Effect of Surface Diffusion on Adsorption–Desorption and Catalytic

Article pubs.acs.org/JPCC

Effect of Surface Diffusion on Adsorption−Desorption and Catalytic Kinetics in Irregular Pores. I. Local Kinetics Aldo Ledesma-Durán, S. I. Hernández, and Iván Santamaría-Holek* Unidad Multidiscliplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México, CP 76230 Juriquilla, Querétaro, Mexico ABSTRACT: By using a mass balance equation, we deduce an effective equation for the concentration of adsorbed particles that considers the surface diffusion of the particles and an adsorption−desorption process inside a pore of irregular shape. This equation, together with the generalized Fick−Jacobs equation for the bulk diffusion, allows us to quantify the augment in the effective flux through a porous material due to the migration of the gas material along the surface. The equation for the surface concentration has a similar structure to the wellknown Fick−Jacobs equation and it takes explicitly into account the shape of the pore through the width and length of the walls, making our model an important tool in the understanding of the interaction of diffusion and adsorption in porous materials where the length of the pores is greater than its width. In this work we predict the profile for the fractional surface coverage as a function of the geometry, the surface and bulk diffusion coefficients, and the isotherm of the process in several illustrative situations that permit us to prove that the effective diffusion coefficients augments with the surface diffusion, that the surface diffusion can give place to internal fluxes in opposite directions between bulk and surface particles, and finally that the diffusivity of adsorbed particles is greater in the narrow regions of the pore, in contrast with what happens to bulk particles. Our description predicts very interesting couplings between bulk and surface diffusion that occur locally and are regulated by the adsorption−desorption kinetics. parameters. Then, the flux of particles as a function of time is measured. The difference between the effective flux and the one that would be expected if only bulk diffusion were present, is taken as the surface diffusion flux. Assuming some particular mathematical form of this surface flux and a particular isotherm for the entire process, it is defined a surface diffusion coefficient as the fitting parameter of the flux curves as a function of time. The experimental data at this description level are mostly analyzed under the optics of two models: (a) the Kapoor-Yang model12 and (b) the pore and surface diffusion models of Do and co-workers.13,14 In both cases, it is assumed that the system is isothermal, that the adsorbed particles are in local equilibrium with the particles of the bulk, and that the bulk and surface fluxes can be distinguished with different diffusion coefficients having a constant value at zero loading.6 In all cases, the surface diffusion coefficient measures the proportionality between the gradient of concentration along the pore and the excess of mass flux due to surface migration, i.e., the flow that is not due to bulk diffusion. A possible drawback when these type of approaches are followed is that the form of the flux depends directly on the assumed mathematical form of the mass flux. This means that the previous models where formulated to account for different types of

1. INTRODUCTION The surface diffusion refers to the process of particle migration in the adsorbed state, and it is responsible for unusual mass transfer rates through porous media. This process may enhance the mass flux compared to the case when only bulk diffusion of particles in an adsorbent media occurs.1−4 In the last case, the adsorbed particles are stagnated on the walls and can leave its position only through desorption to the bulk. However, under certain circumstances and characteristics of the fluid and the substrate,5 such adsorbed particles are allowed to migrate along the surface producing a new flux that causes an apparent diffusivity larger than the diffusivity of a process where the adsorbed phase is immobile.6 Surface diffusion is of great importance because it can be responsible for most of the enhanced mass fluxes of fluids through porous media.7−9 It has been experimentally proved by using nuclear resonance methods that, at microscopic level, the surface diffusion coefficient increases with the surface fractional coverage and the temperature.10,11 Transient experimental methods like the differential adsorption bed, the time lag method, the constant molar flow, and the differential permeation technique, are used to assess the surface diffusion at mesoscopic level (see the Review of Medved et al.6). In most of those experiments, some fluid passes through a given porous media where the reduction of the available volume due to the solid phase is taken into account using the porosity and tortuosity © XXXX American Chemical Society

Received: April 18, 2017 Revised: June 6, 2017 Published: June 21, 2017 A

DOI: 10.1021/acs.jpcc.7b03652 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

process, the surface diffusion, and the geometric irregularity of a bidimensional pore. Together with the generalized Fick−Jacobs equation for the bulk particles, these two equations allow us to obtain all the relevant information for the reduced description of the reaction−diffusion processes inside the pore. In section 4 we establish the relation of these two equations with the apparent diffusion coefficient that can be measured in a membrane, and therefore, we establish how this coefficient depends upon the molecular diffusion of the fluid and the value of the surface diffusion coefficient. Section 5 presents illustrative examples clarifying the scopes and advantages of our model. Finally, in section 7 we comment and discuss the conclusions of our work.

macroscopic experiments, and therefore they do not refer to the microscopic surface diffusion coefficient. In this sense, the surface diffusivity that each model provides for each type experiments could not be consistent with the other one; see Medved et al.6 With the aim of avoiding this issue, in the present work we will split the total mass flux into volumetric (bulk) and surface contributions, in such a way that they follow their own equation and, therefore, can be studied separately. The major difference with the before mentioned traditional schemes is that we will study the effective bulk and surface diffusivities on the basis of direct molecular and surface diffusion coefficients, which can be directly measured at the limit of zero loading from tracer diffusion experiments. As these measurements can be carried out independently by other methods, our model for a single pore does not depend upon the particular experimental set up. As we will see, this means that as long as the shape of the pore and the mean square displacement (msd) of the adsorbed particles at zero loading are known, our methodology allows us to study with detail the dependence of the bulk and surface concentrations on the irregular geometry of the pore and the isotherms. Our analysis follows a procedure similar to that of the reduction scheme of Fick−Jacobs (FJ) for confined volumetric diffusion. In this theoretical model, the mass balance equation can be averaged over the transversal coordinate, giving an equation for the concentration of particles in the bulk only along the coordinate of transport.15−17 However, in the present case we will use two effective or reduced mass balance equations corresponding to the bulk and surface concentrations. The main achievements of this work are to deduce the reduced mass balance equation for the surface concentration that considers surface diffusion and the adsorption−desorption kinetics occurring at the surface of the pore. For the bulk concentration, we will use the generalized FJ approach deduced in LedesmaDurán et al.17 that considers bulk diffusion and chemical interchanges with the walls. Following these lines, we were able to prove that the surface concentration inside a single pore obeys a very similar equation to the generalized FJ equation and that this equation for the surface concentration allows us to quantify the exchange of material that occurs between the bulk phase and the surface inside a pore of irregular shape. Therefore, our model clearly discerns the interrelation that occurs between the specific shape of the pore and the surface diffusion. As we will see, this in turn affects the isotherms, the fractional surface coverage, and more importantly, the apparent diffusion coefficients that can be measured by means of uptake experiments. Because the geometry of various porous structures is known, this study allows us, in principle, to predict the concentration of the material within each pore, which now can be measured and verified by imaging experiments.18−22 We have to mention that, because our main goal in this work is to relate the adsorption and diffusion processes with geometric characteristics of the pore, in this work we focus in a twodimensional pore where the mathematical form of the FJ equations we present allows us to interpret easily the results in terms of the constriction, sinuosity, and surface area of the pore, characteristics that can be also measured in three-dimensional pores. The work is organized as follows. In section 2 we summarize the deduction of the generalized Fick−Jacobs equation in the presence of heterogeneous reactions in pores. Then, section 3 is devoted to derive an equation for the concentration of adsorbed particles that takes into account the adsorption−desorption

