Nanoporous Materials: Types, Properties and Uses

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

NANOPOROUS MATERIALS: TYPES, PROPERTIES AND USES

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY Additional books in this series can be found on Nova‘s website at: https://www.novapublishers.com/catalog/index.php?cPath=23_29&seriesp= Nanotechnology+Science+and+Technology Additional E-books in this series can be found on Nova‘s website at: https://www.novapublishers.com/catalog/index.php?cPath=23_29&seriespe= Nanotechnology+Science+and+Technology

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

NANOPOROUS MATERIALS: TYPES, PROPERTIES AND USES

SAMUEL B. JENKINS EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‘ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Nanoporous materials : types, properties, and uses / editors, Samuel B. Jenkins. p. cm. Includes index. ISBN 978-1-61122-999-8 (eBook)

Published by Nova Science Publishers, Inc. † New York

CONTENTS

Preface Chapter 1

Chapter 2

vii The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid V. D. Borman and V. N. Tronin Energetics and Percolation Properties of Hydrophobic Nanoporous Media V. D. Borman and V. N. Tronin

Chapter 3

Ordered Mesoporous Materials for Drug Delivery Applications Spomenka Simovic and Dusan Losic

Chapter 4

Nanocavity: A Novel Functional Nanostructural Unit G. Ouyang and G. W. Yang

Chapter 5

Recent Advances in the Titania Porous Materials Growth through Micro-Arc Oxidation Arūnas Jagminas

Chapter 6

Preparation and Properties of Nanoporous Materials Prepared from Natural Clay Minerals J. Temuujin,, K.J.D.MacKenzie, Ts.Jadambaaand A.van Riessen

1

45 73 163

191

211

Chapter 7

Magnetic Nanoporous Materials S. Giri

Chapter 8

Surface and Mechanical Characteristics of Mesoporous Anodic Aluminum Oxides Tong Hong Wang, Te-Hua Fang and Shao-Hui Kang

243

Quasi Monocrystalline Porous Silicon (QMPS) – A Potential Material for Optoelectronic and Photovoltaic Applications Mahua Chakraborty (Banerjee), Sukumar Basu and Hiranmay Saha

261

Chapter 9

233

vi Chapter 10

Chapter 11

Index

Contents Low-k Nanoporous Interdielectrics: Materials, Thin Film Fabrications, Structures and Properties Moonhor Ree, Jinhwan Yoon and Kyuyoung Heo

273

Novel Manufacturing and Processing Technologies of Nanoporous Silicon Jia-Chuan Lin and Wei-Chih Tsai

315 355

PREFACE Nanoporous materials consist of a regular organic or inorganic framework supporting a porous structure. Nanoporous materials are separated into three subtypes: microporous materials, mesoporous materials and macroporous materials. In recent years, nanoporous materials have been recognized as promising candidates for the multifunctional applications such as catalysis, ion-exchange, gas storage low density magnetic storage, etc. In addition, nanoporous materials are also of scientific and technological importance because of their ability to absorb and cooperate with atoms, ions and molecules on their sizeable interior surfaces and pore space. This new book proposes and reviews advances being made in the field of nanoporous materials. Chapter 1- After compression of a system formed by a nanoporous media and a nonwetting liquid to the threshold pressure value pc 0 , the liquid fills the pores of a porous media. In accordance with prevailing concepts, passage of the liquid from the bulk to the dispersed state can be described as a percolation-type transition [1]. The percolation-type spatial distribution of pores filled with the liquid is confirmed by the ―devil‘s staircase‖ effect involving the change in the resistance of a porous media (porous glass) upon its infiltration with mercury in the vicinity of the threshold infiltration pressure [2]. The percolation type of infiltration of porous media is also confirmed by the ―viscous fingers‖ effect, in which a wetting liquid is displaced from pores by some other liquid [3]. In this case, a nonuniform front of porous media infiltration is formed. This process is typical of infiltration of macroscopic porous bodies with wetting liquids. The threshold type of infiltration was observed for nonwetting liquids, for grained porous media (zeolites) with a pore size of R = 0.3–1.4 nm and silochromes (R = 4–120 nm) filled with nonwetting liquid metals, and for hydrophobized granular porous bodies with a silicon oxide skeleton (R = 3-50 nm) filled with water, ethylene glycol, or salt solutions [4–21]. The grain size in [4–20] was 1–100 µm. To fill nanometer-size pores with a nonwetting liquid with a surface energy of 0.05–0.50 J/m2, a threshold pressure of pc 0 = 102–103 atm is required. When the liquid passes from the bulk to the dispersed state in a nanoporous media with a specific volume of 1 cm3/g, the energy absorbed by the liquid and returned (accumulated) when the liquid flows out amounts to 10–100 kJ/kg. This value is an order of magnitude higher than for polymer composites or alloys with the shape memory effect, which are widely used now [20]. This forms the basis for devices for mechanical energy absorption and accumulation. Bogomolov [23] was the first

viii

Samuel B. Jenkins

to indicate such a possibility of accumulating mechanical energy. It should be noted that 1 kg of a porous material is sufficient for absorbing the energy of a media having a mass of 1 t and moving at a velocity of 50 km/h. Chapter 2- Energetics of "nanoporous medium--nonwetting liquid" systems is one of the new directions in basic and applied research [1-8]. In the simple model of a porous media in the form of cylindrical channels, this threshold pressure is described by the Laplace-Washburn equation p  2  R  cos , where  is the surface energy of the liquid, R is the pore radius, and  is the contact angle (for a nonwetting liquid,   90 ). For filling nanometer-sized pores by a nonwetting liquid with a surface energy of 005  05 J/m2, it is necessary to apply a threshold pressure of 10  10 atm. When the liquid passes from the bulk of the material to a dispersed state in pores of the nanoporous medium with a specific volume of ~ 1 cm3/g, the absorbed and accumulated (returned when the liquid flows out) energy can reach 10 100 kJ/kg. This value is one order of magnitude higher than the energy observed for widely used materials, such as polymer composites and alloys with the shape memory effect [9-11]. Among the systems under investigation are silochromes, zeolites with liquid metals, hydrophobized silica gels, and zeolites with water and aqueous solutions of organic compounds and salts. In recent years, hydrophobized nanoporous media have become available owing to the development of the method used for modifying the surface of nanoporous media, for example, with alkyl chlorosilanes [6,7,12-22]. To date, nanoporous media with different pore shapes, porosities, specific surface areas, specific volumes, average pore radii, and pore size distributions have been studied [1,6,7,19,20,21-55]. The investigations performed thus far have been concerned primarily with equilibrium properties. Experiments have been carried out at a low compression rate of the system when the rate of 2

3

3 increase in the pressure ( p ) in the liquid--porous media system is (10 -1) atm/s. In the

infiltration-defiltration cycle, there is a hysteresis, so that the threshold pressure of infiltration is higher than the pressure of defiltration. Moreover, the majority of the systems studied are characterized by the phenomenon of nonoutflow of a nonwetting liquid when a part of this liquid remains in the porous medium as the excess pressure decreases to zero. The absorbed energy is determined by the product of the volume of filled pores and the difference between the infiltration and defiltration pressures. In frameworks the model of cylindrical channels, these pressures are described by the Laplace--Washburn equation with different angles of wetting. The phenomenon of nonoutflow of a nonwetting liquid has restricted the practical application of the system. These phenomena have been observed in the systems under investigation irrespective of the type of a modifier of the pore surface. Chapter 3- Conventional drug therapy is associated with a number of challenges, such as poor drug stability and/or solubility in biological environment, lack of selectivity, severe toxicity and unfavourable pharmacokinetics. The application of nanotechnology to medical devices - ―nanomedicine‖ is recognized as an emerging field with huge potential for development of new therapeutic concepts. Research on mesoporous materials for biomedical purposes has experienced an outstanding increase during recent years. Three major types of mesoporous materials for drug delivery application were emerged including: mesoporous silica engineered by organic synthesis and porous silicon, anodically oxidised alumina (AAO) and nanotubular titania fabricated by electrochemical methods. Although still in early stages,

Preface

ix

few in vivo studies clearly show the potential of these materials for drug delivery devices in orthopedics implants, dental implants, and vascular stents, where not only is the controlled release of drugs such as antibiotics or growth factors desired, but also appropriate biointegration is needed. In this chapter we collect and analyze some of the most relevant milestones in the research of mesoporous materials for controlled drug delivery for implantable and systemic delivery systems. To provide a comprehensive overview to the reader, this review firstly analyzes biocompatibility aspects, which are the major prerequisite for application of materials that come into contact with biological systems. Secondly, we consider the basic aspects of the textural properties (surface and porosity) that contribute to the understanding of drug adsorption and controlled release processes. Finally, more sophisticated stimuli-responsive materials are reviewed. This is only beginning of the further research in terms of correlating biomaterial chemistry and tissue responses and new clinical approaches required not only for orthopaedics, but also treatment for a number of other diseases (hearth, cancer, diabetes, Parkinson‘s, Alzheimer‘s etc). Chapter 4-The discovery of the nanocavity structures marked a highlight point in the condensed matter physics and material science of low-dimensional systems. However, it is hard to get a deeper understanding for these unique nanostructures with negative curvature of the fundamental physics underlying from the perspective of classic method. Of great importance is the question of the many physical quantities, such as surface energy, cohesive energy and mechanical modulus, etc., keeping not constant due to under-coordinated atoms in the surface or interface layer of low-dimensional systems [1-3]. The most striking feature is the inner surface atomic state of nanocavity structures with negative curvature are different from those of multilayers, nanocrystals and bulk counterparts. Thus, the surface energy in inner surface of nanocavity structures is the most important quantity, which plays the dominant role and should be responsible for the novel performances. The present chapter focuses on the basic physical principles of surface energy of nanocavity structures and its novel performances based on the nanothermodynamics and continuum mechanics considerations. Deeper insight into the physical mechanism behind and analytical solutions to the unusually mechanical behavior and thermal stability of nanocavities are presented. Correlation between the surface energy in inner surface of nanocavity structures and its effect on local stiffness, sink effect, interface diffusion and nonlinear shrinkage has been established. It is found that the inner surface energy of nanocavity increases with the size of nanocavity decreasing, which is the inverse of the size dependent surface energy of free nanocrystals. Accordingly, the method for nanocavity structures not only reveals the new physics and chemistry of nanostructural surface energy, but also provides general theoretical tools to calculate the surface energy and related properties. Chapter 5- Micro-arc oxidation or so-called plasma electrolytic oxidation (PEO) is an effective way to make much thicker oxide films on the surface of titanium and its alloys surprisingly improving protection properties of this material. PEO operates at potentials above the breakdown voltage of the growing passive film on the Ti anode and is characterized by numerous arcs moving rapidly over the electrode. Due to a high temperature in the microplasma channels, penetrating continually the growing film from the surface to the metal/oxide interface during micro-arc oxidation, the fused components of solution are usually inserted into these films. Furthermore, the quantity of sintered compounds increases with the treatment time producing a ceramic layer with nonuniform distribution of elements, porosity, and

x

Samuel B. Jenkins

properties. On the other hand, the anodizing of Ti under micro-arc conditions in the aqueous solutions of pure acids, such as sulphuric and phosphoric, results in the formation of rough films with a network of pores randomly distributed and nonuniform in size. In this chapter, recent progress in fabrication, design, characterization and potential applications of quite pure, thick porous titania coatings by the PEO way are presented and discussed. We showed also that the growth of PEO coatings in strongly alkaline silicate solutions originates from the formation of low valence titanium oxides at the substrate surface as thick as several micrometers. Chapter 6- In this chapter, we review recent work carried out in our research group on the preparation and characterisation of nanoporous materials from a variety of clay minerals by selective leaching methods. These nanoporous materials are prepared by exploiting the crystal architecture of layer-lattice minerals. According to their layer periodicity, clay minerals are divided into different types, namely 1:1 type (kaolinite, antigorite etc), 2:1 (montmorillonite, vermiculite, pyrophillite, talc etc) and amorphous (allophane etc). The combination of tetrahedral Si2O5 and octahedral Al2(OH)4 or Mg2(OH)4 layers gives a range of unique crystal architectures. Nanoporous materials have been prepared using a simple selective leaching technique. Differences in the solubility of the clay components at different pH values are exploited to leach a particular component from the clay structure and produce pores. Leaching occurs preferentially at moderate temperatures (80-90oC). The use of clay minerals to prepare nanoporous materials allows the pore size and shape to be controlled. It is found that in most cases, preliminary amorphisation of the clay minerals favours the leaching process. However, some minerals such as vermiculite can be acid-treated directly, producing porous silica with the highest specific surface area of all the minerals (670 m2/g), by comparison with the silica of the lowest surface area (15 m2/g) produced from talc. Pore size distribution measurements reveal the presence of micro and nanopores. Amorphisation of the clay minerals can be achieved by either thermal or mechanical treatment (grinding). Preliminary milling destroys the layer structure of the clay mineral, making it difficult to control the pore size and shape. Thermal amorphisation of the clay mineral favours the production of porous materials with high surface areas and controlled pore size distributions. Nanoporous silica, -alumina and composite materials have been prepared from kaolin, montmorillonite, pyrophillite, phlogopite and talc show excellent decolorisation, adsorption and ion-exchange properties. Chapter 7- Currently, nanoporous materials have been recognised as promising candidates for the multifunctional applications such as catalysis, ion-exchange, gas storage, low density magnetic storage, etc. Because of the diverse range of metal organic (inorganic) networks the structural, chemical, and physical properties of nanoporous materials are fascinating from fundamental interest as well as technological applications. The use of transition metal ions within the nanoporous structure opens up the possibility of various applications with improved electrical, optical, and magnetic properties. Among them, the search for the improved magnetic properties is challenging ascribed to their potential applications in developing low density magnetic storage materials, magnetic sensors, and intelligent or multifunctional materials. The magnetic properties of the metal organic (inorganic) networks are strongly influenced by the structures of the materials where magnetic properties have been tuned by designing varieties of porous structures composed of

Preface

xi

different transition metals. In this article the magnetism of nanoporous materials is reviewed based on recent experimental results. Chapter 8- Surface and mechanical properties of porous anodic aluminum oxides (AAO) were achieved by means of scanning electron microscope (SEM), atomic force microscope (AFM), indentation tests and finite elements method (FEM) simulations. A two-step anodized mesoporous anodic aluminum oxide was successfully fabricated vertically and hollowly. Both microindentation and nanoindentation were carried out. The results showed that the nanoporous AAO was hydrophobic with a contact angle of 105° while the nanoporous-filled AAO is in a relatively good wettability. Localized pop-in can be found during nanoindentation due to the collapse of the beneath cylindrical structures. Over a certain load, microindentation may induce radial cracks from the indented edge to outward of the AAO. The underside of the indented AAO sample was milled to figure out the structural changes. The effects of the nanoporous filling on the Young's modulus and the hardness are investigated and discussed. A three-dimensional finite element model was also successfully developed to understand the nanoindentation-induced mechanism. A maximum von Mises stress of 1.058 GPa occurred beneath the indenter. Chapter 9- Research on Porous Silicon (PS) is being pursued since their discovery for use in optoelectronics, solar cells and sensors. Silicon containing nano size (10-15 nm) pores produced by electrochemical anodization has the well-known potential advantage of enhanced light trapping inside the material. Also its low reflectance loss makes it suitable for optoelectronic and photovoltaic applications. However, the lack of stability due to ageing effect because of slow native oxidation could not make PS a promising material so far. To overcome this instability problem, recently an idea of Quasi Monocrystalline Porous Silicon (QMPS), a modified form of PS was conceived. When low porosity (~ 20-30%) porous silicon is thermally annealed in the temperature range, 1050-1100C and in pure H2 ambient, the nanopores get transformed both in shape and size, resulting in QMPS. During annealing, the open pore channels on the surface of PS layer become closed and pore-free smooth surface is formed as monocrystalline silicon, with nano-size voids embedded inside the body that might help in the enhanced optical absorption. The presence of nano voids and favorable electrical properties as that of silicon makes QMPS suitable as active layer for low-cost solar cells. Researchers have also used this material as passive seed layer for epitaxial growth for solar cell fabrications. A few researchers have investigated the structural, optical and electrical properties but fabrication of solar cells using QMPS as an active layer is yet to be achieved. However, the primary theoretical modeling has indicated that about 15 – 16% efficiency solar cells are possible to be targeted. Modeling on optical absorption and carrier concentration of QMPS layers have also been done in detail. Further studies can be performed on the effect of variation of the size of nanopores during formation either by thermal annealing or by laser and thermal annealing on the properties of QMPS. Study of QMPS/p+ Si interface in detail may also be very interesting in order to get a quantitative idea of defect density, nature of trapping states etc. There is an ample scope of studies with QMPS since the material is still in the initial stage of investigation and it can bring forth a sensational advancement in the important areas of optoelectronics and solar cells in terms of high efficiency, low cost and relatively easy device fabrications. Chapter 10- The use of low dielectric constant (low-k) interdielectrics in multilevel structure integrated circuits (ICs) can lower line-to-line noise in interconnects and alleviate power dissipation issues by reducing the capacitance between the interconnect conductor

xii

Samuel B. Jenkins

lines. Because of these merits, low-k interdielectric materials are currently in high demand in the development of advanced ICs. One important approach to obtaining low-k values is the incorporation of nanopores into dielectrics. The development of advanced ICs requires a method for producing low-k dielectric materials with uniform distributions of unconnected, closed, individual pores with dimensions considerably smaller than the circuit feature size. Thus the control of both pore size and pore size distribution is crucial to the development of nanoporous low-k dielectrics. This article reviews recent developments in the imprinting of closed nanopores into spin-on materials to produce low-k nanoporous interdielectrics for the production of advanced ICs. This review further provides an overview of the methodologies and characterization techniques used for investigating low-k nanoporous interdielectrics. Chapter 11- Porous silicon (PS) films consisting of many pores and pillars are widely used to yield efficient visible photoluminescence (PL) and electroluminescence (EL) at room temperature. Such light-emission behaviors are primarily attributed to electron confinement in the nanocrystals that constitute the PS film. Also, the PS film shows a unique electrical property of negative difference conductance (NDC) for the carrier mobility difference in different sizes of pores and pillars. The techniques of PS formation have been developed by many different methods. Among them, electrochemical anodization is the most commonly used. The anodization is performed in a hydrogen fluoride (HF) solution with an anodic current on the sample. However, it is very difficult to control the manufacturing parameters precisely in the manufacturing processes. Especially in nano-scale, the anisotropic etching or selective etching cannot be easily achieved in the wet-etching method. However, it is well known that nano-scaled PS (NPS) shows light emission more readily in the visible range than macro-PS does. Therefore, precise control of the NPS formation is very important. In this chapter, the authors introduce three novel manufacturing technologies. In the first method, the Hall-effect is applied in the electrochemical anodization. By Lorentz force, the carriers can be well swept to the allocated areas. In the second method, a forward biased pnjunction is used for bottom-hole assistance. The NPS films with nano-scaled pores and highaspect-ratio pillars can be formed. In the third method, the authors proposed a simple and mask-free method for the fabrication of patterned PS using a localized electric field. The electric field is applied by patterned electrodes (anode and cathode) which are placed horizontally underneath the sample. No masking-layer or related photo-lithography processes are needed in this method. Strong visible photoluminescence emissions in PS can be obtained by these novel methods. In addition, the surface roughness of NPS leads to a high contact resistance to the metal layer of the applications on EL, and hence a waste of an electrical power. To enhance the EL efficiency and stabilize performance, a good metal contact to the light-emitting layer (NPS layer) is required. In the authors‘ study, a supercritical fluid (SCF) technique is explored to overcome the processing problem on metal contact. The SCF is a liquid or gas material used in a state above the critical temperature and critical pressure where gases and liquids can coexist. The use of a CO2 SCF and silver (Ag) nanoparticles is shown to improve the electrical contact between the metal and NPS. Therefore, the contact resistance can be largely reduced, and the power consumption of EL on the PS structure can be greatly reduced by the method.

Preface

xiii

The morphology, porosity, and photoluminescence of the NPS prepared by the proposed method are investigated. This novel technology will have much benefit in the applications of nano-electronic and opto-electronic devices.

In: Nanoporous Materials Types, Properties and Uses Editor: Samuel B. Jenkins, pp. 1-44

ISBN: 978-1-61668-182-1 © 2010 Nova Science Publishers, Inc.

Chapter 1

THE DYNAMICS OF INFILTRATION OF A NANOPOROUS MEDIA WITH A NONWETTING LIQUID V. D. Borman and V. N. Tronin National Research Nuclear Univrsity MEPHI , Moscow, Russia

1. INTRODUCTION After compression of a system formed by a nanoporous media and a nonwetting liquid to the threshold pressure value pc 0 , the liquid fills the pores of a porous media. In accordance with prevailing concepts, passage of the liquid from the bulk to the dispersed state can be described as a percolation-type transition [1]. The percolation-type spatial distribution of pores filled with the liquid is confirmed by the ―devil‘s staircase‖ effect involving the change in the resistance of a porous media (porous glass) upon its infiltration with mercury in the vicinity of the threshold infiltration pressure [2]. The percolation type of infiltration of porous media is also confirmed by the ―viscous fingers‖ effect, in which a wetting liquid is displaced from pores by some other liquid [3]. In this case, a nonuniform front of porous media infiltration is formed. This process is typical of infiltration of macroscopic porous bodies with wetting liquids. The threshold type of infiltration was observed for nonwetting liquids, for grained porous media (zeolites) with a pore size of R = 0.3–1.4 nm and silochromes (R = 4– 120 nm) filled with nonwetting liquid metals, and for hydrophobized granular porous bodies with a silicon oxide skeleton (R = 3-50 nm) filled with water, ethylene glycol, or salt solutions [4–21]. The grain size in [4–20] was 1–100 µm. To fill nanometer-size pores with a nonwetting liquid with a surface energy of 0.05–0.50 J/m2, a threshold pressure of pc 0 = 102–103 atm is required. When the liquid passes from the bulk to the dispersed state in a nanoporous media with a specific volume of 1 cm3/g, the energy absorbed by the liquid and returned (accumulated) when the liquid flows out amounts to 10–100 kJ/kg. This value is an order of magnitude higher than for polymer composites or alloys with the shape memory effect, which are widely used now [20]. This forms the basis

2

V. D. Borman and V. N. Tronin

for devices for mechanical energy absorption and accumulation. Bogomolov [23] was the first to indicate such a possibility of accumulating mechanical energy. It should be noted that 1 kg of a porous material is sufficient for absorbing the energy of a media having a mass of 1 t and moving at a velocity of 50 km/h. In earlier publications, infiltration of pores in a porous media was described in the mean field approximation as a percolation transition in an infinitely large porous media [1]. The pore volume filled under a pressure p was calculated as of the volume of an infinitely large cluster formed by pores with a radius larger (in accordance with the Laplace pressure) than the minimal radius of the pores accessible to the nonwetting liquid under the given pressure. The mean field approximation using the Bethe lattice makes it possible to qualitatively describe the dependence of the filled volume on pressure in the vicinity of threshold pc 0 only under the assumption of a special asymmetric size distribution for pores [1]. In contrast to second-order phase transitions including the percolation transition [26], the systems under investigation exhibit an infiltration-defiltration hysteresis, as well as (complete or partial) nonoutflow of the nonwetting liquid from the porous media when the excess pressure drops to zero [5-7, 10-21]. It should be noted that the nondefiltration restricts the application of the system for energy absorption and accumulation, while hysteresis controls the absorbed and accumulated energy (returned during defiltration). It was shown in [9, 21] that during slow infiltration of the systems under investigation, the pressure dependence of variation V ( p ) in the volume of liquid in a porous media in the infiltration-defiltration cycle (hysteresis) and the volume of the liquid remaining in pores can be described by percolation theory if we take into account energy barrier A( R, p ) of the fluctuational infiltration-defiltration of the liquid in a pore of radius R. Condition A( R, p )  0 for porous medias with a certain pore size distribution makes it possible to find the pressure that corresponds to the access of a pore of radius R to infiltration in a system of connected pores. For porous media, this condition generalizes the Laplace relation. With increasing pressure, the number of pores accessible to infiltration increases and the pores surrounding the given one may also become accessible. Thus, a cluster of accessible pores is formed in the porous media. For the systems studied in [9, 21], infiltration of porous medias upon a slow change in pressure is observed in the vicinity of the percolation threshold for such a fraction  ( p ) of

the volume of accessible pores, for which the inequality  c 0   ( p) /  c 0  10 2  10 4 holds,

where  c 0 is the percolation transition threshold (  c 0  0.18 for 3D systems [24, 25]). This means that when the grain size of the porous media is L ~ (102-104) R , where R is the mean pore radius in a grain ( R ~ 1-10 nm), correlation length   R    c 0

v

( v  0.8 [24,

25]) becomes comparable to grain size L or exceeds it (   L ). This allows us to treat the infiltration of a grain of the porous media as a spatially uniform process. If the characteristic time  P of variation in pressure is much longer than characteristic hydrodynamic time

 z , of nonthreshold ( A( R, p)  0 ) infiltration of clusters of accessible

pores, the volume of the liquid in the porous media at a given pressure can be calculated if the distribution function for accessible-pore clusters over the number of pores in them is known

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

3

[20]. Infiltration first occurs from the grain surface, and then the liquid flows via clusters of filled pores to the clusters of accessible pores.. Thus, infiltration of grains of a porous media with a non-wetting liquid for  P >>  z can be described as infiltration of clusters formed by accessible pores. In view of the small grain volume, we can disregard the spatial nonuniformity in the formation of clusters of accessible pores. It was shown in [27, 28] that upon rapid compression (with a pressure growth rate of p = 104-105 atm/s) of the systems formed by a silochrome SKh 1.5 granulated porous media and Wood‘s alloy or a Fluka 100 hydrophobized granulated porous media and water, infiltration takes place beyond the percolation threshold at a pressure considerably exceeding threshold pressure pс0. The threshold pressure was p0 =1.6pc0 for the former system [27] and

p0 =2pc0 for the latter system [26]. Infiltration is also associated with irregular oscillations in pressure [27]. It follows hence that when the characteristic time of compression of the system decreases, the mechanism of infiltration of the porous media changes. However, the mechanism of infiltration of the porous media under fast compression remains unclear. To reveal the regularities of infiltration of a nanoporous media with a nonwetting liquid is of fundamental importance for understanding the dynamics of percolation transition and of practical interest for the development of shock-absorbing systems. In Section 2, we will study experimentally the infiltration-defiltration process for systems consisting of a Libersorb 23 (L23) hydrophobic granular nanoporous media and water or an aqueous solution of CaCl2 for pressure compression rates of p > 104 atm/s in the situation when the characteristic time

 P of pressure growth is shorter than the characteristic time  z

of non-threshold hydrodynamic infiltration of clusters of accessible pores. New regularities in threshold infiltration under rapid compression are established, which noticeably distinguish the infiltration process in this case from infiltration of a nanoporous media under slow variation in pressure. It can be expected that upon an increase in compression rate and a decrease in time compared to

 P as

 z , the fraction of accessible pores increases and the system is ―thrown‖ beyond

the percolation threshold. In this case, an ―infinitely large‖ cluster of accessible pores is formed in each grain, and the fraction of such pores increases so that the medium of pores in the grain becomes virtually homogeneous. Consequently, upon a decrease in ratio  P /  z , infiltration must be in compliance with the Darcy law [29] upon an increase in pressure, and the infiltration time of the porous media must decrease. However, it was found that infiltration pressure p0 in the systems under investigation is independent (within the experimental error) of the compression energy and, hence, of the pressurization time. During infiltration of the porous media, the new value of threshold pressure p0 remains unchanged and the filled volume is determined not by the fraction of accessible pores, but by the compression energy. For p  p0 , the liquid does not infiltrate the porous media. Thus, it was found that pulsed compression of the systems studied here leads to the emergence of a threshold infiltration pressure p0 higher than pressure pc 0 of the percolation transition observed for

 P >>  z . It was also found that the area of the infiltration-defiltration hysteresis

4

V. D. Borman and V. N. Tronin

loop under rapid compression is larger than for

 P >>  z . This indicates the emergence of an

additional dissipation mechanism. We can naturally It associate this additional dissipation with a flow of the viscous liquid in a porous media. It was found, however, that the experimental time dependences of pressure and volume for the systems studied here do not change (within experimental error) upon a fivefold change in the viscosity of the liquid. Thus, it is established that the infiltration rate in grains of a hydrophobic nanoporous media is independent of the viscosity of the liquid. In Section 3, a model describing the dynamics of infiltration in a granular porous material is constructed. It is assumed that infiltration in grains occurs independently and a pressuredependent distribution of accessible pore clusters is formed in each grain. Under fast compression, infiltration occurs at a pressure of p0  pc 0 . For the systems studied here, p0  1.2 pc 0 , and more than 70% of all pores become accessible to infiltration.

