Conference-ACSIN-10-The Problem of Surface Tension Definition of Nanodrops

The dependence of the surface tension of Lennard-Jones liquid small drops on the radius and temperature has been calculated by the molecular dynamics method. It is found that the mechanical surface tension is different from the Gibbs’s surface tension for drops of 50 – 2000 molecules and is equal to it for drops of more than 2000 molecules. It is shown that both the mechanical and Gibbs’s surface tensions decrease with the decrease of the equimolar radius of the drop and reach zero at the same R0 depending on temperature. The radii of tension also reach zero at the same R0. The dependence of the ratio of the mechanical surface tension of the drop to the surface tension of the flat surface liquid–vapor (σ/σ∞) on the ratio of the equimolar radius of the drop to R0 is a universal function. The limit of applicability of the surface tension concept to the small radii droplets changes from 50 to 300 molecules depending on temperature. [DOI: 10.1380/ejssnt.2010.197]


INTRODUCTION
The surface tension of liquids is an important parameter for describing thermodynamic processes in multiphase systems [1].The concept of the surface tension in thermodynamics is propounded in J. Gibbs works [2], it is also conventional.According to this conception an additional term σdA (σ is the surface tension, dA is change of the surface area of liquid) is introduced into fundamental thermodynamic equations.Thus, such surface tension σ determines additional energy to the thermodynamic potentials and is called the thermodynamic surface tension; this energy is connected with the change of the surface area of liquid.Alongside with the surface tension J. Gibbs introduces the dividing surface which is called the tension surface.This is a zero thickness mathematical surface to which the surface tension is applied.For the thermodynamic description of systems with flat dividing liquid-gas surface the position of the tension surface is of no importance.If the dividing surface is curved, there is a problem of the tension surface choice.
On the other hand, there is a mechanical definition of the surface tension [3] connected with the substitution of a smooth density profile and a smooth pressure profile by steps.In this case the surface tension is expressed in terms of pressure tensor of real liquid-gas interface where P N (z) is a normal component and P T (z) is a tangential component of the pressure tensor.They depend on z coordinate, which is normal to the dividing surface.Though there is some uncertainty of a choice of microscopic expression for the pressure tensor, the surface tension should not depend on this choice [3,4].
There isn't any reason to presume, that the thermodynamic definition and the mechanical one must lead to different values for flat liquid-gas interface.Really, Baidakov and others have calculated the surface tension of the Lennard-Jones liquid by the molecular dynamic method in two ways [5].The mechanical surface tension has been calculated by the formula (1), and the thermodynamic one has been calculated according to the equation derived from the fundamental equation of the Gibbs capillarity theory.Here u s is the surface density of internal energy.The calculations have shown good coincidence of the mechanical and thermodynamic values of surface tension.
The problem of surface tension definition becomes complicated when we go over to curved surfaces of the liquidvapor interface, in particular, to a liquid drop surrounded by its own vapor.In this case the value of the surface tension will depend on the choice of the tension surface position.J. Gibbs has proposed to choose the tension surface position that the surface tension should have the minimum value.Thus the Gibbs's model assumes that inside the sphere with the tension surface radius there is a fluid with bulk liquid properties.The chemical potential of this liquid coincides with the chemical potential of the vapor surrounding the sphere.Thus, the pressure, density and temperature inside the drop are equal to their values in the bulk liquid at the given chemical potential.The fundamental thermodynamic equation for the system internal energy takes the usual form of Here dU , dS and dm are the change of the internal energy, that of the entropy and that of the mass of the whole system, respectively, dV l , dV v and dA are the change of the liquid drop volume, that of the vapor volume and the change of the drop surface area, P l , P v and T are the pressure inside the drop, the vapor pressure and the system temperature.The tension surface of the Gibbs's drop generally does not coincide with the equimolar surface, therefore some part of the mass of the liquid-vapor system does not belong to any of two phases and constitutes the part of the fluid adsorbed on the surface of the interface.The difference between the radii of the equimolar surface and the tension surface is determined by the parameter δ = R e −R s called the Tolman length.Tolman [6] obtained the surface tension dependence on the radius of the tension surface in the form ) where σ ∞ is the surface tension of the flat liquid-vapor interface.If we assume that δ = δ ∞ = const and δ/r 1, we'll get an approximate formula valid for large drops Thus, the thermodynamic surface tension defined by Gibbs satisfies the equations ( 3) and (4).J. Gibbs pointed out that the surface tension of a small drop must decrease with the decrease of the radius of the drop tension surface, so that at R s = 0 the surface tension should be also equal to zero (σ = 0) [2].
On the other hand, the mechanical surface tension is defined with the aid of the pressure tensor [3]: As R s in this formula is not known beforehand, it is necessary to have one more equation, which will define R s .In papers [4,7] the following equation is proposed for this purpose: This equation is derived from the Buff equations [8], based on the mechanical equilibrium conditions of the drop-vapor system.Using the equations ( 6) and (7) and knowing the profiles of the pressure tensor components one can calculate R s and σ.
In advance it is not obvious that the surface tension calculated by the formulae ( 6) and (7) will coincide with the thermodynamic surface tension used in (3).Here there is a problem of conformity of the surface tension defined by the formulae ( 6) and (7), on the one hand, and the Gibbs surface tension model, on the other hand.Besides, there is a problem of applicability of the surface tension concept to small drops and clusters since the surface tension is a macroscopic thermodynamic parameter, and a nanodrop is a microscopic system.
These problems acquire special importance for nucleation describing.One of the effective methods of nanoparticles production is condensation of the particles from supersaturated vapor.The first stage of the nanoparticles formation is nucleation -that is creation of critical embryos, from which the nanoparticles grow later by joining atoms from supersaturated vapor.There is a so-called Classical Nucleation Theory (CNT), which enables to calculate the nucleation rates depending on the process conditions [9][10][11].This theory is a statistical thermodynamic one really, because it uses the thermodynamic parameters of clusters and small particles, which are the critical embryos.The critical embryos at the moment of their occurrence are the liquid clusters or nanodrops and only later they become the solid nanoparticles as a result of cooling.The surface tension of the critical embryo is an important parameter in the Classical Nucleation Theory, because it defines the work of the critical embryo formation [9,10].Though this theory often leads to unsatisfactory results, it and its modifications are still being used for interpretation of experimental data [12,13].One of the causes of discrepancy between the CNT and experiment can be that the dependence of the surface tension on the critical embryo radius is ignored often.Thus the problem arises how to determine the droplet surface tension dependence on its radius and temperature.

