Thermodynamic Resistance to Matter Flow at The Interface of a

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Thermodynamic Resistance to Matter Flow at The Interface of a Porous Membrane K. S. Glavatskiy and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland, St Lucia, Queensland 4072, Australia S Supporting Information *

ABSTRACT: Nanoporous materials are important in industrial separation, but their application is subject to strong interfacial barriers to the entry and transport of fluids. At certain conditions the fluid inside and outside the nanoporous material can be viewed as a two-phase system, with an interface between them, which poses an excess resistance to matter flow. We show that there exist two kinds of phenomena which influence the interfacial resistance: hydrodynamic effects and thermodynamic effects, which are independent of each other. Here, we investigate the role of the thermodynamic effects in carbon nanotubes (CNTs) and slit pores and compare the associated thermodynmic resistance with that due to hydrodynamic effects traditionally modeled by the established Sampson expression. Using CH4 and CO2 as model fluids, we show that the thermodynamic resistance is especially important for moderate to high pressures, at which the fluid within the CNT or slit pore is in the condensed state. Further, we show that at such pressures the thermodynamic resistance becomes comparable with the internal resistance to fluid transport at length scales typical of membranes used in fuel cells, and of importance in membrane-based separation, and nanofluidics in general.

1. INTRODUCTION It has long been known that interfacial energy barriers exist in the vicinity of the adsorbent surfaces, which must be overcome by adsorbing or desorbing molecules.1 In particular, experiments suggest that diffusion investigated by transient uptake and release measurements on porous systems is much slower than intracrystalline diffusion.2 Such barriers offer an additional transport resistance that depends on the atomistic structure of the adsorbent and the potential energy landscape near the surface and are detrimental to the efficiency of membrane or kinetic separation. The resulting interfacial mass transfer coefficients for CH 4 have been computed by various researchers: for AFI type zeolite structure,3,4 for silicalite,5−7 and for AlPO4-5,8,9 showing significant interfacial resistance even for ideal structures. It may be expected that the stronger interactions and disorder-related structural defects in carbons will lead to even larger barriers. The interfacial resistance of a nanoporous membrane comprising carbon nanotubes (CNTs) is usually analyzed in the context of the hydrodynamic end (or entrance) effect10−12 when the fluid faces a geometrical obstacle, such as the corner of the CNT boundary shown in Figure 1a. The flow laminae bend when they enter the CNT, as shown in Figure 1b. If the fluid has a nonzero viscosity, this bending results in a viscous friction between the different laminae and between the boundary lamina and the wall. If the fluid is inviscid, potential flow occurs, for which the resistance is zero. The expression for the resistance for an incompressible viscous fluid has been derived long ago,13,14 and we will refer to this resistance as the © XXXX American Chemical Society

hydrodynamic resistance. The hydrodynamic effects are typically relevant on the macroscopic scale, which is much larger than the diameter of the CNT, and the conventional hydrodynamic description is expected to fail inside the CNT. Nevertheless, the external hydrodynamic resistance has been modeled both by finite element solution of the Navier−Stokes equation15,16 and by molecular dynamics simulations.16,6 For dense bulk fluids the results agree quite well, which suggests that the main contribution to the pore end resistance is caused by bending of the flow laminae outside the CNT.16 The interfacial resistance is also analyzed in the context of Fickian diffusion, when the net flux is caused by the difference in concentration inside and outside the nanoporous membrane.9,6 As in the case of hydrodynamic analysis, this description is most relevant to dense bulk fluids, for which the density within the nanoporous membrane is not significantly different from that in the bulk. The reason for this is that, in general, matter flow is driven by fugacity difference, rather than concentration difference.17 This distinction is not important in the non-equilibrium one-phase region, where density does not change greatly with position. However, it becomes crucial in the two-phase region, where high-density and low-density fluids coexist with each other. In particular, for such systems there exists an enormous density difference even in equilibrium, but this density gradient does Received: February 1, 2016 Revised: March 23, 2016

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DOI: 10.1021/acs.langmuir.6b00375 Langmuir XXXX, XXX, XXX−XXX

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Langmuir not cause matter flow from one phase to the other in equilibrium. Experimental methods quantifying surface barriers have also been provided by microimaging.18−20 By implying diffusion limitation they suggest that the main reason for surface resistance is blockage of surface pores.21 Alternative interpretations include surface pore narrowing and surface desorption22−24 and deviations from the ideal crystal structure due to defects.25−27 Clearly, these mechanisms suggest that the interfacial resistance of nanoporous materials has a molecular nature and is caused by specific adsorbate behavior at the interface. Such resistance is additional to that due to the hydrodynamic effects that have been analyzed by computational modeling using molecular dynamics and finite element methods.16 In this article we analyze the problem of interfacial resistance from a different perspective, namely, by means of nonequilibrium thermodynamic modeling. It has been known that the interface between two phases poses an excess resistance to matter flow.28,29 This happens due to the difference between specific enthalpies of the two phases, which are separated by this interface. This effect is thermodynamic in nature and is typically ignored, which introduces a gap in understanding the nature of the interfacial resistance. At the same time, it allows one to address the interfacial resistance on both the molecular scale and the scale of field variables. The structure of the pore affects the thermodynamic properties of the adsorbate. In particular, the effects of both pore blockage and pore narrowing can be addressed by varying the size of the pore space available for adsorbate to enter. Furthermore, the thermodynamic analysis is performed in terms of field variables, which allows direct connection to fluid mechanics. Thus, the main purpose of this work is to demostrate the importance of thermodynamic effects for the interfacial resistance of CNTs and slit pores. Due to small CNT diameter, the fluid within the pore is subjected to strong interactions with the pore wall.9,4 This changes the state of the fluid compared to the bulk state at the same conditions and creates a new fluid phase within the CNT. One can therefore speak of the two phases: bulk fluid phase b and adsorbed membrane phase a (cf. notation in Figure 1), which are in equilibrium with each other. In non-equilibrium, the interface between two phases modifies the properties of the system, compared to a single uniform phase.28,29 In particular, energy transport due to phase change and mass transport due to matter flow happen simultaneously and are therefore coupled. This leads, in particular, to additional heat flux in the interfacial region, which is proportional to the enthalpy of adsorption.30 Furthermore, this coupling results in an additional drop of the chemical potential (pressure) across the interfacial region. Because of this, the interfacial region poses an excess resistance to matter flow, which is proportional to the square of the enthalpy of adsorption.28,29 This is the case even for ideal fluid with zero viscosity. We will refer to this resistance as the thermodynamic resistance. It is therefore evident that effects related to the interface of the nanoporous membrane will be manifested essentially in two ways. First, it is the region where the geometry of the flow changes, resulting in the hydrodynamic end-effects, and viscosity-dependent resistance. Second, it is the region, where the fluid changes its phase, from the low-density state to the high-density state, resulting in the thermodynamic resistance. The above arguments show that there exists a need for a more general description of the interfacial resistance of the