According to estimates, infiltration of a grain of a porous media under rapid compression occurs when the fraction of accessible pores is  0  0.28 , which is higher than percolation threshold  c 0  0.18 . In this case, the porous media is beyond the percolation threshold for accessible pores and an infinitely large cluster of accessible pores (whose size coincides with the size of the grain), surrounded by smaller clusters of accessible pores, is formed in each grain of the porous media. Finite-size clusters contain about 20% of all pores in the porous media, while the infinitely large cluster contains 80% of all accessible pores. For this reason, infiltration in a grain of the porous media under rapid compression will be described as rapid infiltration of liquid into finite-size clusters of accessible pores occurring simultaneously in the entire space of pores in a grain, followed by slow percolation of the liquid from these clusters into the growing infinitely large cluster of accessible pores. Obviously, no infiltration front is formed in this case over time intervals of percolation of the liquid into the infinitely large cluster. For  P >  z , infiltration of a liquid into a porous media is described as percolation of the liquid from a cluster of filled pores to a cluster of accessible pores, while for

 P <  z , the

process is the percolation of the liquid from a cluster of filled pores to the infinitely large cluster of accessible pores. We solve a system of kinetic equations constructed for coordinateindependent distribution functions for clusters of accessible and filled pores and which describe these process is solved for slow and fast infiltration. In the case of slow infiltration, a new result is the divergence of the characteristic time

 v of infiltration in pores of a grain at percolation threshold  c 0 via accessible pores (critical

retardation). In the case of fast infiltration, solution of the system of kinetic equation implies that infiltration must occur at a constant pressure p0 . For p  p0 , infiltration should not be observed. Pressure p0 and characteristic time  v are controlled by the characteristic time of pressurization in the vicinity of the new value of infiltration threshold c , which is higher than the known percolation threshold. Quantity

c is a universal characteristic for porous

bodies, and pressure pc  p0 corresponding to it is determined by the size distribution of pores and by surface energies of the liquid and the interface between the liquid and the porous media.

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

5

The solution of the system of kinetic equations leads to another new result, viz., nonlinear response of the medium to external action, which is manifested in the compensation of this action due to percolation of the liquid from the cluster of filled finite-size pores to an infinitely large cluster of accessible but unfilled pores. As a result of such compensation, infiltration must be independent of the viscosity of the liquid. Infiltration must be accompanied by oscillations of pressure and smaller oscillations of the volume. The resultant time dependences of pressure and volume under rapid compression, as well as the dependences of p0, the maximum filled volume, and the total infiltration time on the compression energy, successfully describe the experimental data for systems L23 + H2O and L23 + CaCl2 under investigation (Section 4). The domain of applicability of the proposed model of infiltration dynamics of nanoporous bodies is also considered in this section. This work is the result of rsearch carrrier out in following articles: V. D. Borman, A. M. Grekhov, and V. I. Troyan, Zh. Éksp. Teor. Fiz. 118 (1), 193 (2000) [JETP 91 (1), 170 (2000)], V. D. Borman, A. A. Belogorlov, A. M. Grekhov, G. V. Lisichkin, V. N. Tronin, and V. I. Troyan, Zh. Éksp. Teor. Fiz. 127 (2), 431 (2005) [JETP 100 (2), 385 (2005)], V. D. Borman, A. A. Belogorlov, A. M. Grekhov, G. V. Lisichkin, V. N. Tronin, and V. I. Troyan, Pis‘ma Zh. Tekh. Fiz. 30 (23), 1 (2004) [Tech. Phys. Lett. 30 (12), 973 (2004)], V. D. Borman, A. A. Belogorlov, A. M. Grekhov, V. N. Tronin, and V. I. Troyan, Pis‘ma Zh. Éksp. Teor. Fiz. 74 (5), 287 (2001) [JETP Lett. 74 (5), 258 (2001)], V. D. Borman, A. A. Belogorlov, G. V. Lisichkin, V. N. Tronin, and V. I. Troyan JETP, Vol. 108, No 3, March 2009.

2. EXPERIMENTAL TECHNIQUE AND RESULTS In experiments, the dynamics of infiltration of water and aqueous solutions of CaCl2 in Libersorb 23 (L23) granular nanoporous media with a mean pore radius of R  6.5 nm was studied This porous media is KSK-G silica gel with SiO2 as the skeleton material, whose surface was chemically modified in accordance with the technique described in [29] to impart hydro-phobic properties to the surface. The specific surface of L23 is approximately 200 m2/g, its specific volume is 0.56 cm3/g, and the mean grain size of the powder of the porous media is 10  m. A sample of the porous media 2-10 g in mass was placed in a container permeable to the liquid in a high-pressure chamber with a volume of ~60 cm3. The chamber was filled with a liquid (water or 25% (in mass) aqueous solution of CaCl2). A movable 180mm-long rod 10 mm in diameter was inserted through a seal in the cover of the chamber. In experiments on infiltration in nanoporous media, the liquid-porous media system was subjected to fast compression on the experimental bench shown schematically in Figure 1. Lower slab 1 is fixed by mounts 2 to upper slab 3. Load 5 in 10 kg mass could freely slide over steel ropes 4. Strain gauge 6 bearing high-pressure chamber 7 filled with a liquid and a porous media was fastened to slab 1. The gauge could measure forces from 10 to 104 N with an error less than 5% for forces exceeding 100 N. Rod 8 of the chamber was rigidly connected with the rod of displacement pickup 10 via steel plate 9. During the impact against load 5, rod 8 entered chamber 7, leading to an increase in pressure in the system. Pickup 10 detected displacements of rod 8 of up to 14 cm under impact and changes in the volume ( V ) of up to 11 cm3 (for area 5-0.8 cm2 of the rod 8with an error not exceeding 5%. Gauge 6

6

V. D. Borman and V. N. Tronin

measured force F exerted by the load on the rod and, hence, the pressure in the chamber (p = F/S). The frequency range of the force and displacement pickups with a constant sensitivity was limited by a frequency of 5 kHz. Signals from the pickups were detected via an analogto-digital converter and processed by a computer. The pressuriza-tion rate in experiments was p = (1-8) x 104 atm/s. Energy E of the impact varied from 20 to 100 J.

Figure 1. Experimental bench for studying the dynamics of infiltration of a nonwetting liquid in nanoporous bodies

Infiltration of water (L23 + H2O) and 25% aqueous solution of CaCl2 (L23 + CaCl2) into porous media L23 for a low pressurization rate ( p ≤ 1 atm/s) was also studied for comparison. For this purpose, we used the setup described in [20], which ensured a slow variation in pressure and measurement of the change in volume of the system (i.e., the volume of the liquid infiltrating the porous media at a fixed pressure). Additionally, total compressibility of the chamber and the liquid (  = (4.5 ± 0.4) x 10-3 cm3/atm for water and

 = (3.1 ± 0.3) x 10–3 cm3/atm for an aqueous solution of CaCl2), as well as compressibility  = (1.8 ± 0.2) x 10–3 cm3/atm of the empty porous media, were measured in experiments when the chamber was filled with a liquid without a porous media. The reproducibility of infiltration—defiltration of the CaCl2 solution in the porous media indicated the absence of segregation of the salt in the pores of L23.

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

7

Figure 2 shows the time dependences of pressure in the chamber filled only with the liquid (aqueous solution of CaCl2 with a volume of 60 cm3), as well as the time dependence of pressure and volume in the case when the chamber was filled with the aqueous solution of CaCl2 (with a volume of 55 cm3) and the L23 porous media with a mass of m = 4 g. These curves were obtained for an impact energy of E = 40 ± 2 J. For the L23 + H2O and L23 + CaCl2 systems studied here at pressurization rate p > 104 atm/s, irregular oscillations of pressure took place; similar oscillations were observed in the liquid Wood‘s alloy-silochrome SKh-1.5 system [28] It should be noted that the amplitude of irregular volume oscillations predicted in [28] is much lower in the systems studied here than the amplitude of pressure oscillations. It follows from Figures 2a and 2b that in contrast to elastic compression of the chamber with the liquid, the increase in pressure in the porous media-liquid system is limited by the value of pressure p0 = 205 ± 10 atm averaged over irregular oscillations. Figure 2b shows for comparison the value of pressure pc 0 = 180 atm corresponding of the threshold of infiltration of the CaCl2 solution into pores of L23 for a low pressurization rate p ≤1 atm/s. The threshold values of pressure for the L23 + H2O system are pc 0 = 150 ± 8 atm and p0 = 180 ± 9 atm. Quantity pc 0 is defined as the pressure at which the compressibility of the infiltrated liquid-porous media system is maximum. The characteristic time of the increase in pressure from pc 0 to p0 is t1 = 2 ms, which corresponds to a pressurization rate of p  1.2x104 atm/s. It can be seen that for p > 104 atm/s, the volume of the system decreases at a pressure p0 higher than the percolation transition pressure pc 0 [9, 21] in the case of slow infiltration. The duration of compression of the system is controlled by time t 2 = 23 ms, at which the decrease in volume is maximum. During the time interval from zero to t1, the decrease in the volume of the system is – V = 1.10 ± 0.05 cm3 and is equal (to within the measurement error) to the decrease in volume – V = 1.00 ± 0.05 cm3 due to compressibility of the chamber, liquid, and porous media. In time interval t1 – t2, the value p0 of the pressure averaged over oscillations is constant; consequently, the observed change in the volume (see Figure 2c) is associated not with the compressibility of the chamber and system, but with the infiltration through the pores of the porous media. Thus, infiltration of pores begins at a pressure p0 higher than the percolation transition pressure; maximum infiltration (change in the volume of the system) is attained at instant t2, and the entire process of infiltration occurs at a constant pressure p0 averaged over oscillations. Maximal infiltration at t = t2 is Vm = 1.20±0.05 cm3, which is smaller than the volume Vpore = 2.3 cm3 of pores in the sample; i.e., for impact energy E = 40 J, infiltration of liquid through accessible pores in the sample with the mass of m = 4 g does not occur. According to estimates, the work of compression (Eel = 432 J) in the time interval from 0 to t2 coincides with impact energy E = 40 + 2 J to within the measurement error. Over time intervals t > t2, the increase in the volume of the system and chamber is associated with the removal of elastic stresses and defiltration of the liquid from the pores of the porous media. Dependences analogous to those depicted in Figure 2 are also observed for the L23 + H2O system.

8

V. D. Borman and V. N. Tronin Our measurements make it possible to find the dependence of infiltration pressure p0 ,

the maximum filled volume of pores Vm = V (t 2 ) - V (t1 ) , and infiltration time tin = t2 – t1 under rapid compression of the system on infiltration energy Ein =E – Eel, where Eel is the part of the impact energy spent for elastic compression of the liquid-porous media system and on the increase in the volume of the chamber, Eel = (χ1 + χ2 + χ3)· p0 2/2. Since the value of

p0 is independent of the impact energy (Figure 3a), the value of Eel is constant. Figure 3a shows that infiltration pressure p0 for the L23 + H2O system is independent of energy to within experimental error in the range Ein = 30-80 J. However, a tendency toward an increase in p0 is observed upon an increase in energy. The dependences of the infiltration time for a porous media and maximum filled volume Vm of pores on the specific energy of infiltration are close to linear to within the measurement error (Figures 3b and 3c). The

Vm ( Ein ) curve is plotted for the specific energy of infiltration (Ein/m). The maximum possible filled volume is limited by the specific volume of pores and is proportional to the mass of the porous media. The possible maximum energy absorbed during infiltration is also proportional to the mass of the porous media. For L23, the specific volume of pores is 0.56 cm3/g. Dependence ∆Vm(Ein/m) is limited by this volume, which corresponds to the maximum specific infiltration energy (12 J/g). Analogous dependences are also observed for the L23 + CaCl2 system.

It follows from dependences Vm ( Ein ) and tin(Ein) that total flux (flow rate) J of the

liquid averaged over the infiltration time is independent of energy. Indeed, the maximum infiltration volume for infiltration energy E in can be defined as tin

Vm   J (t )  dt  J  t in 0

For Vm  Ein and tin  Ein, we have J ( E )  const . The same result follows from the expression for infiltration energy:

Ein 

tin

 p(t )  J (t )  dt 0

For p(t )  const  p0 , we have Ein  p0  J  t in , and average flux J either depends on energy only slightly, or is independent of energy altogether to within the measurement error. It follows from Figure 2c that time dependence V (t ) of the sample volume deviates from the linear dependence only in the vicinity of the maximum infiltration time t2. Consequently, the flux is independent of energy (J(t) = const) everywhere except in this neighborhood. Thus, the pressure at which infiltration through nanopores of a disordered

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

9

porous media occurs for the systems under investigation, as well as the average flux of the liquid in pores in a rapidly pressurized system (when p > 104 atm/s), depends weakly on the impact energy or is independent of it altogether, and it is apparently controlled only by the properties of the system.

Figure 2. Time dependence of pressure in a chamber filled with a CaCl2 solution (25%) (a) without a porous media and (b) with it and (c) time dependence of the decrease in the volume of the L23 + CaCl2 system

Figure 4 shows the dependence of pressure on the change in the volume of the L23 + H2O system for various impact energies. These curves are plotted as a result of computer processing of measured dependences p(t) and V (t ) and make it possible to analyze the features of the transformation of the mechanical impact energy during infiltration-defiltration of a non-wetting liquid in the pores of a porous media. Dependence p (V ) obtained for the same system for p < 1 atm/s (curve 6) is also shown in Figure 4 for comparison. Under such conditions, the initial increase in pressure under elastic compression of the system and a small change in the volume in the vicinity of pressure p0 = 150±10 atm is replaced by a decrease in

10

V. D. Borman and V. N. Tronin

the volume during infiltration of the liquid through the pores upon a small ( p pc 0  5% ) change in pressure.

Figure 3. Dependences of (a) pressure and (b) pore infiltration time on the infiltration energy and (c) dependence of the maximum specific filled volume of pores on the specific infiltration energy for the L23 + H2O system. The solid line corresponds to dependence (61), while dashed lines correspond to (55), (56) (see below)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

11

Figure 4. Infiltration–defiltration hysteresis loop for the L23 + H2O system for various energies of impact action: E = 30 (1), 50 (2), 60 (3), 70 (4), and 80 J (5); curve 6 is the hysteresis loop for quasistatic infiltration–defiltration

A further increase in pressure is associated with elastic deformation of the chamber, liquid, and porous media infiltrated with the liquid. When the rod is withdrawn from the chamber, the volume of the system increases and the pressure decreases due to defiltration of the nonwetting liquid from the pores and the removal of elastic stresses. The p (V ) dependences in Figure 4 form hysteresis loops whose areas determine the absorbed impact energy. It can be seen that an increase in the impact energy leads to an increase in the volume of the liquid infiltrating through the pores of the porous media. It follows from Figure 4 that infiltration for different impact energies occurs at a pressure of p0 = 180 ± 10 atm. The infiltration pressure under slow compression ( p < 1 atm/s) increases with the filled volume of the pores. This is associated with the size distribution of pores in the porous media since upon an increase in pressure, smaller pores become accessible to the nonwetting liquid and are filled with it [8, 20]. Apparently, a certain increase in the infiltration pressure upon rapid compression (curves 1-5 in Figure 4) with the impact energy is also associated with the size distribution of pores. Analogous dependences are observed for the L23 + CaCl2 system also. According to the results obtained in [20, 23], the absorbed energy is equal to the heat of the formation and subsequent disappearance of menisci of the liquid in the infiltrationdefiltration processes. These menisci are formed at the mouths of filled (empty) pores adjacent to the empty (filled) pores. The energy equal to the work of formation of the liquidporous media interface and spent during infiltration is returned during defiltration of the liquid. The energy absorption during the infiltration-defiltration process is associated with the heat of formation of menisci.Comparison of the p (V ) dependences for rapid (curves 1-5 in Figure 4) and slow (curve 6) compression of the system shows that infiltration pressure p0 >

pc 0 . It follows hence that the area of the hysteresis loop in the case of rapid compression increases and an additional dissipation mechanism apparently comes into play. The flow of the liquid in pores occurs under a considerable excess of pressure p0 over percolation transition pressure pc 0 , ( p0 – pc 0 ) / pc 0  0.2, when more than 70% of pores are accessible

12

V. D. Borman and V. N. Tronin

to infiltration of the nonwetting liquid. In this case, we can naturally assume that the additional energy dissipation is associated with energy losses in the flow of the viscous liquid in nanopores. To verify this assumption, we performed experiments in which the p(V), p(t), and V(t) dependences were investigated for the L23 + CaCl2 system under fast and slow compression at temperatures varying from 258 to 323 K. The viscosity  of the CaCl2 solution varies by a factor of 5 in this temperature range [30]. However, surface energy  of the solution [31] and surface energy  of the interface between the porous media and the liquid change

upon variation in temperature. These quantities control pressure pc 0 (  ,  ) which determines the percolation threshold of infiltration of the solid. For 3D systems, the percolation threshold is determined by the fraction of pores accessible to infiltration,

 ( Pco)   co  0.18 [3, 9, 21, 25,26]. Here,  ( pc 0 ) 



4    R 3  f ( R)  dR 3 R ( pco )

R( p c 0 ) 





    1  1      pc 0   

f(R) is the size distribution of pores, and  is the ratio of the areas of the menisci and of the pore. This relation was obtained for an infinitely large porous media. For a real porous media with a mean pore radius R and a grain size L, the value of pc 0 changes by approximately

R L  102  103 , which is smaller than the maximum experimental error. It was found that in the temperature interval 258-323 K, the value of pc 0 (and, hence, the values of  and  ) do not change to within the measurement error. In the time interval from 0 to tin, the values of p(t), as well as of V (t ) , obtained for different temperatures for impact energy E = 40±2 J coincide to within the experimental error (Figure 5).

Figure 5. Time dependences of (a) pressure and (b) variation of volume for the L23 + CaCl2 system

with viscosity ranging from 1.27 × 10–3 to   7.13 10 Pa s. The curves in the temperature interval from 260 to 323 K coincide to within experimental error 3

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

13

It follows hence that for the L23 + CaCl2 system under investigation, the infiltration dynamics in the temperature range 258-323 K and the flow of liquid in nanopores are independent of the viscosity of the liquid. Thus, for a pressurization rate of p > 104 atm/s in the systems studied here, infiltration of nanopores of a porous media occurs at a constant pressure p0 , which is higher than percolation transition pressure pc 0 . Pressure p0 weakly depends on the impact energy, exhibiting a tendency toward growth within the experimental error (see Figure 3a). The energy dependence of the filled volume of pores and infiltration time are close to linear dependences to within the experimental error (see Figures 3a and 3b), and the mean flux of the liquid in pores is independent of the impact energy. During infiltration, additional dissipation energy is observed as compared to slow infiltration; however, dependences p(t) and V (t ) do not change with temperature or upon a fivefold variation in the viscosity coefficient of the liquid (see Figure 5). It has also been established that the relative amplitude of oscillations of the volume during infiltration is considerably smaller than the relative amplitude of pressure oscillations.

3. MODEL OF INFILTRATION DYNAMICS FOR A POROUS MEDIA 3.1. Formulation of the Problem Let us consider the dynamics of infiltration of grains in a disordered nanoporous media containing pores of different sizes and immersed in a nonwetting liquid. We assume that infiltration in grains occurs independently. At the initial instant, pores in each grain are empty and the liquid pressure is zero. As the pressure increases and attains a critical value, infiltration begins in the grains of the porous media. The problem involves the calculation of the time dependence of a filled volume, V(t), at a preset pressure p(t) with a characteristic time  p of increasing pressure for various relations between this time and the characteristic hydrodynamic time of infiltration of the porous media. Speaking of infiltration in the porous media, we will henceforth mean in all cases the infiltration of one of its grains, unless the opposite is specially stipulated. Obviously, infiltration may occur in a grain only if the pores form a connected system in it. Porosity  defined as the ratio of the pore volume to the volume of a grain in the porous media must be such that the fraction of connected pores is considerably larger than the fraction of pores that do not belong to the connected system. If size L of the grains of the porous material is much than the maximum size of the pores, the characteristics of a grain of the porous media are indistinguishable from the characteristics of an infinitely large media to within R/L ~ 10-4-10-2. In this case, infiltration through all pores of the gain may occur only when porosity

 exceeds percolation threshold  c , which is a

characteristic of an infinitely large porous media. For 3D systems, the percolation threshold is

c  0.18 [2,25,26]; the connectivity of pores with one another in this case is the result of

the formation of infinitely large clusters of pores for

  c . For porous media with porosity

14

V. D. Borman and V. N. Tronin

 near percolation threshold  c , this cluster is strongly rarefied and contains only about 1% of the total number of pores in the porous media [25,26, 32]. For porous media with a

  c , the number of pores in the infinitely large cluster increases with  , attaining a value of 100% for   1 . porosity

Figure 6 shows the probability that a pore belongs to the infinitely large cluster as a function of porosity

 . It can be seen that with increasing porosity, for   c , the value

of P ( ) tends to unity and, hence, the space of pores in a grain becomes nearly homogeneous due to the growth of the infinitely large cluster of pores. We will henceforth assume that

  c .

Figure 6. Probability P that a pore belongs to an infinitely large cluster as a function of porosity



 fraction 

or

(fraction

of accessible pores). Probability P is normalized to the total number of porosity



The infiltration of a nonwetting liquid under a pressure p in a porous media requires that a certain amount of work be done. For this purpose, it is necessary to overcome a certain critical pressure, which is the Laplace pressure pc ( R) ~  R for an isolated pore (which is assumed for simplicity to be spherical), where R is the pore radius,   

sl

  sg ,  sl

and  being the surface energies of the interfaces between the solid and the liquid and the solid and the gas, respectively. An empty pore in a porous media may be, depending on its radius, in one of two possible states (either accessible or inaccessible to the infiltrating liquid at a given pressure p). The probability that the pore is in these states can be written in the form [21] sg

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

wi  p, R   1  exp  A p, R  T   

15

1

(1)

where

 A  p, R    p 

3   R

    1      1     

A p, R is the work that must be done to fill a pore of radius R with a liquid under pressure p; T is the temperature; and  is the surface energy of the liquid,  is the ratio of the meniscus surface area to the pore surface area. It can be seen from expression (1) that if

A p, R  0 , then probability w ~ 1 and a pore can be filled with the liquid; if, however, A p, R  0 , we have w  0 and the pore becomes inaccessible. Consequently, the homogeneous space of pores with various sizes during infiltration at a preset pressure is divided into pores that can be filled, A p, R  0 (accessible pores) and pores that cannot be filled,

A p, R  0

(inaccessible pores). Thus,

we can assume that the medium subjected to infiltration is a heterogeneous medium consisting of accessible and inaccessible pores playing the role of white and black spheres, respectively, in percolation theory [25]. Such a medium can experience a percolation transition occurring via the formation of clusters of accessible pores followed by infiltration of a nonwetting liquid in such formations. In the general case, percolation threshold  accessible pores does not coincide with

c0

for

 c . However, for   c , in view of the

homogeneity of the pore space, we can consider pores together with the skeleton material surrounding them (thick-wall pores) and analyze percolation through these pores. In this case, the percolation threshold for accessible pores and for porosity obviously coincide ( 

c0

=  c ).

