[membrane science and technology] fundamentals of inorganic membrane science and technology volume 4...

Fundamentals of Inorganic Membrane Science and Technology Edited by A.I. Burggraaf and L. Cot

�9 1996, Elsevier Science B.V. All rights reserved

Chapter 4

Methods for the characterisation of porous structure in m e m b r a n e materials

A. Julbe and J.D.F. Ramsay

Laboratoire des Mat6riaux et Proc6d6s Membranaires (UMR 5635 CNRS- ENSC-UMII), Ecole Nationale Sup6rieure de Chimie de Montpellier, 8 Rue de I'Ecole Normale, 34053 Montpellier cedex 1, France

4.1 GENERAL INTRODUCTION

With the development of more complex and sophisticated inorganic membranes there is a need for a better understanding of membrane structures and their influence on the mechanisms of separation processes. This requirement for a better insight into the relationships between (a) the membrane synthesis route, (b) the membrane microstructure or morphological properties and (c) the permeation properties, has been widely emphasised in the literature. Informa- tion on membrane characteristics is essential for membrane users, manufacturers and scientists to choose an appropriate membrane for a specific application, controlling membrane quality and preparation process parameters or understanding transport mechanisms.

In this section these relationships will be explored in more detail with particular emphasis on the porous properties of membranes and their characterisation. Firstly we will present the general definitions and terminology used to describe porous media. The origin of porosity in inorganic materials will also be outlined and related to a quantitative description of pore structures in

~ [ M E M B R A N E PREPARATION I . . . . , ,,

[ C H A R A C T E R I S A T I O N OF THE POROUS S T R u c T U R E I

68 4 - - M E T H O D S F O R T H E C H A R A C T E R I S A T I O N O F P O R O U S S T R U C T U R E

Morphology related parameters (active and inactive pores)

- Pore size distribution - Specific surface area - Porosity - (Pore shape information ?)...

Morphology and Permation related parameters (active pores only) - Active pore size distribution - Cut-offvalue - Bubble point - Fluid flow. hydraulic pore radius

. ! , O, ONOF,E O ANCE I "

Fig. 4.1. Methodology for membrane characterisafion. Listing methods and related parameters.

idealised model systems. In such model systems, the pore geometry can be defined precisely in terms of pore size, shape, connectivity, etc. This has provided the basis for recent theoretical developments describing diffusion and transport processes in such porous materials. This link between the concept of a model porous structure and the theoretical prediction of diffusivity and mass transport is also crucial in the characterisation of porous materials. Thus each characterisafion technique yields experimental parameters which are related to the pore structure of a material; these parameters are then generally used to define the porous properties on the basis of an assumed model pore structure. This aspect will be emphasised in the description of the different characterisation methods described.

In the characterisafion of membrane materials, it is important to distinguish "static" characterisation techniques leading to morphology related parameters and "dynamic" techniques in which membrane permeability is involved, leading to permeation related parameters and in some cases to morphology related parameters concerning active pores only [1]. Figure 4.1 shows a list of static and dynamic characterisation methods and the corresponding characteristic parameters which can be obtained. There are already in the literature several book chapters and reviews which list a range of characterisation techniques for membranes [1-4] but the selection of the appropriate characterisation techniques is not always obvious. A recent IUPAC report entitled "Recommenda- tions for the characterisation of porous solids" [5] has stressed conclusions of particular significance, which include the following:

4 - - METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 6 9

1. The selection of a method of characterisation must start from the material and from its intended use.

2. The method chosen must assess a parameter "related as directly as possible" to phenomena involved in the application for which the material is used.

3. The complexity of the porous texture of materials is such that, even on theoretical grounds, the concepts which can be used to describe the texture usually entail the introduction of simplifying assumptions.

4. No experimental method provides the absolute value of parameters such as porosity, surface area, pore size, etc.; each gives a characteristic value which depends on the principles involved and the probe used.

5. As a consequence, "perfect agreement" between parameters provided by different methods should not be sought. Instead, there must be an awareness of the specific, limited and complimentary significance of the information deliv- ered by each method of characterisation of a porous solid.

The choice of a characterisation technique is also dictated by the particular sample characteristics such as the nature of the material, whether it is supported or not, its size, shape, isotropy and mechanical resistance as well as the range of pore size. The destructive nature of the technique may also require attention. Thus careful preparation of the sample is often needed (drying, outgassing, cleaning) to eliminate adsorbed species like water or hydrocarbons especially in the case of microporous materials.

Finally it should be noted that the characterisation of membranes is more demanding than most other porous materials. Firstly, the membranes separation layer is generally thin and supported, which requires a sensitive technique capable of analysing a sample in such a form. The characterisation of a powder "equivalent" to the membrane cannot in all cases be considered as ~representative of the membrane texture. Secondly, the structure is frequently anisotropic and moreover often microporous. Assessment of the microporosity is much less advanced compared to meso- and macro-porosity, despite emphasis given to this in the recent IUPAC symposia [6--8]. The current and widespread interest in the characterisation of microporous materials is well illustrated by the numer- ous and varied publications found in these symposia proceedings. These high- light recent developments in characterisation techniques, their applications and limitations. The particular features of importance in membrane studies will be considered in the light of the characterisation techniques to be described.

4.2 D E S C R I P T I O N OF P O R O U S M A T E R I A L S - - D E F I N I T I O N S

4.2.1 Origin of Pore Structure

Porous materials can be formed in several ways, although the following three are most important in the synthesis of membrane structures:

7 0 4 m METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

(1) In the first, the pores may be an inherent feature of crystalline structures (e.g. zeolites, clay minerals). Such intracrystalline pores are generally of molecular dimensions and result in very regular networks often described as "structural" porosity.

(2) Secondly, the pores may be formed by the packing and subsequent consolidation of small particles as may occur in some inorganic gels and ceramics. These processes have been defined (IUPAC) as 'constitutive', because the final structure depends on the original arrangement of the primary particles and on their size.

(3) The third route is defined as"substractive' (IUPAC), in that certain elements of an original structure are selectively removed to create pores. Exam- ples include the formation of porous metal oxides by thermal decomposition of hydroxides, of porous glasses by chemical etching, of activated carbons by controlled pyrolysis, of ceramic foam membranes by burning off a polymer (e.g. polyurethane), of alumina by anodic oxidation of aluminium to give oriented cylindrical pores with a narrow size distribution.

4.2.2 Quantitative Description of Pore Structures

It is useful at this stage to define pore structure and terminology. Here again we m a i ~ y follow the recommendations for the characterisation of porous solids recently proposed by IUPAC [5,9].

Porosity is defined as the fraction r of the total volume of the sample which is attributed to the pores detected by the method used

~= Vp/V (4.1)

The value of this fraction depends on the method used to determine the apparent volume V (geometrical, fluid displacement) and on that used to assess the pore volume Vp. We note some methods (e.g. methods using a gas or a fluid) only have access to "open pores" (through pores and /or blind pores) whereas others may have also access to "closed pores" (e.g. methods using radiation scattering). Evidently for membrane characterisation, methods which distinguish open pores are of particular importance. Furthermore, for any given method, ~ will depend on the size of the molecular probe (fluid displacement, adsorption) or of the scale of measurement (stereology). This latter concept is developed extensively in the theory and methods involving fractal analysis. We must also note the concept of inaccessible or "latent pores" [10] corresponding to open pores with a pore width smaller than the probe molecular size. The determination of true density, apparent density and He-replacement density can be used to determine both open and closed porosity of bulk crystalline materials [10] but in the case of supported layers it is not as simple. Image analysis is commonly used to estimate the porosity of unsupported thin films,

4 - - METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 71

but it must be appreciated that bulk membrane porosity and surface porosity of a porous medium differ markedly.

The specific surface area is defined as the accessible area of solid surface per unit mass of material. It is also dependant on the method employed and the size of the probe used (e.g. adsorbate, molecular probe, wavelength of radiation, etc.). The recorded value will also depend on the assumptions inherent in the simplified models applied to interpret experimental data. This consideration is of particular significance for materials containing micropores.

The pore size, or more precisely pore width, is the distance between two opposite walls of the pore. Pore size which is a property of the utmost importance in membrane applications, is even less susceptible to precise definition as already discussed by IUPAC [5]. The problems encountered for the determination of the surface area are further complicated by the fact that the pore shape is generally highly irregular, leading to a variety of definitions of "the size". Moreover, porous systems generally consist of interconnected networks, and results will depend on the sequence in which pores are accessed by the method. It is partly for this reason that quantitative descriptions of pore systems are generally based on model structures. The following classification of pore sizes based on the average width of the pores has been recommended by IUPAC [9]:

Macropores: width > 50 nm (0.05 ~tm) Mesopores: width between 50 and 2 nm Micropores: width < 2 nm (supermicropores 0.7 nm, ultramicropores < 0.7 nm) This definition is based on different physical adsorption phenomena of gases

in pores of different size. Adsorption interactions of adsorbates are stronger in micropores and modify the bulk properties (density, surface tension) of the adsorbed fluids. The maximum size of ultramicropores corresponds to the bilayer thickness of nitrogen molecules adsorbed on a solid surface (2 x 0.354 nm).

Usually the pores in a material do not have the same size but exist as a distribution of size which can be wide or sharp. We can characterise a film by a nominal or an absolute pore size. In fact this definition rather characterises the size of the particles or molecules retained by the layer. Pore size distribution is classically represented by the derivatives dSp/drp or d Vp/drp as a function of rp (pore radius) where Sp and Vp are respectively the wall area and volume of the pores. The size in question is here the radius, which implies that the pores are known to be, or assumed to be, cylindrical. In other cases, rp should be replaced by the width.

