THEORY OF THE LATTICE BOLTZMANN METHOD FOR MULTI-PHASE AND MULTICOMPONENT FLUIDS

THEORY OF THE LATTICE BOLTZMANN METHOD FOR MULTI-PHASE AND MULTICOMPONENT FLUIDS A Thess Submtted to the Graduate Faculty of the North Dakota State Unversty of Agrculture and Appled Scence By Qun L In Partal Fulflment of the Requrements for the Degree of MASTER OF SCIENCE Major Department: Physcs December 2006 Fargo, North Dakota

ABSTRACT L, Qun, M.S., Physcs, College of Scence and Mathematcs, North Dakota State Unversty, December 2006. Theory of the Lattce Boltzmann Method for Mult-Phase and Multcomponent Fluds. Major Professor: Dr. Alexander Wagner. Although the lattce Boltzmann method has been employed to smulate the dynamcs of some multcomponent systems, a general lattce Boltzmann algorthm that smulates the dynamcs of an arbtrary multcomponent system wth explct thermodynamc consstency s stll needed. In ths thess, I developed a lattce Boltzmann algorthm from a free energy approach that smulated the dynamcs of a system wth an arbtrary number of components. The thermodynamc propertes, such as the chemcal potental of each component and the pressure of the overall system, were ncorporated n the model. I derved a symmetrcal convecton dffuson equaton whch was of the same form for each component. The Naver Stokes equaton and contnuty equaton for the overall system as well as a convecton dffuson equaton for each component were recovered. The algorthm was verfed through smulatons of bnary and ternary systems n one-dmenson. The equlbrum concentratons of components of bnary and ternary systems smulated wth my algorthm agreed well wth the concentratons obtaned by mnmzng the free energy. I also studed Gallean nvarance volatons n lattce Boltzmann methods. The sources of the Gallean nvarance volaton were analyzed from a unfed perspectve

for both pressure and forcng methods. In ths thess, a measure of Gallean nvarance and correctons reducng Gallean nvarance volatons are presented. I valdated my analyss and compared the results of dfferent correctons wth smulatons n one- and two-phase systems. v

ACKNOWLEDGEMENTS I would lke to acknowledge my advser, Dr. Alexander Wagner, for hs valuable advce and gudance throughout ths thess. Sncere thanks are due as well to Dr. Alan R. Denton, Dr. Thomas Ihle, and Dr. Iskander Akhatov, members of my graduate faculty commttee, for ther thoughtful readng of ths thess. I would express my apprecaton to Dr. Rchard T. Hammod, who frst supported me to study n the Physcs Department at the North Dakota State Unversty, to Dr. Danel Kroll, Dr. Charles Sawck, Dr. Crag Rottman, and Dr. Terry Pllng for ther help. Thanks also to Ben Lu, Erc Foard, Goetz Kaehler, Stephan Loew, the students sharng the same offce wth me, for all the help n day to day lfe. v

TABLE OF CONTENTS ABSTRACT..................................... ACKNOWLEDGMENTS.............................. v LIST OF TABLES.................................. LIST OF FIGURES................................. x x CHAPTER 1. INTRODUCTION.......................... 1 CHAPTER 2. FUNDAMENTAL HYDRODYNAMIC EQUATIONS AND THERMODYNAMICS OF PHASE SEPARATION.............. 6 2.1. Hydrodynamc Equatons.......................... 6 2.2. Dffuson Processes............................. 7 2.3. Basc Phase Separaton Theory....................... 10 2.4. Phase Separaton of a System Descrbed by a Landau Free Energy... 15 2.5. Phase Separaton of a System Descrbed by the Flory-Huggns Free Energy 20 2.5.1. Phase separaton of a bnary system................ 21 2.5.2. Phase separaton of a ternary system............... 23 CHAPTER 3. LATTICE BOLTZMANN METHOD................ 27 3.1. Sngle Relaxaton Tme Lattce Boltzmann................ 27 3.1.1. One-dmensonal three-velocty lattce Boltzmann model..... 36 3.1.2. Lattce Boltzmann for a lqud-gas system............. 38 3.2. Lattce Boltzmann for Multcomponent Systems............. 42 v

