Curvature dependence of the interfacial heat and mass transfer coefficients
aa r X i v : . [ c ond - m a t . s o f t ] O c t Curvature dependence of the interfacial heat and mass transfer coefficients.
K. S. Glavatskiy , and D. Bedeaux School of Applied Sciences, Royal Melbourne Institute of Technology, Melbourne, VIC 3001, Australia and Department of Chemistry, NO 7491, Norwegian University of Science and Technology, Trondheim, Norway
Nucleation is often accompanied by heat transfer between the surroundings and a nucleus ofa new phase. The interface between two phases gives an additional resistance to this transfer.For small nuclei the interfacial curvature is high, which affects not only equilibrium quantitiessuch as surface tension, but also the transport properties. In particular, high curvature affectsthe interfacial resistance to heat and mass transfer. We develop a framework for determiningthe curvature dependence of the interfacial heat and mass transfer resistances. We determine theinterfacial resistances as a function of a curvature. The analysis is performed for a bubble of a one-component fluid and may be extended to various nuclei of multicomponent systems. The curvaturedependence of the interfacial resistances is important in modeling transport processes in multiphasesystems.
I. INTRODUCTION.
Mesoscale structures in soft matter can spontaneously form in such systems as surfactant solutions [1]. They arecharacterized by small aggregates of a new phase on a nanometer scale. The growth of nuclei is the first step in amacroscopic phase transformation [2]. These processes have been studied more than a hundred years [3]. Duringnucleation, there is an energy barrier due to the energy costs to create an interface between the two phases [4]. Theclassical theory of nucleation fails to describe the results of experiments adequately [5]. A number of extensions havetherefore been proposed [6]. Some employ kinetic equations [7], other molecular dynamic simulations [8]. The overallpicture is still far from clear [9]. The overall reason for a system to nucleate, however, is to decrease the total system’sGibbs energy. The homogeneous phase inside the nucleus has a lower Gibbs energy than the nucleating phase. Belowthis critical size, the nucleus prefers to shrink. Above this critical size, it starts to grow. The typical size of the criticalnucleus is of the order of nanometers. An important aspects of the nucleation process is a small radius of a nucleus,which corresponds to its high curvature. The curvature increases the surface tension relative to the surface tension ofa flat interface [10].One of the main issues, which make nucleation complicated, is that it is a non-equilibrium process [6]. There existfluxes of heat and mass between the surroundings an the nuclei, which facilitate its growth. The interface betweentwo phases has an additional resistance for heat and mass transfer even for a flat interface [11]. During nucleation theinterfacial curvature is finite and changes the heat and mass resistances of the interface. It is the aim of this paper toinvestigate how the curvature of a small nucleus affects the interfacial resistance.Earlier we developed a tool to analyze the interfacial resistances for flat interfaces [12, 13]. We introduced integralrelations which allowed us to calculate the interfacial resistances knowing only the equilibrium properties of thesystem. This is very useful for our purpose, since nucleation is a non-stationary process, which is complicated toanalyze. Earlier we calculated the resistances of flat surfaces [11] and in this paper we will extend the analysis to aspherical surface.The equilibrium properties of the system are modeled with the help of the square gradient model [14, 15]. The squaregradient theory for the interfacial region originates from the work of van der Waals [16] for liquid-vapor equilibriumof a one-component fluid and the work of Cahn and Hilliard [17] for fluid-fluid equilibrium of binary mixtures. Theintroduction of the gradients of the densities in the thermodynamic description successfully explains macroscopicthermodynamic behavior of two-phase coexistence, in particular, the surface tension [14]. It has been widely used tomodel the surface behavior of planar fluid interfaces [18]. A systematic extension of this theory to non-equilibriumsystems using non-equilibrium thermodynamics in two-phase multi-component systems has been given [19–24].In this paper we establish a method to calculate the interfacial resistances for spherical interfaces. The resultscan be in principle verified in molecular simulations [9] or experiments. Performing a particular measurement ofthe interfacial resistance may be a complicated process, so it is important to understand what data one may expectfrom particular experiments. Here we consider a spherical bubble of a one-component system. However, the analysisis applicable to droplets and multicomponent systems with no restrictions in generality. The paper is organized asfollowing. In Sec. [II] we give a brief description of the key points of the square gradient model. In Sec. [III] wediscuss how the macroscopic properties of the system are connected to the local continuous profiles. We introducethe excess densities and excess resistances, the properties which describe behavior of the entire interface. In Sec. [IV]we present calculations of the interfacial resistances according to the developed model. The curvature dependence ofthe resistances is discussed. Finally, in Sec. [V] we summarize our findings.
