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The Effect of Surface Diffusion on Adsorption-Desorption and Catalytic Kinetics in Irregular Pores. Part II: Macro-Kinetics Aldo Ledesma-Durán, Saúl Iván Hernández-Hernández, and Iván Santamaría–Holek J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b03653 • Publication Date (Web): 12 Jun 2017 Downloaded from http://pubs.acs.org on June 20, 2017

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The Journal of Physical Chemistry

The Effect of Surface Diffusion on Adsorption-Desorption and Catalytic Kinetics in Irregular Pores. Part II: Macro-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 E-mail: [email protected]

∗ To

whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract In this work we show how the effective diffusion coefficient of a membrane, in which heterogeneous reaction-diffusion processes are present, is related to the geometric irregularity of the pores and how it is affected by the surface diffusion of an adsorbed phase. A theoretical expression for this effective membrane diffusion coefficient is deduced starting from a recent generalization of the so-called Fick-Jacobs approximation. Our analysis comprises the interrelation of bulk and surface diffusion with the heterogeneous catalysis and adsorption/desorption interchange of matter between the bulk and the walls of the pore. Therefore, our theoretical framework is a very useful tool in modeling the process of heterogeneous catalysis inside porous materials when the shape of the pores is known. Through some illustrative examples involving the classical Langmuir processes as a reference, we provide a useful methodology correlating the local properties of transport inside a pore of irregular shape with the effective diffusion coefficient of a membrane. This coefficient can be measured experimentally by fitting the spatio-temporal concentration profiles in adsorption experiments. We show the different dependences of this coefficient with the surface diffusion, the shape of the pore, and the average loading of adsorbed particles. A reverse methodology is also sketched, in which we suggest how to use the data emerging from experiments in order to determine the intensity of the surface diffusivity.

1

Introduction

Knowledge of mass transport across porous materials is an important subject for the understanding of fundamental physical concepts associated with the development of technological applications. 1 Depending on the size of their pores, these materials have different applications when used as adsorbents and catalysts in the chemical industry. 2 Their transport properties depend upon the strength which the interaction among host-guest and guest-guest molecules have inside the pore. 3 This interaction can be particularly determinant in mesopores and nanopores where the diameters of the guest molecules are comparable with the diameter of the pore, causing some interesting


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effects such as capillarity, pore blockage, cracks or sieve selectivity. 4 It is also very important for micropore materials where the shape of the pore can control in a significant way the amount of adsorbate material that crosses or enters to a membrane. 5 Therefore, the understanding of how the details of a porous structure and their surface characteristics are interrelated with the adsorbate dynamics in the bulk of the pore results of great importance in the modeling of processes where the catalysis plays a central role. 6 Adsorbate diffusion is, in many cases, a key parameter controlling the efficiency of catalytic reactions. 7 Besides, the relevance of diffusion phenomena concerns both the microscopic scale, where they determine the local reaction rates, and the macroscopic scale, by governing the rates of reactant supply and product removal. 8 This diffusion inside an adsorbent material can occur by two processes. First, by bulk diffusion which is a consequence of collisions either with the pore walls (known as Knudsen diffusion), 9 or between the molecules, and second, by means of surface diffusion where the particles adsorbed to the wall are allowed to migrate along the surface causing an extra flux that should be added to the one resulting from bulk diffusion. 4 The relative importance of these two types of diffusion processes in a porous material, where the ratio surface/volume is usually large, is still a matter of research due to the lack of models able to relate the local properties of the transport inside the pores with the macroscopic measurements of fluxes and concentrations at the ends of the membrane. 10 Nonetheless, as far as we know, the surface diffusion can be particularly important in determining the magnitude of effective fluxes of adsorbate material, especially at low pore loadings. 11,12 Elucidating the different contributions to the diffusion process in small pores is still far from trivial task. 13 This is due to the fact that the variation of the molecular concentration in pores may change the character of the diffusion process in an appreciable extent, causing transitions which range from the monolayer adsorption to complex phenomena like capillary condensation. 4,14 These effects are reflected in terms of the different behaviors of the self-diffusion coefficient as a function of the total concentration. In some cases, this coefficient has a monotonic increasing behavior due to the fact that, at high loadings, the amount of particles with low motility (due to the steric