The mean pore hydraulic radius r H for a porous solid is obtained through the relationship

rH = Vp/S (4.2)

where Vp is the pore volume determined at saturation and S the surface area, e.g. determined by the BET method, rH can be theoretically related to the sample

7 2 4 - - METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

mean pore width w by means of the relation

w = 2rl r H (4.3)

in which 11 is a pore shape factor (11 =1, 1.4, 2, 3 respectively for slit shaped pores, voids between randomly-packed spheres, cylindrical pores and spherical pores) [11].

Tortuosityfactor or tortuosity ~ was first introduced by Carman in 1937 [12,13] by reference to a direction that corresponds to a given macroscopic flow. It was defined as the square of the ratio of the "effective average path length" in the porous medium L e to the shortest distance L measured along the direction of macroscopic flow

"r (ae/a) 2 (4.4)

For cylindrical perpendicular pores (case of Anotec | membranes), the tortuosity is equal to unity. For more complex porous structures the tortuosity is usually higher [14]. The Kozeny-Carman equation describing the laminar flow through porous media can be used to calculate the tortuosity of the membrane, by introducing in the equation the membrane thickness, permeability, internal surface area, pore volume fraction and pore shape. Due to variabilities of the parameters involved, this parameter is usually not used to describe a membrane pore structure, but has been mainly introduced to correct the calculated permeability and obtain agreement with the experimental value. This parameter, reflecting the complexity of porous media (random orientation of the pores in the material), is not clearly defined in the IUPAC reports. Other definitions of tortuosity can be given such as those found in [13] for geometric tortuosity or in [15] relating tortuosity to diffusion coefficients and porosity ~. In this latter case tortuosity is expressed by

"r, = EDp/ Def f (4.5)

where Dp is the diffusion coefficient in a pore (cm2/s) and Deff is the effective diffusion coefficient (cm2/s).

In some specific cases other parameters can be considered as important in the characterisation of membrane morphology like the surface roughness, pore anisotropy and porous network connectivity [16,17]. Concepts of percolation and fractal geometry are also of interest to better describe the statistical and random structures of many porous solids [14,18,19].

4.2.3 Models for Porous Structures

One important, but often not clearly defined variable in the characterisation of porous layers, is the shape (or geometry) of the pores. In order to interpret the characterisation results and relate pore size to physical models, it is often

4 - - M E T H O D S F O R T H E C H A R A C T E R I S A T I O N OF P O R O U S STRUCTURE 73

al a2 a3

, , 2 . : ' . . :. '.: : .. ~i . r,. , . . .

~ 7 . ; ~ : ~ ~ " ~ ~,,~:, .~,.~ .... � 9 .. �9 ~ ~ , , , ~ . ; . .

b c

" ":';~?~ ~":'~ !~";':~ ~: ' " ": "

Fig. 4.2. Some idealised pore structures: (a) cylindrical pores (al,a2: parallel non-intersecthlg capillaries; a~: z = 1; a2:1: > 1; a3: non-parallel intersecting capillaries; (b) slit-shaped pores; (c) voids between packed spheres; (d) closed pores; (e) blind pores; (f) open pores; (g) funnel; (h) ink bottle;

(i) pores with constrictions.

essential to make assumptions about the pore geometry (Fig. 4.2). Classical simple descriptions of pores involve cylindrical, slit or spherical shapes. Simple pore shape geometry can be considered close to reality in some specific cases such as, for example, Anotec | alumina membranes (cylinders), some zeolites (prisms, cavities, windows, etc.), clays and activated carbons (slits). However in most ceramic membranes resulting from oxide sintering pores are voids left between packed particles, they have neither a regular shape, nor regular size and contain constrictions. This kind of material can contain closed pores, blind pores (open only at one end) or through pores (open at two ends). The concept of stochastic geometry of pores has been applied in [20] to simulate by statistical models, the structure of an Anotec | cellular ceramic membrane, a sintered ceramic membrane and a foam membrane. The reported agreement between theory and experiments (laminar flow of pure solvent and during fouling) is very good. A corrugated random pore model has been developed in [21] to predict hysteresis loops and Hg entrapment in mercury porosimetry measurements. Many other elaborate models of membrane structure are described in the literature and a major part is reported in [22].

74 4 - - M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE

In addition, it is not the average pore size which is the determining factor in membrane performance, but the smallest constriction in the porous medium. Indeed some characterisation techniques determine the dimension of the pore entrance rather than the pore size. Such techniques often provide better information about permeation related characteristics, provided that through pores are concerned.

4.3 STATIC CHARACTERISATION TECHNIQUES

4.3.1 Stereology

Microscopy can provide visual details of the membrane surface and cross- section morphology. Image analysis of micrographs can give quantitative data from these direct observations.

The resolution of optical microscopy, usually performed at a magnification of 500-1000 is often sufficient to observe large defects at the membrane surface but higher resolution is needed for the observation of fine texture. Scanning electron microscope (SEM) generates electron beams and forms an image from the emit- ted electrons as a result of interaction between the bombarding electrons and the atoms of the specimen. Since electrons have a much shorter wavelength than light photons, higher resolution information can be obtained from SEM than from optical microscopy. SEM is now routinely used to obtain magnifications of 105; the resolution can reach 5 nm. Field emission scanning electron microscopy (FESEM) enable the SEM resolution to be decreased to around 1.5 nm. This improvement is mainly due to the electron source (cold cathode instead of thermoionic source).With the latter technique, samples can be observed at low accelerating voltage and with a lighter me ta l l i za t ion - which is important in the case of fragile membrane structures. Figure 4.3 compares the SEM and FESEM micrographs of mesoporous and microporous silica membranes.

The best transmission electron microscopes (TEM) have about I nm resolution. However, because of limitations in sample preparation, frequently the interpretation is restricted to about 10 nm. In the TEM, the electrons that form the image must go through the specimen, which limits the thickness of the sample. The technique is thus only able to analyse unsupported thin membranes or stripped membrane layers. High resolution transmission electron microscopy (HRTEM), reaching a resolution of 0.3 nm, enables the organisation of crystalline planes of atoms to be observed. A new approach to HRTEM is the Z-contrast scanning transmission electron microscopy (STEM) which provides a direct image of material atomic structure and composition. The intensity or brightness of the image is proportional to the square of the atomic number (Z) [23].

Atomic force and scanning tunnelling microscopies have recently revolutionised

4 ~ M E T H O D S F O R T H E C H A R A C T E R I S A T I O N O F P O R O U S S T R U C T U R E 75

SEM FESEM ~ " ~ : :"- " ~ ,T~r ~-: *~-7". ~ ~ . ~ , * ~ : r :

~ ~ ; ~ . ~ . ~ , . ~ '

. . , . . . . . .

�9 ' " ~ " ' " ' 4 ! ,

I ~ ~ ~ ~ i

. ~.~;,~;,~: . .,,. . . . .

~ ,4.~:,.,, ~, , . . . .

�9 ,, ~ ]~ , .~

, , ~y~ , ; : -~ , ' .~ .~ , .~ . , , , , ,

~ . . . . . . . . . . . . . . . . . . . "" , ~ : , ~ ' ~ - ~ , ~ - . ,~ ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 4.3. Images of silica membrane surfaces obtained by SEM and FESEM: (a) mesoporous membrane; (b) microporous membrane.

the study of surface structures. These techniques, collectively called scanning probe microscopies, are capable of imaging angstrom to micron sized surface features. They are non-destructive and require little sample pre-treatment. The basic physical phenomenon of scanning tunnelling electron microscopy (STEM) is the "tunnelling effect of electrons in vacuum". A very fine metallic tip-probe is placed at a few atomic distances (d - 5-10/~) of the surface studied, in order to allow a slight recovering of the electronic state functions of the probe and of the sample surface (Fig. 4.4a). A potential difference V (a few mV to a few V) is applied between these electrodes and an electronic current can then go from the probe to the surface (or in reverse, depending of V polarity). These electrons, by the tunnelling effect, get over the potential barrier due to the existing vacuum between the tip and the analysed surface (Fig. 4.4b). By similarity with the tunnelling effect between two flat electrodes, it is possible to estimate the tunnel current intensity I and to predict its variation with the distance probe/sample [24]. At low polarisation tension limit V and low temperature

76 4 - - METHODS FOR THE C H A R A C r E R I S A T I O N OF POROUS STRUCTURE

o a,

v ! ....::.. . . . .

d b) ENERGY

EF

SAMPLE TIP

Fig. 4.4. Physical principle of tunnelling microscopy [24].

I = V exp(-2h -1 q2m~ d) (4.6)

where ~) is the potential barrier due to vacuum. For a typical value ~) = 4eV, Eq. (4.6) establishes that I decreases with one order of magnitude for an increase of only I A in the distance probe/sample d. From this extreme sensitivity, the high resolution of the microscope perpendicular to the studied surface is obtained. Lateral resolution depends on the dimensions of the probe used. The samples used must be smooth on a microscopic scale and, up to now, have a regular structure to allow interpretation of the results. In addition, STEM samples must be conductive. Pores on a nanometer scale can be observed, although the interpretation of the results remain the most difficult part. Atomic force microscopy (AFM), developed in 1986, allows the direct observation of non-conductive materials and is attractive for membrane surface examination in air or even under liquids. There is no need for any specific sample preparation before examination. A diamond or tungsten or quartz tip placed at the end of a flexible lever (cantilever), applies a constant force (<< to typical bonding energy) on the atoms of the surface studied. The surface is displaced under the tip and, as the interaction force between tip and surface varies from place to place, a lever deflection is observed. This extremely weak deflection is measured by STEM or by laser interferometry. An image of the interacting forces between the tip and the surface is obtained by measuring the cantilever deflection as a function of


the lateral tip position. The application of this technique to the membrane field has mainly been confined to the study of polymeric membranes [25,26] but the technique has also been adapted to the characterisation (particle size, shape and surface roughness) of ultrafiltration and microfiltration inorganic membranes [27].

