Structural Characterisation of Natural and Industrial Porous Materials: A Manual 3030474178, 9783030474171

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Sean Patrick Rigby

Structural Characterisation of Natural and Industrial Porous Materials: A Manual

Structural Characterisation of Natural and Industrial Porous Materials: A Manual

Sean Patrick Rigby

Structural Characterisation of Natural and Industrial Porous Materials: A Manual

123

Sean Patrick Rigby Department of Chemical and Environmental Engineering University of Nottingham Nottingham, UK

ISBN 978-3-030-47417-1 ISBN 978-3-030-47418-8 https://doi.org/10.1007/978-3-030-47418-8

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Mum and Dad

Contents

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1 1 1 2 3 8 12 13

2 Gas Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Mechanisms of Adsorption . . . . . . . 2.2 Nature of Experiment . . . . . . . . . . . . . . . . . 2.2.1 Sample Preparation . . . . . . . . . . . . 2.2.2 Choice of Adsorbate . . . . . . . . . . . . 2.2.3 Experimental Conditions . . . . . . . . . 2.2.4 Typical Data Sets and Terminology . 2.2.5 Gas Uptake Kinetics . . . . . . . . . . . . 2.3 What Can I Find Out with This Method? . . . 2.3.1 Surface Area . . . . . . . . . . . . . . . . . 2.3.2 Pore Size Methods . . . . . . . . . . . . . 2.3.3 Pore Connectivity . . . . . . . . . . . . . . 2.3.4 Pore Size Spatial Disposition . . . . . 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 15 15 17 17 17 19 20 22 25 25 36 42 44 46 46

3 Mercury Porosimetry . . . . . . . . . . . . . . . . . . . 3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . 3.2 Nature of Experiment . . . . . . . . . . . . . . . 3.3 What Can I Find Out with This Method? .

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49 49 53 56

1 Introduction . . . . . . . . . . . . . 1.1 Aim of This Book . . . . 1.2 Porous Materials . . . . . . 1.3 Characterisation Science 1.4 What Is a Pore? . . . . . . 1.5 Void Space Descriptors . 1.6 Structure . . . . . . . . . . . References . . . . . . . . . . . . . . .

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3.3.1 Surface Area and Pore Size Distribution (PSD) 3.3.2 Pore Network Geometry . . . . . . . . . . . . . . . . . 3.3.3 Mercury Porosimetry Simulators . . . . . . . . . . . 3.3.4 Pore Network Topology . . . . . . . . . . . . . . . . . 3.3.5 Pore Size Spatial Correlation . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Thermoporometry and Scattering . . . . . . . . . . . . . . . . 4.1 Thermoporometry . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . 4.1.2 Nature of Experiment . . . . . . . . . . . . . . . 4.1.3 What Can I Find Out with This Method? 4.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . 4.2.2 Nature of the Experiments . . . . . . . . . . . 4.2.3 What Can I Find Out with This Method? 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 69 69 72 75 84 84 84 85 87 87

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5 Nuclear Magnetic Resonance and Microscopy Methods . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Theory of Nuclear Magnetic Resonance (NMR) Spectroscopy and Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 NMR Relaxometry and Pulsed-Field Gradient NMR . 5.2.3 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . 5.2.4 Computerised X-Ray Tomography (CXT) . . . . . . . . . 5.2.5 Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Nature of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 NMR Relaxometry and Pulsed-Field Gradient NMR . 5.3.3 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . 5.3.4 Computerised X-Ray Tomography (CXT) . . . . . . . . . 5.3.5 Electron Microscopy (EM) . . . . . . . . . . . . . . . . . . . . 5.4 What Can I Find Out with These Methods? . . . . . . . . . . . . . . 5.4.1 Porosity/Voidage Fraction and Porosity Descriptors . . 5.4.2 Pore Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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90 90 92 95 96 98 101 101 102 105 105 106 107 107 109 111 112

6 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Utilisation of Pore Network Effects in Pore Characterisation 6.3 Combined Mercury Porosimetry and Thermoporometry . . .

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115 115 116 118

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Contents

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6.3.1 6.3.2 6.3.3 6.3.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Considerations . . . . . . . . . . . . . . . What Can This Method Tell Me? . . . . . . . . . . . Key Features of Pore–Pore Co-Operative Effects in This Case Study . . . . . . . . . . . . . . . . . . . . . . 6.4 Integrated Gas Sorption and Mercury Porosimetry . . . . . 6.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Experimental Details . . . . . . . . . . . . . . . . . . . . 6.4.3 What Can I Find Out with This Method? . . . . . 6.5 Combined MRI and Gas Sorption . . . . . . . . . . . . . . . . . 6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Experimental Considerations . . . . . . . . . . . . . . . 6.5.3 What Can I Find Out with This Technique? . . . 6.6 Combined CXT and Gas Adsorption . . . . . . . . . . . . . . . 6.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Experimental Considerations . . . . . . . . . . . . . . . 6.6.3 What Can I Find Out with This Method? . . . . . 6.7 Integrated NMR Cryodiffusometry and Relaxometry, Combined with Gas Sorption . . . . . . . . . . . . . . . . . . . . . 6.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Experimental Considerations . . . . . . . . . . . . . . . 6.7.3 What Can I Find Out with This Method? . . . . . 6.8 Combined CXT and Liquid Metal Intrusion . . . . . . . . . . 6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Experimental Considerations . . . . . . . . . . . . . . . 6.8.3 What Can I Measure with This Technique? . . . . 6.9 Serial Gas Sorption with Different Adsorptives . . . . . . . . 6.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2 Experimental Considerations . . . . . . . . . . . . . . . 6.9.3 What Can I Find Out with This Method? . . . . . 6.10 Scattering Methods and Mercury Porosimetry . . . . . . . . . 6.10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 Experimental Considerations . . . . . . . . . . . . . . . 6.10.3 What Can I Find Out with This Method? . . . . . 6.11 Combined CXT, MRI, and Mercury Porosimetry . . . . . . 6.11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Experimental Considerations . . . . . . . . . . . . . . . 6.11.3 What Can I Find Out with This Method? . . . . . 6.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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122 122 122 123 123 140 140 140 140 146 146 147 147

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148 148 148 149 152 152 153 153 160 160 160 162 163 163 164 164 164 164 165 166 166 169

7 Structural Characterisation in Adsorbent and Catalyst Design . . . . . 173 7.1 Special Considerations for Industrial Materials . . . . . . . . . . . . . . 173 7.2 Relating Pore Structure to Raw Material Properties and Fabrication Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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7.3 Relating Mass Transport to Pore Structure . . . . 7.4 Understanding Product Activity and Selectivity 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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178 184 187 187

8 Pore Structural Characterisation in Engineering Geology . . . . 8.1 Special Considerations for Natural Porous Systems . . . . . . . 8.1.1 Impact of Geological Processes on Pore Structure . 8.1.2 Pore Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . 8.2 Predicting Permeability, Reservoir Producibility, and Bound Volume Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Permeability and Reservoir Producibility . . . . . . . . 8.2.2 Bound Volume Index (BVI) . . . . . . . . . . . . . . . . . 8.3 Characterising Multi-scale, Hierarchical Porous Structures . . 8.3.1 Fractal and Multi-fractal Models . . . . . . . . . . . . . . 8.3.2 Overcondensation Methods . . . . . . . . . . . . . . . . . . 8.3.3 Multi-scale Imaging . . . . . . . . . . . . . . . . . . . . . . . 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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189 189 189 190 192 193

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Chapter 1

Introduction

1.1 Aim of This Book The basic aim of this book is to give an introduction to the many and varied ways of characterising porous solids. It assumes a basic knowledge of geometry, and basic chemistry and physics of the level of a typical undergraduate in the sciences or engineering.

1.2 Porous Materials Porous materials are basically solid objects with interior holes. These interior holes are known as the void space. Porous materials are ubiquitous in the environment around us, including the building materials within which we live and the rocks under our feet. Such materials also form key elements of many modern technologies. The void spaces of porous materials are often made up of holes too small to see with ordinary human senses, and thus, techniques used to study them must make the unseen visible in some way. The forms and patterns of the holes inside materials are often so complex that the first step is to develop a way to classify and describe what is present as the necessary precursor to understanding physico-chemical processes that take place there. The materials that surround void spaces are many and various, including catalysts, adsorbents, rocks, tissue scaffolds, and drug delivery devices. However, this work will concentrate on the rigid end of the spectrum of solids. However, issues relating to elasticity and deformation will still be covered, as required by the context. Table 1.1 lists some common technical terms which comprise the specialist language when referring to porous materials. Further, more method-specific terms will be introduced in later chapters.

© Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_1

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1 Introduction

Table 1.1 Common terminology in the context of porous materials Term

Definition

Pore

Constituent element of an internal void space

Bulk volume

Envelope volume occupied by porous solid

Porosity or voidage fraction

Ratio of internal void space volume to bulk volume of porous material

Open porosity

Void space accessible from the outside of the sample

Closed porosity

Void space inaccessible from the outside; isolated

Through pore

A pore open at both ends

Neck

Narrow joining region between two pores that has a finite length

Window

Narrow joining of zero length between two pores

Pore body

A general widening in the void space adjoined by necks or windows

Sorption

Includes both adsorption and desorption

Ink-bottle pore

Pore arrangement whereby a narrow neck guards access to a larger pore body

Micropore

Pore of size 2 nm and less than 50 nm

Macropore

Pore of size >50 nm

Co-ordination number

Number of pore bonds meeting at a node

Connectivity

Average pore co-ordination number for whole network

1.3 Characterisation Science The characterisation of porous solids is like other sciences in that it seeks regularities and patterns in the physical phenomena associated with different characterisation techniques, and in porous structures. Like other sciences, it begins with classification schemes, such as naming a very commonly occurring structure, where a narrow neck guards the entrance to a larger void space, as an ‘ink-bottle pore’. Characterisation science then examines how such common void space structures manifest themselves in the raw data sets from different experimental techniques. Characterisation science also looks for previously unknown physical effects particular to the various different experimental techniques. It then considers how the presence of these effects impacts upon interpretation of data, and even how these effects might be turned into tools for delivering yet further information on underlying porous structures. As in other sciences, this understanding leads to control in the sense of permitting explicit intelligent design of characterisation experiments to avoid or utilise such physical effects. Like other sciences, characterisation science can also involve seeking symmetries in an attempt to simplify otherwise intractable complexity. One type of such symmetry is the assumption of complete randomness, which corresponds to a translational symmetry wherein one zone of the void space could be translated to the location of

1.3 Characterisation Science

3

another zone and the void space would still look (statistically) the same. This can mean ignoring some (second order) particular features of a void space. Ultimately, this leads to the idea of scientific model construction, whereby a much simpler structure than reality is the product of the characterisation rather than a full facsimile of the void space. A characterisation of a porous solid is, essentially, a theory about the structure of the void space. It must, therefore, not just retrodict (i.e. ‘save the appearances’ of) the characterisation data used to develop it, but also make successful, novel predictions for new data. This book will describe a range of novel predictions in the form of new experimental methods. This book will purposefully cover a wide range of different characterisation methods due to the aim of trying to achieve consilience between the theories behind each method, and a concordance of results for the proposed nature of a void space and its descriptors. Each method has its own advantages and disadvantages, which may complement each other. A key aspect of this book is the chapter on Hybrid Methods which describes attempts to closely integrate different techniques together, and force a confrontation between findings from each, but, thence, also aids in achieving a holistic, self-consistent interpretation of all data.

1.4 What Is a Pore? The void space of a porous solid is the interior free space within the envelope volume of the exterior geometric surface of the solid particle. Void spaces come in a huge variety of different geometric and topological forms. Some sort of description of the particular characteristics of a given void space is necessary to be able to distinguish between these forms, and, ultimately, understand how differences between void spaces impact physical processes within them. Void space geometries can be highly complex and convoluted, and, thus, segmentation into smaller constituent parts is a means to potentially simplify this description. A pore is a segment of the larger void space that can be distinguished from the remainder of the void space in some objective way. While all pore characterisation methods segment the overall void space in some way, they do not necessarily do so in the same way. This lack of uniqueness of segmentation method is where the ambiguity of what a pore is arises from. Probably the most unambiguous segmentation of the void space arises when the solid phase itself divides it up into completely disconnected and isolated elements that then can be readily identified as individual pores. For example, examination of a cross section through a foam structure reveals readily distinguished, isolated, bubble-like pores (as in Fig. 1.1). The development of polymer templating in the fabrication of precipitated silicas has enabled some control to be achieved over the form of the void space. For example, surfactant polymer templates, that form cylindrical rods in solution, can be used to fabricate porous silicas with a void space that apparently consist of an ordered array

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1 Introduction

Fig. 1.1 SEM image of a ceramic foam structure

of parallel, regular cylinders, such as the materials known as MCM-41 or SBA-15. For example, Fig. 1.2 shows an electron micrograph image of such a porous material. At a casual level of inspection, the images in Fig. 1.2 suggest the void space consists of cylindrical pore elements that can be readily identified by eye in the images. Figure 1.2a shows a regular arrangement of roughly circular entrances in a hexagonal array. Figure 1.2b shows an axial cross section of the pore channels. Perceived as a regular array of cylinders, the pores can be described using Euclidean geometry. Hence, the cylindrical pores will possess a particular diameter and length and will be arrayed relative to each other according to a particular pore axial spacing and angle between lines joining the centrelines of neighbouring pores. However, on closer visual inspection of Fig. 1.2, it can be seen that the pores are not perfect cylinders, since there is some distortion from a circular shape in the pore mouths and there are some undulations along the length of the pore walls. Hence, these images show that the description given of a void space depends upon the level of inspection. In order to incorporate the additional features observed in the image into the description of the porous medium requires additional parameters. These might include an eccentricity to describe the deviation from circularity of the pore mouth and cross section, and an amplitude and wavelength to describe the undulations (if periodic) of the pore wall along its length. Different types of void spaces may suggest different sets of descriptors. A further type of templated silica is fabricated using polymer templates that form uniform spheres that stack in solution according to a particular close-packing geometry, such as simple cubic or hexagonal close packing. This leads to a void space consisting of

1.4 What Is a Pore?

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Fig. 1.2 TEM images of SBA-15 templated silica, showing a pore openings and b channel cross sections

spherical pore bodies arranged in an ordered array, and joined by narrow windows (with no length) or necks (which possess some length) located at the contact points where the polymer template spheres touched. Such a regular Euclidean geometry can be described by Euclidean parameters such as the pore diameter, window diameter, and pore centre-to-centre spacing. An example of this type of structure is shown in Fig. 1.3. However, closer inspection of the images in Fig. 1.3 reveals some deviations from the Euclidean order described above. There are variations in the pore body sizes and the degree of similarity in shape to that of an idealised sphere (known as sphericity). The surfaces of the pores are rough, not smooth like a Euclidean sphere. The distribution in pore body sizes might be described by a probability density function (PDF) of pore sizes weighted by number or pore volume. In turn,

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1 Introduction

Fig. 1.3 TEM image of SBA-16 silica material. The square array of white dots in the centre of the image corresponds to pore bodies formed by (subsequently removed) polymer template

this PDF can be characterised by the standard set of descriptive statistics, such as mean, standard deviation, skewness, and kurtosis. Some void spaces are so disordered and heterogeneous that, initially, they seem to defy any possible analogy with simple Euclidean shapes, such as for the controlled pore glass (CPG) shown in Fig. 1.4. The CPG has rough, tubular, worm-like pores that twist and turn, and intersect apparently at random. However, just as any curve, no matter how sharp, can be made to look like a straight line by looking ever more closely at shorter and shorter sections, then part of a CPG pore looks like a cylinder if a short enough length is chosen and small-scale surface roughness is ignored. This process of abstracting part of the void space as a much simpler element is known as ‘pore structure modelling’, and the cylinder as the pore model. The sophistication of the pore structure modelling can be increased by using more complex systems of geometry and more complicated mathematical objects to incorporate ever more features of the real pore structure. As will be seen below in Sect. 1.5, there are several different approaches possible to construct pore models. The mathematical model bridges the divide between the realist approach and the phenomenological approach to pores. The realist approach is when the pores are considered to correspond directly to a particular bit of the real void space, as with the cylindrical pore model in Fig. 1.4b. In the realist approach, the pore is picked out by some aspect of the geometry of the real void space, such as a change in characteristic shape or diameter. However, in what we shall refer to here as the phenomenological

1.4 What Is a Pore?

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Fig. 1.4 SEM of controlled pore glass sample with nominal pore size of 55 nm (a) and close-up of individual pores (b). The red cylinder represents a potential pore region

8

1 Introduction

approach, the pore is picked out, or identified, by the occurrence of a physical process just within that particular region of the void space. This physical process might be the condensation of a vapour, or the intrusion of a non-wetting fluid. The physical process will occur in the particular pore when the control variable for the process achieves a particular value, such as the pressure of the vapour. Each pore so identified is thus associated with a particular value of the process control variable. As will be seen later in the book, the value of the control variable can be related to a geometrical parameter, which is assumed to be (related to) a property of the real void space. However, strictly, it is only related to a theoretical, mathematical model of the void space, which may have variable degrees of similitude with the real void space. This is because the relation between the control variable and the geometrical parameter is obtained from a simplified theoretical description of the physical process that often ignores details of some geometrical features of real void spaces, such as surface roughness, and that physical processes may not have a simple monotonic relationship with void space geometric parameters. This will be discussed in more detail in subsequent chapters.

1.5 Void Space Descriptors The International Union of Pure and Applied Chemistry (IUPAC) developed a classification scheme for pore sizes still commonly used (Thommes et al. 2015). Micropores are defined as holes with the minimum characteristic dimension of 2 nm. Parameters characterising the porous material, known as void space descriptors, generally, come in two broad groups, namely the statistical, and the geometrical and topological. As seen above in Sect. 1.4, the geometrical descriptors are taken from mathematical schemes of geometry such as Euclidean. Gelb and Gubbins (1999) proposed a geometric definition of the pore size distribution (PSD). As shown in Fig. 1.5, they considered the basis of the definition to be the subvolumes of the system accessible to spheres of different radii. The quantity V pore (r) was defined as the volume of the void space ‘coverable’ by spheres of radius r or smaller. Hence, the derivative –dV pore (r)/dr is the fraction of the void volume coverable by spheres of radius r but not by spheres of radius r + dr and is considered a direct definition of the PSD. For disordered solids, there is a need to try to quantify the degree of heterogeneity. Fractal geometry has been used as a way to try and achieve this end. Fractals are objects that possess the special property of ‘self-similarity’ (Avnir 1989). Self-similar objects are made up of repeated copies of themselves over different length-scales, with the copies being either exact or statistical. The characteristic properties of fractal objects, such as surface area, depend upon the size of the ruler/yardstick used to measure them. The potential of fractals to describe complex, heterogeneous porous solids will be discussed in Chaps. 2 and 8.

1.5 Void Space Descriptors

9

Fig. 1.5 Two-dimensional illustration of the geometric derivation of the pore size distribution. Point ‘X’ is only coverable by the smallest (solid) circle, while point ‘Y’ is coverable by the smallest and mid-size (dashed) circles, and point ‘Z’ is coverable by all three circles. By determining the largest covering circle for every point in the void volume, we obtain a cumulative pore volume curve. Reprinted (adapted) with permission from Gelb and Gubbins (1999), Copyright (1999) American Chemical Society

The most comprehensive statistical record of a given void space would be if the structure was digitised with individual, discrete volume elements (voxels) of a sufficiently small size to capture all structural features. The set of binary numbers for the whole porous material sample is known as the phase function Z(r), which takes a value of zero if solid is located in the voxel at position vector r, or a value of unity if the voxel corresponds to void space. While the phase function is comprehensive, it is very difficult to make quick and easy comparisons between different porous solids. However, the phase function has a series of moments that provide descriptors that can be used to make comparisons between different porous solids. The first-order moment of the phase function is the simple arithmetic mean of all the values of the phase function averaged over the whole volume within the envelope of the porous

10

1 Introduction

material, and the descriptor thereby obtained is known as the porosity or voidage fraction, ε: ε = Z (r).

(1.1)

The second-order moment is known as the density–density correlation function and represents the probability of finding the same phase in the voxel at a displacement r from a given voxel containing one phase, or the other. It, thus, takes a value of unity at a displacement of zero. It is given by: R Z (u) =

[Z (r) − ε][Z (r + u) − ε]   . ε − ε2

(1.2)

If the porous structure is isotropic, then the value of the function only depends upon the modulus of the displacement r between voxels. The correlation function can take different forms in different directions where this is not the case. For completely random, isotropic porous structures, the correlation function has a simple exponential form:   R Z (u) = exp −(u/λ)ω ,

(1.3)

where ω is an index and λ is the characteristic length-scale. An example of a porous structure created with a simple exponential correlation function (where ω is unity) is given in Fig. 1.6. A comparison of Figs. 1.6 and 1.4 shows that, to the eye, the statistically reconstructed porous medium looks very similar to the real porous material. However, strictly the second-order moment does not include any topological information about the porous material, and the higher-order moments are required to capture this information. These moments are very difficult to measure, and, so, other topological descriptors are used more often. If certain microscopic details of the void space, such as pore shape, are ignored, the void space can be reduced by some (as yet undisclosed) algorithm to the mathematical object known as a graph, as shown in Fig. 1.7. The graph consists of nodes connected by bonds. In the real porous material, the nodes represent the junctions between pores represented by the bonds. A basic property of a bond or node is whether they are connected to other bonds and nodes, or remain isolated or disconnected. Even if a few bonds and nodes are interconnected on a local basis, forming a local cluster, this cluster may be disconnected from the remainder of the other bonds and nodes in the whole structure. The concept of ‘connectivity’ of a porous structure is a measure of how multiply connected its elements are. For a porous network reduced to a set of n nodes and b bonds, as in Fig. 1.7, a quantitative measure of connectivity called the first Betti number, p1 , is given by (Tsakiroglou and Payatakes 2000): p1 = b − n + p0 ,

(1.4)

1.5 Void Space Descriptors

11

Fig. 1.6 Statistical reconstruction of a porous solid (blue) with a simple exponential correlation function

where p0 is called the zeroth Betti number and is equal to the number of disconnected parts of the network. The interface between solid and void in the porous material forms a multiplyconnected closed surface. A direct measure of the connectivity of this closed surface is the genus denoted G. The genus is defined as ‘the number of (non-self-intersecting) cuts that may be made upon the surface without separating it into two disconnected parts’ (Tsakiroglou and Payatakes 2000). In practice, the genus is equal to the first Betti number for the bond and node graph formed from the same porous material. There is a complication in deriving the genus or first Betti number for real materials as some pores will be truncated at the outer envelope surface of the porous particle, and thus, it is possible to define the genus as either including or excluding these anomalous surface pores.

12

1 Introduction

Fig. 1.7 A multiply-connected 2D porous structure (Gmax = 9, Gmin = 1). Reprinted from Tsakiroglou and Payatakes (2000), Copyright (2000), with permission from Elsevier

The first Betti number, or genus, is scale dependent, in that it will increase with the size of the pore network, since n and p will increase. However, as the lattice size gets larger, the genus often becomes a linear function of the lattice size, and it is then possible to define the descriptor known as the specific genus, , as the mean genus per unit volume. In less complex descriptions of porous media, the number of pore bonds meeting at a given node is called the pore co-ordination number (of that node). If the pore co-ordination number is averaged over all of the nodes within the whole network, this is called pore connectivity.

1.6 Structure The complex materials encountered in industry and in nature often present idiosyncratic issues and problems for characterisation, and standard approaches do not necessarily work or provide the required information. In such cases, individual characterisation programmes need to be developed. In Chaps. 7 and 8, this work will illustrate these issues with a series of case studies of industrial and natural materials.

References

13

References Avnir D (ed) (1989) The fractal approach to heterogeneous chemistry. Wiley, New York Gelb LD, Gubbins KE (1999) Pore size distributions in porous glasses: a computer simulation study. Langmuir 15:305–308 Thommes M, Katsumi K, Neimark AV et al (2015) Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC technical report). Pure Appl Chem 87(9–10):1051–1069 Tsakiroglou CD, Payatakes AC (2000) Characterization of the pore structure of reservoir rocks with the aid of serial sectioning analysis, mercury porosimetry and network simulation. Adv Wat Res 23(7):773–789

Chapter 2

Gas Sorption

2.1 Basic Theory 2.1.1 Mechanisms of Adsorption Adsorption is when molecules (the adsorbate, or adsorptive), derived from a fluid in direct contact with a solid (the adsorbent), accumulate upon the surface of the solid. These molecules accumulate there due to a binding interaction between the adsorbate and adsorbent. The adsorption process has to be exothermic due to the substantial loss of entropy, for the previously dispersed adsorbate molecules, on being localised at the surface. Adsorption is also a dynamic equilibrium process, since molecules are both arriving at the surface and escaping at all times. Adsorption begins to happen when, initially, more molecules arrive than depart, leading to an accumulation at the surface. As more molecules adsorb, the size of the departing flux also increases. The equilibrium coverage is obtained when the size of the arriving flux (determined by the fluid chemical potential, and thus pressure or concentration of adsorbate in the bulk fluid) equals that of the departing flux (largely determined by temperature and occupancy). At the molecular scale, the random vibrations of the solid lattice, due to heat, cause collisions between surface atoms and adsorbed molecules, leading to the transfer of energy and momentum, which can enable adsorbed molecules to escape the attractive force of surface atoms. The overall coverage of adsorbed molecules obtained depends upon the pressure or concentration of the adsorbate in the fluid, and the strength of adsorbate–adsorbent interactions. At higher pressures still, adsorbate molecules may begin to also bind to already adsorbed molecules to create a continuous film of adsorbate. This adsorbed phase is generally denser than the unadsorbed fluid and can even be denser than liquid-phase adsorbate, due to the attractive force of the wall potential encouraging very close packing of adsorbate. As the bulk adsorbate pressure increases, this adsorbed film thickens. What happens next depends upon the nature of the pore. For micropores, where the opposite wall potentials overlap substantially, the influence of the wall extends to the centre of the pore. Hence, the pore-filling process, as pressure increases, © Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_2

15

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2 Gas Sorption

is continuous, whereby the contents of the pore just densify with increasing pressure. For meso- and larger pores, the influence of the pore walls is negligible on the centre line. In this case, there is a much more pronounced distinction between the densities of the adsorbed phase and the bulk fluid filling the rest of the pore. The higherdensity, surface-adsorbed film thickens with increasing pressure until it becomes thermodynamically unstable with respect to the state where the pore is completely filled with high-density adsorbate phase. When the pressure is reduced again, the path followed by the plot of adsorbed amount against pressure does not necessarily follow the same route as that when the pressure was ascending. Where there is a gap between the two isotherm branches, this is known as ‘hysteresis’. The presence of hysteresis signifies the presence of a non-equilibrium process. This will be discussed in more detail below. Besides the presence or absence of hysteresis, isotherms can also come in a number of other forms, as shown in Fig. 2.1. The reversible Type I isotherms are associated with monolayer adsorption or microporous materials. Type II isotherms are also reversible and associated with multi-layer adsorption on non-porous (or very large pore macroporous) materials. Type III isotherms are similar to Type II except that the former has much lower strength of adsorbate–adsorbent interactions than the latter. The additional

Fig. 2.1 IUPAC classification scheme for isotherms (adapted from Thommes et al. 2015)

2.1 Basic Theory

17

steep steps up in amount adsorbed at higher pressures for isotherm Types IV and V are associated with capillary condensation. The presence of capillary condensation is also often indicated by the presence of a hysteresis loop, as in Types IV and V. Type VI isotherms are associated with layered build-up where each layer is completed before the next starts, as indicated by a step.

2.2 Nature of Experiment 2.2.1 Sample Preparation Typically, if exposed to the atmosphere, all samples will adsorb some contaminants from the air, especially water. In order to obtain an accurate measurement of pore structure, it is necessary to remove these contaminants without changing the underlying surface. This typically requires evacuation under vacuum (known as degassing), and often some heating to aid the contaminant desorption process. For some common materials, such as amorphous silica, there is a substantial scientific literature on the influence of thermal treatment on the surface chemistry and properties (Chuang and Maciel 1997). For new samples, it is probably advisable to perform a thermogravimetric analysis (TGA) to determine at what temperature contaminants desorb but the surface remains unmodified. For example, many oxide materials, such as silica and alumina, have their surface bonds terminated in hydroxyl groups. Heating these materials above ~400 °C begins the process of co-condensation of the hydroxyl groups, and their loss as water, thereby changing the surface structure and/or chemistry (Davydov et al. 1964). In such circumstances, TGA can be used to determine the conditions necessary to remove physisorbed contaminants but leave the underlying surface unaffected.

2.2.2 Choice of Adsorbate This section considers single adsorbate experiments. Consideration of multiple adsorbate experiments will be given in Sect. 6.8. Pore structural characterisation methods particularly utilise physical adsorption, known as physisorption. This is because, in order to only characterise pore structural geometry, the probe molecule should be as non-specifically adsorbing as possible, such that it is as ‘blind’ to variation in surface chemistry as possible. The ideal adsorbate would adsorb with equal strength on all types of surface present in the sample. In physical adsorption, the adsorbate

18

2 Gas Sorption

molecules are attracted to the surface by van der Waals forces, such as permanent dipole–dipole interactions, induced dipole–dipole interactions, and dispersion forces. Nitrogen For surface area and structural characterisation, the ideal adsorbent would just be sensitive to adsorbent geometry only, and thus adsorb equally on all types of surfaces with no indication of specific adsorption. However, this ideal is not possible in practice. The choice of adsorbate is typically based upon tradition and ease of availability. This often means the first choice is nitrogen, at liquid nitrogen temperature (of 77 K), due to its easy availability and historical usage. At first glance, nitrogen looks like a highly suitable adsorbate. Liquid nitrogen is cheap and readily available, and the triple bond of the dinitrogen molecule makes it very chemically inert. However, despite its common usage, nitrogen illustrates many of the issues when choosing a suitable adsorbate. The low temperature of its normal boiling point means that mass transport processes are sluggish at isotherm temperatures, which can mean that it takes a very long time for nitrogen molecules to penetrate networks of very small pores. Further, the nitrogen molecule has a quadrupole moment, giving rise to a charge asymmetry that makes it specifically attracted towards polar surface species, such as hydroxyl groups. There is evidence to suggest that this specific adsorption leads to underestimates of surface area in materials, such as partially dehydroxylated silicas, with certain types of chemically heterogeneous surfaces (Watt-Smith et al. 2005). Nitrogen may be excluded from pores that may allow admission of smaller molecules, such as hydrogen. Nitrogen is a rod-shaped molecule, albeit a short one. Hence, there is uncertainty concerning the orientation and packing of molecules on the surface, and, hence, the correct cross-sectional area to use. Argon It is often suggested that argon might serve as a better alternative to nitrogen. The normal boiling point of argon is marginally higher at 87 K. As a noble gas, argon is a symmetrical, monoatomic molecule. In principle, this should make it attractive as a non-specific adsorbent, but the argon atom is quite polarizable, to the extent that it has even been proposed as probe for surface acidity (Matsuhashi et al. 2001). In the past, argon adsorption has been conducted at liquid nitrogen temperatures (77 K), since liquid nitrogen was so much cheaper than liquid argon for cooling (Gregg and Sing 1982). However, since 77 K is below the triple point of argon (88.8 K), the adsorbed state of argon is uncertain and may be a solid rather than liquid-like condensate. This makes determining the appropriate relative pressure difficult. Krypton The typical adsorption experimental method places limitations on what adsorbent samples can be studied with which adsorbate. Krypton is used to measure surface area for low area solids because the low saturated vapour pressure at 77 K allows the ‘dead space correction’ to be measured with the required precision for use with low surface area samples. The ‘dead space’ is the accessible area of the inside of the apparatus

2.2 Nature of Experiment

19

outside the sample. As with argon, 77 K is below the triple point (116 K), but the reference adsorbed state is taken to be the supercooled liquid (Gregg and Sing 1982). Xenon Xenon is a large, easily polarisable monatomic molecule which makes it prone to specific adsorption on polar sites, and thus less attractive as an adsorbent. However, it is particularly interesting as an adsorbate because it is commonly used in NMR spectroscopy and magnetic resonance imaging for pore structure characterisation (see Chap. 5). Hydrocarbons Alkanes are relatively unreactive, especially compared to alkenes and alkynes, and thus potentially usable for structural characterisation. Methane is of particular interest for geological applications, such as shales. The higher alkanes are progressively longer, and flexible, chain molecules. This raises questions of their likely conformation on adsorption on a surface, especially a rough one, which leads to doubts about the correct cross-sectional area to use for the molecule. However, despite the apparent complexity, the multi-layer build-up of longer chain hydrocarbons (like butane) can follow simple models (Watt-Smith et al. 2005).

2.2.3 Experimental Conditions The basic data set for surface area, pore size distribution or other parameter estimation from gas sorption is the equilibrium isotherm. Hence, it is essential to ensure that each individual data point in the isotherm is full equilibrated. In order to achieve this, enough time must be allowed for all of the gas required to enter the void space. In some machines, it is possible to obtain gas uptake kinetics (discussed in more detail below) for each isotherm data point and explicitly observe whether uptake has reached equilibrium. The basic experiment in gas sorption typically involves developing a pressure table with the set of pressure points to obtain for the adsorption and desorption isotherms. The range of these values depends upon the capabilities of the machine used. The amount of gas adsorbing at each pressure can be measured by volumetric or gravimetric methods. Conventional gas sorption apparatus is designed to truncate the adsorption isotherm before the sample chamber completely floods with liquid nitrogen. Due to the limitations on the precision with which pressures close to saturation can be achieved, the increments in pressure often end well short of that needed for pore filling of the sample, especially for samples with large macropores. This means that conventional gas adsorption experiments are not suitable for full characterisation of samples with larger macropores. In the literature, in order to deal with this issue, what is often attempted is to stitch together pore size distributions from gas adsorption and mercury porosimetry (see Chap. 3). However, there is an alternative using only

20

2 Gas Sorption

gas sorption, namely the overcondensation method introduced by Aukett and Jessop (1996). A similar method was also used by Murray et al. (1999). In the overcondensation method, the first stage is to increase the pressure in the sample tube to above the saturated vapour pressure of nitrogen. This pressure increase should produce sufficient condensation such that even the biggest pores are filled with liquid nitrogen at the top of the overcondensation desorption isotherm, which will probably also involve some bulk condensation in the sample tube. This bulk condensation is the happening actively avoided by the conventional experiment with commercial apparatus. The requisite period needed to reach this stage is determined by the sample size and the pore volume. While there is no issue if the volume of condensate is much higher than that needed for complete pore filling, the total duration of the experiment would be much longer in that case. Once complete pore filling had occurred, the pressure is lowered to just below the saturated vapour pressure of nitrogen such the bulk condensate vaporises completely while keeping all the sample internal porosity liquid filled. Once this state has been accomplished, the first data point on the overcondensation desorption isotherm can be measured. This point corresponds to the total pore volume of the sample. The pressure is then progressively lowered in small decrements, and the rest of the desorption isotherm can be obtained in the usual way. This method is potentially most useful for characterising macroporous rocks and so will be discussed in more detail in Chap. 8. The surface tension of many condensed adsorbates is said to be quite high and could potentially lead to mechanical deformation of the sample (Gor et al. 2017). However, the measured strains for common materials are typically very low ~10−3 to 10−6 (Gor et al. 2017). It tends to be only highly porous materials, such as aerogels, that have significant strains ~30% (Gor et al. 2017).

2.2.4 Typical Data Sets and Terminology As will be explained in more detail below, gas sorption data is often history dependent. This means that the amount of adsorbate actually adsorbed on the sample at any given pressure can depend upon how the system got there from the initially completely evacuated sample. The conventional experiment generally involves starting with a completely evacuated sample containing no adsorbate, and then progressively increasing the pressure of adsorbate, in steps, all the way up to the highest possible with the apparatus and only then reducing it all the way back down to the lowest pressure. This type of experiment is aimed at obtaining what are typically referred to as the boundary adsorption and desorption isotherms, as shown in Fig. 2.2. It can be seen from Fig. 2.2 that the amount adsorbed on the desorption boundary curve is sometimes larger than the amount adsorbed on the adsorption boundary curve at the same pressure. This difference, or gap, is called ‘hysteresis’. This hysteresis is a manifestation of the history dependence of the data, since a particular pressure on the desorption isotherm was only obtained after a particular pattern of previous pressure changes. The points, in

2.2 Nature of Experiment

21

Volume adsorbed / (cm3(STP)/g)

(a) 800 700 600 500 400 300 200 100 0 0.8

0.84

0.88

0.92

0.96

1

P/P0

Volume adsorbed / (cm3(STP)/g)

(b) 450 400 350 300 250 200 150 100 0.4

0.5

0.6

0.7

0.8

P/P0 Fig. 2.2 a Schematic of crossing adsorption (open diamonds) and crossing desorption (open squares) scanning curves; b schematic of converging adsorption (open diamonds) and converging desorption (open squares) scanning curves. Arrows are added to indicate the direction of the change in pressure. The boundary curves are shown by the solid line

the isotherm plot where the boundary adsorption and desorption curves first meet at either edge of the hysteresis region, are known as hysteresis closure points. It is conceivable that the adsorption isotherm acquisition might be halted at an ultimate pressure below that of the highest possible with the apparatus, and the direction in the change in pressure reversed then. This partial adsorption process is called a scanning curve. In such a scanning experiment, the adsorption portion of the curve obtained would be the same as for the boundary adsorption isotherm, but the desorption portion may well be very different.

22

2 Gas Sorption

Hence, this part is sometimes referred to as a descending (because pressure is being lowered) scanning curve in its own right. In order to create an ascending scanning curve, it is necessary to follow the whole boundary adsorption isotherm to the highest possible pressure, then follow the boundary desorption isotherm part way down in pressure. If the direction of the change in pressure between steps on the boundary desorption isotherm (i.e. down) is reversed (to being back up) before it reaches the fully evacuated state, then the subsequent curve produced, as the pressure is increased again, is known as an ascending scanning curve. Its form may differ substantially from the equivalent portion of the adsorption boundary curve over the same pressure range (for reasons that will be discussed below), as shown in Fig. 2.2. The form of the ascending and descending scanning curves can differ between samples, and for different starting pressures (where the direction of pressure change is reversed) for the scanning curve for the same sample. In general, certain common forms are observed. Where the ascending or descending scanning curve leaves one boundary curve and joins the other at a similar amount adsorbed, it is called a ‘crossing’ scanning curve (see Fig. 2.2a). Where a descending scanning curve meets the boundary desorption isotherm at the lower (pressure) hysteresis closure point, or an ascending scanning curve meets the adsorption boundary curve at the upper (high pressure) hysteresis closure point, then the scanning curve is called ‘converging’ (see Fig. 2.2b). If, after reversing the direction of change in pressure while following along a boundary isotherm, the direction of change in pressure is reversed yet again somewhere along the scanning curve, it then produces what is known as a ‘scanning loop’, even if the change in direction of pressure change does not actually cause the path of the data set to rejoin the boundary isotherm at exactly the same place it originally left it, though this may occur.

2.2.5 Gas Uptake Kinetics In addition to data suitable for structural characterisation, gas sorption can also be used to obtain information on mass transport. Many gas adsorption apparatus, both volumetric and gravimetric, permit the obtaining of data on gas uptake against time. An example of the typical raw data for gas uptake in two different samples with different porous structures, but of same overall bulk size and shape, obtained using a volumetric apparatus is shown in Fig. 2.3a. The same uptake can be displayed using the pressure in the sample chamber, or converted to equivalent volumetric uptake. The data typically consists of two steps, down in sample chamber pressure and up in volumetric uptake. It should be noted that the time axis in Fig. 2.3a is a logarithmic scale. The first step, occurring over very short time scales (~10 s) corresponds to uptake by the sample porous material itself. It is noted that, as might be expected for samples of similar bulk profiles, the first step changes in pressure are

2.2 Nature of Experiment

23

(a)

(b) Adsorbed gas uptake/(arbitrary units)

25

20

15

10

Sample#2

5

Sample#1

0 0

100

200

300

400

500

600

Time/s Fig. 2.3 a Raw kinetic gas uptake data measured on a volumetric apparatus for a monolithic foam sample without (sample #1) and with (sample #2) mercury entrapped in macropores. b Renormalised (at 10 s) data from a showing response of sample alone

24

2 Gas Sorption

identical for both samples. However, the rate of gas uptake into the porous structure, signified by the second step, is different for the two samples. The raw data can be renormalised, such that the new origin occurs on the end of the intermediate times flat plateau in the raw data, as shown in Fig. 2.3b. Similar data to Fig. 2.3b can be obtained directly using gravimetric apparatus. Where the data is accessible, the shape of the pressure step from a gravimetric apparatus should be examined to ensure it is as close as possible to a right-angled step. In real experiments, the pressure rise occurs over a finite period of time, and this should, ideally, be very short compared to time scale of gas uptake into the sample itself. This means the boundary condition for subsequent analysis is simplified. Where this is not possible, a more complex analysis than discussed below is required (and discussed in standard texts such as Crank 1975). Further, the temperature variation of the sample during gas uptake should also be examined if possible. Since adsorption is exothermic, there is often an exotherm when adsorption first begins (when the most adsorbate is likely to be adsorbed) and, if the heat does not escape the sample quickly, can lead to a rise in temperature of the sample itself which briefly can raise the diffusivity (since molecular diffusion is ∝T 3/2 ) before the sample cools down again. This can make the data look like it has two different diffusion components, namely a fast early uptake followed by a slower later uptake. This can be mistaken as apparent fast uptake into a readily accessible network, followed by slow ingress into a less accessible microporous network. The gas uptake data can be fitted to a range of equations, including the Linear Driving Force (LDF) model, and spherical and cylindrical geometry solutions to the diffusion equation with well-mixed and constant volume boundary conditions (Crank 1975). For gas uptake experiments, the appropriate solution to the diffusion equation can often be approximated by the so-called Linear Driving Force (LDF) model and uptake data fitted to the expression (Do 1998): M(t) = 1 − exp(−kt), M(∞)

(2.1)

where M(t) is the measure of uptake at time t, M(∞) is the amount adsorbed at infinite time, and k is a mass transfer coefficient (MTC). The mass transfer coefficient itself is given by: k=

GD , a2

(2.2)

where G is a geometrical constant (15 for spheres, 8 for cylinders), D is the effective diffusivity of the porous medium/network, and a is the characteristic diffusion length (such as the radius of a spherical or cylindrical adsorbent). The LDF tends to fit best to the upper 50% of the uptake curve.

2.2 Nature of Experiment

25

The full solution for the appropriate diffusion geometry can be used when a better fit to the full uptake curve is required. For a cylinder submerged in a ‘well-stirred’ reservoir, the appropriate solution to the diffusion equation implies the rate of uptake at time t which can be described by (Crank, 1975): ∞    4α(1 + α) Mt =1− exp −Dqn2 t/a 2 , 2 2 M∞ 4 + 4α + a qn n=1

(2.3)

where qn are positive nonzero roots of: αqn Jo (qn ) + 2J1 (qn ) = 0.

(2.4)

For a sphere submerged in a ‘well-stirred’ reservoir, the rate of uptake at time t can be described by: ∞    6α(1 + α) Mt =1− exp −Dqn2 t/a 2 , 2 2 M∞ 9 + 9α + a qn n=1

(2.5)

where qn are positive nonzero roots of: tan qn =

3qn . 3 + αqn2

(2.6)

The foregoing equations are suitable when the sample consists of a homogeneous, isotropic diffusion medium. Some samples may have some sort of internal partition, or patchwise heterogeneity, such that they behave like two separate media for adsorption. In such a case, a composite LDF model may offer a better fit: M(t) = p(1 − exp(−k1 t)) + (1 − p)(1 − exp(−k2 t)), M(∞)

(2.7)

where p is the fraction of component 1, with MTC of k 1 .