2. FICK−JACOBS EQUATION FOR BULK DIFFUSION AND ADSORPTION IN AN IRREGULAR PORE For conceptual and notational completeness, it is convenient to summarize in this brief section the main aspects of the study of volumetric diffusion and adsorption inside an irregular pore in accordance with the FJ approach. According to the FJ framework, if the pore is long enough compared with its width, it may be shown that just one direction along the pore (the longitudinal one) is of interest to the effective transport properties of the material.15 In this case, the concentration changes along the transverse directions of the pore are considered negligible when compared with the changes along the longitudinal coordinate.16 This allows us to study the process in terms of only one equation that incorporates explicitely the geometric relevant aspects of the diffusive transport in a FJ equation.15 The FJ approach takes as its basis the diffusion equation in a two-dimensional pore, and therefore, just like that mean field equation, the FJ equation is valid in the case when the concentration and the fluxes are homogeneous functions of the coordinate longitudinal x, implying with this that the particles have to be much smaller than the width of the pore and, this last one smaller than its length.15 In ref 23 we have shown that the model compares very well with the numerical solutions of the complete diffusion equation even for drastic constriction conditions and large sinuosity. However, when particles inside the bulk pore can also be interchanged with the walls of the pore, the generalized FJ equation deduced in Ledesma-Durán et al.17 considers not only the Fickian diffusion of particles within the irregular volume of the pore but also the material adsorption−desorption through the irregular walls of the pore. This equation for the average volumetric concentration of a substance, Cb, is of the form ∂C b ∂C b ⎤ γ 1 ∂ ⎡ (x ,t ) = ⎥⎦ + rb ⎢⎣D bw w ∂x w ∂t ∂x

(1)

where Db(x) is the effective diffusion coefficient (which depends upon the longitudinal coordinate x) and rb(Cb,x,t) is the rate of adsorption−desorption at the surface. The subscript “b” stands for volumetric (bulk) quantities like concentration or diffusivity. The importance of this generalized FJ equation is that it incorporates explicitly in the equation the three geometrical factors related with the effectiveness of the diffusion and adsorption processes. In the case of the diffusion, the relevant geometrical parameters are the pore width w(x) and the middle line h(x). The first parameter appears directly in the generalized FJ equation, and both appear in the effective diffusion coefficient Db(x), which can be approximated as24 B


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The Journal of Physical Chemistry C D b (x ) =

D0 2

1 + hx (x) +

1 w2 12 x

adsorbed concentration per unit of volume Cs defined according to17

(2)

Subscript x stands for derivation respect to this longitudinal coordinate. On the contrary, in the case of the adsorption, the relevant geometric quantity is the length of the active walls on the surface of the pore. This is quantified through the factor γ(x) appearing in eq 1. If l1 and l2 are the lengths of the two walls of a bidimensional pore, then ∫ L0 γ(x) dx = l1 + l2, where L is the longitudinal length of the pore (Figure 1). Equation 1 constitutes

Cs(x ,t) ≡

w2(x)

1 w(x)

∫w (x)

Ca(x ,y ,t) dy

(3)

1

where w(x) = w2(x) − w1(x) is the width of the pore whose walls are defined by the functions w2 and w1, Figure 1. By use of Cs, the number of adsorbed particles is obtained integrating the concentration over the entire volume of the pore: L

N (t ) =

w2(x)

∫0 ∫w (x)

Ca(x ,y ,t) dy dx =

1

∫0

L

w(x) Cs(x ,t) dx (4)

Volumetric concentrations of adsorbed particles are commonly used17 because they help to estimate reaction constants in the cases when the shape of the pore is unknown, and only internal volumes and average concentrations of the input feeding of the materials are known. However, when one is interested in a mass balance equation for the adsorbed particles, it seems more natural to consider a mass balance equation that uses the concentration defined as the number of particles by unit of surface. In this case, the surface diffusion coefficient can be measured with tracer diffusion experiments along the adsorbed surface in the limit of negligible loading,6,28 and the surface flux can be directly evaluated by measuring the surface concentration (in mol/cm of surface length).6 In view of this, the process can be better described by using a coordinate that runs along the pore wall. It was shown in Ledesma-Durán et al.17 that an appropriate way to incorporate the irregularity of the pore in the description is by introducing the coordinate zi(x) (with i = 1, 2):

Figure 1. Geometry of a pore of length L and width w(x) = w2 − w1. The two walls are represented by the shape functions w1 and w2. The lengths of each wall are l1 and l2, respectively. The coordinates zi run along the two walls.

a generalized Fick−Jacobs equation whose consequences stated in Ledesma-Durán et al.17 could be summarized as follows: (1) the diffusive transport of material could be reduced by the presence of narrow cages in the channel, and (2) if the active sites are distributed homogeneously along the pore walls and the adsorbed phase is inmobile, then the material tends to be adsorbed/desorbed near the bottle necks of the pore.23 In a system where the characteristic times of both processes are of the same order, the last two facts raise interesting consequences about the distribution of material inside pores with throats and cages.23,25,26 This geometric dependence affects the value of the internal effectiveness factor and the mass transfer coefficient,27 useful chemical engineering concepts that can be easily adapted to the FJ methodology.23

zi(x) =

∫0

x

γi(x′) dx′

(5)

Here, γi = 1 + wi , x ′2 is the density of length of the surface i (Figure 1). If the surface concentration si(zi,t) along the wall i is defined as the number of particles adsorbed per unit of length in 2D (or per unit surface in 3D problems), then this quantity measures the number of adsorbed moles along each surface. Mathematically, this is expressed by the relation

3. EQUATION FOR THE CONCENTRATION OF ADSORBED PARTICLES: SURFACE DIFFUSION AND ADSORPTION Before proceeding with the mathematical calculations, we believe it appropriate to notice that the model we present as well as the units used to measure the different quantities appearing in the model are established for a two-dimensional pore. This means that the volume of the pore is measured in cm2 and the surface area in cm. Therefore, the volumetric concentration of the particles in the bulk Cb is given in mol/cm2. Concerning to the concentration of adsorbed particles, this can be measured in two ways: first, we can count the number of particles adsorbed by unit of surface s (in mol/cm of surface) or the number of adsorbed particles in the internal volume Cs (in mol/cm2). As we will see, these two equivalent forms of measuring the number of surface particles will constitute the key for understanding the mathematical dependence of their distribution inside the pore. 3.1. Volumetric and Surface Concentrations. The concentration of particles adsorbed at the surface, Ca(x,y,t), is defined as the amount of particles adsorbed by unit of volume of the pore. This quantity is useful when the internal volume of the pore is known. However, because it is difficult to a priori estimate this volume, in practice it is more convenient to use the average

