Nanoclusters and Microparticles in Gases and Vapors 9783110273991, 9783110273908

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De Gruyter Studies in Mathematical Physics 6 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia

Boris M. Smirnov

Nanoclusters and Microparticles in Gases and Vapors

De Gruyter

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ISBN 978-3-11-027390-8 e-ISBN 978-3-11-027399-1

Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek 7KH'HXWVFKH1DWLRQDOELEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ˆ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The motivation for writing this book was to classify various processes in excited and ionized gases involving nanoclusters and microparticles. These processes are of interest for various applications in gases with participation clusters and small particles and take place in natural phenomena, mostly in the atmosphere. From the standpoint of interaction of clusters or particles with a surrounding gas, one can single out two regimes for these processes, so that the kinetic (dynamic) regime corresponds to a small gas density and the process involving a small particle is similar to collision of atomic particles. In the opposite, diffusion (hydrodynamic) regime for a dense gas, the particle motion subjects to the laws of hydrodynamics. In the kinetic regime a small particle can be in a strong interaction with no more than one buffer gas atom, whereas in the diffusion regime there is a strong interaction of a small particle with many buffer gas atoms. The goal of the book is to classify various processes involving clusters or small particles. In the end, this analysis gives some formulas or simple algorithms to determine the rate of a certain process together with the criterion of validity of this result. On this way we are based on the liquid drop model for the structure of small particles that was introduced in physics by N. Bohr about 80 years ago for the analysis of properties of atomic nuclei and allows one to predict the nuclear fusion that is a basis of atomic reactors and atomic bombs. Next, for collision processes involving clusters and small particles we use the hard sphere model (or the model of billiard balls) which was introduced in physics by J. C. Maxwell 150 years ago and helped for H. Boltzmann and J. C. Maxwell to create the kinetic theory of gases. Of course, this restricts the accuracy of theoretical description of particles and processes. In particular, the cluster may be marked out as an independent physical object due to magic numbers, i.e. numbers of atoms at which the solid cluster has a filled structure. But the liquid drop model corresponds to an average over cluster structures, and then their magic numbers are lost. Nevertheless, one can analyze errors resulted from these models in some cases. From such an analysis one can find that the error in the rates of processes due to the model assumptions usually does not exceed 10–20% and is less than that followed from the accuracy of some parameters for certain processes. From the other hand side, the above models allow us to simplify the theoretical description of the processes involving clusters and microparticles

vi

Preface

that gives the possibility to classify various processes from the general standpoints. We are based on the concepts for processes involving isolated nanoclusters and microparticles in a buffer gas, and the majority of these concepts which were developed 50–100 years ago. In considering each process, we give the derivation of the rate of this process that allows one to establish both the criterion of the validity of the results and the conditions of its realization. The above models give simple algorithms for various processes and regimes where the results are in the analytic or semianalytic form. Processes under consideration include transport processes in gases involving small particles and relaxation in particle motion, processes on the particle surface (combustion of particles, quenching of excited atoms on the particle surface, attachment of atomic particles to the particle surface), charging processes and interaction of a charging particle with an ionized gas, nucleation processes, including conversion of an atomic vapor inside a buffer gas in a gas of clusters, coagulation, coalescence (Ostwald ripening), cluster growth under the action of external fields, and growth of structures, as chain aggregates, fractal aggregates, fractal fibres, aerogels. Each analysis is finished by analytic formulas or simple models which allow us to evaluate the rate of a certain real process with a known accuracy or to estimate this and criteria of validity are given for these expressions obtained.

Contents

Preface

v

List of figures

xiv

1 Introduction

I

1

Properties of small particles and their behavior in gases

2 Nanoclusters and microparticles in gases 2.1 Gas with small particles as physical object . . 2.2 Small particles in the Earth atmosphere . . . 2.3 Methods of removal of dust particles from gas 2.4 Artificial small particles in gas . . . . . . . . 2.5 Electric processes in earth atmosphere . . . . 2.6 Dusty plasma of solar system . . . . . . . . . 2.7 Problems . . . . . . . . . . . . . . . . . . . 3 Cluster properties and their modeling 3.1 Cluster structures . . . . . . . . . . . . . . . 3.2 Phase transition in cluster . . . . . . . . . . . 3.3 Analytical and computer modeling of clusters 3.4 The liquid drop model for clusters . . . . . . 3.5 Spectral properties of clusters . . . . . . . . . 3.6 Problems . . . . . . . . . . . . . . . . . . .

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4 Dynamics of collisions in buffer gas involving clusters 4.1 Hard sphere model in atomic physics . . . . . . . . . . 4.2 Models of atom collisions with cluster or small particle 4.3 Analytic and computer methods in cluster physics . . . 4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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26 26 30 36 38 40 43

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49 49 53 55 58

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II

Contents

Processes involving small particles in gases

5 Transport phenomena in gases involving small particles 5.1 Cluster motion in gas in force field . . . . . . . . . . . . 5.2 Mobility of charged clusters in gas in strong electric field 5.3 Diffusion of clusters in gas . . . . . . . . . . . . . . . . 5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 6 Particle motion in gas flows 6.1 Relaxation of particle velocity in gas flow 6.2 Gas flow in tubes . . . . . . . . . . . . . 6.3 Drift of particles in gas flows . . . . . . . 6.4 Particle departure on periphery of gas flow 6.5 Problems . . . . . . . . . . . . . . . . .

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7 Processes in buffer gas on surface of small particles 7.1 Equilibrium of metal cluster with parent vapor in buffer gas 7.2 Character of cluster growth due to attachment of free atoms 7.3 Quenching of metastable atoms on cluster surface . . . . . 7.4 Character of combustion of small particles . . . . . . . . . 7.5 Kinetic and diffusion regime of particle combustion . . . . 7.6 Recombination of charged clusters in buffer gas . . . . . . 7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Charging of small particles in ionized gases 8.1 Particle charging in dense buffer ionized gas . . . . . . . 8.2 Particle charging in dense gas discharge plasma . . . . . 8.3 Double layer of gas discharge . . . . . . . . . . . . . . . 8.4 Particle charging in rarefied ionized gas with free ions . . 8.5 Particle charging in rarefied ionized gas with trapped ions 8.6 Particle charging and screening in rarefied ionized gas . . 8.7 The charge distribution of particles in ionized gas . . . . 8.8 Charging of small clusters in ionized gas . . . . . . . . . 8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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116 116 121 125 127 131 135 140 142 144

ix

Contents

9 Growth of clusters and small particles in buffer gas 9.1 Types of nucleation processes . . . . . . . . . . . . . . 9.2 Kinetic regime of cluster coagulation . . . . . . . . . . 9.3 Diffusion regime of cluster coagulation . . . . . . . . 9.4 Cluster coagulation in external field . . . . . . . . . . 9.5 Ostwald ripening . . . . . . . . . . . . . . . . . . . . 9.6 Method of molecular dynamics in nucleation processes 9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . 10 Structures formed in aggregation of solid particles 10.1 Fractal aggregates . . . . . . . . . . . . . . . . . . . . 10.2 Growth of fractal aggregates . . . . . . . . . . . . . . 10.3 Growth of particle structures in external electric fields 10.4 Growth of elongated particle structures in electric field 10.5 Aerogels . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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150 150 152 156 158 160 167 168

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187 187 194 198 200 207 210

11 Conclusion

215

Appendix A Physical parameters A.1 Fundamental physical constants . . . . . . . . . . . . . . . . . A.2 Melting and boiling points of elements . . . . . . . . . . . . . .

217 217 218

Appendix B Conversional factors B.1 Conversional factors in formulas for atomic particles and small particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219

Appendix C Transport coefficients of atomic particles in gases

225

Bibliography

229

Index

247

219

List of figures

2.1 2.2

2.3 2.4 2.5

2.6

2.7 2.8 2.9

2.10 3.1 3.2 3.3

Typical size of aerosols and microparticles located in the Earth atmosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eruption of Grimsvotn volcano in Iceland in May 2011 [35, 36]. The eruption flow reaches up to 11 kilometers and along with gases and small particles contains stones and fragments of rocks. . . . . . . . . Smoke from a chimney-stalk as a result of combustion of an organic fuel [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Firework in Venice [108]. . . . . . . . . . . . . . . . . . . . . . . . . The Earth as a spherical capacitor whose lower plate with a negative charge is located on the Earth surface, and the position of the upper plate corresponds to the upper edge of clouds at the altitude of several kilometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charging of the Earth results from lightnings which are electric discharges between clouds and Earth surface [120, 121]. The clouds are formed and destroyed as a result of the moisture circulation in the course of water evaporation from the Earth surface and water returning on the Earth surface in the form of precipitations. . . . . . . . . . . . . . . . Eruption of Chile’s Puyehue volcano in June 2011 [128] that is accompanied by short lightnings inside the erupt. . . . . . . . . . . . . . . . The basis of Saturn rings is a dust created by planet satellites and interacted with the plasma of a solar wind and Saturn magnetic field [142]. As a result of interaction with a solar wind a comet tail with ions and dust particles is separated from the comet nucleus in the direction from the Sun and due to interaction with the magnetic field the tail conserves connection with the comet nucleus in the course of its evolution [154]. Eruption of the volcano Puyehue in Chile in June 2011 [162] where the smoke stream passes through clouds. . . . . . . . . . . . . . . . . . . The interaction potential between nearest atoms of the cluster as a function of distances between atoms for a short-range interaction of atoms. The icosahedral structure. . . . . . . . . . . . . . . . . . . . . . . . . Atom positions in the icosahedral cluster consisting of 13 atoms. a) Side view, b) top view, c) the developed view of a cylinder in which a clustericosahedron is inscribed. . . . . . . . . . . . . . . . . . . . . . . . .

9

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23 24 27 27

28

List of figures 3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

a) Cuboctahedron. b) Cluster structures consisting of 13 atoms that are cut from the crystal lattice with close packing. It is shown positions of atoms-balls located in a central plane. Circles are projections of three atoms-balls of the upper plane onto the central plane and they coincide with the projections of three atoms-balls of the lower plane onto the central plane for the hexahedron cluster structure, while the projections of three atoms-balls of the lower plane onto the central plane for the cuboctahedron cluster structure are shown by crosses. . . . . . . . . . Types of cluster excitations: a) atom oscillations in a cluster, b) configuration excitation corresponded to a change of the atom configuration of the cluster [166]. . . . . . . . . . . . . . . . . . . . . . . . . . . . The temperature dependence for the entropy of transition at melting of the 13-atom Lennard-Jones cluster [171, 173] that is obtained on the basis of the data of computer simulation of the 13-atom cluster if it is found under adiabatic [174] and isothermal [175] conditions. . . . . . The dependence on a number of cluster atoms n for the melting point Tm of sodium clusters [180, 181]. The melting point of bulk sodium is 371 K. Arrows indicate magic numbers of atoms in the sodium cluster. Mass-spectrum of sodium clusters in a beam passed through a thermostat depending on the thermostat temperature [182, 183]. Cluster melting leads to a loss of magic numbers, and hence an oscillation structure of the cluster mass-spectrum disappears. This allows one to determine the melting point for the corresponding atom shell on the basis of variation of its mass-spectrum. . . . . . . . . . . . . . . . . . . . . . . . The dependence on the number of cluster atoms n for the melting point of sodium clusters [182, 183]. The melting point of macroscopic sodium is Tm D 371 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The thresholds of photoionization of a neutral cluster .J / and photodetachment of a negatively charged cluster .EA/ as the method of determination of the energetic gap between the entirely filled and unoccupied electron bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental values for the threshold of photoionization of clusters Hgn and photodetachment of clusters Hg n [215] and the energetic gap followed from these data that separates free and occupied electron bands for Hgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation of experimental data [180, 181] for the melting point of sodium clusters depending on their size that is given in Figure 3.7, by a smooth function according to formula (3.6.1). . . . . . . . . . . . . . Approximation of experimental data for the melting point of sodium clusters [182, 183] given in Figure 3.9 by a smooth dependence of a number of cluster atoms according to formula (3.6.1). . . . . . . . . . The specific surface energy for clusters with a pair interaction of atoms in the case of a short-range interaction between nearest neighbors and the face centered cluster structure of the cluster [163, 209]. . . . . . .

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xii

List of figures 3.15 3.16

4.1

4.2

4.3 4.4

4.5 4.6 5.1

6.1 6.2 6.3 6.4

The specific surface energy for clusters of the hexagonal structure in the case of a short-range interaction between nearest neighbors [163, 209]. The specific surface energy for clusters of the icosahedral structure in the case the truncated Lennard-Jones interaction potential between toms, so that the nearest neighbors partake in interaction only [163, 209]. Open circles corresponds to the icosahedral cluster structure, whereas dark circles relate to the icosahedral structure of the core and to the cubic face-centered structure of a not completed surface shell of atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

47

Interaction potential of two atomic particles as a function of a distance between them (1) and the approximation of this interaction potential (2) within the framework of the hard sphere model. . . . . . . . . . . . . Dynamics of scattering in the hard sphere model in the center of mass frame of reference, so that Ro is the hard sphere radius,  is the impact parameter of collision, # is the scattering angle. . . . . . . . . . . . . Character of particle scattering for the hard sphere model.  is the impact parameter of collision, Ro is the hard sphere radius. . . . . . . . The recombination coefficient of positive and negative ions in air. The solid curve is an approximation of experimental data which are given by open circles [255] and dark circles [256]. The left part of the solid curve corresponds to three body recombination of positive and negative ions in accordance with the Thomson theory [257], and the recombination rate at large pressures is given by the Langevin formula [328]. . . . . Cumulus clouds over a sea [259]. . . . . . . . . . . . . . . . . . . . . Cumulus clouds over a land [260]. . . . . . . . . . . . . . . . . . . .

60 61 62

Character of cluster interaction with atoms of a surrounding gas in the course of cluster motion in a buffer gas. In the kinetic regime of atomcluster interaction, each time one atom only can strongly interact with the cluster under consideration, while in the diffusion regime many atoms simultaneously interact strongly with the cluster. . . . . . . . .

68

Geometry of motion of a gas flow in a cylinder tube with a conical end. Character of motion of a gas flow near the output orifice within the framework of the model of the small transient region. . . . . . . . . . The ratio of the velocity of gas outlet through an orifice to the sound speed at the gas temperature in the flow before the orifice [306]. . . . Geometry of motion of a gas flow in the tube consisting of two cylinder tubes and arc tube with identical circle cross sections. . . . . . . . . .

50

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79 81 83 86

List of figures 7.1

8.1

8.2

8.3 8.4

8.5

8.6

8.7

8.8

Dependence of the temperature of surrounding air T on the temperature To of a particle of activated birch coal of a radius 20 μm with combustion parameters according to formula (7.4.8) for particle combustion: 1 – the asymptote T D To , 2 – the balance equation (7.4.12), 3 – the balance equation (7.4.12) in neglecting particle radiation, 4 – the balance equation (7.7.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The reduced number density of electrons ne D Ne =No (1, 2) and ions nC D NC =No (3, 4) depending on the local reduced potential energy u.R/ of the particle field. uo D 6 for 1 and 3 and uo D 4 for 2 and 4. Lines of identical electric potentials in the plane passed through the particle center if an electron is located in the Coulomb center of a charged particle p and constant electric field. The length units are taken such that r1 D Ze=E D 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . The reduced difference of the number density of ions and electrons n.u/ D .NC  Ne /=No for uo D 2 (1) and uo D 4 (2). . . . . . . . . The correction to the simplified potential energy U.R/ D z.R/e 2 =R of self-consistent field R 1leads to replace this potential energy by the value U.R/  U D R ze 2 dR=R2 . Dark triangles relate to shielding by free ions, whereas open triangles correspond to screening of the particle charge by trapped ions. . . . . . . . . . . . . . . . . . . . . . . . . . The trajectories of the trapped ions captured on the closed orbit in the Coulomb center (left) and the screened Coulomb center (right). 1 is the particle, 2 is the ion trajectory origin, 3 is the finish point of the ion trajectory part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Used ion parameters of a trapped ion formed in the resonant charge exchange event that proceeds at a point R: rmin , rmax are the minimum and maximum distances from the particle center for the trajectory of a captured ion,  is the angle between the direction of ion motion after the resonant charge exchange event and the vector R. 1 is the particle, 2 is the point of resonant charge exchange for ion-atom collisions. . . The part of the screening charge due to trapped ions (8.6.5) for an argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures. Open circles corresponds to the version when free ions dominate in screening of the particle field, and closed circles describe the version when trapped ions dominate. . . . . . . . . . . . . . . . . . . . . . . The reduced radius of action of the particle field in accordance with formula (8.6.7) as a function of the reduced number density of a surrounding argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures. Open circles relate to the first version if free ions dominate in the particle field screening and the ion number densities are given by formulas (8.6.3), and dark circles correspond to the second version with the number densities of ions according to formula (8.6.4).

xiii

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122 125

130

132

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137

138

xiv

List of figures 8.9

9.1

9.2

9.3 9.4 9.5 10.1 10.2

The number densities of free (squares) and trapped (circles) ions in a surrounding argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures as a function of the reduced distance from the particle. Dark signs correspond to formulas (8.4.5), while formulas (8.6.2) relate to open signs. . . . . . . . . . . . . . . . . . . . . . . . The mechanisms of nucleation and cluster growth. a) conversion of an atomic vapor in a gas of clusters; b) coagulation; c) coalescence (Ostwald ripening); d) aggregation. . . . . . . . . . . . . . . . . . . . The probabilities Pdif .u/ and pkin .u/ of a given size of metal clusters for the diffusion and kinetic regimes of cluster motion (a) and the difference of these probabilities (b). . . . . . . . . . . . . . . . . . . . . . . . . Formation of a mist in lowlands at a temperature decrease in a night time [401]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulus cloud on the last stage of its existence [406]. . . . . . . . . Sant Elmo’s fires on the endings of ship masts [407]. . . . . . . . . .

Chain aggregate formed in a plasma with magnesium [408]. . . . . . Fractal aggregate consisting of gold particles and is formed in a colloid solution. [410] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Structure of electrocytes formed in a blood as a result of aggregation of red cells [411]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Fractal aggregate of iron formed in iron evaporation and method of their analysis after collection on a grid. [412] . . . . . . . . . . . . . . . . 10.5 Character of the structure growth for the cluster-cluster mechanism of aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Dependence of a number of gold particles inside the sphere of a radius R for the fractal aggregate of Figure 10.2. The straight line corresponds to formula (10.1.3) and Dˇ D 1:75. . . . . . . . . . . . . . . . . . . 10.7 Character of interaction of a chain aggregate that is modelled by a cylinder with a spherical particle of the same radius. As a result, a spherical particle attaches to the ending of the cylinder particle, increasing its length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Geometry of interaction of a cylinder particle with a spherical one of identical radii, if the axis prolongation passes through a center of a spherical particle (a), and interaction of two cylinder particles if their axes are located in one line (b). . . . . . . . . . . . . . . . . . . . . . 10.9 Fragment of a fractal fiber that is cut out along (a) and across (b) of the fractal fiber [452]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Projection of an aerogel fragment. . . . . . . . . . . . . . . . . . . . 10.11 The fractal dimension D of an aerogel as a function of the correlation distance r defined by formula (10.1.1). . . . . . . . . . . . . . . . . .

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164 170 184 185 188 189 190 191 192

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202 206 208 210

Chapter 1

Introduction

The object under consideration consists of a gas and isolated particles of nanometer or micrometer sizes located inside this gas. In reality this object can have various forms including aerosols, dust, smoke, mist and haze and other particles of micron sizes in the Earth atmosphere which are formed as a result of water vaporization on the Earth surface, combustion of forests and peatbogs, volcano eruptions, action of storms on the Earth surface. In addition, aerosols and dust particles penetrate in the Earth atmosphere being products of power installations, chemical production, metallurgical furnaces etc. An astrophysical dust is a remarkable part of the universe matter. Nanoclusters are generated in specific devices in the form of cluster beams and can be used for preparation of specific films by deposition on a substratum and also can be used in other applications of nanotechnology. Combustion of a black powder in the atmosphere where a black powder is divided in separate grains in air is used in pyrotechnics and in military science. Wide applications of small particles take place and are expected in medicine and pharmacology. Air pollutants in the form of small particles are danger for specific productions and people health that require to remove these pollutants from the atmosphere. Thus, we have a wide circle of the cases where processes in gases involving particles of nanometer and micrometer sizes are important and must be studied. Investigation of such systems and processes with their participation is the goal of this book. According to their character, the processes in gases involving nanoclusters or microparticles may be divided in three groups. Transport of small particles in gases including their diffusion in gases and relaxation of particles in gases, relate to the first group of these processes. The second group of processes involving small particles includes attachment of atoms located in a gas to a small particle. This joins processes of quenching of metastable atoms and excited molecules on the particle surface, chemical processes with participation of small particles where a small particle is a fuel and a surrounding gas contains an oxidant in the form of a gas. This group contains also the process of growth of small particles in a buffer gas as a result attachment to it free atoms or molecules located in a buffer gas. In addition, processes of charging of small particles as a result of attachment of electrons and ions from a surrounding ionized gas we relate to this group processes. As is seen, in these processes an isolated small particle interacts

2

Chapter 1 Introduction

with atomic particles of a surrounding gas. The third group of processes includes processes of growth of clusters, particles and structures. In contrast to the previous case, where small particles were isolated and do not interact with each other, interaction between small particles or these particles with growing particles and structures determines the character of each growth process. These processes include coagulation, coalescence or Ostwald ripening, growth of fractal aggregates, fractal fibres and aerosols. We will study separately each of these processes. These processes have an analogy with processes involving small particles in liquids and solids, and basic concepts of these processes were elaborated 50– 100 years ago for small particles in a dense matter. In contrast to dense systems, two regimes of each process may be realized in gases. Namely, in the diffusion or hydrodynamic regime of the process a small particle under consideration interacts strongly with many atoms or molecules of a buffer gas simultaneously, as it takes place in a dense matter. In the kinetic or dynamic regime that takes place in a rareness buffer gas, a strong interaction between a small particle and surrounding atoms or molecule with only one atomic particle is possible. The criterion of each regime depends on the character of this process. In particular, for processes of the first group in the kinetic regime the mean free path of atoms of a buffer gas in the absence of small particles is large compared to a particle size, and the opposite relation between these parameters takes place for the diffusion regime of the process. In the case of processes of the second group for determination of the process regime it is necessary to compare a size of a small particle with an average distance between atoms or molecules which react with the small particle. Thus, the analysis of a certain process involving a small particle is accompanied by the criteria where a given regime of this process holds true. Analyzing the processes involving small particles, we use simple models both for small particles and dynamics of the process under consideration. We use the liquid drop model for nanoclusters or microparticles according to which they are modelled by a spherical drop that is cut out from a bulk system, so that the parameters of this drop are determined by properties of its bulk material. This model was applied firstly by N. Bohr and collaborators to the atomic nucleus and along with its other properties of atomic nuclei allowed one to describe nuclear fission. Note that the atomic nucleus is a two-component system consisting of protons and neutrons, whereas a nanocluster under consideration is an one-component system, i.e. its description within the framework of this model is simpler. Next, nanoclusters and microparticles become identical physical objects if we use this model. In consideration the dynamics of processes involving nanoclusters and microparticles, we are based on the hard sphere model. This model assumes a sharply varied repulsive interaction potential between colliding particles that allows one within the framework of this model to consider collision of atoms like to

Chapter 1 Introduction

3

billiard balls. Being applied to gaseous atoms, this model allows to J. C. Maxwell 150 years ago to make a principal step in creation the kinetic theory of gases. This model requires a smallness for the interaction region of colliding particles compared to their sizes. In the case of small particles this criterion is fulfilled independently on the character of interaction because a region of a strong interaction is of the order of atomic sizes that is small compared to a cluster size. Being guided by these two models, where the liquid drop model accounts for interaction inside a small particles, while the hard sphere model describes dynamics of collision of a small particle with atomic particles from a gas, we simplify the problem of determination of the parameters for this collision. Because a small particle subjects to classical laws, the rate of a certain process is expressed in the analytical or semianalytical form with indication the criteria of validity of these relations. Thus, simple models allow us to obtain simple results. Of course, it is necessary to analyze which is the accuracy such results with respect to real objects or processes. For example, a solid nanocluster is characterized by magic numbers which correspond to completed structures, and cluster parameters are nonmonotonic functions of the number of cluster atoms. Applying to it the liquid drop model, we average cluster parameters over neighboring cluster sizes. One can check the accuracy of this operation that is made below. One can compare the above approach with the possibilities of computer simulation of cluster properties and processes. In principle, computer simulations open new possibilities, but simultaneously they require more detailed information. For example, let us consider a metal cluster located in a buffer gas. In our approach we consider our object as a particle of a spherical shape that is cut out from a bulk metal, so that its radius is connected strictly with a number of cluster atoms. In computer simulation of this cluster it is necessary to introduce interaction between cluster components, i.e. ion-ion, electron-ion and electron-electron interactions. There are some troubles on this way because the metal cluster under consideration is a quantum system, and to escape these troubles, various models for interaction of atomic particles may be used, so that the parameters of interaction for each model may be obtained by application of these models to bulk metals and determination of interaction parameters from comparison some measured parameters of a bulk metal with calculated ones. Next, this operation is demanded for different models of interaction, and comparison of results of different models allows one to determine the reliableness and accuracy of computer calculations. Without such operations the reliability of the results of cluster computer simulations become problematic. Hence, computer simulation is a strong method allowed one to make complex mathematical operations and take into account interaction of many atomic particles simultaneously. If it is applied to a certain physical object, as nanoclusters

4

Chapter 1 Introduction

or microparticles, additional information is required about interactions in these systems, and the absence of this information makes computer calculations for a given object or process not reliable. In addition, using such computer simulations are available for professionals only. Approaches for cluster processes in this book allows us to obtain the result in the analytical form with indication the criterion of its validity and to estimate the accuracy of this result. But in contrast to computer methods, it is impossible to improve this accuracy. This analysis along with the expressions for the rates of the processes under consideration leads to a more deep understanding of these processes and gives the connection between its parameters. This may be a key for practices. For example, the process of production of the electric power in thermoelectric power station is based on combustion of a fuel, and products of this process in the form of a smoke contain small particles as soot, and it is necessary to extract these particles from a gas stream to prevent the environment from the pollution. For this goal one can use an appropriate existing device, for example, to extract particles by their charging and removal from the stream by an electric field. In principle, this operation may be fulfilled without understanding processes involving small particles and be optimized by change the regime of the used device. But, if we wish to decrease the output of small particles and to set the second step for extraction the particles, it is necessary to understand deeper the processes involving small particles. Moreover, if we consider the extracted particles not only as pollutants, but also as a component for production certain materials, a more detailed study is required for the above processes and products. These tasks raise the production in a new level and creates a new culture of production. For this new culture of production a more deep understanding of these problems is demanded, and this book may assist to this.

Part I

Properties of small particles and their behavior in gases

Chapter 2

Nanoclusters and microparticles in gases

2.1

Gas with small particles as physical object

The object of our analysis is gases with located in them small isolated particles in the liquid or solid state. From the physical standpoint this system is metastable since joining of individual particles into a compact system is thermodynamically favorable. This means that this system is non-equilibrium, and therefore the processes in gases involving small particles are of importance, an these processes establish a current state of this system and are responsible for this evolution. Moreover, this non-equilibrium system is created as a result of external actions, and hence processes involving small particles in gases are responsible not only for development of this system, but also for its formation, and these processes will be considered. Systems consisting of gases with small particles are of practical interest. The best known example of such systems is the Earth atmosphere in which small particles are injected as a result of evaporation of water and other components (sea salt etc.), eruption of volcanoes, forest combustion, sand storm and so on. On the other hand, these systems result from the man production activity. It is necessary in each certain case to understand the character of processes and the methods to act upon them. The object of this consideration are uniform isolate particles of nanometer and micrometer sizes, and hence we set aside some related objects and processes, as biological clusters [1], and also colloid solutions [2, 3, 4] which contain a disperse phase with particle sizes of 5–200 nm, since the specific properties of these and other related systems can be of principle. In this context, it is of importance that the basic concepts and terms for gaseous systems with small particles were obtained firstly in studies of colloid solutions. For example, the term “coagulation” was introduced in physiology as the process of formation of clots as a result of joining of blood corpuscules [5]. We do not consider below the artificial Earth satellites [6, 7] and asteroids which size is outside the considered range of particle sizes and can be operated by other processes. Probes in ionized gases used for measurements of plasma parameters [8, 9] are also outside our consideration, though their size 50–400 μm [10] is on the boundary sizes in this book, and the reason of this that our object is free small particles in gases. A dusty plasma that

8

Chapter 2 Nanoclusters and microparticles in gases

is a system where interaction between small particles is of importance is outside our consideration for the same reason, though some problems of a dusty plasma where interaction between particles is not essential, are analyzed below. So, we consider below gaseous systems with isolated small particles of nanometer and micrometer sizes. In this consideration we are guided by concepts obtained for colloid solutions and other system of a similar type. We apply these concepts to simple analytic models which allows us to calculate fast the parameters of processes under certain conditions with an accuracy 10–20%. It should be noted the important peculiarity of processes with small particles in a gaseous phase. In processes involving small particles in a condensed phase which concepts are taken often as a basis, atomic particles which take part in these processes undergo to a strong action from a matter where they are located, and hence a friction due to interaction with this matter is of importance for displacement of atomic and small particles. Another regime of such processes involving small particles is possible in gases due to their rareness if a gaseous environment does not influence on the character of the process. We demonstrate this on the example of the process of Ostwald ripening where an ensemble of small particles is located in a solid or liquid solution simultaneously with their atomic vapor, and particles are found in equilibrium with this vapor. The Ostwald ripening process [11, 12, 13] consists in growth of particles because as a result of process of evaporation of small particles and attachment of atomic particles to the surface of small particles, small particles of lower sizes evaporate, whereas small particles of larger sizes grow, and the average size of small particles increases in time. According to the classical theory of the ripening process or coalescence [14, 15, 16] that was developed for solid solutions, the motion of atomic particles in a space results from their diffusion in a solid. This means that the diffusion regime of coalescence is realized in this case. But if we deal with the same process in gases or plasmas, attaching atoms can reach the particle without collisions, the result does not depend on the diffusion coefficient of atoms, and this corresponds to the kinetic (Knudsen) regime [17] of this process. It is clear that these regimes are describes by different models. Below in each case we analyze both regimes of processes under consideration and criteria where each of them is realized.

2.2

Small particles in the Earth atmosphere

We now consider various cases where small particles are located in gases. Figure 2.1 contains various types of nanoparticles and microparticles located in atmospheric air with a range of their sizes. For comparison clusters and electric

9

Section 2.2 Small particles in the Earth atmosphere electric probe pollen mist cement dust mould cloud aerosol atmospheric dust coal dust asbestos dust bacteria condensation nuclei in clouds sea salt tabacco smog fire smoke virus Aitken particles nanoclusters

1

2 3 5 10 20

50 100

nanometers

300

1

2 3 5 10 20

50 100

300

1000

micrometers

Figure 2.1. Typical size of aerosols and microparticles located in the Earth atmosphere.

probes include in this Figure. Aitken particles were studied earlier than other particles of the Earth atmosphere (aerosols) [18, 19, 20, 21]. The Aitken particles are located at high altitudes, above clouds, and their basis are radicals of sulfur compounds which result from vaporization of meteorites at high altitudes and from processes on the Earth surface at low altitudes. Due to a small size below 0.1 μm the Aitken particles are responsible for a blue sky color because shortwave photons scatter on these particles effectively. The number density of Aitken particles at altitudes 10–20 km is 102 –104 cm3 [22]. At lower altitudes aerosols, i.e. a water suspension in atmospheric air, are formed as a result of condensation of a water vapor that is injected in the atmosphere owing to water vaporization on the Earth surface [23]. Sulfur SOx and nitrogen NOx oxides and atmospheric ions are nuclei of condensation in aerosol formation. Hence the presence of small additives in atmospheric air determines atmospheric physics and chemistry [24, 25, 26, 27, 28], and these processes depend on certain local conditions in a given region of the atmosphere. Since small admixtures influence on atmospheric processes involving aerosols, ejections in the atmosphere as a result of productions act on the character of atmospheric processes [29]. Accumulation of aerosols leads to formation of mist, smoke, haze [30, 31]. Another type of small particles in the atmosphere is soot that results from combustion of solid and liquid organic compounds in air. Then along with CO2 and

10

Chapter 2 Nanoclusters and microparticles in gases

water molecules which are formed in total combustion of hydrocarbons, the process can proceed as pyrolysis [32] that corresponds to a chemical process of an organic molecule at high temperature in the oxygen absence. This process leads to bonding of carbon atoms and to formation various organic molecules and radicals among which polycyclic aromatic hydrocarbons (PAH) [33] are the most danger. These molecules can be contained in soot along with other radicals and compounds. Soot and other chemically active molecules and radicals are captured subsequently by water aerosols in the atmosphere. The presence of chemically active particles in the atmosphere represents a danger to the health of inhabitants. Penetrating in lungs together with inhaled air, these compounds can cause pneumonia, bronchitis, tuberculosis and heart failure. A risk to cancer catch follows from a long location in impure air. The most strong result of such an action of atmospheric pollution took place in a London smog in December 1952 [34]. An inverse temperature distribution was observed in that time, i.e. a own temperature was below the above temperature, and convective air transport was absent. As a result, a mist in air was motionless and collected smoke from chimney which supported heat in this season. Pollution from chimney was accumulated and was delayed in the atmosphere during 5 days until the inverse temperature distribution was maintained. Then the visibility does not exceed 500 m during 5 days and 50 m during 2 days, and amount of SOx in the atmosphere exceeded the standard value (700 ppb) in 7 times. Next, the mass density of particulate matter for particles with diameter below 10 μm PM10 was in the limits from 3000 up to 14000 μg=cm3 , whereas in a quiet atmosphere it is approximately 30 μg=cm3 . A strong air pollution during a London smog led to tragic consequences—4000 persons were dead during the smog and 8000 persons dead after the smog under its action. Specific conclusions were accepted in 1956 which were connected with using of a gas, electricity and smokeless coal for heating the living apartments in London. This allowed one to escape such consequences subsequently. Volcano eruptions give a certain contribution to formation of toxic components in the atmosphere. Figure 2.2 represents a case of volcano eruption. Through a time a volcano smoke becomes fee from blocks and massive particles, whereas small particles are located in the atmosphere enough long. Small particles of the low atmosphere are washing out by rains, while small particles located above clouds can be found in the atmosphere during several months [37]. In accordance with the character of volcano processes, a volcano ash contains large amount of toxic substances, and the basic part of them are sulfur-containing substances. Therefore, a volcano ash has a negative influence on the environment nature and persons. For example, volcano eruptions in Iceland in 1783 led to the death of 20% habitants of Iceland. A toxic cloud with sulfur-containing compounds

Section 2.2 Small particles in the Earth atmosphere

11

Figure 2.2. Eruption of Grimsvotn volcano in Iceland in May 2011 [35, 36]. The eruption flow reaches up to 11 kilometers and along with gases and small particles contains stones and fragments of rocks.

moved to Europe and led to the death of 23 thousands of persons in the Great Britain [37]. The remembering eruption of the Krakatau volcano in 1883 [38, 39] caused 36 thousands of victims and is equivalent to explosion of 200-megaton bomb that exceeds the action of the atomic bomb in Hiroshima in 1945 in 13 000 times. A volcano ash acts not only on persons, but on equipment also, and can put out of operation the aircraft engine. For this reason 400 flights were cancelled in May 2010 during volcano eruption in Iceland, when a volcano ash was spread over the Europe North [40]. Smoke propagation in combustion of forests and peatbogs has an analogy with the volcano eruption process, but these processes differ by the power and particle

12

Chapter 2 Nanoclusters and microparticles in gases

composition. If sulfur-containing compounds and silicon compounds are the main components of volcano eruptions, products of combustion of forests and peatbogs is a soot where the basis are carbon atoms with chemical bonds between them as a result of pyrolysis, and nitrogen oxides and other mineral compounds are bonded with them. Because of a restricted power, a smoke resulted from combustion of forests and peatbogs does not reach high altitudes of the atmosphere, it is located below clouds, and propagates on distances up to 100 km. In addition, the size spectrum of smoke particles resulted from combustion of forests and peatbogs is more uniform than that from the volcano eruption (see Figure 2.1) and does not contain blocks and large particles as in the case of volcano eruptions. Since a smoke of combustion of forests and peatbogs does not rise above clouds, it is washing out by rain and its residence time in the atmosphere does not exceed several days. A smoke from combustion of forests and peatbogs creates a haze or mist in a daytime with a specific smell.

