Energetics and Percolation Properties of Hydrophobic Nanoporous Media [1 ed.] 9781617284618, 9781616688660

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers, Incorporated,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

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ENERGETICS AND PERCOLATION PROPERTIES OF HYDROPHOBIC NANOPOROUS MEDIA

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Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

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ENERGETICS AND PERCOLATION PROPERTIES OF HYDROPHOBIC NANOPOROUS MEDIA V. D. BORMAN AND

V. N. TRONIN

————————————————

Nova Science Publishers, Inc. New York

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

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.

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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. Additional color graphics may be available in the e-book version of this book.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Tronin, V. N. Energetics and percolation properties of hydrophobic nanoporous media / V.N. Tronin and V.D. Borman. p. cm. Includes index. ISBN  (H%RRN) 1. Porous materials--Transport properties. 2. Hydrophobic surfaces. 3. Percolation (Statistical physics) I. Borman, V. D. II. Title. QC173.4.P67T76 2009 620.1'16--dc22 2010022734

Published by Nova Science Publishers, Inc. † New York

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

CONTENTS Preface Chapter 1

Introduction

1

Chapter 2

The Model of a Porous Medium. Infiltration Fluctuations

5

Work and Thermal Effect in the Infiltration-Defiltration Cycle

11

Chapter 4

Conditions for the Closed Cycle

19

Chapter 5

Temperature Dependences of the Infiltration and Defiltration Pressures

25

Chapter 6

Thermal Effect

33

Chapter 7

Conclusions

39

Chapter 3 Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

vii

References

41

Index

45

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

PREFACE Energetics of "nanoporous medium-nonwetting liquid" systems is one of the new directions in basic and applied research. In the simple model of a porous media in the form of cylindrical channels, this threshold pressure is described by the Laplace-Washburn equation, where is the surface energy of the liquid is the pore radius and the contact angle (for a nonwetting liquid). 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. This new book reviews research on hydrophobic nanoporous media.

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

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Chapter 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 = 2σ / R ⋅ cosα , where

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σ

R is the pore radius, and α is the D contact angle (for a nonwetting liquid, α > 90 ). For filling nanometersized pores by a nonwetting liquid with a surface energy of 0.05 ÷ 0.5 2 3 J/m2, it is necessary to apply a threshold pressure of 10 ÷ 10 atm. When is the surface energy of the liquid,

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

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

2

V. D. Borman and V. N. Tronin

carried out at a low compression rate of the system when the rate of increase

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−3 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 ΔV / 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-

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

Introduction

3

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 - fulldefiltration process initial and finite states of the system are similar, the

∫

internal energy change ΔE 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

∫

∫

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temperature in the hydrophobic silica gel-water systems under investigation during infiltration-defiltration cycle was ϕc ),

ϕ

is its porosity equal to the ratio of pores to the porous media volume,

ϕc

is the percolation threshold, which is the characteristic of the porous media. For 3D systems, the percolation threshold

ϕc = 0.18 [63].

At the same

time, the connectivity of pores with one another is the result of the formation of infinitely large clusters of pores at

ϕ = ϕc .

Figure 1 shows the

dependence of the probability normalized to unity of a pore belonging to the infinitely large cluster on porosity ϕ .[63,64]

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

6

V. D. Borman and V. N. Tronin It can be seen from Figure 1 that, in the vicinity of the percolation

threshold

ϕc , only a small number of pores (~1%) belong to the infinite

cluster; therefore, in this case, only a small fraction of these pores, as well as pores that belong to the finite clusters connected with the boundary of the porous media, can be infiltrated. At increasing porosity and for

ϕ >> ϕ c

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P (ϕ ) → 1 and, consequently, the pore space becomes homogeneous due to the growth of the infinitely large cluster of pores. Under these conditions, the infiltration of the porous media can be described as the infiltration of an infinite cluster of pores. It is this infiltration that will be considered in the present paper. It is assumed that thermal fluctuations at the pressure р in the vicinity of the infiltration threshold of the porous medium bring about the formation of macroscopically small regions in the form of clusters consisting of N pores filled with a liquid. Each cluster arises at the boundary of the porous medium and, in the view of the boundedness of the pore volume, grows through the attachment of the other filled pores to it. We believe that, at the beginning of the growth, each cluster can be considered a system of branched chains consisting of filled pores. In the course of infiltration of the porous medium with the liquid, the external pressure does the work. This process is accompanied by the formation of energetically unfavorable surfaces of both the menisci of the liquid in pores and the liquid--porous medium interfaces. Moreover, the state of the gas in the pores and the elastic state of the porous media change as well. If the adiabatic work of formation of an infiltration fluctuation is δA(N) and the energy of dissipation due to the friction can be disregarded (see [5]), the probability of the formation of a fluctuation can be written as w~wо exp(δS) [65], where δS=-δA/T is the fluctuation of the entropy. Therefore, an increase in the quantity δA (δA>0) with an increase in the number of 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 δА~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) ~ with characteristic size R.