II. LOCAL DESCRIPTION OF A SPHERICAL SURFACE.
For a one-component system the specific local free energy in the interfacial region is a function of both the massdensity ρ ( r ) and the mass density gradient ∇ ρ ( r ): f ( r ) = f ( T, ρ ) + 12 κρ ( r ) |∇ ρ ( r ) | (1)where f is the homogeneous free energy and κ is a parameter of the model, independent of the temperature, whichshould be chosen such that the value of the surface tension reproduces a typical experimental value. In equilibriumthe total Helmholtz energy reaches its minimum given the condition that the total mass is fixed. This requiresminimization with respect to the density of the grand potential Ω = − R d r p ( r ), where p ( r ) = ( µ e − f ( r )) ρ ( r ) and µ e is the equilibrium chemical potential: µ e = ∂ ( ρf ) ∂ρ − κ ∆ ρ (2)where ∆ ≡ ∇ · ∇ is the Laplace operator. In a spherically symmetric system all the quantities depend only on theradial coordinate so that Eq. (2) becomes µ e = µ ( ρ, T ) − κ (cid:18) ρ ′′ + 2 r ρ ′ (cid:19) (3)where µ ( ρ, T ) is the homogeneous chemical potential and prime indicates derivative with respect to the radius. Theactual density profile can be found from Eq. (3).In the interfacial region the pressure becomes a tensor σ αβ ( r ) = p ( r ) δ α β + γ α β ( r ) (4)where γ α β ( r ) ≡ κ ∇ α ρ ( r ) ∇ β ρ ( r ). In a spherically symmetric system σ αβ has a diagonal form with σ ( r ) ≡ p n ( r )being the so-called normal pressure p n ( r ) = p ( r ) + γ rr ( r ) = p ( r ) + κρ ′ ( r ) (5)and σ ( r ) = σ ( r ) ≡ p t ( r ) being the tangential pressure: p t ( r ) = p ( r ).Note that, unlike in the system with planer interface, the normal pressure is not constant with respect to theposition through the interface. This leads to the existence of Laplace pressure [14].In non-equilibrium the thermodynamic properties change not only with position, but also with time. Furthermore,the temperature and the chemical potential are no longer constant. However, a local description of the interfacialregion can still be given with the help of the square gradient model. To extend the equilibrium square gradientmodel to non-equilibrium we will assume that all the thermodynamic densities are given by the same expressions asin equilibrium [24]. In particular the specific Helmholtz energy f ( r, t ) = µ ( r, t ) − p ( r, t ) /ρ ( r, t ) (6)Furthermore, the chemical potential µ ( r, t ) = µ ( ρ ( r, t ) , T ( r, t )) − κ (cid:18) ρ ′′ ( r, t ) + 2 r ρ ′ ( r, t ) (cid:19) (7)where a prime indicates the derivative with respect to r .The non-equilibrium thermodynamic relations need to be supplied by the balance equations. For a one-componentsystem there are four balance equations, for the density, momentum, energy and entropy. For a spherically symmetricfluid with all fluxes along the radial direction with the barycentric velocity having only the radial component v. Thebalance equations are ∂ρ∂t = − r ∂∂r (cid:0) r ρ v (cid:1) ∂ρ v ∂t = − r ∂∂r (cid:0) r ( p n + ρ v ) (cid:1) − pr∂ρu∂t = − r ∂∂r (cid:0) r J e (cid:1) ∂ρs∂t = − r ∂∂r (cid:0) r J s (cid:1) + σ s (8)where σ s is the entropy production, while J e and J s are the total energy and entropy flux respectively. It is convenientto introduce the mass flux J m , the momentum flux J p , the heat flux J q as J m ≡ ρ v J p ≡ p n + ρ v J q ≡ J e − J m ( h + v /
2) (9)where h = u + p/ρ is the specific enthalpy and u is the specific internal energy. In stationary states the left hand sideof all the equations in Eq. (8) is zero, so it takes the following form (cid:0) r J m (cid:1) ′ = 0 (cid:0) r J e (cid:1) ′ = 0 (cid:0) r J p (cid:1) ′ = 2 rp (cid:0) r J s (cid:1) ′ = r σ s (10)Note, that unlike in the case of planar interface, the mass flux and the energy flux in the direction across the interfaceare not constant. They decrease inversely proportionally to the radius squared. This leads to the fact that instationary states both the mass flux and the energy flux become infinite at