2.3

Methods of removal of dust particles from gas

Pollution of the atmosphere in local places is connected with a production activity to a greater extent. Power energetic set up, especially with a solid fuel, as well as chemical production and other forms of the man activity leads to rejection of detrimental impurities in the atmosphere, and solid particles are a remarkable part of rejections [41]. More a century along with development of the production, new methods are created which allow one to reduce rejection of detrimental impurities in the atmosphere. On this way, on the one hand the work is made for monitoring of detrimental impurities in the atmosphere (for example, [42, 43]), and on the other hand, there is a park of devices that prevents the rejection of detrimental impurities in the atmosphere within the framework of a certain set up [44]. There is a restricted number of methods [45] which allow one to remove small particles from air. Filters—porous materials are used for this goal, and when air passes through a filter, small particles stuck in pores. In particular, various filter types are used in microelectronics [46, 47] where poor rooms are used for production of elements of microelectronics since impact of dusty particles into the electronic scheme must be excluded. Therefore, purification of air corresponds to high requirements. In particular, in production of chips an upper boundary for the number density of dust particles of size above 0.3 μm is 3.5 m3 [48]. Two methods are used in removal of small particles from a gas stream, as it takes place in ejection of combustion products in the form of a smoke with dust particles from a factory chimney (Figure 2.3). The first method is used charging of particles in an ionized gas, usually in a specific gas discharge plasma.

Section 2.3 Methods of removal of dust particles from gas

13

Figure 2.3. Smoke from a chimney-stalk as a result of combustion of an organic fuel [49].

Formed charging particles attach to the walls later [50, 51, 52]. Another method is based on a high inertia of particles compared to atoms or molecules because of a large particle mass. Therefore if a gas stream rotates in a set up that is called cyclone [44], where rotation proceeds in a plane perpendicular to the stream direction, the particles occur on the periphery and may attach to the walls. An important element of both schemes is an impactor, i.e. a region near the walls where a gas is motionless [53, 23, 44]. These regions are separated from the gas stream by nets, and when dusty particles occur in this region, they stop and fall down where they are gathered by collectors. There are various devices of this type which are intended for a wide range of air streams [54, 55]. There is a simple method for particle removal from the air stream that is based on the inertia of particles. Then the air stream is moving along a bent tube and the

14

Chapter 2 Nanoclusters and microparticles in gases

movement of air molecules corresponds to a tube shape, whereas small particles collide with a tube surface in tube turn and are gathered by collectors located there. A park of devices for purification of gas streams from dust particles exist during decades. These devices for purification of gas streams are worked up to certain energetic or production facilities and accompany them by solution corresponding ecological problems. Creation of new generations of energetic facilities based on combustion of solid organic fuel, and also chemical and metallurgy facilities rise new problems which require a more detailed understanding of processes in gases involving small particles, and this is a goal of this book.

2.4

Artificial small particles in gas

Faraday [56] developed methods for creating gold nanoparticles of a given size in colloidal solutions, further advanced by Zsigmondy [57]. These systems were the vehicles for studying light scattering by small particles [58] and the character of growth of fractal aggregates [410, 431, 432, 62, 63]. Gold clusters play roles now in medicine [64], lithography [65], chemical catalysis [66] and in nanoelectronic devices [67]. Second, relativistic effects are important for atomic interactions in gold clusters and lead to various form of isomers as a result of structural mixing. This yields a variety of cluster structures, and hence shows the consequence of small energy differences between neighboring configurational states, including that of the lowest state. Therefore in several ways, gold clusters are paragons of clusters with metallic properties. Furthermore, gold clusters exhibit chemical behavior that is strikingly different from the inert character usually associated with bulk gold. A specific form of materials are their powders consisting of separate particles of micron sizes [68, 69]. Powders are convenient for technology (for example, [70, 71, 72]), since amount of used material may be dosed in a simple way. Therefore powders are used both in industry and in medicine. Usually a size of powder particles exceeds microns, otherwise particles stick together [68, 73]. Since particles may be chemically complex and may be porous, they are applied in medicine widely. Nevertheless, powders consisting of particles of submicron sizes are altered because of sticking of individual particles. In our consideration of particles located in gases this is not dramatically. Because in gases particles do not border with each other, they may be of nanometer sizes, and such particles are called clusters (see Figure 2.1). In contrast to powders, clusters have a simple chemical form usually. Hence, clusters are mostly a physical object, and we will consider them from this standpoint. Clus-

Section 2.4 Artificial small particles in gas

15

ters are applied mostly for preparation of films as a result of their deposition on a substratum where they can have a porous structure. As a matter, this corresponds to transport of a material in the form of a cluster beam. In a gas of a high temperature clusters are effective radiators [74, 75], and cluster sources of light are similar to incandescent lamps, but since clusters exist at higher temperatures compared to a wires, clusters sources of light are characterized by a higher efficiency. Next, as a fine physical object [76, 77], clusters are used in specific applications, as an autonomous source of neutrons [78, 79, 80, 81], an effective source of X-ray radiation [82, 83, 84, 85, 86, 87] etc. Various methods are used in generation of clusters in the form of cluster beams. Laser action on a surface is one method of generation of cluster beams [88, 89, 90, 91]. Cluster beams may be obtained in using a magnetron plasma [92, 93, 94, 95, 96, 97] and arc plasma [98, 469]. Cluster beams of easily melted materials may be obtained by the free jet expansion method [100, 101, 102, 103], where a gas or vapor propagated in a vacuum or rarefied gas through a nozzle. In particular, irradiation of a metal surface by a laser beam gives a flux of metal atoms that changes their parameters in the course of passing through a gas and can be converted in a cluster beam under certain conditions. Since clusters are usually uniform atomic systems of a simple chemical composition, they are a convenient model in study of properties of small particles. Another method of generation of small particles in gases is spraying of liquid streams in a gas where this liquid is converted in small drops or aerosols. In military applications [104] this method is used in large volumes. This allows one to obtain a smoke screen that masks movement of troops. Evidently, it was used firstly in a civil war in USA in 19 century and was used widely during first and second world wars. A basic component of a smoke is a nucleus of condensation and it collects a water vapor in aerosols in a humid atmosphere or uses lightly vaporized organic liquids. Existing sources with usage of a barrel of such a material allows one to create a smoke screen in a region of size of 100 km during 15 min [105]. In the first world war this method was used for creation of aerosols on the basis of poison-gases. Evidently, application of spraying of liquid streams in a gas is convenient for perfumery and medicine as a simple method to drive a product or drug to the object. This method with creation of a smoke screen is used also in cinematography and in agriculture, where it is used to prevent plants from a night) frost. Indeed, a not rare night frost at springer is able to destroy shoots or flowers of plants and fruit trees, because during this time a soil is cooled as a result of its radiation. To escape it, a fire is used which smoke prevents output of infrared radiation from the Earth surface and in this manner does not give its cooling. Smoke pots are more effective and allow one to create a smoke screen on large areas.

16

Chapter 2 Nanoclusters and microparticles in gases

A specific method of release of high energy densities is used in thermite reactions that was suggested in 1895 by German chemist H. Goldschmidt and uses a high specific heat release in oxidation of metals. A typical and spread reaction of such a type is the following 2Al C Fe2 O3 ! Al2 O3 C 2Fe C 852 kJ=mol:

(2.4.1)

The peculiarity of the termit reaction is its small rate, so that this reaction proceeds quiet, without explosions. In order to initiate this reaction, some components are added which increase the initial temperature [106]. Since the reagent contains simultaneously both fuel and oxidizer, this process proceeds after its initiation independently on external conditions. Another peculiarity of the termit reaction is connected with a high heat release. For example, in the process (2.4.1) using of 1 g of aluminium allows one to melt 60 g of iron. Nevertheless, a more memorizing application of the termit reaction relates to termit bombs which are known as fire bombs. A large number of such bombs were thrown on London and Moscow in 1941. If this bomb falls on an iron roof, it creates a high temperature that causes combustion of roofs and lead to fires. A simple way to escape it is to throw a burning bomb to the soil where it burns down. The above facts are not connected with generation of particles in the atmosphere. But this connection arises if termites are used in pyrotechnics where they are divided in small particles which burn in air independently. Especially it is of importance in nano-structured super-thermites [107], where the active material consisting of a fuel and oxidant is found in the form of grains of nanometer sizes (tens or hundreds nm). This material includes also a glue that is a bonding agent and a component that causes a weak explosion. As a result, an object decays in separate grains which burn independently in the atmosphere and their combustion creates specific glowing in air (as represented in Figure 2.4 [108]). This glowing lasts during seconds due to specifics of chemical reaction. Because there are various versions for components of the termit material, one can create various pyrotechnical materials, so that they may be improved and new pyrotechnical materials are created even now. Pyrotechnical materials allow one to create high local temperatures as a result of slow chemical reactions involving small particles, and these reactions are not connected with explosions [109]. Because of a low rate of chemical reactions, pyrotechnical materials are conserved in contrast to explosive materials. The above termit pyrotechnical material is only one type of pyrotechnical materials [110, 111] which existed long before discovering of thermites. The basis of pyrotechnical material was a gun-powder that was invented in China more then thousand years before and was transported to Europe through several centuries where under the name gunpowder or black powder it becomes an essential part of

Section 2.4 Artificial small particles in gas

17

Figure 2.4. Firework in Venice [108].

all military battles [112, 113]. Basic components of a black powder are ammonium nitrate, sulfur and activated coal which provide the combustion processes according to the schemes [109] 2KNO3 C S C 3C ! K2 S C N2 C 3CO2

(2.4.2)

or 10KNO3 C 3S C 8C ! 2K2 CO3 C 3K2 SO4 C 6CO2 C 5N2 :

(2.4.3)

Though the specific released energy for these processes is lower significantly than that for effective fuel and explosive materials, its advantage consists in low rates of chemical processes, and it is useful as a weapon element or in pyrotechnical materials for creation of fireworks. During a long history of the use of a black powder its composition was improved [114, 115] mostly due to various additives. In particular, sulfur is excluded from a smokeless black powder, but a more complex organic component is used. In particular, the combustion process in some version of a black powder proceeds according to the scheme [109] 4KNO3 C C7 H8 O ! 3K2 CO3 C 4CO2 C 2H2 O C 3N2 :

(2.4.4)

18

Chapter 2 Nanoclusters and microparticles in gases

Depending on the character of powder applications, this powder may contain specific additions. For example, in using the powder as a pyrotechnical material small additions may determine the color of its glowing. Note that combustion of powders proceed in the form of combustion of separate powder particles.

2.5

Electric processes in earth atmosphere

We below demonstrate the role of small particles in natural processes and phenomena on the example of the atmospheric plasma where charging of water aerosols and their falling toward the Earth surface under the action of gravitation forces leads to charge separation in the Earth atmosphere that is responsible for the action of the Earth electric machine. As an electric system, the Earth is a spherical capacitor where the charged Earth surface is the lower plate (see Figure 2.5). An electric field of the strength of approximately E D 130 V=cm is supported near the Earth 2 surface that corresponds to the negative Earth charge of Q D ER˚ D 5:8  105 C [116, 117] (R˚ D 6300 km is the Earth radius). The Earth electric potential with respect to a surrounding space is about U D 300 kV. Assume that the electric field strength decreases linearly with removal from the Earth surface and the electric field strength is zero at the upper plate, we obtain the distance between the plates of the Earth capacitor L D 2Uo =Eo  5 km. From this it follows that electrical phenomena of the Earth are developed at altitudes of several kilometers. We use a simple model where positive and negative ions are located in the atmosphere, and the difference between the number densities of the positive and negative ions is N . Then the Poisson’s equation gives for the electric field strengths E dE D 4eN; dz

(2.5.1)

where z is the distance from the Earth surface, and the solution of this equation takes the form  Eo z ; LD (2.5.2) E D Eo 1  L 4eN As a result, this model leads to the above connection between the electric potential Uo and the electric field Eo near the Earth surface. Next, according to measurements, the current density over lands is 2:4  1016 A=cm2 and over oceans it is equal in average to 3:7  1016 A=cm2 [116], that corresponds to the total current through the Earth atmosphere of I D 1700 A. This current is created in the quiet atmosphere by ions and charged particles of the atmosphere and leads to discharge of the Earth. A typical discharge time due to

19

Section 2.5 Electric processes in earth atmosphere

+ +

–

+ –

–

+ – +

–

Earth – – + + +

–

– +

–

+

+ –

E

Earth

Figure 2.5. The Earth as a spherical capacitor whose lower plate with a negative charge is located on the Earth surface, and the position of the upper plate corresponds to the upper edge of clouds at the altitude of several kilometers.

currents in it is equal  D Q=E  6 min. To support the negative Earth charge, it is necessary a strong process that leads to the Earth charging. This process results from electric breakdown between clouds and the Earth through lightnings [118], and just lightning carry the negative charge to the Earth surface (see Figure 2.6). Thus, just lightnings provide the Earth charging [118]. In turn, it is necessary a high electric potential between clouds and the Earth surface for generation of lightnings, and this determines the specifics of lightnings [119]. We give above the Earth electric parameters and describe processes which establish the Earth charge. Then the question arises what is the connection between the above electric processes in the Earth atmosphere and the behavior of small particles in gases that is the object of our consideration. The electric atmospheric processes are the secondary processes with respect of transport of moisture through the Earth atmosphere. Amount of transferred moisture through the atmosphere per unit time as a result of water evaporation from the Earth surface and water returning to the Earth in the form of precipitations is 4  1020 g=year or 1:3  1013 g=s. In considering the electric atmospheric processes to be accompanied the water circulation through the atmosphere, we obtain the specific charge transport to be 1:4  1010 C per water gram. This transport is realized if water is located in the atmosphere in the form of negatively charged aerosols. Then the falling of negatively charged aerosols to the Earth surface under the action of the aerosol weight creates electric currents in the atmosphere and leads to the charge separation in the atmosphere. This takes place in clouds (see Figure 2.6 [120, 121]) where high electric fields are created. In particular, the electric potential of the lower cloud edge with respect to the Earth surface is 20–100 MV [122] at a distance between them of 1–2 km. A subsequent breakdown of air leads to transport of the part cloud charge to the Earth. This leads to a decrease of the electric potential of the lower cloud part, and a transferred charge to the Earth is distributed over the Earth

20

Chapter 2 Nanoclusters and microparticles in gases

Cloud lightning evaporation

evaporation

Earth Figure 2.6. Charging of the Earth results from lightnings which are electric discharges between clouds and Earth surface [120, 121]. The clouds are formed and destroyed as a result of the moisture circulation in the course of water evaporation from the Earth surface and water returning on the Earth surface in the form of precipitations.

surface. The Earth charging proceeds in this manner due to small atmosphere regions where clouds are located and relatively high water concentration is supported [120, 123, 124, 125, 126]. The Earth discharging results from transport of ions and charged small particles under the action of the Earth electric fields. This seemingly simple and transparent scheme of electric processes in the Earth atmosphere becomes complicated by action of some real factors. To explain the negative Earth charge, we assume the negative charge of water aerosols. In reality, the aerosol charge is established as a result of attachment of positive and negative atmospheric ions to the aerosol, and the aerosol charge is negative if the mobility of negative atmospheric ions is higher than that of positive ions. But the ratio of the mobilities for ions of a different sign depends on sorts of ions which are located in a given atmosphere point. Ions are formed in the atmosphere from molecules which are admixtures to atmospheric nitrogen and oxygen, and the charge transported to a given point of the Earth surface may be both negative and positive depending on ion sorts formed there. According to measurements [127], a number of lightnings which carry the negative charge to the Earth surface exceeds a number of lightnings which carry the positive charge to the Earth surface in 2:1˙0:5 times. The total current of the negative charge exceeds the total current of the positive charge in average in 3:2 ˙ 1:2 times [127]. In addition, approximately 100 flashes take place per second [122]. This scheme is based on location of water aerosols and positive and negative ions in the Earth atmosphere. Positive ions compensate the negative charge of

Section 2.6 Dusty plasma of solar system

21

aerosols, and their current may compensate the charging electric current of falling charged aerosols that leads to the Earth charging. This problem is simplified if along with liquid negatively charged water aerosols the ice particles are formed in clouds. Then collisions of ice particles with water aerosols becomes of importance. As a result, ice particles are charged positively, and water aerosols are charged negatively [269, 270, 271, 272, 273, 274, 275]. In this case the requirements to positive ions become not strong. Note that clouds are formed at altitude where the atmosphere temperature is close to the freezing temperature of water. If more heavy negatively charged aerosols fall down faster than ice particles, and this leas to the negative charging of a lower cloud edge. Of course, transport of ice particle and water aerosols at these altitudes can change their phase state. It should be noted that this mechanism of generation of an atmospheric electricity may be universal for planets. Indeed, if there are two sorts of particles in an atmosphere, so that particles of the first sort are charged positively, and the particles of the second sort are charged negatively, their fall under the action of their weight to the planet surface creates currents in the atmosphere that leads to generation of high electric potential in the atmosphere. Electric breakdown under action of such potentials causes discharging of the planet. As a demonstration of this possibility, we give in Figure 2.7 eruption of the volcano in Chilly that took place in Chile in May 2011. Eruption of this volcano contains dust particles of a different sort, and their charging may lead to generation of electric fields in the atmosphere that causes origin of lightning. Figure 2.7 exhibits origin of lightnings among erupting volcano ashes.

2.6

Dusty plasma of solar system

One more example of small particles located in a gas relates to a dusty plasma of a solar system [136, 137, 138]. This plasma results from interaction of the solar wind [139, 140, 141] with an interplanetary dust. A solar wind is a plasma stream emitted by a solar corona in an environment, and this plasma consists of electrons and ions. Interaction of a solar wind with a dust of comet tail or with a dust of Saturn, Uranium and Jupiter planets leads to formation of a dusty plasma, and interaction of this plasma with surrounding magnetic fields creates specific structures of a dust plasma as rings of Saturn, Uranium and Jupiter. The number density of electrons and protons of the solar wind at the Earth level is approximately 7 cm3 , the electron temperature is about 2  105 K, the proton temperature equals to 5  104 K, and the stream velocity is 4  107 cm=s. As an example of a dust plasma of the solar system we consider E- and F -Saturn rings which contain ice particles of size from 0.5 μm up 10 μm [143,

22

Chapter 2 Nanoclusters and microparticles in gases

Figure 2.7. Eruption of Chile’s Puyehue volcano in June 2011 [128] that is accompanied by short lightnings inside the erupt.

144], and a typical number density of ice particles is equal 30 cm3 [143]. The source of these particles are the satellite Enceladus for E-ring of Saturn [145, 146, 147] and satellites Prometheus and Pandora for F-ring [148]. Simultaneous measurements within the framework of the Cassini project give the following parameters of dust particles [149, 150] and plasma parameters [151, 152] for E-ring. Note that the specifics consists in a strong interaction of small particles with a plasma of the solar wind which parameters vary under the action of dust particles. This means that the density of a plasma between particles outside action of their field differs from the density of the solar wind plasma. The number density of this plasma is No D 30–100 cm3 for E-ring of Saturn that exceeds the number density of the solar wind plasma that is 0:1 cm3 , and the electron temperature for this plasma is 10–100 eV [153]. The basic ion sorts are OH and H2 OC with the ion temperature of the order of 103 K. Note that the number density of dust particles is Np D 30 cm3 [143] there. As is seen, mixing of the solar wind with a dust that is emitted by Saturn satellites leads to formation of the plasma of Saturn rings (see Figure 2.8) with a higher number density of electrons and ions compared to those of its source, the solar wind. This results from capture of ions by charged dust particles, that stops a flux of the solar wind plasma and hence increases the number density of electrons and ions. The same takes place in comet tails [155, 156] where a

Section 2.6 Dusty plasma of solar system

23

Figure 2.8. The basis of Saturn rings is a dust created by planet satellites and interacted with the plasma of a solar wind and Saturn magnetic field [142].

Figure 2.9. As a result of interaction with a solar wind a comet tail with ions and dust particles is separated from the comet nucleus in the direction from the Sun and due to interaction with the magnetic field the tail conserves connection with the comet nucleus in the course of its evolution [154].

dusty plasma is formed as a result of interaction of the solar wind and a comet dust (see Figure 2.9 [154]), and the magnetic field is the main element of this interaction [157]. The density of a plasma of the comet tail at the Earth level is 103 –104 cm3 [158, 159], that exceeds remarkably the plasma density in the solar wind, whereas the temperature of electrons in the comet plasma is estimated as Te  104 K [159, 160, 161] that corresponds to the electron temperature in the solar wind plasma. Heightened number density of the comet plasma results from the capture of ions by charged dust particles and also is determined by a large lifetime of captured ions. Thus an interplanetary dust interacted with the solar wind forms a specific dusty plasma where electrons of the solar wind attach to dust particles, and these negatively charged dust particles form around them an ion coat, so that the charge density in a formed dusty plasma is higher than that in the created it solar wind.

24

Chapter 2 Nanoclusters and microparticles in gases

Figure 2.10. Eruption of the volcano Puyehue in Chile in June 2011 [162] where the smoke stream passes through clouds.

2.7

Problems

Problem 2.1. Gases and vapors break away the volcano crater and are directed vertically up, piercing the clouds. Small particles both dust particles captured by the vapor stream and other particles formed in the course of its cooling, move together the stream. Assuming the basic vapor sort to be SO2 , the temperature on the exit of the volcano crater to be T D 3000 K and that the stream breaks away with the sound velocity, determine the maximum altitude that may be attain by small particles. A smoke resulted from volcano eruption and directed up, pulls apart clouds and rises above them (see Figure 2.10). The gravitation force prevents from propagation of particles up. Assuming that small particles do not brake in collisions with air molecules, we find the maximum altitude h from the energy conservation m

c2 D mgh; 2

(2.7.1)

where m is the mass of a small particle, c is the initial velocity of a stream after the crater, g is the free fall acceleration. Let us equalize the stream velocity to the sound speed c for SO2 molecules at the temperature T D 3000 K that is

25

Section 2.7 Problems

c D 7:0  104 cm=s. Then we obtain from the above equation hD

c2 D 25 km: 2g

(2.7.2)

This is the upper boundary for the altitude that is attained by small particles for two reasons. First, collisions with air molecules decreases this value, and second, the velocity of small particles near the volcano crater may be lower than the speed velocity of vapor molecules. Nevertheless, this result exhibits the possibility for small particles to reach altitudes above clouds.

Chapter 3

Cluster properties and their modeling

3.1

Cluster structures

A cluster as a system of a finite number of cluster atoms possesses an intermediate position between simplest atomic systems—molecules and macroscopic systems of bound atoms—solids nd liquids. Our goal is to represent in a simple form general peculiarities of the cluster nature, and in this consideration we restrict by the simplest clusters which consist of identical atoms with a pair interaction between them, so that the dependence of the pair interaction potential of atoms on a distance between them is given in Figure 3.1. In this case the interaction inside the cluster takes place between nearest neighbors, as it proceeds in clusters consisting of inert gas atoms or gaseous molecules. Let us restrict by a cluster consisting of 13 atoms that has a completed structure. Figure 3.2 represents the icosahedral cluster structure that is exposed in Figure 3.3, and Figure 3.4 represents completed structures of the cluster consisting of 13 atoms which are cut from the cubic facecentered lattice and hexagonal lattice. As is seen, a cluster consisting of 13 atoms has three closed structures—icosahedron, cuboctahedron and hexahedron, and the optimal structure depends on the interaction potential between atoms. If interaction between nearest neighbors dominates inside the cluster, the icosahedral structure is favorable, because of 42 bonds between nearest neighbors for the icosahedral cluster consisting of 13 atoms, whereas there are 36 bonds between nearest neighbors if this cluster has the cubic face-centered or hexagonal structure. Taking the distance between nearest neighbors for clusters of 13 atoms with the cuboctahedral and hexagonal structure to be equal to the equilibrium distance Ro between atoms for a diatomic molecule (Figure 3.1), we give the lengths of other bonds in Table 3.1. It should be noted that the total number of pair bonds in this 13-atom cluster is a number of 2 combinations C13 D 78, and 36 of them relate to bonds between nearest neighbors. The difference between the cuboctahedron and hexahedron structures of the 13-atom cluster is determined by atoms located in the upper and lower triangles.

27

Section 3.1 Cluster structures U

0

R R0

Figure 3.1. The interaction potential between nearest atoms of the cluster as a function of distances between atoms for a short-range interaction of atoms.

Figure 3.2. The icosahedral structure.

In particular, let us consider the pair Lennard-Jones interaction potential between atoms that has the form [164, 165] "    12 # R R 6 ; C U.R/ D D 2 Ro Ro where Ro is the equilibrium distance between atoms, D is the depth of the potential well for a pair interaction of atoms. The 13-atom Lennard-Jones cluster has the total binding energy of atoms E D 44:34D for the icosahedral structure,

28

Chapter 3 Cluster properties and their modeling

a

b

c Figure 3.3. Atom positions in the icosahedral cluster consisting of 13 atoms. a) Side view, b) top view, c) the developed view of a cylinder in which a cluster-icosahedron is inscribed.

Table 3.1. A number of bonds between atoms of an indicated length for the cluster consisting of 13 atoms and having the cuboctahedron and hexahedron structure [163]. These clusters can be cut from the cubic face-centered and hexagonal crystal lattice correspondingly.

Bond length Rp o Rp o 2 Ro p8=3 Rp o 3 Ro 11=3 2Ro

Cuboctahedron 36 12 0 24 0 6

Hexahedron 36 12 3 18 6 3

E D 40:88D for the cuboctahedron structure, and E D 40:90D for the hexahedron structure. The Lennard-Jones pair interaction potential of atoms is convenient as the model one because it contains simultaneously both a short-range and long-range parts. Nevertheless, this is not suitable for description of a condensed system of inert gases, where the long-range part of interaction is weaker than that corresponds to the Lennard-Jones interaction potential [166]. As is seen, the 13-atom cluster can have different structures and their competition depends on the character of interaction inside the cluster, so that variety of cluster structures is wider than for a bulk crystal structure. In particular, in the above example of the 13-atom cluster the cuboctahedral and hexahedral structures correspond to the cubic face-centered and hexagonal lattice structures and can be cut from them. The icosahedral structure that is optimal for the 13-atom cluster

29

Section 3.1 Cluster structures

a

b

+

+

+

Figure 3.4. a) Cuboctahedron. b) Cluster structures consisting of 13 atoms that are cut from the crystal lattice with close packing. It is shown positions of atoms-balls located in a central plane. Circles are projections of three atoms-balls of the upper plane onto the central plane and they coincide with the projections of three atoms-balls of the lower plane onto the central plane for the hexahedron cluster structure, while the projections of three atoms-balls of the lower plane onto the central plane for the cuboctahedron cluster structure are shown by crosses.

is not realized in an infinite crystal lattice because the distances between nearest neighbors for the icosahedron geometric figure are different. A cluster as a system of bound atoms is considered as a specific physical object due to magic numbers for a solid cluster [167] since this cluster property is specific. This property means that many cluster parameters, such as the atom binding energy, the cluster ionization potential, the cluster electron affinity as a function of a number of cluster atoms have extrema at magic numbers of cluster atoms. In reality, magic numbers correspond to completed atomic or electron shells, and hence clusters with magic numbers are formed more effective than clusters with neighboring numbers of cluster atoms. Correspondingly, magic numbers are observed as maxima in mass-spectra of clusters in excited gases or cluster beams [168], and in mass-spectra of photoionization of cluster beams [169]. In experiments magic numbers are observed in clusters consisting of up to tens thousands of atoms [170]. 13 is the magic number for the above cluster with a pair interaction of atoms where interaction between nearest neighbors dominates [171]. In this case the surface atom has 6 nearest neighbors for the icosahedron structure (see Figure 3.2 and Figure 3.3) and 5 nearest neighbors for the cuboctahedral and hexahedral structures. Adding of one atom to the 13-atom cluster gives 3 new bonds between nearest neighbors. On contrary, detachment of one atom from 13-atom cluster leads to a decrease of bonds for surface atoms. Hence, the 13-atom cluster has

30

Chapter 3 Cluster properties and their modeling

a larger binding energy of the surface atom than clusters consisting of 12 and 14 atoms that means that 13 is the magic number for clusters where interaction between nearest neighbors dominates. This property relates also for other cluster parameters, so that the ionization potential and the electron affinity for the 13-atom cluster is larger than those for clusters consisting of 12 and 14 atoms. There are an infinite number of magic numbers of clusters which correspond to clusters with completed atom shells if interaction between cluster nearest neighbors dominates. The availability of magic numbers is the specific cluster property that allows one to consider clusters as a specific physical object. The analysis of the cluster structure allows one to give a strict cluster definition separating clusters from macroscopic particles. In contrast to a bulk particle, clusters contain a large number of atoms on the surface. One can use it in the cluster definition considering clusters as particles where more 10% atoms are located on its surface. Analyzing clusters where interaction between nearest neighbors dominates, we will be guided by the icosahedral structure of the solid cluster with the optimal cluster structure. Accounting for distribution of atoms over atomic shells, we have that a cluster with m completed shells contain n atoms that is given by [172] n.m/ D

10 3 11 m C 5m2 C m C 1; 3 3

(3.1.1)

and the number of surface atoms of this cluster is n.m/ D n.m/  n.m  1/ D 10m2 C 2:

(3.1.2)

As it follows from this, the boundary between the cluster and macroscopic particle for the above definition corresponds to the number of atomic shells m D 28 and atoms n  8  104 . Note that for a number of atomic shells of the icosahedral cluster m D 13 the number of atoms is n  8  104 , and 21% of atoms is located in the surface layer of the completed cluster structure.

3.2

Phase transition in cluster

A cluster is a convenient object for modeling of a macroscopic system of atoms since it contains a finite number of atoms and one can expect that principal properties of clusters and bulk atomic systems are identical. On the other side, known properties of macroscopic systems of bound atoms can be transferred to clusters. Let us consider from this standpoint the phase transition in clusters, assuming that like to a macroscopic systems a cluster may be solid or liquid in a simple case. The analysis of the phase transition in clusters allows one to understand

31

Section 3.2 Phase transition in cluster

a

atom oscillations

b

configuration excitation

Figure 3.5. Types of cluster excitations: a) atom oscillations in a cluster, b) configuration excitation corresponded to a change of the atom configuration of the cluster [166].

this phenomenon deeper. We below consider the phase transition for the 13-atom cluster with a pair interaction between atoms where interaction between nearest neighbors dominates. It is clear that the solid state of the cluster under consideration contains the maximum number of bonds between nearest neighbors for a given number of atoms, and it corresponds to the cluster icosahedron structure. The liquid cluster state corresponds to excitation of the solid aggregate states. There are two types of excitations given in Figure 3.5. They include a thermal motion of atoms due to their oscillations (Figure 3.5 a), and configuration excitation of clusters (Figure 3.5 b) leads to a change of the number of bonds between nearest neighbors. For the 13-atom cluster the lowest cluster configuration excitation by energy corresponds to transfer of a surface atom over its surface where this atom forms 3 bonds with nearest neighbors, and the configuration excitation of the 13-atom icosahedral cluster corresponds to loss of 3 bonds. The next configuration excitation of this cluster at zero temperature leads to a loss of one more 2 bonds, that is the first configuration excitation of the 13-atom cluster is separated by a large energetic gap both from the solid (ground) atom state and from following configuration cluster states. We call below the lowest excited configuration states of the cluster as the liquid aggregate state, and subsequently we will show the validity of this definition. In this definition the liquid aggregate state of the 13-atom cluster is characterized by a large statistical weight g D 180, that is the product of a number of surface atoms (12) which can be transferred over the cluster surface and a number of positions over the cluster surface (15) where the transferred atom has three bonds with nearest neighbors and does not border with a formed vacancy. For the thermodynamic equilibrium inside the cluster between the ground and excited (solid

32

Chapter 3 Cluster properties and their modeling 10.0 9.5 9.0

Δs

8.5 8.0 7.5 7.0 6.5 0.20

T, D 0.22

0.24

0.26

0.28

0.30

0.32

Figure 3.6. The temperature dependence for the entropy of transition at melting of the 13-atom Lennard-Jones cluster [171, 173] that is obtained on the basis of the data of computer simulation of the 13-atom cluster if it is found under adiabatic [174] and isothermal [175] conditions.

and liquid) configuration states we have according to the Boltzmann formula the following ration of the probabilities for the cluster to be in the liquid (pl ) and solid (ps ) aggregate states     pl E E D g exp  D exp S  ; S D ln g: (3.2.1) ps T T In the case under consideration we have at zero temperature of cluster atoms S D ln 180 D 5:2, E  3D, where D is the energy of one bond. In considering the phase transition in clusters, we assume the statistical weight of the liquid aggregate state to be larger than that for the solid one, and the phase transition from the solid to the liquid state is possible at a final temperature. We determine the cluster melting point Tm as pl .Tm / D ps .Tm /, and according to formula (3.2.1) the cluster melting point is Tm D

E : S

Taking the parameters of the phase transition at zero temperature, we have Tm  0:6D. In reality the above assumption is not valid because the liquid state is more friable compared to the solid one, and the transition entropy grows with an increasing temperature, whereas the energy of configuration excitation decreases

33

Section 3.2 Phase transition in cluster 300

T m, K

250

n 200 50

100

150

200

Figure 3.7. The dependence on a number of cluster atoms n for the melting point Tm of sodium clusters [180, 181]. The melting point of bulk sodium is 371 K. Arrows indicate magic numbers of atoms in the sodium cluster.

weakly due to thermal motion of atoms. In particular, Figure 3.6 gives the entropy of this phase transition. Accounting for this dependence leads to the cluster melting point Tm  0:3D, and it is different weakly for a cluster as a microcanonical ensemble of atoms (the adiabatic case) if the cluster does not exchange by energy with an environment, and canonic ensemble of atoms (the isothermal case) if this cluster is located in a thermostat [174, 175]. From this consideration it follows one more principal property of the cluster phase transition which differs the phase transition in clusters from that for macroscopic systems of atoms. Namely, there is dynamic coexistence of the solid and liquid phases in clusters [174, 176, 177, 178, 179] that is absent for a macroscopic atomic systems. Indeed, within the framework of thermodynamic equilibrium (3.2.1) we assume coexistence of solid and liquid phases, if the ratio of these probabilities is found within the limits a
Tmax is not realized. At the temperature of surrounding air T D Tmax an instability develops, and the particle acquires a higher temperature by jump, and this temperature corresponds to another regime of its combustion.

