F A R - F R O M - E Q U I L I B R I U M GAS K I N E T I C T H E O R Y ; R E A C T I V E S Y S T E M S A N D S O U N D P R O P A G A T I O N By Duncan George Napier B. Sc. (Chemistry), University of Waterloo , 1990 M. Sc. (Comp. Sci. & Systems), McMaster University, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1997 © Duncan George Napier, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted " by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. • Department of CU^TfZ^ The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract There exist many physical situations that involve large departures from thermody-namic equilibrium. These systems may be spatially nonuniform and/or time dependent. Traditional kinetic theory involves a perturbation type of approach that implies small departures from equilibrium. The present thesis is devoted to the development of the-oretical methods valid for systems far removed from equilibrium. The techniques are applied to a study of the departure from equilibrium in reactive systems, and for the propagation of acoustic waves in gases. The rates of gas phase reactions can be calculated from averages of the appropriate reactive cross sections over the distribution of velocities of the reacting species. Reactive processes, especially for reactions with activation energy, tend to remove translationally energetic species and the velocity distribution is perturbed from a Maxwellian. The extent of the departure from a Maxwellian can be estimated from solutions of the Boltz-mann equation. If there is a good separation of elastic and reactive time scales, steady solutions of the Boltzmann equation can be obtained with a procedure analogous to the Chapman-Enskog (CE) method. Nonequilibrium effects for model reactive systems with and without reverse reactions and in the presence and absence of products are stud-ied. The range of validity of the C E method is studied by comparing results from a C E method and an explicitly time-dependent solution method. The C E approach assumes a weak perturbation and is referred to as a Weak Non-Equilibrium (WNE) approach. A Strong Non-Equilibrium (SNE) approach that treats the distribution functions of each of the components as Maxwellians at different temperatures is applied to strongly per-turbed systems in which the reaction causes the temperatures of each species to differ ii from the system temperature. A third approach is a modification of the S N E approach and is referred to as Modified Strong Non-Equi l ibr ium ( M S N E ) . A l l three methods are compared wi th the results of an explicitly time-dependent solution. The C E method was found to be valid only when the ratio of elastic to reactive collision frequencies is greater than 1 0 5 . The propagation of acoustic waves through a gas perturbs the velocity distribution function of the gas. In a collision-dominated gas, the velocity distribution function can be approximated by a Maxwell ian and the phase velocity and attenuation of the sound wave can be determined from a dispersion relation derived from the Navier-Stokes equations of fluid dynamics. In the rarefied region, hydrodynamics is no longer valid and kinetic theory methods must be used. The method of solution of the Bol tzmann equation by Wang-Chang and Uhlenbeck ( W C U ) fails to describe the behaviour of sound waves as the frequency of oscillation approaches that of the interparticle collisions in the gas. A gener-alized Bol tzmann equation ( G B E ) introduced by Alexeev is applied to the sound problem and the results are compared with those of the W C U method and experiment. The k i -netic theory description for sound propagation in a simple gas introduced by Sirovich and Thurber (ST) is applied to binary mixtures of gases and the results compared wi th those of experiment. It is shown that the ST method provides better agreement wi th experiment than the W C U method. The results with the G B E do not converge. i i i Table of Contents Abstract ii Table of Contents vii List of Tables viii List of Figures x Table of Acronyms xv Acknowledgement xvi 1 Introduction 1 1.1 Non-Equilibrium Effects in Chemical Reactions for Uniform Systems . . . 6 1.1.1 The Effect of Products 6 1.1.2 Role of Species Temperatures 7 1.2 High-Frequency Sound Propagation in Rarefied Gases 8 1.2.1 Single Component Systems 8 1.2.2 Mixtures of Gases 9 2 Non-Equilibrium Reactive Systems : Effect of Reaction Products 10 2.1 Introduction . 10 2.2 The C E Approach - Weak Non-Equilibrium (WNE) 12 2.2.1 The Reaction A + C products 13 2.2.2 The Reaction A + A -> B +B 25 i v 2.2.3 The Reaction A + A ^ B + B 32 2.2.4 The Limit of Vanishing Product 40 2.3 Time-Dependent Solution of the Boltzmann Equation 44 2.3.1 The Reaction A + A B + B 44 2.4 Summary and Conclusions 54 3 Non-Equilibrium Reactive Systems : The Role of Species Tempera-tures 58 3.1 Introduction 58 3.2 The Reaction A+C ->• products : Strong Non-Equilibrium (SNE) . . . . 62 3.3 The Reaction A+C -»• products : Modified Strong Non-Equilibrium (MSNE) 66 3.4 Comparison of Non-Equilibrium Effects with WNE, SNE and MSNE For-malisms 68 3.5 Time-Dependent Species Temperatures; Comparison with W N E 72 3.6 Time-Dependent Solution for A + C —> products . . . 76 3.7 Summary of Results 86 4 Sound Dispersion in Single-Component Systems 88 4.1 Introduction 88 4.2 A Simple Hydrodynamic Theory of Sound 92 4.3 Boltzmann Equation for Small-Amplitude Disturbance 94 4.3.1 Method of Wang-Chang and Uhlenbeck 96 4.3.2 Sound Dispersion Relations 100 4.3.3 Solution of Dispersion Relations for the W C U Method 101 4.3.4 Solutions of the W C U Method 103 4.3.5 The Generalized Boltzmann Equation (GBE) Method of Alexeev 110 4.3.6 Comparison of W C U and G B E Methods 114 4.3.7 The Sirovich-Thurber (ST) Method 119 4.3.8 Effects of ST Modifications to the W C U Method 122 4.3.9 High-Frequency Cutoff and Free-Molecule Approximation 131 5 Anomalous Sound Dispersion in Two-Component Systems 137 5.1 Introduction 137 5.2 Boltzmann Equation for Small Amplitude Disturbance: Two-Component System 140 5.3 ST Method for Two Component System 142 5.4 Solution of the Dispersion relation for a Two Component Mixture . . . . 144 5.5 Results for a Two-Component System 148 5.6 Conclusion and Summary 153 6 Summary and Outlook 155 Bibliography 159 Appendices 167 A Isothermal Systems 167 B The Generalized Chemical Kinetic Equation of Alexeev 170 B.0.1 Physical Method of Derivation 170 B.0.2 Reactive System A + A products 171 B.0.3 Generalized Hydrodynamic Equation for a Reactive System . . . 172 B.0.4 Relation between Difference Equations and G B E Result 175 C ST Collision Matrix Elements 180 v i D The linearized B G K Model 183 v i i List of Tables 2.1 Comparison of 77 values calculated from C E , TJCE, a n d asymptotic values from the time-dependent two-temperature approach riasy for the system A+A —> B+B. The ratio of initial product to reactant density n!(0) /n 2 (0) is 1. See Figure 2.9 50 2.2 Comparison of asymptotic values, [AT^t)]^, calculated from integration of the Boltzmann equation and corresponding C E values [ATi(t)]cE, for the reaction A + A B + B 52 3.1 Comparison of asymptotic temperature values, ( ^ r - 1 - ) y , calculated from integration of hydrodynamic equations and corresponding one-term C E fr-r \ W N E values, [ T ) • The mass ratios for components 1 and 2, mi/m2 is 0.5, the density ratio ni/n2 = 1.0 and a/a* = an/a* = a22/a* — an/a*. 76 3.2 Comparison of time-dependent, WNE, SNE and MSNE values for 77 for various an/a* = ayi/a* and aX2/a*. The mass ratio, mi/m,2 = 3.0 and the initial density ratio, n i (0) /n 2 (0) = 2.0, and E*/kBT(0) = 10 81 3.3 Comparison of time-dependent, W N E and SNE values for 77 where a/a* = c n / c * = 022/0* = 0\2J'a*'. The mass ratio, mx/m^ = 3.0 and the initial density ratio n i (0 ) /n 2 (0 ) = 2.0 82 3.4 Comparison of time-dependent, W N E and SNE values for 77 for various an/a* = <722/cr* and cr1 2/cr*. The mass ratio, mx/m^ — 3.0 and the initial density ratio m(0)/n2(0) = 2.0 83 V l l l 3.5 Comparison of time-dependent and MSNE values for r\ for various crn/a* = 022 /c* and oyijo*. The mass ratio, mi/rri2 = 3.0 and the initial density ratio ni(0)/n2(0) = 2.0 84 4.1 The first five axially symmetric Burnett functions 97 C l Some notation used in the present work and equivalent notation in Sirovich and Thurber [128] .'• 182 ix List of Figures 2.1 Variation of 77 and 77 versus e* for A + C —> products. 77 is given by the solid line and 77 by the broken line. The mass ratio m\jm2 and density ratio n i / n 2 are equal to (A) 2 and 8, (B) 10 and 1,(C) 2 and 1, (D) 10 and 1/8 respectively; aE = au — <T12 ='cr2 and aE/aR• = 1. 20 2.2 Variation of A T 7 / T versus density ratio A = ni/n2 for the system A + C —» products. The mass ratio mi/m2 equals (A) 1/8, (B) 1/4, (C) 1 and ( D ) 2; e* = 4 and oEjaR = 1. A T 7 = 0 at mini/m2n2 = 1. . 23 2.3 Variation of A T 7 / T versus M l 5 mass fraction of A, for the system A + C -+ products. The density ratio n i / n 2 equals (A) 1/8, (B) 1/2, (C) 1 and ( D ) 2; e* = 4 and aE/aR = 1 24 2.4 Variation of 77 versus e* for A + A —> B + i? compared with A + A —» products. A + A —> products is shown by curve (a). For A + A —)• B+B, the density ratio n 2 / n i equals (b) 1.0, (c) 0.5, (d) 0.1 and (e) 0.01; aE/aR = 1. The value of 77 for A + A -+ B + B in the limit 8 -> 0 is shown by ( o ). 31 2.5 Variation A T 7 / T versus the log of density ratio S = n2/ni for A + A —> B + B; e* = 3 and aE/aR = 1 33 2.6 Variation of 77/ for A + A # B+B versus e*j. The reverse activation energy, e*, equals (A) 10, (B) 5, (C) 3 and (D) 0; aE/aR = 1. The density ratio n2jn\ equals (a) 0, (b) 0.01, (c) 0.1, (d) 0.5 and (e) 1. The variation of 77 for the system A + A—> B + B in the limit S —> 0 is shown by ( o ). 38 x 2.7 Variat ion of A T 7 / T versus the log of density ratio S = n 2 / n i for i + A ^ B + B. Forward and reverse activation energies are (A) e*f = 3 and e* = 3 and (B) e*f = 3 and e* = 0 , aE/aR = 1 39 2.8 Variat ion versus e* of the ratio of 77 for A + A —> products to r\ for A + A —> B + B , in the l imit <5 —>• 0. oEjoR = 1; the solid line is the ratio of converged values of 77; the broken line is the ratio of two-term solutions. 43 2.9 Rat io of the time-dependent rj(t) and corresponding C E T\CE for the system A + A —y B + B. The reduced activation energy e* equals (a) 13, (b) 9, (c) 6 and (d) 4. The elastic collision cross sections for each curve are listed on Table 2.1 49 2.10 Comparison of time-dependent and C E temperatures for different cross sections for A + A —>• B + B. Time-dependent, A T 7 / T shown by the solid line and corresponding C E values, shown by the broken line. The ratio of elastic to reactive hard sphere collision cross sections, CTE/CR, and reduced activation energy, e*, are equal to (A) 1, 9 (B) 10, 9, (C) 1, 13 and (D) 200, 13 51 2.11 Comparison of time-dependent and C E temperatures for different density ratios for A + A —> B + B. Time-dependent A T T / T is shown by the solid line and the corresponding C E values, by the broken line. The density ratios n2/ni equal (A) 1, (B) 0.5, (C) 0.1 and (D) 0.01; e* = 2 and &E/O~R = 1L 53 2.12 T ime dependence of the ratio rj(t) (A + A -+ B + B) to the C E value of 77 (A + A products). The reduced activation energy, e*, and the ratio of elastic to reactive hard sphere collision cross section, O~E/°~RI A R E equal to (a) 6, 20 (b) 4, 20 (c) 13, 200, (d) 6, 200, (e) 4, 200 and (f) 2, 1; n 2 (0 ) /n i (0 ) = 10~ 5 55 x i 3.1 Variation of 77 versus e* computed from the SNE, W N E and MSNE meth-ods for different mass ratios. The mass ratio, m 1 / m 2 , equals (a) 1, (b) 1.5, (c) 3, and (d) 5. an/a* = a22jo* = aV2/a* = 1 and n i /?7, 2 = 1. Figure (a) shows a one component system, A + A —> products, the solid line, and a two component system A + C —>• products, the broken line 70 3.2 Variation of 77 versus e* for the SNE, W N E and MSNE methods for the reaction A + C —¥ products for the different density ratios. The density ratio, ni/n2, equals (a) 1.5, (b) 2, (c) 3, and (d) 5. o~n/a* = a22/cr* = o~vi/o~* = 1 and m\/m2 = 1 73 3.3 Time-dependence of the hydrodynamic corresponding C E fractional tem-perature separation. The hydrodynamic value of (1 — T 7 / T ) is given by the solid line and the C E value, a\*\ is given by the broken line. The elastic collision cross sections an/a* = a22/a* = a\2ja* are (a) 1, (b) 10 and (c) 100, with E*/kBT(0) = 3, ni(0)/n2(0) = 5 and ml/m2 = 1. . . . 75 3.4 Time dependence of the ratio n(t) to r 7 M S N E , 7 7 ™ and 7 7 S N E for Ti(0)/r 2(0) = 1. The ratio of elastic to reactive hard sphere collision cross section an/cr* = 1000 while C T I 2 / C T * equals (a) 1000, (b) 200, (c) 20, and (d) 1. The ratio CTII/CT22 = 1, E*/kBT(0) = 10, m i / m 2 = 3 and ni(0)/n 2(0) = 2. 80 3.5 Time dependence of the ratio r/(t) to 7 7 M S N E , T ? w n e and .77 M S N E for Ti(0)/T 2(0) = | . The ratio of elastic to reactive hard sphere collision cross sections is 1000 for o n / a * and CT12/CT* equals (a) 1000, (b) 200, (c) 20, and (d) 1. The ratio on/o-22 = 1, E*/kBT(0) = 10, ml/m2 = 3 and ni(0)/n 2(0) = 2. . . 85 4.1 Log-log plot of W C U method results for (A) f3u)~lv$ and (B) aco^vo versus 1/Kn. The number of moments retained are 4, 6, 16, 20 and 36, NS is the Navier-Stokes result and experimental results are denoted by o 108 x i i 4.2 Log-log plot of G B E method results for (A) fiuj~lv0 and (B), aco~lv0 versus 1/Kn. The number of moments retained are 4, 5, 6, 16, 20 and 36 and experimental results are denoted by o 118 4.3 Log-log plot of ST solution results for (A) (3cu~1v0 and (B) au>~lv0 versus 1/Kn. The number of moments retained are 3, 4, 9 and 12, arrows denote the collisionless approximation and experimental results are denoted by o. 124 4.4 Graph of /3LO~1VQ versus 1/Kn for (A) 6-term approximation with A equal to A 3 2 and — v, (B) 9-term approximation with A equal to A 3 3 , —v and 0, and (C) 12-term approximation with A equal to A 4 3 , — v and 0 128 4.5 Graph of acu~1vQ versus 1/Kn for (A) 6-term approximation with A equal to A 3 2 and —v, (B) 9-term approximation with A equal to A 3 3 , — v and 0, and (C) 12-term approximation with A equal to A 4 3 , — v and 0 129 4.6 (A) Bounding region in complex £-plane for roots of dispersion relation and winding plots for 4-term D, 1/Kn equal to (B) 5 and (C) 0.05. 133 5.1 Plot of Re[D] and Im[D] versus « w _ 1 % e for a He-Xe mixture ATH e = 0.45, 1/Kn= 10.5, and RuTxvHfi = 3.23. 146 5.2 Plot of Re[D] and Im[D] versus a u _ 1 % e for a He-Xe mixture X H e = 0.45, 1/Kn = 10.5 where f5to~lvHe equals (A) 3.23 and (B) 2.13. Intersection points of Re[D] and Im[D] are (a), (b) and (c). . 147 5.3 Plot of / versus f3to~lvne for (a) and (c) branches of the trajectory of Re[D] = Im[D] 149 5.4 Plots of j f lu r^He versus 1/Kn for XEe equal to 0.8, 0.7, 0.45 and 0.3. The ST result is the solid line, Johnson and Bowler's calculation is shown by the broken line and experimental results are denoted by o 151 xiii 5.5 Plots of ato^VEe versus l/Kn for various X^e equal to 0.8, 0.7, 0.45 and 0.3. The ST result is the solid line, Johnson and Bowler's calculation is shown by the broken line and experimental results are denoted by o. . . 152 B . l Plot of (1/n—1) versus time for a second-order reaction A + A —> products with E*/keT = 2. The solid line represents the standard hydrodynamic solution, i?, while the dashed line and (+) represent the analytical and numerical solutions, respectively, of the G H E 177 B.2 Semilog plot of the reaction rate constant for A + A —> products versus E * / k B T . The hydrodynamic solution, R, is denoted by O, while the nu-merical and analytical solutions to Eq. (B.4) are denoted by (+) and the dashed line, respectively 178 xiv Table of Acronyms Acronym Definition B G K Bhatnagar-Gross-Krook C E Chapman-Enskog G B E Generalized Boltzmann Equation G H E Generalized Hydrodynamic Equation GJ Gross-Jackson MSNE Modified Strong Non-Equilibrium NS Navier-Stokes SNE Strong Non-Equilibrium ST Sirovich and Thurber W C U Wang-Chang and Uhlenbeck W N E Weak Non-Equilibrium X V Acknowledgement I would like to acknowledge the help, advice and guidance of my supervisor, Dr. Bernard Shizgal. The following work would not have been possible without his dedicated support. I would also like to acknowledge the personal support and encouragement of my wife, Skyelar, over these sometimes difficult years. In addition, I would like to ac-knowledge financial support provided by the Natural Sciences and Engineering Research Council of Canada, the University of British Columbia University Graduate Fellowship, McDowell and Liard Scholarships as well as the Department of Chemistry. x v i Chapter 1 Introduction The Boltzmann equation is the basis for almost all studies of gas kinetic processes. It is believed to be valid in the dilute gas region where only binary collisions between particles are considered. For a gas at equilibrium, the Boltzmann equation predicts that the distribution of velocities in the gas is a Maxwellian. When a gas is perturbed from its uniform state, it has a tendency to return to its uniform state through molecular motion and intermolecular collisions. The tendency towards a uniformity results in the transport of physical properties through the system. For example, ordinary diffusion involves the transport of mass from one region to another as a result of a density gradient, viscosity is the transport of momentum resulting from velocity gradients and thermal conductivity is the transport of thermal energy that is driven by thermal gradients in the gas. A typical application of gas kinetic theory is the calculation of the transport proper-ties of simple gases and mixtures from molecular properties. This is traditionally carried out using the Chapman-Enskog (CE) method described in standard references [1-4]. A solution of the Boltzmann equation is obtained as an asymptotic expansion of the so-lution in Kn where Kn is a Knudsen number, a dimensionless ratio of the microscopic mean free path to the characteristic length scale for the variation of macroscopic vari-ables. The C E method is only valid in the collision-dominated region, Kn —> 0 where the velocity distribution function for the particles is close to a Maxwellian. The gas is driven to equilibrium by collisions between its constituent particles. For a collision dominated system, there is a clear separation between the collisional time scale and the time scale 1 Chapter 1. Introduction 2 for changes of macroscopic quantities. The C E method is known to be invalid for strongly non-equilibrium systems for which the required separation of length and time scales is not obtained [5-9]. For highly rarefied systems, Kn —> oo, the mean free path of the particles is ex-tremely large relative to the macroscopic length scale. The velocity distribution function is significantly non-Maxwellian and the C E method is not applicable. The constitutive relations, such as Fourier's law, which relates the heat flux to the temperature gradient and serves to close the hydrodynamic equations, are no longer valid. A velocity distri-bution function can be obtained in the collisionless limit from Liouville's theorem. This method is used, for example, to determine the velocity distribution function in planetary exospheres [10,11]. Treatment of the intermediate region between collision-dominated and collisionless regions often presents the greatest difficulties. This thesis considers gases that are per-turbed by either a chemical reaction or sound waves. When there is no separation of perturbative and relaxational time scales, conventional non-equilibrium methods break down. This is the so-called intermediate region for which the time scales for the per-turbation and relaxation processes are approximately the same order of magnitude. It is the intermediate region that is the focus of this work. The term far-from-equilibrium refers to conditions under which non-equilibrium treatments that are applicable in the collision dominated and/or collisionless regions break down and no longer describe the system in question. The description of transport for such far from equilibrium systems have important applications in nonlocal heat transport in plasmas [5], hypersonic flows [6], diffusive flows [7,8] as well as astrophysical applications [9], to mention a few. This introductory section provides a broad overview of the subject of the thesis. More details of the particulars of the theoretical approach are given for reactive systems and sound wave propagation in Sections 1.1 and 1.2, respectively. The first half of this work presents Chapter 1. Introduction 3 studies of the non-equilibrium effects associated with reactive systems. The reaction per-turbs the velocity distribution in the gas which in turn perturbs the rate of the reaction. The main objective of these studies is to determine the non-equilibrium rate of reaction where n is the correction to the equilibrium rate of reaction A;(°). Non-equilibrium ef-fects in reacting systems have had a very long history beginning with the classic papers by Kramers [12], Prigogine and Xhrouet [13], and the unpublished work of Curtiss [14]. These early works and subsequent publications by Present [15] and Mahan [16] consid-ered low order expansions of the Chapman-Enskog equations for these reactive systems. Eliason and Hirschfelder [18] and Ross and Mazur [17] provided formal approaches to the study of such non-equilibrium reactive effects. Monchick [19] applied entropy-variational methods to the problem of non-equilibrium rates of reaction. In the early 1970s, Shizgal and Karplus [20-23] provided converged solutions to general single and multicomponent reactive systems. A complete bibliography up to 1970 can be found in these papers. A comparison of the C E method and Grad's moment method [24] has been reported by Eu and Li [25]. Eu and Li have studied the influence of spatial inhomogeneity on the non-equilibrium rate of reaction. Xystris and Dahler [26] considered the influence on chemical reactions of mass and momentum transport. Interest in this problem has continued to the present. The coupling of temperature and density gradients to the rate of reaction has also recently been studied by Nettleton [27,28] and has potential applications in the modelling of inhomogeneous chemically reacting flows. Many recent studies of non-equilibrium effects in reactive systems involve Monte Carlo simulations of the dynamics of relaxation and reaction. Baras and Malek-Mansour [29,30] as well as Popielawski, Cukrowski and co-workers [31-38] have carried out Monte Carlo simulation experiments which they view as a means of verifying the calculated results, since direct Chapter 1. Introduction 4 experimental verification of these non-equilibrium effects have not been reported to date. Qin and Dahler have recently proposed a procedure for determining the spectral char-acteristics of light that is scattered from chemically reacting fluid [39]. The purpose of their method is to determine the effects of reaction on the scattering function and other observable dynamic response characteristics of the mixture. Methods such as light scat-tering may provide a means for direct observation of non-equilibrium effects in chemically reacting systems. The present work was partly motivated by the recent work of Cukrowski et al. [31], who stated that much of the earlier work in this field was in error due to neglect of a separation of species temperatures that occurs in reacting mixtures. This separation of species temperatures was pointed out by Shizgal and Karplus [21], who observed that neglecting this separation in the manner of Pyun and Ross [40] removed an important contribution to the non-equilibrium rate of reaction. The work of this thesis involves a critical evaluation of the work of Cukrowski et al. It is shown that the system studied by Cukrowski is different from that studied by previous workers [13-16,18,17,20-23,40]. This work also provides a detailed study of the role of species temperatures and the order of magnitude of the non-equilibrium reactive effects. The method used to study the role of species temperatures is based on the Strong Non-Equilibrium (SNE) and Modified Strong Non-Equilibrium (MSNE) methods recently developed by Pascal and Brun [41]. The usual C E approach for small departures from equilibrium is termed Weak Non-Equilibrium (WNE). These methods are an outgrowth of the so-called generalized methods developed by Kogan [42] and Alexeev [43-45]. Alexeev has recently proposed a generalized Boltzmann equation (GBE) and applied it to a series of far-from-equilibrium problems, including that of sound dispersion in rarefied gases [46,47]. Modelling correct sound dispersion behaviour in rarefied gases is a difficult problem that was first addressed almost forty years ago by Wang-Chang and Uhlenbeck (WCU) Chapter 1. Introduction 5 [48]. The perturbation is a density fluctuation that propagates as a sound wave through coherent or in-phase interparticle collisions in the gas. Elastic intermolecular collisions drive the system back to equilibrium between cycles of the oscillation. If the frequency of the oscillations is sufficiently high, or if the rate of collisions between gas is particles sufficiently low, far-from-equilibrium conditions arise as the particles of the gas fail to equilibrate between cycles of the oscillation. The W C U method does not agree with experimental results at high sound frequencies and low pressures, as was demonstrated by the very extensive calculations of Pekeris et al. [52,53]. This inadequacy has attracted many proposed solutions, including the modified free-flow treatment of Kahn and Mintzer [54], the modified spectral representation of the series solution of the linearized Boltz-mann equation proposed by Buckner and Ferziger [55] and the division of the Boltzmann collision operator by the drift term that Sirovich and Thurber [56] proposed. These ap-proaches have succeeded in extending the W C U method to varying degrees. The G B E method [46] has recently been proposed by Alexeev as a method for treating the sound dispersion problem over all frequencies and pressures. The G B E of Alexeev is found in this work to break down at high frequencies. There has been a renewed interest in gas kinetic theories of sound dispersion result-ing from predictions by Johnson and Huck [57] as well as Campa and Cohen [58] of the existence of novel sound propagation modes in mixtures of gases having disparate masses. These novel propagation modes have since been observed experimentally in He-Xe mixtures by Johnson and Bowler [59] who refer to them as anomalous sound modes. The anomalies refer to sudden, unexpected changes in sound speed and attenuation over small changes in gas density and sound frequency and the measurement of several, dis-tinct propagating disturbances, all of which have no interpretation based on conventional hydrodynamics. These phenomena are attributed to the decoupling of light and heavy components in the mixture. More recent measurements by Clouter et al. [60], Montfooij Chapter 1. Introduction 6 et al. [61] and Wegdam and co-workers [62-67] have established the existence of these anomalous sound modes for a series of mixtures of gases having disparate masses. The method of calculating sound dispersion properties employed by Sirovich and Thurber for a simple gas is extended to mixtures of gases, and the problem of anomalous sound dispersion behaviour in gases is investigated. 1.1 Non-Equilibrium Effects in Chemical Reactions for Uniform Systems 1.1.1 The Effect of Products The work of Chapter 2 is motivated by Cukrowski and co-worker's [31] treatment of non-equilibrium effects in gas phase reactions. Cukrowski et al. considered model reactions of type A + A ^ B + B, with and without the reverse reaction, and studied the effect of the separation of the species temperature on the rate of reaction. As mentioned previously, the authors contend that earlier treatments [13-16,18,17,20-23,40] are, in their own words, "seriously flawed " [31]. In this work, non-equilibrium effects are considered for model reactive systems A + A # B + B with and without reverse reactions as well as the system A + C —> products for which the products of the reaction are removed. It is shown that when products of a reaction are included, a choice has to be made as to whether a reverse reaction occurs or not. The inclusion of products and the choice of reversibility or irreversibility determines the correction to the reaction rate. An explicitly time-dependent solution for the irreversible reaction A + A —» B + B was also studied and compared to the CE-type solution. The time-dependent method, unlike the C E method, makes no time scale assumptions and gives the region of validity of the C E method quantitatively. It was demonstrated that the ratio of elastic to reactive collision frequencies must be extremely large (greater than 105). It is shown in Appendix A that Chapter 1. Introduction 7 the system studied by Cukrowski is different in several important ways from most of the previous work that it refers to. 1.1.2 Role of Species Temperatures As mentioned before, chemical reactions with an activation energy tend to remove trans-lationally energetic particles of the reactant species. The reactive process causes the temperatures of the two species to differ from the system temperature and this effect can play an important role in the determination of the departure of the rate coefficient from the equilibrium value. In Chapter 2, this effect is examined with the C E method of solution of the Boltzmann equation which treats the reactive processes as a weak pertur-bation. In Chapter 3, extensions of the C E (WNE) approach which involve the expansion of the distribution functions about Maxwellians at different temperatures, referred to as Strong Non-Equilibrium (SNE), and Modified Strong Non-Equilibrium (MSNE) are con-sidered. Departures of the rate coefficients from their equilibrium values are computed and compared for each of the three methods, along with an explicitly time-dependent solution of the Boltzmann equation. It is concluded from the time-dependent studies that the C E method fails when SNE conditions apply and the SNE method fails under conditions when the C E method is appropriate. The MSNE is an improvement on both and seems to work over a range of conditions. The MSNE uses the concept of the generalized C E method introduced by Kogan and Alexeev. This generalized C E method involves retaining high-order terms in the SNE solution so that the SNE solution is consistent with the W N E solution in the limit of weak non-equilibrium. Alexeev [46] recently extended these generalized C E methods to the Boltzmann equation and developed a G B E . From the hydrodynamic approximation of the G B E , Alexeev [46] has in turn developed a generalized hydrodynamic theory of which conventional hydrodynamic theory is a subset. Alexeev claims to have successfully Chapter 1. Introduction 8 applied generalized hydrodynamics to the problem of high-frequency sound propagation in rarefied gases. A brief evaluation of Alexeev's approach to chemically reactive systems is presented in Appendix B. 1.2 High-Frequency Sound Propagation in Rarefied Gases 1.2.1 Single Component Systems Wang-Chang and Uhlenbeck [48] were the first to apply gas kinetic theory to the prob-lem of sound propagation in monatomic gases. Around the time of Wang-Chang and Uhlenbeck's theoretical studies, experimental results obtained by Greenspan [49,50] and Meyer [51] showed that the Navier-Stokes equations gave incorrect results for the speed and attenuation of sound at high frequencies in rarefied gases (a region known as the Knudsen region). The solution of Wang Chang and Uhlenbeck extended the validity of theoretical sound speed and attenuation calculations only slightly further into the Knud-sen region than the Navier-Stokes equation. Alexeev has applied the generalized hydrodynamic equations to the problem of sound dispersion at high frequencies and low pressures and obtained qualitative agreement with experiment [46]. In Chapter 4, the sound dispersion problem is tackled directly from the Alexeev's G B E , in the same manner that Wang-Chang and Uhlenbeck solved the gas kinetic theory of sound dispersion. The work of Section (4.2) shows that the G B E does not accurately describe real properties of sound waves beyond the Navier-Stokes approx-imation. The failure of the W C U method to describe the properties of sound at high frequen-cies and low pressures resulted in a period of active study in the mid-1960s of the gas kinetic theory of sound propagation. Several solutions were proposed, most notably that Chapter 1. Introduction 9 of Sirovich and Thurber (ST) [56]. The ST method succeeded in giving values that agreed with experimental results. The reasons for its success do not seem to have been fully established, and the latter part of Chapter 4 is a clarification of some of the issues surrounding Sirovich and Thurber's solution of the sound dispersion problem: A concise study of the ST method and its limitations is motivated by the need to interpret recent experimental data obtained for sound dispersion in mixtures of gases of disparate masses. 