Guangze Han*,Xiaoyan Wang

Department of Physics,South China University of Technology,Guangzhou 510641,China

Keywords:Stokes–Einstein equation Eyring's equation Slit nanopore Self-diffusion coefficient Simple quasicrystal model of liquid

ABSTRACT Dynamical properties of liquid in nano-channels attract much interest because of their applications in engineering and biological systems.The transfer behavior of liquid confined within nanopores differs significantly from that in the bulk.Based on the simple quasicrystal model of liquid,analytical expressions of self-diffusion coef ficient both in bulk and in slit nanopore are derived from the Stokes–Einstein equation and the modified Eyring's equation for viscosity.The local self-diffusion coefficient in different layers of liquid and the global self-diffusion coefficient in the slitnanopore are deduced from these expressions.The in fluences of confinementby pore walls,pore widths,liquid density,and temperature on the self-diffusion coefficientare investigated.The results indicate thatthe self-diffusion coefficientin nanopore increases with the pore width and approaches the bulk value as the pore width is sufficiently large.Similar to that in bulk state,the self-diffusion coefficient in nanopore decreases with the increase of density and the decrease of temperature,but these dependences are weaker than that in bulk state and become even weaker as the pore width decreases.This work provides a simple method to capture the physical behavior and to investigate the dynamic properties of liquid in nanopores.

1.Introduction

Numerous types of nanoporous media have found their way in industrial use,among them are activated carbons and zeolite structures.There are significant advances in unit operations such as nano filtration,gas separation,water purification,desalination,energy generation and fuel cell technology based on liquid flowing through nanoporous media.The behavior of liquid confined within nanopores(size of a few molecular diameters)significantly differs from that in the bulk.In particular,the confinement and reduced dimension affect the phase transitions.Understanding the effect of such confinement on the behavior of liquid is of crucial interest for both fundamental research and potential applications[1–3].

However,in spite of wide applications in some engineering and biological systems,the dynamical properties in such narrow channels are not well understood so far.To understand at the microscopic level the direct in fluence of confinement on molecular motion and hence on dynamical properties is still a difficult task[4].Nevertheless,modeling the transportof fluids in narrow pores and confined spaces has attracted considerable attentions among scientists and engineers for over a century,due to its importance to a variety of applications of industrial interest[5].Mass transfer can be principally attributed to two different mechanisms:macroscopic transport via convection and microscopic transport from diffusion.In pores of a few nanometers in diameter,diffusion is the primary mechanism[5].Self-diffusion is one of the most important mass transport properties,which is the diffusion of tagged particles in a fluid where all particles are chemically identical,and it is characterized by self-diffusion coefficient.Knowing self-diffusion coef ficients and their characteristics has greatrelevance in the physics of molecular liquid,since this knowledge can serve e.g.for an elaborate check of theoretical approaches.This is in particular true for liquid,because the self-diffusion in liquid arises from a series of dynamic intermolecularinteractions,so itis meaningfulfor insightinto the microscopic interactions in liquid from the dynamic viewpoint.

Many properties of liquid in porous media become inaccessible to experimental measurements when the pore size approaches molecular dimensions.It is thus desirable to develop an effective theoretical method to predict the self-diffusion characteristics in porous media.Nanoporous carbons such as activated carbons are often considered as model materials to investigate the effect of nano-confinement on the properties of liquid.Activated carbons,which are widely used in gas separation,purification,and catalytic reaction,have large internal specific surface area and almost uniform pore size of several molecular diameters.Therefore,activated carbon is chosen for the calculations in this study.

Currently the research methods forthe liquid confined in nanopores are mainly based on computer simulations[4–7].For example,Sofas et al.[8]investigated the transport properties of liquid argon flowing through a nanochannel with non-equilibrium molecular dynamics simulations；Sun et al.[9]investigated the transport properties of Ar–Kr binary mixture confined in a nanochannel by equilibrium molecular dynamics simulation.In this study,a simple quasicrystal model of liquid is used to represent the spatial configuration of liquid in slit nanopores,from which the activation energy for liquid molecule diffusions is deduced.An analytical expression of self-diffusion coefficient will be derived from the Stokes–Einstein equation and Eyring's equation.This expression includes the effect of confinement of nanopores on the dynamic properties of the liquid in the pores,and it will be used to investigate the characteristics of self-diffusion.Comparing with computer simulations,this method has more physical meanings,it is especially simple for numerical calculation,and it can be extended to describe the mass transfer enhanced by external field by including the external potential in the activation energy.

