## Abstract

Water-vapour transport in nanostructured composite materials is poorly understood because diffusion and interfacial exchange kinetics are coupled. We formulate an interfacial balance that couples diffusion in dispersed and continuous phases to adsorption, absorption and interfacial surface diffusion. This work is motivated by water-vapour transport in cellulose fibre-based barriers, but the model applies to nanostructured porous media such as catalysts, chromatography columns, nanocomposites, cementitious structures and biomaterials. The interfacial balance can be applied in an analytical or a computational framework to porous media with any microstructural geometry. Here, we explore its capabilities in a model porous medium: randomly dispersed solid spheres in a continuous (humid) gas. We elucidate the roles of equilibrium moisture uptake, solid, gas and surface diffusion coefficients, inclusion size and interfacial exchange kinetics on the effective diffusivity. We then apply the local model to predict water-vapour transport rates under conditions in which the effective diffusivity varies through the cross section of a dense, homogeneous membrane that is subjected to a finite moisture-concentration gradient. As the microstructural length scale decreases from micrometres to nanometres, interfacial exchange kinetics and surface diffusion produce a maximum in the tracer flux. This optimal flux is flanked, respectively, by interfacial-kinetic- and diffusion-limited transport at smaller and larger microscales.

## 1. Introduction

Demand for synthetic polymers and metals in packaging and structural applications is widely regarded as unsustainable, and has recently motivated research to help replace these non-renewable feedstocks with plant-based fibres and chemicals (Bogoeva-Gaceva *et al.* 2007; Thomas *et al.* 2011). While plant-based fibres often produce excellent mechanical and gas-barrier properties, they are notoriously susceptible to moisture absorption and permeation (Morton & Hearle 1993).

Paper, for example, is a compact network of entangled wood fibres, each of which comprises crystalline and amorphous cellulose domains embedded in a matrix of hemicelluloses and lignin (Roberts 1996). Water molecules can *adsorb* onto the elementary fibrils and *absorb* into the amorphous cellulose, forming hydrogen bonds with hydroxyl groups (Morton & Hearle 1993; Watanabe *et al.* 2006). When perturbed from equilibrium, adsorbed and absorbed water molecules migrate with an overall flux that can be modelled with an effective bulk diffusion coefficient (Morton & Hearle 1993; Luukkonen *et al.* 2001).

However, penetrant flux is well known to depend on the product of solubility and mobility factors, e.g. the product of a Henry solubility coefficient and an effective diffusion coefficient. Tracer solubility in porous media may reflect adsorption and absorption on a range of microstructural length scales. While equilibrium moisture-content isotherms are abundant in the literature, moisture is generally considered to either reside as an adsorbed phase within nanopores or absorb into an effective fluid (Krishna & Wesselingh 1997). Similarly, bulk diffusion may reflect diffusion within solid and fluid domains (Cussler 2006), but may also reflect interfacial (surface) diffusion (Krishna & Wesselingh 1997).

Strategies to reduce water-vapour transport in paper involve calendaring (to reduce porosity), coating with nanofibrillar cellulose and chemical modification to increase hydrophobicity (Spence *et al.* 2011). However, approximately 40 per cent of the cellulose can be amorphous and, thus, even with vanishing porosity, there is opportunity for moisture transport by diffusion within the amorphous cellulose (Morton & Hearle 1993).

Surface and bulk moisture uptake and transport phenomena scale differently with the microstructural length scale. Decreasing this length from micrometres to nanometres significantly increases the surface area and, therefore, may significantly enhance transport by surface diffusion and moisture adsorption. Surprisingly, there is presently no sufficiently complete theoretical model that couples equilibrium moisture uptake to diffusive transport under the influence of a bulk moisture gradient.

Maxwell's theory (also termed the Clausius–Mossotti theory) rigorously addresses diffusion in dilute random dispersions of spherical inclusions in a continuous matrix (Maxwell 1873). With a self-consistent approximation (Hashin 1968), also termed the cell model (Chang 1982), the dilute theory can be extended to higher solid volume fractions, achieving satisfactory agreement with computations (Akanni *et al.* 1987) and experimental measurements (Vagenas & Karathanos 1991). While steady diffusion is governed by the same Laplace equation as thermal and electrical conduction, the problems are distinguished by their interfacial boundary conditions and the equilibrium base state.

