TOPICS IN CURRENT AEROSOL RESEARCH (PART 2) EDITED BY G. M. HIDY Science Center, North American Rockwell Corporation, Thousand Oaks, California 91360 AND J. R. BROCK University of Texas, Austin, Texas PERGAMON PRESS Oxford • New York • Toronto Sydney • Braunschweig Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1972 Pergamon Press Inc. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Inc. First edition 1972 Library of Congress Catalog Card No. 72-179657 Printed in Great Britain by Watmoughs Limited, Idle, Bradford; and London ISBN 0 08 016809 4 PREFACE DESPITE an ever-widening technological interest in aerocolloidal systems, the fundamental science of aerosols, on which practical considerations must rely, largely has been set aside until recently in deference to other problems. However, in the last decade a kind of renais­ sance in aerosol research has taken place in which much of the classical work of the early twentieth century is being extended. New sophisticated theoretical and experimental tech­ niques are being developed, and are being applied to understand better the behavior of aerosol systems. As scientists in many diverse fields expand their activity in aerosol research, the results of investigations appear in a wide variety of journals that reach entirely different small groups of workers. The problems of communication of these scientific results are complicated further by the worldwide character of aerosol science. To help focus attention on the variety of important aerosol research presently being published, and to open a new channel for international communications between workers in this field, we have organized a new series entitled International Reviews in Aerosol Physics and Chemistry. This work will consist of a collection of monographs of book length, and of companion volumes of selected review articles dealing with several aspects of aerosol science, and its relationship to the study of the so-called "particulate state of matter". The scope of the International Reviews will be limited to results which contribute significantly to the state of fundamental knowledge of aerosol behavior. Because International Reviews in Aerosol Physics and Chemistry is designed to concentrate on the fundamental aspects of aerosol science, it should have considerable usefulness to both practicing scientists and to graduate students in such widely diverse fields as physics, physical chemistry, meteorology, geophysics, astronomy, chemical engineering, mechanical engineering, aerospace engineering, environmental sciences, and medicine. This book, the third volume of the series, is the second part of a series of review articles representing selected topics in current aerosol research. It includes two articles dealing in considerable depth with the theory of diffusiophoresis and thermophoresis and with the mathematical treatment of integrodiflerential equations coming from the theory of aerosol coagulation. The manuscript of Professors Derjaguin and Yalamov provides considerable insight into the methods used by these investigators to derive a theory for thermophoretic and diffusio- phoretic forces acting on spheres in the range from free molecule to continuum behavior. The analysis of these investigators differs substantially from other recent theoretical calcula­ tions. In addition, the experimental techniques employed by Derjaguin and co-workers give results that are in disagreement with experimental measurements of other investigators. Therefore, it is important for the scholar of this subject to consider their remarks in the light of results derived in other models, as described in Chapters 5 and 6 of Volume 1, and in the light of conflicting experimental results described in the Derjaguin and Yalamov article as well as in the recent literature on this subject. We feel that it is desirable to present such vii viii PREFACE divergent viewpoints as part of this series, which, it is hoped, will serve as a forum for the development of aerosol science. Dr. Drake's article reviews in detail for the first time the extensive literature on the mathe­ matical solutions for the kinetic model of the coagulation equation. This article expands the discussion in Chapter 10 of Volume 1, covering the older and recent methods to calculate size distribution functions for clouds of particles undergoing collisions, along with gravita­ tional settling, and other physical processes. Although this article is strongly oriented towards the applied mathematician, it should be useful as a guide to the aerosol scientist in treating the coagulation problem. Dr. Drake surveys the collision process not only in the light of aerosol behavior, but also as it bears on the development of drople tsize distributions in clouds containing hydrometeors. It is with pleasure that we introduce this new volume in the series to the expanding audience of readers interested in aerosol science. G. M. HIDY J. R. BROCK INTRODUCTION WHEN considering the motion of isolated aerosol particles in a free volume of a gas homo­ geneous in composition and temperature, it suffices to supplement the mechanical equations of motion by the laws of dependence of the frictional resistance on the velocity and radius of the particles f and, in the case of volatile particles, with the laws of