108

7.5

Chapter 7 Processes in buffer gas on surface of small particles

Kinetic and diffusion regime of particle combustion

We above consider the kinetic regime of particle combustion that is governed by a small parameter  defined by formula (7.4.10). This parameter characterizes the probability of participation in the chemical reaction for an oxygen molecule contacted with the particle surface. The concentration of oxygen molecules at the particle surface c.ro / is zero in the diffusion regime of combustion and is equal to co that is the concentration of oxygen molecules far from the burning particle. In consideration a dense gas   ro , where  is the mean free path of molecules in air, we have for the total number of oxygen molecules J intersected a sphere of a radius R is J D 4R2 D.O2 /Na

dc ; dR

(7.5.1)

where D.O2 / is the diffusion coefficient of oxygen molecules in air, Na is the number density of air molecules, and c.R/ is the concentration of oxygen molecules at a distance R from the particle, so that Na c.R/ is the number density of oxygen molecules. Because oxygen molecules are not formed and absorbed in a space, this relation may be considered as the equation for the concentration c.R/ for oxygen molecules. Solving this equation, we obtain the relation between the concentrations at a given distance R from the particle and far from it co in the form J c.R/ D co  : 4D.O2 /Na R In the diffusion regime where the Smolukhowski formula (4.2.5) J D 4D.O2 /ro Na .O2 / is valid for the rate of reactant oxygen molecules and, this formula coincides with formula (7.3.6). If a distance from the particle surface is small compared to the mean free path of molecules in air , the rate of combustion in accordance with the definition of this quantity and formula (7.4.3) is given by s T J D 4 ro2  (7.5.2) Na c.ro / .To / D 4 ro D.O2 /Na c.ro /; 2 m.O2 / where c.ro / is the concentration of oxygen molecules at the particle surface. This allows us to represent the above formula for the concentration of oxygen molecules in the form ro c.R/ D co  c.ro / ; (7.5.3) R

Section 7.6 Recombination of charged clusters in buffer gas

109

where the parameter  is given by formula (7.4.11). This leads to the following connection between the oxygen concentrations at the particle surface c.ro / and far from particle co c.ro / D

co 1C

(7.5.4)

The kinetic regime of particle combustion corresponds to the limit   1, whereas the criterion   1 is fulfilled in the diffusion regime of particle combustion. One can see that the kinetic regime of particle combustion requires the criterion ro  :  Correspondingly, on the basis of formula (7.5.2) we have s T  co Na .To / D 4 ro D.O2 /Na : J D 4 ro2  2 m.O2 / 1C 1C

(7.5.5)

(7.5.6)

This formula is transformed into the Smolukhowski formula (4.2.5) in the diffusion regime   1 of particle combustion and is converted in formula (7.5.2) in the kinetic regime   1 of particle combustion.

7.6

Recombination of charged clusters in buffer gas

If clusters or small particles are located in gases and they contain both positive and negative charge in this gas, collision of two oppositely charged clusters leads to decrease of the charge in this gas. We below consider collision of two oppositely charged clusters with charges Z1 and Z2 , where the cluster charges are expressed in electron charges. We use the assumption that a contact of these clusters leads to charge transfer, and after this contact the cluster of a smaller charge becomes neutral. In this consideration, we will be guided by metal clusters located in a buffer gas. We first consider the kinetic regime of the process when a contact of clusters results from their direct collisions. We use the assumption that Coulomb interaction of clusters during their contact exceeds their kinetic energy in the center of mass frame of reference, that is Z1 Z2 e 2  "; Ro

Ro D r1 C r2 ;

(7.6.1)

110

Chapter 7 Processes in buffer gas on surface of small particles

where Ro is the sum of cluster radii r1 and r2 , " is the kinetic energy of the clusters in the center of mass frame of reference. Considering clusters as classical particles, we then obtain for the cross section of cluster contact instead of (4.2.1) D

Z1 Z2 e 2 .r1 C r2 / : "

(7.6.2)

This gives for the rate constant of this process under the assumption of the Maxwell distribution of clusters over velocities r  8 1 2 D hvi D Z Z e .r C r / krec p 1 2 1 2 m " s 2 ; (7.6.3) D 4Z1 Z2 e 2 .r1 C r2 / T where D m1 m2 =.m1 C m2 / is the reduced mass of clusters, so that m1 and m2 is the mass of the corresponding cluster, and T is the cluster temperature. In the diffusion regime of cluster recombination we have that collision of clusters with buffer gas atoms does not allow approaching of clusters. If we assume that the first cluster is motionless in this process and draw around it a sphere of a radius R that exceeds cluster radii, we obtain for the total number J of clusters of the second type intersected this sphere per unit time as J D 4R2 w2 .R/N2 ; where 4R2 is the area of this sphere, w2 is the drift velocity of charged clusters of the second type, N2 is their number density. The electric field strength from the first cluster at a location point of the second cluster is E D Z1 e=R2 , and its drift velocity is w D Z2 EK2 , where the mobility K2 relates to the second cluster with a single charge. From this we obtain for the rate constant of recombination taking into account also the motion of the first cluster with respect to the second one krec D 4Z1 Z2 e.K1 C K2 /: Because clusters are of the same type, we have the identity of the mobilities for clusters of a single charge K1 D K2 D K, that gives for the rate of cluster contact which leads to cluster recombination krec D 8Z1 Z2 eK:

(7.6.4)

We now give the criterion of validity for the diffusion regime of cluster recombination. Evidently, formula (7.6.3) holds true if an average distance between

111

Section 7.7 Problems

charged clusters is small compared to the mean free path of clusters ƒ that is determined by formula (6.1.4). This gives the criterion of the diffusion regime of cluster recombination in the form 1=3

Ncl ƒ  1;

(7.6.5)

where Ncl is the number density of charged clusters.

7.7

Problems

Problem 7.1. An atomic silver vapor is converted in a gas of silver clusters in argon at pressure of p D 1 Torr and the concentration of silver atoms with respect to argon atoms is cb D 104 . Being guided by a number of silver clusters as n  104 , determine the critical temperature below which cluster growth proceeds. Let us take the parameters of silver clusters according to the data of Table 7.1 as "o D 2:87 eV, A D 2:0 eV, po D 1:5  106 atm. Defining the critical temperature Tcr such that this provides the equality of the rates of attachment of free atoms and evaporation of bound atoms, we have for the total silver pressure p and the equilibrium pressure according to formulas (7.1.8) and (7.1.11)   "o 2A p D po exp  : C 1=3 Tcr 3n Tcr Solving this equation under given conditions and the total silver pressure p D 104 Torr, we find Tcr D 1080 K. Cluster formation and growth is possible only below this temperature. Problem 7.2. Find the connection between the temperature of surrounding air and the temperature of a particle of activated birch coal with combustion parameters according to formula (7.4.8) for a particle radius 20 μm, where this connection in the kinetic regime of combustion is given in Fig. 7.1. We are based on the balance equation (7.4.12) that includes heat release as a result of particle oxidation and heat loss due to radiation of the particle and thermal conductivity of surrounding air. In contrast to equation (7.4.12), we take into account that heat release is restricted by diffusion transport of oxygen molecules to the particle surface in accordance with formula (7.5.6) for the rate of heat release. As a result, we obtain the following heat balance equation   1 Ea 4 3 ro o exp  D 4 ro .To /.To  T / C 4 ro2 aTo4 : 3 To 1 C .To / (7.7.1)

112

Chapter 7 Processes in buffer gas on surface of small particles

Solving this equation at a certain particle radius, we obtain the dependence of the air temperature T far from the particle as a function of the particle temperature To . The results are given in Figure 7.1 for the particle radius ro D 20 μm, and we below analyze them. As is seen, the dependence T .To / is grouping around the asymptote T D To . At low temperatures where the contribution to the heat balance from particle emission exceeds that from particle combustion, T > To . We have that T D To if these contributions are equal, that for the particle radius ro D 20 μm corresponds to the temperature To D 1150 K. The corresponding temperature of Table 7.3 To D 1134 K is lower because the data of Table 7.3 are based on the kinetic regime of particle combustion, but one can see a small difference between these results. In addition, it follows from comparison of the results in Figure 7.1 the principal role of the diffusion transport of oxygen molecules to the burning particle at high temperature. Indeed, thermal explosion takes place in the kinetic regime of particle combustion, and at the particle radius ro D 20 μm it takes place at the particle temperature To D 1260 K in ignoring particle radiation and at To D 900 K if particle radiation is taken into account. At higher temperatures the solution of the heat balance equation at given assumptions is not stable. As it follows from Figure 7.1, the solution of the balance equation (7.7.1) is stable at any temperatures, i.e. the thermal instability is absent. Problem 7.3. Determine the temperature of burning spherical aluminium particles in air and a time of particle combustion if a particle radius is ro D 10 μm, the air pressure is 1 atm, and the concentration of aluminium atoms corresponds to total oxidation of aluminium with formation of molecules Al2 O3 and total consumption of oxygen. Assume that combustion products are removed far from the particle and the combustion process proceeds in the diffusion regime. After ignition of the aluminium particle, its total combustion corresponds to the scheme 4Al C 3O2 ! 2Al2 O3

(7.7.2)

and we concentrate an attention on the particle combustion when its equilibrium with surrounding air is established. As a result of combustion of an aluminium particle or bulk aluminium in accordance with the scheme (7.7.2), the energy H D 1676 kJ=mol D 401 kcal=mol is released [315] that corresponds to the energy of " D 17:4 eV per one Al2 O3 molecule or the energy of 11:6 eV per one oxygen molecule. We assume a high rate of the chemical process that leads to an energy release on the particle surface or near it and relates to the diffusion regime of the oxidation process. The released energy is transferred to surrounding

113

Section 7.7 Problems

air as a result of thermal conductivity of air or is emitted by the particle in the form of radiation. Since the Wigner-Seits radius [206, 207] for aluminium is rW D 1:58 Å, an aluminium particle of a radius ro D 10 μm contains no D 2:5  1014 atoms and its mass is 4:5  108 g. The concentration of oxygen molecules c.O2 / D 0:21 in air corresponds to the total concentration of aluminium atoms c.Al/ D 0:28 contained in aluminium particles according to the problem conditions where in the end of the process oxygen and aluminium are consumption entirely, and the above aluminium concentration is contained in molecules Al2 O3 which are a product of the combustion process. Hence, if at the beginning air consists of 79% of nitrogen molecules and argon atoms, and also of 21% of oxygen molecules, in the end of the process it consists of 85% of nitrogen molecules and argon atoms, and also of 15% of Al2 O3 molecules. Taking the particle temperature to be T  1000 K, one can estimate that several percent of the released energy is consumed on air heating. We now consider heat equilibrium of a burning particle with surrounding air. We have the heat balance equation by analogy with equation (7.4.12 4D.O2 /N.O2 /ro " D 4 ro .To /.To  T / C 4 ro2 aTo4 ;

(7.7.3)

where the left hand side of this equation describes the released power in the diffusion regime, and the right hand side of this equation accounts for the consumed power as a result of air thermal conductivity and particle radiation. Here D.O2 / is the diffusion coefficient of oxygen molecules in air, N.O2 / D cN is the number density of oxygen molecules, so that c is their concentration, and N is the total number density of air molecules. Next, .To / is the coefficient of air thermal conductivity, a is the grey coefficient of the aluminium surface and in this temperature range we take a D 0:5,  is the Stephan-Boltzmann coefficient, " is the released energy per one oxygen molecule as a result of the process (7.7.2). The temperature dependence for the thermal conductivity coefficient is .To /  To0:7 as well as the reduced diffusion coefficient of oxygen molecules D.O2 /N  To0:7 . We now analyze the heat balance of the combustion process (7.7.2). If we neglect the particle radiation in the course of particle combustion, the difference between the temperature To of the burning particle and the air temperature T far from the particle is given by formula (7.3.5) and has the form To  T D

"D.O2 /N co ; 

where co is the initial concentration of oxygen molecules in air. For the following parameters [312] D.O2 / D 0:15 cm2 =s and  D 2:62  104 W=.cm  K/ at the temperature T D 300 K and the pressure p D 1 atm from this formula it

114

Chapter 7 Processes in buffer gas on surface of small particles

follows To  T  5500 K. Since To  103 K, from this it follows that particle emission is of importance for starting the particle ignition. On the next stage of the combustion process (7.7.2), after establishment of an identical particle and air temperatures To D T we have from the heat balance equation (7.7.3) D.O2 /N.O2 /" D ro aTo4 : Taking D.O2 /N.O2 / D 1:7  1016 cm1 s1  T 0:7 , where T is the air temperature T is expressed in K, we obtain at the equilibrium stage T D To , that gives To D 8900 K. Because we neglect both dissociation of air molecules and atom evaporation from the particle surface, the subsequent results are estimations only. We now determine a time of combustion of the aluminium particle in air in ignoring the processes of evaporation of the aluminium particle and air dissociation. The balance equation for the number n of aluminium atoms inside the particle has the form dn D J; dt

(7.7.4)

where J is the number of oxygen molecules which attain the particle surface per unit time. For a uniform gas this quantity is given by the Smoluchowski formula (4.2.5) [224] J D 4D.O2 /Ncr; where r is a current particle radius, c is the current concentration of oxygen molecules, D.O2 / is the diffusion coefficient of oxygen molecules in air, N is the number density of air molecules, and we assume this value to be independent of the space point. Taking the temperature dependence for this parameter DN  T 0:7 , we have the following relations n c D ; no co

r D ro



c co

1=3

 ;

T To

3:3 D

r ; ro

where n; c; r; T are the corresponding current parameters, and no ; co ; ro ; To are these parameters at the beginning of particle combustion. We assume for definiteness that only 1=3 of the maximal possible released energy is consumed on the particle surface. Hence, from the balance equation (7.7.3) it follows that the temperature of the particle surface is To D 3000 K, and we will be guided by this particle temperature. We have J D Jo

c r T ; co ro To

115

Section 7.7 Problems

where Jo is the rate of the process (7.7.2) at the beginning that is equal Jo D 4:8  1016 s 1 under given conditions. Since r  n1=3 and r  T 3:3 , we obtain the right hand side of the balance equation (7.7.4) in the form c J D Jo co



n no

1:26 ;

and the balance equation (7.7.4) takes the form  1:26 n dn : D Jo dt no Solving this equation, we find a typical time of this process D

no no ; 0:26Jo nmin

(7.7.5)

where nmin is the minimum number of particle atoms where this mechanism of particle combustion finishes. We find the value nmin as the boundary of the diffusion regime of particle combustion that is given by  nmin D

 rW

3 ;

where  is the mean free path of air molecules in air, and rW D 1:58 Å is the Wigner-Seits radius for aluminium. At the pressure p D 1 atm and the temperature T D 3000 K the mean free path of air molecules   1 μm and nmin  31011 under given conditions. On the basis of formula (7.7.5) we obtain   20 s.

Chapter 8

Charging of small particles in ionized gases

8.1

Particle charging in dense buffer ionized gas

If a small particle is located in an ionized gas, plasma electrons and ions attach to the particle and later recombine on its surface. As a result, the particle becomes charged and creates with a surrounding ionized gas a self-consistent electric field that influences the process of electron and ion attachment. As a result of this process, a small particle becomes a sink for charged particles of the ionized gas. The principal results for this problem were obtained a half-century ago and is a basis of the contemporary understanding of this problem. But contemporary study of new physical objects or some aspects of known physical objects lead to a specific glance on some problems of plasma physics. The goal of this paper is to combine old principal solutions of the problem of charging of a small particle in an ionized gas with some algorithms for specific conditions of this process. Charged particles are the component of a dusty plasma [136, 137, 320, 321, 322, 323, 324], and particles located in ionized gases are a sink for charged atomic particles of this gas. The self-consistent field is created around the particle due to surrounding electrons and ions which interact with the particle. It is significant that electrons and ions attach to the particle surface, and their subsequent recombination proceeds onto the particle surface. Due to absorption of electrons and ions by the particle surface the screening of the particle field by electrons and ions differs from the Debye screening [325, 326]. Our goal is to analyze the selfconsistent field near a particle located in an ionized gas that consists of atoms with a small admixture of electrons and ions. Simultaneously this analysis allows one to determine the particle charge that follows from the equality of electron and ion currents towards the particle surface and a radius of action of the particle field. Of course, the character of this equilibrium depends on parameters of the particle and ionized gas, and we consider various conditions for this equilibrium. Modeling a small particle located in an ionized gas as a spherical particle of a radius ro , we use the following criterion for a dense buffer gas ro  ;

(8.1.1)

where  is the mean free path of buffer gas atoms. We characterize the drift of electrons in a dense buffer gas by the diffusion coefficient De and the mobility Ke

Section 8.1 Particle charging in dense buffer ionized gas

117

of electrons in a buffer gas. The rate Je .R/ of intersection a sphere of a radius R by electrons, i.e. a number electrons intersected this sphere per unit time, is given by   dNe 2 Je D 4R De (8.1.2)  we Ne ; we D EKe ; dR where Ne .R/ is the electron number density at a distance R from the particle center, we is the electron drift velocity, and E.R/ is the electric field strength on this distance R from the charged particle. The minus sign in the second term accounts for an opposite direction the electron flux and electric field strength. Within the framework of the Fuks theory [327] we assume the electric field strength relatively small, so that the electron drift velocity is proportional to the electric field strength. Next, we ignore a screening the particle field by an ionized gas, so that the electric field strength is determined by the Coulomb field of the negatively charged particle ED

Ze ; R2

(8.1.3)

where Z is the particle charge in units of electron charges (Z > 0). Since formation and recombination of electrons and ions is absent near the particle, the relation (8.1.2) may be considered as the equation for the number density of electrons Ne that has the form with accounting for formula (8.1.3) ! dNe Ze 2 Ne Je D 4De  ; (8.1.4) 1 T dR and we used the Einstein relation (5.3.1) [285, 286] between the diffusion coefficient and mobility of electrons in a gas Ke D

eDe ; T

where T is the electron temperature. Solving this equation under the condition Ne .ro / D 0 and introducing the reduced potential energy of an electron in the particle field u.R/ D Ze 2 =RT , we obtain Z

 dR0 0 exp u.R/  u.R / 0 2 ro .R /  2  Ze Je T Ze 2 1  exp : D  4De Ze 2 TR T ro

Ne .R/ D 

Je 4De

R

118

Chapter 8 Charging of small particles in ionized gases

Using another boundary condition far from the particle Ne .1/ D No , we obtain the Fuks formula [327] for the rate of electron attachment to a particle of a negative charge Je D

4De No Ze 2 i; h  2  1 T exp Ze T ro

(8.1.5)

where No is the number density of electrons and ions far from the particle. From this one can find the rate of attachment of positive ions to a particle by replacing Z ! Z and the electron parameters by the ion ones. This operation gives JC D

4DC No Ze 2 i :  2 T 1  exp  Ze T ro h

(8.1.6)

The equilibrium particle charge follows from the equality of electron and ion currents to the particle surface that gives ZD

Ke ro T ln ; 2 e KC

(8.1.7)

where we assume the particle to be negatively charged and the electron and ion temperatures to be identical. Since we use that attachment of an individual electron or ion to the particle surface does not change the character of electron and ion motion near the particle, the above consideration requires the following criterion Z  1:

(8.1.8)

In addition, the used assumption of a weak screening the particle field by a surrounding plasma holds true under the criterion rD  ro ;

(8.1.9)

where rD is the Debye-Hückel radius [325, 326] for an ionized gas. We give also the limiting cases of the Fuks formulas (8.1.5) and (8.1.6) for the total flux of atomic charged particles to the surface of a cluster or small macroscopic particle. In the limiting case of a neutral particle Z ! 0 the Fuks formulas (8.1.5) and (8.1.6) are converted into the Smolukhowski formula (4.2.5) [224] Jo D 4DC No ro :

(8.1.10)

119

Section 8.1 Particle charging in dense buffer ionized gas 1.0

ne,n+

0.8

4

3

0.6 0.4 0.2

2 0

1

1 2

3

4

u 5

6

Figure 8.1. The reduced number density of electrons ne D Ne =No (1, 2) and ions nC D NC =No (3, 4) depending on the local reduced potential energy u.R/ of the particle field. uo D 6 for 1 and 3 and uo D 4 for 2 and 4.

In the case of a large particle charge Ze 2 =.ro T /  1 for an attractive interaction potential (a particle and ion have charges of the opposite sign) formula (8.1.6) is transformed in the Langevin formula [328] JC D

4Ze 2 No DC D 4ZeKC No ; T

(8.1.11)

Let us analyze some aspects of particle charging. We rewrite the distribution of the number densities of electrons Ne and ions Ni near the particle which in terms of the reduced potential energy of an electron in the particle field u D u.R/

exp.uo /  exp.u/ exp.uo  u/  1 Ne .R/ D No ; NC .R/ D No ; exp.uo /  1 exp.uo /  1 (8.1.12) where u D ze 2 =.RT / is the reduced particle potential, and uo D u.ro /. Figure 8.1 give the dependencies (8.1.12) at some values of uo . Note that for a plasma consisting of electrons and ions uo  1 because Ke  KC . One can see that at low values of u NC  No ;

Ne D No .1  u/:

From this it follows that only electrons give a contribution to screening of the particle field.

120

Chapter 8 Charging of small particles in ionized gases

Let us consider the character of particle field screening by a surrounding plasma. The Poisson’s equation u.R/ 

1 d 2 .Ru/ 4e 2 ŒNC .R/  N .R/

D R dR2 T

at low u takes the form d 2 .Ru/ Ru.R/ D ; 2 2 dR rD where the Debye-Hückel radius rD is determined by electrons and is given by s T : (8.1.13) rD D 4No e 2 Correspondingly, the electric potential of the particle ' D uT =e has the form   Ze R : (8.1.14) '.R/ D exp  R rD From this one can evaluate the screening charge Z from the plasma around the particle when the criterion (8.1.9) holds true Z 1 Z 1 2 4R .NC  N / D 4R2 No u D Z; Z D ro

ro

where we use formula (8.1.14) for the particle electric potential and assume that the main contribution to this integral is determined by large distances from the particle R  rD  ro . Thus, the shielding charge compensates the particle charge. Let us define the radius of action l of the particle field in a plasma such that the particle interaction energy with electrons or ions is comparable to their thermal energy, i.e., u.l/ D 1. Since u.R/ D Ze 2 =RT and the particle charge is given by formula (8.1.7), we have   Ke l D ro ln ; (8.1.15) KC and l exceeds the particle size. The Fuks theory relates to a dense buffer gas (8.1.1) if the particle charge is large (8.1.8), the electron Te and ion Ti temperatures are identical, and the plasma density is small in accordance with the criterion (8.1.9). One can generalize the

121

Section 8.2 Particle charging in dense gas discharge plasma

Fuks formula (8.1.7) for the particle charge to the case where the Maxwell energy distributions of electrons and ions hold true, but the electron Te and ion Ti temperatures are different. Because the expressions for electron (8.1.5) and ion (8.1.6) attachment rates to the particle surface are independently, these formulas are conserved for different electron and ion temperatures. From the equality of these rates we have for the particle charge [163, 209] ZD

ro Te K .Te / ln : e2 KC .Ti /

(8.1.16)

If the energy distribution function differs from the Maxwell one, in accordance with derivation of formula (8.1.5) the value Te D eDe =Ke is used as the electron temperature [163, 209]. Note that the Fuks theory is based also on the linear dependence of the electron and ion drift velocities on the electric field strength.

8.2

Particle charging in dense gas discharge plasma

If the particle is located in a gas discharge plasma, simultaneous action of an external electric field of gas discharge and the particle field makes the complex character of attachment of electrons and ions to the particle. In particular, Figure 8.2 gives the electric potential ' D Ex C

Ze ; R

that is created by a discharge electric field of strength E and by a particle of charge Z. Here the origin is taken at the particle center, and x is the direction along the discharge electric field. As is seen, equipotential curves are closed at distances from the particle below r1 , where r Ze r1 D ; E One can see that the problem of particle charging in a gas discharge plasma may be reduced to the Fuks theory of particle charging if the criterion ro  r1 holds true. In this case it is necessary to take into account the real dependence we .E/ for the drift velocity of electrons on the electric field strength in high electric fields. For definiteness we below consider the case of a helium plasma of a high electric field strength, so that a typical electron energy exceeds significantly a thermal energy of atoms and ions. But this energy is not so high in order to ignore by ionization processes. Because of small exchange by energy in elastic electron-atomic collisions, a thermal energy of ions is close to a thermal atoms energy in a wide

122

Chapter 8 Charging of small particles in ionized gases 6

4

2

2

2

4

6

8

10

2

4

6

Figure 8.2. Lines of identical electric potentials in the plane passed through the particle center if an electron is located in the Coulomb center ofpa charged particle and constant electric field. The length units are taken such that r1 D Ze=E D 4.

range of electric field strengths where a typical electron energy exceeds an atom thermal energy significantly [329]. Being guided by this range of electric field strengths, we assume the diffusion cross subsection of electron-atom collisions to be independent of the collision energy, as it takes place in the helium case. Then the electron distribution function in a range of electric field strengths under consideration is determined by the Dryvesteyn formula [330, 331], that leads to the following formulas for the electron drift velocity we and its diffusion coefficient (for example, [332]) s  1=4 s  m 1=4 eE eE M e De D 0:29  ; we D 0:90 ; eE  T; me me M me where  D 1=.Na   / is the mean free path for electrons, so that   is the diffusion cross subsection of electron-atom scattering, me ; M are the electron and atom masses, T is the atomic temperature expressed in energetic units. In the

Section 8.2 Particle charging in dense gas discharge plasma

helium case (   6 Å2 , M=me  7700) these relations have the form s s eE eE De ; we D 0:097 ; aD D 28: De D 2:7 me me we

123

(8.2.1)

We have now another dependence on the particle electric field strength for the electron drift velocity and electron diffusion coefficient compared with those in the Fuks case, and hence we solve one more the transport equation (8.1.2) as the balance equation for the electron number density at other dependencies we .E/ and De .E/. Consider the case r Ze ro  r1 D ; (8.2.2) E where E is the electric field strength far from the particle. The rate of electron attachment to the particle surface is given by Je D De

dNe C we Ne : dR

Considering this as an equation for the electron number density, we use the above dependence of the electron diffusion coefficient De and the electron drift velocity we s s s r12 r2 Ze 2 De D Do 1 C 2 D Do 1 C 2 ; we D wo 1 C 12 ; R R R where Do ; wo are the electron diffusion coefficient and electron drift velocity of electrons in a buffer gas far from the particle. In the range R  r1 the expression for the electron attachment rate has the form s   r12 dNe Ne 2 ; C Je D 4R 1 C 2 Do  R dR a where a D Do =wo D 28. Solution of this equation for the case r1  a  ro gives   Z R  dR0 Je R R0 Ne .R/ D  exp exp  q 4Do a a ro R0 r 2 C .R0 /2 1     a R Je (8.2.3)  ln exp 4Do ro a

124

Chapter 8 Charging of small particles in ionized gases

Table 8.1. Parameters of a gas discharge helium plasma and particle charging in it.

E=N , Td 1 2 4 6

Tef eV 0.46 0.92 1.85 2.78

", eV 0.61 1.22 2.43 3.65

r1 , μm 52 48 46 41

Z; 104 5.1 8.6 16 19

and this solution holds true at R < r1 , whereas the sign minus means that the electron flux is directed to the particle. In this case we have Je D

4Do r1 No exp.R=a/; ln.a=ro /

r1  R  a:

(8.2.4)

The particle charge Z follows from equality of rates of electron and ion attachment to the particle, where the latter is given by the Langevin formula (8.1.11). It p is convenient to use this equality as equation for r1 D Ze=E that has the form r  Do r1 1  (8.2.5) exp ; r1  a: a a EKC a ln.a=ro / Note that the particle charge depends weakly on a particle radius ro that is located in a range a  ro  . Table 8.1 contains parameters of particle charging under certain conditions, namely, the pressure p D 1 atm that gives for the mean free path  D 0:6 μm, the particle radius is ro D 1 μm, that gives a D 17 μm. Table 8.1 contains the effective temperature Tef of electrons and the electron average energy " far from the particle which are given by the formulas s s M M eDe E D 0:325 eE; " D 0:427 eE: Tef D we me me The data of Table 8.1 correspond to the criterion r1  a  ro  :

(8.2.6)

Though this sequence of the size parameters holds true, the accuracy of the formulas used is restricted and formula (8.2.5) gives rough values for the particle charge. Note that the particle charge according to the Table 8.1 data is approximately in 30 times than that according to formula (8.1.7). This shows that the presence of an electric field in an ionized gas can increase significantly the charge of a particle located in this gas. We also note that even in a narrow range of parameters (8.2.6) the numerical values of Table 8.1 require the absence of ionization processes near the particle while a large particle charge corresponds also high electric field near the particle, i.e. the above results transfer the tendency in the influence of an external electric field on the value of the particle charge.

125

Section 8.3 Double layer of gas discharge

0,8 0,7

n(u)

0,6 0,5 0,4 1

0,3

2

0,2 0,1 0,0 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

Figure 8.3. The reduced difference of the number density of ions and electrons n.u/ D .NC  Ne /=No for uo D 2 (1) and uo D 4 (2).

8.3

Double layer of gas discharge

In considering the particle charging in a dense plasma where the criterion (8.1.9) violates, we assume the criteria ro  rD ;

rD  

(8.3.1)

to be fulfilled and use firstly formulas (8.1.12) for the electron and ion number densities as an approximation. Then the relative difference of the number densities of ions and electrons is given by n.u/ 

NC .R/  Ne .R/ Œ1  exp.u  uo / Œ1  exp.u/

D ; No 1  exp.uo /

(8.3.2)

and Figure 8.3 gives the dependence n.u/ that is symmetric with respect to the transformation u ! uo  u and has the maximum at u D uo =2, where it is equal nmax D

1  exp.uo =2/ : 1 C exp.uo =2/

(8.3.3)

From this we have n.u/ 1 at any u uo , and at small u this function is n.u/ D u.

126

Chapter 8 Charging of small particles in ionized gases

Let us refuse from the criterion (8.1.9) of a low density of charged atomic particles. Approximating the number densities of electrons and ions by formula (8.1.13), we obtain the Poisson equation as early in the form d 2 .Ru/ Ru.R/ D ; 2 2 dR rD and its solution is '.R/ D

  ze R exp  ; R rD

(8.3.4)

where z is an effective particle charge shielded by ions, and z Z, while if the criterion (8.1.9) holds true, z  Z. One can apply the above results to the layer near walls of the positive column of gas discharge that is named the double layer. In this case electrons and ions of the gas discharge plasma attach to the walls and transfer them the charge, and the plasma far from the walls is quasineutral, and the criterion (8.3.1) holds true. Using the expressions (8.1.13) for the number densities of electrons and ions and assuming uo  1, one can obtain that the relation NC D No is violated only close to the surface of walls. Using this fact in the Poisson equation, we write it in the form Ne d 2u D 1  ; dx 2 No

xD

z ; rD

(8.3.5)

where z is the distance from the walls, and the Debye-Hückel radius is given by formula (8.1.13). In this case the walls are charged negatively that creates the electric field. This field prevents the attachment of electrons to the walls and equalizes the electron and ion currents toward the walls. The reduced electric potential is given by formula (8.1.15) if the electric potential is zero far from the walls. As a result, we have the following equation for the reduced electric field strength

d 2u Ke .Te / exp.u/  1 : (8.3.6) D ; u.1/ D 0; u.0/ D uo D ln dx 2 exp.uo /  1 KC .T / Let us decrease the order of equation (8.3.5) by multiplication this equation by du=dx and integration the equation under the boundary du=dx.x D 1/ D 0. We obtain r du 2.e u  u/ D : (8.3.7) dx e uo  1

127

Section 8.4 Particle charging in rarefied ionized gas with free ions

On the basis of formula (8.3.7) we have from this for the electric field strength Eo at the walls p r p Te du uo  1 Te 2 Te 2 Eo D  1 u f .uo /: (8.3.8) D D erD dx erD e o 1 erD Under real conditions uo  1, and f .uo /  1. Indeed, f .2/ D 0:92; f .3/ D 0:95; f .4/ D 0:97. Therefore in reality we have for the electric field strength at the walls p Te 2 Eo D  ; (8.3.9) erD and this electric field attract ions to the walls and repels electrons from the walls. This formula holds true if the criterion (8.3.1) rD   is fulfilled.

8.4

Particle charging in rarefied ionized gas with free ions

Particle charging in a rarefied ionized gas corresponds to the criterion   l;

(8.4.1)

where l is a radius of action of the particle field, and this criterion is opposite with respect to the criterion (8.1.1). Let us assume that positive ions screen the negative particle charge. Defining by U.R/ the potential energy of a self-consistent field that is established in the course of ion flight in the particle field, we below determine this potential that acts on positive ions. In determination the self-consistent field, we transfer from dynamics of ion motion in the particle field to the statistical mechanics on the basis of the ergodic theorem [333, 334] that gives the space distribution function [326, 335] of free ions in the particle field. In this transfer we take the probability dPi for ion location in a given space region to be proportional to a time range dt during which a given ion is located in this space region [326, 335]. Correspondingly, the ion number density is proportional to the above probability Ni .R/ v dPi . In turn, a time range dt for ion location in a space region of distance between R and R CdR from the particle follows from the motion equation [336] dt D

dR dR ; D p vR v 1  2 =R2  U.R/="

where vR .R/ is a normal velocity component with respect to the force center for an ion at a distance R from the particle; v is the ion velocity far from the particle,

128

Chapter 8 Charging of small particles in ionized gases

" D mi v 2 =2 is the ion kinetic energy far from the particle, so that mi is the ion mass, and  is the impact parameter of collision. On the other hand, we have the ion number density Ni .R/ at a distance R from the particle as R ddPi : Ni v 4R2 dR Normalizing this relation such that in the absence of interaction the ion number density is equal to the ion number density No far from the particle, we have Z .R/ d Ni .R/ D No ; (8.4.2) q 2 U.R/ 0 1 R  2 " where for free ion motion U.R/ D 0 the impact parameter of ion-particle collision is .R/ D R if R is the distance of closest approach. Let us divide ion trajectories in two groups, so that the first group includes the trajectories with ion capture by the particle  c , where c is the impact parameter of collision above which the ion-particle contact is impossible. The connection of this impact parameter of ion capture in the self-consistent field with the particle radius ro as the distance of closest approach is given by [336]

U.ro / : c2 D ro2 1  " If an ion moves along a trajectory of the second group, it goes to infinity after approach with the particle. Summarizing both types of trajectories in the ion number density, we obtain formula (8.4.2) in the form [337] 2r 3 s 2 No 4 U.R/  U.R/ 5 : (8.4.3) Ni .R/ D 1 C 1  c2  2 " R " Averaging over the Maxwell distribution function of ions   2"1=2 " ; f ."/ D No  p 3=2 exp  Ti Ti we obtain in the range of strong ion-particle interaction [337] 2s 3 s 2 2 jU.R/j jU.R/j  jU.ro /jro =R 5 Ni .R/ D No 4 ; C Ti Ti

jU.R/j  Ti : (8.4.4)

We are guided by ion-particle attraction in this formula.

129

Section 8.4 Particle charging in rarefied ionized gas with free ions

We now determine the potential energy U.R/ for a self-consistent field that is created by the particle charge and is screened by free ions. The particle has a negative charge Z such that jZje 2 =ro  Ti . Therefore the ion number density in the region of strong ion attraction is Ni  No , and because the electron number density in this region Ne < No , electrons do not partake in screening the particle field, and hence we below neglect by the presence of electrons in a region of strong ion-particle interaction and assume that the radius of action of a self-consistent field l exceeds the particle radius (l  ro ). As a result, ions form so called ionic coat near the particle and the particle Coulomb field together with the ionic coat creates a self-consistent field of the particle and surrounding plasma. Let us introduce a current charge z.R/ inside the sphere of a radius R that is the sum of the particle charge and a charge of ions located inside this sphere. According to the Gauss theorem [338, 339] we have s dz 4jU.R/j ; D 4R2  Ni .R/ D 4R2  No dR Ti where z.ro / D jZj and we restricted by a region R  ro in formula (8.4.4). The potential energy U.R/ of the self-consistent field is Z 1 z.R/e 2 U.R/ D E.R/dR  ; R R and the used simplification leads to an error U in the potential energy that may be determined in the following order of the perturbation theory. We give in Figure 8.4 the ratio of the accurate value in the right hand side of equation for z.R/ to its approximated value depending on a distance R from the particle and will use below its approximated value. After using this simplification, the equation for a current charge z.R/ takes the form s 4ze 2 dz D 4R2  No ; dR Ti R and has the following solution "

 5=2 #2 R z D jZj 1  ; l

lD

0:66 2=5

No



jZjTi e2

1=5 D

0:66jZj2=5 2=5

No

1=5

; (8.4.5)

Ro

where Ro D jZje 2 =Ti , and Ro  ro . Note that according to derivation of this formula, it holds true until jU.R/j  Ti and if this criterion violates, we use the above formulas as an approximation.