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restrictions or the presence of an adsorbed phase) is increased. 4 At the other hand, a monotonic decreasing behavior is found in systems where the molecular exchange between the adsorbed and the vapor phases is reduced with increasing concentration. 4 Some experimental methods like interference microscopy and the pulsed field gradient nuclear magnetic resonance (PFG NMR) have been applied to zeolites in order to determine diffusivities in nanoporous host-guest systems. 4,15,16 They are able to provide the rate of intracrystalline migration of adsorbed molecules under equilibrium conditions, allowing to measure an effective diffusion coefficient as the fitting parameter of the spatio-temporal concentration profiles. 6,17,18 In terms of the surface diffusion, some experimental techniques, such as scanning tunneling microscopy and field ion microscopy, have allowed the tracking of single molecules on surfaces on the short time scale, while the PFG NMR in surface studies can be used in the scale which ranges from milliseconds to seconds. 2 The aim of this work is to establish the quantitative connection between the effective diffusion coefficient of a membrane Dm , that can be inferred from results of imaging experiments by fitting the curves of the spatio-temporal concentration profiles, with the microscopic aspects of the mass transport along a single pore, which are characterized through the apparent diffusion coefficient of a single cell of material Dt . In works of chemical reaction engineering , the effective diffusion coefficient of a membrane Dm (φ , τ, δ ) quantifies the average reduction of the flux when a fluid crosses a porous material. Hence, it depends on the porosity φ , the tortuosity τ, and the constriction factor δ . 19–21 The first two parameters are purely geometric, and are associated with the total free volume of the membrane, whereas the constriction factor δ , depends on the internal degree of corrugation of the pores and on its microscopic geometric properties. We will show that the constriction factor is the key quantity that incorporates the presence of the above mentioned microscopic processes associated to surface diffusion and heterogeneous chemical kinetics. Thus, we will establish how a relation of the well known form Dm = D0 δ (φ /τ) 19–21 can be derived from a microscopic level involving the apparent diffusion coefficient Dt . In order to accomplish this objective, we will start by using the local transport model presented


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in the Part I: Local-Kinetics, of this research work. In this model, the concentration of an adsorbent inside an irregular pore is seen as the sum of two types of particles: some in the bulk, and some attached to the wall. The idea of this abstract separation is that both kind of particles obey different types of diffusion processes and, therefore, the diffusive part of their respective equation has different mathematical form in each case. Besides, in the case of heterogeneous catalysis, the catalytic reaction is produced just at the surface. These two aspects allows to set two different equations for each phase. We have shown in Part I: Local-Kinetics, that these two equations are able to reproduce the processes of bulk and surface diffusion, as well as the interchange between the liquid-vapor phases in a reaction of adsorption-desorption on the surface of the pore. The bridge between microscopic and macroscopic descriptions is then built by using mass conservation and the results of the local diffusion coefficient derived from the Fick-Jacobs scheme (FJ) for a single pore. This last quantity involves an average that allows to quantify the effects that the microscopic irregularity of the pore has on macroscopic quantities. The effect that surface diffusion and boundary conditions have on the value of the effective diffusion coefficient of a membrane are also discussed in detail. The work is organized as follows. In Section 2 we deduce the effective diffusion coefficient of a membrane from the local apparent diffusion coefficient deduced in Part I: Local-Kinetics. It is also shown that the constriction factor can be interpreted as the effective resistance to flow in a corrugated pore, and that this resistance is affected by the adsorption/desorption processes as well as by surface diffusion. Then, Section 3 is devoted to show the important role that surface diffusion and the possible boundary conditions on experiments have in this flow resistance and therefore, on the value of the effective diffusivity of the membrane. Section 4 demonstrates the excellent quantitative consistency of the macro-transport model deduced in this work with the spatially detailed description of Part I: Local-Kinetics, in several pore geometries. This section also provides a methodology that may be used to infer the value of the molecular surface diffusion coefficient from experimental data of the effective diffusion coefficient and the average loading. Finally, in Section 5 we present a brief discussion and the conclusions of the present work.



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2

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Deduction of the Effective Diffusion Coefficient from the Local Apparent Diffusion Coefficient

In the Part I: Local-Kinetics of this work, we have provided a detailed demonstration on how the combined volumetric and surface diffusion processes, together with the some reaction occurring in the wall (like adsorption-desorption and catalysis), may be described very accurately at the local level by means of the equations: ∂Cb 1 ∂ ∂Cb = Db ω + Rb , ∂t ω ∂x ∂x

(1a)

∂Cs 1 ∂ 4Ds ∂ wCs = + Rs . ∂t w ∂x γ ∂x γ

(1b)

Here Cb (x,t) and Cs (x,t) are the concentrations per unit volume of particles in the bulk and adsorbed in the surface, respectively. Rb and Rs are the volumetric rates of reaction for the adsorptiondesorption and catalytic processes. Ds is the coefficient of surface diffusion, and Db (x) is the position dependent bulk effective diffusion coefficient of the FJ approximation, 22–25 which measures the reduction of the effective transport in terms of the molecular diffusion coefficient D0 and the confinement geometry. Introducing the total mass concentration Ct (x,t) = CB +Cs , combining eqs 1, and introducing the diffusion currents of particles at the bulk and the surface (see Part I: Local-Kinetics), the local apparent diffusion coefficient is given as the sum of the bulk and surface diffusion coefficients Dt (x, λ , Ds ) = D0b + D0s ,