1.2.2 Mixtures of Gases Recent experimental studies have confirmed anomalous sound behaviour in mixtures of gases having disparate masses that was first predicted by Johnson and Huck [57]. Anomalous dispersion behaviour has been observed in the form of multiple coexisting sound modes, degenerate modes and sound mode discontinuities that are not described by conventional hydrodynamic treatments. Johnson [59] and Wegdam [62] have ap-plied Burger's two-temperature flow equations for composite gases [68] in order to de-scribe anomalous behaviour, with some success. Their methods are based on the Grad thirteen-moment method [24] and are therefore limited to hydrodynamic moments. It was reasoned that the ST method, based on its success in describing single-component systems, would also give good results for multicomponent systems. In Chapter 5, the ST method has been extended to multicomponent systems and the results for calculations on a He-Xe system are compared with experimental results and calculations of Johnson and Bowler [59]. The agreement of phase velocities and attenua-tion constants obtained by two-component ST method are found to result in qualitative agreement with the calculation of Johnson and Bowler. Chapter 2 Non-Equilibrium Reactive Systems : Effect of Reaction Products 2.1 Introduction Non-equilibrium effects associated with reactive systems have received considerable at-tention in the literature for a long time [12-23,25-38]. Many calculations have been carried out for reactive systems of the type, A + C ^ products (2.1.1) with the complete neglect of the products. Reactions tend to selectively remove transla-tionally energetic particles of the system. This results in a perturbation of the velocity distribution functions which in turn perturbs the reaction rate. The main objective of this chapter is to calculate the influence of reaction products on the translational distribution functions of the reactants. The extent of the departure of the translational distribution function from equilibrium (a Maxwellian) is expressed as the fractional decrease of the non-equilibrium rate coefficient, k, from the equilibrium value, k^°\ that is, The change in the distribution function is given by the Boltzmann equation. The ap-proach employed by most workers involves the application of the Chapman-Enskog (CE) method of solution of the Boltzmann equation [1]. The reactive process is assumed to be a small perturbation on the system. The C E approach yields a.very special solution of the Boltzmann equation referred as a "normal solution" [1] for which the time dependence 10 Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 11 of the distribution function is implicit through the time dependence of the density, n(t), and the time variation of the temperature, T(t), that is, f(c,t} = f(c;n(t),T(t)). Cukrowski et al. [31] recently carried out a study of the non-equilibrium effects for the reaction A + A -> B + B . (2.1.2) The treatment of Cukrowski et al. is approximate as they assumed that the distribu-tion functions of the two species were local Maxwellians at different temperatures and the reaction was treated as an isothermal process. The Boltzmann equations are then approximated by the time-dependent equations for the number densities and the species temperatures. This approach is similar to previous work on a time-dependent theory of hot atom reactions [72] and temperature relaxation [73]. The results of calculations by Cukrowski et al. were compared with estimates of rj obtained with the C E analysis by Cukrowski, Fritzsche and Popielawski [32] and from computer simulations. The main objective of the work by Cukrowski et al. [31] was to compare the non-equilibrium effects of A + A—> B + B with those of A + C —» products. In this chapter, the non-equilibrium effects of the above reactions are re-examined with particular attention to the effect of the reverse reaction B + B -+ A + A and the density dependence of the products. Shizgal and Karplus demonstrated long ago [21,22] that the non-equilibrium effects in reactive systems are particularly sensitive to whether the system is adiabatic (dT/dt ^ 0) or isothermal (dT/dt = 0). If dT/dt ^ 0, the non-equilibrium effects have an explicit dependence on the heat of reaction [13,35]. It is also shown that in the time-dependent studies of the reaction with products, A + A -+ B + B, the value of dT/dt in the limit of vanishing products requires some very careful analysis [32,33]. The limit of vanishing products, studied by Pyun and Ross [40], and Fitzpatrick and Desloge [70], is of particular interest to this work. The difference between results obtained Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 12 from a C E analysis of the model reaction A + C —> products, where the product species are completely ignored, and results obtained for the model reaction A + A —> B + B with the products are taken into account are studied in detail here. The reversible reaction A + A ^ B + B is also studied in comparison with the other systems. Shizgal and Karplus [22] conducted a C E analysis of the general reversible reaction A + B C + D, which includes the effect of the reverse reaction. Section 2.2.1 describes the C E method of solution of the Boltzmann equations for the distribution functions of the two species involved in the reaction A + C —> products. The extent of non-equilibrium effects in terms of the quantity r\ is determined. The useful-ness of the Shizgal-Karplus temperature, introduced by Cukrowski and coworkers [32] is evaluated. The irreversible reaction A + A -+ B + B and the reversible reaction A + A B + B are considered in like manner in Sections 2.2.2 and 2.2.3, respectively. An important aspect of the analysis is the use of microscopic reversibility in the case of the reversible reaction and the lack thereof for the irreversible reaction. Although these three reactive systems are very similar, the kinetic theory treatment of the non-equilibrium effects in-volves several important subtle aspects that are discussed in detail. In Section 2.3, the results of a time-dependent solution of the Boltzmann equation is presented. The range of validity of the C E approach is determined from this analysis. 2.2 The C E Approach - Weak Non-Equilibrium (WNE) The C E method is used to solve the Boltzmann equation for which reactive processes are treated as a weak perturbation. This has been referred to as Weak Non-Equilibrium (WNE) by Pascal and Brun [41] and alternatives to this treatment are the subject of Chapter 3. In the following sections of this chapter, detailed calculations using the C E Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 13 approach are carried out for the hard sphere elastic and line-of-centers reactive model system. The hard-sphere reactive model has been used in earlier work [20-22]. The differential elastic cross sections are taken to be hard spheres (O~E = d^/4) and the total reactive cross section is the line-of-centers cross section given by, a* = naR(l-E*/E) E > E* (2.2.1) = 0 E < E*. where E* is the threshold energy and oR = dR/4.. The hard sphere elastic and reactive diameters are dE and dR, respectively. 2.2.1 The Reaction A + C -> products The distribution functions for the two species are denoted by 7 and 77 equal to 1 for species A and 2 for species C. For this reaction, the distribution functions, are assumed to be given by two coupled Boltzmann equations of the form, dt dt J JU'J' -fif]ongdQdc + J J[f'j'2-fif2]cTi2gdndc2\ - j J hf2o*gdadc2 (2.2.2) / J[f'2f' -f2f]o-229dndc + J j[f[f2 - hf2]<J12gdndci\ - j J hh^gdtodd (2.2.3) where alr] and 0* are the elastic and reactive cross sections, respectively. The first collision operators on the RHS of Eqs. (2.2.2) and (2.2.3) are the self collision terms for A - A and C-C elastic collisions. The second collision operators take account of A - C elastic collisions and couple the two equations. The reactive terms with the reactive cross section represent the loss of both species. The other quantities have their usual definitions [20-22]. The notation employed in Eqs. (2.2.2) and (2.2.3) is standard [1] Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 14 where the unlabeled distribution and velocity variable refer to the other particle in a binary collision. The term g is the relative velocity, |cx — c2|, of the two bodies involved in the collision. For all applications in the present chapter, ou = o22 = cr12 = O~E and the ratio OR/OE is varied. These time-dependent equations are solved with the assumption that the reactive cross sections are small relative to the elastic cross sections. The parameter e « OR/OE has been inserted into the Boltzmann equations to take this ordering into account. The method of solution of these coupled Boltzmann equations is as described in [21]. The velocity distribution function is perturbed from its equilibrium value, / 7 = / 7 ° ) [ l + 6 ^ 7 ] , (2.2.4) where tp7 is the perturbation from equilibrium and e is the strength of the perturbation. Equations of order 1/e obtained from Eq. (2.2.4) in Eqs. (2.2.2) and (2.2.3) determine that the zero order distribution functions, /(°) = n7(t)[m7/27rkBT(t)]3/2 exp[—m1c^/2kBT(t)\, are Maxwellians characterized by time-dependent number densities, n7(t), and the single temperature, T(t). The C E approach involves the assumption that (to lowest order) the time dependence of the distribution function is given by, d^_dj^/dniy°) djW(dTV0) dt - dn, [dt J + dT [dt) ' (2-2'5) where the time variation of the densities is determined from the Boltzmann equations, and given by, For this reaction, there is a loss of energy owing to reactive collisions and the variation of the temperature (to lowest order) is given by Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 15 C 7 (c 7 ) — (2.2.10) With the substitution of Eq. (2.2.4) into the Boltzmann equations, Eqs. (2.2.2) and (2.2.3) and use of Eqs. (2.2.5-2.2.7), the C E equations for the perturbations tp7 are obtained by equating terms zero-order in e, . / / / i ( 0 ) / ( 0 ) [ ^ + ^ - ^ i - ^)augdQdc +J j fl{0)f2[0)[i;l1-ij1}o12gdndc2 + J J fi0)A0)bl>2 - H<?i2gdndc2 = f^Cricr) (2.2.8) / //f/ ( 0 )h/4 + V''-V'2 - ^o-22gdndc + J j fi^f^^-H^gdndc, + J J /fVf^l - AW12gdQdCl = /2(0)C2(c2),(2.2.9) where the inhomogeneous terms are defined by where £ 7 = m 7 c 7 / 2 £ ; £ T and, R?(c7) = U f^o*gdndcv. (2.2.11) The solution of these coupled linear integral equations is obtained with the expansion of the unknown functions -07 in Sonine (Laguerre) polynomials, S 7 ^ ( £ 7 ) defined in [20-22]. With the expansions, ^ ( £ r ) = X>! T )S 7 J )(^), i=l the Boltzmann equations, Eqs. (2.2.2) and (2.2.3) are reduced to the set of algebraic equations for the coefficients a-7' given by, JT [(nJlS?, S?] + n 1 n 2 {5? ) , S[»}) af + nin2{S?, Sf}af] = n^af, (2.2.12) 3=1 and N E [{n22[S?, Sf] + n i n 2 { S i \ Sf}) af + nxn2{S%\ Sf}af] = nxn2af\ (2.2.13) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 16 where n The bracket matrix elements [S^\S^] are nin2ay' = -S0tA07> - S^-1 £ 4 T ) + 4 (2.2.14) 7 (2.2.15) x [Sf + S{l)' - S7l) - S^o^gdQdc^dc, and the brace matrix elements {S^S^} are n7nv{S7j\S^} = J J J fWfWsPisW - S^o^gdQd^dc, (2.2.16) n 7 n , { 5 f , 5 « } = J j j' fWf}j0)SW[sW -Sf}oirigd£ldcvdc7. (2.2.17) The bracket and brace matrix elements were evaluated for a hard sphere elastic cross section from analytic expressions obtained by Lindenfeld and Shizgal [94]. The integrals are the Sonine moments of the reactive collision frequencies and evaluated for the line of centers reactive cross section (Eq. (40) of [21]). The concentration dependence in the middle term of the RHS Eq. (2.2.14) is particularly important. It is this term, arising from (dT/dt)^ ^ 0 (see Eq. (2.2.10)), that gives rise to large effects when the two reactants are dissimilar, that is for unequal mass ratios or densities. The density is defined in terms of fy°\ that is, (2.2.18) (2.2.19) so that / f^ip7dc7 = 0 and hence d0 = 0. The temperature is defined by (2.2.20) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 17 where n = ri\ + n2 and J f ^ m ^ d d + j /i°W^2dc2 = 0, (2.2.21) and hence niaS1} + n2oi2) = 0. (2.2.22) Consistent with this definition of the temperature is that the two equations in Eqs. (2.2.12) and (2.2.13) with i = 1 are the negative of one another; their sum being equal to zero is a reflection of conservation of energy when both species are taken into account. Consequently, the set of equations that is solved is the set with Eq. (2.2.22) replacing either of the two equations with i = 1 in Eqs. (2.2.12) and (2.2.13). The solution of Eqs. (2.2.12) and (2.2.13) together with Eq. (2.2.22) as discussed above yields the expansion coefficients and the fractional decrease in the equilibrium rate of reaction as given by, SISfi0)f2i)o*gd£ldc1dc2 -UJhfa'gdndddct V JfJfi0)A0)o*gcmdc1dc2 JJJf^f^^i + ^gdndc^c, L i = l i=l For the hard sphere cross section models, since 7] is directly proportional to OR/OE, this ratio can be set equal to 1 and the t] calculated is then multiplied by ORJOE- The cal-culation of the mean energy of each species yields the definition of species temperatures, T7, which is T7 = T[ l — ai7)], (2.2.24) where T is the temperature in the Maxwellians. Cukrowski et al. [32,31] refer to these species temperatures as the Shizgal-Karplus temperatures. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 18 Cukrowski et al. [32] suggested that the non-equilibrium rate coefficient can be ap-proximated with the use of the species temperatures as given by Eq. (2.2.24) with the determined to lowest order, that is, from the solution of Eqs. (2.2.12) and (2.2.13) with only one term in the expansion of i/>7. An estimate of the fractional change in the rate coefficient is determined with Maxwellians characterized with the species tempera-tures, that is, ( — » where A;(Ti,T2) is the rate of reaction between two species having component tempera-tures Ti and T 2 [74] and KTuT2) __ ( TeffX ^ exp[-^fe] *«>) " l r ( o ) J exph-jy . (2-2-26) where T e f f represents the "effective" temperature m i T 2 + m 2 T i J-eff — (2.2.27) m i + m 2 Clearly, this result can only be useful if the two species are distinct and the species temperatures differ from the temperature T. It is anticipated that their procedure will be invalid when the two species have similar masses and number densities. Cukrowski et al. have employed this approach in several studies of non-equilibrium effects reported in [82] and references therein. The relationship between r\ and fj can be understood by recognizing that the expansion of the local Maxwellians, T ( ° ) ( T 7 ) , at the species temperatures about the temperature T leads to an expansion in Sonine polynomials of the form, F ^ ( T 7 ) = / ( 0 ) ( T ) 1 + E 2=1 A T y S?(mc2/kBT) (2.2.28) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 19 where A T 7 = T - T 7 . Equation (2.2.28) illustrates the use of the ratio F ^ ( T 7 ) / f ^ ( T ) as the generating function for the Sonine polynomials. In this way, the approximation by Cukrowski et al. [32], consists of an expansion of the distribution function in Sonine polynomials with the expansion coefficients as powers of It is correct only to lowest order when only one term is retained in the expansion. In this way, the use of the Shizgal-Karplus temperature as employed by Cukrowski et al. yields a non-equilibrium correction of the form, V | ( ^ ) > / 4 n g ( ^ ) > M (2.2.29) It is expected that this approximation may be useful when the terms in dominate the solution in Eq. (2.2.23). The variation of the correction 77 versus the system variables was reported at length in [21]. Here the comparison with 77, Eq. (2.2.29), is of interest. Figure 2.1 shows the variation of n (solid curve) obtained from Eq. (2.2.23) with retention of a sufficient number of terms to provide convergence to four significant figures, in comparison with the Cukrowski et al. [32] approximation, Eq. (2.2.29) (dashed curve). The agreement between the converged solution, 77, and the approximation, Eq. (2.2.29), varies considerably according to the ratios of the masses and densities of the reactants A and C. With Eq. (2.2.14) and the definition of given by Eq. (2.2.18), it can be shown that a\j) and a\j) are proportional to - 1. The ratio 2 ^ is 16, 10, 2 and 5/4 in Figures 2.1A to 2.ID, respectively. As the quantity ^ £ —» 1 the agreement between 77 and 77 becomes poorer. This is due to the fact that as —> 1, —> 0 resulting in A T 7 —> 0 and 77 —>- 0. This result holds for all orders of approximation. Consequently, 77 is a reasonable approximation to 77 in Figure 2.1 A for which = 16. The agreement worsens as tends to unity in Figure 2.IB, 2.1C and 2.ID. For Figure 2.ID where the temperature perturbation is extremely small, the contribution becomes insignificant, Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 20 Figure 2.1: Variation of rj and r) versus e* for A + C —>• products, -q is given by the solid line and fj by the broken line. The mass ratio m\jmi and density ratio n i / n 2 are equal to (A) 2 and 8, (B) 10 and 1,(C) 2 and 1, (D) 10 and 1/8 respectively; oE = an = aX2 = 022 and OE/O-R = 1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 21 A T T PS 0 and the approximation fj by Cukrowski et al. is not valid. The dependence of -q on the density is one of the important objectives of the present work, in particular with respect to the density of the products for the reactions considered in Sections 2.2.2 and 2.2.3. If only two terms in the expansion of the distribution func-tions are retained, approximate expressions can be obtained which show explicitly the concentration dependence. If only equal mass ratios are considered, O~E = CR, and defin-ing A = ni/ri2, common collision frequency factors on both sides of Eqs. (2.2.12) and (2.2.13) are eliminated. With only two terms retained in the expansions, Eqs. (2.2.12) and (2.2.13) are approximated by the set of 3x3 equations, -2(1 + A ) I i ( l + A) ^ - ( 2 A + f ) _ i 2 15 8 \ V -1(1 + A ) f - ( 2 / A + f ) ; The solution of this set of equations is, / „(D \ v 42) j ( l - A i i / 2 \ l + A i 2 / 4 V 2^/4 ) af af ( 1 X ? [ ^ + 3 l i i /2] 60(1 a<2> = a ( 2 ) 2 60(1 + A ) 2 -Aaf . . A [ ( 4 + l l A ) i 2 + 2 ( l - A ) i x ] [(ll + 4 A ) i 2 - 2 ( l - A ) i i 60(1 + A ) 2 ' The fractional decrease in the rate of reaction 2 is then given by, 1 V 120(1 + A ) 2 A 0 For the line-of-centers reactive cross section, (2.2.30) (2.2.31) '(1 - A ) 2 ( 2 i 2 + 3L4i /2 ) i i + (2A 2 + 11A + 2)i2,/2] . (2.2.32) A0 = 2exp(-e*) A1 = -2{e* + l/2)exp(-e*) i 2 = . ( ( e*)2_ e*_ 1 / 4 ) e x p (_ e* ) (2.2.33) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 22 where e* = E*/kBT. The two term result given by Eq. (2.2.32) is a very good approxima-tion to the converged result. Figures 2.2 and 2.3 show the variation of = A T T / T ver-sus density and mass ratios, respectively. The energy conservation relation = — A a ^ imposes the constraint that temperature perturbations A T 7 for A and C must be opposite in sign. This property is common to all curves in Figures 2.2 and 2.3. All the curves also show that there is no temperature perturbation, A T 7 = 0, when ^ A = 1, as discussed in connection with the results in Figure 2.1. Figure 2.1 shows that as A —> 0, the tem-perature of the major species, T 2 —> T. However, the distribution of the minor species is strongly perturbed and A T i is large. The point A = 1 in Figure 2.2C (mi = m 2) corresponds to the system A + A —>• products. In this case, the species A and C become indistinguishable and are characterized by a single temperature, Ti = T 2 = T. The variation of temperature perturbation A T 7 / T versus mass fraction of A, Mi = m\j{m,\ + m 2), is shown in Figure 2.3. A striking feature of Figure 2.3 is that for a reaction involving reactants with dissimilar masses (ie. Mi' —» 0 or M i —> 1) there are large temperature perturbations. As the density ratio A increases in Figures 2.3A-2.3D, the mass fraction for which A T 7 = 0 shifts towards Mi = 0 in accordance with the dependence m i A / m 2 = 1. Figure 2.3A has the smallest A and as a consequence, A T 7 = 0 at Mi = 8/9. With increasing A , the mass fractions for which A T 7 = 0 are 2/3, 1/2 and 1/3 for Figures 2.3B-2.3D, respectively. Figures 2.3B (n i /n 2 = 1/2) and 3D (ni /n 2 = 2) show the symmetry resulting from the interchange of species A and C. Figure 2.3C corresponds to the system A + A —> products for which rii = n2, and A T 7 = 0 at mi = m2-In a subsequent paper, Cukrowski et al. [31] discussed further the role of the species temperatures in the determination of rj. They concluded that the results of previous Figure 2.2: Variation of A T T / T versus density ratio A = n i / n 2 for the system A + C —> products. The mass ratio m i / m 2 equals (A) 1/8, (B) 1/4, (C) 1 and (D) 2; e*. = 4 and (7E/°'R = 1- A T 7 = 0 at m 1 n 1 / m 2 n 2 = 1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 24 Figure 2.3: Variation of A T 7 / T versus M i , mass fraction of A, for the system A + C —> products. The density ratio nxjn-i equals (A) 1/8, (B) 1/2, (C) 1 and (D) 2; e* = 4 and O~E/O-R = 1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 25 workers [13,15,20] for the one component reactive system A + A —> products, is in error owing to the neglect of the effect given by Eq. (2.2.24). However, as has been shown, this temperature effect disappears when the reactants are identical and plays no role in the non-equilibrium effects. In Appendix A it is shown that the system studied by Cukrowski et al. differs from the one component system above. Their work corresponds to this reaction occurring in a large excess of a second (nonreactive) component (with the mass of A) that acts as an infinite heat bath so that the temperature of the system does not change. This was considered at length by Shizgal and Karplus [22]. For A + A —> products for which n\ = n 2 , the two term solution for rj becomes This is twice the result obtained from Eq. (2.2.32) because in this limit the two compo-nent system has twice the number of elastic collisions as the one component system [21]. Moreover since = d^ = 0 and A T = 0, there is no perturbation of the temperature due to the reaction and no effect on the reaction rate. 2.2.2 The Reaction A +. A -> B +B In this section, the non-equilibrium effects in a model reactive system introduced by Cukrowski et al. [32,31] are of interest. The species A and B can be taken to be different internal states of the some species and the reaction can be considered as an inelastic collision involving no mass transfer. Non-equilibrium effects for such processes have been considered previously [78,79]. The main objective here is to determine the influence of the products on the non-equilibrium effects in comparison with the results of Section 2.2.1. The effect of products has been discussed by Pyun and Ross [40] and by Fitzpatrick and lim rj = A->1 A\ 32An' Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 26 Desloge [70]. For this reaction, the Boltzmann equations are similar to those in Section 2.2.1, except that the reactive term for the products is a gain term evaluated with the reactive cross section for the forward reaction. The coupled Boltzmann equations for this system are, dfi = 1 dt e dJl = I dt e f /[/]/' ~ fif]angdndc + J j[f[f2- hf2]o-l2gd£ldc2 J J fifo-*gdttdc J /[/2/ - f2f}cr22gdCldc + J j[f'j2- fJ2]ol2gdQdCl J J f'j'a'gdndc!. (2.2.35) (2.2.36) In Eq. (2.2.36), it is important to note that the reactive collision integral represents the production of product B species and the integration is over the velocity space of one of the two reactant A species. The primes on the distribution function refer to A species that enter into a collision. All primed quantities in both Eq. (2.2.35) and (2.2.36) cor-respond to gain terms while all unprimed ones correspond to loss terms. The primes on the last term of the right-hand side of Eq. (2.2.36) are consistent with this convention. An important aspect of this model is that with the complete neglect of the reverse re-action, microscopic reversibility [74] for the reactive collisions cannot be employed. This reactive collision production rate is determined analogous to the calculations by Whipple [75], Riley and Matzen [76] and Shizgal and Lindenfeld [94]. The C E method of solution parallels the discussion in Section 2.2.1. The only modifi-cation is the nature of the reactive collision term for the product B in Eq. (2.2.36) which differs from the loss term of the reactant A in Eq. (2.2.35). The C E method is applied as in Section 2.2.1, with the lowest order reactive collision frequencies evaluated using Maxwellians and given by, / ! 0 ) i # 0 ) dc 2 = / / fWfWSg'dtod^dc, (2.2.37) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 27 where the prime is used to distinguish this product collision frequency with the loss terms in Eq. (2.2.11). The reactive collision probability is determined by changing the integration to one over the velocity of one of the product B species. With transformations to relative velocity g and center of mass velocity G, / 2 ( 0 ) ^ ( 0 ) dc 2 = e - e * ( ^ ) 2 / Jfff^o*g'dndgdG'. (2.2.38) where G is conserved in a reactive collision. This integral can be converted to an inte-gration over the relative velocity of the products with energy conservation in differential form (g'dg' = gdg) and with the reduced masses of the reactants and products equal, /J 0 )i# 0 )dc 2 = e-- ( ^ ) > / fMa^dgdG. (2.2.39) This result is specialized to the line-of-centers cross section so that / i ° ) ^ d c 2 = 7 r 4 C - - ( ^ ) > / / ( V g d G > (2.2.40) and /<°>i#°> = / /(Vc, (2.2.41) where the integral is the hard sphere collision frequency [1] for unit total cross section. It is this form of the reactive production collision term (for the line-of-centers reactive cross section) that determines the form of the C E equation for this model reaction. The work by Cukrowski et al. [32,31] incorrectly considered this reactive collision frequency for product B equal to the loss term for reactant A. The C E method of solution is applied based on the assumption, Eq. (2.2.5). The time variation of the densities is determined from the Boltzmann equations, and given Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 28 For this reaction, there is no energy conservation in reactive collisions and the time rate of change of the temperature is non-zero and evaluated to lowest order with the use of the explicit form of the reactive collision frequency in Eq. (2.2.41). By evaluating the rate of change of the energy of each species from the Boltzmann equations and adding the results one gets, where the species designation on the Ai integrals is dropped since the masses are the same for both reactant and product. The quantities denoted by Ai(0) refer to the Ai integrals for zero threshold energy and arise from moments of the reactive collision frequency for species B, Eq. (2.2.41), that is, Cukrowski et al. [32] assumed that translational energy is conserved in reactive collisions; the energy lost by the reactant is gained by the product and that dT/dt = 0. Their result is in contradiction to Eq. (2.2.43). With the substitution of Eq. (2.2.4) into the Boltzmann equations, Eqs. (2.2.35) and (2.2.36) and use of Eqs. (2.2.5), (2.2.42) and (2.2.43), the C E equations for the perturbations tp7 are given by, (2.2.43) e-e*M0). (2.2.44) + tpi - ip}crugdttdc + J J / ^ / ^ [ V ' I - ip1]o-l2gdttdc2 j j /i(0)/2(0)W - MongdMkz = / ^ G i C d ) , (2.2.45) + tp2 ~ ip]o22gdQdc + j j ff fi0)[il>'2 - foWwgd&ldcx J j / i ( 0 ) / f [V>i - A]o-ngdndCl = / i 0 ) G 2 ( c 2 ) , (2.2.46) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 29 where the inhomogeneous terms are defined by 1 (dnA{0) 1,3 ^ fdT\(0) G2(c2) = T l i 1 / dn2 W - H - ® ( f ) W - ^ (2.2.47) (2.2.48) n2 \ dt The C E equations, Eqs. (2.2.45) and (2.2.46) are analogous to Eqs. (2.2.8) and (2.2.9) in Section 2.2. The only distinction is the form of the reactive collision frequencies and the inhomogeneous terms Eqs. (2.2.47) and (2.2.48) in comparison with Eq. (2.2.10). The perturbations of the distribution function are expanded in Sonine polynomials S^(^) as in Section 2.2.1 and result in the set of algebraic equations similar to Eqs. (2.2.12) and (2.2.13) given by, £ [(n?[S{°, SP] + nin2{sf\ S[j)}) af + m r f , S^af] = n\&\ (2.2.49) 3=1 and N £ [(r%[S%\ S(2j)] + nin2{si\ S?}) af + nxn2{S%\ S[j)}af 3=1 where and nlpf* = -S0lA0 - 8li'-^{A1 - Ax(G)e~e\ + A% n = - n ^ 2 ) . (2.2.50) (2.2.51) n\^] = -80iAa + <JW — [ ^ i - Ai(0)e-e*] + A(0)e'e\ (2.2.52) n The fractional decrease in the forward reaction rate only involves the distribution function of species A and is given by oo V = -2Y,41)Al/A0 . (2.2.53) As was done in Section 2.2.1, it is useful to consider the approximate result with only two terms in the expansion of the distribution functions in Sonine polynomials. Setting Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 30 O~E = o-R, defining 8 = 712/711, and eliminating common collision frequency factors from both sides of Eqs. (2.2.49) and (2.2.50), a 3 x 3 set of equations is obtained ( ^ 1 " ^ hS \( a<1} \ ( ^[At(0)e-<* + SAJ \ i 2 / 4 -2(1 + 5) ±(1 + 5) -(2 + f5) V 4(1 + 5) fS -(28* + f8)J ^ (af \ ^5 8 u v 42) j v - i 2 ( 0 ) e - £ 7 4 The solution of this set of equations is, al1' = a l 1 ' = , ( 2 ) r31 [-04x(0)e- e* + 8AX) + (A2S + A2(0)e"e*)] 60(5 + 1) 2 L 2 - 6 Q ( / + 1 ) 2 [ W 0 ) e - £ * + - (25 + 15/4)i 2 + ^i2(0)e-f-*] -af/8 ~60(5 + l ) 2 5 [ ( i l ( 0 ) e " £ t + + ( 2 + f 5 ) i 2 ( 0 ) e ~ £ * ~ 4 ^ (2.2.54) (2.2.55) With Eq. (2.2.55), an expression is obtained for 77 comparable to Eq. (2.2.32) which is not reported for the sake of brevity. It is important to note that the correction, 77 (Eq. (2.2.53)), depends only on the expansion coefficients, af of species A. The two term solution is accurate to within 5 percent of the converged values for the conditions discussed here. This two term analytic solution provides for a useful interpretation of the density dependence. The variation of 77 versus e* for several density ratios is shown in Figure 2.4. The curve (a) of Figure 2.4 shows the variation of 77 for the reaction A + A —> products. The curves (b)-(e) are for decreasing density ratio, 5 = n 2 / n i , of product to reactant. The 77 values calculated for A + A —» B + B are larger than that for A + A —> products due to the contribution of the af terms, and the non-equilibrium effects generally increase with a decrease in 8. Note that 77 attains a limiting value as 8 —> 0 (dashed curve and open circles). The peak near e* = 1 for the system A + A —>• B + B is caused by the Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 31 Figure 2.4: Variation of 77 versus e* for A + A -» B + B compared with A + A —¥ products. A + A —> products is shown by curve (a). For A + A —> B + B, the density ratio n2/rii equals (b) 1.0, (c) 0.5, (d) 0.1 and (e) 0.01; oE/oR = 1. The value of n for A + A -» B + B in the limit 8 —>• 0 is shown by ( o ). Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 32 relatively large temperature difference, A 7 \ = a\ , between the two components A and B . This temperature difference does not occur for the system A + A —> products and results in small values of 77 for e* ^ 2. There is good agreement in the values of 77 for the systems A + A —> products and A + A —> B + B &s 8 —» 0 and e* becomes large. Figure 2.5 shows the variation of A T 7 / T versus log 8 for the reaction A + A —>• B + B wi th e* = 3. O f primary interest here is the behaviour in the l imit of vanishing reactant or product. In the l imi t of vanishing reactant, ri\ —>• 0, 8 —> 0 0 , the product is in large excess and acts as a heat bath at temperature T , so that A T 2 —>• 0 as 1 /82 as shown in Figure 2.5. The distribution function of the reactant is not a Maxwell ian and A T \ —» 0 as 1/8. This l imi t corresponds to the isothermal reactive systems of reference [21]. In the l imi t of vanishing product, n 2 —> 0, 8 —¥ 0 and the perturbation of the distribution function of the product is large; the expansion coefficients af^ vary as 1/8 (see E q . (2.2.55)). Similarly, A T 2 —> co as 1/8 so that A T 2 / T shown in Figure 2.5 is large and negative. B y contrast, as 8 —>• 0, AT\/T remains finite. Since the correction 77 shown in Figure 2.4 depends only on a\x\ it remains finite. A comparison of these results with the results for the reaction A + A —> products is presented in Section 2.2.4. 