Thermal and electrical conduction in dispersed composites have been studied extensively. For example, Hasselman & Johnson (1987) and Cheng & Torquato (1997*b*) calculated the effective thermal conductivity of sphere packings with continuous flux and discontinuous temperature interfacial boundary conditions, appropriate for thin insulating interfaces arising from poor adhesion or surface coating. Similar calculations were undertaken with continuous interfacial temperature and discontinuous flux boundary conditions (Cheng & Torquato 1997*a*), appropriate for composites with highly conductive interfaces. Bonnecaze & Brady (1990) developed a computational methodology to rigorously calculate the electrical conductivity of dispersions, and later used this approach to calculate the effective conductivity of random sphere packings (Bonnecaze & Brady 1991), demonstrating good agreement with the Maxwell model at particle volume fractions up to 60 per cent when the sphere conductivity is low. Hashin (1968) compared the effective electrical conductivity obtained from the Maxwell model with experimental data from Stepanow (1912). The comparison shows that experimental data lie between the effective diffusivity given by the Maxwell model and the Maxwell model with phase inversion (Torquato 1991).

Ochoa-Tapia *et al.* (1993) addressed surface diffusion on impenetrable inclusions, assuming that the adsorbed phase is in local equilibrium with a continuous gas phase. Diffusion in the continuous phase without surface diffusion was addressed by Akanni *et al.* (1987) using Monte Carlo simulations. By comparing computations with theoretical effective diffusivity models, they concluded that the self-consistent Maxwell model is satisfactory for a variety of porous media. Later, Vagenas & Karathanos (1991) fitted experimental data from moisture transport in granular food materials to effective diffusivity models, also demonstrating the general applicability of the Maxwell approach.

More generally, a diffusing tracer will partition between the dispersed, continuous and interfacial regions. Disturbing thermodynamic equilibrium by applying a mean concentration gradient may violate the assumption of local interfacial equilibrium. This occurs when the microstructural length scale (e.g. inclusion size) is small enough for the interfacial diffusion fluxes to become comparable to the interfacial exchange fluxes: a situation that is more likely to prevail in nanostructured materials. Under these conditions, the effective diffusivity depends on the equilibrium moisture content and particle size: parameters that do not influence the effective diffusivity according to presently available theories of molecular diffusion in composite materials (Ruthven 1984). A well-known example is the parallel pore-fibre diffusion model for moisture transport in paper materials (Gupta & Chatterjee 2003; Ramarao *et al.* 2003).

Here, we formulate an interfacial balance that couples diffusion in dispersed and continuous phases to adsorption and surface diffusion. This model accommodates equilibrium partitioning between the dispersed, continuous and interfacial domains, allowing two of the six kinetic coefficients to be expressed in terms of equilibrium adsorption and absorption (solubility) isotherms. While the interfacial balance can be applied to porous media with any geometry, we demonstrate its key features using a model porous medium: randomly packed beds of spheres in the self-consistent Maxwell approximation.

The paper is set out as follows. We begin by deriving the general interfacial boundary conditions, drawing on kinetic equations and the principle of detailed balance from non-equilibrium thermodynamics. After linearizing the model about local thermodynamic equilibrium, identifying the minimal set of dimensionless parameters and estimating their magnitudes, we focus on a model porous medium, furnishing the effective diffusivity for packed beds of spheres, using self-consistent Maxwell approximation for finite solid volume fractions. Results examine the concentration disturbances, and how the effective diffusivity depends on the independent dimensionless parameters. We then implement the microscale model on the macroscale, computing water-vapour transport rates (WVTRs) in barriers with finite thickness and differential moisture content.

## 2. Theory

### (a) Diffusion and interfacial balance equations

Without loss of generality, we consider a dispersed solid phase, a continuous gas phase and their interface (figure 1). We apply the model to a single solid sphere with radius *a* in an unbounded gas. However, the model and, in particular, its boundary conditions are intended to apply to arbitrarily shaped interfaces in porous media and, indeed, in other multi-phase fluid (gas–liquid and liquid–liquid) systems.

In the solid and gas phases, steady-state tracer conservation demands
2.1
where *n*_{s} and *n*_{g} are the solid and gas concentrations (number densities). On the interface,
2.2
where *n*_{i} is the interface number density, *D*_{i} is the interface diffusivity (taken to be constant here) and **∇**_{i}=**∇**−** n**(

**⋅**

*n***∇**) with

**the outward unit normal (from the solid to gas phase). Note that 2.3 are the interfacial exchange fluxes (from phase**

*n**j*to phase

*k*) with

*k*

_{jk}≥0 the respective kinetic coefficients. On the solid and gas sides of the interface, 2.4 and 2.5 where

*D*

_{s}and

*D*

_{g}are the solid and gas diffusion coefficients (taken to be constants here). Note that equations (2.4) and (2.5) balance each diffusion flux with the accompanying net interfacial exchange flux.