(2)

where the local diffusion coefficients of the bulk and surface processes are D0b ≡ Db

1 , 1+λ


(3a)

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D0s ≡

4Ds λ . γ2 1 + λ

(3b)

Here, we have denoted explicitly the dependence of the apparent diffusion coefficient on the rate of conversion λ , which is positive for the most common isotherms and is defined by λ≡

∂Cs . ∂Cb

(4)

If the isotherm Cs (Cb ) of the process is known, the parameter λ measures the rate at which the particles in the bulk turn into the adsorbed phase. The predictions and applications of eqs 3 were carefully analyzed in Part I: Local-Kinetics. A series of interesting effects regarding the local behavior of the adsorbed and bulk particles inside an irregular pore were predicted. Among the most interesting effects, to mention an example, we have shown that for a Langmuir process of the form RL = k+ (CvCb − KLCs ),

(5)

the bulk particles tend to distribute homogeneously inside the volume of the pore, while the adsorbed phase (that is allowed to diffuse along the surface), tends to accumulate in the wider regions of the pore. In the last equation, k+ is a rate constant, KL the equilibrium constant and Cv is the concentration of available sites, see Part I. We have to remark that although we have used the Langmuir isotherm in the model examples along this work, our formalism can be easily extended to more general and realistic adsorption/desorption processes. Let us emphasize that, if within the FJ approximation the ratio Db (x)/D0 measures the diminution of the effective mass flux due to the corrugation of the pore, 22–26 in the generalized FJ approximation used here, the ratio Dt (x; λ , Ds )/D0 measures the change (diminution or enhancement) of the effective mass flux due to the corrugation, the surface diffusion, and the presence of heterogeneous chemical reactions in the pore. Thus, the relevance of the apparent diffusion coefficient Dt given by eq 2, is that it summarizes a powerful theoretical description of mass transfer in pores.



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2.1

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Relating the Microscopic Dt (x) with the Macroscopic Dm

The general relation between the effective diffusion coefficient of a membrane Dm and the apparent diffusion coefficient in a pore, Dt (x, λ , Ds ), can be established by considering that, since Dm is a constant, it is expected that its relation with Dt (x, λ , Ds ) must be given through an adequate average. As it has been explained in the introduction, Dm quantifies the average change of the mass flux when a fluid crosses a porous material. Its standard mathematical form (which has been confirmed by experiments) is 19–21 φ Dm = D0 δ , τ

(6)

where D0 is the molecular diffusion coefficient of the particles in the bulk phase in absence of confinement, and φ and τ are the porosity and tortuosity of the membrane, respectively. The symbol δ is called the constriction factor since it quantifies, in an averaged form, the effect of the internal corrugation of the pores on the mass flux. In particular, the inverse 1/δ is a measure of the intrinsic resistance of a pore to the flow. The effective diffusion coefficient Dm is very important because it enters in the macro-transport diffusion equation ∂ 2C ∂C = Dm 2 , ∂t ∂x

(7)

and therefore, it has a very simple experimental meaning. In eq 7, the symbol C stands for the total concentration of a gas measured over the entire volume of a membrane cell (L ·W in 2D), 27 that is C≡

n . LW

(8)

Here, n is the mol number of particles, and L and W are the longitudinal and the transverse (macroscopic) lengths of a membrane cell (see Fig. 1 in Part I: Local-Kinetics). If the concentration C(x,t) is known as a function of the external conditions in which the membrane is immersed, then one could use Dm in eq 7 as a fitting parameter that allows to quantify the effect that the porous


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material has in the effective transport. The value of Dm that we model in this work can be also obtained from an experimental setup by adjusting the spatio temporal concentration profiles inside the pore with Eq. 7. As we have explained in the introduction section, this type of measurements can be carried out by using PFG NMR and interference microscopy techniques. 7,9,18,28,29 In the following lines we describe an argument that allows us to model this effective diffusion coefficient based in simple conservation arguments. Although the strict mathematical deduction of this Dm is given in Ref. 27 for the case of bulk diffusion, we present here a more physical approach based in the comparison between the effective flux of particles inside the pore Ht , and the flux that one would expect over the entire membrane H. In this deduction, the flux of particles includes all the processes occurring at the pore, i.e., bulk and surface diffusion, adsorption and desorption. However, the mathematical deduction we present here is equivalent to that of Ref. 27 In stationary conditions, the total mass flow H across the membrane can be determined from eq 7, after identifying the diffusion current: j = −Dm dC/dx. An integration of this current along the transversal and longitudinal coordinates yields the macro-transport relation H = −Dm