nanoporous membrane, which takes into account that the fluid inside and outside the CNT may be in different phases. In this article we develop such a descripton and investigate the effect of this thermodynamic resistance and its dependence on the bulk conditions, while comparing it with the hydrodynamic resistance. The structure of this work is as follows. In section 2, we introduce the system and compare the thermodynamic resistance with the hydrodynamic resistance. In section 3, we introduce the theory to calculate the thermodynamic resistance. We combine the density functional theory within the adsorbed phase with the square gradient theory in the interfacial region to model the thermodynamic resistance. In section 4, we present the results of calculation and discuss their implications, and finally in section 5, we give some concluding comments.

2. HYDRODYNAMIC AND THERMODYNAMIC RESISTANCES Consider a fluid which fills a nanoporous membrane. The membrane consists of numerous CNTs, which form narrow pores separated by a solid material. We consider the membrane to be a regular collection of CNTs, which are parallel to each other and perpendicular to the membrane surface. Each CNT can be treated separately. The total resistance of the membrane consists of the resistance of the CNT body and the resistances of its entrance and exit. In this work we study the resistance at the CNT boundary, which is shown in Figure 1a . The resistance of the body of the finite-length CNT is proportional to its length and can be calculated as RCNT,L = rCNT,∞L, where rCNT,∞ is the resistivity (the resistance per unit length) of the infinite-length CNT and L is the length of the CNT. In this way all of the boundary effects are attributed to the boundary resistance. Due to the change of geometry of the system and physics of the molecular interactions near the CNT boundary, there exists an interfacial region, where the character of the flow changes from that of the bulk to that of the adsorbed fluid. This region has a nonzero dimension, which we will refer to as the interfacial width. If the interfacial width is less than half of the length of the CNT, it is possible to separately consider the interfacial region and the body of the CNT; we address here such cases only. However, if the width of the interface is larger than half the CNT length, the interfacial region becomes indistinguishable from the body of the CNT. This case requires special treatment and will not be considered in this work. The fluid flow can be described by a volumetric flow rate Q ≡ Su, which is equal to the volume of the fluid, passing through the system per unit of time. Here S is the cross-section area and u is the fluid velocity. Also, the fluid flow can be described by the flux J ≡ ρu, where ρ is the fluid density. The use of volumetric flow rate is convenient when the fluid is incompressible and the system cross-section changes with position. In contrast, the use of the flux is convenient when the fluid is compressible. The flow of fluid from the bulk phase to the channel of the CNT experiences two types of phenomena. The first is a hydrodynamic resistance due to the changing cross-section of the available flow domain. The second process is the resistance between two different phases, the bulk and the adsorbed. This resistance is caused by the difference between the partial enthalpies in two phases. To distinguish these phenomena, we consider the interfacial region in Figure 1a to comprise the two interfacial regions, corresponding to the above phenomena. In B


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is independent of either the CNT radius, a, or the fluid density, ρ. In the case of adsorption of a gas, for which the requirement of incompressibility is violated, eq 4 may be considered as an upper bound for the pressure drop. The thermodynamic resistance is due to the change of the fluid phase in the system, shown in Figure 1c . The phase of the fluid is determined by the type of interactions between walls and fluid. We represent the bulk phase by the fluid confined between the (virtual) “passive” walls. There are no interactions between the passive walls and the fluid, as well as there is no explicit excluded volume effect due to finite size of the molecules. This fluid is homogeneous and has the bulk density. The fluid within CNT is confined between the “active” walls, which are the real walls of the CNT, and interact strongly with the adsorbate. The fluid within the CNT is in the adsorbed state and is strongly inhomogeneous due to the interactions with the walls. For the purpose of the current study these inhomogeneities are not relevant, and we will consider the adsorbed fluid as homogeneous with the adsorbed density, ρa, which is larger than ρb. Due to difference in the nature of interactions with the passive and active walls, there exists a nonzero enthalpy of adsorption.31 This results in an additional resistance Rμ to the fluid transport through the interface between the bulk and the adsorbed phase.28 In the case of constant temperature, T, the drop in chemical potential is proportional to the flux;29 i.e., Δμ = −RμTJ

(2)

where the minus sign indicates that the direction of the flux is the same as the direction of the chemical potential drop. To compare the thermodynamic and the hydrodynamic resistances, we translate them to the same units. The chemical potential drop is related to the drop of pressure, as ΔPμ = ρbΔμ, while the adsorbate volumetric flow rate is related to the flux as Q = Jπa2/ρa. Therefore, the pressure drop across the interface due to the phase change is

Figure 1. Interface of a single CNT. (a) Full view. (b) Hydrodynamic perspective: bending of the stream lines in bulk fluid. (c) Thermodynamic perspective: phase change of the fluid.