δσ R

for an isolated pore

δσ = σ sl − σ sg ,where σ sl

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

is the surface

The Model of a Porous Medium. Infiltration Fluctuations

7

σ lg = σ is the surface energy of the sl solid–gas interface, δσ = σ cosα , α is the wetting angle.

energy of the solid–liquid interface,

At a pressure lower than the critical value p in , the adiabatic work satisfies the inequality δA(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 δА~Т and thermal fluctuations in the system can lead to the formation of clusters from N pores. At a pressure р> pin , the infiltration of individual pores becomes energetically favorable because the quantity δA(N) is negative. Since the work is δА~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

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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

δA( p )

be the work expended for

providing the fluctuation infiltration of one pore. According to [5], the

expression for the work δA( p ) with due regard for the formation of menisci can be written in the form

δA( p ) = − pV + σS m + δσ ( S − S m ) Here,

(1)

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

σ

the surface area of the menisci, and

is the surface energy of the liquid.

For a spherical pore with the radius R, the work

δA( p )

can be

represented in the form [5]

δA = A( p, R ) ⋅ 4 3 ⋅ π ⋅ R 3 A ( p, R ) = − p +

3 ⋅ δσ R

,

⎡ ⎞⎤ ⎛σ ⋅ ⎢1 + η ⋅ ⎜ − 1⎟⎥ ⎠⎦ ⎝ δσ ⎣

(2)

where η is the ratio of the meniscus surface area to the pore surface area. Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

8

V. D. Borman and V. N. Tronin

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

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:

δA = A( p, R, L ) ⋅ 4 3 ⋅ π ⋅ R 3 A ( p, R, L ) = p (

,

3x 3 ⋅ δσ − 1) − 2 R

σ ⎤ ⎡ , ⋅ ⎢x − δσ ⎥⎦ ⎣

(3)

where

x=

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

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

The Model of a Porous Medium. Infiltration Fluctuations

9

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

R and, at L → ∞ , transforms into the Laplace pressure 2δσ . This means that, in the work expended for infiltrating the pore, ~

radius

pout

R

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 σ~δσ~0.5 kJ/m2, R~10 nm, and , χ~10-4 atm-1), when the pressure deviates from the value рI by δр=-10-2 рI, the work changes from δА~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 (δp/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 leads to the appearance of an additional Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

contribution to the work δA. The value of this contribution

δAg 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

μ0

multiplied by the density of the gas

ρ g at the pressure p , i.e.,

δAg ~ μ 0 ρ g V . For water, we can write μ 0 ~ 2.2 x10 3

J and kg

kg . The infiltration of the porous media occurs at a pressure of m3 7 J the liquid pin ~ 2 x10 . In this case, the additional contribution to the m3

ρ 0 ~ 1.21

work due to the presence of the gas in the pores is given by

δAg ~ μ 0 ρ gV ~ pinV

μ0 ρ g pin

~ 10 − 2 δA . Therefore, the influence of the

gas in the pores on the infiltration of the porous media can be ignored.

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

10

V. D. Borman and V. N. Tronin Since the pore can be either filled (probability w ~ 1 , δA( p ) < 0 ) or

empty (probability w = 0 , δA( p ) > 0 ), the normalized probability can be written in the form [11]

[

( T )]

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wi ( p ) = 1 + exp δA

−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 δA = 0 determines the pressure necessary for the infiltration of a pore on the shell of the infinite cluster of filled pores.

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

Chapter 3

WORK AND THERMAL EFFECT IN THE INFILTRATIONDEFILTRATION CYCLE Let us calculate the work Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

arbitrary degree of infiltration

θ

δA

and the thermal effect ΔQ for an

of the porous medium. The thermal effect

ΔQ in filling of a porous media by nonwetting liquid comprises the thermal effect ΔQ p due to the formation of the liquid-solid interface , the thermal effect ΔQw related to formation-disappearance of menisci and the thermal effect ΔQu related to the compressibility of the nonwetting liquidnanoporous media system .

ΔQ = ΔQ p + ΔQw + ΔQu

(5)

The ΔQ p , ΔQw and ΔQu values can be calculated using thermodynamic relations [12], which determine the thermal effect in formation of the surface

ΔQ s

ΔQs = −

dσ ΔS dT

Here, ΔS is the change in the system surface. Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

(6)

12

V. D. Borman and V. N. Tronin To calculate the thermal effect ΔQ 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 , nτ

τ = 2.2 [63]. 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 θ ~ θ c . Taking it into consideration, we can represent the thermal effect δQ related to filling of one pore as:

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not filled at

δQ = −T

dδσ dσ ( S − zS г ) − T zS гW (θ ) dT dT ,

(7)

S = 4πR 2 is the area of the surface of a pore with radius R , z is a number of nearest neighbours, W (θ ) 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 (θ ) , we obtain 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:

Energetics and Percolation Properties of Hydrophobic Nanoporous Media, Nova Science Publishers,

θ

Work and Thermal Effect in the Infiltration-Defiltration Cycle

13

θ

dδσ ΔQ p (θ ) = −T < ( S − zS z ) > θP (θ )dθ dT ∫0 ∞

< S − zS z >= ∫ dR ( S − zS z ) f ( R )

(8)

0

ΔQw (θ ) = −T

θ

dσ < zS zW (θ ) > dθ dT ∫0

Here, f (R ) is the function normalized to unity of the size distribution for

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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

(δR )