3.3. Smulatons of Multcomponent Systems.................. 52 3.3.1. Smulatons of bnary systems................... 52 3.3.2. Smulatons of ternary systems................... 56 CHAPTER 4. GALILEAN INVARIANCE OF THE LATTICE BOLTZMANN METHOD..................................... 60 4.1. Lattce Boltzmann Gallean Invarance Volatons and Correctons... 60 4.2. Gallean Invarance Correctons for Sound Waves............. 62 4.3. Test of Correcton n Lqud-Gas Phase Separaton............ 66 REFERENCES.................................... 70 APPENDIX A. FLORY-HUGGINS MODEL OF POLYMER SOLUTION.... 75 APPENDIX B. GALILEAN INVARIANCE IN CONTINUUM.......... 80 APPENDIX C. ANALYTICAL SOLUTION OF ONE-DIMENSIONAL SOUND WAVE....................................... 83 C.1. Sound Wave Equaton Wthout Vscosty................. 83 C.2. Sound Wave Equaton wth Vscosty................... 85 APPENDIX D. THE DERIVATION OF PRESSURE TENSOR......... 89 APPENDIX E. GALILEAN INVARIANCE EXAMINATIONS.......... 91 APPENDIX F. CODE FOR THEORETICAL SOLUTION OF A BINARY SYSTEM..................................... 95 APPENDIX G. CODE FOR THEORETICAL SOLUTION OF A TERNARY SYSTEM..................................... 98 APPENDIX H. CODE FOR LIQUID-GAS SYSTEM (HOLDRYCHQ)..... 105 APPENDIX I. CODE FOR LIQUID-GAS SYSTEM (FORCINGQ)....... 109 v

APPENDIX J. CODE FOR BINARY SYSTEM SIMULATION......... 113 APPENDIX K. CODE FOR TERNARY SYSTEM SIMULATION........ 118 v

LIST OF TABLES Table Page 3.1 The algorthms for the LB smulaton of a non-deal flud........ 35 E.1 Velocty product summaton(d2q7).................... 91 E.2 Velocty product summaton(d2q9).................... 92 x

LIST OF FIGURES Fgure Page 2.1 The phase separaton of a system and the shape of ts bulk free energy densty functon............................... 12 2.2 The local stablty of a system and the shape of the free energy densty functon................................... 15 2.3 The bnodal and spnodal ponts and the shape of the free energy densty functon of a system............................. 16 2.4 The domans of phase separaton wth the spnodal and bnodal lnes.. 17 2.5 The theoretcal bnodal and spnodal lnes for a monomer bnary system 23 2.6 Bnodal lnes of a polymer system represented n a Cartesan and a trangular coordnate............................ 26 3.1 Comparson of bnodal ponts obtaned by Holdych and ForcngQ to the theoretcal values.............................. 40 3.2 The comparson of thermodynamcs obtaned by HoldychQ to the theoretcal values n a lqud-gas system.................. 41 3.3 The comparson of thermodynamcs obtaned by ForcngQ to the theoretcal values n a lqud-gas system.................. 42 3.4 The comparson of the bnodal ponts obtaned by smulaton to the theoretcal solutons n a bnary system.................. 54 3.5 The comparson of thermodynamcs obtaned by smulaton to the theoretcal values n a bnary system.................... 55 3.6 The comparson of bnodal ponts obtaned by smulaton to the theoretcal values n a ternary system................... 58 3.7 The comparson of thermodynamcs obtaned by smulaton to the theoretcal values n a ternary system................... 59 4.1 The comparson of the Q correcton to the theoretcal value of a sound wave..................................... 64 4.2 The comparson of error of relatve error of a sound wave smulaton wth and wthout Q correcton.......................... 65 x

4.3 Comparson of the relatve errors of a sound wave obtaned by smulaton wth and wthout Q correcton vs. u 0................... 66 4.4 Gallean nvarance errors E(u 0 ) of a lqud-gas system.......... 67 x

CHAPTER 1 INTRODUCTION The lattce Boltzmann (LB) method s a mesoscopc lattce smulaton method whch has been appled frutfully to many research areas such as turbulence [15, 63], mesoscale blood flow [16], nterfacal waves [8], magneto-hydrodynamcs [12], and multcomponent systems [51, 65]. Hstorcally, LB was derved from the lattce-gas automata (LGA) [17, 18, 60] n the 1990s, although the two methods are ndependent [25, 26, 41, 42]. The LGA traces partcle movements on a lattce, and can recover the Naver-Stokes equatons [18], thus smulatng hydrodynamcs. LGA s uncondtonally stable and are very good to smulate mcro-flow wth large ntrnsc fluctuatons. However, LGA exhbts strong Gallean nvarance (GI) volatons, and they are lmted to small Reynolds numbers [35]. To overcome these defcences, LB was developed [13, 43]. Instead of tracng the movement of partcles, LB traces the evoluton of a densty dstrbuton functon, whch depends on poston and velocty. The velocty s dscretzed such that, n one tme step, the denstes move to the neghborng lattce stes to whch ther assocated veloctes pont. Ths movement s called streamng. Between streamng steps, collsons occur at lattce stes and change the densty dstrbuton functon. Qan et al. ntroduced Bhatnagar, Gross and Krook (BGK) s [5] sngle relaxaton tme approxmaton to smplfy the descrpton of the collson [41]. The 1