the origin. In order to make this possibleone has to introduce a source or sink for heat and mass at a spherical surface close to the center of the bubble. Usingpreviously derived integral relations for the transfer coefficients for heat and mass transfer through a surface we onlyneed equilibrium profiles. We therefore refrain from a further analysis of stationary states.To obtain the expressions for the interfacial resistances, we need to consider non-equilibrium. For a proper descrip-tion of a non-equilibrium process the Gibbs relation is required. Following [24], we write the Gibbs relations for aspherical system as T dsdt = dudt + p dvdt − v v 1 r ∂∂r (cid:0) r γ rr (cid:1) (11)where s , u , v ≡ /ρ are the specific entropy, internal energy and volume respectively, which are related to the otherthermodynamic quantities in a manner, which is similar to Eq. (6) and Eq. (7). Furthermore, d/dt is the substantial(barycentric) time derivative: d/dt = ∂/∂t + v ∂/∂r . Note, that Eq. (11) is not restricted to the stationary statecondition. All the quantities depend in general both on position and time. However, the arguments ( r, t ) were omittedto simplify the notation.Combining the above equations we obtain the expression for the local entropy production in the interfacial region: σ s = J q (cid:18) T (cid:19) ′ (12)The entropy production is always positive and therefore the heat flux is given by the linear constitutive relation (cid:18) T (cid:19) ′ = r qq J q (13)In the context of the square gradient theory the local resistivity profile r qq is represented by the two terms[20, 25],a homogeneous term and a square gradient term: r qq ( r, t ) = r qq, ( T, ρ ) + A ( T, ρ ) | ρ ′ | (14)where ρ and T also depend on position and time. In principle, the homogeneous term r qq, is given by a kind ofequation of state for the resistivity. Given the lack of knowledge about the temperature and the density dependencewe model this term by a linear interpolation between two known values of the bulk resistivities: r qq, ( ρ ) = r iqq + ( r oqq − r iqq ) ρ − ρ i ρ o − ρ i (15)where r iqq and r oqq are the resistivities of the coexisting homogeneous inner and outer phase with a flat interface, takenfor instance at the temperature of the outer boundary of the box. ρ i and ρ o are similarly densities of coexistinghomogeneous inner and outer phases with a flat interface at this temperature.It was shown earlier [11] that the existence of the square gradient contribution is consistent with the second lawof thermodynamics and gives a more accurate description of the interfacial resistances for the planar interface thanthe expressions of the kinetic theory. The coefficient A in the square gradient contribution may depend on the localtemperature and density. In the previous work for a planar interface [25] it was modeled as A = α r oqq + r iqq max[ ∇ ρ ( r )] (16)where α is a dimensionless coefficient of the order of unity and max[ ∇ ρ ( r )] is the maximum value of the densitygradient for the planar interface. This maximum corresponds to the inflection point of the density profile at thetemperature considered.We note that r iqq , r oqq , ρ i , ρ o , and max[ ∇ ρ ( r )] are parameters of the resistivity profile r qq ( r, t ). In the context of thesquare gradient model r qq depends only on T ( r, t ), ρ ( r, t ) and ρ ′ ( r, t ), the local values of the temperature, the densityand the density gradient. Thus, the above parameters are just constants. In particular, these values do not dependon the curvature of the interface and should be calculated for a flat interface. A dependence of these parameters onthe surface curvature would make the theory non-local. As this is inherently inconsistent with the square gradientdescription, we will not study this here. Note, that due to this, the values of r qq, in the origin and at the outerboundary are not equal to r iqq and r oqq respectively. III. EXCESS RESISTANCE OF A SPHERICAL SURFACE.