Several other techniques referred to as microscopy and based on several different phenomena can be found in the literature. These include acoustic microscopy based on the interactions of acoustic waves with materials [28] the projection microscopy which is still under development and gives a hologram image of the sample illuminated by a beam of low energy electrons [29]. For membrane applications a scanning electrochemical microscope has been developed based on the measurement of the local flux of electroactive ions across the membrane. The ability to detect ~ 1 ~tm radius pores separated by 50-100 ~tm has been demonstrated with mica membranes [30].

Image analysis can be performed on sample micrographs to obtain pore density, pore size, pore area and porosity. The analyser used is generally a video camera linked with a computer, allowing enhancement of the image by adjustment of the contrast and colour overlays of the features (such as pores) prior to measurement. Because some subjectivity is involved in adjusting the contrast and extent of overlay, a statistical analysis of replicated measurements is required. Mean and standard deviations of the populations are necessary to calculate the confidence interval and adequately estimate the sample surface characteristics. Quantitative image analysis is a useful technique which enables reliable numerical data to be obtained from sample surface micrographs [31-33] or digitised negatives [34]. A fractal analysis, studying the behaviour of a geometric parameter over a range of scales of observations has been proposed [35] from SEM images of silica membranes. The principal advantage of quantitative microscopy as a tool for characterising macropores in porous solids is the ability to measure both open and closed porosity and the shape, location and orientation of pores but only two dimensional measurements can be performed [36]. Image analysis is well adapted to estimate the porosity of a supported membrane, this parameter is often difficult to obtain by other techniques. Up to now this method has been applied to a large extent to study the porous texture of organic membranes.

Nuclear magnetic resonance imaging has become a powerful tool in medical diagnostics and can also be applied to a variety of ceramic problems, including the imaging of liquids in pores. The simplest information obtained is the spatial variation of porosity which can be obtained by careful selection of the pulse sequence and delays in the imaging experiment. This technique needs signifi- cant expertise to obtain reliable data [37]. The dispersion of paramagnetic tracers in porous media can also be studied by NMR imaging and give access to the 3D tracer distribution [38].

78 4 m METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

4.3.2 Intrusive Methods

4.3.2.1 Mercury porosimetry

In this method, mercury (which is a non-wetting liquid) is forced into the pores of a dry sample. For each applied pressure, the volume of mercury entering the sample porous structure is determined very accurately (e.g. by measuring the variation of capacity induced by the reduction in height of the Hg column connected to the measuring cell). The relationship between pressure P and pore radius rp is given by the modified Laplace equation (Washburn equation)

rp = - 27 cos0/P (4.7)

where rp is the radius of a capillary shaped pore and 7 the surface tension at the liquid/air interface. As mercury does not wet the membrane, the contact angle 0 is greater than 90 ~ and cos0 has a negative value. A widely accepted value for oxides is 0 H g / o x i d e = 140 ~ and ~r lg /a i r = 0.48 N/m. These values can vary with the experimental conditions (sample material, temperature) and may affect the results. Very high pressures are needed for pores in the nanometer range. Indeed a pore radius of 1.5 nm corresponds to a pressure of about 450 MPa, which may damage the ceramic layer. Typical results for a commercial tubular 0~A1203 asymmetric support (SCT-US Filter) are given in Fig. 4.5.

With thin supported ceramic membranes, the pore volume due to the membrane is relatively small and better results are obtained if a major part of the support is scraped off. Specific preparation of samples (e.g. support embedded in a resin) can change the results [39]. If the membrane weight is known and if its pore size can be well differentiated from that of the support the method can be used to determine the porosity of a supported layer.

In parallel with mercury porosimetry in which a non wetting liquid is used, we can mention the suction porosimetry in which a wetting liquid like water (0 _< 0 < ~/2) is held within the porous solid [5]. In this case the Laplace equation predicts that it will experience a reduced hydrostatic pressure, inversely proportional to the radius of pores in which menisci are formed. The lower limit of pore size accessible to this technique is around a few tens of microns.

4.3.2.2 Gas adsorption/desorption isotherms (physisorption)

This technique is one of the most important and extensively used methods in the characterisation (porous volume, specific surface area and pore size distribution) of porous inorganic materials [40,41]. Nevertheless, real solid/gas interfaces are complex, leading to uncertainties in the assumptions made, and different mechanisms may contribute to physisorption (e.g. monolayer-multilayer ad-


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Diameter (/an) Fig. 4.5. M e r c u r y po ros ime t ry analysis of a commercia l (SCT-US Filter) tubular 0~A1203 asymmet r i c

m e m b r a n e suppor t (Micromeritics ASAP 2000). (a) Cumula t ive intrusion vo lume as a function of the

appl ied p r e s s u r e / p o r e diameter; (b) differential intrusion vo lume as a function of the pore diameter .

sorption, capillary condensation or micropore filling). Consequently attention has to be paid to the problems and ambiguities arising in connection with the reporting and interpretation of physisorption data. Details of definitions and methodology on this particular technique is given in Ref. [9].


The adsorption and desorption isotherms of an inert gas (classically N 2 at 77 K) on an outgassed sample are determined as a function of the relative pressure (Prel = P/Po, i.e. the ratio between the applied pressure and the saturation pressure. The adsorption isotherm is determined by measuring the quantity of gas adsorbed for each value of p/po by a gravimetric or a volumetric method (less accurate but simpler). A surface acoustic wave device can also be used as a mass sensor or microbalance in order to determine the adsorption isotherms of small thin films samples (only 0.2 c m 2 of sample are required in the cell) [42,43].

The adsorption isotherm starts at a low relative pressure. At a certain minimum pressure, the smallest pores Will be filled with liquid nitrogen. As the pressure is increased still further, larger pores will be filled and near the saturation pressure, all the pores are filled. The total pore volume is determined by the quantity of gas adsorbed near the saturation pressure. Desorption occurs when the pressure is decreased from the saturation pressure. The majority of physisorption isotherms may be grouped into six types [9]. Due to capillary condensation, many mesoporous systems exhibit a distinct adsorption--desorption behaviour which leads to characteristic hysteresis loops (Type IV and V isotherms) whose shape is related to pore shape. Type I isotherms, characterised by a plateau at high partial pressure, are characteristic of microporous samples. A typical isotherm, representative of a mesoporous sample is given in Fig. 4.6, with a schematic representation of the adsorption steps.

Mesopore size calculations are usually made with the aid of the Kelvin equation in the form

ln(p/po) = -fyV/rKRT (4.8)

wi thf is the geometrical factor depending on the shape of the meniscus formed by the liquid in the capillary (f = 1 for slit-shaped pores, f = 2 for cylindrical pores); I is the surface tension of the liquid condensate at the absolute temperature T; V is the molar volume of the liquid at the absolute temperature T; and rK the Kelvin radius, dimension characteristic of the capillary (radius of a cylinder or slit width).

The pore radius rp of a cylindrical pore may be calculated from

rp = r K + t (4.9)

where t is the thickness of the adsorbed layer of vapour in the pores, before capillary condensation occurs (t is estimated from calibration curves with similar non-porous solids). For a parallel-sided slit, the slit width Wp is given by

W p = r K + 2t (4.10)

The Kelvin equation is useful to calculate the distributions of pore volume and area as a function of pore diameter in the mesoporous range (isotherms of Type IV). This can be performed by the BJH (Barret, Joyner and Hallenda)

4 -- METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 81

IV

DESORPTION

C II and III , . . .

Relative pressure (p/p o) 1

" ' [ ' " . . . . . v . v : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I .1_. 2rp II

I l l ........................................................................................................ i i[ i2rK [.:+:.:.:.:.:.:.: :.:.:.:.:,-.:.-.-.:.-,--.-.:.-.:.:.:.:.:.:.:.:.:,:.:.:.!.k.:,:.:.:.:.:.:.:.:':.:,: |

I V

Fig. 4.6. A typical adsorption-desorption isotherm for a mesoporous sample showing corresponding steps of adsorption [1]" rp: pore radius; rK: Kelvin radius; t: t-layer thickness.

method which considers opened cylindrical pores and may be applied to the desorption branch. Nevertheless, the Kelvin equation is based on thermodynamic considerations which are not valid for micropores and are only valid for larger mesopores [41,44]. Thus it is claimed that Kelvin type adsorption models overestimate micropore filling pressures and are unreliable for pore size distribution determinations below 75 A [45]. Nevertheless the method has recently been applied to considerably lower limits (1.3-1.7 nm) [10] which highlights current uncertainty and the complexity which still exists with this problem.

The oversimplified BET (Brunauer, Emmet and Teller) theory, valid for relative pressures between 0.05 and 0.35, allows the calculation of the specific surface area of solids and the estimation of the interactions between the solid and the vapour (from the value of the constant c). The BET equation is mainly applicable for Type II and IV isotherms. The specific surface area deduced from Type I isotherms has no physical meaning because the notion of a monolayer is not applicable in the case of micropores [9,41,46,47].

82 4 - - METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

Case of microporous systems The classical methods for interpreting adsorption data rely on equations that

are of uncertain validity for micropores and small mesopores mainly because of increased adsorbate-adsorbate and adsorbate-adsorbent interactions. Many papers in the literature deal with this problem of characterisation of micropores from adsorption experiments [6-8,10].