In a porous media, pores are in contact. For this reason, the value of pc 0 ( R) defined by the condition A p, R  0 is determined by the contacts of a given pore with its neighbors and, hence, on fraction  of menisci. Consequently, we can define the pores accessible at such a

pressure p as pores whose radii satisfy the condition pc 0 ( R)  p . Upon a change in pressure, some of the formerly inaccessible pores become accessible and are filled with the liquid (if it can reach them). The approach of the liquid flow to the given pore is governed by percolation theory and occurs via the formation of accessible pore clusters both of a finite and an infinitely large size [9, 21]. Thus, the dynamics of infiltration in a grain of a porous media can be represented as the formation of the medium for infiltration (i.e., a system of clusters of accessible pores followed by infiltration in a part of these clusters). Since the infiltration in a grain of the porous media detected experimentally occurs when percolation length   R    c 0

v

(

v  0.8 ) becomes comparable to grain size L or exceeds it   L ), infiltration in the grain

16

V. D. Borman and V. N. Tronin

can be treated as a uniform process occurring simultaneously in the entire pore space of the grain and resulting in the formation of clusters of filled pores. Thus, the problem of infiltration of a porous media can be formulated as the problem of calculating the coordinate-independent distribution functions for clusters of accessible and filled pores over the number of pores, followed by calculation of volume V(t) of the liquid in the porous media under pressure p(t). As before [8], we assume that the size distribution for pores is narrow ( R R  1 ) so that the percolation transition is independent of R R

3.2. Basic Equations The times in which accessible and filled pores form are substantially different. Indeed, in accordance with expression (1), the formation of accessible pores is controlled by the time of pressure variation in the system, while the time of filling is the hydrodynamic time of infiltration of the liquid through the clusters of accessible pores. These times may differ by orders of magnitude; for this reason, the pores accessible at instant t can be divided into accessible and filled and accessible but unfilled. Consequently, to describe the infiltration dynamics, it is necessary to trace the formation processes of clusters of accessible pores and clusters of filled pores separately. In deriving kinetic equations for distribution functions f (n, t ) and F (n, t ) of accessible and filled pores, we will assume that the transformation of an accessible pore into a filled one only leads to the disappearance of the accessible pore (i.e., the infiltrated medium does not change in the course of filling). It should be noted that the change in the medium being infiltrated will be taken into account below as the filled volume is calculated in the mean field approximation. The formation of clusters in the problem of spheres (black and white spheres) was described in [34], where the distribution function for clusters of white spheres over the number of spheres in these clusters was introduced. A change in the distribution function in this model occurs as a result of coalescence of clusters of white spheres. Following [33], we will describe the dynamics of infiltration of the liquid in a grain assuming that the medium for infiltration is inhomogeneous and consists of accessible and inaccessible pores. In this case, accessible pores play the role of white spheres and their fraction is defined as 

 ( p)   w( R, p)dRf r ( R) R 3

(2)

0

where f r ( R) is the size distribution function for pores and quantity of w( R, p ) is defined by relation (1). In describing the dynamics of infiltration of a non-wetting liquid in a porous media, pressure is a function of time; consequently,  also depends on time. Bearing this in mind, we can write the system of kinetic equations defining the time evolution of the distribution functions for clusters of accessible and filled pores over the number of pores in them in the form

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

17

 f (n  m, t ) f (m, t ) F (n, t ) n1 S ( (t ))   F (m, t )   F (n, t )  F (n, t ) t  (m, n  m) m1  (n, m)  pc (n) m 1

(3)

 f (n, t ) 1 1 n 1 q  {  m (n  m) q f (m, t ) f (n  m, t )  n q f (n, t )  m q f (m, t ) 2n q f (n, t ) S ( )}  t  d 2 m1 m 1 (4) n 1

 F (m, t ) m 1

 f (n  m, t ) f (m, t ) S ( (t ))   F (n, t )  F (n, t )  (m, n  m) m1  (n, m)  pc (n)

where

S ( )    (   c ) ,     c 0 ,  d  (

 1 dp 1 )  ( (t ))1  p ,  p  ( ) ; (5) t pdt

 p is the characteristic time of pressure variation;  pc is the characteristic percolation time for an infinitely large cluster of accessible pores from filled clusters,

 d has the meaning

of the characteristic time in which accessible pores form upon time variation of pressure; q,  , and  are critical indices ( q  0.8 ,   0.2 [33] and   0.6 for 3D systems [8]);

S ( (t )) is the effective part of an infinitely large cluster of accessible pores (i.e., the fraction of pores belonging to the infinitely large cluster and accessible to infiltration); and ( x ) is the Heaviside function. Equation (3) defines the distribution function for clusters of filled pores at an arbitrary instant. The first term describes the formation process of a cluster of n pores as a result of infiltration into clusters of n - m accessible pores via clusters of m filled pores over characteristic time

 (m, n  m) . The second term corresponds to the attachment of any

cluster of accessible pores to the cluster of n filled pores during infiltration over characteristic

 (n, m) . The third term describes infiltration of the infinitely large cluster of accessible pores from filled clusters over characteristic time  pc (n) . Equation (3) disregards the

time

variation of distribution function F ( n, t ) due to coalescence of clusters of filled pores with one another, which corresponds to the assumption of invariability of the medium in the course of infiltration. Function F ( n, t ) for a nearly complete infiltration will be calculated below in the mean field approximation. Equation (4) defines the time evolution of the distribution function for accessible-pore clusters due to their coalescence with one another (first two terms), attachment to the infinitely large cluster (third term), and infiltration-defiltration of the liquid from these clusters (three last term).Times

 (n, m) and  pc (n) appearing in Eqs. (3) and (4) can be

estimated from the following considerations. Let V(m) be the volume of a cluster of m accessible pores, V(n) be the volume of a cluster of n filled pores, j(n) be the flux from n filled pores, S(n, m) be the area of contact between clusters of m accessible and n filled pores, and S ( n) be the area of the contact of the cluster of n pores with the infinitely large cluster.

18

V. D. Borman and V. N. Tronin

Then we can write

 (n, m) 

V (m) V ( n) ,  pc (n)  hese quantities depend j (n) S (n, m) j ( n) S ( n)

on the size distribution of pores. Since we are interested only in the dependences of times  (n, m) and  pc (n) on the number of filled and accessible pores in the clusters, we will estimate the values of these quantities assuming that all pores in a cluster are of the same size coinciding with the average size of a pore in the porous media ( R ). In this case, we have

V ( m) 

4 3  R m , S (n, m)  4 R 2 (nm)q , S (n)  4 R 2 nq (q' is the critical index). Using the 3

known expression for the flux in a porous medium, j  kn p

 L ( kn is the penetration

factor of the medium ,~ -viscosity) [28], we obtain

 (n, m)   0 ( p)n  q m1q  pc (n)   0 ( p)n  q1

,

(6)

where

 0 ( p) 

4 RL 3kn ( p  pc 0 ( R ))





and pressure pc 0 ( R ) ~  R is defined by the condition  A pc ( R ), R  0 Equations (3) and (4) allow us to calculate the distribution functions for clusters of accessible and filled pores over the number of pores in them for a preset variation of pressure p(t). Equation (4) contains the terms with essentially different physical meaning. The first three terms in kinetic equation (4) cannot be interpreted as a collision integral since these terms vary only with    (t ) and p(t). These terms are on the order of

 d proportional to

 p , which is not the intrinsic time of the system, and reflect the variation of distribution f (n, t ) of accessible pores only upon the variation of pressure and, as a consequence, of quantity    (t ) . If   const , these terms are equal to zero. For function

   (t ) , these terms must appear in Eq. (4) simultaneously with (f  )(d dt ) . Thus, derivative f t on the left-hand side of Eq. (4), as well as derivative F t , defines the variation of distribution functions f (n,  (t ), t ) and F (n,  (t ), t ) due to the change in the external pressure and due to infiltration–defiltration of the liquid through accessible pores. Equations (3) and (4) contain an integral of motion corresponding to the conservation of the total number of pores accessible to infiltration taking into account the fact that part of these pores have already been filled. Indeed, multiplying Eqs. (3) and (4) by n, summing over n, and adding the resultant expression, we obtain

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid    d ( nF (n, t )   nf (n, t ))   t n 1 dt n 1



n

q 1

f (n, t ) S ( )

19 (7)

n 1

We can write probability P ( ) that an accessible pore belongs to the infinitely large cluster as  P ( )   d  n q 1 f (n,  ) S ( )  n 1

where

(8)

 d is the fraction of accessible by unfilled pores. Relation (8) is analogous to the

expression derived in [33] for the problem of spheres in percolation theory. Considering that the distribution functions for clusters of accessible and filled pores depend on time both explicitly and due to the change in pressure (and, hence, in quantity  (t ) , using expression (8), and setting

 (0)  0 , we obtain 



n 1

n 1

 nF (n, t )   nf (n, t )   ( p(t ));

(9)

This relation corresponds to conservation of the total number of pores accessible to infiltration under pressure p at instant t. In deriving Eq. (9), we used the normalization of function f (n, t ) taking into account the fact that some of accessible pores may belong to an infinitely large cluster, 

 nf (n, t )   n 1

d

(1  P( ))

In this case, distribution function F (n, t ) for clusters of filled pores is normalized to the total number of filled pores (including the filled pores formed from the infinitely large cluster of accessible pores). Equations (3), (4), and (9) contain the times corresponding to different processes occurring during infiltration of a porous media: characteristic time  p of variation of external pressure, characteristic time

 d of the formation of accessible pores, characteristic time

 z ~  (n, m) of the formation of the cluster of filled pores (angle brackets denote averaging over the ensemble of clusters of accessible and filled pores), characteristic time

  ~   (n) of defiltration of the liquid to the infinitely large cluster of accessible and empty pores, and characteristic time  v ~ ( volume. For 3D systems,



 nF (n, t )

t ) 1 of variation of the total filled

n

c 0  0.18 and  ~ 0.6 ; consequently,  p   d in all cases in

20

V. D. Borman and V. N. Tronin

accordance with relation (5). Since infiltration of the volume occurs due to variation of the external pressure, we have  v  max( d , z ) .

We will consider two cases corresponding to slow (  p   v   z   d ) and fast (

 v   z   p   d ) variations of pressure. The solutions to systems of equations (3), (4), and (9) are significantly different in these cases.

3.3. Kinetics of Infiltration for Slow Variation of Pressure Let us consider the case of a slow variation in pressure, when  p   v   z   d . We will be interested in infiltration of a porous media over time intervals t ~  v and will calculate the time dependence of the filled volume under pressure p. In Eq. (4), the first term on the right-hand side plays the leading role since it is on the order of  d1 , while the second term is on the order of  z   d . Since  p   z , a change in pressure rapidly leads to the 1

1

formation of accessible pores (over time intervals t >  d ) followed by infiltration of the liquid (over time intervals t   z ). In accordance with relation (9), the fraction of accessible pores decreases upon infiltration. An increase in pressure leads to the formation of and filling of pores that have become accessible. In view of condition  p   z , infiltration of the solid media upon a slow variation in pressure occurs near the percolation threshold over accessible pores, remaining below this threshold. For this reason, S ( )  0 in Eq. (4), and the terms containing distribution function F (n, t ) for filled pores (and, hence quantity  z   p ) are small as compared to the terms containing

 d and can be discarded. In this case, Eq. (4)

assumes the form of the equation used in [33]; the solution to this equation is known:

f 0 (n, t ) 

C(t)n (t) , Z (t )

n (t )  n  exp(r (t )

1/ a

n),

(10)



Z (t )   n n (t ) n 1

Here, function C(t) is controlled by the normalization of distribution f 0 (n, t ) , varies over time intervals t ~  v and determines the filled volume. The critical indices for 3D systems are given by [24, 25, 33]

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

21

1

1 2   2.2 , a  0.9 , r   u  q (1  u ) q du ,q=0.83 20

(11)

Distribution function F0 (n,  (t )) for    c 0 is defined over time intervals

t ~  v , p   v   z   d by the steady-state solution to Eq. (3) in the absence of an infinitely large cluster of accessible pores, S ( )  0 .In the continual limit this equation can be solved [36]. For a slow variation in pressure, the distribution function for filled pores is proportional to distribution function (10) for accessible pores [36]:

F0 (n,  (t ))  C1 (t ) F0 , F0 =

n (t ) Z (t )

(12)

where  n (t ) and Z (t ) are defined in (10). Function C1 (t ) varies over time intervals

t ~  v and controls the variation of the filled volume. It should be noted that distribution function (12) for filled pores was used by us earlier to describe experiments on infiltration of a nonwetting liquid through a porous media upon a slow variation in pressure [20]. We will derive the time dependence of the fraction of filled pores in the given case using relation (9). Substituting relations (10) and (12) into (9), we obtain

C (t )  C1 (t )   ( p(t ))

(13)

On the other hand, substituting relations (10) and (12) into Eq. (4) and considering that

 v   z   d , we obtain

C (t )C1 (t ) dC  dt v

(14)

 f0 (n  m) f (m)   nF0 (n)( mF0 (m)   F0 (n) 0 )  v n1  (m, n  m) m1  (n, m) m1

(15)

Where [36]

1



n 1

Figure 7 shows the dependence of

 v  z on  in the vicinity of percolation threshold

 c 0 , which was calculated using relation (15) for q  0.83 and a  1 . This dependence is successfully approximated by the expression

v 1   z (1    c 0 ) 

(16)

22

V. D. Borman and V. N. Tronin

where

  0.4 . Thus, the value of  v  z is always greater than unity and  v  z   near

the percolation threshold. (Figure 7).

Figure 7. Dependence of relative infiltration time

 v  z on fraction 

of accessible pores for slow (I)

and fast (III) infiltration (infiltration in domain II is impossible). The solid and dot-and-dash curves correspond to dependences (16) and (38), respectively

3.4. Kinetics of Infiltration under a Fast Variation of Pressure Let us consider fast variation in pressure, when  v   z   p   d . In this case, we will be interested, as before, in the behavior of the infiltrated porous media over time intervals

t ~  v . As in the case of slow infiltration, the first term on the right-hand side of Eq. (4) plays

the major role since it is on the order of  d , while the second term is on the order of 1

 z1   d1 . Since  p   z , a change in pressure leads to rapid formation of accessible pores (over time intervals t ~  d ). For times

 z  t , filled pores are absent. Thus, over time

intervals  z  t   p   d , system of equations (3), (4) assumes the form  f 1 n1 q   m (n  m)q f (m,  ) f (n  m,  )  n q f (n,  ) mq f (m,  ) 2n q f (n,  ) S ( );  2 m1 m 1

F (n, t )  0

(17)

(18)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

23

It can be seen that over time intervals t satisfying the inequality  v   z  t   p   d , the formed accessible pores now have time to be infiltrated; as a result, the porous media is in a state above the percolation threshold over accessible pores for    c 0 with

F (n)  f (n) . Over time intervals  v  t   z   p   d , the process of infiltration of the porous media begins in accordance with Eqs. (3) and (4) (in these equations, the effective part of the infinitely large cluster of accessible pores is S ( )  0 . Over these time intervals, in view of condition t   z   d , the time derivative in Eq. (4) can be set at

(d dt )(f  ) . By virtue of condition  z   d , the sums of the terms in Eq. (4)

 d  z In this case, Eq. (3) is satisfied automatically. Thus, over time intervals t such that  v  t   z   p   d , the containing F (n, t ) is zero in the zeroth and first orders in

equation for distribution function f (n, t ) for accessible pores coincides with the first equation of system (17),(18), while the equation for F (n, t ) assumes the form n 1

[ F (m)m q (n  m) q 1 f (n  m,  )  F (n)n

q

m 1

The equation for f (n, t ) at S ( )  0 near

1  dc

m

q 1

f (m,  )

m 1

  F (n)n q1S ( )  0

(19)

 c 0 (   c 0 ) has a solution differing from

f0 (n, t ) in Eq. (10) only in the value of critical index a [33]. Function C(t) appearing in Eq. (10) controls the variation of the filled volume and varies over time intervals t ~  v ; consequently, this function can be assumed to be constant for

 v  t   z   d .

Equation (19) with known distribution function (10) for accessible pores is a homogeneous equation for function F ( n) . A nonzero solution to this equation exists only if the determinant of matrix Anm vanishes:

det( Anm )  0 

Anm   nm (n  m) q 1 f 0 (n  m,  )m q   nm (m q  k q 1 f 0 (k ,  )  mq1S ( )) k 1

 nm

1, n  m  0, n  m

(20)

Matrix Anm has the form of a triangular matrix with zeros above the principal diagonal. The determinant of such a matrix is equal to the product of the diagonal elements,

24

V. D. Borman and V. N. Tronin 

det A   (1) m (m q  k q 1 f 0 (k ,  )  m q1S ( )) k 1

m

and does not vanish. Consequently, Eq. (20) has no solutions for finite values of n and m. For n  and m  , the contact areas of two clusters are controlled by a single critical index; consequently, q  q  1 . Passing in Eq. (20) from summation to integration, considering that f0 (n  m)

n~ m

 (n  m) , and setting 

lim k q 1 f 0 (k ,  )  2 (k )  dxx q 1 f 0 ( x,  ) k 0

0

we obtain from Eq. (20) 



0

1

lim Anm  lim  nm m q (2  dxx q 1 f 0 ( x,  )   dxx q 1 f 0 ( x,  )  S ( ))

n , m 

where

n , m 

(21)

 nm is the Kronecker delta. This leads to the following equation that defines the value

of  c corresponding to a nonzero distribution function for filled pores: 



0

1

2 dxx q 1 f 0 ( x,  )   dxx q 1 f 0 ( x,  )  S ( )  0

(22)

Expressions (2), (5),(10), and (22) for S ( )  0 show that the value of  c is controlled by percolation threshold  c 0 and the critical indices appearing in Eq. (22). If function

f 0 ( x,  ) is defined by expressions (10), the integrals appearing in Eq. (22) can be expressed in terms of the gamma function and the Whittaker functions [35]. In this case, solving numerically Eq. (22) for q  0.83, a  0.9 [34], and

c 0  0.18 , we obtain  c =0.28.

Thus, Eq. (19) has the solution F (n)  0 for c 0    c and F (n)  0 for   c . Consequently, we can state that a new state of the system is formed for   c over time intervals  z   p   d for t   p   d . Further infiltration of the porous media over time intervals t ~  v may occur via its passage to this state, which emerges in the case under investigation due to the infinitely large cluster of accessible pores. It follows from Eq. (3) that pressure pc corresponding to the transition of the porous media to the new state can be determined, in accordance with formula (2), from the relation 

 w( R, p )dRf ( R) R = 3

c

0

r

c

(23)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

25

It follows from relations (1) and (23) that pressure pc , in contrast to  c , depends on the characteristics of the porous media and the liquid, such as the size distribution function for pores, the surface energies of the liquid and porous media, and coefficient  |. By way of example, Figure 8 shows the dependence of pc / pc 0 on half-width R R of the size distribution of pores at

   1 3 ,   1.2 . Pressure pc 0 in this case can be found from

the relation 

 w( R, p

c0

)dRf r ( R) R 3 =c0

0

Figure 8. Dependence of ratio

pc pc 0

of threshold pressures on halfwidth

R R of the size

distribution of pores.

Figure 8 shows that pressure pc corresponding to the transition of the porous media to the new state is higher than pc 0 and increases with the halfwidth of the size distribution of pores; pc  pc 0 for R R  0 . Let us now derive the equation describing the time dependence of the fraction of volume of pores filled with the liquid in the case of infiltration of a porous media in the vicinity of  c . For this purpose, we write Eq. (4) in the form

F (n,  , t ) 1    A ( )F (m,  , t ) t  0 ( p) m1 nm

(24)

26

V. D. Borman and V. N. Tronin Matrix Anm is defined by relation (20) and its eigenvalues are defined by the equation

det( Anm   nm )  0 . For finite values of n and m, we have 

det( Anm   nm )   (  (m q  k q 1 f 0 (k ,  )  m q1S ( )))

(25)

k 1

m

and, hence, the eigenvalues of matrix Anm are negative for finite n and m. For n  and

m  , we obtain, in accordance with relation (21) 



0

1

   ( )  mq (2  dxx q 1 f 0 ( x,  )   dxx q 1 f 0 ( x,  )  S ( ))  z (   c ) (26) Angle brackets indicate averaging over an ensemble of clusters for m  1 , and z and  are constants. Numerical calculations for q  0.83, a  1 , and   0.2 give z  0.8 and

  0.8 . Thus, the spectrum of eigenvalues of matrix Anm for n  and m  acquires a small (in the vicinity of

  c ) positive eigenvalue corresponding to relaxation

time   ~ (   c )  z while the remaining eigenvalues are finite for 

are on the order of 

  c , negative, and

1 z .

Using relation (26), we can write Eq. (24) in the form

F (n)  ( )  F (n)   Anm F (m) t  0 ( p) m

(27)

Matrix

Anm 

has eigenvalues that

 ( n)  0 ,

 ( ) 1 Anm     0 ( p)  0 ( p) nm

 ( n) ~

1

z

, which do not vanish for  =  c . Considering

  d    , we obtain the equation t t dt  F (n,  ) d  F (n,  )  ( )   F (n)   Anm F (m,  ) t dt   0 ( p) m

containing terms varying over substantially different time intervals:

(28)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

F F ~ , t  z

A

nm

F (m,  ) ~  (n) F ~

m

27

F

z

For time intervals t ~  v , we can obtain the following estimate:

F (n, t ) ~

F



~

F

 c

, c  pc (

p  p

,

(29)

) p  pc

 ) p  pc is calculated for pressure pc defined by relation (23), which p p 1  1 shows that  ( c )   c   c 0 ~ 3 , x  c  1 . Consequently, pc ( ) p  pc ~ 5 and, x pc 0 p x Compressibility (

5  hence, pc ( ) p  pc ~ ( (c )) 3 . Since the value of  c for which the new state of the porous p media being infiltrated is formed is higher than  c 0 , we have pc  pc 0 and   pc ( ) p  pc ~ 102 . Therefore,  c ~  p 3   p , for   1 . Using these estimates in the p  zeroth and first orders in z and Eq. (28), we obtain c 5

d  F (n,  )  ( )  F ( n,  ) dt   0 ( p)

(30)

F (n,  )   Anm F (m,  ) t m

(31)

In fact, the procedure described above corresponds to obtaining a solution to Eq. (27) by expanding it in the eigenfunctions of operator A. Equations (30) and (31) describe substantially different processes. Equation (31) describes the kinetics of formation of finitesize clusters of filled pores over time interval  z   0  around an infinitely large cluster, 1

while Eq. (30) describes a slow ―macroscopic‖ infiltration of the liquid into an infinitely large clusters of accessible pores through finite-size clusters of filled pores over time intervals

 v ~  0 (   c )    z for  ~ c (  ~ 0.8 ). The left-hand side of Eq. (30) describes the variation in the distribution function for filled pores as a result of external action, while the right-hand side describes the variation in distribution function F (n,  ) as a result of infiltration through an infinitely large cluster of accessible pores. Consequently, Eq. (30) also

28

V. D. Borman and V. N. Tronin

corresponds to the condition of compensation of the external action by the system, according to which the change in the distribution function for filled pores due to the external action is compensated by the response of the system in the form of flow of the liquid to an infinitely large cluster of empty pores. Analysis of Eq. (31) taking into account the change in the distribution function for accessible pores will be carried out below when we will consider oscillating modes of infiltration. Equation (30) makes it possible to determine fraction

0 of the pores for which

infiltration of a porous media can be initiated. It follows from Eq. (30) that the following estimate is valid in the case considered here:

F



 c1 

 ( ) F  0 ( p)

(32)

Relations (26), (30), and (32) lead to the conclusion that the value of

0 for which

infiltration of the porous media may begin is defined by the relation

( ( c )) which shows that fraction



2 3

2 3

 0 ( p)  z ( 0   c ) p

(33)

0 of pores is defined by rate  p1 of the pressure growth. Since

0  1 , the value of 0 is close to c . Pressure p0 corresponding to the beginning of p

infiltration is defined by the relation analogous to (23): 

 w( R, p )dRf ( R) R = 3

0

r

0

(34)

0

Thus, Eq. (33) makes it possible to find the fraction of pores for which the system compensates the external action, and infiltration of the liquid in the porous media begins, leading to a macroscopic change in the volume of the liquid in the porous media The change in the volume of the system consisting of a porous media and a liquid occurs over time intervals t ~  v   z   d due to infiltration of the liquid into an infinitely large

cluster of accessible pores through finite-size clusters of filled pores. We will derive an equation describing the time dependence of the fraction of volume of pores filled with the liquid in the vicinity of  0 . For this, we write distribution function F (n, t ) in the form

F (n, t )  x(t ) F1 (n, t ) .

(35)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

29

Here, quantity x(t ) varies over time intervals t ~  v   z , while the variation of

F1 (n, t ) occurs over time intervals t ~  z   v . Since the new stationary state appears for

 c >  c 0 , we will calculate the filled volume assuming that the space of accessible pores is homogeneous and all pores of the porous media are accessible to infiltration. Using relation (9), we normalize the distribution functions for accessible and filled pores to the total volume of accessible pores, assuming that it is equal to unity. This is due to the fact that the value of p0 increases with the compression energy (see below) so that all pores in the porous media become accessible and   1 . It follows from Eq. (9) that with such a normalization, all pores accessible to infiltration (including those belonging to the infinitely large cluster) are taken into account. In this case, quantity x(t) is the fraction of filled pores at instant t. 