130

Chapter 8 Charging of small particles in ionized gases

1,00 0,98

(1–ΔU/U)

1/2

0,96 0,94 0,92 0,90 0,88 R/l 0,86 0,0

0,2

0,4

0,6

0,8

1,0

Figure 8.4. The correction to the simplified potential energy U.R/ D z.R/e 2 =R of self-consistent field leads to replace this potential energy by the value U.R/  U D R1 2 2 ze dR=R . Dark triangles relate to shielding by free ions, whereas open triangles R correspond to screening of the particle charge by trapped ions.

In determination the particle charge Z, we assume that each contact of an ion or electron with the particle surface leads to the charge transfer event. Taking into account that electron and ion currents to the particle surface are originated in a region with a weak particle field, we use the above values for the rates of ion Ji and electron Je attachment to the particle surface [336] s s     8Ti 8Te jZje 2 jZje 2 2 2 Ji D No   ro exp  1C   ro ; Je D No ;  mi r o Ti  me r o Te (8.4.6) where No is the number density of electrons and ions far from the particle, Ti , Te are the ion and electron temperatures correspondingly, me ; mi are the electron and ion masses. Equalizing the rates of ion and electron attachment to the particle surface and introducing the parameter x D jZje 2 =.ro Te /, we obtain   Te 1 Te m i : (8.4.7)  ln 1 C x x D ln 2 Ti m e Ti Taking xTe  Ti , we have the following equation for x [294] xD

1 Ti mi : ln 2 2 x Te m e

(8.4.8)

Section 8.5 Particle charging in rarefied ionized gas with trapped ions

131

In particular, for an example of a gas discharge argon plasma with the parameters Te D 1 eV and Ti D 400 K formula (8.4.8) gives x D 2:86 or jZj=ro D 2:0 nm1 . Note that we are guided by a large particle charge jZj  1, so that attachment of one electron or ion does not influence on the subsequent process of attachment of electrons and ions. For the above example this corresponds to ro  0:5 nm. We note also that formulas (8.4.7) and (8.4.8) require that the potential energy of an electron on the particle surface is jZje 2 =ro . This holds true if the radius of action of the particle field l exceeds the particle radius ro . The above consideration is based on the criterion (8.4.1) that has the form Na l   1; where   is diffusion ion-atom cross section of scattering that is assumed to be independent of the collision velocity. If an atomic ion is located in a parent gas, the diffusion cross section of ion-atom scattering is expressed through the cross section res of ion-atom resonant charge exchange as   D 2res [340], and the criterion (8.4.1) takes the form 2Na lres  1:

(8.4.9)

In particular, for an argon plasma the cross section of resonant charge exchange at the collision energy in the laboratory frame of reference 0:01 eV is equal to res D 83 Å2 [208], and the regime under consideration is realized at the pressure p  0:1 Torr, if the particle radius is ro D 1 μm and the ion temperature is comparable to the room one. One can seem that criteria (8.1.1) and (8.4.1) relates to two opposite regimes of particle screening in an ionized gas. In reality, l  ro and, in particular, for the above example of the argon plasma with parameters Te D 1 eV , Ti D 400 K, ro D 1 μm and jZj D 2000 we have l D 90 μm and l D 36 μm for No D 109 cm3 and No D 1010 cm3 correspondingly. In addition, the cross section of electron-atom scattering at low energies is small compared to the ion-atom cross section of scattering. For example, in the helium case the electron-atom diffusion cross section lies in the range of 5–7 Å2 , while the cross section of resonant charge exchange at the ion energy of 0:01 eV is 43 Å2 . From this it follows that there is a large range of particle sizes between the cases of (8.1.1) and (8.4.1).

8.5

Particle charging in rarefied ionized gas with trapped ions

Along with free ions, trapped ions, i.e. captured in closed orbits, may be of responsible for particle screening [341]. Though the probability of resonant charge

132

Chapter 8 Charging of small particles in ionized gases 2

2

3

1

1

3

Figure 8.5. The trajectories of the trapped ions captured on the closed orbit in the Coulomb center (left) and the screened Coulomb center (right). 1 is the particle, 2 is the ion trajectory origin, 3 is the finish point of the ion trajectory part.

exchange is small for an ion propagated through a region of the particle field action, this small probability is compensated by a large lifetime of a trapped ion in a closed orbit. The trajectories of trapped ions are different for the Coulomb particle field and a screening Coulomb fields [336, 342], as it is demonstrated in Figure 8.5. The role of the captured ions in screening of the particle charge was studied widely, in particular , in [136, 343, 344, 345, 346, 347, 348, 349, 350, 351]. We below represent a simple and practical version [337] with using that the cross section res of resonant charge exchange is independent of the collision energy and exceeds significantly that for elastic ion-atom scattering. Therefore colliding ion and atom in the resonant charge exchange process are moving along straightforward trajectories [352, 353] (the relay-race character of charge transfer). Accounting for these facts allows us to restrict a number of parameters that determine the ion capture in a closed orbit as it is given in Figure 8.6. We below determine formation of trapped ions in appropriate range of parameters and spread the results as a model in the other parameter range. We use the criterion Ro D

jZje 2  ro ; Ti

(8.5.1)

and take into account that a captured ion may become free if it is located near the boundary of pthe particle field action, i.e. if R  l, whereas ions captured at distances R > ro Ro are moving along stable closed trajectories. Hence, trapped

Section 8.5 Particle charging in rarefied ionized gas with trapped ions

133

Figure 8.6. Used ion parameters of a trapped ion formed in the resonant charge exchange event that proceeds at a point R: rmin , rmax are the minimum and maximum distances from the particle center for the trajectory of a captured ion,  is the angle between the direction of ion motion after the resonant charge exchange event and the vector R. 1 is the particle, 2 is the point of resonant charge exchange for ion-atom collisions.

ions influence the screening of the particle field at the criterion p l > ro Ro :

(8.5.2)

In this case a subsequent charge exchange process transfers the ion in closed p orbits that are nearby to the particle. If a distance from the particle is less than ro Ro , a subsequent charge exchange event leads to ion capture by the particle. Thus, under the criterion (8.5.2), the kinetics of a trapped ion consists of a series of ion transitions in nearby closed orbits of this ion, and in the end the ion attaches to the particle surface. Because of many acts of subsequent events of charge exchange, the number density of trapped ions may be enough high and can exceed the number density of free ions in the particle field for a rarefied plasma. Therefore in this limit trapped ions determine the screening of the particle field. Because the cross section of resonant charge exchange exceeds significantly that for elastic ion-atom collision, colliding ion and atom in this process are moving along straightforward trajectories, and a forming ion acquires the energy " and motion direction of the former atom [352, 354]. Assuming the ion trajectory does not touch a particle surface (the distance of closest approach rmin exceeds the particle radius ro ), we have from the orbital momentum conservation [336] for ion transition into a stable close orbit [337] R2 U.R/  U.ro / 1C ; 2 ro "

(8.5.3)

134

Chapter 8 Charging of small particles in ionized gases

where U.R/ is the particle potential energy for an ion at a capture distance R. For simplicity we consider the range of parameters jU.ro /j  jU.R/j  ";

(8.5.4)

at which a trapped ion cannot go to infinity and is captured into a closed orbit. If an ion is formed with an energy " at a distance R from the particle, the probability Ptr .R; "/ of capture of a free ion in a closed orbit or the probability ptr .R; "/ of transition of a trapped ion in another closed orbit are given by r Z cos o p ro Ro d cos  D cos o D 1  ; R ro Ro Ptr .R; "/ D ptr .R; "/ D R2 0 (8.5.5) in accordance with the parameters indicated in Figure 8.6. Spreading this result on a wide range of distances and accounting for the possibility for an ion to leave a closed orbit at the boundary of the region of the particle field action, we represent the probability of ion capture in a closed orbit as r   p l ro Ro ; l > 1  Ptr .R; "/ D 1  ro Ro (8.5.6) R2 R In order to find the connection between the number density of free Ni and trapped Ntr ions in the region of particle field action, we use the balance equation Na res Ni Ptr vi D Na res Ntr vtr .1  ptr /;

(8.5.7)

where vi ; vtr are the relative velocities in the charge exchange process for a free or trapped ion and atom, Ptr ; ptr are the probabilities of ion transition in a closed orbit for a free and trapped ion correspondingly. For ion distances R from the particle punder consideration we have jU.R/j  ", and the velocity of a free ion is vi D 2jU.R/j=mi . The kinetic energy of a trapped ion is in average jU.R/j=2 according to the virial theorem [355] if ion is located p in the Coulomb particle field. p This gives for the velocity of a trapped ion vtr D jU.R/j=mi , and vi =vtr D 2. Since Ptr D ptr , we obtain from the balance equation (8.5.7) ! r p r  l R2 2 ro Ro ro Ro ; Ntr .R/ D Ni .R/ 1C 1 1 1 ro Ro R2 R2 R (8.5.8) p l  R Ro ro and the last term takes into account existence of closed orbits for a trapped ion only at R < l. Though our derivation corresponds to a middle distances from the

135

Section 8.6 Particle charging and screening in rarefied ionized gas

particle and is spread to the particle field boundary, the results may give reliable evaluations. From this it follows that at a low number density No of a surrounding plasma trapped ions dominate in screening the particle field, whereas at a high plasma density free ions determine the screening of the particle field.

8.6

Particle charging and screening in rarefied ionized gas

Simultaneous participation of free and trapped ions in screening the particle field complicates the calculation of screening parameters. We use below a simple algorithm taking into account only free or trapped ions, and under conditions, where the contributions of free and trapped ions in the particle field screening are comparable, we use both these versions. A proximity of the results of these limiting versions justifies this algorithm. We below represent this algorithm. As in the case of a dense buffer gas, we introduce a current charge z.R/ for a total charge in a sphere of a radius R and use a simplified expression for the potential energy U.R/ of a self-consistent field U.R/ D

z.R/e 2 : R

In the case where free ions dominate in screening of the particle field, the number densities of free Ni and trapped Ntr are equal according to formulas (8.4.4), (8.4.5) and (8.5.8) v " u  5=2 #2 u 4Ro R t Ni .R/ D No 1 C ; 1 R l (8.6.1) p   2 2R 2 R ; Ntr .R/ D Ni .R/  ˆ.R/ 1  ro Ro l and the function ˆ.R/ is given by r 1 ro Ro ˆ.R/ D 1 2 R2

r 1C

ro Ro 1 R2

! :

In the other limiting case when trapped ions dominate we are based on the equation for a current charge z.R/ dz D 4R2  Ni .R/; dR

136

Chapter 8 Charging of small particles in ionized gases

with the boundary condition z.ro / D jZj, and use the number density of trapped ions Ntr in this equation instead of Ni .R/. The accuracy of this approximation is given in Figure 8.4 by open triangles for trapped ions. Solution of this equation gives " p 2=9  9=2 #2  R jZjro Ro z D jZj 1  ; l D 1:05 : (8.6.2) l No ˆ.9l=11/ From this we have for the number density of free and trapped ions for the second version when trapped ions dominate v " u  9=2 #2 u 4R R o Ni .R/ D No t1 C ; 1 R l (8.6.3) p   2 2R 2 R ; Ntr .R/ D Ni .R/  ˆ.R/ 1  ro Ro l and the particle charge Z given by formulas (8.4.7), (8.4.8) is independent of the version because electron and ion currents (8.4.6) are started outside the particle field. On the basis of the above expressions for the number densities of free and trapped ions we find the screening charge due to free Qi and trapped Qtr ions for each version according to formulas Z l Z l 2 4Ni .R/R dR; Qtr D p 4Ni .R/R2 dR; (8.6.4) Qi D ro Ro

ro

and according to definition of the size l of the particle field action we have Q D Qi C Qtr D jZj:

(8.6.5)

From this we find the part of the screening charge that is determined by trapped ions

D

Qtr : Qi C Qtr

(8.6.6)

Figure 8.7 gives the dependence of the part of the screening charge due to trapped ions (8.6.6) on the reduced number density of a surrounding argon plasma far from the particle for the example of an argon gas discharge plasma under consideration with the electron Te D 1 eV and ion Ti D 400 K temperatures. As is seen, two versions where the number density of ions is determined by formulas (8.6.1) and (8.6.3) give nearby results. In particular, at the reduced number

137

Section 8.6 Particle charging and screening in rarefied ionized gas 1,0 0,9 0,8 0,7 0,6 ȟ

0,5 0,4 0,3 0,2 2

–1 N0r , cm

0,1

0

0,0 1

10

100

1000

Figure 8.7. The part of the screening charge due to trapped ions (8.6.5) for an argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures. Open circles corresponds to the version when free ions dominate in screening of the particle field, and closed circles describe the version when trapped ions dominate.

density No ro2 D 100 cm1 of the plasma the contribution of trapped ions into the charge screening is D 0:53 and D 0:50 for the first and second version correspondingly. In accordance with two versions under consideration where the number density of ions are given by formulas (8.6.3) and (8.6.4), on the basis of formulas (8.4.5) and (8.6.2) one can find the reduced radius of action of the particle field lfree where free ions dominate and for this action radius ltrap if trapped ions dominate in the particle screening. We then have lfree A D ; ro .No ro2 /2=5 A D 0:66

.jZj=ro /2=5 ; .Ro =ro /1=5

BD

ltrap B D ; ro .No ro2 /2=9

1:05 .jZj=ro /2=9 .Ro =ro /1=9 ˆ

(8.6.7)

Figure 8.8 gives the reduced radius of action of the particle field in accordance with formula (8.6.7) as a function of the reduced number density of a surrounding argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures. In particular, for No ro2 D 100 cm1 formulas (8.6.7) give for the reduced radius of the particle field action l=ro correspondingly 28 and 29 for the first and second versions.

138

Chapter 8 Charging of small particles in ionized gases

90

l/r0

80 70 60 50 40 30 20

2

N0r0 , cm–1

10 1

10

100

1000

Figure 8.8. The reduced radius of action of the particle field in accordance with formula (8.6.7) as a function of the reduced number density of a surrounding argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures. Open circles relate to the first version if free ions dominate in the particle field screening and the ion number densities are given by formulas (8.6.3), and dark circles correspond to the second version with the number densities of ions according to formula (8.6.4).

Note the scaling law that follows from the above analysis. As is seen, the parameters of particle field screening depend on the number density of a surrounding plasma No and particle radius ro through the combination No ro2 . The equal contribution of free and trapped ions for the above example of an argon plasma takes place roughly at the reduced plasma density No ro2 D 100 cm1 , whereas trapped ions disappear at the reduced plasma number density No ro2 D 103 cm1 for the example under consideration. In addition, Figure 8.9 gives the space distribution for the number density of free and trapped ions at an indicated reduced number density No ro2 D 100 cm1 of a surrounding plasma. Thus, the particle charge in the case where the mean free path of ions in a surrounding gas is large compared to a radius of action of the particle field depends weakly on the character of shielding of the particle charge by a surrounding plasma because the electron and ion currents to the particle surface are created in a region where ion-particle interaction is absent, and formula (8.4.8) allows one to determine the particle charge. A simple algorithm used is based on comparison of two limiting versions where free or trapped ions give the main contribution to the particle screening. In addition, parameters of the particle screening are expressed through the parameter combination No ro2 .

139

Section 8.6 Particle charging and screening in rarefied ionized gas

5

Ni/N0, Ntr/N0

4 3 2 1 0 10

R/r0 15

20

25

30

Figure 8.9. The number densities of free (squares) and trapped (circles) ions in a surrounding argon plasma with the electron Te D 1 eV and ion Ti D 400 K temperatures as a function of the reduced distance from the particle. Dark signs correspond to formulas (8.4.5), while formulas (8.6.2) relate to open signs.

Let us consider the limiting case of a low plasma density if this plasma does not screen the particle field and l D Ro . Trapped ions dominate in this case, and their number density according to formula (8.5.8) is given by p   p R 4 2 No R3=2 1 ; Ro R  Ro ro : (8.6.8) Ntr .R/ D p p Ro  ro Ro From this formula it follows that the maximum number density of ions takes place at the distance Rmax D 0:6Ro from the particle and is equal to Ntr .R/ D 0:59No

Ro : ro

This number density exceeds significantly the number density No of electrons and ions far from the particle. We have from formula (8.6.8) for the charge of ions inside a sphere of a radius Ro p 4 2Ro4 No e No Ro4 Qtr .Ro / D  0:86 : ro ro

140

Chapter 8 Charging of small particles in ionized gases

This leads to the following criterion for a smallness of the particle charge Qtr  jZje, that is No Ro3 

T i ro : e2

(8.6.9)

In particular, for the above example of an argon plasma with Te D 1 eV; Ti D 400 K and a particle radius ro D 1 μm we have Ro  210 nm, and the criterion (8.6.9) gives No  3  105 cm3 , so that under laboratory conditions the particle field screening is of importance. Note that the above consideration is better for a nonequilibrium plasma Te  Ti . Indeed, we have two p boundaries for trapped ions, so that ions are captured in closed orbits at R ro Ro , and the other boundary of existence of trapped ions corresponds to R < l, so that in the limiting case of a rarefied plasma R < Ro . The above formulas for the number density of trapped ions are the better, the stronger these two boundaries are separated. In particular, for the above example of an argon p plasma at pTe D Ti we have for the ratio of distances for these boundaries Ro =ro  ln.mi =me /  3:3, while in the case p of a nonequilibp rium plasma Te D 1 eV; Ti D 400 K we have Ro =ro D jZje 2 =.Ti ro /  p Te =Ti  ln.mi =me / D 18 in the limit of a low number density of this plasma.

8.7

The charge distribution of particles in ionized gas

In the case under consideration of particle charging in a dense ionized gas the particle charge is large, and a change of the particle charge by one does not act on interaction of the charged particle with electrons and ions of a surrounding plasma. Let us introduce the charge distribution function f .Z/ of particles that is the probability to have a negative charge Z for the particle and is normalized as X f .Z/ D 1: (8.7.1) Z

Because the particle charge results from attachment the electrons and ions to the particle surface, the kinetic equation [374] for the charge distribution function f .Z/ has the following form in the stationary case [163] f .Z/ŒJe .Z; Z C 1/ C JC .Z; Z  1/

D f .Z  1/Je .Z  1; Z/ C f .Z C 1/JC .Z C 1; Z/;

(8.7.2)

where Je .Z; Z C 1/; JC .Z; Z  1/ are the rates of attachment of electrons and ions from a surrounding plasma to a particle of charge Z which are described by

141

Section 8.7 The charge distribution of particles in ionized gas

formulas (8.1.6) and (8.1.7) correspondingly and have the form Je D

4De No ro Zx ; exp.Zx/  1

JC D

4DC No ro Zx ; 1  exp.Zx/

xD

e2 ; ro T

(8.7.3)

where x  1. We note that the maximum of the distribution function Z D Z corresponds to the relations f .Z/ D f .Z C 1/ and Je .Z; Z C 1/ D JC .Z; Z  1/. This means that the balance equation (8.7.2) in the stationary case may be separated in the two independent equations such that f .Z/Je .Z; Z C 1/ D f .Z C 1/JC .Z C 1; Z/; and in the Fuks case Je .Z; Z C 1/ D expŒ.Z  Z/x : JC .Z C 1; Z/ Introducing the variable z D Z  Z, we reduce the balance equation to the form f .z C 1/ D exp.zx/: f .z/

(8.7.4)

Assuming z to be a continuous variable because of x  1 and solving this equation, we obtain in old variables # " .Z  Z/2 x ; x  1; (8.7.5) f .Z/ D C exp  2 where the normalization constant is C D

r

x : 2

From formula (8.7.5) it follows that the width of the distribution function is large Z  x 1=2  1. This solution describes the character of the establishment of equilibrium for the charge distribution function of particles that results from electron and ion attachment to particles. In analyzing the screening and charging of an isolated particle in an ionized gas, we restrict only the part of these problems related to a large particle charge. In this case fluctuations of the particle charge are relatively small, and the particle charge is grouped around its average charge that is considered as the particle charge. Next, we assume the ionization rate of plasma atoms or molecules surrounding the particles to be large compared to the recombination rate due to attachment of

142

Chapter 8 Charging of small particles in ionized gases

electrons and ions to particles, and the presence of particles in the plasma does not violate the ionization balance of this plasma. This is fulfilled in a laboratory plasma, but is violated in an astrophysical plasma. In addition, the basic concepts under consideration will be worked out more than fifty years ago, and now reliable simple models and realistic simplified formulas are of interest for the mass analysis of these problems, that was the goal of this analysis.

8.8

Charging of small clusters in ionized gas

Above we consider the case when a particle located in an ionized gas has a large charge and its charge distribution function includes many possible charges of this particle. We now consider an opposite case in accordance to the criterion xD

e2  1; r o Te

(8.8.1)

where the electron temperature Te exceeds the ion temperature T or is equal to it. In the kinetic regime of particle charging according to the criterion inversed to (8.1.1) ro   we obtain that the potential energy of a singly charging particle exceeds significantly a thermal energy of an electron or ion. From this it follows that small particle may be neutral or have a single positive or negative charge since the probability of a charge increase for a singly charged particle in collision with electrons or ions contains an additional factor exp.e 2 =ro T /. Note that the case under consideration relates to small clusters. In particular, for the electron and ion temperatures T D 300 K we have x D 1 at ro D 56 nm, i.e., the criterion (8.8.1) may be valid for nanometer clusters only. Being guided by neutral clusters Mn and charged clusters MnC ; Mn consisting of n atoms, we have the following processes of charging and charge lost for such clusters e C Mn ! Mn ; AC C Mn ! MnC C A;

e C MnC ! Mn ; AC C Mn ! Mn C A;

(8.8.2)

where the processes of cluster charging and neutralization result from cluster collisions with electrons e and positive plasma ions AC . From this we have the following set of balance equations for the number density of neutral clusters N0 ,

Section 8.8 Charging of small clusters in ionized gas

143

singly negatively N and positively NC charged clusters in the stationary case dN0 D 0 D k.0; 1/Ne N0  k.0; 1/Ni N0 C k.1; 0/Ni NC C k.1; 0/Ne NC ; dt dNC D 0 D k.0; 1/Ni N0  k.1; 0/Ne NC ; dt dN D 0 D k.0; 1/Ne N0  k.1; 0/Ni N ; dt where Ne ; Ni are the number densities of electrons and ions correspondingly, and the argument in the rate constant of the process indicates the cluster charge change. In determination the rate constant of processes we assume the classical character of collisions and that each electron-cluster or ion-cluster contact leads to charge transfer. Taking the Maxwell energy distribution function of electrons and ions, we obtain the following expressions for the rate constants s s 8Te 8T k.0; 1/ D  ro2 I k.0; 1/ D  r 2;  me  mi o s s 8Te  ro2 8T  ro2 k.1; 0/ D I k.1; 0/ D ;  me 1 C e 2 =.ro Te /  mi 1 C e 2 =.ro T / (8.8.3) where me ; mi are the electrons and ion masses respectively, Te ; T are the electron and ion temperatures correspondingly. From this we have for the ratio of the number densities of charged clusters to those of neutral clusters clusters s 1  e2 Te mi Ne N 1C D ; N0 T me Ni ro T s  1 N T me Ni e2 D : (8.8.4) 1C N0 Te mi Ne r o Te It is necessary to add to this the normalization condition for clusters containing a given number n of atoms N0 C N C NC D Nn ;

(8.8.5)

where Nn is the total number density of charged and neutral clusters consisting of n atoms. According to the above result, the more rarefied is a buffer gas, the larger charge has a micronsized particle at a given size, and the larger contribution to the ion

144

Chapter 8 Charging of small particles in ionized gases

coat is from trapped ions. Such conditions take place for a dusty plasma of a solar system [136, 137, 138], and we consider it briefly. Let us apply the results to a dusty plasma of the Saturn E-ring. On the basis of its parameters ro D 1 μm, Te D 30 eV, Ti D 103 K and formula (8.4.7) we obtain for the particle charge jZj D 2  105 , and according to formula (8.5.1) we have for a radius of the action of the particle field l (a size of the ion coat) Ro  0:3 cm. This value is comparable to a distance between nearest neighbors since the number density of particles is Np D 30 cm3 [143] l  Ro . But these estimations contradict to observable data. Indeed, a given number density of trapped ions Ni  106 cm3 exceeds the observable value No  102 cm3 . Nevertheless, the observable number density of a dusty plasma exceeds the number density of charged particles in a solar wind  0:1 cm3 . But the plasma flux due to a solar wind cannot provide an equilibrium with a dusty plasma, and the number density of a dusty plasma is lower significantly than that under equilibrium conditions. Indeed, on the basis of observable parameters of the Saturn rings we have that the dusty plasma consists of negative charged particles, trapped positive ions and the conserved part of a solar wind. The average charge equals Z D 3, i.e. a size of the region of the particle field action is comparable with the particle size and is small compared with a distance between nearest particles.

8.9

Problems

Problem 8.1. The basic ion sort in weakly ionized helium at a gas temperature of T D 300 K and a given external electric field is HeC . Determine the charge of a microparticle located in this helium plasma and give the criterion that the particle field is relatively small. The mobility of HeC in helium at the normal number density of atoms (Na D 2:69  1019 cm3 ) is [366] KC D 10:4 ˙ 0:3 cm2 =.V  s/. We use that the diffusion cross section of electron scattering on the helium atom is independent on the collision energy and is close to   D 6 Å2 in a wide energy range, so that the electron mean free path in helium under the normal number density of helium atoms is  D 1=.Na   / D 0:62 μm. In this case, when the mean free path of electrons in a gas is independent of the electron velocity, the electron drift velocity we is equal at small electric field strengths E and at equilibrium between electrons and a gas s 8 eE we D ; 3  me T

145

Section 8.9 Problems

where me is the electron mass, T is the gas temperature that is expressed in energetic units. This gives for the electron mobility under normal conditions Ke D we =E D 2:6103 cm2 =.Vs/ that leads to Ke =KC D 250 and ln.Ke =KC / D 5:5. From this we find the particle charge according to formula (8.1.8) Z D ro =a, where a D 10 nm. The criterion of a weakness of the particle field has the form eE  T , and since the electric field strength at the particle surface is E D Ze=ro2 , on the basis of formula (8.1.8) for the particle charge Z we obtain this criterion in the form Ke  ro :  ln KC As is seen, this criterion is stronger that the criterion (8.1.1) for a high density of a buffer gas. In particular, in the helium case this criterion has the form Na ro  1016 cm2 : Problem 8.2. Determine the specific charge of aerosols (water drops in the atmo sphere) in a region where basic negative ions are NO 2 and NO3 , and the basic positive ion in this atmosphere region is H C  H2 O. Determine a typical charging time, if the ion number density in this atmosphere region is 103 cm3 and a typical particle radius is ro D 1 μm. Under normal conditions we have for the ion mobilities [357, 358, 359, 360]  2 2 C K.NO 2 / D 2:5 cm =.V  s/, K.NO3 / D 2:3 cm =.V  s/, and K.H  H2 O/ D 2 2:8 cm =.V  s/. From this it follows that aerosols are charged positively Z > 0. Taking for definiteness the mobility of negative ions to be K D 2:4 cm2 =.V  s/ and representing formula (8.1.7)) for the aerosol charge Z in the form Z D ro =a, we obtain at temperature T D 300 K for the parameter of this formula a D 0:36 μm, and the equilibrium particle charge is Z D 2:8. On the basis of formulas (8.1.5) and (8.1.6) we have the equation of evolution of the aerosol charge   dZ 4No Ze 2  K expŒZe =.r T /

; D JC  J D K C  o dt expŒZe 2 =.ro T /  1 (8.9.1) where JC ; J are the numbers of positive and negative ions attached to the aerosol per unit time and given by formulas (8.1.6) and (8.1.5). Note that for the equilibrium particle charge Ze 2 =.ro T / D 0:15, that allows us to expand exponents of formula (8.9.1) in a series over this small parameter. Substituting in this equation values of the above parameters, we reduce it to the form dZ D 0:013s 1 .1  Z=2:7/: dt

146

Chapter 8 Charging of small particles in ionized gases

From this we find a typical time of establishment of the equilibrium particle charge   4 min. Problem 8.3. Compare the electron current on the particle surface (8.1.5) with that for a model where electrons are under the thermodynamic equilibrium in the field of a highly charged particle. Neglect the screening of the particle field by a surrounding ionized gas in the region of its action. In the case of a large particle charge the potential particle energy is large compared to a thermal electron energy and we have according to formula (8.1.5) for the total rate of electron attachment to the particle surface   4De No Ze 2 Ze 2 Je D exp  ; T T ro for Ze 2 =.ro T /  1. In the case of thermodynamic equilibrium this rate is determined by electrons which are located close to the particle surface, and then we have for the rate of electron attachment to the particle surface s s   8T 8T Ze 2 2 2 Ie D  ro ; Ne .ro / D  ro No exp   me  me T ro where we use the Boltzmann formula for the electron number density near the charged particle,   Ze 2 ; Ne .ro / D No exp  T ro so that No is the number density of electrons far from the particle. Note that the same expression for the rate of electron attachment we obtain in the kinetic regime of electron-cluster interaction (formula (8.4.6)), considering collision between electron and a charged cluster. From this we have for ratio of currents onto the particle surface s Ie T ro ro T D  : Je De 2 me Ze 2 As is seen, different parameters determine the rate of electron attachment to the particle surface within the framework of the thermodynamic equilibrium for electrons and the Fuks model that accounts for violation thermodynamic equilibrium in the particle vicinity due to , gives the dependence on other parameters for. Thus, violation of the thermodynamic equilibrium is of principle for electron attachment to a particle located in an ionized gas.

Section 8.9 Problems

147

Problem 8.4. A microparticle is located in a weakly ionized dense gas (8.1.1) that satisfies to criterion (8.1.3). Ascertain conditions where the electric field strength near the particle Eo exceeds significantly the electric field strength E of gas discharge. This gas discharge plasma is heated through interaction of electrons and ions with an electric field. Then collisions with a strong interaction of ions with atoms of a buffer gas lead to a remarkable energy exchange of colliding particles, whereas the electron energy " in collisions with atoms varies by the value "  me "=M [329], where M; me are the masses of the atom and electron correspondingly. Assuming the cross sections of electron and ion collisions with atoms to be of the same order of magnitude, we find that in a certain range of the electric field strengths a typical electron energy Te exceeds a thermal energy of ions, and this range of the electric field strengths E will be consider below. We now use formula (8.1.6) for the ion flux onto the particle surface being guided by the helium case where the diffusion cross section of electron-atom scattering weakly depends on the collision energy. Then the distribution function of electrons in neglecting inelastic collisions is given by the Druuyvesteyn formula [330, 331]. In particular, the average electron energy in a constant electric field of strength E is [332, 356] s M " D 0:43 eE; me where  D 1=.Na   / is the mean free path of an electron in a buffer gas. Using formula (8.1.8) for the particle charge Z as an estimation and accounting for a typical electron energy Te  ", we obtain for the electric field strength at the particle surface Eo that is created by the particle charge s Ze Te  M E ln.Ke =KC /: Eo D 2  ro ero ro me Comparing the electric field strengths of gas discharge and at the particle surface, we have r 1 E r o me  : Eo  M ln.Ke =KC / As is seen, the first factor is large due to criterion (8.1.1), whereas two subsequent factors are small. Hence, the electric field strength due to a particle charge at the particle surface may be large or small compared to the electric field strength of gas

148

Chapter 8 Charging of small particles in ionized gases

discharge. In the first case the particle field equalizes the electron and ion currents to the particle surface. Note that the region of the particle field action assumes to be not large, and then the energy distribution functions of electrons far and near the particle are close. Problem 8.5. On the example of a microparticle located in a dense gas discharge plasma of helium show that in the stationary case of particle charging the electric field strength near the particle exceeds significantly the electric field of gas discharge. Let us take into account only elastic electron-atom scattering, that leads to the Druuyvesteyn formula [330, 331] for the energy distribution function of electrons  2 " f ."/ D C exp  2 : (8.9.2) "o Here p " is the electron energy, C is the normalization coefficient, "o D eEe M=3me , where E is the electric field strength of gas discharge, e is the mean free path of electrons in helium, me ; M are the electron and atom masses correspondingly. The boundary electron energy "b is estimated as "b  eEo e ;

Eo D

Ze 2 ; ro2

where Z is the established particle charge. The rates of electron and ion attachment to the particle are equal correspondingly Je D  ro2  we Ne ;

Je D  ro2  wi Ni ;

where we ; wi are the electron and ion drift velocities respectively, Ne ; Ni are the number densities of electrons and ions near the particle surface. In the helium case the cross section and, correspondingly, the mean free path of electrons in helium do not depend on the electron energy, and the drift velocity of electrons in the electric field of a high strength E (for example, [356, 209]) is given by s  m 1=4 eE e e we D 0:90 ; M me where e D 1=.Na e / is the mean free path of electrons in helium, so Na is the number density of atoms, e is the cross section of electron-atom collisions.

149

Section 8.9 Problems

Assuming HeC to be the basic sort of ions, one can express the ion drift velocity in a strong electric field through the cross section of ion-atom charge exchange res [352] r 2eEi wi D ; M where the mean free path of ions in helium is i D 1=Na res . Taking the diffusion cross section of electron-helium atom collision as [208] e D 6 Å2 and the cross section of resonant charge exchange in the helium case at the collision energy 0:1 eV to be res D 35 Å2 . We hence have for the ratio between the electron and ion drift velocities s we res M D 1:13 D 230: wi e me Taking near the particle surface the number density of ions Ni  No , we have from equality of electron and ion currents to the particle surface on the basis of the distribution function (8.9.2) Ne =Ni  exp."b ="o /, where "b  eEo e is the boundary electron energy, and the particle is attained by electrons with a higher energy only, where jEo j D Ze=ro2 is the electric field strength on the particle surface under the action its charge Z. On the other hand side, the boundary electron energy is s p M "b D ln 230"o D 2:3"o D 2:3eEe D 110eEe : 3me Comparing this with the above expression for the boundary electron energy "b , we obtain Eo  110E; i.e. the electric field strength at the particle surface due to the particle equilibrium charge exceeds the electric field strength of gas discharge by two orders of magnitude in the helium case.