Several methods have been proposed to determine the microporous volume accessible to a given gas, namely the t-plot, the nonane preadsorption technique and the Dubinin-Radushkevich equation. The simplest technique to determine a sample microporous volume is the t-method of de Boer and its extensions. This method compares the isotherms of porous solids with standard isotherms obtained with non-porous reference solids of similar composition. In the original t-method, the amount adsorbed is plotted against t which is the multilayer thickness calculated from the standard isotherm. Any deviation in shape of the given isotherm from that of the standard is detected as a deviation of the "t-plot" from linearity. For the assessment of microporosity, the thickness of the multilayer is irrelevant and Sing proposed that this is preferably replaced by c~ s = na/n~ where n as is the amount adsorbed at a fixed P/Po = s. [9,48]. Usually s is set at 0.4 (which is the limit between meso and micropores). Figure 4.7 shows typical types of t-plot and ms plots [10]. The nonane preadsorption technique consists in measuring the adsorption before and after micropore blocking with nonane which is held inside the pores even after degassing at room temperature [49]. The Dubinin methods are based on the theory of micropore volume filling [50,51]. The total microporous volume accessible to a given adsorbate Wo can be obtained from the DR equation

v ( cm~/g ) M i c r ~ 1 7 6 1 7 6 �9

Microporous "':'i ....... :'i" ! volume ' i; ..~...,..' �9 " " " : . . . ' " " �9 i

t (A) 20

Fig. 4.7. Typical t-plot obtained from adsorption isotherms on mesoporous, microporous and non- porous samples.

4 m M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE 83

W = W o exp [-(A / ~Eo) 2] (4.11)

where A = RT ln(po/p) is the adsorption potential and Eo and [~ are specific parameters of the system studied (viz energy of adsorption and affinity coefficient) and which need to be known.

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r H, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores!). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slit- shaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution.

An alternative method proposed by Horvath and Kawazoe for slit shaped pores [53] and recently extended to cylindrical pores [54], is to calculate an average potential function inside the micropores, relating the average fluid- fluid and solid-fluid interaction energy of an adsorbed molecule to its free energy change upon adsorption. This method is an improvement over the Kelvin approach in that it acknowledges the strong fluid-solid attractive forces in micropores, but does not take into account pore wetting [45]. Another method, based upon an improved molecular description of adsorption (modern statistical mechanics) using a nonlocal Density Functional Theory (DFT) [55], has also been applied to derive pore size distributions in activated carbons (slit- shaped pores). The method leading to pore size distributions in the whole range of micro to mesopores is being extended to cylindrical pore geometries for modelling silica and zeolites.

It must be concluded that the quantitative determination of micropore size is still an ambiguous problem: new theories, models, mechanisms and simula- tions are still under study [56-58]. Therefore isotherm interpretations must be used carefully and can be considered as useful mainly for qualitative studies. No reliable method has been developed for the determination of the micropore size distribution. At present the most promising approach appears to be that of pre-adsorption linked with the use of various probe molecules of known size and shape [59--61]. For example, this approach has been applied successfully for silica compacts characterisation in [61] using spherical symmetrical inert molecules, such as neopentane and trimethylsiloxysilane [(CH3)3SiO]4Si with diameters of-~6.5 and 11.5 A respectively. In general the limited availability of volatile probe molecules with diameters extending above 10 ~ puts a restriction on the applicability of this method. Furthermore effective pore sizes determined by this technique depend on the kinetic and thermodynamic properties of the


adsorbate in the porous material (rate of diffusion, total porous volume, electrostatic interactions, etc.).

To make further progress it will be necessary to employ well defined microporous and small pore mesoporous adsorbents and have available non-porous reference solids of the same surface structure. This will be possible with the current progress in material synthesis procedures (e.g. organised amorphous structures obtained in the presence of templating agents [62,63]. We have to note that zeolites are often not suitable reference materials because the strong electrostatic fields within the crystalline cavities can polarise the gas molecules. Therefore the adsorption equilibrium in these materials is not solely a function of the size of the adsorbate.

The analysis of thin supported films by N2 adsorption is often difficult due to the very small percentage of pore volume contributed by the thin layer relative to that of the support. Usually it is necessary to scrape off most of the bulk support layer to increase the pore volume percentage of the thin film. Recent technical improvements in pressure sensors on commercial apparatus (reaching now a sensitivity of 5.10 -s mmHg) or new sophisticated detection techniques using surface acoustic waves may in some cases solve this problem.

4.3.2.3 Calorimetric Determinations

Immersion calorimetry The measurement of the heat of immersion of a "dry" material in different

liquids can permit a rapid and accurate determination of the surface area and pore size distribution below 10 A. The enthalpy change is related to the extent of the solid surface, to the presence of micropores and to the chemical and structural nature of the surface. The technique has been mainly applied to carbons [64]. The immersion liquid is usually water for hydrophilic oxides like mineral oxides, or an organic liquid (benzene, n-hexane) for hydrophobic solids like carbons. One of the limitations of this technique is that the specific enthalpy of immersion of the open surface must be determined with a non-porous standard material of surface composition similar to the porous solid studied. The non-microporous part of the surface area can be determined by prefilling the micropores with an absorbate prior to immersion. Information on the size of micropores can be obtained from the kinetics and enthalpy of immersion into a set of liquids with increasing molecular size [5].

Thermoporometry Thermoporometry is a thermal method which is based on the thermal analy-

sis of the liquid-solid phase transformation of a capillary condensate held inside the porous body under study. The technique was developed by Brunet


al. in the 1970s [65] and is based on the principle that the equilibrium state of the solid, liquid and gaseous phases of a highly dispersed pure substance is determined by the curvature of the interfaces. In a liquid filled porous material, the solidification temperature of the liquid depends only on the liquid/solid interface (which is always almost spherical) in the pore [66,67]. The smallest size of a stable crystal (R~) is inversely proportional to the solidification temperature depression

AT = T - To (4.12)

where To is the normal phase transition temperature of the liquid and T is the temperature where the phase transition is actually observed when this liquid is contained in the pores. In finely porous materials, the liquid therefore crystal- lises or melts at the temperature where the pore radius r equals Re + t, where t is the thickness of a liquid-like film adhering to the solid matrix of the porous material and which does not undergo a change of state [68]. The melting or solidification thermogram can be monitored in a differential scanning calorime- ter (DSC). The exothermic heat effect measured from the solidification of a liquid in a porous medium is schematically shown in Fig. 4.8. From the solidification thermogram it is possible to determine:

- the pore radius distribution (between 1.5 and 150 nm), from the depression of the solidification temperature, AT, due to the Gibbs-Thompson effect,

- the pore volume, from the energy Wa evolved in the phase transformation. This determination must take into account that the heat of the phase transformation i sa function of freezing point depression,

- the pore surface, from the simultaneous measurements of AT and Wa.

1

' i / . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . I ~ . . . . . . . . . . . . . . . . . ! i I I

0 T(~

Fig. 4.8. Schematic il lustration of the exothermic heat effect from the solidification of a l iquid in a

p o r o u s mater ia l as a funct ion of t empera ture . (A) N a r r o w pore size distribution; (B) broad pore

size distr ibution [1].

86 4 w METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

The relation between pore size r(nm) and the extent of temperature depression AT (Fig. 4.8) is obtained from the equations derived by Brun from Gibbs- Duhem and Laplace equations. For cylindrical pores with water inside (0 < AT < -40 and t = 0.8 nm), it leads to

during solidification: r(nm) = (-64.67/AT) + 0.57 (4.13)

during melting: r(nm) = (-32.33/AT) + 0.68 (4.14)

In the same way, the apparent energy of solidification Wa can be measured and related to AT by two other equations. The corresponding pore volume d V between T and T + AT is given by

dV=dW/PWa (4.15)

where p is the density of water corresponding to AT. By combining the various equations, a pore volume distribution as a function of pore radius can then be derived from the thermograms [65].

The difference between the solidification and melting temperatures in cylindrical pores is due to the fact that the shapes of the interfaces present during these transitions are different. In spherical shaped pores however, there is no difference and the same thermodynamic equation can be used to describe both solid -+ liquid and liquid --> solid transitions. Consequently by analysing both the melting and solidification curves, one determines a pore shape factor. In thermoporometry the shape factor for a porous material [68,69] can vary generally between I (spherical pores) and 2 (cylindrical pores).

In membrane filtration, water-filled pores are frequently encountered and consequently the liquid-solid transition of water is often used for membrane pore size analysis. Other condensates can however also be used such as benzene, hexane, decane or potassium nitrate [68]. Due to the marked curvature of the solid-liquid interface within pores, a freezing (or melting) point depression of the water (or ice) occurs. Figure 4.9a illustrates schematically the freezing of a liquid (water) in a porous medium as a function of the pore size. Solidification within a capillary pore can occur either by a mechanism of nucleation or by a progressive penetration of the liquid-solid meniscus formed at the entrance of the pore (Figure 4.9b).

Thermoporometry is a method which measures cavity sizes and not inlet sizes. It has been mainly used for the characterisation of organic mesoporous membrane texture [70-73] but has been also applied to inorganic alumina symmetric membranes [73] with a good reliability. However the solidification of water in small pores may sometimes damage the membrane structure due to the expansion of the condensate and consequently different results can be obtained after several runs [74].

4 m METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 87

r3 r l r2

• • , ~ BULK PHASE

/~/U D-LI,KE LAYER, t

. . . : . . . . . . . ~-.- .~. : ' r

:-'-:'" :" :.'-": :",":-7.:..

t PENETRATION NUCLEATION

Fig. 4.9. Schematic d rawing of the extent of depression of the solidification temperature: (a) in relation to the pore diameter; L = liquid (water); S = Solid (ice); r = pore radius (rl > r2 > r3) [2]. (b)

detail of the solidification processes [68].