II. FORMULATION OF THE PROBLEM
Thus the problem is to ascertain the difference between the thermodynamic and mechanical definitions of surface tension, and to calculate their dependences on the drop radius and the system temperature.
The most adequate methods for solving this problem are the direct numerical simulation methods: that of Monte Carlo and that of the molecular dynamics.The molecular dynamics method has been long used to calculate the surface tension for the flat division surface between liquid and vapor [4,5].It was also used many times to calculate the surface tension of the liquid drops [7,[14][15][16][17][18][19].From this point of view the most interesting papers are [7,14], in which the basic methods of molecular dynamics calculations of small drops surface tension are developed.At the same time all these calculations have been done in a narrow range of system parameters and give no possibility of precise determination of the dependence of the drops surface tension on their radii.Besides, these http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 8 (2010) papers do not concern the problem of applicability of the surface tension concept to the droplets of small radii.
In this paper the systematic calculations of the small drops by the molecular dynamics method are presented.The calculations had been done for a wide range of temperatures and system sizes and permitted to do some generalized conclusions.The limit of applicability of the surface tension concept to the small radii droplets has been established.

III. THEORY
In our study we dealt with the molecular systems, containing one nanodrop, surrounded by its own vapor, which was in thermodynamic equilibrium with the drop.In the obtained systems the pressure P (r) is the tensor with two components different from zero.P N (r) and P T (r) are the normal and tangential components of the pressure tensor calculated by molecular dynamics method.The mechanical surface tension is expressed in terms of the pressure tensor by the formulae ( 6) and (7).σ and R s were calculated with these formulae.
A concept of surface tension arises when a real drop with continuous density profile is replaced by a homogeneous liquid phase, inserted into a spherical container, which is separated from surrounding vapor by a thin film, compensating the difference of pressure inside and outside the sphere.The radius of the sphere is R s .Sometimes the internal liquid phase is called a comparison phase.The difference of the pressures between the comparison phase and the surrounding vapor obeys the Laplace formula: where P l and P g are the pressure of the liquid in the comparison phase and that of the vapor, respectively.The choice of the pressure and density of the liquid in the comparison phase is arbitrary.According to this choice values of σ, R s , and also the equimolar radius R e will change.It is common practice to distinguish mechanical and thermodynamic routes to the droplet surface tension [4,7].The mechanical route assumes equal forces and moments of forces in real and model systems.In this case equations of mechanical equilibrium of a drop are used.The thermodynamic route to the surface tension is used in thermodynamic equations.It is assumed, that these two surface tension definitions must bring different values σ and R s , at least for small droplets, though the nature of this difference is not quite clear.J. Gibbs defined surface tension of a drop in conditions of equal chemical potentials of the comparison phase and real system [2].This Gibbs's definition is often identified with the thermodynamic route.
It has been established, that formula (7) assumes that the pressure in the comparison phase is equal to the pressure in the center of the real drop.At the same time, the chemical potential of the comparison phase does not correspond to the chemical potential of the vapor surrounding the drop.Such correspondence can be observed solely for large enough drops, containing homogeneous liquid phase in their centers, if the radius of the phase exceeds the effective molecular interaction radius.If the real drop is very inhomogeneous due to its small size, the chemical potential of the molecule in the center of the drop depends not only on the corresponding pressure and density, but also on distribution of molecules, interacting with the given one, i.e., on pressure and density profiles.Thus the mechanical surface tension determined by formulae ( 6) and ( 7) cannot be applied in thermodynamics equations where the chemical potential of the comparison phase is equal to the chemical potential of vapor.
To calculate the surface tension of drops according to J. Gibbs definition, it is necessary to equate the chemical potential of the comparison phase to the chemical potential of the surrounding vapor, i.e., the comparison phase pressure and density must be identical to those of the bulk liquid for a given chemical potential.
It has been established, that the formula ( 7) is not correct for the calculation of the drop surface tension according to J. Gibbs definition, while the formula ( 6) is valid.Formula (8) can also be applied.That's why, if we know the state equation of the bulk liquid, i.e., the liquid pressure dependence on the density, we can calculate the comparison phase chemical potential for various pressures.Then we estimate the chemical potential of the vapor around the drop and equate it to the comparison phase potential.We get P l and ρ l of the comparison phase.Further, using formula (8) and the equation ( 6), we calculate σ and R s for the Gibbs surface tension.