their growth and evaporation inasmuch as they affect the change in particle mass with time. For fairly large and non-volatile aerosol particles it is evidently sufficient to use Stokes' law derivable from the phenomenological hydrodynamics of a viscous liquid, and then the problem becomes a purely mechanical one. Complications requiring a molecular-physical approach may be of different origins. (1) If the Knudsen number for the aerosol particle {Kn s Xi/R where λ^ is the mean free 9 path of a given species of gas molecule and R is the particle radius) is not small enough, Stokes' law must be substituted by other laws derived from the kinetic theory of gases. The resistance to motion is then dependent not only on the particle radius, but also on the law of interaction between the gas molecules and the particle surface characterized most simply by the coefficient of diffuseness of reflection q of the molecules when they strike the particle surface. Thus molecular-physical reasoning becomes necessary in this case. (2) The instance of volatile particles in the regime Kn > 0 involves the problem of motion of a body of variable mass, complicated by the fact that in the general case determination of the rate of growth or evaporation of the aerosol particle requires the application of gas- kinetic calculations as well as diffusion equations. This was shown long ago by Langmuir(1) for the limiting case of very high Knudsen numbers and by Fuchs(2) for moderate Knudsen numbers, the latter on the basis of an examination of the concentration jump of vapour molecules at a phase boundary. It should be specially stressed that the lower limit of Knudsen numbers, down to which evaporation and condensation follow Langmuir's kinetic law, is proportional to the conden­ sation coefficient; in other words, the upper limit radii of drops which follow this law are inversely proportional to the condensation coefficient. Since the presence of a cetyl alcohol monolayer on the surface of a water drop may decrease(3) the condensation coefficient to 0.00003, purely kinetic conditions of condensation (or evaporation) in an ordinary atmo­ sphere are ensured in this case even for drops of the order of 1 cm in radius. Thus even the movement of such large drops maybe governed by molecular-physical regularities besides the laws of mechanics and aerodynamics. The question whether the range of applicability of Stokes' law will simultaneously become narrower remains open because the accommodation coefficient for the tangential impulse and for the energy of incident molecules evidently does not decrease in the presence of adsorption monolayers so much as the condensation coefficient does. (3) When an aerosol particle approaches the surface of a solid or liquid, there arises a direct molecular-force interaction which may result in the particle depositing on the surface t Restricted in all cases to spherical particles. 5 6 TOPICS IN CURRENT AEROSOL RESEARCH even though the inertia forces be negligibly small. Unlike deposition on obstacles by inertia, the coefficient of deposition due to molecular-surface forces grows with decreasing velocity. The pertinent theory was developed recently by one of us.(4) Another molecular-physical aspect which must be taken into account when considering the deposition of aerosol particles on obstacles is the change in resistance to approaching the surface compared to Stokes' formula. Part of this change may be related to the fact that at a certain stage the thickness of the gas interlayer reaches the same order as the mean free path of the gas molecules. (4) The mechanics of aerosols becomes very complicated at high number concentrations when the movements of individual particles are no longer independent of one another. For example, a cloud of particles may settle much faster than its isolated component particles under the same gravity acceleration. However, this complication is not of a molecular- physical nature. In all the cases considered, molecular-physical phenomena do not directly affect the driving force acting on the aerosol particles, which may be of purely mechanical or of electrostatic origin (charged or polarized particles in an electric field). (5) The situation is fundamentally different when the aerosol particle is suspended in a non-uniform gas. If a heat flow is passing through the gas, the aerosol particle is caught up by it and begins to move. Hence it may be said that a thermophoretic force is applied to the particle, acting in the direction of decreasing temperature. The magnitude of the resulting velocity depends on the nature of the interaction between the gas molecules and the particle surface if the Knudsen number is much larger than unity. With small Knudsen numbers (Kn <^ 1) the thermophoretic velocity is much less sensitive to the nature of this interaction and so is the gas slip on the particle surface, a phenomenon inherently connected with thermophoresis. Similarly, if there is a diffusion flow of another gas or vapour in the gas in question, a diffusiophoretic force appears causing diffusiophoresis of the aerosol particle. A phenomenon close in nature to thermophoresis, known as radiometrie effect, occurs if the particle is in the path of thermal radiation. Here the temperature gradients arising around the