2.Self-diffusion Coefficient of Liquid

2.1.Stokes–Einstein equation

The most common basis for estimating diffusion coefficients in liquid is the Stokes–Einstein equation,which is based on the Nernst–Einstein equation in the hydrodynamic theory[10].Coef ficients calculated from this equation are only in about 20%accuracy.Nonetheless,it remains the standard,against which alternative correlations for molecular diffusivity are judged[11].For a dilute solution of A in B,the Stokes–Einstein equation ofdiffusivity is DAB=kT/6πηBRA[10],where T is the temperature in Kelvin,k is the Boltzmann constant,ηBis the viscosity of pure solvent,and RAis the solute radius.This diffusion coefficient is valid when the ratio of solute to solvent radius exceeds five.It is reasonable because the Stokes–Einstein equation is derived by assuming a rigid solute sphere diffusing in a continuum of solvent.When the solute radius is less than five times that of the solvent,this equation is revised to DAB=kT/4πηBRA.If molecules A and B are identical(that is,for self-diffusion)and if they can form a cubic lattice with adjacent molecules just touching:2RA=(Vm/NA)1/3,where Vmis the molar volume and NAis Avogadro's number.The Stokes–Einstein equation is revised to[10]

This equation has been found to agree with self-diffusion data for a number of liquids in about 12%accuracy.Eyring's activated-state theory also gives a similar result[12].In this study,Eq.(1)is chosen as the expression of self-diffusion coefficient of liquid.

2.2.Viscosity of liquid

Viscosity η is a key parameter in Eq.(1),which is one of the important transport properties in chemical industry and process design.Reliable viscosity data are usually determined by experiments,but it is impossible that the experiments are made for all fluids.Hence,many theoretical,semitheoretical and empirical equations of liquid viscosity have been proposed[13,14].Most of these methods for estimation of viscosity of pure liquids and liquid mixtures are based on the principle of corresponding states,the absolute rate theory of Eyring[15],or on molecular dynamic calculations.Eyring's absolute rate theory is still a good theory for liquid viscosity[15,16].It can represent the viscosity-temperature relation and is a simple method to predict liquid viscosity.The expression of viscosity proposed by Eyring is

where h is the Planck constant,R is gas constant,and Eais the molar activation energy.The molar activation energy is the key parameter in this equation,but is difficult to obtain.Liquid molecules keep on thermal moving,including vibrations and rotations.As shown in Fig.1,if there exists a vacancy adjoining a tagged molecule T,then it is possible for this molecule to escape from the cage formed by its neighboring molecules and jump into this vacancy.The least energy needed for this escaping and jumping is the activation energy or the potential barrier for the jumping.

Fig.1.Illustration of a tagged liquid molecule jumping to its neighboring vacancy.

In derivation of Eq.(2),Eyring did not consider the probability of creating a vacancy adjoining the tagged molecule T.However,because the number of the vacancy in liquid is very small,this probability has to be taken into account.If the molar energy required to create a vacancy in the liquid is Ev,the Boltzmann distribution indicates thatthe probability to create a vacancy is α=exp(-Ev/RT)[17].Eyring's equation can be modified to[18]

Substituting Eq.(3)into Eq.(1)leads to the self-diffusion coefficient of liquid

Eq.(4)says that,apart from temperature,the self-diffusion coef ficient of liquid depends on the activation energy and the energy needed to create a vacancy in the liquid.

3.Simple Quasicrystal Model of Liquid and Activation Energy in Slit Nanopores

3.1.Simple quasicrystal model of liquid and self-diffusion coefficient in the bulk

The Lennard-Jones potential(L-J potential)is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules.A form of the potential energy was first proposed in 1924 by Lennard-Jones[19]

where ε is the depth of the potential well,σ is the finite distance at which the potential is zero,and r is the distance between the particles.Let r0denote the distance at which the potential reaches its minimum,?(r0)=- ε；the distances are related as r0=21/6σ.The term r-12is the repulsive term acting at short ranges and the term r-6is the attractive long-range term.For the interaction between two polar molecules,the potential energy is[20]

This potential energy is sometimes known as the Stockmayer 12-6-3 potential.Symbol p is the dipole moment of a molecule.Parameters in Eqs.(5)and(6)can be determined with the empirical relations[10],ε/k=0.77Tc,σ =0.841 × 10-8Vc1/3,where Tcand Vcare the critical temperature and critical volume,and all the quantities are in SI units.