IV. RESULTS OF CALCULATIONS
In our work, the surface tension of droplets was calculated by the molecular dynamics method.The calculations were made for the system containing 100 -4500 molecules in a cubic cell with periodic boundary conditions.Interaction between molecules was specified by Lennard-Jones potential with truncation at r = 5r 0 .Reduced variables were used: distance-r = r * /r 0 , temperature-T = kT * /ε, energy-U = U * /ε, densityρ = ρ * r 0 3 , time-t = t * /r 0 (ε/m 0 ) 1/2 , pressure-p = p * r 0 3 /ε, surface tension-σ = σ * r 0 2 /ε, and chemical potential-µ = µ * m 0 /ε.Here ε and r 0 are the Lennard-Jones potential parameters, m 0 is the molecule mass.Variables labeled by an asterisk are dimensional.A special procedure was used to obtain an equilibrium system composed of a liquid drop in the center of the cell and vapor occupying the remaining space.We call such drop obtained in numerical experiment real.The size of the drop depended on the number of particles in the cell and the mean density of the system.Density profiles, chemical potential of the system, equimolar radii and surface tension radii of drops, mechanical and Gibbs surface tensions were calculated.The details are given in [19].
Figure 1 presents typical profiles of drop density ρ(r).The equimolar radii of the drops were calculated according to these dependences by the formula [7] Here ρ l , ρ g are the comparison phase density and vapor density respectively.It should be noticed, that the http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/)   equimolar radii for the mechanical and Gibbs surface tension did not differ much.Figure 2 presents typical profiles of the pressure tensor components P N (r) and P T (r) used for calculations of the tension surface radius R s and σ the surface tension.
Figure 3 presents the nanodrops mechanical surface tension dependences on the equimolar radius R e for various temperatures.One can see, that the mechanical surface tension decreases greatly with the R e decrease and becomes equal to zero at a certain R e = R 0 , which depends on temperature.With R e increase the mechanical surface tension tends to the value of the surface tension of the liquid-vapor flat surface.
To calculate surface tension of drops according to J. Gibbs definition, first, one should know the dependence of the chemical potential of the bulk liquid phase on pressure, and, second, the chemical potential of vapor.To solve the first problem, we have made special series of bulk liquid phase chemical potential calculations by molecular dynamics method at the temperature T = 0.65.First, we obtained an equilibrium system consisting of a flat liquid phase layer surrounded by vapor from two sides.The layer thickness exceeded the molecules interaction radius three times.The computations have shown that the vapor above such flat separating surface is an ideal gas, i.e. the ideal gas equations p = ρT (ρ is vapor concentration) holds accurate to 1%.The chemical potential of the vapor and, therefore, that of the liquid layer was calculated by formula µ 0 = −(5/2)T ln(T ) + T ln(p) suitable for classical ideal gas.Then calculations were made for a similar system with two repulsing walls on the opposite sides of the cell that served as plungers compressing the flat liquid layer.Reducing the cell volume, we obtained liquid phases at different pressures.Thus the equation of liquid state at a given temperature has been obtained.The chemical potential of the liquid phase was calculated by the formula The approximation curve of dependences P l and ρ l on µ l was established.
The chemical potential of vapor has also been calculated by formula (10) with µ l replaced by µ g and P l -by P g .The equation of vapor state has been obtained from the molecular dynamics data for equilibrium systems containing a liquid drop surrounded by its own vapor, i.e., from the calculations of the surface tension of the drops.Vapor pressure and density were taken far away from the drop at the distance exceeding the interaction radius of molecules.After estimation of the chemical potential of the vapor around the drop we equated it to the comparison phase chemical potential, got P l and ρ l of the comparison phase.We used the approximation curve of the dependence of these values on the liquid chemical potential.Further, using the formula (8) and the equation ( 6), we calculated σ and R s for the Gibbs surface tension.tension can be observed with R e decrease.At a certain equimolar radius R 0 both surface tensions go to zero.This radius corresponds to the drop, containing about 50 molecules.The radius of the tension surface R s also goes to zero.With the further equimolar radius decrease the surface tension becomes negative, though the drop holds stable in the conditions of the numerical experiment.Figure 4 shows that the Gibbs surface tension coincides with the mechanical one for relatively big drops containing 2000 and more molecules.For drops containing smaller quantity of molecules the Gibbs surface tension exceeds the mechanical one up to their disappearance at R e = R 0 . V.