Ni(t ) =

∫0

li

si(z ,t ) dz =

∫0

L

si(x ,t ) γi(x) dx

(6)

where li = zi(L) is the length of wall i. Summation of eq 6 over the number of walls yields the relation between the exact surface concentration si(z,t) and the average adsorbed concentration Cs, that is 2

w(x) Cs(x ,t ) =

∑ si(x ,t ) γi(x) i=1

(7)

In the particular cases of symmetric (w′1 = −w′2) and serpentine (w′1 = w′2) pores, one finds γ1 = γ2 = γ/2, and therefore, s1 = s2 = s. Dropping out the subscript i from the last equation one finds that

s(x ,t ) =

w(x) Cs(x ,t ) γ (x )

(8)

This equation establishes that the proportionality factor between the average adsorbed concentration Cs(x,t), and the surface C


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The Journal of Physical Chemistry C concentration s(x,t) of the adsorbed particles in an irregular symmetric pore is the term w(x)/γ(x). 3.2. The Mass Conservation Equation. Surface diffusion and adsorption can be described by using the mass conservation equation for surface concentration s(x,t). If Ds(z) (in cm2/s) is the surface diffusion coefficient and rs(z) is the surface reaction rate at each wall (in mol/cm s), then the mass balance equation for s(x,t) is ∂s ∂⎡ ∂s ⎤ = ⎢⎣Ds(z) ⎥⎦ + rs ∂z ∂z ∂t

(9)

∂Cs 1 ∂ ⎡ 4Ds ∂ ⎛ wCs ⎞⎤ ⎢ = ⎜ ⎟⎥ + R s w ∂x ⎣ γ ∂x ⎝ γ ⎠⎦ ∂t

(12b)

4. APPARENT DIFFUSION COEFFICIENT AND FRACTIONAL SURFACE COVERAGE In the Introduction we have discussed that the presence of surface diffusion may result in an enhancement of the total mass flux across a porous media.3,7−9 From the macro-transport point of view, this striking effect is frequently accounted for by introducing the notion of an apparent diffusion coefficient, arising from a combination of the volumetric and surface diffusion flows. In the present section, we will provide a derivation of this apparent diffusion coefficient by using eqs 12. Proceeding from this mesoscopic description, we will demonstrate that the apparent diffusion coefficient is, in general, a local quantity that depends on the geometrical features of the pores. Hence, this result conduces us to introduce a distinction between the local description and a macro-transport one, for which the introduction of an effective diffusion coefficient of a membrane or porous medium is more convenient. Two conditions associated with the conservation of mass should be imposed on the transport of the feeding material across the pore. First, we have that the total diffusive flux per unit of area Jt should be the sum of the corresponding bulk Jb and surface Js fluxes:12−14

(10)

where we have taken into account that the total wall length of symmetric pores is the sum of the two wall’s length. If the pore is not symmetric, then eq 10 should be stated for each wall. This equation represents the adsorbed concentration as a function of the position and time for a symmetrical pore, in which surface diffusion and adsorption−desorption processes are present. The functional form of this equation is similar to that of eq 1 for Cb, which corresponds to the bulk phase. Nonetheless, a significant difference is that the diffusion term contains a correction factor inversely proportional to the density length γ. Compared with previous descriptions,6 eq 10 has the advantage of describing the spatial and temporal evolution of the average adsorbed concentration Cs, in terms of the surface tracer diffusion (measured with Ds) and the local reaction at the wall of the pore, rs. The surface reaction rate rs can be measured as the net flux of adsorbed particles through the wall, i.e., as rs = J·n̂, where n̂ is the normal vector to the wall pointing outside.17 The mathematical form of eqs 1 and 10 allows us to conclude that the relation between the volumetric reaction rates Rβ (in mol cm2/s) and the reaction rates just at the surface rβ (in mol/cm s) is17 w(x) Rβ(x ,t ) rβ(x ,t ) = γ (x )

(12a)

The convenience of using eqs 12a and 12b instead of eqs 1 and 10 depends upon the quantities one may be monitoring in the system.17 If the rate of adsorption/desorption of particles just at the surface (rs and rb) can be measured, then it is more convenient to use eqs 1 and 10. Nonetheless, this rate is usually unknown, and therefore, a functional form or a kinetic model for the volumetric reaction rates Rb and Rs is needed. In this case, the value of the parameters of the model are fitted by looking at the transient profiles of concentration that can be obtained by solving eqs 12. In any case, the reduced eqs 12 incorporate the irregularity of the pore through the geometric factors related with the width of the pore w(x) and the density of length wall γ(x). The surface diffusion coefficient may depend on the position in the general case. This is due to the presence of kinks, defects, or strains on the surface29 or, indirectly, through the dependence of the diffusion coefficient on the loading.18,30 Furthermore, it is important to notice that eqs 12 are valid, at least in principle, for pore spaces where the particles are much smaller than the width of the pore. However, we believe that our formalism can be extended to describe much narrower systems where the effects of finite size of the particles are included. Nonetheless, this is something that must be confirmed in future works comparing modifications of the FJ equations that consider the finite size of the particles31,32 and simulations or experiments that appear to suggest that, in the limit of narrower pores, the distinction between bulk and surface particles can be neglected.33,34

This equation is valid for both walls as long as the pore is symmetric. If the pore is not symmetric, eq 9 should be written for the surface concentration at each wall. In this equation, rs measures all the possible chemical interchanges of the surface particles, and therefore, depending on the complexity of the adsorption/desorption process, the rate of change of the surface concentration can be modeled in the simplest case by a linear process of adsorption or, more generally, as dependent on another chemical species. It is important to emphasize that in this model, the diffusion coefficient Ds is measured as the proportionality factor between the flux and the difference of concentration along the surface. Therefore, it can be estimated by direct measurement of the msd of the particles along the solid surface in the limit of very low loading.28 In the next lines, we will explain why the surface diffusion coefficient Ds used in eq 9 has a more fundamental physical meaning than those used in traditional mass transfer models.6 Equations 5, 7, and 9 can be combined to deduce a mass conservation equation for the average adsorbed concentration Cs. The result is ∂Cs rs 1 ∂ ⎡ 4Ds ∂ ⎛ wCs ⎞⎤ ⎢ = ⎜ ⎟⎥ + γ ∂t w ∂x ⎣ γ ∂x ⎝ γ ⎠⎦ w

∂C b ∂C b ⎤ 1 ∂ ⎡ = ⎥ + Rb ⎢⎣D bw w ∂x ∂t ∂x ⎦

Jt = Jb + Js

(13)

In a similar way, because the volumetric Cb and the adsorbed Cs concentrations are measured along the same volume (the void space of the pore), at every point in the space, mass conservation imposes the condition:

(11)

for β = b, s. Therefore, in terms of volumetric rates, eqs 1 and 10 take the final form D


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The Journal of Physical Chemistry C C t = C b + Cs

(14)

Dt ≡ D b

where Ct is the total concentration of the fluid inside the pore. Now, according to eq 14, the temporal change of Ct obeys the ∂C ∂C ∂C relation ∂t t = ∂tb + ∂t s , which, after using eqs 12, can be written in the form 4Ds ∂ ⎛ wCs ⎞⎤ ∂C t ∂C 1 ∂ ⎡ ⎢D b w b + = ⎜ ⎟⎥ w ∂x ⎣ ∂t ∂x γ ∂x ⎝ γ ⎠⎦