2.3 What Can I Find Out with This Method? 2.3.1 Surface Area The surface area of an adsorbent can be obtained from analysis of experimental data using a model for monolayer or multi-layer physical adsorption. Finding a surface area involves using a model of adsorption to determine the monolayer capacity, and an assumption about the effective area occupied by a single molecule on the surface. An adsorbate surface might be conceived of as like a chessboard, with each square

26

2 Gas Sorption

corresponding to a site for a single molecule to adsorb, and, thence, the monolayer capacity is the number of squares on the board. The two most commonly used adsorption models are the Langmuir and Brunaer–Emmett–Teller (BET) equations. Mathematical derivations of these equations are given elsewhere (Gregg and Sing 1982). The key issues to bear in mind during the use of these models are the common assumptions made in their derivations. First, it is assumed that adsorption has reached equilibrium, and, as mentioned above, this is a key consideration for the experimental conditions. Second, it is assumed that the surface is homogeneous, with all surface adsorption sites having the same heat of adsorption, and, third, that the surface of the adsorbent is flat. This is unlikely to be true for any adsorbent, except a few exotic types like graphite. The implications of this will be discussed below. Fourth, both aforementioned models assume no lateral interactions between molecules. This assumption is most closely approximated when the heat of adsorption is much greater than the latent heat of vaporisation of the adsorbate. The Langmuir model assumes that adsorption only arises in a monolayer, where all molecules are in contact with the surface. The Langmuir isotherm equation is: BP V , = Vm 1 + BP

(2.8)

where V is the amount adsorbed, Vm is the monolayer capacity, P is the pressure, and B is an empirical constant related to heat of adsorption, given approximately by: B ≈ eq1 /RT ,

(2.9)

where q1 is equivalent to the isosteric heat of adsorption of the first adsorbed layer (monolayer), assumed to be equal for all adsorption sites. The shape of plots of Eq. (2.8) is similar to Type I isotherms according to the IUPAC classification scheme (Fig. 2.1). To obtain the monolayer capacity, Eq. (2.8) is linearised, such that: 1 1 1 1 = + . v vm B P vm

(2.10)

Hence, if the data obeys the Langmuir model, a plot of 1/V against 1/P should be a straight line. The monolayer capacity is the reciprocal of the intercept of that straight line. The constant B is obtained from the ratio of the intercept to the slope. Once the monolayer capacity has been obtained, the specific surface area can be obtained by multiplying that capacity (in molecules per gram) by the effective crosssectional area (CSA) of a single molecule. The ISO (2010) recommends a value of 0.162 nm2 for nitrogen. The monolayer capacity can also be thought of as the complete filling of narrow micropores, which is an idea that will be discussed below. The standard CSA value for nitrogen is calculated assuming the adsorbed phase is

2.3 What Can I Find Out with This Method?

27

a liquid-like, close-packed phase. However, if the binding of the molecule to the surface is weak, as indicated by a BET constant less than 10, then the molecules may be more mobile and the bed of adsorbed molecules will expand. This means the area occupied by a single molecule will be higher than the standard cross-sectional area of 0.162 nm2 . The variation of effective cross-sectional area with BET constant (and thus q1 ) for a range of different adsorbates has been given by Karnaukhov (1985). The BET model expands on the Langmuir model by allowing for multi-layer adsorption, where the adsorbate can adsorb upon itself, as well as on the adsorbent surface. In the standard BET model, it is assumed that adsorption sites for molecules in the second and subsequent layers are directly on top of previously adsorbed molecules in layers below. It is assumed that molecules in the second and higher layers only interact with molecules directly (vertically) above and below them, and that the heat of adsorption is then equal to the specific latent heat of vaporisation. The generalised BET equation is given by: C x 1 − (N + 1)x N + N x N +1 V = , Vm 1 − x 1 + (C − 1)x − C x N +1

(2.11)

where x is the relative pressure, N is the maximum possible number of adsorbed layers and C is an empirical constant given by: C ≈ e(q1 −q L )/RT ,

(2.12)

where qL is the latent heat of vaporisation. Low values ( 0.3) in the BET plot, as shown in Fig. 2.4b. This means the model is overpredicting the amount adsorbed at higher pressures. This can happen because the real sample violates the third model assumption listed above and is not flat. Real surfaces are rough on the molecular scale. Concavities in the surface (and, overall, even disordered porous materials are concave) mean that the space for adsorbing the second and subsequent layers declines with distance from the surface,

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2 Gas Sorption

Fig. 2.5 Example of a two-component BET model (Eq. 2.14) composite isotherm (thick line) and its two constituent isotherms (thin lines). Reprinted with permission from Watt-Smith et al (2005), Copyright (2005) American Chemical Society

and the maximum molecular capacity of the model decreases. For rough surfaces that possess the special property of self-similarity over different length-scales, and are, thus, fractals, the decline in maximum capacity is given by the equation: Ai = i 2−D , A1

(2.16)

where A1 is the area in the first adsorbed layer, Ai is the area in the ith adsorbed layer, and D is the surface fractal dimension (2 ≤ D ≤ 3). This effect can be incorporated into the BET model and a fractal version of the BET equation derived (Mahnke and Mögel 2003). This is given by:  log(V ) = log(Vm ) + log

 Cx − (3 − D) log(1 − x), 1 − C + Cx

(2.17)

where the symbols are as defined above. Examples of plots of Eq. (2.17) with various values of the parameter D are given in Fig. 2.6. It can be seen that the various isotherm curves are very similar in the standard BET region, around the initial adsorption knee associated with the formation of a statistical monolayer, but diverge as more layers are adsorbed at higher pressures. It is noted that, as the surface becomes rougher, reflected in a higher fractal dimension, the amount adsorbed is decreased, due to the decline in maximum possible occupancy in successive adsorbed layers. As D → 3,

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Fig. 2.6 Form of Eq. (2.17) for various values of the fractal dimension D

the isotherm shape tends to that of the Langmuir isotherm (Eq. (2.8). This is because the very roughest surfaces can only fit a single layer of adsorbed molecules in the very convoluted crevices of their surfaces. Case Study on Surface Area #1—Use of Different Isotherm Models The foregoing discussion has highlighted that the standard BET surface area determination method has a number of issues, related to the assumptions made in the theory, when studying real materials. In particular, if the BET approach is just treated as a curve-fitting exercise, then there is no way of knowing how accurate the surface area is likely to be. However, there are a number of ways that a more sophisticated approach can be taken to improve accuracy and reliability. Figure 2.7 shows the raw experimental adsorption isotherm data for a glass sample fitted to the BET model (Eq. 2.13) using the approach described in the International Standard (ISO 2010), and two component homotattic patch models (Eq. 2.14) with either fractal BET (Eq. 2.17) components, or Langmuir and Henry’s law components. The standard BET model was fitted over the range of relative pressures from 0.05 to 0.3, and the homotattic patch models were fitted over the range of relative pressures from 0 to 0.4. However, in Fig. 2.7 the lines for the fitted models have been extended beyond the fitted range to see how well the models predict the upper reaches of the isotherm. It can be seen that the standard BET fit deviates from the experimental data at relative pressures only just beyond the fitted range. Indeed, it overpredicts the amount adsorbed so it is unphysical in this range. In contrast, the Langmuir and Henry’s law components homotattic model continues to fit the experimental data well beyond the fitted range up to relative pressures of ~0.5–0.55. However, for

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relative pressures of ~0.6 and beyond it underpredicts the experimental data, and this might be because of the onset of capillary condensation, which is not accounted for in the Langmuir or Henry’s law models. However, the two-component fractal BET, homotattic patch model, fits the data even up to relative pressures beyond 0.8, despite only being fitted below relative pressure of 0.4. The extent to which an isotherm model fits the experimental data beyond the fitted range, but still within the relevant physical range for the model, can be used as a way to distinguish between models. In this case, the two-component fractal BET model seems the best model based on potential to fit observations beyond the fitted range. Case Study on Surface Area #2—Use of Different Adsorbates While repeating experiments and the fitting process provides estimates of the random error present in the values of surface area for a new class of chemically heterogeneous porous material measured by the common nitrogen BET method, it provides little indication of the lurking systematic errors. As explained in the Introduction, the strategy to ensure accuracy advocated here is consilience, or the harmonisation of evidence from different sources. The potential use of different adsorbates represents a simple example of the variation that can be introduced to make manifest the accuracy of the methods.

Fig. 2.7 Multi-layer region, and beyond, of conventional nitrogen adsorption isotherm for glass sample. The solid line represents a fit of the standard BET model according to the ISO (2010) method. The dashed line is a fit for 0 < x < 0.4 of a homotattic patch model (Eq. 2.14) with two fractal BET components (Eq. 2.17). The dotted line is a fit for 0 < x < 0.4 of a homotattic patch model with Henry’s law and Langmuir (Eq. 2.8) components

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The simple case study presented here is the comparison of the characterisation parameters obtained for partially dehydroxylated silica surfaces. Adsorption isotherms for a range of different adsorptives can be obtained for the same silica. Adsorption isotherms for nitrogen (at 77 K), argon (at 87 K), propane (at 199 K), butane (at 273 K), and hexane (at 273 K) on G1 silica were obtained. These data have been fitted to the fractal BET model (Eq. 2.17). An example of the fit to the isotherm for propane is shown in Fig. 2.8. The range of fits was constrained to be in the region before the hysteresis loop so that there will be no capillary condensation. The fractal dimensions from the fits are shown in Table 2.1.

Fig. 2.8 Propane adsorption isotherm obtained at 273 K for sol-gel silica G1 (multiplication). Also shown is a fit of the data to Eq. (2.17) (solid line)

Table 2.1 Surface fractal dimensions obtained from the adsorption isotherms for various adsorptives for sol-gel silica G1

Adsorptive

Surface fractal dimension (±0.01)

Range of fit in relative pressure

Nitrogen

2.38 ± 0.01

0.05–0.60

Argon

2.25 ± 0.01

0.05–0.60

Propane

2.15 ± 0.01

0.05–0.57

Butane

2.25 ± 0.01

0.05–0.37

n-Hexane

2.00 ± 0.01

0.02–0.59

Cyclohexane

2.00 ± 0.01

0.004–0.6

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The equivalent surface fractal dimension value obtained from Porod analysis of small angle X-ray scattering (SAXS) (see Chap. 4) data was 2.27 ± 0.11 (Watt-Smith et al. 2005). It is noted that the fractal dimension for nitrogen is significantly higher than that from SAXS, whereas that for the hexanes is significantly lower, while those for the rest of the adsorbents is within experimental error and sample variability. Where the same value of a characteristic adsorbent structural parameter has been obtained by two different physical processes (gas adsorption and X-ray scattering) using two different theories of analysis (BET and Porod), this suggests that the theory describing the adsorption is physically correct. In this case, the adsorption model was the BET mechanism, which includes just vertical Van der Waals interactions, and neglects capillary condensation. However, deviations from theory can also be informative about adsorption mechanisms and/or the nature of the adsorbent. As shown using Eq. (2.16), the main effect of the concavities associated with a rough fractal surface, incorporated into the fractal BET equation, is the reduction in number of adsorption sites with increasing adsorbed multi-layer thickness. An increasing fractal dimension is associated with a stronger manifestation of this effect. However, other aspects of the adsorption mechanism can also give rise to an enhanced version of this effect. For example, if adsorption on the surface occurs predominantly on isolated, finite patches, the stable form of the adsorbed phase could be more like a pile of oranges in a supermarket, whereby the adsorbed layers form a pyramidal structure (Watt-Smith et al. 2005). The pyramidal structure also has reduced numbers of sites with each successive layer, and this reduction follows a similar power law to Eq. (2.16) (Watt-Smith et al. 2005). In Table 2.1, nitrogen has a higher than expected fractal dimension that might be due to this effect. Thermal pre-treatment of silicas can lead to partial dehydroxylation of the surface. The quadrupole moment of nitrogen means it has much greater adsorption affinity for the patches of the surface that remain covered in polar hydroxyl groups. Hence, the adsorption on silicas for nitrogen could arise predominantly on the remaining hydroxylated surface patches leading to an overestimate of fractal dimension, and, also, a concomitant underestimate of surface area. Further, as will described in more detail in Chap. 6, there are other discrepancies in the behaviour of nitrogen and argon, in particular in how they wet surfaces, with argon not wetting heavy metals like mercury. The underestimate of fractal dimensions from adsorption, as occurs for the hexanes in Table 2.1, can also be revealing concerning the idiosyncrasies of the mechanism of adsorption. A reduction in surface fractal dimension can arise if the earliest stages of adsorption preferentially fill in the concavities (ruts) in the surface, such that the resultant subsequent surface for adsorption is smoother than before (Pfeifer et al. 1991). This effect could be confirmed by freezing in place the first adsorbate acting as a filler and measuring the surface fractal dimension of the resultant with a different adsorbent (Pfeifer et al. 1991). Micropore Volume Strictly speaking the BET model is not even applicable to microporous solids because the mechanism of adsorption in the BET physical model is not that which occurs

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in such materials. In microporous solids, the surface potentials of the two opposite walls of a pore overlap meaning a clear distinction between the adsorbed phase and bulk gas is not possible. Filling of micropores occurs by continuous densification of the whole pore fluid (albeit with a gradient in density across the pore) rather than localisation in a specific. much denser multi-layer. The simplest method to obtain the micropore volume, for an adsorbent giving rise to a Type I isotherm, would be to fit the data to the Langmuir model (Eq. 2.8). The adsorption capacity parameter could then be converted to a micropore volume by, for example, assuming the adsorbed phase has the same density as the bulk liquid adsorbate at the isotherm temperature. For cases where the sample contains both microporosity and mesoporosity, then a two-component, homotattic patch model (Eq. 2.14) could be fitted to the data with a Langmuir (or other suitable isotherm) component to represent micropore adsorption, and a BET component to represent mesopore multi-layer adsorption. The micropore volume only can then be obtained from the capacity parameter of the Langmuir component. The classical t-plot and α s -plot approaches to determine micropore volume were developed when computers to do the above fitting were much less available (Gregg and Sing 1982). The classical t-plot is based on comparing the shape of the observed isotherm on a given adsorbent with that expected from a standard isotherm for adsorption on a non-porous solid. If micropores are present, an excess adsorption will occur at low pressure that will not be accounted for by the standard isotherm. The classical t-plot consists of a plot of the experimentally measured amount adsorbed against the average thickness of the t-layer (multi-layer) obtained at a given relative pressure from a standard isotherm. The t-layer thickness is generally obtained from a so-called universal t-layer equation, such as those of Halsey (1948), or Harkins and Jura (1944), where the latter is given by: 0.5  t = 13.99/(0.034 − log( p/ p0 )) .

(2.18).)

In principle, this plot should be a straight line through the origin if there are no micropores, and with a nonzero intercept equivalent to the microporous volume, if there are. However, the classical t-plot will only have a significantly sized region that is linear if the standard isotherm is sufficiently representative of the material being tested. The aforementioned homotattic approach has more flexibility to account for sample heterogeneity, and, thence, deviations from the standard isotherm shape. Specific Pore Volume for Meso-/Macroporous Materials If the gas adsorption isotherm turns over to create a flat plateau at the top, then the amount adsorbed can be easily turned into an estimate of total specific pore volume. The ultimate amount adsorbed can be multiplied by a molar volume for the likely adsorbed state. This is often assumed to be bulk liquid adsorbate at the isotherm conditions. The value thereby obtained is typically known as the Gurvitsch volume (Gregg and Sing 1982). However, many adsorption isotherms are still hyperbolic in form (i.e. rising vertically) at the highest relative pressures achieved in the conventional experiment. The

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conventional experiment seeks to avoid filling the whole sample holder with bulk condensed nitrogen, and thus stops short of saturation pressure. In contrast, overcondensation experiments seek to achieve this state, with a view to determining the pore-filling volume of condensed phase (Murray et al. 1999). In the overcondensation experiment, the first step is to raise the pressure in the sample tube to higher than the saturated vapour pressure of nitrogen. This pressure rise should encourage sufficient condensation such that even the pores of the largest size are filled with liquid nitrogen at the start of the overcondensation desorption isotherm. The requisite period to achieve this stage is determined by the sample size and the pore volume. Once complete pore filling had been achieved, the pressure can be lowered to just below the saturated vapour pressure of nitrogen such the bulk condensate vaporised completely while keeping all the sample internal porosity liquid filled. Once this stage has been accomplished, the first data point on the overcondensation desorption isotherm can be measured. This point corresponds to the total pore volume of the sample. The pressure is then progressively lowered in small steps, and the rest of the desorption isotherm can be obtained in the usual way.

2.3.2 Pore Size Methods In addition to total micropore volume, it is possible to obtain the micropore size distribution, and, also, the size distribution of meso- and macropores from gas sorption. The key aspect of the analysis method to obtain pore size distributions is the physical description/theory used for the pore-filling and/or capillary condensation process(es). For micropores, the pore wall potentials will overlap to some extent in the middle of the pore. This means that the pore tends to fill by a gradual densification process with increased pressure, rather than the step change associated with capillary condensation. The former is associated with a Type I isotherm, while the latter process is associated with isotherm Types IV and V. Kelvin–Cohan Equation-Based Methods The earliest approach, based upon classical continuum thermodynamics, and derived using the Young–Laplace equation, was the Kelvin equation, which written for a cylindrical pore is given by Gregg and Sing (1982):

p ln p0

=

κγ V¯ cos θ, rC RT

(2.19)

where κ is a geometry parameter and depends on the pore and meniscus type. For a cylindrical pore open at both ends, where the adsorbed film forms a cylindrical sleeve-shaped meniscus, κ = 1, while for a pore with one dead end, or for desorption, condensation occurs at a hemispherical meniscus such that κ = 2. γ is the surface tension, V¯ is the partial molar volume, R is the universal gas constant, θ is the gas–liquid–solid contact angle (typically assumed to be zero), and r c is the pore

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core radius. The characteristic size parameter in the Kelvin equation corresponds to the empty core of the pore, since gas will generally (for large (>10) BET constant surfaces) already have adsorbed in a film on the pore surface. The actual pore size, r p , is given by: r p = rc + t,

(2.20)

where t is the so-called t-layer thickness of the adsorbed film given by Eq. (2.18). The geometry factor κ in the Kelvin equation gives a simple account of the cause of hysteresis in sorption isotherms for capillary condensation in isolated, through cylindrical pores open at both ends. In such pores, condensation occurs from a cylindrical-sleeve meniscus along the pore wall, whereas evaporation occurs from a hemispherical meniscus at the pore ends. When this is occurring, the values of the geometry factor κ mean that the relative pressure for the desorption isotherm should be the square of the corresponding relative pressures for the adsorption isotherm. However, while Cohan (1938) showed this was the case for some materials, it is not generally the case. The value of the geometry factor κ can vary between 1 and 2 for different samples. It is generally assumed, for nitrogen, that the adsorbed condensate is perfectly wetting of the surface such that the contact angle is zero, and thus the cos θ term is unity (Gregg and Sing 1982). However, some studies (Androutsopoulos and Salmas 2000) have allowed the cos θ term to be a free-fitting parameter, and, by doing so, it has been shown that a corrugated, cylindrical pore geometry can give rise to all of the IUPAC standard hysteresis loop types (Thommes et al. 2015), thereby suggesting differences in pore wetting can explain differences in isotherm shape. Hence, forms of hysteresis loops are not definitive indicators of pore shape, as has been suggested (Gregg and Sing 1982). From Eq. (2.19), the condensation and evaporation pressures for the same adsorbate in the same pore, or the condensation pressures for the same adsorbate in open/closed pores of the same diameter, or identical pores with different wetting properties, can be related via the ratio: ln

P P0

= 1





P k1 cos θ1 P 1 ln = ln , k2 cos θ2 P0 2 δ P0 2

(2.21)

where the subscripts 1 and 2 refer to either condensation and evaporation, respectively, or two different pores of the same radius. For condensation and evaporation, for a through cylindrical pore with a fully wetting surface, k 1 = 1, k 2 = 2, and the cos θi terms both equal unity. Hence, in that case, the power δ is equal to 2, and the relative pressure for evaporation is the square of the relative pressure for condensation. This case corresponds to the well-known Cohan (1938) equations mentioned above. For less wetting surfaces, cos θi would be less than unity, and, thus, the power would be less than two. Previous work has shown that the power δ to superpose adsorption and desorption branches for non-local density functional theory (NLDFT) kernels calibrated with fumed silica is 1.8. For a fully wetting equilibrium sorption system, with no hysteresis, the power would be unity. Studies of templated KIT-6 (Kleitz

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2 Gas Sorption

et al. 2010) and disordered silicas (Hitchcock et al. 2014) have suggested that δ = 1.5 for three-dimensional, interconnected networks. The PSD obtained from gas sorption is typically a probability density function for size weighted by pore volume. In order to obtain a pore size distribution, the Kelvin equation is typically incorporated into the Barrett–Joyner–Halenda (BJH) (Barrett et al. 1951) algorithm, or similar (Gregg and Sing 1982). The algorithm takes into account that, at a given pressure step, for a sample with a wide range of pore sizes, then some pores may be full of condensate and no longer adsorbing at all, while some other pores may just have a surface film that simply gets thicker with increased pressure, and some pores may have reached the critical pressure given by Eq. (2.19) such that capillary condensation is occurring. Irrespective of whether the isotherm is actually obtained from adsorption or desorption, the BJH algorithm starts the calculations at the top (highest pressure) point of the isotherm and makes the assumption that the sample is completely pore-filled with condensate at this point. Hence, the very first pressure step is associated only with condensation/evaporation for the very largest pores in the sample. Thereafter, the calculation involves two contributions to the amount of adsorbate adsorbing/desorbing. These two contributions are the change in thickness of the multi-layer film for pores without condensate in the pore core, and the condensate for pores reaching the critical pressure for capillary condensation/evaporation. Hence, for pore structures with isotherms of the form of that shown in Fig. 2.4a where there is a flat plateau at the top of the isotherm the porefilling assumption of the BJH algorithm is correct. However, for scanning curves of the form shown in Fig. 2.2b, where some of the void space is left empty at the top of the isotherm, then the BJH calculation will neglect the multi-layer film variation in pores not filled at the top of the isotherm. This will be taking place in the real sample for pressures in the range of the scanning curve, and, thus, will be confused with the adsorbate for smaller pores, and, thence, the pore volumes for smaller pores will be overestimated by the BJH algorithm. This problem can be solved, at least for desorption isotherms, using the aforementioned overcondensation method that enables complete pore filling to be achieved even for macroporous materials. The universal t-layer equations used in the BJH algorithm are generally obtained for non-porous solids whose surfaces are approximately flat, such that the growth of the multi-layer thickness is steady with increasing thickness. However, pores in a mesoporous solid represent extreme confinement, and so the pore surface will have a small radius of curvature. Broekhoff and De Boer (BdB) (1967) introduced a pore size distribution analysis that included the impact of the strong surface curvature on the multi-layer thickness. The power δ in Eq. (2.21) to superpose adsorption and desorption branches for the BdB method is 1.5, which is noted as the same as that for amorphous silica (Hitchcock et al. 2014). The fact that the Kelvin equation is derived using bulk thermodynamics means it also has limitations when applied to very small pores. The problem is best appreciated from a consideration of the adsorbate density variation across the pore. For a large pore, the deepest parts of the wall potential well are largely confined to the proximity of the wall, and thus does not extend beyond the multi-layer film. In this case, it is reasonable to suggest that the adsorbed film is liquid-like and the pore core

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before condensation is occupied by adsorbate of gas-like density. Hence, the density profile shows a step change at the edge of the adsorbed film. The meniscus is also of sufficient size that it can be treated as a continuum. The early adsorption below the bulk saturated vapour pressure arises from the curved meniscus geometry. However, as the pore size declines the macroscopic concepts of a well-defined meniscus, and surface tension, become less appropriate. This is because as the pore gets smaller, the walls become closer together, and the wall potentials begin to overlap substantially. This creates a greater attractive force in the centre of the pore, and the adsorbate density becomes larger than the bulk gas phase, even at relatively low pressures. This means that models based upon a clean phase transition between ‘empty’ and ‘full’ pores become inappropriate. Hence, more sophisticated adsorption models take this into account. Density Functional Theory (DFT) Some authors (Landers et al. 2013) have suggested that the Kelvin–Cohan equations tend to underestimate the pore size below ~20 nm. However, other authors (Kruk et al. 1999) have found that, with minor correction, it is accurate down to a few nanometres. Nevertheless, at the time of writing, the most commonly used models are based upon density functional theory (DFT). This method is based on the following approach. The density functional is calculated for pores of a given surface chemistry (surface potential), size and geometry, at a particular pressure. The density functional can be converted to an amount adsorbed. This is repeated for different pressures to construct a full isotherm for pores of a single characteristic size. This complete process is repeated in turn for pores of different sizes. The complete set of calculated isotherms for a range of different sizes of pore, of a particular geometry and chemistry, is called the ‘kernels’. The experimental isotherm obtained for a real sample is then assumed to be a composite isotherm made up of the particular contributions from each kernel isotherm in the set. The procedure to obtain the pore size distribution thus consists of a fitting process, whereby the sizes of the individual contributions from each kernel are determined. The histogram of these contributions constitutes the pore size distribution. DFT kernels are available for slit and spherical pores besides cylinders. If it is known in advance (such as from microscopy data) that pores in particular size ranges have differing pore geometry, then this can be built into fitting procedure (Landers et al. 2013). The fitting of the kernels to the experimental isotherm is a so-called ill-posed problem, as it can have infinite solutions. Different software can use different regularisation techniques to overcome this issue and thus give rise to different solutions (Landers et al. 2013). The DFT pore size distribution can appear smoother than that from the BJH method, since the obtaining the former involves a certain degree of regularisation, while the latter directly uses the isotherm points (Jagiello and Jaroniec 2018). The precision of the pore size distribution obtained also depends upon the number density of kernels over a particular pore size range available for the fit, and in some commercial software packages this number tends to decline for larger pores. If the available kernel number density is low, this can result in artefacts in the PSD in the form of abrupt steps in pore volume at the position of the available kernels.

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Unfortunately, the process is not completely a priori predictive of the experimental isotherm since it is necessary to calibrate the fluid–fluid interaction parameter by comparing DFT predictions for bulk fluid liquid–vapour equilibrium, gas density, saturation pressure, and surface tension. However, Jagiello and Jaroniec (2018) suggested the bulk fluid as reference may not be appropriate for highly confined fluid within pores. In addition, the adsorbate–adsorbent interaction potential must be calibrated by adjusting this parameter such that the DFT kernel for adsorption on a non-porous solid matches that for adsorption on a suitable reference sample. For example, the reference sample for silica surfaces is a fumed silica (Ravikovitch et al. 1995). DFT kernels are typically available for silica and carbon surfaces. For surfaces, like metal organic frameworks (MOFs), where explicit kernels are not available, it is difficult to predict which would be appropriate in advance. One strategy might be to try different kernels and consider the level of agreement between different pore size methods. As with the BJH algorithm, it is necessary to make an assumption about the mechanism of capillary condensation on the branch of the isotherm being used. The options in DFT are the equilibrium or spinodal (metastable) kernels. The development of ordered, templated silica materials, such as MCM-41 and SBA-15, has enabled the correct choice for arrays of parallel straight cylinders to be determined. Neimark and Ravikovitch (2001) have shown that, for DFT calibrated against fumed silica, the experimental adsorption isotherm for SBA-15-type materials tends to match the spinodal DFT kernel, while the desorption isotherm tends to match the equilibrium kernel (as there are no pore-blocking effects). However, for pores smaller than 5 nm, the non-local DFT (NLDFT) spinodal isotherms deviate from the experimental adsorption isotherms. This is because the NLDFT does not account for the nucleation phenomenon (Landers et al. 2013). Below a pore size of ~4 nm, the capillary condensation becomes reversible and matches the equilibrium kernel. In the earliest versions of DFT algorithms, the surfaces of the model pores used to determine the kernels were smooth. This tended to lead to the adsorbed film building up in successive complete layers in turn, such that the shape of the kernels had the step-like form with each step corresponding to the completion of a full layer. This shape is unlike real materials, which tend to have smooth build-up of adsorbate, and led to artefacts in the PSD. The cause of the issue is that real materials tend to have surfaces with heterogeneous chemistry and/or surface roughness, which lead to a broad range of strengths of interaction across the various adsorption sites, and, thence, more progressive occupation. The artefactual layering tended to cause artificial gaps in the PSD at ~1 and ~2 nm (Landers et al. 2013). Later versions of DFT models have incorporated surface heterogeneity to avoid this problem. One approach is quenched solid density functional theory (QSDFT). This suggests that the inner part of the pore wall has a constant gradient in solid density with the roughness parameter δ representing the half-width of the region with variable density. It is suggested this represents the so-called corona of microporous wall that surrounds the main mesoporous channel in many templated silicas (Landers et al. 2013). The roughness parameter can be independently fixed using XRD data, or used as another free-fitting parameter. It should be noted that the pore width is defined in a slightly

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different way between NLDFT and QSDFT, so the PSD might not be comparable (Landers et al. 2013). A different approach was proposed by Jagiello and Jaroniec (2018). They derived DFT kernels for a straight cylindrical pore with a sinusoidal modification of the pore diameter along the long axis of the cylinder. The parameters of the model were the amplitude and wavelength of the sinusoid. DFT has been especially used to study cavitation. The presence of cavitation is often manifested by a sharp desorption step (always) at around relative pressures of ~0.4–0.45 for nitrogen. Cavitation occurs when the metastable condensed phase within a pore body of an ink-bottle pore evaporates before the neck empties, in contrast to pore blocking where both body and neck empty together. It has been proposed that cavitation occurs because the adsorbed phase becomes ‘stretched’ such that it breaks, and is thus also known as the tensile strength effect (Gregg and Sing 1982). Since the relative pressure at which the cavitation effect occurs is thought to be largely determined by the properties of the adsorptive itself, rather than the adsorbent, if it occurs, then the desorption isotherm will contain no information about pore size at that pressure. It is, thus, useful to know when it is happening. A test for the presence of the cavitation effect is to compare the PSD obtained from the adsorption and desorption branches of the isotherm for two different adsorptives (such as nitrogen and argon). If the PSDs from the two hysteresis loop branches agree for one adsorptive but not the other, then cavitation is probably present in the one with disagreement. Monte Carlo and Other Simulation Methods The key limitation of the Kelvin–Cohan and (NL)DFT approaches is they are restricted to regular geometries, when many interesting materials are disordered and amorphous. In contrast, the mean-field version of DFT and Monte Carlo simulations can cope with more irregular geometries in disordered solids. The mean-field DFT (MFDFT) approach is lattice-based and thus can be used with heterogeneous structures (Kierlik et al. 2002). MFDFT is a highly simplified description of the underlying physical process and is not suitable for quantitative studies of adsorption. However, it can describe the essential processes involved in gas sorption. In particular, MFDFT studies have suggested that the hysteresis for disordered porous solids results from the complex energy landscape that arises for the various different conformations possible for adsorbed ganglia in such heterogeneous materials. The adsorbed phase can become kinetically trapped in a suboptimal energy minimum on both the adsorption and desorption isotherms, and, hence, neither represents the true equilibrium isotherm (Kierlik et al. 2002). This suggests that using the simplified approaches described above can only give approximate pore sizes. Monte Carlo (MC) simulations consider the adsorption process at the scale of individual molecules. Simulations are performed under a particular set of thermodynamic conditions, with the Grand Canonical (GC) ensemble being the most commonly used (Gelb and Gubbins 1998, 1999). Monte Carlo simulations require explicit statement of interaction potentials between molecules and surfaces. In the simulation,

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2 Gas Sorption

the void space is populated with molecules and the energy of the system is calculated. Molecules are then introduced, removed, or moved at random, and the energy of the system recalculated after each rearrangement step. The simulation proceeds until the minimum energy configuration of molecules is found, subject to the external constraints imposed upon the system. The adsorbed amount can then be calculated. In principle, this could be done for every pressure point in an isotherm, and for a range of different possible pore structures, and the results compared with experiment. However, GCMC simulations are extremely computationally expensive in CPU time, and thus this approach is not practicable with current computers. Instead, GCMC can be used to test and validate simpler theories that are more computationally efficient. Gelb and Gubbins (1998) tested the BET analysis of simulated nitrogen adsorption isotherms obtained for model CPG materials and found it gave rise to a good estimate of surface area for larger pore materials. However, for pore sizes less than 4 nm, the BET method tended to overestimate surface area due to high monolayer densities created by the strong surface curvature. Gelb and Gubbins (1999) conducted a similar test of the BJH PSD analysis method. For simulated CPG structures, they found that the PSDs were in surprisingly good agreement with a geometric PSD (described in Chap. 1), but the BJH PSDs were slightly sharper and systematically shifted, by about 1 nm, to lower pore sizes.

2.3.3 Pore Connectivity Gas sorption can be used to determine topological information for porous solids. This is based upon the application of percolation theory to the understanding of the cause of hysteresis. One reason for the presence of hysteresis is the so-called pore-blocking, or ‘ink-bottle’ effect. During desorption, liquid-like condensate can only evaporate if it is right next to vapour, and thus the liquid has somewhere to desorb to. However, if, as shown schematically in Fig. 2.9, a large pore with liquid condensate below its critical pressure is located behind a narrow pore neck still full of stable condensate, then the liquid in the large pore cannot desorb. Desorption from the large pore can only occur once the pressure has dropped below that to make the liquid condensate

Fig. 2.9 Schematic diagram depicting a pore network where the pore-blocking effect would arise

2.3 What Can I Find Out with This Method?

43

in the small pore neck unstable and, thence, evaporate. At this point, the vapour– liquid meniscus would retreat to the end of the larger pore and evaporation can then commence from it. Therefore, pore blocking leads to a delay in desorption to below the equilibrium phase transition pressure. In a real, randomly disordered porous solid, there may exist many different pathways, from a given pore to the surface of the material, down which a meniscus may reach the pore, allowing evaporation of condensate. The higher the number of different pathways, the easier it will be for the condensate to evaporate at higher pressures, since it is then less likely to be blocked by a small pore neck. Hence, the larger the pore network connectivity (average number of pores meeting at an intersection), the narrower the hysteresis is likely to be. In a large pore network, one small neck size can govern access to many larger pores. Once this neck desorbs, the other pores can simultaneously desorb. This effect is responsible for the often sharp knee, and steep step, observed in some Types IV and V desorption isotherms (see Fig. 2.10). Methods for determining pore connectivity, thus, depend upon converting the hysteresis width into a pore connectivity. Percolation theory can be used to assess the likelihood, given a certain fraction of the pores being below their critical evaporation pressure, that a vapour pathway will exist to a given pore. Detailed descriptions of the methodology can be found in work by Seaton (1991) and are discussed in more detail in Chap. 6. The sharpness of the knee in the desorption isotherm depends upon the abruptness of the percolation transition. These both thus, in turn, depend upon how many pores empty of condensate as the percolation transition is approached, relative to during the transition itself. Strictly, the lattice size of the pore network is the side length in terms of numbers of individual pores traversed when passing from one side of the lattice to the opposite. If the sample size is small, the lattice size is also typically small, and, thus, the surface-area-to-volume ratio of the lattice is large. In such a case, many more pores are surface pores, such that they have direct access to the vapour phase. This means that it is more likely, for samples with small particle sizes, that the condensate may desorb from many surface pores before a pathway is formed spanning across the whole pore network to the centre of the particle. In this case, desorption will commence even before the pressure corresponding to the knee in the desorption isotherm, and the knee will be more rounded. In a very large random lattice, only a very small fraction of the largest pores will be on the surface, and thus, the desorbed amount prior to the percolation knee will be negligibly small. Hence, the sharpness of the desorption knee is a potential proxy indicator for the lattice size. However, even for a very large lattice, if there is a significant spatial correlation in pore size, such that there are vast regions of larger pores adjoining the surface, then the premature desorption of condensate from large surface pores can be much more significant. In such a case, a large spatially correlated lattice would appear like a small random lattice and have a well-rounded percolation knee (Fig. 2.10). The size of the pore-blocking effect (as manifested by the width of the hysteresis) can be reduced by the presence of vapour pockets within the pore network at the top of the isotherms, rather than the whole network being pore-filled. This is because these internal vapour pockets provide menisci for desorption even for the pores deep within

44

2 Gas Sorption

Fig. 2.10 Schematic diagram showing the impact of pore connectivity on gas sorption hysteresis width

the bulk of the porous solid. The presence of unfilled pores at the top of descending scanning curves thus means that the hysteresis of scanning curves is typically much thinner than that for boundary curves. This effect can be used to validate percolation models of desorption (Liu et al. 1993).

2.3.4 Pore Size Spatial Disposition In the development of percolation analysis [such as the work of Seaton (1991)], it was presumed that the adsorption isotherm would give rise to the pore size distribution unaffected by pore–pore co-operative effects. However, the adsorption has since been shown to be influenced by such effects too, and thus sensitive to pore connectivity. The two pore–pore co-operative effects that influence gas adsorption are ‘advanced condensation’ (or advanced adsorption) and (network) ‘delayed condensation’. The operation of advanced condensation can be illustrated by the consideration of a simple through ink-bottle pore consisting of a set of two identical, narrow, cylindrical pore necks co-axial with, and at either end of, a larger cylindrical pore body. Condensation would occur in the pore necks according to the Kelvin equation for a cylindrical sleeve meniscus. The filled pore necks would then complete the hemispherical meniscus across both the ends of the pore body. If the radius of the pore body is less than twice that of the pore necks, then the condensation pressure for a cylindrical-sleeve meniscus for the necks would also exceed that for the pore body via a hemispherical meniscus. This means the pore body would then fill at the same pressure as the pore necks even though it is larger. Conversely, if the pore

2.3 What Can I Find Out with This Method?

45

body is over twice the size of the pore necks, it will only fill (via condensation at the hemispherical menisci) at a higher pressure. Hence, if the pore body size is less than the critical ratio to the neck size, then it would not be separately discernible by nitrogen adsorption. If the necks and bodies of the ink-bottle pore were each separate cylinders, they would all each fill at the pressure corresponding to that for a cylindrical-sleeve meniscus in the Kelvin equation. The advanced condensation effect thus breaks the unique relationship between condensation pressure and pore size. The presence of the advanced condensation effect would narrow the apparent PSD obtained from the adsorption isotherm to an extent dependent on the relative placement of necks and bodies of different sizes (Esparza et al. 2004; Matadamas et al. 2016). However, one key difference between advanced condensation and pore blocking is that the former would both operate in the direction of the centre of the porous particle, and away from it, while pore blocking only operates towards the centre of the particle. In order to demonstrate this difference, consider two simple pore structure models, one being the through ink-bottle pore mentioned above, where the narrow necks are either side of a pore body, and a second consisting of a through funnel-type structure where two large pore bodies have a narrow neck sandwiched between them. In both cases, the narrow neck(s) could act as an initiation point for advanced adsorption, but only in the case of the ink-bottle pore would the pore bodies be pore-blocked. The presence of both effects would thus indicate an ink-bottle arrangement, while the presence of advanced condensation alone would suggest a funnel-like arrangement. In a real material, these arrangements could manifest in a core-coat-type structure, where either small pores are concentrated in the core of a porous particle, and larger pores in a surrounding coat, or vice versa. If the particle size and size of the core and coat are large such that the particle could be fragmented into finer particles smaller than the core or coat size, then the original pore size arrangement could be detected from the change in the isotherms following fragmentation. If both adsorption and desorption isotherms shift position and shape following fragmentation, then pore blocking was occurring, but if only the adsorption isotherm changes then just advanced condensation was occurring. Another type of pore–pore co-operative effect that can potentially impact the adsorption isotherm is delayed condensation. This effect arises because the condensation pressure in a given pore is actually determined by the pore potential present, rather than the pore size. At the molecular scale, condensation of adsorbate happens in confined geometries at pressures lower than the bulk because the attractive forces from a highly curved solid wall are then strong enough to help concentrate the adsorbate molecules there in space, and induce more transitory densification of the adsorbate, such that it nucleates condensation more easily. The more solid wall in close proximity to the pore cavity, the greater the pore potential therein and thus the lower the pressure needed to achieve condensation. An isolated spherical cavity would have solid wall around the whole periphery. However, a spherical pore body, of the same characteristic radius, but with adjoining cylindrical pore necks, would have ‘holes’ in the solid wall where the necks join the body. Hence, in the latter the amount of solid within the same radius of the centre of the pore body is smaller than

46

2 Gas Sorption

in the former, and thus the pore potential is lower in the cavity with necks. Assuming adsorbate could access it somehow, capillary condensation would be at a lower pressure for the cavity with completely solid walls. Since the number of holes in the pore wall increases with connectivity, then the condensation pressure would depend upon pore connectivity for a series of cavities with nominally the same size but different numbers of adjoining necks. Depending upon the neck sizes, some might fill with condensate before the pore body, and this condensate would effectively fill in some of the holes in the wall, thereby reducing the necessary condensation pressure in the body. Hence, the particular pressure for condensation within a pore body, thus, depends upon the connectivity to pore necks and their particular sizes, as well as the size of the body itself. This suggests that the gas adsorption isotherm alone is insufficient to obtain an accurate pore size distribution for complex, irregular porous solids. The delivery of additional information by integrating gas sorption with other techniques will be discussed in Chap. 6.

2.4 Conclusions Since gas sorption is an indirect characterisation method, it cannot specify in advance all the choices needed to obtain a pore size distribution. Hence, some assumptions are needed to make progress that need validating against subsequent findings or independent methods. Pore size methods ultimately based upon the Kelvin equation are arguably valid down to at least ~10 nm and maybe even down to ~5 nm. Since the PSD methods that use NLDFT are still based upon a model of the void space consisting of an array of independent pores, any gain in the accuracy of the description of the condensation mechanism in a single pore is dwarfed by the much larger error introduced by pore– pore co-operative effects like advanced adsorption.