On a macroscopic level the interfacial region is described by the so-called excess quantities. In equilibrium they allowone to consider the entire interface as a single entity. The use of excess quantities can be extended to non-equilibriumand this idea has been proven to be useful in many applications [26], showing that non-equilibrium interface can alsobe considered as a single entity.Excess quantities are defined using local continuous profiles. A discussion of the technical details of this definitionhas been presented in [24]. A general theory of the non-equilibrium interface in terms of the excess densities incurvilinear coordinates using the non-equilibrium local description was presented in [12]. Here we briefly summarizethe main points for a spherical interface.A key quantity in a macroscopic description of the interface is the excess of a thermodynamic density (mass density,energy density, entropy density, etc.). While the density is measured per unit of volume, the excess of a density ismeasured per unit of surface area. It depends on the position R of so-called dividing surface, which may be chosenarbitrarily inside the interfacial region. It is one of the properties of the surface, that while this choice affects the valuesof different thermodynamic properties, it does not affect the thermodynamic relations. This is true in equilibriumand has been recently verified in non-equilibrium [21, 24, 27]. For a thermodynamic density per unit of volume φ itsexcess in spherical coordinates b φ is defined as b φ ( R ) ≡ R Z L dr r φ ex ( R, r ) (17)where L is the radius of the spherical box, R is the position of a dividing surface and φ ex ( R, r ) ≡ φ ( r, t ) − φ s,i ( r, t ) Θ( R − r ) − φ s,o ( r, t ) Θ( r − R ) (18)where Θ is the Heaviside function (1 for positive and 0 for negative values of the argument), while φ i and φ o are valuesof the homogeneous densities inside and outside the bubble extrapolated to the dividing surface R . For equilibriumprofiles φ i and φ o are independent of r and t and we take φ i to be equal to the value of φ in the center of the bubbleand φ o to be equal to the value of φ at the outer boundary.There exist various choices of the dividing surface and a common one is the the equimolar dividing surface R ρ . Incase of a one-component fluid it is defined as b ρ ( R ρ ) = 0. The other choices of the dividing surface are the surface oftension and the inflection point. The further analysis does not depend on a particular choice of the dividing surface,and we will not specify it until the results. Each of the dividing surfaces can be used to define the size of the bubble.We will now consider stationary states. One of the relevant quantities for heat and mass transport across theinterface is the excess entropy production b σ s . The local entropy production is a density per unit of volume, sothe excess entropy production is defined using Eq. (17). It can be shown [12] that the excess of the local entropyproduction which is given by Eq. (12), is b σ s = J iq ∆ 1 T − J m (cid:18) ∆ e µT − e h i ∆ 1 T (cid:19) = J oq ∆ 1 T − J m (cid:18) ∆ e µT − e h o ∆ 1 T (cid:19) (19)where ∆(1 /T ) ≡ /T o − /T i and ∆ e µ/T ≡ e µ o /T o − e µ i /T i are the jumps between the values of the correspondingfunctions extrapolated from the two bulk regions to the interfacial region and evaluated at the dividing surface.Furthermore, the superscripts i and o indicate the values of the corresponding homogeneous quantities inside andoutside the bubble, which are extrapolated to the dividing surface R . All the quantities in Eq. (19) depend on thechoice of the dividing surface. Furthermore, e