In principle, the kinetic coefficients could be calculated from kinetic theory. However, the ratios *k*_{jk}/*k*_{kj} are related to equilibrium partitioning; so, for example, having ascertained *k*_{gi} from the kinetic theory of gases, *k*_{ig} can be ascertained from an equilibrium adsorption isotherm. In general, *k*_{jk} are expected to depend on the temperature and all tracer concentrations *n*_{j}. However, at sufficiently low densities (or surface concentrations *n*_{i}), *k*_{jk} are temperature-dependent constants; so each flux is proportional to *n*_{j} evaluated at the interface. Note that the adsorption rate obtained from the statistical rate theory (Elliott & Ward 1997) for small perturbations from equilibrium is equivalent to the rate furnished by the law of mass action.

### (b) Equilibrium

We denote the equilibrium number densities *n*^{0}_{s}, *n*^{0}_{g}, *n*^{0}_{i} with . At equilibrium, equations (2.2)–(2.4) require , and , but it is easily shown that only two of these balances are independent. Thus, specifying one equilibrium phase composition, , say, uniquely specifies the other two phase compositions (at a given temperature). This is equivalent to equating the equilibrium chemical potential of the tracer in the three phases. Accordingly, and , giving

Note, however, that the principle of detailed balance (Onsager 1931) demands , , and , furnishing thermodynamic constraints 2.6 and, thus, three independent constraints from equilibrium adsorption and absorption isotherms. Accordingly, the number of independent kinetic coefficients can be reduced from six to only three when equilibrium isotherms are available.

The total equilibrium uptake in a packed bed of spheres with sphere volume fraction *ϕ* can be written as
2.7
2.8
where, for example, is furnished by a BET isotherm (Brunauer *et al.* 1938) or one of its variants (Anderson 1946; de Boer 1953; Guggenheim 1966), and is furnished by solubility data, such as Henry's law or a Flory–Higgins-type solution model (Huggins 1941; Flory 1942).

### (c) Scaling and parameter estimates

With the application of a bulk concentration gradient, which for a dilute bed of spheres furnishes a far-field boundary condition
2.9
where ** B**=〈

**∇**

*n*

_{g}〉 is the average gradient of

*n*

_{g}, equations (2.1)–(2.4) furnish the perturbations , and , where . Therefore, to linear order in these perturbations, 2.10 where the partial derivatives are evaluated at equilibrium. Next, scaling position with

*a*, volume concentration perturbations with

*n*

^{0}

_{g}, and the interfacial concentration perturbation with

*n*

^{0}

_{i}, the model equations (with ) can be written in dimensionless form, 2.11 2.12 2.13 2.14 with dimensionless parameters , , , , ; and , , . All these dimensionless parameters (

*Π*s) are the ratio of a characteristic diffusion flux or interfacial exchange flux to the equilibrium gas-interface exchange flux.

Note that the thermodynamic constraints (2.6) require *Π*_{ig}=1, *Π*_{is}/*Π*_{si}=*H* and *Π*_{gs}/*Π*_{sg}=*H*, which reduce the foregoing set of independent dimensionless parameters to
2.15
with interfacial boundary conditions
2.16
2.17
2.18

Gas kinetic theory suggests that *k*_{gi} and *k*_{gs} are of the order of the thermal velocity for water molecules at room temperature. Thus, when *k*_{gi}=*k*_{gs}, we have *Π*_{sg}∼*H*^{−1}, and, for example, with *D*_{i}∼10^{−8} m^{2} s^{−1}, *D*_{s}∼10^{−10} m^{2} s^{−1} and *D*_{g}∼10^{−5} m^{2} s^{−1}, we find *Π*_{i}∼10^{−10}*a*^{−2}*K*, *Π*_{s}∼10^{−12}*a*^{−1} and *Π*_{g}∼10^{−7}*a*^{−1}. For particles with size in the range *a*∼0.01–1 μm, for example, , and . Finally, we expect *Π*_{si}≪1 when the tracer kinetic velocity in the solid is much less than in the gas.