The strongly repulsive molecular forces at short range have an important effectto create the short-range order,which is the main characteristic of liquid structure[21].Based on this characteristic,a simple quasicrystal model of liquid is proposed in the present work,in which the closely packed body-centered cubic structure is used to represent the spatial configuration of liquid.This configuration is not only easy to simulate but also very close to the real structure of simple liquid.Fig.2 shows that in this model the liquid has layered structure with each layer shifting a half of the intermolecular distance relative to its neighboring layers.Thus each molecule locates in the center of the cube consisting of its eightnearest molecules.Based on this simple quasicrystal model of liquid,the intermolecular potential energy of liquid molecules in bulk both in equilibrium position and during its jumping,ψff(a,b)and ψvff(a,b,x),can be expressed analytically,which are derived in detail in Appendices A1 and A2.

The mechanical energy of a liquid molecule consists of kinetic energy and potentialenergy.The motion of a liquid molecule includes vibration and rotation,and the kinetic energy associated with these motions is averagely 3RT per mole according to the Equipartition theorem[22].The potential energy per mole is(NA/2)ψff(a,b),where digit 2 is used to eliminate the double counting ofthe intermolecularpotentialenergy.The total mechanical energy per mole is then(NA/2)ψff(a,b)+3RT.The least energy needed to create one mole vacancy in liquid is the negative of this value,

As shown in Fig.1,from symmetry,when the tagged molecule is at the middle point of the jumping path(x=a/2),its potential energy attains the maximum value.Thus the molar activation energy is

The viscosity and the self-diffusion coefficient of bulk liquid can be obtained by substituting Eqs.(7)and(8)into Eqs.(3)and(4),namely

3.2.Potential energy of liquid molecules in slit nanopores

Porous materials have importantapplications in adsorption technology[23].According to their dimensions,the pores are classified as macropore,mesopore and micropore.The micropores with nanodimension have large specific surface area,which are the main adsorption sites[24].The shapesofmicroporesare mainly slit,wedge,cylinder,and pillar.The slit nanopores are the main form in activated carbons,and are commonly chosen as the theoretical model[3].The potential energy of a fluid molecule in a slit pore only depends on the perpendicular distance between the fluid molecule and the material surfaces.As shown in Fig.3,an individual pore is represented by two semi-in finite parallelslabs separated by width H.The layers in each slab are separated by a uniform spacing Δ.The surface is assumed to be smooth and rigid.By integrating the L-J potential between one fluid molecule and each atom of the individual planes,the potential energy between a solid wall and a fluid molecule is derived as

which is called as the Steele 10-4-3 potential[25],whereρsis the atomic number density of solid wall,and z is the normal distance between a fluid molecule and one of the solid wall.The cross interaction parameters of the potential depth and effective diameter are obtained by the typical Lorents–Berthelot combining rules εsf=(εssεff)1/2,σsf=0.5(σss+ σff).The minimum of Steele 10-4-3 potential occurs approximately at a distance z0=(2/5)1/6σsf=0.858σsf.This potential has become the most popular model of carbon materials for applications ranging from fundamental study of adsorption phenomena to pore size characterization by commercial laboratory equipment.The atomic number density and the layer spacing for carbon materials are ρS=114.0 nm-3,and Δ =0.335 nm.

The literature showed that the liquid confined in slit nanopores is distributed in layers[26].It is reasonable to assume that the spatial configuration of liquid in a slit nanopore can be modeled with Fig.4.Comparing with the separation H,the other two dimensions of the slit nanopore are assumed to be in finite.A molecule in the slit pore will experience actions from the two walls of the pore.The potential energy of the fluid molecule with the walls of the slit pore is

Fig.2.Simple quasicrystal model of liquid with body-centered cubic structure,(a)in three-dimensional view；(b)in two-dimensional view from above.

Fig.3.The model of slit nanopore.

Fig.4.Layer distribution of liquid molecules in slit nanopore.