the first process, depicted in Figure 1b, the domain of the fluid flow shrinks from the bulk cross-section to the cross-section of the CNT. The fluid is considered to be homogeneous so that its density ρ is uniform and equal to the density of the bulk phase, ρb. The resistance to the flow is determined by the hydrodynamic effects only. In the second process, depicted in Figure 1c, the fluid flows from the bulk phase to the adsorbed phase along the channel of uniform cross-section. In the stationary state the fluid flux J remains the same, while the value of the density changes from ρb in the bulk fluid to the crosssectionally averaged value ρa in the adsorbed fluid. The hydrodynamic resistance causes a pressure drop across the interface in the system in Figure 1b . The pressure drop between the infinite half-space and the pore of radius a for an incompressible fluid with viscosity η is given by the Sampson expression13,14 η ΔPη = C 3 Q (1) a

( −ΔPμ) = Rμ

ρa ρb T πa 2

Q

(3)

The prefactors in front of the flow rate, Q, in eq 3 and eq 1 may be compared. We introduce the hydrodynamic resistance, Rη, to the matter flux as the factor η/a3 from eq 1 scaled by the factor Tρaρb/πa2 from eq 3: Rη ≡

η πa 2 a3 Tρa ρb

(4)

Then, Rμ/Rη is a dimensionless number representing the relative magnitude of the thermodynamic resistance with respect to the hydrodynamic resistance. This number should be compared with the Sampson constant, C, from eq 1. When Rμ/Rη ≪ C, the thermodynamic effects are negligible for the resistance of the CNT boundary. In this case the resistance to the flow is caused mainly by bending of the flow laminae of the viscous fluid. In contrast, when Rμ/Rη ≫ C the hydrodynamic effects are negligible. In this case the resistance to the flow is caused by the interfacial condensation during the flow.

The value of the coefficient C ≈ 3 is obtained after the integration of a function, which depends on the fluid vorticity, and is mainly a result of the particular geometrical configuration. For a membrane, consisting of a stack of CNTs, the value of C would be different, which would take into account the geometrical details of CNT packing. Note that eq 1 is strictly valid only for incompressible flow, and the constant C

3. THERMODYNAMICS OF THE DIFFUSE INTERFACE 3.1. Thermodynamic Resistance. We model the interfacial resistance in the context of the diffuse interface29 and the Gibbs’ surface.32 In the theory of diffuse interface the density profile across the interfacial region is considered to be a smooth C


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phase, and J is the constant mass flux. Furthermore, Δ indicates the difference between the values of the two neighboring phases. It follows from eq 8 that the force−flux relations in the interfacial region are given by

continuous function ρ(z), which changes asymptotically from ρb to ρa. In practice, the transition happens only over a short interfacial distance, w, of the order of a nanometer. Beyond that region the density profile is indistinguishable from either ρb or ρa. The interfacial resistance is the excess resistance of the interfacial region over that of the surrounding phases (bulk and adsorbed). In the interfacial region one can introduce three different local resistivities: the actual resistivity profile, r(z); the resistivity of the bulk phase, rb, extrapolated to the interfacial region; and the resistivity of the adsorbed phase, ra, extrapolated to the interfacial region. The excess resistance is then defined as R = r̂ ≡

∫ dz [r(z) − rbΘ(zs − z) − raΘ(z − zs)]

Δμ = R mm J + R qm( Jq , b + JsbTb) T ΔT − 2 = R mq J + R qq( Jq , b + JsbTb) T −

where Rmm, Rqq, and Rmq = Rqm are the phenomenological resistances of the interface. In contrast to eq 7, there are two force−flux equations: for the temperature drop and for the chemical potential drop across the interface. Furthermore, each of the equations contains two terms: due to flow of heat and due to flow of matter across the interface. The resistances Rmm, Rqq, and Rmq are the material properties of the adsorbate, while the values of the fluxes Jq,b and J depend on the boundary conditions Δμ and ΔT. Comparing eq 9 with eq 7 and eq 8 with eq 6, it is possible to show34 that the interfacial resistances Rqq, Rqm, and Rmm are the excesses of the corresponding local resistivities rqq, rqm, and rmm, defined by eq 5: R qq = rqq̂ , R qm = rqm ̂ , R mm = rmm ̂ , with

(5)

where Θ is the Heaviside step function, while zs is the position of the “dividing surface”. The dividing surface is a mathematical surface which separates the bulk and adsorbed phases and is placed somewhere in the interfacial region. For the system studied a natural choice for the dividing surface is the position of the CNT boundary. A temperature or pressure gradient results in flow of the fluid from one phase to the other. A measure of irreversibility due to this flow is the entropy production, σ, which relates the thermodynamic forces (the gradients of temperature and pressure) to the resulting fluxes. For a one-component nonviscous fluid in the interfacial region the expression for the local entropy production is33,34 ∂ 1 σ(z) = Jq ∂z T

rqm(z) = rqq(z)[h(z) − hb] rmm(z) = rqq(z)[h(z) − hb]2

(6)

where Jq(z) ≡ Je − Jh(z) − Ju (z)/2 is the measurable heat flux. Here Je and J are the total energy flux and the total mass flux, respectively, which in stationary states are constant. Furthermore, h is the specific enthalpy and u is the fluid velocity. Typically, the streaming kinetic energy u2/2 is much smaller than the specific enthalpy, h, and may be neglected.29 The entropy production (eq 6) results in the force−flux relation between the gradient of the temperature and the heat flux

Rμ ≡ R mm − R qm 2/R qq (7)

(11)