W ΔC, L

(9)

where we have used constant Dirichlet boundary conditions at the entrance and exit of the pore, that introduce the total concentration difference: ΔC = C(L) −C(0). The total concentration Ct is defined inside the void space of the pore and it can be written in the form Ct

n , lz w

where lz is the real length of the pore (see Part I: Local-Kinetics) and w =

(10) 1 L

w(x)dx is the

average width of the pore. Thus, the mass flow Ht across the pore can be written in the form Ht −Dt

w ΔCt , lz


(11)


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where ΔCt = Ct (L)−Ct (0). Notice here that boundary conditions are irrespective of which concentration is used, and therefore ΔCt = ΔC as long as both concentrations have the same units. In eq 11 we have introduced the notation Dt for the averaged apparent diffusion coefficient to emphasize that this average is not simply an integration over the longitudinal direction. The average Dt has to consider a weight factor that should depend on the pore width. Although the detailed discussion of this point will be given below, it is intuitive to see that one may write the relation Dt = D0 δt = D0 δ , and therefore, that the constriction factor δt should contain all the effects of internal constriction (corrugation) of the pores in the macroscopic relation for Dm . The eq 11 can be rewritten in terms of the concentration C and the mass flow H over the entire volume of the cell, by noticing that the total concentration Ct should be larger than C according to the geometrical relation Ct =

L W C, lz w

(12)

and, similarly, that the mass flow Ht across the pore is related to the mass flow over the entire volume by Ht =

W H. w

(13)

Substitution of eqs 12 and 13 into eq 9 yields the relation L H = −D0 δt w 2 ΔC, lz

(14)

where we have also used the relation Dt = D0 δt , and the equality ΔCt = ΔC. Considering now that δ ≡ δt and the definitions of the porosity φ = w/W , and of the tortuosity τ = lz2 /L2 given in, 19,27 the last equation can be written in the form H = −D0 δ

φW ΔC, τ L

(15)

which, after compared with eq 9, yields eq 6. It is worth to stress that the constriction factor δ ≡ δt appearing in eq 15, takes into account the geometrical aspects of the corrugation of the pore and 10 ACS Paragon Plus Environment

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the effects of surface reactions in the pore. This fact is evidenced by noticing that the explicit expression for the constriction factor δt can be calculated by using mathematical induction over all the slices forming the pore, 27 or by directly integrating eq 9 over the x coordinate. The last procedure was followed by Bradley 22 for the pure diffusive case. Here, we have extended its application to the case of heterogeneous reactions by using Dt given by eq 2 instead of Db , in order to deduce the following mathematical expression for the constriction factor: 1 = δ

D0 Dt (x) w(x)

X

w(x)X .

(16)

Using eqs 2 and 3 in eq 16, yields the generalized constriction factor 1 = δ

1 + λ (x) s [Db + λ (x) 4D ] w(x) γ2

w(x)X .

(17)

X

Eqs 6 and 17 are very important since they connect the intra-pore aspects of the transport associated with the geometry, the surface diffusion, and the chemical processes at the surface, with the macroscopic or global properties of the transport across the membrane or the porous material. In order to close this section, it is convenient to point out that in the derivation of the effective diffusion coefficient Dm , it was assumed a net flux of particles across the membrane caused by the concentration difference C(L) − C(0). However, this assumption is not essential since it can be proved by performing numerical experiments that the same diffusion coefficient holds for saturation boundary conditions where both extremes of the membrane are exposed to the same external concentration. 27 This is due to the fact that the internal flux calculated with the use of the Fick-Jacobs scheme is very precise also locally for long pores and, therefore, it can be used even when the internal concentration differences inside the pore tend gradually to zero, as it occurs under saturation conditions when there is no net gradient. 27