4.3.2.4 Nuclear magnetic resonance

Nuclear magnetic resonance, which is sensitive to short-range order, has been recently used to obtain information on the structure of pores. Two main techniques can be found in the literature [75]" one is based on the study of NMR relaxation times of a fluid inside pores and the other on the chemical shift of 129Xe trapped in the material.

N M R spin-lattice relaxation measurements The use of the low-field NMR spin-lattice relaxation technique has recently

been successfully demonstrated [75-80] as a pore structure tool for porous materials saturated with a solvent (usually water). The basic principle is that the portion of pore fluid near a pore wall undergoes spin-lattice (T1) and spin-spin (T2) relaxation in a magnetic field at a faster rate than the bulk fluid. This, coupled with the fast diffusional exchange of fluid between regions within


"--'- Surface affected phase

4---- Bulk fluid phase

P o r e w a l l

Fig. 4.10. Schematic diagram of pore fluid during an NMR experiment [80].

a pore, yields a unique average relaxation time constant for each different geometrical environment (i.e. pore size) within a porous material. This relaxation time constant is dependent on the ratio of surface fluid to non-surface (or bulk) fluid in the pore, which is a measure of the pore hydraulic radius.

From the two-fraction fast-exchange model, the measured T 1 can be related to the pore hydraulic radius rH by

1 / T 1 = o~ + ~ / r H (4.16)

where c~ is 1/Tlbulk is determined from the spin-lattice experiment on the fluid only and ~ is a surface interaction parameter determined with partially saturated samples with different fluid contents. The physical model associated with this equation is illustrated in Fig. 4.10. The thickness of the surface affected phase is typically 0.3 + 0.1 nm.

A saturated porous medium, such as a membrane, should relax as a combination of its different geometrical regions, i.e. pores. A pore size distribution can be determined via NMR relaxation experiments if the observed relaxation curve can be deconvoluted into its component parts, and if the relationship between pore size and relaxation time constants can be determined. The technique can be applied to macro, meso and microporous samples if the pore geometry is known or assumed. Examples of pore structure characterisation of flat organic and inorganic membranes are given respectively in [77] and [76], NMR results tend to compare favourably with those of other more classical techniques The maximum size of the membrane samples to be analysed by this experimental method is limited by the homogeneity of the magnetic field (around 10 cm).

129Xe NMR

Another adaptation of the NMR technique involves the measurement of the chemical shift of 129Xe adsorbed in a sample. The recent development of this last technique has mainly been concerned with the study of the porous structure of microporous materials such as zeolites [81,82], mesoporous silica [11,83], chlat- rates [84], organic polymers and supported metal catalysts [82]. 129Xe is an inert


spherical molecule with a diameter of 0.44 nm and a 1/2 spin nucleus. Polari- sation of its spherical electronic shell, during adsorption or due to collision with other Xe atoms or with the surface of a solid, affects the NMR chemical shift. Consequently by studying the resonance line (shape and position) as a function of the quantity of adsorbed xenon, information can be obtained on the porous structure of sorbents as well as on the rate of exchange between Xe atoms in the gas phase and those adsorbed on the sorbent surface. Nevertheless in most cases, the interpretation of the results is not easy and depends also on the chemical nature of the materials, homogeneity, crystallinity and on the ease of diffusion of Xe molecules in the pore structure. In the case of zeolites, the interpretation of 129Xe NMR chemical shift ~Xe is usually described using the following relationship [82]

PXe

~Xe = ~o + ~s + ~E 4- y ~(Xe-Xe) ' d P x e (4.17) 0

where 8o is the reference (chemical shift of xenon gas at zero pressure), 8s is the contribution due to collisions of xenon atoms with the zeolite walls, 8E arises from the electric field created by cations and PXe is the density of Xe adsorbed in the cavities. The dependence of 8 upon the quantity of Xe is usually explained in terms of the contribution due to collisions between Xe atoms in the micropores, resulting from the increase in the local Xe density compared to the equilibrium one. The slope of the straight line for the relationship 8Xe = flPXe) is an important parameter which indicates the range of pore sizes. This slope is usually large for microporous materials or zeolites and near zero for mesoporous silica. From a qualitative point of view, the shape of the NMR peaks can, in tortuous materials, be considered as an indication of the homogeneity of the material pore structure. Indeed two distinct signals can be obtained [85] by mixing two samples of sol-gel derived microporous silicas prepared from different TEOS sols [86]. Figure 4.11 compares the pore hydraulic radius distributions (obtained by the MP method from N2 adsorption isotherms) of isolated and mixed samples and the corresponding NMR peaks.

For a quantitative exploitation of the technique, several fundamental problems relate to the extent and rate of motion which the Xe atoms undergo in the porous solid and the interpretation of the average chemical shift [81]. Even empirical correlations which have been proposed in the literature [87,11] relating the chemical shift (extrapolated at PXe "-O) to pore size cannot be considered as general and often give inconsistent results from one type of material to another [75,83].

NMR techniques are powerful but not yet entirely general. They have probably scope for more development, are non destructive and can provide microscopic information on pore networks. These techniques have future scope in

90 4 -- METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE

1. .Nz":ADSORPTION ] i

0,3

~ 0,2

-~0,1

o

129Xe NMR

SAMPLE A ]

o 4 s t2 16 Pore hydraulic radius r. (A)

t

t �9 "150 " ' ' 100 50.6xe(ppm)

fSAMPLE B i

' 1sO " " 1 o 0 ' "SO3xe(ppm)

"~ 0,3

=~ 0,2 ~,

0 4 8 12 -16

Pore hydraulic radius rH (/x.)

"~ 0,2

0,1

0 4 8 12 16

Pore hydraulic radius r. (A)

SAMPLEA+B [ i

"~1~ . . . . 160 . . . . 506xe(pp m

Fig. 4.11. Comparison of the hydraulic radius distributions of pores (obtained by the MP method from N2 adsorption isotherms) of separate and mixed silica samples and the corresponding 129Xe

NMR spectra [85].

particular for a better characterisation of microporous materials. A better un- ders tanding of the physical chemical phenomena is required for the technique to be fully exploited. In this respect, measurements at below ambient temperature [75] and at progressive degrees of pore filling are informative.


4.3.3 Non-intrusive Methods

4.3.3.1 Radiation scattering

Radiation scattering from solids can arise from variations of scattering length density (see below) which occur over distances exceeding the normal intera- tomic spacings. Such variations occur when solids contain pores, and details of the porosity and surface area can be obtained from measurements of the angular distribution of scattered intensity. The appropriate angular range (see below) where this information is contained is given by

d ~ ~/20 (4.18)

where d is the pore size and ~, the wavelength of radiation, which may be X-rays, neutrons or light. Since the theory and analysis of radiation scattering from porous solids has been published extensively [88-90], it will consequently not be detailed here.

In a scattering experiment, a monochromatic beam of electromagnetic radiation (light, X-rays) or neutrons, of wavelength ~o, intensity Io and wavevector Ko, is directed on a sample and the scattered intensity I(Q) is measured as a function of angle, 20, to the incident direction (Fig. 4.12). Here Q is the momen- tum transfer (Q = 4~zsin0/~,). Typical wavelengths for different types of radiation and the corresponding spatial resolution of density fluctuations or inhomogeneities such as pores, which may be determined experimentally are given in Table 4.1.

The form of the scattering curve obtained from radiation scattering measurements depends on the complexity of the system. Arelatively simplecase which is often encountered is that of a two-phase system, composed of particles (or pores) having a homogeneous scattering density, dispersed in another continu- ous medium, such as a liquid (or solid for the case of pores). Here an analysis of the scattering can provide details of the spatial and orientational correlations of the particles (or pores) [91]. A simplified analysis of small angle scattering frequently involves the Guinier approximation which is valid for randomly distributed (uncorrelated) systems of particles (or pores) with a relatively

Io, ko r

Is,k~

~ ~ . . " ~ 20 \\

SAMPLE

Fig. 4.12. Schematic representa t ion of the incident and scat tered beams in a SAS exper iments .


TABLE 4.1

Typical wavelengths for different types of radiation and corresponding spatial resolution of inhomogeneities

Type of radiation Typical wavelength Inhomogeneity Spatial resolution (nm) (nm)

Light 400-600 X-rays 0.1-0.4 Neutrons 0.1-2.5

Refractive index 200-20 000 Electron density 5-50 Scattering length density 50-500

narrow distribution in size. The Guinier equation (4.19) describes the form of the scattering I(Q) in the initial region of the curve (see Fig. 4.13)

I(Q) -- e x p ( - Q2. R~)/3 (4.19)

where Rg is the radius of gyration of the particle or pore (NB: for spheres of radius R, R = 1.29 Rg). This relationship is valid when Q.Rg < 1 and applies to inhomogeneities of any shape. At higher Q, (Q.Rg > 4), I(Q) decays approximately as Q-4, which is predicted by the Porod law [92] viz.

"" al I ~ ~1 I ~ - ' . \ I

b) a) I i I _ ~ \ I , It ~ \l

rZGuinier region I ~ l~ i~ . . i . . [

t - ~ 1 --" ................ \ \ ~ . . . . ~ j !

,, 1

!- \ l Mass ffactal \ p- range \ ,Porod / ~ \ reg!on.'. [-- ~ \(Surface fractai / \ range)

/ I I I I ~ J l o g Q

Fig. 4.13. Schematic representation of a particle aggregate (a) having a range of self-similarity between approximately al and a2 (mass fractal structure). The form of the scattering expected is

depicted in (b) [91].