References Androutsopoulos GP, Salmas CE (2000) A new model for capillary condensation—evaporation hysteresis based on a random corrugated pore structure concept: prediction of intrinsic pore size distributions. 1. Model Formulation. Ind Eng Chem Res 39(10):3747–3763 Aukett PN, Jessop CA (1996) Assessment of connectivity in mixed meso/macroporous solids using nitrogen sorption. Fundamentals of adsorption. Kluwer Academic Publishers, MA, pp 59–66 Barrett EP, Joyner LG, Halenda PP (1951) The Determination of Pore Volume and Area Distributions in Porous Substances. I. Computations from Nitrogen Isotherms. J Am Chem Soc 73(1):373–380 Broekhoff JCP, De Boer JH (1967) Studies on pore systems in catalysis X: calculations of pore distributions from the adsorption branch of nitrogen sorption isotherms in the case of open cylindrical pores. J Catal 9:15–27 Chuang IS, Maciel GE (1997) A detailed model of local structure and silanol hydrogen banding of silica gel surfaces. J Phys Chem 101:3052–3064

References

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Cohan LH (1938) Sorption hysteresis and the vapor pressure of concave surfaces. J Am Chem Soc 60:433–435 Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford Davydov VY, Kiselev AV, Zhuralev LT (1964) Study of surface and bulk hydroxyl groups of silica by infra-red spectra and D2O exchange. Trans Farad Soc 60:2254–2264 Do D (1998) Adsorption analysis: equilibria and kinetics. Imperial College Press, London Esparza JM, Ojeda ML, Campero A, Dominguez A, Kornhauser I, Rojas F, Vidales AM, Lopez RH, Zgrablich G (2004) N-2 sorption scanning behavior of SBA-15 porous substrates. Colloids Surf A 241:35–45 Gelb LD, Gubbins KE (1998) Characterization of porous glasses: simultion models, adsorption isotherms, and the brunauer-emmett-teller analysis method. Langmuir 14:2097–2111 Gelb LD, Gubbins KE (1999) Pore size distributions in porous glasses: a computer simulation study. Langmuir 15:305–308 Gor GY, Huber P, Bernstein N (2017) Adsorption-induced deformation of nanoporous materials—a review. Appl Phys Rev 4:011303 Gregg SJ, Sing KSW (1982) Adsorption. Surface area and porosity. Academic Press Inc., London Halsey GD (1948) Physical adsorption on non-uniform surfaces. J Chem Phys 16:931–937 Harkins WD, Jura D (1944) Surfaces of solids. XII. An absolute method for the determination of the area of a finely divided crystalline solid. J Am Chem Soc 66:1362–1366 Hitchcock I, Malik S, Holt EM et al (2014) Impact of chemical heterogeneity on the accuracy of pore size distributions in disordered solids. J Phys Chem C 118(35):20627–20638 International Standards Organisation (ISO) (2010) BS ISO 9277:2010 Determination of the specific surface area of solids by gas adsorption—BET method. ISO, Switzerland Jagiello J, Jaroniec M (2018) 2D-NLDFT adsorption models for porous oxides with corrugated cylindrical pores. J Colloid Interface Sci 532:588–597 Karnaukhov AP (1985) Improvement of methods for surface area determinations. J Colloid Interface Sci 103(2):311–320 Kierlik E, Monson PA, Rosinberg ML, Tarjus G (2002) Adsorption hysteresis and capillary condensation in disordered porous solids: a density functional study. J Phys Conden Matter 14:9295–9315 Kleitz F, François Bérubé F, Guillet-Nicolas R, Yang C-M, Thommes M (2010) Probing adsorption, pore condensation, and hysteresis behavior of pure fluids in three-dimensional cubic mesoporous KIT-6 silica. J Phys Chem C 114(20):9344–9355 Kruk M, Jaroniec M, Sayari A (1999) New approach to evaluate pore size distributions and surface areas for hydrophobic mesoporous solids. J Phys Chem B 103:10670–10678 Landers J, Gor GY, Meimark AV (2013) Density functional theory methods for characterization of porous materials. Colloids Surf A 437:3–32 Liu HL, Zhang L, Seaton NA (1993) Analysis of sorption hysteresis in mesoporous solids using a pore network model. J Colloid Interface Sci 156(2):285–293 Mahnke M, Mögel HJ (2003) Fractal analysis of physical adsorption on material surfaces. Colloids Surf A 216:215–228 Matadamas J, Alferez R, Lopez R, Roman G, Kornhauser I, Rojas F (2016) Advanced and delayed filling or emptying of pore entities by vapour sorption or liquid intrusion in simulated porous networks. Colloids Surf A 496:39–51 Matsuhashi H, Tanaka T, Arata K (2001) Measurement of heat of argon adsorption for the evaluation of relative acid strength of some sulfated metal oxides and H-type zeolites. J Phys Chem B 105(40):9669–9671 Murray KL, Seaton NA, Day MA (1999) An adsorption-based method for the characterization of pore networks containing both mesopores and macropores. Langmuir 15:6728–6737 Neimark AV, Ravikovitch PI (2001) Capillary condensation in mms and pore structure characterization. Micropor Mesopor Mater 44:697–707 Pfeifer P, Johnston GP, Deshpande R, Smith DM, Hurd AJ (1991) Structure analysis of porous solids from preadsorbed films. Langmuir 7(11):2833–2843

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Ravikovitch PI, O’Domhnaill SC, Neimark AV, Schuth F, Unger KK (1995) Capillary hysteresis in nanopores: theoretical and experimental studies of nitrogen adsorption on MCM-41. Langmuir 11:4765–4772 Seaton NA (1991) Determination of the connectivity of porous solids from nitrogen sorption measurements. Chem Eng Sci 46(8):1895–1909 Thommes M, Katsumi K, Neimark AV et al (2015) Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report). Pure Appl Chem 87(9–10):1051–1069 Walker WC, Zettlemoyer AC (1948) A dual-surface BET adsorption theory. Y Phys Colloid Chem 52:47–58 Watt-Smith M, Edler KJ, Rigby SP (2005) An experimental study of gas adsorption on fractal surfaces. Langmuir 21(6):2281–2292

Chapter 3

Mercury Porosimetry

3.1 Basic Theory Mercury porosimetry was originally, principally aimed at macroporous materials because of the limitations on upper pore size that can be probed with conventional gas sorption methods (Gregg and Sing 1982). Mercury porosimetry is based upon the phenomenon that mercury is a non-wetting liquid for most porous materials and thus needs a pressure exceeding the saturated vapour pressure to force it into the pores. This is because forcing mercury to enter small pores greatly increases the mercury surface area in contact with the material, and the surface tension force (γ ) opposes this unfavourable expansion in contact. A raised hydrostatic pressure is, thence, needed to overcome this resistance. Mercury porosimetry is most often conducted as a quasi-equilibrium experiment. This means the applied hydrostatic pressure in the mercury (pHg ) is increased in a small step, and the mercury will enter pores where the absolute pressure is now large enough to overcome the surface tension.The pressure needed to intrude mercury is calculated from a special case of the Young–Laplace equation (Gregg and Sing 1982):  1 1 , − p = −γ + r1 r2 

p

Hg

g

(3.1)

where pg is the residual gas pressure in the evacuated porous sample, and r 1 and r 2 are the radii of curvature of the mercury–gas meniscus. For intrusion into a cylindrical pore, the meniscus is a segment of a sphere such that: r1 = r2 = r p cos θ,

(3.2)

where θ is the mercury–gas–solid contact angle (defined as >90° for non-wetting fluid). Combining Eqs. (3.1) and (3.2) gives an expression:

© Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_3

49

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3 Mercury Porosimetry

p Hg − p g = p =

−2γ cos θ , rp

(3.3)

which is commonly referred to as the ‘Washburn equation’. This provides a simple relationship to convert the mercury pressure into a pore size. The surface tension is typically taken as 480 mN m−1 . It is also common practice to assume a contact angle of ~130–140°. However, for mercury porosimetry not to be just a relative method, the appropriate values of surface tension and contact angle for a given sample must be obtained. Contact angle typically depends upon surface chemistry and roughness (Wenzel 1949). The contact angle can be obtained from the sessile drop experiment (Giesche 2006). In such an experiment, the sample is typically powdered and compacted into a flat dish. A drop of mercury is then placed on top of the compacted powder, and the shape is examined under a microscope to determine the contact angle. If the dish is tilted, and the drop starts to progress across the powder surface, it is often found that the advancing contact angle is different to the retreating contact angle. This is called ‘contact angle hysteresis’. However, it is suggested that the values of surface tension and contact angle at the radii of curvature for mercury menisci present in very small pores may be very different to the macroscopic values measured by such as the sessile drop experiment. Hence, alternative methods of calibration of the γ cos θ term in the Washburn equation have been proposed. One option is to use a standard material with regular pores for which it is possible to obtain mercury intrusion and extrusion pressures, and the pore size by independent means. For example, Liabastre and Orr (1978) studied the morphology of controlled pore glasses using electron microscopy and mercury porosimetry. These workers measured the diameters of the pores in the glasses by direct observation from microscopy. They compared these values with the corresponding values obtained from mercury intrusion and extrusion porosimetry via Eq. (3.3), assuming fixed values of contact angle and surface tension. Subsequently, Kloubek (1981) utilised these data to determine the relationships for the variation of the product γ cos θ as a function of pore radius, for both advancing and retreating mercury menisci. Kloubek (1981) obtained expressions of the form: γ cos θ = A +

B , r

(3.4)

where A and B are constants whose values are dependent on whether the mercury meniscus is advancing or retreating. The values of these constants found for the controlled pore glasses, and their pore size ranges of application are given in Table 3.1. Table 3.1 Values of the constants A and B for use in Eq. (3.4) with silica materials, and their ranges of applicability

A

B

Range of validity (nm)

Intrusion

−302.533

−0.739

6–99.75

Retraction

−68.366

−235.561

4–68.5

3.1 Basic Theory

51

The values of the constants A and B can be found by the substitution of Eq. (3.4) into Eq. (3.3) and the calibration of the pressure at which the intrusion or extrusion of mercury occurs for a particular porous solid against an independent measure of pore size. The values of A and B determined for mercury porosimetry in controlled pore glasses can also be used to analyse the raw porosimetry data for a new sample. If an a priori superposition of the intrusion and retraction curves for the new material is subsequently achieved, then the values of A and B determined for retraction from controlled pore glasses must also be appropriate for the new material. If no such superposition is obtained, then, either the aspect of a surface that controls the value of γ cos θ for retraction differs between the new material and the controlled pore glasses (CPGs) in some way (in which case the values of A and B will differ), or there is some other cause of hysteresis (which will be discussed below). Such a superposition has been achieved for porosimetry data for various silica materials, such as fumed and sol-gel silicas (Rigby and Edler 2002). Fumed and sol-gel silicas have similar surface chemistry to controlled pore glass. As will be described in Chap. 4, measures of the surface roughness of porous materials can be obtained independently using smallangle X-ray scattering (SAXS). It has been found that the silica materials where superposition of the porosimetry curves was achieved using the Kloubek (1981) correlations had similar surface fractal dimensions (from SAXS) to CPGs (Rigby and Chigada 2010). Further, while superposition of the porosimetry curves was not achieved for silica surfaces with significantly different surface fractal dimensions to that of CPGs (2.20 ± 0.05), superposition has been achieved for other types of surface chemistry where the surface did have similar surface roughness [e.g. silica–alumina (Rigby et al. 2017)]. This might suggest that mercury porosimetry contact angle hysteresis is dominated by surface roughness effects, at least for silaceous materials. Alternative sets of values of the parameters in Eq. (3.4) have been suggested for alumina (Rigby 2000a) and are given in Table 3.2. More recently, Wang et al. (2016) have used molecular dynamics to develop correlations for the variation with surface curvature (pore radius) of the γ cos θ term for carbon surfaces. Even where the Kloubek (1981) correlations can achieve superposition of at least some part of the porosimetry curves, sometimes other parts of the curves will not overlap. This suggests a residual hysteresis with another cause other than contact angle effects. A number of theories have been proposed for this residual hysteresis related to structure, which will be discussed below. However, it is noted that Kloubek (1981) showed that hysteresis can arise in mercury intrusion and extrusion from a cylindrical pore with a conical dead end, even without contact angle hysteresis. The degree of hysteresis depends upon the sharpness of the cone apex angle. Table 3.2 Parameters for use in Eq. (3.4) for aluminas

A

B

Range of validity (nm)

Intrusion

−302.533

−0.739

6–99.75

Retraction

−40

−240

4–68.5

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3 Mercury Porosimetry

For intrusion into pore geometries that are more complex than a simple cylinder, the theory is less straightforward, and most real materials do not have pores that are regular cylinders. For example, the so-called canthotaxis effect can arise when mercury enters a spherical pore body from a cylindrical pore bond (Felipe et al. 2006). As the mercury meniscus reaches the mouth of the cylindrical pore bond, the intrusion of the main body of mercury is halted as the dome of the leading meniscus must then expand to reach the sides of the spherical pore body, since they recede away from the axis of the cylindrical bond, such that it can re-establish the normal advancing contact angle. This expansion of the meniscus requires an increase in the capillary pressure even though the characteristic size of the pore body is larger than the neck, and this extra pressure barrier is larger the greater the ratio of the bodyto-neck size. The impact of the canthotaxis effect is that, for void spaces consisting of wide pore bodies accessed via narrow necks, the apparent shielding effect of the necks will be even more pronounced than expected otherwise. A common pore geometry encountered in porous media, which is not cylindrical, is the random packing of particles, such as a packed bed of spheres. For example, as seen in Fig. 3.1, sol-gel silicas consist of a packing of the precipitated spherical sol particles, and fumed silicas, such as Cab-O-Sil or aerosil, also consist of packings of spherical particles. The intrusion into such packings is controlled by the narrow windows that exist between the spherical pores. A well-known theory for the breakthrough pressure, when mercury penetrates the windows of a spherical

Fig. 3.1 AFM image of a sol-gel silica G2. The sol particles have been partially sintered by calcination

3.1 Basic Theory

53

Fig. 3.2 Schematic diagram illustrating the ‘pore shielding’ effect. For the parallel pore bundle on the left, each pore is detected, but in the ‘ink-bottle’ pore on the right the large pore is shielded behind the small pore

packing, was presented by Mayer and Stowe (1965). While this theory has been criticised for assuming that the intersection of the mercury surface with the solid spheres lies wholly within the plane of minimum cross section of the pore, the error this introduces is small (Bell et al. 1981). More recently, the physical processes involved in mercury intrusion have been simulated directly by molecular dynamics and lattice Boltzmann methods (Martic et al. 2002; Hyväluoma et al. 2007). However, while these techniques can cope with complex void space geometries, they are very time consuming and limited to representation of only very small sample volumes due to computing limitations. Since mercury enters pores in order of decreasing size on intrusion, then, if larger pores only have access to the exterior via paths through smaller pores, there is the potential for what is known as the ‘pore-shielding’ or ‘pore-shadowing’ effect (see Fig. 3.2). In the case of a large pore located beyond a small pore, the mercury pressure must be raised to that required to enter the small pore before it can finally intrude the larger pore. This effect is often (Diamond 2000) presented as a serious flaw of mercury porosimetry, but can actually be used constructively, as will be seen below. The shielding phenomenon means that mercury intrusion is a so-called percolation process, like nitrogen desorption. The point of inflexion in the mercury intrusion curve is generally identified as the percolation threshold (Katz and Thompson 1986). The position and form of the percolation threshold are determined by the connectivity of the void space, the size of the pore network, and the spatial correlation in pore sizes. The sharpness of the percolation transition is related to the pore network size. For small networks, the surface pores represent a large fraction of the whole, and so early intrusion of large pores, directly accessible from the surface, tends to blur the transition.

3.2 Nature of Experiment For mercury porosimetry experiments, it is imperative that a sample is clean. This is because the advancing mercury meniscus can accrete molecular debris on the surface leading to changes in the contact angle. The sample must also be evacuated to remove residual air which would otherwise be compressed as mercury progressively entered

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the sample and oppose that intrusion. The evacuated sample is placed in a sample holder called a mercury penetrometer. The sample is then immersed in mercury. Mercury will only enter pores larger than ~14 µm at atmospheric pressure and thus mercury will, typically, surround the sample particles. This enables the sample bulk volume to be determined. Thereafter, the hydrostatic pressure in the mercury can be increased in small steps and the amount of mercury entering the sample determined by monitoring the drop in the mercury reservoir level. Once the maximum mercury pressure possible with the machine used has been achieved, the pressure may be decreased again in small steps. An example of a dataset exhibiting many of the features typical of mercury porosimetry data is shown in Fig. 3.3. From Fig. 3.3a, it can be seen that the high pressure part of the intrusion and extrusion curves are very similar in form. Indeed, this is confirmed in Fig. 3.3b where the data has been analysed using the Kloubek (1981) correlations, whereby the superposition of the top part of the curve has been achieved. The ultimate highest pressure possible in a porosimetry experiment is generally determined by the apparatus used. Typical values possible are 30,000 and 60,000 psia (212 and 414 MPa, respectively), which correspond to pores sizes of ~6–8 nm and ~3–4 nm, respectively, depending on Washburn equation parameters assumed. However, as with gas sorption, it is possible to conduct scanning curve and loop experiments. Since high pressures are used in mercury porosimetry, the sample must be mechanically stable. It is possible for samples to be inelastically crushed by the mercury pressure even before any actual intrusion occurs. Since the intrusion of mercury is monitored by the drop in the level of the mercury reservoir, then reduction of the sample envelope volume by crushing is indistinguishable from internal intrusion. If the reduction in sample volume due to crushing is irreversible, then, when the pressure is reduced, it will look like this volume of mercury is entrapped in the sample. In this case, the mercury retraction curve will tend to be a horizontal line. There are a number of tests to check if a sample is suffering mechanical damage. First, if some mercury does leave the sample on extrusion, it is possible to increase the pressure again and re-intrude the mercury. If the sample is undamaged, then the mercury should follow exactly the same path back into the sample and thus reproduce the original intrusion curve. However, if mechanical damage has occurred, the mercury intrusion will be significantly different to the first time. Second, it is possible to weigh the sample before and after porosimetry. If true mercury entrapment has occurred, the volume of it can be obtained from the mercury retraction data. The expected mass increase for the sample, arising from the entrapped mercury, can be calculated. If there has been shrinkage of sample and thus only apparent entrapment, then no mercury will actually be present and the sample weight after porosimetry will not have increased as expected. Third, the sample can be coated in rigid resin, or an impermeable membrane, such that mercury cannot enter the sample. Any apparent intrusion would likely then be simply due to internal structural collapse and thus associated with apparent entrapment. An example of a data set from such an experiment is shown in Fig. 3.4. It can be seen that with a membrane around the sample

3.2 Nature of Experiment

55

Fig. 3.3 Example of a mercury porosimetry raw data set, a exhibiting some of the common features such as hysteresis and entrapment and b following analysis using the Kloubek (1981) correlations

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Fig. 3.4 Mercury porosimetry curve (volume variation as a function of mercury pressure) of sample X1 (solid line) without membrane; (filled diamond) with membrane and (open circle) with membrane (corrected data). Reprinted from Alié et al. (2001), Copyright (2001), with permission from Elsevier

there is almost complete, apparent ‘entrapment’ of the seemingly (but impossibly) intruded mercury. There is also an alternative protocol for the mercury porosimetry experiment where the pressure is varied to keep the flow rate of mercury into the sample constant, and these pressure change requirements interpreted in terms of structure (Yuan and Swanson 1989). The necessary equipment is less widely available, and the raw data set more difficult to interpret than conventional quasi-equilibrium mercury porosimetry experiments. The technique will not be discussed any further here.

3.3 What Can I Find Out with This Method? 3.3.1 Surface Area and Pore Size Distribution (PSD) If the sample weight is known, then the specific volume of mercury entering the sample over each pressure step can be known. Hence, it is possible to infer the volume of pores in the range of sizes corresponding to the pressures at either end of the pressure step. This gives rise to a histogram of pore volume between a set of bin sizes corresponding to the pressure steps used. By making an assumption about pore geometry, it is possible to determine a surface area, since, for example, the surface-area-to-volume ratio of a cylindrical pore is two over the radius.

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When the sample is crushed by mercury pressure, the pore size distribution can still be obtained but an alternative data analysis to the Washburn equation (or similar) is required. For example, if the pores can be treated like a thin-walled pressure vessel, then the Euler buckling equation can be used, such as that for cubic pores of size L (Alié et al. 2001): L=

KE , P 0.25

(3.5)

where P is the pressure required to crush the pores, and K E is given by:  0.25 K E = nπ 2 E I ,

(3.6)

where E is the elastic modulus and I is a geometric factor. An example of a data set (for same material as in Fig. 3.4) where both the mechanisms of sample crushing and intrusion are present is shown in Fig. 3.5. The break in slope at about ~30 MPa corresponds to the transition between mechanisms. The scanning curve to ~30 MPa has a similar shape to that for the sample covered in membrane, as shown in Fig. 3.4, which is due to the sample crushing. The re-intrusion from atmospheric pressure to 200 MPa, and subsequent retraction, is due to actual intrusion into the sample.

Fig. 3.5 Mercury porosimetry curve (volume variation as a function of mercury pressure) of sample X1 (solid line) until 200 MPa; (filled square) until 30 MPa (‘open square’, last point was recorded 12 h after the end of the experiment) and (multiplication sign) until 200 MPa on sample partially compacted until 30 MPa. Reprinted from Alié et al. (2001), Copyright (2001), with permission from Elsevier

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3.3.2 Pore Network Geometry The mercury retraction curve is a potential source of information on the void space structure of a porous solid. However, the retraction of mercury from even highly ordered materials is a very complex process. For example, under certain circumstances, retraction even from a straight, cylindrical capillary can be associated with mercury entrapment, as shown in Fig. 3.6. If the two menisci of intruding mercury, entering from both ends of the tube, are allowed to coalesce, they will form a continuous thread of mercury from one end to the other then. However, for mercury to be able to retreat immediately, on reversing the direction of the change in pressure, it must possess a free meniscus. If mercury retains a free meniscus, within the cylinder, it can reverse direction and extrude without entrapment. In that case, the cap of mercury will first flatten, as the contact angle first adjusts to the retraction value, resulting in a slight reduction in intruded volume with pressure, as shown in the scanning curve for a CPG sample in Fig. 3.7. The position of the contact line will only retreat once the extrusion contact angle has been established, which is manifest by a larger decrease in mercury volume with decreasing pressure, as seen for the scanning curve in Fig. 3.7 for pressures below 17.9 MPa. Without a free meniscus, one must be created before retraction of the mercury thread can commence. This occurs by the process known as ‘snap-off’. The creation

Fig. 3.6 Close-up photograph of mercury ganglia (black) entrapped within a straight cylindrical pore within silica glass following mercury porosimetry. Reprinted from Hitchcock et al. (2014), Copyright (2014), with permission from Elsevier

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Fig. 3.7 Mercury intrusion up to 414 MPa (filled diamond) and subsequent retraction back to ambient (open square) for a CPG with nominal pore size of 24 nm. Also shown is a scanning curve up to 48.1 MPa (multiplication sign). Reprinted from Hitchcock et al. (2014), Copyright (2014), with permission from Elsevier

of the free menisci requires energy, and the mercury pressure must be lowered, beyond that expected merely from contact-angle hysteresis, to stretch the mercury thread until it breaks. This tends to result in a right-angled knee in the retraction curve at the point the thread breaks, as seen at 13.8 MPa in the retraction curve from 414 MPa in Fig. 3.7. The delay effect when snap-off is required has been simulated using mean-field density functional theory (Rigby et al. 2011). If the process occurs in a straight, regular cylinder, there will be no particular feature of the pore space that picks out one point along the cylinder rather than another. Hence, in that case, snap-off may occur in multiple positions simultaneously, such that mercury slugs intermediate between two snap-off positions become disconnected. For mercury to retreat from the sample, it must retain connection, via a continuous mercury-filled pathway, to the external mercury bath. As shown in Fig. 3.6, when that connection is broken the mercury becomes entrapped. In more complex geometries than straight cylinders, the shape of the void space may possess particular regions more conducive to the snap-off process than others. The snap-off process is especially enhanced by the presence of narrow pore necks joining wide pore bodies, such as shown in Fig. 3.8. Close-up imaging studies of the necks during mercury retraction shows that the mercury develops an hourglass shape as the mercury pressure is reduced, which eventually thins to nothing, and eventually breaks (Tsakiroglou and Payatakes 1998). If larger pore bodies are sandwiched between two pore necks, where snap-off has occurred, mercury in the body can

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Fig. 3.8 Mercury intrusion and entrapment (black) in micromodels etched in glass. Reprinted from Wardlaw and McKellar (1981), Copyright (1981), with permission from Elsevier

become entrapped. Experiments on glass micro-models containing pore networks composed of pore bodies of different sizes, interspersed with narrow pore necks, as shown in Fig. 3.8, suggest that the amount of mercury becoming entrapped increases as the ratio of the pore body-to-neck size increases. Some authors have suggested that snap-off occurs above a particular ‘snap-off ratio’ of these sizes, typically ~6 (Matthews et al. 1995). Hence, mercury porosimetry curves with both steep intrusion curves occurring at high pressure, and high entrapment, are associated with void spaces with narrow necks interspersed between wide pore bodies. Studies of mercury retraction from even more complex networks in glass micromodels show that mercury also tends to become entrapped in heterogeneities whereby isolated regions of large pores are surrounded by a sea of small pores, as shown in Fig. 3.9 (Wardlaw and McKellar 1981). From Fig. 3.9, it can be seen that mercury has snapped off at the boundary between regions of larger pores surrounded by smaller pores leaving mercury entrapped preferentially in larger pores. Glass micromodels permit the direct observation of the convoluted geometries adopted by mercury ganglia during intrusion and retraction within complex void

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Fig. 3.9 Glass micromodel with non-random heterogeneity, a filled with mercury and b following mercury retraction. Reprinted from Wardlaw and McKellar (1981), Copyright (1981), with permission from Elsevier

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spaces, and thus provide potential validation of analytical models of the behaviour of mercury menisci. For example, Tskairoglou and Payatakes (1990, 1998) have developed analytical relationships for mercury intrusion in the lenticular throats (necks) created when etching two-dimensional networks in glass sheets. They also developed expressions for the retraction and snap-off pressures for retraction of mercury via various possible meniscus geometries that can be adopted in simulations of mercury retreating from pore body-pore bond networks. The aforementioned studies in glass micromodels suggest that mercury can become entrapped in a variety of pore geometries. There have been attempts (Day et al. 1994) to develop a classification scheme for mercury porosimetry data, along the lines of the typology for gas sorption isotherms and hysteresis loops mentioned in Chap. 2, which links the form of the data to the nature of the underlying void space. However, the boundary curves from the conventional mercury porosimetry experiment do not provide enough information to decide which particular underlying pore geometry is giving rise to mercury entrapment. Therefore, more information is required, which can be analysed using pore network simulators.

3.3.3 Mercury Porosimetry Simulators The presence of the pore-shielding effect in mercury intrusion, and the indirect nature of mercury porosimetry characterisation data, means that data analysis to obtain an unshielded PSD requires of model of interpretation. Structural models attempt to simplify the complex structure of real disordered materials by incorporating only the main features that most affect the observed form of the mercury porosimetry data. The simplification process often requires the adoption of a series of assumptions. Structural models for interpretation of mercury porosimetry data come in a number of different forms. The earliest models replaced the complex shapes of real pores within disordered materials with regular Euclidean shapes. This was to enable the use of simple analytical expressions for mercury intrusion such as the Washburn equation. The first simulators were based upon two-dimensional pore bond networks (Fatt 1956; Androutsopoulos and Mann 1979), where the individual pores are represented by simple geometries such as straight cylinders, or slits, arranged in lattices, such as square planar, similar to shown in Fig. 3.9. Later models used random, three-dimensional pore bond networks (Portsmouth and Gladden 1991). More complex pore geometries were represented using networks consisting of both pore bodies and pore bonds (Tsakiroglou and Payatakes 1990; Matthews et al. 1995; Felipe et al. 2006). For example, the Pore-Cor model had cylindrical pore bonds and cubic pore bodies, as shown in Fig. 3.10. The more complex that a structural model becomes then the more descriptors that are required to fully characterise it. For example, the simplest, square planar pore bond networks have a fixed pore connectivity (of 4) and are described by a pore diameter distribution for the pore bonds. The spatial arrangement of pore sizes is at

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Fig. 3.10 Three-dimensional representation of a Pore-Cor unit cell exhibiting a random structure having 25% porosity, and 0.01 µm minimum, 0.1 µm median and 1 µm maximum throat diameters. Reprinted from Laudon et al. (2008), Copyright (2008), with permission from Elsevier

random. For random, three-dimensional pore bond networks, the pore co-ordination number at each node can be varied. The pore co-ordination numbers can be constant across nodes, or have a distribution of their own. For networks of pore bodies and pore bonds, each may have a separate distribution of sizes. Further, pore sizes can be spatially correlated, as shown in Fig. 3.9, and described by a correlation function. A correlation function describes how the probability that two given pores have the same size varies with the distance between them. As more descriptors are required to characterise a structural model, more input data is required to specify the parameters of these descriptors. Unfortunately, the simple mercury intrusion curve, up to the maximum ultimate pressure possible with the porosimeter used, will only provide the data necessary to specify the parameters for the simplest of structural model type, such as the square planar network

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(Androutsopoulos and Mann 1979). As the underlying model complexity increases, more sources of data are required. Fortunately, the mercury porosimetry technique is flexible enough to provide these. For example, scanning curve and loop experiments can be performed, as will be discussed in the next section, where it will be shown how they are used to derive model pore connectivity.

3.3.4 Pore Network Topology Simulations of mercury intrusion and retraction on random pore networks have shown that, given a particular mechanism, the level of entrapment predicted depends upon both the pore connectivity and spread of pore sizes within the distribution (Portsmouth and Gladden 1991). This means that the level of entrapment alone is not an unambiguous indicator of pore connectivity. Hence, more observables for a given sample are needed to extract an estimate of pore connectivity. Additional observables can be extracted from mercury porosimetry scanning loop, or ‘mini (hysteresis) loop’ experiments. The idea of the ‘mini (hysteresis) loop’ was originally invented by Reverberi et al. (1966). The concept of a mini-loop is shown schematically in Fig. 3.11 and can be carried out during either the boundary intrusion or retraction process. Scanning loops in mercury porosimetry are analogous to those in gas sorption. On the intrusion boundary curve, the pressure is increased to some maximum (say point B in Fig. 3.11), at which point the direction of the pressure changes is reversed, and some mercury potentially retracts from the sample (along

Fig. 3.11 A schematic representation of mini-hysteresis loops on the overall intrusion and extrusion curves. Reprinted from Portsmouth and Gladden (1991), Copyright (1991), with permission from Elsevier

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BC in Fig. 3.11). The direction of the pressure changes is reversed once more (at point C), before the retraction process reaches ambient pressure, and is increased to the previous maximum pressure (at point B), and can continue until the ultimate maximum possible with the machine (point D). On the retraction boundary curve (DH in Fig. 3.11), the pressure can be reduced to some point (like F in Fig. 3.11) and there the direction of the pressure changes reversed such that re-intrusion can commence, and this proceeds up to a higher pressure point less than the ultimate possible (such as point G), where the direction of pressure changes is reversed once more, and potential extrusion can occur until the pressure re-reaches point F. As shown in Fig. 3.11, the shapes of the mini-loops thereby obtained can be characterised by various parameters, which provide supplementary observables for constraining the properties of the real material. The vtot is the total volume of mercury that intrudes into, or extrudes from, the sample over the pressure range of the mini-loop on the boundary intrusion, or extrusion, curve, respectively. The, typically, lower volume of mercury that intrudes into, or extrudes from, the sample during the mini-loop process itself is denoted vm . The difference between vtot and vm is denoted vtrap . Since intrusion boundary curve mini-loops start at the higher pressure end, where the mercury has already breached some of the shielding provided by narrow necks, the re-intrusion arm of the mini-loop is stripped of some of the shielding effects that would still be present in the boundary curve itself. Hence, if a series of mini-loops, covering, seamlessly, a pressure range of the boundary intrusion curve, is obtained, the resultant set of re-intrusion curves can be used to obtain a partially deshielded pore size distribution. Simulations of mercury intrusion and retraction on random pore networks have also shown that the form of the variation of the ratio of vm /vtot , with the dimensionless (using the minimum network pore size as the reference), loop average pore size for a given mini-loop (see Fig. 3.12), depends uniquely upon the pore network connectivity. It has been found that the shape of the function of vm /vtot with the dimensionless loop average pore size is independent of the number of mini-loops used to define the function and the shape of the pore size probability density function of the underlying network. Hence, obtaining the combination of the entrapment, and the variation of the ratio of vm /vtot , with the dimensionless loop average pore size, allows the pore connectivity to be deduced for a given porous material. This example, thus, shows how increasing the sophistication of the mercury porosimetry experiment performed can provide the greater information necessary to obtain ever more detailed descriptions of real porous media.

3.3.5 Pore Size Spatial Correlation A further way to increase the information from mercury porosimetry experiments is to modify the sample in certain key ways that actually take advantage of the fact that, in porosimetry, mercury must penetrate from the exterior of the sample. Such sample modifications include sample size change and covering the exterior of the

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Fig. 3.12 Variation of the mini-loop ratio V m /V tot with average mini-loop radius. Reprinted from Portsmouth and Gladden (1991), Copyright (1991), with permission from Elsevier

sample in some sort of mercury-proof coating (such as the membrane mentioned above in Sect. 3.2). Mechanically robust samples, such as ceramic catalyst pellets or rock cores, can be fragmented to create samples with different particle sizes. It has been found that the mercury intrusion curve shape and level of entrapment vary with particle size variation from ~millimetres down to ~10–100 s micrometres for catalyst pellets (Rigby 2000b). As mentioned in Sects. 2.3.3 and 3.1, the shape of a percolation transition depends upon the overall pore network lattice size (within a sample particle) and degree of spatial correlation of pore sizes within it. If a pore network consists of a completely random arrangement of pore sizes, and the pore size is well below the lattice side-length (e.g. nm), then a change in lattice size by fragmentation of a millimetre-sized pellet to a ~10s micron-sized powder fragment will not change the shape of the intrusion curve. However, if the pellet is macroscopically patchwise heterogeneous, namely it has large-scale spatial correlation in pore size, then fragmenting a pellet may de-shield a substantial number of larger pores, leading to a change in the shape of the percolation transition. It is likely the mercury intrusion knee would become more rounded and shift towards larger pore sizes on fragmentation of the sample. If the sample material has large-scale spatial correlation in pore sizes, analogous to seen within the glass micromodels in Fig. 3.9, then fragmentation can reduce or remove mercury entrapment. If the sample is fragmented to length-scales of the order, or less, of the heterogeneities, then the shielding by the ‘sea’ of small pores surrounding the ‘island’ of larger pores is removed. Hence, the potential for snap-off of the mercury meniscus at the boundaries of the larger and smaller pores is reduced,

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while the potential for entrapment is decreased. The drop in the level of entrapment following sample fragmentation can be used as a proxy measure for the degree of spatial correlation of pore sizes over length-scales larger than the fragment particle size. These data have been used in conjunction with simulations of mercury retraction in models with different patterns of spatial correlation to use entrapment to constrain models of the spatial arrangement of pore size (Rigby 2000b).

3.4 Conclusions Mercury porosimetry is an indirect characterisation method that, in the absence of independent supporting information from other methods, requires assumptions on pore and network geometry before proceeding. However, when supplemented with pore shape and form of network information from microscopy, and contact angle and surface tension data from preliminary experiments, can deliver absolute pore size distributions from boundary curves arising from the standard experiment. Further, mercury porosimetry is a very flexible technique that also allows for sophisticated experimental protocols, such as scanning and mini-loop experiments, that can vastly supplement the information from the basic boundary curves. These experimental types increase the number of void space descriptors that can be obtained to include such parameters as pore connectivity.

References Alié C, Pirard R, Pirard J-P (2001) Mercury porosimetry applied to porous silica materials: successive buckling and intrusion mechanisms. Colloids Surf A 187–188:367–374 Androutsopoulos GP, Mann R (1979) Evaluation of mercury porosimeter experiments using a network pore structure model. Chem Eng Sci 34(10):1203–1212 Bell WK, Van Brakel J, Heertjes PM (1981) Mercury penetration and retraction hysteresis in closely packed spheres. Powder Technol 29(1):75–88 Day M, Parker IB, Bell J, Fletcher R, Duffie J, Sing KSW, Nicolson D (1994) Modeling of mercury intrusion and extrusion. In: Rodríguez-Reinoso F, Rouquerol J, Unger KK, Sing K (eds) Characterisation of Porous Solids III (COPS III). Stud Surf Sci Catal 87:225–234 Diamond S (2000) Mercury porosimetry—an inappropriate method for the measurement of pore size distributions in cement-based materials. Cem Concr Res 30(10):1517–1525 Fatt I (1956) The network model of porous media. 1. Capillary pressure characteristics. Trans Am Inst Min Met Engrs 207(7):144–159 Felipe C, Cordero S, Kornhauser I, Zgrablich G, López R, Rojas F (2006) Domain complexion diagrams related to mercury intrusion-extrusion in Monte Carlo-simulated porous networks. Part Part Syst Charact 23(1):48–60 Giesche H (2006) Mercury porosimetry: a general (practical) overview. Part Part Syst Charact 23:9–19 Gregg SJ, Sing KSW (1982) Adsorption. Surface area and porosity. Academic Press, London

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Hitchcock I, Lunel M, Bakalis S, Fletcher RS, Holt EM, Rigby SP (2014) Improving sensitivity and accuracy of pore structural characterisation using scanning curves in integrated gas sorption and mercury porosimetry experiments. J Colloid Interface Sci 417:88–99 Hyväluoma J, Turpeinen T, Raiskinmäki P, Jäsberg A, Koponen A, Kataja M, Timonen J, Ramaswamy S (2007) Intrusion of nonwetting liquid in paper. Phys Rev E 75:036301 Katz AJ, Thompson AH (1986) Quantitative prediction of permeability in porous rock. Phys Rev B 34(11):8179–8181 Kloubek J (1981) Hysteresis in porosimetry. Powder Technol 29(1):63–73 Laudon GM, Matthews GP, Gane PAC (2008) Modelling diffusion from simulated porous structures. Chem Eng Sci 63(7):1987–1996 Liabastre AA, Orr C (1978) Evaluation of pore structure by mercury penetration. J Colloid Interface Sci 64:1–18 Martic G, Gentner F, Seveno D, Coulon D, De Coninck J, Blake TD (2002) A molecular dynamics simulation of capillary imbibition. Langmuir 18(21):7971–7976 Matthews GP, Ridgway CJ, Spearing MC (1995) Void space modeling of mercury intrusion hysteresis in sandstone, paper coating and other porous media. J Colloid Interface Sci 171:8–27 Mayer RP, Stowe RB (1965) Mercury porosimetry—breakthrough pressure for penetration between packed spheres. J Colloid Interface Sci 20(8):893–911 Portsmouth RL, Gladden LF (1991) Determination of pore connectivity by mercury porosimetry. Chem Eng Sci 46(12):3023–3036 Reverberi A, Ferraiolo G, Peloso A (1966) Determination by experiment of the distribution function of the cylindrical macropores and ink bottles in porous systems. Ann Chim 56(12):1552–1561 Rigby SP (2000a) New methodologies in mercury porosimetry. In: Rodriguez-Reinoso F, McEnaney B, Rouquerol J (eds) Characterization of Porous Solids VI (COPS-VI). Stud Surf Sci Catal 144: 185–192 Rigby SP (2000b) A hierarchical structural model for the interpretation of mercury porosimetry and nitrogen sorption. J Colloid Interface Sci 224(2):382–396 Rigby SP, Chigada P (2010) MF-DFT and experimental investigations of the origins of hysteresis in mercury porosimetry of silica materials. Langmuir 26(1):241–248 Rigby SP, Edler KJ (2002) The influence of mercury contact angle, surface tension and retraction mechanism on the interpretation of mercury porosimetry data. J Colloid Interface Sci 250:175–190 Rigby SP, Chigada PI, Wang J, Wilkinson SK, Bateman H, Al-Duri B, Wood J, Bakalis S, Miri T (2011) Improving the interpretation of mercury porosimetry data using computerised X-ray tomography and mean-field DFT. Chem Eng Sci 66(11):2328–2339 Rigby SP, Hasan M, Stevens L, Williams HEL, Fletcher RS (2017) Determination of pore network accessibility in hierarchical porous solids. Ind Eng Chem Res 56(50):14822–14831 Tsakiroglou CD, Payatakes AC (1990) A new simulator of mercury porosimetry for the characterization of porous materials. J Colloid Interface Sci 137(2):315–339 Tsakiroglou CD, Payatakes AC (1998) Mercury intrusion and retraction in model porous media. Adv Colloid Interface Sci 75(3):215–253 Wang S, Javadpour F, Feng Q (2016) Confinement correction to mercury intrusion capillary pressure of shale nanopores. Sci Rep 6:20160 Wardlaw NC, McKellar M (1981) Mercury porosimetry and the interpretation of pore geometry in sedimentary rocks and artificial models. Powder Technol 29:127–143 Wenzel RN (1949) Surface roughness and contact angle. J Phys Chem 53(9):1466–1467 Yuan HH, Swanson BF (1989) Resolving pore-space characteristics by rate-controlled porosimetry. SPE-14892 4(1):17–24

Chapter 4

Thermoporometry and Scattering

4.1 Thermoporometry 4.1.1 Basic Theory

Melting and Freezing Point Depression Effect Thermoporometry is also known as cryoporometry. Typically, the former term is used in conjunction with experiments conducted using differential scanning calorimetry (DSC), while the latter term is used in conjunction with experiments performed using nuclear magnetic resonance (NMR) methods. Thermoporometry is based on the physical principle that the melting and freezing points of fluids imbibed within porous media are shifted to lower temperatures with decreasing pore size. In order to utilise this effect to characterise pore structures, a means must be found to measure the melting or freezing point, and the amount of fluid undergoing the phase transition at that point. In addition, the pores must be big enough that there exists a relatively well-defined phase transition point. When a fluid is imbibed within the close confinement of the voids within a porous material it typically exists as very small droplets or ramified ganglia. In such close confinement, the radius of curvature of the meniscus between the fluid, and the solid walls and/or vapour phase will be very high. This means that the surface tension forces will be relatively strong and exert a force equivalent to an external pressure on the fluid. From simple thermodynamics, an increase of pressure on a liquid will depress its melting point. The ultimate energetic reason for this behaviour is that, within a porous solid, the much smaller crystals of ice have a much greater surface area compared to the bulk, which means the finely divided solid phase has excess energy and, thence, shifts the liquid–solid equilibrium towards the liquid phase. For fluids within porous solids, the freezing and melting points do not typically occur at the same temperature, and hysteresis is present. In general, the ice phase will melt at a temperature higher than that which the liquid originally froze at. Three theories have been suggested that can account for the presence of hysteresis (Petrov © Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_4

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and Furó 2006). First, the freezing of a liquid can be kinetically limited at the equilibrium temperature and requires a seed of ice crystal for (heterogeneous nucleation of) freezing to occur then. If the temperature is reduced below the equilibrium temperature, the freezing eventually occurs homogeneously at the spinodal point. Second, freezing can be delayed by a pore-blocking effect akin to those present in gas desorption and mercury intrusion, which will be discussed in more detail below. The third theory is discussed next. Single Pore Hysteresis Petrov and Furó (2006) suggested that the freezing–melting hysteresis often observed in thermoporometry arises from a free energy barrier between metastable and stable states of pore-filling material. The equilibrium state corresponds to a minimum in the Helmholz free energy. The Helmholz free energy of the system is made up of the volume free energies of the solid and liquid phases (dependent upon their chemical potentials), the free energies of the solid–liquid and liquid-wall interfaces, a contribution that represents the effect of the surface-induced perturbation in the liquid, and work terms. The first three contributions mentioned above depend upon the thickness of the liquid layer adjacent to the pore wall, which can range from zero to the pore diameter. At relatively low temperatures, the liquid-layer thickness dependent part of the Helmholz free energy has two local minima, one corresponding to a fully molten pore, and one to a pore with a frozen core, leaving a liquid-like layer at the pore surface. This liquid-like layer arises from the surface-induced perturbation mentioned above. The system is kinetically limited as there exists an energy barrier between these two states that the available thermal fluctuations cannot surmount. As the temperature is increased, the size of this barrier decreases, leading to the potential for transition between the states and pore core melting to begin. The corresponding freezing temperature of the system depends upon whether there is an ice seed available next to the liquid phase, or it requires supercooling and homogeneous nucleation. According to this phenomenological description, Petrov and Furó (2006) showed that the freezing point depression is given by: υγsl T 0 S , Tf ∼ =− H V

(4.1)

while the melting point depression is given by: υγsl T 0 ∂ S Tm ∼ , =− H ∂ V

(4.2)

where υ is the molar volume, γ sl is the surface free energy, T 0 is the bulk melting point, H is the latent heat of melting, S is the surface area of the pore, and V is the volume of the pore. Using Steiner’s formula for equidistant surfaces, Petrov and Furó (2006) showed that Eq. (4.2) can be rewritten as: 2κ V υγsl T Tm ∼ 2κ = Tf , =− H S 0

(4.3)

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where κ is the integral mean curvature of the pore surface given by:  κ=

1 2S

   1 1 ∫ dS, + r1 r2 S

(4.4)

where r 1 and r 2 are the principal radii of curvature. For a cylindrical pore, 2κV/S = 1/2 (since for a cylinder, ∂S/ ∂V = 1/r and S/V = 2/r), and thus, this means that the predicted numerical difference of a factor of 2 in T m and T f can be used to determine whether a pore has a cylindrical geometry. Equations (4.3) and (4.4) implicitly imply that, upon freezing, the ice front advances axially along the pore from the hemispherical meniscus at the end where the initial liquid is in contact with solid, while melting commences from the liquid-like film next to the pore wall and propagates radially from the surface towards the pore core. Advanced Melting Phenomenon Some porous materials, such as templated silicas, have an ordered, ‘wine-rack’ type structure, with parallel arrays of straight, regular cylindrical pores, for which the single pore hysteresis theory would apply directly. However, many porous media have much more complex void spaces with interconnecting pores. In such materials, additional physical effects arise, and, in particular, the phenomenon known as advanced melting. The simplest geometry in which this effect may arise is a through ink-bottle pore, consisting of a large, cylindrical pore body sandwiched between two smaller, co-axial pore necks, such as illustrated in Fig. 4.1. At very low temperatures, the pore cores of the body and necks would be frozen, leaving a liquid-like layer at the pore walls. On increasing the temperature, first, the liquid–solid menisci in the pore necks would advance radially, thereby melting all of the liquid in the necks. This would then mean that the ends of the pore body would have a liquid–solid meniscus spanning across the whole pore cross-section, such that two hemispherical menisci existed at the pore ends. As described in the previous section, the phase transition occurs at a lower temperature axially via a hemispherical meniscus, than radially via a cylindrical-sleeve meniscus. Hence, the pore body will now melt at the lower temperature corresponding to the hemispherical meniscus. From Eq. (4.3), if the pore body diameter does not exceed that of the pore necks by more than a factor of 2, then both the necks and body will melt at the same temperature. This effect is analogous to the advanced adsorption effect described in Sect. 2.3.4. Since the nature of the void space being characterised is obviously not known in advance, the particular meniscus geometry via which melting occurs cannot be known for each pore. Hence, if a monotonic relationship between melting temperature and pore size is presumed, then the advanced melting effect will make the pore body seem to have the same size as the necks.