µ ≡ µ + v / e h ≡ h + v / J iq and J oq . These fluxes are different, and their difference is determined by the enthalpy of the phase change: J iq − J oq = J m (cid:16)e h o − e h i (cid:17) .The form of the excess entropy production (19) suggests the force-flux relations∆ 1 T = R qq J νq − R νqm J m ∆ e µT − e h ν ∆ 1 T = R νmq J νq − R νmm J m (20)where ν is either i or o . Note, that R qq is independent of ν . The coefficients R qq , R νqm , R νmq and R νmm are theresistances of the interface to the heat and mass transfer. They determine the jumps of the temperature and thechemical potential across the bubble interface. Like all other interfacial quantities, these resistances depend on thesize of the bubble. It is the aim of this paper to investigate this dependence.In the context of linear irreversible thermodynamics these resistances are determined by equilibrium properties ofthe interface. Analysis in [12] and [11] applied to a one-component system gives the following expressions for theinterfacial resistances: R qq = E [ r qq ] R νqm = R νmq = E (cid:2) r νqm (cid:3) R νmm = E [ r νmm ] (21)where E [ φ ] denotes the excess of a quantity which is not a density. For the resistances it is defined as E [ φ ] ( R ) ≡ R Z L dr r φ ex ( R, r ) (22)where φ ex is still given by Eq. (18) and r qq ( r ) is given by Eq. (14). Furthermore, we have introduced as short handnotation: r νqm ≡ r qq ( h ν − h ) r νmm ≡ r qq ( h ν − h ) (23)Note, that in contrast to the multicomponent system, these quantities are not independent: they are proportional tothe heat resistivity r qq . Note furthermore, that for equilibrium profiles e h = h . Furthermore, h i is equal to the actualvalue of h in the center of the bubble and h o is equal to the actual value of h at the outer boundary. IV. RESULTS AND DISCUSSION.
We consider cyclohexane and use the van der Waals equation of state at the temperature T = 330 K. The vander Waals parameters A = 2 .
195 [J m /mol] and B = 14 . × − [m /mol]. The parameter of the square gradientmodel κ = 1 . × − [J m /kg ], which gives the value of the surface tension of the planar interface γ = 0 . L = 80 nm with a bubble being formed in the center. Toavoid boundary effects, when the size of the bubble is close to the size of the container, the maximum bubble sizeconsidered is equal to 65 nm. As it was mentioned above, there exist a minimum size of the bubble, due to the finitecompressibility of the liquid. For cyclohexane this size is equal to approximately 18 nm. To avoid effects of instability,the minimum bubble size considered here is equal to 25 nm. This range of bubble sizes is on the one hand good toconsider large curvatures, and on the other hand it gives sufficient data to extrapolate them to planar interfaces.The typical profiles of the density are given in Fig. 1. The curves in Fig. 1 represent the bubbles of different size.Gradual increase of the total mass of the fluid leads to a gradual decrease of the bubble radius.The local resistivity profiles which are modeled by Eq. (14) from these density profiles are presented in Fig. 2.Eq. (14) contains one parameter α qq , which determines the significance of the square gradient contribution to thelocal resistivity and therefore the magnitude of the peak in the interfacial region. It was shown in [11] that in orderto satisfy the second law of thermodynamics this parameter should differ from zero. In this calculations we used thevalue α qq = 9. The thermal conductivity of the gas and the liquid phase ℓ iqq = 0 . ℓ oqq = 0 . r νqq = ( ℓ νqq T ) − . ρ [ k g / m ] FIG. 1: Density profiles for various bubble sizes.
