the ayer number. v v b nvpt,v RT a v + b + a b n v v + b, here T,v, p are reated by the SRK equation 3. IV. RESULTS AND DISCUSSION 7 A. The vapor and iquid at equiibrium Equiibrium properties are needed for the cacuation of the driving forces. The equiibrium properties are aso important in themseves, as this particuar system has not been studied before. The Lennard-Jones spine potentia in this study has a cutoff distance equa to.5 reduced units, whie.73 reduced units were used earier. Equiibrium resuts were found with two simuation techniques: The Monte Caro simuation and the equiibrium moecuar dynamics. Figure 5 shows the vapor-iquid coexistence curve with resuts from both studies. The reduced density is potted as a function of the reduced temperature for the two phases. The points at higher temperatures triange symbos are the re- J q J q 664-6

7 TRANSFER COEFFICIENTS FOR EVAPORATION OF A PHYSICAL REVIEW E 75, NEMD for this work Xu, et a. EMD for this work T *.8.6 data from EMD data from MC ρ T s /T c FIG. 5. The iquid-vapor coexistence curve for the Lennard- Jones fuid with a ong-range spine potentia. suts of the Monte Caro simuations. Those at ower temperatures circe symbos are from the resuts of the moecuar dynamics simuations. We see that the sets of data agree within some percent in the region where they overap. The coexistence curve data were fitted by the foowing formua:. 8 L V = T c T T c The universa exponent is given by =.3. Furthermore is a system-dependent constant for which we found =.8±. in reduced units. The critica temperature was found to be T c =38.9±. K. Figure 6 gives the reduced vapor pressure of the system, as a function of the reduced temperature. To find a fit to this pot, we used the Causius-Capeyron equation: p * T = p * exp vaph, 9 RT where p * is a constant for which we found p * =7±. Furthermore vap H is the enthapy of evaporation. Approximating vap H to be constant, we found vap H=649±8 J/mo. For the short-range spine potentia the vaues were found to be p * =8.4 and vap H=55 J/mo. P * data from MC data from EMD T * FIG. 6. The vapor pressure as a function of temperature for the Lennard-Jones fuid with a ong-range spine potentia. FIG. 7. The equation of state for the iquid-vapor surface for a fuid with a ong-range spine potentia; the surface tension as a function of temperature at equiibrium EMD and at nonequiibrium NEMD. Resuts for a short-range spine potentia are aso shown. The simuations gave a critica temperature T c =38.9 K, a critica density c =.35 4 mo/m 3, and a critica pressure p c =4.5 6 Pa. These vaues are coser to the vaues of rea argon, T c =5.7 K and p c = Pa, than those obtained for the short-range potentia, where we found T c =. K, and p c =3.3 6 Pa, respectivey. The reationship between the surface tension and the surface temperature is the equation of state for the surface. Foowing Guggenheim,, we potted in Fig. 7 the quantity CS =/ c /3 kt c versus T CS =T s /T c. These variabes were derived from the aw of corresponding states and must be distinguished from the reduced surface tension and temperature in Tabe I. The pot shows the corresponding states surface tension versus corresponding states surface temperature. The circe symbos are from nonequiibrium moecuar dynamics simuations NEMD and the pus symbos are from equiibrium moecuar dynamics simuations EMD. We can see that they agree we. This proved again,6 that the surface is a separate thermodynamic system, which is in oca equiibrium, even in the presence of a arge temperature gradient. This is the condition for using nonequiibrium thermodynamics. The resuts from the system with the short-range spine potentia are shown in the figure using trianges. The system with a short-range potentia have a critica temperature, T c, and density, c, that differ consideraby from the vaues given above. By potting the resuts in CS units, the resuts from the earier investigation agreed with our data. This shows that the principe of corresponding state for the surface tension can be appied successfuy to Lennard-Jones spine systems for both short- and ong-range potentias. The data for the surface tension were fitted to the equation of state for the surface = T c T T c, 3 where =.6 is a universa constant. We obtained =.334±.3 N/ m. The vaue of the constant for the 664-7