system evolves by means of one streamng and one collson per tme step. The macroscopc physcal quanttes mass and momentum are gven by the velocty moments of the densty dstrbuton functon. Because the standard BGK model descrbes the collson of deal gases, the standard LB algorthm can only smulate deal gas dynamcs. To smulate non-deal fluds, the attractve or repulsve nteracton among molecules, whch s referred to as the non-deal nteracton, should be ncluded n the LB model. There are two approaches to ncorporate non-deal nteractons. One s to mmc mcroscopc nteracton forces drectly, whch s often referred to as Shan-Chen s approach [20, 44, 45, 46]; the other s to derve the non-deal nteracton from the total free energy of the system and then ncorporate t a posteror [2, 38, 48, 51], whch s often referred to as the free energy approach. By usng the free energy approach, the chemcal potental of each component s utlzed explctly n the smulaton, but n Shan-Chen s approach, t s not. Therefore, a free energy approach for multcomponent smulatons s better suted to study the thermodynamcs of a multcomponent system. In ths thess, I study the free energy approach only. Free energy approaches ntroduce the non-deal nteracton nto LB ether through a pressure term [38, 48, 49] or a forcng term [44, 53, 55, 57]. Both approaches are equvalent as far as the recovery of the hydrodynamc equatons s concerned. They gve very smlar smulaton results but show a slght dfference n the stablty of the algorthms. The forcng approach, however, leads to non-neglgble hgher order terms for systems wth large densty gradents [55]. The analyss of the effect of these hgher order terms s beyond the scope of ths thess. The LB algorthm has been extended to smulate the thermodynamcs and 2

hydrodynamcs of a multcomponent system. The challenge for the LB smulaton of a multcomponent system les n the fact that momentum conservaton s only vald for the overall system but not for each component separately, and therefore dffuson occurs between the components. In addton, for a vald smulaton scheme, any representaton of the three components should gve the same smulaton results. That s, the scheme should be symmetrc. For a bnary system of components A and B wth denstes ρ A and ρ B, the smulaton usually traces the evoluton of the total densty ρ A +ρ B and the densty dfference ρ A ρ B [51]. Although ths scheme s successful n the smulaton of a bnary system [38, 56], ts generalzaton for the LB smulatons of systems wth an arbtrary number of components s asymmetrc. For nstance, to smulate a ternary system of components A, B, and C wth denstes ρ A, ρ B and ρ C, the total densty of the system, ρ A + ρ B + ρ C, should be traced, and the other two denstes to be traced may be chosen as, e.g., ρ B and ρ A ρ C [29]. Ths approach s lkely to be asymmetrc because the three components are treated dfferently as s the case of Lamura s model [29]. If an LB method s not symmetrc, t wll lose generalty although t may stll be adequate for certan applcatons. In ths thess, I establshed a manfestly symmetrc scheme whch s generally vald for a smulaton of a system of an arbtrary number of components. Several papers consderng LB smulatons of multcomponent systems exst n the lterature. Hudong Chen et al. successfully smulated the amphphlc flud wth the Shan-Chen s approach, by treatng the surfactant molecules as dpoles [11]. X. Shan et al. [45] establshed a multcomponent LB model that gave the convecton dffuson equaton for each component. Whle Shan-Chen s approach s manfestly symmetrc, 3

t has the dsadvantage that thermodynamc propertes of the system are not easly accessble. Lamura et al. [29] successfully smulated the amphphlc ternary system wth a free energy approach, but they chose the densty parameters n an asymmetrc way. Some researchers have dscussed LB smulatons of the dffuson process, whch s an ntrnsc character of the dynamcs of the multcomponent system, but they dd not gve any ready recpe for the LB smulaton of an arbtrary multcomponent system. For example, B.Deng et al. [14] presented an LB method to smulate the convecton dffuson equaton wth a source term for a one-component system, but they have not dscussed the smulaton of the convecton dffuson process for a multcomponent system. A. Akthakul et al. [1] developed a multcomponent free energy LB model by usng Enskog Chapman expanson and appled t to the study of the convecton dffuson process of a polymer ternary system. The convecton dffuson equaton they derved cannot guarantee that the chemcal potental of each component would be strctly constant at equlbrum. My LB model to smulate the dynamcs of an arbtrary multcomponent system s a varant of the free energy approach, and the chemcal potental of each component s ncorporated explctly. Therefore, the thermodynamc quanttes n equlbrum (that s, the overall pressure of the system and the chemcal potental of each component) should be constant macroscopcally. In ths thess, a symmetrc form of the convecton and dffuson equaton for each component has been derved whch guarantees that the pressure and chemcal potental wll be constant n equlbrum. To test the valdty of the model, I smulated phase separaton n bnary and ternary systems and checked that the systems reached a constant overall pressure and a constant chemcal potental 4