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Fig. 4.1 Schematic depiction of the operation of advanced melting. In a through ink-bottle pore, the melting first occurs in the cylindrical necks in the radial direction (as indicated by vertical arrows), originating from the non-freezing layer at the surface of the pores. Melting in the larger, cylindrical pore body then occurs at the same temperature, but originating from the complete hemispherical meniscus formed at the ends of the body once the necks have fully melted, and proceeding in an axial direction (as indicated by horizontal arrows)

4.1.2 Nature of Experiment

Choice of Experimental Technique In order to perform a thermoporometry experiment, it is necessary to measure the temperature of the phase transition of the probe fluid and the amount of the probe fluid undergoing that transition. In general, these measurements are made using either differential scanning calorimetry (DSC) or nuclear magnetic resonance (NMR). DSC requires apparatus with the necessary cooling apparatus and data processing system. The DSC detects melting or freezing by the flow of latent heat into, or out of, the sample. A typical DSC has the capability to measure heat flow rates with a resolution of ±0.5 μW and an accuracy of ±2 μW. Thermoporometry experiments with DSC are run as a pseudo-equilibrium process, since the temperature variation is usually made in a continuous ramp, rather than stepwise (as per NMR below). This means that a sufficiently low temperature ramp rate must be selected to ensure that all of the probe fluid that melts at a particular temperature has had time to do so

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before the temperature has changed significantly. Typically, DSC measurements are carried out at a low scanning rate of ~0.1 ◯ C min−1 . However, preliminary studies involving varying the scanning rate are essential to ensure that this value is the most appropriate. The absolute volume of pores melting at a particular temperature can be obtained from the heat flow by dividing by the latent heat of fusion. NMR experiments are typically conducted on liquid-state spectrometers tuned for the 1 H nucleus (e.g. in the hydrogen nucleus in water). The temperature of the NMR probe typically uses a controlled flow of cool nitrogen gas, evolved from liquid nitrogen, in combination with a heating element below the sample in the gas flow stream, to maintain a particular temperature. Typical systems are able to adjust and maintain a temperature within ±0.1 K for the range of 123–423 K. The chosen NMR pulse sequence is usually a simple spin-echo sequence, such as a basic form of the Carr-Purcell Meiboom-Gill, (CPMG) sequence. The echo time of the pulse sequence must be selected such that it acts as a relaxation time filter, wherein the signal from the solid ice decays to undetectability, but signal is retained for the liquid. The NMR signal obtained will then be proportional to the amount of nuclei within liquid molecules. By this means, NMR can be used to measure the fraction of the void volume occupied by probe fluid that is in the molten state. The experiment is generally conducted in a stagewise equilibrated manner. The sample is first cooled to a temperature beyond which supercooling is overcome, and the whole sample is frozen. The temperature can then be increased in small steps and allowed to equilibrate at each step before the NMR signal strength is measured at each temperature, to give rise to a melting curve, such as that shown in Fig. 4.2. The freezing curve can be obtained in an analogous way with temperature decrements. If required, the absolute volume of pores melting or freezing at a particular temperature can be obtained by comparing the change in the size of the NMR signal strength to that of a standard of known volume of probe fluid. Such an internal standard might be a capillary of known volume of bulk fluid. Sample preparation for dry samples involves fluid imbibition. The simplest approach is to immerse the sample in the probe fluid and allow capillary action to draw the liquid into the pores. If the sample is left for sufficient time, this process is usually enough to displace all of the air originally within the sample. This has been explicitly tested for typical systems such as deionised water in mesoporous silicas. Hollewand and Gladden (1995) found that the water uptake by the ambient immersion method was not significantly different to that by vapour uptake into a previously evacuated sample. In thermoporometry, it is generally required to have a bulk film left around the sample particles. Hence, when the sample is removed from the probe fluid bath, only excess water is removed but leaving sufficient to form a bulk phase film on the exterior of the sample. This bulk phase can then act as a ‘seed site’ for heterogeneous nucleation of ice growth. Partially saturated samples can be obtained by using vapour-phase adsorption but this will be discussed in more detail in Sect. 6.6 on hybrid methods. Non-wetting fluids, such as mercury, can also be used as probe fluids in thermoporometry but obviously require a different procedure for sample preparation. Mercury porosimetry (as described in Chap. 3) can be used to produce samples for

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Fig. 4.2 NMR cryoporometry data for a powder sample from batch S1 showing the substantial initial supercooling for the freezing curve (open diamond) of the first cycle, and the separate freezing (open square) and melting (filled square) curves for the subsequent second cycle. Also shown is the prediction of the melting curve (multiplication sign) for the second cycle obtained using the experimental data from the corresponding freezing curve and Eq. (4.3). The lines shown are to guide the eye. Reprinted from Perkins et al. (2008), Copyright (2008), with permission from Elsevier

thermoporometry if the mercury entrapment is sufficiently large and/or in the right places that it can be used as a probe fluid. Choice of Probe Fluid If the purpose of the experiment is to obtain the full pore size distribution for the whole void space of the sample, then it is necessary to have a probe liquid that can wet the whole void space. For example, water will readily wet the pore space of sol-gel silicas and expunge all the air. A key consideration, in the choice of probe fluid for thermoporometry, is the nature of the liquid–solid phase transition. In order for the different sizes within, especially, a narrow, pore size distribution to be resolved properly, the phase transition for the probe fluid should be as abrupt as possible. This means that the probe fluid changes from a well-defined solid phase to a clearly liquid phase over a narrow temperature range. Some probe fluids, such as hydrocarbons, can have much more complex phase transitions with intermediate phases between hard ice and mobile liquid. For example, cyclohexane is reported to have a so-called mushy-ice phase with molecular mobility part way between liquid and solid (Dore et al. 2004). This tends to smear out the phase transition from liquid to solid (or vice versa) in a particular pore of a given size making the temperature range over which it appears to happen overlap with those for

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other pores with different pore sizes. Hence, in order to select the appropriate probe fluid, it is necessary to consult the phase diagram for that fluid to assess the likely complexity of the phase transition. Two probe fluids can be used simultaneously in the same sample, for reasons that will be explained below. If a partial saturation of one probe fluid has been obtained, such as by mercury porosimetry or adsorption, as mentioned above, then some of the sample void space may remain accessible to another probe fluid. It is, thus, possible to attempt dual-liquid thermoporometry. For example, a sample containing entrapped mercury in some pores can be immersed in a bath of a second probe fluid, such as water or hydrocarbon liquid, as described in Sect. 4.1.2. What happens in this procedure depends upon the relative wetting properties of the two probe fluids, and any kinetic limitations on migration of the first probe fluid. For example, if a mercurycontaining mesoporous silica is immersed in water, the water is so much more wetting of the polar silica surface than mercury that it will force the mercury out over the course of minutes by capillary action (Mousa et al. 2019). However, this may still be sufficient time to initially freeze the system with the water in the initially empty porosity only. Since mercury melts at a lower temperature than water the melting curve for the mercury can be obtained while the water is still ice. Subsequently, the melting curve for the water can be obtained at a higher temperature. In contrast, if a less polar solvent is selected for the second probe fluid, then the mercury is retained much longer even without freezing the system.

4.1.3 What Can I Find Out with This Method?

Pore Shape In order to obtain pore size distributions by indirect methods, the first assumption made is often that of pore shape, such as cylinder or slit. However, as described in Sect. 4.1.1, basic theory suggests that the width of the hysteresis between melting and freezing curves can be used to determine the pore geometry in thermoporometry. For example, Fig. 4.2 shows the melting and freezing curves obtained for a fragmented sample of a sol-gel silica sphere, denoted S1. Also shown in Fig. 4.2 is the melting curve predicted using the freezing curve data and the form of Eq. (4.3) appropriate for cylindrical pore geometry. The data in Fig. 4.2 shows that Eq. (4.3) produced a good prediction for the shape and position of the melting curve. Hence, the model of Petrov and Furó (2006) suggests that the hysteresis probably has a single pore origin, and that the pores in S1 are of cylindrical geometry. Pore Size Distributions In order to determine absolute pore sizes from thermoporometry, it is necessary to know the values of the characteristic parameters in Eq. (4.1). However, it is very difficult to obtain appropriate values of the surface free energy in confined geometries.

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Hence, it is common practice to simply calibrate the constant of proportionality, known as the Gibbs–Thompson parameter, in the expression: Ti =

k GT i , x

(4.5)

where T i is the melting or freezing point depression, k GTi is the corresponding Gibbs–Thompson parameter, and x is the size of the crystallite of frozen probe fluid i. The size parameter x is related to the pore size, d, by the equation: x = d − 2t,

(4.6)

where t is the thickness of the unfrozen liquid-like layer that exists at the pore surface when the core is frozen. This t-layer is typically 1–2 molecular diameters thick, and arises even for liquids like mercury. Various ways have been adopted to calibrate k GT . The most common method is to measure the pore size of a model material independently. Common model materials include templated silicas, such as MCM-41 or SBA-15, which have a wine-racklike structure of parallel, regular, cylindrical pores (Schreiber et al. 2001). The pore size for the model material is determined by electron microscopy or gas sorption. However, different sets of workers have obtained different values of k GT even for nominally the same pore/meniscus geometry. For example, Schreiber et al. (2001) suggested that the value of k GT for melting in silicas with a through cylindrical geometry is 52 K nm, but, in contrast, Gun’ko et al. (2007) suggested a value of 67 K nm for melting in similar silicas. Even for the supposedly simple pore geometries of templated model materials, the meniscus geometry will not be known directly if only a simple melting curve is obtained, as there is no absolute reference, not even that which would be provided by a heterogeneously nucleated freezing curve. Hence, in order to disambiguate the correct meniscus geometry for the phase transition, additional information is required. For example, in order to obtain k GT for mercury, Bafarawa et al. (2014) used controlled pore glasses (CPGs), which are model materials having cylindrical, wormlike pores with a narrow pore size distribution, as seen from the very steep mercury intrusion curve shown in Fig. 4.3. Partial entrapment of mercury occurs following porosimetry on CPGs allowing them to be used for calibrating k GT . However, since the mercury only partially occupied the void space it was not clear whether the melting would arise from a hemispherical meniscus for mercury at a dead end, or via a cylindrical-sleeve meniscus for mercury slugs entrapped part way along a longer pore, like a tube train in a London underground tunnel. This ambiguity was resolved by using mercury porosimetry scanning curves to manipulate the disposition of the entrapped mercury. Mercury was entrapped in the same CPG using porosimetry experiments involving either a full intrusion to 414 MPa which fully saturated the sample with mercury at the top of the intrusion curve, or a partial intrusion to only 48.2 MPa, as shown in Fig. 4.3. The same amount of mercury was entrapped in each case. This mercury was

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Fig. 4.3 Mercury intrusion and extrusion curves for porosimetry experiments on samples of CPG1 with ultimate pressures of 414 MPa (open diamond—intrusion, open square—extrusion) and 48.2 MPa (open triangle—intrusion, multiplication sign—extrusion). The lines shown are to guide the eye. Reprinted from Bafarawa et al. (2014) under Creative Commons CC-BY license

then used as the probe fluid for a subsequent DSC thermoporometry experiment, and the resultant melting curves are shown in Fig. 4.4. From Fig. 4.4, it can be seen that both DSC data sets contain a bulk liquid melting peak at ~38–39 °C. The shoulders evident on the main bulk peak may be due to mercury on the exterior surface of the sample located within cracks and gaps of varying sizes/geometries, plus larger blobs of mercury liquid more like bulk. Figure 4.4 also shows the DSC melting curve for the mercury entrapped following the porosimetry experiment with an ultimate intrusion pressure of 414 MPa. These data showed a relatively narrow peak at the bulk melting point of mercury (~−39 °C), and a wider, more asymmetric peak with its mode at −40.9 °C, together with a slight tail towards lower temperatures, arising from the entrapped mercury. The melting point depression, relative to the bulk liquid value, for the pore fluid was 1.9 °C. The atomic diameter of mercury is ~0.3 nm. From the known pore size of the CPG, the Gibbs–Thomson parameter obtained was ~45 K nm (based on diameter). If the observed melting occurred from a cylindrical-sleeve meniscus, this implies that the corresponding Gibbs–Thomson parameter for freezing/melting via a hemispherical meniscus would be 90 K nm. However, if the melting was, instead, occurring via a hemispherical meniscus, this would imply that the Gibbs–Thomson parameter would be 22.5 K nm for melting from a cylindrical-sleeve meniscus. This ambiguity was resolved using the data for the sample prepared using the porosimetry scanning curve shown in Fig. 4.3. In the corresponding thermoporometry data, it was found that the mode of the melting peak for mercury entrapped within the

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Fig. 4.4 DSC melting curves for a macroscopic droplet of bulk mercury and b entrapped mercury following porosimetry experiments on samples of CPG1 with ultimate pressures of 414 MPa (solid line) and 48.2 MPa (dashed line). Reprinted from Bafarawa et al. (2014) under Creative Commons CC-BY license

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sample was at a temperature of ~−42.0 to −42.1 °C, corresponding to a melting point depression of 3–3.1 °C, which was significantly bigger than for the sample following full saturation with mercury. The peak for the scanning curve sample occurred over the same range of temperatures as the broad shoulder on the peak for the full intrusion experiment (up to 414 MPa). The melting temperature for the entrapped mercury for the scanning curve sample is such that the melting point depression is about twice that of the modal melting peak for the mercury entrapped following the complete saturation experiment. Bafarawa et al. (2014) proposed that the scanning curve was more likely to produce hemispherical menisci than the full intrusion such that it was likely that the melting peak position for the scanning curve experiment corresponds to that for a hemispherical meniscus, and the melting curve peak position for the full saturation experiment corresponds to that for a cylindrical-sleeve meniscus. Hence, overall, the CPG thermoporometry suggested that k GT for mercury is ~45 K nm (for diameter) when melting arises from a cylindrical-sleeve meniscus, and 90 K nm for freezing/melting arising from a hemispherical meniscus. An alternative method to resolve the ambiguity surrounding the k GT value is to use dual-liquid porosimetry. If a relatively low meting point probe fluid, such as mercury (mpt. −40 °C), is initially entrapped in a sample it can form dead ends in previously through pores, such as shown in Fig. 4.5. If a higher melting point probe liquid (such as water, mpt. 0 °C) had also been imbibed into the remaining void space then, when it is still an ice phase, it can act as a complementary dead end, and thus possible site

Fig. 4.5 Light microscopy image of mercury ganglia (black) entrapped in straight glass capillary. Reprinted from Hitchcock et al. (2014), Copyright (2014), with permission from Elsevier

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for a hemispherical meniscus, for the melting of the low melting point probe fluid. Thus, the shape of the melting meniscus of the low melting point probe fluid can be controlled. Pore Connectivity and Spatial Heterogeneity As mentioned above, the causes of hysteresis in thermoporometry can have single pore or co-operative pore–pore origins. Of relevance to pore connectivity is that a pore-blocking effect, analogous to that arising in gas desorption and mercury intrusion, can occur for heterogeneously nucleated freezing in thermoporometry. If bulk ice surrounds a porous solid which has a complete layer of narrow necks at the surface guarding access to larger voids deeper within the solid, then the molten probe fluid within the larger voids will only freeze once the ice front advances down the necks. This will occur at a lower temperature than would otherwise be needed to freeze the larger voids, and hence pore blocking by the necks can be said to arise. Hence, the advance of the ice front into a porous solid is an invasion percolation process, and the same percolation theory as applied to gas sorption data in Sect. 2.3.3 can also be applied to thermoporometry data. If the melting process within cylindrical pores only occurs radially from a cylindrical-sleeve meniscus, then the melting curve can supply the unshielded pore size distribution. However, the freezing curve will be affected by both single pore and pore-blocking structural hysteresis. In the case of materials, where there is a strong macroscopic correlation in the spatial distribution of different pore sizes these can be deconvolved by fragmentation of the sample to particles of size smaller than the size of the different pore size regions. For both whole and fragmented samples surrounded by a bulk ice film, freezing will be heterogeneously nucleated and occur by the hemispherical menisci arising at pore mouths. Hence, there would be no single pore hysteresis and the difference in the path of the freezing curves would be solely due to percolation effects. An example of this can be seen in Fig. 4.6, where there is a shift in the position of the freezing curve between whole and fragmented samples. In this case, the fragmented sample freezing curve can be used to obtain the de-shielded pore size distribution. In Fig. 4.6, the drop in intensity, I, of the whole pellet curve corresponds to the percolation threshold. A large width for the distance between the freezing curves for whole and fragmented samples would be associated with low connectivity between different pore size regions. A steep freezing curve for the whole pellet sample but a gentle sloping curve for the fragmented sample would mean that there were relatively lots of small, different pore size regions, as opposed to a few large ones, which would be indicated by a more rounded percolation knee in the whole pellet freezing curve. The advanced melting effect can also reveal the presence of large-scale correlations in pore size. If a pore network contains regions of small pores neighbouring regions of bigger pores, the former can facilitate melting in the latter, thus, hiding their existence in the melting curve for the whole sample. However, if the similar pore size regions are spatially extensive, such that the sample can be fragmented into powders with particle sizes smaller than the size of these regions, then the small pores are removed from their former proximity to the larger pores and can, thus,

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Fig. 4.6 NMR cryoporometry (second freeze/thaw cycle) freezing curves for whole (filled square) and powder (filled triangle) samples for sol-gel silica denoted S1. Also shown (solid line) is the prediction for the whole pellet freezing curve obtained using the mercury porosimetry intrusion curve for whole pellets, and the Gibbs–Thomson equation calibrated using the NMR cryoporometry freezing curve and the mercury intrusion curve for a powder sample from batch S1. Reprinted from Perkins et al. (2008), Copyright (2008), with permission from Elsevier

no longer facilitate advanced melting. This means that the melting curve for a fragmented sample will show melting over a broader temperature range. This effect has been observed for sol-gel silicas, such as that for which the data is given in Fig. 4.7. Scanning curves and loops can also be used in thermoporometry to probe pore geometry and spatial arrangement (Rigby et al. 2017). For example, many materials contain large macroporous and pervasive channels that provide good mass transport access to many more, smaller side channels. It is often desired to know the entrance sizes of these side channels, as various processes can act to narrow or blind such openings. A study has been made of the mesoporosity branching off macropores in a silica-alumina catalyst (Rigby et al. 2017). Figure 4.8 shows the NMR cryoporometry boundary melting curves for bulk ice and ice contained in the pores of the silicaalumina catalyst. Figure 4.8 shows that the melting temperature is depressed for the water in the void space of the sample. A scanning curve consisting of melting up to an ultimate temperature of 272 K, followed by freezing is also shown in Fig. 4.8. It can be seen that the gap between the melting and freezing branches of the scanning curve is initially (around the point where the temperature rise is reversed) very narrow, perhaps being only ~0.5 K, and the hysteresis remains narrow until the curve reaches a temperature of ~269 K. However, for the region below a temperature of ~267 K on the

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Fraction of molten phase

1.2 1 0.8 0.6 0.4 0.2 0 263

265

267

269

271

273

275

Temperature /K Fig. 4.7 An overlay plot of typical data sets, consisting of freezing and melting boundary curves, for whole (stars) and fragmented (diamonds) samples from batch S1 sol-gel silica. Reprinted from Hitchcock et al. (2011), Copyright (2011), with permission from Elsevier

Fig. 4.8 NMR cryoporometry boundary melting curves for bulk ice (solid line) and water imbibed within the void space of the silica-alumina catalyst (dashed line). Also shown in a melting–freezing scanning curve (multiplication sign and dotted line) up to 272 K. The double-headed arrow indicates that the width of the hysteresis between the freezing and melting branches of the scanning curve has grown to ~6 K at a melting temperature of 267 K. Reprinted with permission from Rigby et al. (2017), Copyright (2017) American Chemical Society

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Fig. 4.9 Schematic diagram of proposed model for pore network (black) of the silica-alumina catalyst. Reprinted (adapted) with permission from Rigby et al. (2017), Copyright (2017) American Chemical Society

melting curve, the hysteresis width has grown to ~6 K. Therefore, at a temperature of 267 K, the freezing point depression is about double that of the melting point depression. Using a Gibbs–Thompson parameter of 52 K nm (for melting from a cylindrical-sleeve meniscus) and a non-melting layer thickness of 0.4 nm, the pore diameter at this point is ~9.4 nm. The freezing point depression at the turning point for the scanning curve was ~2 K, suggesting a pore entrance size of ~53 nm (assuming the Gibbs–Thompson parameter for freezing via a hemispherical meniscus is twice that of a cylindrical-sleeve shape). The NMR cryoporometry scanning curve data shown in Fig. 4.8 are consistent with the pore network model shown in Fig. 4.9. If all of the probe fluid imbibed in the model was initially frozen, then melting would start from a cylindrical-sleeveshaped meniscus at the boundary between the non-freezing surface layer and the frozen bulk in the smallest through pore (as in Fig. 4.1). Once this pore was fully molten, hemispherical menisci would be formed at its junction with the neighbouring medium-sized pores. Melting in these medium-sized pores could then occur via hemispherical menisci (as shown in Fig. 4.1). If the rise in temperature was halted with the smallest and medium-sized pores molten but with the largest pores remaining frozen then, when the direction of the temperature change was reversed, freezing could then commence from the hemispherical menisci remaining at the junctions between the medium-sized and largest pores. Hence, both the melting and freezing processes in the medium-sized pores would arise from hemispherical menisci, and, thus, it would be expected that there should not be any hysteresis. However, once the temperature had decreased sufficiently for the smallest pores to freeze, this would arise via a hemispherical meniscus, instead of the cylindrical-sleeve shape during melting, and thus hysteresis would be expected. Hence, in summary, while the medium-sized pores would be anticipated to freeze and melt with little or no hysteresis, hysteresis would be expected between the freezing and melting of the smallest pores. This scenario is similar to what was shown in Fig. 4.9, wherein the early part of the freezing scanning curve had very little hysteresis but this increased

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substantially at lower temperatures. Hence, if the large pores in this scenario are a pervasive macropore network, and the medium-sized pores are the entrances to the mesoporous side branches, then the scanning curve can obtain their size, as above.

4.2 Scattering 4.2.1 Basic Theory

Small-Angle X-Ray Scattering (SAXS) When a material is irradiated with X-rays they are primarily scattered from the electrons. The spatial distribution of electrons within a material is, typically, not homogeneous, and variations in electron density (number of electrons per unit volume) arise across the sample. When the typical sizes of these spatial heterogeneities are similar to the wavelength, λ, of the incident X-rays, then the scattering will be mostly observed at angles greater than 10z , as in normal X-ray diffraction patterns. However, where the heterogeneities in the electron density occur over distances between the orders of ~0.5 and ~400 nm, then an appreciable intensity of X-rays is scattered at small angles. Therefore, SAXS enables the determination of information for structures over longer length-scales than the normal interatomic distances found for dense materials. The scattered X-ray intensity, I(q), arising from a disordered porous solid depends upon the scattering wave vector, q, whose magnitude is given by: |q| = (4π/λ) sin θ,

(4.7)

where 2θ is the angle through which the X-rays are scattered. q has units of reciprocal length, and, thus, 1/q can be thought of as the ‘ruler’ for the measurement.

4.2.2 Nature of the Experiments In order to obtain the scattering pattern, the sample must be suspended in the path of the X-ray beam. The shape of the sample affects the scattering pattern, and thus this will need to be taken into account when processing the raw data. Hence, it is usual that the sample is somehow contrived to be a simple geometry. For example, if the sample is already a simple shape like a sphere, such as a sol-gel silica bead, then it can sometimes be suspended in the beam as it is. For other samples with, initially, more irregular shape, they may be fragmented to a powder form and, either, put in a narrow, glass capillary tube, or stuck to a thin strip of sticky tape, to form a line shape. The scattering pattern may need to be corrected for background and detector response. The raw data, following initial processing by the SAXS equipment, consists of an

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ordered data set of scattered intensity I(q) variation with the wave vector q. Since the X-rays penetrate the whole of the sample, they can be used to probe isolated porosity, as well as that connected to the surface probed by methods like thermoporometry or gas adsorption. However, if contrast between the void space and the solid is poor, a fluid can be imbibed in the pores to improve scattering contrast. The probe fluid, thus, needs to have a large difference in electron density compared to the solid, and, of course, will only penetrate accessible porosity.

4.2.3 What Can I Find Out with This Method? General theoretical expressions derived for I(q) depend upon the value of the product qR, where R is the characteristic size of the structural aspect that is leading to the scattering (such as the solid-pore interface). X-ray scattering can be used to determine properties, such as surface area or roughness, and pore or solid particle size. At larger values of q, the scattering can provide information on the pore sizes, whereas at lower values of q, it can provide information on the pore surface. Guinier et al. (1955) has demonstrated that, for isolated scattering particles, at the limit of small values of q, when the group qRg is not much bigger than unity, the intensity of scattered X-rays is given by:   I (q) = I (0) exp −q 2 Rg2 /3 ,

(4.8)

where I(0) is the scattered intensity at q = 0 and Rg is the radius of gyration of the scattering structures. Rg can, therefore, be obtained from the slope of a plot of the ln I against q2 . R can be estimated from Rg if the shape and distribution of the scatterers is known. For example, if the system of scatterers can be considered to be made up of identical spherical objects with radius r 0 , then the relation r 0 = 1.3Rg can be used to find r 0 (Guinier et al. 1955; Venkatrama et al. 1996). As shown in the microscopy studies discussed in Chap. 1, many sol-gel silica are made up of a close packing of spheres (Reyes and Iglesia 1991). Therefore, such a packing of spheres is frequently utilised as representation of silica materials (Reyes and Iglesia 1991). Another approach for the analysis of SAXS data was given by Debye et al. (1957). Debye et al. (1957) demonstrated that a porous structure, where the pores have a completely random distribution of size and shape, has an exponential correlation function, γ = exp(−r/a). The correlation function measures the degree of spatial correlation between two instances of the same phase (solid or void), as a function of their distance apart, r. For such a system, the scattered intensity would decay with q according to: −2  I (q) = I (0) 1 + a 2 q 2 ,

(4.9)

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where I(0) is the scattered intensity at zero deviation of the beam, and a is the correlation length, which can be considered a measure of the size of the objects performing the scattering. When qa > 1, a plot of I −1/2 against q2 should yield a straight line if the system obeys Eq. (4.9). The descriptor a is obtained by taking the square root of the result of dividing the gradient by the corresponding intercept. For random particle packings, the typical size of gaps between particles (the pores) is the same size as the particles. a can be thought of as a typical particle or pore size. A typical example of such a plot for scattering data for a sol-gel silica is shown in Fig. 4.10. The size of a from the plot is 9.38 ± 0.02 nm. Scattering from a solid surface often obeys what is known as the Porod (1951) law, where the intensity of X-ray radiation scattered by a surface is typically proportional to a negative power of the q vector, such that (Guinier et al. 1955; Venkatrama et al. 1996): I ∝ q −η ,

(4.10)

where η is the power. Usually, this dependence is observed only when q satisfies the inequality qξ  1, where ξ is the characteristic length of the structure producing the scattering. From the value of η, the nature of the structure under examination can be deduced. If the exponent is in the range 1 < η < 3, then it is possibly a mass fractal of dimension (Dm )SAXS = η, whereas if the exponent is in the range 3 < η < 4, it is probably describing a surface fractal of dimension (DS )SAXS = 6 − η. When η = 4, Eq. (4.10) leads to the original Porod’s law, where (DS )SAXS = 2 and the surface is smooth. The coefficient of proportionality in Eq. (4.10) is related to the surface area of the solid but requires calibration in order to obtain it.

Fig. 4.10 Debye-type fit of Eq. (4.9) to SAXS data for a sol-gel silica G1. Reprinted from Rigby and Edler (2002), Copyright (2002), with permission from Elsevier

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4.3 Conclusions Since it can use the pre-existing pore fluid of already wet samples, thermoporometry has the potential advantage that it can be used without a sample pre-treatment step involving drying, which may modify the sample creating artefacts in the characterisation data. Thermoporometry can involve just obtaining the basic boundary curve data but can also make use of more elaborate experiments involving use of scanning curves, which will potentially deliver much richer information on the void space. Both melting and freezing processes in thermoporometry are potentially impacted by pore–pore co-operative effects which can be co-opted to deliver information on pore network properties. In particular, a proper understanding of the advanced melting effect, and when it occurs, is key to the correct interpretation of, and obtaining accurate pore sizes from, thermoporometry data. X-ray scattering experiments, similar to thermoporometry, often do not need any sort of sample pre-treatment or addition of a new probe fluid. With scattering, this means it can probe isolated, disconnected porosity, as well as that porosity accessible from the exterior surface of the sample. Scattering can be used to obtain descriptors of surface area, surface roughness, primary particle size, and pore size.

References Bafarawa B, Nepryahin A, Ji L, Holt EM, Wang J, Rigby SP (2014) Combining mercury thermoporometry with integrated gas sorption and mercury porosimetry to improve accuracy of pore-size distributions for disordered solids. J Colloid Interface Sci 426:72–79 Debye PA, Anderson HR,.Brumberger H (1957) Scattering by an inhomogeneous solid. II. the correlation function and its application. J Appl Phys 28(6): 679–683 Dore J, Webber B, Strange J, Farman H, Descamps M, Carpentier L (2004) Phase transformations for cyclohexane in mesoporous silicas. Phys A 333:10–16 Guinier A, Fournet G, Walker CB, Yudowitch KL (1955) Small-angle scattering of X-rays. Wiley, New York Gun’ko VM, Turov VV, Turov AV, Zarko VI, Gerda VI, Yanishpolskii VV, Berezovska IS, Tertykh VA (2007) Behaviour of pure water and water mixture with benzene or chloroform adsorbed onto ordered mesoporous silicas. Cent Eur J Chem 5(2):420–454 Hitchcock I, Holt EM, Lowe JP, Rigby SP (2011) Studies of freezing–melting hysteresis in cryoporometry scanning loop experiments using NMR diffusometry and relaxometry. Chem Eng Sci 66:582–592 Hitchcock I, Lunel M, Bakalis S, Fletcher RS, Holt EM, Rigby SP (2014) Improving sensitivity and accuracy of pore structural characterisation using scanning curves in integrated gas sorption and mercury porosimetry experiments. J Colloid Interface Sci 417:88–99 Hollewand MP, Gladden LF (1995) Transport heterogeneity in porous pellets-I. PGSE NMR studies. Chem Eng Sci 50:309–326 Mousa S, Baron K, Softley E, Fletcher RS, Kelly G, Garcia M, Mcleod N, Rigby SP (2019) Elimination of ambiguity in analysis of thermoporometry using dual probe liquids. In: Düren T et al (eds) Characterisation of porous solids XII (COPS-XII) Perkins EL, Lowe JP, Edler KJ, Tanko N, Rigby SP (2008) Determination of the percolation properties and pore connectivity for mesoporous solids using NMR cryodiffusometry. Chem Eng Sci 63:1929–1940

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Petrov O, Furó I (2006) Curvature-dependent metastability of the solid phase and the freezingmelting hysteresis in pores. Phys Rev E 73:011608 Porod G (1951) Die Röntgenkleinwinkelstreuung von dichtgepackten kolloiden Systemen. Kolloid Zeit 124:83–114 Reyes SC, Iglesia E (1991) Effective diffusivities in catalyst pellets: new model porous structures and transport simulation techniques. J Catal 129(2):457–472 Rigby SP, Edler KJ (2002) The influence of mercury contact angle, surface tension and retraction mechanism on the interpretation of mercury porosimetry data. J Colloid Interface Sci 250:175–190 Rigby SP, Hasan M, Stevens L, Williams HEL, Fletcher RS (2017) Determination of pore network accessibility in hierarchical porous solids. Ind Eng Chem Res 56(50):14822–14831 Schreiber A, Ketelsen I, Findenegg GH (2001) Melting and freezing of water in ordered mesoporous silica materials. Phys Chem Chem Phys 3:1185–1195 Venkatrama A, Boateng AA, Fan LT, Walawender WP (1996) Surface fractality of wood charcoals through small-angle X-ray scattering. AIChEJ 42(7):2014–2024

Chapter 5

Nuclear Magnetic Resonance and Microscopy Methods

5.1 Introduction This chapter covers the use of the nuclear magnetic resonance (NMR) effect in pore structure characterisation. NMR is covered in this chapter since it forms the basis of one of the most important imaging modalities, magnetic resonance imaging (MRI). Several other imaging methods will also be considered, namely computerised X-ray tomography (CXT), scanning electron microscopy (SEM), transmission electron microscopy (TEM), and helium ion microscopy (HIM). Imaging refers to techniques that aim at a direct (or pseudo-direct) visualisation of the void space of porous media. Imaging techniques typically produce the images as two-dimensional (2D) or three-dimensional (3D) lattices of ‘picture elements’, often contracted to ‘pixels’. In 3D, the image elements consist of volume elements, which is contracted to ‘voxels’. The size of these pixels or voxels determines the smallest elements of the structure imaged that can be distinguished and is known as the image ‘resolution’. The particular signal intensity in each pixel, from a range known as the ‘greyscale’, determines the phase it belongs to, such as solid or void. In order to directly image the void space within a porous material, it is necessary for the resolution to be smaller than the pore size. However, there are techniques that can obtain spatially resolved information on the void space even when the resolution is larger than the pore size, and these will be described below.

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5.2 Theory of Nuclear Magnetic Resonance (NMR) Spectroscopy and Imaging Techniques 5.2.1 NMR Spectroscopy The NMR effect is based on the fact that many nuclei of various different isotopes behave somewhat like little compass needles, and, thus, when placed in a strong, external magnetic field, B0 , will line up with it like a compass needle does with magnetic north. This situation has the lowest energy state for the nucleus. However, since nuclei are so small they are continuously ‘buffeted’ by thermal energy and, at any given time, some of the compass needles will be flicked ‘upside down’ by thermal fluctuations, into a higher energy state. The difference in energy between the aligned and anti-aligned states, E, is proportional to the magnetic field strength. The constant of proportionality depends upon the nature of the nucleus and is known as the gyromagnetic ratio: γ =

2π E . h B0

(5.1)

where h is Planck’s constant. Nuclei possess a magnetic field because they can be crudely considered as like a spinning top of charge, and an electric charge in motion produces a magnetic field. Unlike a macroscopic spinning top, which can spin at any angle on its axis, the apparent axis of rotation of a nucleus can only take up very particular orientations relative to the external magnetic field. The number of orientations depends upon the size of the spin possessed by the particular type of nucleus. For example, a hydrogen nucleus (proton) has a spin value, I, of ½, which means it can take up 2I + 1 = 2 orientations, one labelled such that m = +1/2 is with, and m = −1/2 is against, the imposed field. However, for a hydrogen nucleus, the supposed spin axis is not lined up exactly with the external magnetic field but is tilted slightly to one side. This means that the nucleus undergoes the quantum mechanical equivalent of precession of the axis of rotation, known as Larmor precession, at a (Larmor) frequency, ω, of: ω = γ B0 .

(5.2)

In a macroscopic sample, there is a large ensemble of nuclei (nuclear spins), and some will be aligned with the field and some against. Overall the net magnetic field due to the nuclei will add up to an overall total known as the magnetisation. At equilibrium, this magnetisation will be aligned with the external magnetic field since the majority of the constituent nuclear magnets will be pointing that way. However, a nucleus contained within an atom is partially shielded from the full strength of the external magnetic field by the electrons surrounding it. This is because electrons have a relatively strong magnetic field of their own. The amount of shielding felt by the atomic nucleus depends upon the number, and typical distance away, of the electrons surrounding it, which, in turn, depends upon the forms of the electron

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shells, which themselves depend upon the type of bonding occurring with other nearby atoms within the same molecule, or between the molecule and a neighbouring solid surface. The shielding of the external magnetic field by the electrons results in a slight change in the field strength felt by the nucleus, which by Eq. (5.2) will change the Larmor precession frequency slightly. This effect is quite small, and thus is typically expressed in parts per million (ppm), but can be measured. The changes in frequency are usually measured relative to the frequency of a reference species, which for hydrogen nuclei (protons) is usually tetramethylsilane (TMS). Since the changes are impacted by the chemical environment of the nucleus, they are known as the chemical shift. The xenon-129 isotope is a spin-1/2 nucleus. Xenon-129 NMR spectroscopy can be used for pore structure characterisation because the chemical shift of the xenon atom is very sensitive to the local environment and to chemical factors such as the composition of the material, the type and amount of co-adsorbed molecules, and the form and size of host void spaces. In the absence of strong adsorption, the chemical shift of xenon, δ, is given by: δ = δ0 + δs + δ X e ,

(5.3)

where δ 0 is the chemical shift of xenon gas extrapolated to zero pressure, and δ s is the term due to interaction with the pore surface. The second term can depend upon geometry of the xenon environment on the surface. The term denoted δ Xe arises from intermolecular collisions inside the pore. This term is, thus, proportional to xenon density and, hence, increases with the xenon concentration. In order to relate Eq. (5.3) to pore size, a further assumption about what is occurring inside the porous medium is required. If it is assumed that there is a fast exchange of xenon between the surface and bulk gas, which is typically associated with weak Henry’s law type interactions with the surface, then the chemical shift of xenon-129 in mesoporous materials would be given by: δ=

δa 1+

V K S RT

,

(5.4)

where δ a is the chemical shift of xenon in the adsorbed phase, V is the pore volume, S the surface area, T the temperature, and K the Henry’s law constant. In this case, chemical shift is not dependent upon the equilibrium pressure. From Eq. (5.4), if K and δ a are known, then the volume-to-surface ratio, V/S, can be obtained. However, since gas molecules can diffuse a long way in the course of the NMR experiment, then the chemical shift, and thence the V/S, would be averaged over a large region of void volume, including not just a single pore but many. If the xenon is used to probe a porous solid containing adsorption sites that will strongly bind xenon, then Eq. (5.3) must be amended to reflect this, such that: δ = δ0 + δs + δ X e + δsas , where δ sas represents the impact of the strong adsorption sites.

(5.5)

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5.2.2 NMR Relaxometry and Pulsed-Field Gradient NMR The nuclear spin system can absorb energy at the Larmor frequency, typically corresponding to radio frequencies, which will flip nuclei from alignment with the field, to against it. The net magnetisation may then not be aligned with the field. The system is then in an excited state, and will want to return to the lower, ground state. The return to the ground state is facilitated by fluctuations in the overall pervading magnetic field, which are caused by the motions of nearby (also magnetic) nuclei resulting from say Brownian motion of the fluid molecules they are within. The time it takes for one particular nucleus to return to the lower energy level is unknown beforehand, as it occurs at random, but the overall ensemble of nuclei present in the sample will relax with a half-life (time constant) known as the spin-lattice relaxation time, denoted T 1 . The spin-lattice relaxation time for hydrogen nuclei within water molecules imbibed in the void space of a porous solid depends upon the size of the cavity in which the molecule is located during the time of the experiment. This is because, as mentioned above, the relaxation is caused by random fluctuations in the magnetic field arising from molecular motion. Molecules of probe fluid located close to the surface of the porous solid will be temporarily bound by the attractive forces of the solid surface, and their motion will be thereby reduced, which will change the rate of fluctuation of the magnetic field they generate. For liquids in porous solids, this results in an increase in the rate of relaxation for the surface layer of fluid molecules (such that it has a time constant of T 1s ). However, the surface molecules are also in rapid diffusional exchange with bulk molecules in the pore cores that are tumbling faster, and thus have a different relaxation rate (with time constant T 1b ). The observed relaxation rate for the ensemble of nuclei within the void space is, thus, the volumeweighted average of those for the two different locations, such that (Brownstein and Tarr 1977):   1 λS 1 λS 1 = 1− + , T1 V T1b V T1s

(5.6)

where λ is the thickness of the surface-affected layer. For cylindrical pores, of diameter d, S/V = 2/d. Hence, the observed T 1 is a function of pore size with bigger values associated with larger pores. Besides T 1 there is another NMR time constant used for pore characterisation, known as the spin-spin relaxation time, and usually denoted T 2 . At equilibrium the nuclei will be precessing around the external magnetic field at the relevant Larmor frequency, but all out of phase with each other. Hence, at any moment, the components of their individual magnetic fields perpendicular to the main field will be arrayed at random in all directions, and add up to zero net field. However, the ‘resonance’ aspect

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of the name NMR comes from the fact that when the ensemble of nuclei is hit with a radio-frequency pulse, and some are flipped upside down, the Larmor precession of the nuclei is put into phase with each other, and a net magnetisation emerges along a direction other than the main field that precesses along with the nuclei. This is akin to the situation where an array of spinning tops each with a line painted down one side, all had every line pass the same direction at the same time during their individual rotations. Given the nuclei are, in effect, little magnets in motion together, they will generate an alternating electric current, oscillating with the Larmor frequency, if placed inside a wire coil. However, this situation does not last following the end of the radiofrequency pulse. This is because the local magnetic field subsequently experienced by each nucleus is not the same in both space and time. For real magnets, it is impossible to have a completely uniform field over an extended area, and thus the different nuclei, located at the various possible positions within the external magnetic field, will each precess at very slightly different frequencies. This makes the nuclear precessions get out of phase with each other, which progressively dissipates their large net magnet field that is cutting the coil as it rotates, and thereby creating the current therein, causing that to diminish. This process has a time constant known as T 2 *. The magnetic field experienced by a given nucleus also fluctuates randomly in time and space due to the random thermal motions of the molecules containing neighbouring nuclei. This means the precession speed fluctuates at random, eventually leading to a further dephasing process characterised by a time constant called T 2 . Since this latter process depends on molecular motion, it too is influenced by the presence of solid walls in porous media, and thus the T 2 follows an analogous equation to (5.6) such that:   1 λS 1 λS 1 D(γ GTE )2 , (5.7) = 1− + + T2 V T2b V T2s 12 but with an additional third term which is due to diffusion-induced relaxation, and G is the local field gradient, and T E is the echo-time parameter in the spin-echo type NMR experiment used to measure T 2 . For mesoporous samples, the third term in this equation is typically very small compared to the others, and is neglected. The change in precession frequency with magnetic field strength can also be used to monitor the motion of molecules. If a known, linear gradient in the magnetic field, g, is applied across the macroscopic sample, then the Larmor precession frequency will be a function of distance r such that (Hollewand and Gladden 1995): ω = γ B0 + g.r.

(5.8)

Hence, those nuclei located in the strong part of the field gradient will precess faster than those located in the weaker part of the field gradient. Hence, the nuclei will start to get out of phase with each other, and their net magnetic field will decline. Therefore, the electric current they generate in the surrounding coil will dissipate. The gradient is then switched off for a period, but the offset in the phase of precession of different nuclei that it created is conserved. If the gradient is then switched back

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on but with the direction reversed, those nuclei previously precessing quickly now precess slowly, and vice versa, so the previously lagging nuclei can now can catch up with their newly slow counterparts, and the net magnetisation grows again in strength, thereby recreating a larger current again. However, if any nucleus has moved during the course of the experiment, it will not experience the particular field strength needed to exactly reverse the initial dephasing of precession. Hence, the signal strength, in the form of the maximum current generated after gradient reversal, will be reduced compared to the case if the nuclei had not moved. The extent of the movement of the molecule that caused a particular decline in maximum recovered signal strength can be determined from diffusion theory. If either the duration of the gradient or the gradient strength is varied, a different level of signal attenuation will result such that the ratio of the maximum signal actually achieved, I, to that obtained without diffusion present, I 0 , is given by:     δ I D , = exp −γ 2 g 2 δ 2  − I0 3

(5.9)

where δ is the duration that the magnetic field gradient of strength g is left on to achieve (or reverse) dephasing,  is the time left for diffusion to occur between the two periods the gradient is switched on, and D is the measured self-diffusivity of the molecules. Random motions during the duration of actual application of the magnetic field gradients are hard to reverse, and so the duration of the gradients is kept as short as possible, hence, the name pulsed-field gradient NMR. The main part of the diffusion, leading to the decline in signal strength, should occur between the application of the pulses. The diffusion of molecules can be used to probe different aspects of the pore structure depending upon the relative size of pores and the root mean square (rms) displacement of molecules during the course of the period  given by the Einstein equation:  2 1/2 √ = 6D. r

(5.10)

If the rms displacement is much larger than the pore size, then the observed diffusivity DA is controlled by the tortuosity, τ , of the pore network, such that: DA =

D0 . τ

(5.11)

where D0 is the self-diffusivity of bulk probe liquid. The tortuosity is a measure of the extent of deviation from straight-line paths between points in the porous medium imposed by the pore walls. A more convoluted pathway of twists and turns would result in a higher tortuosity, and a slower observed diffusivity. However, if the rms displacement is of similar order to the size of individual cavities comprising the void space, then the observed diffusivity depends upon the pore size. At very short diffusion times the molecules will not sense the pore walls at all (and the

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observed diffusivity will have its bulk value), but, as the diffusion time increases, more and more molecules will eventually bump into the pore wall, and thereby have the displacement they would have otherwise have made curtailed. The apparent diffusivity observed, thus, appears as a function of diffusion time, such that: 3/2

D() = D0 −

4D0 S 1/2  . 9π 1/2 V

(5.12)

If the observed diffusivity (from Eq. 5.9) is measured as a function of increasing diffusion time it will appear to decline. The surface area-to-volume ratio of pore cavities can be obtained from the gradient of a fit of these data to Eq. (5.12). An analogue of Eq. (5.12) would also apply to the situation when a probe fluid was confined to the mesopore network of a small powder particle or crystallite where the rms displacement exceeded the particle size but some sort of confinement kept the fluid within the particle. In that case, D0 would be replaced by DA for the internal pore network, and S/V would be the ratio of the external surface area to bulk volume of the powder particle. In the derivation of Eq. (5.9), it is assumed that the diffusion is isotropic. If the probe fluid is confined to a geometry within an impermeable boundary in some direction, then the self-diffusion will be restricted in that direction. For example, fluid confined to narrow cylindrical channels may only move in pseudo-one-dimension (if the channel cross-section is very narrow compared to the length). Further, fluid confined to a narrow slit may only move in two directions and would be highly restricted in displacements in the third. If the molecules do not make any displacement along the direction of the field gradient, then the motion cannot be detected. Many samples consist of porous powder particles containing channel or slit pores will, thus, have very many of these pores orientated in random directions relative to that of the applied pulsed gradient. The restrictions placed on the displacement direction of the molecules manifest as deviations from straight lines on the so-called log-attenuation plot, which is a semi-logarithmic plot of Eq. (5.9) with log I plotted as a function of γ 2 δ 2 g2 ( − δ/ 3) (often denoted ξ ). Different geometric constraints lead to logattenuation plots with differing degrees of curvature, as described in more detail by Callaghan (1991).