Chapter 9

Growth of clusters and small particles in buffer gas 9.1

Types of nucleation processes

In considering the nucleation processes in a gas, we will be guided by a system consisting of a buffer gas (inert gas) with an admixture of a metal vapor. If the partial pressure of an atomic metal vapor located in a buffer gas exceeds the saturated vapor pressure at a given temperature, formation and growth of clusters is possible from a vapor excess. Though the partial pressure of an atomic vapor is lower significantly in comparison with that of a buffer gas, just the metal component is joined in clusters because the binding energy of metal atoms in clusters exceeds significantly that for buffer gas atoms, and for this reason a buffer gas is not altered in the course of cluster growth. Next, cluster growth proceeds through formation the condensation nuclei, and therefore the cluster growth process is called as the nucleation process. Mechanisms of cluster growth in a buffer gas are represented in Figure 9.1. All these processes were studied starting from 19 century for other systems and media, and this explains the terms of these processes. But since these terms were introduced for other media, a different sense may be used for these terms in applications to gaseous systems. We follows to definitions of the Landau-Lifshitz course of physics and give below its definitions. The first mechanism of cluster growth given in Figure 9.1 a) corresponds to conversion of an atomic vapor in a gas of clusters that proceeds in the case if the pressure of an atomic vapor exceeds the saturated vapor pressure at a given temperature. Then the first stage of the cluster growth process is formation of nuclei of condensation, and then cluster growth results from attachment of free atoms to nuclei of condensation or clusters as it was considered in Section 7.2. Then the kinetic regime of cluster growth is realized if the criterion (7.1.6) holds true, and the diffusion regime of conversion of an atomic vapor in a gas of clusters if the inverse criterion is valid. Coagulation is the second mechanism of cluster growth given in Figure 9.1 b). This term was introduced in physiology where “coagulation is a complex process by which blood forms clots” [367] and we define coagulation as a process of formation of a single liquid drop as a result of contact of two liquid drops. Sometimes this process is named “coalescence” [368], but coalescence is a synonym of the

Section 9.1 Types of nucleation processes

151

a) Attachement of atoms

b) Coagulation

c) Coalescence

d) Aggregation

Figure 9.1. The mechanisms of nucleation and cluster growth. a) conversion of an atomic vapor in a gas of clusters; b) coagulation; c) coalescence (Ostwald ripening); d) aggregation.

Ostwald ripening process [369] as it was used in the classical theory [14] and book [279] for the diffusion regime of this process. This mechanism is represented in Figure 9.1 c). If solid particles are joined during their contact, but conserve partially their initial shape, this process we call aggregation and represent in Figure 9.1 d). Indeed, according to definition [370] “aggregation is the collecting of units or parts into a whole”. In addition, structures formed as a result of this process are called aggregates (chain aggregates, fractal aggregates), and we devote to these structures the next section.

152

9.2

Chapter 9 Growth of clusters and small particles in buffer gas

Kinetic regime of cluster coagulation

Coagulation as a mechanism of nucleation (see Figure 9.1 b)) results in association of two liquid drops at their contact in a buffer gas according to the scheme Mnm C Mm ! Mn ;

(9.2.1)

where M is a drop (cluster) atom. The coagulation process is typical for the Earth atmosphere where joining of water drops leads to formation of mist and atmosphere pollution [30, 310, 23, 373]. Below for definiteness we will consider an example of the coagulation process with growth of liquid metal drops in a buffer gas. Note that in contrast to previous cases of cluster-gas interaction, separation of regimes of this interaction for cluster coagulation takes place both for motion of an individual cluster in a gas and for relative motion of two clusters. The criterion of the diffusion regime of cluster approach has the form Ncl ƒ3  1;

(9.2.2)

where Ncl is the number density of clusters, ƒ is the mean free path of a cluster in a gas with respect to its motion direction in accordance with formula (6.1.4). The number density of clusters is Ncl D Nb =n, where Nb is the number density of bound atoms in clusters, n is the average number of cluster atoms. To separate in this criterion the dependence on the number density Na of buffer gas atoms and a typical number n of cluster atoms, we introduce the concentration of bound atoms in clusters is cb D Nb =Na , and then on the basis of formula (6.1.4) for the mean free path ƒ of a cluster in a gas we have the criterion (9.2.2) for the diffusion character of cluster approach in the form r n 1 4 m 2  4=3 ; cl D r ; (9.2.3) 2=3 3 ma W cb Na cl2 where m is the mass of a buffer atom, ma is the mass of a metal atom, rW is the Wigner-Seits radius. In particular, in the case of coagulation of liquid copper clusters in argon we have cl D 2:3  1016 cm2 , and at the concentration of bound copper atoms in argon cb D 0:01 criterion (9.2.3) for the diffusion regime p of cluster approach has the form Na n3=4  31023 cm3 cb . From this criterion it follows that in reality both regimes, the diffusion and kinetic ones, may be realized. Moreover, in the course of cluster coagulation the transition is possible from kinetic regime of cluster approach to the diffusion one. For the scheme (9.2.1) the rate of cluster growth may be connected with the rate of cluster association on the basis of the Smolukhowski equation [374] both

Section 9.2 Kinetic regime of cluster coagulation

153

for the kinetic regime of cluster approach and for the diffusion regime of their approach Z Z @fn 1 (9.2.4) D fn k.n; m/fm d m C k.n  m; m/fnm fm d m: @t 2 Here k.n  m; m/ is the rate constant of the association process, the factor 1=2 takes into account that collisions of clusters consisting of n  m and m atoms are present in the above equation twice, and theR size distribution function of clusters fn is normalized according to the relation fn d n D Ncl with Ncl - the number density of clusters. R1 Let us prove that the number density of bound atoms Nb D 0 nfn d n is conserved in the course of coagulation within the framework of the Smolukhowski equation (9.2.4) according to which all the atoms which partake in coagulation remain in a given space region. Indeed, multiplying the Smolukhowski equation (9.2.4) by n and integrating it over d n, we obtain Z Z dNb @fn d D nfn d n D nd n dt dt @t Z Z 1 D  nk.n; m/fn d nfm d m C nk.n  m; m/fnm fm d nd m; 2 and in the second integral n > m. Replacing n  m by n in right-hand side of this equation, we reduce a new equation to the form dNb =dt D 0. This means that within the framework of the Smolukhowski equation (9.2.4) the density of bound atoms in clusters is conserved. We now define the average cluster size as Z 1 Z 1 Nb nD ; Nb D nfn d n; Ncl D fn d n: (9.2.5) Ncl 0 0 We have dNb dn dNcl D Ncl Cn D 0; dt dt dt that give the following equation of cluster growth dNcl dn D n D 0: dt Ncl dt Let us consider a simple case if the rate constant of association of two clusters is independent of cluster sizes k.n; m/ D kas , and then the Smolukhowski equation (9.2.4) takes the form Z 1 Z n @fn 1 fm d m C kas fnm fm d m: D kas fn @t 2 0 0

154

Chapter 9 Growth of clusters and small particles in buffer gas

Integration of this over d n gives Z 1 @fn dNcl 1 dn D D  kas Ncl2 @t dt 2 0 where in the second term of the right hand side of this equation we replace n  m be n. As a result, we obtain the equation of cluster growth in the form dn 1 D kas Nb : dt 2

(9.2.6)

Note a general character of this equation that holds true for any mechanism of cluster growth but requires a weak size dependence for the rate constant of association. Solution of the above equation gives for the average cluster size 1 n D kas Nb t; 2

(9.2.7)

under assumption that at the beginning the cluster size is relatively small. We now determine the size distribution function for clusters under the above assumption. It is convenient to use instead of the size distribution function fn of clusters the concentration cn of clusters of a given size that is defined as cn D fn =Nb , where Nb is the number density of bound P atoms in clusters. The normalization condition for the cluster concentration is n ncn D 1, and the kinetic equation (9.2.4) in terms of cluster concentration takes the form Z 1 Z @cn 1 n cm d m C cnm cm d m; (9.2.8) D cn @ 2 0 0 where the reduced time is  D Nb kas t. This equation has the following solution cn D

 n exp  : n n2 1

(9.2.9)

The concentration of clusters of a given size is satisfied to the normalization condiR1 tion 0 ncn d n D 1 and to formula (9.2.7) for the average cluster size n. Indeed, substituting the expression (9.2.9) into the kinetic equation (9.2.4), we obtain formula (9.2.7) in the form n D =2 that corresponds to equation (9.2.7). Correspondingly, relation (9.2.9) gives for the size distribution function of clusters in this process fn D

 n exp  ; n n2

Nb

and this distribution function is normalized by the condition Nb D

(9.2.10) R1 0

nfn d n.

155

Section 9.2 Kinetic regime of cluster coagulation

The above expressions are based on the assumption that the cluster association rate is independent of a cluster size and is valid both for the kinetic and diffusion regimes of cluster coagulation. Let us consider the kinetic regime of cluster coagulation where the association rate according to formulas (4.2.3) and (4.2.4) depends weakly on a cluster size. Substituting the expression (4.2.3) for the association rate into kinetic equation (9.2.6) and averaging this rate constant on the basis of the size distribution function (9.2.9), we obtain the following equation for variation of the average cluster size dn D ko Nb n1=6 I; dt s Z Z xCy 1 1 1 1=3 1=3 2 .x Cy / I D exp.2x  2y/xdxydy D 5:5: 2 0 xy 0 (9.2.11) This leads to the following expression for the average cluster size [375, 376] n D 6:3.Nb ko t /1:2 :

(9.2.12)

Since n  1, this formula holds true under the criterion ko Nb t  1: In addition, the kinetic regime of coagulation corresponds to small cluster sizes where the criterion holds true that is opposite with respect to criterion (9.2.3), namely 2=3

n

cb

4=3 2 cl

:

(9.2.13)

Na

If a metal vapor located in a buffer gas is converted in a gas of metal clusters, we have that on the first stage the cluster growth process results in attachment of free metal atoms to condensation nuclei and metal clusters. A assuming clusters to be liquid, we have cluster growth on the second stage of the nucleation process as a result of coagulation. Let us show that these stages are separated. Indeed, the first stage corresponds to the scheme (7.2.1) and is based on a large value of the parameter (7.2.2). In the end of this stage, the average cluster size in accordance with formula (7.2.3) is n  G 3=4 , and a time of this stage is at  G 1=4 =.Nb ko /, where Nb is the number density of bound atoms. A typical time of an increase of the cluster size as a result of coagulation is according to formula (9.2.12) coag 

n5=6 G 5=8  : Nb ko Nb ko

156

Chapter 9 Growth of clusters and small particles in buffer gas

Taking the average cluster size in the end of the coagulation stage to be comparable with that after conversion atomic metal vapor in a gas of metal clusters, we obtain for the ratio of a time of the coagulation stage coag to a typical time at of the first stage of the nucleation process, that is coag  G 3=8 : at Since G  1, the processes of conversion of a metal vapor in a gas of clusters and the subsequent coagulation process are separated.

9.3

Diffusion regime of cluster coagulation

The diffusion regime of cluster coagulation with respect to their approach takes place if the criterion (9.2.2) holds true. It is convenient to represent this criterion in the form (9.2.3), and then equation of cluster growth as a result of coagulation (9.2.6) includes the rate constant of association of two liquid clusters with radii r1 and r2 in accordance with formula (4.2.6) that has the form k12 D 4.D1 C D2 /.r1 C r2 /; where D1 , D2 are the diffusion coefficients of indicated clusters in a buffer gas. We now consider two regimes of cluster coagulation according to cluster motion in a buffer gas for the diffusion regime of coagulation with respect to cluster approach in accordance with the criterion (9.2.3). For the diffusion regime of cluster motion we have for the rate constant of cluster association kas using formula (5.3.2) for the cluster diffusion coefficient for the diffusion regime of cluster motion   2T 1 1 kas D C (9.3.1) .r1 C r2 /;   r1 ; r2 : 3 r1 r2 Since the main contribution to the association rate constant gives nearby cluster sizes, we have for the association rate if the average cluster size is ro kas D

8T ; 3

ro  ;

(9.3.2)

and this rate constant is independent both of cluster sizes and of the size distribution function for clusters. Using formula (5.1.3) for the gas viscosity, one can represent the association rate constant kas in this regime in the form r r T T 8T 128 (9.3.3) D p g D 4:8 g ; ro  : kas D 3 m m 15 

157

Section 9.3 Diffusion regime of cluster coagulation

In the kinetic regime of cluster motion in cluster coagulation the rate of cluster association depends on the cluster size. For simplicity, we assume all the clusters to have an identical size ro that varies in the course of cluster growth. Then on the basis of formulas (4.2.6) and (5.3.2) we have for the cluster association rate in this regime of cluster growth r kas D 3

2T 1 ; m Na ro

 ro  ;

nD

ro rW

3

2=3



cb

4=3 2 cl

:

(9.3.4)

Na

Substituting this rate constant in equation (9.2.6) for evolution of the cluster size, we obtain as a result of solution of this equation  nD

8T m

3=8 

cb t rW

3=4

2=3

;

n

cb

4=3 2 cl

:

(9.3.5)

Na

Thus, we conclude that growth of neutral clusters as a result of coagulation is described by formula (9.2.12) in the kinetic regime of cluster approach that is governed by criterion (9.2.13). In the diffusion regime of cluster coagulation, evolution of the average cluster size in time is given by formula (9.2.7) with the rate constant of cluster association according to formula (9.3.2) for diffusion regime of cluster motion in a buffer gas. In the case of the diffusion regime of cluster motion, i.e., a small cluster concentration at a current cluster size and the kinetic regime of cluster motion in a buffer gas variation of the cluster size in time is given by formula (9.3.5). The indicated formulas exhaust the channels of cluster evolution in their coagulation. It should be noted that the above expressions relate to a neutral gas where clusters grow, and charging of clusters as it takes place in an ionized gas can change the character of cluster growth. We demonstrate this for the diffusion regime of cluster approach in an ionized gas where the charge equilibrium is established, and the cluster charge is large. In particular, we consider the case when the mean free path of ions i is small compared with a cluster radius ro , i.e. i  ro , and then the cluster charge is given by formula Z D xro Te =e 2 , and x is determined from equation (8.4.8). In this case the cluster charge is large and the average charge coincides with the most probable cluster charge, and correspondingly a cluster radius ro is connected uniquely with its charge. Accounting for repulsion of joining clusters, we have the following equation of cluster growth instead of (9.2.6)   Z 2e2 dn D kas Np exp  ; dt ro T

158

Chapter 9 Growth of clusters and small particles in buffer gas

where T is the temperature of cluster motion that coincides with the gaseous temperature, and kas is given by formula (9.3.4) in the case under consideration. Solving this equation in a range of parameters where cluster charging influences its growth significantly, we obtain for evolution of the cluster radius ! 3 rW x 2 Te2 e2T ro D 2 2 ln (9.3.6) kas t ; x Te 3ro2 e 2 T where the factor under logarithm is large, the ion temperature coincides with the gas temperature T , kas is the rate constant of association for neutral clusters in accordance with formula (9.3.4), and the reduced cluster charge x is given by formula (8.4.8). As is seen, we have another character of cluster growth in this case compared to the case of neutral clusters according to formula (9.3.4), and cluster growth in this case proceeds slowly. In addition, we have different regimes of cluster growth depending on cluster charge parameters.

9.4

Cluster coagulation in external field

Let us consider the diffusion regime of cluster coagulation if the rate constant of cluster association according to formula (9.3.2) and does not depend on a cluster size. Then a typical time of cluster growth is estimated as   1=.Np kcoag / where Np D Nb =n is the number density of clusters-particles, and a typical time of cluster growth is proportional to a typical cluster size n if the total number density of cluster atoms Nb is conserved in time. An external field causes directed motion of clusters, and if clusters are located in an external field, the rate constant of cluster association increases with cluster growth, and this mechanism of cluster growth may be dominated even in low external fields. For definiteness, we below consider aerosol growth in the Earth atmosphere due to fall of aerosol particles under the action of the gravitation field when the velocity of the particle falling is given by formula (5.3.4). Then the rate of particle association is kas  w  ro4 ; where w is the difference of the velocities of particle falling, and a typical time of particle growth depends on a particle size as   1=.Np kcoag /  n1=3 , i.e. this time increases with an increase of a particle size, and even for a weak field starting from some particle size this regime of particle growth dominates. We now determined the rate of particle association if these particles are located in a buffer gas and are falling under the action of gravitational forces. We assume

159

Section 9.4 Cluster coagulation in external field

that each contact of these particles leads to their association. Then the rate of association of two particles of radii r1 and r2 is given by kas D .w1  w2 /.r1 C r2 /2 D

2g 4 Œ.r1  r24 / C r1 r2 .r12  r2 /2 ; 9

r1 r 2 ;

where the drift velocity w of a falling particle is given by formula (5.3.4), so that  is the density of a particle material,  is the viscosity of a buffer gas. Taking the size distribution function (9.2.10) for particles and normalizing it to one, we have f .n/ D exp.n=n/=n2 ; we obtain for the average rate constant of association Z hkas i D kas .n1 ; n2 /f .n1 /f .n2 /d n1 d n2 Z 4=3 4=3 1=3 1=3 2=3 2=3 D C.jn1  n2 j C 2n1 n2 jn1  n2 j/  f .n1 /f .n2 /d n1 d n2 ; C D

4 2 rW g : 9

This gives for an average rate constant 2 ro4 g J; 9 Z x Z 1 exp.x/dx exp.y/dyŒ.x 4=3  y 4=3 / J D

hkas i D

0

0

C 2x 1=3 y 1=3 .x 2=3  y 2=3 /

Z 1 Z 1 exp.x/dx exp.y/dyŒ.y 4=3  x 4=3 / C 0

x

C 2x 1=3 y 1=3 .y 2=3  x 2=3 /  2:8: As a result, we obtain from this hkas i 

2ro4 g : 

(9.4.1)

Let us consider association of aerosol particles in the Earth atmosphere and compare the two mechanisms for the diffusion regime of particle growth. In the first one approach of associating particles results from their diffusion in air and the rate constant is given by formula (9.3.5). In the second case association of

160

Chapter 9 Growth of clusters and small particles in buffer gas

aerosol particles is determined by their falling in the gravitation field, and the fall velocity depends on the particle size. Then the rate constant of association is given by formula (9.4.1). Evidently, at low particle sizes the diffusion mechanism of association dominates, whereas the drift mechanism becomes the basic one at large particle sizes. The boundary particle size rb is determined by the equality of the rate constants (9.3.5) and (9.4.1) which give  rb D

12T g

1=4 :

(9.4.2)

At room temperature from this formula it follows for atmospheric air rb  1 μm. Since rb  , where   0:1 μm is the mean free path of air molecules in air, both mechanisms of association of aerosol particles may be realized in the diffusion regime of particle association. As is seen, this mechanism of cluster growth is of importance for large clusters or particles. Therefore in considering coagulation of particles under the action of an external electric field of strength E, we take the diffusion regime of particle-gas interaction. In this case the particle mobility is given by formula (5.1.7) Kdif D e=.6 ro /, and the drift velocity of a large particle is given by w D ZEKdif . Because the particle charge is large Z  1 and according to formula (8.1.7) it is proportional to the particle radius ro , we find that the particle drift velocity in this case is independent of a particle radius. This means that particles of different sizes have identical drift velocities, and the rate constant of association of two particles according to the mechanism under consideration is zero.

9.5

Ostwald ripening

The Ostwald ripening process for cluster growth [11, 12, 13] given in Figure 9.1 c) results in interaction of clusters with a parent atomic vapor. Since clusters are found in equilibrium with a parent atomic vapor, the total rate of evaporation from the surface of clusters is equal to the total rate of atom attachment to the cluster surface. Though this balance is fulfilled in general, it is not valid for clusters of a given size. Namely, the evaporation rate for large clusters is lower than the rate of atom attachment to such clusters, and such clusters are grown. On contrary, the opposite relation takes place for small clusters, and small clusters disappear. As a result, the average cluster size increases in this process. The Ostwald ripening process is typical for growth of grains and particles in solid and colloid solutions and leads to coarsening of nonuniform and mixtures in the liquid and solid states [377, 378, 379, 380, 381]. It is of importance also for island evolution in film growth [382]. Below we are based on the classical theory

161

Section 9.5 Ostwald ripening

of the ripening process in the diffusion regime [14, 15, 16]. This theory allows us to understand the nature of process of the cluster growth [279, 383] and use it for the various types of transport in a system. Based on the classic theory of the Ostwald ripening process, we will be guided by growth of metal clusters in gases. So, we have individual clusters located in a buffer gas, and these clusters form an atomic vapor that is in equilibrium with clusters through processes of atom attachment to clusters and cluster evaporation. For simplicity, we take the concentration of bound atoms in clusters cb D Nb =Na to be large compared with the concentration of free metal atoms cm D Nm =Na , i.e. cb  cm ;

(9.5.1)

where Na is the number density of buffer gas atoms, Nb is the total number density of bound atoms located in clusters, and Nm is the number density of free metal atoms. In this case clusters are a large reservoir for free metal atoms. The criterion of the diffusion regime for cluster growth as a result of Ostwald ripening has the form Nm 3m  1 ; or 3 ; cm  Na2 m

(9.5.2)

where m D 1=.Na m / is the mean free path of free metal atoms in a buffer gas, and m is the gas-kinetic cross subsection of metal atoms in a buffer gas. Let us consider the equilibrium between metal clusters and an atomic vapor. For clusters consisting of n atoms this equilibrium corresponds to equality for rates of atom attachment and cluster evaporation corresponds to the number density of free metal atoms Nm .n/ that is given by formula (7.1.11)   2A Nm .n/ D Nsat .T / exp ; 3n1=3 T where Nsat .T / is the saturated number density of metal atoms over a plane metal surface at temperature T , A is the specific surface energy of the cluster in accordance with formula (3.6.3). Let us introduce the size distribution function fn of clusters, so that fn d n is the number density of metal clusters contained from n up n C d n atoms. This distribution function is the solution of the kinetic equation that includes the processes of cluster evaporation and atom attachment to clusters. The rate of atom attachment to a given cluster is determined by formula (4.2.5) J D 4Dm ro Nm ;

162

Chapter 9 Growth of clusters and small particles in buffer gas

where Dm is the diffusion coefficient for metal atoms in a buffer gas, ro is the cluster radius that according to formula (3.4.1) is ro D rW n1=3 . The equilibrium between the atom attachment and cluster evaporation processes leads to the relation Z Z 4Dm ro Nm fn d n D 4Dm ro Nm .n/fn d n: As is seen, for a given cluster size the equilibrium between the rates of attachment and evaporation processes may be violated. Let us introduce the critical cluster size ncr such that Nm D Nm .ncr /;

(9.5.3) 1=3

and the critical cluster radius according to formula (3.4.1) is rcr D rW ncr . The rates of atom attachment and cluster evaporation for clusters of the critical size are coincided. The critical cluster size divides clusters in two groups, so that for smaller clusters n < ncr the evaporation rate is larger than the rate of atom attachment to clusters, and such clusters lost atoms. On contrary, the opposite relation between the evaporation and attachment rates takes place for larger clusters n > ncr , and these clusters acquire atoms in total. As a result, a number of atoms in clusters of the first group decreases in time, and in clusters of the second group it increases in time, that leads to growth both the average cluster size and the critical cluster size. The total rate of atoms attachment Jat to clusters of this group and the difference J for cluster evaporation and atom attachment to clusters of the first group are given by Z ncr n1=3 fn d n; Jat D 4Dm rW Nm 0

Z ncr 2A 1=3 1=3 1=3 J D 4Dm rW Nm n fn d n exp  ncr /  1 : (9.5.4) .n 3T 0 It is convenient to transfer to a new variable u D ro =rcr D .n=ncr /1=3 , that is the reduced cluster size, and introduce the probability Pdif .u/, so that Pdif .u/du is the probability that the cluster size ranges between u and u C du. According to the definition Z 1 Pdif .u/du D 1: 0

Since

Z 0

1

fn d n D Ncl D

Nb ; n

163

Section 9.5 Ostwald ripening

where n is the average cluster size, we have fn d n D

Nb Pdif .u/du; n

fn D

ncr Nb Pdif .u/: n 3u2

(9.5.5)

In the case of the diffusion regime of for the Ostwald ripening process we have for the distribution function Pdif .u/ on the basis of a strict theory [14, 279, 383] 34 e u2 expŒ1=.1  2u=3/

25=3 .u C 3/7=3 .3=2  u/11=3 u2 expŒ1=.1  2u=3/

D 69:4 ; .u C 3/7=3 .3=2  u/11=3

Pdif .u/ D

(9.5.6) u < 3=2;

and Pdif .u > 3=2/ D 0. This function has the maximum at umax D 1:135 at which Pdif .umax / D 2:152, i.e. this maximum relates to large clusters, and Pdif .1/ D 1:726. The size distribution function Pdif .u/ is represented in Figure 9.2 (a). Since the difference of the rates for atom attachment and cluster evaporation are equal for clusters of the first and the second group, we have the relation Z

Z

1 0

.1  u/Pdif .u/du D

1:5 1

.u  1/Pdif .u/du:

(9.5.7)

Thus, the size distribution function Pdif .u/ must satisfy to equation (9.5.7). Next, a part of clusters of the first and second groups is Z ˛D

0

Z

1

Pdif .u/du D 0:425;

1˛ D

1:5 1

Pdif .u/du D 0:575:

We also have for the average cluster size R1 R1 3 u Pdif .u/du 0 nfn d n n D R1 D ncr 0R 1 D 1:135ncr ; 0 fn d n 0 Pdif .u/du and the distribution function at the critical point f .ncr / D

Nb du 0:507Nb : Pdif .1/ D n dn n2cr

Let us analyze the dynamics of evolution of the critical size. Because of the criterion (9.5.1), a typical time of variation of the critical cluster radius is relatively large, and in the course of this process the critical cluster radius may be considered as constant. Equilibrium between bound and free metal atoms results in equality

164

Chapter 9 Growth of clusters and small particles in buffer gas

2.0

1.5

1.0

0.5

(a)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.06 0.04 0.02

0.5

1.0

1.5

–0.02 –0.04

(b) Figure 9.2. The probabilities Pdif .u/ and pkin .u/ of a given size of metal clusters for the diffusion and kinetic regimes of cluster motion (a) and the difference of these probabilities (b).

of the total rate of cluster evaporation and atom attachment, and variation of the critical size follows from equation Z ncr d nfn d n D J; dt 0 that gives d ncr J 1:97ncr J : D D dt ncr f .ncr / Nb Assuming in formula (9.5.4) for J the possibility to expand exponent for sizes given the main contribution to the integral, we obtain for the rate of decrease of

165

Section 9.5 Ostwald ripening

small clusters per unit volume J D 4Dm rW Nm

2A 3T

Z

ncr 0

.1  u/fn d n

Z 1 8A .1  u/Pdif .u/du D Dm rW Nm Nb 3T n 0 0:627ADm rW Nm Nb D : ncr T From this we obtain for variation of the critical cluster size d ncr A D 1:24 Dm rW Nm : dt T

(9.5.8)

Correspondingly, the average cluster size n grows in time as [14, 279] A n D 1:4 Dm rW Nm t: T

(9.5.9)

We note also the automodel character of the size distribution function of clusters P .u/ that means the form of distribution function (9.5.6) is conserved in time, while the average cluster size (9.5.9) varies in time. To transfer from the diffusion regime of the Ostwald ripening process to the kinetic regime [17], we must replace in formulas (9.5.4) the factor 4Dm rW n1=3 by the factor ko n2=3 , that gives  

Z ncr 2A 1=3 2=3 1=3 J D Nm ko n fn d n exp  ncr /  1 .n 3T 0 "  1=3 # Z n 2A ncr 1=3 1 fn d n: ko Nm n (9.5.10)  3T 0 ncr We note that the size distribution function of clusters in the kinetic regime pkin .u/ differs from that in the diffusion one Pdif .u/, but satisfies to equation (9.5.7) that means the rate difference J for clusters of the first and second groups differ by a sign only Z 1:5 Z 1 .1  u/pkin .u/du D .u  1/pkin .u/du: 0

1

In order to take into account the difference between the size distribution functions of clusters in the diffusion Pdif .u/ and kinetic pkin .u/ regimes, we represent the connection between these distribution in the form pkin .u/ D C u exp.au/Pdif .u/

166

Chapter 9 Growth of clusters and small particles in buffer gas

and take the parameters C and a such that the above relation like to (9.5.6) and normalization condition would be fulfilled. Then we obtain C D 3:35 and a D 1:18, and the size distribution function of clusters in the kinetic regime has the form pkin .u/ D 3:35u exp.1:18u/Pdif .u/ D 232 exp.1:18u/ u3 expŒ1=.1  2u=3/

; .u C 3/7=3 .3=2  u/11=3

u < 3=2I

pkin .u > 3=2/ D 0:

(9.5.11)

The maximum distribution function is pkin .1:126/ D 2:147, and at the critical size it is equal pkin .1/ D 2:125. This size distribution function is represented in Figure 9.2 (a). One can expect that the dependencies Pdif .u/ and pkin .u/ are different because they have a different form. In reality, these probabilities are close, and Figure 9.2 (b) gives the difference of these probabilities. Next, from this we have the following relation between the average cluster size n and the critical cluster size ncr n D 1:123ncr : Repeating the above operations for the kinetic regime of cluster growth, we obtain formula (9.5.10) as J D 0:049

A ko Nm Nb ; T n2=3 cr

and the analog of equation (9.5.8) for the kinetic regime of cluster growth has the form d ncr A 1:90ncr J D 0:094  ko Nm n1=3 D cr : dt Nb T Solution of this equation gives for the average cluster size in the case of the Ostwald ripening process 

A n D 0:014 ko Nm t T

3=2 :

(9.5.12)

Thus, the asymptotic behavior of cluster growth leads to the automodel character of cluster growth for the Ostwald ripening mechanism both in the diffusion and kinetic regime of cluster growth. This growth character starts when a typical cluster size exceeds significantly its initial value. According to formulas (9.5.9) and (9.5.12) the rate of cluster growth in the diffusion and kinetic regimes is characterized by the different time dependence.

Section 9.6 Method of molecular dynamics in nucleation processes

9.6

167

Method of molecular dynamics in nucleation processes

In analyzing the processes of nucleation and cluster growth, we are based on analytic methods. These processes correspond to transition of a system of particles form the state of continuous spectrum to their bound state. One can expect the convenience to use for this goal computer methods, mostly the method of molecular dynamics with using an appropriate interaction potential of particles that allows one to study the dynamics of evolution of an ensemble of classical particles. Method of molecular dynamics gave some results which are important for understanding of various aspects of physics. In particular, as it was discussed above, coexistence of phases in the phase transition solid-liquid was discovered [176, 178] as a result of computer simulation of Lennard-Jones clusters by methods of molecular dynamics. We note also investigation of hot liquid LennardJones clusters [385, 386, 387] by methods of molecular dynamics according to which the structure of the surface for large clusters differs from that constructed on the basis of concepts of classical thermodynamics [388, 389, 390, 391, 392]. We add to this that computer codes exist (for example, [393, 394, 395]) for the analysis certain problems by methods of molecular dynamics. Hence the method of molecular dynamics becomes available like to the analytical methods. But it is necessary the understanding of physical problems in the course of computer simulation, as well as in applications of analytical formulas. This makes more strong requirements to qualification of an investigator because a formal usage of methods of computer simulations to some physical problems may lead to contradictions. We analyze from this standpoint computer calculations [396, 397, 398, 399, 400] of nucleation rates for an atomic vapor by the method of molecular dynamics. The troubles in the computer simulation of nucleation processes whose nature is connected with transition from a free state or a state of continuous spectrum in a bound state, is determined by a large statistical weight of states of continuous spectrum. In order to overcome partially this problem in direct calculations of nucleation rates in a supersaturated buffer gas, it is taken a high vapor density and a small size of a cell where a bound state of atoms is formed. But this lead to nonrealistic parameters of a nucleating vapor and additional errors in the computer model. For example, nucleation of germanium in argon was studies in [399], so that a number of germanium atoms was taken 8000, and the ratio of the number of argon atoms to that of germanium atoms ranged from 1 up 5, and these atoms were located in a cell of a linear size from 15 nm up 60 nm. We will not analyze the choice of the interaction potentials between atoms that is problematic, but only consider such calculations from the standpoint of the results obtained. If we use

168

Chapter 9 Growth of clusters and small particles in buffer gas

the model (7.2.1) for this process and take in accordance with this calculation the argon temperature to be 300 K, the cell length to be 20 nm and the ratio of the germanium numbers to argon numbers to be one, we obtain the value G D 14 of the parameter (7.2.2). This allows one to use the model (7.2.1) for this process, and according to formulas (7.2.3) we have a typical time of this cluster growth process cl D 1:5  1010 s, the average cluster size in the process end is n D 3, and the maximal cluster size according to formula (7.2.3) is equal 9. According to the computer simulation [399] the nucleation time is cl D 6:3  1010 s under these conditions, the average cluster size is n D 5, and the maximum cluster size is equal 36. The difference of these results is because of this system is nonuniform due to a high atom number density that is equal for germanium atoms 11021 cm3 at the beginning. This corresponds to initial germanium pressure of 30 atm that cannot be realized under stationary conditions. Thus, the main lack in using methods of molecular dynamics for calculations of nucleation rates consists in a nonrealistic initial pressure of the vapor at which other mechanisms may be responsible for the vapor nucleation process. For example, the initial pressure of a copper vapor in evaluations [400] was 5:2 61 atm, whereas the saturated vapor pressure for copper at the melting point Tm D 1358 K is 3:6  107 atm [315], and the values used in this calculations are not available in reality. In addition, new mechanisms of nucleations become important at high pressure and the heat regime of the nucleation process is of importance. Therefore a transfer of results obtained at high vapor pressures to a range of low pressures may lead to contradictions. This means that direct calculations of the nucleation rates by the method of molecular dynamics cannot be used formally in modeling of the real nucleation processes. This conclusion cannot be considered as a lack of computer simulation in comparison with analytic methods in the analysis of nucleation processes. Computer methods are a fine instrument whose application must be based on a deep understanding of processes to which they are employed. In the same manner we can go to contradiction if we apply analytical formulas outside a range of their validity. But computer simulation of physical processes is a contemporary strong method, and its importance is confirmed by many physical results obtained by this method. Therefore computer methods can give a contribution to study the nucleation processes, but computer methods require the correct use.

