Review
Surface diffusion in porous media: A critical review

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Abstract

Four transient experimental methods for the study of surface diffusion of gases and liquids in porous media are critically reviewed. Each method is based on the measurement of a physical quantity that is sensible to a change in the surface diffusivity so that experimental data can be used to extract a value of the surface diffusivity. After a brief sketch of the experimental set-up, the present theoretical models proposed to describe the kinetics of the methods by a balance equation or a system of such equations are presented. It is then explained that when a (usually numerical) solution of the model is fitted against experimental data, values of the associated kinetic parameters, especially the surface and pore diffusivities, can be obtained. It is also explained how values of these two diffusivities can be separated from each other, which is often a challenging task. The advantages and disadvantages of the four methods are pointed out in each case, and we eventually provide a comparison between the methods. A discussion of the concentration and temperature dependencies of the surface diffusivity precedes the review of the methods.

Graphical abstract

Various transport mechanisms of adsorbate particles through a pore.

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Research highlights

► Measurement of surface diffusivity in porous media. ► Critical review of four transient methods. ► Corresponding mass transport models presented. ► Concentration and temperature dependence of surface diffusivity discussed.

Introduction

Surface diffusion was originally introduced to study unusually large mass transfer rates of gases through porous media. It turned out that, besides the bulk diffusion in pores, particles can migrate across a foreign material along the pores surface. As is well known, bulk diffusion is driven by a concentration gradient of the migrating particles that diffuse through a bulk mobile phase stagnating in pores, and there is no interaction between the particles and the inner walls of the pores (except, perhaps, collisions). However, particles may adsorb on these walls and diffuse on the surface while being adsorbed. The particle migration in the adsorbed state (in the potential field of adsorption) is called surface diffusion (see Fig. 1). It can be experimentally observed and probed in porous media by nuclear magnetic resonance methods, yeilding the surface diffusivity in dependence on the surface coverage [1], [2].

It has been well established by now that surface diffusion contributes significantly to the total mass transfer rate in porous materials both in gas–solid and liquid–solid adsorption systems. For example, in reversed-phase liquid chromatography systems, the pore diffusivity is about one order less than the apparent diffusivity (that includes contributions from both the pore and surface diffusivities) so that more than 90% of the particles migrate by surface diffusion [3], [4]. In gas–solid systems the surface diffusion contribution may be even more profound [5]. Thus, surface diffusion is a topic of great interest to adsorption science and is perhaps one of the least understood phenomena.

Surface diffusion should be considered as one of the mechanisms that substantially contributes to the transport of particles through porous materials, especially as opposed to bulk diffusion (sometimes Knudsen diffusion is considered as an additional mechanism occurring separately from surface and bulk diffusions). It is therefore the aim of this review to give a concise account of four selected measurement methods that may be applicable, possibly after appropriate modifications, to the study of surface diffusion in such materials. Our main focus will be on the description of diffusion models corresponding to the kinetics of a given experimental method. Such an exposition demonstrates how various models are related to and complement each other, thus allowing one to analyze experimental data by employing a specific model or a combination of models that are conveniently tailored to the transport of an adsorbate through a microporous adsorbent. Indeed, the models can be used to extract values of kinetic parameters, including the surface diffusivity, from an appropriate analysis of experimental data. The procedures that enable one to separate the surface diffusivity from the bulk and/or other diffusivities are indicated. In addition, we provide a list of advantages and disadvantages of the methods as well as a comparison between the individual methods, mainly from the viewpoint of the evaluation of experimental data. However, before the review of the methods, we mention basic ideas on surface diffusion and discuss the dependence of the surface diffusivity on temperature and concentration.

Section snippets

Basic ideas on surface diffusion

The layer of adsorbed particles, along with possible patches of bare pore surface where no particles are adsorbed, is usually referred to as the stationary phase (see Fig. 1). It is more dense and better organized than the bulk mobile phase so that it has different physical and chemical properties than the bulk phase. Detailed information on the composition, structure, and physical and chemical properties of the stationary phase is important for appropriate understanding and description of

Surface diffusivity

The macroscopic surface diffusion flux, Js, is equal to the molar amount of particles adsorbed on the surface that passes through a unit area during a unit time interval, and may be expresses as a product of the surface concentration, Cs, and velocity, vs, of the adsorbed particles, Js = Csvs. According to irreversible thermodynamics, the driving force of surface diffusion is the gradient of the chemical potential, μs, of the adsorbed particles. Hence, the flux satisfies the Fickian equationJs=-Ls

Method 1: The differential adsorption bed (DAB) method

In DAB experiments one measures the dependence of the fractional uptake, Fup, on time. Fup at a time t is defined as the amount adsorbed at t divided by the amount adsorbed at equilibrium; the uptake is initially equal to 0 and reaches 1 at final equilibrium. The DAB method was proposed by Mayfield and Do [54] and some of its latest application can be found in [55], [56]. The method basically operates as follows.

An adsorbate is let proceed to an adsorption/collection section of an apparatus.

Method 2: The time lag method

The origins of the method go back to the works of Daynes [65] and Barrer [66]. In this method a pellet (or an ensemble of parallally placed pellets) is situated between two chambers, one of which (the upstream chamber) is filled with an adsorbate particles and the other (the downstream chamber) is initially isolated from the pellet and is empty (evacuated). The filled chamber is, at an initial time, opened and the diffusion of adsorbate particles through the pellet begins. For some time, the

Method 3: The constant molar flow (CMF) method

The CMF method was proposed by Do [80] and later applied to experiments by Prasetyo and Do [81] and Do et al. [82], and Park [83], [84], [85], [86], for example. A CMF rig consists of a supply reservoir and a sample cell (the adsorption section) separated by a variable leak valve that very accurately controls the flow of an adsorbate from the reservoir into the adsorber in the adsorption section. At an initial time a constant molar flow, ν, of adsorbate is introduced. The bulk concentration, Cb

Method 4: The differential permeation (DP) method

The DP method [10], [93], [94], [95], [96], [97], [98] was originally developed to study the transport of gases, and it is thus presented here (its related versions and/or recent applications can be found in [99], [100], [101], [102], for example). The theoretical models that have been proposed to describe the kinetics associated with the method are rather rich: besides pore and surface diffusions, they involve such transport mechanisms as viscous flow, capillary condensate flow, and

Brief discussion of selected experimental results

In Sections 4 Method 1: The differential adsorption bed (DAB) method, 5 Method 2: The time lag method, 6 Method 3: The constant molar flow (CMF) method, 7 Method 4: The differential permeation (DP) method we provided results on the surface diffusivity at zero loading Ds0 for the adsorption of selected gaseous species on activated carbon. The results are summarized in Table 1. Several conclusions may be drawn from the table.

The values of Ds0 at temperatures ranging from 273 to 303 K have the

Conclusions

In this review we considered four methods for the study of surface diffusion of particles in porous media: the differential adsorption bed (DAB), time lag, constant molar flow (CMF), and differential permeation (DP) methods. The kinetic models associated with each method were presented. We pointed out that fitting the models results to experimental data, values of the pore diffusivity Dp and the surface diffusivity Ds0 can be obtained. When comparing the methods, various aspects may be taken

Acknowledgements

This research was supported by the Ministry of Education, Youth and Sports of the Czech Republic, under the project No. MSM: 6840770031, the VEGA under Grant No. 1/0302/09, and the FCVV Initiative under Grant No. I-06-399-01.

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