(15)

λ≡

D b0 ≡ D b

(18)

which is a sum of the fluxes (19a)

and 4Ds ∂ ⎛ wCs ⎞ ⎜ ⎟ γw ∂x ⎝ γ ⎠

(19b)

Equation 18 can be written in the form of eq 17 by using the mathematical relation

∂Cβ ∂x

=

∂Cβ ∂C t ∂C t ∂x

(23a)

4Ds

λ γ2 1 + λ

(23b)

From its definition, it follows that the conversion rate λ could be positive or negative. According to the IUPAC classification, most important adsorption isotherms are monotonic increasing functions of the concentration implying that λ ≥ 0. In view of this fact and from eqs 23, it follows that the apparent surface diffusion coefficient contribution D0s is positive, making the total apparent diffusion coefficient, Dt, larger than the mere volumetric diffusion coefficient D0b. However, some evidence for supercritical fluids indicates that λ may be negative37,38 implying that the desorption dominates for high concentration regimes. In this case the apparent surface diffusion coefficient contribution D0s is negative and, therefore, the magnitude of the total apparent diffusion coefficient decreases. In the next, we will assume λ ≥ 0, for which the apparent bulk diffusion coefficient D0b decreases with λ, whereas the apparent surface diffusion coefficient D0s increases. Equation 22 is a very important result because it relates the diffusion coefficients measured from the experiments with the (mesoscopic) bulk and surface tracer diffusivities, Db and Ds, as long as the isotherm of the process and the internal geometry of the pore are known. In addition, eq 22 demonstrates that the local character of the diffusion coefficients D0b and D0s in eqs 23, as they depend on the concentration profiles and the geometry of the pore, is conferred to the total apparent diffusion coefficient Dt(x,t), defined through eq 20. This dependence is given through the FJ bulk diffusion coefficient Db, which depends in turn on the porosity, tortuosity, and constriction of the pore,36 and through Ds, which is assumed constant at zero loading but which can depend locally on the specific pore architecture (curvature, stress, defects, etc.) In the case when both diffusion processes are slower than the adsorption−desorption kinetics, it is possible to use the isotherm to find λ and consequently estimate Dt from eq 22. Besides, if both diffusion processes are the slower steps when compared with the kinetics of the adsorption process, then the isotherm allows us to find an approximate solution for the distribution of volumetric and surface concentrations by making the right side of eq 15 equal to zero. In this case we have to solve

where we have introduced the apparent diffusion coefficient Dt. The last two expressions consider exclusively the amount of fluid at both sides of a membrane but do not consider the internal distribution in the volume and surface of the pore; i.e., it does not distinguish between absorbed and bulk particles. This implies that Ct and Jt are quantities that can be more easily measured in macro-transport experiments.12−14 As we will show in the second part of this work, the spatio-temporal profiles of the total concentration Ct are the key quantities that will allow us to compare the results of our model with microimaging experiments19,35 through an effective diffusion coefficient. This coefficient defined in the second part of this work is deduced in the same way as in ref 36 with the difference that, in the present work, we include also adsorption and desorption processes. To do so, we first establish a way of measuring the net motility of the particles Dt(x) inside the pore, a motility that evidently depends on the zone of the pore where it is measured and that it does not distinguish between bulk or surface particles as we describe in the following lines. Comparison between eqs 15 and 16 and the use of eq 13 allow us to identify the total diffusive flux per unit of area Jt:

Js = −

1 1+λ

and the local surface diffusion coefficient

(17)

∂C b ∂x

(22)

where, for convenience in the analysis of the results, we have introduced the local bulk diffusion coefficient

Ds0 ≡

⎡ ∂C 4Ds ∂ ⎛ wCs ⎞⎤ Jt = −⎢D b b + ⎜ ⎟⎥ γw ∂x ⎝ γ ⎠⎦ ⎣ ∂x

(21)

Dt (λ ,x) = D b0 + Ds0

and, therefore, that the total diffusive flux per unit of area Jt satisfies Fick’s law

Jb = −D b

∂Cs ∂C b

After using eq 14, we have that

(16)

∂C Jt = −Dt t ∂x

(20)

We emphasize here the fact that, in general, Dt is a function of position and time because it depends on the Fick−Jacobs diffusion coefficient Db given by eq 2, and on the length density of the pore γ defined after eq 5. If the adsorption−desorption isotherm is known, Cs = Cs(Cb), then the partial derivatives in the previous equation can be estimated from the conversion rate λ:

where we have used Rb = −Rs in an equimolar adsorption− desorption reaction. Physically, this means that for each adsorption event, a particle leaves the bulk and enters the surface adsorption field, and vice versa. Following the arguments of section 2, it is natural to assume that, in the projected Fick− Jacobs scheme, the change in time of the total concentration satisfies the diffusion equation ∂C t ∂C t ⎤ 1 ∂ ⎡ = ⎢⎣wDt ⎥ ∂t w ∂x ∂x ⎦

4D ∂C ∂C b + 2s s ∂C t γ ∂C t

for β = b, s. Substitution into

eq 18 allows us to identify the apparent diffusion coefficient E


Article


Figure 2. Local diffusion coefficients given by eqs 22 and 23 as a function of the loading θ for (a) the Langmuir isotherm, eq 29, and for the (b) sublinear and (c) superlinear Freundlich adsorption isotherms, eq 31. These local diffusion coefficients were calculated for a rectangular pore. For the Langmuir processes we used α = 50/3, k+ = 1, and KL = 0.3 in eq 27.

∂C 4Ds ∂ ⎛ wCs ⎞⎤ ∂ ⎡ ⎢D b w b + ⎜ ⎟⎥ = 0 ∂x ⎣ ∂x γ ∂x ⎝ γ ⎠⎦

adsorbed phase saturates the wall of the pore, by following two possible adsorption−desorption isotherms. Assuming for simplicity that the surface diffusion coefficient Ds is also constant and that the reaction steps are faster than the diffusion ones, one can use directly the isotherm of the process, Cs(Cb), to estimate λ by means of eq 21. Langmuir Isotherm. In this scheme the adsorption− desorption reaction depends on the nearness of bulk particles to the surface of the pore, and hence it is proportional to the volumetric concentration Cb and to the density of vacant sites sv in the surface, which can be expressed in terms of the total density of vacant sites s0v and the surface concentration of adsorbed particles s in the usual form sv = s0v − s. In the Langmuir approximation the reaction rate rL is thus determined by the formula

(24)

for Cb or Cs knowing the isotherm Cs(Cb). In this limit, the estimation of the apparent diffusion coefficient Dt can be determined by obtaining the stationary solutions for Cb, Cs (or Ct using eq 14), given the boundary conditions. Then the apparent diffusion coefficient can be calculated by using eq 20. In both limits, the expressions we provide allow us to determine the value of the effective diffusion coefficient Dm of a membrane composed by many pores. This determination requires the averaging of the local description along the x coordinate within the width of the membrane. Taking as a basis the mass conservation law at global level, that is, by considering the mass flux at the ends of the porous material, like in macrotransport theories, one can show that the expression for Dm is given by τ 1 = ϕ Dm

1 ⟨w(x)⟩ Dt (x) w(x)

rL = k+(s v C b − KL s)

(26)

where KL = k−/k+ is the equilibrium constant. Using eq 11, we have that the volumetric rate of adsorption−desorption is given by the well-known formula

(25)

where τ and ϕ are the tortuosity and porosity coefficients, respectively. The averages indicated by the symbols ⟨...⟩ are normalized integrations taken along the width of the membrane. The product of the averaged terms constitute, in turn, the definition of the generalized constriction factor δt, which plays a key role in diffusion problems across porous materials. A more detailed derivation and analysis of the implications of this important formula are given in part II of this work.