Next we consider the dependence of the interfacial resistances R qq , R oqm and R omm on the curvature. We do this −3 r [nm] r qq [ ( m s ) / ( J K ) ] FIG. 2: Resistivity profiles for various bubble sizes. for three choices of the dividing surface: equimolar surface (em), surface of tension (st) and the inflection point (ip).The dependencies are presented in Fig. 3, Fig. 4 and Fig. 5 respectively. In addition, the value of the correspondingresistances for the planar interface is indicated (the point of zero curvature).Furthermore, a quadratic fit of the form R ab = R ab, (cid:18) R ab, R + R ab, R (cid:19) (24)where ab stands for either qq , or qm , or mm , is applied to the data and plotted by a solid line. The fit includes thezero-curvature value of the resistances. The values of the coefficients are given in Table I, Table II and Table III. Inaddition, the actual value for the planar resistance is given. TABLE I: The values of the R qq resistance for the planar interface andthe coefficients of the quadratic fit (24) for different dividing surfacesdividing R qq, ∞ , R qq, , R qq, , R qq, ,surface (m s)/(J K) (m s)/(J K) nm nm em 3.0920 × − × − - 0.1469 1.4838st 3.1044 × − × − - 0.4852 - 8.1855ip 3.1146 × − × − - 0.8534 - 4.2083 −11 R qq [ ( m s ) / ( J K ) ] emstip FIG. 3: Excess resistance R qq as a function of the bubble curvature for different dividing surfaces: equimolar surface (em),surface of tension (st), inflection point (ip). Symbols represent the data from Eq. (21), lines represent the quadratic fit Eq. (24).TABLE II: The values of the R qm resistance for the planar interface andthe coefficients of the quadratic fit (24) for different dividing surfacesdividing R qm, ∞ , R qm, , R qm, , R qm, ,surface (m s)/(mol K) (m s)/(mol K) nm nm em -1.2939 × − -1.2939 × − × − -1.3138 × − × − -1.3294 × − R mm resistance for the planar interfaceand the coefficients of the quadratic fit (24) for different dividing surfacesdividing R mm, ∞ , R mm, , R mm, , R mm, surface (m s J)/(mol K) (m s J)/(mol K) nm nm −7 R q m o [ ( m s ) / ( m o l K ) ] emstip FIG. 4: Excess resistance R oqm as a function of the bubble curvature for different dividing surfaces: equimolar surface (em),surface of tension (st), inflection point (ip). Symbols represent the data from Eq. (21), lines represent the quadratic fit Eq. (24).em 4.4428 × − × − - 1.4042 - 51.2815st 4.7242 × − × − - 0.3404 - 48.5718ip 4.9452 × − × − - 1.0753 - 42.6580 It is interesting to observe, that the resistance curves for the different dividing surfaces have a common point ofintersect. It is easy to understand that there could exist such point R ∗ , which we will call the static point . This isthe point where the resistance is the same for different dividing surfaces. In other words, at this point the excessresistance does not change when we change the dividing surface. For a resistance R ab which depends on the position R of the dividing surface this condition is expressed as dR ab /dR = 0. Using Eq. (22) this gives the condition for thestatic point dR ab dR = r o − r i + 2 R ∗ R ab ( R ∗ ) = 0 (25)The position of the static point is determined by the value of the difference between the bulk resistivities r o − r i andthe value of the excess resistance. The heat resistivity r qq of the gas phase is higher than the one of the liquid phase,so that r oqq < r iqq . Furthermore, R qq resistance is always positive. This makes the static point for R qq resistance tobe positive. For the system studied, the static point is situated at approximately 107.7 nm, giving the value of R qq −4 R mm o [ ( m s J ) / ( m o l K ) ] emstip FIG. 5: Excess resistance R omm as a function of the bubble curvature for different dividing surfaces: equimolar surface (em),surface of tension (st), inflection point (ip). Symbols represent the data from Eq. (21), lines represent the quadratic fit Eq. (24). excess resistance approximately 3.0885 × − (m s)/(J K). r oqm resistivity has a higher value for the liquid than forthe gas, so that r oqm > r iqm . Furthermore, R oqm resistance is always negative. This makes the static point for R oqm resistance to be positive as well. For the system studied, the static point is situated at approximately 27.4 nm, givingthe value of R oqm excess resistance approximately -1.3155 × − (m