The well-known BET adsorption isotherm (Brunauer *et al.* 1938) furnishes
2.19
where (for a flat, non-fractal interface) is a constant ( is the specific binding-site area), and the fractional occupancy
2.20
Here, *c*≥0 is a dimensionless parameter, and the relative humidity (RH) , with *n*^{*}_{g} being the gas-phase saturation concentration, is obtained from the saturation vapour pressure.

### (d) Tracer concentration perturbations

From linearity and symmetry, the solutions of Laplace's equation for a sphere in an unbounded medium (equation (2.1)) have the form
2.21
2.22
2.23
where with *θ* the polar angle, and ** B**=

**∇**〈

*n*

_{g}〉

*a*/

*n*

^{0}

_{g}is the dimensionless gas-phase concentration gradient. Note that

*λ*

_{0},

*λ*

_{1},

*λ*

_{2}are scalar constants;

*λ*

_{1}is termed the dipole strength, which characterizes the

*r*

^{−2}far-field disturbance to the gas-phase concentration.

At the interface (*r*=1),
and
so the interfacial balance and boundary conditions (equations (2.12)–(2.14)) furnish explicit formulae for *λ*_{0}, *λ*_{1} and *λ*_{3}, each in terms of the six independent dimensionless variables in equation (2.15). These are provided as electronic supplementary material.

### (e) Local effective diffusivity

Averaging the flux in a dilute random bed of spheres gives −*D*_{e}**∇**〈*n*_{g}〉, where the effective diffusivity *D*_{e}=*D*_{g}(1−3*λ*_{1}*ϕ*). Note that a self-consistent approximation for beds with finite solid volume fraction gives
2.24
where, in a composite with a continuous tracer concentration across the interface, *λ*_{1}=(1−*χ*)/(2+*χ*) with *χ*=*D*_{s}/*D*_{g} (Maxwell 1873). Other special cases arising from the general solution are provided as electronic supplementary material. Note that, as a consequence of applying the Maxwell self-consistent model, tracer concentrations are influenced by the mean concentration gradient. Explicit particle–particle interactions must be accounted for by using much more computationally expensive particle-based computations, in a manner similar to that of Bonnecaze & Brady (1991), for example.

### (f) Membrane-averaged permeability

If a membrane with thickness *L* is subjected to a gradient in the gas-phase moisture content from one side to the other, , then the steady-state flux *J*=const. satisfies
2.25
where *P* is the membrane permeability, and *D*_{e} depends on the (local) average gas-phase tracer concentration *n*^{0}_{g}. Thus, scaling distance with *L* and concentrations with *n*^{*}_{g}, a convenient computational form for solving equation (2.25) is
2.26
where *z**=*z*/*L* and the relative humidity .

Equation (2.26) is easily solved using a standard ‘shooting’ algorithm: with the specified value of *x*(*z**=0), iteratively vary *P* to achieve the specified value of *x*(*z**=1)=*x*(*z**=0)+Δ*x* when integrating equation (2.26) from *z**=0 to 1. Here, we integrated equation (2.26) using the Matlab function ode45. According to membrane phenomenology, , where is an overall solubility parameter (dimensionless Henry constant) and is an overall mobility diffusion coefficient.

Later, we report fluxes as the WVTR (g m^{−2} d^{−1}), normalized for a membrane with thickness 100 μm and 50 per cent RH at room temperature on one side and zero RH on the other. Explicitly, WVTR equals *P* multiplied by 0.5×*n*^{*}_{g}×(18/*N*_{A})×10^{4}×24×3600≈9.9×10^{9} for humid air at room temperature (all variables in SI units, ). Thus, as a useful point of reference, WVTR when *ϕ*=0 (pure moist air) under these conditions is 2.5×10^{5} g m^{−2} d^{−1} (*D*_{g}≈2.5×10^{−5} m^{2} s^{−1}).