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3

Surface Diffusion and Boundary Conditions

The first application of the precedent results is the analysis on the relative importance between the surface and the bulk diffusion coefficients defined in eqs 3. As it was shown in Part I: LocalKinetics of this work, the relative importance between these two diffusivities causes interesting effects at local level, like for instance, the formation of feedback cycles between adsorbed and bulk particles even in a macroscopic stationary state. Here, we will elucidate how these effects are considered at the level of macro-transport across the membrane. In particular, we will propose a methodology that can be used in the future to understand the results of some experiments, like the different diffusion coefficient inferred by means of PFG NMR experiments, and which measure the rate of intracrystalline migration of adsorbed molecules under equilibrium conditions, 4,15,16 and the diffusivity that can be measured from sorption-rate experiments in which the overall rate of transport under non-equilibrium conditions is determined. 11 The first point we have to emphasize in this analysis is that the local character of the diffusion coefficients defined in eqs 3, D0b and D0s , is influenced by the external conditions imposed on the pore, that is, on the boundary conditions. The consequence of this influence is that the (local) apparent diffusion coefficient Dt and, therefore, the membrane diffusion coefficient Dm , are also affected by these boundary conditions. Reaction engineering experiments are usually performed following two basic protocols. In one of these protocols a net flux of particles across the membrane is experienced, whereas in the other protocol, a pore saturation process is examined. The first case corresponds to a chemical reactor configuration, whereas the second case corresponds to saturation conditions in which the fluid enters at both sides of the membrane. 5 In the following, we will solve the local and macro-transport models in order to show explicitly the effects above mentioned. For this purpose we will assume a rectangular pore, that is, when the effects of corrugation and tortuosity are not present. More general cases will be analyzed in subsequent sections. 12 ACS Paragon Plus Environment

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Local diffusion coefficients.

We will first solve the local model consisting of eqs 1 in the case

of Langmuir adsorption kinetics, for which Rs = −Rb = RL with RL given by eq 5. The boundary conditions for the concentration in the chemical reactor configuration are C(0) = C0 and C(L) = 0, whereas for the saturation conditions are C(0) = C(L) = C0 .

Figure 1: Influence of the surface diffusion on the adsorption dynamic on a rectangular pore of constant width w(x) = 1 and length L = 5, under the boundary conditions indicated. The colors correspond to different values of Ds /D0 . Solid lines correspond to the total concentration Ct , and to the apparent diffusion Dt . Dashed lines correspond to the bulk concentration Cb , and effective bulk diffusion coefficient D0b . Dashed-dotted lines correspond to the surface concentration Cs , and the effective surface diffusion coefficient D0s . The profiles were obtained by assuming an adsorptiondesorption kinetics dictated by the Langmuir process, eq 18. The parameters of this process were α = 5/3 and KL = 0.3 . . In order to have consistent boundary conditions for the bulk and the surface concentrations, Cb (x,t) and Cs (x,t), we will assume a rapid equilibration of both bulk and surface concentrations at the pore ends, in such a way that concentration of the adsorbed phase, Cs (x = 0) and Cs (x = L), can be inferred from the value of the bulk concentration, Cb (x = 0) and Cb (x = L), by using the Langmuir isotherm Cs (Cb ) emerging from eq 5. In Part I: Local-Kinetics, we have shown that this isotherm can be written in the form Cs (Cb ) =

αCb , 1 + KL−1Cb 13

ACS Paragon Plus Environment

(18)


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where KL is the equilibrium constant. The long-time stationary state numerical solution of the equations for the bulk and the surface concentration profiles, Cb (x,t) and Cs (x,t), together with the corresponding surface, bulk and apparent diffusion coefficients D0s , D0b and Dt , are respectively shown in Fig. 1 for three different values of the ratio between the molecular surface and bulk diffusion coefficients: Ds /D0 = 0.0, 0.5 and 2.0. These results show that the boundary conditions and the surface diffusion modify significantly the spatial dependence of the diffusion coefficients, as well as the concentration profiles or, equivalently, the form of the fractional surface occupancy θ (x) ≡ Cs /Cs0 . 30 The apparent diffusion coefficient Dt depends on the position in the net flux case, since the effective surface and bulk local diffusion coefficients depend upon the loading profile characterized by θ (x). Using this quantity, the eq 2 yields the following expression for the apparent diffusion coefficient Dt (x) = D0

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

(19)

From this example we may infer the important fact that, even in a rectangular pore where there are no geometrical effects due to the corrugation, the loading inside the pore may be spatially non-homogeneous depending on the external conditions imposed on the membrane.

Effective diffusion coefficient of the membrane. From the previous examples we may conjecture that the macroscopic diffusion coefficient Dm of a membrane composed of rectangular of pores, will also depend on the boundary conditions used, that is, on the method used to measure it. This can be calculated explicitly for the present case by substituting eq 19 into in eqs 17 and 6. One obtains D0 φ = Dm

1 + α[1 − θ (x)]2 s 2 1 + αD D0 [1 − θ (x)]

,

(20)

X

where, for simplicity in the formula presentation, we have used that the tortuosity for symmetric pores is τ = 1. The extension to non-symmetric pores is straightforward. In the Table 1 we present