Besides the interactions with the walls,the tagged molecule also experiences interactions from its surrounding fluid molecules.For a slit nanopore full of liquid,the total potential energy of a molecule in the slitpore isψff(a,b)+ψsf(z).The expressionψff(a,b)willhave different terms according to the position of the tagged molecule in the pore.For example,for the tagged molecule in one of the two first layers(adjoining a wall),ψff(a,b)=(a,b)+(a,b)+(a,b)；for molecules in the second layers(second next to a wall),ψff(a,b)=(a,b)+(a,b)+(a,b),and so on.It is obvious that the diffusive motion normal to the pore walls is negligible.The potential energy with the pore walls changes only the energy needed to create a vacancy,so from Eq.(10),the self-diffusion coefficient for the diffusive motion parallel to the pore walls is

3.3.Local and global self-diffusion coefficients in slit nanopores

The self-diffusion coefficient of liquid in slit pores,Eq.(13),depends on the potentialenergy of liquid molecule,and in turn this potentialenergy depends on its position in the pore.Molecules in different layers have different potentials,and hence have different local self-diffusion coefficients.The global self-diffusion coefficient in the pore can be determined by averaging the local self-diffusion coefficients.As derived in Appendix A3,the relationship between the global diffusion coefficientand the local diffusion coefficient Dkis

where φkis the area fraction of layer k.

Confinement leads to local diffusion at nanoscale,which has been observed[27].For a slit nanopore that can only accommodate one layer of liquid,ψvff(a,b,x)=(a,x)and ψff(a,b)=(a,b),the local self-diffusion coefficient is the same as the global self-diffusion coefficient.For the slit nanopore which can only accommodate two layers of liquid,molecules in these two layers have the same potential energy,ψvff(a,b,x)=(a,x)+(a,b,x)and ψff(a,b)=(a,b)+(a,b).The global self-diffusion coefficient is also equal to the local self-diffusion coefficient.

However,as shown in Fig.5,for the slit nanopore accommodating three layers of liquid,the potential energy of molecules in the first layer is ψvff(a,b,x)=(a,x)+(a,b,x)+(a,b,x)and ψff(a,b)=(a,b)+(a,b)+a,b).The potential energy in the second layer is ψvff(a,b,x)=(a,x)+(a,b,x)and ψff(a,b)=+(a,b).With these expressions,the local self-diffusion coefficient D1in the two first layers and D2in the second layer can be found with Eq.(13).For a regular slit pore,the cross area of the layer is proportional to its width,so the area fraction is equal to width fraction.In this case,the global self-diffusion coefficient for Fig.5 can be found with Eq.(14),which is

Fig.5.Slitpore filled with three layers of liquid molecules,with the distances exaggerated for the sake of clarity.

With this same method the local and global self-diffusion coef ficients in slit nanopores filled with four layers, five layers,and so on can also be found.

4.Calculations and Discussions

4.1.Viscosities and self-diffusion coefficients in the bulk

For the simplicity of calculation,the intermolecular distances are chosen as a=b=r0,and the distance of the first layer to the wall is z0,here r0and z0are the minimum potential distances.The width of the slit pore is measured in terms of molecular layers.For example,there are 3 layers in the slit pore in the case of Fig.5,so the width of the slit pore is H=2z0+2b；if there are n layers of liquid molecules in the slit,the width of the slit is then H=2z0+(n-1)b.Two liquids are arbitrarily chosen for the calculation,one is benzene(C6H6),which is a nonpolar liquid；the other is toluene(C7H8),which is a polar liquid.The model parameters are listed in Table 1,and the calculation temperature is 25°C.

With these parameters,viscosities and self-diffusion coefficients of liquids C6H6and C7H8in the bulk are calculated with Eqs.(9)and(10),which are also listed in Table 1,together with the experimental data.These theoretical predictions are close to the experimental data.In the present model the liquid molecules are treated as sphere and regularly arranged in layers,which is an idealization.Therefore,it is not expected that the theoretical predictions are precisely equal to experimental data.

Table 1 Model parameters and comparisons of calculations at 25°C with experimental data[28,29]

4.2.Effects of confinement on self-diffusion coefficients

Available experimental and simulation data in literature indicate that molecular transport is strongly affected by the fluid-wall potential.This potential leads to a significant reduction in the diffusion coefficient compared to that obtained with the Knudsen formulation[5].Fig.6 shows the local self-diffusion coefficients in different layers for a slit nanopore filled with seven layers of liquid.The self-diffusion coef ficientofthe two firstlayers adjacentto the walls is very smallcomparing to the other layers,and itgradually increases with the middle layer having the maximum value.This means thatthe two walls ofthe slitexerta relatively strong force to the liquid molecules,increasing the viscosity and hence decreasing the self-diffusion coefficient.This conclusion is also confirmed by the simulation results[27].