3.2. Square Gradient Theory. The density profile in the interfacial region changes significantly over a small distance. This makes the relations between the thermodynamic quantities depend not only on the local value of the density but also on the density gradient. This observation has been successfully implemented in the square gradient approximation for the liquid−vapor interface35−38 as well as for the fluid−solid interface.39−41 In this approximation the order parameter (the density) varies smoothly across the interface from the bulk value of one homogeneous phase to the bulk value of the other homogeneous phase. A homogeneous fluid is metastable or unstable in the intermediate values of the density and cannot exist. To ensure that equilibrium is possible, the free energy density in the interfacial region consists of two terms

where rqq is the local heat resistivity. Note that for a onecomponent fluid there is no constitutive relation between the gradient of the chemical potential and the local mass flux, as there is no corresponding term in the entropy production. In other words, a one-component fluid has no intrinsic local resistivity to matter flow. In contrast, such resistance emerges for the entire interfacial region; this occurs because a fluid undergoing non-equilibrium phase change is involved in two simultaneous processes: mass flow between phases of different densities and energy flow between phases of different specific enthalpies. On the macroscopic scale a measure of irreversibility is the interfacial entropy production σ̂, which is the excess of the local entropy production over that of the surrounding phases and which is defined similarly to the excess resistance (eq 5). The interfacial entropy production has an additional term, compared to eq 6:28,29 1 1 σ ̂ = Jq , b Δ − J (Δμ + sbΔT ) T Ta

(10)

where h(z) is the equilibrium enthalpy profile, while hb is the bulk value of the enthalpy. Equation 10 suggests that emergence of local resistance to the mass transport in the interfacial region is due to nonzero enthalpy of adsorption. For a typical adsorption process the temperature is kept constant on both sides of the membrane and within the membrane, so ΔT = 0. Nevertheless, during the phase change the temperature within the interfacial region may vary.28,35 We will neglect this effect, assuming that we may still use eq 9 with uniform system temperature, T. For ΔT = 0, we obtain eq 2 with

2

∂ 1 = rqq Jq ∂z T

(9)

f (z) = fEOS (T , ρ(z)) + f∇ (ρ(z), ρ′(z))

(12)

where prime indicates the derivative with respect to position z. Here f EOS is the homogeneous expression for the free energy, which has a double-well structure as a function of the density, with the local minima corresponding to the densities of coexisting phases.39,42 Furthermore, f∇ is a contribution which depends on the density gradients and, in the first approx-

(8)

where sb is the specific entropy of the bulk phase, Jq,b is the heat flux in the bulk phase, Ta is the temperature of the adsorbed D


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Langmuir imation, is proportional to the square of the density gradient.43 In the coexisting region f EOS(ρ) leads to large negative energetic excess, while f∇(ρ,ρ′) leads to large positive energetic excess. Having these terms together minimizes the grand potential ∫ [f(z) − μeq ρ(z)] dz with the chemical potential μeq being constant throughout the entire interfacial region. For a liquid−vapor interface one uses an equation of state (EOS) expression for f EOS(ρ), e.g. ,van der Waals EOS.29 In contrast, for a solid−fluid interface it is more convenient to use a fourth-order double-well polynomial as an approximation for f EOS. The are two reasons for this. First, it is indeed a good approximation for the actual EOS.40 Second, from the microscopic point of view the solid phase is not homogeneous and it is hard to devise an analytic expression for the free energy, which is valid for both the fluid density and the averaged solid density. The double-well free energy density explicitly accounts for the coexisting densities ρa and ρb, as well as the coexisting chemical potential, μe, and the pressure, pe: fW (T , ρ) = −pe (T ) + μe (T )ρ + W (T , ρ)

κ

ρ (z ) =

w≡

κ ⎛ d ρ (z ) ⎞ ⎜ ⎟ 2 ⎝ dz ⎠

2

+

ρa − ρb 2

⎛ z − zs ⎞ ⎟ tanh⎜ ⎝ w/4 ⎠

(18)

4 2 (ρa − ρb )

κ G

(19)

If the EOS has a van der Waals form, then the interfacial thickness is determined uniquely. Indeed, in this case the value of G can be directly evaluated from the parameters of the EOS numerically, while the value of κ is determined from the surface tension γ = κ ∫ |ρ′(z)|2 dz. If, however, the EOS is given by eq 15, then G depends on the interfacial thickness as a parameter; so does κ:

(13)

κ = (γw)

(14)

G=

3/4 (ρa − ρb )2

⎛γ⎞ 24 ⎜ ⎟ ⎝ w ⎠ (ρ − ρ )4 a b

(20)

However, it may also be difficult to evaluate the surface tension between the bulk and the adsorbed phase. Typically, the value one can expect to obtain from experiments is the surface tension between the bulk and the porous system, which includes both the adsorbed phase and the adsorbent. Alternatively, one can obtain the surface tension between the bulk gas and the bulk liquid. Both values may be a poor approximation for the surface tension between the bulk gas phase and the adsorbed fluid phase. Therefore, from the practical point of view, it is more convenient to view w as an independent parameter of the model. 3.3. Density Functional Theory. The fluid within the CNT is subjected to strong interactions with the CNT walls. Because of this the fluid molecules prefer to layer along the walls, which results in strong inhomogeneities of the fluid density within CNT. We are interested in the average density within the CNT, for which we require the density profile in the radial direction. The latter is obtained from density functional theory (DFT).43−45 In this section we give a brief overview of the method. The starting point in DFT is similar to eq 12. The local specific free energy of the fluid depends on the radial position x and consists of several contributions, which depend on both the local and nonlocal density at position x:

(15)

where W(T,p,ρ) has the same form as in eq 14, except that ρa and ρb are now functions of both T and p. The gradient contribution is approximated by the square gradient term alone: f∇ (ρ , ρ′) =