RL = k+(CvC b − KLCs)

(27)

where Cv = C0s − Cs is the volumetric concentration of vacant sites and, hence, C0s is the saturation concentration. The adsorption isotherm is determined by setting RL = 0. The result is Cs(C b) =

Cs0C b KL + C b

(28)

Now, the conversion rate λ defined in eq 21 can be written in terms of the fractional surface occupancy θ = Cs/C0s and the

5. PREDICTIONS ON DIFFUSION AND ADSORPTION: INTERPLAY AT LOCAL LEVEL In this section, we will illustrate the main effects predicted by our formalism on the apparent diffusion coefficient Dt at the local level. To do this, we consider several representative cases of the interplay among an irregular geometry, surface diffusion, and adsorption−desorption kinetics. The corresponding study at the global level is presented in part II of this work. Our description predicts very interesting couplings between bulk and surface diffusion that occur locally and are regulated by the adsorption− desorption kinetics. 5.1. Rectangular Pore of Periodic Boundaries: The Effect of Pore Loading. To exemplify the effects that loading has on the local diffusion coefficients D0b and D0s defined in eqs 23, it is convenient to leave the role of the pore geometry for subsequent subsections. To achieve this objective, we have to analyze the case of a rectangular pore, for which w(x) is constant and therefore Db = D0 and γ = 2 are also constants. Thus, the relative importance of the surface and bulk diffusion processes can be evaluated as the

C0

parameter α ≡ Ks , by the formula λ = α(1 − θ)2. For the Langmuir adsorption isotherm, the apparent diffusion coefficient defined in eq 22 becomes Dt = D0

α(1 − θ )2 1 + D s 1 + α(1 − θ )2 1 + α(1 − θ )2

(29)

Figure 2a shows the diffusion coefficients Dt, D0b, and D0s , defined in eqs 22 and 23, in terms of the loading θ. The result predicts that the bulk diffusion coefficient D0b will increase with the loading θ whereas the surface diffusion coefficient D0s will decrease. This means that the local surface diffusion inside a saturated region of adsorbed particles does not contributes appreciably to the total diffusive flux Jt defined in eq 17. This result is consistent with the fact that in the saturated state, a dynamical arrest occurs. Freundlich Adsorption Isotherm. In this case, the volumetric rate of adsorption−desorption satisfies the relation F


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Figure 3. Adsorption−desorption process in three trapezoidal pores when there is surface diffusion (bottom) and when the adsorbed phase is fixed (top). The different colors correspond to different pores. We consider the Langmuir process of eq 27 with the same parameters of Figure 2. Left panel: concentration profiles of the total, bulk, and surface volumetric concentrations obtained by solving eq 12 with net flux boundary conditions. Middle panel: surface concentration obtained from Cs using eq 8. Right panel: fluxes at the entry of the pore using eqs 13 and 19. The sketched pores in the legend have width w(x) = 1 + 2ax where a = −0.05, 0.0, and 0.1 for the blue, orange, and red pores, respectively. The length of the pores is L = 5. The value of D0 is 1 in both cases and Ds = 1 in the second case.

RF =

1 (KFC bn − Cs) k−

always space for the migration from one region to other. These behaviors for the local surface diffusion coefficient will be similar for isotherms of types II and III, whereas behavior similar to the Langmuir process should occur for isotherms of types I, IV, and V. 5.2. Trapezoidal Pores under Nonequilibrium External Conditions. Langmuir Isotherm. Now we will study the effect that a bottleneck or an opening have in the Langmuir adsorption−desorption dynamics when surface diffusion is present. This case generalizes recent results on adsorption− desorption dynamics for systems in which surface diffusion is absent.23 The first assessment of the influence of the geometry on the diffusion process can be performed through the trapezoidal geometries sketched in the inset of Figure 3. The assumed boundary conditions correspond to a net flux of mass across the pore. The material is adsorbed at the two walls, and the particles are allowed to diffuse also in the surface with a constant molecular diffusivity Ds. Volumetric Cb and adsorbed Cs concentrations are plotted in Figure 3. Similar to the case studied in ref 30, the flux entering into the pore is larger for the open pore (red) than for the closed one (blue), even when in the first pore some material returns to the right side due to the surface flux. Therefore, the mass transfer coefficient is larger in the open pores as it was in the case of immobile adsorbed phase.23 Besides, the total flux is bigger when there is surface diffusion than when the adsorbed phase is fixed. This example is also useful to illustrate the difference between adsorbed concentration, Cs(x,t), plotted in the left panel of Figure 3, and the surface concentration at the middle panel, s(x,t). In these pores, Cs is greater for the closed pore but lower for the second because, according to eq 8, the first one depends inversely upon the width of the pore (which is lower in the blue pore) whereas s only depends of the length of the wall. Finally, by seeing the fluxes at the right panel on Figure 3, it can be seen that the surface diffusion allows the adsorbed phase to migrate even to the outside. Therefore, the profile of the total concentration is less bulky than when there is no surface diffusion. 5.3. Sinusoidal Closed Pore with Periodic Boundaries. Langmuir Isotherm. Here we will analyze an example that helps one to understand how the distribution of particles in the bulk and the surface of a pore is affected by the surface diffusion. In particular, we will show the emergence of an interesting local hydrodynamic evaporation−condensation-like dynamics.