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the values of Dm /(φ D0 ) obtained from eq 20, for the two boundary conditions considered and for different values of the surface diffusion coefficient. From the Table 1, it follows that the net Table 1: Value of Dm /(φ D0 ) using eq 20 for the processes studied in Fig. 1. See the text for details. Ds /D0 Net Flux Saturation

0 0.65 0.85

0.5 0.81 0.92

2 1.36 1.14

effect of the surface diffusion on the effective diffusivity of a membrane depends on two factors, the boundary conditions and the relative importance of the molecular surface diffusion coefficient Ds as compared with D0 . For net flux conditions, increasing the surface diffusion from Ds = 0.0 to Ds = 2.0 duplicates the value of Dm . For saturation conditions the increase of Dm is only of about 1.3 times. Hence, net flux experiments seem to be more sensitive to changes on the surface diffusion than saturation experiments. In the following section we will show that the basic behavior obtained for a rectangular pore can change drastically when the pore is irregular, due to the relation between Ds and Db or D0 , as it follows from the eq 17.

4

Effective Diffusion Coefficient and Spatio-Temporal Profiles

In this section we want to show the excellent agreement between the predictions of the local model for Ct (x,t) = Cb + Cs , defined through the microscopic model stated in eqs 1, and those arising from the macro-transport model for C(x,t), defined through eq 7. To make this comparison quantitatively equivalent, it is convenient consider symmetric pores for which τ = 1 and φ = 1. We have to emphasize that these assumptions are only for the purpose of focusing the analysis of the effect of the internal corrugation on the flow reduction across the pore, independently of the tortuosity and the total porosity of the membrane. Thus, eq (7) takes the simpler form ∂C ∂ 2C = D0 δ 2 , ∂t ∂x 15 ACS Paragon Plus Environment

(21)


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with δ given by eq 16. This effective diffusion equation describes the spatio-temporal concentration profile of all the particles of a given chemical species, both in bulk and surface without regard to the specific internal geometry of the pore. Therefore, it is particularly important where the experimental method only permits to measure the total mass flux, and could be used as a theoretical modeling tool in experiments measuring diffusion and adsorption kinetics inside zeolites by means of, for instance, the PFG NMR method. 13,16,18,31

Figure 2: Adsorption dynamic for three different pores with the same internal volume and tortuosity. The boundary conditions at the ends are of saturation. The time considered in these cases is tmax = 50. The length of the three pores is L = 10 and the width of the pores is w(x) = 1 (blue), w(x) = 1 + 0.4 sin(12πx/L) (orange), and w(x) = 1 + 0.5 sin(12πx/L) + 0.12 sin(24πx/L) (red). Eqs 1 are solved in order to obtain Ct = Cb +Cs This solution is plotted in dashed lines. In contrast, the macroscopic approximation obtained by solving eq 21 with the same boundary conditions, is plotted in solid lines. The process is of the Langmuir type in eq 5, with k+ = 0.1, Cs0 = 0.5 and KL = 1.0. In the top row Ds = 0, and at the bottom Ds = D0 . The molecular diffusion coefficient is D0 = 1. The values of the effective constriction δ for each case are given in Table 2. Some transient times are shown.

Concentration profiles. The comparison is done by solving eqs 1 and 21 by assuming a Langmuir adsorption kinetics (eq 5), and under saturation boundary conditions. The results are shown in Fig. 2 for the three different pores sketched in the inset. The dashed lines represent the concentration profiles obtained by solving the coupled system of equations given in eqs 1, whereas the solid lines represent the solutions for the total concentration obtained from solving the macroscopic eq 21. We have performed these calculations in absence (top) and the presence (bottom) of surface diffusion, i.e. with Ds = 0 and Ds = D0 , respectively. In all cases the concentration profiles increase from zero until a maximal stationary concentration. 16 ACS Paragon Plus Environment

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Table 2: Different constriction factors δb and δ obtained for the processes described in Fig. 2. The parameter δ was calculated using eq 16. The parameter δb can be calculated using the same equation, but by substituting Dt (x) by the space dependent diffusion coefficient Db (x) of the basic FJ approximation (when there is no adsorbed phase and the walls are reflective). See the text for details. Pore Ds /D0 0 1