2.2.3 The Reaction A + A # B + B In this section the reaction of Section 2.2.2 is considered but allowing for the reverse reac-tion. This changes the nature of the Boltzmann equations since microscopic reversibility is valid and forward and reverse reactive collision terms are related. The formalism for the kinetic analysis for this system was treated in detail in [21]. Microscopic reversibility relates the cross sections for the forward and reverse reactive cross sections [74] as given by p)o-}g) = t4o*rg2r (2.2.56) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 33 Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 34 where p is the reduced mass of the pair entering a reactive collision, and the subscripts f and r refer to the forward and reverse reactions, respectively. For the line-of-centers reactive collision cross section model for both forward and reverse reactions, this implies products are equal, the corresponding reactive hard sphere diameters have to be equal, df = dr. If the forward and reverse threshold energies are unequal, the "heat of reaction", The gain and loss terms in the reactive collision integral terms have been combined by using microscopic reversibility, Eq. (2.2.56). The equality of the reduced masses has been used. The reactive collision terms have the same formal appearance as the unlike elastic collision terms. It is important to mention that the integration over c in the reactive collision term is over the velocity of species A in Eq. (2.2.57) and over the velocity of species B in Eq. (2.2.58). The application of the C E method of solution of the Boltzmann equations follows the discussion in Sections 2.2.2.1 and 2.2.2, and reference [21]. The important differences are that, that djpf = d?Tpr. Since for the present model the reduced mass of the reactants and A.E = e*f — e* has an important effect on the extent of the non-equilibrium. The coupled Boltzmann equations for this system are of the form, (2.2.59) and (2.2.60) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 35 are the equilibrium estimates of the density and temperature time derivatives employed in Eq. (2.2.5). The resulting C E equations are similar in form to Eqs. (2.2.8) and (2.2.9), with the inhomogeneous terms replaced with defined by f(o)H?) = Sl[Am + ^^_R?] (2.2.61) where the reactive collision frequencies include both forward and reverse rates, given by /i(0)M0) = / / [ / i 0 ) 7 ( 0 ) ' - / r / ( °V /^c (2.2.62) and /i0 )40 ) = / /[/i ( 0 )7 ( 0 )' - fPfWKgrdndc (2.2.63) The are the moments of these reactive collision frequencies with Sonine polynomials. These can be related to the A\^ quantities for the forward reaction of Section 2.2.1. The moments of reactive collision frequencies are written in a more compact notation n?ii 1 } - [1 - (^)'''exV(AE/kBT)}A^(e*f) (2.2.64) , 2 and nlA^ = [1 - (^) exp(-AE/kBT)]Af\e;) (2.2.65) where = Ej/kBT and e* = E*/kBT, Ej and E* are the forward and reverse activation energies respectively. The C E equations for this model system are very similar to those in Sections 2.2.1 and 2.2.2 and given by, £ [(n 2 [Sf \ S?] + nin2{S?,S^}) af + n^-fsf\ S^}af] = nfaf \ (2.2.66) j=i and £ [{r%[sH\ S?} + nin2{sP,S?}) af + nin2{S%\ S?}af ] = n\af. (2.2.67) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 36 where and n 2kBT n\af = -8Qin\Af + 5 ^ ^ ^ + n{A\". (2.2.68) „ 2 ~ ( 2 ) _ r 2J(2) r n 2 L\E 2 1(2) 17 (2 ) ^ 9 fiQ^ n 2 a i —-^oi^A) - 2kBT + n 2 A i • (2.2.b\)) The A | 7 ' integrals which involve both forward and reverse reactions can be written in terms of the Af integrals for either the forward or reverse reaction as a consequence of microscopic reversibility. The approximate solution with only two terms in the expansions leads to the 3 x 3 equation of the form, -¥ ' - 2 ( 1 + 5) ±5 ±(1 + 5) - ( 2 + f 5 ) f 5 v - 1 ( 1 + 5) f 5 - ( 2 5 2 + f 5 ) ; A / af ^ v 42) j ( i d ) A\' + 1 AE A 1+5 2kBTRXQ (1) 5 2 A< 2 ) J (2.2.70) The system is solved to give a\ = af = [31(Af(l + 5) + A f ( ^ ; ) ) + 4(4X) - AfS)5] (2) _ a (2) 60(5+ 1) 2 2 '60(5 + 1) 2 ( i ) [2(Af(l + 5) + - ^ ( 8 * + 1 5 ) - 7 A * > S '2kBT' AE (2.2.71) i( 2 )A2l 60(5 + l ) 2 5 [2(41}(1 + 5) + ^ ( ^ ^ - * ( 2 ) ' f i X " 1 R U 2 - ^ ) ) - ^ ( 8 5 + 15)5 2 - 7 5 A 2 i j ] . The two-term approximations of rjf and rjr (the fractional decrease for forward and reverse reactions, respectively) can be obtained from the solutions of Eq. (2.6.70) with Eqs. (2.2.64) and (2.2.65). The corrections to the reactive rate coefficient are given by, N E Vf = -Z<4M1)/4>1\ (2-2.72) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 37 and Tk = -T,<iM2)/42)- (2-2.73) i=l The two term solutions give very good approximations to the converged values of rj. This reversible reactive system differs from the systems studied in Sections 2.2.1 and 2.2.2 in that the reaction can attain chemical equilibrium whenever the forward and reverse rates are equal, that is when k^pn\ — k^n2, and the equilibrium density ratio is given by 5 2=exp(—AE/k BT). At chemical equilibrium the distribution functions are Maxwellian and rjf = rjr = 0 [17,21]. For other density ratios, rjf and rjr are finite and can be either positive or negative, the sign depending on the factors (1 —<52exp(AE/kBT)) and (1 - l / 5 2 e x p ( - A E / k B T ) ) in Eqs. (2.2.64) and (2.2.65). These features are demonstrated in Figure 2.6 where the variation of rjf versus e*f is shown for several values of e* (Figures 2.6A-2.6D) and various density ratios (curves (a)-(e)). The correction rjf equals zero for values of e*f where chemical equilibrium exists, that is, S2 = exp(—AE/k B T). For <5=1, curve (e), rjf = 0 for e*f = e* as seen in Figure 2.6. For > e*, rjf < 0 and for e*f < e*, rjf > 0 as discussed above. For the other density ratios, curves (a)-(d), rjf > 0 for values of S2 > exp(—AE/k BT). The maxima in rjf in Figure 2.6 near e*f = 1 are due to temperature effects resulting from the separation of the temperatures of the two species. The dashed curves and open circles correspond to the limit of vanishing product, n2 —> 0. This limiting behaviour is discussed in Section 2.2.4 in comparison with the results in Section 2.2.1 and 2.2.2. The variation in the temperature perturbation A T 7 / T versus the density ratio 8 = n 2/nx is shown in Figure 2.7 for two pairs of values of e*f and e*. Figure 2.7 shows that as one species disappears (log 8 —> oo or log S —> — oo), the temperature perturbation for that species becomes extremely large, while the temperature perturbation of the more abundant species approaches a finite limiting value. This behaviour is due to and Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 38 Figure 2.6: Variation of r]f for A + A # B + B versus e*j. The reverse activation energy, e*, equals (A) 10, (B) 5, (C) 3 and (D) 0; O-E/GR = 1. The density ratio n 2 / n i equals (a) 0, (b) 0.01, (c) 0.1, (d) 0.5 and (e) 1. The variation of?? for the system A + A B + B in the limit 8 —> 0 is shown by ( o ). Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 39 - 4 - 2 0 2 l o g 5 Figure 2.7: Variation of A T 7 / T versus the log of density ratio 8 = n2/ni for A + A =± B + B. Forward and reverse activation energies are (A) = 3 and e* = 3 and (B) e*f = 3 and e* = 0 , aE/aR = 1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 40 A T i varying as 8 and A T 2 varying as 1/8 as given by Eqs. (2.2.71). The coefficients dp and A T 7 vanish at chemical equilibrium, that is for 82=exp(—AE/kBT). The sign of A T 7 follows the sign of n as discussed above. Figure 2.7A, for which e*j- = e*, shows a symmetry for A T 7 versus log 8 whereas Figure 2.7B, for which ^ e*, is asymmetric. Note that A and B are interchangeable by interchanging the labels 1 and 2 and with e*. The dependence of the perturbation to the reaction rate, rjf, on the system parameters is more complicated than for the two irreversible systems. The parameters that drive the equilibrium towards A or B are now not merely the forward activation energy but also the reverse activation energy as well as the ratio of densities of the components A and B. The irreversible system A + A —> B + B in the limit of vanishing products, 8 —>• 0, is shown as a dashed curve and open circles in Figures 2.6A-2.6D. The reversible and irreversible systems do not coincide in the limit of vanishing product, with the notable exception of the case of vanishing product in Figure 2.6D. One obtains identical n and A T 7 values for A + A v= B + B and its irreversible equivalent in the limit of 8 —» 0 and Ai? = e*f (i.e. e* = 0). These conditions represent a reversible process with no reverse activation threshold for a vanishing back reaction B + B —> A + A. This yields a special set of circumstances where e* = 0, = AE and af^ = 0 in Eq. (2.2.70), giving rise to identical forms for the matrix Eqs. (2.2.54) and (2.2.70). This yields identical results for dp to all orders of approximation. A detailed discussion of the effect of products is presented in the next section. 2.2.4 The Limit of Vanishing Product It is important to note that each of the three systems described here involve specific assumptions in the formalisms that lead to the form of the Boltzmann equations and the application of the GE method. The different solutions derived in Sections 2.2.1-2.2.3 Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 41 are dictated by the explicit expressions for (dT/dt)^ in the implementation of the C E method. The elastic collision operators for the three systems are formally similar. The time rate of change of the temperature is given by Eqs. (2.2.7), (2.2.43) and (2.2.60) for the reactions A + A - » products, A + A —> B + B and A + A =± B + B, respectively. In the limit of vanishing product Eq. (2.2.60) does not coincide with Eq. (2.2.43), and similarly Eq. (2.2.43) does not coincide with Eq. (2.2.7) (for mi = m 2). The different results for (dT/dt)^ lead to the different inhomogeneous terms in the C E equations, Eq. (2.2.14), (2.2.51) and (2.2.52), and (2.2.68) and (2.2.69) for the three different reactions. These are similar except for the terms in Su that arise from (dT/dtY°\ These factors do not coincide in the limit of vanishing products. Hence, the results for the corrections, r\ do not correspond in the limit of vanishing products. The dependence of 77 versus e* for increasingly small density of product for A + A —> B + B was shown in Figure 2.4. The limit of vanishing product shown by the open circles and dashed curve does not coincide with the curve (a) for A + A —> products. An approximation to the dashed curve is obtained with the two-term results, Eqs. (2.2.55) and for 8 —> 0 one gets = " 6 ^ [ ( 3 l i l ( ° ) e ~ £ * + 24(0)e^)4/2 ' +2(4(0)e"e*- 15/44 + ^ 4(0)e"e*)4/4] (2.2.74) which differs from the result, Eq. (2.2.34) for the system A + A -+ products. The terms in A,(0) in Eq. (2.2.74) are the terms relating to the production of product, B. This is a manifestation the nonvanishing effect of vanishing products discussed by Fitzpatrick and Desloge [70]. The nonvanishing temperature effect of vanishing product B for A + A —> B + B is caused by the fact that af (Eq. (2.2.55)) is not zero for n 2 —> 0. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 42 For A + A -+ B + B, in the limits 8 -> 0 and e* large, lim a2l) « i 2 / 8 <5->0,e*->oo and A 2 lim r? « —^— . 5->0,e*->oo 32A0 In the large e* limit, the terms in A(0) are neglected relative to the terms in Ai which vary as (e*)zexp(—e*). In Figure 2.8, the ratio of the corrections for these two reactions versus e* in the <5 —>• 0 limit is shown. Although there is no agreement for small e*, the ratio is very close to unity for e* ^ 8, consistent with the analysis leading to Eq. (2.2.74). The dashed curve is the two term result. The results for these two cases discussed above are identical in the limit of large e*. This result holds to all orders of approximation. Hence, a region is observed for which the system A + A —» B + B corresponds to the system A + A —> products. Similarly, the limit of vanishingly small reverse reaction for the system A+A =± B + B does not coincide with the irreversible reaction A + A —)• B + B. The irreversible case or —> 0 violates the principle of microscopic reversibility (Eq. (2.2.56)) and cannot be applied in this case. As with A + A —> B + B, the limit of 5 —> 0 results in the coefficients CLp becoming infinite. The matrix equations Eq. (2.2.54) and (2.2.70) corresponding to the two term solutions of the reversible and irreversible systems differ only in the inhomogeneous component arising from the form of (dT/dt)^\ Eq. (2.2.60). There appears to be no limiting procedure whereby the results for the system A + A x= B + B go over to A + A —> B + B. Figure 2.6 shows that the perturbations become significantly large if 5 departs far enough from the equilibrium value. It has been demonstrated that the choice for (dT/dt)^ in the GE method of solution of the Boltzmann Equation for the reactive system leads to somewhat different results. The C E method in each case is carried out consistent with the basic assumptions of the method [1,13-16,18,17,19-23]. It Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 43 0 4 8 12 16 20 * Figure 2.8: Variation versus e* of the ratio of rj for A + A —> products to 77 for A + A —> B + B, in the limit 5—^0. OE/CTR = 1; the solid line is the ratio of converged values of 77; the broken line is the ratio of two-term solutions. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 44 provides the asymptotic solutions of the Boltzmann equations on the reactive time scale. The transient time variation on an elastic time scale is assumed to be extremely short so as not to affect the long time dependence. A verification of the C E results requires an explicit time-dependent solution of the Boltzmann equation for different values of the reactive to elastic relaxation time scales. When the time scales are well separated it is expected that the time-dependent calculation will give results in agreement with the C E results. 2.3 Time-Dependent Solution of the Boltzmann Equation The C E method provides a very special solution of the Boltzmann equation which gives the long time behaviour on the reactive time scale. The time dependence is implicit through the variation of the density and the temperature. This section is concerned with the explicit time-dependent solution of the Boltzmann equation. Comparison is made of the time-dependent solutions with the C E results in Section 2.2 for different values of e* and O-R/CTE- Detailed analysis of the results for the reaction A + A —> products and A + A —> B + B, especially in the limit of vanishing B are made. Similar studies for model reactive systems with the neglect of the products have been reported [23,69]. 2.3.1 The Reaction A + A -> B + B The distribution functions are expanded about local Maxwellians, Ff, characterized by time dependent number densities, n7(t), and temperatures, T 7(t). The velocity distribu-tion function is written Fy = F^0)[i + ^(x,t)], (2.3.1) Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 45 where tp7(x, t) is the time-dependent perturbation from Maxwellian. The present ap-proach parallels a study of temperature relaxation in binary gases [73] and the applica-tion to hot atom reactions [81]. With Eq. (2.3.1) in Eqs. (2.2.35) and (2.2.36) one has that, dt where and = / jW^Ff -F^F^]9o1Tldndcv + j J F^F^y + - V>7 - V y W ' ^ c y + / / [Fi0)'Ff[^ + <] - F^F^ - Vy]] i ^ d f i d c v + Rjic,), (7,r;) = l ,2 (2-3.2) Rl(Cl) = -40)(Cl) - 2J f Fi0)F^^go*dndc (2.3.3) R2(c2) = R2{0\c2) + 2JJ F{0)lF^'^igo*d^dc. (2.3.4) The reactive collision frequency R2(c2) for the product is a function of c2 and its evalu-ation requires the transformation to product velocities as discussed in Section 2.2.2 and elsewhere [17,75]. With the expansion in Sonine polynomials, one has that, M M = Z#\t)SlH$), (2-3.5) where the expansion coefficients are time dependent. It is important to note that = 0 owing to the definition of the species temperature, T T . The coupled set of Boltzmann equations is reduced to the set of equations of the form, 1 dbP • • • N = Ai7) + D 4 7 ) 6 j r ) + C ^ ) ] , i = 2,3..JV (2.3.6) Ni dt j = 2 Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 46 where, \f = n^Sifsfy-Af/n, Xf = n2<S?,Sf >+Df/n2 B\? = n,[S?,Sf]+nv<S^,Sf>-^-R.fSjl . 3 \ 1 dT 7 1 dn7 C\? = nv<Sf,S^>+-Rt?5vl (2.3.7) The angle integrals, < S^\S^ > in Eq. (2.3.7) are the two-temperature matrix elements of the collision operators ninr,<S^\S^> = j j J FWFW[S®' -S^}S^aJvgdndcydCri (2.3.8) and evaluated for the hard sphere cross section using analytic expressions found in [80]. The integrals A\f are the moments of the equilibrium reactive collision frequencies, Rf, evaluated as in reference [21] and given by Eq. (2.2.18). The corresponding integrals for product species B are given by, D?] Floy(Tl)FW(Tl)S<?\T2)a*gdndcdc^ and can be evaluated as in [21] for the special case m-i = m2 and one has that, ^=^(|),/,E(i-i|)i"4G)W)«-'-. Note that Kk(0) represents the zero threshold energy integral, Kk in Eq. (41) in ref-erence [21], and e* is (—E*/kBTi). The two-temperature Df reduces to Ai(0)e~e* for Ti = T 2 . The quantities R\? are the matrix elements of the reactive collision operator in Eq. (2.3.3), (2.3.4). These have been found to be of second order in Sonine polyno-mials and can be neglected. The quantities are the integrals of products of Sonine Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 47 polynomials that arise from dF^/dt, and dS^/dt. These details of the calculation have been discussed in reference [80]. All the coefficients in Eq. (2.3.6) are time dependent since the temperatures characterizing these matrix elements as well as the densities are time dependent. Hence Eq. (2.3.6) is coupled to the equations for the densities |41).+ 2i;6 i (t)A} drii/dt = —dri2/dt — — n a 3 J=2 (2.3.9) and the temperatures dTi 2n2Ti dt < 5 J 1 ) , 4 0 ) > - A (i) nin2 \^W+<Si\S[2j)>bf riin2 3 1 1 3 (2.3.10) dt 2mT2 <si1\s[0)> > D l (2) riin2 / n 2 V dt N + 2Z[[<S{2\siJ)>+2 p(2) R ^ + 2 + 3=2 [< Sil),S[j) > - 2 D nin 2 riin2- 3 (2.3.11) riin2 The perturbation of the distribution function and the fractional change in the rate of reaction, r)(t), are determined from the numerical time integration of Eq. (2.3.6), and the equations for the densities and temperatures, Eqs. (2.3.9)-(2.3.11). The calculations by Cukrowski et al. [31] did not include the departure from the local Maxwellians, that is, the perturbation of the distribution functions was assumed to be zero. Hence their analysis is based solely on the hydrodynamic equations, Eqs. (2.3.9)-(2.3.11) with (t) = 0, and in a sense cannot be considered as a solution to the Boltzmann equation. These authors use the time dependent species temperatures (referred to as the Shizgal-Karplus temperatures) to determine a time-dependent correction to the rate of reaction as given by Eq. (2.2.53). Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 48 For the numerical integration of Eqs. (2.3.6)-(2.3.11) a dimensionless time vari-able, tE, is defined given by ^ ^ ^ ( O J / ^ T T ^ T ^ O ) ^ ] 1 / 2 . The initial conditions assume Maxwellian distribution of reactants and products and therefore all coefficients bf = 0 at t = 0. The set of equations Eqs. (2.3.6)-(2.3.11) are integrated with a fourth order Runge-Kutta integration procedure. Figure 2.9 shows the time variation of r\{t)lr)CE versus the dimensionless elastic collision time. The value of TJCE varies with time implic-itly through n(t) and T(t), Eqs. (2.3.9)-(2.3.11). The different curves are for different choices of the activation energy and the ratio of elastic and reactive collision hard sphere diameters, that is, for different values of the elastic and reactive time scales. When these time scales are well separated, it is expected that the long time or asymptotic value of the time-dependent result, that is lim^oo rj(t) —> r)asy coincides with the C E result. For the topmost curve (a) in Figure 2.9, with e* = 13 and OEJOR = 200, this separation of relaxation times is satisfied and the time-dependent result coincides with the C E result. The subsequent curves, from top to bottom, are for decreasing reduced activation en-ergy. In each case, except for the bottom curve, a steady value of rj is obtained but does not coincide with T]CE, even for arbitrarily large ratios for OE/O~R. Table 2.1 lists the values of O~E/O~R, e*, the ratio of relaxation times and the ratio r)(t)/r)cE- For example, for the curve Figure 2.9(d), e* = 4 and the ratio T](t)/rjcE ~ 0.55 even for arbitrarily large O~E!O~R. This result is unexpected and was not observed in earlier studies for a one-component system [23,69]. The phenomenon arises from the separation in temperatures, A T 7 ( i ) , which is shown in Figure 2.10 for different values of e* and OEJOR. These attain steady values which agree remarkably well with the corresponding C E values when the required separation of relaxation times is obtained. The ratio TE/TR decreases in Fig-ures 2.10A-2.10D. As a consequence of this steady temperature separation, the quantity < Sf,Sf > a A T 7 (see [80]) differs from zero. This two-temperature matrix element occurs in Xf = n 7 < 5 W , 5 7 0 ) > -Af/rij in Eq. (2.3.6), which becomes independent Figure 2.9: Ratio of the time-dependent r)(t) and corresponding C E TJCE for the system A + A —> B + B. The reduced activation energy e* equals (a) 13, (b) 9, (c) 6 and (d) 4. The elastic collision cross sections for each curve are listed on Table 2.1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 50 curve e* O~E/°~R ??CE T/asy/T/CE 2 1 7.3(0) 8.15(-4) 2.18(-2) 0.049 (d) 4 1 5.5(1) 1.53(-2) 2.24(-2) 0.514 (d) 4 20 1.1(3) 1.20(-3) 2.19(-3) 0.546 (d) 4 200 1.1(4) 1.20 (-4) 2.20(-4) 0.545 (c) 6 20 8.1(3) 1.50(-3) 1.86(-3) 0.803 (c) 6 200 8.1(4) 1.52(-4) 1.86(-4) 0.817 (b) 9 20 1.6(5) 4.96(-4) 5.21(-4) 0.953 (b) 9 200 1.6(6) 5.00(-5) 5.21(-5) 0.960 (a) 13 200 8.8(7) 8.27(-6) 8.29(-6) 0.997 Table 2.1: Comparison of rj values calculated from C E , TJCE, and asymptotic values from the time-dependent two-temperature approach r}asy for the system A + A —> B + B. The ratio of initial product to reactant density nx(0)/n2(0) is 1. See Figure 2.9. of OE/CTR for large OE/CTR. It is the finite values of < Sty^S^ > which leads to the r)(t)/rjcE values in Figure 2.9 and Table 2.1 which are less than unity. For larger values of e*, < SW, S7°) > vanishes more rapidly than A • 7', and has a reduced effect so that for e* > 13 shown in Figure 2.9 and Table 2.1 r)(t)/r)cE ~ 1- The physical explanation for this result is that rjasy includes the effect of the reaction and temperature relaxation [72,73]. There is a perturbation of the distribution function that arises from both pro-cesses. The C E result, rjcE, does not include this two-temperature effect. Figure 2.11, shows the time variation of A T 7 with decreasing product density. The dashed curves are the corresponding C E results with the time variation implicit through n7(t) and T(t). The reduced activation energy e* is equal to 2 and OE/<JR is unity so that the elastic and reactive time scales are similar. The agreement between the C E results for all but the shortest times in Figure 2.11 is somewhat unexpected since TE/TR ~ 1 unlike the conditions of Figure 2.10. Although there is no steady state, the ratio [ATija^^/fATiJcB becomes time-independent on the t$ time scale. It is useful to notice the increasingly large perturbation of T 2 as n2 —>• 0. This can be inferred from Eq. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 51 Figure 2.10: Comparison of time-dependent and C E temperatures for different cross sections for A + A —> B + B. Time-dependent, A T 7 / T shown by the solid line and corresponding C E values, shown by the broken line. The ratio of elastic to reactive hard sphere collision cross sections, crE/o-R, and reduced activation energy, e*, are equal to (A) 1, 9 (B) 10, 9, (C) 1, 13 and (D) 200, 13. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 52 e* CJE/O-R n2(0)/ni(0) [AT^/JATJCE 1 [AT^/JAT^CE 2 2 1 0.5 1.000 0.992 2 1 0.1 1.000 1.000 2 1 0.01 1.000 1.000 4 1 1.0 0.998 0.870 4 20 1.0 1.000 0.990 4 200 1.0 1.000 0.999 1 First order approximation. 2Converged solution. Table 2.2: Comparison of asymptotic values, [ATi(rj)]asy, calculated from integration of the Boltzmann equation and corresponding C E values [ATi(t)]CE, for the reaction A + A B + B. (2.3.10) where dT2/dt varies as 1/8 and becomes infinite as 8 -+ 0 for nonzero values of rii. Table 2.2 shows that for several values of e*, o~E/o-R = 1 and the initial density ratio, rii(0)/n2(0) the steady state [ATi]asy/[ATI]CE ratios which are very close to unity. It is clear that there is very good agreement irrespective of the value TE/TR, unlike the behaviour shown in Table 2.1. The explanation for this behaviour is that the equation for Ti, Eq. (2.3.9), to lowest order (bf = 0) evolves only on the reactive time scale. In Ta-ble 2.2, the first entry corresponds to the first order approximation to [ATi]asy/[ATi]cE whereas the second entry corresponds to a higher order approximation with sufficient number of terms to give agreement to four decimal places. To first order, the time-dependent and C E results coincide. The departures from unity in the last column of Table 2.2 arise from the perturbations from the local Maxwellians. The difference between treating products as vanishingly small and nonexistent is shown in Figure 2.12. The time variation of the ratios for n(t) for A + A —> B + B (in the limit of vanishing products) are graphed in comparison with TJCE for A + A —> products Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 53 0 4 8 12 16 20 0 4 8 12 16 20 Figure 2.11: Comparison of time-dependent and C E temperatures for different density ratios for A + A —> B + B. Time-dependent A T 7 / T is shown by the solid line and the corresponding C E values, af by the broken line. The density ratios n-ijux equal (A) 1, (B) 0.5, (C) 0.1 and (D) 0.01; c* = 2 and oE/oR = 1. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 54 (with complete neglect of products). The time-dependent results in Figure 2.12 are for 8 = ^ = 10~5 for various e* and OE/O~R values. The time-dependent equations cannot be integrated for the initial condition with no product since the temperature T2 is unde-fined. This is in sharp contrast to the work of Cukrowski et al. [32,31] which show that T2(0) = T(7/6 + e*/3) for n2 -»• 0, obtained with the assumption (dT/dt)^ = 0. Since this isothermal constraint does not apply, T2(0) is indeterminate for n2 —> 0. The ratio of the C E results for these reactions is shown in Figure 2.8. The lack of agreement at small e* is the nonvanishing effect of vanishing products. The time-dependent results in Figure 2.12 are completely consistent with the C E result. There is agreement for n for the two reactive models in the limit of vanishing product for large e*. For small e*, the time-dependent results demonstrate, as does the C E approach, the nonvanishing effect of vanishing product. This effect is not a consequence of the assumptions of the C E method, but arises from the different nature of the two reactive systems. 2.4 Summary and Conclusions In this chapter, the perturbation of the distribution function from the Maxwellian was calculated from the Boltzmann equation using a C E method of solution. The model reactive systems A + C —> product, A + A —» B + B and A + A == B + B were studied. A hard sphere elastic cross section was chosen for all elastic collisions and a line-of-centers reactive section with a reduced activation energy e* = E*/ksT was used. The correc-tions to the rate coefficients from the equilibrium values were calculated and the role of the departure of the species temperatures from the system temperature was studied in detail. It is shown that for small e* the results for the reaction A + A -+ B + B do not coincide with A + A —> product which has been termed the nonvanishing effect of vanishing products [40,70]. For large values of e*, the results for these two reactions do Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 55 Figure 2.12: Time dependence of the ratio n(t) (A + A B + B) to the C E value of 77 (A + A —>• products). The reduced activation energy, e*, and the ratio of elastic to reactive hard sphere collision cross section, OE/(JR, are equal to (a) 6, 20 (b) 4, 20 (c) 13, 200, (d) 6, 200, (e) 4, 200 and (f) 2, 1; n2(0)/ni(0) = IO"5. Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 56 agree. It was shown that the results for the reversible reaction A + A ^= B + B do not coincide with those for A + A —>• B + B when the reverse reaction is made negligible. The reason for this is that microscopic reversability holds for the reversible reaction and not for the irreversible reaction. A two-temperature time-dependent solution of the Boltzmann equation for the reac-tions A + A —>• B + B was carried out to study the validity of the C E method. It has been shown that when the ratio of the elastic to reactive time scales is sufficiently small, the time-dependent calculations show a steady state. The steady state coincides with the C E result for sufficiently large e* and OE/OR. However, for moderate values of e*, the steady state value does not agree with the C E result owing to the departure of the species temperature, T 7 , from the system temperature, T. A similar study was carried out for the limit of vanishing B and the results compared with that for A + A —» product. Agreement was found at large e* but not at small e*, consistent with findings in the com-parison of the C E results for the two systems. The formalism presented here can be applied to reactive systems with realistic elastic and reactive cross sections. The collision matrix elements that are required in the solu-tion of the Boltzmann equation can be evaluated for arbitrary collision cross section sets. This was done by Shizgal and Karplus [20] for the microscopic reactions H 2 (v 1 ) + H 2 (v 2 ) , where v x and v 2 are the vibrational states. The non-equilibrium kinetics of translational hot 0( 3 P) formed from photodissociation of 0 2 and O 3 in the Earth's atmosphere was carried out several years ago by Lindenfeld and Shizgal [84] with a realistic cross section set. The quantitative results for some particular reactive system will differ from the re-sults for the model systems presented here but one should find a very similar dependence on the system variables. The behavior of model systems, permit a better understanding of the results for physically interesting systems with a fixed set of parameters. In par-ticular, the results obtained in this paper in terms of the effects of the products on the Chapter 2. Non-Equilibrium Reactive Systems : Effect of Reaction Products 57 non-equilibrium corrections, the role of the species temperatures and the usefulness of the C E approach wi l l be qualitatively similar in other systems. The magnitudes of the corrections w i l l be different but the trends discussed here wi l l be similar. For realistic systems where the non-equilibrium effects might be large, the time-dependent approach would be preferred over the C E method which is restricted to very small departures from equil ibrium. There are important applications to disequilibrium of the vibrational states for molecular systems for which the formalism of this paper w i l l have practical applica-tions [41]. The use of local Maxwellians characterized by the species temperature, T 7 , in the present time-dependent calculations can also be employed in a CE- type solution of the Bol tzmann equation. A similar procedure was used by Pascal and B r u n [41] for the calculation of transport coefficients in molecular systems. The different temperatures for translational and vibrational degrees of freedom in their paper correspond to the different species temperatures in this work. They referred to their approach as Strong Non-Equi l ib r ium (SNE) . A detailed analysis of S N E methods of solution of the Bol tz -mann equations for reactive systems is the subject of the next chapter. Chapter 3 Non-Equilibrium Reactive Systems : The Role of Species Temperatures 3.1 Introduction The non-equilibrium effects for a model reactive system, A + C —> products, that arise from the perturbation of the velocity distribution from the Maxwellian are studied in this chapter. As in the previous chapter, the objective of the present chapter is the calcula-tion of the fractional decrease of the non-equilibrium rate coefficient from the equilibrium value. In Chapter 2, the effect was examined with the C E method of solution of the Boltz-mann equation which treats the reactive process as a weak perturbation. This approach is referred to as Weak Non-Equilibrium (WNE) in the context of the present chapter. The reactive process causes the temperatures of the two species to differ from the sys-tem temperature and this effect can play an important role in the determination of the departure of the rate coefficient from the equilibrium value. The method investigated in the present chapter is an extension of the C E approach and involves the expansion of the distribution functions about Maxwellians at different temperatures and is referred to as Strong Non-Equilibrium (SNE). The SNE method exhibits a flaw in that it generally does not give the W N E case in the limit of equal temperatures for each component. This situation is remedied by an approach which is a modification of SNE and is referred to as Modified Strong Non-Equilibrium (MSNE). The three methods are described and departures of the rate coefficients from their equilibrium values are computed for each case and compared, along with an explicitly time-dependent solution of the Boltzmann 58 Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 59 equation. The usual approach to the description of systems close to equilibrium is based on the C E of solution for neutral and ionized systems [71]. This method is known to be invalid for strongly non-equilibrium systems for which the required separation of length and time scales is not obtained [5-9]. Several groups have considered the description of transport for such systems far from equilibrium; in particular the nonlocal heat transport in plas-mas [5], hypersonic flows [6], diffusive flows [7,8], as well as astrophysical applications [9]. For spatially inhomogeneous systems, the C E method is valid when the mean free path of the constituents is much less than the typical length scale [71]. For reactive systems, a C E method is applicable when the elastic time scale is much less than the reactive time scale [23]. Kogan [42] and Alexeev et al. [43,46,47] have considered extensions of the C E method to chemically reactive systems far removed from equilibrium. Aleexev has referred to this method as the generalized C E method. The development of techniques to describe such systems is an important endeavour. This chapter, although restricted to a spatially homogeneous reactive model system, is a detailed study of alternatives to the C E method in situations similar to the above when the required separation of time scales does not exist. This work is also motivated by the general methods of solution of the Boltzmann equation introduced by Pascal and Brun [41] in their study of transport processes in molecular systems. The main purpose of this chapter is to reexamine the non-equilibrium effects associ-ated with a simple reactive system of the type, A + C —r products, where the reactive process perturbs the distribution functions of both species from the equilibrium Maxwellian distributions and the rate coefficient differs from the equilibrium rate coefficient. As in the previous chapter, 77 = (A;(°) — k)/k^\ the fractional decrease of Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 60 the non-equilibrium rate coefficient k from the equilibrium rate coefficient k^°\ obtained from different treatments will be compared. In Chapter 2, the distribution functions were expressed in terms of Maxwellians at the same temperature plus a small correction that arises because of the reactive process. The departure of the distribution functions from Maxwellian was then determined with a C E type approach [1] that has been de-scribed at length in the literature [20-22] and the previous chapter. The C E approach has been referred to as W N E by Pascal and Brun [41] in their study of transport processes in molecular systems. In their case, the distribution function for the molecular system is expanded about a single temperature even though the translational and vibrational temperatures may differ. The different species temperatures in the present chapter is somewhat analogous to different translational and vibrational temperatures in molecular systems. It is shown that the W N E method is valid provided that the elastic cross sec-tions for A-A, C-C and A-C collisions are all the same order of magnitude and all much larger than the reactive collision cross section. The usual C E approach to this problem [13-23] is based on the treatment of the reactive process as a small perturbation on the elastic collision processes that restore the system to equilibrium. This is equivalent to the recognition that reactive cross sections are typically much smaller than elastic cross sections. In a mixture, it is possible for a series of time scales to coexist, due to different values for the collision cross sections among the different components. In Section 3.2, such a situation is considered for a two-component system in which the elastic cross sections for A - A and C-C collisions are similar and much larger than the cross sections for elas-tic A - C and reactive A - C collisions. The different magnitudes of these cross sections result in different time scales and are used to obtain CE-like solutions. For this situ-ation, distribution functions are expanded about Maxwellians at different temperatures and the departure from equilibrium of these local Maxwellians is calculated. Pascal and Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 61 Brun refer to this approach as SNE and in their application expand the distribution function about equilibrium Boltzmann distributions at different translational and vibra-tional temperatures. There are many different physical situations far from equilibrium for which the coupling of different species is weak and thus they are characterized with distribution functions at different temperatures. This is most notable in plasma systems where electron-ion and electron-neutral collision rates are relatively small and electron temperatures can be considerably different from ion and/or neutral temperatures. The transport theory of plasmas [74] takes this feature into account. These phenomena are evident in the ionosphere where departures from equilibrium occur owing to the influ-ence of electromagnetic fields on charged particles and also because of chemical reactions [74,91,92]. The transport theory of the drift of ions in an electric field involves the spec-ification of different temperatures for the ion velocity distributions for velocities parallel and perpendicular to the electric field [93]. For chemically reactive systems similar to the model systems considered in this work, Cukrowski et al. [32,31] have discussed at length the effects of different species temperatures. In Section 3.3, a modification of the SNE approach is presented and a Modified Strong Non-Equilibrium (MSNE) method is introduced. The deficiencies of the SNE method were discussed by Pascal and Brun since their results with SNE do not coincide with W N E when the vibrational and translational temperatures coincide. With the MSNE formal-ism, previously referred to as the generalized C E method [41-43,46,47], the results with for SNE coincides with W N E in the limit of equal translational and vibrational tempera-tures. This MSNE approach is applied to the model reactive systems A + C - » products and the results with W N E and SNE compared. In order to understand the range of valid-ity of the three methods, a rigorous time-dependent solution of the Boltzmann equations is considered. The time-dependent analysis is used to study the establishment of the steady states assumed in the WNE, SNE and MSNE approaches. A comparison of the Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 62 three formalisms is discussed and compared with the explicit time-dependent results in Sections 3.5 and 3.6. 3.2 The Reaction A + C -+ products : Strong Non-Equilibrium (SNE) For purposes of studying systems where species temperatures can play an important role, it is useful to consider the expansion of the distribution functions about local Maxwellians characterized by densities n T and different species temperatures, T T , that remain to be specified. This is equivalent to the assumption that the rate of collisions between unlike species is very slow relative to the self collision rate. This is an approach that is considered in plasma systems [74], in ionospheric applications [91,92] and is also employed in the determination of mobility of ions in neutral gases [93]. The Boltzmann equations, Eqs. (2.2.2) and (2.2.3), are rewritten in the form, dh 1 dt e dt e J J [f'if ~ fifWngdQdc] + J f [f'j'2 - fif2]o-ugd£ldc2 - j J hf2o-*gdSldc2 (3.2.1) / / [f2f ~ f2f]o-22gdttdc + j j[f'j'2 - /i/2]<7i20dir2dci - JI fif2o-*gdQdCl (3.2.2) where the term in 1/e now multiplies only the self-collision integral term. The. collision term between unlike species is considered to be much smaller than the like species collision term. The exchange of energy between components is slow owing to a small o7V cross section or a disparate mass ratio. The distribution function is now written in the form, / 7 = F f [ i + ^ 7] (3.2.3) where = n7(t)[m7/27rkBT1(t)}^ 2exp[-m7c 2/2kBT7(t)} is the local Maxwellian, the solution to the equation of order 1/e that results with the substitution of Eq. (3.2.3) into Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 63 Eqs. (3.2.1) and (3.3.2). The local Maxwellians are characterized by different temper-atures, T7(t). The time dependence of the species densities and temperatures to lowest order is given by dt I [ dt ~ III F i ] F2Q) a* 9d^dcidc2 (3-2.4) 'dTA{0) _ 2T\ dt I 3nx j J j F?)F?\y12-yl)o'12gdtodcldc2 (3.2.5) + j J jFi0)Ff[^-yl}a*gdndCldc2 'dTA^ = 222 dt J 3n i r 3 J J JFf F2{0\y'2 - y22)al2gdndCldc2 (3.2.6) UJF^if ^ - y2]o-*gdndCldcf} where y2 = m7c2/2kBT7. Equations (3.2.5) and (3.2.6) include the energy exchange between components owing to elastic collisions as well as that due to reactive collisions. With the substitution of the expansion, Eq. (3.2.3), into the Boltzmann equations, Eqs. (3.2.1) and (3.2.2), the uncoupled set of integral equations to zero order e is 11 F^F^l^ + 4 ~ <Ay - (f>]cr77gdQdc = F^G7(c7) 7 = 1 , 2 (3.2.7) where n , \ 1 fdn7\{0) 1 ,3 2. (dT7\{0) = -n7{-df) -T7^2-^{-df) - j j [FWFW - F^F^]a12gdQdcv + JJ ^ gdSldc- (3.2.8) where y2 = m7c2/2kBT7. The inhomogeneous terms, G T (c T ), involve the unlike elastic collision terms evaluated to lowest order. These terms vanish only when the two tempera-tures are equal; Tx = T2. These terms do not appear in the corresponding inhomogeneous Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 64 terms, Hy(cj) (see Eq. (2.2.10)) in the W N E approach. Employing the expansions M * r ) = E & W ( $ (3-2-9) i=2 in Eq. (3.2.7) one gets the set of algebraic equations for the coefficients given by, E ^ ^ , ^ ] ^ ^ ! ^ , 7 = 1,2 (3.2.10) i=2 where the square bracket matrix elements [S7l\ S^] are the one-component matrix el-ements employed in Chapter 2 (Eq. (2.2.15)). The coefficients (3^ depend on the two temperatures T\ and T 2 and are defined by, r W | 7 ) = / F^GjS^dCy. (3.2.11) The integrals in Eq. (3.2.8) involve the lowest order matrix elements of the unlike species collision operator that were evaluated elsewhere [80]. These are discussed again in the next section. The definition of the species densities and temperatures gives n 7 = j F^dcy (3.2.12) and | n 7 A ; B T 7 = \ j F^m^dc, (3.2.13) respectively. These definitions in turn imply that JF^UjdCy = 0 (3.2.14) and J F^rrijC^dCj = 0, Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 65 so that &o = b(i — 0. For the SNE, the fractional decrease in the equilibrium rate of reaction is given by, VSNE = -2ZEbtr)AP(T1,T2)/A0(Tl,T2). (3.2.15) 7=1i=2 where AP(TUT2) = I f F^F^S^gdSld^do, The set of equations, Eq. (3.2.10), is coupled to the time-dependent equations for n 7 and T 7 , Eqs. (3.2.4)-(3.2.6). The solution of Eq. (3.2.10) requires specifying an initial condition, integrating Eqs. (3.2.4)-(3.2.6) and inverting Eq. (3.2.10) at each time. This is very similar to W N E , for which the correction 7 7 W N B , given by n in Eq. (2.2.23), varies implicitly with time through n7(t) and T(i). However, the matrix equations Eq. (2.2.12) and (2.2.13) can be inverted for chosen values of n 7 , m 7 and e* = E*/kBT and the variation of 7 i W N E can be studied [20-22]. In the SNE case, the matrix equation, Eq. (3.2.10), can be inverted for a set of values n 7 , m 7 and e* = E*/ksTes. The T 7 values are set to those that result from the W N E solution and are given by Eq. (2.2.24). In Section 3.4, the variation of 7 7 S N E is shown versus the system parameters for this choice o f T 7 . In the work of Pascal and Brun [41], the results with the SNE do not correspond to the results with the W N E when the vibrational and translational temperatures are equal. The reason for this is that the collision operators that are inverted are not the same in the two formalisms owing to the assumptions of the model. This can be understood by comparing Eq. (3.2.7) with Eq. (2.2.12) and (2.2.13) where in the former case, only the self-collision (1-1 and 2-2 collision) operators are inverted whereas in the latter case, both self-collision and unlike-collision (1-2 and 2-1 collision) operators are inverted. For the special case where the two species are identical and Tx = T2 the W N E and SNE coincide. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 66 The two unlike-collision operators add and are equal to the self-collision operator, so that Eqs. (2.2.12) and (2.2.13) are equivalent. The number of collisions in the W N E is thus twice the number of collisions in the SNE because in the former case one effectively counts the number of self-collisions twice [21]. 3.3 The Reaction A + C —> products : Modified Strong Non-Equilibrium Pascal and Brun [41] introduced a modification to SNE in order that the results would agree with the W N E results in the limit Tv;b = T t r a ns - This involves including a higher or-der term in the perturbative analysis of the Boltzmann equations leading to Eqs. (3.2.7). If the terms linear in e are retained on the LHS of the Boltzmann equations, Eq. (3.2.1) and (3.2.2), then in place of Eq. (3.2.7) one gets analogous to Eqs. (2.2.8) and (2.2.9) in W N E but with the important distinction that are as defined by Eqs. (3.2.8) in the SNE approach. The unlike species collision operators on the LHS of Eqs. (3.3.1) and (3.3.2) are not self-adjoint in contrast to the situation for W N E . This aspect of this approach has been discussed at length by Pascal and Brun following the original work by Kogan et al. [42] and Alexeev et al. [46]. An important consideration is that the solution of the homogeneous equation with the adjoint collision (MSNE) i f ^ Fi0)Ff b ecause the species temperatures differ. The inhomogeneous terms Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 67 operators, corresponding to Eqs. (3.3.1) and (3.3.2), must be orthogonal to the inhomo-geneous terms F f G 7 in order for unique solutions to exist [1,41]. Consequently, it has been assumed [41,42,46] that one can neglect the non self-adjoint part of the collision operators in Eqs. (3.3.1) and (3.3.2). This procedure is adopted here and justification for it is provided later in connection with specific applications. Pascal and Brun referred to this approach as modified strong non-equilibrium. With the expansion ^ W = E c W ( 4 ) (3.3.3) i=l in Eqs. (3.3.1) and (3.3.2), one gets the set of algebraic equations for the coefficients cf given by, E [(n?[S?\ S f 0 ] + n i n 2 < S?, Sf >s) cf + n i n 2 < S?, Sf >s cf] = n.nrff, 3=2 (3.3.4) and E Sf] + n,n2 < S®, Sf >s) cf + nxn2 < S?,Sf >s cf] = rhn^f. 3=2 (3.3.5) The angle bracket matrix elements < S^, >s are the self-adjoint part of < S^, > and depend on the two temperatures T\ and T2. The non self-adjoint part of these matrix elements vanish for T\ = T 2 . If the difference between the temperatures is small, then the neglect of the non self-adjoint part of the operators is justified. The expression for 77MSNE is identical to the expression Eq. (3.2.15). As discussed in Section 3.2, the set of equations, Eqs. (3.3.4) and (3.3.5), is coupled to the time-dependent equations for n T and T 7 , Eqs. (3.2.4)-(3.2.6). Their solution requires specifying an initial condition, integrating Eqs. (3.2.4)-(3.2.6) and inverting Eqs. (3.3.4) and (3.3.5) at each time. However, this time-dependent problem is avoided by using the Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 68 Ty values from the W N E calculated from Eq. (2.2.24). The following section provides a comparison of the results of the WNE, SNE and MSNE formalisms. 3.4 Comparison of Non-Equilibrium Effects with W N E , SNE and M S N E Formalisms In this section, the results for the W N E method of Section 2.2.1 along with the SNE and MSNE results described in Sections 3.3.2 and 3.3.3 are presented and compared for the model line of centers reactive cross section cr*ot = 7rd2(l — E* jE) and elastic hard-sphere cross section, aJV = dyrj/A. The required matrix elements of the elastic collision operators have been evaluated for equal temperatures [20,21] and unequal temperatures [80]. The calculations of the moments of reactive collision terms have also been described elsewhere [20,21]. For SNE and MSNE, the species temperatures must also be specified and one pro-ceeds as follows. Consistent with the assumption that the elastic cross section is much larger than the reactive cross section, the time scale for elastic collisions is very short rel-ative to the time scale for reactive collisions and the species temperatures quickly attain quasistationary values. In this way, Eqs. (2.2.12) and (2.2.13), (3.2.10), and (3.3.4) and (3.3.5) are inverted for the calculation of n in the W N E , SNE and MSNE formalisms, respectively. The results depend critically on the species temperatures used in the SNE and MSNE methods. The species temperatures Tx and T 2 used in the SNE and MSNE methods are those obtained from the inversion of the W N E equations, Eqs. (2.2.12) and (2.2.13) and calculated with Eq. (2.2.24). The values of T T are then used in Eqs. (3.2.10) and Eqs. (3.3.4) and (3.3.5) for the SNE and MSNE calculations, respectively. This ap-proach is justified in Section 3.6 by integrating the system of hydrodynamic equations, Eqs. (3.2.4)-(3.2.6), and demonstrating their equivalence with the solution of the C E Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 69 system of equations (WNE). The non-equilibrium corrections to the reaction rate, rj, are shown in Figures 3.1(a)-3.1(d). For W N E , there is only one temperature, T, the system temperature, and 7? W N E (T) = [k(T) - k^(T)]/k^(T) and the reduced reactive activation energy is e* = E*/kBT. In the case of SNE and MSNE, the species temperatures Ti and T 2 , are used to obtain n(Ti,T2) defined as the fractional change from k^0\T) given by, n(TuT2) = [fc(Ti,T2) - k^(T)}/k(°\T), (3.4.1) where T = [niT\ +n2T2]/(ni + n2). Note that the fractional decrease in the reaction rate coefficient, Eq. (3.4.1), is expressed relative to the one-temperature system equilibrium rate coefficient. This serves for the purpose of comparison with the W N E . Equation (3.4.1) differs from the values for ?7S N E and r ? M S N B in Eq. (3.2.15). Writing Eq. (3.4.1) in terms of expansion coefficients, the two-temperature expression for the fractional change in the rate constant is v(TuT2) = ( k W K f ) - M -J:JZbfAf(TuT2)/U (3.4.2) V A0 / T=1 i = 2 For SNE and MSNE, e* is expressed in terms of the component temperatures and is equal to E*/kBTeS, where TeS is given by Eq. (2.2.27). For all of the numerical calcu-lations, 5 to 7 terms were retained in expansion of the distribution functions in Sonine polynomials and provided values of n to three significant figures. Figures 3.1(a)-3.1(d) show the variation of n versus e* for WNE, SNE and MSNE for a set of mass ratios, m\jm2 with the density ratio n\jn2 = 1 and cru/o* = o22/a* = a12/o* = 1. The n value for the W N E is calculated with Eq. (2.2.23) and for the SNE and MSNE with Eq. (3.4.2). Figure 3.1(a) shows the case for which nxjn2 = 1 and Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 70 Figure 3.1: Variation of rj versus e* computed from the SNE, W N E and MSNE methods for different mass ratios. The mass ratio, m i / m 2 , equals (a) 1, (b) 1.5, (c) 3, and (d) 5. cn /c* = . C 2 2 / c * = cr1 2/cr* = 1 and n i / n 2 = 1. Figure (a) shows a one component system, A + A —> products, the solid line, and a two component system A + C —> products, the broken line. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 71 m\jrri2 = 1, that is for A — C, and the reaction reduces to A + A —> products. In Figure 3.1(a), the corrections obtained with the WNE, SNE and MSNE methods are all equal, and shown with the solid curve. The dashed curve in Figure 3.1(a) is the result that is obtained with the W N E and MSNE for a two component system, A + C —> products, in limit A —> C. The solid curve in Figure 3.1(a) is the result that is obtained for a one component system, A + A —> products. For W N E and MSNE in the limit A —> C the 1-1 and 1-2 collisions are indistinguishable, and hence the number of collisions is twice the number of collisions for a one-component system A + A —> products. This gives a value of r] for the one-component system that is twice the rj value for the two-component system in the limit of identical species for the W N E and MSNE methods. For the SNE method, only 1-1 collisions are included in the collision operator that is inverted, Eq. (3.2.10). For Figures 3.1(b)-3.1(d) the mass ratios are 1.5, 3 and 5, respectively, and differences are observed in the behaviour of n obtained from WNE, SNE and MSNE. Figure 3.1(b) shows the case where the masses are slightly different (m 1 /m 2 = 1.5). The SNE curve shows similar behaviour to the solid curve in Figure 3.1(a), but the W N E and MSNE curves are closer to the dashed curve corresponding to the two component system. As the mass ratio increases it is observed that an increase in n values, especially for e* ;$ 2. This is the result of increased temperature split between Ti and T 2 caused by the dispar-ity between the masses of the two components. This temperature splitting increases the departure of the system from equilibrium. Figures 3.1(b)-3.1(d) show that for e* > 2, the SNE result does not agree with the W N E and MSNE results. At larger e* values (e* > 4), there is good agreement between the W N E and MSNE results, whereas the SNE result agrees with the MSNE result for small values of e* ~ 0. Figure 3.2 shows the change in rj versus e* for a set of density ratios with mi/m2 = 1 and on/cr* = cr22/<7* = o^/o* = 1. The density ratio between the two components, Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 72 ni/n2 increases from 1.5 to 5 in Figures 3.2(a) to 3.2(d), respectively. As the density ratio increases, rj increases. The behaviour is similar to that observed by varying the mass ratios in Figure 3.1. As ni/n2 increases from Figure 3.2(a) to 3.2(d), the differ-ence between the two component temperatures increases and larger n values result. The increase in 77 is particularly large for e* < 2 due to the large influence of the different species temperatures. As the density ratio n i / n 2 departs from unity and becomes very different from 1, it is observed that the effect of the larger temperature separation that results in larger n values. Again, it is observed that for e* > 4, the W N E and MSNE are in agreement. One notable difference between Figure 3.1 and 3.2 is that when the density ratios become large, the SNE and MSNE do not agree for e* ~ 0. 3.5 Time-Dependent Species Temperatures; Comparison with W N E The W N E approach involves the expansion of a distribution function about the Maxwellian at one temperature, although different species temperatures are calculated from Eq. (2.2.24) with the solution of the C E equations, Eqs. (2.2.12) and (2.2.13). For the W N E , the departure of the species temperatures from the system temperatures, T, to lowest order is determined by retaining only the af coefficients in Eqs. (2.2.12) and (2.2.13), and also using Eq. (2.2.22). The result for component 1 is T — Ti \ W N E nx 1 M 2 - ^ n (e* + l/2)e~£*. (3.5.1) T J n AMiM2 The hydrodynamic equations, Eqs. (3.2.4)-(3.2.6), give the time-dependent behaviour of the temperatures and densities of each component. The conditions under which the W N E method gives a good approximation of the hydrodynamic result is deduced by integrating the coupled set of differential equations Eqs. (3.2.4)-(3.2.6) and comparing them with the results of the C E solution. This provides an estimate of the range of validity of Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 73 Figure 3.2: Variation of n versus e* for the SNE, W N E and MSNE methods for the reaction A + C —> products for the different density ratios. The density ratio, rt i /n 2 , equals (a) 1.5, (b) 2, (c) 3, and (d) 5. an/a* = 022/cr* = 012/0* = 1 and m i / m 2 = 1. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 74 the C E method. Figure 3.3 shows the results of the integration of the hydrodynamic equations, Eqs. (3.2.4)-(3.2.6), for different choices of the time scale parameter T E / T R = cr* exp[-e*]/a1 2 and initial condition Tx(0) = T2(0). The results shown in Figures 3.3(a) - 3.3(c) are for an increasing separation in the elastic and reactive time scales. After a brief transient, the temperatures approach an asymptotic dependence which varies on a much longer time scale. It is shown that in the limit T E / T R —> 0 the steady solu-tions of Eqs. (3.2.5) and (3.2.6) coincide with W N E to the lowest order, Eq (3.5.1). As the separation between the elastic and reactive time scales increases, the agreement between the hydrodynamic (solid curves) and the W N E results (dashed curves) improves. The ratio of elastic to reactive hard sphere collision cross-sections, o\\/o* = 0-22/0* = 012/0*, are 1, 10 and 100 in Figure 3.1(a)-3.1(c), respectively. The agreement improves as Oyi/o* increases as is clear from the behaviour in Figure 3.3; Figure 3.3(c) shows exact agreement on the reactive time scale. It is important to notice the different time and temperature scales in Figures 3.3(a)-3.3(c) which depend on the value of on/o*. The departure of T 7 from T for the two species are not equal since the density ratio ni/n2 = 5 in Figure 3.3. A comparison of the results with the W N E , that is, Eq. (3.5.1), with the long time asymptotic, quasi-steady results, from the numerical integrations in Figure 3.3, denoted by (f-jr 1^) 7 , is shown for a range of different system parameters in Table 3.1. As the ratio of elastic to reactive time scales becomes sufficiently large, the ra-(T—T \asv 1 IT-T \ W N E tio ( ) / \ ~T) approaches unity. This aspect of the C E approach has been discussed before [23,73]. The basic conclusion is that the separation of elastic and reac-tive time scales must be of the order of 10~3 to 10~4 for the C E approach to be valid. For this situation, the departure from equilibrium will be correspondingly very small. The W N E species temperatures have been employed to define the two-temperature matrix elements for calculating 7 7 S N E and T J M S N E as discussed in Sections 3.3.2 and 3.3.3. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 75 Figure 3.3: Time-dependence of the hydrodynamic corresponding C E fractional tem-perature separation. The hydrodynamic value of (1 — T 7 / T ) is given by the solid line and the C E value, o' 7 \ is given by the broken line. The elastic collision cross sec-tions ou/o* = o22/o* = o12/o* are (a) 1, (b) 10 and (c) 100, with E*/kBT(0) = 3, 7i1(0)/n2(0) = 5 and mi/m2 = 1. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 76 E*/kBT o/o* 1 1 3.68(-l)2 3.355(-2) 4.484(-2) 0.748 2 1 1.35(-1) 1.919(-2) 2.529(-2) 0.759 4 1 1.83(-2) 5.529(-3) 6.502(-3) 0.851 1 100 3.68(-3) 5.129(-4) 5.157(-4) 0.995 2 100 1.35(-3) 3.125(-4) 3.140(-4) 0.995 4 100 1.83(-4) 7.651(-5) 7.668(-5) 0.995 1 1000 3.86(-4) 5.167(-5) 5.171(-5) 0.999 2 1000 1.35(-4) 3.165(-5) 3.168(-5) 0.999 4 1000 1.83(-5) 7.721 (-6) 7.721(-6) 1.000 2(-n) = x l O -Table 3.1: Comparison of asymptotic temperature values, (TrTl j , calculated from in-, \ WNE tegration of hydrodynamic equations and corresponding one-term C E values, ( r 1 j The mass ratios for components 1 and 2, m 1 / rn 2 is 0.5, the density ratio ni/n2 = 1.0 and o/o* — on/o* = 022/0-* = 012/0*. 3.6 Time-Dependent Solution for A + C —>• products The CE-type solutions discussed in Sections 2.2.1,3.3.2 and 3.3.3 give the long-term behaviour on the reactive time scale. The solutions obtained are a special set of normal solutions that describe the quasistationary perturbed system. This section examines an explicit time-dependent solution of the Boltzmann equation that does not assume very different reactive and elastic cross sections. The validity of the W N E , SNE and MSNE methods versus the four time scales given in terms of an, a2 , cr12 and o* is of interest here. If o~n and a 2 2 are large it is expected that initial non-Maxwellian distributions will become Maxwellian at different temperatures on a short time scale. On a longer time scale defined by o~i2, the two components will equilibrate. Finally, the reactive process, considered generally as the longest time scale, perturbs the distribution function from the Maxwellian. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 77 The time-dependent distribution function is expressed as F 7 (x, i ) = F f [1 + V» 7(x,t)], (3.6.1) where the local Maxwellians, F7°\ vary implicitly in time, t, through the number density, n7(t), and temperature, T 7 (£) and also ipy(x,t), the time-dependent perturbation from Maxwellian. This method is analogous to the study of temperature relaxation in binary gases [73] and the application to hot atom reactions [72]. The substitution of Eq. (3.6.1), into Eqs. (3.2.1) and (3.2.2) with e = 1, gives, dt - j j FfFfgo-^dSldc, + J j FfFfl^ + Vy - V>7 - VV W ' M c 7 ' -JJFf'Ff^g^dndCr, (7,77) = 1 ,2. . (3-6.2) The perturbations from the local Maxwellians are expanded in Sonine polynomials ^{x,,t) = J2tt\t)S^(x% (3.6.3) i=2 where the expansion coefficients are explicitly time-dependent. The set of Boltzmann equations, Eq. (3.6.2), is thus reduced to the set of coupled equations of the form, 1 fllll) N ^ = ^ + D W + W ] , i = 2,3..JV (3.6.4) where, = nv<sy,Sf>--A. (7) Btf = Th[S®.S?]