9.7

Problems

Problem 9.1. Metal atoms located in a buffer gas are converted in metal clusters according to the scheme (7.2.1). To prove that the contribution of formation of

169

Section 9.7 Problems

diatomics in the lost of metal atoms is small in comparison with attachment of free metal atoms to clusters. According to the scheme (7.2.1) of cluster growth the balance equation for the number density of free metal atoms N has the form Z dN 2 (9.7.1) D KN Na  N ko n2=3 fn d n; dt where Na is the number density of buffer gas atoms, K is the rate constant of three body process [the first process in the scheme (7.2.1)], ko is the reduced constant of cluster processes according to formula (4.2.2), fn is the size distribution function R of clusters that is normalized according to the relation Ncl D fn d n, where Ncl is the number density of clusters, and fn d n is the number density of clusters with a number of bound atoms from n up to nCd n. The first term in the right-hand side of equation (9.7.1) is the variation of the number density of free metal atoms due to collisions with buffer gas atoms which lead to formation of metal diatomics, and the second term describes attachment of free metal atoms to clusters according to the scheme (7.1.1). We note that metal diatomics formed on the first stage of the cluster growth process are subsequently nuclei of condensation for forming clusters, so that the total number density of formed clusters is Z Z Ncl D fn d n D KN 2 Na dt: (9.7.2) Denoting the initial number density of metal atoms to be No , we consider the initial stage of the cluster growth process when the number density of bound atoms in clusters No  N is small compared the number density N of free metal atoms. In accordance with the character of the cluster growth process, the total number of bound metal atoms in clusters of size n is fn d n  dt, where dt is a time interval when diatomic molecules are formed and subsequently they are converted in clusters of size n. Since according to equation (7.1.1) we have dt  d n=n2=3 , the size distribution function of clusters has the form [209]) fn d n D

C ; n2=3

(9.7.3)

and from equations (7.1.1) and (9.7.2) we obtain C D

KNa No ; ko

(9.7.4)

under the assumption that the number density of free metal atoms does not vary during a given time interval. Using formula (9.7.4) in evaluation the second term

170

Chapter 9 Growth of clusters and small particles in buffer gas

Figure 9.3. Formation of a mist in lowlands at a temperature decrease in a night time [401].

of the right-hand side of equation (9.7.1), we have Z N ko n2=3 fn d n D KNo2 Na nmax .t/; where the maximum cluster size at a given time nmax .t/ in accordance with solution of equation (7.1.1) has the form  nmax .t/ D

No ko t 3

3 ;

nmax .t/  1:

From this it follows that the ratio of the first and second terms of the balance equation (9.7.1) is equal by order of magnitude to the maximum cluster size that is attained at this time. Since the maximum cluster size nmax is large, the main contribution to the loss of free metal atoms is given by the process of atom attachment to clusters. Problem 9.2. The air temperature was equal to 15ı C at evening and its humidity was 70%, while at morning the air temperature decreases up to 5ı C, and a part of atmospheric water is converted in mist. Determine a typical aerosol size in a valley bottom through 5 hours after a temperature decrease assuming that a wind is absent at altitudes up to 10 m. Formation of a mist in a humid air at a temperature decrease is a typical situation (see Figure 9.3) and takes place if the air temperature becomes below the temperature at which the partial pressure of an atmospheric water vapor is equal to

171

Section 9.7 Problems

the saturated vapor pressure of water at this temperature. Under given conditions the saturated vapor pressure for a water vapor at the temperature 15ı C equals to 1:706 KPa, and is 0:873 KPa at the temperature 5ı C [315]. Hence, a water part in a condensed phase is 0:28 g=m3 that corresponds to the number density of bound water molecules of Nb D 9:4  1015 cm3 . In addition, according to formula (3.4.2) the Wigner-Seits radius of water molecules in liquid drops is equal rW D 1:92 Å. Assuming a large number density of condensation nuclei in air at the first stage of the nucleation process, we obtain the basic time of growth of water drops as a result of coagulation. Hence the size distribution of drops is independent on the first stage of drop growth. Taking water drops to have an identical size at each time and assuming the mist is conserved during 5 hours, i.e., the velocity of drop falling does not exceed 2m/hour, we obtain on the basis of formula (5.3.4) for the velocity of drop falling that the drop size does not exceed ro D 1:8 μm. Since ro > , where the mean free path of air molecules   0:1 μm, we will be guided by the diffusion regime of coagulation of water drops. Then on the basis of formula (9.3.1) we have for the rate of drop growth in this regime kas D

8T ; 

ro  ;

(9.7.5)

so that at the temperature T D 278 K we have kas D 6:0  1010 cm3 =s. Then formula (9.2.7) gives for an average drop size which they have through t D 5 hours 1 n D Nb kas t D 5  1010 ; 2 that according to formula (3.4.1) corresponds to the drop radius ro D 0:71 μm. Since this value exceeds the mean free path of air molecules in air (0:1 μm), the diffusion regime of drop growth holds true. In addition, this size is less that the critical size (ro D 1:8 μm) at which drops fall onto the Earth. Let us determine the mean free path of light photons l in mist assuming that drops scatter as small balls and the drop radii ro are identical. We have lD

1 ro D ; 3 2 N   ro Nb   rW

(9.7.6)

where N D Nb =n is the number density of drops, and n is a number of molecules in one drop. Under given conditions we obtain l D 3:4 m, and the value l characterizes a distance on which objects are visible in this mist. As is seen, a formed mist resulted from a temperature decrease of a moist air is optically dense. This mist contains small drops and may be scattered both by a wind and by water evaporation as a result of an air heating.

172

Chapter 9 Growth of clusters and small particles in buffer gas

Problem 9.3. Determine the parameters of growth of clusters SiO2 in the region of combustion of SiH4 and methane CH4 in atmospheric air if the concentration of SiH4 in methane is 10% under conditions of the complete burning of methane and total consumption of oxygen. Find the residence time of SiO2 -clusters before departure from the hot flame region where clusters are liquid, if the residence time of clusters in a flame is 0:01–0:1 s. A fuel in the form of methane with an admixture of SiH4 goes in the combustion region in a flow of this gas where the fuel is mixed with air and is burnt. This process at a given concentration of SiH4 ([402]) proceeds at atmospheric pressure according to the scheme 0:1SiH4 C 0:9CH4 C 2O2 C 8N2 ! 0:1SiO2 C 0:9CO2 C 2H2 O C 8N2 : (9.7.7) Taking the temperature of the combustion chamber to be 2000 K that is above the melting point of SiO2 [315], we obtain that liquid clusters or liquid drops grow in the combustion chamber. At atmospheric pressure and this temperature the number density of air molecules is Na D 4  1018 cm3 , and since the WignerSeits radius for SiO2 equals to rW D 2:2 Å, and the concentration of molecules SiO2 under given conditions is cb D 0:01, we deal with the diffusion regime of drop growth if the drop size ro  1 μm since the mean free path of air molecules under given conditions is   1 μm. On contrary, the kinetic regime of drop growth takes place at ro  1 μm. For the diffusion regime of drop association and for the kinetic regime of motion of drops in air we obtain as a result of solving of equation (9.2.6) for drop growth with using formula (9.3.1) for the association rate of two drops that a typical drop size n through a growth time t is r   2T 2 ef cb t 3=4 3 ; ef D (9.7.8) ; ro  ; nD 3 rW m where cb is the concentration of bound atoms S i in air, ef is the reduced rate of drop association that is equal ef D 2:6  1013 s1 under given conditions. Since a typical residence time of molecules in a flame (i.e. a typical time of a gas flow through the combustion chamber) is in a range [402] 0:01–0:1 s, according to formula (9.7.8) a typical drop size in the process end is 45–80 nm. This size corresponds to the kinetic regime of drop motion in air that was used above. Problem 9.4. Mercury clusters grow in argon at the pressure of 1 Torr and temperature of 500 K as a result of coagulation. Find the criterion of validity of the kinetic regime for cluster growth. Determine the average cluster size through 1 s

173

Section 9.7 Problems

and 10 s after the beginning of cluster growth at the concentration of mercury atoms in argon cb D 103 . Under given conditions the number density of argon atoms is Na D 1:9  1016 cm3 , and the parameter cl that is given by formulas (6.1.5) and (9.2.3) is cl D 1:9 Å2 . This gives the criterion of the kinetic regime of cluster coagulation in the form q

n 2=3

 9  109 :

cb

In particular, for the concentration cb D 103 of bound mercury atoms we have the boundary cluster size for transition between the kinetic and diffusion coagulation regimes n D 9  107 . Since the Wigner-Seitz radius for a mercury cluster is rW D 1:80 Å, we have the boundary cluster radius ro D 80 nm. Being guided by the kinetic regime of cluster growth as a result of coagulation, we take the reduced rate constant for mercury clusters ko D 2:3  1011 cm3 =s according to the data of Table 7.1. At the concentration of bound mercury atoms cb D 103 we have on the basis of formula (9.2.12) for the cluster size in the kinetic regime of cluster coagulation n D 9:6  103 through time t D 1 s and n D 1:5  105 through time t D 10 s. Since the Wigner-Seitz for a mercury cluster is rW D 1:80 Å, these cluster sizes correspond to its radii 3:8 nm and 9:6 nm respectively. Problem 9.5. Copper atoms are located in argon under the pressure of 10 Torr, and the concentration of copper atoms in argon is c D 1%. A typical cluster size is n  1  106 . Find the temperature at which the rates of cluster growth due to coagulation and coalescence become equal. We take for copper rW D 1:47 Å according to Table 3.2, so that clusters under consideration has a radius of 15 nm. Next, according to the Table 4.2 data‘ko D 4:2  1011 cm3 =s at T D 1000 K, and the temperature p depenT . Bedence for this rate constant according to formula (4.2.2) is ko  ing guided by the temperature T  1500 K, we have at this temperature for the number density of argon atoms Na and bound copper atoms Nb the values Na D 6:4  1016 cm3 ; Nb D 6:4  1014 cm3 , and ko D 5:1  1011 cm3 =s. Under these conditions the parameter cl of formula (9.2.3) is cl D 2:58 Å, and the criterion (9.2.13) for the kinetic regime of coagulation is n  5  106 , i.e. the criterion (9.2.13) holds true under given conditions, and the kinetic regime of cluster growth as a result of coagulation is realized in this case. Hence, the right

174

Chapter 9 Growth of clusters and small particles in buffer gas

hand side of equation (9.2.12) has form in the case of coagulation dn D 1:9  105 s1 : dt Analyzing the coalescence process, we have the criterion (9.5.2) for the number density of free copper atoms leads to the kinetic regime of Ostwald ripening under the criterion Nm  105 cm3 , and being guided by the kinetic regime of cluster growth, we use formula (9.5.12) for the rate of cluster growth that gives dn D 7:1  109 cm3 =s  Nm : dt From this it follows that the rates of cluster growth due to the kinetic regimes of coagulation and coalescence are equalized at Nm D 2:7  1013 cm3 . We now determine the temperature T of argon at which this number density of copper atoms is attained. The equilibrium number density of free copper atoms is given by formulas (7.1.8), (7.1.11) and from these formulas it follows     2A po "o 2A D : Nm D Nsat .T / exp exp  C 1=3 T T 3n1=3 T 3n T Using the parameters for copper atoms "o D 3:4 eV; A D 2:2 eV; po D 1:5  106 atm according to data in Table of Figure 7.1. From this it follows that the indicated number density of free copper atoms is reached at the temperature about T D 1500 K. Problem 9.6. A droplet of W F6 of a radius 20 μm is inserted in argon at the pressure of 1 atm and temperature in a range 4000–5000 K. In the end W F6 is decomposed in W -clusters and F2 such that the concentration of bound tungsten atoms becomes 1%, and fluorine firstly in the form of atoms and then molecules propagate over all the space and then attach to chamber walls. Determine a typical cluster size if clusters are located in a chamber time t D 0:01 s. This droplet contains the tungsten mass approximately 7  108 g (the specific weight of W F6 is 3:4 g=cm3 [315] or approximately 2  1013 tungsten atoms. In considering the transformation of molecules W F6 into tungsten clusters, we use a simplified model for decomposition of these molecules [403, 404] assuming destruction of molecules W F6 to go subsequently, step by step, in collisions with argon atoms, i.e. according to the scheme W Fk C Ar ! W Fk1 C F C Ar; For simplicity we assume identical the binding energies "F for each bond W Fk  F . Then the rates of decay of molecules W F6 and radicals W Fk .k D 1 5/

175

Section 9.7 Problems

in collisions with argon atoms are independent of k and are determined by the formula  "  F ; d D Na kgas exp  T where Na is the number density of argon atoms, T is a current argon temperature, kgas is the gas-kinetic rate constant for collisions of molecules and radicals with argon atoms, and we assume the rate d to be independent of the number of fluorine atoms k in the molecule or radicals. This simple model [403, 404] allows us to analyze the kinetics of decomposition of molecules W F6 in argon. We now determine the binding energy for one fluorine atom "F in the molecule W F6 with accounting for the Gibbs energy for transformation of the molecule W F6 into metallic tungsten and molecules F2 that is approximately 16:8 eV per molecule [315], the dissociation energy 1:66 eV for fluorine energy and the binding energy 8:4 eV per tungsten atom in bulk tungsten. This gives the binding energy for one fluorine atom "F  4:9 eV in the molecule W F6 . From this we have for a typical time of decay of molecules W F6 in hot argon d D

6 exp."F =T / ; Na kgas

d D 8  103 s at the argon temperature T D 4000 K, and d D 5  104 s at the argon temperature T D 5000 K. Decay of molecules W F6 in collisions with argon atoms in hot argon causes its cooling. Let us consider the stationary regime of W F6 decomposition if an energy for decay of these molecules is taken from surrounding argon as a result of thermal conductivity. Let molecules W F6 and their radicals be located in a spherical region of a radius o and assuming this region to be cooled up to a temperature T at the first stage of this process. The heat flux is equal rT , where  is the thermal conductivity coefficient. Hence the power that is determined by thermal conductivity of argon is equal P D 4 r 2 rT D 4o .To  T /; where r is the distance from the region occupied by detaching molecules, o is a radius of the region occupied by these molecules. On the first stage of decomposition of molecules W F6 their decay requires the power P D "F d NW 

4o3 ; 3

176

Chapter 9 Growth of clusters and small particles in buffer gas

where NW is the total number density of tungsten atoms in molecules W F6 . Equalizing this heat absorption to heat release as a result of argon thermal conductivity, we obtain the difference of temperatures for the region occupied by molecules W F6 and surrounding argon To  T  D

"F Na kgas NW o2 : 3 exp."F =T /

(9.7.9)

Under given conditions (o D 20 μm, cW D NW =Na D 0:01, p D 1 atm) at the temperature T D 5000 K (Na D 1:5  1018 cm3 , kgas D 6:5  1010 cm3 =s;  D 1:2  103 W=.cm  K/) we obtain To  T  0:1 K. This estimate means that thermal conductivity is a fast process under given conditions. As a result of this process the temperatures inside the reaction zone and outside it are equalized. We now determine a typical cluster size through t D 0:01 s at T D 5000 K. We use that cluster growth is determined by coagulation, and its rate is given by formula (9.3.5). Indeed, according to formula (9.2.3) we have cl D 1:62  1016 cm2 in this case, and the criterion (9.3.5) gives n  106 . Hence formula (9.3.5) gives for the average cluster size n D 1:3  107 that corresponds to the cluster radius ro D 38 nm (rW D 1:61 Å for tungsten). Next, the gas-kinetic cross subsection of collision between argon and tungsten atom at this temperature is g D 3  1015 cm2 , and the mean free path of tungsten atoms is   2 μm, i.e. the criterion   ro is fulfilled for the case (9.3.5). Note that the cluster growth process under given conditions goes simultaneously with detachment of fluorine atoms from tungsten atoms. Hence, clusters resulted from this process may contain fluorine atoms along with tungsten atoms. Let us determine the degree of expansion of the cluster region on the basis of formula for its radius r dr 2 D 6Ddt; where D is the cluster diffusion coefficient that is given by formula (5.3.2) D D Do =n2=3 , and under given conditions we have Do D 12 cm2 =s. Next, according to formulas (9.3.4) and (9.3.5) we obtain 3n1=3 d n No dt D ; 4=3 N to W 4no where to D 0:01 s is a time of cluster location in an aggregation chamber, i.e. it is a time of cluster growth, no D 107 is the final average cluster size, NW is a current number density of bound tungsten atoms, and No is its initial value. Because NW D No o3 =r 3 , where o is the initial radius of the cluster region, we

177

Section 9.7 Problems

find for the final radius r of the cluster region r5 D

30Do to o3 2=3

:

no

Using the above values of the parameters of this formula (Do D 12 cm2 =s; to D 0:01 s; o D 20 μm, and no D 107 ), we obtain for the cluster region radius in the end of the cluster growth process r  40 μm, that is comparable with the drop radius o given. Problem 9.7. MoF6 is inserted in an afterglow arc argon plasma at the temperature of 3200 K and pressure of 1 atm. At the beginning inserted molecules occupy a cylinder region in a plasma flow of a radius 100 μm and the concentration of Mo atoms is 0:1% on the first stage of the nucleation process. The number density of electrons and ions is 108 cm3 firstly. An afterglow plasma is moving in a cylinder tube of a radius 2 cm and length of 1m, and there is an orifice on the tube axis and 6 orifices on the tube periphery of a radius 0:3 mm. Analyze the character of cluster growth with accounting for the charging process, if the temperature is 1000 K at the tube end. Determine the concentration of Mo atoms after ejection near the central orifice. This Problem is analogous to the previous one, but the region occupied by clusters has another geometry that is connected with the gas flow. We first analyze gas dynamics of the argon flow. The number density of argon atoms at the exit is Na D 7:3  1018 cm3 and the velocity of the output flow according to formula (6.2.15) is c D 4:3  104 cm=s that corresponds to the flow intensity J D 8:6  1021 s1 . From this we determine the velocity of the argon flow at the hot flow part as v D 100 cm=s and near the exit is equal approximately 30 cm=s, and the total time of molecule passage through a tube is equal about 0:02 s. This corresponds to the total flow of Mo atoms 2:2  1015 s1 or the flowing molybdenum mass per unit time 0:35 μg=s. In analyzing the kinetics of the process of MoF6 transformation in Mo-clusters, we use the scheme of the chemical process given in the previous Problem. Then we have the following parameters "F D 4:3 eV, "Mo D 6:3 eV, and a typical time of decay of molecules MoF6 in hot argon is given by d D

6 exp."F =T / : Na kgas

This formula gives d D 0:009s at the argon temperature T D 3200 K. If we assume that simultaneously with detachment the fluorine atoms from molecules and

178

Chapter 9 Growth of clusters and small particles in buffer gas

radicals MoFk , there is an additional channel for destruction of these molecules according to the scheme MoFk C MoFp ! Mo2 FkCp1 C F: Therefore a typical time of transformation of MoF6 molecules in Mo-clusters would be lower. In order to analyze the heat balance for a region where molecules MoF6 are destroyed, we take into account that the heat absorption proceeds in a cylinder region due to detachment of fluorine atoms from molecules and radicals, and the power of this process per unit length is P D "F d NMo  o2 ; where o is the radius of a cylinder region that is occupied by molybdenum atoms, the rate of fluorine atom attachment is d D 1=.6d ), and the number density of molybdenum atoms is NMo D Na cb . Heat release results from thermal conductivity of argon, and its power per unit length of the tube is Prel D 2rT D

2T ; ln.R=o /

where T is the temperature difference in the tube cross section, and r is the tube radius. Equalizing these powers, we obtain for the temperature difference instead of formula (9.7.9) for the spherical geometry T D

"F Na kgas NMo o2 ln.R=o / : 2 exp."F =T /

(9.7.10)

Under the conditions of this Problem at the temperature T D 3200 K (o D 0:1 mm, R  1 cm, cb D NMo =Na D 0:001, Na D 2:4  1018 cm3 , kgas D 5:6  1010 cm3 =s;  D 9:3  104 W=.cm  K/) we obtain T  0:1 K, i.e. heat release proceeds fast, and the temperatures are identical in regions with the chemical reaction and without it. In considering the cluster growth process in a neutral gas, we neglect cluster evaporation during the cluster growth process. In order to give the criterion (9.2.3) for the coagulation mechanism of this process, we use the value of the WignerSeits radius rW D 1:60 Å according to Table 3.2 and evaluate the parameter cl D 5:3 Å, so that the criterion (9.2.3) has the form n  104 . This criterion corresponds to the diffusion character of cluster approach. On the other hand, we have the mean free path for argon atoms at the temperature 3200 K  D 13 μm, and the cluster diffusion coefficient in argon is evaluated in the kinetic regime if

Section 9.7 Problems

179

n  1013 . In this case a number of atoms in one cluster at the end of the process is given by formula (9.3.5), and taking a time of cluster growth to be 0:02 s, we obtain on the basis of this formula n D 1:4  106 , and this cluster size justifies the regime chosen. This corresponds to the cluster radius ro D 18 nm. In addition, the number density of clusters near the exit with the argon temperature T D 1000 K is Ncl D 5:3  109 cm3 . We now evaluate expansion of the region occupied by clusters due to their diffusion in argon. The diffusion coefficient is given by formula (5.3.2), and in the kinetic regime of cluster motion it depends on a cluster size as D  n2=3 , and since according to formula (9.3.5) the average cluster size grows in time as n  t 3=4 , we have variation of the cluster diffusion coefficient in time as D  t 1=2R . Taking the expansion of the cluster region in the radial direction as x 2 D 2 Ddt , we have from this Z x 2 D 2 Ddt D 4D./; where  D 0:02s is the residence time for atoms and clusters inside the tube. Under the above parameters we have according to formula (5.3.2) Do D 11 cm2 =s and D./ D 9  104 cm2 =s, so that x  40 μm that though is less than the initial size of the cluster region, but is comparable with it. Problem 9.8. Under parameters of the previous Problem analyze processes of cluster charging in an afterglow plasma where the number density of electrons and ions is 108 cm3 . We first find a time of recombination of electrons and ions as a result of their departure to walls. In the simple Schottky case of an ionization equilibrium [405] this time is equal R2 =.5:78Da /, where R is the tube radius, Da is the ambipolar diffusion coefficient. In this case where the temperatures of electrons and ions are identical, we have Da D 2Di , where Di is the ion diffusion coefficient. Assuming that ArC is the basic ion sort, we have the mobility [208] Ki D 1:0 cm2 =.Vs/ for ions ArC in argon at T D 1000 K under the normal number density of atoms that corresponds to the diffusion coefficient Di D 0:087 cm2 . At the tube end we have Di  1 cm2 =s and Da  2 cm2 =s, and this gives a time of plasma departure on walls ion D 0:3s that exceeds by the order of magnitude the residence time of this plasma in the tube. Therefore in this case departure of electrons and ions to walls does not influence the ionization equilibrium inside the tube. We also analyze the similar channel of a plasma loss due to attachment to clusters assuming the cluster region is a cylinder of a radius o . If electrons and ions penetrate in the cluster region, they perish. The plasma flux to this region

180

Chapter 9 Growth of clusters and small particles in buffer gas

is j D Da rNe , and the plasma is quasineutral, i.e. the number densities of electrons Ne and ions are identical. The plasma flux per unit length of the tube is J D 2Da rNe ; where  is a distance from the tube axis. Considering this as an equation for the number density of electrons and ions and solving it with the boundary condition Ne .o / D 0, we obtain Ne ./ D

No ln.=o / ; ln.R=o /

J D

2Da No ; ln.R=o /

(9.7.11)

where No is the number density of electrons and ions in the absence of the cluster region, R is a tube radius, and R  o . Because n D R2 No electrons and ions relates to tube unit length, from this we find a time of plasma destruction due to departure to the cluster region D

n R2 ln.R=o / : D J 2Da

As is seen, this lifetime is large compared to that due to attachment of electrons and ions to walls and is under given conditions   5s, i.e. this channel may be neglected. In analyzing the ionization equilibrium of clusters with a surrounding ionized gas, we evaluate the cluster charge on the basis of formula (8.4.8) in the final tube part, that gives x D 4:2. From this on the basis of the parameters given (T D 1000 K; ro D 18 nm) we obtain for the average cluster charge ZD

ro T x D 3:3: e2

From this it follows for the ratio of the interaction potential of two clusters at their contact to a thermal energy

D

Z 2e2 D 5; 2ro T

exp. / D 6  103 :

As is seen, charging of clusters influences now the cluster growth process. One can evaluate the cluster size in the end of the process for the limiting case of a large charge. Because the rate constant of association (9.3.4) under Problem parameters is equal kas ro D 0:14 cm2 =s, formula (9.3.6) gives for the cluster radius near the final tube part ro D 4:5 nm. This value may be considered as a lower boundary for the cluster radius. Indeed, in this case the average cluster size

181

Section 9.7 Problems

is equal Z D 0:8, and the assumption is fulfilled in deduction of formula (9.3.6) that the cluster charge is large and is connected unambiguously with the cluster radius. We will be guided by an average cluster radius between two limiting cases, that is ro  10 nm, and this size corresponds to the average cluster charge Z D 2:5. Hence, a cluster contains in average 2:4  105 atoms, and the number density of bound atoms in a cold region Nb D 7:3  1015 cm3 corresponds to the number density of clusters Ncl D 3  1010 cm3 . Thus we obtain that the average charge density of clusters ZNcl D 8  1010 cm3 that exceeds significantly the number density of charged particles in a surrounding plasma. From this it follows that plasma transport from a region that is free from clusters to the cluster region is of importance for cluster charging now. In this consideration we find the penetration length ı in the cluster region for plasma ions on the basis of formula p p ı D 2Da t D 4Di t ; and we account for the relation between the coefficient of ambipolar diffusion Da and ion diffusion Di if the electron and ion temperatures are identical. We have an intermediate case for the rate of ion absorption by the cluster because a size of the region of the cluster field is comparable to the mean free path of ions. We use formula (8.1.11) for the rate of ion attachment to a cluster that gives 1 D 4ZeKi Ncl t for a time of absorption of an atomic ion. Using the Einstein relation (5.3.1), we then obtain for the penetration length s T ı ; (9.7.12) Ze 2 Ncl and under conditions of the Problem this formula gives ı D 25 μm. Thus, the basic part of cluster region under given conditions contains neutral clusters. We now find the depth of plasma penetration inside the cluster region from other consideration, namely, we compare the plasma flux from a cluster free region to the cluster region and the flux of ion attachment to clusters. The first one is given by formula (9.7.11) and is equal J D

2Da No : ln.R=o /

182

Chapter 9 Growth of clusters and small particles in buffer gas

Taking the penetration depth ı for attachment of electrons and ions to clusters and using as above the Langevin formula (8.1.11) for the rate of ion attachment to an individual cluster, we obtain for the attachment rate Jat per unit tube length Jat D 2o ı  4ZeKi Ncl Ni ; where Ni is the number density of electrons and ions in the region where their attachment to clusters occurs. Equalizing these rates, we obtain the depth of the cluster region where attachment of electrons and ions to clusters occurs ıD

T No 2 2o e ZNcl Ni

ln.R=o /

:

(9.7.13)

In order to use this formula we need in the electron and ion number density Ni inside the cluster region. One can use this relation in another manner by equalizing the penetration depths according to formulas (9.7.12) and (9.7.13). Then we find the ratio of electron and ion number densities in the upper layer of the cluster region and outside it s T Ni 1 D : No 2o ln.R=o / Ze 2 Ncl Assuming that cluster layers are not mixed, we obtain that in upper layer small charged clusters are located (Z D 2:5; Ncl D 3  1010 cm3 ) and we find from this formula Ni =No  0:02. This analysis shows that cluster charging process have a complex character in a cluster plasma. In particular, for conditions under consideration, where clusters are located in a narrow cylinder region of an ionized gas flow inside a cylinder tube, only clusters from an upper layer of the cluster region are charged. This influence on cluster growth as a result of their joining, and hence these clusters have a lower size than neutral clusters in a basic part of the cluster region. Problem 9.9. Growth of aerosol particles in a cumulus cloud results in their coagulation in the diffusion regime. Estimate the rate constant of association of charged water aerosols and the character of their growth being guided by average aerosol parameters (see Problem 4.1), namely, the average aerosol radius is ro D 8 μm, and the number density of aerosol particles is Np D 1  103 cm3 . Using formula (9.4.1) for the rate constant of particle association due to coagulation in the gravitation field and taking into account repulsion of particles of charge Z during their contact, we have for the association rate constant   2ro4 g Z 2e2 ; exp  kas D  2ro T

183

Section 9.7 Problems

and the particle charge is given by formula (8.9.1) and is equal for this particle charge Z D 12. Note that the factor under the exponent is determined by an order of magnitude and hence the subsequent results may be considered as estimates. On the basis of the above formula we have the rate constant kas D 9:2  107 cm3 =s of particle association at ro D 8 μm, and the exponent in this expression is 2 2 expŒZ e =.2ro T / D 0:6. This gives for a typical time of doubling of the molecule number constituted the aerosol according to formula (9.2.7) 

2  40 min: kas Np

When the number of aerosol molecules is doubled and the aerosol radius becomes ro D 10 μm, the rate constant of aerosol association is kas D 7:4107 cm3 =s, and the exponential factor equals expŒZ 2 e 2 =.2ro T / D 0:2, that gives a typical time  for doubling of the number of aerosol molecules is   70 min. The subsequent doubling of the number of aerosol molecules leads to the following parameters kas D 5:6  108 cm3 =s, expŒZ 2 e 2 =.2ro T / D 0:006, and   40 hours. According to these calculations, aerosol growth is stopped on a certain stage of this process. We determine above that the rate of aerosol growth finishes at ro  10 μm. But this aerosol size may be shifted because formula (8.9.1) gives an estimation. Next, we did not take into account convective transport of air in the atmosphere, so that mixing of aerosol particles with a flux of warm wet air accelerates the aerosol growth process. This stage of the growth process is represented in Figure 9.4. Let us now consider this process from another standpoint. We assume that formula (8.9.1) is correct, but it relates to a lower aerosol size, and the lifetime of aerosols with a radius of 8 μm is 2 hours. Then the primary aerosol particle has a radius of 6:3 μm and obtain the charge Z D 11 in collision with an ice particle. When an aerosol particle acquires a radius of 8 μm, it has a charge Z D 22. Next, when its radius becomes ro D 10 μm, i.e. a number of molecules in an aerosol particle is doubled compared with the radius ro D 8 μm, its charge is Z D 44, and the subsequent doubling of a number of aerosol molecules proceeds through 49 hours. One can see that growth of aerosol particles is stopped practically. Nevertheless, we repeat an estimated character of formula (8.9.1) that is a basis of the above analysis. In this case aerosol particles partaken in charge transfer includes approximately 12% of atmospheric water. Problem 9.10. Assuming that aerosols of a radius 10 μm, of a charge Z D 40 and of the number density of N D 1  103 cm3 constitute a cumulus cloud (see previous Problem), find the velocity of propagation of charged aerosols on a

184

Chapter 9 Growth of clusters and small particles in buffer gas

Figure 9.4. Cumulus cloud on the last stage of its existence [406].

lower edge of the cloud. Take the parameters of a prethunderstorm weather with an electric potential of a lower cloud edge has a high with respect to the Earth surface that is 20–100 MV [122], and the altitude of L D 1 km over the Earth surface for the lower cloud edge. We assume the cloud width to be large compared to its altitude over the Earth surface. Take for definiteness the electric voltage between the lower cloud edge and the Earth surface to be U D 50 MV and in the gap between the Earth surface and cloud charged particles are absent. Then the electric field strength in this gap is E D 500 V=cm, and this electric field is created by charged aerosols. Assuming the cloud layer is think compared to the gap width, we find the charge density  of the cloud is e E ;  3  108 D 4 cm2 and because the number density of aerosol particles is 1  103 cm3 and the mean aerosol charge is 44, we find the thickness of a layer occupied by aerosols l  60 m. This estimation shows that the charged layer is less than the cloud height.

185

Section 9.7 Problems

Figure 9.5. Sant Elmo’s fires on the endings of ship masts [407].

We now determine the drift velocity of aerosols under the action of the electric field of a lower cloud edge of the strength E D 500 V=cm. According to formula (5.1.7) we have for the drift velocity of aerosol particles w under the action of the cloud electric field wD

eEZ D 0:01 cm=s; 6 ro 

where ro is a radius of aerosol particles, Z is their average charge, and we use given parameters of aerosol particles. This value is small compared to the velocity of aerosol falling under the action of the gravitation force w  1 cm=s. Note that convection fluxes in atmospheric air influence stronger on displacement of neighboring cloud layers than those under the action of the cloud electric field and gravitation force.

186

Chapter 9 Growth of clusters and small particles in buffer gas

Note that if charged aerosols propagate to the Earth surface and occupy a space near it, a high electric field is created in the atmosphere near the Earth. This field may cause corona discharge near stretched objects that is accompanied by a weak glowing of air near these objects. This phenomenon is known during thousands years under different names. Because it was observed often by sailors, last centuries this phenomenon was named as Sant Elmo’s fires (the patron saint of sailors), and an example of this phenomenon is given in Figure 9.5. As is seen, the character of electric phenomena in the Earth atmosphere is more complex than this is given in Figure 2.6. Nevertheless, according to this Figure as a basis we consider electric phenomena in the atmosphere as a secondary process of water circulation in the atmosphere. Moreover, charging of the Earth surface results from electric discharges-lightnings, and these discharges proceed through the falling of charged water aerosols in clouds, and the typical size of these aerosols is 10 μm. According to the above estimations, under optimal conditions charging of the Earth through this mechanism uses 15–20% of evaporated water. But along with this scheme of electric processes in the atmosphere, other processes may accompany them. In particular, fluxes of warm wet air are a source of water molecules which attach to water aerosols and increase their size. Forming aerosols-drops acquires a heightened velocity of aerosol falling. Second, a part of charged aerosols does not partake in creation lightnings. These aerosols fall on the Earth surface quiet and transfer their charge to the Earth surface. Hence, the electric processes in the Earth atmosphere are reacher than that followed from a simple scheme.

Chapter 10

Structures formed in aggregation of solid particles 10.1

Fractal aggregates

Let us consider a spread physical situation when liquid clusters are formed in a hot vapor and then they move in a cold space region where they become solid. This takes place if clusters are formed in flames and plasmas. We now study the stage of cluster growth when they are solid and join with each other as a result of contacts. Then solid clusters or particles conserve their shape to a great extent and their joining leads to formation some structures which are the object of our analysis. In this study we use the model for solid clusters or particles, so that they are spherical particles of an identical radius ro , and the area of their contact in forming structures is relatively small. Therefore the results of this analysis describes real systems of this kind qualitatively. It is of importance in this consideration the assumption that two contacted particles form a strong chemical bond. In reality, this corresponds to a certain stage of the chemical processes when surface atoms, molecules or radicals of contacted particles are chemically active. In particular, in the case when clusters of metal oxides are formed from a metal vapor located in a hot buffer gas this means that the oxidation process proceeds also during contacts of joining particles. Thus, under considering conditions, association of two solid particles results in formation of a chemical bound may occur in points of their contact. Association of a group of solid particles leads to formation some structures where incident particles conserve mostly their initial shape. Because in physics these structures are named aggregates, we will call this process as the aggregation process (see Figure 9.1 d)) though it can have other names in other sciences. Let us analyze the growth of aggregates in gases and plasmas. If the aggregate contains a small number of large particles and it is formed in an external electric field, it has an elongated shape as it takes place for an example given in Figure 10.1. Usually such elongated aggregates, chain aggregates, are formed in a plasma or in gases located in an external field. If aggregation of small solid particles proceeds in absence of external fields, i.e. in an isotropic medium, these particles associate with transformation of their contact in a chemical bond, and an area of the particle contact is small compared with area of particle surfaces. As a result, these particles form fractal aggregates

188

Chapter 10 Structures formed in aggregation of solid particles

Figure 10.1. Chain aggregate formed in a plasma with magnesium [408].

in various media including gases [409]. As an example, Figure 10.2 represents a fractal aggregate obtained in a solution. The convenience of solutions for this goal consists in the possibility to obtain the particles of nearby sizes by a change of the solution acidity. In this case the charge of particles in the solution depends on its acidity, and because growth of these particles results from attachment of atomic ions to particles, one can govern the particle size by the solution acidity. This allows one to obtain almost identical radius of these particle (the ratio of the radius fluctuation to the particle radius is approximately 5%). In particular, in the case of Figure 10.2, spherical gold particles of 14:5 nm in diameter (each particle contains approximately 105 gold atoms) were produced from Na.AuCl4 / that is located in the solution of tri-sodium citrate. Citrate ions absorbed by the a growing gold particle prevent the particles from attachment the subsequent atomic ions of gold to them, and the above particle size is determined by a solution acidity. Because attachment of citrate ions to gold particles depends on a particle size, in the course of growth of gold particles from Na.AuCl4 / all formed gold particles have almost identical size, and a gold excess in the solution is consumed on growth of new particles. When all Na.AuCl4 / molecules are converted in gold particles of a given size, a small amount of pyridine is added to the solution that removes citrate ions from the particle surface, and gold particles can associate with each other. As a result, a fractal aggregate is formed that is represented in Figure 10.2. Formation of fractal aggregates has an universal character, and Figure 10.3 gives an example of formation of fractal aggregates in the living world. Figure 10.3 contains the electrocyte structure [411] that is formed

189

Section 10.1 Fractal aggregates

0.5 —m

Figure 10.2. Fractal aggregate consisting of gold particles and is formed in a colloid solution. [410]

in the course of growth of clots in blood. This is the coagulation process in a physiological solution was studied essential earlier than other similar processes. We will be guided by fractal aggregates formed in gases and plasmas, and Figure 10.4 gives an example of a fractal aggregate that is formed as a result of iron evaporation [412]. This study opened investigation of fractal aggregates formed in gases and plasmas, and also given the method of the analysis for formed structures. This study was a basis of the Witten-Sander model [409], which took research of fractal aggregates to a new level of investigation and led to comprehensive computer simulations of the aggregate growth processes. We first consider experimental methods of generation and study of fractal aggregates. The above method consists in evaporation of metals in inert gases with subsequent collections of an evaporating soot-like deposit (for example, [413, 414, 415, 416]). This deposit is analyzed by Transmission Electron Microscope. In this case metal particles are formed on the first stage of the process as a result of strong heating of a metal-containing surface. On the second stage, these particles are joined in fractal aggregates. A similar method may be based on evaporation of metals by power pulse laser. In the region of a laser torch a dense atomic metal vapor is formed as a result of metal vaporization. This metal vapor is converted in liquid particles (clusters) of nanometer sizes in a region of a high temperature, and when they occurs in a more cold regions, the particles become solid and are joined in fractal aggregates. Because of a high power, a size of elemental metal particles (we will call them monomers) in this case is higher than that at thermal

190

Chapter 10 Structures formed in aggregation of solid particles

Figure 10.3. Structure of electrocytes formed in a blood as a result of aggregation of red cells [411].

evaporation of atoms and is approximately 20 nm [417]. Another method of generation of fractal aggregates in gases consists in burning of silicon tetrachloride in a hydrogen or hydrogen-oxygen flame (for example, [418, 419, 420]). The same process is possible as a result of burning of silicon tetramethane S iH4 in air or oxygen. Various mechanisms of growth of fractal aggregates from small hard particles are possible [421, 422, 423, 434, 425, 426]. We will be guided by cluster-cluster aggregation mechanism [427, 428], where the structure growth results from subsequent joining of solid particles in small clusters and then small clusters are joined in large clusters as it is shown in Figure 10.5 [428]. Moreover, within the framework of the cluster-cluster mechanism of aggregation (CCA) we assume that the probability of sticking of two solid particles as a result of their contact is of order of one. In addition, the aggregation process proceeds in a buffer gas, and we consider the diffusion regime of approach of associated particles. All these conditions lead to certain properties of the aggregates formed, and below we study the properties of such aggregates. Usually it is used the assumption in modeling of these structures that the aggregate consists of identical monomers [421, 422, 423, 434, 425, 426]. The formed aggregate has fractal properties [429, 430], and therefore it is called the fractal aggregate. Thus the fractal aggregate is a friable structure whose density drops as the aggregate size increases. Chemical bonds connect nearest particles, but the total

191

Section 10.1 Fractal aggregates

Figure 10.4. Fractal aggregate of iron formed in iron evaporation and method of their analysis after collection on a grid. [412]

structure is random. Nevertheless, the fractal aggregate is characterized by a latent parameter which is called the fractal dimension D of the structure [429, 430]. The fractal dimension may be defined in a different manner. In the first case it characterizes the correlation between positions of individual particles. Indeed, let us define the correlation function as C.r/ D

1X h.r C r0 /.r0 /i .ri /.ri C r/ D ; n h.r0 /i i

where n is a number of a points, i is a point number, and  is the density that equals to one if a point is occupied and is equal to zero if a point is free. In addition, a test point is occupied, and an averaging is made over many test points. The dependence of the correlation function on a distance r from a test particle is C.r/ D

const ; r d D˛

(10.1.1)

where d D 3 is the space dimensionality, and this is the first definition of fractal structures. Another definition on the basis the same concept we use for the mass of particles M that is contained in a sphere of a radius R and is given by the following

192

Chapter 10 Structures formed in aggregation of solid particles

Figure 10.5. Character of the structure growth for the cluster-cluster mechanism of aggregation.