Assuming that function F1 (n, t ) is normalized to unity (

 nF (n, t )  1), we can represent n 1

1

the distribution function for accessible pores in the form

f (n)  (1  x(t )) f 0 (n,     c ), 

 nf (n,    n 1

0

(36)

 c )  1

Substituting expressions (35) and (36) into Eq. (4) and considering that, by virtue of (31), the value of

  0  c  const over time intervals t ~  v   z   d (and, hence,

F1  0 ), we obtain  F1

F x(1  x) dx x(1  x) x 1   ( 0 ) F1 (n)   A F ( m) dt t  0 ( p)  0 ( p) m nm

(37)

Consequently, for t ~  v   z , using expression (30), we obtain from relation (37)

 ( p) dx x(1  x)  , v  0 dt v  ( 0 )

(38)

Using relations (26) and (33), we obtain the characteristic volume infiltration time

v 

2   0 ( p)  ( ( 0 )) 3  p  (0 )

(39)

30

V. D. Borman and V. N. Tronin It follows hence that if condition (30) describing the compensation of an external action

by the system is satisfied, characteristic volume infiltration time characteristic time  p of pressure growth and by the difference

 v is controlled by

0  c ; consequently, it is

independent of the viscosity of the liquid. Thus, in the case of fast variation of pressure (  z   p   d ), infiltration of the porous media occurs via rapid infiltration of finite-size clusters, occurring simultaneously in the

entire volume of the grain (over time intervals t ~  z ) and slow infiltration (over time intervals t ~  v   z ) of the liquid into the infinitely large cluster of accessible pores through finite-size clusters of filled pores. As a result, the new state of the system consisting

of the nonwetting liquid and the porous media is formed over time periods t   v as a result of the nonlinear response of the system to the external action. Equation (38) describing the infiltration of a porous media was proposed phenomenologically in [28].

3.5. Oscillating Modes of Infiltration System of equations (4), (5) for

  c may have solutions oscillating with time. To

prove this formally, we will seek the solution to system (4), (5) for

  0 in the form

F (n, t )  F0 (n)   F (n, t ),  F ( n, t )  F0 ;

(40)

f (n, t )  f 0 (n)   f (n, t ),  f (n, t )  f 0 ; Here, f 0 ( n) and F0 (n) are defined by formulas (10) and (12),

F0 (n) 

( c  Y ) n (t ) Y n (t ) , f 0 ( n)  Z Z (t )

(41)

where Y is the fraction of filled pores. Substituting expressions (40) into Eqs. (4) and (5) and using the tau-approximation, in the first order in deviations  F ( n, t ) and  f ( n, t ) , we obtain

 f (.n, t )  

f 0 ( n)

 F (n, t )  

F0 (n)

d

z

 f (n, t ) 

F0 (n)

 f (n, t ) 

f 0 ( n)

z

z

 f (n, t ) 

 F (n, t ) 

f 0 (n)

z

S ( )



 F (n, t ) 

 F (n, t ),

S ( )

 pc

 F (n, t ) (42)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid where

31

  0  c 0 . The equation for  F (n, t ) corresponds to Eq. (30) written in the 

approximation. System of linear equations (42) has oscillating solutions if the eigenvalues of the matrix of this system contain imaginary parts. This is the case when the condition

A G AG G 2 (1  )2  2  2 2  2  0 B B B B

(43)

is satisfied, where

A

F0 (n)

z

,B 

f 0 ( n)

d

,G 

f 0 (n)

z



S ( )



It follows from relation (43) that the condition for the emergence of oscillations can be satisfied only when A  0, B  0 , and G  0 . Assuming that  pc ~  z and using relation (43), we obtain the following condition for the existence of oscillating modes of infiltration of a porous media: W ( x, Y ,  , n)  (1  x

Y 2 S ( ) Y S ( ) S ( ) 2 )  2 x(1  )  2 x 2 (1  )  x2 (1  )  0 (44)  Y f 0 (n)  Y f 0 ( n) f 0 ( n)

where x   d

 z . For preset values of the total fraction  of filled and accessible pores, fraction Y of filled pores, and ratio of times x   d  z  1 , inequality (44) defines the number of pores constituting the cluster whose infiltration-defiltration results in the emergence of an oscillating mode. The solution of linear system of equations (42) is cumbersome and is not given here. Characteristic period T0 of oscillations coincides in order of magnitude with the infiltration-defiltration characteristic time for a cluster of filled pores. Analysis of condition (44) shows that in the case of slow infiltration (  v   p   z   d ),

  c 0 and S ( )  0 , oscillations are absent. In the case of rapid infiltration (  v   z  t   p   d ), in the vicinity of the transition to the new state, we have  ~ 0 , S ( )  0 , and the fraction of filled pores is small ( Y  1 ). In this case, condition (44) can

when

be satisfied for a certain value of number n. Figure 9 shows a typical graph of function W ( x, Y , n) and eigenvalues of the matrix of Eqs. (42) as functions of number of pores n in the cluster. It can be seen that the real parts of the eigenvalues of the matrix of system (42) are negative for any number of pores in the cluster. Condition (44) is satisfied for a cluster consisting of n  nk  2 pores; for n  2 , the eigenvalues of the matrix of system of equations (42) acquire an imaginary part that vanishes for n  2 (see Figure 9). The existence of the imaginary part in the eigenvalues of the matrix of system (42) corresponds to the emergence of an oscillating infiltration mode. Consequently, for x  0.01,

32

V. D. Borman and V. N. Tronin

Y  0.01 , and 0  0.28 , an oscillating mode appears as a result of infiltration-defiltration of a cluster containing n ~ 2 pores. It follows from expression (6) that this process occurs over time intervals of approximately 2 0 ; consequently, when the conditions of fast infiltration are satisfied, oscillations with a period T0 ~ 2 0 may appear and will accompany infiltration of the porous media. Substituting Eq. (40) into (9), taking into consideration that condition (44) is satisfied for

n ~ nk  2 , and retaining the oscillating part, we obtain

 (t )  0   (t ),  (t )  0  (t )  nk ( f (nk , t )   F (nk , t )

(45)

W ( x, Y , n) , as well as (b) of real and (c) imaginary parts of eigenvalues of matrix (44) on number n of pores in a cluster, plotted for x  0.01, Y  0.01 , and 0  c  0.28 Figure 9. Dependences (a) of function

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

33

For pressure oscillations, we obtain from Eq. (3) 

w( R, pc ) dRf r ( R) R 3 ) -1  p 0

 p(t )   (t )( 

(46)

 f (nk , t ) and  F (nk , t ) of the distribution

Thus, the emergence of oscillations

functions leads to the emergence of pressure oscillations with a period T0 ~ 2 0 . Let us find the time dependence of the volume in an oscillating regime. For this purpose, we substitute Eqs. (45) into Eq. (38). The solution of the resultant equation has the form

0

x(t )  1

0  x(0) x(0)

exp(

1

v

t

 (

0

(47)

 (1   ( 0   c )

 1

) (t )) dt )

0

Here, x (0) is the fraction of pores filled by the instant of passage of the system to a new state and quantity

 v is defined by relation (31). Since the characteristic period of oscillations

is T0 ~ 2 0   v , the value of

t

  (t )dt  0 over

time intervals t   v and, hence,

0

oscillations of the relative volume must be smaller than pressure oscillations (46). In our experiments, a liquid Wood‘s alloy (Tm = 345°C) and a Silokhrom SKh-1.5 porous media were placed in a high-pressure chamber. The mass and size of the Silokhrom grains were m = 1 g and 300 µm, respectively. The pore diameter in Silokhrom SKh-1.5 ranged from 130 to 260 nm. The pressure in the chamber was produced by mechanical action on a rod that could enter the chamber through gaskets. A decrease in volume of the Wood‘s alloySilokhrom system upon moving the rod inside the chamber produced excessive pressure. The change in volume was measured using a displacement pickup. The pressure was measured by a strain gauge that was mounted on a support under the high-pressure chamber. The gauge could detect strength from 0 to 1000 kg in the frequency range up to 10 kHz with an accuracy of=10%. The filling critical pressure was determined from the V(p) dependence of the filled pore volume on pressure p for a quasi-static pressure buildup with a characteristic time of=10 s. For the system under study, this value was found to be pc0 = 120 atm. In the experiments with dynamic filling, the time-dependent pressure in the chamber was measured for the pulsed mechanical action on the chamber rod. The measured compressibility of Silokhrom SKh-1.5 was  = 1.6 x 10-3 atm-1. Since the compressibility of the chamber with volume Vch =120 cm3 was  = 1.4 x 10-5 atm-1., a change in the chamber volume filled with the Wood‘s alloy (the compressibility of the Wood‘s alloy is ~10–6 atm1) was vastly larger than the change in the Silokhrom volume in the dynamic experiments on the time scale of t2, the liquid leaks away from the porous media. At these times, the p(t) curves also display oscillations (upper and middle panels in Figure 10b). It follows from the data in Figure 10 and from the additional experiments that, at a fixed duration of the p0 pulse, there is a critical pressure p0c = 300 atm below which the filling oscillations are absent. Note that the increase in the p0 pulse duration from 10 to 20 ms also results in the disappearance of oscillations.

3.6. Physical Pattern of Infiltration of a Nonwetting Liquid into a Porous Media Thus, in accordance with the model considered here, we obtain the following pattern of infiltration of a nonwetting liquid through pores of a disordered porous media (see Figure 7). Infiltration is described as a spatially nonuniform process with the help of distribution functions f (n, t ) and F (n, t ) for clusters of accessible empty and filled pores, respectively. These functions satisfy kinetic equations taking into account the pair ―interactions‖ of accessible-pore clusters with clusters of accessible filled pores. In the case of a slow infiltration, for which characteristic time  p of pressure growth is longer than the characteristic infiltration time in the vicinity of percolation threshold

 c 0 from the standard

percolation theory, the critical retardation effect should be observed (see Figure 7). In this case, the characteristic infiltration time  v   for   c 0 and, hence for pressure

p  pc 0 . Under these conditions, all pores accessible at this pressure are being filled. Under a rapid pressure increase, in which time  p is shorter than characteristic time

z

of infiltration through clusters of accessible pores, these pores have no time to be filled before

 c 0 over time scale t ~  z an infinitely large cluster of accessible but empty pores is formed. In this case, P ( )  0 (see Figure 7). Steady-state distribution functions f 0 ( n) and F0 (n) are formed over time intervals  d and  z , the attainment of percolation threshold

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid respectively, such that

35

 d   z . It follows from the solvability condition for the system of

kinetic equations for function F0 (n) that this function differs from zero only when the pressurized system attains a new infiltration threshold in the fraction of empty accessible pores

c  0.28 ), which is higher than the familiar percolation threshold c 0  0.18 . The

c  0.28 is a new characteristic of the dynamics of infiltration of the porous media. In the range of  from 0.18 to 0.28, infiltration of the porous media should not be observed over a characteristic time of pressure growth  p   z (see Figure 7). threshold value of

Figure 10. Plots of pressure in the chamber vs. time: (a) a column filled with a liquid Wood‘s alloy; (b) a column filled with a liquid Wood‘s alloy and a porous media (Silokhrom SKh-1.5, 1 g); and (c) numerical simulation. The panels correspond to different values of p0 max and t (see text)

Analysis shows that for t   z , distribution function f (n, t ) for clusters of accessible pores is quasistationary, f (n, t )  f 0 (n,  (t )) , while the kinetic equation for distribution function F (n, t ) for clusters of filled pores for n  and

  c has a small positive (in

 the vicinity of the new threshold) eigenvalue  ~ (   c ) ,   0.8 . This eigenvalue

controls the characteristic time of the increase in the macroscopic volume of the pore space

36

V. D. Borman and V. N. Tronin

filled with the liquid (  v ~ 1 ). Remaining eigenvalues

n are negative and correspond to

characteristic relaxation frequencies for finite-size clusters of filled pores. Consequently, the infiltration of a grain of a porous media is a rapid (with characteristic time

 z   v ) process

of the formation of finite-size clusters of filled pores around an infinitely large cluster of accessible pores. The liquid flows through these clusters to the infinitely large cluster of accessible pores, filling it over a macroscopic time t   v   z .

Figure 11. Time dependence of fraction x of filled volume for the L23 + CaCl2 system (circles) and the corresponding curve calculated by Eq. (38) (solid curve)

The solution to the kinetic equation for distribution function F (n, t ) shows that the increase in distribution function F (n, t ) (and, hence in quantity

 ) during infiltration time

 v is compensated by the variation of F (n, t ) due to infiltration of clusters of accessible pores as a result of their interaction (percolation of the liquid) with clusters of filled pores, as well as due to percolation of the liquid to the infinitely large cluster of accessible pores from finite-size clusters of filled pores. Such a compensation can take place since the system consisting of the nonwetting liquid and the porous media is ―thrown‖ beyond new infiltration threshold

c , and characteristic infiltration time  v for   0  c is independent of the

viscosity of the liquid (which ensures this dynamic compensation) and is determined by time

 p of the pressure growth so that  v   p (0  c )  ,  0.4 . Thus, fraction 0 of accessible pores and, hence, the pressure remain unchanged during infiltration of a porous media. For a fraction of accessible pores close to (or higher than) new threshold 0 , the system of kinetic equations for the distributions functions f (n, t ) and F (n, t ) of clusters of accessible empty and accessible filled pores has oscillating solutions. The characteristic scale

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid of the period of oscillations is on the order of infiltration time

37

 z . The oscillations must be observed during

 v and correspond to the periodic infiltration-defiltration of the liquid in

finite-size clusters.

4. DISCUSSION OF RESULTS AND COMPARISON WITH EXPERIMENT Experiments show that the characteristic times of variation of volume and pressure for the system of a nonwetting liquid and a nanoporous media considered here are

 v ~ 25 ms

and  p ~ 5 ms, respectively (see Figure 5). Let us estimate characteristic times

 z and  0 .

For this purpose, we write

v ~

V ( L) , SJ

(48)

where V ( L) is the volume of a grain of size L, S is the surface area of the grain through which the liquid infiltrates, and J is the flow rate of the liquid. Using the linear relation between the velocity of the liquid flow and pressure gradient,

k p J  ~n  L

(49)

where kn is an unknown coefficient having the meaning of the penetrability factor of the medium [29], and assuming that V ( L) 

4 3 L and S  4 L2 , we obtain 3

v ~ Assuming that time

L2 3kn p

(50)

 v is given, we can obtain unknown coefficient kn from relation

(49):

L2 kn ~ 3 v p Then definition (6) for  0 

4 RL gives 3kn p

(51)

38

V. D. Borman and V. N. Tronin

0 ~

4R v L

For R ~ 10 nm and L ~ 1 µm, the value of for hydrody-namic time Using relation

 z ~  0 ( p) n m q

1 q

(52)

 0 is approximately 1 ms. Using relation (6) for

 ~ 0 ~ 0.3 , we obtain  z ~ 10 ms.

 d   (0 )1  p , for   0.6 , we obtain  d ~ 0.1 ms. Thus, the inequalities

typical of rapid infiltration (  v   z   p   0   d ) are observed in experiments with the systems under investigation, which allows us to use the developed model of infiltration for interpreting experimental data. For a slow variation in pressure ( p ~ 1 atm/s),  p ~ 100 s and, hence  p   z   d . This justifies the use of relations (10) and (12) for describing the results of experiments on slow infiltration of a nonwetting liquid through a nanoporous media [9, 21]. We can estimate the value of quantity 0 emerging due to compensation of the external action by the nanoporous media being infiltrated and determining the infiltration threshold using relation (33), which gives 2 0 0  c  (  ( c ) 3 )  z p

1

for

(53)

 0 ~ 1 ms and  v ~ 25 ms, we obtain 0  c  0.003  0.283 . Thus, infiltration of a

porous media in these experiments may begin in the vicinity of the transition of the system to a new infiltration state for

0  c / c ~ 102 . It should be borne in mind that in view of

the finite size of grains, percolation threshold

 c 0 (and, hence, quantities such a 0 and c

determined from it) also have values differing from the values of the corresponding quantities in the case of an infinitely large medium. This difference can be estimated assuming that infiltration through a grain of size L begins when correlation length  ~

R ,   0.8 [33] becomes equal to the grain size.  ( ) 

Consequently,

R L

1

c 0 ( L)  c 0  ( )

(54)

3 This gives  c 0 ( L)   c 0 ~ 10 , indicating the possibility of using the infinitely large

medium approximation for describing the result of experiments. It follows from relations (22) and (23) that positive eigenvalue

  0 , which

determines the characteristic infiltration time of the porous media, is formed in the limit

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

39

n  . The numerical solution of Eqs. (4) and (5) shows that the formation of a positive eigenvalue of system of equations (4), (5) begins for a number of pores n  100 in the cluster. At the same time, a grain of size L ~ 1 10 µm in the system under investigation L3 contains a number of pores n ~ 3 ~ 106  109 , which allows us to use the relations R obtained in Section 3 in the infinitely large medium approximation for describing experimental dependences. Infiltration of a porous media occurs for the fraction

0  0.28 of accessible pores. For

the systems under investigation, the infiltration pressure is p0  1.2 pc 0 . It follows from expression (1) that the probability of a pore being accessible to infiltration is 0.93; consequently, 93% of all pores in a grain of the porous media become accessible to infiltration at room temperature. It can be seen from Figure 6 that for

0  0.28 , about 70%

of pores in the porous media belong to the infinitely large cluster of accessible pores. The remaining 23% of pores (that do not belong to the infinitely large cluster) form finite-size clusters. These clusters surround the infinitely large cluster and, being infiltrated over time interval  z ~ 10 ms, form finite-size clusters of filled pores through which the liquid flows to the infinitely large cluster of accessible pores, infiltrating it over the characteristic time

 v ~ 25 ms. Thus, infiltration of a grain of a rapidly pressurized porous media can be treated

as a uniform process of infiltration of the liquid though finite-size clusters of accessible pores (about 20% of pores in the entire porous media) occurring simultaneously in the entire space of pores in the grain with characteristic time

 z ; it is followed by percolation of the liquid

from these clusters to the growing infinitely large clusters of accessible pores, containing about 70% of all pores in the porous media with a long characteristic time

 v ~ 25 ms.

Our experiments show that the time dependences of infiltration pressure and filled volume do not change upon a fivefold change in the viscosity of the liquid. Infiltration pressure pc at which the porous media passes to a new state can be determined from Eq. (34), which does not contain viscosity; consequently p0 is independent of viscosity. Numerical solution of Eq. (34) for

0  0.28 gives p0  200 atm, which corresponds to

experimental data (see Figure 5). It follows from Eq. (39) and relation (40) that the volume infiltration time is independent of the viscosity of the liquid due to dynamic compensation of the external action by the nonwetting liquid-nanoporous media system. Figure 11 shows the time dependence of fraction x of the filled volume for the L23 + CaCl2 system, calculated using Eq. (38). It can be seen that the experimental and theoretical dependences almost coincide. It should be noted that Eq. (38) is valid over time intervals t ~  v   z ~ 10 ms. Consequently, the coincidence of the theoretical and experimental time dependences of the filled volume at shorter times is accidental. In the model developed here, the filled volume and the infiltration time of a porous media are functions of the compression energy. These dependences can be derived from Eq. (38).

40

V. D. Borman and V. N. Tronin

Indeed, multiplying Eq. (38) by p0 and taking into account that E 

xmax



pdx , where xmax is

0

the maximum fraction of filled volume, we obtain

E

xmax

 0

 in

p0 dx   dt 0

p0

v

x(1  x) 

p0  in 2 v

(55)

Pressure depends on the compression energy only slightly and has a tendency to increase within the experimental error (see Figure 3a). Since p0 and

 v are independent of the

compression energy in the zeroth approximation, relation (55) shows that infiltration time

 in ~ E . Integrating Eq. (38) with respect to time, we obtain xmax 

 in

dt

 0

x(1  x) 

v

 in E 2 v

(56)

It follows hence that in the model developed here, the maximum filled volume is a linear function of the compression energy; consequently, the flow rate of the liquid during infiltration of the porous media is independent of energy. Dependences (55) and (56) for the infiltration time and for the maximum filled volume on the compression energy describe the experimental data to within the measurement error (see Figures 3b and 3c). The value of pressure p0 corresponding to the beginning of infiltration also depends on energy. Indeed, Eqs. (29) and (33) lead to

 0 ( pc )  pc ( ) p  p ) ,   0.8  p p

(0  c ) ~ (

c

(57)

Integrating this equation by p , we obtain

 dpp

c

 0 ( pc )   (p )  (p ) ( ) p  p  0 c  pc d  0 c E  p p p p c

 dp(

0

  c )  ( 0   c ) pc

(58)

(59)

Here, E is the compression energy per unit volume of the nanoporous media. Using expressions (58) and (59), we obtain from Eq. (57)

 ( p ) E  1 0 ( E )  c  ( 0 c )  p pc 1

(60)

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

41

Suppose p0 ( E )  pc   p( E ) , we obtain from Eq. (34) 

w( R, pc )  ( p ) E  1  p( E )  (  dRf r ( R) R3 )-1 ( 0 c ) p  p pc 0 1

(61)

Figure 12. Time dependences of infiltration pressure p for the L23 + H2O system for various compression energies E = 10 (1), 35 (2), and 60 J (3). The mass of the porous media is 4 g and the threshold pressures are

pc 0  150

It follows hence that

atm and

pc  180 atm.

 p( E )  E1 ( 1) and therefore p0 ( E )  pc  E1 ( 1) .

Consequently, when the external action with various compression energies is compensated, the attained excess over threshold value of 0 and, hence, the excess of infiltration pressure

p0 over the threshold pressure depend on the compression energy. This ensures an infiltration rate (with characteristic time  v ) satisfying relation (33). The dependence of pressure p0 on the compression energy is shown in Figure 3a. It can be seen that experimental data are described by relation (61) correct to the measurement error. In accordance with Eq. (32), reflecting the condition of compensation of the external action by the system, the time dependences of the infiltration pressure and filled volume in our model do not change with the viscosity of the liquid for this system (see Figure 5). Thus, for systems L23 + H2O and L23 + CaCl2, infiltration of the porous media under rapid compression is a nonuniform process in which clusters of filled pores ensuring percolation of the liquid to an infinitely large cluster of accessible but empty pores are formed after a new threshold

c of infiltration through accessible pores is attained that exceeds the

known percolation threshold

 c 0 . In accordance with the proposed model, the absorption of

the compression energy occurs at a constant pressure p0 , which exceeds pressure pc

42

V. D. Borman and V. N. Tronin

corresponding to the new threshold

c . This pressure p0 is determined by the condition of

compensation of the external action by the increase in the number and size of clusters of filled pores at a given rate of pressure growth, which ensures percolation of the liquid to the infinitely large cluster of accessible pores at a constant pressure. Such an infiltration regime takes place if characteristic time  p of pressure growth is shorter than the characteristic time of infiltration of the cluster of accessible pores. This is ensured (under an increase in pressure) by the attainment of value

  c for the fraction of accessible pores without infiltration of

accessible pores and by the formation of an infinitely large cluster of accessible pores. On the other hand, when the compensation condition is satisfied, the value of  p may be smaller than characteristic infiltration time which contains factor 

2 3

 v or close to it, in accordance with Eq. (39),

( 0 ) . In this pattern, if pressure p is (as a result of rapid pressur-

ization) such that the inequality pc 0  p  pc is satisfied, infiltration does not take place in the porous media. The existence of such an infiltration mode (the presence of a pressure ―gap‖ in infiltration) is confirmed by the experimental results depicted in Figure 12. It can be seen that the maximum pressure p  160 atm was attained during the infiltration of the porous media (curve 1). This value is higher than pc 0  150 atm, but lower than threshold

p0  180 atm. To within the experimental error, the change in the volume of the system coincides with the change in the volume due to deformation. The infiltration regime at a constant pressure is not observed if the compression energy exceeds the maximum value (E = 60 J; curve 3 in Figure 12) determined by the specific energy of infiltration of the porous media. In these conditions, the decrease in the flow rate of the liquid during infiltration of the porous media may attain a value of J  J min  J ( E )  const and the required energy absorption rate is not ensured for the characteristic time of pressure growth. For this reason, the response of the system consisting of a nonwetting liquid and a nanoporous media is an increase in pressure to the maximum value, followed by defiltration (curve 3 in Figure 12). It should be noted that the value of J min cannot be calculated in the mean field approximation under the assumption of invariability in the medium during infiltration since initial kinetic equations (3) and (4) disregard the interaction of clusters of filled pores.

ACKNOWLEDGMENTS The authors are grateful to L.A. Maksimov, who read the manuscript and made some valuable remarks, and to I.V. Tronin for fruitful discussions and assistance.

REFERENCES Sahimi, M. (1993). Rev. Mod. Phys., 65, 1393.