5.2.3 Magnetic Resonance Imaging Magnetic field gradients can be used for labelling nuclei with their position for the purposes of spatially resolved imaging as well as following diffusion. Magnetic field gradients can be applied in multiple directions to achieve full three-dimensional information. The position of a nucleus can be encoded via the field gradients through differences in precession frequency and the phase of the rotation, as shown in Fig. 5.1. The positional encoding permitted via the field gradients enables just the signal emanating from only a small region of the sample volume (a voxel) to be isolated

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Fig. 5.1 Schematic diagram illustrating how magnetic field gradient (Gz ) can encode position by variation in Larmor frequency

and measured. Imaging utilises so-called soft pulses of radio-wave radiation that only cover a narrow range of frequencies, rather than so-called hard pulses that will excite all frequencies. If a magnetic field gradient is applied along the axis of a sample, then a soft pulse would enable only a slice of the sample to be excited, and thus selected for characterisation. The spatially encoded information can be assembled into lattices of signal from voxels within two-dimensional slices, or full three-dimensional volumes, of the sample. Sequences of radio-frequency (rf) pulses and gradients to achieve spatial encoding of NMR signal can be combined with the sequences that are used for relaxation time or diffusion measurements, in a technique called ‘pre-conditioning’. This technique allows an individual relaxation or pulsed-field gradient experiment to be conducted in each individual voxel of the image lattice, and, thus, a map of the corresponding characteristic parameter to be obtained.

5.2.4 Computerised X-Ray Tomography (CXT) There are several different types of CXT, but this section will be confined to absorption imaging. The basic principle of absorption CXT is that if X-rays, from a suitable source, are passed through an object some of the X-ray energy will be transmitted and some absorbed (and some can be scattered). Materials containing higher densities of electrons will absorb more X-rays. Hence, solids tend to absorb more than fluids like gases, and heavy elements absorb more than light elements. The absorbed amount also depends upon path length, such that the transmitted intensity I follows Beer’s law: I = I0 exp(−μx),

(5.13)

where I 0 is the X-ray intensity incident on the target material, x is the path length, and μ is the extinction coefficient, dependent upon the electron density of the material. Since the X-rays must pass through the sample to be detected, this puts a limit on the path lengths (and thus size/thicknesses) of particular materials that can be imaged.

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The transmitted X-rays can be intercepted by some sort of detection device, such as a screen. The transmitted X-rays form a two-dimensional shadow picture where image intensity is determined by the fraction of X-rays transmitted. A full threedimensional reconstruction of the object casting the shadow can be obtained by passing the X-ray beam through the object over a wide range of different angles to create a set of 100–1000 s of projections. For non-living materials, the sample is usually placed on a turn-table so that it can be rotated to allow the beam to pass through it from many different angles, as shown in Fig. 5.2. Computer algorithms then use this set of shadow pictures to determine the most likely three-dimensional spatial arrangement of density within the test object to give rise to that particular set of projections. The achievable resolution R of CXT images is given by (Cnudde et al. 2011): R=

  1 d + 1− s, M M

(5.14)

where s is the spot size of the X-ray source, d is the pixel size, and M is the magnification. The magnification is given by (Cnudde et al. 2011): M = (total source − detector distance)/(total source − object distance). (5.15) Hence, the sample size potentially limits the resolution by determining the upper limit for the distances possible in Eq. (5.15). Laboratory micro-CXT equipment can achieve resolution down to ~100s nanometres. Higher resolutions can be achieved on synchrotrons, but the availability is limited. In general, the resolution is limited at low magnification by the detector pixel size, while, at higher magnifications, the X-ray spot size becomes limiting.

Fig. 5.2 Schematic depiction of the principles of computerised X-ray tomography

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5.2.5 Electron Microscopy Electron microscopy is based on the principle that all matter has a wave-particle duality. Beams of electron particles, thus, have a wavelength associated with them. The wavelength of electrons is much smaller than that of light. The resolution possible with a given microscopic technique is limited by the wavelength of the probe radiation. Hence, the small wavelength of electrons gives electron microscopes a better resolving power than light microscopes. The wavelength of visible light means it is too large to resolve mesoporosity and microporosity, but electron microscopes can do this. Besides imaging, electron microscopy can also provide information on structure (through analysis of scattering patterns) and chemistry (through spectroscopy). However, these techniques will not be discussed any further here. Scanning Electron Microscopy (SEM) SEM can provide 2D images down to resolutions of several nanometres. A schematic diagram of the principle components of a SEM apparatus is given in Fig. 5.3. In SEM, a beam of electrons is scanned across the surface of the sample, and interacts with the atoms of the sample. This gives rise to returning electrons that carry information about the sample. There are two main types of returning electrons, namely the back-scattered electrons (BSEs), and secondary electrons (SEs). BSEs are electrons that originated from the incident beam but are reflected from the sample by elastic scattering, so there is no significant energy loss. The amount of scattering increases with the atomic mass of the target and so the intensity of BSEs contains information on the composition of the target. The SE are produced when an electron from the incident beam excites an electron from the sample, such that it approaches the surface of the sample and can escape, and thereby reach the SE detector. Dual-Beam Microscopy More recently, SEM has been combined with a focused-ion beam (FIB) to provide 3D data. FIB consist of beams of ions, typically gallium or argon, used to etch away the surface of the sample like the jet of a water pressure washer removes solid deposits on pathways. The combination of FIBs with SEM (FIB-SEM) can be used as a form of destructive 3D nanotomography. A schematic diagram showing the basic principles of FIB-SEM is shown in Fig. 5.4. In FIB-SEM, a small trough is etched from the surface of the sample, as in Fig. 5.5, and a SEM image is taken of the exposed internal surface. The FIB is then used to etch away the initial surface to reveal the next layer below. A SEM image of that surface is then taken. This process is repeated, with the FIB acting like a nanoscopic version of open-cast mining, and the SEM taking images at each stage. The process is akin to taking a sliced loaf of bread and progressively removing the slices from the bag one at a time and photographing the top external surface of each successive remaining slice as it is revealed. During the collection of FIB-SEM slices, minor drift can occur that requires the additional application of alignment (rotation) and shearing corrections (Ma et al. 2019). If the images of each slice are re-assembled into a stack, they will form a pseudo-3D image. Obviously, it

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Fig. 5.3 Schematic diagram of principle components of a SEM. Reprinted from Bultreys et al. (2016), Copyright (2016), with permission from Elsevier

is not a full 3D image because information is missing from the volumes behind and between the image planes. The FIB_SEM method requires drying of the sample before imaging, and this can lead to cracks due to shrinkage (Keller et al. 2013). Hence, special drying methods, such as high pressure drying, may be needed for some samples. The dual-beam method can also lead to image artefacts for some samples. The ion beam etching process can lead to debris being deposited within pores, thus, obscuring them (Saif et al. 2017). The ion milling can also lead to vertical stripes in the images, known as the ‘waterfall effect’ (Keller et al. 2013).

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Fig. 5.4 Schematic diagram of the principles of FIB-SEM. The ion beam etches the trench and the electron beam takes the images. Reprinted from Bultreys et al. (2016), Copyright (2016), with permission from Elsevier

Transmission Electron Microscopy (TEM) and Electron Tomography Transmission electron microscopy (TEM) involves the beam of electrons passing through the sample to form an image. Electron tomography, also known as 3D TEM, uses similar basic principles to X-ray tomography, except that the ‘shadow pictures’ are created by a beam of electrons. Helium Ion Microscopy (HIM) Helium ion microscopy uses a beam of helium ions as the probe radiation instead of a beam of electrons. Helium ions have the advantage over electrons of a reduced diffraction effect due to a reduced wavelength, but also a reduced sputtering effect compared to other ions (Hill and Rahman 2011). HIM can achieve nanometre resolutions. The ‘optics’ of the HIM are very similar to those of the SEM shown in Fig. 5.3. The large penetration depth of helium ions (e.g. ~250 nm into silicon for a 30 keV He ion) means it can be used for scanning transmission microscopy of thin layers.

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Fig. 5.5 Example of a trench dug by a gallium beam for dual-beam microscopy

5.3 Nature of Experiments 5.3.1 NMR Spectroscopy The strength of the NMR signal is determined by the difference in the populations of the two energy levels mentioned above. At thermal equilibrium, the populations follow a Boltzmann distribution. It, thus, follows that the difference in populations can be increased by increasing the size of the energy difference between the energy levels. This means that the signal-to-noise ratio for NMR can be improved by increasing the size of the magnetic field. NMR spectrometers typically use (superconducting) electromagnets to provide the external magnetic field. The size of an NMR magnet is usually referred to according to the resonance frequency of hydrogen nuclei. Hence, a 4.7 T magnet is referred to as ‘200 MHz’. NMR magnets range up to gigahertz. For a very few types of well-ordered porous material, NMR spectroscopy can be used to determine the pore structure using solid-state NMR on the solid itself. For example, solid-state silicon-29 double-quantum dipolar recoupling NMR can be used to probe the distance-dependent dipolar interactions between naturally abundant silicon-29 nuclei in zeolites. For zeolites, NMR data can be combined with the unit cell parameters and space group to solve structural models (Brouwer et al. 2005). Similar experiments can be used to resolve the structure of metal-organic frameworks.

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Pore structure characterisation by NMR usually requires a probe fluid that can be imbibed within the externally accessible porosity. Only some isotopes are NMR active, but normal hydrogen is one of those, so just this enables a range of potential probe fluids, such as water or hydrocarbons. The NMR relaxation rate of nuclei within the solid itself tends to be so fast that the signal from the solid decays before it can be even measured easily. Even with a strong magnet, the NMR effect is still, relatively, quite weak so a high concentration of nuclei is needed within a sample to give a strong signal. Normal xenon gas can be used to characterise porous materials with high internal surface area and porosity, such that the amount of xenon is relatively high, as with materials like zeolites. However, for other porous systems, it may be necessary to use hyperpolarised xenon, which has an enhanced NMR signal. The NMR signal strength is directly proportional to the number of nuclei present, known as the spin density. Hence, if the void space of a porous solid contains a probe fluid with an NMR active nucleus, such as hydrogen nuclei in water, then the voidage fraction will be, to a first approximation, proportional to the NMR signal strength. However, as mentioned above, the NMR relaxation times of fluids within porous media are also affected by the pore size, and the relaxation rate impacts the measured signal strength.

5.3.2 NMR Relaxometry and Pulsed-Field Gradient NMR

Relaxation As mentioned above, the spin-spin relaxation has two components, one due to permanent spatial heterogeneities in the strength of the external magnetic field, and one due to the random motion of the molecules, and it is the latter that is sensitive to pore sizes. Since the effects of magnetic field heterogeneity are permanent in time, they can be reversed using the so-called spin-echo experiment. Hence, some version of the spin-echo experiment is used to measure spin-spin relaxation time. More detail can be found in Callaghan (1991). In the spin-echo experiment, the NMR signal strength I follows an exponential decay, such that:  I = I0 exp

 −t , T2

(5.16)

where I 0 is the true spin density, t is the experimental echo-time, and T 2 is the spinspin relaxation time. Hence, this shows that the NMR spin density can be measured simultaneously with NMR relaxation time in the spin-echo experiment. Spin-lattice relaxation is measured using either the so-called saturation recovery or inversion recovery experiments, which also simultaneously supply the spin density. In order to convert a relaxation time into a pore size, the value of the term λ/T is (sometimes combined together as ρ s and known as the surface relaxation strength, or surface relaxivity) must be known. The value of the surface relaxivity, and even the

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individual values of λ and the surface relaxation time, can be obtained from drying experiments (D’Orazio et al. 1990). A porous sample is allowed to progressively dry out such that the probe fluid saturation decreases over time, and the relaxation time is measured at a series of different saturations. If the solid surface is wetting of the probe fluid, it will tend to dry out by the so-called thinning-film mechanism. In that case, the liquid stays in contact with all of the internal surface as it dries, and the thickness of the liquid film, thus, declines with saturation simultaneously throughout the sample. In those circumstances, in Eq. (5.6), the fraction of the fluid considered in the bulk phase (with relaxation time T ib ) declines linearly with the saturation level. If T ib  T is , as is usually the case, then the expected variation of relaxation time with fractional pore saturation with probe fluid (V/V 0 ) is: Ti =

Tis V0 V . λ S V0

(5.17)

Hence, if the drying is via a film-thinning mechanism, and the above approximation holds, then the surface relaxivity can be obtained from the gradient of observed relaxation time against fractional saturation. The total specific accessible pore volume V 0 can be obtained from an independent method, such as gravimetry, and the specific surface area S can be obtained from gas adsorption BET analysis. In such circumstances, relaxometry is not a completely independent pore size measure in its own right. However, this is not an issue if spatially resolved data from imaging is required, as this is not possible from nitrogen adsorption on its own anyway. It should be noted that, in the foregoing method, it was assumed that the surface relaxivity was a constant for the whole surface of the sample, irrespective of pore size. For chemically heterogeneous samples, this may not be the case, since surface relaxivity might correlate with pore size. For example, this might be the case if the porous sample consisted of a composite, with different components having different chemistry and different pore sizes. However, there is an even more critical issue, if the sample contains significant amounts of paramagnetic impurities, such as metal ions with odd electrons, such as Fe2+ , Cu2+ , Ni2+ , and Mn2+ . Highly paramagnetic species can increase the rate of relaxation of the probe fluid dramatically, such that it is so short, that it cannot be measured. Even in lower concentrations that do not destroy the NMR signal prematurely, the relaxation time variation can represent variations in surface concentrations of paramagnetic species rather than pore size. For porous materials that contain paramagnetic ions distributed homogeneously through the structure but are also mass fractals, an alternative relaxometry approach is possible (Devreux et al. 1990). If a porous material is a mass fractal, then the mass, M, will be distributed with distance r in space according to: M(r ) ∼ r D .

(5.18)

The spin-lattice relaxation rate of the sample will be facilitated by a process, called dipolar coupling, for which the time constant increases with the distance as r 6 . Hence, the recovered magnetisation at time t after perturbation will be that of the

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spins that are contained in a sphere of radius r ~ t 1/6 . Hence, the observed recovery of magnetisation m will show a time dependence such that: m(t) ∼ t D/6 .

(5.19)

Such a power law has been observed in silicon-29 magic angle spinning (MAS) solid-state NMR studies of silicas containing paramagnetic chromium species. Diffusion Both the relaxation time and diffusivity determination rely upon measurements of signal decay. The effects of relaxation and diffusion could be correlated for porous media, given void spaces with small pores often have higher tortuosity, which leads to reduced diffusivity. For the stimulated echo type of pulsed-field gradient experiment, the general expression for the signal intensity is: I = I0

 i



 pi exp −Di γ 2 g 2 δ 2  − 3δ   , i pi

(5.20)

where pi is given by: pi

  2td1 td2 2td1 td1  = pi exp − , = pi exp − − + − T2i T1i T2i T1i T1i

(5.21)

where t d1 and t d2 are delay times within the NMR pulse sequence of the stimulatedecho experiment. From Eq. (5.20), it can be seen that, as the diffusion time  is increased, the signal will become more weighted towards the slower relaxing regions, as the quicker relaxing regions will decay away. In order to extract the diffusivities of all regions irrespective of relaxation time, a rich data set of many data points obtained over a range of diffusion times is needed to simultaneously fit Eqs. (5.20) and (5.21) with several potential components. The nature of the probe fluid selected can also make a difference to the tortuosity measured using PFG NMR. As mentioned above, the random thermal motion of molecules means that they will exchange between the bulk-like fluid in the core of the pore and the layer(s) of molecules in close proximity of the wall. Migration into this latter surface layer means molecules come into range of the surface potential and will interact. This surface interaction is responsible for the reduction in molecular tumbling that leads to the increase in relaxation rate, but can also be strong enough to retard the more general migration of molecules through the void space. The extent of this retardation depends upon the type of molecule, which determines the type and number of interactions possible with the surface. For example, silica surfaces possess polar hydroxyl groups, which can interact via dipole-dipole electrostatic attraction or even hydrogen bonding with the hydroxyl groups in water. These effects mean studies have been able to show that the tortuosity factor obtained depends upon the liquid used (D’Agostino et al. 2012). In addition to the transient adsorption effect mentioned above, it should be also born in mind that, according to Eq. (5.10), different

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molecules will diffuse different distances in the same diffusion time, and thus the averaging of the diffusivity of the sample will be different.

5.3.3 Magnetic Resonance Imaging Spatially resolved maps of the macroscopic spatial variation in spin density, and thence porosity, can be obtained using magnetic resonance imaging. The degree of spatial resolution possible for such maps depends upon a variety of factors. The NMR imaging machine must have sufficiently strong magnetic field gradients. As the resolution is made finer, the NMR signal arises from a smaller and smaller volume, and thus is diminished in strength, especially if the sample has low porosity to begin with. Obtaining sufficient signal-to-noise to obtain a good quality fit to Eq. (5.16) for a high resolution image would thus require increasing the number of scans performed. The same NMR experiment can be repeated and the results co-added to improve the quality of the data. The signal-to-noise ratio (S/N) increases with the square root of the number of scans taken. A high resolution image for a low porosity sample can make the acquisition time necessary to obtain the number of scans needed for a good S/N very long indeed (many hours to days). In order to obtain spatially resolved maps of porosity, pore size, and tortuosity, it is usually necessary to use a probe fluid of some sort. The fluid must have an NMR active nucleus, and be wetting of the pores.

5.3.4 Computerised X-Ray Tomography (CXT) The typical highest resolution that can be achieved by laboratory-based equipment is ~1–2 microns down to several hundred nanometres, depending on the machine used (Bultreys et al. 2016). The high beam intensity of synchrotron sources allows images to be obtained very quickly which means porous structures evolving relatively quickly can be studied (Bultreys et al. 2016). In order to avoid noise in the CXT image, there needs to be a high count of incoming X-ray radiation at the detector. To ensure a high number of photons contributing to the image, the acquisition time can be extended. As explained above, the many projections, taken from different angles, needed for tomographic reconstruction are obtained, for inanimate subjects, by re-orientating the sample on a rotating turntable. Hence, to ensure, there is no blur in the image from undesired sample movement, the sample must be immobilised. This means only samples which can be firmly fixed into place can be imaged. If the material being imaged has a low electron density, then the contrast between void space and solid can be low. The contrast between void and solid can be enhanced by imbibing a probe fluid of high electron density into the void space. This is only a possibility for voids accessible from the exterior. The probe fluid must be wetting of

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the pores. Common probe fluids include di-iodomethane, and solutions of bromide, or iodide salts (Moradllo et al. 2017).

5.3.5 Electron Microscopy (EM) The sample preparation required for EM depends upon the original form of the sample. If the sample is already a fine powder or thin film it may not need reducing in size, even for TEM. First, the sample must be small enough to fit into the sample holder, and, for TEM-based methods, it must be thin enough to allow transmission of electrons. Sample size can be reduced using saws, water jets, microtomes, or ultrasonic disc cutters. When it is necessary to obtain a flat plane to examine, the sample can be embedded in resin and sectioned. Dimpler machines allow the sample thickness to be reduced to the thickness, whereby it becomes transparent to the electron beam. The dimpler creates a depression or furrow in the sample. Eventually, an argon ion miller can be used to increase the depth of the depression or furrow such that a hole in the bottom of the sample is created. However, in the process of generating the hole the dimpler also, simultaneously, creates highly thinned sample around the circumference of the hole, which can then be extracted and used in TEM. Samples that cannot be thinned by dimpling or milling can be reduced in size by ultramicrotomy. This involves cutting thin slices with a diamond or glass knife. Samples are generally first embedded in a resin of similar hardness to make cutting easier. Thin sheets of sample can also be cut out by focused ion beam (FIB). A strip of surface of the sample can be covered with metal to protect it from the ion beam. Then, the beam can be used to cut trenches either side of the metal strip, and, finally, a thin sheet of sample can be extracted from under the metal film. This thin sheet will be thin enough for electrons to penetrate. For soft or molten samples, this can all be done at liquid nitrogen temperatures to freeze the system as a solid, if necessary. Plasma cleaners should always be used in the final stages of sample preparation to remove hydrocarbon deposits and other remnants of the size-reduction process, to provide a clean surface to be imaged. Samples (especially insulators) can be coated with conductive materials such as carbon, platinum, or gold to prevent charging during imaging (which manifests itself as intense glows in the image). Once prepared samples can be placed on a grid. These are made of materials such as copper, graphene oxide on holey or lacey carbon, and nickel. The particular grid support used will depend upon the nature of the sample. Powder samples can be suspended in solvents such as IPA, acetone or water, and then dropped onto the grid and dried. However, when eventually seeking the particles to be imaged, it should be remembered that particles tend to gather at the edge of drying liquid droplets, and give rise to the so-called coffee cup-ring effect.

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5.4 What Can I Find Out with These Methods? Many of the standard analyses conducted on 2D and 3D images can be done using commercial software toolboxes, some of which are available to download for free from the Internet. However, these have their limitations and considerable operator input may be required. Data obtained from imaging methods is divided into 2D and 3D data sets obtained from the various methods described above, such as SEM, and tomography, respectively. In particular, the 3D methods eliminate stereological error present in 2D methods. The dual-beam microscopy methods are, in some ways, intermediate between 2D and 3D methods. The individual data components acquired are 2D SEM images but for a succession of planes in the sample that can be re-assembled into a stack in the third dimension.

5.4.1 Porosity/Voidage Fraction and Porosity Descriptors Indirect Methods—NMR/MRI As mentioned above, the spin density (together with the T 1 or T 2 relaxation time) maps can be obtained from standard NMR experiments. The voidage fraction of a sample can be obtained by comparing the measured spin density against that for a calibration sample with a known volume of the same probe fluid to obtain the volume of probe fluid in the sample. The ratio of this volume to the bulk volume of the sample will give the voidage fraction. Direct Imaging Before imaging data can be used for even the simplest void space descriptor determination, various image artefacts must be removed. The different imaging modalities result in a range of potential artefacts. In CXT, the common artefacts are due to noise, blurring, beam hardening of polychromatic beams (containing a spectrum of photon energies), or drifts in the incident intensity due to source heating. Noise can be removed by passing the image data through a denoising filter, of which there are several options (Schlüter et al. 2014). Image blur can be removed through edge enhancement (Schlüter et al. 2014). Beam hardening is manifested in the image as streaking around high-attenuation features, and a gradation in image intensity with distance from the centre of the sample. These are very difficult to remove, but the beam can be pre-hardened by passing through a filter before the sample itself. Once an image of a porous material has been obtained, the next stage is to decide on the significance of a particular pixel intensity or greyscale value, such that it can be allocated which pixels correspond to void space and which to solid. This process is called image ‘gating’ or ‘segmentation’. The simplest approach is to create a histogram of pixel intensities and see if there is a bimodal distribution, such that one

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mode can be clearly assigned to void and one to solid, with a clear dearth of pixels (a deep valley between the two peaks in the distribution) at intermediate intensities that can be designated the cut-off between the two. However, this clear bimodal distribution does not always arise, and then gating becomes more difficult and more subjective. Where the appropriate gating level is not obvious various algorithms have been developed to correct biases in the histogram (Schlüter et al. 2014). In addition, a range of different algorithms have also been developed to perform the thresholding/gating procedure (Schlüter et al. 2014). However, these can each deliver slightly different answers. One approach to address this issue might be to try several different algorithms and see how sensitive the results are to the particular algorithm each time. If the results are insensitive to choice of algorithm, then one can have confidence in the values thereby obtained. Once the image has been thresholded between solid and void, readily available image analysis software can be used to count up the void pixels to generate a porosity or voidage fraction value. As mentioned in Chap. 1, another name for the porosity is the zeroth moment of the phase function, where the phase function Z at a position x in the image is defined as:

Z (x) =

1 if x belongs to void space . 0 otherwise

(5.22)

Hence, the porosity ε is given by: ε = Z¯ .

(5.23)

The first moment of the phase function is known as the correlation function R and is given by:  R(u) = [Z (x) − ][Z (x + u) − ]/ − 2 .

(5.24)

The correlation function characterises the spatial distribution of solid and void space through the distribution of the probability that a pixel at a displacement u from a given pixel at position x are of the same phase (i.e. void space or solid). If the porous structure is isotropic, then just one correlation function is needed to describe the structure, and the displacement is simply the straight line distance between pixels (rather than for just one spatial component). The correlation function also provides a key criterion for assessing the statistical reliability of void space descriptors, namely the correlation length. The correlation length is the value of |u| when the corresponding value of R first becomes zero. The correlation length is the characteristic length-scale beyond which the value of void space descriptors becomes independent of the volume of void space used to obtain the measurement. In order to obtain a representative (i.e. ‘typical’) value of a descriptor, the measurement volume must exceed the correlation length. It is possible, for very heterogeneous systems, for the correlation length to, effectively, exceed the

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particle size. This means the whole particle must constitute the measurement volume to obtain the overall average porosity, and a smaller sample volume will, necessarily, be incorrect. It is also possible that a heterogeneous sample can consist of different phases that have their own internal correlation lengths. A key issue with microscopy and imaging methods is that the field of view, or imaged volume, may not exceed the correlation length, and thus the void space descriptors, thereby obtained may be statistically unrepresentative of the whole. This is often, but not the only reason why imaging and indirect methods obtain different estimates of supposedly the same void space descriptor, like porosity.

5.4.2 Pore Size Indirect Methods—NMR/MRI As mentioned above, the T 1 or T 2 relaxation time can be obtained simultaneously from standard NMR experiments. Relaxation times can be converted to pore sizes using the two-fraction fast-exchange model if a pore geometry is assumed to convert the pore surface-area-to-volume ratio into a pore dimension. Direct Imaging An overall average ‘pore size’ can be obtained from the estimates of total surface area S and pore volume V derived from the whole gated (void/solid) image using the simple average hydraulic diameter d given by: d=

4V . S

(5.25)

Obtaining a pore size distribution from a direct image of a void space first requires choosing a definition of what constitutes ‘a single pore’. In the case of model porous media, such as SBA-15 silica which has an hexagonal array of parallel, regular cylindrical void spaces, looking like a miniature ‘wine rack’, it is relatively, objectively possible to identify each such void space as an individual pore. However, it is much more difficult to do so for disordered, irregular void spaces. A variety of algorithms are available to segment the void space up into individual pores. One example of a segmentation algorithm to extract individual pores from the continuum of the whole void space is called ‘morphological thinning’ first proposed by Lin and Cohen (1982). The process involves: (1) Identification of surface voxels as solid voxels with one or more nearest neighbour voxels that are void; (2) These nearest neighbour void voxels are thinned, and labelled with the number of round in which they were thinned; (3) Thinned voxels can now count as potential solid surface voxels for next round of thinning, and so nearest neighbour voids are identified;

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(4) (5) (6) (7)

These nearest neighbour void voxels are thinned and labelled; The thinning continues until all voxels are labelled (as per Fig. 5.6); Pore centres are then identified as local maxima in the thinning label; These local maxima are then grown to form pores by finding all connected voxels at the same thinning level; (8) The pores are grown further by finding all connected voxels at the next lowest level of thinning; (9) This process is repeated iteratively until all voxels belong to a pore. In morphological thinning, the necks do not need to be separately identified as they are always voxels that were thinned at an earlier stage than the pores they join. Once individual pores have been identified by image segmentation, they often show an irregular morphology very unlike Euclidean spheres and cylinders. Hence, there is not an immediately obvious dimension to assign as the pore size, equivalent to the diameter of a sphere or cylinder. At this stage in the process, there is yet another set of options for deciding on the particular characteristic parameter to describe a pore. One simple option is to determine the surface area and volume of individual pores, and then apply Eq. (5.25). Another option is to find the maximum inscribed sphere size. In this approach, spheres of increasing size are placed within the pore space, and the largest size is taken that will all still fit within the designated pore volume. However, the problem with this approach is that it neglects surface corrugations in particularly irregular pores.

Fig. 5.6 Schematic diagram illustrating the thinning process at successive stages. Reprinted from Baldwin et al. (1996), Copyright (1996), with permission from Elsevier

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The chord length-distribution (CLD) function p(z) can be used to quantify the characteristic dimensions of a disordered porous solid (Torquato and Lu 1993). Chords are the lengths between intersections of random lines with the two-phase (solid-void) interface. In particular, the CLD function p(z)dz is the probability of finding a chord of length between z and z + dz in one of the phases, say void. Network Connectivity and Specific Genus Some measure of the topology of a porous solid is essential to understand physical processes affected by void space interconnections, including gas sorption and mass transport. Tsakiroglou and Payatakes (2000) recommended that, in general, to describe the topology of the complex, multiply connected, closed surface (between void and solid) in space, the concept of its ‘deformation retract’ is useful. This construction is formed by shrinking the surface continuously until it approaches a bond-and-node network. In lattice-based models of porous solids, such as pore bond networks, it is easy to define the co-ordination number as the number of pores meeting at a given node, and the connectivity as the overall co-ordination number averaged over all network nodes. If individual pores have been segmented from the continuous void space of a direct image, then similar definitions can also be applied. A more direct measure of the connectivity of a multiply connected closed surface as found in images is the genus. The genus G of a multiply connected closed surface is defined as ‘the number of (non-self-intersecting) cuts that may be made upon the surface without separating it into two disconnected parts’ (De Hoff et al. 1972; Tsakiroglou and Payatakes 2000). The genus is a function of the measurement volume so another quantity, the specific genus , defined as the mean genus per unit volume, is typically used as a descriptor of pore interconnectivity. The specific genus is estimated from the constant slope of a plot of the mean genus versus sample volume. It should be noted that the specific genus depends upon the separation of network nodes, as well as their connectivity. The plot to obtain the specific genus can be derived from examining serial sections of a porous medium (as obtained from FIBSEM). The moments of the phase function mentioned above can be extended to three or more points, which would then include topological information.

5.5 Conclusions NMR and imaging methods are generally less destructive than other methods, such as mercury porosimetry, where it is difficult to remove mercury in order to conduct a subsequent experiment. NMR methods have the advantage that they can obtain both structural and transport information, using different experimental techniques, on the same sample, using the same probe fluid. Imaging methods potentially provide a more direct characterisation of the porous system than other methods like gas sorption or thermoporometry, and include spatial information, albeit with limits on resolution.

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However, NMR pre-conditioned imaging methods, such as relaxation time weighted MR images, provide a way to still obtain some spatial information and map small scale void space characteristics below the resolution limit of more direct imaging. While more abstract models of interpretation are not needed in imaging, there is often still some sort of data processing and analysis, and the potential for artefacts.

References Baldwin CA, Sederman AJ, Mantle MD, Alexander P, Gladden LF (1996) Determination and characterization of the structure of a pore space from 3D volume images. J Colloid Interface Sci 181(1):79–92 Brouwer DH, Darton RJ, Morris RE, Levitt MH (2005) A solid-state NMR method for solution of zeolite crystal structures. J Am Chem Soc 127(29):10365–10370 Brownstein KR, Tarr CE (1977) Spin-lattice relaxation in a system governed by diffusion. J Magn Reson 26:17–25 Bultreys T, De Boever W, Cnudde V (2016) Imaging and image-based fluid transport modeling at the pore scale in geological materials: a practical introduction to the current state-of-the-art. Earth Sci Rev 155:93–128 Callaghan PT (1991) Principles of nuclear magnetic resonance microscopy. Oxford University Press, Oxford Cnudde V, Boone M, Dewanckele J, Dierick M, Van Hoorebeke L, Jacobs P (2011) 3D characterization of sandstone by means of X-ray computed tomography. Geosphere 7(1):54–61 D’Orazio F, Bhattacharja S, Halperin WP et al (1990) Molecular diffusion and NMR relaxation of water in unsaturated porous silica glass. Phys Rev B 42(16):9810–9818 D’Agostino C, Mitchell J, Gladden LF, Mantle MD (2012) Hydrogen bonding network disruption in mesoporous catalyst supports probed by PFG-NMR diffusometry and NMR relaxometry. J Phys Chem 116(16):8975–8982 De Hoff RT, Aigeltinger EH, Craig KR (1972) Experimental determination of the topological properties of three-dimensional microstructures. J Microsc 95:69–91 Devreux F, Boilot JP, Chaput F, Sapoval B (1990) NMR determination of the fractal dimension in silica aerogels. Phys Rev Lett 65:614–617 Hill R, Rahman FHMF (2011) Advances in helium ion microscopy. Nucl Instr Methods Phys Res A 645(1):96–101 Hollewand MP, Gladden LF (1995) Transport heterogeneity in porous pellets-I. PGSE NMR studies. Chem Eng Sci 50:309–326 Keller LM, Schuetz P, Erni R, Rossell MD, Lucas F, Gasser P, Holzer L (2013) Characterization of multi-scale microstructural features in Opalinus Clay. Micropor Mespor Mater 170:83–95 Lin C, Cohen MH (1982) Quantitative methods for microgeometric modelling. J Appl Phys 53:4152–4165 Ma L, Dowey PJ, Rutter E, Taylor KG, Lee PD (2019) A novel upscaling procedure for characterising heterogeneous shale porosity from nanometer-to millimetre-scale in 3D. Energy 181:1285–1297 Moradllo MK, Hu Q, Ley MT (2017) Using X-ray imaging to investigate in-situ ion diffusion in cementitious materials. Constr Build Mater 136:88–98 Saif T, Lin Q, Butcher AR, Bijeljic B, Blunt M (2017) Multi-scale multi-dimensional microstructure imaging of oil shale pyrolysis using X-ray micro-tomography, automated ultra-high resolution SEM, MAPS Mineralogy and FIB-SEM. Appl Energy 202:628–647 Schlüter S, Sheppard A, Brown K, Wildenschild D (2014) Image processing of multiphase images obtained via X- ray microtomography: a review. Wat Res Res 50(4):3615–3639

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Torquato S, Lu B (1993) Chord-length distribution function for two-phase random media. Phys Rev E 47:2950–2953 Tsakiroglou CD, Payatakes AC (2000) Characterization of the pore structure of reservoir rocks with the aid of serial sectioning analysis, mercury porosimetry and network simulation. Adv Wat Res 23(7):773–789

Chapter 6

Hybrid Methods

6.1 Background Many of the pore characterisation methods described in Chaps. 2–5 are often used alone, or merely in parallel, on the same material. For example, two separate pore size distributions may be obtained from two different samples of the same batch of catalyst using mercury porosimetry and gas sorption, respectively. Even where two, or more, techniques are used in parallel, the resultant data is often used only in a complementary fashion, such as to obtain the pore size distribution and pore connectivity independently from gas sorption and PFG NMR, respectively. Where an attempt is made to reconcile the findings for notionally the same descriptor using two parallel methods, this often proves difficult. For example, pore size distribution peaks even from gas desorption and mercury intrusion, supposedly both invasion percolation processes, do not always match in position or shape due to uncertainties in the correct contact angle. However, if the techniques are used in parallel, it is often not clear whether observed differences arise from intrabatch variability in test materials or from deviations from basic theory of operation of the technique. A more consistent interpretation of data from two techniques can be obtained by serial application of both techniques to the same sample, in what will be referred to in this work as ‘hybrid methods’. Hybrid methods can also potentially involve the serial application of the same technique on more than one separate occasion to the same sample, but either side of the utilisation of another different technique. Reconciling data from serial experiments with different techniques on the very same sample requires a much more rigorous understanding of the physical processes involved, since any discrepancies cannot be simply dismissed as intrabatch variation. This greater understanding of the physical processes underlying the different characterisation methods necessary for use of hybrid methods also leads to the expansion of the information-providing capabilities of the techniques to deliver both more accurate and new void space descriptors. The integration of two different physical processes from different characterisation methods also provides more data from which to derive broader knowledge of the © Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_6

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void space. A particular requirement, when applying hybrid methods to complex, disordered porous solids, is an understanding of the emergent effects arising when moving from single, isolated pores to full interconnected networks. It will be seen below that hybrid methods can particularly help in understanding network emergent effects. In this chapter, the various capabilities of the different, potential, hybrid combinations of the aforementioned individual techniques will be described by way of a set of case studies. The case studies will show both how the hybrid methods can be applied and what they can potentially deliver. Each case study will be summarised by a set of key features relevant to the particular hybrid technique utilised.

6.2 Utilisation of Pore Network Effects in Pore Characterisation Complex, disordered, porous solids, with interconnected networks of pores of different sizes, are still the predominant systems met in catalysis, pharmaceutics, and geology. As described previously, the basic theories for interpreting many characterisation methods, such as gas sorption, mercury porosimetry, and cryoporometry, consider only single pores. These theories, as such, thus cannot account for effects emergent due to the ensemble of differently sized pores present within the network of a disordered solid. A number of such emergent network, or pore–pore co-operative, effects are known. These include pore shielding/pore shadowing in mercury porosimetry, pore blocking, advanced condensation (also known as the ‘cascade effect’) and delayed condensation in gas sorption, and pore shielding and advanced melting in cryoporometry. The presence of these complicating effects has led some to completely dismiss indirect methods in the past. Rather than making interpretation of characterisation data for disordered materials simply more difficult and ambiguous than for simpler, ordered materials, these network effects can, instead, provide the necessary additional information needed to accurately interpret indirect characterisation data for more complex materials. Hence, these effects should be viewed as tools to be utilised, rather than barriers to knowledge. In order to use network effects for pore characterisation, their mechanisms of operation need to be fully understood. It is the purpose of this chapter to provide such understanding. The simplest example of a pore network is the so-called ink-bottle pore, and it can be used to demonstrate the way network effects can reveal additional information about complex solids. The simplest ink-bottle pore consists of a narrow, cylindrical pore neck guarding the entrance to a larger, cylindrical pore body with one dead end, as shown in Fig. 6.1a. In pore shielding/pore shadowing, intruding mercury in porosimetry, or a freezing front in cryoporometry, can only enter the pore body when the applied pressure exceeds, or the temperature is below, respectively, that required to enter the neck. Pore blocking in gas desorption arises because a free meniscus is required for condensate to desorb when it is below its condensation pressure. Thence,

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Fig. 6.1 Schematic diagrams of multiple pore configuration types known as a dead-end ink-bottle (a), a through ink-bottle (b), and a T-junction (c)

in the ink-bottle pore, the condensate in the body cannot desorb until the liquid in the pore neck also desorbs, but this will only occur at a lower pressure than the critical pressure for the body. Hence, desorption is delayed to a lower pressure, when both the pore body and the neck desorb together. A slightly different form of ink-bottle is the through variety depicted in Fig. 6.1b. In this case, the cylindrical pore body is open at both ends, but at each end the body is guarded by a smaller cylindrical pore neck, such that the long axes of the pore necks and body are all co-linear. This sort of structure might occur even for model, templated materials, such as SBA-15, where corrugations arise in the cylindrical pores. Advanced condensation and advanced melting can arise in such structures. Advanced condensation will occur in gas adsorption when the critical pressure for condensation via a cylindrical-sleeve meniscus in the necks exceeds that required for condensation via a hemispherical meniscus in the pore body, such that the whole ensemble fills together. This is because condensation in the pore neck completes a full hemispherical meniscus at the end of the pore body. It is recalled from the Cohan equations (in Chap. 2) that capillary condensation from a cylindrical-sleeve meniscus in a through cylindrical pore will occur at the same pressure as from a hemispherical meniscus in a dead-end pore that is twice as big as the through pore. Hence, the Cohan equations predict that advanced condensation will arise when the neck is at least half the diameter of the body. Advanced melting of the pore body, simultaneously with the necks, can occur during cryoporometry when the temperature required for radial melting from the non-freezing layer in the pore neck exceeds the temperature needed for axial melting in the pore body. This is because melting of the ice core in the pore neck completes a hemispherical solid–liquid meniscus at the end of the frozen pore body. The critical body-to-neck size ratio not to be exceeded for advanced melting to occur is also two.

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Delayed condensation arises in pore network configurations, such as the example shown in Fig. 6.1c, where there is a junction such that the main pore does not have completely solid walls. The hole in the wall of the pore lowers the pore potential within the pore below that for an equivalent pore of the same overall geometry but with completely solid walls. The decline in pore potential leads to the need for a higher pressure for capillary condensation. Hence, increased pore connectivity will raise condensation pressure. The ability of these aforementioned network effects to deliver new information on more complex pore structures arises because of the idiosyncrasies of the mechanisms of the particular effects arising for different techniques. For example, the poreshielding effect in mercury intrusion is directional. The shielding only manifests in the data if the pore neck of an ink-bottle arrangement is directed more towards the exterior of the sample and, thus, is reached in the mercury intrusion pathway before the body. In contrast, in advanced melting, in cryoporometry, the spatial direction of the neck relative to the body does not affect whether the effect occurs or not. However, a case study below shows the combined information from two methods is needed to realise what is actually occurring for both techniques. Hence, hybrid methods are the key to the positive utilisation of the emergent network effects mentioned above.

6.3 Combined Mercury Porosimetry and Thermoporometry 6.3.1 Introduction The key advantage of combining mercury porosimetry with thermoporometry is that it can potentially de-shield a mercury intrusion pore size distribution without the need for complicated pore network modelling as is often used (Androutsopoulos and Mann 1979; Matthews et al. 1995). This is simply because mercury tends to become entrapped in the shielded pores, and can be used as the probe fluid in thermoporometry.

6.3.2 Experimental Considerations Combined mercury porosimetry and thermoporometry can be most readily performed using differential scanning calorimetry in series with porosimetry. Ideally, the sample mass should be re-weighed after removal from the residual mercury in the penetrometer to ensure that the mass increase tallies with that expected from the entrapment level evident in the porosimetry data. After the mercury porosimetry experiment, the discharged sample, or a suitably sized portion thereof, can be transferred to a DSC pan and sealed. Ideally, a small droplet or film of mercury should also be transferred

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with the sample to provide a bulk reference peak in the DSC data. Further, the time between discharge of the sample from the porosimeter and the first freezing run in the DSC should be minimised to ensure there is no significant migration of entrapped mercury following the porosimeter reaching atmospheric pressure again on retraction. For a new sample, it should also be rerun on the DSC some time after the first experiment to check that the mercury does not move. If the entrapped mercury peak(s) are in the same position both immediately after discharge from the porosimeter and at a later time, then it is unlikely the entrapped mercury is migrating. It is conceivable that mercury in partially saturated samples may also migrate during the freeze–thaw process itself. This can be tested for by employing repeated freeze–thaw cycles and checking that the melting peak position in the DSC data does not shift from one such cycle to the next (Rigby 2018).

6.3.3 What Can This Method Tell Me?

Case Study: Ink-Bottle Versus Funnel-Shaped Pore Spatial Arrangements This example will compare the mercury intrusion and mercury melting curve pore size distributions for two catalyst samples denoted A and B. The mercury porosimetry intrusion and extrusion curves analysed by the Kloubek (1981) correlations, as described in Sect. 3.1, for samples A and B are shown in Fig. 6.2. For sample A, it is noted that, while the intrusion of the smallest pores in the sample is reversible, most of the mercury becomes entrapped in the rest of the pores. In contrast, for sample B mercury entrapment begins to occur immediately on retraction, and the slope of the retraction curve is always less than that for the intrusion curve, such that mercury is probably retained in all pore sizes. Both samples have a high mercury entrapment, which makes them especially amenable to mercury thermoporometry, but even lower entrapment levels can supply useful information. Figure 6.3 shows comparisons of the cumulative pore size distributions obtained from mercury intrusion porosimetry and mercury DSC thermoporometry for catalyst samples A and B. The pore volumes have been renormalised such that the total cumulative volume is unity in all cases and such that the variable on the vertical axis represents the fraction of all pores with sizes smaller than the corresponding horizontal-axis value. The horizontal-axis variable is the reduced pore size, obtained by dividing the measured pore size by the median pore size (i.e. the pore size where half of the other pores are either smaller or larger). This variable has been chosen to facilitate direct comparison of the shapes of the distributions. From Fig. 6.3a, for sample A, it can be seen that there are deviations between the PSDs obtained from porosimetry and thermoporometry at both low and high pore sizes. The deviation at low pore sizes is due to the reversibility of mercury intrusion in these pores, meaning there is no mercury left in these pores to detect with thermoporometry. For the deviation at high pore sizes, it is noted that the PSD tail is wider for porosimetry than it is for thermoporometry. In contrast, in the data for

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(a)

3.5E-04

Cumulative volume/(ml/cm2)

3.0E-04 2.5E-04 2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 1

10

100

1000

Pore radius/nm

(b)

Fig. 6.2 Mercury porosimetry intrusion (solid line) and extrusion (plus sign) curves analysed by the Kloubek correlations for catalyst samples, a A and b B. Reprinted (adapted) with permission from Malik et al. (2016). Copyright (2016) American Chemical Society

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1

Fractional cumulative volume

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1

1

10

100

Reduced pore size

(b) 1.0

Fractional cumulative volume

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1

1.0

10.0

Reduced pore size Fig. 6.3 Comparison of renormalised cumulative pore size distributions from mercury intrusion porosimetry (dashed line) and mercury thermoporometry (solid line) for catalyst samples A (a) and B (b)

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catalyst B in Fig. 6.3b, it is the large pore size tail of the thermoporometry data that is wider than that for porosimetry. Since both advanced melting and pore shielding would lead to a narrowing of the PSD, this suggests advanced melting is present for catalyst A, while pore shielding is present for catalyst B. According to the advanced melting theory described above, both effects would only arise together if the ratio of pore body sizes to pore neck sizes was less than the critical value, which is two for cylindrical pores, and the necks were closer to the outside, along the mercury intrusion pathway, than the bodies. Since the effects do not occur concurrently in either sample, these conditions must not apply. The thermoporometry data for catalyst B can be thought of as de-shielding the large pore size end of the distribution from mercury intrusion without the need for any sort of pore modelling requiring the sort of assumptions about the pore space described in Chap. 3. The low pore size tail in the thermoporometry PSD for sample B is shifted to lower pore sizes than that for mercury intrusion. This may be due to a similar advanced melting only effect to that seen in the high pore size tail for sample A. The pores in the small-sized tail of the PSD in sample B may be arranged in a funnel-type shape like the pores of the large-sized tail of sample A. A funnel shape is the ink-bottle geometry of Fig. 6.1a but in the mirror-image orientation relative to the direction of penetration of mercury. In the funnel-type arrangement, mercury intrudes pore sizes in order of decreasing size, meaning no shielding occurs, but the narrower pore can still lead to advanced melting in the wider neighbour.