4 - - M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE 93

I (Q) ~ ( S / V ) . Q-4 (4.20)

where ( S / V ) is the surface to volume ratio of the particle (or pore). This law is obeyed when the interface between the two phases is smooth in the scale of Q-space measured. A more general expression which applies to surfaces which are irregular or have a curvature on a scale smaller than the reciprocal Q space is given by

I (Q) ~ S . Q-(6-D~) (4.21)

where Ds is the surface fractal dimension. For a 'smooth' surface Ds is 2 (the Euclidean 2D dimension) whereas for fractally rough surface Ds may approach 3. This leads to power law exponents which may range between 3 and 4. Such situations frequently arise for systems containing very small pores, such as amorphous microporous oxide gels.

Scattering techniques are non intrusive, non-destructive and have particular advantages in the characterisation of the surface and porous properties of materials containing either closed or molecular sized pores and where outgassing pre-treatment may result in irreversible changes in microstructure. Meas- urements may be carried out, indeed, in the presence of a gas or with hydrated materials and there is no need to evacuate the sample.

Both small angle X-ray (SAXS) and neutron scattering (SANS) are established techniques and their experimental application is similar. However, limitations on sample size, thickness and containment are much more restricted with X-rays because of absorption of radiation. One problem which can arise with neutrons is the subtraction of the flat incoherent contribution which can be quite large in the case of hydrogenous materials. This disadvantage can be partially offset by the possibility of using isotopic substitution. SANS is particularly powerful because the penetrating power of neutrons makes it possible to study material microstructure in the wet state. Instrumentally, both SAXS and SANS require a source of radiation, collimation system, sample containment and a detection system.

Important information on the contribution of porosity contained in closed and open pores, pore morphology, pore size distribution and/or surface texture can be derived from X-ray and neutron scattering. This is achieved by condensing probe molecules having a similar scattering cross section to the solid, in the open pores ~ a technique known as contrast matching [93,94]. This approach has been developed in detail using SANS where isotopic effects have been exploited (e.g. H20/D20) to study oxide gels and porous carbons [94,95].

Small angle scattering has been used extensively to analyse a wide range of porous materials (ceramics, carbons, oxide gels, cement, bone materials, etc.) [96-99]. An important and recent development concerns the investigation of materials which contains an oriented porous texture, such a fibres and layer-

9 4 4 - - M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE

Neutron beam

ACF

FI

~~ 6 b

! 04 10:(Y ~'

.-. % . 0 I-- 0 i i

0

,02

1 J ~ ~

10-3 10-2 10'1 QIj, -~

Fig. 4.14. SANS of oriented activated carbon fibres. (a) Schematic representation of a sample of carbon fibres oriented with their axes parallel to the incident neutron beam; (b) corresponding anisotropic scattering along the two axes of the 2D detector. (i) SANS along the vertical axis, (ii) SANS

along the horizontal axis of the detector [101].

like materials. Frequently the pores are highly anisotropic and aligned with respect to a specific particle orientation forming the porous texture. For these materials unique microstructural information can be derived from small angle scattering measurements with both neutrons and X-rays. These details are not obtainable from bulk measurements, such as adsorption isotherms. This application of SANS has recently been demonstrated with ceramic alumina fibres [100] and microporous carbon fibres [101]. Such measurements on materials having anisotropic pore structures require the detailed analysis of the equivalent anisotropic scattered intensity, measured on a two-dimensional detector, as shown for a sample of aligned activated fibres (ACF) in Fig. 4.14. The application of small angle scattering to determine the structure of membrane material in situ is illustrated by recent investigations of Anotec | alumina membranes containing oriented cylindrical pores [102] and those on oriented porous sol-gel layers deposited on metal substrates [103]. In future such applications will increase the growing availability of SAS facilities at synchrotron and neutron sources.

4.3.3.2 Wave propagation

Ellipsometry If light is reflected at the boundary of two different optical media, the

polarisation of the electromagnetic vibration is changed according to the Fres- nel equations [104]. The change of the status of polarisation is characterised by


two ellipsometric angles representing the phase shift and the change of the amplitude. When analysing a thin layer supported on a substrate with a knowr~ index (e.g. Si), the measurement of these two angles at extinction gives information on the layer thickness and refractive index n. This latter parameter is sensitive to the film porosity. Knowing the index of refraction for the constitu- ents of the film, various optical models can be used to calculate the volume fraction of solid [105]. The effective medium theories, i.e. theories of Brugge- man, Garnett and Lorentz-Lorenz, for a two phase system are based on the following equation

(n 2 - n~) /(n 2 + 2n 2) = Va[(n 2 - n~) /(n 2 + 2n2)] + (4.22)

Vb [(n~- n 2)/(n~ + 2n~)]

where Va and Vb are the volume fractions of phase a and b and na and nb are the refractive indices of these phases respectively. The measured refractive index of the film is n and nh is a host refractive index which is assigned different values according to the model used. In the Bruggeman effective medium approximation, which can be successfully applied to films with a random configuration of two phases, n h is set equal to n. The Garnett theory (in which n h = na o r nb) is generally applied to a film consisting of one phase completely surrounded by the other. The Lorentz-Lorenz theory is obtained by setting nh equal to 1, i.e. empty space and is the most frequently applied to estimate thin film porosity [106,107]. The volume fraction of solid Vs is obtained through the simplified equation

V s = [(y/2 - 1 ) (y/2 + 2)]/[(n 2 + 2) (n 2 - 1)] (4.23)

in which na is the refractive index of the solid skeleton. Other relationships can be found in literature between refractive index n and density p like the Glad- stone-Dale equation [108,109]

p = Kx(n- 1) (4.24)

Ultrasonic methods The attenuation and velocity propagation of ultrasound in porous solids

depends on the pore size and porosity. The theory of the method is very complex and has been developed mainly for two phase systems containing heterogeneities (e.g. solid/voids). Ultrasonic methods have applications in the area of non-destructive testing of materials, particularly metals and ceramics, where such heterogeneities may be interfaces, pores, inclusions, grain bounda- ries, or compositional variations [110]. Experimentally measured quantities are the backscattered ultrasonic signal or characteristics of the forward-scattered wave, such as propagation velocity or attenuation. Ultrasonic techniques can be applied to assess the width of voids, d, and porosity, 8, from variations in the


ultrasonic velocity, v, and attenuation 0c a [111]. The method is appropriate when o.d > 0.2, where r~ is the longitudinal wave number in the matrix given by

r~ = ~ / v (4.25)

co(= 2~:v) being the circular frequency of the sound of frequency r~. Typically, when v > 107Hz, the method is applicable for the non-destructive examination of voids in solids, when d > 1 ~tm [5].

Recent developments have extended the ultrasonic techniques to the characterisation of thin layers of metals, and polymers deposited on substances to obtain measurements of the thickness/density product. Using techniques where the film are immersed in a fluid, such as water, measurements have been made, by the low frequency normal incidence double through-transmission method, with film thickness ranging from 20 to 200 ~tm [112] a range which is of particular relevance to membrane systems.

An example of acoustic microscopy investigations (by microechography, acoustic signature V(z) and acoustic imaging) of porous silicon layers (290 ~tm) is given in [113]. Information which can be obtained concerns the elastic properties of the material, the layer thickness and density as well as a mapping of the surface and subsurface of the material, revealing roughness and defects.

Surface acoustic waves (SAW), which are sensitive to surface changes, are especially sensitive to mass loading and theoretically orders of magnitude more sensitive than bulk acoustic waves [43]. Adsorption of gas onto the device surface causes a perturbation in the propagation velocity of the surface acoustic wave, this effect can be used to observe very small changes in mass density of 10 -12 g / c m 3 (the film has to be deposited on a piezoelectric substrate). SAW device can be useful as sensors for vapour or solution species and as monitors for thin film properties such as diffusivity. They can be used for example as a mass sensor or microbalance to determine the adsorption isotherms of small thin film samples (only 0.2 cm 2 of sample are required in the cell) [42].

4.3.3.3 Ion-beam analysis

The application of ion beam analysis techniques to determine pore size and pore volume or density of thin silica gel layers was first described by Armitage and co-workers [114]. These techniques are non-destructive, sensitive and ideally suited for the analysis of thin porous films such as membrane layers (dense support is needed for backscattering). However, apart from a more recent report on ion-beam analysis of sol-gel films [115] using Rutherford backscattering and forward recoil spectrometry, ion beam techniques have not been developed further despite their potential for membrane characterisation. This is probably due to the limited availability of ion beam sources, such as charged particles accelerators.

4 - - METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 9 7

, .............. ........

~ lOOOO - PEAK

rO

E,~ = 3.3 MeV

I ~ ~ " . ~ i OXYGEN

sooo "-" . ~ ; / szuco~ , : �9 k 4, EDGE

~-v,l '~" - " . " " "~ . . . . . . . . . .~

0 1 2

ENERGY OF BACKSCATTERED PARTICLES (MelO

Fig. 4.15. Experimental arrangement and RBS spectrum of porous (~ ) and non-porous ( ..... ) silica thin films [ 114].

The work of Armitage and co-workers was carried out using both tandem Van der Graaff and linear accelerators as sources for focused proton and o~-particle beams. The method involves observing the energy distribution of ions elastically backscattered after exciting a resonance in one of the nuclei of the sample. In this work the resonances for protons (2.66 MeV) and ~-particles (3.05 MeV) scattered from 160 were employed. An illustration of the experimental arrangement with an example of a spectrum obtained from a porous SiO 2

sample with 3.3 MeV ~-particles is s h o w n i n Fig. _4.!5, together with the spectrum from a sample of non-porous SiO 2 glass. It will be noted that the resonance peak for the porous sample is broader. This peak is associated with the 3.05 MeV resonance arising from the elastic scattering of o~-particles from 160. A description is given in [114] of the methods of calculation of pore size from such backscattering data, with various assumptions for the pore and interpore path length distributions.