particle heated by the radiation are the direct cause of the driving force. Since the temperature field surrounding the particle is of complex configuration, the general theory of the radiometrie effect is more complicated and will not be discussed here. A good up-to-date summary of the work done in this field will be found in reference 5. We shall point out only that the study of radiometrie forces was started already last century<.6_8) However, this work could lead to no general conclusions because the results were found to be greatly dependent on the experimental conditions. The first to describe the phenomenon of thermophoresis were Tyndall(9) and Rayleigh,<10> who observed the repulsion of dust particles by heated bodies, resulting in the formation of an optically void zone (the so-called dark layer). Aitken,(11) Lodge and Clark,(12) and Watson(13) measured the thickness of the black layer. The theoretical studies devoted to determination of the thickness of the dark layer(14~15) were carried out at an insufficiently sophisticated level because, owing to great mathematical difficulties, their authors resorted to disputable simplifications of the physical phenomena occurring near the particle. For this reason the agreement found between theory and experiment in order of values seems to us most likely accidental. The appearance of a gradient of temperature inside the particle and around it due to non-symmetrical illumination and heating gives rise to forces similar to thermophoretic forces which cause transfer of the particle. Theoretical description of this phenomenon is INTRODUCTION 7 complicated by the fact that the temperature field arising around the particle is very complex, depending greatly on its shape, size, transparency, and refractive index. The two latter factors are strongly dependent on the properties of the substance of the particle which are, generally speaking, not known exactly for such small sizes. For this reason an exhaustive theory of photophoresis can be built only after the theory of thermophoresis is complete, and will obviously be much more complex than the latter. Since the theory of thermophoresis has only recently been completed to any definite degree, it is advisable to defer its application to photophoresis for the time being. The thermophoresis of aerosol particles is closely related to the thermotranspiration of gases through slits and capillaries, the former being the movement of aerosol particles with respect to a gas in the field of a temperature gradient, and the latter the movement of a gaseous medium through a capillary with respect to its walls in the presence of a longitudinal thermophoresis. It will first be useful to get an insight into the kindred mechanism of thermotranspiration (or thermo-osmosis) of gases, which is simpler from the point of view of both physics and mathematics. Knudsen(16~20) was the first to study the thermotranspira­ tion of gases in narrow capillaries [Kn = (Xja) > 1, where a is the capillary radius]. Suppose the constant temperatures 7\ and T are maintained in vessels at the ends of a narrow capil­ 2 lary. Then, according to Knudsen's approximate calculation, when a steady state is reached at which the gas flow through the capillary equals zero, the pressures p and/? that are l 2 established will satisfy the relation It also followed from Knudsen's experiments that the radiometrie force acting at low pressures (Kn > 1) on a radiometer vane is directly proportional to the pressure. The same qualitative conclusion was obtained theoretically by Knudsen on the basis of elementary approximate gas-kinetic calculations. Einstein,(21) who derived an estimative formula for the force proceeding from qualitative considerations, came to a similar conclusion. Einstein's method was used by Cawood(22_23) for calculating the force acting on an aerosol particle of a size comparable with or smaller than the free path of the gas molecules. A qualitative theory of thermophoresis was also given in later works by Clausius(24) and Stetter.(25) However, experimental studies*26_28) which appeared at that time and slightly later indicated that in the case of large Knudsen numbers the measured velocity of thermo­ phoresis is twice or more than twice as high as that calculated theoretically .(22~25) It should be pointed out that the studies mentioned126-28* were carried out without due allowance for the convection phenomena that may have occurred in the experimental units. A more detailed analysis of all the experimental work done in the field of thermophoresis is given in papers by Derjaguin and Rabinovich, and in a thesis by Rabinovich,<29) where it is shown that in the case of small particles the velocities of thermophoresis found experimentally are on the average twice as high as those obtained theoretically in references 22-25. The first observation of diffusiophoresis was published by Aitken(,11) who discovered a black zone around an evaporating water drop. Later, Watson(13) confirmed Aitken's data. Subsequently, diffusiophoresis became a subject of theoretical and experimental analysis by Derjaguin, Dukhin, and other authors, (30~33) and of experimental studies by Facy,(34) who simultaneously attempted to derive a formula for the velocity of diffusiophoresis. 