Table 10.1. Fractal dimensions of fractal aggregates formed as a result of evaporation of atoms or molecules from a surface with the subsequent association of atoms and molecules in particles and then in fractal aggregates when particles become solid [412]. The mean fractal dimension is 1:60 ˙ 0:07.

Material Fe Fe Zn Zn SiO2 D˛ 1:69 ˙ 0:02 1:68 ˙ 0:02 1:67 ˙ 0:02 1:68 ˙ 0:02 1:55 ˙ 0:02 Dˇ 1:52 ˙ 0:04 1:56 ˙ 0:02 1:50 ˙ 0:04 1:60 ˙ 0:04 1:56 ˙ 0:06 formula, if the mass of individual particle is mo  M D mo

R ro

Dˇ ;

(10.1.2)

where ro is the radius of an elemental particle (monomer). This gives for a number of particles n located inside the sphere of a radius R  nD

R ro

Dˇ :

(10.1.3)

If we model a fractal aggregate in average as a spherical particle, formula (10.1.3) gives a number of particles inside a fractal aggregate of a given size. We note that the fractal dimension of fractal aggregates as a latent parameter of a random system may be determined with a certain accuracy that is limited even if the statistics in choice of averaged objects trends to infinity. The accuracy indicated in Table 10.1 is determined by averaging the results for different objects. Next, in the case of the above fractal aggregate which results from cluster-cluster

193

Section 10.1 Fractal aggregates ln n 8

6

4

2

ln R 0

1

2

3

4

5

Figure 10.6. Dependence of a number of gold particles inside the sphere of a radius R for the fractal aggregate of Figure 10.2. The straight line corresponds to formula (10.1.3) and Dˇ D 1:75.

aggregation in the diffusion regime of motion of associating particles in a buffer gas the fractal dimension is less than 2. Then this object is transparent, and the projection of this object on a plane has in the two-dimensional space the same fractal dimension as this structure in the three-dimensional space. This is fulfilled for fractal aggregates given in Figure 10.2 and Figure 10.4, and the indicated fact simplifies treatment of photographies of the fractal aggregates, because one can use circles on the plane of photography instead of a sphere with a given occupied point in its center for determination both the correlation function and the occupation density inside the sphere. Moreover, one can use other figures for determination the fractal dimension Dˇ , as rectangles used in this operation in Figure 10.4. We also demonstrate in Figure 10.6 the method of determination of the fractal dimension Dˇ which is made for the fractal aggregate of Figure 10.2. Figure 10.6 uses the analysis of 100 aggregates located on a grid of 1 μm in size, and measurements [410, 431, 432, 433] gives Dˇ D 1:77 ˙ 0:1. Computer simulation is a convenient method to analyze the properties of fractal aggregates with identical monomers as the basis of these aggregates under assumption that the probability of joining of monomers or fractal aggregates is one after their contact. On the basis of averaging of the results of computer simulations [435, 436, 437, 438, 439, 440] for the diffusion regime of cluster association gives for the fractal dimension of fractal aggregates D D 1:77 ˙ 0:03. This is in accordance with the fractal dimension of gold fractal aggregates according to Figure 10.6 that is D D 1:75 ˙ 0:10. A rarefied structure of the fractal aggregate follows from the character of their formation because of periphery monomers screen internal monomers of a growing fractal aggregate from association with other fractal aggregates and hence do

194

Chapter 10 Structures formed in aggregation of solid particles

not allow to penetrate them deeply inside a test fractal aggregate. This effect is stronger for the diffusion regime compared to the kinetic one because in the kinetic regime aggregates may penetrate deeper each in other. Statistics of computer simulation for formation of fractal aggregates in the kinetic regime of their growth [435, 436, 437, 438, 439, 440], where joining fractal aggregates move along straightforward trajectories, gives the average value of the fractal dimension D D 1:93 ˙ 0:06. As a fractal aggregate increases, its density drops, and a large fractal aggregate is unstable [441]. This instability can result from thermal oscillations of a fractal aggregate and under the action of gravitation forces [441]. The experience exhibits that in reality the maximum size of observed fractal aggregates is R D 1–10 μm, and the maximum number of monomers in an aggregate is equal 103 –104 . In consideration fractal aggregates resulted from association of small fractal aggregates in the diffusion regime, we assumed the probability of sticking of aggregates at their contact to be one. If the sticking probability of association of two fractal aggregates at their contact is small, a formed structure is denser, and this mechanism is called the reaction limited cluster aggregation (RLCA). Computer simulation under these conditions [442, 443, 444] leads to the fractal dimension of the forming fractal aggregate of D D 2:02 ˙ 0:06. This process was studied in details in colloid solutions where under conditions of RLCA-mechanism of aggregation the fractal dimension of forming fractal aggregates averaged over measurements [432, 445, 446, 447, 448] is 2:05 ˙ 0:05. In this case the rate of growth of fractal aggregates may be operated by a salinity of the solution, and if this salinity is high, so that the aggregation process is hampered and lasts hours and days. This process corresponds to the RLCA-mechanism of growth of fractal aggregates. Moreover, according to study [448] in a fresh solution where fractal aggregates are formed during some minutes, the fractal dimension of a formed fractal aggregate is approximately 1:75, that corresponds to the CCA-mechanism. If this solution is conserved during several days, fractal aggregates densify, and their fractal dimension exceeds 2. Evidently, an identical process of structure densification proceeds when fractal aggregates are located in gases. This process is accompanied by an increase of a number of chemical bonds in these aggregates and leads to their restructuring.

10.2

Growth of fractal aggregates

In considering the growth of fractal aggregates as a result of their association, we will be based on the above models where a fractal aggregate is considered as a spherical particle. Then a contact of two fractal aggregates during their motion in

Section 10.2 Growth of fractal aggregates

195

a buffer gas leads to joining of these aggregates, and a forming fractal aggregate is characterized by the same fractal dimension, and we will model this aggregate by a spherical particle also. This corresponds to the assumption that a fractal aggregate that is formed after association of two fractal aggregates, changes their shape and becomes a spherical particle to a greater or lesser extent. This assumption holds true for growth of compact liquid drops, where after each event of association of two drops as a result of their contact, a formed drop becomes spherical under the action of the surface tension. This restructuring does not proceed in the case of growth of fractal aggregates, so that using the liquid drop model for fractal aggregates leads to a lower rate of aggregate growth since the cross section of aggregate association at a given mass of colliding aggregates has a minimum at their spherical shape. Thus, modeling of fractal aggregates by spherical particles, we use formulas of the liquid drop model with additional formula (10.1.3) for dependence of a number of monomers in a fractal aggregate on its radius. Then formula (4.2.3) takes the form s   2 n1 C n2 1=2  1=D 8T 1=D 2 k.n1 ; n2 / D n1 C n2 ;  .r1 C r2 / D ko  n1 n2 s 8T  2 r ; (10.2.1) ko D mo o where mo is the mass of an elemental particle, ro is its radius, D is the aggregate fractal dimension, and we assume all the monomers to be identical. Introducing the average number of monomers for two colliding aggregates n D .n1 C n2 /=2 and taking for CCA-regime of aggregate growth the fractal dimension to be D D 1:77, we obtain from this k.n1 ; n2 / D C ko n0:63 :

(10.2.2)

Indeed, the parameter C.n1 ; n2 / is conserved at permutation n1 $ n2 and has a minimum at n1 D n2 , where it is C D 5:66. This depends weakly on the ratio n1 =n2 and at n1 =n2 D 2 or at n1 =n2 D 1=2 we have C D 5:83. We take below C D 5:7. Then the equation (9.2.6) takes the following form dn D 5:7ko Np n0:63 ; dt where Np is the number density of elemental particles (monomers) that assumes to be independent of fractal aggregate sizes. The solution of this equation has the following form instead of formula (9.2.12) n D 7:4.ko Nb t /2:7 ;

(10.2.3)

196

Chapter 10 Structures formed in aggregation of solid particles

and this formula holds true for times ko Nb t  1. This means that the first stage of the growth process with formation of solid particles which is subsequently a basis of fractal aggregates, proceeds fast, whereas a time of growth of fractal aggregates is determined by the last stage of the growth process. We now give the criterion of validity of the kinetic regime for growth of fractal aggregates which was considered above. To obtain this criterion, it is necessary to rewrite formulas for relaxation of clusters to relaxation of fractal aggregates in a gas. We have that the relaxation time of a fractal aggregate of a radius R in the kinetic regime is equal according to formula (6.1.2)  D2 R 3mo rel D r  ; r D p ; ro 8 2 mT Na ro2 From this we obtain for the mean free path of a fractal aggregate instead of formula (6.1.4)  D=22 p 3 mo R ; ƒkin D : (10.2.4) ƒ D ƒkin p ro 4 mNa ro2 Next, the criterion of validity of the kinetic regime of aggregate approach in the growth of fractal aggregates according to formula (9.2.2) has the form Nag ƒ3  1 that gives  D=26 R 3  1; (10.2.5) Np ƒkin  ro where Nag is the number density of fractal aggregates, Np is the number density of elemental particles-monomers. In considering the diffusion regime of growth of fractal aggregates, we first determine the diffusion coefficient of fractal aggregates in a gas. Then we model the fractal aggregate by a spherical particle and assume its hydrodynamic radius to be equal to its geometric radius. Under these assumptions we have on the basis of formula (5.3.2) for the diffusion coefficient of motion of a fractal aggregate in a gas in the kinetic regime p p 3 T Dp ro2 3 T Dkin D p D ; Dp D p ;   R: (10.2.6) R2 8 2 mNp R2 8 2 mNp ro2 For the diffusion regime of motion of a fractal aggregate in a buffer gas formula (5.3.2) gives for the diffusion coefficient of a fractal aggregate Ddif D

T ; 6R

  R:

(10.2.7)

197

Section 10.2 Growth of fractal aggregates

Basing on these formulas for the diffusion coefficients of a fractal aggregate in a gas, we consider the diffusion regime of approach of fractal aggregates in the course of these aggregation, that corresponds to the opposite criterion with respect to criterion (10.2.5)  2D=6 p 3 mo R 1=3 Np  ; (10.2.8) p 2 ro 4 mNa ro where Np is the number density of monomers of fractal aggregates in a space, ro is a monomer radius, mo is its mass, R is the fractal aggregate radius, and Na is the number density of buffer gas atoms. Growth of fractal aggregates is described by formula (4.2.5), and the rate of association of two fractal aggregates is given by the Smolukhowski formula (4.2.6) that is kag D 4.D1 C D2 /.R1 C R2 /;

(10.2.9)

where D1 ; D2 are the diffusion coefficients in a buffer gas for colliding fractal aggregates, and R1 ; R2 are their radii. For simplicity, we take this formula in the form kag D 16DR, where R is the average radius of fractal aggregates at this time, and D is their diffusion coefficient in a buffer gas. From this we have for the aggregation rate in the kinetic regime of aggregate motion in a buffer gas kag D 16Dp ro2 =R;

  R:

(10.2.10)

Because equation (9.2.6) takes the form now dn 1 D kag Np ; dt 2 and the average number of monomers in a fractal aggregate according to formula (10.1.3) is equal  D R nD ; ro this gives for evolution of the average number of monomers in fractal aggregate

D=DC1 8D nD   R: (10.2.11) Dp ro Np t DC1 In the diffusion regime of motion of fractal aggregates in a buffer gas we obtain for the aggregation rate r T 8T (10.2.12) kag D D 4:8 g ;   R; 3 m

198

Chapter 10 Structures formed in aggregation of solid particles

that is identical to formulas (9.3.2) and (9.3.3) since in this case the result is independent of the size and mass of associating particles. This gives a simple dependence of the average number of monomers in fractal aggregate on time as a result of solution of equation (9.2.6) r T 4T nD (10.2.13) Np t D 2:4 g Np t;   R: 3 m Thus, formulas (10.2.3), (10.2.11) and (10.2.13) describe growth of fractal aggregates in time for different regimes of growth of fractal aggregates. We repeat assumptions used in these derivations. First, we assume an identical size of monomers which constitute fractal aggregates. Second, fractal aggregates are modelled by spherical particles, and the hydrodynamic radius of these particles coincides with their geometric radius. Due to these assumptions, the results obtained may be considered as estimations. In addition, fractal aggregates occupy a small part of a space where they are located and do not interact with each other.

10.3

Growth of particle structures in external electric fields

The role of electric fields in growth of particles and structures increases with an increasing particle size. We demonstrate this on the example of metal particles located in an uniform electric field of a strength E. Then interaction potential U of two spherical particles of a radius a at a distance R between them is estimated as U 

d2 ; R3

where the induced dipole moment is d D ˛E, and the polarization of a metallic particle is given by [449] ˛ D a3 . From this it follows U 

E 2 a6 ; R3

and this exhibits a strong dependence of the interaction potential on a particle size. This means that electric fields through induced dipole momenta accelerate the rate of growth of structures as a result of association of solid particles and structures in a plasma. Below we consider this problem in detail. We now determine the association rate of two solid spherical metal particles located in a buffer gas. For definiteness, we consider the diffusion regime of growth where the mean free path  of buffer gas atoms is small compared to a

Section 10.3 Growth of particle structures in external electric fields

199

particle radius a, so the rate of association of two spherical particles is given by formula (9.3.2) U 

E 2 a6 ; R3

and the force F that acts on a test particle from another one located on a distance R from this is F 

E 2 a6 : R4

Equalizing it to the Stokes force (5.1.2) that acts on a test particle moving with a velocity w, we obtain the velocity of a test particle with respect to a nearest particle in an external electric field under the action of induced dipole momenta w

E 2 a5 : R4 

From this we obtain the rate of association of these particles as a result of interaction of induced dipole momenta kind  wR2 

E 2 a5 : R2 

Evidently, a typical distance between associating particles R that is responsible for the association process is determined by the number density Nag of particles, and we have the following estimation for the rate of association due to interaction of induced particle momenta under the action of an external electric field 2=3

kind 

E 2 a5 Nag : 

(10.3.1)

Comparing this with the rate constant of association of two spherical particles according to formula (9.3.2) kas D 8T =.3/ in the absence of an external field, we obtain 2=3

E 2 a5 Nag kind  : kas T

(10.3.2)

From this it follows that the particle association in an external electric field due to interaction of induced dipole momenta dominate for high electric field strengths, large sizes of associated particles and their high density. As a result of this mechanism of particle association, structures are formed which are like to chain aggregates.

200

Chapter 10 Structures formed in aggregation of solid particles

Note that in the above analysis we assume that a fractal aggregate has the same polarizability as a compact metal particle of the same radius, i.e. we assume the conductivity of this particle to be enough large. Because the polarizability of a growing particle increases sharply with an increasing particle size, for large fractal aggregates the growth mechanism through interaction of induced moments becomes dominated. Hence, in growth of structures of solid particles of a high conductivity (metal particles) in a buffer gas in an external electric field fractal aggregates grow on the first stage of the growth process as it takes place in the absence of electric fields, and we are based on the cluster-cluster aggregation mechanism given in Figure 10.5. When fractal aggregates reach a certain size a where the ratio (10.3.2) is of the order of one, the mechanism of the aggregate growth changes. We below determine the aggregate size where this transition occurs. Let us take at the beginning monomers of a radius ro and their number density is Np . Then the aggregate number density according to formula (10.1.3) is Nag D Np

 r D o

a

;

and we take the fractal dimension of the fractal aggregate formed in cluster-cluster aggregation to be D D 1:77. Then equalizing the ratio (10.3.2) to one, we find for the size a of fractal aggregates at which the mechanism of their growth is changed aD

T

!0:31 :

2=3 1:77 ro

E 2 Np

(10.3.3)

A typical time of aggregate growth up to the transient size 3 D 4T Np

10.4



a ro

1:77 D

3 4T 0:69 Np1:21 E 1 :1ro0:97

:

(10.3.4)

Growth of elongated particle structures in electric field

Thus, elongated structures result from association of spherical particles if the association process proceeds in an external electric field and interaction of induced dipole momenta of joining particles determines the rate of this process. Moreover, when these structures are formed, attachment of new spherical particles proceeds to the endings of a formed elongated structure [450]. Indeed, Figure 10.7 exhibits the character of the interaction potential of these particles, and maximum attraction of a joining spherical particle takes place near the endings of the elongate

Section 10.4 Growth of elongated particle structures in electric field

201

ȡ

21

Region of repulsion 1

Region of attraction –21

–1

Region of attraction z cylinder aerosol

1

21

Figure 10.7. Character of interaction of a chain aggregate that is modelled by a cylinder with a spherical particle of the same radius. As a result, a spherical particle attaches to the ending of the cylinder particle, increasing its length.

structure. Next, as this structure grows, the rate of association process increases. We now estimate the rate constant of growth of a cylinder metal particle of length 2l and radius a if it is located in a gas of spherical metal particles of a radius a. Since attraction of a spherical particle to a cylinder one proceeds mostly near endings of the cylinder particle, their association takes place near the cylinder endings. The induced charge in a metal cylinder particle located in an external electric field with field direction along the particle axis is equal [449] with an accuracy up to a factor under the logarithm dq Ez D ; dz 2 ln.l=a/ where z is the distance from the cylinder center, 2l is the length of this cylinder particle. As is seen, the charge sign is different for left and right parts of the cylinder particle, and its average charge is zero. The dipole moment of the cylinder particle is determined mostly by its regions far from the center. If a spherical particle is located near the axis of the cylinder particle at a distance R from its ending (see Figure 10.8 (a)), we have for the interaction potential of spherical and cylinder particles Z U.R/ D 

l l

dq  d ; .l  z C R/2

202

Chapter 10 Structures formed in aggregation of solid particles 21

a

R

21

21

b

R

Figure 10.8. Geometry of interaction of a cylinder particle with a spherical one of identical radii, if the axis prolongation passes through a center of a spherical particle (a), and interaction of two cylinder particles if their axes are located in one line (b).

where d D a3 E is the dipole moment of the spherical particle. For the range of parameters l  R  a we have from this U.R/ 

E 2 a3 l : R ln.l=a/

(10.4.1)

From this we have for the force that acts on a spherical particle and compels it to move to the ending of the cylinder particle F 

E 2 a3 l ; R2 ln.l=a/

l  R  a:

Equalized this to the Stokes force (5.1.2) of friction, we estimate the velocity of motion of a spherical particle with respect to the ending of the cylinder particle w

E 2 a2 l ; R2  ln.l=a/

l Ra

and the rate constant of attachment of a spherical particle to the ending of a metal cylinder particle is kcyl  R2 w 

E 2 a2 l ;  ln.l=a/

l  a:

One can make this estimation on the basis of equation wD

dR E 2 a2 l  2 : dt R  ln.l=a/

Its solution gives for a typical time of particle approaching 

R3  ln.l=a/ : E 2 a2 l

(10.4.2)

Section 10.4 Growth of elongated particle structures in electric field

203

Taking a typical distance of the cylinder particle ending from a nearest spherical 1=3 particle to be R  Np and introducing the rate constant of attachment of a spherical particle to the ending of the cylinder one as kcyl  1=.Np /, one can obtain again formula (10.4.2). Note that in derivation formula (10.4.2) we assume 1=3 R  l. This means that this formula holds true if Np l  1. We now represent the criterion of this scenario. We have a gas of spherical particles of a radius a with a metal conductivity (then the polarizability of an individual particle is [449] ˛ D a3 ), and these spherical particles are located in a dense buffer gas in an uniform electric field. Along with spherical particles, there are also particles of the cylinder shape which are formed as a result of attachment to them spherical particles and the axis of cylinder particles is directed along the electric field. Under these conditions we estimate the rate constant of joining of spherical particles due to interaction of induced dipole momenta kind that is given by formula (10.3.1) and the rate constant of attachment of spherical particles to the endings of cylinder particles kcyl that is given by formula (10.4.2). We now represent the criterion where association of spherical particles proceeds through attachment to cylinder particles. Denoting the number density of spherical particles as Np and the number density of cylinder particles as Nl , we obtain this criterion in the form Nl kcyl  Np kind , and on the basis of formulas (10.3.1) and (10.4.2) we obtain this criterion in the form [163] 2=3

a3 Np Nl  : Np l ln.l=a/

(10.4.3)

Note that in the case if spherical particles form a compact structure, Np a3  1, so that if spherical particles are located in a gas, we have Np a3  1. Hence, the criterion (10.4.3) holds true if the most part of a particle material is contained in cylindrical particles (the mass of an individual cylinder particle is higher in l=a times than that of a spherical one). Therefore, if we have a gas of of spherical particles, their joining in an external electric field proceeds similar to cluster-cluster aggregation if the electric field is absent. Namely, at the beginning they form small cylinder particles joining due to interaction of induced dipole momenta, and then small cylinder particles join in larger particles of almost cylinder shape. We now consider growth of cylinder particles in the diffusion regime of this process. Let us take a gas of cylinder metal particles of length 2l and radius a, so that l  a, and these particles are located in a dense gas and in an electric field of a strength E, so that ˛E 2  T;

(10.4.4)

204

Chapter 10 Structures formed in aggregation of solid particles

where the polarizability of a metal cylinder is [449] ˛D

8l 3 ; 3 ln.l=a/

l  a:

For simplicity, we assume associative particles to be of an identical length 2l at a given time. Since according to the criterion (10.4.4) the particle axes are directed along the electric field, we consider association of two cylinder particles whose axes are located almost along the same line. Taking a distance between their endings to be R (see Figure 10.8 (b)), we construct the interaction potential of two cylinder particles as a result of interaction of induced charges. At large separations R  l the interaction potential is U.R/ D 

2d 2 8l 3 2˛ 2 E 2 2 D  D 2˛E ; R3 R3 3R3 ln.l=a/

R  l;

where d D ˛E is the induced dipole moment. If R D 0, i.e. particles form one particle of length 4l, this interaction potential is U.0/ D 6˛E 2 ; because the interaction potential is a monotonic function of a distance R between the endings, one can combine these formulas as U.R/ D 6˛E 2 f .R= l/;

f .x/ D

1 ; .1 C cx/3

cD

1 Œ9 ln.l=a/ 1=3 : 2 (10.4.5)

From this we obtain for the force between cylinder particles F .R/ D 

3c U.R/: l.1 C cx/

In equilibrium this force is equal to the friction force of cylinder particles moved along the axis, that is given by the following formula [451] instead of Stokes formula (5.1.2) F .R/ D 6wRh ;

Rh D

2l ; 3 ln.l=a/

where the hydrodynamical radius Rh in the perpendicular direction is twice compared to that in the axis direction. From this we determine the rate of association of cylinder particles kas in the standard method that we used above taking

Section 10.4 Growth of elongated particle structures in electric field 1=3

R D Nl we obtain

205

, where Nl is the number density of cylinder particles. As a result, 1=3

kas D

U.Nl 1=3 .lNl /2

/ ln.l=a/ 1=3

 Œ1 C c=.lNl

:

(10.4.6)

/

Note that because cylinder particles are formed from spherical ones by growth along the electric field, we have a Nl  Np ; l and the number density of spherical particles Np is conserved in the course of the growth of cylinder particles. For simplicity, we now consider the case of a low number density of particles Nl l 3  1, that corresponds to Np al 2  1:

(10.4.7)

For estimations we obtain a simplified expression for the association rate of two identical cylinder particles in the limit of a small number density of these particles Nl l 3  1. We have for the interaction potential U.R/ of cylinder particles at large distances R  l between them within the framework of the geometry of Figure 10.8 (b) jU.R/j D

2˛ 2 E 2 l 6E 2 2D 2 D  ; R3 R3 R3 ln2 .l=a/

R  l:

From this we determine the force F between particles and equalizes it to the friction force F  lw= ln.l=a/ for particle motion in a viscous media F 

l 6E 2 R4 ln2 .l=a/



lw ; ln.l=a/

R  l:

This gives for the relative drift velocity of cylinder particles w

lw l 5E 2  ; 4 R  ln.l=a/ ln.l=a/

R  l:

From this we have for the rate constant of association of two identical cylinder particles kas  R2 w 

l 13=3 a2=3 E 2 2=3 Np ;  ln.l=a/

Nl l 3  1;

(10.4.8)

206

Chapter 10 Structures formed in aggregation of solid particles

a 10 μm

100 μm

b

Figure 10.9. Fragment of a fractal fiber that is cut out along (a) and across (b) of the fractal fiber [452].

where we use that cylinder particles are formed from spherical ones and this leads to the relation Nl D aNp = l, where Np is the number density of spherical particles at the beginning. Note that if we substitute in formula (10.4.8) l D a, we obtain formula (10.3.1). As is seen from formula (10.4.8), the rate constant and the rate of growth of cylinder particles increases with an increasing size of these particles. Hence, a typical time of growth of cylinder particles is based on formula (10.3.1) or formula (10.4.8) with l D a that gives 

 5=3

a5 E 2 Np

:

(10.4.9)

Note that though in modeling chain aggregates formed and grown in an external electric field by cylinder particles, we lost some peculiarities of this process, such an analysis gives a general character of the structure growth. Because the growth process accelerates with an increasing particle length, finally we obtain a group of fibres which diameter is comparable with that of initial spherical particles. It is clear that these fibres are interwoven in the course of the process, and in the end we have a fractal fiber as it is shown in Figure 10.9. Thus, the structures resulted from association of particles in an external electric field, differ from those in the absence of the electric field. We note in conclusion that the thread-like structures are known in aerosol physics [327, 30]. Moreover, it is known that these structures are connected with an external electric field. A classical example of this [454, 455] is that combustion of a magnesium tape leads to formation of spheri-

207

Section 10.5 Aerogels

cal aerosols consisting of magnesium oxides, while a smoke resulted in the same manner in arc contains aerosols of a fiber structure. It shall be noted that subsequent interweaving of fractal fibres as a result of joining their intermediate links lead to formation the fractal tangles [456] which are interwoven fractal fibres. Thus, different structures may be formed under action of external fields.

10.5

Aerogels

If solid particles are located in a buffer gas in the absence of external fields, they join in fractal aggregates. If we forget temporarily that large fractal aggregates are unstable, we obtain that fractal aggregates will grow, until neighboring fractal aggregates are touched with each other, and their joining leads to formation of a continuous rareness structure of solid particles. Let us consider a system that at the beginning consists of solid particles of a radius a and of the number density Np . Then these particles are joined by formation the chemical bonds in points of their contacts. Assume that in the course of growth of fractal aggregates the number density of particles in a space is conserved. Then the final size of fractal aggregates according to formulas (10.1.2) and (9.2.13) is  Rc D

3 4Np

1=.3D/

aD=.3D/ :

(10.5.1)

This is the correlation radius [457, 458, 459], and a formed structure is an aerogel. Thus, an aerogel has fractal properties at small distances R < Rc and becomes uniform at large distances R > Rc . In particular, this means that if we cut an aerogel piece of a size r < Rc in the form of a ball, where the center is occupied by a particle, the number density of particles in accordance with formulas (10.1.2) and (9.2.13) is given by  N  Np

Rc r

3D :

(10.5.2)

If the radius of a ball that is cut out an aerogel exceeds the correlation radius Rc , its average density is Np . The aerogel as a technical object [457] is produced in other way and has a more high density, but it has the above physical properties. Practically, an aerogel is a porous structure with a rigid skeleton as it is represented in Figure 10.10. Considering the aerogel to be composed from identical particles of a radius ro and assuming that the contact area occupies a small part of the particle area, we

208

Chapter 10 Structures formed in aggregation of solid particles

10 nm Figure 10.10. Projection of an aerogel fragment.

obtain for the specific area S of the aerogel internal surface SD

3 ;  o ro

o D mo Np ;

(10.5.3)

where mo is the mass of an individual particle. Let us consider some aspects of production of a technical aerogel. The classical method of aerogel production was developed by S.S.Kistler [460, 461, 462, 463] over seventy years ago. Note that though an aerogel may be prepared from various metal oxides and organic compounds [457, 464], the most part of existed aerogels consists of SiO2 . The classical method of aerogel production is based on the reaction of Na2 SiO3 salt with the water solution of hydrochloric acid, that proceeds according to the scheme Na2 SiO3 C 2HCl ! SiO2  nH2 O C 2NaCl:

(10.5.4)

The salt NaCl precipitates and is removed from the solution, and the remaining solution is washing and filtered. Water in this solution is replaced by either ethanol or methanol which are characterized by a low critical temperature and pressure compared to water. As a result, an aerogel is formed, and in its pores molecules of the solution may be located. In order to remove these molecules from aerogel pores and to obtain a rigid skeleton, this substance is dried in an autoclave under supercritical conditions for the solution. Some steps of this technological process may be modified in contemporary technology of the aerogel production, but a general concept of the above scheme is conserved. In particular, hydrolysis of tetramethoxysilane is more technological method for formation of silica that proceeds according to the following scheme [465, 466] Si.CH3 O/4 C 4H2 O ! SiOH4 C CH3 OH;

SiOH4 ! SiO2  2H2 O: (10.5.5)

Section 10.5 Aerogels

209

It should be noted that the production of aerogels, especially of a high quality, requires a large time. For example, production of aerogels in the 98 liter autoclave (the temperature of 270ı C and pressure of 90 bar) for the CERN Cherenkov detectors lasted almost one week as the total cycle and gave about 50 liters of aerogel per one cycle [467, 468]. Because the aerogel production is connected with flammable substance, it is a danger production, so that a large-scale production of aerogels is problematic. The lower boundary for the classical method of the silica aerogel production is approximately 0:02 g=cm3 . The contemporary methods allow one in principle to produce an aerogel which specific gravity is as low as the normal air specific gravity that is 0:003 g=cm3 . Let us return to formation of an aerogel in a gas as a result of association of solid particles with accounting for the stability of fractal aggregates from which this fractal aggregate is formed. Let Rcr be the critical radius of fractal aggregates at which it is conserved the stability that according to formula (10.1.2) corresponds to the number density of particles in fractal aggregates of   a D 3 Ncr  a : Rcr As it follows from the aerogel nature, the critical radius of fractal aggregates which constitute an aerogel is the correlation radius Rc . Indeed, in analyzing the growth of fractal aggregates which are a basis of an aerogel as a result of clustercluster aggregation, we find that if an aggregate reaches a critical size, it becomes unstable. This means that new bonds formed in joining of fractal aggregates are broken. But if fractal aggregates in the course of destruction of chemical bonds between some particles remain close to each other, new chemical bonds are formed, and this process continues until strong chemical bonds arise between particles of interacting fractal aggregates. This leads to formation a heightened density of particles compared to the case of cluster-cluster aggregation. As a result the fractal dimension defined on the basis of correlation function (10.1.1) for large scales corresponds to a compact particle system D D 3, while at small scales it is identical to that of a fractal aggregate D D 1:77 with this mechanism of aggregation. The dependence of the fractal dimension of an aerogel on a distance r of correlation is given in Figure 10.11. This character of aerogel formation as a result of aggregation in a buffer imposes a requirement to the rate of aggregation. It is necessary that restructuring in the course of bond formation between joining aggregates lasts faster than a following fractal aggregate can partake in the process of aggregate growth. Since the restructuring process is independent of the aggregate density in a space, whereas the rate of the process of joining with other aggregates is proportional to this value, aerogel growth proceeds reliable at low density of aggregates.

210

Chapter 10 Structures formed in aggregation of solid particles

D 3 1.77 r

0

RC

Figure 10.11. The fractal dimension D of an aerogel as a function of the correlation distance r defined by formula (10.1.1).

One can see that the process of growth of fractal structures, as fractal fibres and aerogels, proceeds in two stages, so that the first stage is the conversion of a gas or vapor of atoms or molecules in liquid drops where this process proceeds in a buffer gas, and the second stage is joining of solid particles resulted from the first stage, in fractal structures. Therefore, various methods of cluster generation [469] are the basis of formation and growth of fractal structures.