(30)

where KF = k+/k− is the inverse of the equilibrium constant and n is the order of the process for which the adsorption isotherm is given by Cs = KFC bn

(31)

Recalling that θ = Cs/C0s and using eq 31, we may deduce the expression λ = nαFθ1 − 1/ n, with αF = KF1/ n(Cs0)1 − 1/ n. Substituting these results into eq 22 we have Dt = D0

nαFθ1 − 1/ n 1 + D s 1 + nαFθ1 − 1/ n 1 + nαFθ1 − 1/ n

(32)

In the linear case (n = 1), eq 32 reduces to the constant D +K D expression Dt = 10 + KF s , which is the limiting form of the F

effective diffusion coefficient previously derived by using macrotransport theory.39 The tendencies of the results predicted for the Freundlich adsorption isotherm show a strong dependence on the order of the adsorption−desorption process n. In Figure 2b we have assumed n < 1. The results show that the local apparent diffusion coefficient Dt is a decreasing function of the fractional surface occupancy for Ds > D0, whereas it increases with θ for Ds < D0. For Ds = D0 it remains constant. Also, the local adsorbed surface diffusion D0s is a decreasing function of θ independently of the value of Ds, whereas the local volumetric diffusion D0b increases in all cases. In Figure 2c we have assumed n > 1. In this case the local apparent diffusion coefficient Dt is an increasing function of the fractional surface occupancy for Ds > D0, whereas it decreases with θ for Ds < D0. It should be emphasized that in this case the local adsorbed surface diffusion D0s is an increasing function of θ, independent of the value of Ds, whereas the local volumetric diffusion D0b always decreases. These results indicate that the surface diffusion will be a decreasing function of the fractional surface occupancy for slow and saturating adsorption processes. On the contrary, surface diffusion will be enhanced for strong nonsaturating adsorption processes as those described by Freundlich adsorption, with orders n ≥ 1. Only in this case is the behavior of the total apparent diffusion coefficient dictated by the adsorbed surface diffusion. Physically, these effects may be understood by considering that for n ≥ 1 there is always available surface for the particles to be adsorbed on, and therefore, there is G


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Figure 4. Concentration inside a sinusoidal closed pore with periodic BC. The process is of the Langmuir type in eq 27 with the same parameters as in Figure 2. The pore has length L = 5 and width w(x) = 1 + 0.4 sin(6πx/L). The different colors correspond to different values of Ds/D0. Left panel: concentrations obtained by solving eq 12, and the diffusion coefficients obtained by using eqs 22 and 23. Middle panel: surface concentration obtained by using eq 8, and the stationary local fluxes as a function of the position by using eqs 13 and 19. Right panel: disposition of bulk and adsorbed particles inside this irregular pore. See text for details.

Figure 5. Two types of linear adsorption processes involving the volumetric and surface concentrations of adsorbed particles, obtained by numerically solving eqs 33 and 34. We assumed Klv = 0.3, Kls = 0.3⟨w(x)/γ(x)⟩ = 0.136, D0 = 1, Ds = 1, and k− = 1. The pore used has length L = 5 and width w(x) = 1 + 0.5 sin(4πx/L) + 0.12 sin(8πx/L). Red lines correspond to the solutions of eq 33 whereas the blue lines to the solutions of eq 34. The dashed and dotted lines correspond to increasing transient times until the stationary state, indicated by the solid lines. The top row shows, from left to right, the ratchet pore, the bulk concentration Cb, and the fluxes Jt, Jb, and Js as a function of the position. The bottom row shows the apparent diffusion coefficient Dt and the local coefficients D0b and D0s for both adsorption processes. It also shows the surface concentration and reaction rate obtained from using eqs 8 and 11, respectively.

Langmuir process, the trend shown by the effective diffusion coefficients indicates that local diffusivity for the bulk particles is larger in the bottlenecks, whereas the mobility of surface particles is larger in the wider region of the pore, as is illustrated in the local diffusion coefficients plotted in Figure 4. Notice also that increasing the molecular surface diffusivity increases the value of the local surface diffusion coefficient D0s (dotted lines) and deviates the tendency of the apparent diffusion coefficient Dt (solid lines) from the bulk one D0b (dashed lines). In Figure 4, the bulk and surface fluxes by unit of area defined in eqs 19 were represented. The effective flux of bulk particles Jb points toward the wider region of the pore, as indicated by the dashed lines, whereas the opposite occurs for the surface flux Js, which points toward the bottleneck, as indicated by the dotted lines. The sum of these two contributions yields a vanishing total mass flux Jt given a null net flux in the stationary state, indicated by the solid lines. This interesting effect depends upon the form of the isotherm, as we will illustrate in section 5.4. 5.4. Linear Surface and Volumetric Adsorption Processes in a Ratchet Pore. In the previous section, we proved that the Langmuir process in the volumetric concen-

Consider a closed sinusoidal pore having a large sequence of cages and bottlenecks. In this example the pore is filled with a fixed amount of fluid and, then, the process of adsorption− desorption starts. The numerical solutions of eqs 12 and 15 are calculated by assuming again the Langmuir adsorption kinetics (eq 27), and by imposing periodic boundary conditions, because we want to search for the peculiarities of the coupled diffusion and chemical dynamics inside the pore. In the upper row of plots in Figure 4, we present the volumetric and the surface concentrations of the fluid showing that the bulk particles tend to be distributed homogeneously along the volume, whereas the adsorbed phases tend to be aggregated in the wider regions of the pore. This result suggests the emergence of an internal process of counterdiffusion between bulk and adsorbed particles that occurs inside the pore, which is corroborated in the lower row of Figure 4. Because the surface diffusion has no effect on the bulk distribution of particle for the Langmuir adsorption process, the variations in the spatial distribution of the total concentration, and therefore, on the effective apparent diffusion coefficient, are only due to the different arrangement of particles along the surface. For the H


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The Journal of Physical Chemistry C trations, eq 27, leads to an almost constant volumetric concentration, and to a nonhomogeneous distribution of the adsorbed particles. In view of this, in the present section we study the conditions that are necessary to form a uniform monolayer of adsorbed particles. To complete this program, we will consider two different linear adsorption processes inside the ratchet pore. The first process is a linear volumetric isotherm dictated by the relation (eq 30): R lv =

1 (KlvC b − Cs) k−

∂s

where we have used the surface flux Jz = −Ds ∂z (eq 9). Therefore, because γ ≠ 0, the homogeneity of s(x,t) for Rsl requires that Jz = 0 and, therefore, that s(z) is constant. As can be seen in Figure 5, this homogeneity is linked to the homogeneity at which the surface reaction at the surface r occurs.

6. PREDICTIONS ON LOCAL HETEROGENEOUS CATALYSIS: ISOMERIZATION REACTION The theoretical framework based on eqs 12 allows for an accurate modeling of heterogeneous catalysis due to the flexibility in choosing boundary and initial conditions that include nonequilibrium conditions and the election of specific positions for active sites inside the pores. This makes our framework a useful tool in modeling heterogeneous catalysis in mesoporous and macroporous systems. Let us illustrate this capability by considering a simple isomerization reaction at the pore walls.27 In this example, the particles of A enter the pore with concentration CAb , and some are adsorbed at the wall, forming the reactive AS with concentration

(33)

The second case is also a linear adsorption process in which the isotherm depends on the surface particle concentration of adsorbed particle s, instead of the average concentration Cs. It is opportune to recall that these quantities are related by eq 8. Thus, the second process is modeled according to R ls =

1 (KlsC b − s) k−

(34)