8 TOPICS IN CURRENT AEROSOL RESEARCH It should be pointed out that Facy's calculations were based on the untenable assumption that the particles in a diffusing mixture remain at rest with respect to the barocentric co­ ordinates, i.e. with respect to the centre of gravity of the gas-vapour mixture. It will be seen that this assumption is at odds with reality. The approach adopted in reference 31 (for large particles, i.e. small Knudsen numbers) is more correct. However, it will be shown below that the boundary conditions accepted in that work need refining to give the correct solution to the problem. Chronologically, the first correct result was obtained in a paper by Derjaguin and Bakanov,(33) who realized that it would be easier to calculate the diffusiophoretic force for small particles and therefore large Knudsen numbers, since in this case the presence of the particle does not disturb the velocity distribution of the gas-mixture molecules striking the particle surface compared to its distribution at the same point in the absence of the particle, which can be found from the formulae of Chapman and Enskog. In this way formulae were obtained for the force and rate of diffusiophoresis of small particles for the case of purely specular reflection of gas molecules. Two years later, analogous calculations were published by Waldmann,(35) who restricted himself, however, to the zero approximation at which the velocity of diffusiophoresis in a coordinate system fixed to the centre of gravity of the mixture vanishes when the masses of the diffusing gas molecules become equal. The correct formula for the velocity of thermophoresis of small particles was first derived by Bakanov and Derjaguin(36) in a similar way by calculating the total momentum per second transmitted to the aerosol particle by the gas molecules striking its surface. The key to the solution of this problem lies in the fundamental observation that in the case of a small particle the gas-molecule velocity distribution is disturbed very little by the presence of the aerosol particle compared to the distribution at the same point in the absence of the particle which can be found from the Chapman-Enskog formulae.f(37) The authors of reference 36 considered three principal instances of interaction between the gas molecules and the aerosol particle surface: (a) specular reflection, involving only a change in sign of the normal components of the molecular velocity; (b) diffuse reflection with no change in absolute value of the molecular velocity, the directions of the velocities of the reflected molecules being independent of their directions before striking the particle and distributed in space according to the law of the cosine of the angle between the normal to the surface at the point in question and the direction of the velocity; (c) what is known as diffuse "evaporation" of the gas molecules after striking the particle surface, in which the only condition fulfilled is that of equality of the number of molecules striking an element of the surface to the number evaporating from the same surface element according to the "cosine law". If the particle is moving uniformly, the thermophoretic force is balanced by the force of viscous resistance, and the total momentum per second transmitted to the particle in all three cases mentioned equals zero. From this condition the following expression was obtained in reference 36 for the velocity of thermophoresis: 15 cX ^ = - Ï 6 ^ T g r a d T ' (2) where μ = 8 for instances (a) and (b), and μ = π + 8 for instance (c), c is the mean velocity of the molecules, λ is their free path, and Tis the absolute temperature. t A similar idea has been expressed earlier by Einstein(21) for the case of thermophoresis. INTRODUCTION 9 In cases (a) and (b), equation (2) gives velocities about three times as high as Einstein's estimative formula(21) and two and a half times as high as Cawood's formula.(22_23) In case (c) the result is about 27 % lower than for (a) and (b). Therefore even here the result is still more than twice as high as the velocities obtained by Einstein and Cawood. It should be noted that for the instance of specular reflection [case (a)], formula (2) was obtained in reference 36 by another method as well, based on regarding the gas-aerosol particle system as a mixture of two gases, the aerosol particles playing the part of the second component of the gas mixture. Here the number concentration of the second component is very small compared to that of the gas, whereas the mass and size of the particles are much greater than the mass and size of the gas molecules. Under such conditions the separation of the mixture will be analogous to the thermodiffusional separation of a binary gas mixture. Making use of the Chapman-Enskog formula(37) for the mean velocity of the molecules of the second component with respect to the centre of inertia of the totality of gas molecules, Bakanov and Derjaguin easily derived the following formula for the velocity of thermo- phoresis of aerosol particles : n2 U = D K grad T, (3) T i2 T where n is the number of gas molecules in 1 cm3, n is the number of aerosol particles in x 2 1 cm3, n = n + n , D is the coefficient of mutual diffusion of the gas-aerosol particle x 2 12 system, and K is the thermodiffusion ratio. T After calculating