This method, using a microfocused beam, has unique advantages over other techniques which could be very useful in membrane characterisation. Thus in the above work examples of measurements made on gel layers as a function of sampling depth (from -3 ~tm to 100 ~tm) and as a function of distance across the sample were illustrated. It will also be noted that the technique is equally appropriate for measurements in the micro and mesoporous ranges.

4.3.3.4 Positron lifetime spectroscopy

When a positron (generated e.g. by a 22Na source) enters a condensed medium, it may be annihilated directly with an electron, or it may capture an

98 4 B M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE

120

100

80

%,)

~ 40

20

.

o

@

s o , 0 0 , s o Pore diameters, Nz adsorption results (~

Fig. 4.16. Positronium lifetime dependence on pore diameter. Solid line is the theoretical relation found from the SchrOdinger equation for spherical pores [118].

electron to form a hydrogen-like atom, called a positronium. Ortho- or para- positroniums decay spontaneously by direct annihilation but with very different time scales (0.1 ns for the para- and 110 ns for the ortho-positronium in air). The lifetime of orthopositronium (measured with a scintillation counter detect- ing the I radiation) depends on its interaction with surrounding molecules (pick-off quenching caused by interactions with unpaired molecular electrons). Consequently, a positronium decays faster inside a pore than in air. The various measured lifetimes can be attributed to several kinds of trapping sites: disloca- tions, intersections of two or three crystallite interfaces and internal surface of larger voids.

The experimental set-up consists of a positron source (22Na), a scintillation counter, to detect the I radiation from the positronium decay, and electronic peripheral equipment to analyse the time spectrum of the positron annihilation.

Positronium lifetime spectroscopy is particularly well suited for studying defects in crystals and structural fluctuations in amorphous materials and can give an estimate of free volumes in condensed matter [116]. It is a useful technique to estimate the free volume of polymeric membranes [117]. In a study on silica gels, the decay lifetime has been found (Fig. 4.16) to be proportional to the pore diameters (measured by N2 adsorption) between 30 and 100 ~ [118]. Information on pore size distribution and surface area may also be obtained by means of calibration curves.

4.4 DYNAMIC CHARACTERISATION TECHNIQUES

4.4.1 Rejection measurements

Rejection measurements with reference molecules like dextrans, proteins or polyglycols are often used by membrane manufacturers. A parameter extensively used for membranes characterisation is the "cut-off" value, which is


defined as the lower limit of solute molecular weight for which the rejection is at least 90%. It is argued that these rejection measurements have the closest parallel to operating conditions. But it must be realised that rejection measurements always depend on the type of solute (shape and flexibility of the macro- molecular solute, charge of the solute), the membrane (its interaction with the solute) and the process parameters used (pressure, cross-flow, velocity, geometry of the test cell, concentration and type of solute, pH). In particular charge effects, concentration/polarisation, pore blocking and fouling phenomena will affect rejection measurements considerably.

Recent studies based on comparison between gel permeation chromatogra- phy and ultra/micro-filtration [119] have shown that whatever the chemical nature and shape of the model macromolecule used, it is possible to predict the cut-off value of a membrane by considering the hydrodynamic volume of the macromolecule. This parameter provides an appropriate definition of the effective solute size to be considered in hydrodynamic models.

4.4.2 Liquid Displacement Techniques

4.4.2.2 Liquid~gas methods (bubble point, liquid expulsion permporometry)

This very simple and established method has become a standard technique used by suppliers to measure the largest active pores (as well as cracks or pinholes) in a membrane. The principle is to measure the pressure needed to force air through a liquid-filled membrane. The bottom of the filter is in contact with air and, as the air pressure is gradually increased, air bubbles penetrate through the membrane at a certain pressure. The pressure and pore radius are related by the Laplace equation

rp = 2y cos0/AP (4.26)

where rp is the radius of a capillary shaped pore and y the surface tension at the l iquid/air interface (~/(water/air)---- 72.3 10 -3 N/m; ~/(t-butanol/air)- 2 0 . 0 10 -3 N/m; "~(fluorocarbon/air) = 16.0 10 -3 N/m).

An air bubble will pass through the pore when its radius is equal to that of the pore (Fig. 4.17), assuming that the contact angle is zero. Penetration will first occur through the largest pores and since the pressure is known, the pore radius can be calculated from Laplace equation.

With water as the wetting medium, the water / air surface tension is relatively high and it is necessary to apply a high pressure if small pores are present (145 bars for a pore radius of 0.01 ~tm); water can then be replaced by another liquid (e.g. alcohols or hydrocarbons). Nevertheless, as the method is dependent on the type of liquid used (different wetting effects), i-propanol is often used as a


b u b b l e P2<PI ,~ ,,~..~, ,,. �9 , " . . , i . ,,

" Wetting �9 liquid

. . . . . " ".;. , . ,o L. i . _ i

Porous m e m b r a n e Pl

Fig. 4.17. Principle of liquid expulsion permporometry.

Gas flow

D.now /

, 4 / / "Wet"

B u b b l e p o i n t / I flow c u r v e l

~ , ~ ~ f f Mean flow p o r e ...1

Pressure ( Pore d i a m e t e r s

Fig. 4.18. Maximum and mean flow pore size determinations by bubble pressure point test [3].

s tandard liquid. The rate at which the pressure is increased and the pore length can also influence the measurements.

The mean pore size and pore size distribution can be evaluated by perform- ing this measurement by a stepwise increase of the pressure. In this case the gas flow across the wet defect-free membrane is recorded (Fig. 4.18) as a function of the applied pressure difference across the sample ("wet curve"). The point of first flow is identified as the "bubble point". This continues until the smallest detectable pore is reached. Then the flow rate response corresponds to the situation in a completely dry sample. The measurement of gas flow through the same membrane in a dry state gives a linear function of the applied pressure difference ("dry curve"). The pressure at which the "half-dry" curve intersects with the "wet" curve can be used to calculate an average pore diameter. Pore number distributions can also be derived from flow distribution curves.

This method has been approved as a ASTM procedure [120-122] and used in the commercial computer controlled Coulter Porometer (Coulter Electronic Ltd) for pore sizes much larger than 0.44 lxm. However the theoretical basis used for the evaluation of the accumulated data neglects the specificity of gas flow in pores and incorrectly considers the flowing gas as an incompressible fluid. The assumption of gas flow dependency only on AP distorts the resulting

4 - - M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE 101

dimensionless distributions [39]. A more realistic description of the situation, in which progressively smaller pores contribute to inert flow due to the applied pressure difference, is given in [123].

4.4.2.3 Liquid-liquid displacement porosimetry (or biliquid permporometry) [124-128]

In this technique, which is a combination of bubble pressure and solvent permeability methods, a liquid A wetting the membrane is displaced by a fluid B (non miscible liquid less wetting than A). The principle is based upon the Laplace equation determining the mechanical equilibrium at the interface

AP = 2ycos0/r (4.27)

where AP is the pressure drop through the membrane, r is the pore radius, 0 is the contact angle between the two liquids and the solid and t~ is the interfacial tension between the two liquids. The liquid A will be displaced by B only when the pressure applied to B has reached the value

AP = 2y/r (Cantor's equation) (4.28)

The technique consists in measuring the B (e.g. water) flow rate (J) through a membrane impregnated with A (e.g. isobutanol or mixtures of alcohols and water) as a function of the pressure difference AP. We have to note that it is possible to modify the method from "pressure controlled" to "flow controlled" in order to reduce the test time and increase its flexibility [126]. At a certain minimum pressure the largest pores become permeable, while the smaller pores still remain impermeable. This minimum pressure depends mainly on the type of membrane material (contact angle), type of permeate (surface tension) and pore size. When all pores are filled with B, the liquid flux J through the membrane becomes directly proportional to the pressure.

A typical flux versus pressure curve is shown in Fig. 4.19. The recorded Pi and Ji values introduced in the Laplace equation directly give the pore equivalent pore radius ri and the distribution of permeability (Ji/Pi) vs. pore radius.

r(max) r(mi.)

Fig. 4.19. Flux versus pressure for a membrane having a distribution of pore sizes.


Pore number N i, a r e a A i ( A i = ~ N i r2) and their distributions are indirectly obtained by using the Hagen-Poiseuille relationship [126]

Ji = [lr, Ni r4 Pi] /[8~Ax] ( 4 . 2 9 )

where ~t is the absolute viscosity of the testing liquid and Ax is the equivalent length of the capillary pores. If pores are assumed to be voids between packed spheres, the Kozeny-Carman equation has to be used to express the flux as a function of pressure. This technique can be applied to ultra and micro-filtration membranes and is very close to the situation that can take place in filtration as only the through pores are considered and the membranes are studied in wet conditions.

4.4.3 Fluid Flow Measurements

These techniques involving the measurement of membrane permeability to a fluid (liquid or gas) lead to a mean pore radius (usually the effective hydraulic radius rH) whose quantitative value is often highly ambiguous. The flux of a fluid through a porous material is sensitive to all structural aspects of the material [129]. Thus, in spite of the simplicity of the method, the interpretation of flux data, even for the simplest case of steady state, is subject to uncertainties and depends on the models and approximations used.