δb 0.00 1.00

δ 0.88 1.00

δb 0.84 0.84

δ 0.71 0.82

δb 0.72 0.72

δ 0.64 0.71

Once we have found the concentration Ct = Cb + Cs from the microscopic profiles, we have calculated the apparent diffusion coefficient according to eq 2, and then we evaluated the value of the constriction factor given in eq 16. The results are summarized for each pore and each surface diffusion coefficient in the Table 2. As it can be corroborated from such Table, the generalized constriction factor δ (which is proportional to the effective diffusion coefficient Dm defined in eq 6) decreases when the pore is more corrugated. In the cases considered here, this coefficient is larger when surface diffusion is present. This means that the existence of a mobile adsorbed phase causes a diminution of the pore resistance to the mass flux. 11 The solutions of eq 21 for C(x,t) contain only one (constant) adjustable parameter δ which, together with the boundary conditions at the exterior of the pore, constitute all the required information in order to get an approximate concentration profile. From Fig. 2 it is clear that the comparison with the microscopic concentration (dashed lines) is remarkable for all the geometries and diffusion parameters considered here. Diffusion coefficients and surface coverage. The apparent diffusion coefficient Dt (x) and the surface coverage θ (x) are shown in Fig. 3 for two cases: with and without surface diffusion. Concerning the diffusion coefficient (top row), the main result is that the surface diffusion increases the value of the apparent diffusion coefficient. This result is consistent with the experimental evidence that the surface diffusion increases the net flux throughout the pore, with respect to the case when the adsorbed phase remains immobile. 11 17 ACS Paragon Plus Environment


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Figure 3: Adsorption dynamic for the three different pores and the two surface diffusion coefficients studied in Fig 2. Top. Value of the apparent diffusion coefficient in eq2, deduced from the profiles depicted in Fig 2. Bottom: Loading θ (x) as a function of the position. The average value of the loading θ (x)X for each process is shown in the inset. The bottom row of Fig. 3 shows the stationary loading profiles for each one of the previous pores and values of the surface diffusion coefficient. From our results, it follows that the surface diffusion increases the irregularity of the loading profiles, consistently with the results of Part I: Local-Kinetics of this work. What is important here is that, for the same chemical parameters of adsorption and desorption, the measured value of the average loading θ (x)X change with the shape of the pore and the value of the surface diffusion coefficient. This important result is emphasized in the insets of the Fig. 3. In the next section we will explain that the dependence of Dt (and therefore, the dependence of Dm ) in the average concentration θ (x)X can be used as a key ingredient in order to understand the microscopic aspects occurring inside the pore, when the experimental plot of effective diffusivity of the membrane Dm versus the average loading θ (x)X is measured in the experiments.


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4.1

Inferring the Surface Diffusion from the Effective Diffusion Coefficient of a Membrane.

In the previous sections we have presented and illustrated a theoretical formalism that consistently brings together the micro- or mesoscopic description of mass transfer across a pore, with the macroscopic or average mass transport properties of the entire porous material. This bridge considers surface diffusion and chemical reactions at the walls and leads directly to the evaluation of the average loading factor θ X and of the effective diffusion coefficient Dm . The former quantity can be measured from mass uptake experiments, whereas the later can be experimentally assessed by using interference microscopy experiments or the PFG NMR technique. 18,31 Furthermore, we have shown that our models approach with excellent agreement the value of Dm if the microscopic information contained in eqs 1 is known. Because of its tremendous practical importance, one may rise the question if it exists the possibility of a reverse procedure, that is, the possibility to infer the intensity of the surface diffusion coefficient Ds from the experimental data on the porosity φ , the tortuosity τ, the average loading θ X , and the effective diffusion coefficient Dm . The answer is yes and, as we will show below, this connection can be established indirectly by using the microscopic model postulated trough eqs 1, in order to construct the plots of Dm versus θ x for several values of Ds . In order to illustrate this reverse procedure, let us follow the same methodology as in the last section. First we estimate the diffusion coefficient Dm of a membrane assuming a Langmuir process. We therefore solve the eqs 1 with the volumetric rate RL dictated by the eq 5. The solution for Cs (x,t) allows us to asses the spatial dependences of the fractional coverage θ (x). Then, we substitute θ (x) into the apparent diffusion coefficient Dt in eq 2, and calculate the constriction factor δ given by eq 16. In order to emphasize the importance of the internal geometry, we have repeated this procedure for two types of pores, one rectangular and one sinusoidal. The results are shown in Fig. 4. As it is expected, a variation of the equilibrium constant of the Langmuir process, KL , changes in turn the average fractional coverage inside the pore. Since the two pores compared here have the 19 ACS Paragon Plus Environment


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Figure 4: Modified Constriction factor δ = Dm /(D0 φ ) as a function of the average fractional coverage for two pores of length L = 5, and width w(x) = 1 (red) and w(x) = 1 + 0.2 sin(6πx/L) (blue). The process studied here is of the Langmuir type with the same reaction parameters as in Fig. 2, except for the equilibrium constant KL that we change from 0.01 to 5 in order to obtain different values of the average loading θ X . Different colors correspond to the two different pores shown in the inset. Different symbols correspond to different values of Ds /D0 .