The Dynamics of Infiltration of a Nanoporous Media with a Nonwetting Liquid

43

Thompson, A. H., Katz, A. J. & Raschke, R. A. (1987). Phys. Rev. Lett., 58, 29. Feder, J. (1988). Fractals (Plenum, New York, Mir, Moscow, 1991). Bogomolov, V. N. (1978). Usp. Fiz. Nauk, 124, 171, [Sov. Phys-Usp. 21, 77]. Yu, A. Fadeev, & Eroshenko, V. A. (1995). Ross. Khim. Zh., 39(6), 93 , Eroshenko, V. A. & Yu. A. (1996). Fadeev, Zh. Fiz. Khim., 70(8), 1482 [Russ. J. Phys. Chem. 70(8), 1380]. Matthews, G. P., Ridgway, C. J. & Spearing, M. C. (1995). J. Colloid Interface Sci., 171, 8. Kloubek, J. (1994). J. Colloid Interface Sci., 163, 10. Borman, V. D., Grekhov, A. M. & Troyan, V. I. (2000). Zh. Éksp. Teor. Fiz., 193, [JETP 91 (1), 170 (2000)]. Kong, X., Surani, F. B. & Qiao, Y. (2005). J. Mater. Res., 20, 1042. Qiao, Y. & Kong, X. (2005). Phys. Scr., 71, 27. Surani, F. B., Kong, X. & Qiao, Y. (2005). Appl. Phys. Lett., 87, 251906. Qiao, Y. & Kong, X. (2005). Phys. Scr., 71, 27. Kong, X. & Qiao, Y. (2005). Appl. Phys. Lett., 86, 151919. Suciu, C. V., Iwatsubo, T. & Deki, S. (2003). J. Colloid Interface Sci., 259, 62. Han, A., Kong, X. & Qiao, Y. (2006). J. Appl. Phys., 100, 014308. Surani, F. B. & Qiao, Y. (2006). J. Appl. Phys., 100, 034311. Surani, F. B., Han, A. & Qiao, Y. (2006). Appl. Phys. Lett., 89, 093 108. Chen, X., Surani, F. B., Kong, X. (2006). et al., Appl. Phys. Lett., 89, 241 918. Kong, X. & Qiao, Y. (2006). J. Appl. Phys., 99, 064 313. Borman, V. D., Belogorlov, A. A., Grekhov, A. M., Lisichkin, G. V., Tronin, V. N. & Troyan, V. I. (2005). Zh. Éksp. Teor. Fiz., 127(2), 431 [JETP 100 (2), 385]. Nielsen, L. E. & Lande, R. F. (1993). Mechanical Properties of Polymers and Composites (Marcell Dekker, New York,). Bogomolov, V. N. (1995). Phys. Rev. B: Condens. Matter, 51, 17040. Borman, V. D., Belogorlov, A. A., Grekhov, A. M., Lisichkin, G. V., Tronin, V. N. & Troyan, V. I. (2004). Pis’ma Zh. Tekh. Fiz., 30(23), 1 [Tech. Phys. Lett. 30 (12), 973 (2004)]. Shklovskii, B. I. & Efros, A. L. (1979). Electronic Properties of Doped Semiconductors (Nauka, Moscow, Springer, Berlin, 1984). Isichenko, M. B. (1992). Rev. Mod. Phys., 64, 961. Surani, F. B., Kong, X., Panchal, D. B. & Qiao, Y. (2005). Appl. Phys. Lett., 87, 163 111. Borman, V. D., Belogorlov, A. A., Grekhov, A. M., Tronin, V. N. & Troyan, V. I. (2001). Pis’ma Zh. Éksp. Teor. Fiz., 74(5), 287, [JETP Lett. 74 (5), 258 (2001)]. Basniev, K. S., Kochina, I. N. & Maksimov, V. M. (1993). Underground Hydromechanics (Nedra, Moscow,) [in Russian]. The Chemistry of Grafted Surface Compounds, Ed. by G. V. Lisichkin (Fizmatlit, Moscow, 2003) [in Russian]. Handbook of Physical Quantities, Ed. by In: I. S. Grigoriev, & E. Z. Meilikhov, (Énergoatomizdat, Moscow, 1991; CRC Press, Boca Raton, FL, United States, 1997). Handbook of Chemistry and Physics, Ed. By In: D. R. Lide, (CRC Press, London, 1994). Yu. Yu. Tarasevich, Percolation: Theory, Applications, and Algorithms (URSS, Moscow, 2002). Abrikosov, A. A. (1979). Pis‘ma Zh. Éksp. Teor. Fiz. 29 (1), 72 (1979) [JETP Lett. 29 (1), 65].

44

V. D. Borman and V. N. Tronin

Morse, P. M. & Feshbach, H. (1953). Methods of Theoretical Physics (McGraw-Hill, New York, Inostrannaya Literatura, Moscow,), Vol. 2. V. D. Borman, A. A. Belogorlov, G. V. Lisichkin, V. N. Tronin, and V. I. Troyan JETP, Vol. 108, No 3, March 2009.

In: Nanoporous Materials Types, Properties and Uses Editor: Samuel B. Jenkins, pp. 45-71

ISBN: 978-1-61668-182-1 © 2010 Nova Science Publishers, Inc.

Chapter 2

ENERGETICS AND PERCOLATION PROPERTIES OF HYDROPHOBIC NANOPOROUS MEDIA V. D. Borman and V. N. Tronin National Research Nuclear Univrsity MEPHI , Moscow, Russia

1. INTRODUCTION Energetics of "nanoporous medium--nonwetting liquid" systems is one of the new directions in basic and applied research [1-8]. In the simple model of a porous media in the form of cylindrical channels, this threshold pressure is described by the Laplace--Washburn equation p  2  R  cos , where  is the surface energy of the liquid, R is the pore radius, and  is the contact angle (for a nonwetting liquid,   90 ). For filling nanometer-sized pores by a nonwetting liquid with a surface energy of 005  05 J/m2, it is necessary to apply a threshold pressure of 10  10 atm. When the liquid passes from the bulk of the material to a dispersed state in pores of the nanoporous medium with a specific volume of ~ 1 cm3/g, the absorbed and accumulated (returned when the liquid flows out) energy can reach 10 100 kJ/kg. This value is one order of magnitude higher than the energy observed for widely used materials, such as polymer composites and alloys with the shape memory effect [9-11]. Among the systems under investigation are silochromes, zeolites with liquid metals, hydrophobized silica gels, and zeolites with water and aqueous solutions of organic compounds and salts. In recent years, hydrophobized nanoporous media have become available owing to the development of the method used for modifying the surface of nanoporous media, for example, with alkyl chlorosilanes [6,7,12-22]. To date, nanoporous media with different pore shapes, porosities, specific surface areas, specific volumes, average pore radii, and pore size distributions have been studied [1,6,7,19,20,21-55]. The investigations performed thus far have been concerned primarily with equilibrium properties. Experiments have been carried out at a low compression rate of the system when the rate of 2

3

46

V. D. Borman and V. N. Tronin

3 increase in the pressure ( p ) in the liquid--porous media system is (10 -1) atm/s. In the

infiltration-defiltration cycle, there is a hysteresis, so that the threshold pressure of infiltration is higher than the pressure of defiltration. Moreover, the majority of the systems studied are characterized by the phenomenon of nonoutflow of a nonwetting liquid when a part of this liquid remains in the porous medium as the excess pressure decreases to zero. The absorbed energy is determined by the product of the volume of filled pores and the difference between the infiltration and defiltration pressures. In frameworks the model of cylindrical channels, these pressures are described by the Laplace--Washburn equation with different angles of wetting. The phenomenon of nonoutflow of a nonwetting liquid has restricted the practical application of the system. These phenomena have been observed in the systems under investigation irrespective of the type of a modifier of the pore surface. The revealed difference between the infiltration and defiltration pressures and the absorption of the mechanical energy observed in the infiltration--defiltration cycle due to the pressure hysteresis, as a rule, have been explained by the hysteresis of the contact angle; however, the mechanism responsible for the appearance of the latter hysteresis has remained unclear [4,7,19,20,21,22,56]. It has been established that the infiltration and defiltration pressures depend on the temperature and that, for the porous medium with a disordered structure of pores, the defiltration pressure increases (by several factors) with an increase in the temperature, whereas the infiltration pressure decreases only slightly (by less than 10%) or remains constant [21,22,24]. This means that, during infiltration and defiltration, the phenomenological contact angles differently depend on the temperature. For zeolites, the revealed temperature dependences exhibit a more complex behavior; moreover, the volume ( V ) memory effect can be observed with an increase in the temperature and its subsequent decrease. It is worth noting that this effect is one order of magnitude (in V  V ) stronger than that observed for known alloys and composites [10,11]. At present, there exist several hypothesis regarding the nature of the contact angle hysteresis. This hysteresis has been attributed to the rough surface of pores, the chemical inhomogeneity of the surface, and the dependence on the direction of the liquid motion [19,20]. In the framework of the concept that the porous medium is a system of cylindrical channels, the absorbed energy is expended for forming a liquid-porous media surface, which appears in the course of infiltration and disappears during defiltration at different pressures due to different contact angles [8,51,57]. In case of the closed hysteresis loop in the infiltration - full-defiltration process initial and finite states of the system are similar, the



internal energy change E cicle  dE  0 and the work done to perform the filling of the porous medium should be equal to the thermal effect

 dA   dQ . The measurements carried

out in [19,58] showed that rise of temperature in the hydrophobic silica gel-water systems under investigation during infiltration-defiltration cycle was 0) with an increase in the number of

Energetics and Percolation Properties of Hydrophobic Nanoporous Media

49

pores N in the cluster leads to a decrease in the fluctuation probability. This corresponds to the thermodynamic stability of the initial state of the system. For А~Т, the infiltration fluctuation can increase. In this case, the system becomes unstable and the liquid begins to infiltrate the porous medium. The infiltration of a nonwetting liquid under the pressure p in a porous media requires a certain amount of work to be done to fill the pores of the porous media. For this purpose, it is necessary to overcome a certain critical pressure, which is the Laplace pressure

pc ( R) ~

 R

for an isolated pore with characteristic size R.   

the surface energy of the solid–liquid interface,  gas interface,

lg

sl

  sg ,where  sl is

  is the surface energy of the solid–

   cos  ,  is the wetting angle. sl

At a pressure lower than the critical value p in , the adiabatic work satisfies the inequality A(N)>0 at any value of N and the fluctuation probability decreases with an increase in N. Therefore, the fluctuation probability is equal to zero for any macroscopically large number of pores. Fluctuations of finite length arise, but no infiltration of the porous media occurs. At a pressure in the vicinity of the critical value p in , the work is А~Т and thermal fluctuations in the system can lead to the formation of clusters from N pores. At a pressure р> p in , the infiltration of individual pores becomes energetically favorable because the quantity A(N) is negative. Since the work is А~N, the fluctuation probability at a pressure р> p in is w~1. The pressure difference р- p in causes the liquid to move in the porous media. Now, we consider a porous medium immersed in a nonwetting liquid under an external pressure p , which does the work in the course of infiltration of the porous media. Let

A p be the work expended for providing the fluctuation infiltration of one pore. According to [5], the expression for the work A p  with due regard for the formation of menisci can

be written in the form

A p    pV  S m   (S  S m )

(1)

Here, V is the volume of the pore, S is the surface area of the pore, S m is the surface

 is the surface energy of the liquid. For a spherical pore with the radius R, the work A p  can be represented in the form

area of the menisci, and [5]

A  A p, R   4 3    R 3 , A  p, R    p 

3       1      1 R    

where  is the ratio of the meniscus surface area to the pore surface area.

(2)

50

V. D. Borman and V. N. Tronin

Figure 1. The dependence of the probability normalized to unity of a pore belonging to the infinitely large cluster on porosity 

A similar expression can be written for a cylindrical pore. For example, we write the following expression for the work expended for providing the fluctuation emptying of a cylindrical pore with the length L and the radius R :

A  A p, R, L   4 3    R 3 , A  p, R, L   p( where x 

3x 3    1)  2 R

    x  ,   

(3)

L . 2R

It follows from relationship (2) that the infiltration of the pore becomes energetically favorable at a pressure higher than the critical value p in . The critical pressure is determined by the equality of the work on the fluctuation infiltration of the pore to zero. An analysis of the critical pressure for a spherical pore was performed in [5]. Expression (3) implies that, devastation of the pores becomes energetically favorable at a pressure less than the critical pressure p out , which depends on the pore length L and the pore radius R and, at L   ,

2 . This means that, in the work expended for R infiltrating the pore, the change in the surface energy of the pore dominates over the energy expended for forming menisci. For characteristic values of the parameters of the porous media and the liquid ~~0.5 kJ/m2, R~10 nm, and , ~10-4 atm-1), when the pressure deviates from the value рI by р=-102 рI, the work changes from А~T to ~1 eV. In this case, the change in the probability of infiltration fluctuation at the temperature T = 400 K is equal to ten orders of magnitude. Therefore, for systems with the aforementioned characteristic parameters, the probability w changes abruptly from 0 to 1 in a narrow pressure range (p/p~10-2) in the vicinity of the pressure p in . It should be noted that the inclusion of the gas filling the pores in the analysis transforms into the Laplace pressure p out ~

Energetics and Percolation Properties of Hydrophobic Nanoporous Media

51

leads to the appearance of an additional contribution to the work A. The value of this contribution Ag for one pore with the volume V can be estimated under the assumption that it is equal in the order of magnitude to the specific evaporation energy of the liquid

0

multiplied by the density of the gas  g at the pressure p , i.e., Ag ~  0  g V . For water,

kg J and  0 ~ 1.21 3 . The infiltration of the porous media m kg 7 J occurs at a pressure of the liquid pin ~ 2 x10 . In this case, the additional contribution m3 we can write

to

the

 0 ~ 2.2 x10 3

work

due

Ag ~  0  gV ~ pinV

to

the

0  g pin

presence

of

the

gas

in

the

pores

is

given

by

~ 10 2 A . Therefore, the influence of the gas in the pores on

the infiltration of the porous media can be ignored.

Since the pore can be either filled (probability w ~ 1 , A p   0 ) or empty (probability

w  0 , A p  0 ), the normalized probability can be written in the form [11]



 T 

wi  p   1  exp A

1

,

(4)

Note, that relations (3) and (4) explain the obtained in [6] experimental data for the dependence of infiltration pressure and defiltration pressure on the pore size. Pores are not isolated in a porous media, but they are connected with one another by throats (mouths), in which menisci are formed during infiltration of a certain pore. Thus we can assume that the medium subjected to infiltration is the heterogeneous medium wich consists of full and empty pores playing the role of white and black spheres, respectively, in the percolation theory [63]. Such medium can experience percolation transition occurring via the formation of clusters of accessible pores and followed by infiltration of nonwetting liquid into such formations. [60]. In addition to that, filling of the macroscopic volume of a porous media occurs by infiltration in the infinitely large cluster of accessible pores [60]. Below, we will consider the infiltration of pores located on the shell of an infinite cluster consisting of filled pores. In this case, the condition A  0 determines the pressure necessary for the infiltration of a pore on the shell of the infinite cluster of filled pores.

3. WORK AND THERMAL EFFECT IN THE INFILTRATIONDEFILTRATION CYCLE Let us calculate the work infiltration

A and the thermal effect Q for an arbitrary degree of

 of the porous medium. The thermal effect Q in filling of a porous media by

nonwetting liquid comprises the thermal effect Q p due to the formation of the liquid-solid

52

V. D. Borman and V. N. Tronin

interface , the thermal effect Qw related to formation-disappearance of menisci and the thermal effect Qu related to the compressibility of the nonwetting liquid-nanoporous media system .

Q  Q p  Qw  Qu

(5)

The Q p , Qw and Qu values can be calculated using thermodynamic relations [12], which determine the thermal effect in formation of the surface Q s

Qs  T

d S dT

(6)

Here, S is the change in the system surface. To calculate the thermal effect Q s let us suppose that each pore in a porous media has

z nearest neighbours and pores contact each other by throats, each of which has an area S z . If an empty pore contacts a full one, the meniscus is formed in the throat. A pore in a porous media can be filled only if liquid can reach it. In compliance with above-mentioned assumptions this condition can be satisfied by formation of an infinitely large cluster of filled pores. In this case, only those pores which belong to the shell of the infinite cluster will be filled. It is possible to show that the contribution of filled finite-size clusters (which liquid can reach via filled clusters contacting with the surface of the porous media)to the filled volume is small. Distribution f (N ) of the number of pores in clusters of finite size near the percolation threshold is determined by the scaling dependence f ( N ) ~

1 ,   2.2 [63]. N

From this it follows that the bulk of the cluster contains one or more pores, which are mostly not associated with the surface of the porous medium. Liquid cannot reach such pores and, consequently, they are not filled at

 ~  c . Taking it into consideration, we can represent the

thermal effect Q related to filling of one pore as:

Q  T

d d ( S  zS г )  T zS гW ( ) , dT dT

(7)

S  4R 2 is the area of the surface of a pore with radius R , z is a number of nearest neighbours, W ( ) is the difference (averaged over the ensemble of pores) between the numbers of menisci before and after the infiltration of the pore per the nearest neighbor of the infinite cluster. Considering that filling of a porous media is the result of formation of the infinitely large cluster of filled pores and taking into account the normalized probability P ( ) , we obtain

Energetics and Percolation Properties of Hydrophobic Nanoporous Media

53

that the quantity of heat per a pore released in the process of filling the porous media to the degree of filling  can be written as: 

d Q p ( )  T  ( S  zS z )  P( ) d dT 0 

 S  zS z   dR( S  zS z ) f ( R )

(8)

0



d Qw ( )  T  zS zW ( )  d dT 0 Here, f (R) is the function normalized to unity of the size distribution for pores. For a disordered porous media average values in (8) can be calculated in the framework of a specific model of a porous media. We will use the model of randomly arranged spheres in which pores represent randomly arranged spherical holes [66]. This model does not take into account correlations in location of pores with different radii in accordance with assumption made about the narrowness of size distribution for pores (R)  R . In accordance with the model, the average number of nearest neighbours z , associated with porosity of medium  , and the area of a throat can be written in forms [66]

z  z  8 ln( 1   )

Sz 

(9)

9 2 2 R 256

Using expression (8) and taking into account that (R)  R , we obtain from (9):

Q p ( )  T



d (1   )4R 2  P( )d dT 0 

d Qw  T 4R 2  W ( )d dT 0

z

9 2 1024

(10)

To calculate W ( ) we consider an empty pore located on the perimeter of the infinite cluster of filled pores. Let us suppose that this pore contacts the infinitely large cluster of filled pores via n throats. Thus, menisci are formed in all n mouths and menisci are absent in the remaining z-n throats. After filling this pore, menisci which were there at the beginning of infiltration disappear and the number of menisci will be equal to z-n. In this case we can write W ( ) as:

54

V. D. Borman and V. N. Tronin z

W ( )   ( P( )) n (1   ) z  n1 n 1

z  2n z! z n!( z  n)!

(11)

The first factor under the summation sign determines the probability that an empty pore contacts the infinite cluster of filled pores n times, the second factor is the probability of finding the empty pore close to the infinite cluster, provided that this pore is surrounded by zn empty pores and therefore has z-n throats. The third factor determines the difference between a relative number of menisci after (z-n) and before (n) the filling of the pore. The binomial coefficient takes into account variants of allocation of n menisci on number of pore nearest neighbours. Note that the obtained expression coincides with the full perimeter of the infinitely large cluster calculated in [67], if the third factor is substituted for unity. The sum in Eq. (11) can be calculated analytically

W ( )  ( 2  2  P( )  1  P( ))( P( )    1) z 1  (1   ) z 1

(12)

Relations (9), (10), (12) determine the thermal effect during filling a porous media with porosity  to the degree of filling  . The work A expended for infiltrating the porous media to the volume determined by the fraction

 and the corresponding work expended for infiltrating one pore Ain ( ) can be

calculated from the thermodynamic relationship A 



dS [65]. By using expression (2)

for spherical pores, we obtain

Ain ( , p)   p

4R 3   (1   ) 4R 2   4R 2W ( ) 3

A  A p  Aw A p ( )   (1   )4R

(13)  2

 P( )d , 0

Aw   4R

 2

 W ( )d 0

The sum of heat (10) and work (13) determines the change in the energy of isothermal infiltration of the porous medium: E  E p  E w 



(14)

d d E p  (  T )(1   )4R 2  P( )d , E w  (  T ) 4R 2  W ( )d dT dT 0 0

It follows from (14) that the change in energy of the system during filling the porous medium is determinded by the specific surface energies  and  , geometric properties of the porous medium and the evolution of the infinite cluster of filled pores, which depends on the properties the disordered porous medium (Figure 1).

Energetics and Percolation Properties of Hydrophobic Nanoporous Media

55

For the calculation of the work A( ) and the thermal effect Qv arising upon the defiltration of the liquid from the porous medium, it should be noted that, in the infiltrated porous media, the defiltration of the liquid leads to the formation of empty pores surrounded by at least one filled pore connected through other filled pores with the surface of the porous medium. As in case of filling, the formation of an empty pore goes with the change in surface energy of the liquid-solid and liquid-gas interfaces as well. The change in surface energy is associated with formation-disappearance of menisci [5]. Taking into account this fact, the work expended for emptying the pore in the porous medium A( ) with the degree of infiltration

 and the thermal effect Qv associated with

the defiltration of the liquid from one pore in the porous media can be written in the form

Aout ( )  p

4R 3   (1   )4R 2   4R 2W1 ( ) 3

A p ( )   (1   )4R v

 2

  d , A

v

w

  4R

0

Qv  T

 2

 W ( )d  1

0

d d ( S  zS г )  T zS гW1 ( ) dT dT

(15)

Relations (7) and (15) differ in sign of the last term and the functions W ( ) and W1 ( ) which determine the difference per one nearest neighbour between the number of menisci before and after infiltration (defiltration) in pores. In contrast to the case of infiltration, the defiltration of the liquid occurs initially through the formation of individual empty pores and clusters of empty pores with a decrease in the pressure and, after the infinite cluster of empty pores is formed, through the formation of pores on the shell of this cluster. Upon the defiltration when the low degree of infiltration is reached, the liquid can be retained in the porous medium if it is contained in individual pores or clusters of filled pores surrounded by empty pores with smaller sizes from which the liquid defiltrated at higher pressures. However, as the number of neighbors of empty pores increases, according to relationships (15) (see below Figs. 1-3), the defiltration of the liquid becomes energetically more favorable; i.e., it should proceed at higher pressures. Therefore, the quantity W1 ( ) should be defined as the difference (averaged over the ensemble of pores) between the numbers of menisci before and after the emptying of the pore on the shell of the system of empty pores. Taking it into consideration, calculation of W1 ( ) gives:

W1 ( )  (2 2  3  1)  (1   ) z 1

(16)

The thermal effect, the work, and the change in the energy during the defiltration, when the degree of infiltration varies from 1 to  , can be written in the form similar to relationships (10), (13), and (14):

56

V. D. Borman and V. N. Tronin Q v p ( )  T





d d (1   )4R 2  d , Q v w  T 4R 2  W 1( )d dT dT 0 0

A p ( )   (1   )4R v

 2

 d , A

v

w

  4R

0

 2

 W ( )d 1



E

v

p

(17)

0



d d  (  T )(1   )4R 2  d , E v w  (  T ) 4R 2  W1 ( )d dT dT 0 0

Expressions (17), like relationships (10), (13), and (14), are valid for the case of an isothermal process. This implies that they can be used for describing experiments in the case where the characteristic time of heat transfer (removal)  Q is considerably shorter than the characteristic time system:  Q
6) are used for fabrication MCM materials where pore size can be controlled by varying alkyl chain length of the surfactant [23]. The synthesis of MCM-48 requires more specific

Ordered Mesoporous Materials for Drug Delivery Applications

77

conditions than that of MCM-41: with the common alkyltrimethylammonium surfactants, MCM-48 will not be formed until the surfactant/silica molar ratio is beyond 1 [25-29].

Figure 2. Two different types of micelle–silica interaction and the ordered mesoporous materials obtained after each mechanism. Reproduced from reference [24] with permission.

Use of amphiphilic triblock copolymers to direct the organization of polymerizing silica species has resulted in the preparation of well-ordered hexagonal mesoporous silica structures (SBA-15) with cubic Pm3n (SBA-1, HMM-3), cubic Im3m (SBA-16), and 3D hexagonal P63/mmc (SBA-2, SBA-12) cagelike structures [16,17]. The SBA-15 materials produce highly ordered, two-dimensional hexagonal (space group p6mm) silica-block copolymer mesophases. Calcination at 500°C gives porous structures with pore sizes from 4.6 – 30 nm,

pore volume fractions up to 0.85, and silica wall thicknesses of 3.1 - 6.4 nm. SBA-15 can be readily prepared over a wide range of uniform pore sizes and pore wall thicknesses at low temperature (35 °C to 80 °C), using a variety of poly (alkylene oxide) triblock copolymers and by the addition of cosolvent organic molecules. The block copolymer species can be recovered for reuse by solvent extraction with ethanol or removed by heating at 140 °C for 3 hours; in both cases, yielding a product that is thermally stable in boiling water [37, 38]. A family of mesoporous molecular sieves (denoted MSU-G) with vesiclelike hierarchical structures and unprecedented thermal (1000 ºC) and hydrothermal stabilities (more than 150 h at 100 ºC) associated with high SiO2 cross-linking was prepared through a supramolecular assembly pathway that relies on hydrogen bonding between electrically neutral gemini surfactants of the type CnH2n+1NH (CH2)2NH2 and silica precursors derived from

78

Spomenka Simovic and Dusan Losic

tetraethylorthosilicate. The vesicle shells are constructed of one or more undulated silica sheets that are about 3 nm thick with mesopores (average diameters from 2.7 to 4.0 nm) running both parallel and orthogonal to the silica sheets, which makes the framework structure bicontinuous and highly accessible [39].

Figure 3. Transmission electron micrographs of (A) ordered molecular sieve MCM-41 and (B) disordered molecular sieve KIT-1. Reproduced from reference [34] with permission.