s)/(mol K). r omm resistivity has a lower valuefor the outer phase than for the inner phase, so that r omm < r imm . R omm is also positive. Extrapolation of the dataindicates that R omm changes the sign at approximately 9 nm, where a stable bubble does not exist. Within the domainof these curvatures r i − r o ≈ . × (m s J)/(mol K) is always larger than 2 R omm /R , which makes Eq. (25) tohave no solution for R omm . This means that the static point for R mm resistance does not exist.We also note, that the resistances do not necessarily depend monotonously on the curvature. While the heatresistance for the surface of tension and the inflection point decrease monotonously with increasing curvature, theheat resistance for the equimolar surface has a minimum value of approximately 3.0812 × − (m s)/(J K) when thesize of the bubble is approximately 21.7 nm, which corresponds to the curvature 0.046 nm − . In order to understandthis behavior, it is useful to consider the expression (14) for the local heat resistivity. For the equimolar surface theexcess of the first term is equal to zero, E [ r qq, ] ( R em ) = 0. Thus, excess of the heat resistance is entirely due to thesquare gradient contribution. It is the combination of contribution from the A | ρ ′ | factor and the ( R/r ) factor tothe excess, which makes the curvature dependence of the heat resistance for the equimolar surface to have a convexshape. The heat resistance for the other dividing surfaces, surface of tension and inflection point, has additional1terms. Indeed, the change ∆ R ab of the resistance due to the change δ of the dividing surface is, according to Eq. (22)∆ R ab ≈ dR ab dR δ = (cid:18) r o − r i + 2 R R ab ( R ) (cid:19) δ (26)which increases with the curvature. Thus, the resistance for the surface of tension or the inflection point will divergefrom the resistance for the equimolar surface when the curvature is increasing. We observe exactly this behavior inFig. 3, Fig. 4 and Fig. 5. V. CONCLUSIONS.
We have presented a framework to calculate the interfacial heat and mass resistances of curved surfaces. The methodof determining interfacial resistances is in the context of the Gibbs excess quantities. In particular, the resistances arerepresented as the excesses of local resistivity profiles. Local resistivities are calculated with the help of the squaregradient model, an approach which has shown to be useful for the description of the interfaces.Calculation of the interfacial resistances requires only equilibrium information about the system. In particular, thelocal resistivity profiles, which are the input quantities for the calculation of the excesses, are calculated with the helpof the equilibrium density profiles.We have investigated how the interfacial resistances depend on the interface curvature. It was shown that theychange with the curvature at least quadratically. In a closed system there exist restrictions on the minimum size ofa stable bubble [28] because of the non-zero compressibility of the liquid. Thus, the curvature of a stable bubble ina closed system has an upper bound, which limits the magnitude of the resistance. In open systems, even thoughall bubbles and droplets are unstable, there is no restriction on the nucleus size [29], so the excess resistance is notlimited. However, when the curvature of the system becomes extremely high, the interfacial region fuses with theinner phase and the notion of the excess resistance is undefined. Further research is needed to address such highcurvatures.We have found, that the resistances for different dividing surfaces are different. The interfacial resistance cannot bemeasured on their own without specifying the dividing surface position. This behavior of the interfacial resistancesis analogous to the fact that most of the Gibbs excess densities depend on the choice of the dividing surface[24].However, the form of the force-flux relations (20) is the same for all choices of the dividing surface, just like the Eulerrelation between the Gibbs excess densities is the same for all choices of the dividing surface.
Acknowledgments
Dick Bedeaux wants to thank Øivind Wilhelmsen for extensive discussions. [1] S. Komura.
J.Phys.: Condes. Matter , 19:463101, 2007.[2] C. Domb, M.S. Green, and J.L. Lebowitz, editors.