D and K by the Chapman-Enskog method(37) under the conditions 12 T indicated above and under the assumption that the interactions of the gas molecules with one another and with the aerosol particles, which are taken into account, occur according to the laws of collision of elastic spheres (analogous to specular reflection), a velocity was obtained from (3) which coincided with that resulting from (2). Equation (2) not only satisfactorily describes the experimental data obtained in the papers mentioned above(26_28) but also agrees excellently with the more recent data of Derjaguin and Rabinovich(29» 38) obtained in an apparatus having a narrow slit with a longitudinal temperature gradient which possesses many advantages over the modified Milliken conden­ ser used in references 26-28 and with the experimental results of Derjaguin and Storozhilova (39-40) obtained by the so-called jet method in which a horizontal mixed aerosol jet is under the influence of a preset vertical temperature gradient caused by a constant temperature difference between two horizontal plates. The jet method is considered the most accurate available experimental procedure because its error is not more than 10%. It may be pointed out that after Bakanov and Derjaguin{36) a similar formula for the velocity of thermophoresis was obtained by Waldmann(41) by the same method based on calculating the total momentum per second imparted to the particle by the gas molecule. The second method of calculation in reference 36, based on regarding the gas-aerosol particle system as a binary mixture in the process of thermodiffusion, was repeated by Mason and Chapman(42) 4 years later without mentioning reference 36. If the theory of thermophoresis of small aerosol particles (Kn > 1) became quite satisfac­ tory owing to reference 36, the same cannot be said of the theory for large particles (Kn <ζ 1), though both theoretical and experimental results were available. The theory of thermophoresis of large particles is far more complex than that of small particles, though the first theory of thermophoresis was proposed by Epstein(43) precisely for 10 TOPICS IN CURRENT AEROSOL RESEARCH large particles, and its correctness was not questioned until recently, f However, in his calculations Epstein used as his boundary condition the velocity of thermal slip of the gas along the particle surface, derived rather approximately by Maxwell.(44) It should be pointed out that Maxwell(44) was the first researcher to consider the forces arising in a non-uniformly heated gas. He found the expression for the tensor of stresses caused by the temperature gradient. True, to avoid some of the major mathematical difficul­ ties involved Maxwell took into account the tensor terms containing second derivatives with respect to the temperature and the velocity, but neglected squares and products of first derivatives. Subsequently, Burnett*50) took these terms into account as well. The complete form of the stress tensor is also given in Chapman and Cowling's monograph(.37) Maxwell carried out a complete calculation of the stress tensor for the special case of molecular interaction where the force of interaction is inversely proportional to the fifth power of the distance between the interacting molecules. Calculations for other laws of interaction can be found in references 51-56. The existence and form o fthis tensor of tem­ perature stresses show that under the influence of a temperature gradient generally, there should appear flows in the gas the nature of which depends on the shape o fthe gas-container wall or gas-suspended particle surface and on the boundary conditions at them. The same paper by Maxwell deals with the tangential stresses at the gas-solid wall surface. In contrast to the rest of the paper, Maxwell here applied a very rough approximation, considering the velocity distribution of the molecules striking the phase boundary surface to be the same as in the bulk of the gas. Such an assumption is tenable for small particles and gives correct results in this case, as has already been mentioned. However, Maxwell applied it to the opposite case of the flat surface of a body much larger than λ. Out of physical considerations, Maxwell assumed that under the action of the tangential stress a velocity jump should appear in the Knudsen layer between the gas and the surface. Further, Maxwell found the following expression for this velocity jump which he called thermal slip: vJ ^L, (4) v v being the kinematic viscosity. Maxwell then used this formula to derive an expression for the thermomolecular pressure drop Ap across the ends of a capillary between which a temperature drop ΔΓ is maintained, 2 Ap = 6^-gradT, (5) ρτ where p is the density and η is the viscosity of the gas. However, the experimental data of Knudsen(ie_17) and other workers (57) showed a pres­ sure drop approximately twice as large as that corresponding to formula (5). For unknown reasons, aerosol specialists usually ignore this discrepancy, which is wrong, of course, because in doing so they contradict the principle of generality and uniformity of physical laws. t Some very inaccurate attempts to build the theory for large particles prior to Epstein could also be quoted. They are all in gross contradiction with experimental data.