+r^<S<>\s!p>-DW ^ 1 dT 7 1 dn 7 ^ ijlT\~dT ~ rZf^dT lJ C[f = nv< 5 ^ , 5 ^ > +E\? (3.6.5) Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 78 The angle integrals, < Sf, S^ > in Eq. (3.6.5) are the two-temperature matrix elements of the collision operators as given in Eq. (2.3.7), and evaluated for the hard sphere cross section. The integrals A-1^ are the moments of the equilibrium reactive collision frequencies, Eq. (2.2.18). The quantities Hijk are the integrals of products of three Sonine polynomials and the terms that occur in Eq. (3.6.5) arise from dF^/dt, and dS^/dt. These details of the calculation have been discussed in references [80,72]. The quantities D[f and E^ are the matrix elements of the reactive collision operator. The matrix elements Bij and dj and quantity \ f are implicitly time-dependent through the density, given by, AP + J2bf(t)Af+bf(t)^ i=2 (3.6.6) dT7 _ 2nnT7, ^ ( 1 ) ( 0 ) ]_Ah) dt ~ 3 > n7 N drii/dt = dri2/dt — and the temperature, given by, + E ( [< 5 T 1 } ' s ? > -DnW] + [< ^ s v ] > -DnW]) • (3-6.7) Hence, the set of equations Eq. (3.6.4), is linear but with nonconstant coefficients. The explicit time-dependent scheme is a rigorous solution of the Boltzmann equation and there is no assumption about the ordering of the various terms. This scheme is used to test the validity of the W N E , SNE and MSNE approaches. For the numerical integration of Eqs. (3.6.4)-(3.6.7) a dimensionless time variable, tE, is defined given by tci 2nS 0 )(0)/[27rA; BT(0) e f f/m 1] 1/ 2. The initial distribution functions are assumed to be Maxwellian (that is, bf (0) = 0) with specified initial densities, n 7(0). The system of coupled equations Eqs. (3.6.4), (3.6.6) and (3.6.7) were integrated over time with a fourth order Runge-Kutta procedure. For each time step in the numerical integration of Eqs. (3.6.4), (3.6.6) and (3.6.7), values of Ti , T 2 , n x and n2 were calculated Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 79 and used to evaluate the matrix elements in Eq. (2.2.12) and (2.2.13), Eq. (3.2.10) and Eqs. (3.3.4) and (3.3.5). The matrix equations were then solved and the fractional changes in the equilibrium reaction rate constant for each of the three formalisms, 7 i W N E ; 7 ] S N E and 7 7 M S N E are computed. The fractional change in the reaction rate constant at each time step was computed from -q(t) — — Z ) 7 = 1 Z ^ ^ W ^ I ^ M o where the collision integrals also vary with time through the species temperatures, T7(t). The time-dependent solution to r](t) eventually attains an asymptotic value, denoted by r? a s y. As for the time-independent calculations, 5 to 7 terms were retained in the Sonine polynomial expansion of the distribution function, providing a value of n to three significant figures. The C E approach of Section 2.2.1 assumes that the reactive and elastic time scales are very well separated. This is inherent in the assumption that the time dependence is implicit in the time dependence of the densities and the temperature, as given by Eq. (2.2.5). This assumption has been studied in references [23,73,95] and it was shown from explicit time-dependent calculations that the ratio of time scales should be of the order of 10~4 to 10 - 5 for this approach to be valid. Since the non-equilibrium effects scale as the ratio of reactive to elastic cross sections, the corrections from equilibrium obtained with the W N E method are expected to be very small. The effect of different 1-2 time scales was studied for the time evolution of rj(t)/rj, where n is the value obtained from the WNE, SNE and MSNE methods (Eqs. (2.2.23) and (3.2.15)). Figures 3.4(a)-3.4(d) show the time evolution of systems with decreasing values of the 1-2 elastic collision cross section ratio, c r 1 2 / c r * . Both components are initially described by Maxwellians at the same temperature, that is, Ti(0) = T2(0), with the reduced activation energy, E*/kBT(0) = 10. The ratio of the elastic collision cross section to reactive collision cross section for the result in Figure 3.4 has been chosen sufficiently large so that the ratio of time scales, T E /VR , is small and the C E type solution of the Boltzmann equation is valid. There is a clear separation of Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 80 1.0 0.8 0.2 0.0 1 1 1 1 I 1 1 WNE (b) -/ --SNE _ -1 , 1 , 1 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 t. 1.0 0.8 £ 0.6 0.4 0.2 0.0 -I 1 T-MS NE~- i1 r SNE WNE (<0 1.0 0.8 -1 £ 0.6 0.4 0.2 -0.0 -1 1 r MSNE • SNE WNE -(d) 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 t. Figure 3.4: Time dependence of the ratio n(t) to r ? M S N E , r 7 W N E and 77 S N E for Ti(0)/T 2(0) = 1. The ratio of elastic to reactive hard sphere collision cross section ou/cr* = 1000 while al2/o* equals (a) 1000, (b) 200, (c) 20, and (d) 1. The ratio ffii/a22 = 1, E*/kBT{0) = 10, m 1 / m 2 = 3 and m(0)/n2(0) = 2. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 81 Figure ou/o* ^asy ^ W N E ^asy/^SNE ^asy ^ M S N E 4(a) 1000 1000 2.89(-5) 1.000 0.278 1.00 4(b) 1000 200 6.74(-5) 0.931 0.649 1.00 4(c) 1000 20 9.77(-5) 0.633 0.943 1.00 4(d) 1000 1 2.89(-5) 0.082 0.994 1.00 Table 3.2: Comparison of time-dependent, W N E , SNE and MSNE values for n for various crii/°~* = 0-22/c* and 012/0*. The mass ratio, rni/rri2 = 3.0 and the initial density ratio, rai(0)/n2(0) = 2.0, and E*/kBT(0) = 10. time scales in Figure 3.4 with a very fast initial transient followed by a steady asymptotic result, which can differ from unity depending on the method used. Figure 3.4(a) with mi/iri2 = 3 and fii(0)/n2(0) = 4 shows the case for which cross sections for 1-1 and 1-2 collisions are equal, on/o* = cr22/o* = o~i2/cr* = 1000. After the initial transient, one obtains identical results for r ? W N E and 7 7 M S N B equal to nASy. In Figures 3.4(b), 3.4(c) and 3.4(d), the ratio on/cu is 5, 50 and 1000, respectively. With increasing 0\\jo\2 the results with MSNE remain in agreement with 7 7 ^ , however, the W N E result does not agree when the SNE result approaches rf^. In Figure 3.4(d), virtually exact agreement is found between 7 ? a s y and 7 i S N E for owjoyi = 1000. The results in Figure 3.4 are sum-marized in Table 3.2. It is shown that when cr 1 2/a* = on/cr* = <72 2/cr*, the variable 7 / a s y is in agreement with results obtained with W N E and MSNE. As cr 1 2/a* decreases, there is disagreement with the W N E result while agreement with the SNE result improves. There is uniform agreement with the MSNE result for all values of cr^/V*. The elastic and reactive time scales in Figure 3.4 and Table 3.2 differ by more than a factor of 10~5 are therefore sufficiently well separated to ensure that MSNE and the time-dependent results agree as shown in Table 3.2. Table 3.3 illustrates the variation of 7 ? a s y with decreasing T B / T R with all elastic cross sections equal, analogous to the situation of Figure 3.4(a). It is observed that when the separation of elastic and reactive time scales is large, T E /VR = 10~7, there is good Chapter 3. Non-Equilibrium Reactive Systems : The R,ole of Species Temperatures 82 E*/kBT(0) a/a* r E / r R n^ V ,WNE ,SNE 2 1 1.35(-1) 2.196(-2) 5 20 3.37(-4) 4.182(-3) 2 1000 1.35(-4) 1.600(-5) 10 1 4.54(-5) 2.283(-2) 10 200 2.67(-7) 1.441 (-4) 10 1000 4.54(-8) 2.900(-5) 0.100 0.648 0.748 0.767 0.991 1.000 0.222 0.265 0.283 0.220 0.278 0.278 Table 3.3: Comparison of time-dependent, W N E and SNE values for 77 where a/a* = an/a* = cr22/a* = cr 1 2/a*. The mass ratio, m i / m 2 = 3.0 and the initial density ratio ni(0)/n2(0) = 2.0. agreement between n^ and ^ W N E . When there is only one elastic relaxation time scale, and the reactive time scale is long, it is anticipated that the W N E result will become valid whereas the SNE result is inapplicable. This behaviour is verified in Table 3.3. It is shown that r)asy/rjw®E approaches unity with decreasing T E / T R whereas rjasy/r}SNE remains significantly less than 1. If on the other hand, <jn = 022 7^ cr12 and there are two elastic time scales which differ from the reactive time scale, it is expected that the W N E results would be inap-propriate and the SNE results to be correct. This behaviour is illustrated in Table 3.4 where T H / T R and T I 2 / T R are varied by changing the values of E*/kBT(0), an and cr22 appropriately. In Table 3.4, with E*/kBT(0) = 10 and an/cr* = 1000 and aV2/a* = 200, ^ a s y ^ W N E = ggg^ w h e r eas for an/a* = 1000 and aV2/a* = 20, ^ /rjWNE = 0.082. When the separation of the time scales for 1-1 and 1-2 elastic collisions increases with c r n / c r i 2 becoming large, the ratio 7 7 a s y / r / S N E is close to unity. In Table 3.4, where for an/cr* = 1000 and a^/a* = 1, n^/nSNE = 0.999. Note that the 77^ and 77SNE values are close only when the time scales elastic and reactive collisions are sufficiently well separated and the ratio of elastic time scales, T U / T 1 2 , is of the order of 10~3. Table 3.5 compares ?7asy and 7yM S N E over a series of different time scales for 1-1 elastic, 1-2 elastic and reactive collisions. Table 3.5 shows disagreement between ?7asy and 77MSNE Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 83 E*/kBT(0) ou/o* V12/0-* n i / T R 3 W r R 4 ^asy y^WNE ^asy y^SNE 2 200 200 6.77(-4) 6.77(-4) 1.953(-4) 0.174 0.256 10 1000 200 4.54(-8) 2.27(-7) 6.74(-5) 0.931 0.649 2 200 20 6.77(-4) 6.77(-3) 2.718(-4) 0.279 0.800 5 200 20 3.37(-5) 3.37(-5) 1.252(-3) 0.362 0.868 5 200 1 3.37(-5) 6.74(-3) 1.684(-3) 0.035 0.972 10 1000 20 4.54(-8) 2.26(-6) 2.897(-5) 0.082 0.994 10 200 1 2.27(-7) 4.54(-5) 5.167(-4) 0.311 0.996 10 10000 1 4.54(-8) 4.54(-5) 1.053(-5) 0.001 0.999 3 (an/a*)e-£* 4 (a12/a*)e-e* Table 3.4: Comparison of time-dependent, W N E and SNE values for r\ for various C n / c * = ^22 /^* and au/cr*. The mass ratio, m 1 / m 2 = 3.0 and the initial density ratio ni(0)/Vi2(0) = 2.0. when E*/ksT(0) — 2 and on/o* = 1 and oyijo* = 1. In this case, the separation of colli-sion time scales for elastic and reactive processes is small, with T H / T R = T ^ / T R = 0.135, and r ^ y / r ^ N E = 0.686. Table 3.5 shows that the ratio ^ / ^ M S N E a p p r o a c h e s unity when the reactive time scale is long compared to the 1-1 and 1-2 elastic time scales. Table 3.5 shows that 77ASY and 7 7 M S N E agree when both 1-1 and 1-2 elastic collision and the reactive collision time scales are well separated. The MSNE result agrees with the time-dependent result over a range of ou/cr* and o^/cr* values. The MSNE method is accurate so long as the elastic and reactive time scales are well separated, and in contrast to the W N E and SNE method, its applicability is not dependent on the relative length of the 1-1 collision and 1-2 collision time scales. It is interesting to note that the initial transient behaviour in Figure 3.4, which occurs on the elastic time scale, is not strongly affected by the change in the 1-1 elastic and 1-2 elastic collision time scales. Figures 3.4(a)-3.4(d) evolve on the same general time scale, with a sharp rise in 77 on the elastic collisional time scale due to the perturba-tion from the chemical reaction followed by a slow relaxation on the reactive time scale. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 84 E*/kBT(0) on/a* r i i / r R 6 rf** ^asy ^ M S N E 2 1 1 1.35(-1) 1.35(-1) 2.196(-2) 0.686 2 1000 1000 1.35(-4) 1.35(-4) 1.600(-5) 0.834 2 200 200 6.77(-4) 6.77(-4) 1.953(-4) 0.891 5 20 20 3.37(-4) 3.37(-4) 4.182(-3). 0.946 2 200 20 6.77(-4) 6.77(-3) 2.718(-4) 0.971 5 200 20 3.37(-5) 3.37(-4) 1.252(-3) 0.980 5 200 1 3.37(-5) 6.74(-3) 1.684(-3)' 0.986 10 200 200 2.27(-7) 2.27(-7) 1.441(-4) 0.997 10 1000 20 4.54(-8) 2.26(-6) 2.897(-5) 1.000 10 1000 1000 4.54(-8) 4.54(-8) 2.900(-5) 1.000 10 10000 1 4.54(-9) 4.54(-5) 1.053(-5) 1.000 6 0n / t7*)e- e * %{cjl2/o*)e-^ Table 3.5: Comparison of time-dependent and MSNE values for TJ for various o~\\/o~* = 0-22/cr* and a^/a*. The mass ratio, m i / m 2 = 3.0 and the initial density ratio ni(0)/n2(0) = 2.0. In order to observe a pronounced effect due to changing the 1-2 collisional time scale, a system in which the two component temperatures are initially different is examined, that is Ti(0) 7^ ^ (O). The resulting temperature equilibration perturbs the distribution function from the Maxwellian in addition to the reaction. Temperature relaxation in the absence of a reaction has been studied previously [95,73]. Figure 3.5 shows the time-evolution of systems with parameters identical to to Figure 3.4, except the initial temperature condition, Ti(0)/T2(0), is 1.5. The rate of tempera-ture relaxation between the two components is governed by the 1-2 collision cross section. The temperature perturbation results in an overshoot of 77(^ /77 above unity followed by relaxation to the steady result, that is close to MSNE result in all cases. The 1-2 elastic collision parameter, a^/a*, is 1000, 200, 20 and 1 in Figures 3.5(a) through 3.5(d) re-spectively, with cru/<7* = 1000. The time scales and the magnitudes of the overshoot for Figures 3.4(a)-3.4(d) are different and are determined by the 1-2 elastic collision cross Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 85 Figure 3.5: Time dependence of the ratio n(t) to 7 7 M S N E , T ? w n e and T ? m s n e for Ti(0)/T2(0) = §. The ratio of elastic to reactive hard sphere collision cross sections is 1000 for an/a* and au/a* equals (a) 1000, (b) 200, (c) 20, and (d) 1. The ratio <W<722 = 1. E*/kBT(0) = 10, ml/m2 = 3 and ni(0)/n2(0) = 2. Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 86 section. The sharp initial slopes that gives rise to the large overshoots in Figures 3.5(a)-3.5(c) arise from large values of the collision matrix elements, < S7l\S^ >, of Eqs. (3.6.4)-(3.6.7). The collision matrix elements are large in magnitude when the species temperatures are significantly different and the 1-2 collision cross section, o\2 is large rel-ative the reactive cross section. Physically, this represents rapid temperature relaxation driven by collisions between particles of two components at very different temperatures. Agreement between the asymptotic steady state and the three non-equilibrium methods appears to be similar to the comparisons shown in Table 3.2 and Figure 3.4. The results show agreement between values of n^ and 7y W N E when o\\jo\2 = 1 (Figure 3.5(a)) and disagreement when ou/oi2 is large (Figure 3.5(d)). Figures 3.5(b), 3.5(c) and 3.5(d) show the time evolution of rj(t), rf^®®, r? S N E and ^ M S N E where c r n / a 1 2 is 5, 50 and 1000, respectively. The steady state values of rj(t) show progressively poorer agreement with r y W N E in Figures 3.5(a) to 3.5(d). In contrast, the values for r)(t) in the steady state are closer to r 7 S N E when o-n/oi2 >> 1. The value of 7 7 S N E disagrees with that for n^7 in 5(a) and agreement between rya s y and ?? S N E improves when o-n/oi2 = 1000 in 5(d). Figures 3.5(a)-3.5(d), also show the ratio /f]MSNE [s v e r v close to unity. This is because the elastic and reactive time scales are sufficiently well separated to ensure that MSNE and the time-dependent results agree as in Table 3.2. 3.7 Summary of Results The perturbation of the distribution function from Maxwellian for species of the reaction A + C —» products has been studied. For elastic collisions, a hard sphere elastic collision cross section was used while a line-of-centers reactive cross section with activation energy e* was used as a model for reactive cross sections. The perturbation to the distribution Chapter 3. Non-Equilibrium Reactive Systems : The Role of Species Temperatures 87 function was computed using three formalisms, the WNE, SNE and MSNE methods. These three formalisms are based on C E type solutions of the Boltzmann equation. The W N E method assumes the distribution functions of all species are characterized by one system temperature while the SNE and MSNE methods specify a temperature for each species of the system. Unlike the system studied by Pascal and Brun [41], W N E and SNE agree in the limit in which the distribution functions of all components are characterized by per-turbed Maxwellians at the same temperature. The results of W N E , SNE and MSNE methods do not, in general, agree and an explicitly time-dependent solution of the Boltz-mann equation was used to validate the solutions obtained by the three methods. The time-dependent method makes no assumptions about the ordering of the terms of the perturbation expansion of the Maxwellian and is considered a reliable check of the solu-tions produced by the W N E , SNE and MSNE methods. It has been demonstrated that W N E method is appropriate for systems characterized by two time scales, the elastic and reactive time scales, when both differ by a factor of greater than 10~5. The SNE method is accurate when there are three distinct time scales, and the 1-1 or self collision, the 1-2 or nonself collision and reactive collision time scales are all well separated. It has also been shown that the MSNE method is more widely applicable than either of the previous two methods and is accurate when the reactive time scale is much longer that the elastic time scale. The formalisms in the present chapter should find useful applications to non-equilibrium effects in spatially nonuniform systems and the study of nonlocal transport. Chapter 4 Sound Dispersion in Single-Component Systems 4.1 Introduction The previous chapters considered kinetic theory methods for reactive systems far re-moved from equilibrium. The SNE approach discussed in Chapter 3 was developed by several workers to provide a methodology for describing reactive systems that are far from equilibrium. The SNE methods and similar generalized Chapman Enskog methods were developed by Kogan [42], Brun [41] and in particular, Alexeev [43-47], who recently suggested that in some applications a generalized Boltzmann equation (GBE) should be considered. Although his work appears to grow out of the study of reactive systems, he has applied his formalism to the study of shocks, and dispersion of sound waves. In Appendix B, the G B E has been applied to a chemically reactive system in the spirit of Chapters 2 and 3. The methodology used is analogous to generalized methods that have recently been used to treat shock structures [98]. A second-order chemical kinetic rate law is treated in a manner prescribed by Alexeev [46]. A correction to the standard chemical rate of reaction for a second-order process is obtained. An analysis of this rate-law correction shows that the correction is an adjustable parameter and not due to collisional effects. In this chapter, the kinetic theory of dispersion of sound waves in monatomic gases is considered. In Chapter 5, the discussion is extended to mixtures of gases and the study of several anomalous effects reported recently. There are potential applications of acoustic 88 Chapter 4. Sound Dispersion in Single-Component Systems 89 waves to the study of reactive systems [39] and there is considerable overlap with recent light scattering experiments [62-66]. The Navier-Stokes equations apply to a description of sound propagation in the limit where conventional continuum hydrodynamics holds. Experimental results obtained by Greenspan [49,50] and Meyer [51] over forty years ago demonstrated that continuum equa-tions of hydrodynamics for small amplitude periodic disturbances are not valid for rarefied gases and for high frequency oscillations. The breakdown of the hydrodynamic approach occurs as the frequency of the applied oscillation approaches the collision frequency of the gas. Kinetic theory appears to be better suited to describing sound behaviour in the rarefied and high-frequency regions. One of the first kinetic theory treatments was for Maxwell molecules by Wang Chang and Uhlenbeck (WCU) who solved the Boltzmann equation with a moment expansion of the velocity distribution function about an equi-librium Maxwellian [48]. Maxwell molecules refer to particles that interact via a force law F(r) ~ r~ 5 , where r is the intermolecular separation. They expanded the velocity distribution function in a finite number of Burnett functions and reduced the Boltzmann equation to a system of linear equations. Burnett functions are products of associated Laguerre polynomials and spherical harmonics. For a solution to exist, the determinant of this set of equations must vanish, giving a dispersion relation. The phase velocity and attenuation of the sound wave have been calculated and compared with experiment. Successive truncations of the Burnett expansions correspond to successive terms in a power series representation of the roots of the dispersion relation [48]. The expansion parameter is the ratio of the disturbance frequency to the frequency of collisions in the gas (the Knudsen number, Kn). Comparison with experimental data shows that the W C U approach extends the validity of the dispersion relations beyond the range of con-tinuum models but is still restricted to systems in which the frequency of the disturbance is not significantly larger than the collision frequency, that is systems that are close to Chapter 4. Sound Dispersion in Single-Component Systems 90 equilibrium. Pekeris and co-workers [52,53] solved the Boltzmann equation with up to 483 Burnett functions and obtained a dispersion relation. In the rarefied (Knudsen) re-gion, Kn > 1, the results of Pekeris for the phase velocity and attenuation showed poor convergence and did not agree with experimental results, demonstrating the limitations inherent in the W C U method. The main challenge for kinetic theory is to obtain a description of sound that is valid over the entire range Knudsen numbers. Kahn and Mintzer [54], in contrast to the W C U method, started with the collisionless limit and used a finite polynomial expansion about the free flow solution. This approach was based on the observation that particles in a system may not collide for several cycles of the disturbance at low pressures for high frequency sound waves. This implies that the collisions are not sufficient to restore equilibrium and therefore there is no local thermodynamic equilibrium in the gas. The results presented in [54] were quite good over the range of experimental values. It was later shown that this unexpectedly good agreement with experiment in the continuum region was due to calculation errors [101,102]. Hanson and Morse [101] showed that Kahn and Mintzer's [54] asymptotic evaluation of the integral JQ°° x n + 1 exp (—^Tpf ~ ^)^x w a s incorrect. Hanson and Morse computed the dispersion relation with the corrected value and showed that calculated phase velocities and attenuations only agreed with experi-mental results in the high-frequency/collisionless limit. The results obtained by Hanson and Morse show poor agreement with experiment [49-51] outside the collisionless region as well as non-physical effects, such as growing or amplified sound modes. Kahn and Mintzer's free-molecule model [54] makes the unjustified assumption that a disturbance propagates as a plane-wave in a collisionless gas. Sirovich and Thurber [56,105] proposed a model of the linearized Boltzmann colli-sion operator that gave good agreement with experimental dispersion relations over the entire range of Knudsen number. The model used an approximation scheme proposed Chapter 4. Sound Dispersion in Single-Component Systems 91 by Gross and Jackson [106]. Almost all work to that point describes the properties of single-component monatomic gases with Maxwell or hard sphere interatomic potentials. It is notable that a description of sound propagation at high frequencies has been ad-equately described by repulsive Maxwell molecule potentials and with neglect of internal degrees of freedom for diatomics. Experimental results obtained by Meyer [51] show that the nondimensionalized phase velocity and attenuation rate are relatively insensitive to the interparticle potential when Kn >> 1. This can be attributed in part to the fact that collisions play a only minor role in the Knudsen region. It has been shown that gas kinetic models with purely repulsive interparticle potentials [56,55,128] give reasonably good agreement with experimental results [49-51] over a wide range of Kn. In order for a disturbance to propagate as a coherent plane-wave in a gas, the disturbed particles have, on average, to collide at a higher frequency than that of the oscillation. At very high frequencies, the perturbing frequency exceeds the mean collision frequency and coherent propagation is carried mainly through the high-energy tail of the distribution function, which also accounts for the observation that sound tends to propagate at higher velocity at higher frequencies. Since high-energy collisions tend to depend mainly on the repulsive part of the interparticle potential this may perhaps explain the accuracy of calculations using model repulsive potentials. Gas kinetic approaches to the theory of sound yield a wealth of little-studied and poorly-understood phenomena that do not appear in conventional hydrodynamic treat-ments. Kinetic treatments predict sound modes in addition to the ones given by hydro-dynamics. These modes are often heavily damped and referred to as "spurious" although their significance and interpretation is not well understood. As the sound frequency in-creases and exceeds the mean collision frequency, the sound modes eventually disappear altogether [104]. It has been determined that some clarification is needed of fundamental properties Chapter 4. Sound Dispersion in Single-Component Systems 92 of gas kinetic solutions of the Boltzmann equation for a small-amplitude periodic distur-bance, especially where Kn >> 1. Much of the previous related work focussed entirely on the development of kinetic models and contained extensive comparisons with contin-uum/hydrodynamic treatments of small periodic disturbances. As a result, many of the features arising from kinetic theory with no analogue in hydrodynamics, such as 'spuri-ous' propagation modes, single-particle modes and sound propagation cutoffs, were left largely unaddressed (for example refer to [56]). In the following sections, the reasons for the failure of the W C U method at high frequencies/low pressures are examined, together with proposed solutions to the break-down of the W C U method. The recent theory of Alexeev [43-47] based on the G B E is examined along with the much earlier work of Sirovich and Thurber [56,105,124]. The purpose of this work is to understand the behaviour of sound waves at high frequencies and low pressures, and in other cases where hydrodynamic theories are no longer valid. One such case, to be studied in the next chapter, is the anomalous sound dispersion behaviour observed experimentally in mixtures of gases near the intermediate zone be-tween the hydrodynamic and collisionless regions [59-67]. The experimental systems are those of disparate mass mixtures which exhibit 'fast' and 'slow' sound modes that do not correspond to any known effect from conventional hydrodynamics. 4.2 A Simple Hydrodynamic Theory of Sound A theory of sound propagation in one dimension can be obtained from the standard equations describing conservation of mass, linear momentum and entropy for an adiabatic system dv, dt X + pvx dp_ dpvx dt dx dvx dP = 0 = 0 (4.2.1) (4.2.2) P dx dx Chapter 4. Sound Dispersion in Single-Component Systems 93 dS dS n , A n . pTt+pv*^ = 0 - (42-3) where p,vx,a,nd S are the density, velocity and entropy per unit mass, respectively and P is the hydrostatic pressure [111]. The rate of change of density in Eq. (4.2.1) can be re-written in terms of a pressure change ^ = ( ^ 1 ^ (4 2 4) at \ d p ) s d t { ] where the subscript S denotes isentropic conditions. Applying the adiabatic condition P = Cp1 (where C is some constant, and 7 is the ratio of specific heats) gives ( & ) . - £ m lkBT The density, flow velocity, pressure and entropy are perturbed from their equilibrium values (po,vx0, P0 and So, respectively) and are written P = Po + P (4.2.6) vx = v'x (vx0 = 0, stationary fluid) (4-2.7) S = S0 + S' (4.2.8) p = Po + P' (4.2.9) where the primed quantities are taken to be small. If Eqs. (4.2.6)-(4.2.9) are substituted into the conservation equations Eq. (4.2.1)-(4.2.3) and only primed terms that are first-order are retained, a linearized set of hydrodynamic equations is obtained ( m \ dP' dv' + p ^ = 0 (4.2.10) jkBT) dt r dx dv' dP' ^ + & •= 0 ( 4 2 - n ) f = 0 (4.2.12) Chapter 4. Sound Dispersion in Single-Component Systems 94 If Eq (4.2.10) is differentiated with respect to t and Eq. (4.2.11) with respect to x, and the two results are equated one obtains a wave equation d2P' fjkT\ d2P' -8F=(V)fei-- < 4' 2 1 3 ) characterized by a sound speed, N = J ^ ' (4.2.14) V m referred to as the adiabatic or Laplacian speed of sound. This is the Euler approximation to sound dispersion and since there is no energy dissipation, the wave does not decay in amplitude, and the attenuation is zero. 4.3 Boltzmann Equation for Small-Amplitude Disturbance The time-evolution of the velocity distribution function of a gas in the absence of external forces is ( ! + c - v ) / = J[ / ] (4.3.1) where J[f] is the Boltzmann collision integral Af] =jj if'fi ~ fh}o-gdtldCl (4.3.2) where o(g, fi) is the differential collision cross section of the collision process. If an oscillation is applied to the gas, the velocity distribution function departs from the Maxwellian by a small term /(c,r,i) = /(°)(c)[ l + Mc,r,t)]. (4.3.3) If Eq. (4.3.3) is substituted into Eq. (4.3.1) and only linear terms in h are taken, followed by use of the identity fW'fW' = fWf(°\ it is found that the linearized Boltzmann Chapter 4. Sound Dispersion in Single-Component Systems 95 equation is of the form (it + C ' V) k = ^4-3'4^ where K is a nondimensional form of the linearized Boltzmann collision operator and K[h] = ^J J f(0)[h{c') + / i i(ci) - h(c) - /ii(ci)]^o-dndci (4.3.5) The dimensionless time variable nt = t(^p-)ll2 jl scales inversely as some parameter /, the microscopic length scale of the system. The microscopic length scale of the system corresponds to a mean free path for the particles in the system and this is given as I = l/n<7 0, where o0 is some constant cross section of the order of magnitude of the collision cross section. A pressure fluctuation propagates as a plane wave (normal mode) in the ideal fluid approximation [104] and the perturbation h is written as /i(c,r,t) = / i(c)e i ( k - r - w t ) (4.3.6) where the wave vector k = k|A;| defines the direction of propagation, and the frequency of the oscillation is to. With Eq. (4.3.6) and the x axis of the space coordinate, r, along k, Eq. (4.3.4) is now (E - eQh = K[h] (4.3.7) where E = - u ( ^ y / 2 u j = ~ i c ° / K (4-3-8) e = -ikl (4.3.9) & = (4.3.11) Chapter 4. Sound Dispersion in Single-Component Systems 96 Equation (4.3.7) is the linearized Boltzmann equation for a rarefied gas system perturbed by small-amplitude oscillations. The following sections discuss methods for solving Eq. (4.3.7) in order to obtain the dispersion relation for the system. The dispersion relation is the equation relating the wave number, k, to the harmonic frequency, oo. 4.3.1 Method of Wang-Chang and Uhlenbeck The disturbance is a plane wave propagating in direction k, taken to lie along the polar-axis x. The perturbation h is written as an expansion in the axially-symmetric Burnett functions ipni(£,,0), n.l=0 where 9 is the angle between £ and the polar axis and (4.3.12) ^,e) = NnleLln^2(e)Pi(cose). (4.3.13) The Legendre polynomials, P[(p),/j, = cos#, are defined by L'/2J W = ^ E ( - i ) r m=0 m 21 - 2m \ l—2m (4.3.14) where m ml (I — m)\ (4.3.15) and [a\ is the largest integer less than or equal to a. The associated Laguerre polynomials, L ^ + 1 / / 2 ( £ 2 ) are given by n [ q > ^ m\T{m + l + l)(n-m)r (4.3.16) m=0 Chapter 4. Sound Dispersion in Single-Component Systems 97 V'oo = 1 v-io = >/i (I - e 2) Table 4.1: The first five axially symmetric Burnett functions The normalization factors in Eq. (4.3.13) are y/Hn\{2l + l ) x 1 / 2 ,2r(n + / + |) such that the basis functions tpni are normalized according to (Vw,Vv/<) = J e-eiPraiZWn'i'i&dt (4.3.18) = 0~nn'b~ll' •. The first five orthonormal axially symmetric Burnett functions are listed in Table 4.1. Equation (4.3.7) is solved by substituting the expansion Eq. (4.3.12), multiplying by each basis function and integrating over £. The result is a set of simultaneous linear equations in ani N,L Ednl- YJ fcCnl.n'l' + Knl,Wv) = 0 (4.3.19) n'l'=0 where the indices n' and V are truncated at N and L, respectively, giving (N +1) x (L + l) expansion terms in Burnett functions. In matrix notation this is written {EI - eC - K)a = 0 . The matrix I is the identity matrix, CnUn'V = (ipnh^n'l') (4.3.20) = 7 T " 3 / 2 / Tpnltxlpri'l'dt J—oo Chapter 4. Sound Dispersion in Single-Component Systems 98 and Kni,n>v = (Vw,if[VvH) (4.3.21) X bPn'l'i^!) + Ipn'l'it'-z) ~ V V ( ' ( £ l ) ~ Ipn'I'(^^gCrdQd^d^ x [^(^i) + ^ ( C 2 ) - ^ ( 6 ) - ^ ( 6 ) ] x [Vv*'(£i) + ipn'i'(i'2) - - ir,n'i'(£2)]go-dQd£id£2 is the matrix of the linearized Boltzmann collision operator. For Maxwell molecules, the axially-symmetric Burnett functions, ijjni Eq. (4.3.13), are eigenfunctions of the collision operator K. For this special case, Knl,n'l' = Xni5nn'8ll> where Xni are the eigenvalues of collision operator, K. The eigenvalues for Maxwell molecules are given by [94] \ n l = sm{6)F{B) (4.3.22) Jo dO x c o s 2 n + ( U J P i (cos u)) +sin2n+l u ) P i (sin2n+7 u)) - ^ + s^°) where F(9) gives the angular distribution of the collision cross section F(8) = b/sin9\db/d9\ (4.3.23) and b is a dimensionless impact parameter for a repulsive power law potential and is an implicit function of 9 given by 8 = TT - 2 / dz[l -z2- {z/bY}-1'2 . (4.3.24) Jo Chapter 4. Sound Dispersion in Single-Component Systems 99 The integral Eq. (4.3.22) was evaluated by Gaussian quadrature using Legendre weights and points. Tables of Xni are also available [53]. A variety of schemes have been developed for computing the matrix elements for ar-bitrary interparticle potentials (reference [94] and references cited therein). The precise form of the matrix of the collision operator will differ according to the model