Fig.6.Local self-diffusion coefficient in different layers,with seven layers in the slit nanopore.

Fig.7 shows the variation of the global self-diffusion coefficientwith the width of slit nanopore,with the predicted self-diffusion coefficient in the bulk forcomparison.The globalself-diffusion coefficientincreases with the pore width forboth liquids and approaches the bulk values,because as the width of slit pore increases,the molecular force from the walls decreases.The calculated values of global self-diffusion coefficient are about 30%to 50%of the bulk values for layers 4 to 7.These are also verified by the computer simulations[6,27].The slope of the curve of nonpolar liquid(C6H6)is larger than that of polar liquid(C7H8),which means that the in fluence of carbon walls on nonpolar liquid molecules is stronger than that on polar liquid molecules.This is the reason that the activated carbon prefers to absorbing nonpolar substance over polar substance[1,2].This is understandable,because the intermolecular force between polar molecules is stronger than thatbetween nonpolar ones,which is not beneficial for adsorption.

Fig.6 also reveals that as one approaches the wall,the local selfdiffusion coefficient decreases drastically,with order of magnitude of the calculation being about 10-12m2·s-1.As shown in Fig.7,when the width of the slit is very narrow and can only accommodate one or two layers of liquid,the order of the magnitude of the calculated global self-diffusion coefficient is about 10-14m2·s-1.This value is close to the values of solids,which means a phase transition,so the liquid freezes to solid[30,31].

Fig.7.Variation of the global self-diffusion coefficient with the slit pore width.

4.3.Dependence of self-diffusion coefficient on density

The dependence of self-diffusion coefficient on liquid density reflects the effects of interaction strength and collision frequency.It is of great importance to examine the contributions of density effects on self-diffusion coefficients.The density of liquid can be represented by intermolecular distance and is inversely proportional to this distance.Fig.8 depicts the variation of self-diffusion coefficient with density for liquid C6H6in slit nanopores of five widths.As expected,the selfdiffusion coefficient increases with the distance,hence the decrease in density,for that in bulk and in five different pore sizes.However,the self-diffusion coefficients in the pores have weaker dependences on the density than that in the bulk liquid,and this dependence becomes even weaker as the pore width decreases.The increase in intermolecular distance means the decrease in intermolecular force,so the viscosity decreases and the mobility of molecules increases.This is the reason thatthe self-diffusion coefficient increases with the decrease of density.For the diffusion in the slit nanopores,because the liquid molecules close to the wall experience stronger force from the wall than that from other molecules and the forces from the wall become even stronger when the pore is narrower,which has weaker dependence on the change in the density,the dependence ofthe pore diffusivity upon density is much weaker than that of the bulk diffusivities.

Fig.8.Variation ofglobalself-diffusion coefficientin slitpore with intermoleculardistance,hence with density(n—width of pore).

4.4.Dependence of self-diffusion coefficient on temperature

The dependence of self-diffusion coefficient on temperature reflects the competition of kinetic energy against interaction potential energy.To characterize the states of the interactions that depend on temperature,it is of great importance to examine the contributions of temperature effects on self-diffusion coefficients.Fig.9 shows the dependence of the global self-diffusion coefficient of liquid C6H6in slit nanopores on temperature,together with the bulk value.The calculations are carried out for three pore widths.As their bulk counterpart,the global selfdiffusion coefficient in nanopores also increases with temperature,but the dependence is weaker than that of bulk value；the narrower the slit pore,the weaker the dependence.As temperature increases,the kinetic energy of liquid molecules and hence the global self-diffusion coefficient increases.However,the liquid molecules in slit nanopores experience the force from the slit walls as well,the narrower the slit pore,the stronger the force,so the dependence on the temperature for the liquid in slit pore is weaker than its bulk counterpart,and it becomes even weaker as the pore width decreases.The change tendencies of self-diffusion coefficient for C7H8on density and temperature are the same with that for C6H6,which are not listed here.

Fig.9.Variation of the global self-diffusion coefficient in slit nanopore with temperature.