The synthesis of MCM-41, a noncrystalline molecular sieve which exhibits short wormlike mesoporous channels interconnected in a three-dimensional, disordered way through numerous seams (or branches), while the channel widths are as uniform as the ordered mesoporous molecular sieves has been reported [34]. The uniform pore-size distribution with peak width ≤0.3 nm at half-height has been confirmed using N2 adsorption obtained at liquid N2 temperature (Figure 3). The three-dimensionally branching local structure of this material has been confirmed using a transmission micrograph of nanosize platinum wires intergrown through the branches inside the mesoporous channels. The uniform pore widths are very similar to the MCM-41, but the channel arrangement interconnected in a three-dimensional disordered way distinguishes this new molecular sieve most conspicuously from the MCM-41, which exhibits a hexagonal arrangement of straight channels. Besides, the disordered structure has higher thermal stability and hydrothermal stability than the MCM-41 structure. The material, designated as KIT-1 (Korea Advanced Institute of Science and Technology Number 1), was synthesized by an electrostatic templating route using sodium

Ordered Mesoporous Materials for Drug Delivery Applications

79

silicate, HTAC1 micelles, and ethylenediaminetetraacetic acid tetrasodium salt (EDTANa4). The silicate was hydrothermally polymerized surrounding HTA micelles in aqueous solution at 370 K, similar to hydrothermal synthesis of MCM-41 using repeated pH adjustment. The surfactant−silicate mesostructure thus obtained is compared with MCM-41 using transmission electron micrographs in Figure 3. While the MCM-41 shows the hexagonal structure in the transmission electron micrograph, no structural orders have been found from the micrograph or the electron diffraction pattern for the KIT-1. Very similar transmission electron micrographs were also obtained when the EDTA salt was substituted by sodium salts of other organic acids such as adipic acid, benzenedisulfonic acid, etc. The surfactant was removed readily from products by calcination in air under static conditions at 823 K. FSM-16 is a mesoporous silica material which has an ordered one-dimensional pore system synthetised from a crystalline single layer polysilicate kanemite (NaHSiO5 X 3H2O) with an aqueous solution of surfactants as pore formation templates [35, 36]. A folded sheet mechanism has been proposed for FSM -16 formation, in which surfactant molecules are intercalated between canemite sheets after which adjacent silicate layers meet together in a periodic way and condense to form a three-dimensional highly ordered mesoporous material. TEM image of FSM–16 structure is presented in Figure 4.

Figure 4. TEM image of FSM – 16 material. Reproduced from reference [14] with permission.

Important efforts have been devoted to controlling the pore size of mesoporous materials [40-44]. The introduction of a swelling agent into the structure directing template is the common method. Common swelling agents are large organic hydrocarbons such as dodecane [40], 1,3,5-trimethylbenzene [41] , tertiary amines [43] , triisopropylbenzene [44]. The introduction of swelling agents has been shown to lead to the increase in pore volumes of around 30%, but loss of long-range order of the mesoporous structure is observed. Mixing surfactant blends to tailor the pore size of mesoporous silica shows nanometer-level control over the pore diameters, although the longest surfactant chain governs the largest pore size achievable [45, 46].

80

Spomenka Simovic and Dusan Losic

Mann and his co-workers [47] have demonstrated that a bacterial superstructure, consisting of a tread of coaligned multicellular filaments of Bacillus subtilis, can be used as a macrotubular array template to extend the length scale of MCM-41 silica mesostructure

patterning. By dipping air-dried bacterial tread into a typical MCM-41 synthesis mixture, they have succeeded in preparing ordered macroporous silica fibers with uniform MCM-41 type mesopore channels in the walls of the fibers. The macroporous framework consist of 0.5 µm wide channels with curved walls 50-200 nm thick and composed of either silica or mesoporous silica. Hollow silica tubules of 0.3 to 3 μm in diameter which exhibited coaxial cylindrical MCM-41 type mesopore channels in their walls were reported by Lin and Mou [48]. The S+ Ipreparation of this hierarchical structure was accomplished via careful control of the surfactant−water content and the rate of silica condensation at high pH values. Transparent silica spheres with diameters from 0.1 to 2 mm and mesoporous structure, similar to that of the hexagonal MCM-41 (pore diameters in the range of 1 to 5 nm), were reported by Stucky and co-workers [49]. The ambient reaction temperature preparation of the spheres was accomplished via the S+ I- electrostatic templating pathway, using an oil-in-water emulsion formed from the hydrophobic tetrabutyl orthosilicate (TBOS) silica precursor and the water-immiscible BuOH. Schüth et al. [50] synthesized hexagonal mesoporous MCM-41 products with fibrous, spherical, and/or sheet-like morphology. These materials were obtained from acidic MCM-41 reaction mixtures in the presence of simple organic co-surfactants (primarily mesitylene). The formation of an oil-in-water emulsion interface by the co-surfactant was responsible for the observed morphologies. The preparation of materials with a particular morphology was controlled by the rate of agitation of the reaction mixture [50]

Figure 5.Comparative sizes of alendronate (0.83 nm), ibuprofen (1.01 nm) and bovine serum albumin (10 nm × 6 nm) used in the adsorption and release tests. Reproduced from reference [24] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

81

Using the acidic S+X-I+ electrostatic assembly pathway and quiescent reaction conditions Ozin et al. [51] have obtained hexagonal MCM-41 products with disk-like, spiral, and spheroid particle morphology. The formation of these curved particle architectures was attributed to growth of a silicate ―liquid crystal embryo‖ with hexagonal cross-section that, under different initial reaction conditions, is subject to increasing degrees of curvature [51]. From the drug delivery point of view, pore diameters 2-10 nm are an issue when large molecules such as proteins have to be loaded. This is the case of serum albumins, one of the major components in plasma proteins in humans and the upper mammals, which is usually composed of a single-chain of 582 amino acids with an average length of 10nm and width of 6 nm and more than 10 times larger than conventional drugs (Figure 5) [24]. The research group of Vallet-Regi has developed a method for tailoring pore sizes of SBA-15 ordered mesoporous materials with the aim of confining large size molecules such as proteins. This method is based on increasing time for the hydrothermal treatment to enlarge the pore diameters [24, 52, 53]. In order to increase the pore size of the SBA-15 mesoporous materials, the time of hydrothermal treatment was prolonged for 3, 5 or 7 days by placing the precursor solution in Teflon®-lined stainless steel autoclaves at 373 K to avoid pressure changes during the treatment time (Figure 6). After that, the washing and drying procedures were the same as used for conventional SBA-15. Mesoporous materials synthetised using this novel method were named SBA15-3d, SBA15-5d and SBA15-7d, where the figure stands for the total time of hydrothermal treatment in the autoclave (Figure 6). Due to the increase of pressure, a swelling effect of the surfactant template chains leads to an increase in a pore diameter of SBA-15 from 8.2 nm up to 11.4 nm.

Figure 6. Scheme of hydrothermal procedure for the synthesis of large-pore SBA-15 ordered mesoporous silica employing conventional Teflon® beakers and Teflon®-lined stainless steel autoclaves. Reproduced from reference [24] with permission.

82

Spomenka Simovic and Dusan Losic

2.2. Biocompatibility of Mesoporous Silica Materials Biocompatibility is the prerequisite for the application of any drug delivery device or system with non-toxicity and non-inflammation effect as major requirements. Biocompatibility of implant biomaterials is a more complex issue that involves cellular response and tissue integration of implanted biomaterials. Infection and inflammation are associated with the large variety of wound occurrences ranging from traumatic skin tears and burns to chronic ulcers and complications following surgery and device implantations [54]. If the wound setting manages to overcome any microorganism invasion by a sufficient immune response then the wound should heal. If not, the formation of an infection can seriously limit the wound healing process. The main goal in treating various types of wound infections is to decrease the bacterial load in the wound to a level that enables wound healing processes to take place [54]. Wound healing is a dynamic, interactive process involving soluble mediators, blood cells, an extracellular matrix, and parenchymal cells. Wound healing has three phases that overlap in time: inflammation, tissue formation, and tissue remodelling [55]. Tissue injury causes the disruption of blood vessels and extravasation of blood constituents. The blood clot re-establishes hemostasis and provides a provisional extracellular matrix for cell migration. Platelets not only facilitate the formation of a hemostatic plug but also secrete several mediators of wound healing, such as platelet-derived growth factors that attract and activate macrophages and fibroblasts. Numerous vasoactive mediators and chemotactic factors are generated by the coagulation and activated-complement pathways and by injured or activated parenchymal cells. These substances recruit inflammatory leukocytes to the site of the injury. The migration and accumulation of leucocytes constitutes a hallmark of inflammatory response. In addition, macrophages, fibroblasts, and blood vessels move into the wound space. The macrophages provide a continuing source of growth factors necessary to stimulate fibroplasia and angiogenesis; the fibroblasts produce the new extracellular matrix necessary to support cell in-growth; and blood vessels carry oxygen and nutrients necessary to sustain cell metabolism. After migrating into wounds, fibroblasts commence the synthesis of an extracellular matrix. The provisional extracellular matrix is gradually replaced with a collagenous matrix, perhaps as a result of the action of transforming growth factor ß1.Once an abundant collagen matrix has been deposited into the injury, the fibroblasts stop producing collagen, and the fibroblast-rich granulation tissue is replaced by a relatively acellular scar [55]. Therefore, fibrosis or fibrous encapsulation is a usual outcome of the soft tissue repair around implants [55, 56], which may adversely affect the function of the implant. Inflammatory cells, particularly neutrophils and macrophages, recruited during the initial inflammatory response, provide a source for cytokines and chemokines that stimulate the migration and proliferation of repair cells such as fibroblasts and endothelial cells. These repair cells are responsible for angiogenesis (neovascularization), matrix deposition, and remodelling, which are processes required for wound healing and tissue regeneration [55]. The lack of toxicity of various silica bioglass formulations has been deduced from both in vivo and in vitro studies [57, 58]. In vivo testing of solid bioglass implants in the soft tissues of rats and rabbits for up to eight weeks indicate the biocompatibility. The sol–gel glass neither caused the inhibition of fibroblast growth nor elicited a marked inflammatory

Ordered Mesoporous Materials for Drug Delivery Applications

83

response. More recent histopathological study of silica xerogel [59, 60] indicated an absence of tissue irritation at the site of the implantation and absence of histological changes in the liver, kidney, lymph nodes and uterus during the implantation period. A fibrotic capsule was formed around the implant. However, this process is not considered pathological, but rather a physiological response that does not prevent implant function. Micro-organisms have a strong tendency to adhere to inert biomaterial surfaces and form biofilms by surface fixation and exopolymer production [61-63]. The first and critical step in an infection process is the bacterial adherence on the material surface. Surface chemical composition, charge, hydrophobicity, roughness, and other physical characteristics are the main factors influencing bacterial adherence and adhesion of protein, as well as other molecules on the implant surface [64, 65]. Basically, porous surface configuration results in the bacteria adhering more aggressively. The initial bacterial adherence often occurs at the surface irregularities [66]. The bacterial adherence onto different multifunctional silica-based bioceramics has recently been evaluated [61]. Staphylococcus aureus and Staphylococcus epidermidis were chosen because they cause the majority of implant-related infections. Two SiO2 mesoporous materials (MCM-41, SBA-15), an ordered SiO2-CaO-P2O5 mesoporous glass (OMG), and a biphasic magnetic bioceramic (BMB), were incubated with S. aureus and S. epidermidis for 90 min, and subsequently sonicated to quantify the number of adhered bacteria on each material. It was found that S. aureus and S. epidermidis adhered significantly less to BMB samples when compared to MCM-41, SBA-15, or OMG. However, when the material pores accessible for bacteria in each material were taken into account, the lowest bacterial adherence was found in MCM-41, and the highest in SBA-15. The results show that bacterial adherence is higher on mesoporous bioceramics, mainly due to the intergranular porosity and grain size morphology rather than to the mesoporous structure. Osteointegration is a fundamental factor in the performance of implantable bone scaffolds. The phenomenon known as bioactivity is a major advantage of silica based devices over currently used polymer-based devices. Bioactivity of silica based devices for bone tissue engineering is based on their ability to deposit a crystallized apatite phase similar to the inorganic phase of bones on the surface [13, 67-70]. Bone tissue in-growth processes at the implant site are accompanied by the formation of apatite phase from calcium and phosphate ions. The silanol groups located on the walls of silica mesoporous are able to react with physiological fluids to produce nanometre-sized carbonated apatite. The phenomenon was demonstrated in 2005 for the first time, implicating a new application of mesoporous materials as bone regenerators [69]. In vitro bioactivity studies by soaking three different mesoporous materials, SBA-15, MCM-48 and MCM-41, in simulated body fluid were carried out, revealing that an apatite-like layer is formed on the surface of SBA-15 and MCM-48 materials after 30 and 60 days, respectively, allowing their use in biomedical engineering for tissue regeneration. MCM-41 also exhibits a bioactive behaviour when its walls are doped with phosphorus [69] or when small amounts of bioactive glasses are added [71]. The textural and topographical properties of the mesoporous materials play an important role in bioactivity, i.e. the kinetics of the apatite formation can be modified [66]. The inorganic scaffold in ordered silica mesoporous materials and in bioactive glasses contains silanol groups that can be functionalised with an enormous variety of organic molecules. This process uses the ability to introduce different species in the mesoporous matrices that can be subsequently released in a controlled manner in combination with the inherent bioactivity to open new fields of application in bone tissue engineering. The

84

Spomenka Simovic and Dusan Losic

mesoporous matrices act as cellular scaffolds with embedded proteins, peptides or growth factors that would be released to the medium promoting cell proliferation and differentiation (Figure 7). The use of such bioactive porous ceramics as scaffolds for tissue engineering is still at a very preliminary stage of research; however, it can be foreseen that their use will be a routine procedure in perhaps a few years [47, 13].

Figure 7. Scheme of the possible bone regeneration from a silica mesoporous material. Reproduced from reference [67] with permission.

Detailed considerations of silica based devices for bone tissue engineering and controlled release are given in the following sections.

2.3. Applications: Bone Tissue Engineering 2.3 1. Bioactivity Silica-based mesoporous materials have high surface areas and high surface density of silanol groups (Si–OH). It is well known that silanol groups, which are present in conventional bioactive glasses, are able to react with physiological fluids to produce nanometer-sized carbonated apatites [68, 72-75]. In 2006 several in vitro bioactivity assays were performed by soaking three mesoporous materials, SBA-15, MCM-48, and MCM-41, into simulated body fluid (SBF) [74]. SBA-15 and MCM-48 developed a nanocrystalline carbonate hydroxyapatite layer onto their surfaces after 30 and 60 days, respectively. No evidence of formation of an apatite-like layer was observed onto the MCM-41 surface. The different bioactive behavior of SBA-15, MCM-48, and MCM-41 matrices is related to the concentration of silanol groups, and also to the textural and structural properties of the mesoporous matrices. MCM-41 has lower concentrations of silanol groups (c.a.2×10−3 mmol SiOH⋅m−2) than SBA-15 and MCM-48 (c.a.13×10−3 mmol SiOH⋅m−2), which could explain the absence of bioactivity after 60 days. In addition, large and accessible pores favour ionic

Ordered Mesoporous Materials for Drug Delivery Applications

85

diffusion into the mesoporous framework and consequently favour apatite formation. SBA15, with the pore size ~ 9 nm and 3D-pores structure of interconnected microporosity, and MCM-48, with 3D-cubic pores system exhibited bioactive responses. MCM-41, which has 2D-pores system with pore sizes close to 2.5 nm did not show bioactive behavior after 60 days. MCM-41 exhibits a bioactive behaviour when its walls are doped with phosphorus [69] or when small amounts of bioactive glasses are added [75]. However, the formation rate of hydroxyapatite layer on the surface of SBA-15 and MCM-48 is significantly slower (30–60 days) in comparison with the conventional sol-gel bioactive glasses (3 days) [76].

Figure 8. Bioactivity study comparing a conventional sol-gel glass in the SiO2-CaO-P2O5 system and SBA-15 mesoporous material. A) FTIR study corresponding to the surface of both materials before and after several times soaked in simulated body fluid (SBF). B) SEM and EDS studies corresponding to the surface of both materials after 3 and 30 days in SBF, respectively. Reproduced from reference [76] with permission

86

Spomenka Simovic and Dusan Losic

Figure 8 illustrates the bioactivity studies using Fourier transform infrared spectroscopy (FTIR) and scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS) techniques of a conventional sol-gel glass SiO2-CaO-P2O5 versus SBA-15 mesoporous matrix. The results indicated that the band at 600cm−1, which corresponds to amorphous calcium phosphate, began to split into a doublet at 560–600 cm−1 after 3 days in the case of conventional sol-gel glass and after 30 days in the case of SBA-15. Moreover, SEM-EDS studies revealed the formation of spherical agglomerate of particles consisting of needle-like crystals with a Ca/P ratio of 1.5 onto the materials‘ surfaces after different periods of time, confirming the formation of a calcium-deficient nanocrystalline apatite layer. These findings suggest that high surface areas and porosities are not sufficient to achieve desirable bioactivity.

2.3.2. Mesoporous Bioglasses The research group of Zhao et al. reported synthesis of bioglass SiO2–CaO–P2O5 ordered mesoporous material with the outstanding textural properties (surface area of 195–427 (m2g−1) and pore volume of 0.46–0.61 cm3/g) and ordered porous arrangements of silica-based mesoporous materials [77,78]. The synthesis of ordered mesoporous glasses was carried out using the evaporation-induced self-assembly (EISA) method in the presence of a non-ionic triblock copolymer surfactant (EO20PO70EO20) as the structure-directing agent and in the presence of tetraethyl orthosilicate (TEOS), triethyl phosphate (TEP), and calcium nitrate, Ca (NO3) 2⋅4H2O as SiO2, P2O5, and CaO sources, respectively, resulting in 2D-hexagonal pore arrangement after calcinations at 700°C [77]. The research group of Vallet-Regí has demonstrated that the textural and structural properties of this new family of mesoporous ―templated bioglasses‖ can be controlled by changing the CaO content. A progressive evolution from 2D-hexagonal to 3D-bicontinuous cubic structure (Figures 9 and 10) with an increase in the textural properties was observed when CaO content decreases [76]. These structural modifications can be explained in terms of the influence of the Ca2+ ions on the silica condensation: Ca2+ ions act as network modifiers by decreasing the network connectivity. Consequently, the inorganic/organic volume ratio of the micelle is increased with the Ca2+ content, thus increasing the curvature ratio of the surfactant micelles and contributing to the formation of hexagonal phases rather than cubic ones. The possibility of tailoring both structural and textural features of mesoporous glasses is undoubtedly an attractive advance towards the development of biomaterials that are able to fulfil the essential requirements for specific biomedical applications. It has been demonstrated that the new family of mesoporous glasses has enhanced bioactivity, with faster apatite formation in comparison with conventional bioactive sol-gel glasses. The kinetics of apatite formation in mesoporous glasses is governed by their textural and structural properties, as opposed to conventional bioactive glasses, where the formation kinetics depends on compositional and textural properties.

Ordered Mesoporous Materials for Drug Delivery Applications

87

Figure 9. TEM images and their corresponding FT diffractograms of mesoporous glasses in the SiO2CaO-P2O5 system with different CaO amounts. (A), (B) TEM images taken with the electron beam parallel and perpendicular to the pore channels of a 2D hexagonal structure corresponding to a mesoporous glass with 58SiO2-37 % CaO-5%P2O5 composition; (C), (D) TEM images taken in the 100 and 111 directions of 3D bicontinuous cubic structure with Ia–3d space group corresponding to a mesoporous glass of 85SiO2-10%CaO-5%P2O5 composition. Reproduced from reference [76] with permission.

88

Spomenka Simovic and Dusan Losic

Figure 10. Evolution of structural characteristics as function of the CaO content in the SiO2–CaO–P2O5 mesoporous glasses by TEM study. Images and FT diffractograms with schematic representing the pore arrangement structure of mesoporous glasses. Reproduced from reference [76] the permission.

Figure 11. In vitro bioactive study in SBF of a mesoporous glass with 10% in mol of CaO content and 3D-cubic structure by FTIR and TEM techniques. Reproduced from reference [80] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

89

By tailoring the structural and textural properties at the nanometric level, a variety of bioactive responses can be observed. Mesoporous glasses with low CaO content (10% in mol) 85SiO2-10%CaO-5%P2O5 and 3D-bicontinuos cubic structure have shown the fastest HCA formation observed up to date in a silica based bioactive material. This behaviour could be explained in terms of the mesoporous structure and the textural properties, i.e. the 3D pore system provides not only high surface area and porosity, but also easier interchange of ions, increased mass transport and diffusion processes and hence, a higher crystallization rate (Figure 11) [68]. Bioactivity of mesoporous glass with 3D bicontinuous cubic structure and composition is depicted in Figure 12. Transmission electron microscopy (TEM) revealed only 4 hours for the formation of the crystalline apatite. The bioactive response is faster than for that of mesoporous glasses with higher CaO content and 2D-hexagonal structure. Figure 12 also shows that the Ca2+ concentration and pH remain almost constant to the values of 7.45 and 100 ppm, respectively during soaking time [68]. This is important since the high local pH values are cytotoxic.

Figure 12. Bioactive in vitro study corresponding to an ordered mesoporous glass with 3D bicontinuous cubic structure and 85SiO2-10%CaO-5%P2O5 composition. (Inset) schematic representation of 3D cubic pore arrangement A) Variation of calcium content and pH values of SBF after soaking the materials during different periods. B) HRTEM study corresponding to the surface of mesoporous glasses after 4 hours soaked in SBF. Low magnification image indicates the needle-like crystals formation on the surface of mesoporous glass. Higher magnification and FT transform showing the (002), (211), and (112) reflections corresponding to d-spacings of 0.34, 0.28, and 0.27 nm of an apatite phase. Reproduced from reference [80] with permission.

Mesoporous glasses with 2D-hexagonal pore arrangements and high CaO content (37% in mol) provided a biomimetic mechanism where a sequential transition from amorphous calcium phosphate (ACP) to octacalcium phosphate (OCP) and to calcium deficient carbonate hydroxyapatite (CDHA) maturation, similar to the in vivo bone biomineralization process was observed by TEM imaging (Figure 13) [79]. This is the first time such an ACPOCP-CDHA maturation sequence has been observed in bioceramics. The open channel array and the textural parameters of 2D-hexagonal structure together with high CaO content allow an intense Ca2+–H3O+ exchange that results in local acid pH values that favor the OCP phase formation. ―Templated bioglasses‖ with 2D-hexagonal mesoporous arrangement and high

90

Spomenka Simovic and Dusan Losic

CaO content (37% in mol) allow an intense Ca2+–H3O+ exchange due to the open channel array, the high textural parameters and the relatively high CaO content. The resulting surface silanol groups eventually condense into an acid hydrated silica layer with local pH values of 6.5 during the first stage, which favour the OCP formation. The local acid pH does not occur in the surface of conventional bioglasses; and it can explain the reason why OCP has never been observed.

Figure 13. Bioactive in vitro study corresponding to an ordered mesoporous glass with 2D hexagonal structure and 58SiO2-37%CaO-5%P2O5 composition. (a) Schematic representation of 2D hexagonal pore arrangement. (b) Variation of pH as a function of time at the bioceramic surface and SBF, highlighting the pH range where OCP phase is more stable. (c) HRTEM study after different times of incubation showing a sequential transition from ACP to OCP and to CDHA. In each instance, a low magnification image, a higher magnification and its corresponding FT transform are shown. Reproduced from reference [80] with permission.

Biomimetism in templated bioglasses is a new property derived from their specific textural and structural characteristics. A sequential transition from amorphous calcium phosphate (ACP)–octacalcium phosphate (OCP)–calcium deficient carbonatehydroxyapatite (CDHA) maturation, similar to the in vivo biomineralization process, has been observed for ―templated bioglasses” [79]. Usually, all bioceramics obtained so far develop a CHDA phase through the direct crystallization of previously precipitated ACP [81], without formation of

Ordered Mesoporous Materials for Drug Delivery Applications

91

OCP phase which is formed in natural bone biomineralization process [82-85]. OCP is a metastable phase and will appear only if the pH in the crystallization system is below 7. The influence of pore size of mesostructured CaO−SiO2 biomaterial on promoting HAP crystallization and growth has been reported by Deng et al. [86] It has been found that by tuning the pore size below or above the critical size, the HAP nucleation process can be switched off or on inside the mesopores. The pore size of 2.7 nm was below the threshold of the critical nucleation size, and the nucleation was suppressed within the mesopores due to its thermodynamic instability. However, the prohibited nucleation enabled Ca ions to freely transport to the outer surface to form a dense layer of crystalline HAP rapidly within one day. A number of composite silica-HAP materials with enhanced bioactivity have been fabricated. For instance, a novel material hexagonal mesoporous silica-hydroxyapatite (HMSHA) has been developed based on self-assembly of nanohydroxyapatite in mesoporous silica in situ [87]. The mesoporous amorphous calcium silicate (MACS) was synthesized using mesoporous silica SBA-15 as both the template and silicon source, and Ca (NO3)2 as the calcium source. The MACS shows a well-defined mesoporous structure with high sp. surface area. Owing to the high specific surface area and pore volume, the MACS had a significantly enhanced bone-forming bioactivity compared with the conventional amorphous CaSiO3 and develop a carbonate-containing hydroxyapatite (HCA) layer on the surface after being immersed in SBF for 4 h [88]. Andersson et al, reported synthesis of silica-calcium phosphate (hydroxyapatite or tricalcium phosphate) composite materials with high in vitro bioactivity [89]. Hierarchically 3D porous mesoporous bioactive glasses that scaffold with different chemical compositions have been prepared by a combination of polyurethane sponge and non-ionic block copolymer EO20PO70EO20 (P123) surfactant as co-templates and evaporation-induced self-assembly process. Scaffolds have the interconnected macroporous network (pore diameter of 200–400 m) and mesoporous walls (mesopore size of 4.9 nm). In vitro bioactivity was dependent on the chemical composition. The introduction of Ca and P into the network facilitates the formation of hydroxylcarbonate apatite layer on the surfaces [90].