DISCUSSION OF RESULTS
It has been established in our paper [19] that the dependence of the nanodrops mechanical surface tension on the equimolar radius is a universal function relative to the temperature, at which the numerical experiment is conducted (see Fig. 5).It has been found that approximation of this dependence by the polynomial of the ratio R 0 /R e can be expressed as follows: ) In addition, the drop equimolar radius R 0 , at which σ = 0, greatly depends on the temperature and becomes infinite in the critical point liquid-vapor (see Fig. 6).The critical point is equal 1.199.The approximation of the dependence of R 0 on the temperature can be expressed by the expression The surface tension of the flat dividing surface liquidvapor σ ∞ also depends on the temperature and becomes zero in the critical point.The Gibbs surface tension dependence on the equimolar radius R e is the same in its quality as that for mechanical surface tension, though it differs a little in its quantity.For relatively large drops the surface tension approaches the value of the surface tension of the flat dividing surface liquid-vapor.For quite small drops the surface tension becomes zero or even negative.But the drop does not evaporate at that time.Though the drop will evaporate after all, if we increase the number of molecules in the simulation sell at the same pressure, because that leads to the increase of fluctuations of the force, acting on the nanodrop from the vapor.That's why the state of a nanodrop with a negative surface tension can be considered as a metastable one in respect of small fluctuations.
On the other hand, the negative surface tension and the negative radius of tension surface R s have no physical Volume 8 (2010) meaning and become formal mathematical parameters of the model.Apparently, it testifies that the notion of surface tension which is used in macroscopic theory can not be applied to such small drops.The limit of applicability of the surface tension concept to the droplets changes from about 50 molecules at T = 0.65 to about 300 molecules at T = 0.95.
Nevertheless the work of formation of a critical embryo is expressed by these values.This work is a very important parameter in the classical nucleation theory.As long ago as in his time J. Gibbs derived the simple expression for this work: W = (1/3)σs, where s = 4πR s 2 is the area of tension surface [2].In paper [18] it is shown that the work of droplets formation in the nucleation process is a positive value, even for the droplets with negative σ. Apparently we must admit that the work of the critical embryo formation is determined not only by its surface tension, but also by some other factors.

VI. CONCLUSION
Thus, it has been established that both mechanical and Gibbs surface tensions decrease greatly with the decrease of tension surface radius R s and become equal to zero at a certain R e = R 0 .A universal dependence of mechanical surface tension is observed, i.e., the dependence of σ on the temperature is expressed by the dependence of parameters σ ∞ and R 0 on the temperature.The value of the Gibbs surface tension coincides with the value of the mechanical surface tension for relatively large nanodrops (> 2000 molecules) and exceeds it for relatively small nanodrops (< 2000 molecules).The limit of applicability of the surface tension concept to the droplets changes from about 50 molecules at T = 0.65 to about 300 molecules at T = 0.95.

FIG. 2 :
FIG. 2:The profiles of the pressure tensor components for the system, containing N = 200 molecules in the cell at the temperatures T = 0.65; 0.7; 0.75; 0.8; 0.85 (from the top).(solid lines are PN , dashed lines are PT ).

Figure 4 0 FIG. 4 :
FIG. 4:The dependence of the mechanical (2) and Gibbs (+) surface tensions on the equimolar radius Re at the temperature T = 0.65 .