5.Conclusions

The simple quasicrystal model of liquid,in which the close-packed body-centered cubic structure is used to simulate the spatial configuration ofliquid,can be employed to describe the diffusion characteristic of liquid in nanoporous media.With the modified Eyring's equation of viscosity and the activation energy calculated from the quasicrystal model of liquid,the self-diffusion coefficient of liquid confined in slit nanopores is analytically expressed with the Stokes–Einstein equation.This coefficient equation reveals the following arguments:(1)the local self-diffusion coefficient decreases as it approaches the slit walls,the global self-diffusion coefficient in slit nanopore is always less than the bulk value(by as much as 50%),and at large pore size the selfdiffusion coefficient approaches the bulk value；(2)the diffusion coef ficients of liquid molecules confined in a slit nanopore are dependent on the density and temperature in the pore,the diffusion coefficients generally decrease with the increase in density and with the decrease in temperature,but the dependences are much weaker than the bulk values and become even weaker as the pore widths decrease.

Nomenclature

a,b,r distance between particles,m

D diffusivity,m2·s-1

Eamolar activation energy,J·mol-1

Evmolar energy creating a vacancy,J·mol-1

h Planck constant,J·s

H,z,Δ distance,m

k Boltzmann constant,J·K-1

NAAvogadro's number,mol-1

p electric dipole moment,C·m

R gas constant,J·mol-1·K-1

RAradius of particle A,m

T temperature,K

Vmmolar volume,m3·mol-1

α probability

ε parameter of potential energy,J

η viscosity,Pa·s

ρ atomic number density,m-3

σ parameter of interparticle distance,m

?,φ,ψ potential energy between particles,J

Appendix A

A.1.Intermolecular potential energy of liquid molecule in equilibrium position

Because the Lennard-Jones potential decreases rapidly with separation,for example,?(3r0)/?(r0)=0.0027,so only the molecules within 3r0to the tagged molecules T are taken into account in this expression.As shown in Fig.A1(a),under this truncation there are 24 molecules in the same layerwith molecule T having effective actions on T.The potential energy of the tagged molecule from these same layer molecules is

As shown in Fig.A1(b),there are also 24 moleculesin each neighboring layer in the effective action region,so the potential energy of tagged molecule T with one of its two neighboring layers is

As shown in Fig.A1(c),there are 21 molecules in each next but layer in the effective action region,the potential energy of molecule T with one of its two next but layers is

If the surrounding of the tagged molecule is full of liquid molecules,there are totally 114 molecules within the effective action region,which are from five layers(one the same layer,two neighboring layers and two next but layers).Then the total potential energy of any molecule in the liquid is

Fig.A1.The space structural relationships of tagged molecule T with its neighboring molecules(a)with the same layer；(b)with the neighboring layer；(c)with the next but layer,with separations exaggerated for the sake of clarity.

A.2.Intermolecular potential energy of liquid molecule during its jumping

In a pure liquid at rest,the motions of each individual molecule are largely confined at its original position.If there is a vacancy beside the molecule and the molecular energy is sufficiently large,the molecule can jump to the vacancy.Fig.A2 shows the jumping process of tagged molecule T to its right neighboring vacancy.As shown in Fig.A2(a),at the position x relative to its original position,there are 24 molecules in the same layerwith T within the effective interaction region.The potential energy of the jumping molecule from its same layer is

In Fig.2(b),there are 26 molecules in each neighboring layer within the effective action region,so the potentialenergy with one of its neighboring layers is

In Fig.A2(c),there are 16 molecules in each nextbutlayer within the effective action region,so the potential energy with one of its next but layers is

Apart from the vacancy,when the surrounding of the jumping molecule is full of liquid molecules,there are totally 108 molecules within the effective action region,which are also from five layers.Then the total potential energy of the jumping molecule at position x is

A.3.Relationship between local and global diffusion coefficients

For one dimensional mass diffusion along direction x,Fick's law is

where Jiis the mass flux of species i,and Ciis concentration.In Fig.A3,we assume that the diffusion is laminar diffusion,the cross area of the k th layer is Ak,the flux and diffusion coefficient of species i in layer k are jikand Dikrespectively.Then the diffusive mass rate through layer k is Akjik,so the average flux is

Fig.A2.The space structural relationships of tagged molecule T with its neighboring molecules during its jumping,with separations exaggerated for the sake of clarity.

Fig.A3.Laminar diffusions in different layers.

where φk=Ak/∑Akis the area fraction of layer k.For steady diffusion,the concentration can only change along the direction of the diffusion,so it has the same value along the perpendicular direction d Cik/d x=d Ci/d x.Applying Eq.(A9)to each layer and substituting it into Eq.(A10)leads to

Comparing Eq.(A11)with Eq.(A9)leads to the relationship between the global diffusion coefficient and local diffusion coefficient,