2.3.3. Mesoporous Bioglasses Microspheres Synthesis methods for SiO2-based mesoporous materials described in previous sections lead to irregularly shaped particles, and consequently variable surface area inconvenient for therapeutic scaffolds where reproducible drug release is required. Control over both the external morphology and the mesostructure of the material leads to the development of more reliable and reproducible drug delivery systems [91]. Microspheres are widely accepted from a clinical point of view because of their ability to form suspensions that can be injected or dispersed into bone cements [92-94]. The synthesis of bioactive mesoporous microspheres in the system SiO2−CaO−P2O5 with an ordered mesopore arrangement has recently been reported [91]. The microspheres (SEM image is presented in Figure 16) have been synthesized by means of the evaporation-induced self-assembly (EISA) method followed by an aerosol-assisted route. This process is based on the use of a piezoelectric ceramic, which is placed below the precursor solution of the material. When the piezoelectric transducer is excited near its own resonance frequency, a geyser is formed at the surface of the liquid. This geyser produces ultrafine droplets, which

92

Spomenka Simovic and Dusan Losic

form an aerosol. N2 gas is used as the carrier to convey the aerosol to the preheating zone, where the solvent evaporation occurs triggering the micellar self-assembly. Thereafter, the droplets go through the pyrolisis zone. The residence time of the particles in the hightemperature zone is controlled by the gas flow. Particles are collected outside the furnace with an electrostatic filter, consisting of a thin tungsten wire suspended in the center of a tubular stainless steel collection plate. Despite size, poly dispersity (Figure 14) spheres synthesized by this technique are not aggregated independently of the surfactant used during the synthesis. Pure SiO2 microspheres exhibit mesopore ordering independently of the surfactant used. However, the addition of cationic surfactants such as CTAB, seems to form hexagonal arrangements with higher ordering degree than those resulting from the addition of nonionic triblock copolymers (P123 and F127). Though pure SiO2 microspheres evidence a mesoporous arrangement that depends on the surfactant added, as well as high surface area and porosity values, only SiO2−CaO−P2O5 mesoporous microspheres develop an apatite like phase in contact with SBF, showing that the presence of CaO (P2O5 only modifies the kinetic growth of the new formed apatite phase) and high textural values (surface area and porosity) are mandatory for the expression of bioactive behavior. A new layer of apatite crystallizes on the surface of the mesoporous materials after 1 day of immersion in SBF, whereas no changes could be observed in materials obtained without surfactant addition. The mesoporous structure closely depends on the structure-directing agent as well as its interaction with the Ca2+ cations during the mesophase formation. Among the different tested surfactants, the triblock copolymer F127 leads to hexagonal ordered structures for low CaO contents, P123 leads to wormlike mesoporous structures for any CaO content, whereas the ionic surfactant cetyltrimethyl ammonium bromide (C16TAB) does not produce accessible mesopores at the external surface, for any CaO content.

Figure 14. SEM micrograph of SiO2−CaO−P2O5 microspheres; Magnification 5000×. Inset shows particle size distribution determined by DLS measurements. Reproduced from reference [91] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

93

2.4. Silica Based Mesoporous Materials as Drug Delivery Systems In addition to the bioactivity which provides ability to silica based mesoporous materials to act as scaffolds and drug carriers for bone tissue engineering, outstanding textural parameters [13,95] enable their capability to act as carriers for controlled drug release [96] which is the basis of their promising characteristics as drug delivery systems [13,95]. In this sense, different chemical surface functionalizations for a more efficient loading and delivery of drugs [97] proteins [98] and DNA [99] have been reported. Moreover, complex systems such as stimuli responsive drug delivery systems [100,101] and composites of silica nanoparticles associated with biopolymers [102-104] have been developed from mesoporous materials. In addition, the fabrication of silica-based nanocapsules has opened new perspectives to the technology of biomolecules immobilization [105] and bacteria encapsulation [106]. Silica based mesoporous materials have unique features that make them excellent candidates for controlled drug-delivery systems [13]: a) An ordered pore network, which is homogeneous in size and allows fine control of the drug load and release kinetics; b) A high pore volume to host the required amount of pharmaceuticals; c) A high surface area, which implies high potential for drug adsorption; d) A silanol-containing surface that can be functionalized to allow better control over drug loading and release. Since 1991, when Mobil Oil Corporation synthesised the silica-based MCM-41, highly ordered mesoporous materials have been attracting attention for applications in fields such as catalysis, lasers, sensors, solar cells and so forth have been proposed and/or developed [1-3]. In 2001 mesoporous silicas were proposed for the first time as a drug delivery system and ibuprofen, a common anti-inflammatory, was loaded into the pore channels of MCM-41 and subsequently released in a sustained manner [96]. Since then, this research field has experienced a significant growth and much effort has been devoted to tailoring the nanostructure and textural properties (i.e. pore diameter, surface area and pore volume) of mesoporous materials that allow control over drug loading and release. Due to the outstanding features of their surface and porosity, ordered mesoporous materials have shown to be excellent candidates for two biomedical applications: 1) bone tissue regeneration and 2) local and oral drug delivery systems. In fact silica-based mesoporous materials are able to incorporate high dosages of drugs into the mesopores [96]. Moreover, their silanol-containing surface can be functionalised, allowing better control over the drug release, which depends on the chemical nature of the functional group attached to the surface [13,95]. Since 2001, when MCM-41 was proposed for the first time as controlled delivery system [96], much research effort has been directed towards changing the chemical properties of mesoporous carriers at the nanometer scale to achieve better control over loading and release of molecules. Drug loading is usually performed by impregnation methods, i.e. soaking the mesoporous carrier into a concentrated molecule solution. Various therapeutics relevant to oral and bone implant drug delivery have been successfully confined into mesoporous silicas.

94

Spomenka Simovic and Dusan Losic

Drug examples are: ibuprofen [96,107-109], amoxicillin [110], gentamicin [111,112], erythromycin [108, 112], vancomycin [81], naproxen [113], aspirin [114,115], diflunisal [116], captopril [117], itraconazole [118], and alendronate [97,119]. Other guest molecules consisted of biologically active species, such as proteins (bovine serum albumin (BSA) [24, 120,121] and certain amino acids (L-tryptophan (L-Trp) [122]).

2.4.1. Drug Loading in Mesoporous Materials In oral therapy for drugs with short half lives, mesoporous silica materials are particularly suitable for prolonged drug release or once-daily therapy [13]. The important issue is the amount of drugs that can be delivered in a certain time period. For instance, in famotidine loaded in mesoporous silica materials, between 40 and 500 mg of drug could be adsorbed and released from a 1.5-g mesoporous tablet [13,123]. The indicated dose of famotidine is 40 mg per day for the treatment gastric ulcers and 500 mg per day for the treatment of ZollingerEllison syndrome [124]. Another example is captopril-loaded MCM-41[13]; captopril is an orally active inhibitor of the angiotensin-converting enzyme and is used for the treatment of hypertension and congestive heart failure. The recommended daily dosage for captopril ranges between 50 and 100 mg per day. Up to 32 % of captopril was incorporated into MCM41 matrices, indicating that a 300-mg tablet could contain the maximum daily dosage, and drug release was complete within 24 h in simulated stomach fluid. Table 1. Mesoporous matrices proposed for implantable (bone) drug-delivery systems and daily dose for alternative administration Mesoporous matrix SBA-15 gentamicin SBA15/PLGA gentamicin SBA-15 erythromycin SBA-15 amoxicillin SBA-15-NH2 alendronate MCM-41-NH2 alendronate MCM-41 ibuprofen

Drug Daily Dose 150-300 mg[a] 150-300 mg[a] 1.5-3 g[b] 1.5-2 g[b] 5-10 mg[b] 5-10 mg[b] 0.9-1.2 g[b]

Dosage [g][c] 2 4.50 3.4 2.50 2 2.5 7.0

[a] Dose for gentamicin recommended for intravenous administration. [b] Doses orally administered. These doses take into account the bioavailability of the drugs, which are 60, 80, 0.7, and 92 % for erythromycin, amoxicillin, alendronate, and ibuprofen, respectively. [c] Dosages contained into 10 g mesoporous silica, which is the approximate amount of silica-based material to graft a bone defect in a femur fracture. Smaller periodontal defects usually require 3-5g silica-based glass graft. Reproduced from reference [13] with permission

The drug release pattern from implantable delivery systems is different: drug release must extend over weeks. Table 1 lists some of the loaded drugs into 10g (amount appropriate for grafting to a bone defect resulting from, for example, a femur fracture) of mesoporous material. Bacterial infection is the most common complication after orthopaedic implant surgery [125]. Acute infection or chronic osteomyelitis develops in 5-33 % of surgeries. Bone marrow swells during infection and reduces blood supply due to the compression of blood vessels in

Ordered Mesoporous Materials for Drug Delivery Applications

95

the bone marrow. The infection can also spread to the surrounding muscle tissues. To reduce chances of serious complications, antibiotic therapy is indicated for 6-8 weeks after surgery. Systemic antibiotic applications (intravenous, intramuscular, oral) can be toxic and ineffective in reaching infected tissues near the implant. The design of therapeutic implant devices that contain actives, such as antibiotics (e.g. gentamycin) has a number of advantages, such as fewer side-effects and efficient therapeutic effects.

2.4.2. Drug Release Profiles and Release Kinetics The most common release profiles for mesoporous materials are shown in Figure 15.

Figure 15. Typical drug delivery release profiles of mesoporous materials. Profile a) is commonly observed for nonfunctionalized matrices, which initially display a burst effect followed by a very slow drug release. This kind of profile can be useful when an immediate high dose is required, for example, for acute infections or inflammations. Profile b) is associated with diffusion or dissolution processes, and generally follows first-order kinetics with respect to drug concentration. Profile c) corresponds to zero-order kinetics; that is, the release process is only dependent on time. This profile is highly desirable for long-term drug-delivery systems. Stimulus-responsive system (d) where the release rate can be controlled by external stimuli (e.g., pH, temperature, magnetic field). Reproduces from reference [13] with permission

In traditional drug delivery systems, the drug release kinetics is governed by dissolution (in the case of poorly soluble drugs) and/or diffusion [24,126].The most common diffusion mechanism is the Higuchi process that relates the total amount of delivered drug with the square root of the releasing time, applicable to many drug delivery forms [127] . For drug or protein-loaded implant mesoporous silica matrices in physiological environments, release kinetics are not dependent on the dissolution of the matrix which is practically insoluble in the plasma pH conditions [128], but rather by diffusion of the drug through the mesopore channels with large surface areas. Under these conditions, the drug or protein release mechanisms can be described by both Fick's laws (Eq. (1) ) of the diffusion of fluids and the Noyes–Whitney equation (Eq. (2) ), describing the diffusion of molecules through porous matrices

Spomenka Simovic and Dusan Losic

96

(1) (2) where J is the flux through a continuous infinite porous matrix, Ct is the drug concentration at the time t, D is the diffusion coefficient and x is the direction of flux in one-dimensional approach. CS is the molecule solubility in the matrix in the tested conditions, S is the surface area in contact with the adsorbed molecules and the constant K gives information of the diffusion coefficient, D, and the dimensions and tortuosity of the porous framework. The combination of Eqs. ( 1 ) and ( 2 ) in the appropriate manner yields that the relative amount of released drug, Qt/Q0, shows the following equation [129]:

(3) where Qt and Q0 are, respectively, the drug or protein amount at time t and the initial amount of drug or protein in the porous matrix and K′ and k are the release constants that are independent of the drug or protein concentration in the ordered matrix as well as in the solvent accessible area. The drug or protein release is influenced by the surface area exposed to the molecules inside the mesopores. It has been found that for larger surface areas release follows a firstorder kinetics since the exponential part of the equation becomes larger. This is the case of MCM-41 materials where the surface area is around 1000 m2/g. When surface areas are lower, the exponential part of Eq. (3) is smaller than the linear part and the global effect is that the drug release follows zero-order kinetics, as can be found for SBA-15 materials. This effect is also found for functionalized matrices, where essentially surface areas are reduced but the overall effect is maintained [24]. When the mesoporous matrices show large mesopore size and the released molecule is big enough, it can be assumed that solvent accessible area does not significantly change along the release period and that no equilibrium conditions are achieved. Usually, for materials with high surface areas and small mesopores, the diffusion of the protein to the release environment is a surface-dependent process that is determined by the Noyes–Whitney equation. We can assume that the drug release mechanism is due to diffusive transport through the ordered array of mesopores since the matrix is virtually insoluble in an aqueous environment. In materials with high surface areas and small mesopores, the diffusion of drug molecules to the liquid media is a surface-dependent phenomenon that can be predicted by first-order kinetics [97]. According to the Fick‘s first law, the equation 3 can be rearranged: Qt/Qo = e-k1t

(4)

where Qt and Qo represent the drug amount at the time t and initial amount, respectively, k 1 is the first order release constant independent on the drug concentration and contains information about solvent accessibility and drug diffusion through mesopores.

Ordered Mesoporous Materials for Drug Delivery Applications

97

When the surface area is smaller and pore size larger, it can be assumed that solvent accessible area does not significantly change during release period and no equilibrium conditions are achieved. Drug diffusion in this case can be explained by zero order release [97,129]: Qt/Qo = k0 t

(5)

where k0 is zero order constant independent on drug concentration and solvent accessible area.

2.4.3. Drug Delivery Systems In Bone Tissue Engineering As discussed in the previous section, silica-based mesoporous materials have shown a bioactive behaviour [95, 13]. This attribute opened the possibility of using these materials for bone tissue regeneration and dental reconstruction. Since bioactive kinetics of conventional mesoporous materials such as SBA-15, MCM-41, or MCM-48 is too slow to be acceptable from a clinical point of view, advances have been made, and highly bioactive mesoporous materials in the system SiO2−CaO−P2O5 synthetised [13,96-98]. These bioactive templated glasses have been confirmed as macro-mesoporous materials for scaffolds in tissue engineering [92, 93], and the first drug delivery tests [94] have shown promising results.

Figure 16. Different bioceramics types employed in bone substitution or regeneration, showed as grains, dense pieces, porous pieces, injectables and thin films. Reproduced from reference [67] with permission.

98

Spomenka Simovic and Dusan Losic

Bioceramics for bone tissue engineering may be in granulated form, or with predefined shapes, in either porous [130] or dense [131] pieces (Figure 16). The demand for bioceramics in injectable form is also increasing [131] since it greatly simplifies surgical practice, and can even be reclassified as non-invasive. For applications that require certain mechanical properties of metals and alloys, the trend is to coat the metallic prostheses with a ceramic layer [133] to achieve a better and faster osteointegration and to reduce the release of ions from the implant alloy. Bioceramic materials should be non-toxic, non- inflammatory and should promote osteogenesis and consequently implant-bone attachment. An added value to the production of these ceramics would be inclusion of drugs such as antibiotics, antiinflammation drugs or anticarcinogens. As already explained, post-implant surgery infections are the major reasons for implant failures. In conventional therapy patients are subjected to oral and parentheral therapy several weeks after surgery. However, oral and parentheral therapy is not efficient since many drugs degrade upon application prior to reaching target tissue around implant, i.e. the drug concentration in target area around implant is low. If drug(s) are included in the implant, and subsequently released in the target area, the local drug concentration is increased and therapy is more efficient. Various drugs can be encapsulated, e.g. not only drugs required to prevent infection/inflammation, but also anticancer drugs and certain peptides, proteins and growth factors that enhance bone-growth. A scheme of the multiple possibilities of bioactive ordered mesoporous silicas as implant materials with drug and protein release capabilities is presented in Figure 17.

Figure 17. Scheme of the multiple possibilities of bioactive ordered mesoporous silicas as implant materials with drug and protein release capabilities. Reproduced from reference [24] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

99

The porous design in the ceramic material, i.e. number, size, shape, distribution, connectivity and potential functionalisation of the pore walls are key parameters that influence drug release profile from the mesopores. The dimensions of the drug molecules that might be of interest in clinical applications for bone implants are in the range of one nanometre (Figure 18) or several nanometers in the case of larger proteins

Figure 18. The size of several drug molecules Reproduced from reference [67] with permission.

The bioactive glass containing gentamicin was in vivo tested in New Zealand rabbit femurs during 1, 4, 8 and 12 weeks to study their biological response. The bone response to the implant was of perfect osteointegration, cortical and sponge bone tissue growth and restoration and partial resorption of the implant of medium length. Figure 19 shows the gentamicin levels detected in different organs (liver, kidney and lung) and proximal and distal bone as a function of implantation time. The local gentamicin levels detected in bone tissue were above the minimal inhibitory gentamicin concentration (cmi) and there was a progressive decrease in gentamicin levels in the bone tissue with time, but the concentration were above the cmi till the end of the experiment [24]. The hybrid structures (mesoporous silica, hydroxyapatite and PLGA) were proposed for bone repair and controlled release of gentamicin [112].

100

Spomenka Simovic and Dusan Losic

Figure 19. Implantation of gentamicin-loaded bioactive glass pieces. Radiographs of rabbit femur after 1 and 12 weeks of implant fixation and gentamicin concentration in different organs and bones vs. implantation time. Reproduced from reference [24] with permission.

2.4.4. Parameters that Influence Drug Release from Ordered Mesoporous Silica Based Materials One of the most common employed methods for drug loading of mesoporous silica is impregnation. The loading solvent must be selected according to the solubility of the drug to obtain the highest possible amount of drug adsorbed. For example, when using polar drug molecules such as amoxicillin or gentamicin, a polar solvent like water has to be used. In the case of non-polar drugs like ibuprofen, the solvent has to be also non-polar, like hexane. There are also intermediate cases, such as erythromycin. In this last case, acetonitrile can be employed as loading solvent polar, like hexane. There are also intermediate cases, such as erythromycin where acetonitrile can be employed as loading solvent. The analysis of nitrogen isotherm before and after drug loading gives valuable information about the variation of the textural parameters due to the drug confinement in the mesopores. After drug loading there is always a reduction of the pore diameter and volume, evidencing the confinement of the drug into the mesopore channels. Ibuprofen molecules were confined into MCM-41 mesoporous material [24].The surface area and total pore volume of the starting MCM-41 were 1157 m2/g and 0.98 cm3/g, respectively. After loading with ibuprofen, these values decrease to 688 m2/g and 0.39 cm3/g. The major factors that govern drug adsorption and release from the mesoporous silica materials are: 1. The textural properties (i.e.pore diameter, surface area, and pore volume) 2. Surface chemistry of silica walls modiffied using different organic groups

Ordered Mesoporous Materials for Drug Delivery Applications

101

2.4.4.1. Pore diameter The process of drug loading into the mesoporous silica is based on the adsorptive properties of mesoporous materials. The pore size of mesoporous materials determines the size of the molecule that can be adsorbed into the mesopores. Pore diameters slightly larger than the drug molecule dimensions (pore/drug size ratio>1) can allow drug loading [96]. Hata et al. reported for the first time that the pores of mesoporous silicas and solvent polarity are fundamental factors for drug loading. The authors encapsulated the anticancer drug taxol into FSM-type mesoporous silica when the pore size was larger than 1.8 nm [134]. The amount of drug loaded in the porous matrix depended significantly on the solvent used, the pH value during the process and the drug concentration in the solution. Drug release from mesoporous silica was evidenced for the first time by the confinement of ibuprofen in two MCM-41 matrices with various pore diameters (1.5, 1.6, 1.9, 2.5 nm) [96,110]. The loading of ibuprofen in MCM-41 was size dependent and reached a value of about 30 %. The full amount of ibuprofen loaded in the MCM-41 matrix was released within three days [135]. Figures 20 and 21 show the different amounts of ibuprofen loaded in four MCM-41 matrices with different pore sizes [67]. The concept of pore size as a kinetic-release controller is applicable to both 2D hexagonal structures, such as MCM-41, and 3D cubic structures, such as MCM-48 mesoporous materials. The research group of Vallet-Regi has studied incorporation of ibuprofen into MCM-48 which has a cubic structure with Ia3d symmetry and a pore size of 3.6 nm, and into LP-Ia3d, which possesses the same symmetry and a larger pore size of 5.7 nm [108]. Ibuprofen has faster release from LP-Ia3d, which indicates the pore size effect.

Figure 20. Release patterns of adsorbed ibuprofen in MCM-41 matrices with four different pore sizes. Reproduced from reference [67] with permission.

102

Spomenka Simovic and Dusan Losic

Andersson et al. [136] reported the influence of the pore connectivity, geometry, and matrix degradation in aqueous media, as well as the pore-size effect on drug release kinetics. Structures containing cagelike or corrugated pores (which are essentially equivalent to larger pores connected through smaller windows) seem to be excellent matrices for long-term drug delivery.

Figure 21. Plot of the maximum load of ibuprofen in four MCM-41 matrices with different pore sizes. The inset table provides data of the ibuprofen load that can be host in every of four matrices together with their release periods. Asterisk (*) indicates incomplete delivery; Reproduces from reference [167] with permission. Reproduced from reference [67] with permission.

Qu et al. [117] reported the influence of mesoporous matrix dimensions on the release of captopril from several 2D hexagonal structures (MCM-4112, MCM-4116, and SBA-15). The drug release was dependent upon surface area available: when the microstructure of the matrix consists of small spherical particles (120-250 nm), the drug release is faster than that observed in bigger rodlike particles (20 m in length), independent of the mesopore size. The Vallet-Regi research group also reported important differences at the macroscopic level in the kinetic release of amoxicillin from powder and disk-shaped samples [110]. Another point to be considered is the matrix structure. Although all mesoporous materials are diverse, their structural types, in terms of channel and cavity topology, are very few; we can distinguish three large groups: 1) MCM-41 [67] or SBA-15 [30] both with unidirectional pores and, in the case of SBA-15, sometimes with a secondary micropore system that interconnect the hexagonal channels; 2) MCM-48 [26] from the M41S family, which exhibits a three-dimensional pore system, formed by intersected longitudinal pores along the three directions in space; and 3) structures with a porous system basically formed by pseudospherical cavities interconnected by different pore configurations, which could

Ordered Mesoporous Materials for Drug Delivery Applications

103

essentially be described as short channels or even windows between cavities, with diameters similar to those of micropores [29]. According to the studies reported, the effect of pore connection on drug release kinetics is not significant [137]. Pore diameter is a limiting factor when the adsorption of large molecules, such as proteins, is considered. Serum albumins, such as bovine surum albumine (BSA) exhibit an average length of 10 nm and width of 6 nm [138,139]. The adsorption of globular proteins, such as bovine serum albumin (BSA) onto MCM-41 matrices, which exhibits pore diameters in the 2–5 nm range, has been reported [24,140]. The protein was adsorbed on the outer surface, not inside of the pores. It has been recently reported that increased mesopore diameters of SBA-15 materials ranged from 8.2 to 11.4 nm can be achieved by employing hydrothermal treatments ranging from 1 to 7 days, respectively. BSA loading tests revealed that there was a pore size effect in terms that the larger pore diameters provide better loading. SBA-15 with pore diametes 8.2 nm, loaded 151 mg/g of BSA and pore diameters of 9.5 nm, 10.5 nm, and 11.4 nm increased loading to 234 mg/g, 242 mg/g, and 270 mg/g, respectively. Release patterns exhibited and initial burst profile where more than 90 % of the adsorbed protein was quickly released during the first 24 hours, followed by a sustained release and the complete delivery was achieved after 192 hours of assay [24].

2.4.4.2. Surface Area Surface area critically determines the amount of drug loaded: the higher the surface area, the higher the amount of drug loaded. This was confirmed when several MCM-41 matrices with different surface areas were employed as ibuprofen delivery systems [135]. MCM-41 mesoporous materials exhibiting surface areas of 768, 936, 1087, and 1157 m2/g loaded 110, 190, 230, and 340 mg/g of ibuprofen, respectively, evidencing an increase in the drug loading with the surface area increase. Furthermore, two mesoporous materials, MCM-41 and SBA15 which exhibit the same structure (2D-hexagonal, p6mm symmetry) but different surface areas (1157 m2/g for MCM-41 and 719 m2/g for SBA-15) were tested as alendronate delivery systems [97]. The amount of alendronate loaded into MCM-41 (139 mg/g) was higher than that into SBA-15 (83 mg/g), confirming the dependence of drug loading on surface area. The alendronate release mechanism from MCM-41 and SBA-15 is due to diffusive transport through the ordered array of mesopores. After 24h of essay the percentage of alendronate released is in the same range (≈55 %) for both host matrices. Release profiles from both mesoporous carriers (NaCl 0.9 %, pH 7.4) exhibited an initial burst effect when ca. 20 % and ca. 50 % of alendronate loaded were quickly released from MCM-41 and SBA-15, respectively. Afterwards, alendronate was released to the delivery medium in a sustained manner, following first-order kinetics for MCM-41 and zero-order or linear kinetics for SBA15 materials. Combining high and regular porosity of mesoporous materials with the presence of organic groups has recently been reported [141,142] The final compounds are metal-organic frameworks (MOFs) MIL (Materials Institute Lavoisier); materials with large SBET and consequently as high drug loading and controlled release. These materials are denoted are shown in Figure 22 A. MIL-100 and MIL-101 have surface areas (SBET) 3-5 times larger than MCM-41 (3340 and 5510 m2g-1), respectively. MIL-100 is able to adsorb 350 mg g-1IBU, and MIL-101 loads 1400 mgg-1 under the same conditions [13]. MCM-41 and MIL-100 materials showed similar

104

Spomenka Simovic and Dusan Losic

ibuprofen release kinetics, and the drug content in MIL-101 is 4 times larger than in MCM41. High drug adsorption levels are a consequence of high surface area of the external micropores. The IBU release profiles are compared in Figure 22 B for both the MIL materials and MCM-41. As it is suggested that most of the drug is adsorbed in the outermost micropores of MIL materials and only a small amount is retained in the closed mesopore cavities, the drug release follows different delivery kinetics from that of MCM-41. Two sections can be observed in the release profiles of the MIL materials, corresponding to the release from micropores and from mesopores. In MCM-41, as the drug is essentially loaded into the mesopore channels, the release profile only shows one exponential profile. The release from MCM-41 and MIL-100 is very similar even though the total surface area is different, whereas the release for MIL-101, with more open mesopores, is faster (Figure 22).

Figure 22. a)Structures of MIL-100 and MIL-101 built up from trimers of chromium octahedra as well as 1,3,5-BTC (benzene-1,3,5-tricarboxylic acid) and 1,4-BDC (benzene-1,4-dicarboxylic acid), respectively. b) IBU release as a function of time for MIL-101, MIL-100, and MCM-41. Reproduced from reference [13] with permission.