In a form similar to that in the previous subsection, the numerical solution of eqs 12 and 15 is calculated using periodic boundary conditions at the ends of the pore, which is initially filled with a fixed bulk concentration. We will assume that Ds is constant. The numerical solution of the equations are shown in Figure 5 were details are provided. The upper row shows a scheme of the pore indicating the preferential migration direction of the particles, in the bulk and in the surface. Also shown is the average bulk concentration obtained for both adsorption processes, as well as the corresponding fluxes. The bottom row shows in turn the three diffusion coefficients, Dt, Db0, and D0s , for both adsorption isotherms. Also shown is the surface particle concentration s and the reaction rate r as a function of the position along the pore. Our result shows that the volumetric process Rvl tends to make the bulk concentration Cb homogeneous, whereas for the surface isotherm the surface concentration s is fixed. It should be emphasized that from global and effective terms, both adsorption processes are not distinguishable because the parameters used in these two reactions make the effective diffusion coefficient Dt the same (Figure 5). Therefore, both schemes of reaction lead to the same global or average results. Nonetheless, the local behavior of the particles shows huge differences between Rlv and Rls adsorption models. This is clearly illustrated in the right panel of Figure 5. The volumetric isotherm Rvl yields nonvanishing stationary bulk Jb and Js surface mass fluxes (red lines), whereas the surface isotherm Rsl predicts a zero value for all fluxes (blue lines). In a similar way, the reaction rate r becomes strictly zero for the surface isotherm (blue solid line) at long times, whereas the volumetric isotherm predicts a nonhomogeneous reaction rate (red solid line). In an entirely similar form as given in subsection 5.3, the longtime stationary state of the volumetric isotherm leads to internal counterfluxes between bulk and adsorbed particles, thus motivating a local hydrodynamic loop process of adsorption− desorption that depends on the form of the pore. The homogeneity of layer of adsorption corresponding to the surface process, Rsl , can be proved analytically by considering that ∂s ∂s ∂z = ∂z ∂x . Because this derivative is zero for Rsl , this means that ∂x J γ − z =0 Ds 2

kA

CAs : A + S XoooY A·S. Afterward, the reactive AS reacts at the pore k −A

walls forming the product BS, with concentration CBs , that is, the kS

isomer B but attached to the wall: A·S XooY B·S. Finally, product B k −S

is desorbed and transported out of the pore with concentration kD

CBb : B·S XoooY B + S. Thus, the set of reaction velocities are k −D

Rads for the adsorption R ads = kads(C bACS − K adsCsA )

(36a)

Rcat for the catalysis R cat = kcat(CsA − KcatCsB)

(36b)

Rdes for the desorption R des = kdes(CsB − KdesC bBCS)

(36c)

Here, CS = − − denotes the concentration of vacant sites in the Langmuir−Hinshelwood approximation. In this simplified example we can consider that the molecular diffusion coefficients D0 and Ds are nearly the same for both species, because there is no change of the molecular mass. The dynamics of the catalytic process can therefore be described by the following two sets of evolution eqs 12, one for the concentrations CAb and CAs and other for CBb and CBs : C0s

CAs

CBs

∂C bA ∂C A ⎤ 1 ∂ ⎡ ⎢D bw b ⎥ − R ads = ∂t w ∂x ⎣ ∂x ⎦

(37a)

A ∂CsA 1 ∂ ⎡⎢ 4Ds ∂ ⎛ wCs ⎞⎤⎥ ⎜ ⎟ + R ads − R cat = ∂t w ∂x ⎢⎣ γ ∂x ⎝ γ ⎠⎥⎦

(37b)

∂C bB ∂C B ⎤ 1 ∂ ⎡ ⎢D bw b ⎥ + R des = ∂t w ∂x ⎣ ∂x ⎦

(37c)

B ∂CsB 1 ∂ ⎡⎢ 4Ds ∂ ⎛ wCs ⎞⎤⎥ ⎜ ⎟ − R des + R cat = ∂t w ∂x ⎢⎣ γ ∂x ⎝ γ ⎠⎥⎦

(37d)

The numerical study was performed by considering three pores having identical ratchet forms (see the top panel at the left in

(35) I


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Figure 6. Effect that the position of active sites has on the production of a product B, when a catalytic reaction takes place at the surface of an irregular pore. Different colors correspond to the different pores shown in the schemes. The three pores have the same geometry as in Figure 5. The reactions in the heterogeneous catalytic process are given by eqs 36, with parameters Kads = Kcat = Kdes = 0.3, kads = kcat = kdes = 1. We plot the results when Ds/D0 are 0 and 0.1 (see the insets). Left: pores and total fluxes of species A and B at x = 0, were calculated using eqs 13 and 19 for each chemical species. Right: surface concentration profiles of the three pores for both surface diffusion coefficients considered. See details in the text.

entrance of the pore and the continuous production of B, species A is constantly entering into the pore, producing B inside the pore, and because the concentration of B at x = 0 is negligible, they exit the pore at a rate determined by the internal effectiveness of the process. More noticeably, in all the cases the surface diffusion increases the amount of B produced. This is due to the fact that the surface diffusion allows the particles to migrate along the adsorption field to regions near active sites and to the exit, increasing in turn the number of particles that are necessary to reach saturation inside the pore. The plots in the right column of Figure 6 are the predictions of mass distribution at the surface, once the stationary state is established. This mass distribution is shown in terms of the total stationary surface concentration profiles, sAS+BS(x), for the three pores considered and for the two values of the molecular surface diffusion coefficient considered, Ds = 0 (top plot) and Ds = 0.1D0 (bottom plot). For the three pores considered, the surface concentration profiles sAS+BS(x) are very similar along the pore length (see dotdashed curves), and for both cases of surface diffusion Ds = 0 (top plot) and Ds = 0.1D0 (bottom plot). Notice, however, that mass distribution is smother when surface diffusion is present. Nonetheless, the relevant differences emerge when looking at the total amount of B produced by the three pores. This production is quantified by sBS, which is shown by the dashed lines. For the two values of the molecular surface diffusion coefficient considered, eqs 36 and 37 predict an enhanced production of B when the adsorption sites are located at the cages (red line), and less production when they are located at the bottlenecks (orange line).

Figure 6), but differentiated from the other two by the position of the active catalytic sites: one located at the cage (red), the throat of the pore (blue), and the bottleneck (orange). For each pore, two cases were considered. In the first case we have assumed only bulk diffusion of particles, and in the second case we have also considered surface diffusion of the adsorbed species. Additionally, we have considered the pores open at x = 0 and closed at x = L, in the presence of saturation conditions for A, that is, CA(0,t) = 1. For the B species, we have assumed that it is free to exit at x = 0 and is rapidly removed, in such a way that Cb(0,t) = 0 is obeyed. Finally, we have consistently assumed that all fluxes, JAb , BS JBb , JAS s and Js , are zero at x = L because the pores are closed. As before, the boundary conditions at the pores walls are given by the isotherm associated with surface concentration, in this case, the Langmuir one given in eq 28. The first interesting results are shown by the mass fluxes of A, B BS JA+AS = JAb + JAS s , and B, JB+BS = Jb + Js evaluated at the entrance of the pore: x = 0. The mass fluxes are plotted as a function of time in the bottom of the first panel of Figure 6. The total flux of A, JA+AS(0,t), is positive in all three cases of the catalytic sites, meaning a net entrance of A to the pore. For t ≃ 1, JA+AS(0,t) reaches a similar maximum, but for times t > 1 the total flux JA+AS(0,t) splits into three curves depending on the location of the catalytic sites. It is larger when the catalytic sites are located at the cage of the pore (red line), and lower when the sites are located at the bottleneck (orange line). The time trends are qualitatively similar for both cases of surface diffusion Ds = 0 and Ds = 0.1D0 (see the insets), although the stationary long-time behavior reached shows that surface diffusion increases the net flow of A. Notice that a larger presence of A in the pore corresponds to a larger production of B in the long-time stationary state. The same plots shows that the net flow of species B, JB+BS(0,t), is negative, meaning that the species B exits the pore at x = 0. The time trends of JB+BS(0,t) for the three pores considered are similar, until t = 1 is reached. Afterward, the fluxes split until reaching a constant (negative) stationary state for long times. The pore with active sites at the cages (red line) shows the larger flux that exits the pore. The pore with active sites at the bottlenecks (orange line) shows the lower flux that exits the pore. Physically, the different signs for the fluxes at x = 0 can be understood as follows: due to the saturated atmosphere at the