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same internal volume and tortousity (τ = 1), then according to eq 6, their difference in diffusivity depends only of their constriction factor δ = Dm /(D0 φ ). The red solid lines of Fig. 4 show the behavior of δ for the rectangular pore and for four different values of the surface diffusion coefficient (symbols), and seven values of the equilibrium constant KL (position of the symbols). In this case, for low values of Ds /D0 , the value of Dm first decreases with the loading until a minimum is reached and then it increases. Exactly the opposite behavior happens for Ds /D0 > 1. However, for an irregular pore like the sinusoidal one shown in blue dashed lines, the data of the effective diffusion coefficient has larger changes with increasing average loading. In this case, the effective diffusion coefficient Dm can have different shapes according to the value of the surface diffusion coefficient. The reason for this is that the diminution of the volumetric diffusion coefficient caused by the geometric irregularity of the pore, augments the importance of the surface processes, due to the augment of the effective wall area. 5,23 In this way, the effective transport inside pores with a high fractional coverage increases the dependence of Dm in the ratio Ds /D0 . We have to stress that the tendencies of the effective diffusion coefficient Dm as a function of the pore geometry and Ds shown in Fig. 4, are not all the expected behaviors of he macroscopic Dm in eq 6. We have to notice that the use of our microscopic model in eqs 1 can give place to a variety of behaviors that depend upon the type of limiting process inside the pore. A detailed study of this important application of eqs 1 will be done in a future work. Nonetheless, in this section we have shown how to construct the plots of Dm versus average loading in terms of very precise equations relating the process of bulk and surface diffusion, adsorption/desorption, and their relation with the geometric irregularity of the pore. As this coefficient can be directly related to quantities of great importance in chemical reaction engineering, like the mass transfer coefficient and the internal effectiveness factor, this study represents a possible bridge between these macroscopic quantities and the before mentioned internal factors. 5



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5

Discussion and Conclusions

Starting from the very general theoretical formalism of Part I: Local-Kinetics for describing mass transport in porous materials when heterogeneous chemical reactions and surface diffusion effects are considered, we have presented a novel theoretical foundation and interpretation of the well known empirical expression for the effective diffusion coefficient of a membrane. This theoretical explanation establishes a solid bridge between the microscopic and macroscopic properties of mass transport across a porous material. In particular, we have shown the link among the effective diffusion coefficient Dm and the ratio between the bulk and surface molecular diffusion coefficients Ds /D0 , the external boundary conditions at which the porous material is immersed, and the average loading of adsorbed particles inside the pore θ x . One relevant contribution of our model is that it allows to quantify in a precise way the influence and the relation of these factors with the irregular geometry of the pore. This is very important since it implies that on using one dimensional diffusion models, the apparent diffusivity of a pore is modulated in a non-linear way by a series of interconnected mechanisms that depend on the pore corrugation. In particular, the internal corrugation of the pore increases the resistance to flow as well as the adsorption-desorption kinetics. These two effects, together with the diffusion of the particles at the surface, tend to reduce the intrinsic resistance of the pore in comparison with the case when the adsorbed phase is stagnant at the wall, making the whole process strongly interrelated and non-linear. A second contribution of our work is that it provides a clear methodology that allows to link the microscopic concentration profiles emerging from eqs 1 with their counterparts emerging from macroscopic experiments. Therefore, our work becomes an useful tool for modeling and understanding the different experimental plots of Dm versus the average loading θ X , that appear in mass uptake and similar experiments. It should be stressed that, although this work allows us to accurately identify the value of Dm if the macroscopic information is known and not vice versa, it can be used as an auxiliary tool in order to identify some characteristic features of the internal pro22 ACS Paragon Plus Environment

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cesses within the pore, if one has the experimental curve of Dm versus the average loading θ X . Following this line, we have proved this versatility of the model by showing how the concavity of the Dm vs. θ X -curve depends on ratio Ds /D0 entering in eqs 2 and 17. To summarize, we have presented a very complete and powerful theoretical framework that copes with all the basic mechanisms associated with mass transport across a porous material in which adsorption-desorption, heterogeneous catalysis, and surface diffusion may be present. Although there are several generalizations and studies in turn that will be analyzed in future works, the formalism is a useful tool in many simulations, design, and experimental protocols on porous materials performance.

Figure 5: TOCimage

Acknowledgement We are grateful to UNAM-DGAPA-PAPIIT project IN116617 for financial support. ALD acknowledges CONACyT for financial support under fellowship 221505. SIH is grateful to project LANCAD-UNAM-DGTIC-276.

References (1) Deutschmann, O.; Knözinger, H.; Kochloefl, K.; Turek, T. Heterogeneous Catalysis and Solid Catalysts; Wiley Online Library, 2009.



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