2.4.4.3. Pore Volume When the confinement of really large molecules is targeted, the pore volume available to host the guest molecule plays an important role in loading [143]. Mesostructured cellular foams (MCFs) have been tested as host matrices for the adsorption of several enzymes and proteins [144,145]. These matrices seem suitable to be used as delivery systems for large molecules, such as proteins. The amount of BSA loaded was higher in MCF (240 mg/g) than in SBA-15 (150 mg/g), following the same trend as pore volume, being 1.9 cm3/g and 1.1 cm3/g for MCF and SBA-15, respectively.

2.4.4.4. Surface Functionalization The pore walls in ceramic matrices can be functionalised with a wide range of chemical species in order to modify their adsorption and release properties. The lack of order in the

Ordered Mesoporous Materials for Drug Delivery Applications

105

space configuration of the tetrahedral lattice of silica matrices results in a large number of connectivity defects, that is, not all the tetrahedrons are connected to another four tetrahedrons sharing oxygen atoms. If an oxygen atom positioned at a tetrahedron corner is not shared with a neighbouring tetrahedron, a silanol group is formed. The presence of large amounts of connectivity defects in mesoporous materials is a direct consequence of their formation mechanism. The interaction between the tensioactive agent and the silicate oligomers in dissolution takes place through the silanol groups SiOH or through the corresponding anion, SiO¯. The concentration of SiOH groups in the material after eliminating the tensioactive agent depends also on the method chosen for this removal. Usually, calcination reduces the concentration of defects, since it promotes the condensation of Si--OH groups, in particular those that share hydrogen bonds. On the other hand, the solvent extraction of the tensioactive agent does not significantly modify the amount of silanol groups. The Si--OH groups exert a remarkable influence on the properties of the material. Generally, their affinity for polar molecules increases with the silanol concentration, but these groups can also react with a large variety of chemical products yielding covalent bonds of the Si-O-R type. This fact allows them to attach or anchor different chemical species on the material surface, i.e. to functionalise their surface [134]. If the H atom in silanol is replaced by chemical species R, which can be linked to the oxygen atom by a covalent bond, a whole family of hybrid materials can be obtained. The most common cases are those in which R is an organic functional group, that is, chemical species of the type of Si-O-Si-R. Functionalisations can be carried out using in situ [146] and ex situ [147,148] techniques. The functional groups can be attached or anchored to the scaffold of the mesoporous material during the material synthesis, in a one-pot method, but also during a later post-synthesis stage. The main difference between methods is the additional stage of the functionalising precursor. In the former the alcoxy groups are hydrolysed and condensed with the silica scaffold precursors, which are usually also silicon alkoxydes; in the latter, the condensation reaction takes place between the functionalising precursor and the silanol groups present in the pore walls. In one-pot methods the organic groups R are linked to silicon atoms in the walls and to the inner part of the silica wall; therefore the functionalisation degree is reduced. Post-synthesis methods ensure that the modifying agents are in the outer surface of the pores, leading to a larger functionalisation degree [107]. The studies reported so far indicate that the postsynthesis functionalization method (i.e. functionalization after the removal of surfactant) leads to better results than the co-condensation method [107,121]. Recently Zeng et al. introduced a controversy in this issue, stating that owing to a higher degree of order and the uniform distribution of functional groups, a better control of drug release can be obtained by using the co-condensation method, rather than postsynthesis or solvothermal processes [114, 149]. The ionic interaction between the carboxy groups in ibuprofen and the amino groups on the silica matrix allows controlled release of ibuprofen from amino-functionalized mesoporous silica [107,121,150]. Ibuprofen molecules adsorbed in MCM-41 silica with and without amino group functionalisation have been characterised by 13C NMR spectroscopy. The 13C MAS single-pulse or cross-polarisation NMR spectra, as well as the 1H MAS NMR spectra show an extremely high mobility of the ibuprofen molecules when the matrix is not modified. It suggests the absence of any interactions between the ibuprofen molecules and the silica surface, despite the presence of a COOH function. This might be explained by the association of ibuprofen molecules into cyclic hydrogen-bonded dimmers: the dimer

106

Spomenka Simovic and Dusan Losic

configuration is formed by an intermolecular hydrogen bond through the carboxy groups. Babonneau et al. [151,152] showed by NMR studies that the ibuprofen molecules exhibit high mobility, which is incompatible with the behavior of molecules linked at the pore wall of mesoporous silica. On the contrary, this mobility agrees with dimers physisorbed on the surface and also explains the predominant initial burst effect observed for these systems. The dimmers are also present in the crystallographic structure of the crystalline S-ibuprofen isomer [153]. In contrast, when the silica matrix is functionalised by amino groups, the 13C NMR experiments indicated a more restricted mobility of the ibuprofen molecules, suggesting possible interactions between the amino groups and the carboxylic groups. Similar behaviour was found for encapsulated benzoic acid, and this opens the possibility to develop heteronuclear correlation NMR methods (e.g., 29Si, 13C) through using 13C-labeled benzoic acid to better characterise the possible interactions of the encapsulated molecules with silica surfaces with and without modification [151,152]. Schematic presentation of ibuprofen-silica interaction is given in Figure 23.

Figure 23. IBU linkage to a silica mesoporous material. A) Nonfunctionalized matrix: IBU can be linked through weak hydrogen bonds between its carboxylic acid group and the silanol groups (left) and is also found as a physisorbed dimer molecule at the surface (right). B) Amino-functionalized matrix: IBU is linked to the pore wall through stronger ionic bonds between carboxylate and ammonium groups. Reproduced from reference [13] with permission.

The polarity of the surface of the SBA-15 material has been modified by anchoring hydrophobic long-chain hydrocarbons (C18) on the surface. This treatment decreases the

Ordered Mesoporous Materials for Drug Delivery Applications

107

interaction of ibuprofen with the modified surface, which results in a very fast delivery of ibuprofen from this system. 13C and 1H NMR experiments evidence differences in the mobility of ibuprofen molecules adsorbed into the matrices, but they are not reflected in the overall release pattern, which obeys a diffusion model [112]. An effective control of the release rate of the antibiotic erythromycin has been achieved by modification the surface of SBA-15 with hydrophobic long-chain hydrocarbon moieties of different lengths. For the sample containing the largest amount of CH2 groups, the release rate decreases by a factor of nearly one order of magnitude compared to that of unfunctionalised SBA-15 [81]. Tang et al. reported the adsorption of famotidine on carboxylic acid functionalized MSU materials [107]. The carboxylation was carried out by the acid-catalyzed hydrolysis of cyano to carboxylic acid groups, and the results obtained by these authors demonstrate that the adsorption of famotidine depends mainly on the degree of functionalization. Figure 24 collects some of the most-used functional groups together with several drugs employed in these systems so far.

Figure 24. Pore-wall functionalization in silica mesoporous materials and structures of several drugs used in these systems. Reproduced from reference [13] with permission.

2.4.4.4.1. Functionalization Using Amino Groups Functionalization using amino groups has been a widely employed strategy to attain better control over molecule loading and release. The Vallet-Regi research group recently reported amino-functionalized MCM-41 and SBA-15 mesoporous silica-based materials containing alendronate for bone repair or regeneration [97] (Figure 25). The amino-modified materials showed a drug loading almost 3 times higher than that of the unmodified materials.

108

Spomenka Simovic and Dusan Losic

This different behavior could be explained by the stronger attracting interactions between phosphonate groups in alendronate and amino groups of functionalized matrices compared to the weaker interactions through hydrogen bonds between phosphonate groups and silanol groups of unmodified materials Drug release revealed that amino functionalization of mesoporous silica allowed a better control on the drug release: the amount of alendronate released after 24 hours was 28% and 58% for amino-modified and unmodified MCM-41, respectively. In the case of SBA-15, the 11% and 56% of the total alendronate loaded were released from amino-modified and unmodified materials, respectively, after 24 hours of assay.

Figure 25. Left: Maximum load of alendronate in ordered mesoporous materials. Right: Chemical bonding between alendronate and amino-functionalized mesoporous materials. Reproduced from reference [97] with permission

Recent reports demonstrated that the amount of alendronate loaded linearly increased as the functionalization degree increased, ranging from 88 mg/g, for the lowest functionalization degree (0%), to 121 mg/g, for the highest functionalization degree (47%) [119] as depicted in Figure 26. Moreover, functionalization degree also allowed better control over alendronate release that is, the higher the functionalization degree, the smaller the percentage of alendronate released (Figure 26),. The organic modification of mesoporous matrices has been employed to modify release kinetics of proteins, such as BSA [24]. As previously described, SBA-15 materials exhibiting different pore diameters were obtained by varying the time of hydrothermal treatment during their synthesis. These materials were then modified using amino groups to favor attraction with amide groups of protein. Organic functionalization always leads to lowering the pore diameter and consequently, as BSA size is on the limit of SBA-15 mesopore dimensions, after amino functionalization, the amount of BSA loaded decreased compared to unmodified materials. Organic functionalization of SBA-15 had a noticeable effect on the BSA kinetic release rate. As a consequence of organic functionalization, the initial burst effect observed in BSA releases from unmodified matrices (ca. 90 %) was drastically reduced after amino

Ordered Mesoporous Materials for Drug Delivery Applications

109

modification (ca. 30%). Moreover, the release of protein from the mesopores of aminomodified materials was incomplete, whereas the total amount of loaded BSA was released after 192 hours from unmodified materials.

Figure 26. Amount of alendronate loaded and percentage of alendronate released as a function of functionalization degree. Reproduced from reference [119] with permission.

Functionalization of large-pore MCF materials with amino groups has been demonstrated to increase the affinity for BSA [24]. There was an increase in BSA adsorption after aminomodification MCF-NH2 (270 mg/g) compared to unmodified MCF (240 mg/g), despite the decrease in pore diameter due to the functionalization. Amino functionalization has a strong influence on BSA release rate. Unmodified MCF exhibited an initial burst effect when almost 60 % of the protein loaded was quickly released to the delivery medium. This burst effect was drastically reduced to ca. 10% after amino modification. After 24 hours of assay, 62 % of the loaded BSA was released from unmodified MCF material and this percentage decreased to 22 % after functionalization with amino groups.

2.4.4.4.2. Functionalization Using Hydrophobic Groups A second strategy for effective control of drug loading and release is functionalization of the silica matrix surface with hydrophobic species. In this case, the drug-surface interactions are not necessarily increased, but the drug transport out of the matrix is seriously impeded as the aqueous medium does not easily penetrate inside the pores. This strategy has recently been developed by several authors. SBA-15 mesoporous matrix has been functionalized with different alkyl chains, octyl (C8) and octadecyl (C18), and erythromycin, a hydrophobic antibiotic was selected to carry out adsorption and delivery assays [81]. As previously described, organic modification led to a decrease of the effective pore diameter and surface

110

Spomenka Simovic and Dusan Losic

area of SBA-15. The decrease in surface area resulted in a decrease in the amount of erythromycin loaded, 130 mg/g and 180 mg/g for SBA-15-C8 and SBA-15-C18, respectively, compared to unmodified SBA-15 (340 mg/g). The octadecyl-functionalized sample exhibited a release rate one order of magnitude lower than that of nonfunctionalized SBA-15. Comparable results were reported using mesoporous materials modified by silylation as ibuprofen delivery systems, showing a lower drug loading after silylation [154] and a decrease in the delivery rate. Similar results were obtained by other authors with mesoporous materials modified by silylation. Captopril [124] has been incorporated and showed a lower drug loading and better controlled release.

Figure 27. Molecular model of interaction between L-Trp and (A) unmodified SBA-15, (B) SBA-15 functionalized with short-chain quaternary amines (C3N+Me3), and (C) SBA-15 functionalized with long-chain quaternary amines (C3N+Me2C18). (D) L-Trp release profiles from functionalized samples are also displayed. Reproduced from reference [76] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

111

Functionalized SBA-15 has been proposed as a delivery system of L-tryptophan (L-Trp) [76,122], a hydrophobic amino acid present in the three-dimensional structure of many peptides, proteins, and growth factors of interest in bone tissue regeneration technologies [155,156]. Unmodified SBA-15 loaded less than 5 mg/g of L-Trp, probably due to the extremely different chemical nature of hydrophobic amino acid and hydrophilic SBA-15. The small amount of L-Trp adsorbed into SBA-15 could be the result of interactions through hydrogen bonds between deprotonated carboxylic group of amino acid and silanol groups covering the silica walls (Figure 27A). SBA-15 matrix was modified using quaternary amines with different alkyl lengths (C3N+Me3 and C3N+Me2C18), as illustrated in Figures 27B and 27C [157]. Functionalization with short alkyl chains (C3N+Me3) allowed coulombic attracting interactions between deprotonated carboxylic groups of amino acid (–COO−) and protonated quaternary amines (–N+R4) covering the mesoporous surface. In this case, the amount of L-Trp loaded into SBA-15-C3N+Me3 matrix was higher (43mg/g) than in unmodified SBA-15 ( 30 nm diameter) prevented. Their later work demonstrates the feasibility of using nanoporous AAO capsules for the encapsulation of cells- that nanoporous alumina capsules (pore diameter 46-75 nm) incorporated with insulinsecreting MIN6 cells can act as effective semipermeable devices allowing transport of glucose and insulin while impeding the transport of larger proteins such as immunoglobulin G [238,240,241]. Figure 41B shows fast release of glucose and hindered release of IgG over 4 days. The release profile of insulin thorough porous alumina from encapsulated MIN6 cells in response to an increase in glucose level is shown on Figure 41C. The experimental trials suggest not only that encapsulated cells do release preformed insulin in response to glucose, but that they are also able to synthesize new insulin. Furthermore, it is important to emphasise that cells retain their viability in the close proximity of AAO membrane showing future promises of this approach for clinical immunoisolation. Flamme et al. [241] reported porous alumina based immunoisolation capsules as an alternative treatment for diabetes. Data suggest that porous alumina membrane can be designed to enable transport of insulin and glucose while impeding the transport of IgG. The encapsulated cells retain viability and respond dynamically to glucose input signals. Daily subcutaneous insulin injections cannot match physiological biphasic behaviour of normal insulin release, nor can meet the demands of food intake, exercise and stress. Organ transplantation is currently the only available method that provides physiological regulation of insulin secretion. Patients who receive successful transplants are committed to lifelong immunosuppressive therapy, which may lead to increased risk of infection and malignancy. Cellular encapsulation or immunoisolation would allow patients to benefit from cell based therapies without the need for immunosupression.

4. NANOTUBULAR TITANIA 4.1. Fabrication and Properties Nanotubular titania or titania nanotube arrays fabricated by self-ordering process with electrochemical anodization have also attracted remarkable attention in recent years due to their unique combination of wide band gap semiconductor properties, nanotube geometry, biocompatibility and large surface area. This material has been used for diverse applications including photocatalysis for hydrogen production, solar cells, energy storage, catalysis, water purification, sensors, membranes, tissue engineering, implants, and drug delivery [20, 21,243,244]. Titania nanotube structures schematically shown in Figure 42A-B compose of vertically oriented, highly ordered nanotubes with hexagonal arrangement and controllable

Ordered Mesoporous Materials for Drug Delivery Applications

131

nanotube diameters (10 -300 nm) and thickness (0.5 µm to 300 µm). The lengths of the nanotubes correlate with the efficiency of film formation, with the longest nanotubes and the highest efficiency being found for nanotubes formed under controlled voltage. The reason for separation into tubes, as opposed to a nanoporous structure is not yet entirely clear. Possible explanation is due to the electric field and local-heating-enhanced dehydration; the titania nanotubes could separate from each other, where directions of volume contraction of the hydroxide layer are normal to the walls. Titania nanotubes can be prepared in a number of forms including nanotube layer on bulk titanium, self-supporting nanotube arrays and nanotube membranes, and their typical structures are summarized in Figures 42 D-G. It is apparent from the cross- sectional SEM images in the inset of the figure that the nanotubes are well separated into individual entities with an average tube diameter at the top of the layer in the range of 160–200 nm. The electrochemical fabrication of self-ordered titania nanostructures was introduced in 1999 by Zwilling et al. by anodization of titanium in a fluoride electrolyte [245]. Since then, a number of anodization approaches, mainly focused on finding the optimal electrolyte and anodization conditions have been explored to achieve a self-ordering regime for titania nanotube growth [20,21,243,244,246-250]. These studies demonstrated that structural parameters of TiO2 nanotubes including inner diameter, wall thickness, length and TiO2 crystallinity can be controlled by adjusting electrochemical conditions such as composition of Ti substrate, electrolyte, pH, temperature, anodization voltage, current and anodization time (Figure 42C). In general, anodic TiO2 nanotube layers can be formed in aqueous and nonaqueous electrolytes containing small amounts of fluoride ions (HF or HF mixtures, NaF or NH4F). When anodization is carried out in acidic, neutral, or alkaline aqueous electrolytes, short tube length were formed (500 nm to 2 µm) which are described as the first generation of titania nanotubes [21,243]. The next generation of nanotubes were grown with thicknesses of more than 250 µm in non-aqueous (such as glycerol-based) fluoride-containing electrolytes [20, 21,243]. To keep dissolution low and grow long tubes, low electrolyte acidity and low fluoride concentration are desired. The highest tube lengths were obtained in organic viscous electrolytes. This condition under anodization voltage of 80-120 V yields vertically oriented, hexagonally close-packed TiO2 nanotube layers. New self-ordering titania morphologies, such as bamboo-type nanotubes, nanolace, branched tubes, inner tubes and multilayer nanotubes have been shown by altering the voltage during the formation of nanotubes [9495]. Although there is considerable research on surface modification of sister materials such as porous silicon and AAO, surprisingly the surface modifications of TiO2 nanotubes has not yet been widely explored [96]. Chemical inertness is the main weakness of this material when placed in contact with biological systems and surface modification is a possible solution of this problem important for many biomedical applications. Plasma surface modification using allylamine (AA) as a precursor has been applied by Vasilev et al to generate a thin and chemically reactive polymer (AAPP) film rich in amine groups on top of the TiO2 nanotube surface [253]. This initial polymer film was used for further surface functionalization by attachment of desired molecules where two modification techniques were used to demonstrate the flexibility for building of new functionalities on titania nanotube surface: electrostatic adsorption of poly(sodium styrenesulfonate) (PSS) as an example of layer-by-layer assembly (LbL), and covalent coupling of poly(ethylene glycol) (PEG) (Figure 43).

132

Spomenka Simovic and Dusan Losic

Figure 42. A-B) Scheme of nanotube titania structures fabricated by electrochemical anodization of titanium. C) Influence of electrolyte on nanotube diameters, TiO2 nanotube structures fabricated by electrochemical D) the cross-sectional structures of titania nanotube layer formed on Ti foil, E-F) the top surface showing the nanopores and nanotube structures, G) the bottom surface showing the nanotube structures detached from underlying titanium substrate. (A-C Reproduced from reference [17] with permission D-G) D.Losic et al. unpublished work).

In addition using surface modification via plasma polymerization allows control over pore diameter at the surface from 140 nm to below 20 nm which is important feature for controlled drug release application. These approaches for tailoring the pore diameters, surface chemistry and wettability of TiO2 nanotubes have potential to significantly improve the properties of this attractive biomaterial and promote the development of new biomedical

Ordered Mesoporous Materials for Drug Delivery Applications

133

devices such as drug eluting medical implants with multiple functions (orthopaedic implants, dental, coronary stents) providing an elegant route to prevent infection, control clotting, or decrease inflammation of these implants.

Figure 43. Scheme for the tailoring of the surface functionalities on TiO2 nanotube films: A) bare TiO2 nanotube arrays, B) plasma polymerization of n-allylamine polymer (AAPP) (1), C) electrostatic adsorption of poly(sodium styrenesulfonate (PSS) on AAPP (3) and D) covalent attachment of polyethylene glycol (PEG) on APPP (3). Reproduced from reference [253] with permission

4.2. Biocompatibility Being biocompatible, titania (native oxide on the surface of Ti) and its alloys particularly Ti-6Al-4V are extensively used in orthopedic and dental implants since the 1970s [125,229,254-256]. The majority of biocompatibility studies of nanotubular titania has been related to their application for orthopaedic and vascular implants. As stated previously, the aim of current orthopaedic biomaterials research is to design implants that induce controlled, guided and rapid healing [6,7]. Approximately 500000 total hip and knee replacements are performed each year in the United States alone. 25% of hip replacements surgeries were revisions due to failure of the previous implant. If implant fixation is not sufficient, loosening and osteolysis can occur. To overcome the problem, implant bone material should stimulate rapid bone regeneration in order to fill deficient bone and fix the implant firmly to the bone. The level of bone growth depends on the surface characteristics of the implant. The influence of nanoporous surface topography on bone in-growth has been described previously. Nonetheless, the dissolution of bioceramic coatings over time and separation from the metallic substrate pose a problem.

134

Spomenka Simovic and Dusan Losic

Figure 44. SEM images of marrow stromal cells on titanium and nanotubular surfaces for up to 7 days of culture. Cells show spherical morphology on titanium (A) compared to spreading morphology on nanotubular surface (B) after 1 day of culture. After 4 days of culture, cells still show spherical morphology on titanium surface (c) compared to spreading and clustering morphology on nanotubular surface (D). After 7 days of culture, some of the cells on titanium seem to be spreading (E), however the cells show high degree of spreading and have started communicating on nanotubular surface (F). High magnification SEM image after 7 days of culture (G) on nanotubular surface shows that cell extensions are protruding into the nanotubular architecture. Reproduced from reference [255] with permission.

Ordered Mesoporous Materials for Drug Delivery Applications

135

Several studies by Desai and co-workers have demonstrated that the nanotubular titania surface is a favourable template for bone cell growth and differentiation and provides clear evidence that osteoblast activity can be significantly enhanced using controlled nanotopographies [125,254,255]. These surfaces supported higher cell adhesion, proliferation and viability up to 7 days of culture when compared to titanium surfaces [125,257]. Popat et al. [125,255] demonstrated the ability of titania nanotubes (pore diameter 80 nm) to promote adhesion, matrix production, proliferation and viability of rat marrow stromal cells up to 7 days compared to titanium surfaces. In vivo biocompatibility was proved by implanting surfaces subcutaneously in rats and performing histological analysis during 4 weeks- chronic inflammation or fibrosis was absent [255]. Increased chondrocyte adhesion on anodized titanium with nanotube structures compared with unanodized titanium was also reported [258]. Figure 44 shows SEM images of MSCs after 1, 4 and 7 days of culture on titanium and nanotubular surfaces. As expected, the cells are spherical after day 1 on both titanium and nanotubular surfaces (Figure 43A-B, respectively). After 4 days of culture, the MSCs still show a spherical morphology on titanium surfaces (Figure 43C); however they show a spreading morphology on nanotubular surfaces (Figure 43D). By day 7, the MSCs on titanium surfaces are still isolated with minimal spreading (Figure 43E), whereas the MSCs on nanotubular surfaces have formed a network indicative of cell–cell communication (Figure 43F). These results indicate that the MSCs are able to spread faster on nanotubular surfaces compared to titanium surfaces within 7 days of culture. High magnification SEM images were taken after 7 days on nanotubular surfaces to visualize the cell extensions. Figure 43G-H show high magnification SEM images of an MSC extension probing the nanotubular architecture. The length of the extension is many times greater than the cell diameter. These extensions seem to help the cell anchor itself to the nanotubular structure. By doing so, the cells can adhere and spread on the surface, resulting in enhanced long-term differentiation. Cells cultured on nanotubular surfaces also demonstrated higher ALP activity without causing adverse immune responses under in vivo conditions. Furthermore, the calcium and phosphorous concentrations were 50% higher on these surfaces suggesting that matrix deposition was pronounced on nanotubular surfaces. Thus, this research provides evidence that osteoblast activity can be significantly enhanced using controlled nanotopographies. It is envisioned that the incorporation of such nanoarchitectures on implant surfaces will further facilitate the culture and maintenance of differentiated cell states, and promote long-term osteointegration. Figure 45 shows a light microscopy image of sections of healthy tissue and the tissue surrounding the implant stained with haematoxylin and eosin. Figure 45A shows histology sections of healthy tissue. There is no fibrous scar tissue present in the tissues surrounding the titanium implant and comparable to healthy tissue (Figure 45B). Figure 44C shows light microscopy images of tissue sections surrounding the nanotubular titania implant. Similar to titanium, there is no fibrous scar tissue formation around the implant. The tissue appears to be healthy and normal. The in vivo results suggest that the nanotubular surfaces do not cause any adverse immune response under in vivo conditions.

136

Spomenka Simovic and Dusan Losic

Figure 45. Histological analysis of tissue (a) control—normal healthy tissue; (b) surrounding titanium implant; (c) surrounding nanotubular implant; results indicate no fibrous scar tissue formation for both titanium and nanotubular implant and the tissues are very similar to the control tissue, dotted line shows where the implant was in contact with tissues. Reproduced from reference [255] with permission.

Bacterial adhesion is the first step in infection development after implant surgery. Initial bacterial adhesion is believed to be not only dependent on the physicochemical properties of the bacteria but also on the properties of biomaterial surfaces [125]. Popat et al. [125] demonstrated that gentamicin-loaded titania nanotubes are effective in minimizing initial bacterial adhesion. Additionally, the effect of nanotubular architecture was evaluated using an osteoblast precursor cell line. These surfaces supported higher cell adhesion and proliferation up to 7 days of culture when compared to titanium surfaces. Cells cultured on nanotubular surfaces demonstrated 50 % higher ALP activity compared to titanium surfaces. The calcium concentration was 50 % higher on these surfaces suggesting that matrix deposition was upregulated on nanotubular surfaces.

Figure 46. Decreased S. epidermis adhesion on nanotubes filled with gentamicin (NT-G) compared to titanium (Ti) and nanotubes (NT); n=10, *p