ρa + ρb

where the factor 1/4 is introduced for convenience and w can be viewed as the interfacial thickness. The value of the interfacial thickness can be evaluated from the parameters κ and G as

and the parameter G is a measure of the energy barrier between two coexisting phases. The first two terms in eq 13 represent the values of the free energy at coexistence, i.e., when the density is equal to either ρa or ρb. For a one-component fluid both ρa and ρb, as well as μe and pe, are functions of the temperature only. The last term, W, is the approximation for the free energy in the region between ρa and ρb. In this region the free energy f W (as well as the other thermodynamic functions) is a function of two variables, the temperature and the density. We also assume that G is independent of the temperature. We assume that a similar polynomial approximation can be made for an interface between the gas and the adsorbed fluid. From a thermodynamic point of view we have a twocomponent system in this case. Because of this, equilibrium between the gas and the adsorbate is characterized not by the temperature alone but also by one more variable. It is convenient to choose the gas pressure, p, as this variable. Furthermore, the EOS free energy density (and the other thermodynamic potentials) in the interfacial region is a function of three variables, T, p, and ρ: fEOS (T , p , ρ) = −p + μe (T , p)ρ + W (T , p , ρ)

(17)

For the f EOS given by eq 15, the solution of eq 17 is a hyperbolic tangent

where ⎛ρ ⎞2 ⎛ ρ ⎞2 W (T , ρ) = G⎜⎜ − 1⎟⎟ ⎜⎜ − 1⎟⎟ ⎝ ρa ⎠ ⎝ ρb ⎠

d f (ρ ) d2ρ(z) = EOS −μ 2 dρ dz

2

f (x) = v(x) + fig (ρ(x)) + fhs (ρ ̅ (x)) + flj ({ρ}(x)) (16)

(21)

Here v is the wall potential. Typically it is chosen to be the 10− 4 Steele potential,46 which is the result of additive LennardJones interactions from the semi-infinite wall. Furthermore, f ig = kBT[ln(ρ(x)Λ2) − 1] is the contribution from ideal gas, which depends on the local density, where Λ is the thermal de Broglie wavelength. Furthermore, f hs is the hard-sphere free energy, which is responsible for repulsive interactions between

where κ is a measure of the surface tension between the coexisting phases. It may be a function of local ρ but is typically taken to be a constant parameter. At equilibrium the density profile is such that the grand potential reaches its minimum, leading to the Euler−Lagrange equation E


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Langmuir the fluid particles of nonzero size. The hard-sphere free energy is a function of a nonlocal density, ρ̅, which is an average density over some finite volume, typically of the order of the particle size. We use the version of DFT called the fundamental measure theory,47,48 implemented by Bonilla et al.45 It uses specific functions for f hs, as well as ρ̅, which take into account the geometrical configuration of the particles. The specific expressions for both f hs and ρ̅ can be found elsewhere.47,48 Finally, f lj is the Lennard-Jones part of the free energy, which is responsible for the attractive interactions between the particles. Is is also nonlocal and depends on fluid density everywhere in the system. In particular, f lj = ∫ ρ(r′)U(|r−r′|) dr′, where U is the attractive part of the Lennard-Jones potential and integration is performed over the system volume. At equilibrium the fluid inside the CNT has a uniform value of μ. The actual density distribution ρ(x) minimizes the grand potential Ω = ∫ ρ(x) ( f(x) − μ) dx. To find the actual solution, we perform an iterative procedure, which uses a trial density profile, calculates the free energy density according to eq 21, and then uses it to calculate the next iteration of the density profile.49 To obtain the adsorption isotherm, we choose a particular value of the chemical potential (pressure) of the gas phase, which is in equilibrium with the adsorbed phase. Performing the above procedure, we find the density profile, which corresponds to the chosen chemical potential. The adsorbed density is then obtained by averaging the resulting density profile, ρa = (2/a2)∫ ρ(x)x dx. Furthermore, by performing the same procedure in the absence of the external field, i.e., for v(x) = 0, we obtain ρb. Next, we perform the above procedure for a range of chemical potentials (or, equivalently, the range of pressures) and temperatures. In this way we obtain the equation of state for ρb(p,T) and ρa(p,T) for the absorbed fluid. From the set of adsorption isotherms we calculate the enthalpy of adsorption.50 In particular, the differential enthalpy of adsorption is Δadsh = −kBT 2

∂φ ∂T

ρa − ρb = const

profile. For the free energy given by eq 15 the density profile is the hyperbolic tangent, as in eq 18, leading to the specific entropy: s (z ) =

⎛ z − zs ⎞ sa + sb s − sb ⎟ + a tanh⎜ ⎝ w/4 ⎠ 2 2

(23)

Since the chemical potential is constant, the specific enthalpy is h(z) − hb =

⎛ z − zs ⎞⎤ Δadsh ⎡ ⎟⎥ ⎢1 + tanh⎜ ⎝ w/4 ⎠⎦ 2 ⎣

(24)

where Δadsh is given by eq 22. The local heat resistivity in the interfacial region depends on the equilibrium density profile and can be modeled in a way which is similar to the local free energy, as in eq 12. In particular, we can write that rqq(z) = rqq ,EOS(ρ(z)) + rqq , ∇(ρ(z), ρ′(z))

(25)

where rqq, EOS is the homogeneous part of the heat resistivity, while the second term accounts for the density inhomogeneity in the interfacial region. In view of a lack of a detailed equation of state for heat resistivity, we assume that rqq,EOS is obtained from linear interpolation with respect to the density between the values for the corresponding phases rqq,b ≡ rqq(ρb) and rqq,a ≡ rqq(ρa). In particular, for the density profile in eq 18, we have the expression rqq ,EOS =

rqq , a + rqq , b 2

+

rqq , a − rqq , b 2

⎛ z − zs ⎞ ⎟ tanh⎜ ⎝ w/4 ⎠

(26)

The gradient contribution is proportional to the square of the density gradient |ρ′(z)|2, as in eq 16. Based on the density profile in eq 18, we write ⎛ z − zs ⎞ ⎟ rqq , ∇ = αqq(rqq , a + rqq , b) cosh−2⎜ ⎝ w/4 ⎠

(27)

where we have introduced a dimensionless factor αqq, which controls the magnitude of the gradient contribution. This factor should be nonzero; otherwise the second law of thermodynamics is violated.34 Previous studies suggest that it has the value αqq ≈ 10.