8 GE et a. PHYSICAL REVIEW E 75, r qq ( - m s/jk) 4 3 Xu, et a. KT for Xu, et a. KT for this work r µq (-6 m s/mok) Xu, et a. KT for Xu, et a. KT for this work.5.5 FIG. 8. The surface transfer resistivity r qq to heat fux for a Lennard-Jones system with a ong-range spine potentia circes and for a short-range spine potentia trianges. Resuts from KT are aso shown for both systems. short-range potentia was =.48±.3 N/m. The surface entropy is minus the derivative of the surface tension with respect to the temperature. This gives s s = T c T T c, 3 where = /T c =3. 4 J/m. In our earier paper we found =.8 4 J/m for the short-range spine potentia. B. Surface resistivities Figures 8 give pots of resistivities for the equivaent sets of variabes, when the measurabe heat fux refers to the vapor side Figs. 8, 9, and and to the iquid side Figs. 8,, and. A coefficients were potted as a function of corresponding states surface tension CS =/ /3 c kt c. Earier resuts, and resuts cacuated from kinetic theory using Eq. for both sets of data, are aso shown, for comparison. r g µq (-7 m s/mok) 4 3 Xu, et a. KT for Xu, et a. KT for this work FIG.. The surface transfer resistivity r q for couping of the mass fux to the heat fux on the iquid side. Resuts are shown for a Lennard-Jones system with a ong-range spine potentia circes and for a short-range spine potentia trianges. Resuts from KT are aso shown for both systems. Figure 8 shows the main resistivity to heat transfer. This coefficient is the same, whether we use the measurabe heat fux on the iquid side or on the vapor side as the fux, cf. Eq. a. The resuts obtained here the circes were somewhat smaer than predicted by kinetic theory the stipped ine. For the system with the short-range potentia, given here by the trianges and the dotted ine, the agreement with kinetic theory was better. Figure 9 shows the couping resistivity, r g q, for couping of the mass fux to the measurabe heat fux on the vapor side. The vaues are now a factor of 3 arger than the vaues predicted by kinetic theory the stipped ine. For the shortrange potentia, trianges and a dotted ine, there was again agreement between the NEMD resuts and kinetic theory. The absoute vaue of the couping resistivities on the iquid side are about times arger than the vaue of r g q, see Fig.. Equation b gives the reation between the couping coefficients on both sides of the surface. Figure shows that the couping resistivity r q is aso affected by a change in the range of the potentia. The NEMD vaues the circes r g µµ (-4 Jm s/mo K) 4 3 Xu, et a..5.5 g FIG. 9. The surface transfer resistivity r q for couping of the mass fux to the heat fux on the vapor side. Resuts are shown for a Lennard-Jones system with a ong-range spine potentia and for a short-range spine potentia. Resuts from KT are aso shown for both systems..5.5 g FIG.. The surface transfer resistivity r to a mass fux when the heat fux on the vapor side is used as a variabe, for a Lennard- Jones system with a ong-range pine potentia circes and for a short-range spine potentia trianges

9 TRANSFER COEFFICIENTS FOR EVAPORATION OF A PHYSICAL REVIEW E 75, r µµ (-3 Jm s/mo K) 6 4 Xu, et a. q *s,g (J/mo) -5 - Xu, et a. KT for Xu, et a. KT for this work FIG.. The surface transfer resistivity r to a mass fux when the heat fux on the iquid side is used as variabe, for a Lennard- Jones system with a ong-range spine potentia circes and for a short-range spine potentia trianges. are consideraby ess negative than the prediction by kinetic theory the stipped ine. For the short-range potentia there was agreement between the NEMD vaues triange and the kinetic theory prediction dotted ine. We concude therefore that kinetic theory predicts resuts for the short-range potentia we, but it fais to predict the vaues obtained with the ong-range potentia. For the ong-range potentia the couping resistivities on the vapor side are much arger than the prediction from kinetic theory. Figures and give the two aternative mass transfer g g resistivities r and r. Again r is sma compared to r. The resuts for the iquid side have a better accuracy. For the short-range potentia, for which the agreement with kinetic theory was so good, we used the mass transfer resistivity to obtain the condensation coefficient. For the ong-range potentia there is no such agreement and it is therefore not appropriate to obtain a condensation coefficient in this manner. A resistivities approach zero for zero surface tension cose to the critica point as they shoud. Away from the critica point the resistivities increase their vaue as the surface tension increases. They a go to a maximum at the tripe point. The tripe point has the owest temperature of the investigation. The nature of the interaction potentia has an impact on a system properties, incuding the equiibrium properties. It is therefore difficut to compare directy the resuts for the short- and the ong-range potentia. As kinetic theory accounts for differences in temperature and vapor concentration, we have chosen to discuss the size of two of the coefficients in comparison with this theory. We have then seen that a more reaistic potentia, the ong-range potentia, gives a heat transfer resistivity which is sighty smaer than the prediction by kinetic theory and a couping resistivity which is 3 times arger than the prediction by kinetic theory. This is a substantia difference. This might have been expected, as kinetic theory is deveoped for hard spheres. A Lennard- Jones fuid with ong-range potentia