## 3. Results

### (a) Equilibrium moisture content

The gravimetric moisture content (GMC) of a porous medium with volume fraction *ϕ*=0.90 is shown in figure 2*b* as a function of the RH for several characteristic microscales *a*. Note that the GMC is the total equilibrium moisture content *n*^{0} given by equation (2.7) minus the average gas-phase contribution (1−*ϕ*)*n*^{0}_{g}. In this example, the solute has a low solubility (*H*=0.01) and a moderate surface affinity (*c*=10) with high surface-site density (*a*_{m}=0.4 nm). We have scaled the absolute moisture content *n*^{0} with the saturation concentration *n*^{*}_{g}, which is a constant at a fixed temperature. With large *a*, GMC increases linearly with RH because the dominant uptake is due to solid absorption. However, as the specific surface area increases (smaller *a*), GMC increasingly reflects surface adsorption, according to the BET isotherm shown in figure 2*a*. Clearly, the equilibrium GMC is a strong function of the gas-phase concentration, particularly close to saturation. Below, we will examine how equilibrium solubility and adsorption influence the tracer flux.

### (b) Tracer concentration disturbance

The tracer concentrations and their respective disturbances for spherical inclusions in an unbounded continuum are shown in figures 3 and 4, respectively. Here, we have set all the dimensionless parameters to 1, except where noted in the figure captions. The accompanying dipole strengths are also reported in the figure captions. Note that the applied gradient is directed from left to right, and that a positive dipole strength (*λ*_{1}>0) increases the effective diffusivity of the composite relative to the gas-phase diffusivity (*D*_{e}/*D*_{g}>1).

In striking contrast to classical diffusion, the concentration disturbance is generally discontinuous across the interface. Interestingly, the internal disturbance can attain a positive gradient, as seen in figure 3*d*, where the equilibrium tracer concentration in the inclusion is very high (*H*=10^{3}). Here, the internal field achieves equilibrium with the external field, and the magnitude of the internal gradient is scaled with the solubility parameter *H*≫1. In figure 3*b*,*e*, the internal field is uniform, but for distinctly different reasons. In figure 3*b*, the particle is thermodynamically penetrable, but there is zero kinetic exchange between the solid and gas or between the solid and interface (*Π*_{sg}=*Π*_{si}=0) (see the electronic supplementary material). In figure 3*e*, however, the tracer solubility in the solid is zero (*H*=0); so it is thermodynamically impenetrable, despite having a finite solid-phase diffusivity and finite kinetic exchange coefficients. Next, in figure 3*a*, there is zero interfacial mobility (*Π*_{i}=0), but the particle is thermodynamically and kinetically penetrable. In figure 3*c*, the kinetic exchange coefficients are large (*Π*_{sg}=*Π*_{si}=10^{3}), producing a continuous tracer concentration across the interface. Finally, in figure 3*f*, a high interfacial mobility (*Π*_{i}=10^{3}) with zero internal mobility (*Π*_{s}=0) is demonstrated to produce a dipole strength that enhances the effective diffusivity. Even with zero internal mobility, the internal field generates a gradient to achieve internal equilibrium with the external field (if given infinite time).

Note that (see the electronic supplementary material) is the limit in which the kinetic exchange rates involving the solid are much slower than those involving the gas. Under these conditions,
Without surface diffusion (*Π*_{i}=0), the solid- and gas-phase diffusion fluxes at the interface balance, requiring at *r*=1. Therefore, when *Π*_{si}≫1 with *HΠ*_{si}≫1, we find , which in dimensional concentration perturbation variables can be written . Also, when *Π*_{si}≪1 with *HΠ*_{si}≪1, ; in dimensional variables, this is , which reflects local equilibrium between the interface and gas phases.

### (c) Effective diffusivity

First, to establish the role of the equilibrium adsorption isotherm , figure 5 shows how the effective diffusivity varies with *Π*_{i} when the other dimensionless parameters all equal 1 with various values of . In this representation, varying *Π*_{i} with constant *Π*_{s} and *Π*_{g} should be considered as changing the surface diffusivity. Under these conditions, there is a moderate increase in *D*_{e} when *Π*_{i}∼*H*.

In figure 6, we vary *Π*_{i} but adjust *Π*_{s} and *Π*_{g} to illustrate how *D*_{e} varies with the particle size when the diffusivities are fixed. Accordingly, plotting as a dimensionless particle size on the abscissa with and highlights how the effective diffusivity becomes limited by interfacial exchange kinetics when is small, and limited by diffusion in the continuous phase when is large and *H* is small. Note, however, that, when *H* is large, the external field produces a large gradient within the dispersed phase, thereby overcoming diffusion limitations in the continuous phase.