(22)

where φ is the fugacity of the fluid. Note that the derivative is taken at the constant excess density, rather then the constant adsorbed density. In the case of adsorption in the CNT the excess density is equal to the difference between the adsorbed and the bulk densities. If both the bulk and the adsorbed fluid are in the low-density state, then the adsorbed density is much larger than the bulk value, and the difference ρa − ρb is with good approximation equal to ρa. However, when condensation occurs, either in the adsorbed phase or in the bulk phase, this approximation is no longer valid and the actual excess density should be considered. In the calculations below we have obtained the enthalpy of adsoprtion directly using DFT, with no approximation. 3.4. Resistance Modeling. To evaluate the local interfacial resistivity, we need information about the local enthalpy profile and the local heat resistivity profile. The local specific enthalpy is h(z) = μ + Ts(z). Here μ is the equilibrium chemical potential, which is constant throughout the interface, while s(z) is the position-dependent local specific entropy. Assuming that the square gradient coefficient from eq 16 does not depend on the temperature,51 the local entropy is represented only by the EOS contribution:36 s(z) = sEOS(ρ(z)). We modulate the specific entropy profile with the density

4. RESULTS AND DISCUSSION We calculate the interfacial resistance for the adsorption of methane and carbon dioxide in a CNT and a slit pore for the range of pressures of the bulk phase from 0.1 Pa to 50 bar for four different temperatures. The temperatures for methane are 173, 273, 298, and 373 K, while the temperatures for carbon dioxide are 273, 298, 373, and 473 K. The temperatures are chosen to be below and above the critical point (which are approximately 190 K for methane and 304 K for carbon dioxide) and to match the conditions for the parameters measured elsewhere (see below). The CNT has radius 16.87 Å with uniform walls having carbon atom density of 0.382 Å−2. The width of the slit is 16.87 Å, with density of wall atoms being the same as the CNT. They interact with the gas via the Lennard-Jones potential, with the parameter values given in Table 1. The fluid−wall interactions follow the Steele potential.46,52 The LJ parameters of methane correspond to TraPPE force field,53 while those of carbon dioxide correspond to the singleatom SAFT-γ force field.54 The choice of the force fields was dictated by two factors. First, DFT uses a single-atom force F


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pressure. As the bulk pressure increases, both methane and carbon dioxide depart from ideal behavior, and at a certain pressure carbon dioxide experiences condensation in the adsorbed phase. This happens both below and above the critical point of the bulk fluid. Nevertheless, the bulk carbon dioxide behaves very much as ideal gas. At 273 K, increasing the bulk pressure beyond about 0.02 bar forces the adsorbed carbon dioxide to condense. This happens at three stages, corresponding to multilayer adsorption, evident from the radial density profiles shown in the inset. Eventually the density of the adsorbed phase approaches that of the condensed bulk phase. In contrast, methane does not experience sharp condensation in the adsorbed phase over the range of pressures used, when its temperature is slightly below the critical point value of the bulk fluid, although a multilayer structure is evident from the density profiles in the inset. Its density experiences a smooth transition from the ideal adsorbed value to the value of the bulk fluid. The bulk methane experiences condensation at higher pressures at 173 K, as expected. Similar behavior is observed for adsorption of carbon dioxide and methane in the slit pore. These adsorption isotherms are shown in Figure S1 of the Supporting Information. We next calculate the enthalpy of adsorption, according to eq 22. This requires calculating several isotherms. Note that the temperature derivative is calculated at constant excess density, ρa − ρb, rather than constant bulk density. It is then plotted against the bulk pressure that corresponds to the bulk density. The values of the enthalpy of adsorption inside the CNT for a range of pressures are presented in Figure 3. Similar data for the adsorption inside the slite pore are presented in Figure S2 of the Supporting Information. At very low pressures the enthalpy of adsorption is noisy, which is due to low loading of the molecules inside the CNT at these pressures. This is not evident from the adsorption isotherms, as the pressure variation of the loading is much larger than the temperature variation of the loading. The actual value of the enthalpy of adsorption at low pressures is essentially constant, as the fluid is in a lowdensity state, both in the bulk and in the adsorbed phase. Increasing the density does not affect the ability of new molecules to adsorb, so the enthalpy of adsorption is mainly determined by the interactions with the walls of the CNT. At higher pressures, when the fluid in the adsorbed phase starts to condense, the magnitude of the enthalpy of adsorption experiences steep changes. The behavior of the enthalpy of adsorption corresponds to a multistage condensation, which is also clearly visible on the adsorption isotherm (Figure 2). From the molecular point of view this corresponds to the transition from one-layer adsorption to two- and three-layer adsorption. The regions of steep decrease of the enthalpy of adsoprtion correspond to the plato regions of the adsorption isotherms. Furthermore, the regions of smooth increase of the enthalpy of adsorption correspond to transitions between different adsorption stages. We then calculate the density profile across the interfacial region according to eq 18, assuming an interfacial thickness w = 1 nm. Strictly, the actual value of the interfacial thickness does not have a precise value,35 and for the purpose of this work it is sufficient to have an approximate estimate. In Figure S5 of the Supporting Information we show a typical axial density distribution of methane adsorbed in a CNT, obtained from grand-canonical Monte Carlo simulation conducted in our group, from which it is evident that the thickness of the interface is of the order of 1 nm. For comparison, two-