6.3.4 Key Features of Pore–Pore Co-Operative Effects in This Case Study • Incidence of pore shielding is sensitive to the particular order of pore sizes encountered along access pathway from exterior, whereas advanced melting is not. • Advanced melting has a critical pore body-to-neck size ratio above which it does not occur, but pore shielding does not.

6.4 Integrated Gas Sorption and Mercury Porosimetry 6.4.1 Background Gas sorption and mercury porosimetry are often obtained for (samples of) the same material and analysed in parallel. However, as will be clear from Chaps. 2 and 3, since both techniques are indirect, they often require a number of assumptions in order to analyse the raw characterisation data. There is often no independent way to validate

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these assumptions. Discrepancies are often observed between nitrogen sorption and mercury porosimetry pore size distributions and other descriptors. There is also a plethora of ad hoc explanations available for these discrepancies, such as structural damage by mercury porosimetry and contact angle uncertainties. In particular, the same sample is also not often used for both techniques, allowing intrabatch variability to be appealed to as another potential cause of discrepancy. However, the application of a series of multiple gas sorption and mercury porosimetry experiments to the same sample means that there are more constraints on reconciling the data (Rigby et al. 2004a). Indeed, given that mercury tends to get entrapped only in a subset of pores within a sample, then the impact on gas sorption isotherms obtained after entrapment relates to just these same pores. This means that integrating the experiments in this way forces a tight reconciliation of the data from both experiments.

6.4.2 Experimental Details For integrated experiments, a full adsorption–desorption isotherm is measured, after which the sample is retrieved and transferred to the mercury porosimeter. Mercury intrusion data is generated for the desired pressure range, followed by retraction back down to atmospheric pressure. Once the mercury porosimetry experiment is finished, the sample is immediately discharged from the sample cell, and recovered and returned to the physisorption sample cell. Once in position on the instrument analysis port, a dewar of liquid nitrogen is immediately raised around the sample to freeze the mercury within the pores in which it had been entrapped. It is important to do this immediately following the mercury porosimetry run to prevent the possibility of mercury ‘bleeding’ from the pore network with time. The sample is then left for approximately 30 min to ensure all the mercury within the sample has frozen solid after which the sample was evacuated to less than 5 μm Hg and held under vacuum for 30 min. A second isotherm can then be measured using the same parameters as the first. Previous work (Nepryahin et al. 2016a; Rigby 2018) has shown that the mercury ganglia in partially saturated samples do not migrate during repeated freeze–thaw cycles.

6.4.3 What Can I Find Out with This Method?

Case Study: Calibration of Pore Size Determination for Disordered Solids As mentioned above, mercury entrapment tends to arise within a particular subset of pores within a given porous material, or it can be manipulated to be so by using mercury porosimetry scanning curves. In some materials, mercury entrapment is quite low, such that it is confined to a very small set of pores. This enables the gas sorption and mercury porosimetry processes occurring within a much smaller and

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well-defined part of a much larger disordered network to be isolated from the wider behaviour of the whole network. Hence, the data analysis can be simplified even for disordered solids and thus presents a clearer situation in which to test sorption theories. The validation of theories of adsorption has tended to have taken place in model, templated porous materials which have relatively simple, well-ordered porous structures, such as SBA-15 silica. Notwithstanding that these apparently ‘ordered’ materials often tend to be more complex structurally than expected, they also tend to be missing many common aspects of the void spaces of disordered materials, such as macroscopic spatial correlations in local pore size and distributions in pore coordination number. As will be seen below, it is thus not clear how well theories of gas sorption developed for relatively simple pore geometries apply to the more complex geometries found in natural and industrial materials. However, the isolation of the sorption behaviour in a small subset of pores within a disordered material is possible with integrated experiments. The sol-gel silica sphere denoted S1 is unusual in that mercury becomes entrapped in intermediate pore sizes, rather than the largest pore sizes, as for most materials. This makes it an interesting test material for theories of gas sorption via integrated experiments. The form as received from the commercial manufacturers is a 2–3 mm silica sphere, as is common for materials made by the sol-gel droplet method. However, if the spheres are fragmented to a powder with particle sizes ~60–90 μm and the mercury porosimetry data is analysed using the Kloubek (1981) correlations, then the hysteresis is completely removed, as shown in Fig. 6.4. This shows that the Kloubek (1981) correlations are appropriate for the analysis of other silicas besides the CPGs used to obtain them. It is noted that S1 has the same surface fractal dimension as these CPGs. The superposition of the intrusion and extrusion curves suggests the hysteresis in the original raw data is purely of contact angle origin. If the corresponding raw mercury porosimetry data for whole sphere samples is also analysed using the Kloubek (1981) correlations, then the superposition is only achieved for the smaller pore sizes (see Fig. 6.5). However, it was noticed that the larger pore region of the retraction curves for both whole and powder samples, and the intrusion curve for whole pellet samples, had a similar shape, and it was only in a narrow range of pore sizes just below ~10 nm that there were differences. This can be made clearer if the incremental volumes, at corresponding pore sizes, for the powder intrusion curve are subtracted from those for the whole pellet extrusion curve, as shown in Fig. 6.6. In Fig. 6.6, it can be seen that these incremental volumes generally cancel each other out except for the relatively sharp peak around 7 nm. Hence, it is in this narrow range of pore sizes that mercury becomes entrapped in S1. Gas sorption isotherms can also be obtained before and after mercury porosimetry on whole pellets from batch S1, and examples are shown in Fig. 6.7. From Fig. 6.7, it can be seen that if the sorption isotherms from after porosimetry are adjusted by + 54 cm3 (STP) g−1 then the top of the isotherms matches the top of those from before porosimetry. If the sorption isotherms from after porosimetry are adjusted by +3 cm3 (STP) g−1 , then the bottom of the hysteresis loop region of the isotherms matches the corresponding region of the isotherms from before porosimetry. This suggests

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Fig. 6.4 Mercury intrusion (multiplication signs) and retraction (open square) curves for a fragmented sample from batch S1 (with powder particle size of ~60–90 μm). The line shown is to guide the eye. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

Fig. 6.5 Mercury intrusion (filled square) and retraction (filled triangle) curves for a whole pellet sample from batch S1. Also shown is a mercury intrusion (filled diamond) curve for a fragmented sample from batch S1 (with powder particle size of ~60–90 μm). The line shown is to guide the eye. The ultimate intrusion volume for the whole pellet sample has been renormalised to that of the powder intrusion curve to facilitate comparison. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

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Fig. 6.6 Variation with pore radius of the difference in incremental volumes (using the same set of pore size bins) between the powder intrusion curve and whole pellet extrusion curve shown in Fig. 6.5. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

that the discrepancy between the isotherms from before and after porosimetry arises at intermediate relative pressures, as might be expected if entrapment was happening in intermediate pore sizes as the porosimetry data suggests. The difference in adsorbed amount between the adsorption isotherms from before and after mercury entrapment can be made more explicit by subtracting the incremental volumes at the corresponding relative pressures, and the result for S1 is shown in Fig. 6.8. From Fig. 6.8, during adsorption in S1, the capillary condensation in the pores that eventually contain entrapped mercury occurred at a peak centred around a relative pressure of 0.939. Combining this data with the integrated mercury porosimetry data, and taking account of the peak width (at half-maximum), suggests that pores of radius 7.32 ± 0.37 nm fill at a relative pressure of 0.94 ± 0.02. In contrast, NLDFT theory (Neimark and Ravikovitch 2001) would suggest a pore radius of 21.0 nm or 17.3 nm assuming adsorption occurs at the adsorption spinodal or at equilibrium, respectively, in a long cylindrical pore. From the approach of Broekhoff and De Boer (1967), a relative pressure of 0.94 ± 0.02 corresponds to an open cylinder of radius 14 nm, which is also higher than from mercury porosimetry. A similar plot to Fig. 6.8 can also be prepared from the desorption data and is given in Fig. 6.9. It can be seen that the difference in evaporation between before and after mercury porosimetries has a similar form to the adsorption plot with a main peak and smaller subsidiary at lower pressure. The main peak in Fig. 6.9 occurs at a relative pressure of 0.903. According to Broekhoff and De Boer (1968) theory, this evaporation pressure would correspond to an open cylindrical pore of radius 13 nm, which is still larger than measured using mercury porosimetry. However, while the power, that the peak adsorption pressure needs to be raised to in order to equal the peak desorption pressure, from NLDFT is 1.8 (Hitchcock et al. 2014a, b) or from the Cohan (1938) equations is 2.0, that from the Broekhoff–de Boer (BdB) theory is 1.5, which does match the experimental observation for S1. Hence, while BdB theory overpredicts the pore size, it does correctly predict the hysteresis width.

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Fig. 6.7 Nitrogen sorption isotherms where the isotherms from after porosimetry (multiplication sign) have been adjusted upwards by a +3 cm3 (STP) g−1 and b +54 cm3 (STP) g−1 . The solid line is the nitrogen sorption isotherms from before porosimetry. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

Previously, it was suggested that the overprediction of the pore size for adsorption could occur because the build-up of the (quadrupolar) nitrogen multi-layer (t-layer) film thickness might be suppressed for S1, relative to other silicas and materials, possibly due to dehydroxylation (and thus loss of polarity) of the surface during thermal pre-treatment (Rigby et al. 2008a, b). However, scanning curves provided evidence for an alternative hypothesis.

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Fig. 6.8 Variation with relative pressure of the difference in incremental volumes between adsorption isotherms from before and after mercury entrapment in a sample from batch S1. The lines shown are to guide the eye. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

Fig. 6.9 Variation with relative pressure of the difference in incremental volumes between desorption isotherms from before and after mercury entrapment in a sample from batch S1. The lines shown are to guide the eye. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Rigby et al. (2008b), Copyright (2008)

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Information from the boundary sorption curves in an integrated sorption experiment can be supplemented with scanning sorption curves for the pores that entrap mercury (Hitchcock et al. 2014a, b). Scanning curves that are associated only with the pores where mercury becomes entrapped can be generated by subtracting the experimental gas sorption data points following mercury porosimetry from the equivalent data points before mercury porosimetry. These can have very different forms to the scanning curves obtained without mercury present. Figure 6.10a shows the descending scanning curves, starting at relative pressure 0.948 on the boundary adsorption isotherm, for all filled pores in S1 (solid line) and for those which entrap mercury (open squares). Also, Fig. 6.10b shows the ascending scanning curves, starting at relative pressure 0.894 on the boundary desorption isotherm, for all empty pores in S1 (solid line) and for the pores which entrap mercury (open diamonds). The ascending and descending scanning curves, for the pores which entrap mercury, cross directly between the boundary curves, whereas those for all pores show some significant change in condensate immediately upon reversing the direction of the pressure change. The result for the entrapment pores is that which might be expected if the pores that fill with mercury were open cylinders, but would also arise if the mercury was entrapped in the pore body of a through ink-bottle pore geometry, and adsorption and desorption were governed solely by the neck size due to the presence of both the advanced condensation and pore-blocking effects. If the sorption processes within the pores that get filled with mercury are actually controlled by neighbouring necks, these would be smaller in diameter than the pore body but might also be very short. The predictions of the condensation and evaporation pressure for NLDFT theories and similar are usually made for long pores. However, mean-field density functional theory (MFDFT) simulations suggest condensation in short necks occurs at higher pressures than might be expected for longer pores, due to the reduced pore potential arising from less solid present (Rigby and Chigada 2009a). The size of the pore necks can be obtained from the mercury intrusion curve for the whole pellet sample, since the entrapment is lost on fragmentation, so these pores must be giving rise to the shielding that caused entrapment. This suggests the pore neck size is ~7 nm. The close similarity between the pore body and pore neck sizes means that these are effectively ‘invisible’ to gas sorption because of the advanced condensation effect, and are only made manifest by mercury porosimetry. Key Features • Can use mercury entrapment to isolate sorption behaviour of a small subset of pores thereby allowing much simpler data interpretation along the lines of model, templated, ordered porous solids. • Mercury porosimetry is more sensitive to pore corrugations than nitrogen sorption due to advanced condensation/adsorption. Case Study: Determining the Mechanisms of Gas Desorption Due to the potential presence of the so-called cavitation, or ‘tensile strength’, effect (Gregg and Sing 1982), gas desorption pressure may be governed by the properties

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Fig. 6.10 a Desorption scanning curve for all pores (solid line) and pores that became entrapped with mercury (open squares) for S1. Arrows have been added to indicate the direction of the change in pressure. b Adsorption scanning curve for all pores (solid line) and pores that became entrapped with mercury (open diamonds) for S1. Arrows have been added to indicate the direction of the change in pressure. Reprinted from Hitchcock et al. (2014a), Copyright (2014), with permission from Elsevier

of the adsorbate, rather than the pore structure being probed. Hence, it is important to know if this process is contributing to the form of the gas sorption isotherm. Integration with mercury porosimetry enables the mechanism of desorption to be tested. For some samples, such as sol-gel silica G1, the mercury porosimetry curves, following removal of contact angle hysteresis by data analysis using Kloubek (1981) correlations, suggest the entrapment occurs only in the very largest pores within the

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sample (as in Fig. 6.11). The largest pores are those most likely to be subjected to pore-blocking or cavitation effects on desorption. However, whereas the loss of the largest pores will impact the top of the adsorption isotherm, where these pores fill, the region of the desorption isotherm that will be affected depends upon the evaporation mechanism. If cavitation is the mechanism, the desorption isotherm around ~0.4–0.5 will be affected, whereas for pore blocking it will be higher pressure regions of the isotherm that are affected. As shown in Fig. 6.12, for sample G1 loss of the largest pores to mercury entrapment led to the loss of the top of both the adsorption and desorption isotherms, as might be expected for pore blocking (Rigby and Fletcher 2004). Key Features • The form of the nitrogen sorption isotherms after mercury entrapment depends upon nitrogen adsorption mechanism in lost pores. Case Study: Mercury Entrapment Mechanism As described in Chap. 3, mercury entrapment can arise from a variety of mechanisms dependent on the pore geometry of the material. Mercury entrapment can be caused by macroscopic heterogeneities in the spatial distribution of pore size, which leads to all the mercury being left within isolated regions of large pores surrounded by a continuous sea of smaller pores by becoming completely disconnected at their boundary. However, in contrast, as seen in the glass micromodel experiments shown

Fig. 6.11 Mercury porosimetry intrusion (open square) and retraction (open circle) curves, analysed using Kloubek (1981) correlations, for a whole pellet sample from batch G1 (where ‘1E + 2’, for example, represents 1 × 102 ). Reprinted (adapted) with permission from Rigby and Fletcher (2004). Copyright (2004) American Chemical Society

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Fig. 6.12 Superposition of the hysteresis loop regions of the nitrogen sorption isotherms obtained both before and after mercury porosimetries for a whole pellet sample from batch G1 (before porosimetry: adsorption (open circle) and desorption (open square); after porosimetry: adsorption (plus sign) and desorption (multiplication sign)). The equilibration time used in the experiments was 5 s. The lines shown are to guide the eye. Reprinted (adapted) with permission from Rigby and Fletcher (2004). Copyright (2004) American Chemical Society

in Fig. 3.8, in entrapment occurring in pore bodies separated by narrow pore necks the residual mercury entrapment may not fill an entire pore body. If that mercury is frozen in place, then it will look like large pores have turned into smaller pores (Rigby et al. 2006a). Hence, if the PSD is acquired before and after mercury entrapment there will be an apparent loss of larger pores and growth in the volume of smaller pores, as shown in Fig. 6.13 for sample P4. Case Study: Pore Length Distribution Pore length is a descriptor for which very few methods are described in the literature, but it is important for determining characteristics like coking resistance of catalysts. Of course, one can examine pores individually, or by image analysis methods, in electron micrographs or tomographic images, but this is currently impractical and statistically unrepresentative for disordered mesoporous solids. Percolation analysis of gas sorption hysteresis from integrated experiments can be used to determine the pore length distribution for disordered mesoporous materials. As mentioned in Sect. 2.3.3, the width of H2 hysteresis loops is related to the connectivity of the pore network. Thin loops are typically associated with high connectivity, and wide loops with low connectivity, as the former presents many opportunities for pore blocking to be circumvented, while the latter does not. In order to turn this generalisation into precise quantitative estimates of pore connectivity, it is necessary to transform the gas sorption data variables into percolation variables. The key issue with this process is that gas adsorption provides a pore size distribution weighted

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Fig. 6.13 a BJH cumulative PSDs derived from nitrogen adsorption isotherms obtained before (solid line) and after (filled circle) mercury porosimetry to 414 MPa on a sample from batch P4. The ultimate pore volume of the cumulative distribution obtained after porosimetry has been renormalised such that it is the same as that before porosimetry. The inset shows schematically how mercury entrapment causes loss of accessible pore volume and larger pores, and partial pore filling leads to the apparent creation of smaller pores. b The variation, with varying pore size, in the difference between the differential PSDs from before to after porosimetry. Reprinted with permission from Rigby et al. (2006a). Copyright (2006) John Wiley and Sons

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by volume of pores of a given size, whereas percolation theory works in numbers of pores (or, strictly, bonds). Actually, it is, thus, necessary to change a position on the adsorption or desorption isotherm, expressed as a volume fraction of pores empty or filled, into a position in terms of a number fraction of pores empty or filled. Strictly, the number of pores would be the incremental volume of pores of a given size, from the volume-weighted distribution, divided by the average volume of a pore of that size. This would require knowing in advance the average length of pores of a given size, but this is not generally known. Hence, it is generally assumed that there is no correlation between pore diameter and pore length, so the average pore length l¯ is the same for all pore sizes (Seaton 1991). This means the number of pores, ni , for a given incremental volume, V i , just depends upon pore diameter, as everything else is a constant, such that: n=

4Vi li π di2

.

(6.1)

Given that, in the percolation analysis, the key parameters are actually ratios of numbers of pores, then it is not even necessary to know this universal average pore length as it occurs in both the numerator and denominator of these ratios, and, thence, cancels out. The subsequent percolation analysis involves determining the actual number fraction of pores that have emptied at a given point on the desorption isotherm, denoted F, compared to the number fraction from the adsorption isotherm distribution that would have emptied in the absence of pore blocking, denoted f. The point of inflexion in the variation of F with f is defined as the percolation threshold (corresponding roughly to the desorption knee), but would shift in position along the f-axis with changes in connectivity, Z, and change in shape with lattice size, L. However, for random pore bond networks, with lattices of connectivity Z and side length L (measured in pore bond lengths), a near-universal scaling relation G has been found such that:   L β/ν Z F = G (Z f − 3/2)L 1/ν ,

(6.2)

where G has the form shown in Fig. 6.14. Finding the network lattice parameters is then just a matter of recasting the experimental data in the form of the groups in Eq. (6.2), and varying Z and L until a match between the experimental data and the general function G (shown in Fig. 6.14) is obtained. The potential for obtaining descriptors for the pore length distribution arises because the percolation analysis also makes the assumption of a completely random system, in the sense that pore sizes are allocated to lattice bonds at random, and there is no spatial correlation in pore size. If, as mentioned above, mercury gets entrapped in the largest pores, then since the largest pores are distributed at random, then so will be the mercury. If it is assumed that entrapped mercury effectively just removes a bond from the network (but not any nodes), then a particular level of mercury entrapment should be associated with a proportionate decline in pore connectivity. However, this prediction was not found to be correct when it was assumed

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Fig. 6.14 Generalised percolation scaling relation G

that there was no correlation between pore diameter and length. The discrepancy could be removed if it was allowed that there was a distribution of pore length for a given diameter. It can be assumed that the average length of pores follows a power law with diameter such that: li = kdiα ,

(6.3)

where k is a geometrical constant dependent on the overall lattice type. In this case, the ratio of pore connectivity measured before and after mercury entrapment is given by Rigby et al. (2004b): N j ZA = N ZB j

V A, j d (2+α) j VB, j d (2+α) j

.

(6.4)

The power in the pore length distribution given by Eq. (6.3) can be obtained by adjusting its value until the constraint specified by Eq. (6.4) holds. Hence, a descriptor for the pore length distribution can be obtained from percolation analysis of gas sorption data from an integrated experiment, provided the material matches the assumptions made in the approach, namely that its void space approximates a random pore bond network, and the mercury entrapment mechanism is such that it removes complete pore bonds only. In subsequent work (Rigby et al. 2005), a method was described that used mercury porosimetry scanning curves to vary the amount of entrapped mercury and integrated

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gas sorption experiments were used, with percolation analysis, to obtain the distribution of pore lengths, and not just the variation of average pore length with diameter. For larger fractions of mercury entrapment, nodes in the pore network will start to become disconnected and the integrated experiment then becomes sensitive to the distribution in pore co-ordination number. Key Point • The need for percolation parameters to be in terms of pore numbers means that gas sorption can be used to probe pore length distributions. Case Study: Use of Different Adsorptives and the Detection of the Network-Delayed Condensation Effect The gas sorption runs in the series of integrated experiments can potentially be performed with several different adsorptives before and after the mercury porosimetry. For example, the usual isotherm temperature for argon (of 87 K) is, like that of nitrogen (of 77 K), also below the freezing point of mercury (234 K), and thus isotherms can be obtained while the mercury is still frozen after entrapment. If the transfer of samples between mercury porosimetry and gas sorption is done carefully to avoid exposure to atmospheric air, then the surface of the sample will not acquire a film of atmospheric moisture, and any change in the accessible surface following entrapment can be probed. Nitrogen and argon sorption isotherms have been obtained before and after mercury entrapment in a variety of silica materials (Rigby 2018). The (apparent) surface roughness has been probed by analysing the gas adsorption isotherms using the fractal BET Eq. (2.17). It was found that, within experimental error, both before and after mercury entrapment the surface fractal dimension from argon matched that obtained from SAXS for the fresh sample (Hitchcock et al. 2014a, b). However, while the nitrogen adsorption surface fractal dimension was higher than from argon adsorption and SAXS before mercury entrapment (due to patchy adsorption described in Chap. 2), it fell in value following mercury entrapment. It was suggested this was because mercury entrapment leads to the creation of many new smooth, flat frozen metal surfaces and the fractal dimension measured by nitrogen declines because nitrogen begins to adsorb on these flat surfaces (Hitchcock et al. 2014a, b). In contrast, it was suggested that there was no change in the argon surface fractal dimension following entrapment because the argon fails to wet the mercury surfaces in the lower pressure part of the isotherm used for fractal BET analysis. If argon is failing to wet frozen mercury surfaces in the multi-layer region of the isotherm, it might also be expected to affect capillary condensation too. Indeed, this has been found to be the case (Rigby et al. 2017a). Integrated gas sorption experiments with both argon and nitrogen were performed in a test meso-/macroporous material, wherein the mercury only became entrapped in large macropores within which the gases would not condense during conventional experiments on the empty

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sample. Hence, when empty of mercury, mesopores intersecting with these macropores behaved like through pores. However, when the macropores were full of mercury, it turned these particular mesopores into dead-end pores ending in mercury walls. In integrated experiments, it was found that while the condensation pressure for nitrogen in these mesopores moved to lower pressures following mercury entrapment, that for argon did not. It was proposed that this was because the nitrogen wetted the mercury wall and thus formed a hemispherical meniscus at the dead end of the mesopores, giving rise to a lower condensation pressure, whereas the argon did not wet the mercury and thus still perceived the pore more as a through pore. Given the aforementioned findings demonstrate a different wettability of frozen mercury by nitrogen and argon, this effect can be used to probe for the networkdelayed condensation effect. The network-delayed condensation effect arises because a through cylindrical mesopore with gaps in its walls due to intersections with other side mesopores would have a lower pore potential along its axis than a cylinder of the same characteristic dimensions but with entirely solid walls. Similarly, a spherical pore with gaps in its walls due to connecting cylindrical mesopores would have a lower pore potential at its centre than a sphere with entirely solid walls (assuming the adsorbate could access somehow). The lower pore potential for pores with holes in the walls leads to a lower capillary condensation pressure than for the solid-walled pore, even though the characteristic pore dimension (diameter) is the same. Hence, increasing pore connectivity can lead to increasing overestimation of pore diameters, due to the network-delayed condensation effect. However, if a porous material contains pores that have intersections in their side walls with other pores that fill with entrapped mercury following porosimetry then the impact of delayed condensation in these pores can be determined. As mentioned above, nitrogen will wet frozen mercury while argon will not. Hence, filling side pores with frozen mercury effectively removes the impact of the holes in the wall for nitrogen, but not for argon. Therefore, while the capillary condensation pressure for the pores will decrease for nitrogen, following porosimetry, it will not for argon (as shown schematically in Fig. 6.15). The impact of network-delayed condensation can be assessed by looking for shifts to lower pressure in the adsorption of nitrogen following mercury entrapment that are not also observed for argon (see Fig. 6.15). This difference in behaviour is best made clear by comparing the cumulative difference in amount adsorbed plots for nitrogen and argon. This plot is obtained by first subtracting each of the incremental volumes adsorbed at particular relative pressures after entrapment from the corresponding values before entrapment and then, second, adding together these values in series with increasing relative pressure. For a sol-gel silica material Q1, this gave rise to plots of the form given in Fig. 6.16. From Fig. 6.16, it can be seen that the plot for nitrogen shows a very pronounced negative peak at lower pressure in the capillary condensation region of the isotherm, followed by an even stronger positive deviation at higher pressure. In contrast, the plot for argon shows very little negative deviation at low pressure, before the stronger positive deviation at high pressure. The negative deviation at low pressure is very much more marked for nitrogen than for argon. For nitrogen, the negative deviation at lower relative pressure extended down to ~−20 to 30 cc (STP) g−1 , whereas, for

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Fig. 6.15 Schematic diagram showing the anticipated impact of entrapped mercury in side pores on the form of cumulative difference in incremental amount adsorbed (before mercury minus amount afterwards) plots for nitrogen and argon. The cumulative difference plot below also includes the broadening expected due to the presence of a range of pore sizes where the effect might happen. Reprinted from Rigby et al. (2017a), Copyright (2017), with permission from Elsevier

argon, the cumulative difference plot even remains mostly positive. The substantial negative peak before the positive peak in Fig. 6.16a for nitrogen means capillary condensation shifted from higher pressures to lower pressures for some pores, whereas having, more or less, only a positive peak for argon in Fig. 6.16b reflects simply the loss of pores filled with mercury. This difference was attributed to the existence, before entrapment, of the network-delayed condensation effect in the pores that gave rise to the negative peak for nitrogen (Rigby et al. 2017b). Key Features • The variation in wetting of mercury by nitrogen and argon allows a number of techniques to probe capillary condensation of these gases in disordered porous solids, especially network-delayed condensation effects.

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Fig. 6.16 Cumulative difference in amount adsorbed plots for adsorption of nitrogen (a) and argon (b) on typical samples from batch Q1. Reprinted from Rigby et al. (2017a), Copyright (2017), with permission from Elsevier

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6.5 Combined MRI and Gas Sorption 6.5.1 Introduction MRI offers the potential to allow the spatial resolution of gas sorption and to precondition the imaging sequence with a range of contrast mechanisms including relaxation time to measure adsorbed ganglia size and diffusometry to measure adsorbate self-diffusion. However, to successfully combine MRI and adsorption, the adsorbate must contain an NMR active nucleus, such as 1 H in 1 H2 O, or 19 F in C19 4 F8 .

6.5.2 Experimental Considerations As mentioned in Chap. 5, NMR is a relatively weak effect, the low pressure parts of the adsorption isotherm may only have a small amount adsorbed, and hence the signal strength may be low. This can be remedied by taking multiple scans in the MRI acquisition, but this may take a very long time (~days) and so the adsorbate–adsorbent sample must be stable for long periods. Some NMR-active nuclei have relatively low natural abundance (especially compared to 1 H), and this may require special imaging techniques given the relatively low concentration of adsorbate in adsorption studies.

6.5.3 What Can I Find Out with This Technique?

Case Study: Adsorption Kinetics Spin-echo and SPRITE 1 H MRI techniques have been used to image transient kinetic uptake profiles of water or light hydrocarbons in zeolites (Bär et al. 2002). Due to the higher density of the adsorbed phase, strictly the concentration profiles obtained over time were those of adsorbed phase. The use of the spin-echo single-point imaging technique permitted 13 C MRI studies of the transient kinetic uptake of CO2 into zeolites, even despite the low natural abundance of carbon-13 (Cheng et al. 2005). In these studies, the use of zeolites meant that the surface area-to-volume ratio of the adsorbent was very high, and thus the volume concentration of adsorbed phase was high. In mesoporous and macroporous solids, the specific surface area, and thus adsorbed phase density, would be much lower, and there would be more gas phase. However, hyperpolarised gases, such as xenon-129, can be used to study transient low-density, gas-phase mass uptake into mesopores, since confinement in such pores leads to a chemical shift in the xenon within the pores, which can be distinguished from bulk gas (Pavlovskaya et al. 2015). Previous work has studied the uptake of xenon into a packed bed of alumina pellets, as shown in Fig. 6.17

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Fig. 6.17 Transport-weighted hyperpolarised xenon-129 images a are shown as function of transport time, τ . The τ values used are also shown. Images used for transport coefficient calculations using the LDF model are shown in b. This set of images was produced from images shown in a by subtracting each τ image from τ = 10.02 s image. Reprinted with permission from Pavlovskaya et al. (2015). Copyright (2015) John Wiley and Sons

(Pavlovskaya et al. 2015). The voxel intensities in the set of images in Fig. 6.17a, effectively, provide a gas uptake curve for each location in the sample corresponding to a voxel. This can be fitted to models, such as the LDF model, to derive mass transfer coefficients for each voxel location (see Chap. 2). Case Study: Spatially Resolved Surface Area and Heat of Adsorption Spatially resolved isotherms at 291 K have been obtained for alumina and zinc oxide materials (as shown in Fig. 6.18) from fluorine-19 spin-warp imaging of the adsorption of octafluorocyclobutane/freon-C318 (C4 F8 ) (Beyea et al. 2003). The images obtained (such as those in Fig. 6.19) provided isotherm data for each voxel, which could be fitted to the standard BET model, and the specific surface area and BET constant (related to heat of adsorption) obtained for each voxel location. Since the hysteresis loop region was also obtained, in principle, the BJH PSD could also have been obtained, though this was not reported. Case Study: Detection of Advanced Condensation While it has been seen above that spatial, network effects, such as advanced condensation, can be detected indirectly using integrated mercury porosimetry, they can be observed more directly using magnetic resonance imaging to visualise the adsorption process. The advanced condensation effect removes the correspondence between a particular condensation pressure and a unique pore size. This can be demonstrated if there is a method to measure the size of pores containing condensate independent of the gas sorption experiment itself. As mentioned in Chap. 5, NMR relaxometry provides a non-invasive way to measure the size of ganglia of liquids. Hence, the size of pores filled during adsorption at a given pressure can be measured using the NMR

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Fig. 6.18 Schematic drawing of the layout of the porous test ‘material’ for imaging of adsorption. Five materials were placed within a phantom to create a single ‘mock’ material which exhibits a space-varying microstructure. The materials used were a nanoparticulate Al2 O3 powder, a nanoparticulate ZnO powder, partially sintered ceramics made from the powders and Vycor glass. A sealed vial of C4 F8 gas was included as a fixed reference. Reproduced from Beyea et al. (2003), with the permission of AIP Publishing

relaxation time. If relaxometry is combined with imaging, the spatial relationship of pores that fill can also be determined. The advanced condensation effect has been demonstrated using MRI for a mesoporous, sol-gel silica material denoted G2, wherein the spatial correlations in local pore size were over a sufficiently large length-scale that they could be discerned with the resolution possible for MRI of ~100 μm, as shown in Fig. 6.20. For water adsorption in the material, it was found, employing the fractal BET equation to evaluate the form of the multi-layer region with the surface fractal dimension, that multi-layer build-up was similar for water and nitrogen (Hitchcock et al. 2010). This agreed with simulations of water adsorption within hydroxylated silica nanopores which suggested that adsorption at low pressure occurred via pervasive multi-layer build-up, rather than isolated ganglia (Bonnaud et al. 2010). From the T 2 relaxation time image for a fully saturated sample, a histogram of individual pixel relaxation times could be obtained, as given in Fig. 6.21. It can be seen that there is a tail of larger pores with relaxation times above 70 ms. Relaxation time images were obtained for partially saturated samples at different relative pressures of water vapour along the adsorption isotherm, as shown in Fig. 6.22. Histograms of the relaxation times for each voxel were obtained and can be compared between relative pressures. As might be expected, the modal value of relaxation time shifts to larger values, with increasing water vapour pressure, as larger pores are filled by capillary condensate. However, if the histograms at relative pressures of 0.965 and 1.0 are compared it can be seen that some of the very largest pores are filled even at low relative pressure. Since these relaxation times are the very largest seen in the histogram for the fully saturated sample, they must correspond to completely filled pores even at intermediate vapour pressures.

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Fig. 6.19 Two-dimensional NMR images of the C4 F8 gas density in the porous material phantom shown in Fig. 6.18. Images are shown at equilibrium gas pressures of 80, 238, and 253 kPa, and were obtained with in-plane spatial resolution of 750 μm × 750 μm and ~6-mm slice thickness). Differences in local sample-to-sample gas density reflect differences in local gas adsorption due to variations in the porosity, pore microstructure, and surface chemistry. At low pressures, the stronger signal in the Vycor glass is a reflection of its higher surface area, while at high pressures the stronger signal in the ceramics is a reflection of the greater porosity. Reproduced from Beyea et al. (2003), with the permission of AIP Publishing

The emergence of the tail of high relaxation times, corresponding to the filling of some of the very largest pores at intermediate relative pressure, suggests that there is the presence of the advanced condensation effect, shown schematically in Fig. 6.23. The MR images themselves, such as that shown in Fig. 6.24, show the spatial disposition of pore filling and show that the regions of filling of the largest pores (peak tips in the figure) are always contiguous with those of smaller pores, as might be expected if filling of the latter is facilitating the filling of the former, in line with the advanced condensation effect.

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Fig. 6.20 Spin–spin relaxation time image of an arbitrary slice through a sol-gel silica sphere following adsorption of water at a relative pressure of 0.98 and temperature of 296 K. The brighter pixels correspond to large values of relaxation time, while the darker regions correspond to lower values. Reprinted (adapted) with permission from Hitchcock et al. (2010). Copyright (2010) American Chemical Society

Fig. 6.21 Histogram of spin–spin relaxation times (T 2 ) for image of fully water-saturated silica gel sphere G2. Reprinted (adapted) with permission from Hitchcock et al. (2010). Copyright (2010) American Chemical Society

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Fig. 6.22 Histograms of spin–spin relaxation times from MR images of partially saturated silica pellet at relative pressures of water vapour of a 0.960, b 0.965, and c 1.0. The frequencies are normalised to the largest value in the particular histogram and thus are not directly comparable between histograms. Reprinted (adapted) with permission from Hitchcock et al. (2010). Copyright (2010) American Chemical Society

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Fig. 6.23 Schematic diagram illustrating the presence of advanced condensation at isotherm point with relative pressure of 0.965. The boxes represent pores of different sizes from the distribution present, and the dark (blue) shading represents a full pore, while the light (blue) shading represents an empty pore. Reprinted (adapted) with permission from Hitchcock et al. (2010). Copyright (2010) American Chemical Society

6.6 Combined CXT and Gas Adsorption 6.6.1 Background Since CXT has different requirements, to MRI, for the probe fluid, then it can be used to image adsorption of a different range of adsorbates. The image contrast on adsorption is much improved in CXT for adsorbates containing atoms with high electron density, such as iodine. However, adsorption of lower electron density adsorbates, such as carbon dioxide, can also be imaged. The ability to detect the adsorbate depends upon sufficient X-ray absorption contrast being achieved with the adsorbent. The average absorbance of a packed bed of adsorbent particles will vary because of differences in the composition of materials, particle porosity, and particle packing density.

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Fig. 6.24 Spin–spin relaxation time (T 2 /ms) image for an arbitrary slice through a pellet from batch G2 exposed to a relative pressure of water vapour of 0.965. Reprinted (adapted) with permission from Hitchcock et al. (2010). Copyright (2010) American Chemical Society

6.6.2 Experimental Considerations CXT data acquisition can be a lot quicker than MRI if X-ray absorbance of adsorbed phase is high and signal-to-noise ratio is higher.

6.6.3 What Can I Find Out with This Method?

Case Study: Adsorption Kinetics and Spatially Resolved Isotherms CXT at ~2 mm resolution has been used to monitor carbon dioxide adsorption in a composite fixed bed of both microporous carbon and zeolite adsorbent pellets (of size ~ mm) (Joss and Pini 2018). The difference, in the change in the X-ray absorbance with increasing adsorbate pressure, between the two absorbents, was used to distinguish them. The adsorbate showed different adsorption isotherms in

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each type of adsorbent pellet, and so the shape of the isotherm obtained in a particular voxel revealed the type of pellet located at that voxel location.

6.7 Integrated NMR Cryodiffusometry and Relaxometry, Combined with Gas Sorption 6.7.1 Background The advantage of conducting a thermoporometry experiment using NMR (then often referred to as cryoporometry) permits its combination with a variety of NMR tools, especially relaxometry and diffusometry. This provides a number of, largely, independent means to measure the same parameter, such as filled pore sizes in adsorption, or obtain additional descriptors, such as the tortuosity of adsorbed fluid ganglia. In addition, both structural and transport information are obtained for the same sample with the same probe fluid.

6.7.2 Experimental Considerations The most general combined experiment might involve initially preparing a partially saturated sample with a particular water vapour sorption history. Once the condensate is equilibrated, it can be used as the probe fluid for NMR cryoporometry, NMR relaxometry, and/or NMR diffusometry (using PFG NMR). NMR relaxometry and cryoporometry provide independent means to measure the size of adsorbed ganglia, and, by extension, the filled pore sizes. As mentioned in Chap. 5, each of these methods has its limitations, but the combined method provides a way to assess their impact. Pore size distributions from cryoporometry are potentially skewed towards smaller pore sizes by the advanced melting effect. Relaxometry pore size distributions are potentially artificially narrowed by diffusion-averaging. There are a number of potentially complicating issues to consider with combination experiments. When conducting diffusometry experiments on partially saturated samples (such as from adsorption or drying experiments), there is the possibility that migration of probe molecules may be through the gas phase as well as the liquid phase (Naumov et al. 2007). Gas-phase transport is much more rapid than liquidphase transport. Hence, the likely occurrence of significant gas-phase transport can be detected by calculating the root mean square (rms) displacements and checking if they are consistent with liquid-phase only mass transport, whereby there is no indication of diffusion apparently occurring faster than for bulk liquid. Further, in cryoporometry, it is suggested that the probe fluid for partially saturated samples may migrate between pores during freezing (Kaufmann 2010). However, this can be detected by measuring repeated freeze–thaw cycles for the same sample and seeing

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whether the melting curve shifts position between cycles, as would be expected if there was any probe fluid creep between pores occurring (Rigby 2018).

6.7.3 What Can I Find Out with This Method? Case Study: Study of Advanced Adsorption and Advanced Melting The power of the combination of adsorption with NMR cryoporometry, relaxometry, and diffusometry has been exemplified in studies of a disordered sol-gel silica, denoted S1 (Shiko et al. 2012). The cryoporometry and relaxometry potentially provide independent measures of the size/location of adsorbed condensate ganglia, while the diffusometry provides information on the interconnectivity of the adsorbed phase. The findings help to answer the question which technique gives the most accurate pore size distribution for a given material. The water sorption isotherms for this material are shown in Fig. 6.25. Figure 6.26 shows the cryoporometry melting curves measured for the adsorbed water ganglia of water vapour within a single pellet sample of S1 obtained at several different relative pressures in the range 0.81-0.94. For each different relative pressure, the denominator of the molten volume fractions was the total pore volume of the pellet, and therefore, the ultimate molten volume fractions reached for experiments below total liquid saturation are less than unity. From Fig. 6.26, it can be seen that, as relative pressure is increased, the melting curves shift to higher temperatures and their slopes become more precipitous (i.e. spread over a smaller temperature range). It is particularly highlighted that the melting curves for relative pressures of 0.91 and 0.92 generally superpose upon each other up to ~269.5 K, but thereafter they diverge. The 0.92 relative pressure curve rises more sharply than that for the 0.91 relative pressure curve, as made clearer in the inset in Fig. 6.26. However, it is also Fig. 6.25 Normalised water adsorption (filled circle) and desorption (open circle) isotherms for a sample of 30 pellets from batch S1, obtained at 294 K. The water uptake was measured gravimetrically. Reprinted from Shiko et al. (2012), Copyright (2012), with permission from Elsevier

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Fig. 6.26 NMR cryoporometry melting curves for the adsorbed phase in a single pellet sample of S1 (Sample 1) at different relative pressures of water vapour. The inset shows a close-up view of the steep parts of the melting curves for relative pressures of 0.91 and 0.92. The lines shown are to guide the eye. Reprinted from Shiko et al. (2012), Copyright (2012), with permission from Elsevier

clear that the 0.91 relative pressure curve shows a notably larger increase in signal intensity over the higher temperature range ~270.3–270.7 K than the 0.92 relative pressure curve, despite the higher ultimate intensity of the latter. Figure 6.27 shows the variation in spin–spin relaxation time, T 2 , of the adsorbed ganglia with relative pressure of water vapour for the same pellet sample as used to obtain the data in Fig. 6.26. It can be seen that after an initial rise up to a relative pressure of 0.84, the T 2 value stays roughly constant until a relative pressure of 0.91, after which it begins to rise steeply. The rise in T 2 between relative pressures of 0.91 and 0.92 is consistent with larger pores filling as pressure is increased, as would be expected from the conventional view of progressive adsorption in ever-larger pores with increasing pressure. In contrast, this is, apparently, inconsistent with the melting curves for cryoporometry experiments at these pressures in Fig. 6.26. These appear to suggest that more of the largest pores are filled at the lower relative pressure because its melting curve is still comparatively steep at the highest melting temperatures (in the inset), whereas the melting curve for the higher relative pressure is comparatively flat over the same range of higher temperatures. However, this apparent discrepancy can be explained by the presence of a substantial advanced melting effect in the

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Fig. 6.27 Variation of NMR spin–spin relaxation time (T 2 ) for the adsorbed phase, obtained at the top of the melting curves (all at 273 K), with relative pressure (P/P0 ) of water vapour for the sample of S1 used to obtain the data in Fig. 6.26 (Sample 1). The errors in the T 2 values are smaller than the size of the symbols. Reprinted from Shiko et al. (2012), Copyright (2012), with permission from Elsevier

cryoporometry data, and this view is supported by the complementary diffusometry data. Figure 6.28 shows the trend in unrestricted diffusion tortuosity with relative pressure for the same pellet sample as used to obtain the data in Fig. 6.26. As the equilibrium adsorption pressure increases, the tortuosity is roughly constant until a relative pressure of 0.91, and thereafter it declines rapidly with increasing relative pressure. This drop in tortuosity will be associated with an underlying increase in the connectivity of the adsorbed ganglia, since the fluid can then self-diffuse around more easily. The increase in interpore connections, in turn, will facilitate pore-to-pore cooperative effects like advanced condensation in adsorption and advanced melting in cryoporometry. This is consistent with the anomaly of the lower saturation melting curves being less steep than the higher saturation curves at higher melting temperatures (and thus higher pore sizes), as the steepness will increase if more melting of larger pores is actually facilitated by connections to smaller pores making advanced melting at much lower temperatures possible.

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Fig. 6.28 Variation of unrestricted diffusion tortuosity for the adsorbed phase, obtained at the top of the melting curves (all at 273 K), with relative pressure (P/P0 ) of water vapour for the sample of S1 used to obtain the data in Fig. 6.26 (Sample 1). Reprinted from Shiko et al. (2012), Copyright (2012), with permission from Elsevier

The much steeper melting curves for fully saturated samples from cryoporometry compared with the adsorption isotherm suggest that, for this material, advanced melting effects are more prevalent than advanced adsorption effects. Key Points • The full integration of several different methods provides direct, mutually validating evidence. • A combination of methods can reveal previously concealed effects like advanced melting and advanced adsorption. • Advanced melting and advanced adsorption depend strongly on pore connectivity.