Figures 7 and 8 are similar to figures 5 and 6, except that we have substantially increased the solid volume fraction, and adjusted and to small values, with *Π*_{s}/*Π*_{g}=*D*_{s}/*D*_{g}=10^{−5} more representative of physical systems, also setting *Π*_{sg} and *Π*_{si} to small values. Together, these changes substantially increase the general magnitude of the effective diffusivity. Note, however, that limiting the kinetic exchange between the solid, interface and gas phases now attenuates the flux with large particles (figure 8). For small particles, the effective diffusivity is, as expected, limited by kinetic exchange when the particles are small enough, but is still sensitive to the solubility parameter *H*. In figure 7, the effective diffusivity attains very large values when *Π*_{i}>1. This reflects the significant surface flux, promoted by high interfacial mobility and specific surface area.

### (d) Membrane-averaged flux (water-vapour transport rate)

Having examined the concentration disturbance and effective diffusivity in uniform membranes, i.e. membranes subjected to a weak bulk concentration gradient, we now turn to the flux in membranes that are subjected to a variety of tracer concentration gradients, as, for example, achieved by using a membrane as a barrier between humid and dry air. Here, we consider barriers separating humid air with varying RH on the one side and either perfectly dry or almost saturated (99% RH) gas on the other. In this manner, we access the roles of the average moisture content and average moisture-content gradient on water-vapour transport.

For conventional paper barriers, WVTRs are *O*(10^{3}) g m^{−2} d^{−1}, which, when *D*_{g}≈2.5×10^{−5} m^{2} s^{−1}, corresponds to *P*/*D*_{g}=*O*(10^{−2}). Wood fibres typically have a high aspect ratio and ribbon-like structures with approximately 5 μm width and 2 μm thickness. Barriers from entangled nano-dimensioned polymeric ribbons, termed nano-paper, have lower WVTR values in the 100–900 g m^{−2} d^{−1} range, depending on the surface chemistry (Huang *et al.* 2012). This reflects the smaller microscale (tube diameters are approx. 200 nm width and approx. 20 nm wall thickness) and the important roles of surface adsorption and diffusion.

Note that careful consideration must be given to external mass transfer resistances. Because WVTR values for porous materials are often several orders of magnitude higher than for hydrophobic polymer- and metal-based barriers, such as polyethylene and aluminium, WVTR measurements can be susceptible to external mass-transfer resistances. These modify the boundary conditions for equation (2.25), via , where and are the external relative humidities and mass-transfer coefficients (depending on experimental geometry and convective flow) on each barrier surface. Here, we report results with *δ*_{±}=0, corresponding to zero external mass-transfer resistance, giving and .

Note that Nilsson *et al.* (1993) measured WVTR for a variety of papers, reporting effective diffusivities by carefully separating the external mass transfer resistance from the intrinsic paper resistance. From these data, intrinsic WVTR values for uncoated, uncalendered papers span the 10^{3}–10^{5} g m^{−2} d^{−1} range, much higher than overall WVTR values, but, nevertheless, comparable to the intrinsic WVTR values predicted by the present theory (see figures below).

Membrane profiles with 100 and 50 per cent RH on the left-hand side and zero RH on the other are shown in figure 9. The shapes of these profiles are representative of those with other particle sizes, but the accompanying WVTRs are very different, as are the general magnitudes of the moisture content. In general, WVTR is sensitive to the RH and the difference from one membrane side to the other (figure 10), particularly when *a* is large. Moreover, WVTR increases with the surface diffusion coefficient when *Π*_{i}∼1 (see below). Here, WVTR is practically independent of the RH difference, but increases by approximately one order of magnitude with an accompanying four orders of magnitude increase in the surface diffusion coefficient (see below). The figures below help us to quantify how the various parameters influence WVTR.

Figure 10 shows how WVTR depends on the RH difference, ΔRH. In figure 10*a*, RH is zero (dry air) on the left-hand side, and varies from zero to 1 (saturated air) on the other. In figure 10*b*, however, RH is 1 on the left-hand (saturated) side, and varies from zero to 1 on the other. These scenarios correspond to two ways of conducting WVTR experiments. As expected, both yield the same WVTR when |ΔRH|=1. Recall, WVTR is scaled in a manner that reflects the intrinsic permeability (mobility and solubility influences). Thus, absolute WVTRs, in samples with the same thickness, increase as shown in the figure, but multiplied by a factor |ΔRH|.