Phase Transitions and Critical Phenomena , volume 1-20. AcademicPress, 1972-2001.[3] J. Feder, K.C. Russell, J. Lothe, and G.M. Pound.
Adv. Phys. , 15:111, 1966.[4] D. Kashchiev.
Nucleation. Basic Theory with applications . Butterworth-Neinemann, 2000.[5] H. Vehkamaki.
Classical Nucleation Theory in Multicomponent Systems . Springer, Berlin, 2006.[6] D. Reguera.
J.Non-Eq.Therm. , 29:327, 2004.[7] V.G. Dubrovskii and M.V. Nazarenko.
J. Chem. Phys. , 132:114507, 2010.[8] S.M. Kathmann, G.K. Schenter, B.C. Garrett, B. Chen, and J.I. Siepmann.
J. Phys. Chem C , 113:10354, 2009.[9] A.Lervik, F.Bresme, and S.Kjelstrup.
Soft Matter , 5:2407, 2009.[10] W. Helfrich.
Z. Naturforsch. , 28c:693, 1973.[11] K. S. Glavatskiy and D. Bedeaux.
J. Chem. Phys. , 133:234501, 2010.[12] K. S. Glavatskiy and D. Bedeaux.
J. Chem. Phys. , 133:144709, 2010.[13] E. Johannessen and D. Bedeaux. Integral relations for the heat and mass transfer resistivities of the liquid-vapor interface.
Physica A , 370:258–274, 2006.[14] A. J. M. Yang, P. D. Fleming, and J. H. Gibbs. Molecular theory of surface tension.
J. Chem. Phys. , 64:3732, 1976.[15] J. S. Rowlinson. Translation of J.D. van der Waals’ ”The Thermodynamic Theory of Capillarity Under the Hypothesis ofa Continuous Variation of Density”.
J. Stat. Phys , 20:197–244, 1979.[16] J. D. van der Waals. Square gradient model.
Verhandel. Konink. Akad. Weten. Amsterdam , 1(8):56, 1893. [17] J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. , 28:258,1958.[18] Andrea G. Lamorgese, Dafne Molin, and Roberto Mauri.
Milan J. Math , 79:597–642, 2011.[19] K. S. Glavatskiy.
Multi-component interfacial transport as described by the square gradient model; evaporation and con-densation . Springer Thesis. Springer, Berlin, 2011.[20] D. Bedeaux, E. Johannessen, and A. Røsjorde. The nonequilibrium van der Waals square gradient model. (I). The modeland its numerical solution.
Physica A , 330:329, 2003.[21] E. Johannessen and D. Bedeaux. The nonequilibrium van der Waals square gradient model. (II). Local equilibrium of theGibbs surface.
Physica A , 330:354, 2003.[22] E. Johannessen and D. Bedeaux. The nonequilibrium van der Waals square gradient model. (III). Heat and mass transfercoefficients.
Physica A , 336:252, 2004.[23] K. S. Glavatskiy and D. Bedeaux.
Phys. Rev. E , 77:061101, 2008.[24] K. S. Glavatskiy and D. Bedeaux.
Phys. Rev. E , 79:021608, 2008.[25] K. S. Glavatskiy and D. Bedeaux. Nonequilibrium properties of a two-dimensionally isotropic interface in a two-phasemixture as described by the square gradient model.
Phys. Rev. E. , 77:061101, 2008.[26] S. Kjelstrup and D. Bedeaux.
Non-Equilibrium Thermodynamics of Heterogeneous Systems . Series on Advances in Statis-tical Mechanics, vol. 16. World Scientific, Singapore, 2008.[27] T. Savin, K. S. Glavatskiy, S. Kjelstrup, H. C. ´Ottinger, and D. Bedeaux. Local equilibrium of the gibbs interface intwo-phase systems.
EPL , 97:40002, 2012.[28] K. S. Glavatskiy, D. Reguera, and D. Bedeaux.
J. Chem. Phys. , 138:204708, 2013.[29] Edgar M. Blokhuis.