potential used. This work focuses on Maxwell molecules since for this potential the matrix rep-resentative of the collision-operator is diagonal. Previous theoretical calculations and experimental data suggest the precise form of interatomic potentials plays only a small role in the phase velocity and attenuation of sound as shown by Meyer's [51] experimental data for the sound dispersion behaviour of a series of very different gases (argon, air and water vapour). The matrix elements of C defined Eq. (4.3.20) are evaluated from the recursion relation [48] (4.3.2.5) 1/2 and the matrix C is given by (4.3.26) Chapter 4. Sound Dispersion in Single-Component Systems 100 {{21 + l)(2t - 1)) 4.3.2 Sound Dispersion Relations fi(n+l)n' 8(1-1)1' The dispersion relation that is sought relates the phase velocity and attenuation of a disturbance to its frequency. The form of the dispersion relations depends on whether the oscillations are free or forced. Taking the +x-axis to be the direction of propagation, the wave number is k = 8 + ia (4.3.27) and the space and time-dependence of the perturbation Eq. (4.3.6) is given by ^i(kx-iot) __ g-ax+i(/3x-ujt) ^ g 2g^ Forced oscillations decay over distance, and the wave number, A; is a complex number for which the real component 8 corresponds to the phase of the wave and the imaginary component a corresponds to the attenuation (8 and a are real and positive). The applied frequency UJ is real. The following properties of forced normal-mode oscillations must therefore be satisfied in physical systems Im[k] = a (4.3.29) > 0 v = co/Re[k] (4.3.30) = CJ/P where v is the propagation or phase velocity of the normal mode. Free oscillations, on the other hand, decay over time and in this case, co = 8 + ia (4.3.31) Chapter 4. Sound Dispersion in Single-Component Systems 101 where the perturbation is now given by ,i(kx—u>t) at+i(kx—j3t) (4.3.32) and Im[u! a (4.3.33) < 0 v Re[uj]/k (4.3.34) B/k 4.3.3 Solution of Dispersion Relations for the W C U Method A solution for the system of homogeneous equations, Eq. (4.3.19) exists only when the secular determinant vanishes, The determinant Eq. (4.3.35) yields the dispersion relation for the system. Phase ve-locities and attenuation constants at a given frequency or wavelength disturbance are computed from the roots of Eq. (4.3.35). The dispersion relation obtained from Eq. (4.3.35) can often be computed numerically with greater efficiency [132] by solving the eigenvalue problem corresponding to Eq. (4.3.19), which for the case of free sound is where D(e) = eC + K and D, C and K are real matrices. The eigenvalues E(e), are transformed into the dimensionless form of dispersion relation co(k), using the relations Eq. (4.3.8) and Eq. (4.3.9). The real and imaginary parts of co give the wave phase and attenuation, respectively, as a function of the wave number k. EI-eC-K\ = 0. (4.3.35) D(e).a = E(e)a. (4.3.36) Chapter 4. Sound Dispersion in Single-Component Systems 102 The eigenvalue matrix for forced sound is obtained by multiplying Eq. (4.3.19) through by the inverted matrix C - 1 giving F(E)a = e(E)a. (4.3.37) with F = (EI - K ) C _ 1 . Appling the transformations Eq. (4.3.8) and (4.3.9) to the computed eigenvalues e(E) gives the forced sound modes k(u>). The real and imaginary parts of k are the phase and attenuation, respectively, of the wave given as a function of frequency, co. This method was exploited by Pekeris [52] for solution of the W C U dispersion relations up to 483 terms. Successive truncations of the matrix Eq. (4.3.19) to (N + 1) x (L + 1) terms gives a dispersion relation in power series expansions in e [48]. Wang Chang and Uhlenbeck demonstrated that for the special case of Maxwell molecules, successive approximations obtained by adding new terms merely add higher order terms in e to the previous ap-proximation [48]. This was demonstrated with specific examples of 3, 5 and 8 term series solutions. The 3-term truncation uses the Burnett functions tpoo,ipoi and ipw. Adding polynomials tpo2 and i/in gives a 5-term truncation and the further addition of polynomials Vtoj V,2O and -0 1 2 gives the 8-term truncation. The choice of these polynomials corresponds respectively to the Euler, Navier-Stokes and Burnett approximations to hydrodynamics. Wang Chang and Uhlenbeck note that the power-law dependence in e of the dispersion relation can be proven for all cases and arises from the selection rule of the matrix C of Eq. (4.3.26), which is Cni,n'v — 0, unless 2n' + I' = 2n + / ± 1. The selection rule ensures a band-diagonal matrix that does not extend farther from the diagonal as one adds more elements in going from an nth order approximation to an n + 1th order approximation. They showed that each new term in the series added higher order terms to the hydrody-namic dispersion relation and noted that the dispersion law would be unsuitable for cases where the ratio of the mean-free path to wavelength was large. Ford and Foch [104] later Chapter 4. Sound Dispersion in Single-Component Systems 103 confirmed this by showing that higher order approximations to sound modes can also be obtained by writing E as a power series expansion in e. At low frequency or in collision-dominated systems, roots that exist about the hydrodynamic modes rapidly converge as successive terms are added. At high frequencies and low collision rates, as the gas ap-proaches the Knudsen region, Kn > 1, the power series converges more slowly, if it does so at all. This implies that in the Knudsen region, sound modes are strongly perturbed from the hydrodynamic roots. Results obtained with the W C U method are shown in the next section. The data from earlier work [104,52] was reproduced in order to confirm the results and corroborate the conclusions of these studies as well as to provide a basis for comparison with other approaches, the discussion of which is deferred to later sections. 4.3.4 Solutions of the W C U Method For the case of forced sound studied by Greenspan and Meyer [49-51], the frequency of oscillation, to is real and the wave number takes a complex value of the form k = (3 + io> where (3 is the phase and a is the attenuation. In the physical situation addressed, a > 0. The ratio a/u is termed the attenuation rate by Greenspan [49,50] and LO/(3 is the phase velocity of the plane wave. Changes in phase velocity and attenuation with frequency are presented in the dimensionless forms av0/to and (3v0/ui, respectively. Calculations for the simplest case, that of Maxwell molecules, are presented. The lowest-order W C U matrix is the set of equations that conserve mass, momentum and energy and is identically the Euler approximation to hydrodynamics discussed in Sec-tion 4.2. The three conserved quantities, corresponds to retaining three expansion terms represented by the Burnett functions ip00,ip0i and ^ 1 0 , respectively. The perturbation is Chapter 4. Sound Dispersion in Single-Component Systems 104 written as the polynomial expansion h = a00ip00 + a01ip0l + a io^ 10 (4.3.38) and the hydrodynamic sound modes arise from these conserved quantities ^ o o ^ o i and -010 which correspond to the eigenfunctions of the linearized collision operator K with zero eigenvalues. The solution to matrix Eq. (4.3.19) to lowest order (3 x 3) can be used to illustrate some features of the dispersion relations. The matrix elements of the collision operator for conserved quantities are zero, the collision term K[h] vanishes and the 3x3 determinant is \EI-eC\ =0 (4.3.39) where I is the identity and eC 0 # 0 k 0 V 0 J\e 0 (4.3.40) J where the result Eq. (4.3.26) for C has been used. The secular determinant gives a cubic equation in E, E(E2 — V ) = 0 6 (4.3.41) with three solutions E E = ±' (4.3.42) (4.3.43) The solution E. = 0 is spurious since the applied frequency is not identically zero. Equa-tions (4.3.42) and (4.3.43) can be written in terms of k and u using Eqs. (4.3.8) and Chapter 4. Sound Dispersion in Single-Component Systems 105 (4.3.9) to give to 1 v0 \h2kBT 6 m (4.3.44) which is consistent with Eq. (4.2.14) for an ideal gas, 7 = 5/3. In the Euler approxima-tion there is no dissipation and therefore no damping term, resulting in a = 0. It is instructive to examine the four-term expansion for the W C U method in order to observe some general characteristics of the solution. In addition to the three conserved quantities, the term ipu is added to the expansion. For the case of Maxwell molecules, the resulting 4 x 4 matrix is eC — K 0 0 0 \ 0 0 0 0 V 0 0 ^11,11 / (4.3.45) where the collision operator element KntU = An = —2/5. The secular determinant is divided through by EA giving 2 1 5 /e \ 2 1 1 f e \ 3 \EJ 3\EJ E + 3 \EJ ° (4.3.46) The term -g is a dimensionless complex number of magnitude i e i ^ UJ a\2 2kBT UJ) V Tn (4.3.47) (a, 8 and UJ are real) where B/uj and a/to are the reciprocal of the phase velocity and the attenuation coefficient, respectively. The hydrodynamic limit is the limit of low frequencies, UJ —> 0 or 1/E —> oo. The quantity e/E goes to a constant in the asymptotic limit UJ —> 0 consistent with the phase velocity and attenuation rate in the hydrodynamic Chapter 4. Sound Dispersion in Single-Component Systems 106 limit. Equation (4.3.46) in this limit is 2 1 / e ^ 2 which gives the phase velocity 5 3 \EJ = 0 <4'3-48> co , b2kBT = v0 (4.3.50) consistent with Eqs. (4.2.14) and (4.3.44) for an ideal gas. Equation (4.3.48) also gives the correct attenuation factor in the hydrodynamic limit - = 0, (4.3.51) to which is also consistent with the Euler approximation to hydrodynamics. The Knudsen limit is that of oo —> oo or 1/E —> 0. The quantity e/E goes to a constant in this limit, as shown by experimental results [49,50], and Eq. (4.3.46) becomes .5 ( i H (!)'•"-» and the dispersion relation in the limit of large applied frequencies is CO av0 -- ± \ CO 2 ( l ± y f ) y f (4-3.53) 0 . (4.3.54) The phase velocity of sound at high frequencies is typically greater than than that at low frequencies, and /3v0/to < 1. This is attributed to the fact that slower-moving particles are less able to transmit a high-frequency disturbance. The fastest-moving particles tend to have higher collision frequencies and are able to transmit the disturbance more effectively [99]. This leads to the choice of the conjugate pair of solutions from Eq. (4.3.53) lim ^ - ±0 .7826 Ul—>0O jjj Chapter 4. Sound Dispersion in Single-Component Systems 107 as the 4-term approximation to the phase velocity in the Knudsen limit. This agrees qualitatively with experimental results [49-51] which show that the phase velocity of a gas increases to a fixed asymptotic limit (\8v0/co\ ~ 0.5) in the Knudsen region. Higher-order dispersion relations were obtained by successive truncations to the W C U matrix. The matrix equation Eq. (4.3.19) was truncated at n' = N and I' = L and the phase velocity and attenuation as well as the numerical convergence was studied. For higher-order truncations in the Burnett function series approximation, numerical meth-ods of solution were used to obtain the zeroes of the secular determinant. The zeroes of Eq. (4.3.35) were computed numerically for successively higher-order truncations using the Newton-Raphson method. The numerical double-precision "zero of the poly-nomial" was less than 10~15 on substitution of the final Newton-Raphson iterate into the polynomial. The results from the root-searching were checked for consistency by cross-comparison with results obtained from the numerical solution of the eigenvalue problem Eq. (4.3.37). The zeroes of the secular determinant for forced sound gave k(to). The real part of k corresponds to 8, the phase, and the imaginary part of k is a, which corresponds to the attenuation. The phase velocity, to/a, and attenuation rate, 8/w, obtained for a series of N and L truncations and the results were compared with those from experiment [49-51]. Matrix truncations of higher order than the Euler approximation result in dispersion re-lations with more solutions than there are sound modes. For systems with multiple roots, the sound modes were identified by their low-frequency behaviour [57]: as UJ —» 0, to/8 (corresponding to the hydrodynamic phase velocity) goes to a known positive constant while a/uj (the hydrodynamic attenuation rate) goes to zero. Figure 4.1A shows 8UJ~~1VQ, the reciprocal of the dimensionless phase velocity, plotted against 1/Kn for a series of successive truncations. Chapter 4. Sound Dispersion in Single-Component Systems 108 1.0 0.5 i i i M M n—i i i 11111 1 — n I i i i i i n i i i i i i n 36 16 "6 20 -o 0 o A N S i i i i i i i 111 L I I I 0.01 0.10 1.00 10.00 100.00 1/Kn 1.00 c — i — i i 111111 1—i i 111111 1—i M I I I I 1—i i 111111 1—i i i ( i U J 0.10 0.01 J I 0.010 0.100 1.000 1/Kn 10.000 100.000 Figure 4.1: Log-log plot of W C U method results for (A) /3cu~lv0 and (B) auj~lVQ versus 1/Kn. The number of moments retained are 4, 6, 16, 20 and 36, NS is the Navier-Stokes result and experimental results are denoted by o. Chapter 4. Sound Dispersion in Single-Component Systems 109 The numeric labels on the curves correspond to the number of Burnett functions ex-pansion terms retained in the W C U matrix. The 4-term expansion is truncated at R = 1, L = 1, the 6-term is truncated at R = 2, L = 1, 16-term at R = 3, L = 3, 20-term at R — 4, L = 3 and 36-term at R = 5, L = 5. The Navier-Stokes result form conventional hydrodynamics fails to reproduce the experimental result and is shown by a dashed line labeled NS. Phase velocities obtained form the W C U method converge quickly in the limit of small Kn and coincides with the hydrodynamic result (j3u)~lv0 = 1) for this limit. The phase velocity becomes larger than the hydrodynamic phase velocity around 1/Kn < 10, and reach asymptotic values that vary between 0.8 and 0.5 around Kn ~ 1. The results agree qualitatively with experimental results for argon (circles on Figure 4.1 A) but after 36 terms, still do not appear to converge in the Knudsen region. Pekeris reports convergence using 483 terms [52] The corresponding attenuation rates are shown in Figure 4.IB. The 3-term or Euler approximation (not plotted here) predicts no attenuation. The 4-term approximation (N = 1, L = 1, Eq. (4.3.45)) of the W C U method gives attenuation rates that are significantly smaller than all higher-order approximations (curves labeled 6, 16, 20 and 36 in Figure 4.IB). This behaviour is consistent with the hydrodynamic interpretation of the effect of higher-order terms. The moments used in the four-term approximation are "000, "001? "010 and tpn which correspond to mass, velocity, energy and heat flux. The 4-term solution gives the behaviour of sound in a thermally conducting, frictionless gas. The influence of heat conduction and viscosity are of the same order of magnitude and there-fore both must be accounted for a consistent solution. (The Prandtl number measures the importance of viscosity and heat conductivity and has been found experimentally to be of order unity for all gases). The 4-term solution is analogous to early treatments of sound propagation (Stokes, c. 1845) [107] that included the effect of viscosity (which corresponds to to the next highest moment, ipm) but not heat conduction. Lamb [108] Chapter 4. Sound Dispersion in Single-Component Systems 110 obtained a hydrodynamic solution for the case involving both viscosity and and heat conduction and the results showed that sound waves in that case are more strongly at-tenuated than when one of the effects is neglected. Lamb's results also showed that the hydrodynamic propagation velocity remained virtually unaffected by the influence of dissipative effects. These results are consistent with 6-term and higher order W C U results for the hydrodynamic region (curves 6, 16, 20, and 36). The phase velocity in the hydrodynamic region is unaffected by higher moments and virtually all attenuation in the hydrodynamic region arises from low-order moments (that, is heat conduction and viscosity). All attenuation rates obtained for 5-term and higher-order truncations coincide with the Navier-Stokes result (dashed line) in the hydrodynamic limit. The attenuation rates obtained from the W C U method show poor agreement and poor convergence with ex-perimental results as Kn ^ 1. The W C U method predicts that the attenuation rate for sound vanishes as in the limit of large Kn, while experimental results show that the attenuation rate approaches constant value ?s 0.22 in that limit. 4.3.5 The Generalized Boltzmann Equation (GBE) Method of Alexeev Recently, Alexeev [43-47] has proposed a G B E which has been claimed to be valid over all Kn. The method of Alexeev is in sharp contrast to the efforts of other workers who proposed approximate solutions based on various treatments of the collision operator [54-56]. Alexeev has suggested that the difficulty with the solutions of the Boltzmann equation in the collisionless limit, Kn —)• oo, lies not with the method of solution of the Boltzmann equation but with the Boltzmann equation itself [46,47]. Alexeev has attempted to solve the problem of sound dispersion at high-frequencies and low-pressures by rewriting the LHS of the Boltzmann equation, Eq. (4.3.1). The time-evolution of the velocity distribution function, f(c), of a gas in the absence Chapter 4. Sound Dispersion in Single-Component Systems 111 of external forces is typically written [1] jj-t=JV\ (4-3.55) where J[f] is the Boltzmann collision integral, Eq. (4.3.2). Alexeev has proposed instead writing Eq. (4.3.55) as where r is the mean free time between collisions. An oscillation perturbs the Maxwellian by a small term, and the velocity distribution function is / (c ,r , i ) = / (°) (c)[ l + / i(c,r,i)]. (4.3.57) If Eq. (4.3.57) is substitued into Eq. (4.3.56) and only linear terms in h are taken, use of the identity fWfW = / (° ) / (°) gives the linearized form of the G B E , which is found to be A periodic perturbation h of the form Eq (4.3.6) is substituted into Eq. (4.3.57) in the manner of the W C U giving the linearized G B E (E-e^x)-^(E-^x)2]h = K[h] (4.3.59) where the nondimensionalized collision frequency is 1 The behaviour of sound as described by the G B E is obtained solving Eq.(4.3.59) for the dispersion relation. Chapter 4. Sound Dispersion in Single-Component Systems 112 Alexeev has presented the generalized hydrodynamic solution which he has suggested extends hydrodynamic theory past the intermediate region and into the Knudsen region [46]. Alexeev called these new hydrodynamical equations the generalized hydrodynamic equations. These generalized hydrodynamic equations are 3 and 5-term approximations to the generalized Boltzmann equation, which Alexeev has called the generalized Euler and generalized Navier-Stokes equations, respectively [46]. Presented here is the solution to Alexeev's G B E , which, if consistent to all orders in h, should address the failure of the W C U solution to agree with experimental results over all Kn. As was done in previous sections, Eq. (4.3.59) is solved by substituting the expansion Eq. (4.3.12), multiplying by each basis function and integrating over £. The result is a set of simultaneous linear equations in an\. The generalized Boltzmann approach gives the matrix equation EI - 7i(eC + K) - 7 2 ^ D a = 0 (4.3.60) where 2E/v - 1 , 1 4 = wiTIM ( 4' 3' 6 2' and the elements of C and K are defined in Eqs. (4.3.20) and (4.3.21), respectively. The elements of the matrix D, Dnl,n>l> = (lpnl^li>n'l') (4.3.63) are obtained in an analogous manner to the calculation of Cni,n'u and using the expression Eq. (4.3.25) twice D — = ( / + 1 ) ( ( 2 / + 3)(2; + i ) j ( 4- 3- 6 4) Chapter 4. Sound Dispersion in Single-Component Systems 113 x (1 + 2) n + l + l Onn'0(1+2)1' i(2Z + 5)(2Z + 3) j - (1 + 2) (l + l) (l + l) (l + l) n (2/ + 5)(2/ + 3) ; 1/2 1/2 \n-l)n' 3(1+2)1 Onn'Ow (21 + 3)(2/ + l ) ; n 1/2 (2/ + 5)(2/ + 3) / b~(n+l)rib~W n 1/2 X (1 + 2) ~ (1 + 2) (2l + 3)(2l + l)J ( n + l + l V(2/ + 5)(2/ + 3), n - 1 (2Z + 5)(2Z + 3) y 1/2 0~(n-l)n'3(1+2)1' 1/2 "In— 2)n'8(1+2)1' n 4 s ( n + l + \ + (/ + ! ) ' 2 -1/2 + i(2Z + 3)(2Z + l ) , 71 5(n-l)n'5ii' 1/2 + (l + l) V(2Z + 3)(2Z + 1), ' n + Z + 1 \ 1 / 2 (2/ + l ) ( 2 Z - 1)J b~nn'b~il' X 1/2 (2Z + 1) (2Z- 1), 0~nn'0~i IV n 1/2 (2/ + 1)(2Z - 1), b~(n-\)n'b~W 1/2 - (I -I) - I (21- 1)(2Z + 1), ' n+1 (21- l ) ( 2 Z - 3 ) / ' (n-l)n '<fy-2)! ' 1/2 ^(n+l)n' 5(J_2)J' rc + Z 1/2 (2/ + 1)(2Z - 1) / Chapter 4. Sound Dispersion in Single-Component Systems 114 x - / + I n + l + 3 \ 1/2 V(2Z + 1)(2Z- 1)/ n + I (21 + l)(2l - l) j ^ ( n + l ) n / < ^ ' 1/2 1 \ 1/2 - 0 + 1) (2/ - 1)(2Z - 3) n <5(n+l)n"5((-2)C \ 1/2 ,(2/ + 5)(2Z + 3)_ % 7 .+2 )n" fy -2 )C A solution for the system of homogeneous equations, Eq. (4.3.60) exists only when the secular determinant vanishes, £ I - 7 i ( e C + K) - 7 2 - D | =0 (4.3.65) The zeroes of the determinant yield a dispersion relation for E in terms of e. 4.3.6 Comparison of W C U and G B E Methods The G B E , Eq. (4.3.60), can be truncated at 3-terms to give the Euler approximation to generalized hydrodynamics. The collision matrix, K is equal to zero. From Eq.(4.3.20) for C, (4.3.21) for K and (4.3.64) for D, a 3 x 3 matrix ( 7 2 ^ 7 i v ^ 7 2 ^ ^ 7!(eC + K) + 7 2 - D = 7 i 72l7 1 V 7 2 V e 7 / 7 i 3 e (4.3.66) 72277 / is obtained. The dependence of the dispersion relation upon the Knudsen number, Kn, was of particular interest. For the case of the G B E , Kn appears explicitly in the dispersion relation. Since Kn = \E/v\, the dispersion relation of the linearized G B E for E 7^ 0, to Chapter 4. Sound Dispersion in Single-Component Systems 115 lowest order, can be written as a 6th order polynomial in E/e by dividing through by E giving The G B E is reported to describe phenomena over the full range of Knudsen number and therefore the asymptotic properties of the dispersion relation Eq. (4.3.67) are of interest. The limit of hydrodynamic limit Kn —¥ 0 was discussed in the analysis of the W C U method above. The Knudsen region, for which Kn —> oo is conveniently treated using the form of Eq. (4.3.67). ' In the limit Kn —>• oo, Eq. (4.3.67) is a cubic equation in X = (e/E)2 £ I - 7 i ( e C + K) - 7 2 - D = 0. 5 o 5 , 1 (4.3.68) with roots X = X = A + B-l ±^(A-B)i-±(A + B + ±) (4.3.69) (4.3.70) where A and B are constants B A (4.3.71) (4.3.72) Chapter 4. Sound Dispersion in Single-Component Systems 116 Collecting real and imaginary components of Eqs. (4.3.69) and (4.3.70), and taking a and 8 to be both real, explicit values for the nondimensionalized phase velocity, /3VQ/UJ and attenuation rate av0/co were obtained. Equation (4.3.69) gives a nonpropagating mode — = ±72/3 - A- B,fl (4.3.73) UJ V 2 ^ = 0. . (4.3.74) UJ Nonpropagating modes also appear in hydrodynamic treatments, where they are known as heat conduction modes [48]. Since a > 0, the growing or amplified mode a < 0 is considered spurious and j is set to 5/3 giving lim av0/uj ~ 1.4117. Kn—>oo Collecting real and imaginary parts of Eq. (4.3.70) gives 82 -a2 , 1 A J S J ^ v l = --(- + A + B)l (4.3.75) and < " * S = ± ^ ( X - B ) 2 , (4.3.76) UJ respectively. Squaring Eq. (4.3.76) gives l/ = {Wj M { A - B ) < ( 4 - 3 - 7 7 ) When Eq. (4.3.77) is substituted into Eq. (4.3.75), a quadratic in terms of the square of the phase velocity is obtained, Chapter 4. Sound Dispersion in Single-Component Systems 117 where C = ( A ± £ ± 4 / 3 ) 7 / 4 = -0.1631 and D = 3(A-5)V/64 = 0.1095. The result is a conjugated pair of roots that ultimately give rise to four solutions for the phase velocity & = ±(c±yvTwy> ( 4 3 79) for the case of an ideal gas 7 = 5/3. Two of the roots are spurious since they give imaginary phase velocities. The remaining two roots give the asymptotic phase velocity lim — w ±0 .650 Kn—>oo UJ The corresponding attenuation rate is obtained from Eq. (4.3.28) av0 UJ UJ 0VO where for an ideal gas D (4.3.80) lim — « 0.509 . The above treatment agrees with the generalized Euler equation [46], for which Alexeev also reports 0VO/UJ = 0.650 and av0/uj = 0.509 in the limit Kn —> 00. The Euler ap-proximation to the G B E gives attenuation rates that only agree qualitatively with the experimental results of Meyer for the case of greatest rarefaction, where PV0/UJ w 0.45 and avo/uj w 0.22. Numerical solutions were computed using the Newton-Raphson method and disper-sion relations obtained for a series of successive truncations for the G B E over a range of Kn. Figure 4.2A plots BUJ~1V0 versus 1/Kn, obtained from the dispersion relation for the G B E (Eq. (4.3.65)) with 3, 4, 5, 20 and 36-terms. Experimental results are shown with circles. All results converge to the hydrodynamic phase velocity for large Kn. In the Knudsen region, the result does not appear to converge and as more terms are added to the matrix (9, 20, 36-term truncations), the solutions appear to become unstable and Chapter 4. Sound Dispersion in Single-Component Systems 118 A '—20 _i I i i mil i i i i mil i i i I mil I i i J ' i i m i 0.001 0.010 0.100 1.000 10.000 100.000 1/Kn 1.00 p — i — i i 11 iiij 1—r 4 0.10 5 o.oi L-I I 11 mi 1—i i i I I I I I 1—i i i um B J I I I 11 III I I I I 11 III I I I I mil I i i i i n i l i i i i i 0.001 0.010 0.100 1.000 10.000 100.000 1/Kn Figure 4.2: Log-log plot of G B E method results for (A) (3LJ~1VQ and (B), aco^vo versus 1/Kn. The number of moments retained are 4, 5, 6, 16, 20 and 36 and experimental results are denoted by o. Chapter 4. Sound Dispersion in Single-Component Systems 119 no longer agree with experiment. The solutions obtained from using 3,5 and 6 terms agree qualitatively with experimental results (circles). Figure 4.2B shows a plot of acu~lv0 versus 1/Kn for a series of successive trunca-tions of the G B E As with the result for phase velocities in Figure 4.2A, the low-order (3, 5 and 6-term) solutions for attenuation rates in Figure 4.2B show only qualitative agreement with experiment in the asymptotic limit 1/Kn —> 0. Agreement of the G B E with experiment becomes worse as more terms are added. As more terms are added, the G B E no longer agrees with either experiment or the W C U method in both hydrody-namic and Knudsen regions. The solutions of the G B E appear to converge only slowly in the hydrodynamic region, but in the Knudsen region appears to diverge as more terms are added (16, 20 and 36-term solutions in Figures 4.2A and 4.2B). The W C U method fails in the Knudsen region but is consistent with experimental results and conventional hydrodynamic behaviour in the hydrodynamic region Kn « 1. These findings do not support Alexeev's claim that the GBE-derived G H E represents an interpolation of hydrodynamical solutions into the collisionless region. This work suggests that the G H E results presented by Alexeev appear to be unconverged series solutions of the G B E . When carried to high orders, the G B E suffers the same defects as W C U , namely nonphysical behaviour and poor convergence. It is concluded that the G B E fails to describe sound propagation behaviour over the entire range of Knudsen number. 4.3.7 The Sirovich-Thurber (ST) Method Citing convergence problems and obvious discrepancies between theoretical and experi-mental results [52,101], Sirovich and Thurber long ago proposed abandoning the W C U method in favor of a model of the linearized Boltzmann collision operator suggested by Chapter 4. Sound Dispersion in Single-Component Systems 120 Gross and Jackson [105,56]. When E q . (4.3.12) is substituted into the R H S of E q . (4.3.1), the result is oo (E- e&Wf, 0) = K[h] = anlKnUnn4n>v (4.3.81) n,l,n' ,l'=0 where Kni,n>v is defined in E q . (4.3.21). The straightforward (JV + 1) x (L + l ) t h order approximation to the Boltzmann collision operator would be K{NL)[h}= 52 anlKnl,nll4n,v (4.3.82) n,l,n',l'=0 where (N + 1) x (L + 1) is the number of Burnett functions retained in terms of n and I indices, respectively. Gross and Jackson [106] suggested that for a more accurate approximation of the full Bol tzmann collision operator, E q . (4.3.81), is N,L oo K{GJ)[h}= 52 anlKnl,n'l'1pn'l' + X £ . 0„,Vnl ( 4 - 3 - 8 3 ) n,(,n',('=0 n,l>N,L where A is some constant that approximately preserves the remainder of the collision term and a useful choice might be A/V+I,L+I- Equation (4.3.83) is transformed by adding N,L N,L 0 = X 52 anlipnl - X 52 am^ni (4.3.84) ra,(=0 n,l=0 giving N,L (E-e£x-X)h= 52 am(Kni,n'i' ~ XSnnlSw)ijjnni. (4.3.85) n,l,n',l'=0 A n essential aspect of the method of Sirovich and Thurber is to factor out e from the L H S of E q . (4.3.85) (E — X \ N , L e I - £ r I h = 52 anl(Knl,n>l> ~ X5nni5iV)lj)niV (4.3.86) V 6 ) n,l,n',l'=0 and then divide by the velocity-dependent term in brackets giving n,l,n',l'=0 ( £ E _ 0 N,L ^ i^ eh= 52 am(Kni,n'v - XSnr,/Sw) n - (4.3.87) Chapter 4. Sound Dispersion in Single-Component Systems 121 where £ = (4.3.88) The derivation of the dispersion relation from E q . (4.3.87) appears to be physically more correct than from E q . (4.3.19) because of the term (£ x — £) as discussed later. Equat ion (4.3.87) is solved by substituting the series expansion on the L H S of E q . (4.3.87), mul t i -plying by each basis function and integrating over £. The result is a set of simultaneous linear equations in an{. In matrix notation this is written (eI + K)a = 0 (4.3.89) where the matr ix elements of K correspond to N,L Krs,nl = 2)2(Kr",n'l' ~ A<Wfr.t') ( 4>nh ff ? ) (4.3.90) n'V V ?x 4 / = E Prs,n'l' I VViJ n'V n'V where and 8rs,n'V — Krs,n'V ~ X5rn5si' (4.3.91) i W i ' = { ^ n u ^ ~ \ (4-3.92) = [ e~y<*f1'di (4.3.93) E q . (4.3.90) is not an eigenvalue problem since the matrix K of Eq . (4.3.92) depends on E and e. The integrals Rni,n',v, m Eq . (4.3.92) can be written in a closed form as a series involving the Plasma Dispersion function Z® = K-1'2 j (4-3.94) Chapter 4. Sound Dispersion in Single-Component Systems 122 which can be computed numerically to high accuracy [129,130]. The expression derived for the integral Eq. (4.3.92) agrees with the result of Sirovich and Thurber [128]. The derivation is presented in Appendix C. The ST method introduces two modifications to the WGU approach. The first is the use of the GJ approximation of the collision operator in which the eigenvalues of terms of higher order than ipNL are represented by the single eigenvalue AJV+I This procedure collapses the eigenvalue spectrum of all eigenvalues of higher order than XNL to the single value A/y+i L+I- The second aspect, shared by the treatments of Foch and Ford [104], Buckner and Ferziger [55] and Skvortsov [131], involves the division by the drift term in Eq. (4.3.86) yielding Eq. (4.3.87). Sirovich and Thurber employed 3, 5, 8 and 11 moments in their model Boltzmann equation and reproduced the results of Euler, Navier-Stokes and Burnett approximations to hydrodynamics as well as a good agreement with experimental results in the rarefied region. The ST method also gives the same solutions as Grad's 13-moment method in the hydrodynamic region but the two do not agree in the rarefied region [127,128]. A detailed comparison of the W C U and ST methods of solution is presented in the next section. 