4.4.3.1 Liquid permeability

The steady state volume flux Jv of an incompressible fluid though a porous medium of cross-sectional area Ac, thickness I and porosity ~ under a pressure differential AP, can be expressed in terms of the mean velocity in the pores Up

Jv = Ace Up = A c K c e r~H AP/2rll (4.30)

If ~ and Kc are known or can be estimated, rH and hence A the specific surface area of the porous medium can be determined. According to the well known Kozeny-Carman treatment

K c = ~c/ , i : 2 ( 4 . 3 1 )

where [5c is a pore shape factor (varying between I for cylindrical pores and 2/3 for pores in the form of slits) and z > I is the tortuosity factor that takes account that the fluid follows tortuous paths through the porous solid. If the porous medium is pictured as a bundle of N cylindrical capillaries of radius re -- 2rH and length le

= 1:l, and if we ignore the effect of tortuosity on the flow pattern, we obtain

2 le /Acl = Nlr,~ T,/A c ~, = N ~ r e (4.32)

and then

4 --METHODS FOR THE CHARACTERISATION OF POROUS STRUCTURE 103

Jv = Nnr2e AP/811 ]e = a c t~ r 2 [3 c AP/2n~21 (4.33)

The first part of the relation (4.33) is the well known Poiseuille equation for cylindrical capillaries. The main problem is the lack of theoretical basis for assigning a value to "r a priori (an empirical value of 1.5 is suggested by Kozeny). This obviously precludes the use of these equations as an independent method for determining rH or A. An example of experimental determination of Kc for y alumina membranes is given in Ref. [130]. The corresponding tortuosity is very high in agreement with the observed microstructure of this type of membrane made of plate-shaped crystallites.

4.4.3.2 Gas permeability

The situation is rather more complex in the case of a gas (assumed to be non adsorbed by the membrane material) because compressibility and molecular effects, which predominate at low pressure, introduce a pressure dependence. Nevertheless the interpretation of results can yield more information than obtained with liquid media.

The measurement of the permeability of non adsorbed gases is classically used to determine the range of pore size in membranes (macro, meso or micropores). Indeed by plotting the permeability as a function of gas pressure, a straight line is usually obtained whose slope gives an indication of the gas transport mechanism in the membrane. A quantitative description of pore structur e can be attempted from the results.

One method which is known under the name of permeametry [131] or Poiseuille-Knudsen method [124] is based on the law of gas permeability in a porous media in the two flow regimes" molecular flow (Knudsen) and laminar or viscous flow (Poiseuille). According to Darcy's law, the gas flux through a membrane with a thickness I can be written as J = KAP/I , where K is the permeability coefficient and AP (AP = P1 - P2) the pressure difference across the membrane. If the membrane pore diameter is comparable to the mean free path of the permeating gas, K can be expressed as a sum of a viscous and a non-viscous term

K= Ko + BoP/n (4.34)

where P = (P1 + P 2 ) / 2 is the mean pressure, 11 is the gas viscosity and Ko and Bo are the characteristics of the membrane and the gas. Using the theoretical equations suggested by Carman and the average molecular velocity v of the gas with a molecular weight M

v = ( 8 R T / ~ M ) 1/2 (4.35)

the mean hydraulic pore radius of the membrane can be expressed by

104 4 u M E T H O D S FOR THE C H A R A C T E R I S A T I O N OF POROUS STRUCTURE

r H = (16/3)(Bo/Ko)(2RT/gM) 1/2 (4.36)

Ko can be obtained by extrapolation of K versus P at P = 0 and Bo is obtained by multiplying the slope of the same function by the gas viscosity rl. The experiment involves the measurement of the membrane permeability as a function of AP, in a pressure range which involves successively molecular flow and viscous flow. By plotting the membrane permeability coefficient as a function of the mean pressure, a straight line can be obtained whose slope and origin can be related to the mean pore hydraulic radius of the membrane. A special apparatus has to be used which is appropriate for the measured pore range, membrane size, thickness and geometry. This technique can be used to evaluate pores ranging from between several A to several gm.

The modelling of gas permeation has been applied by several authors in the qualitative characterisation of porous structures of ceramic membranes [132- 138]. Concerning the difficult case of gas transport analysis in microporous membranes, we have to notice the extensive works of A.B. Shelekhin et al. on glass membranes [139,14] as well as those more recent of R.S.A. de Lange et al. on sol-gel derived molecular sieve membranes [137,138]. The influence of errors in measured variables on the reliability of membrane structural parameters have been discussed in [136]. The accuracy of experimental data and the mutual relation between the resistance to gas flow of the separation layer and of the support are the limitations for the application of the permeation method. The interpretation of flux data must be further considered in heterogeneous media due to the effects of pore size distribution and pore connectivity. This can be conveniently done in terms of structure factors [5]. Furthermore the adsorption of gas is often considered as negligible in simple kinetic theories. Application of flow methods should always be critically examined with this in mind.

4.4.3.3 Permporometry [140-143]

This technique, developed by Eyraud [140] modified by Katz et al. [143] and recently by Cuperus et al. [141], is based on the controlled blocking of pores by capillary condensation of a vapour (e.g. CC14, methanol, ethanol, cyclohexane), present as a component of a gas mixture, and the simultaneous measurement of the gas flux through the remaining open pores of the membrane. The capillary condensation process is related to the relative vapour pressure by the Kelvin equation. Thus for a cylindrical pore model and during desorption we have

In P/Po = -27V cos0/rK RT (4.37)

with r = rK + t (t has to be determined, t = 0.5 nm for cyclohexane [141]). Careful control of the relative vapour pressure permits the stepwise blocking

of pores. Starting from a relative pressure equal to 1, all the pores of the


membrane are filled, hence unhindered gas transport through the membrane is not possible. When the vapour pressure is reduced, pores with a size larger than that corresponding to the vapour pressure are emptied and become available for gas transport. By measuring the gas transport through the membrane upon decreasing the relative vapour pressure, the size distribution of the active pores can be found, in the limit of validity of the Kelvin equation (mesopores). The experiments are usually conducted during the desorption process because the equilibrium time is much longer during adsorption [142]. The calculation of the size distribution of the active pores requires a well defined transport regime. In the absence of an overall mechanical pressure gradient and by using the principle of counter diffusion of two different gases (e.g. 02 and N2), a diffusion of Knudsen type can be usually assumed and the corresponding equation applied [141,142]

Jk = N~rp D k AP /RT xlA 2

with D k = -~ rp ~/8RT /~M w (4.38)

with Jk the Knudsen gas flux (mol/m2/s), N the number of pores, rp the pore radius (m), Dk the Knudsen diffusion coefficient, AP the pressure difference across the membrane (Pa), Mw the gas molecular weight (g/mol), A the membrane surface (m2), 1: the tortuosity, and l the layer thickness (m).

a N2+ Oz

Ethanol + 1

! i'", mbrane 0 ~ + !

Ethanol v

I / l l l | l | l

ImI 11i- 02 ~

mlnmi o | * *

Oz _

[~_~__ Empty pore: Filled pores

Membrane matrix

O<p/p o < l

I I I l l - " Tranport in all pores

Fig. 4.20. Experimental arrangement of permporometry (a) and principle of technique (b) [2].


A stepwise reduction of the relative pressure means that a range of pores will become accessible and contribute to the gas flux. Therefore number of pores as well as flow averages distributions can be obtained as a function of pore sizes. A schematic diagram of the experimental equipment employed [141] is given in Fig. 4.20 together with the principle of the analysis. During the experiments, there is no difference in hydrostatic pressure across the membrane and gas transport proceeds only by diffusion, the flow of one of the two non-condensa- ble gases being measured (for example that of oxygen can be measured with an oxygen selective electrode). The technique has been applied in the characterisation of active pores in alumina membranes with a good reliability [142]. The method does not require a high mechanical stability of the membrane as most of the other techniques and can be applied in the limit of validity of the Kelvin equation, i.e. for pore sizes above 1.5 nm.

4.5 C O N C L U S I O N A N D R E C O M M E N D A T I O N S

The features of the more important characterisation techniques described here are summarised in Table 4.2. As is evident from this review, there is no technique which is universally applicable for the characterisation of the porous properties of all materials. The choice is made on the basis of many criteria, such as the range of pore size, the nature of the material and its form, together with the application envisaged. Frequently, more than one technique is required in a detailed examination. In the case of membranes, particular problems are encountered because of their form and the small quantity of active material involved. Furthermore, other complexities arise in the case of microporous thin films, although, as we have noted here, currently this is an area of active progress. This involves the development of new techniques and advances in phenomenological theories to describe the properties of such nanostructured supported materials.

The techniques given in Table 4.2 are well established and have been sub-di- vided into those which are described as either static or dynamic. We feel this distinction is of particular importance in the characterisation of the porous structure of membranes. Here the performance is determined by the complex link between the structural texture and transport behaviour. An insight into this cOmplexity is frequently provided by dynamic techniques, which are not restricted by the limited quantity of membrane material and are sensitive to the active pathways through the porous structure. Further developments are required in this area both in the improvement of existing techniques and introduction of new techniques. Progress will also come from advances in the theory and modelling of flow behaviour in such porous media, which involve percolation theory and fractal geometry for example. With the refinement of such


theories they may be used directly in the interpretation of experimental data. Of the established static techniques, which we have considered here, that

involving gas adsorption isotherm measurements remains one of the most powerful and widely applicable. It is indeed very accessible with the availability of automated commercial equipment and the variety of data treatment facilities available. Nevertheless, it is still circumscribed by the assumptions implicit in the choice of a pore shape model in the case of mesoporous materials. Its application to microporous structures has recently advanced considerably, al though there are here certain reservations which still exist concerning the general application of theories to describe adsorption in such small pores in ill defined structures.

A variety of other static characterisation methods have been described in this chapter which are not listed in Table 4.2. Many of these are new and in a state of rapid evolution, as for example those involving NMR and radiation scattering. Whilst appropriate for research investigations they do not seem yet to be appropriate as a means of general characterisation. However with the rapid progress under way in these areas, some of these techniques we feel may in the future be ideally suited to membrane characterisation.

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