Table 1. Lennard-Jones Parameters of the Interacting Atoms atom

ε/kB, K

σ, Å

C CH4 CO2

28 148 361.69

3.4 3.73 3.741

field, even if it does not predict all the properties of real fluid accurately. Second, we have chosen the force fields for which the simulation data of thermal conductivity are available.55 The CNT radius and the slit width were chosen to be in mesopore range, to be able to observe condensation of carbon dioxide inside the CNT. At the same time methane reveals different behavior, which allows us to compare the results for different adsorbates. Tables S1−S3 in the Supporting Information summarize values of all parameters used in this work. 4.1. Thermodynamic Properties. The adsorption isotherms have been calculated using DFT, which is described in section 3.3. The results for the CNT are presented in Figure 2. We can see that at low pressures the density of the adsorbed phase is 2 orders of magnitude higher than the density of the bulk phase. Nevertheless, both the bulk phase and the adsorbed phase reveal ideal behavior, with density proportional to

Figure 2. CNT adsorption isotherms for (a) carbon dioxide, and (b) methane. Inset: density profiles along the radial direction of the CNT (a) at 298 K and 0.2 bar, 0.8 bar, and 2 bar; (b) at 173 K and 0.8 bar, 2 bar, and 12 bar. G


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the adsorbed phase for high densities. For the low densities of the adsorbed phase we extrapolate between Aimoli et al. data and the kinetic theory result. The values of the mass transport resistance in the CNT for the range of pressures used here are presented in Figure 4.

Figure 3. Variation of the enthalpy of adsorption in the CNT with bulk pressure, for (a) carbon dioxide and (b) methane.

dimensional DFT modeling of argon adsorption at 87 K in a graphitized carbon black pore suggests that interfacial thickness can be about 2 nm in that case,56 the difference attributable to the different potential parameters of argon. 4.2. Transport Properties. 4.2.1. Thermodynamic Resistance. To calculate the mass transfer resistance, Rμ, we use eqs 5 and 10, as well as eqs 25−27. We use the value of the thermal amplitude αqq = 10 obtained earlier for a fluid−vapor interface.34 The specific value of the coefficient may not be accurate; however the previous study shows that it must be of this order of magnitude in order to not violate the second law of thermodynamics.34 We next evaluate values of the thermal resistivities rqq,b and rqq,a far from the interface. As we can see from the isotherms in Figure 2, the bulk phase is almost everywhere in the gas state with small density. It is therefore reasonable to use kinetic theory to estimate the values of the bulk thermal resistivity. According to kinetic theory, rqq,gas = T−2(cvT)/(3√2πd2), where c = 1.5kB is the specific heat per molecule, vT = 8kBT /mπ is the mean thermal velocity of the molecule with mass m, and d is the molecular diameter. The adsorbed phase can have high density, which is beyond the range of applicability of kinetic theory. Aimoli et al. evaluated the fluid thermal conductivities at high pressures using molecular dynamics.55 We use and interpolate their data to estimate the thermal conductivity of

Figure 4. Variation of the mass transfer resistance in the CNT with bulk pressure, for (a) carbon dioxide, and (b) methane.

Similar data for the mass transport resistance into the slit pore are presented in Figure S3 of the Supporting Information. We see that for low pressures it is essentially constant, as expected, since neither the enthalpy of adsorption nor the thermal resistivity depend on the pressure in this region. For larger pressures we have a strong dependence on the pressure, for both the enthalpy of adsorption and the adsorbed thermal resistivity. This results in variation of the mass transport resistance of the entire interfacial region, as it is proportional to the magnitude of the enthalpy of adsorption (cf. eq 10). 4.2.2. Comparison with the Hydrodynamic Resistance. We next consider the relative importance of Rμ with respect to the external hydrodynamic resistance, Rη, given in eq 4. Using the hydrodynamic resistance as the reference, it is important to mention its range of applicability. The Sampson expression (eq 1) is developed for an incompressible fluid and is therefore not applicable for gas adsorption. This is the case for low pressures, where the Sampson expression may be considered as an upper bound for compressible fluids. Rather, it is applicable for high pressures, when both bulk and the adsorbed phases are in the liquid state. In principle, it is possible to obtain an alternative to H


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the fluid at low pressures the actual ratio Rμ/Rη is larger than that presented in Figure 5. At bulk pressures larger than about 0.1 bar, or when the fluid in CNT starts to condense, the relative resistance exceeds the Sampson value of C ≈ 3, and thermodynamic effects start to dominate. At high density in the CNT the interactions between adsorbate molecules become much stronger, which results in rapid increase of the absolute value of the heat of adsorption, as well as the interfacial resistance. At this range of pressures Rμ/ Rη becomes comparable with the value of C = 3, and the thermodynamic resistance increases relative to the hydrodynamic. Thus, the importance of the thermodynamic resistance is correlated with the fluid phase in the CNT. However, since eq 1 may be inaccurate for compressible fluids, the relative importance of the thermodynamic resistance may be manifested even at lower pressures. At high pressures the fluid within the CNT is in the condensed state, with strong interactions between the molecules. Indeed, the enthalpy of adsorption is calculated as the partial derivative of fugacity with respect to the temperature at constant excess adsorption, and at high pressures this difference is nearly independent of pressure; this results in almost infinite slope of the fugacity vs density difference curve. Therefore, the magnitude of the adsorption enthalpy is large and so is the mass transfer coefficient. At this range of pressures the thermodynamic resistance clearly dominates over the hydrodynamic resistance. This becomes even more important if we take into account that these are the only conditions when eq 1 is applicable. 4.2.3. Comparison with the Internal Resistance. It is interesting to compare the value of Rμ with the total internal resistance of the CNT, rCNT,∞L, where L is the length of CNT. The internal resistivity rCNT,∞, is related to the corrected diffusivity, Do.57,58 The length of CNT for which the total internal resistance is equal to the interfacial resistance is L = RμDoρa /R̅ , where R̅ = 8.31 J/(K mol) is the universal gas constant. We take the values of Do from literature molecular simulation results.59−61 They are available for 298 K, so we present the results for this temperature only. The simulation data for various diameters of CNT have been interpolated for the value of 3.374 nm used here. For the purposes of this work, this is sufficiently accurate, as we are interested in the general pressure variation and the order of magnitude, rather then the precise value. The values of the transport diffusivity58 for carbon dioxide vary only weakly with pressure and are practically independent of the pore size for pore diameters between 2.71 and 5.42 nm.60 We interpolated the values of transport diffusivity between 4 × 10−5 m2/s at the density 1 mol/m3 and 5 × 10−5 m2/s at the density 105 mol/m3, which are the averaged values for the CNT diameters of 2.71 and 5.42 nm. The values of the corrected diffusivity57 for methane also do not show significant dependence on the pore size,60 though they show some dependence on the pressure; and we have used the values for the pore diameter of 1.356 nm61 as an approximation. The interpolated values of the corrected diffusivity for the pressure range used here are presented in the insets of Figure 6. In Figure 6 we plot the effect of the pressure on the length of the CNT and of the slit pore, the total internal resistance of which is equal to the interfacial resistance. This length may be compared with a typical value of the membrane thickness in the fuel cell, which lies between 10 and 500 μm.62 For methane the