7. DISCUSSION AND CONCLUSIONS In this work we have presented a theoretical formalism able to evaluate quantitatively the important effects that surface diffusion has on the net transport of matter throughout a porous material, even in conditions when adsorption−desorption and heterogeneous catalysis are present, like some zeolites or nanostructured materials. The formalism is based on a consistent projection scheme of the mass balance equations inside irregular catalytic pores with nonvanishing surface diffusion and reduces considerably the difficulty of quantitatively assessing the J


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the geometry of the pore. This interrelation is shown in Figure 5, where we have studied the condition under which a monolayer of adsorbed particles occurs. In Figure 6, we illustrate how our formalism quantifies the local catalytic activity of an irregular pore depending on the positions of the active sites along its surface. Consistently with the previous results addressed, the larger internal effectiveness occurs when the active sites are located at the cages of the pore. We have to remark that in this model we have focused in the theoretical formalism that allow us to link all the processes occurring in the heterogeneous catalysis with the geometrical irregularity of the pore. Therefore, we have used two-dimensional pores to emphasize the physical aspects of the problem and simplify the mathematical description. In this way, our simplified 2D model emphasizes, in the local scale, the relation of the processes with the width of the pore, its real length, and the surface area where the adsorption/desorption processes take place and, in the macroscopic scale, as we will see in part II, with macroscopic properties such as the porosity and the sinuosity of the porous material. Therefore, although the strict application of our model for real three-dimensional pores in this stage is restricted to wedge pores or pores with cylindrical symmetry,40 it is clear that both methodology and qualitative predictions of our model are also valid in more general 3D systems. To prove this, in future works, the mathematical form of the FJ equations has to be extended to three-dimensional spaces complicating the mathematics, as can be seen in the works using Fick−Jacobs in general 3D pores,41,42 but not the formalism we have proposed in this work. The complementary analysis of the predictions arising from considering a global description outlined in eq 25, as well as the detailed deduction of the corresponding global quantities playing important technological role, like the effective diffusion coefficient of a membrane, is presented in a companion article, part II of the present work. Our concluding remarks are that the theoretical formalism presented in this article goes beyond previous theoretical approaches on the subject, because it offers qualitative and quantitative explanations of the mechanisms underlying the interplay among surface and bulk diffusion and adsorption and desorption kinetics. In view of this, we believe that our model approach can be an important tool in describing the diffusivity and surface permeability in nanoporous materials measured in imagening experiments. The most important achievement of this work is that it allows one to quantify all the relevant subprocesses taking place in heterogeneous catalysis in porous media, where the shape of the pores is determinant in the amount of the product wanted. As a consequence of this, the present formalism allows us to model or design porous materials with optimized effectiveness of mass transport and production in the same direction as the works in refs 43−46.

performance of this type of systems under several equilibrium and nonequilibrium conditions consistent with their practical use. Additionally, one of the virtues of our formalism is that it establishes transparent connections among volumetric quantities that can be more easily measured in experiments, like the effective diffusion coefficient of a membrane and the average loading with their local counterparts such as local diffusion coefficients, mass fluxes, and reaction rates at the surface. Technically, in the simplest case the model consists of two coupled reaction−diffusion equations for the surface and volumetric concentrations, eqs 12, that exclusively depend on the longitudinal coordinate of the pore. The coupling emerges from the reaction terms that account for adsorption and desorption processes. In the case of bulk and/or catalytic reactions, the basic description is easily extended by the corresponding chemical kinetics, like in eqs 36. The entire description uses the consistent definition and distinction between the local and global apparent diffusion coefficients that incorporate the effects of surface diffusion. This aspect of our analysis is important because it allows evaluating the spatial and temporal dependence of the total mass concentration in the pore and the corresponding mass flux, and more importantly, it allows proposing consistent models for the apparent diffusion coefficients (local and global), from the knowledge of the adsorption−desorption isotherms and of the bulk and surface molecular diffusivities of the material of the pore (eqs 18−23). The predictions and implications of our formalism were studied, analyzed, and illustrated in a series of examples with increasing complexity, in which we have studied the influence of the local surface diffusion on the total diffusion coefficient Dt and other experimentally relevant quantities. In the first case of a rectangular pore, we found that the magnitude and trend of Dt as a function of the pore loading depends upon the relative importance of the molecular diffusivities Ds/D0. The results are summarized in Figure 2. We also proved that the loading inside any pore depends upon the particular external conditions under which the porous material is submerged, that is, the boundary conditions. For net flow consistent boundary conditions, mass distributions at the bulk and the surface were evaluated for conical pores in the presence and the absence of surface diffusion. Figure 3 shows that the presence of surface diffusion makes the surface concentration, s(x,t), of adsorbed particles decrease along the pore axis, in such a way that the total mass flux Jt across the pore is enhanced. The influence of the adsorption isotherm on the surface and volumetric concentrations, fluxes, and diffusion coefficients was also evaluated for the particular case of a Langmuir process in a vanishing mass flux experiment, Jt = 0 for a sinusoidal pore. This example is useful to understand that the adsorbed concentration by unit volume, Cs, is a very different quantity from the surface concentration of adsorbed particles, s(x,t). In this case, for a constant molecular surface diffusivity, Ds, Figure 4 shows an interesting prediction for which the bulk particles tend to be distributed almost homogeneously. In contrast, the mass distribution of adsorbed particles, characterized by s(x,t), is larger in the cages than in the bottlenecks. The interesting implication of this is that a recirculation of particles between the surface and the bulk appears. In fact, the presence of surface diffusion facilitates the migration of particles along the surface from cages to bottlenecks, whereas the particles at the bulk migrate in turn from the bottlenecks to the cages. These effects depend on the peculiar adsorption kinetics and its relation with

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AUTHOR INFORMATION

Corresponding Author

́ *I. Santamaria-Holek. E-mail: [email protected]. ORCID

Aldo Ledesma-Durán: 0000-0003-3258-5616 Notes

The authors declare no competing financial interest. K


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ACKNOWLEDGMENTS We are grateful to UNAM-DGAPA-PAPIIT project IN116617 for financial support. A.L.D. acknowledges CONACyT for financial support under fellowship 221505. S.I.H. is grateful to project LANCAD-UNAM-DGTIC-276.

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