6.8 Combined CXT and Liquid Metal Intrusion 6.8.1 Introduction This section describes methods that serially combine metal intrusion with CXT, whereby the sample containing metal is imaged, and not just use the two techniques in parallel on same material. The method of X-ray imaging intruded metals within

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the void space of porous materials was developed by Cody and Davis (1991), and Hellmuth et al. (1999). Besides mercury, the intrusion of a number of other liquid metals, such as gallium and low melting point alloy (LMPA) (composed of a number of metals such as bismuth) has been used in combination with imaging.

6.8.2 Experimental Considerations If mercury is used as the probe fluid, the mercury can be entrapped in the sample using a standard porosimetry experiment. The amount and position of mercury within the sample can be controlled and manipulated to some extent by using scanning curve and mini-loop experiments to fill only a select fraction of pores. If full Xray tomography is used, the full three-dimensional position information about the location of the mercury can be obtained. A key issue is that the transference of the sample from the porosimeter sample tube to the CXT machine needs to be done as quickly and carefully as possible such that there is no time for mercury migration within the sample or extrusion from the sample. In extreme cases, the sample could be frozen in liquid nitrogen to freeze the mercury in-place immediately upon discharge from the porosimeter. The stability of the entrapped mercury can be assessed by taking CXT images of the same sample repeatedly at several intervals over the course of an extended period such as a week and comparing the spatial distribution of mercury between images. If the probe fluid is gallium or molten LMPA, the sample temperature must be raised above the melting point of the metal to intrude, and dropped back below it again to freeze it in-place while still under pressure. The spatial extent and amount of information that can be obtained from the CXT of metal-containing sample depend upon the X-rays being able to penetrate the sample containing metal. For example, mercury is a very electron dense element, and LMPA contains other dense elements. Hence, the X-rays may not be able to penetrate a very long path length of metal to emerge with sufficient intensity to be able to image. The ultimate path length of metal depends upon the total porosity of the sample and what fraction of that is filled with metal.

6.8.3 What Can I Measure with This Technique?

Case Study: Specific Porosity Mapping The simplest use of the entrapped metal is as just a contrast agent to help distinguish pores from matrix. The particular pores filled with metal will depend upon the ultimate pressure used in the intrusion step, which will limit the minimum pore size that could have been filled, and, additionally for mercury, whether it gets entrapped in the intruded pores. Whatever the nature of the porous structure, mercury at the edge

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of the sample is likely to stay connected to the bulk and, thus, eventually exit the sample before the end of the porosimetry experiment. For samples where all, or virtually all, of the mercury becomes entrapped even for scanning curves, this type of experiment can be used to map the spatial distribution of just the porosity that is externally accessible only via pathways above a certain size. For samples where the mercury only becomes entrapped within pores of a certain type, the spatial distribution of these particular pores can be specifically mapped. For example, mercury intrusion into an alumina foam with macroporous voids separated by mesoporous walls (see Fig. 1.1) resulted in the mercury only becoming entrapped in the macropores. Figure 6.29 shows the mercury porosimetry curves analysed using the Kloubek (1981) correlations. It can be seen that the intrusion in the mesopores was largely reversible and entrapment predominantly occurred in the macroporosity. The confinement of the entrapped mercury virtually exclusively to the macroporosity was confirmed by comparing the Gurvitsch volumes from gas adsorption before and after entrapment. Following discharge from the porosimeter, the sample was imaged by CXT, and an example of an image is shown in Fig. 6.30. It can be seen that, towards the limb, the image contains many ellipsoidal white shapes. However, in the more central regions of the image, it is more uniformly white, and this may be because the X-ray beam could not penetrate the central region of the sample due to too much mercury being present. The identity of the white ellipsoidal regions can be identified by comparing the CXT image with electron micrographs of the foam, such as that in Fig. 6.31. Some of the macropores in the electron micrograph look like they are isolated, but the nature of the image as a 2D section of a 3D material makes it impossible to know for sure.

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Fig. 6.30 Computerised X-ray tomography (CXT) image of a monolithic fragment following mercury entrapment. The bright white ovoid regions correspond to mercury-filled bubble pores. Reprinted from Nepryahin et al. (2016b) under Creative Commons Attribution 4.0 International License

However, the pores filled with mercury in the CXT image must be those connected and externally accessible. Case Study: Determination of Mercury Intrusion and Entrapment Mechanisms For samples where the pores are bigger than the CXT resolution, the conformation of intruded mercury can be studied. For example, CXT images of entrapped mercury droplets in cement paste showed that the surface of the droplets had a rough morphology similar to the interior surface of voids of the cement paste, and thus the entrapped mercury droplet had been snapped off before mercury could retreat from the surface ruts (Zeng et al. 2019). However, the images do show that the mercury intrusion probes (sufficiently large) surface roughness, as well as the main void. If the mercury gets entrapped in most pores it enters as it intrudes, then the percolation pathways can be traced by entrapped mercury. The pore network geometry of an α-alumina consisted of large pore bodies, that entrapped mercury, separated by

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Fig. 6.31 Scanning electron micrographs of the foam showing macroporous cells or ‘bubbleshaped’ pores. Reprinted from Nepryahin et al. (2016b) under Creative Commons Attribution 4.0 International License

narrow pore necks that controlled intrusion. CXT images, such as that in Fig. 6.32, showed that the entrapped mercury could act as a tracer for the path followed by the mercury into the sample. The CXT image showed ‘necklace-like’ structures which were chains of intruded pores left full of entrapped mercury. From such images, it is clear that intrusion is not pervasive across the whole matrix, and some regions, such as the lower central part of the pellet in Fig. 6.32, are left void. A closer inspection of further images, such as that shown in Fig. 6.33, shows that some of the configurations of the mercury ganglia, entrapped within pore bodies, are reminiscent of those obtained in glass micromodels with similar pore geometry (see Fig. 3.8). This confirms that the glass micromodels are good models for the behaviour of mercury in real materials. Case Study: De-shielding of Mercury Porosimetry Pore Size Distribution It is often asserted that mercury porosimetry only measures pore necks, and thus cannot deliver information on wider pore bodies shielded behind these necks (Diamond 2000). However, such ink-bottle pores are also likely to give rise to mercury entrapment in the body. Further, LMPA intrusion experiments can be frozen at any stage by simply lowering the experiment below the melting point of the alloy. The CXT images containing entrapped mercury or frozen LMPA can be analysed to determine the size of the pores containing the metal. The size distribution intruded by metal can be compared with that obtained from the mercury intrusion curve to determine what sizes of pore bodies are shielded by what distribution of pore necks. If the experimental data from coupled metal intrusion and CXT (and SEM) is combined with modelling, a more sophisticated de-shielding of the metal intrusion can be obtained (El-Nafaty and Mann 2001; Ruffino et al. 2005). A pore network that is a potential (statistical) model for the porous medium can be constructed on

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Fig. 6.32 Reconstructed greyscale image slice of α-alumina pellet partially intruded with mercury. Reprinted from Rigby et al. (2011), Copyright (2011), with permission from Elsevier

computer, and the metal intrusion to a given stage simulated therein. The imaging process can then be simulated on the model to create predictions for the form of the image data. For example, virtual slices can be taken through the pore network model to simulate the planes of the reconstructed image stack from CXT (or serial sections from SEM or FIB-SEM). These predictions can then be compared with the experimental data from the imaging (as in Fig. 6.34). This comparison tends to be on a statistical basis and consist of comparing descriptors such as the intruded porosity and size distribution of intruded pore features. If the images match, then the model is a good representation of the real porous material. If the match is poor, then the key structural parameters of the model can be varied, and the simulation of the experiment and imaging repeated, until a good match is obtained, and a good model is thereby obtained. The pore size distribution of the model would then be the de-shielded PSD for the real material. The advantage of this sort of modelling process is that it could potentially incorporate representation of pore structural features in the real material not directly visible in the images due to them being below imaging resolution limits. These subresolution features may still affect the form of the visible features, and thus the former may be probed indirectly with the imaging. Comparisons of FIB-SEM

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Fig. 6.33 a False colour image of entrapped mercury (blue/yellow) in α-alumina pellet matrix (green) partially intruded with mercury and b close-up of configurations adopted by entrapped mercury within individual pore bodies. Reprinted from Rigby et al. (2011), Copyright (2011), with permission from Elsevier

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sections through nanoporous networks with simulated sections through random pore bond networks show good agreement in terms of the morphology of features in the images (see Fig. 6.35) (Rigby et al. 2017a).

Fig. 6.34 SEM photographs of sectioned LMPA impregnated FCC catalyst particles at various scales of magnification: a 300× , b SEM image of selected particle shown a (1000×), and c approximated 2D mapped structure for selected particle in b El-Nafaty and Mann (2001), Copyright (2001), with permission from Elsevier

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Fig. 6.35 a View of a random, cylindrical pore bond network model with 512 nodes and 1726 cylindrical pores. b View of simulated random planar section through network in a. The discrete black and white ovoids correspond to small and large pores, respectively, and the continuous grey matrix corresponds to solid. Reprinted from Rigby et al. (2017a), Copyright (2017), with permission from Elsevier

6.9 Serial Gas Sorption with Different Adsorptives 6.9.1 Background Where the sorption isotherm can be halted at a particular point with the adsorbate kept in place, then an isotherm for a second adsorbate can be obtained subsequently. This would give rise to the potential for a series, or succession, of isotherms with alternating adsorbates. In most cases, what makes the halting of the adsorption of one adsorbate possible is kinetic limitations, in particular, a very slow rate of desorption.

6.9.2 Experimental Considerations The serial adsorption of nonane and nitrogen relies upon the kinetic limitation of the slow desorption of nonane from micropores due to its size. Following an initial nitrogen sorption experiment, the sample, still full of nitrogen gas, is transferred to a desiccator chamber and suspended above a beaker of liquid nonane. The sample is then left for a week to absorb the nonane vapour. Following nonane pre-adsorption, the sample can be transferred back to the physisorption apparatus and a suitable sample-specific pre-treatment applied. For example, evacuation to vacuum at 70 °C for 12 h has been shown previously to remove nonane just from the mesopores, and

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larger, of a ZSM-5-based catalyst (Chua et al. 2012). The appropriate pre-treatment conditions for a new sample can be developed as follows. After filling the sample with nonane, it can be thermally treated at a trial temperature for a series of successive periods of time. After each successive time period, nitrogen sorption isotherms can be obtained. The heat treatment can continue until the hysteresis loop region of the nitrogen isotherms for the nonane-filled sample matches that for the fresh sample, following suitable adjustment (by a constant amount) of the amount adsorbed to account for nonane still retained in the micropores. For example, Fig. 6.37 shows the superposition of nitrogen isotherms obtained for the empty zeolite and the same sample following pore filling with nonane and then evacuated under vacuum at 70 °C for 12 h. Figure 6.36 shows that this specific pre-treatment regime gives rise to substantial overlap of the hysteresis loops. This observation shows that nonane had been successfully completely removed from the mesopores, and hence, the residual nonane is situated only inside the micropores, wherein it prevents access for nitrogen (Fig. 6.36). Serial adsorption of nitrogen–water–nitrogen can be performed because it is possible to freeze the adsorbed phase water in place. This type of experiment can be performed using gravimetric apparatus (Gopinathan et al. 2013). First, a conventional nitrogen sorption experiment is conducted at 77 K on the dry sample. Then, following re-evacuation of the sample, water adsorption up to a particular relative pressure can be performed at a temperature around room temperature. To ensure

Fig. 6.36 Nitrogen sorption isotherms obtained (at 77 K) for the fresh parent zeolite (solid line) and for the same sample following nonane pre-adsorption and evacuation to vacuum at 70 °C for 12 h, with the amount adsorbed adjusted upwards by a constant value of 80.4 cm3 (STP)g−1 for all data points (multiplication). Reprinted from Chua et al. (2012), Copyright (2012), with permission from Elsevier

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complete equilibrium of the water adsorption process, the sample can be maintained at the desired ultimate relative pressure for approximately 12 h. At the end of the partial saturation, the sample chamber can be covered with an insulation jacket, and a liquid nitrogen dewar half-filled with liquid nitrogen can be placed below it. The atmospheric water vapour present in the reactor can be removed by outgassing at a rate of 1000 Pa min−1 (10 mbars min−1 ). This low rate of degassing ensures that atmospheric water vapour is removed, but the condensate remains in-place. Subsequently, the remaining half of the dewar can be filled with liquid nitrogen to fully freeze the sample, and the mass reading allowed to stabilise. The nitrogen adsorption isotherm for the same sample after partial saturation with water can then be obtained by switching the machine from vapour mode back to gas mode, and performing the experiment.

6.9.3 What Can I Find Out with This Method?

Micropore Volume and Accessibility Serial adsorption of nitrogen–nonane–nitrogen can be used to measure micropore volume and the rest of the network only accessible via microporosity. Since nonane gets entrapped in the microporosity, the change in amount of nitrogen adsorbed following nonane pre-adsorption can be attributed to the loss of access to the micropore volume (and any larger pores entirely surrounded by micropores). Pore Neck Size Distributions for Small Sizes in Cavitating Systems The cavitation effect in desorption severely limits the information that single adsorbate sorption studies can provide on the pore body size distribution when the bodies are guarded by necks with particular sizes below the cavitating limit of ~4 nm. Indeed, all that can be deduced is whether blocking by necks below 4 nm exists or not. However, if water is pre-adsorbed to a series of different relative pressures, such that it fills progressively larger pore necks, and the adsorbed amount of nitrogen measured at each successive stage, then the volume of pore network guarded by necks of a given size even in the range below 4 nm can be deduced (Morishige and Kanzaki 2009). In the absence of advanced condensation, the neck volume distribution weighted by size can be obtained from the water isotherm. Case Study: Advanced Adsorption For liquid nitrogen at 77 K and water at 298 K, the values of molar density, surface tension, etc., are such that the constant of proportionality, between the logarithm of the relative pressure and the reciprocal pore size, in the Kelvin equation is virtually unity for both adsorbates. Therefore, for a parallel bundle (i.e. wine-rack type) pore structure with a completely wetting surface for all adsorbates, both nitrogen and water should condense in any given pore size at virtually the same relative pressure. Hence, for scanning adsorption isotherms to particular ultimate pressures both adsorbates

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should fill the same set of pores. This theoretical consideration can be used to test the adsorption behaviour of water and nitrogen against each other in the same system using serial adsorption. For example, Gopinathan et al. (2013) showed that the serial adsorption of nitrogen and water on fresh and coked samples of a commercial hydroprocessing catalyst of the CoMo type reveals that the maximum ratio of pore body size to pore neck size for the occurrence of advanced adsorption depends upon the nature of the adsorbate– adsorbent interaction, as suggested by MFDFT simulations of adsorption (Rigby and Chigada 2009b), and not just the geometry as suggested by the Cohan (1938) equations. It was found that coke deposition within the catalyst caused a decrease in the diameter of larger pore body sizes such that adsorption of water in neighbouring, smaller pore necks could then facilitate advanced condensation of water at that same pressure in the narrowed pore bodies. However, for the fresh catalyst, the same, non-coked large pore bodies required water relative pressures higher than that for the coked sample to fill by condensation and thus remained unfilled with water at lower pressures. For water sorption in these samples only to an ultimate pressure that filled the coked large pores but not the uncoked precursor pores, in the nitrogen adsorption experiment that followed, while the larger pore bodies remained largely empty, and could be filled with nitrogen for the fresh sample, in the coked sample some larger pore bodies were completely inaccessible, thereby leading to a drop in nitrogen adsorption even at relative pressures above the water ultimate pressure. This implies that the aforementioned pore bodies and pore necks in the coked sample must fill independently for nitrogen but not for water, and thence, advanced condensation occurs more readily (for larger ratio pore body–neck pairs) for the latter. These findings suggest pore potential also governs whether advanced condensation occurs rather than just geometry through pore body-to-neck size ratio.

6.10 Scattering Methods and Mercury Porosimetry 6.10.1 Background As seen in Sect. 6.4, an independent means of measuring the size of entrapped mercury ganglia following porosimetry can aid in understanding the physics of mercury retraction and, thence, in interpretation of the raw data. The other proposed methods, serial gas sorption and thermoporometry, have their own uncertainties and disadvantages, as have been detailed. Small-angle X-ray scattering (SAXS) of samples containing entrapped mercury provides an alternative method to measure the size of mercury ganglia (Rigby et al. 2008a).

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6.10.2 Experimental Considerations The advantage of SAXS for studying samples after mercury porosimetry is that the sample preparation required is minimal and the experiment time is very rapid, such that the mercury will have very little time to potentially migrate. The large electron density contrast between mercury and many porous materials, like silica or carbon, means that the scattering is dominated by the mercury interfaces. The experiment would become more difficult where mercury is entrapped in porous catalysts consisting of heavy metal oxides like CoMo catalysts.

6.10.3 What Can I Find Out with This Method? Analysis of the SAXS data using methods such as the Debye plot (see Sect. 4.2.3) enables the size of entrapped mercury ganglia to be determined and, thence, the size of pores in which they are entrapped. This can be used to validate independent measures of this from gas sorption or thermoporometry, and thus aid understanding of mercury entrapment or phase transitions (e.g. melting/freezing) in porous media (Rigby et al. 2008a).

6.11 Combined CXT, MRI, and Mercury Porosimetry 6.11.1 Background As mentioned in Chap. 3, mercury porosimetry is an indirect method and needs a model of interpretation. As highlighted in Chap. 5, many types of porous materials possess macroscopic heterogeneities in the spatial distribution of nanopore sizes over length-scales above that of the field of view of microscopy methods, and MRI is able to map these heterogeneities using relaxation time contrast techniques. An example of such a map is given in Fig. 6.37. However, the potential uses of the MRI data can go beyond simply imaging the heterogeneities since it can also be employed directly in the construction of a model of interpretation for mercury porosimetry. As described in Chap. 5, a relaxation time model can be used to covert NMR relaxation times into pore sizes. Hence, each voxel region in an image such as Fig. 6.37 can be allocated a pore size. The image lattice then becomes a model for a void space similar to that shown in Fig. 3.9. The pore size allocated to each voxel represents that of all the pores in an underlying network in that region, such that each voxel is like one of the patches of pores of a particular size in Fig. 3.9. Mercury intrusion and retraction can then be simulated on the model. However, assumptions about the mechanisms of these processes are required, such as that governing the snap-off of the mercury meniscus at the boundaries of pore

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Fig. 6.37 Spin–spin relaxation time images of perpendicular 2D slices through the centre of a pellet from batch G2 of mesoporous sol-gel silica spheres. The pixel resolution is 40 μm, and the slice thickness is 250 μm. Reprinted from Rigby et al. (2006c), Copyright (2006), with permission from Elsevier

size heterogeneities. These assumptions need validation. Since the heterogeneities give rise to mercury entrapment, as shown in Fig. 3.9, then this observable can be used to validate the model. As CXT can image on the same length-scales as MRI, then complementary CXT can be used to map the macroscopic distribution of entrapped mercury and, thereby, provide data for the validation of the assumptions of the porosimetry simulator.

6.11.2 Experimental Considerations The sample used must be amenable to all three techniques. Hence, the sample must be mechanically stable to high pressure to enable mercury porosimetry to be used. The sample must also not have high concentrations of paramagnetic species embedded in the surface of the pore walls; otherwise, the NMR relaxometry measurements will not reflect pore size variation but only variation in concentration of paramagnetics. The amount of mercury entrapped must be low enough that the X-rays can fully penetrate the sample to obtain the images.

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6.11.3 What Can I Find Out with This Method? As mentioned above, this method can be used to validate models of mercury retraction in materials with macroscopic spatial heterogeneities in pore size. For example, simulations of mercury retraction can be performed wherein mercury snaps off at the boundary between two regions of different pore sizes (i.e. the MR image voxels) if the difference in pore sizes exceeds a specified ‘snap-off ratio’, as proposed by Matthews et al. (1995). The simulations will thereby provide predictions of the pattern of spatial distribution of entrapped mercury for different snap-off ratios, as shown in Fig. 6.38. As can be seen from Fig. 6.38, as the snap-off ratio is increased, the spatial distribution of entrapped mercury becomes more clumped. The predicted spatial distribution of entrapped mercury can be compared with that observed experimentally using CXT. As shown in Fig. 6.39, the CXT images of the silica spheres following mercury porosimetry show that the entrapped mercury is spatially distributed in irregular clumps. The characteristic size of the clumps can be determined for both the simulated and real images using a 2D correlation function. It was found that the correlation lengths were similar for experimental images and images simulated with higher snap-off ratio. Hence, the spatial arrangement of entrapped mercury resulting from simulations with higher snap-off ratio seems more like what is observed experimentally. Therefore, it seems likely that the snap-off ratio is relatively high for the real mercury retraction mechanism. Hence, the hybrid technique improved understanding of mercury retraction in systems with macroscopic heterogeneities in the spatial arrangement of pore sizes. This validated mechanism can then be incorporated into mercury porosimetry simulators for samples where it is not possible to obtain quantitative MR images (e.g. due to paramagnetics). Thence, porosimetry and CXT could be used to deduce the spatial distribution of heterogeneities in pore sizes that would have, otherwise, been provided by MRI.

6.12 Conclusions It has been seen that the idiosyncratic shortcomings of individual characterisation techniques can often be overcome by combining them with other complementary techniques. Some serial combinations of techniques are synergistic, such that, together, they deliver more than the individual techniques alone could, even when these are used in parallel. The direct confrontation of characterisation data and descriptors from the different component techniques, forced by hybrid methods, means a much better fundamental understanding of each technique, and, thence, better quality of characterisation, is obtained. Hybrid characterisation techniques allow the selective study of sorption processes occurring within a small subset of pores located within a much larger disordered network, thus reduce ambiguity of interpretation, and, thence, allow more direct testing of theories, much like ordered,

6.12 Conclusions Fig. 6.38 Variation of the spatial arrangement of entrapped mercury (black pixels) with SOR equal to 1.0 (a), 1.5 (b), and 2.0 (c) for simulations of mercury porosimetry on models constructed from MR images of an ‘equatorial’ slice from a typical pellet taken from batch G2. The pellet diameter is ~3 mm, and the pixel resolution is 40 μm. Reprinted from Rigby et al. (2006c), Copyright (2006), with permission from Elsevier

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Fig. 6.39 a 3D microcomputed X-ray tomography image of a typical pellet taken from batch G2 following mercury intrusion to 414 MPa and retraction back to ambient. A segment of the pellet image has been excised by computer to display the interior of the sample. The black areas on the excision planes correspond to regions containing entrapped mercury. The overall pellet diameter at its equator is ~3 mm; b the same 3D X-ray image as in a but where the main matrix (silica) has been changed, by computer software, to a 50% transparent material allowing more mercury entrapment features deeper within the pellet structure to become visible. Reprinted from Rigby et al. (2006c), Copyright (2006), with permission from Elsevier

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templated model materials do for single pores. Hybrid methods also allow the development of characterisation methods that take into account the network pore–pore co-operative processes that arise in disordered materials, thereby much improving the accuracy and range of descriptors thereby obtained.

References Androutsopoulos GP, Mann R (1979) Evaluation of mercury porosimeter experiments using a network pore structure model. Chem Eng Sci 34(10):1203–1212 Bär NK, Balcom BJ, Ruthven DM (2002) Direct measurement of transient concentration profiles in adsorbent particles and chromatographic columns by MRI. Ind Eng Chem Res 41(9):2320–2329 Beyea SD, Caprihan A, Glass SJ, DiGiovanni AJ (2003) Nondestructive characterization of nanopore microstructure: spatially resolved Brunauer-Emmett-Teller isotherms using nuclear magnetic resonance imaging. J Appl Phys 94(2):935–941 Bonnaud PA, Coasne B, Pellenq R (2010) Molecular simulation of water confined in nanoporous silica. J Phys: Condens Matter 22:284110 Broekhoff JCP, De Boer JH (1967) Studies on pore systems in catalysis X: calculations of pore distributions from the adsorption branch of nitrogen sorption isotherms in the case of open cylindrical pores. J Catal 9:15–27 Broekhoff JCP, De Boer JH (1968) Studies on pore systems in catalysts: XII. Pore distributions from the desorption branch of a nitrogen sorption isotherm in the case of cylindrical pores A. An analysis of the capillary evaporation process. J Catal 10(4):368–376 Cheng Y, Huang QL, Eic M, Balcom BJ (2005) CO2 dynamic adsorption/desorption on zeolite 5A studied by 13 C magnetic resonance imaging. Langmuir 21(10):4376–4381 Chua LM, Hitchcock I, Fletcher RS, Holt EM, Lowe J, Rigby SP (2012) Understanding the spatial distribution of coke deposition within bimodal micro-/mesoporous catalysts using a novel sorption method in combination with pulsed-gradient spin-echo NMR. J Catal 286:260–265 Cody GD, Davis A (1991) Direct imaging of coal pore space accessible to liquid metal. Energy Fuels 5(6):776–781 Cohan LH (1938) Sorption hysteresis and the vapor pressure of concave surfaces. J Am Chem Soc 60:433–435 Diamond S (2000) Mercury porosimetry—an inappropriate method for the measurement of pore size distributions in cement-based materials. Cem Concr Res 30(10):1517–1525 El-Nafaty UA, Mann R (2001) Coke burnoff in a typical FCC particle analyzed by an SEM mapped 2-D network pore structure. Chem Eng Sci 56(3):865–872 Gopinathan N, Greaves M, Wood J, Rigby SP (2013) Investigation of the problems with using gas adsorption to probe catalyst pore structure evolution during coking. J Colloid Interface Sci 393:234–240 Gregg SJ, Sing KSW (1982) Adsorption, surface area and porosity. Academic Press, London Hellmuth KH, Siitari-Kauppi M, Klobes P, Meyer K, Goebbels J (1999) Imaging and analyzing rock porosity by autoradiography and Hg-porosimetry/X-ray computertomography-applications. Phys Chem Earth A 24(7):569–573 Hitchcock I, Chudek JA, Holt EM, Lowe JP, Rigby SP (2010) NMR studies of cooperative effects in adsorption. Langmuir 26(23):18061–18070 Hitchcock I, Lunel M, Bakalis S, Fletcher RS, Holt EM, Rigby (2014a) Improving sensitivity and accuracy of pore structural characterisation using scanning curves in integrated gas sorption and mercury porosimetry experiments. J Colloid Interface Sci 417:88–99 Hitchcock I, Malik S, Holt EM et al (2014b) Impact of chemical heterogeneity on the accuracy of pore size distributions in disordered solids. J Phys Chem C 118(35):20627–20638

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Joss L, Pini R (2018) 3D mapping of gas physisorption for the spatial characterisation of nanoporous materials. Chem Phys Chem 20(4):524–528 Kaufmann J (2010) Pore space analysis of cement-based materials by combined Nitrogen sorption—Wood’s metal impregnation and multi-cycle mercury intrusion. Cement Concr Compos 32(7):514–522 Kloubek J (1981) Hysteresis in porosimetry. Powder Technol 29:63–73 Malik S, Smith L, Sharman J, Holt EM, Rigby SP (2016) Pore structural characterization of fuel cell layers using integrated mercury porosimetry and computerized X-ray tomography. Ind Eng Chem Res 55(41):10850–10859 Matthews GP, Ridgway CJ, Spearing MC (1995) Void space modeling of mercury intrusion hysteresis in sandstone, paper coating and other porous media. J Colloid and Interface Sci 171:8–27 Morishige K, Kanzaki Y (2009) Porous structure of ordered silica with cagelike pores examined by successive adsorption of water and nitrogen. J Phys Chem C 113(33):14927–14934 Naumov S, Valiullin R, Galvosas P, Kärger J, Monson PA (2007) Diffusion hysteresis in mesoporous materials. Eur Phys J Special Topics 141:107–112 Neimark AV, Ravikovitch PI (2001) Capillary condensation in MMS and pore structure characterization. Micropor Mesopor Mater 44:697–707 Nepryahin A, Fletcher R, Holt EM, Rigby SP (2016a) Structure-transport relationships in disordered solids using integrated rate of gas sorption and mercury porosimetry. Chem Eng Sci 152:663–673 Nepryahin A, Robin SF, Elizabeth MH, Sean PR (2016b) Techniques for direct experimental evaluation of structure–transport relationships in disordered porous solids. Adsorption 22(7):993–1000 Pavlovskaya GE, Six JS, Meersman T, Gopinathan N, Rigby SP (2015) NMR imaging of low pressure, gas-phase transport in packed beds using hyperpolarized xenon-129. AIChE J 61 (11):4013–4016 Rigby SP (2018) Recent developments in the structural characterisation of disordered, mesoporous solids. JMTR 62(3):296–312 Rigby SP, Chigada PI (2009a) MF-DFT and experimental investigations of the origins of hysteresis in mercury porosimetry of silica materials. Langmuir 26:241 Rigby SP, Chigada PI (2009b) Interpretation of integrated gas sorption and mercury porosimetry studies of adsorption in disordered networks using mean-field DFT. Adsorption 15(1):31–41 Rigby SP, Fletcher RS (2004) Experimental evidence for pore blocking as the mechanism for nitrogen sorption hysteresis in a mesoporous material. J Phys Chem B 108(15):4690–4695 Rigby SP, Fletcher RS, Riley SN (2004a) Characterisation of porous solids using integrated nitrogen sorption and mercury porosimetry. Chem Eng Sci 59(1):41–51 Rigby SP, Watt-Smith MJ, Fletcher RS (2004b) Simultaneous determination of the pore-length distribution and pore connectivity for porous catalyst supports using integrated nitrogen sorption and mercury porosimetry. J Catal 227:68 Rigby SP, Watt-Smith MJ, Fletcher RS (2005) Integrating gas sorption with mercury porosimetry. Adsorption 11:201–220 Rigby SP, Evbuomvan IO, Watt-Smith MJ, Edler KJ, Fletcher RS (2006a) Using nano-cast model porous media and integrated gas sorption to improve fundamental understanding and data interpretation in mercury porosimetry. Part Part Sys Charac 23(1):82–93 Rigby SP, Watt-Smith MJ, Fletcher RS (2006b) Integrating gas sorption with mercury porosimetry. Adsorption 11(1):201–206 Rigby SP, Watt-Smith MJ, Chigada P, Chudek JA, Fletcher RS, Wood J, Bakalis S, Miri T (2006c) Studies of the entrapment of non-wetting fluid within nanoporous media using a synergistic combination of MRI and micro-computed X-ray tomography. Chem Eng Sci 61(23):7579–7592 Rigby SP, Chigada PI, Evbuomvan IO et al (2008a) Experimental and modelling studies of the kinetics of mercury retraction from highly confined geometries during porosimetry in the transport and the quasi-equilibrium regimes. Chem Eng Sci 63(24):5771–5788

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Rigby SP, Chigada PI, Perkins EL, Watt-Smith MJ, Lowe JP, Edler KJ (2008b) Fundamental studies of gas sorption within mesopores situated amidst an inter-connected, irregular network. Adsorption 14(2–3):289–307 Rigby SP, Chigada PI, Wang J, Wilkinson SK, Bateman H, Al-Duri B, Wood J, Bakalis S, Miri T (2011) Improving the interpretation of mercury porosimetry data using computerised X-ray tomography and mean-field DFT. Chem Eng Sci 66(11):2328–2339 Rigby SP, Hasan M, Hitchcock I, Fletcher RS (2017a) Detection of the delayed condensation effect and determination of its impact on the accuracy of gas adsorption pore size distributions. Colloids Surf A 517:33–44 Rigby SP, Hasan M, Stevens L, Williams HEL, Fletcher RS (2017b) Determination of pore network accessibility in hierarchical porous solids. Ind Eng Chem Res 56(50):14822–14831 Ruffino L, Mann R, Oldman R, Stitt EH, Boller E, Cloetens P, DiMichiel M, Merino J (2005) Using x-ray microtomography for characterisation of catalyst particle pore structure. Can J Chem Eng 83(1):132–139 Seaton NA (1991) Determination of the connectivity of porous solids from nitrogen sorption measurements. Chem Eng Sci 46(8):1895–1909 Shiko E, Edler KJ, Lowe JP, Rigby SP (2012) Probing the impact of advanced melting and advanced adsorption phenomena on the accuracy of pore size distributions from cryoporometry and adsorption using NMR relaxometry and diffusometry. J Colloid Interface Sci 385:183–192 Zeng Q, Wang X, Yang P, Wang J, Zhou C (2019) Tracing mercury entrapment in porous cement paste after mercury intrusion test by X-ray computed tomography and implications for pore structure characterization. Mater Charac 151:203–215

Chapter 7

Structural Characterisation in Adsorbent and Catalyst Design

7.1 Special Considerations for Industrial Materials Pore structure characterisation is important in adsorbent and catalyst design because porous adsorbents and catalysts are often used to substantially increase the active surface area of the particles above simply the external geometric surface area. Simply reducing particle size itself is often not an option, since it increases pressure drop through the packed bed of particles. Since smaller pores can provide larger surface area to volume ratio, but often lead (as in Knudsen diffusion) to lower rates of mass transport, there is often a balance to find between absolute active surface area and its accessibility. Besides pore size, a number of other void space characteristics can affect mass transport, such as mean porosity, pore connectivity, and correlations and heterogeneities in the spatial distribution of pore size. Hence, accurate pore structure characterisation is essential. Pore structure characterisation of industrial materials is used for a number of different purposes, such as: – – – –

New product development Quality control of production Answering customer queries Performance rationalisation.

Each purpose places a different set of constraints on the methods used. New product development and post hoc performance rationalisation often require detailed characterisation studies to fully understand the key physico-chemical processes occurring within pores. In contrast, characterisation methods used for quality control often need to be suitable to use for high throughput of samples at production facilities. This chapter will consider a series of case studies that demonstrate how pore structure characterisation methods can help the intelligent design of adsorbents and catalysts.

© Springer Nature Switzerland AG 2020 S. P. Rigby, Structural Characterisation of Natural and Industrial Porous Materials: A Manual, https://doi.org/10.1007/978-3-030-47418-8_7

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7.2 Relating Pore Structure to Raw Material Properties and Fabrication Method Catalysts and adsorbents come in a number of different forms, including powders, pellets, membranes, thin films, and monolith washcoats. Raw materials are generally in the form of powders or crystallites, and solutions of metal salts. Ideally, it would be possible to select the raw materials, based on their key characteristics, and the parameters controlling the conditions of the fabrication process to achieve a predictable pore structure. The physico-chemical processes involved in product forming are often highly complex themselves, such that they are sometimes not well understood. However, there are various methods that attempt to provide models of porous solids by simulating forming processes with constraints from characterisation data. The simplest model for a porous solid formed from consolidation of powder particles is the spherical packing model. The most idealised case would be a regular packing [such as hexagonal close packing (hcp)] of uniformly sized spheres, where the gaps sizes and connectivity of void space regions is well-defined by Euclidean geometry. This can only be achieved where there is efficient sliding of particles past each other, and no particle deformation, during consolidation. However, Avery and Ramsay (1973) did find that the compaction of some silica and zirconia powders to increasing pressures led to ever denser compacts, with increasing co-ordination number, which had porosities and surface areas very similar to that expected for the corresponding regular packing. At industrial scales, many powder raw materials have awkward handling properties, such as poor flowability. One way to improve this situation is to use dry granulation, or roll compaction, to generate consolidated ribbons that are milled to granules, and sieved to give a particular particle size range, that is, then, used as the feed for final compaction. However, the existence of a range of roll-compactor designs, and the wide choice of operational parameters, makes predicting the resulting compacted tablet properties difficult. There are also a range of operational parameter choices for the final compaction process, such as the type of mould (die), the use of lubricants, the direction of compaction (uni- or bi-directional), and the compaction pressure. Particular aspects of the final pore structure can result from the nature of the compaction process itself. Die wall friction can cause density variation and lamination. CXT imaging has revealed the splintering and cracking of feed particles within the tablet, that generate macroporosity, due to the action of the compaction stresses upon the points of contact, such as shown in Fig. 7.1. From Fig. 7.1, the irregular shapes of individual feed particles can be picked out, largely because of variations in average density between particles, and because of the regions of low density surrounding the feed particles. The variations in average density arise because, in preparing roll-compacted feed, under-size or oversize particles from ribbon milling are often recycled back into the roll-compaction feed, and this may happen several times for some parts of the material, resulting in variation in feed particle density. Some particles may not efficiently pack leaving voids between particles, or particles

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Fig. 7.1 CXT image of cross-sections through the mid-plane of a cylindrical compact formed from milled and size-sorted, roll-compacted feed. Lighter voxel intensities correspond to higher density. The scale bar corresponds to 1000 µm.

may splinter at points of contact with other particles where stresses are more intense, leading to fissures between particles. Pellets of powder feed can also be formed using methods other than tabletting/compaction, but which themselves can lead to their own idiosyncratic void space characteristics. Wet granulation involves combining together powder feed and water, and agitation in some sort of mixer or pan. Sometimes, instead of just powder, the feed consists of already partially consolidated ‘seeds’, that will form the cores of new granules, together with the powder. As the seeds roll around in the agitator they accumulate layers of new powder via bonding aided by the liquid bridges between particles. These layers are often separated by voids, or lower density regions, that can be visualised using MRI, such as shown in Fig. 7.2 (Timonen et al. 1995).

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Fig. 7.2 Two-dimensional slice image through spherical alumina granule immersed in ethanolwater liquid mixture. The layering is evident from the alternating bands of light and dark voxels. Reprinted (adapted) from Timonen et al. (1995), Copyright (1995), with permission from Elsevier

Pellets can also be formed by extrusion of pastes made of particle feed and a liquid, typically water. However, an issue with paste extrusion is lamination. Lamination is the creation of a plane of weakness within the extruded pellet, which may be caused by preferential orientation of feed material, or by an incomplete bonding of feed material. Lamination voids can be visualised using MRI, as shown in Fig. 7.3 (Mantle et al. 2004). For one-pot syntheses, the evolution of the void space can be followed through all the stages of fabrication, even those while the void space is still pore filled with liquid. The pore structure of materials synthesised by the sol-gel method can be followed from the initial precipitation of the sol, the gelation, and subsequent drying steps using NMR methods that utilise the pore water as the probe fluid, such as MRI, relaxometry (Smith et al. 1992), and cryoporometry (Shiko et al. 2012). Once formed,

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Fig. 7.3 ‘Cutaway’ representation from a three-dimensional 1 H data set of an extruded paste formulation. Lighter areas indicate the presence of water and dark areas indicate the presence of voids/laminations. A lamination, shown by a dark area on the 1 H image representation, is seen to propagate through the ZX and XY planes of the image. Reprinted (adapted) from Mantle et al. (2004), Copyright (2004), with permission from Elsevier

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powder pellets are also often dried and then calcined (baked) at high temperatures to improve pellet strength. The drying process for pellets can also be followed using NMR methods (Hollewand and Gladden 1994). Drying can particularly impact the void space of thin layers due to cracking. For example, the platinum-carbon catalyst for the cathode side of proton-exchange membrane (PEM) fuel cells is fabricated as an ink that is applied as a thin layer (~50 µm) to the polymer membrane by some method, such as screen printing or ink-jet printing, and then dried (Malik et al. 2016). The drying can lead to crack formation, like the drying of mud on the bottom of a pond, as highlighted by the mercury that gets entrapped within them following porosimetry (evident as sickleshaped white regions in Fig. 7.4). The ink application process can also leave its trace in the void space structure. Combined mercury porosimetry and CXT studies have shown that the regularity of the mesh, forming the screen used in printing, can leave a macroscopic periodicity in externally accessible voidage, as seen from ‘polka-dot’ pattern of mercury entrapment in Fig. 7.4. During the calcination of pellets, the constituent powder particles may fuse in complex sintering processes, which affect properties such as surface area, but are difficult to predict. The different stages of catalyst fabrication may each impact on the void space structure, and even lead to synergistic effects, whereby a later stage of fabrication leads to an exaggeration of a smaller effect introduced at an earlier stage.

7.3 Relating Mass Transport to Pore Structure The common approach for relating mass transport to pore structure involves obtaining an as accurate as possible pore structural model from characterisation data, and then simulating the mass transport processes on the model (Rieckmann and Keil 1999). The construction of the model tends to be exhaustive, in that it attempts to represent as many aspects of the void space as possible, including porosity, pore size distribution, pore connectivity, spatial correlations in pore size, etc. However, this pre-supposes that a sufficiently large representative volume of the porous material can be represented in the model. This can be an issue when the characteristic lengthscales of heterogeneities can be the size of the macroscopic pellet itself, as illustrated schematically in Fig. 7.5. In Fig. 7.5, three-dimensional MRI shows heterogeneities in the macroscopic spatial distribution of porosity over length-scales of the order of ~100 s µm to millimetres, while the FIB/SEM image shows the heterogeneity also present in the nanoscopic network structure. In a catalyst, with a highly disordered void space over a range of length-scales, one or more aspects of the pore structure may be more important than the rest in determining the rate of mass transport. These key aspects could be deduced from simulations of mass transport on a comprehensive model of the whole void space, assuming such a model could be constructed. However, if the number of pores evident in the SEM image of the typical mesoporous material shown in Fig. 7.5 is replicated for similar volumes across the sample, then, in total, it would contain a

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Fig. 7.4 CXT reconstructed image slices of section of a section of fuel cell layer following mercury porosimetry (a). The brighter white regions correspond to high density regions with high X-ray absorption and the dark regions correspond to low density regions with low X-ray absorption. The image slice has a horizontal line in the upper zone which corresponds to the track across which the linear intensity profiles were obtained after porosimetry (b). Reprinted (adapted) with permission from Malik et al. (2016). Copyright (2016) American Chemical Society

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Fig. 7.5 Schematic diagram illustrating multi-scale structural heterogeneity in a catalyst particle. In the MR image, the white voxels correspond to high porosity, while the dark voxels correspond to low porosity. A focussed-ion beam is used to drill a trench in the surface of the sphere, and SEM examination of the back wall of that trench reveals nanopores (black) in the silica matrix (grey). Reprinted (adapted) from Rigby et al. (2017), Copyright (2017), with permission from Elsevier

truly astronomical number (probably ~1014 ) of pores. Even the current generation of supercomputers could not then successfully model complex mass transport processes for the whole pellet. Further, the macroscopic heterogeneities, evident in the MR image, also suggest that a model of a smaller volume of the pellet may not be representative of the whole. However, an alternative strategy, to exhaustive modelling, is to try to ‘filter’ out just the key void space features impacting upon mass transport by special experimental methods. Scientific models are meant to be simplifications of the real world that only include those aspects of the system that substantially impact the physical process(es) of interest. Some of the techniques introduced in Chap. 6 on Hybrid Methods can be used to select the key aspects of the void space to model. For mass transport in pores, this type of ‘filtration-like’ strategy, to find the key features to model, can be illustrated by an analogy with the traffic flow along the road system of a country, like the UK. In the analogy, the different classes of roads, including (in the UK) six-lane motorways, dual-carriageways, A-roads, B-roads, and country lanes, represent the different sizes of pores present, while the absence of motorways in areas like the Scottish Highlands contrasting with the vast converging motorway network around London, represents heterogeneities in the spatial distribution of large pore sizes. The

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traffic flow on the UK road network is analogous to the flow of molecules through the pore network of a catalyst. The importance of particular roads to the efficiency of traffic flow in the UK can be seen when certain sizes of roads, or roads in a given region, are shut down due to roadworks, and the traffic is forced to find alternative routes. If a specific road is closed for roadworks, and traffic flow is severely disrupted, then the particular importance of that road is made clearly manifest. In a porous solid, a similar test, for the importance of particular features of the pore network, can be conducted by selectively blocking particular subsets of pore regions, types, and/or sizes by various means. The largest pores in a complex void space can be eliminated from the accessible network by filling with entrapped mercury (Nepryahin et al. 2016). Mercury porosimetry scanning curves mean that only pore necks with a size greater than corresponding to the ultimate applied mercury pressure will have been penetrated. This means mercury will only become entrapped in voids accessible via necks above a certain size. For many samples, mercury entrapment can amount to ~90–100% of the original mercury intruded, meaning most voids are removed. In any case, the particular subset of pores for which access is now denied due to mercury entrapment can be measured using integrated gas sorption (as in Fig. 7.6), and the spatial pervasiveness of such pores can be mapped using CXT (as in Fig. 7.7, and in Watt-Smith et al. 2006). The commercial methanol synthesis pellets shown in Fig. 7.7 were fabricated by tabletting using a pre-compacted feed (Nepryahin et al. 2016). The pre-compacted feed was milled and sieved to achieve a desired size range, with over and under-sized particles recycled to the pre-compaction step. Hence, some feed, for the tabletting

Fig. 7.6 Gas adsorption pore size distributions for the subsets of pores that progressively become inaccessible from the exterior as the mercury entrapment is increased for progressively higher ultimate pressures (as indicated in legend) for scanning curves. Mercury intrusion pressures of 75.7, 103, and 227 MPa correspond to neck sizes of 8, 6, and 3 nm, respectively. Mercury intrusion pressures of 75.7, 103, and 227 MPa gave rise to entrapment levels of 6.8, 18.5, and 53.5% of the void space, respectively. Reprinted from Nepryahin et al. (2016) under Creative Commons CC-BY licence

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Fig. 7.7 CXT images of the mid-plane cross-section of typical samples of 5 mm diameter cylindrical commercial methanol synthesis pellets with entrapped mercury filling 0% (a), 6.8% (b), and 18.5% (c) of void space. Reprinted from Nepryahin et al. (2016) under Creative Commons CC-BY licence

step, had been pre-compacted several times, such that a variation in density occurred. The presence of large dark patches in Fig. 7.7b suggests some regions of the pellet remain largely unintruded with mercury (which appears brighter white) up to pressures of 103 MPa (corresponding to a neck size of 6 nm). The dark patches have irregular, angular shapes reminiscent of the pre-compacted feed particles, and thus this finding suggests some feed particles must be (largely) shielded by necks smaller than ~6 nm. The smaller neck sizes in these feed particles presumably result from experiencing greater numbers of pre-compaction rounds before final tabletting. In terms of the above analogy, it is equivalent to certain regions of a country only be served by the smallest types of roads, such as B-roads and country lanes. The cracks

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showing up as bright white lines in Fig. 7.7b have been completely filled with mercury. This is equivalent to shutting down the trunk roads across the UK. The light grey regions are where mercury has shut down pores accessible via necks greater than 6 nm, equivalent to loss of the A-roads in the UK. The impact, on mass transport, of shutting down the pores and regions showing as bright white and light grey in Fig. 7.7 can be measured using rate of nitrogen adsorption experiments performed before and after mercury entrapment. The loss of the pores, due to the mercury entrapment visible in Fig. 7.7b, resulted in a mass uptake rate that was only 52% of what it was when there was no such blockage, as in Fig. 7.7. This is a measure of the importance of these lost void space features to overall mass transport. An alternative experiment for progressively examining the importance of different subsets of pores to mass transport is using NMR cryodiffusometry (Perkins et al. 2008). A sample can be imbibed with a probe liquid, such as water or cyclohexane, which is then completely frozen. An NMR cryoporometry melting curve experiment can then be started, but once the system reaches equilibrium at each temperature step the self-diffusivity of the liquid present can also be measured. Starting with the smallest pores in the sample, as the temperature is raised, at each step, ever larger pores become molten, and the diffusivity measured will progressively incorporate their contribution to the mass transport. Hence, if a particular subset of pores are particularly important to mass transport, then, once they become molten, the liquid self-diffusivity might be expected to exhibit a sharp increase. The cryoporometry experiment melting curve can also be made spatially resolved using MRI, since the solid phase will be invisible in the imaging due to possessing too short a relaxation time (Balcom et al. 2003). A further method that can measure the importance of particular subsets of pores to mass transport is combined serial adsorption of nitrogen and water, with rate of adsorption measurements made for nitrogen before and after filling a subset of pores with water. Other things being equal, as water vapour is increased, then condensed water will tend to fill the sample starting with the smallest pores first. Hence, the impact of the loss of ever larger pores on mass transport can be tested by progressively increasing the saturation with water, and freezing it in-place, between nitrogen uptake experiments. If a particularly important subset of pores is filled with water at one particular vapour pressure, the subsequent rate of nitrogen uptake ought to decline markedly. The spatial distribution of the adsorbed water can be mapped using MRI (Hitchcock et al. 2010). Once the key subsets of pore sizes, or void space regions, which make the biggest contribution to mass transport, have been identified by one or more of the above methods, the origins of these features from the fabrication process can be sought. For example, the variation in the density of the roll-compacted feed for the pellets shown in Fig. 7.7 results from the recycling of oversized and under-sized milled fragments. This link offers the potential to decide whether the impact these features make on product performance is sufficient to justify changes in the fabrication process to remove them.