therefore agrees even ess than the fuid with a short-range potentia with the premises of kinetic theory FIG. 3. The heat transfer to the heat fux on the vapor side for the Lennard-Jones fuid with a ong-range spine potentia circes and a short-range spine potentia trianges. C. The heat of transfer The couping coefficient for heat and mass transport deserves extra attention, because of its importance to predict the heat fux into the two phases adjacent to the surface. For a physica interpretation, the couping coefficient is most convenienty expressed via the heat of transfer. When the heat fux on the vapor side is used to define the heat of transfer, we have q *s,g = J q g J T= = r s,g q s,g r. qq 3 Using the heat fux on the iquid side, we have q = *s, J q = r s, q s, J T= r, 33 qq where we note that r s,g qq =r s, qq. When these definitions are combined with Eq. b, wefind q *s, q *s,g = vap H. 34 During evaporation, there is a heat sink in the surface, given by the enthapy needed to transfer a partice from the iquid to the vapor phase. The surface may coo down, but at stationary state the heat must come from the surroundings. The heats of transfers give the fractions of the enthapy of evaporation which must be taken from the respective phases. A arge vaue for say q *s, means that a arge part of the required heat is taken from or returned to the iquid side, depending on the sign of the mass fux. Their absoute ratio q *s, /q *s,g is equa to r s, q /r s,g q and the couping resistivities are given in Figs. 7 and 8. We find that the absoute vaue of the heat of transfer is arger on the iquid side than on the vapor side, for the ong-range potentia 3 times as we as for the short-range potentia 9 times. So the iquid provides 75% of the required heat for evaporation of partices with a ong-range potentia and 9% for a short-range potentia. The minus sign of the heat of transfer on the vapor side see Fig. 3 expresses that heat is transferred in a direction opposite to the mass fux

10 GE et a. The heat of transfer on the vapor side from this simuation and from the previous resuts were compared with the prediction of kinetic theory in Fig. 3. We see again, that the system with the ong-range potentia is not at a predicted by the corresponding kinetic theory coefficients. For the shortrange interaction potentia the agreement is reasonabe. V. CONCLUSION We have used moecuar dynamics simuations to study a one-component fuid interacting with a ong-range Lennard- Jones spine potentia, r c =.5. We presented equiibrium properties of the system, and transfer coefficients for heat and mass transport. We found that the reationship between surface tension and surface temperature are the same in equiibrium and in nonequiibrium. This once more proved that, the surface can be regarded a separate system and that this PHYSICAL REVIEW E 75, system is in oca equiibrium. This is a prerequisite for using nonequiibrium thermodynamics. We cacuated the fim resistivities and compared them with the vaues obtained for the short-range potentia systems and with kinetic theory. Whie the predictions of kinetic theory seemed to agree with the resuts obtained for the short-range potentia, they do not agree with our resuts. For the ong-range potentia the couping coefficient on the vapor side was 3 times arger than the prediction by kinetic theory, and cannot be negected in an accurate cacuation of the heat fuxes in the system. This points to a need for more reaistic data for transfer resistivities, so that such systems can be accuratey modeed. ACKNOWLEDGMENT The authors are gratefu for the Storforsk Grant No /V3 from the Norwegian Research Counci. A. Røsjorde, S. Kjestrup, D. Bedeaux, and B. Hafskjod, J. Cooid Interface Sci. 4, 355. J. Xu, S. Kjestrup, D. Bedeaux, A. Røsjorde, and L. Rekvig, J. Cooid Interface Sci. 99, R. Tayor and R. Krishna, Muticomponent Mass Transfer Wiey, New York, D. Bedeaux and S. Kjestrup, Int. J. Thermodyn. 8, E. Johannessen and D. Bedeaux, Physica A 33, J. M. Simon, S. Kjestrup, D. Bedeaux, and B. Hafskjod, J. Phys. Chem. B 8, J. W. Cipoa, Jr., H. Lang, and S. K. Loyaka, J. Chem. Phys. 6, F. Duan, V. K. Badam, F. Durst, and C. Ward, Phys. Rev. E 7, M. Bond and H. Struchtrup, Phys. Rev. E 7, J. W. Gibbs, The Scientific Papers of J. W. Gibbs Dover, New York, 96. A. Røsjorde, D. W. Fossmo, D. Bedeaux, S. Kjestrup, and B. Hafskjod, J. Cooid Interface Sci. 3, 78. D. Bedeaux and S. Kjestrup, Physica A 7, S. R. de Groot and P. Mazur, Non-Equiibrium Thermodynamics Dover, London, T. Ytrehus, Mutiphase Sci. Techno. 9, B. Hafskjod and S. K. Ratkje, J. Stat. Phys. 78, A. Z. Panagiotopouos, Mo. Phys. 6, R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids McGraw-Hi, New York, B. D. Todd, P. J. Daivis, and D. J. Evans, Phys. Rev. E 5, B. D. Todd, D. J. Evans, and P. J. Daivis, Phys. Rev. E 5, E. A. Guggenheim, J. Chem. Phys. 3, Voker C. Weiss and Woffram Schroer, J. Chem. Phys.,