10.6

Problems

Problem 10.1. Determine a typical time of relaxation for a SiO2 fractal aggregate of a radius 1 nm in air in the diffusion regime of this process if a radius of monomers constituted this fractal aggregate is 10 nm. This fractal aggregate results from cluster-cluster aggregation, so that its fractal dimension is D D 1:77. Let us use formula (6.1.1) for the relaxation time of a particle in the diffusion regime of particle-gas interaction i rel D

M ; 6Ro 

Ro  ;

where M is the particle mass, Ro is its radius. On the basis of formula (10.1.2) we represent this formula in the form rel D

mo RoD1 : 6 roD 

(10.6.1)

Section 10.6 Problems

211

Using the parameters under given conditions (mo D 9  1018 g), we obtain rel  0:1 μs. This time is less significantly than that for a compact particle of the same size. Problem 10.2. The first stage of combustion of methane CH4 with an admixture SiH4 in air in a flame is described in Problem 9.3. On the second stage products of combustion with solid SiO2 -particles remain in a cold chamber during t D 1 min, and solid particles are joined in fractal aggregates. Determine parameters of fractal aggregates in the end of location in a cold region. According to results of Problem 9.3, solid SiO2 -particles are injected in a cold region being located in nitrogen at atmospheric pressure and room temperature with the concentration of SiO2 -molecules cb D 0:01. We take for definiteness a radius of SiO2 solid particles to be a D 40 nm that corresponds to their mass mp D 6  1016 g, and since the Wigner-Seits radius for SiO2 equals to rW D 2:2 Å, this particle contains 6  106 SiO2 -molecules. Because the number density of molecules in cold region is Na D 2:41019 cm3 , and because the concentration of SiO2 -molecules is equal cb D 0:02, the number density of bound SiO2 -molecules is 4:8  1017 cm3 , so that the number density of SiO2 -particles (monomers) is Np D 8  1010 cm3 . Under these conditions we have from formula (10.2.4) ƒkin D 2:2  106 cm, 1=3 and the parameter Np ƒkin  0:01. From this it follows that the criterion (10.2.8) holds true at any sizes of fractal aggregates, i.e. the diffusion regime of approach of fractal clusters is realized. Next, the mean free path of nitrogen molecules in the buffer gas is  D 1:2  105 cm, i.e. =a D 3, and the diffusion regime of growth of fractal aggregates takes place, that is, formula (10.2.13) describes the growth of fractal aggregates in time. Thus, according to formula (10.2.12) the rate constant of growth of fractal aggregates is kag D 61010 cm3 =s, and the number of monomers in a fractal cluster in the end of the process according to formula (10.2.13) is equal n D 3103 , that corresponds to the aggregate size R D 4 μm for the CCA-regime of cluster growth, whereas the average distance 1=3 D 35 μm. between nearest aggregates is Nag Problem 10.3. Fractal aggregates grow under conditions of previous Problem in an external electric field of strengths 100 V=cm and 1 kV=cm. Determine the boundary size of a fractal aggregate a at which the transition proceeds to the regime where the mechanism of aggregation under the action of an electric field occurs. Find a typical time of growth of a fractal aggregate up to this size. We use formula (10.3.3) for the boundary size a of a fractal aggregate with the following parameters: the monomer radius is ro D 40 nm, the number density

212

Chapter 10 Structures formed in aggregation of solid particles

of monomers is Np D 8  1010 cm3 , the temperature is T D 300 K. Then formula (10.3.3) gives a D 7:2 μm at the electric field strength E D 100 V=cm and a D 1:7 μm at E D 1 kV=cm. A number of monomers given by formulas (10.1.3) and (10.3.3) depends on the electric field strength E as n  E 1:1 . Correspondingly, according to these formulas the number of monomers in fractal aggregates is equal 1  104 and 800 for the electric field strengths E D 100 V=cm and E D 1 kV=cm respectively. An external field does not influence on aggregate growth up to this size, and hence a typical time of aggregate growth up to this size is given by formula (10.3.4) and equals to a  7 min at E D 100 V=cm and a  0:5 min at E D 1 kV=cm correspondingly. Since subsequent growth of cylinder particles proceeds fast and keep within these values, the indicated times may be considered as the total times of growth of cylinder particles or fractal fibres [452, 453] (Figure 10.9) in an electric field. Problem 10.4. Under prethunderstorm conditions, the electric field strength near the Earth surface reaches 1 kV=cm, that is three orders of magnitude exceeds that for a quiet atmosphere. A smoke from a chimney-stalk of a type of Figure 2.3 throws into the atmosphere. This smoke consists of solid particles of an average size 0:1 μm and the basis of these particles are compounds of carbon, sulfur, ammonium and metal oxides. Content of this soot in the atmosphere after mixing with air is 10mg of soot per gram of air. Ascertain the possibility of aggregation of these soot particles in atmospheric air. In this consideration we assume soot particles to be chemically active, so that they can form chemical bonds in points of their contact. Then for aggregation of these particles it is necessary that a typical time of their joining is less than a time of redistribution of these particles over atmospheric air. The latter is determined by convective mixing of the air stream from a chimney-stalk with surrounding air that in turn depends on certain conditions in atmospheric air. Not having a chance to determine this value, we accept it by eye as to be in the range mix  1–10 min. Below we determine a time of soot aggregation in a quiet atmosphere and compare it with this value. In these estimations we consider a soot particle as a porous spherical particle with a typical density of   1 g=cm3 . Under given conditions the mass of an individual particle is mo  4  1015 g, and the initial number density of soot particles is Np  3  109 cm3 . On the basis of these parameters formula (10.3.3) gives for a particle radius a  2 μm starting from which the electric field influences the aggregation process. A typical time of the structure growth up to this size according to formula (10.3.4) is equal   4 min. In these estimations we assume the diffusion regime of particle growth with the fractal dimension D D

Section 10.6 Problems

213

1:77 for a growing fractal aggregate according to the cluster-cluster aggregation mechanism. As is seen, since an aggregation time is comparable with a typical time of air mixing, formation of elongated particle structures under the action of an electric field is problematic though in principle this possibility exists. The final structures have the form of soot flake or form elongated soot structures. Problem 10.5. Small particles of metal oxides (Ti2 O3 ; Fe2 O3 ) are formed in a plasma torch that is supported by irradiation of a metal surface (Ti; Fe) in air at the atmospheric pressure by a focused laser beam in accordance with experiments as it takes place in experiments [417, 470]. After a plasma torch particles of a metal oxide of a radius of 10 nm pass in a cold chamber where the concentration of bound metal atoms in air is c D 103 . Assuming that the maximum radius of stable fractal aggregates to be Ro D 5 μm, find what part of a cold chamber occupies a forming aerogel and what is a typical time of its formation. Under given conditions the basis of an aerogel are fractal aggregates of a radius of 5 μm. We first determine the parameters of these fractal aggregates. According to formula (3.4.2), the Wigner-Seits radius is equal rW D 2:33 Å for Ti2 O3 and rW D 2:29 Å for Fe2 O3 . The solid clusters of a radius 10 nm include 7:9  104 of Ti2 O3 molecules and 8:3  104 of Fe2 O3 molecules. Correspondingly, these solid clusters have the mass 1:9  1017 g in the Ti2 O3 case and 2:2  1017 g in the Fe2 O3 case. The number density of solid clusters is under given conditions Np D 3:1  1011 cm3 for Ti2 O3 solid clusters and Np D 3:0  1011 cm3 for Fe2 O3 solid clusters. Next, according to formula (10.1.3) fractal aggregates of a radius of 5 μm consist of 6:0  104 monomers—solid clusters of a radius 10 nm, where we use the fractal dimension D D 1:77 for fractal aggregates. In addition, fractal aggregates of a radius 5 μm contain 4:7  109 Ti2 O3 molecules and 5:0  109 Fe2 O3 molecules. At a given concentration of bound metal atoms in air c D 103 at the temperature T D 300 K and pressure p D 1 atm, the number density of fractal aggregates of a radius 5 μm is equal 4:9  106 cm3 in the Ti2 O3 case and 5:2  106 cm3 in the Fe2 O3 case. As is seen, all these parameters for Ti2 O3 and Fe2 O3 molecules are close and do not exceed the accuracy of these values. Therefore below we will consider these parameters for Ti2 O3 and Fe2 O3 molecules to be identical. In determination a typical time of formation of fractal aggregates of a radius 5 μm, we will be guided by the diffusion regime of growth of fractal aggregates, where the rate constant of structure growth is equal kas D 6:0  1010 cm3 =s according to formula (9.3.2). A typical time of this process given by formula

214

Chapter 10 Structures formed in aggregation of solid particles

(9.2.7) is equal D

2n ; kas Np

where Np is the number density of solid clusters, and this value is approximately 11 min. The following step of aerogel formation is the process of joining of fractal aggregates in an aerogel where these fractal aggregates form a close-packed structure. Then for a typical time of aerogel growth is given by a latter formula where the number density of fractal aggregates Nag is used instead of the number density of solid clusters Np . In particular, this formula gives for a typical time   0:5 year for growth of an aerogel of a radius 1 mm. Note that the number density of metal atoms in a forming aerogel is 1:5  1019 cm3 , that exceeds the concentration of metal atoms in air at the beginning by almost three orders of magnitude. Thus, aerogel growth is accompanied by concentrating of metal molecules in a restricted space region.

Chapter 11

Conclusion

In consideration the behavior of isolate small particles, nanoclusters and microparticles, located in excited or ionized gases we are found on the stage when the principal concepts are developed more or less, mostly in dense systems, and it is necessary to work out simple and reliable algorithms for certain conditions related to applied problems. Note that this consideration is connected with different physical objects. Namely, microparticles related to certain materials or being treated in a certain manner, can exist in the form of powders [68], whereas if nanoclusters are contacted with each other, they are sticken together and lose their individuality. Therefore nanoclusters exist in the form of cluster beams or being distributed in gases. In spite of this difference, these physical objects may be described by identical models, and here the liquid drop model and the hard sphere model are used for this goal. Then nanoclusters and microparticles are considered as liquid or solid spherical particles, and the analysis of some properties or processes has a simple and universal character that is identical both for nanoclusters and microparticles. Another side of this generalization is a loss of the accuracy and some information. In particular, the cluster property of extrema of cluster parameters at magic numbers of cluster atoms which correspond to optimal cluster structure is lost for these models. Of course, the accuracy of models in the analysis of certain cluster properties is evaluated, and the accuracy of application of the above models for clusters is of the order of ten percent. The principal peculiarity of processes involving nanoclusters and microparticles in gases is the regime of a process with respect to interaction of a small particle with atoms of a buffer gas. Indeed, in a rarefied buffer gas only one atom can interact strongly with a small particle at given time, and this regime is similar to collisions of atomic particles in gases, that corresponds to the kinetic regime. In another limiting case for the diffusion regime of interaction between a small particle and buffer gas atoms many atoms interact strongly with atoms each time simultaneously. Hence, the behavior of a small particle in such a dense buffer gas is described by hydrodynamics laws. For example, the kinetic regime of transport of a small particle in a buffer gas is realized if the particle radius is small compared to the mean free path of buffer gas atoms in a buffer gas, and the diffusion regime of particle transport corresponds to the opposite criterion. Thus, the results

216

Chapter 11 Conclusion

of the above analysis are algorithms for parameters of processes involving a small particle which are accompanied by criteria where the formulas obtained are valid.

Appendix A

Physical parameters

A.1

Fundamental physical constants

Table A.1 Electron mass Proton mass Atomic mass unit Ratio of the proton mass to the electron mass Ratio of atomic and electron masses Electron charge

Plank constant Light speed Fine structure constant Bohr radius Rydberg constant Bohr magneton

Avogadro number Stephan-Boltzmann constant Molar volume Loshmidt number Faraday constant

me D 9:10938  1028 g mp D 1:67262  1024 g 1 m.12 C / Ma D 12 D 1:66054  1024 g mp =me D 1836:15 Ma =me D 1822:89 e D 1:602176  1019 C D 4:8032  1010 CGS e 2 D 2:3071  1019 erg  cm h D 6:62607  1027 erg  s „ D 1:05457  1027 erg  s c D 2:99792  1010 cm=s ˛ D e 2 =.„c/ D 0:072973 1=˛ D „c=e 2 D 137:03600 ao D „2 =.me e 2 / D 0:529177 Å Ry D me e 4 =.2„2 / D 13:6057 eV D 2:17987  1018 J B D e„=.2me c/ D 9:27401  1024 J=T D 9:27401  1021 erg=Gs NA D 6:02214  1023 mol1  D  2 =.60„3 c 2 / D 5:604  1012 W=. cm2  K4 / R D 22:414 l=mol L D NA =R D 2:6868  1019 cm3 F D NA e D 96485:3 C=mol

218

A.2

Appendix A Physical parameters

Melting and boiling points of elements

Table A.2 Element Li Be Na Mg Al K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Rb Sr Y Zr Nb Mo

Melting point, K 454 1560 371 923 933 336 1115 1814 1941 2183 2180 1519 1812 2750 1728 1358 693 303 312 1050 1795 2128 2750 2886

Boiling Elepoint, K ment 1615 Tc 2744 Ru 1156 Rh 1363 Pd 2730 Ag 1032 Cd 1757 In 3103 Sn 3560 Sb 3680 Cs 2944 Ba 2334 La 3023 Ce 5017 Pr 3100 Nd 2835 Pm 1180 Sm 2680 Eu 961 Gd 1655 Tb 3618 Dy 4650 Ho 5100 Er 4912 Tm

Melting Boiling Elepoint, K point, K ment 2430 4538 Yb 26077 4423 Lu 3237 3968 Hf 1828 3236 Ta 1235 2435 W 594 1040 Re 430 2353 Os 505 2875 Ir 904 1860 Pt 301 944 Au 1000 1913 Hg 1191 3737 Tl 1072 3669 Pb 1204 3785 Bi 1294 3341 Po 1441 3000 Fr 1350 2064 Ra 1065 1870 Ac 1586 3539 Th 1629 3396 Pa 1685 2835 U 1747 2968 Np 1802 3136 Pu 1818 2220 Am

Melting Boiling point, K point, K 1097 1466 1936 3668 2510 4876 3290 5731 3695 5830 3459 5880 3100 5300 2819 4700 2041 4098 1337 3129 334 630 577 1746 600 2022 544 1837 527 1235 300 900 970 1800 1320 3470 2023 5061 1845 – 1408 4404 917 – 913 3500 1450 2600

Appendix B

Conversional factors

B.1

Conversional factors in formulas for atomic particles and small particles

Table B.1. Conversional factors in formulas related to atomic particles.

1.

Formula  p v D C "=m

2.

vDC

3.

" D C v2

4.

! D C"

5. 6. 7.

! D C = " D C = !H D CH=m

8.

rH D C

9.

p D CH 2

N%

/

p T =m

p "m=H

Factor C

Units used

5:931  10 cm=s 1:389  106 cm=s 5:506  105 cm=s 1:289  104 cm=s 1:567  106 cm=s 6:693  107 cm=s 1:455  104 cm=s 3:299  1012 K 6:014  109 K 2:843  1016 eV 5:182  1013 eV 1:519  1015 s1 1:309  1011 s1 1:884  1015 s1 1.2398 eV 1:759  107 s1 9655s 1 3:372 cm 143:9 cm 3:128  102 cm 1:336 cm 4:000  103 Pa D 0:04 erg=cm3 7

" in eV, m in m/ e " in eV , m in m/ e " in K, m in m/ e " in K, m in M/ a T in eV, m in M/ a / " in K, m in me " in K, m in M/ a v in cm=s, m in m/ e v in cm=s, m in M/ a v in cm=s, m in m/ e v in cm=s, m in M/ a " in eV " in K  in μm  in μm H in Gs, m in m/ e H in Gs, m in M/ a " in eV, m in m/ e , H in Gs " in eV, m in M/ a , H in Gs " in K, m in m/ e , H in Gs " in K, m in M/ a , H in Gs H in Gs

me is the electron mass unit .me D 9:108  1028 g/, is the atomic mass unit .Ma D 1:6605  1024 g/:

M/ a

220

Appendix B Conversional factors

Explanations to Table B.1 1. The particle velocity is v D mass.

p

2"=m, where " is the energy, m is the particle

p 2. The average particle velocity is v D 8T =. m/ with the Maxwell distribution function of particles on velocities; T is the temperature expressed in energetic units, m is the particle mass.

3. The particle energy is " D mv 2 =2, where m is the particle mass, v is the particle velocity. 4. The photon frequency is ! D "=„, where " is the photon energy. 5. The photon frequency is ! D 2c=, where  is the wavelength. 6. The photon energy is " D 2„c=. 7. The Larmor frequency is !H D eH=.mc/ for a charged particle of a mass m in a magnetic field of a strength H . p 8. The Larmor radius of a charged particle is rH D 2"=m=!H , where " is the energy of a charged particle, m is its mass, !H is the Larmor frequency. 9. The magnetic pressure pm D H 2 =.8/.

221

Appendix B Conversional factors in formulas Table B.2. Conversional factors in formulas for clusters or small particles. Factor C

N% Formula 1. 2. 3. 4. 5.

6. 7.

8.

9. 10. 11. 12. 13. 14. 15. /

Used units

m in M/ a ,  in g=cm3 n D C.ro =rpW /3 4.189 ro and rW in Å 2 T =m 4:5714  1012 cm3 =s rW in Å, T in K, ko D C rW m in M/ a w D Cro2 = 0:2179 cm=s ro in m,  in g=cm3 ,  in 105 g=.cm  s/ 5 2 cm =s K in cm2 =.V  s/, D D CK T 8:617  10 T in K K in cm2 =.V  s/, 1 cm2 =s T in eV D in cm2 =s, T in K K D CD=T 11604 cm2 =.V  s/ 2 D in cm2 =s, T in eV 1 cm =.V  s/, p 2 21 2 rW in Å, Na in cm3 , Do D C T =m=.Na rW / 1:469  10 cm =s T in K, m in M/ a 2 0:508 cm =s rW in Å, Na D No/ , T in K, m in M/ a 54:69 cm2 =s rW in Å, Na D No/ , T in eV, m in M/ a p 2 Ko D C =. T mNa rW / 1:364  1019 cm2 =.V  s/ rW in Å, Na in cm3 , T in K, m in M/ a . rW in Å, Na D No/ , 0:508 cm2 =.V  s/ T in K, m in M/ a 2 54:69 cm =.V  s/ rW in Å, Na D No/ , T in K, m in M/ a Do D C T =.rW / 7:32  105 cm2 =s rW in Å, T in K,  in 105 g=.cm  s/ 2 Ko D C =.rW / 0:085 cm =.V  s/ rW in Å,  in 105 g=.cm  s/ p p 2 ƒo D C ma =.Na rW m/ 750 cm Na in cm3 , rW in Å, m, ma in M/ a x D C =.ro T / 1:44 ro in nm, T in eV 16:71 ro in m, T in K p p 6:86  108 cm3 =s T in K, m in M/ kas D C T =. mNa ro / a , Na in cm3 , ro in μm 3:68  1011 cm3 =s T in K, kas D C T =  in 105 g=.cm  s/ 4 9 3 9:8  10 cm =s ro in m,  in g=cm3 , kas D C ro =  in 105 g=.cm  s/ rW D C.m=/

1=3

0:7346 Å

Ma D 1:66054  1024 g is the atomic mass unit. / No D 2:689  1019 cm3 .

222

Appendix B Conversional factors

Explanations to Table B.2 1. The Wigner-Seits radius for a cluster or spherical particle within the framework of the liquid drop model according to formula (3.4.2); ma is the atom mass,  is the mass density of a cluster material. 2. The number of atoms for a cluster or spherical particle n for the liquid drop model according to formula (3.4.1). 3. The reduced rate constant for collisions involving clusters according to formula (4.2.2). 4. The free fall velocity w according to formula (5.3.4) for a spherical particle of a radius ro , i.e., w D 2gro2 =.9/, where g if the free fall acceleration,  is the mass density for a particle material,  is the viscosity of a media where the particle moves. 5. The Einstein relation (5.3.1) for a charged particle located in a gas D D K T =e, where D; K are the diffusion coefficient and mobility of a charged particle, T is the gas temperature. 6. The Einstein relation (5.3.1) for the mobility of a charged particle moved in a gas K D eD=T . 7. The reduced diffusion coefficient Do of a particle in a gas in the kinetic regime, so that the particle diffusion coefficient D of a particle consisting of n atoms at the normal number density Na D No D 2:687  1019 cm3 of gas molecules according to formula (5.3.2) equals to Dn D Do =n2=3 , where p 2 Do D 3 2T = m=.16No rW /, T is the gas temperature, m is the gas atom mass, rW is the Wigner-Seits radius. 8. The reduced zero-field mobility of a spherical particle Ko in the kinetic regime, so that K is the mobility of a particle consisting of n at the normal number density of gas atoms Na D No D 2:687  1019 cm3 is equal p accord2 2=3 ing to formula (5.1.6) Kn D Ko =n , where Ko D 3e=.8No rW 2 mT //, and other notations are indicated in the previous point. 9. The reduced diffusion coefficient do of a particle in a gas in the diffusion regime, so that the particle diffusion coefficient D of a particle consisting of n atoms at the normal number density Na D No D 2:687  1019 cm3 of gas molecules according to formula (5.3.2) is equal to Dn D do =n1=3 , where do D T =.6 rW /,  is the gas viscosity, rW is the Wigner-Seits radius, other notations are given above.

Appendix B Conversional factors in formulas

223

10. The reduced zero-field mobility of a spherical particle o Ko in the diffusion regime, so that K is the mobility of a particle consisting of n at the normal number density of gas atoms Na D No D 2:687  1019 cm3 is equal according to formula (5.1.7) Kn D o =n1=3 , where o D e=.6 rW /, and other notations are indicated above. 11. The mean free path according to formula (6.1.4) ƒ D ƒo n1=6 for a cluster or small particle, consisting of n atoms of a mass ma , if the particle moves p p 2 m/, in a buffer gas with mass m of its atoms. Here ƒo D 3 ma =.4Na rW where Na is the atom number density, rW is the Wigner-Seits radius. 12. The parameter x D e 2 =.ro T / according to formula (8.7.3) is responsible for particle charging, where ro is the particle radius, T is the temperature of attached atomic particles to a small particle. 13. The rate constant of association for the kinetic regime of particle coagulation p that is equal kas D 3 2T =m=.Na ro / in accordance with formula (9.3.4), where m is the mass of gas atoms, Na is the number density of gas atoms, ro is the particle radius. 14. The rate constant of association in the diffusion regime of particle coagulation that is equal kas D 8T =.3/ in accordance with formula (9.3.3), where  is the viscosity coefficient of a buffer gas. 15. The rate constant of association in the diffusion regime of particle coagulation due to different velocities of particles in their falling in the gravitation field of the Earth, that is equal kas D ro4 g=/ according to formula (9.4.1), where ro is the particle radius,  is the density of a particle material, g is the free fall acceleration,  is the viscosity coefficient of a buffer gas.

224

Appendix B Conversional factors

Table B.3. Conversional factors in formulas for clusters or small particles moving in air at the temperature T D 300 K and pressure p D 1 atm, so that the number density of air molecules is Na D 2:45  1019 cm3 . Coefficient C

N% Formula 1. 2. 3. 4. 5. 6. 7. 8.

w D Cro2 D D C =ro2 K D C =ro2

D D C =ro K D C =ro kas D C =ro kas D 6:0  1010 kas D Cro4

0:01178 cm=s 1:93 cm2 =s 5:98  105 cm2 =.V  s/ 1:19  107 cm2 =s 4:6  107 cm2 =.V  s/ 9:0  108 cm3 =s 3 cm =s – 5:3  1010 cm3 =s

Used units  in g=cm3 , ro in μm ro in nm ro in nm ro in μm ro in m ro in nm –  in g=cm3 , ro in μm

Explanations to Table B.3 1. The free fall velocity w of a spherical particle of a radius ro in accordance with formula (5.3.4), so that w D 2gro2 =.9/, where g if the free fall acceleration,  is the mass density for a particle material,  is the air viscosity. 2. The diffusion coefficient D of a small spherical particle in air in the kinetic regime in accordance with formula (5.3.2). 3. The zero-field mobility of a spherical particle K in the kinetic regime in accordance with formula (5.1.6). 4. The diffusion coefficient D of a small spherical particle in air in the diffusion regime in accordance with formula (5.3.2). 5. The zero-field mobility of a spherical particle K in the diffusion regime in accordance with formula (5.1.7). 6. The rate constant of association for the kinetic regime of particle coagulation in accordance with formula (9.3.4), where ro is the particle radius. 7. The rate constant of association in the diffusion regime of particle coagulation that in accordance with formula (9.3.3). 8. The rate constant of association in the diffusion regime of particle coagulation due to different velocities of particle falling in the gravitation field of the Earth in accordance with formula (9.4.1).

Appendix C

Transport coefficients of atomic particles in gases Table C.1. Self-diffusion coefficients D for atoms and molecules in a parent gas at the normal number density of atoms or molecules (N D 2:687  1019 cm3 ) [312, 313]. D; cm2 =s He Ne Ar Kr Xe

1.6 0.45 0.16 0.084 0.048

D; cm2 =s H2 N2 O2 CO

D; cm2 =s

1.3 0.18 0.18 0.18

H2 O CO2 NH3 CH4

0.28 0.096 0.25 0.20

Table C.2. Thermal conductivity coefficient of gases  at the pressure p D 1 atm in units 104 W=.cm  K/ [313]. T; K

100

200

300

400

600

800

1000

H2 He CH4 NH3 H2 O Ne CO N2 Air O2 Ar CO2 Kr Xe

6.7 7.2 – – – 2.23 0.84 0.96 0.95 0.92 0.66 – – –

13.1 11.5 2.17 1.53 – 3.67 1.72 1.83 1.83 1.83 1.26 0.94 0.65 0.39

18.3 15.1 3.41 2.47 – 4.89 2.49 2.59 2.62 2.66 1.77 1.66 1.00 0.58

22.6 18.4 4.88 6.70 2.63 6.01 3.16 3.27 3.28 3.30 2.22 2.43 1.26 0.74

30.5 25.0 8.22 6.70 4.59 7.97 4.40 4.46 4.69 4.73 3.07 4.07 1.75 1.05

37.8 30.4 – – 7.03 9.71 5.54 5.48 5.73 5.89 3.74 5.51 2.21 1.35

44.8 35.4 – – 9.74 11.3 6.61 6.47 6.67 7.10 4.36 6.82 2.62 1.64

226

Appendix C Transport coefficients of atomic particles in gases

Table C.3. Viscosity coefficient of gases in units 105 g=.cm  s/ at atmospheric pressure [312]. T; K

100

200

300

400

600

800

1000

H2 He CH4 H2 O Ne CO N2 Air O2 Ar CO2 Kr Xe

4.21 9.77 – – 14.8 – 6.88 7.11 7.64 8.30 – – –

6.81 15.4 7.75 – 24.1 12.7 12.9 13.2 14.8 16.0 9.4 – –

8.96 19.6 11.1 – 31.8 17.7 17.8 18.5 20.7 22.7 14.9 25.6 23.3

10.8 23.8 14.1 13.2 38.8 21.8 22.0 23.0 25.8 28.9 19.4 33.1 30.8

14.2 31.4 19.3 21.4 50.6 28.6 29.1 30.6 34.4 38.9 27.3 45.7 43.6

17.3 38.2 – 29.5 60.8 34.3 34.9 37.0 41.5 47.4 33.8 54.7 54.7

20.1 44.5 – 37.6 70.2 39.2 40.0 42.4 47.7 55.1 39.5 64.6 64.6

Appendix C Transport coefficients of atomic particles in gases

227

Table C.4. Transport coefficients for atomic particles in a gas in the first ChapmanEnskog approximation. Factor C

N% Formula

3.

8:617  10 cm =s 1 cm2 =s K D CD=T 11604 cm2 =.V  s/ 1 cm2 =.V  s/, p D D C T = =.N 1 / 4:617  1021 cm2 =s

4.

p K D C. T N 1 /1

5.

DC

6. 7.

p  D C T m=2

D CE=.T N/

1. 2.

/

D D CK T

Units 5

p

T =m=2

K in cm2 =.V  s/, T in K K in cm2 =.V  s/, T in eV D in cm2 =s, T in K D in cm2 =s, T in eV 1 in Å2 ,N in cm3 , T in K, in Ma 1:595 cm2 =s 1 in Å2 , N D 2:687  1019 cm3 , T in K, in Ma 171:8 cm2 =s 1 in Å2 , N D 2:687  1019 cm3 , T in eV, in Ma 68:11 cm2 =s 1 in Å2 , N D 2:687  1019 cm3 , T in K, in Ma 7338 cm2 =s 1 in Å2 , N D 2:687  1019 cm3 , T in eV, in Ma 4 2 1:851  10 cm =.V  s/ 1 in Å2 , N D 2:687  1019 cm3 , T in K, in Ma 171:8 cm2 =.V  s/ 1 in Å2 , N D 2:687  1019 cm3 , T in eV, in Ma 7:904  105 cm2 =.V  s/ 1 in Å2 , N D 2:687  1019 cm3 , T in K, in Ma 7338 cm2 =.V  s/ 1 in Å2 , N D 2:687  1019 cm3 , T in eV, in Ma 1:743  104 W=.cm  K/ T in K, m in Ma , 2 in Å2 7:443  105 W=.cm  K/ T in K, m in Ma , 2 in Å2 5:591  105 g=.cm  s/ T in K, m in a.u.m., 2 in Å2 1:160  1020 E in V=cm, T in K,  in Å2 , N in cm3 16 1  10 E in V=cm, T in eV,  in Å2 , N in cm3 2

Ma D 1:66054  1024 g is the atomic mass unit.

228

Appendix C Transport coefficients of atomic particles in gases

Explanations to Table C.4 1. The Einstein relation (5.3.1) for the diffusion coefficient of a charged particle in a gas D D K T =e, where D; K are the diffusion coefficient and mobility of a charged particle, T is the gas temperature. 2. The Einstein relation (5.3.1) for the mobility of a charged particle in a gas K D eD=T . 3. The diffusion coefficient of an p atomic particle in a gas in the first ChapmanEnskog approximation D D 3 2T = =.16N 1 /, where T is the gas temperature, N is the number density of gas atoms or molecules, is the reduced mass of a colliding particle and gas atom or molecule, 1 is the average cross section of collision. 4. The mobility of a charged p particle in a gas in the first Chapman-Enskog approximation K D 3e 2=.T /=.16N 1 /; notations are the same as above. 5. The gaspthermal conductivity in the first Chapman-Enskog approximation p  D 25 T =.32 m2 /, where m is the atom or molecule mass, 2 is the average cross subsubsection of collision between gas atoms or molecules; other notations are the same as above. 6. The p gas viscosity in the first Chapman-Enskog approximation  5 T m=.242 /; notations are the same as above.

D

7. The parameter of ion drift in a gas in a constant electric field D eE=.T N/, where E is the electric field strength, T is the gas temperature, N is the number density of atoms or molecules,  is the cross subsubsection of collision.

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Index

Aerogel 207 Aerosols 9 Aerosol association 170 Aerosol charge 62, 145 Aerosol growth 171, 182 Aggregation 151, 187 Aggregation in electric field 200 Aitken particles 9 Arrenius law 103 Atmospheric electric machine 21 Atmospheric ions 59 Attachment of atoms to cluster 151 Attachment of clusters to walls 88 Attachment mechanism of cluster growth 151 Black powder 17, 18 Boiling point 218 Braking force 67, 69 Chain aggregate 151, 188 Chapman–Enskog approximation 68, 69 Charge distribution of clusters 118 Charge separation in the atmosphere 19 Charging of clusters 118, 121 Charging of particles 13, 18 Chimney-stalk 13 Close packed structure 26 Clouds 19–21 Cluster 14, 15, 26 Cluster charge 118, 121 Cluster charging 142, 178 Cluster-cluster diffusion limited aggregation (CCA) 190 Cluster flow near exit orifice 85 Coagulation 151, 152 Coalescence 151, 160

Combustion of fuel 12, 13 Combustion of small particles 103 Combustion on particle surface 103 Coexistence of cluster phases 33 Comet tail 32, 33 Competition of fcc and icosahedral structures 26 Computer simulation 8, 55 Configuration excitation 31 Conversional factors 219 Conversion of atomic vapor in gas of clusters 150 Correlation radius 207 Cuboctahedral cluster 28, 29 Cumulus clouds 61, 183 Cyclone 13 Debye–Hückel radius 118, 120 Decompositon of molecules 174 Density functional theory 57 Dielectric-metal transition 43 Diffusion coefficient for clusters in a gas 72 Diffusion cross section 53 Diffusion regime of atom-cluster processes 8, 68 Diffusion regime of particle combustion 108 Diffusion regime of drop evaporation 96 Diffusion regime of quenching 100 Double layer 125, 127 Drift velocity of charged clusters 71 Druyvesteyn formula 147 Dusty plasma 21 Earth charging 20 Einstein relation 117

248 Electric atmospheric processes 18 Electric potential of charged particle 120 Elongated particle structures 202 Entropy of phase transition 32 Equilibrium cluster charge 121 Equilibrium of clusters with vapor 96, 161 Ergodic theorem 127 Evaporation rate for clusters 96 Face-centered cubic (fcc) structure 27 Fall velocity of particles 74 Fcc-cluster with central atom 46 Fcc-cluster without central atom 46 Filters 12 Flame 172 Flow through tube 78 Flow through orifice 85 Fluctuation of the melting point 44, 45 Fractal aggregate 158, 187, 210 Fractal dimensionality 192 Fractal fiber 206 Fractal tangle 207 Free fall cluster velocity 74 Fuks formula 118 Gas-kinetic cross section 51, 52, 78 Hard sphere model 5, 49 Hexagonal cluster 26, 28, 47 Hexagonal structure 26 Highest occupied molecular orbital (HOMO) 42 Icosahedral cluster 27, 28, 47 Impactor 13 Ion current to cluster 118 Ionic coat 129 Jelium model 58 Kinetic equation for cluster charging 140 Kinetic regime of atom-cluster processes 8, 68

Index Kinetic regime of drop evaporation 96 Kinetic regime of quenching 100 Kinetic regime of particle combustion 109 Knudsen number 69 Langevin formula 119 Lennard-Jones interaction 27 Lightning 19 Liquid cluster state 31 Liquid drop model 5, 38 Lowest occupied molecular orbital (LUMO) 43 Magic numbers 5, 29 Mass-spectrum of clusters 34 Mean free path of atoms 52 Mean free path of clusters 78 Mean free path of fractal aggregates 196 Melting point 218 Melting point of cluster 33, 34 Metal clusters in buffer gas 168 Mist 170 Mobility of clusters 69 Molecular dynamics method 167 Monomer 195 Mott–Hubbard correlation energy 42 Nearest-neighbor interaction 26 Noncentered fcc-cluster 46 Nuclei of condensation 9, 97, 150 Number densities of electrons and ions in particle field 119, 125 Ostwald ripening 8, 160 Parameter of cluster growth 97 Penetration length 181 Phase coexistence 33 Phase transition 30 Plasma torch 213 Poiseuille formula 78 Pollutants 10–12 Powder 14

249

Index Probe 7 Pyrotechnics 17 Quenching of metastable atoms on cluster surface 100 Rate constant of atom attachment to cluster 92 Rate constant of atom-cluster collision 53 Rate constant of cluster association 153 Rate constant of cluster-cluster collision 54 Rate constant of mutual neutralization of charged clusters 109 Rate of cluster evaporation 96 Rate of cluster growth 92 Reduced rate constant of collisions involving clusters 54 Relaxation of fractal particle 210 Relaxation time of moved particle 76– 78 Resonant charge exchange 131 Restructuring 195 Reynolds number 72, 80 Sant Elmo’s fires 185 Saturated vapor pressure 94 Saturn rings 23, 144 Scaling law for particle field screening 137 Screening charge 129, 136

Screening particle field 129 Self-diffusion coefficient 225 Self-consistent cluster-plasma field 127 Separation of charges in atmosphere 19, 75 Size distribution function 154, 163, 166 Smog 10 Smoke 12–14 Smoluchowski equation 153, 154 Smoluchowski formula 55, 93, 103 Solar wind 21 Soot 9, 10 Specific area of internal surface 208 Specific mean free path 70 Specific surface energy 46, 96 Spectrum of cluster 40 Statistic mechanics for ions in particle field 127 Sticking probability 100 Stokes formula 68 Stokes number 85, 86 Thermites 16 Trapped ions 132 Thermal conductivity coefficients 225 Viscosity coefficient of gas 226 Volcano processes 10, 24 Volcano smoke 10 Wigner–Seits radius 39, 40