eq 1 for a compressible fluid, and in this case the parameter C might depend on the fluid density, as well as the radius of the CNT. In particular, it is expected that the hydrodynamic resistance for a gaseous fluid is much less than for a liquid.56 This will significantly decrease the value of C, down to several orders of magnitude lower than the original value. Such analysis is not the focus of the present work, and in view of lack of such expression we analyze our results, taking that C is a constant with a numerical value of 3. Figure 5 depicts the ratio of the thermodynamic resistance at the CNT−bulk fluids interface to the external hydrodynamic

Figure 5. Variation of the mass transfer coefficient for the CNT scaled with the Sampson resistance with bulk pressure, for (a) carbon dioxide and (b) methane.

resistance, given as, Rμ/Rη. Similar data for the resistance ratio into the slit pore are presented in Figure S4 of the Supporting Information. We see that, in contrast to the mass transfer coefficient, the relative resistance depends strongly on the pressure for the whole range of pressures. This is mainly due to the density factor in the expression for Rη. For small pressures the fluid is in the low-density state in both phases, so the thermodynamic effects of the adsorption are expected to be low. The fluid molecules interact weakly, so the enthalpy of adsorption is a small constant and is independent of the pressure, and the resulting resistance is therefore small. Rμ/ Rη increases due to the density factor, which distinguishes J and Q. For these pressures the relative resistance is much less than the Sampson value of C ≈ 3; however, due to compressibility of I


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fluid when entering the pore. Because these processes are affected by different phenomena, their relative importance may not always be the same. As can be seen from Figure 5, the relative magnitude of the thermodynamic resistance with respect to the hydrodynamic resistance increases with the pressure, and beyond pressures of about 0.1 bar it is the thermodynamic resistance which dominates. Clearly, the thermodynamic resistance increases in importance as the pressure increases. The main factors which influence the thermodynamic resistance are the density of the bulk phase and the enthalpy of adsorption from the bulk phase into the CNT. We have also estimated the length of the CNT which has the same internal resistance as the interfacial resistance. It follows from Figure 6 that, for operating conditions of 1 bar, this length may be comparable with typical membrane thicknesses. For such values of the CNT length, the interfacial resistance is comparable with the internal resistance for CO2, while it is smaller for methane. Thus, the relative importance of interfacial resistance compared to the internal resistance varies with the species involved in the transport. These results should be of value in applications of nanoporous materials in general, where fluid infiltration plays a central role, particularly in nanofluidics and membrane transport, where system size is of nanoscale to microscale dimension.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00375. Lennard-Jones parameters of the interacting atoms, external conditions, other parameters, results for slit pore, and information about the interfacial profile (PDF)

■

Figure 6. Pressure variation of the membrane length for which the interfacial resistance is equal to the internal resistance, for (a) the CNT of radius 16.87 Å and (b) the slit pore of width 16.87 Å. Inset: values of the corrected diffusivity interpolated from published data.59,61

AUTHOR INFORMATION

Corresponding Author

*Tel.: +61 7 3365 4263. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

interfacial resistance becomes comparable with the internal resistance at moderate pressures. The thermodynamic resistance is also larger than the hydrodynamic resistance at these pressures. This behavior is similar for both the CNT and the slit pore. In contrast, for carbon dioxide the role of the interfacial resistance with respect to the internal resistance becomes important even at low pressures. The effect is less pronounced for the slit pore.

■

ACKNOWLEDGMENTS We thank Lang Liu for conducting the GCMC simulation from which we obtained the interfacial thickness used here. This research has been supported by a grant (No. DP150101824) from the Australian Research Council, under the Discovery Scheme.

■

5. CONCLUSIONS We have presented here an analysis of the interfacial resistance of a nanoporous membrane comprising carbon nanotubes. The purpose of the analysis was to take into account that the fluid inside and outside the CNT may be in different phases. This changes the fluid behavior at the pore entrance, which at certain conditions may affect the overall membrane resistance. We have shown that in addition to the standard hydrodynamic resistance one must also consider the resistance caused by thermodynamic effects. The hydrodynamic and the thermodynamic resistances of the CNT boundary are caused by different phenomena. The hydrodynamic resistance is present due to bending of flow laminae of the viscous fluid when entering the CNT. The thermodynamic resistance is present due to condensation of the

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Thermodynamic Resistance to Matter Flow at The Interface of a

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