Clearly, permeability can increase substantially with moisture content, particularly when one side is close to saturation. More interesting, perhaps, is the qualitative change upon varying the particle size *a*. Decreasing the microstructural length scale increases the specific surface area, increasing surface moisture content and generally enhancing transport by surface diffusion. With very small *a*, surface diffusion is limited by interfacial exchange kinetics, making WVTR practically independent of RH. Note that the curves in figure 10 with large particle radii (*a*=100 and 1000 nm) resemble literature data for paper (Yang *et al.* 2011), typically exhibiting a substantial increase in WVTR with increasing RH. Thus, according to the model, moisture transport occurs with competition between surface diffusion—on surfaces with an *O*(0.1 μm) to *O*(1 μm) microstructural length scale—and interstitial gas-phase diffusion. These inferences are consistent with the microstructural length scale ascertained from the equilibrium moisture content (figure 2).

WVTRs when RH(*z*=0)=0.5 and RH(*z*=*L*)=0 are shown in figures 11–13. Other relevant parameters are provided in the captions. Figure 11 examines how the Henry solubility parameter *H* influences WVTR. Here, *Π*_{sg}=*H*^{−1}, so increasing *H* is accompanied by a reciprocal decrease in the interfacial exchange kinetics between the gas and solid phases.

Note that WVTR is a non-monotonic function of particle size *a*, as demonstrated more clearly in figure 12, where *H*=0.01. Here, ostensible maxima are flanked by kinetically hindered transport (local minimum in WVTR) with small particles, and diffusion-limited transport with very large particles. Increasing the adsorption capacity, by increasing the BET adsorption parameter *c*, increases the maximum WVTR, also shifting it to larger particle sizes. Note that the increase in WVTR with decreasing particle size at asymptotically small particle size can be attributed to the equilibrium surface moisture content, not its influence on the effective diffusivity.

Figure 13 shows how the BET isotherm parameter *c* influences WVTR, as might be achieved by chemically modifying the surface, tuning the hydrophobicity but not influencing *Π*_{sg}. Clearly, increasing the hydrophilicity (increasing *c*) significantly increases WVTR, particularly for small particles. Figure 14 shows how the solid volume fraction influences WVTR. As generally expected, and as often reported in the literature (Spence *et al.* 2010), WVTR decreases with increasing solid content, reaching finite values at zero porosity. Again, WVTR is not a monotonic function of particle size. Small particles promote WVTR by increasing the surface moisture content, promoting higher surface diffusion, but limiting WVTR owing to interfacial kinetics, as previously identified in figure 12, for example.

Finally, figure 15 shows how WVTR varies with the surface diffusivity. Note that we have scaled *D*_{i} with . In this manner, WVTR increases monotonically with the abscissa (with fixed particle size *a*). However, increasing *D*_{i} by four orders of magnitude increases WVTR by only two orders of magnitude. When *D*_{i} is small, WVTR plateaus to a value at which the flux is dominated by diffusion through the gas phase, and, therefore, is independent of particle size. When *D*_{i} is large, however, WVTR plateaus to values that also reflect particle size.

## 4. Summary

We have developed an interfacial boundary condition for diffusion in composite materials that unifies thermodynamic equilibrium, surface diffusion and interfacial exchange kinetics. The boundary condition can be easily integrated into continuum, microscale computations to capture geometrical influences on the bulk effective diffusion coefficient. Here, we applied the boundary condition to a model porous medium comprising penetrable solid spheres dispersed in a continuous gas. We demonstrated how the model parameters influence the effective diffusivity in a uniform porous medium, and then applied it to membranes with finite thickness, subjected to finite moisture gradients. The model demonstrates that water-vapour transport increases with RH, and that water-vapour transport is sensitive to the microstructural length scale. Nanostructured porous media are subject to interfacial kinetic limitations, whereas the flux in microstructured porous media is limited by classical diffusion in the continuous and dispersed phases. Our calculations suggest that water-vapour transport is maximized when the characteristic length scale is of the order 100 nm. Note that we have not considered Knudsen diffusion in the continuous gas phase. This will probably reduce WVTR in porous media where the pore size is less than 100 nm. The model is not limited to moisture transport in gas-solid porous media, such as textiles, catalysts, biomaterials and engineering structures (Xi *et al.* 1994); indeed, it may apply to a broader class of diffusion in porous media, including, for example, molecular diffusion in water-saturated porous structures.

## Acknowledgements

Supported by the NSERC Innovative Green Wood Fibre Products Network.

- Received March 20, 2012.
- Accepted May 15, 2012.

- This journal is © 2012 The Royal Society