4.3.8 Effects of ST Modifications to the W C U Method Modification of the straightforward approach of Wang-Chang and Uhlenbeck by others [54-56] appears to be motivated primarily by the failure of the W C U method to predict correct attenuation behaviour in the Knudsen region. The ST method gives excellent agreement with experimental results for the dispersion of sound in monatomic gases, even at fairly low orders (3, 4 terms) [56]. The results obtained by Sirovich and Thurber were reproduced in this work and the results are presented in Figure 4.3. It demonstrates fairly good agreement with experiment at low Chapter 4. Sound Dispersion in Single-Component Systems 123 orders (3 or 4 Burnett terms) and nonvanishing attenuation factors the Knudsen region. The convergence of the ST solutions is markedly better than those of the W C U method (Figure 4.1). Figures 4.3A and 4.3B show that retaining only 12 terms in the ST method gives very good agreement with the experimental data. A survey of the literature suggests that little is known as to why the ST method succeeds in describing the behaviour of sound at high frequencies. The applicability of the ST method also does not seem to have been extended beyond simple monatomic gases. The purpose of the work from this point is to determine the validity of some assertions of Sirovich and Thurber regarding their method. The ST method is then applied to a more complicated system, namely that of a gas mixture. It is instructive to illustrate the impact of the Gross-Jackson (GJ) treatment on the dispersion relation by examining some of the lower-order solutions. For a Maxwell gas, the 3-term approximation with the choice of Burnett functions V'oo,'0oi a n d "01 o gives a collision operator of the following form A K{3)[h] = -Xh+^J [a0 0 x 1 + a o i & k (4.3.95) The 3 x 3 GJ matrix is ( e + (A0o — A)i?oo,oo ' (Aoo — X)RQO,OI (AOO — A)i?oo,io ^ ( aoo ^ V \ « i o j = 0 (4.3.96) (Aoi — A)i?0 1 > 0o e + (Aoi — A)i?oi i 0i ( A 0 i — A)i?o 1 ) 1 0 (Aio — A)i?!0,00 (Aio — A)i?io,01 e + (Aio — A)i?io,lO J where Rni,n'i' is given by Eq. (C.6) in Appendix C and A is some appropriate constant that preserves the form of the truncated collision operator. The first few values of Rni n*y are R •00,00 z(0 (4.3.97) Chapter 4. Sound Dispersion in Single-Component Systems 124 >° 3 ca 100.0 1/Kn sp 0.10 h -3 100.0 1/Kn Figure 4.3: Log-log plot of ST solution results for (A) (3u>~lv0 and (B) aco^vo versus 1/Kn. The number of moments retained are 3, 4, 9 and 12, arrows denote the collisionless approximation and experimental results are denoted by o. Chapter 4. Sound Dispersion in Single-Component Systems 125 Roi,oo Roi,oi RlOfiO Rw,oi Rw,w #00,01 = V2 [l + R 00,10 — -Roi.io — 3^ I2 -= + 2 ] z ( 0 + f - k " I _ Setting A = A 2 2 to approximate the truncated part of the operator can be shown to give a reasonable result for velocity and attenuation rates. In this case, A0o = A 0 i = A n = 0 and the dispersion relation can be computed from the roots of the secular determinant which is e3 + e2XU0 + ^ 2 H 0 + *3M0 =0 (4.3.98) where *> = i ? + ^ ( i p - i ) - | w and Z(£) is the Plasma Dispersion function of Eq. (4.3.94). The complex roots of the complex function Eq. (4.3.98) were computed numerically using the Newton-Raphson method. The phase velocity and attenuation rate for 3, 4, 9 and 12-term GJ solutions are shown in Figures 4.3A and 4.3B. The results agree well with experimental data [51] for forced sound in argon. In contrast to the W C U method which gives dispersion relations over the entire range of Kn, the ST method fails to give solutions above a frequency referred to here as the cutoff frequency. For the 3-term solution, the cutoff Chapter 4. Sound Dispersion in Single-Component Systems 126 is approximately Kn = 0.7. For higher order approximations it is closer to Kn ~ 0.1 (Figures 4.3A, 4.3B). A further discussion of high-frequency cutoffs is deferred to Section 4.3.9. The ST method makes two modifications to the W C U method. The first modification uses the GJ approximation (Eq. (4.3.83)) that can be shown to affect only solutions obtained to low order. Sirovich and Thurber apply Eq. (4.3.83) to approximately preserve the truncated portion of the collision operator. The value of A is unspecified, but Sirovich and Thurber base their argument on an approximate spectral representation, and consider A to be the value for which which all eigenvalues AJV+I L+I and higher are approximated with truncated spectrum. The case of Maxwell molecules suggests another possible value for the choice of A . If one replaces A with — v, a dimensionless collision frequency, the RHS of Eq. (4.3.85) is the kernel of the Hilbert form of the Boltzmann equation, N,L (E-e£x + v)h = J2 am{KnUn,v - v6nni5u> + v5nn>b~w)i>n>v (4.3.100) n,l,n',l'=0 N,L = E anlKnltn'l''lPril' n,l,n',l'=0 where K*ipni is the kernel K*ipnl = K ^ + u^m (4.3.101) Equation (4.3.100) is, for the case of Maxwell molecules, identically the formulation of the sound dispersion problem suggested by Ford and Foch [104] and by Skvortsov [131]. It is useful to note that the 3-term approximation to the ST collision operator, Eq. (4.3.95), with A = — v is identically the B G K model operator (Appendix D). Setting A = 0 gives the W C U collision matrix on the RHS of Eq. (4.3.85). The ST method is, in effect, Chapter 4. Sound Dispersion in Single-Component Systems 127 dividing the collision operator into two terms and moving one term over to the drift side prior to division of the RHS by the drift term. Some calculations demonstrate the effect of changing A . Figures 4.4 and 4.5 show the effect the choice of A on the phase velocity and attenu-ation, respectively, for a set of ST solutions truncated at 6 (N=2, L=l) , 9 (N=2, L=2) and 12 (N=3, L=2) terms. The corresponding values A = \N+I L + I in step Eq. (4.3.83) are A 3 2 = -0.8949i/, A 3 3 = -1.058i/ and A 4 3 = —1-112/v. There is some small variation in the phase velocity and attenuation depending on whether A = AJV+IL+I or A = —u for 6 and 9 term approximations. No solutions were found for setting A = 0 in the 6-term case (Figures 4.4A and 4.5A). The 9-term solution shows some dependence on the choice of A , especially in the Knudsen region. The choice of A = 0 for the 9-term solution moves the cutoff from Kn R J 0.2 to Kn R J 1. The 12-term solutions appear to be insensitive to the value A (Figure 4.4C and Figure 4.5C). This suggests that the influence of A on the solution declines as more terms are retained. The intended purpose of ST for introducing A in Eq. (4.3.83) is to increase the convergence of low-order solutions. As more terms are added, and the solutions converge, the inclusion of the term A becomes unnecessary. This suggests that the choice of A is to a large degree arbitrary and that as more terms are used in the expansion and as the expansion converges, the use of the term A becomes redundant. The second modification of Sirovich and Thurber is the division by the drift term in Eq. (4.3.87) and this appears to significantly affect the nature of the solution. Equation (4.3.81) is of the form L[h] = K[h] (4.3.102) where for forced sound, (L = E — e(E)£x) and K is the Boltzmann collision operator. Chapter 4. Sound Dispersion in Single-Component Systems 128 1.0 3 0 . 5 h T~| 1 1 1 1—I I I I I —\I I I I I A _1 I ' I I o . i l.o \— o.s h - i — i — i — i — i i 111 l . O l O . O 1 / K n l O O . O - i — i i i i I A,=0 B 33 i i I I I _j i i i_ _i i i i ' • ' • O . l 1 O 3 ca 0 . 5 T~] 1 1 1 1—I-l . O l O . O 1 / J C n l O O . O -I 1 1—I- - i 1—i—r C I- 1 1 1 ' . 0 1 1 0 l O . O l O O . O 1 / K L n Figure 4.4: Graph of versus 1/Kn for (A) 6-term approximation with A equal to A « and -v (B) 9-term approximation with A equal to A 3 3 , -v and 0, and (C) 12-term approximation with A equal to A 4 3 , -v and 0. Chapter 4. Sound Dispersion in Single-Component Systems 129 0.10 O . O l O . I O O . O l O . I O O . O l _1 I I L J I ! I • • • • I O . 1 1 O 1 O O l O O . O 1 / K n Figure 4.5: Graph of ow" 1 ^ versus 1/Kn for (A) 6-term approximation with A equal to A 3 2 and —v, (B) 9-term approximation with A equal to A 3 3 , — v and 0, and (C) 12-term approximation with A equal to A 4 3 , — v and 0. Chapter 4. Sound Dispersion in Single-Component Systems 130 The problem is to determine the eigenvalues and eigenfunctions of Eq. (4.3.102) and ulti-mately, e(E). Grad showed that the operator £ x on the LHS has a continuous eigenvalue spectrum [120]. The operator K on the right hand-side has a discrete eigenvalue spectrum for Maxwell molecules. In the present study, Eq. (4.3.81) is discretized by expressing h in terms of a series expansion eigenfunctions of the RHS operator. This approach is convenient in that it gives easily-treated representations of the discretized operators, defined by the matrices K and C (Eqs. (4.3.20) and (4.3.21)). The discretization of h in eigenfunctions of the RHS operator does not guarantee that the LHS operator, with a continuous spectrum, will or can be accurately represented in a discretized form. It appears that in the collisionless limit, the discrete representation is inadequate and the W C U method fails. This raises the question as to why the W C U method works in the hydrodynamic limit. The reason for this appears to be that the hydrodynamic region is the collision-dominated region and terms of the collision operator, which has discrete eigenvalues, dominate the solution. Buckner and Ferziger [55] approached the problem by expanding h in a complete set of eigenfunctions of that was the sum of both the discrete and continuous spectra the linearized Boltzmann equation. They obtained reasonably close agreement with experi-mentally determined dispersion relations for Knudsen numbers in the range 10~2 to 100. Their result is characterized by good agreement with experiment at the limits where Kn is large or small, but poorer agreement with experiment in the intermediate region Kn « 1 [102]. Solutions obtained by the W C U method in the Knudsen region appear to arise from a class of solutions that do not correspond to the physical system. The 483 moment solution of Pekeris [103] shows the attenuation a/to going as 1/UJ in the Knudsen region. This implies for the W C U method, a is no longer a function of UJ for large values of to, a fact that is inconsistent with the definition of a dispersion relation. Chapter 4. Sound Dispersion in Single-Component Systems 131 Sirovich and Thurber have modified the W C U method by first dividing Eq. (4.3.102) through by L and then discretizing the equation. This procedure gives a very different result than the W C U method. The difference can be noted by examining the forms of the respective matrix equations and the dispersion relations. The dispersion relation obtained from four-term ST method with A = 0, analogous to the 4-term W C U is (4.3.103) The difference is self-evident from a comparison between Eq. (4.3.46) obtained from the W C U method and Eq. (4.3.103) from the ST method as well as the phase velocity and attenuation factors that result from the respective solutions, as shown in Figures 4.1 and 4.3. Division by the drift term prior to discretization appears to preserve the correct physical character of the solution. 4.3.9 High-Frequency Cutoff and Free-Molecule Approximation It can be demonstrated that under some conditions, there are no roots to the secular determinant for the ST matrix. If one considers a determinant of the ST matrix D(0 = \el-C(0\ (4-3.104) The existence of roots of the secular equation can be proven by obtaining the winding number of the determinant for any given value of E (for the case of free sound, e would be the adjustable parameter). The winding theorem may be stated as follows : "The number of zeroes of a complex function D(£) of a complex variable £ in a region of the complex £-plane for which D is an analytic function of £ is equal to the number of times Chapter 4. Sound Dispersion in Single-Component Systems 132 the representative point D circles the origin of the complex D plane as £ is carried around a boundary region in the £ plane " [104]. If the winding theorem is applied above the so-called cutoff frequency, it can be shown that there are no plane wave solutions to the linearized Boltzmann equation for sound. The precise value of the cutoff frequency depends on the collision model and for the case of Maxwell molecules is around Kn ~ 10. The winding theorem can be used to locate regions for which there are no roots of the secular determinant. The sound modes for forced sound waves lie in the range — 1 < 0vo/u> < 1 and 0 < av0/cu < 1. The domain of interest is the upper half of the complex £-plane. A large semicircle as shown in Figure 4.6 (A) bounds the solutions. A trajectory around a large semicircle requires computing D ( £ ) for |£| large. Hand computation for low orders is feasible (Eq. (4.3.98)), since Z(£) has the asymptotic form m ~ - E JfvrVL Zr2 n-1 (4.3.105) n=0 1 \2> The D(£) trajectory resulting from a path £ very close to and parallel with the real axis can also be computed by hand for low orders, since near the real axis Z(£ + i<f>) (£, (j) both real) has the asymptotic form lim Z{1 + i(f>) ~ -2e? f ex2dx + iTvl/2e~^ (4.3.106) #->o+ Jo Figures 4.6(B) and 4.6(C) show winding plots for the 4-term approximation denoting the existence (Figure 4.6(B)) of four roots below the cutoff frequency, corresponding to 1/Kn w 0.18 and none above it (Figure 4.6(C)). There is some evidence to suggest that the cutoff represents the high-frequency limit at which normal modes no longer physically exist. In collision-dominated systems, the impulse is carried as a plane wave by coherent collective modes. This is because parti-cles have an opportunity to collide numerous times between oscillations. In collisionless or high-frequency situations, the impulse is carried by incoherent single-particle modes. Figure 4.6: (A) Bounding region in complex £-plane for roots of dispersion relation D(£) and winding plots for 4-term D, 1/Kn equal to (B) 5 and (C) 0.05. Chapter 4. Sound Dispersion in Single-Component Systems 134 There will, in the later case, be an insufficient number of collisions between oscillations to restore the system to equilibrium. Kahn and Mintzer recognized this problem and suggested that the reason for the breakdown is that Maxwellian velocity distribution function constitutes a poor first approximation in the Knudsen region [54]. A simple model of a forced periodic disturbance in a collisionless system can be con-structed as follows. The collisionless gas kinetic equation for a system with a disturbance in the re-direction is df df Equation (4.3.107) has a solution of the general form f = f{0)[l + h(t-x/£x)] . (4.3.108) The solution can be interpreted as a perturbed velocity distribution in which particles are emitted with the Maxwellian characteristic of the source but have a harmonic component along the axis of propagation (in this case the x-axis) h = Aeluj{t-& (4.3.109) where A represents an amplitude of the perturbation. An impulse, P', generated by the disturbance is POO P'{x,t) =ATT1/2 e^-^e-^d^ (4.3.110) Jo where the limits on the integral emphasize that only particles traveling in the +x direction are considered. In Eq. (4.3.110), time and space dependencies are separable and •P'W = ™ # ("-HI) /"OO = Air1/2 / e^e-^d^ (4.3.112) Jo Jo cos(—) + isin(—) %d$x (4.3.113) Chapter 4. Sound Dispersion in Single-Component Systems 135 The impulse is assumed to decay through the random motion of the particles in the collisionless system. The magnitude of the impulse P' at x then depends on the magni-tude of the impulse at the origin P'(0) and the distance x over which it has passed. A Beer-Lambert law for the decay of the impulse can be written P'(x)=P'{<d)e-^k{x)dx (4.3.114) where k is the propagation constant that describes the amplitude and phase of the im-pulse. Following the work of Meyer, [51], the term k is identically the wave number k = 0 + ia where _ k = « » r w / n o ) i ( 4 3 1 1 5 ) ox dln[P'(x)} dx dP'(x) dx P'(x) " When P'(x) from Eq. (4.3.113) is substituted into Eq. (4.3.115), an expression for k is obtained f°° aH(fr)+^"(fr)]c-f^ r ^ = dx ^ x 2 ng) | 0 o o [ cos (^)+zs in (^) ]e -^^ The dimensionless propagation speed (3vo/u and attenuation rate avo/u> aire obtained from the real and imaginary parts of Eq. (4.3.116) which, following some manipulation are found to be (3Vo /(cosg)I'(cosg) + /(sin^)J'(sin^) PJ ~u7 ~ 7(cos^) 2 + / (s in^) 2 V 2 ovo = / '(cosg)I(smf) - I'(cosf )/(sinf) rj co /(cos^f ) 2 + / (s in^) 2 V2 7 Sx (4.3.117) where r°° 2 I{g{£x,w,x)) = g(£x,u),x)e-t*d£x (4.3.118) Jo Chapter 4. Sound Dispersion in Single-Component Systems 136 and n 1 9 1 , 1 = - 7 T • (4.3.119 LO OX Meyer and Sessler computed (3/LO and a/cu numerically for the case of LOX large [51] and obtained — « 0.45 (4.3.120) to — « 0 . 2 2 (4.3.121) LO which was in excellent agreement with their experimental result. Figure 4.3 shows that the asymptotic values for phase velocity and attenuation, Eqs. (4.3.120) and (4.3.121) correspond closely to the converged values obtained for the collision-dominated system at the cutoff point. The fact that phase velocities and attenuation rates near the cutoff agree with the free-molecule result suggests that the observed cutoff may represent the limit of the collisionless region. The accuracy as well as the limitations of the ST method have been established for one-component systems. The next chapter attempts to apply the ST method to the complicated sound dispersion behaviour that has been observed in mixtures. Chapter 5 Anomalous Sound Dispersion in Two-Component Systems 5.1 Introduction Recent experimental studies of sound propagation in mixtures of gases have demonstrated a new set of dispersion behaviours that are not described by conventional hydrodynamic treatments [59-67]. These anomalous properties have been measured in mixtures of gases of disparate mass. Some experiments have shown the existence of sound modes with propagation velocities and attenuation rates that exhibit significantly different charac-teristics than hydrodynamic modes. This phenomenon is attributed to the decoupling of the light and heavy components when the frequency of sound approaches the order of the collisional frequency of the gas mixture. Related phenomena, such as sound mode degeneracies and propagation gaps at critical frequencies and gas composition have also been studied in these systems [59,63,67]. The possibility of multiple sound propagation modes in gas mixtures was originally raised by Liboff [112]. Huck and Johnson [57] were the first to suggest the possibility of measuring fast and slow sound propagation modes in a gas mixtures. They termed this 'double sound propagation'. The term 'fast sound' describes the mode that prop-agates exclusively through the light component was coined by Bosse et al. [116] to de-scribe a peak in the selfstructure factor for Li in molecular dynamics simulation of liquid Lio.8Pbo.2- This high-frequency peak was attributed to rapid Li density fluctuations in a background of heavy Pb. Campa and Cohen [58] later predicted the existence of fast 137 Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 138 sound modes in mixtures of dilute binary fluids using a B G K model of a modified En-skog fluid [58,117,118]. Their calculations indicated the existence of fast sound modes that propagate only through the light component at much higher velocity than ordinary sound. These propagation modes do not involve the heavy component at all. Experimen-tal data has since revealed the existence of analogous slow sound modes. Unlike the work of Huck and Johnson [57] (and later that of Wegdam et al. [62]), Campa and Cohen's [117] work suggested for the first time the existence of nonhydrodynamic propagation modes associated with nonconserved quantities. The first experimental measurement of fast-propagating free sound waves was in He-Ne mixtures using neutron scattering [61]. Rayleigh-Brillouin light scattering revealed slow sound in He-Xe mixtures [64]. Both fast and slow sound modes were then later observed through light scattering in H 2 -Xe and H 2 -Ar mixtures [62,65,66]. Other studies reported fast sound in H 2 - A r and slow sound in H 2 - S F 6 [60]. Foch et al. [119] calculated phase velocities of sound in mixtures of noble gases and compared. them with experimental results but failed to observe anomalous dispersion behaviour . Since no conventional hydrodynamic model of sound dispersion has yet ac-counted for anomalous sound dispersion, all current analysis of experimental data has thus far involved the use of two-temperature hydrodynamics [59,62,114,115]. This treat-ment is based on Burger's representation of a homogeneous multi-component fluid in which densities, flow velocities and temperatures are individually computed for each com-ponent [68]. Burger's treatment is based on the thirteen-moment approximation of Grad [24] for which each species velocity distribution function is a local Maxwellian distribu-tion with respect to a species flow velocity and species temperature. A set of generalized hydrodynamic equations, allowing for separate species temperatures, are obtained by taking appropriate moments of the coupled Boltzmann equations for the mixture. For Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 139 comparison with experimental work on He-Xe mixtures, Johnson used transport proper-ties calculated for a model interatomic potential. Recent experimental data obtained from Rayleigh-Brilloiun scattering has revealed a wealth of anomalous sound dispersion phenomena in multicomponent gases, especially at large Kn. All current approaches thus far suffer from the defect that agreement with experiment is quantitative only for small Kn. This conclusion is apparent from a compar-ison of Johnson and Bowler's [59] measured and calculated attenuation rates for He-Xe mixtures as well as Schram and Wegdam's [62] measured and calculated values for the phase velocity in H 2 -Xe mixtures. The treatment of Cohen and Campa has only been applied to simple model systems (BGK treatment of hard-sphere Enskog fluids) [118]. Gas kinetic approaches are an obvious method for exploring anomalous sound dispersion phenomena due to the fact that anomalous sound behaviour is frequently observed near the limits of the hydrodynamic region [59,62]. The purpose of this work is to apply a gas kinetic theory of mixtures to small-amplitude oscillations near or in the Knudsen region of a gas. Experimental data suggests that oscillatory disturbances in gases exhibit complicated behaviour in this region [59-67]. Experimental measurements of anomalous sound dispersion have been conducted in gases near ambient temperatures and pressures that are still within the region of validity of kinetic theory. It seems reasonable to use gas kinetic theory as a tool for exploring anomalous sound dispersion properties of real systems. This work attempts to examine and interpret non-hydrodynamic behaviour observed in real systems using methods of gas kinetic theory. The ST method appears to successfully reproduce the behaviour of simple gas systems [49-51] but have never been applied to mixtures gases of disparate mass and a the rich set of dispersion behaviours recently reported. Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 140 5.2 Boltzmann Equation for Small Ampli tude Disturbance: Two-Component System Consider a dilute mixture of gases labeled 1 and 2 where the distribution functions are given by two coupled Boltzmann equations ( ^ + c - V ) / i = Mfi) + J1MJ2) (5-2.1) ^ + C ' v J / 2 = J 2 2 ( / 2 ) + J 2 1 ( / b / 2 ) (5.2.2) (5.2.3) where J „ 7 is the Boltzmann cross collision term, Jrn = J J[fi,fy - fnf^o-nygdttdcr,. (5.2.4) The distribution functions, fv are close to a Maxwellian distribution, characterized by the number density n 7 and a common temperature T. The distribution function for species 77 is written as a Maxwellian perturbed by a small term hn, ' fv(c,v,t) = f^(c)[l + hv(c,v,t)}. (5.2.5) The linearized Boltzmann equation is I — + c • V J = ([KJJK^ + KniLni\hn + Ljjhy) (5.2.6) where the KM operator corresponds to the linearized self-collision term Kmhn = — f f 40)K + h'v, - h v - hv,]gomdQdCTI, (5.2.7) while linearized cross collisions are given by, Lmhv = — f f ff[h'v - hn]avlgd^dc7 (5.2.8) Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 141 and where L 7 7 / i 7 = — f f / 7 0 ) [ ^ 7 - h^a^gdQdej (5.2.9) '2kBT\1'2 firm (5.2.10) /ST = ' Arr (5-2.11) I 1 mvm7 (mv + rrij) and cr0m and a^1 are self and cross-collision constant cross-sections. The oscillatory disturbance for each component is h(cv,r,t) = ^ ( c „ ) e i ( k : p - w t ) (5.2.12) where the x axis lies along k. With Eq. (5.2.12) in the Boltzmann equations Eq. (5.2.6), one has that ( E V K , = K ^ + Q ^ I L ^ + L^hj] (5.2.13) where Q(VI) - _ ^TL n 7 (T^ 7 / 1/2 nvaj' \ p J Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 142 5.3 ST Method for Two Component System The perturbation of each component n from the Maxwellian is given by oo K,(tr,,») = E (5.3.1) n,l=0 and the self-collision operators for each component are identical as in the one-component case, N,L oo E oS^a^S + ^ + i ^ - i E «!Mf (5.3.2) n,l,n',l'=0 n,l>N,L N,L = E anl (Knljri'l' ~ ^N+l L+\0~nn'&ll')^nl,n'l' + AjV+1 L + l ^ n,l,n',l'=0 where K(J^v = ( ^ \ K ^ l ) . (5.3.3) The linearized cross-collision operators L T ? 7 and L 7 T may be written in an analogous fashion, N,L L ^ h r , = 52 anl {Lnl,n<V ~ f'N+l L+An'^ll')^}' + PN+1 L+lhq (5.3.4) n,l,n' ,l'=0 N,L L^^hj = 52 anl (Lntjn'l' ~ VN+l L+l^nn'^ll')^}' + vN+l L+\hv n,l,ri,l'=0 where 4 l i ' = ( ^ , ^ S ) (5.3.5) The eigenvalues of Km, L n i and L T T have been calculated by Lindenfeld and Shizgal [94]. Equations (5.3.2), and (5.3.4) are substituted into Eq. (5.2.13) ( X - S [QX - Ai\r+i L+i - Q ( 7 ? 7 V i v + i L+I ~ MJV+I L+I]) hv = (5.3.6) n,l,n'l'=0 Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 143 where GnUi'V — KnLn'l' ~~ ^N+l L+l^nn'Sw (5.3.7) HnCn'l' — ^nljl'l' ~ L+\0~nn'0~W' (5.3.8) and HnCn'l' —. ^nfi'V " UN+1 L+An'^W • (5.3.9) The ev is factored out of the LHS of Eq. (5.3.6) - e , (IO, " £,) ^ = E ( [ G a ^ + O ^ ^ S , ] ^ ^ (5.3.10) n,l,n',l'=0 where g = Ev - (Aiv+i L+i + Q{T,I)[PN+I L+I + "N+l L+I]) (5 3 11) Dividing by the velocity-dependent term in parenthesis on the LHS of Eq. (5.3.10) and multiplying by each basis function and integrating over £ gives a coupled set of linear equations N,L N,L E E ([^L.n 'c + HLI,1>I< RlJ,n'vani (5.3.12) n,l=0 n'l'=0 +Q^HT\ltnH'RiJ,nn'anl') ~ €lal (1) and N,L N,L E E ([G%1,> + Q^H^V] Kflnlv^ (5.3.13) n,!=0n'('=0 + (2) Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 144 where R, rs.n'I' 1®> [4"rc]n _ (5.3.14) The coupled, homogeneous set of Equations (5.3.12) and (5.3.13) is written in matrix form ( GMRW + QMHMRW QMHMRW . Q(21) RW RV + Q(21) ij( 2 2 ) RW ( a.™ \ v « ( 2 ) y +1 6iO, (1) e 2a (2) (5.3.15) where the elements of the submatrices G ( 1 1 ) , G ( 2 2 ) , H ( 2 2 ) , i f ( 1 2 ) , i f ( 2 1 \ R^\ R^ are given by Eqs. (5.3.7)-(5.3.9) and (5.3.14), and subarrays and are the expan-sion coefficients. The dispersion relation is obtained from the secular determinant D = 0 where D(u) Q ( 2 i ) H ( 2 i ) GWRW + QWHWRW + e2I (5.3.16) 5.4 Solution of the Dispersion relation for a Two Component Mixture The roots of the dispersion relation were computed for some physical systems using the simple model of Maxwell molecules discussed previously. Phase velocities and attenua-tions were calculated using masses and collision cross sections corresponding to He-Xe mixtures for which experimental data is available [59]. The experimental work of John-son and Bowler [59] measured phase velocities and attenuations over a range of helium Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 145 mole fractions Xne- The acoustical properties, (3/co and a/co are expressed relative to the adiabatic speed of sound in helium, v^e. Straightforward application of the conventional Newton-Raphson root-finding method to the determinant of Eq. (5.3.16) for He-Xe mixtures does not give satisfactory re-sults. Figure 5.1 shows a plot of the real and imaginary parts of D, Re[D] and Im[D], respectively, for a/cov^e ranging from 0 to 0.8 for Xu_e = 0.45, 1/Kn — 10.5 and p1co~1/vne = 3.23. The determinant D approaches the abscissa very slowly and has multiple roots in the region /3/wyHe « 0(1) a/co~lv^e « 0(1). The determinant D has several roots in the region between a/co^vue between 0.2 and 0.6 where its value is very close to zero. The Newton-Raphson method was found to be inadequate under these conditions and was abandoned in favour of a root-finding procedure that essentially uses the method of bisections. A root can be determined for a given co and f3 to high accuracy by first obtaining the set of values of a for which Re[D] = Im[D] = I. Each value of a for a given co and (3 corresponds to a branch of the solution. The parameter I is the distance of the point of intersection of the real and imaginary parts of D from the zero. The value of (3 is varied slightly and new values / are obtained. Figures 5.2A and 5.2B show values of the determinant D in the region aco~lv^e be-tween 0.2 and 0.8 where (3co~1vne is 3.23 and 2.13, respectively. The intersection points between the curves Re[D] and Im[D] at Re[D] = Im[D] = / in Figure 5.2 correspond to points on the trajectory I((3,a). In Figure 5.2A, three branches ((a),(b) and (c)) of i" are visible for a w _ 1 « H e equal to 0.20, 0.27 and 0.51. As /3co~lvne goes from 3.23 in Figure 5.2A to 2.13 in Figure 5.2B, branch (b) vanishes, and branches (a) and (c) move to au~lvne equal to 2.35 and 5.55 respectively. The branches are plotted against (3 and the point at which / vanishes corresponds to solutions of D = 0. Figure 5.3 shows the trajectories of two branches, (a) and (b), of / for Xne = 0.45, Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 146 Figure 5.1: Plot of Re[D] and Im[D] versus OLLO 1vHe for a He-Xe mixture X^e = 0.45, 1/Kn = 10.5, a n d . / ^ " 1 ^ = 3.23. Figure 5.2: Plot of Re[D] and Im[D] versus a.uo~lVHe for a He-Xe mixture Xne = 0.45, 1/Kn = 10.5 where 3uj~~lVHe equals (A) 3.23 and (B) 2.13. Intersection points of Re[D] and Im[D] are (a), (b) and (c). Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 148 1/Kn = 10.5 plotted versus (5. The point at which the trajectory / crosses the abscissa in Figure 5.3 ( vertical dotted line) gives the value of (3/covue of the dispersion relation. Branch 1 was selected as the hydrodynamic root over Branch 2 as its trajectory is over much smaller values of a, corresponding to less-strongly damped modes. Branch 2 is an example of what is often termed a "spurious" mode. Johnson and co-workers did not find clear evidence of non-hydrodynamic modes although from their data they conclude that non-hydrodynamic modes do influence the experimental result, especially at large Kn [59]. There is clear experimental evidence from other workers [62] of nonhydrodynamic sound modes in disparate-mass gas mixtures. The data presented by Wegdam et al. [62] is for free sound waves and is not as complete as that of Johnson as only phase velocities for a limited number of mole fractions X H e are reported. Mixtures of monatomic gases with disparate masses have been shown to exhibit com-plicated sound dispersion behaviour [59,62] that deviates significantly from the behaviour of single-component systems discussed in Chapter 4. Phenomena such as the decrease of phase velocity with increasing Kn [59] and the decoupling of sound into "fast" and "slow" modes [62] have been observed experimentally. Conventional hydrodynamics does not predict these effects and as a result, these effects present an opportunity to test gas kinetic models. Calculations of phase velocities and attenuation factors were computed using the ST method for a Maxwell molecule model system of a He-Xe mixtures and the results are presented in the following section. 5.5 Results for a Two-Component System Dispersion relations were obtained for a series of mixtures of He and Xe, with X H e rang-ing from 0.3 to 0.8 over a range of 1/Kn from 5 to 30. Nine Burnett functions (N=2, Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 149 2.0 3.0 4.0 5.0 6.0 7.0 Figure 5.3: Plot of I versus 0UJ lvEe for (a) and (c) branches of the trajectory of Re[D] = Im[D). Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 150 L=2) were used for each component and collision cross sections for He and Xe with a Maxwell power-law potential were calculated from standard references [133]. Figure 5.4 shows curves for phase velocities for several helium mole fractions X H e - For X H e = 0.8 and 0.7, the ST method (solid line) and 2-temperature hydrodynamic method of Johnson and Bowler [59] (dotted line) agree fairly well with experimental phase velocities (cir-cles). For Xfte = 0.45, Figure 5.4 shows that neither the ST result nor the calculation of Johnson and Bowler agrees with experiment for smaller Kn. As the mole ratio of He is decreased to 0.3, Johnson and Bowler's result shows good agreement with experiment, that is a steady phase velocity, while the ST method predicts a small decline in the phase velocity for 1/Kn below 10. Attenuation factors for He-Xe mixtures versus 1/Kn are shown in Figure 5.5. Agree-ment between experimental results (circles) and those calculated by the ST method (solid line) in this case is better for ATHe small than for cases where Xne is close to 1. For Xfte = 0.8 and XEe = 0.7, the ST method severely underestimates damping. For ATH e = 0.45 and X#e = 0.3, agreement is much better. The results of Johnson and Bowler (dotted line) are consistent with the analysis for phase velocity. Johnson and Bowler's results are fairly close to the experimental result, except when Xu_e = 0.45. The values of a and (3 computed by the ST method tend to show better agreement with experiment as the number of Burnett functions is increased from 4 (N=l, L=l) to 6 (N=2,L=1) and then to 9 (N=2, L=2) (shown in Figures 5.4 and 5.5) terms per com-ponent. Numerical calculations using more than 9 moments gave numerical overflows, which were assumed to be the result of evaluating a large determinant with over 400 elements. While further work needs to be done to accurately establish the numerical convergence of the results presented in Figure 5.4 and 5.5, some qualitative observations about the effect of increasing the numbber of moments upon the dispersion relations can be made. Mixtures in which helium is the dominant species (X^e = 0.8 and Xne = 0.7) Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 151 JL 10 20 1/Kn 30 10 20 1/Kn 30 Figure 5.4: Plots of / f o - ^ H e versus 1/Kn for XHe equal to 0.8, 0.7, 0.45 and 0.3. The ST result is the solid line, Johnson and Bowler's calculation is shown by the broken line and experimental results are denoted by o. Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 152 Figure 5.5: Plots of acu~lvu_e versus 1/Kn for various X H e equal to 0.8, 0.7, 0.45 and 0.3. The ST result is the solid line, Johnson and Bowler's calculation is shown by the broken line and experimental results are denoted by o. Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 153 tend to show faster convergence for 0 and a than mixtures in which xenon is present in larger amounts (Xne = 0.45 and Xue = 0.3). Also, more moments are required to obtain converged values of 8 and a as Kn increases. The latter observation is common to both single-component systems of Chapter 4 and the two-component systems in the present chapter. The calculations performed on the He-Xe system show that the ST method for a real system is feasible. The agreement with experiment still leaves much to be desired, however. There may be a few reasons for this. First and foremost, the convergence properties of the dispersion relation should be ascertained to ensure that the series is in fact convergent. Secondly, more realistic interaction potentials are probably worth investigating, although they would involve more computation. Thirdly, use of a basis set other than Burnett functions may help the convergence of the dispersion relation and this possibility should be investigated. 5.6 Conclusion and Summary In spite of the overall success in modelling the acoustical properties of gases, there are some major defects common to gas kinetic treatments. In hydrodynamic calculations, any thermodynamic and transport coefficients for any system can be obtained from em-pirical data and inserted into the equations. In gas kinetic theory, one cannot do this without modifying the transport properties. Internal degrees of freedom can also be incorporated into hydrodynamic equations through phenomenological data and should, in principle, be modelled using inelastic Boltzmann collision terms. Another deficiency arises from the effects of modelling interatomic potentials. Phase velocities and attenuation coefficients provide only a simple check of the va-lidity of the models discussed in this and the preceding chapters. Comparisons with Chapter 5. Anomalous Sound Dispersion in Two-Component Systems 154 light-scattering data provide a still more comprehensive test of these calculations. For the detailed modelling of Rayleigh-Brillouin lineshapes obtained from light scattering, thermodynamic and transport coefficients affect the line-broadening in the peaks. For example, the Maxwell potential yields a system in which the thermal diffusion coeffi-cient is zero. It is known that thermal diffusion causes significant line-broadening in Rayleigh-Brillouin scattering and a model that neglects this effect will predict signifi-cantly narrower spectral lines than are observed experimentally [110]. As a result, light scattering calculations that assume Maxwell potentials between the particles can never give quantitative results. Chapter 6 Summary and Outlook This thesis has considered corrections to the equilibrium rate of reaction from the perturbation of the velocity distribution function in a number of model systems. The effects of reversibility and irreversibility as well as the effect of reaction products on the nonequilibrium reaction rates have been studied. The role of the departure of species temperatures from the equilibrium values in reacting species have also been examined. The region of validity of the C E and related SNE and MSNE methods have been estab-lished using explicitly time-dependent calculations of the departure of the reaction rate and temperature from their equilibrium values. A critical evaluation was provided of the work of Cukrowski et al. [31], which questioned much of the earlier work in the field. An aspect of this subject that is lacking is that there has been, to the author's knowl-edge, no direct experimental observation of these nonequilibrium effects. Only recently have there been experimental developments in this direction that may permit direct ver-ification of nonequilibrium reactive effects in realistic systems. The direct measurements of velocity distribution functions in reacting systems have been reported recently by sev-eral groups [85]. These generally involve the determination of the details of the velocity distribution function of a product of a reaction (often hydrogen) from the Doppler profile of an emission line. Similar methods have been used to characterize the distribution of atomic hydrogen about the Earth at altitudes above 500km [86]. The methods discussed in this paper have been used by Lindenfeld and Shizgal [87] to study the escape of atomic 155 Chapter 6. Summary and Outlook 156 hydrogen from Earth; a process with an activation energy analogous to the model re-active systems of this work. In particular, the cooling of the escaping hydrogen over the background major species which is atomic oxygen has been estimated and measured from the Doppler profiles of Lyman-/? emission [88]. This is an excellent example of the departure of the temperatures of different species from the system temperature. It is speculated that significant translational non-equilibrium could exist in reacting gas-phase systems with large temperature and density gradients such as flame fronts [134]. Unpublished calculations by the author on model reactive systems suggest that large gradients alone are insufficient to cause significant nonequilibrium effects in reac-tion rates. It is suggested that nonequilibrium effects could be enhanced by a feedback mechanism in a multistep reactions. Such a situation could involve a system in which a translationally excited products of one reaction step are consumed in other reaction steps of the reaction sequence. The theoretical methods developed in this thesis are anticipated to be of practical use in the interpretation of such experiments when the results become available. The W C U , G B E and ST methods for calculating sound dispersion properties in sim-ple gases have been examined. The W C U method fails to converge and gives incorrect results outside the hydrodynamic region. Solutions of the sound dispersion problem ob-tained from G B E method are shown to diverge in the limit of high frequencies and low pressures. Sound dispersion properties calculated using the ST method give fairly good agreement with experiment. The importance of the Gross-Jackson modification as well as the significance and interpretation of cutoff frequencies of the ST method have been examined. The ST method is applied to the problem of anomalous sound dispersion in gas mixtures. The results show qualitative agreement with the calculations of previous workers [59] and some agreement with experiment. Chapter 6. Summary and Outlook 157 The methods of Chapters 3 and 4 have applications to ongoing work on light scatter-ing in gas systems. The theory of sound waves applies directly to the study of propagating density fluctuations [135], a phenomenon that is studied in light scattering experiments. The spectrum of density fluctuations is experimentally obtained by Rayleigh-Brillouin light scattering [136]. A beam of monochromatic light, having frequency w0 and wave vector k 0 is scattered by a neutral gas giving a broadended spectrum of scattered light centered about LO0. When the gas pressure and angle of observation relative to k 0 are appropriately selected, a fine structure in the spectrum consisting of three lines appears. The Rayleigh line is undisplaced and centered about frequency coo, while a symmetrically displaced doublet at frequencies co0 — UJ-Q and LOQ + OJQ make up the Brillouin doublet, where u>B is the frequency shift. The Rayleigh peak is undisplaced but broadened by thermal fluctuations. The Brillouin doublet is displaced as a result of acoustical waves which diffract the incident beam and cause a Doppler shift in its frequency. Light scattering measurements at large angles of observation involve fluctuations in the region where the wavelength is comparable to the mean free path. The predictions of continuum theory have long been known to give poor agreement with experimental results in this region (reference [137], for more recent results, see [110]). As the viewing angle increases, the width of the Brillouin peaks increase by as much as a factor of ten, a fact that continuum theories have not been able to account for. Gas kinetic meth-ods that are identical to methods applied to the theory of sound have been applied to the interpretation of wide-angle light scattering measurements [138]. The calculations [138] have confirmed the accuracy of commonly used gas kinetic models. Most recent measurements of anomalous sound dispersion have been obtained from measurements of the Rayleigh-Brillouin spectrum of light scattered from mixtures of gases of disparate masses [62-66]. Qin and Dahler [39] have recently shown that the spectrum of scattered light from a mixture can be influenced by reactions. There is the potential for studying Chapter 6. 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Sugawara, S. Yip, L. Sirovich, Phys. Fluids., 11, 925 (1968). [136] J. P. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill Inc, New York (1980). [137] T . J . Greytak and G. B. Benedak, Phys. Rev. Let., 17, 179 (1966). [138] M . Nelkin and S. Yip, Phys. Fluids, 9, 3, 1966; O. L. Deutsch and S. Yip, Phys. Fluids, 17, 1, 252 (1974); G. Tenti, C. D. Boley, R. C. Desai, Canadian J. Phys., 52, 4, 286 (1974); G. Tenti and R. C. Desai, Canadian J. Phys., 53, 1279 (1975). Appendix A Isothermal Systems Cukrowski et al. [31] suggested in a recent paper that the neglect of the different species temperatures in the treatment of the one component reactive system A + A products (A.l) is in error. Cukrowski et al. [31] argued that even for this one component system, there must be a contribution from the ai expansion coefficient owing to a difference in the temperature of species A and the temperature in the Maxwellian. In order to introduce this effect they replaced reaction (A.l) with the reaction, A + A^B + B. (A.2) For this system, they set dT/dt = 0 so that this term is no longer included in the inhomogeneous function (compare with Eqs. (2.2.10) and (2.2.11) in the C E equation). This assumption is not correct as discussed in Section 2.2.2. The elastic A — A collision operator that was used in their work is given by Eq. (20) of the earlier paper [32], and to first order is of the form, lf[f{l)] = IJ[f{iyfi0Y-f(l)fi%n9dnd^ (A3) This is not the same linearized operator used in the other works [13,20,40] (see also Eqs. (2.2.2), (2.2.3), (2.2.35) and (2.2.36)) since one of the distribution functions of the same species has been set equal to the Maxwellian. With the substitution, = f^<j>, the 167 Appendix A. Isothermal Systems 168 elastic A — A collision operator is &<f>] = / ' / /(0)A(°V - ^gdSldc-i (A4) and is not the correct linearized one component collision operator. The correct operator is the one given by the first term on the LHS of Eq. (2.2.8). The operator in Eq. (2.2.8) conserves both the number density and the energy whereas the operator given by Eq. (A.4) conserves only the number density. With the neglect of dT/dt and the way in which the operator was linearized, the system studied by Cukrowski et al. [32] corresponds to the reaction Eq. (A.l) occurring in a large excess of a second species (with the same mass as A) that acts as a heat bath. This is the isothermal system studied in the paper by Shizgal and Karplus [22]. The C E equation considered by Cukrowski et al. [32] is of the form, The Boltzmann equation for the product is not included so that there is no density dependence considered. It is important to notice that in this equation the cross section that appears in the collision operator on the LHS is for A - A elastic collisions, and not A-B collisions if B were in large excess and acted as a constant temperature heat bath. For the system studied by Cukrowski et al. [32] (and in [21]), the temperature of the one component reactive species is different from the bath temperature as given by Eq. (2.2.24). The leading term in the expansion is indeed o,\ and a first approximation to the correction to the equilibrium rate of reaction is given by, / / /<o)/i°y - <f>Ki9<ffwc, = / , n = A2jAAo (A.6) which with Eq. (2.2.33) is given explicitly by, 77 = -[ e * + ^] 2exp(-6*) (A.7) Appendix A. Isothermal Systems 169 which is given by Eq. (3.2) of Cukrowski et al. [31]. The C E equation for the one-component system, Eq. (A.l), is given by The important distinction between this equation and Eq. (A.5) is that the energy is not a summational invariant of the first equation whereas it is for the second equation. The coefficient ai is nonzero for the solution of Eq. (A.5) but it is zero for Eq. (A.8). The lowest order estimate of the fractional decrease of the rate coefficient from the equilibrium value is obtained with a2 and is given by, which appears as Eq. (3.3) in Cukrowski et al. [31]. This result is twice the result given by Eq. (2.2.34) as explained in Section 2.2.1. The results Eqs. (A.7) and (A.9) are for different systems and cannot be compared directly. Appendix B The Generalized Chemical Kinetic Equation of Alexeev The system studied is the model reactive system A + A —y products (B.l) with the complete neglect of products. The system is a simple one for which many cal-culations have been carried out. In this case, an isothermal constraint is applied and nonequilibrium effects are neglected. Alexeev's "physical method" [46] is used to obtain a hydrodynamic rate equation for the rate of change of the density of the system. The resulting "generalized hydrodynamic" result is compared to the conventional hydrody-namic result. B.0.1 Physical Method of Derivation For a gas of particles having velocity distribution function / i(c, x, t), the change in the number of particles having velocity c over the small time interval At in the absence of external forces is where c, x and t are velocity, displacement and time coordinates, respectively. The terms on the right-hand side of Eq. (B.2) are the Boltzmann collision operators which represent the gain and loss through elastic and reactive collisions, JE and J[, respectively. For spatially homogeneous system, Alexeev expands the LHS of Eq. (B.2) in a Taylor / i(c,x, t) - / i(c,x - Atc,t- At) = (JE + Ji) At (B.2) 170 Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 171 series [46] f1-(f1-At^ + ^ ^ + ...) = (JE + JI)At (B.3) The G B E is obtained by setting the time interval, At, to be of order r, the mean time between collisions. All terms of order r 2 and higher are ignored. The result is the standard G B E of Alexeev H-TIF = JE + JI- {BA) Equation (B.4) is the linear form of the G B E that is obtained from the physical method of derivation [46]. The nonlinear form of the G B E is [46,47] dfi_d_ dt dt dfi T—— dt = JE + JI (5.5) and both forms of the G B E for the model system A + A —> products were examined. B.0.2 Reactive System A + A —> products An isothermal gas of elastic hard spheres with a line-of-centers reactive model was used to model any measurable effects that would characterize the G B E . The total reactive collision cross section is given by <T* = naR(l - E*/E) E > E* (B.6) = 0 E < E*. where E* is the threshold energy and GR = d2R/A. The hard sphere elastic and reactive diameter was dR. This simple system can be solved exactly and adequately describes the reaction kinetics of some real systems. The model A + A —> products describes a system in which bimolecular collisions between particles exceeding the activation threshold results in the escape of particles from the system. Reactive collisions change the density as well as mean time between Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 172 collisions in this hypothetical system. This case corresponds to an example of which r is not a free parameter [47]. The hydrodynamic expression for the rate change of density of the reacting system is obtained by integrating the velocity distribution function, fi over all velocities, c. With f fide = nx Eq. (B.3) is dn\ or d2ni (ar)2 d 3 ni f, , . where ar is the time step, At, written as the product of r, mean time between collisions, and some constant scaling factor, a. Alexeev's "physical method" is the special case of a = 2. In contrast to Eq. (2.44) of reference [46], Eq. (B.7) is not truncated after two terms. The series in Eq. (B.7) can be solved exactly for the system A + A —> products. Integrating the right-hand side of Eq. (B.7), for a Maxwellian velocity distribution, fi = ni(t)[rn/2irkBT]3/2 exp[—mc2/2kBT], the elastic Boltzmann collision operator van-ishes, JfEdc = J J J [f[f - fif}gadQdcdci = 0. (B.8) The result for the reactive collision operator using the line-of-centers collision cross section is JJRdc = J J j ftfga'dndcda = -n\aR {^^j ' exp[-E*/kBT}, (B.9) the familiar Arrhenius reaction-rate law. B.0.3 Generalized Hydrodynamic Equation for a Reactive System The following discussion treats a "generalized hydrodynamic" reactive system comprised of particles whose velocity distribution is almost Maxwellian. The generalized hydro-dynamic equation for the homogeneous case of the model system A + A —>• products Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 173 is dm (IT d2rii (or) 2 d?nx 9 /8k B T\ 1 ^ 2 , , ' For the model system, the mean time between collisions is the number of particles in the system divided by the rate of collisions. The rate of collisions, is ///fjgadndcdc, = n\o ( ^ ^ j ' (£ -11) where o is the total collision cross section for A and the mean time between collisions is 1 / 7T77J \ V 2 Re-writing ni, £ and r in dimensionless form, rii = T^V (B-13) (™l)o f = - (B.14) • i=— (B.15) T[) where the zero subscript refers to the values at some inital time t — 0. Combining Eqs. (B.13) and (B.14), f = 1/ni. (5.16) Rewriting Eq. (B.10) (omitting the "hat" " over dimensionless units) where i? = ^ e x p [ - £ * / £ ; s T ] . Writing r in terms of n\ (Eq. (B.16)) is observed that Eq. (B.17) has an exact solution ri\ in t of the form ni = — ( 5 . 1 8 ) rt + <j) y J Appendix B. The Generalized Chemical Kinetic Equation of Alexeev If A This is consistent with observed reaction rates for second-order chemical kinetics, except that the rate constant no longer corresponds to R, but to some "corrected" generalized hydrodynamic value, r. The left hand side of Eq. (B.17) can be summed by substituting in the relation 1 = (-l)N — = (-l)NNhNnN+l (B19) Substituting the iVth order derivative Eqns. (B.19) into the Taylor series Eq. (B.17), the G H E , the rate equation becomes oo -n\ £ a ^ " 1 / = -n\R (5.20) N=l Applying the summational identity the result £ = Y~- (B.21) N=l 1 U l R= (5.22) 1 — ar or R (B.23) 1 + aR v ' is obtained. If the physically reasonable assumption is made that the rate of reaction is always less than the rate of binary collisions in the gas, then 0 < i ? < l . I f i ? < l , then ar < 1, and the series Eq. (B.21) is convergent. Equation (B.22) represents the correction to the rate of reaction as a result of solving the chemical kinetic equation as a difference equation. Note that the correction is independent of the density (and hence the mean free path) and depends only on the hydrodynamic reaction rate and a free parameter a, corresponding to a "step size". There is no obvious justification for choosing a = 2 as in [46]. Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 175 B.0.4 Relation between Difference Equations and G B E Result The above treatment is just one of a number of possible difference equations that one can solve analytically for this system. For example an alternative treatment to Eq. (B.10), ni(t) - ni(t - or) = -nx{t - ar)2Rar (£.24) also has an exact solution of the form Eq.(B.22). Both sides are expanded as Taylor series and using the identity for the iVth derivative on the RHS, d~P- = ( - ! ) " ( N + l ) l r » n r 2 ( £ . 2 5 ) one obtains R=(l- ar)r. (5.26) The forward difference equation ni(t + ar) - m(t) = -ni{t)2RaT (5.27) when evaluated in an analogous manner gives the reaction rate correction i ? = — ^ — (5.28) 1 + ar v ' or r = — ^ — . (5.29) 1 - aR v . For the backwards differences the corrected rate , r Eq. (B.29), is smaller that the hydrodynamic rate, R (4.2.28), while for the forward difference equation, r is larger than R. Note that Eqs. (B.23) and (B.29) are, as expected, equivalent for the case ar << 1, since 1/(1 + ar) — l — ar + ar2 — ar3 + — The corrections to the rate equation that as a Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 176 result of the generalized treatment appear to arise from numerical inaccuracies resulting from the choice of the parameter a in the finite-difference chemical kinetic equation, and not any physical phenomenon. The standard generalized hydrodynamic expressions for this system (Eq. (B.4) and (B.5)) can be easily solved by the method analogous to the one above, or by standard numerical methods. The nonlinear case Eq.(B.5) is solved for the model system by observing that ^ = -^^t- Solutions of the linear and nonlinear equations, Eqs. (B.4) and (B.5) respectively, both coincide giving the "corrected" rate R=(l + ar)r. (5.30) Figure B . l shows the reciprocal of the density evolution in time for the hydrody-namic and generalized equations. The solid line is the standard hydrodynamic result for E*/kBT = 2 while the points are the numerical solution of Eq. (B.5) and the dotted line the analytic solution. The boundary conditions for solving the second order ordinary dif-ferential equation numerically by finite differences were n\ = no at t = 0 and dny/dt = 0 for large t. The slope of the solid line corresponds to R, while the slope of the dotted line corresponds to r . Figure B.2 shows a semilogarithmic plot of numerical and analytical solutions for the "corrected" rate r versus the threshold energy, E*/kBT. This is compared the hy-drodynamic value, R, which has an Arrhenius form and gives a straight line. The dotted line shows the analytical solution (Eq. (B.23)) and agrees well with the curve for the numerical result of Eq. (B.4) (shown by +). An exact solution of the generalized hydrodynamic equation for the model system A + A —> products based on Alexeev's "physical method of derivation" [46] was carried out. The exact solution contains a single adjustable parameter for the given system. This Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 177 Figure B . l : Plot of (1/n — 1) versus time for a second-order reaction A + A —> products with E*/1CBT = 2. The solid line represents the standard hydrodynamic solution, R, while the dashed line and (+) represent the analytical and numerical solutions, respectively, of the G H E . Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 178 -4—> O o •4-> 0 -K o -v. 0 o ~~I 1 1 1 1 <^ Hydrodynamic (R) -(- Generalized Hydrodynamic (r)| Analytic (2-term, r) O E 7 k B T Figure B.2: Semilog plot of the reaction rate constant for A + A —> products versus E*/1CBT. The hydrodynamic solution, R, is denoted by O , while the numerical and analytical solutions to Eq. (B.4) are denoted by (+) and the dashed line, respectively. Appendix B. The Generalized Chemical Kinetic Equation of Alexeev 179 parameter is shown to be equivalent to the step size of the Boltzmann difference equation. This parameter controls all deviations from the known hydrodynamic result. This demon-strates that this deviation does not arise from physical effects but rather from numerical errors arising from the choice of step size. Based on these observations, it was concluded that Alexeev's G B E and G H E do not describe the model system A + A —> products . Appendix C ST Collision Matrix Elements The matrix elements defined in Eq. (4.3.92) / i>n'l' 1/2 f°° e ^Ipnllpn'V 4 W where E - X ( C l ) (C.2) are evaluated with Legendre polynomials P;(/i) and associated Laguerre polynomials I/^+1//2(^2) defined in Eqs. (4.3.14) and (4.3.16), respectively. The product of two axially symmetric Burnett functions is given by „ „' [1/2J I//2J . / , „ / , A V" rPM'P''M' (C2\P+P''+rn+m' Ic -2(m+m') VnlVnH' - Anl.n'l' 2^ 1^ 2^ Z_y 1 nl,n'l' ( £ j (U) p = 0 p ' = 0 m = 0 m ' = 0 where A, nLn'i 1 n\n'\(2l + 1)(2/' + l)T(n + / + 3/2)T(n' + /' + 1/2)TT 2*+*'+I n\n\ and -ppm,p'm' / -i \p+p'+m+m' ( m \ ( I' \ I 21- 2m \ I 21'-2m' x n \ I n i T(p + l + 3/2)r(p' + /' + 3/2) (C.3) (C.4) (C.5) 180 Appendix C. ST Collision Matrix Elements 181 W i t h E q . (C.3), the Rni,n'i' integrals, can be written in the form n n' U/2J L*'/2J Rnl,n'l' = h.nl,n'V E E E E ^^n'™ %p+p1+m+m',1+1'-2{m+m') • (C.6) p=Qp'=0 m=0 m'=0 where Zij is (C.7) The integral Zij is evaluated by setting d £ = d£xd£yd£z and £ 2 = 4 r 2 + £ y + £ 2 the integrand of E q . (C.7) is expanded into a polynomial in ^ x , £ y and £ z of the form • g=0 r=0 A'irA2{q-r) Z<i(%-q) In E q . (C.8), d ^ and POO Aj = TT'112 / x>e~x dx. J—00 ' 7 r - 1 / 2 r ( i ± i ) n even n odd Since ^x _ t j - l 1 ^ / ^x i-l = er1+e (C.8) (C.9) ( C I O ) ( C . l l ) mul t ip ly ing E q . ( C . l l ) by 7r~2 exp (—£2) and integrating over al l £ x gives the recurrence relation = Aj-.1+t(Aj-2 + t(Aj-3+ £(...))) (C.12) Appendix C. ST Collision Matrix Elements 182 Present Work Sirovich & Thurber E -T2 e — ik ,n'V Ifvj zn -i2^inn t The truncation constant is normalized to 1 in S&T notation while their choice of reduced velocity £1 results in a factor -^ =. Table C l : Some notation used in the present work and equivalent notation in Sirovich and Thurber [128]. which when evaluated gives L 0 - i ) / 2 j _ ^ ( ( > « + E r l ~ 2 k A 2 k , (c.i3) fc=0 where Z(£) = Zo(£) is the well-known Plasma Dispersion function [129]. The integral Rni,n'i' is equivalent to the result obtained by Sirovich and Thurber [128], Appendix B. Table C l gives notational equivalents between this and their work. Appendix D The linearized B G K Model The B G K approximation to the Boltzmann equation is % + c-Vf = K[f] (D.l) where v is the collision frequency, and the gain part of the collision term is approximated by a local Maxwellian / L M - The nonequilibrium velocity distribution function, / is writ-ten in terms of the equilibrium velocity distribution function /(°? which is perturbed by a small term h f = f{0)(l + h) (D.2) The constitutive relations for density, n, momentum, mc, and translational energy, 3kBT/2 give n = nW(l + h) = J fW(l + h)dc (D.3) c = C = J f(0){l + h)cdc (c ( 0 ) = 0 stationary fluid) (D.4) T = T(°)( l + r) = Jf^(l + h)^fdc (D.5) where n(°V = / f(0)hdc (D.6) C = J f{0)hcdc (D.7) 183 Appendix D. The linearized BGK Model 184 The local Maxwellian distribution function is written in terms of the perturbed mass, momentum and energy / ( m V/2 r ( c ~ Q 2 i (D.9) and substituting in Eqs. (D.3), (D.4) and (D.5) into Eq. (D.9) gives /LM in terms of the equilibrium properties and T^°\ f = n(l + </>) m \ 3 / 2 2irkBn°)) V I 1 3/2 exp[ mc exp m ( 2 exp[ i l + T TJ | - 2 A ; B T ( ° ) J ^ L 2kBTW1^lkBTW A linearized form of Eq. (D.10) is obtained for for small values of </>, ( and r by substi tuting the identities 1 (D. ; i+r) 1 1 3 15 2 = 1 T H T (l + r)3/2 2 4 2 into Eq. (D.10) giving m c exp[ mc2T 2kBT(°)] ^ L 2/ t B T(°) -x e x p [ - m C 2 ( 1 - r ) l e x p [ C - C ( 1 - T ) -X 6 X P L 2kBTW J 6 X P [ A;BT(°) -= / ( o ) ( i + 0) ( I - | T ) (l + mc2T 1 + (D.ll) (D.12) (D.13) (D.H) 2A; B T(°); ^ ' A;BT(°) where exponentials in in r and £ of Eq. (D.13) are expanded in a Taylor series and quadratic and higher-order terms are dropped. Expanding Eq. (D.14) and neglecting quadratic and higher-order terms in </>, r and ( gives / L M ^ / ( 0 ) m c 2 T mc • C 3 0 + ^ T W " 2 T + 2kBT^ (D.15) The B G K model collision operator is obtained from Eqs. (D.2) and (D.15) for K[h] " ( / L M - / ) K[h] mc 2fcBT(°) 2 (D.16) Appendix D. The linearized BGK Model 185 - / I + G> + 2 £ . C + T ( > - £ ) which is essentially Eq. (4.3.95).
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Far-from-equilibrium gas kinetic theory : reactive systems and sound propagation Napier, Duncan George 1997
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Title | Far-from-equilibrium gas kinetic theory : reactive systems and sound propagation |
Creator |
Napier, Duncan George |
Date Issued | 1997 |
Description | There exist many physical situations that involve large departures from thermodynamic equilibrium. These systems may be spatially nonuniform and/or time dependent. Traditional kinetic theory involves a perturbation type of approach that implies small departures from equilibrium. The present thesis is devoted to the development of theoretical methods valid for systems far removed from equilibrium. The techniques are applied to a study of the departure from equilibrium in reactive systems, and for the propagation of acoustic waves in gases. The rates of gas phase reactions can be calculated from averages of the appropriate reactive cross sections over the distribution of velocities of the reacting species. Reactive processes, especially for reactions with activation energy, tend to remove translationally energetic species and the velocity distribution is perturbed from a Maxwellian. The extent of the departure from a Maxwellian can be estimated from solutions of the Boltzmann equation. If there is a good separation of elastic and reactive time scales, steady solutions of the Boltzmann equation can be obtained with a procedure analogous to the Chapman-Enskog (CE) method. Nonequilibrium effects for model reactive systems with and without reverse reactions and in the presence and absence of products are studied. The range of validity of the CE method is studied by comparing results from a CE method and an explicitly time-dependent solution method. The CE approach assumes a weak perturbation and is referred to as a Weak Non-Equilibrium (WNE) approach. A Strong Non-Equilibrium (SNE) approach that treats the distribution functions of each of the components as Maxwellians at different temperatures is applied to strongly perturbed systems in which the reaction causes the temperatures of each species to differ from the system temperature. A third approach is a modification of the SNE approach and is referred to as Modified Strong Non-Equilibrium (MSNE). All three methods are compared with the results of an explicitly time-dependent solution. The CE method was found to be valid only when the ratio of elastic to reactive collision frequencies is greater than 105. The propagation of acoustic waves through a gas perturbs the velocity distribution function of the gas. In a collision-dominated gas, the velocity distribution function can be approximated by a Maxwellian and the phase velocity and attenuation of the sound wave can be determined from a dispersion relation derived from the Navier-Stokes equations of fluid dynamics. In the rarefied region, hydrodynamics is no longer valid and kinetic theory methods must be used. The method of solution of the Boltzmann equation by Wang-Chang and Uhlenbeck (WCU) fails to describe the behaviour of sound waves as the frequency of oscillation approaches that of the interparticle collisions in the gas. A generalized Boltzmann equation (GBE) introduced by Alexeev is applied to the sound problem and the results are compared with those of the WCU method and experiment. The kinetic theory description for sound propagation in a simple gas introduced by Sirovich and Thurber (ST) is applied to binary mixtures of gases and the results compared with those of experiment. It is shown that the ST method provides better agreement with experiment than the WCU method. The results with the GBE do not converge. |
Extent | 7995086 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0061687 |
URI | http://hdl.handle.net/2429/7436 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1997-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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