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7.4 Understanding Product Activity and Selectivity The overall observed adsorbate pick-up, or catalyst reaction rate, for a given pellet, is known as its activity, with units of moles per unit time. Where there is competitive adsorption between different molecules, or competing chemical reactions, the differences in molecular species concentrations created by mass transport limitations may result in a particular selectivity towards one molecule or reaction, rather than another. Mass transport limitations lead to the necessity for high concentration gradients across pellets in order to drive the diffusive flux, and the steepness of these gradients may vary between different molecules due to differences in individual molecular diffusivities. For catalysts, the ratio of the actual, observed pellet activity, compared to what would have occurred had the whole pellet operated throughout at the periphery concentration, is called the pellet effectiveness factor. A higher concentration of one molecule rather than another, and a greater dependency of a given rate on that particular concentration, can lead to one chemical reaction being favoured over another. If a reaction is diffusion limited and the rate is controlled by mass transport, one obvious solution to improve activity and pellet effectiveness is to reduce the otherwise required diffusion path length by decreasing particle size. However, in packed bed columns reduced pellet particle size results in increased pressure drop down the column, and thus more energy loss. Further, the diffusivity of the pellet might be improved by increasing porosity by having lower density pellets. However, low density pellets tend to be mechanically weak, with little tensile or compressive strength, leading to friability and breakage, thereby creating small fragments and powder which increase column pressure drop. Hence, there is the need to seek the optimal compromise between pellet effectiveness, pellet size, and strength. Many catalyst systems have pore structures that evolve during use for various reasons. For example, running at high temperatures means that catalysts consisting of nanoparticulate crystallites of metals dispersed across the internal surface of a porous support can sinter. Since the catalytic surfaces tend to be of high energy there is a thermodynamic driving force to reduce this area. This happens by the metal crystallites fusing to form larger particles. These larger particles can change pore geometry and reduce the void size. In addition, some types of side reactions, as mentioned above, can lead to the deposition of liquids or solids, such as coke or metal sulphides, within the pores, that can fill and block said pores. The development of nano-casting methods (Rigby et al. 2004), using a variety of nanoscopic templates to control void space geometry in the fabrication of porous materials offers the possibility that, if optimal void space geometries can be designed by process simulation on computer, then they can actually be subsequently fabricated (Prachayawarakorn and Mann 2007; Wang and Coppens 2008). Developments in sophisticated synthesis methods have allowed the fabrication of ever more complex nanoscopic porous structures, such as a Menger sponge (Mayama and Tsujii 2006). However, a common drawback, to the well-ordered, controlled porosity materials, is that they tend to be unstable, and, therefore, degrade, at the conditions used in

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typical industrial processes, such as high temperature, and in the presence of water. This means more robust but disordered materials are still very much used. Further, as mentioned above, the designer, ordered structure can be modified by catalyst sintering or coking processes. There is, thus, little advantage to having the ideal pore structure at the outset, if this is quickly modified during use to something less optimal. Some studies adopt what might be called a ‘realist’ approach to pore structure modelling where they attempt to explicitly represent the void space itself within the model. However, many approaches are purely ‘phenomenological’ and, as such, treat porous media as a macroscopic continuum that dispenses with describing the actual microscopic details of the void space structure completely, and only represents their impact in terms of a lumped empirical parameter, such as the tortuosity factor. This approach obviously limits the understanding possible of the physical processes taking place at the pore scale, like coking. Since phenomenological methods do not rely much on pore structure characterisation they will not be discussed further here. As mentioned in Chap. 3, structural models for porous solids come in a range of different classes, namely: (i) Pore bond networks (ii) Pore body and neck networks (iii) Particle packing models. Each type of model has a particular set of potential void space descriptors associated with it, as listed in Table 7.1. If void space descriptors are obtained using indirect techniques, such as mercury porosimetry, then often a model of interpretation is used. This then raises the question whether the void space descriptors so obtained are independent of the method Table 7.1 Void space descriptors that can be incorporated into different major structural model types

Model type

Descriptors

Pore bond networks

Lattice size Porosity Pore shape Pore bond size distribution Connectivity Spatial correlations in pore sizes

Pore body and neck networks

Lattice size Porosity Pore shape Pore neck size distribution Pore body size distribution Connectivity Spatial correlations in pore sizes

Particle packing models

Porosity/packing efficiency Pore size distribution Particle co-ordination number

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of derivation. The descriptors will lie somewhere along the greyscale between phenomenological and realist, with the level of realism determined by how many different confrontations, with various different types of experimental characterisation data, the model can successfully survive. For example, in the Chap. 6 on Hybrid Methods it was reported that Ruffino et al. (2005) used a pore bond network model that simulated serial sectioning of samples containing frozen LMPA to mimic SEM or CXT images, as well as the metal intrusion curves from the porosimetry experiments used to generate the samples. Androutsopoulos and Salmas (2000) used a corrugated, cylindrical pore model, derived from gas sorption data, to predict mercury intrusion curves with reasonable success (Salmas and Androutsopoulos 2001). Tsakiroglou et al. (2004) simultaneously deconvolved nitrogen sorption and mercury porosimetry data to obtain a common set of void space descriptors. The void space model used to do this was a random network of potentially variably shaped pore bodies and necks, each with separate size distributions, and correlation functions describing the level of correlation between body and neck sizes, and a variable mean network co-ordination number. Once constructed, there are several different mathematical approaches that can be used to simulate coupled mass transport, and reaction or adsorption, processes, but these are outside the scope of this book. Very few studies indeed make a priori predictions of catalyst activity and selectivity from realist pore structural models, and then also ‘close the loop’ in terms of validation of the model by comparing those predictions with actual experimental data. Rieckmann and Keil (1999) obtained the pore size distribution of the bidisperse, meso-/macroporous silica-alumina supports for a palladium catalyst using nitrogen adsorption and mercury porosimetry, and the mesopore connectivity using the percolation analysis proposed by Seaton (1991) (as described in Chaps. 2 and 6). They then used these data to construct a three-dimensional, random, cubic network of interconnected cylindrical pores, on which they simulated coupled diffusion and reaction processes for the selective hydrogenation of 1,2-dichloropropane to propane and hydro-chloric acid in a single-pellet reactor. It was found that the original process model had to be modified with the incorporation of an adjustable surface diffusivity to obtain good agreement between predictions and experiment. However, Rieckmann and Keil (1999) did not present their raw pore structure characterisation data, so it was not possible to see whether the various uncertainties and errors that can arise in gas sorption and mercury porosimetry characterisations were present, and could offer an alternative hypothesis for the lack of agreement between experiment and simulations, rather than the surface diffusion flux. Chen et al. (2008) modelled a nanoporous membrane using a simple cubic, smooth cylindrical pore bond network. The descriptors of the model included the pore bond network and the average pore co-ordination number (connectivity). It was suggested that pore shape was not important because molecular dynamics simulations have indicated that fluid transport in pores is hardly influenced by pore shape if the correct average size, transport length, and concentration are applied (Duren et al. 2003). It was found that the network model could successfully predict the selectivity for separation of a helium-argon gas mixture provided it had the correct thickness (lattice

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size) and average pore size. Mourhatch et al. (2010) studied the change in heliumargon permeation selectivity of a similar membrane following periods of chemical vapour deposition of different periods. The model made reasonable predictions of the change in separation selectivity with vapour deposition time.

7.5 Conclusions Pore structure characterisation has a number of uses in application to catalysts and adsorbents that make different demands on the methods. Catalyst and adsorbent fabrication processes are generally too complex to be able to predict final pore structure based upon raw material properties and processing parameters. However, pore characterisation methods can be used to provide characteristics to permit correlation. Various characterisation methods permit the identification of key void space features that impact mass transport rates. A full a priori accurate prediction of diffusionlimited product performance from pore structural characterisation data has yet to be achieved without the use of post hoc adjustable parameters.

References Androutsopoulos GP, Salmas CE (2000) A new model for capillary condensation—evaporation hysteresis based on a random corrugated pore structure concept: prediction of intrinsic pore size distributions. 1. Model formulation. Ind Eng Chem Res 39(10):3747–3763 Avery RG, Ramsay JDF (1973) The sorption of nitrogen in porous compacts of silica and zirconia powders. J Colloid Interface Sci 42:597–606 Balcom BJ, Barrita BC, Choi C, Beyea SD, Goodyear DJ, Bremner SW (2003) Single-point magnetic resonance imaging (MRI) of cement based materials. Mater Struc 36:166–182 Chen F, Mourhatch R, Tsotsis TT, Sahimi M (2008) Pore network model of transport and separation of binary gas mixtures in nanoporous membranes. J Mem Sci 315(1–2):48–57 Duren T, Jakobtorweihen S, Keil FJ, Seaton NA (2003) Grand canonical molecular dynamics simulations of transport diffusion in geometrically heterogeneous pores. Phys Chem Chem Phys 5(2):369–375 Hitchcock I, Chudek JA, Holt EM, Lowe JP, Rigby SP (2010) NMR studies of cooperative effects in adsorption. Langmuir 26(23):18061–18070 Hollewand MP, Gladden LF (1994) Probing the porous structure of pellets- An NMR study of drying. Magn Reson Imag 12(2):291–294 Malik S, Smith L, Sharman J, Holt EM, Rigby SP (2016) Pore structural characterization of fuel cell layers using integrated mercury porosimetry and computerized x-ray tomography. Ind Eng Chem Res 55(41):10850–10859 Mantle MD, Bardsley MH, Gladden LF, Bridgwater J (2004) Laminations in ceramic forming— mechanisms revealed by MRI. Acta Mater 52(4):899–909 Mayama H, Tsujii K (2006) Menger sponge-like fractal body created by a novel template method. J Chem Phys 125(12):124706 Mourhatch R, Tsotsis TT, Sahimi M (2010) Network model for the evolution of the pore structure of silicon-carbide membranes during their fabrication. J Mem Sci 356(1–2):138–146

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Nepryahin A, Fletcher R, Holt EM, Rigby SP (2016) Structure-transport relationships in disordered solids using integrated rate of gas sorption and mercury porosimetry. Chem Eng Sci 152:663–673 Perkins EL, Lowe JP, Edler KJ, Tanko N, Rigby SP (2008) Determination of the percolation properties and pore connectivity for mesoporous solids using NMR cryodiffusometry. Chem Eng Sci 63:1929–1940 Prachayawarakorn S, Mann R (2007) Effects of pore assembly architecture on catalyst particle tortuosity and reaction effectiveness. Catal Today 128(1–2):88–99 Rieckmann C, Keil FJ (1999) Simulation and experiment of multicomponent diffusion and reaction in three-dimensional networks. Chem Eng Sci 54(15–16):3485–3493 Rigby SP, Beanlands K, Evbuomwan IO, Watt-Smith MJ, Edler KJ, Fletcher RS (2004) Nanocasting of novel, designer-structured catalyst supports. Chem Eng Sci 59(22–23):5113–5120 Rigby SP, Hasan M, Hitchcock I, Fletcher RS (2017) Detection of the delayed condensation effect and determination of its impact on the accuracy of gas adsorption pore size distributions. Colloids Surf A 517:33–44 Ruffino L, Mann R, Oldman R, Stitt EH, Boller E, Cloetens P, DiMichiel M, Merino J (2005) Using x-ray microtomography for characterisation of catalyst particle pore structure. Can J Chem Eng 83(1):132–139 Salmas CE, Androutsopoulos GP (2001) Pore structure analysis of an SCR catalyst using a new method for interpreting nitrogen sorption hysteresis. Appl Catal A 210(1–2):329–338 Seaton NA (1991) Determination of the connectivity of porous solids from nitrogen sorption measurements. Chem Eng Sci 46(8):1895–1909 Shiko E, Edler KJ, Lowe JP, Rigby SP (2012) Probing the impact of advanced melting and advanced adsorption phenomena on the accuracy of pore size distributions from cryoporometry and adsorption using NMR relaxometry and diffusometry. J Colloid Interface Sci 385:183–192 Smith DM, Deshpande R, Brinker CJ, Earl WL, Ewing B, Davis PJ (1992) In-situ pore structure characterisation during sol-gel synthesis of controlled porosity materials. Catal Today 14(2):293– 303 Timonen J, Alvila L, Hirva P, Pakkanen TT, Gross D, Lehmann V (1995) NMR imaging of aluminium oxide catalyst spheres. Appl Catal A 129(1):117–123 Tsakiroglou CD, Burganos VN, Jacobsen J (2004) Pore structure analysis by using nitrogen sorption and mercury intrusion data. AIChEJ 50(2):489–510 Wang G, Coppens M-O (2008) Calculation of the optimal macropore size in nanoporous catalysts and its application to DeNO(x) catalysis. Ind Eng Chem Res 47(11):3847–3855 Watt-Smith MJ, Rigby SP, Chudek JA, Fletcher RS (2006) Simulation of nonwetting phase entrapment within porous media using magnetic resonance imaging. Langmuir 22(11):5180–5188

Chapter 8

Pore Structural Characterisation in Engineering Geology

8.1 Special Considerations for Natural Porous Systems 8.1.1 Impact of Geological Processes on Pore Structure In contrast to industrial materials, most of the processes responsible for the pore structure of rocks are natural and, thus, uncontrolled. The structure of the void space of rocks plays a critical role in many geological processes, such as migration and retention of water, gas and hydrocarbon migration and storage, and evolution of hydrothermal systems. However, artificial reservoir engineering interventions, such as fracking and in situ combustion, can also change pore structure. Hence, characterisation of porous rocks can aid predictions of hydrocarbon production or carbon dioxide sequestration. Diagenesis refers to the physical and chemical processing that rocks undergo over geological time. Just as it would be desirable for industrial materials to be able to directly relate parameters of the raw materials and fabrication process to resultant void space descriptors, a key aim in geology is to be able to relate geological factors, like mineral grain size, sorting, degree of compaction, and type and amounts of diagenetic cements, to rock pore structure. Burial depth is a key factor, especially relevant to porosity, since the effects of increasing overburden and temperature promote compaction and cementation (Lai et al. 2018). Deviations from this general trend highlight the operation of other processes (Lai et al. 2018). The long geological history of many rocks means a series of diagenetic modifications can have been made to their void spaces over time. Diagenetic modifications can alter the amount and distribution of pores, often creating smaller and more disconnected spaces. Various types and degrees of diagenesis reshape the void space in distinctive ways. Fracturing and dissolution are the main pore volume-enhancing factors, whereas cementation and compaction are the main porosity-reducing factors. Cementation decreases porosity by filling in void spaces and decreases

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permeability by shrinking pore necks and lowering connectivity. For example, porosity reduction in tight sandstones is primarily the result of cement precipitation (Sakhaee-Pour and Bryant 2014). It is claimed that past, particular geological processes experienced by given rocks can manifest in the specific form of their pore characterisation data. For example, the sorting of mineral grains in rocks is likely to give rise to particular modes in mercury intrusion curves representing the relevant grain size (Lai et al. 2018).

8.1.2 Pore Types Geological porous media have voids over a vast range of length-scales from huge faults down to microporosity. However, this work will only consider void spaces likely to be encountered from the rock core scale (~cm) downwards. While many new industrial materials have designer void spaces whereby particular pores are engineered at different length-scales, natural materials have much more variety due to the heterogeneity of rocks. This means that classification schemes of pore types are much more common when characterising geological media. This is because particular pore types are often associated with particular rock or mineral species, or result from particular geological processes. Many rocks have multi-modal pore size distributions where each size mode is associated with a particular mineral type. For example, pores below one micron in size are most often associated with clays (Lai et al. 2018). However, given the heterogeneity in natural materials, there is not one general classification scheme for all such materials. There are certain desirability criteria that a classification scheme should meet to be useful. The classification scheme should distinguish between pores in the commonly occurring range of sizes. The IUPAC classification scheme of micropores, mesopores, and macropores (see Chap. 1) is often unsuitable for rocks due to the large range of pore sizes found in the macropore category making it undiscriminating (Loucks et al. 2012). The scheme should, where possible, incorporate some inherent characteristics, such as permeability and wettability. Loucks et al. (2012) have suggested that, unless supported by explicit independent data on formation mechanism, pore classification schemes should be descriptive rather than interpretative. Examples of interpretative classifications include ‘organophyllic’, ‘organophobic’, and ‘carbonate dissolution’. Characterising mudrocks and shales has become particularly important recently due to the development of shale gas and shale oil extraction. Loucks et al. (2012) proposed a tripartite classification scheme for matrix pores. Two types of pores are associated with the inorganic mineral phase and one type with the organic matter. The mineral pores were subdivided into intraparticle (intraP) and interparticle (interP) pores. These two types of pore also correspond to the classification scheme proposed for tight sandstones by Sakhaee-Pour and Bryant (2014). These workers classified tight sandstones, based on the fraction of interparticle porosity, into intergranular (interparticle) dominant, intermediate, and intragranular (intraparticle) dominant.

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Mudrock pores are formed by both depositional and diagenetic processes. They may undergo multiple stage origins involving initial deposition, compaction, cementation, and dissolution processes (Loucks et al. 2012). The geometry of interparticle pores depends strongly on their history, which includes both primary pore preservation and diagenetic alteration. In older, more deeply buried mudrocks, interparticle pores are usually reduced in abundance by compaction and cementation. Grain-edge pores are located around the edge of particles and are possibly due to the matrix separating from the particle during compaction, especially around organic matter (Loucks et al. 2012). Interparticle pores are most common between clay aggregates, between clay and mineral particles, or between ductile clay and rigid particles. In sandstones, quartz grains can often show overgrowths with coatings of, such as, iron oxide or clays. The presence of clays in sandstones significantly affects porosity and permeability. Quartz itself can also be precipitated as overgrowths, reducing larger pores to narrow slit-/slot-like pores (Lai et al. 2018). Pore throats (or necks), which typically arise where grains meet, also have their own classification scheme (Zou et al. 2012; Lai et al. 2018). The pore throat types in sandstones include ‘contracted throat’, ‘shot-like’, ‘flake-like’, ‘curved’, and ‘pipe-like’. For example, the large residual pores between primary particles tend to be connected by narrow necktype and narrow flake-like throats, whereas interparticle and intraparticle dissolution pores tend to be connected by thin, pipe-shaped necks. Intraparticle pores occur within particles that might be grains or crystals. Most intraparticle pores are formed by diagenetic processes, but some may be primary in origin. Examples of this type of pore include (1) moldic pores formed by partial or complete dissolution, (2) preserved intrafossil pores, (3) intercrystalline pores within pyrite framboids, (4) cleavage-plane pores within clay and mica mineral grains, and (5) intragrain pores (e.g. within peloids and faecal pellets) (Loucks et al. 2012). Intraparticle pores are more common in younger rocks. The shape of intraparticle pores depends upon their origin. For example, in clays and micas pores are slit-shaped, while moldic pores take upon the form of their precursor. Intraparticle pores can form by dissolution due to migration of corrosive fluids. Unstable feldspars often undergo a significant degree of dissolution, which manifests as skeletal feldspar grains or even as a moldic pore where the complete grain has dissolved away. Micropores are generally associated with intermolecular layer pores in clays and are difficult to detect except by gas sorption. Pores begin to form within organic matter when thermal maturation, as shown by vitrinite reflectance, has reached roughly 0.6% or above, since below this figure pores are rare (Loucks et al. 2012). Organic matter pores are generally ovoid, bubble-like in form, which can make them look isolated in two-dimensional sections, but demonstrate connectivity in three dimensions. The limited evidence available suggests that organic matter pores occur, and thus form, more often in Type II kerogen than Type III kerogen (Loucks et al. 2012). Some authors (e.g. Chen et al. 2015) have suggested that pore shape is a characteristic feature of organic matter pores compared to mineral-associated pores. They suggested the cross sections of the former are typically round or elliptical, while the latter are more typically slit-like.

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8.1.3 Sampling This book will confine itself to the characterisation of rock core samples, rather than field scale methods. Even so, rock cores are of the order of centimetres in diameter and in length. Unlike industrial materials, where the aim is to produce uniformity of product, natural porous samples tend to have much larger levels of heterogeneity over many length-scales, from the pore size up to the overall core dimension. For most of the characterisation methods described in previous chapters, there is a limit to the sample volume that can be tested. The typical sample sizes for each technique are given in Table 8.1. In order to turn rock cores into a form suitable for most gas sorption or mercury porosimetry sample tubes, it is necessary to fragment the monolithic whole to chips or powders. This raises issues around whether the particle or grain size of the samples makes a difference. For rocks containing pervasive isolated porosity, as well as connected porosity, grinding the cores to different particle sizes may break into different fractions of the isolated pores. Hence, void space descriptors, such as (accessible) surface area, can increase with decreasing particle size. Where particles are segregated by size, care should be taken that the grain size of certain minerals that are particularly hard, say, does not mean they all end up in one particle size bin, whereas more brittle minerals become fragmented and end up distributed across all bin sizes. Imaging methods can be used to study larger samples, but there is a trade-off between sample size and/or field of view, and image pixel resolution. In principle, imaging can be carried out in body scanners designed for human subjects, but the resolution is then typically centimetres. Imaging with CXT at much higher resolution is called microfocus imaging. The limit of imaging resolution can skew the statistical distributions of void space descriptors if it means missing features too small to be resolved. For example, particle size distributions for rock grains from CXT can give higher averages than microscopy methods because the CXT misses smaller particles (Cnudde et al. 2011). Table 8.1 Typical sample sizes for characterisation techniques

Characterisation technique

Typical sample size

Gas sorption

~0.1–1 g

Mercury porosimetry

~0.1–1 g

Thermoporometry on DSC

~ 3 mm fragment

NMR in narrow bore liquid state NMR spectrometer

~ 4-mm diameter fragments, up to ~1 cm long

NMR imaging in wide bore magnet

~ cm

CXT

~mm to ~cm

Microscopy

ng–g

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As mentioned in Chap. 5, imaging data allows the measurement of correlation lengths for properties visible in images, such as density or porosity. The correlation length is the characteristic length-scale beyond which the value of the mean porosity or density becomes a constant over the volume averaged. Hence, the correlation length would suggest the minimum size of the field of view necessary for results to be statistically reliable. The scale of sampling with different techniques will also determine which pores are seen (Loucks et al. 2012).

8.1.4 Sample Preparation As mentioned above, many rocks undergo compaction as a result of the increased pressure experienced due to burial to increasing depths. However, the rocks may deform during recovery, due to either damage or simple elastic rebound as the pressure is released. This means that the in situ pore structure at depth may be very different to that at ambient conditions in the laboratory. In particular, pressure relief often results in the opening up of microcracks (Sakhaee-Pour and Bryant 2014). The impact of the transfer of samples from depth to the laboratory depends upon the characterisation technique used. For example, mercury porosimetry involves application of an isotropic, compressive stress to the sample, which may close microcracks back up before intraparticle intrusion begins. In contrast, gas sorption will not generally involve sufficiently high pressures to achieve this effect. Many pore characterisation methods, such as microscopy, gas sorption, and mercury porosimetry, require a ‘clean’ dry surface before the experiment can begin, in order to avoid artefacts due to surface contaminants, especially atmospheric moisture. However, freeze-drying, or thermal pre-treatment under vacuum, or another atmosphere such as flowing nitrogen, may modify the sample. This is a particular issue with rocks, especially those containing clay minerals since moisture can be trapped between clay mineral interlayers (Holmes et al. 2017). Once the moisture is driven out of these pores, the structure may collapse and not be subsequently accessible. Shrinkage due to drying, or expansion due to ice formation in freeze-drying, can also occur. However, pre-adsorbing water in order to attempt to hold the layers open may result in substantial swelling of the clays beyond their natural state. High pressure freezing can be an alternative to conventional freeze-drying that prevents the artefacts introduced by the latter (Keller et al. 2013). The highly chemically heterogeneous nature of rock samples can mean that, in gas sorption, the nature of the adsorptive used can change the result, and this can depend upon the nature of the sample pre-treatment. For example, shale rocks can contain both organic carbon and inorganic mineral phases that both contain pores. Adsorptives with low polarity, like cyclohexane, may bind well to non-polar organic carbon surfaces, while showing weak binding to polar inorganic mineral surfaces. Conversely, high polarity adsorptives like water may be strongly attracted to highly polar inorganic surfaces but be excluded from low polarity organic carbon pores. There may also be molecules with intermediate polarity, like weakly quadrupolar

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nitrogen, that will bind to some extent to both types of surfaces. In the most convenient type of case, such as Rempstone shale (Rigby et al. 2020) nitrogen can be used to measure the total surface area, while water can be used to measure the inorganic surface area and cyclohexane the organic surface area. In such a case, the areas from water and cyclohexane add up to that from nitrogen. However, since this type of finding relies upon the relative binding affinitiess of the three adsorptives for the two types of surface being ‘just right’ this straightforward result does not always arise. Just as the choice of probe fluid for gas sorption requires careful consideration for rocks, the same is also the case for cryo-/thermo-porometry. This is because rocks very commonly contain very large pores, well into the macropore range (>50 nm) according to the IUPAC definition. Given that the Gibbs–Thomson parameter (i.e. the constant of proportionality between pore size and melting/freezing point depression) for many common probe fluids (e.g. water and cyclohexane) is ~10s K nm, then the melting/freezing point depression for typical sandstone reservoir rocks with micronsized pores will be very small. However, there are known probe fluids that will wet common reservoir rocks and have well-defined phase transitions, and which also have sufficiently large Gibbs–Thomson parameters to make thermoporometry feasible for large pore rocks. These probe fluids include octamethylcyclotetrasiloxane, which has a Gibbs–Thomson parameter of ~140 K nm (Vargas-Florencia et al. 2007). Samples for microscopy are often prepared as polished thin sections to enable clear imaging of mineral grains. However, the sample preparation can produce surface topographic irregularities due to the different hardnesses of the different mineral grains (Loucks et al. 2012). This can make identification of authentic pores, relative to artefacts of this effect, difficult. The identification of real pores can be made easier by pre-injection with a contrast agent such as resin or low melting point metal alloy (such as Wood’s metal) (Loucks et al. 2012). Alternatively, ion-milling methods can reduce artefacts, as it produces only minor topographic variations. Flat surfaces for sections permit the proper quantification and typing of pores. However, ion-beam milling can produce other artefacts of its own. For example, rock samples with a matrix rich in clay or with large clay particles can dessicate and produce shrinkage cracks on drying. Further, it is possible to get redeposition of milled material within pores, which confounds identification, and also the phenomena known as current striations and ‘curtaining’, whereby linear and/or flaring structures forming minor relief are created, leading to artefacts (Loucks et al. 2012).

8.2 Predicting Permeability, Reservoir Producibility, and Bound Volume Index 8.2.1 Permeability and Reservoir Producibility Prediction of permeability in disordered porous solids tends to be based on the critical path theory (Ambegaokar et al. 1971). This suggests that, if a porous medium has a

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very broad range of resistances to mass transport distributed randomly across the void space, then the overall observed rate will be controlled by a particular intermediate resistance. This is because the mass transport flux will tend to avoid high resistance regions, while transport would be easy through low resistance regions, thus meaning that it is intermediate resistances that end up as rate-controlling. The critical path is the passage through the material that minimises resistance to mass transport, and thus must pass through the critical controlling resistance. In porous solids, the mass transport resistance is inversely proportional to pore size, since, for example, Knudsen diffusivity is proportional to pore size. For tight reservoir rocks, the Klinkenberg permeability has been found to strongly correlate with the square of the largest pore throat radius (Lai et al. 2018). The mass transport-controlling pore size in the critical path tends to coincide with the percolation threshold for an invasion percolation process. This is because the critical pore size in each case is the minimum size met along the path from one side of the material to the other (or to the centre). This means that characterisation methods based on invasion percolation processes, like mercury intrusion porosimetry and gas desorption, can be used to measure the critical pore size for critical path theory. Hence, the mercury intrusion percolation threshold pore size is a key parameter in many correlations for determining permeability, such as that of Katz and Thompson (1986). While, as shown in Chaps. 3 and 6, the percolation threshold depends upon pore network connectivity, the permeability correlation often needs a tortuosity correction factor to account for deviations of the real pore structure from an idealised, homogeneous, completely random pore network. Given different rock types will deviate from the idealised network to differing degrees, based on variation in formation mechanisms, each rock type tends to have a different tortuosity factor. Since prediction of transport processes requires information on void space connectivity, as well as pore size, then studying just two-dimensional (2D) polished thin sections does not provide the necessary information (Cnudde et al. 2011). Serial sectioning coupled with image analysis to reconstruct the three-dimensional (3D) structure would provide more information on connectivity. However, serial sectioning misses the void space between section planes. 3D CXT or MRI can provide the requisite detail if the characteristic sizes of the pores are above the lower resolution limit of the imaging. The producibility of a reservoir depends upon the pervasiveness and accessibility of the pore network, and the flow properties of the connected pathways through the rock. This pathway may encompass fractures and matrix pores. Methods that allow the mapping of such pathways, such as combined mercury porosimetry and CXT, integrated with methods that directly assess the transport properties of these pathways, such as rate of gas sorption, are especially useful for understanding this parameter (Nepryahin et al. 2016a, b). This method has shown that the introduction of the pores that become filled with mercury for the shale shown in Fig. 8.1 is responsible for a factor of ~1000 increase in gas mass transport through the shale (Rigby et al. 2020).

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Fig. 8.1 a 3D CXT reconstruction of a treated shale chip following mercury porosimetry. The near-surface entrapped mercury within the chip is shown in red. b Full 3D distribution of entrapped mercury within the shale chip. c Gated 2D CXT reconstructed slice showing entrapped mercury in red and shale matrix as grey, and d 2D spatial distribution of entrapped mercury for the same 2D slice as shown in c. Reprinted from Rigby et al. (2020) under a Creative Commons CC-BY license

8.2.2 Bound Volume Index (BVI) The recovery of oil is divided into primary, secondary, and tertiary recovery. The primary recovery of oil is that obtained as a result of the natural reservoir pressure pushing the oil to the surface. Once this pressure has been dissipated, it must be replaced by gas or liquid injection to continue production. The injection of brine to force oil to the surface is one form of such secondary recovery. In the ideal case, the injected brine acts like a plunger in a syringe and pushes all the oil to the production well. However, in real-world cases, the brine may ‘overtake’ the oil and reach the production well first, leaving behind oil. Unfortunately, once the injected brine has found a route from injection to production well bypassing the oil, it continues to follow that route, and oil production declines and ends. The fraction of the original that remains as residual oil left in-place is called the ‘bound volume index’ (BVI). The fraction of oil capable of being mobilised and produced is the ‘free-flow index’ (FFI).

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Oil in sandstone rocks is often a non-wetting fluid, just as mercury is for most materials. Hence, it is frequently found that the amount of mercury that gets entrapped in a sample following porosimetry corresponds to the BVI (Appel et al. 1998). This is because the types of physical process, like ‘snap-off’ of menisci, that lead to mercury entrapment also lead to oil being bypassed by water. Further, as discussed in Chap. 3, the connectivity of a void space affects the level of entrapment of mercury, with lower connectivity leading to higher entrapment. As discussed in Chap. 5, pulsed field gradient (PFG) NMR can measure the self-diffusivity of fluids in void spaces, and the diffusivity increases with increased connectivity. Hence, if the self-diffusivity of a fluid molecule (e.g. typically water in brine) is measured in reservoir rocks, it is often found there is a variation of diffusivities that can be modelled as a bimodal distribution. The fraction of the slow moving component (low diffusivity) in such a case is often found to correspond to the BVI (Appel et al. 1998). Further, if the distribution of spin–spin (T 2 ) relaxation times is measured for the fluid (again typically brine) imbibed within a porous rock, and if there is a tail of low T 2 , then the volume fraction of this phase also often corresponds to the BVI (Appel et al. 1998). The cut-off value in T 2 to identify this ‘tail’ is obtained from the empirical findings of a series of tests on many rocks of a similar type.

8.3 Characterising Multi-scale, Hierarchical Porous Structures 8.3.1 Fractal and Multi-fractal Models The high complexity of rocks means there is a need for mathematical tools that can describe such complexity. Often, apparent heterogeneity hides an underlying order. Fractals are objects that possess the property of ‘self-similarity’ which means they consist of repeated copies of a template repeated at ever decreasing length-scales. This self-similarity can be exact, as in objects like Koch curves (see Fig. 8.2) or on a statistical basis. The Koch curve is made up of a basic generator shape that is reduced in size by a common factor at each generation. Most natural materials are fractals of the statistical type. Another basic property of fractals is that, in contrast to Euclidean shapes like planes or cubes, the key geometric characteristics, such as surface area, vary with the size of the ruler used to measure them. For fractals, the measured surface area, A, for an object, with some overall linear size measure of R, follows a power law with the size of the ruler r, such that:  D R r 2, A∝ r

(8.1)

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Fig. 8.2 Koch curve. a Basic generator and resultant shape from b iteration 1 and c iteration 2

where D is the fractal dimension. The constant of proportionality in Eq. (8.1) is known as the lacunarity and depends upon the shape, and for a uniform circular geometry would be equal to π . In the Euclidean geometry of flat planes, D would be equal to 2, indicating it is two-dimensional. In Eq. (8.1), when D = 2, the r 2 (ruler or yardstick size) terms cancel top and bottom, and the normal Euclidean result for area is recovered. For rougher surfaces, where the surface irregularities extend into the third dimension, D would exceed 2. For highly irregular and rough volume-filling surfaces, D would tend to 3. Besides surface area, the mass or volume of an object can also be fractal, and the mass or volume varies from the normal Euclidean result for three-dimensional solid shapes which involves scaling with ~R3 . For idealised, mathematical fractals, the surface area would follow an expression of the form of Eq. (8.1) across all length-scales from the infinitely small to the infinitely large. However, for real-world objects the range of fractal behaviour is confined to lie between two finite length-scale cut-offs. For real objects, the upper length-scale cut-off for fractal behaviour corresponds to a correlation length above which the geometric parameter becomes Euclidean, and no longer depends upon the ruler length. The lower length-scale cut-off often corresponds to the size of the primary particle making up the larger length-scale structure, which is analogous to the basic generator shape in Fig. 8.1a. Natural materials, such as rocks, experience formation processes that operate at different length-scales and can give rise to different geometric behaviours over these different length-scales. For example, a rock might be formed from primary grains that have a fractally rough surface. These primary particles may pack together to form the matrix of a sedimentary rock with an internal void space, formed from the gaps between the particles, that is fractal. The rock may also be penetrated by a fracture network that is also fractal in spatial arrangement. The surface fractality of the primary particles must have an upper limit of the particle size. The fractality of the matrix pore network volume will have an upper limit in size of a few primary

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particle diameters, whereas the fractality of the fracture network may extend from the largest pore size up to the size of the facies. A structure consisting of different fractal scaling regimes over different ranges of length-scale is known as a ‘multi-fractal’. Multi-fractal models have been used to characterise the varying degree of heterogeneity in different types of rocks and act as a fingerprint. In a very few cases, a particular fractal dimension is associated with a specific formation mechanism, or a particular type of fractal gives a particular fractal dimension, such as the Menger sponge has a mass fractal dimension of 2.72 (Avnir 1989). The surface fractal dimension for rocks is often obtained from the pore size distribution using variants on the expression (Pfeifer and Avnir 1983): −

dV ∝ r 2−D , dr

(8.2)

where V is the specific pore volume for pores larger than radius r. Equation (8.2), or its integrated variants, is often fitted to pore size distributions. However, when fitting to a multi-fractal model there are a number of issues. In the most unconstrained approach, changing the fractal dimension applying over different length-scales gives rise to a whole set of additional free parameters in the form of the length-scale cut-offs between each fractal regime. It must be always born in mind that any curve can ultimately be fitted to a series of different straight lines. Hence, in order for the multi-fractal model to possess legitimacy, the fitting process must be statistically rigorous, and the standard statistical criteria for defining reasonable straight-line fits must be used. If statistical rigour is not maintained, then the fitting process becomes subjective, and the resulting set of fractal dimensions and cut-offs obtained purely artefactual. Even when statistical rigour is maintained, the cut-offs for different fractal regimes should be considered hypotheses for independent testing. As mentioned above, regime cut-offs should correspond to a physical aspect of the system, such as the overall grain size of primary particles, and this can be validated using data from other structural characterisation methods, such as microscopy, and/or the effect on mass transport processes (Rigby 1999). It is often suggested that, for a fractal scaling to be physically meaningful, it must extend over at least an order of magnitude variation in length-scales (Watt-Smith et al. 2005). Alternatively, Pfeifer and co-workers (Pfeifer 1984; Pfeifer et al. 1983) specified a criterion for the minimum ratio of length-scale cut-offs for the fractal scaling regime to be valid. In particular, for the rulers/yardsticks used to satisfy the condition of ‘minimal self-similarity’ (or exclusion of non-recurrent irregularities), then 2 2 /rmin >∼ 2. Further, they also suggested an intraconvertability of yardstick rmax and overall length-scale cut-offs such that: Rmax = Rmin



2 rmax 2 rmin

1/2 .

(8.3)

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8.3.2 Overcondensation Methods One of the reasons mercury porosimetry is so popular for characterising rocks is that it is one of the very few techniques that can probe the large range of length-scales from ~100s µm to nanometres with just one experiment. However, as described in Chap. 3, the overcondensation method in gas sorption can also probe macroporosity. It has the advantage that high pressures of ~400 MPa are not needed to probe nanoporosity, and thus there is less risk of crushing the sample. The gas overcondensation method has shown that conventional gas sorption experiments tend to miss much of the macroporosity present in shales, as shown in Fig. 8.3. As shown in Fig. 8.3, overcondensation can probe both macroporosity and microporosity. The boundary overcondensation desorption curves also permit the use of ascending gas adsorption scanning curves that can probe pore morphology. In particular, crossing ascending scanning curves even for the very wide hysteresis seen in shales suggest the presence of the large voids with narrow windows seen for organic matter in microscopy data (Rigby et al. 2020). The shape of the conventional isotherm hysteresis loop might lead to a suggestion of slit-shaped pores (Gregg and Sing 1982), but the overcondensation isotherm reveals this shape is incorrect, and thus inferences from it are flawed.

Fig. 8.3 Ascending adsorption scanning curves (starting at relative pressures of 0.6 and 0.8) and corresponding descending desorption curves (indicated by descending arrow) for scanning loops starting on the overcondensation boundary desorption isotherm (solid line), and conventional isotherms for thermochemically treated Rempstone shale sample (filled circle). It can be seen that the conventional isotherm misses a lot of porosity and leads to a misleading hysteresis loop shape. Reprinted from Rigby et al. (2020) under a Creative Commons CC-BY license

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8.3.3 Multi-scale Imaging The formation and diagenetic processes experienced by rocks mean that they generally have heterogeneity in structure over a vast range of length-scales from the reservoir to the molecular scale and even from the core scale downwards. While some indirect characterisation methods, like mercury porosimetry, can probe a large fraction of the range of these length-scales (~100s µm to ~nm for porosimetry), multiple imaging modalities are typically required for more direct characterisation (Saif et al. 2017). Three-dimensional TEM (also known as electron tomography) and dual-beam microscopy can probe down to nanometre length-scales, but the sample volume and/or field of view are limited to 100s nanometres or a few micrometres at that resolution. Hence, microscopy methods are often combined with CXT methods which probe larger length-scales from 100s nanometres up to the size of the core, although multiple CXT imaging at different resolutions may be required to bridge this entire range. Automated image acquisition methods have been developed such that the particular location of the fields of view at high resolution can be found accurately within the wider fields of view for lower resolution imaging. For a start, the trench used to obtain a FIB/SEM image at nanoscale can be located within a SEM image of a much wider field of view (Saif et al. 2017). If the field of view of small-scale imaging does not approach the absolute correlation length in the structure at the next larger length-scale, then it is necessary to identify all the statistically different features/heterogeneities in the larger length-scale image and sample them at the lower length-scale such that the individual parameters for the heterogeneities are known and can be incorporated at the larger length-scale representation (Ma et al. 2019). Due to the large numbers of different regions that might be found at the lower length-scale, it may not be possible to image every different heterogeneity region at smaller scales. In that case, some sort of assumption about grouping types of large-scale features (e.g. particular grain types) together as a single ‘phase’ having similar properties may be required to reduce the amount of information needed. These phases can then be identified in larger-scale images and allocated representative parameters from the lower length-scale sampling process. This process requires some sort of statistical testing to ensure that the parameters chosen are representative of the group. However, where the degree of heterogeneity is so large, such that multi-scale imaging alone becomes impracticable, another strategy is required. An alternative approach, especially where trying to understand structure–transport relations, is to use a phenomenological ‘filtering’ to identify the key aspects of the void space that control mass transport. This approach progressively removes different subsets of the void space, assesses their impact on mass transport, and thereby establishes the relative importance of particular void space features, such as pores in a particular size range. This methodology can be implemented using a variety of experimental techniques. Integrated rate of gas sorption and mercury porosimetry uses entrapment of mercury into ever smaller pores to determine their importance to Knudsen diffusion during low-pressure gas uptake (Nepryahin et al. 2016a). Implementation of the

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strategy using NMR cryodiffusometry starts with a sample with the probe fluid completely frozen. The probe fluid in ever-larger pores is progressively melted in stages, and the increase in molecular self-diffusivity at each stage is measured with PFG NMR (Nepryahin et al. 2016b). Imaging can still augment these experimental techniques by providing spatial mapping of the particular subset of the void space being investigated, as shown for the particular macroporosity containing entrapped mercury in the shale sample in Fig. 8.1.

8.4 Conclusions It has been seen that there are special considerations that apply more specifically to geological, than to other types of samples. In particular, geological samples raise issues of sample preparation and the statistical representativeness of the characterisation data obtained from imaging techniques.

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