for each component n equlbrum. The theoretcal value of the volume fracton of each component n equlbrum can be obtaned by numercally determnng the lowest free energy of the system (Appendx G). The smulaton results showed that the volume fracton of each component n equlbrum n LB agreed well wth the value predcted by mnmzng the free energy. Therefore, my model has been valdated, at least n equlbrum. However, n my LB model each component has the same vscosty and moblty. Therefore, an extenson of the model to concentraton dependent vscosty and moblty s stll needed. I also studed the Gallean nvarance (GI) of the LB algorthm. Although every vald physcal model should be Gallean nvarant, the LB models, ether for deal gases or for non-deal fluds, are not perfectly Gallean nvarant. In some applcatons, the GI volaton errors of LB are neglgble, yet n others they cause problems. The source of GI volaton n LB for deal gas models has been analyzed by Y. Qan et al. [40]. The sources of GI volaton n LB for free energy based non-deal flud LB models have been analyzed, and correctons have been suggested by several groups [21, 22, 31, 57]. In ths thess, the sources of GI volaton for an deal gas and a non-deal flud are analyzed from a unfed perspectve. A formula was presented that can check whether or not an LB model s Gallean nvarant and can dentfy, to the lowest order, the terms that cause the GI volaton. LB smulatons of sound waves and phase separaton are performed n one dmenson to valdate my analyss and to compare dfferent correcton methods. The content of the second part of ths thess has been publshed [57]. 5

CHAPTER 2 FUNDAMENTAL HYDRODYNAMIC EQUATIONS AND THERMODYNAMICS OF PHASE SEPARATION 2.1. Hydrodynamc Equatons Three fundamental equatons govern the hydrodynamc processes n a flud [30, 28]. The contnuty equaton descrbes the conservaton of mass: t ρ + J = 0, (2.1) where t s tme, J s the mass flux whch s defned as J ρu, ρ s the mass densty of the flud, and u s the macroscopc velocty of the flud. The Naver-Stokes equaton descrbes the conservaton of momentum: t (ρu α ) + β (ρu α u β ) = β P αβ + β σ αβ + ρf α, (2.2) where σ αβ s the stress tensor, F α s the component α of an external force on a unt mass n a unt volume, and the Ensten summaton conventon s used. For Newtonan fluds, the stress tensor s gven by [39] ( σ αβ = η β u α + α u β d ) 2 δ αβ u + µ B δ αβ u, (2.3) 6

where η s the shear vscosty, and µ B s bulk vscosty; P αβ s the pressure tensor; and d s the spacal dmenson of the system. The energy equaton (commonly called the heat equaton) descrbes energy conservaton [28]: ρ t e + ρu α α e = α (k α T) P αβ ( β u α ) + σ αβ ( β u α ), (2.4) where e s the nternal energy per unt mass of the flud and k s the thermal conductvty. For a perfect gas e = C V T, where C V s the specfc heat at constant volume and T s the temperature. However, n ths thess, I only studed sothermal processes durng whch the system can freely exchange heat energy wth an external envronment. The heat transfer process was replaced wth a constant temperature condton n the smulatons. Therefore, the smulatons satsfy Eq. (2.1) and Eq. (2.2) but do not satsfy Eq. (2.4). For a multcomponent system, besdes the overall macroscopc flud flow, dffuson also contrbutes to the mass transport process. The next secton s devoted to the dscusson of the dffuson process n a multcomponent system. 2.2. Dffuson Processes In multcomponent systems, there are two mechansms for mass transport: convecton and dffuson. Convecton s the flow of the overall flud, whle dffuson occurs where the average veloctes of components are dfferent. The velocty of the overall flud s a macroscopc quantty because t s conserved, but the average veloctes of the components are not. The macroscopc velocty of the flud u can be expressed 7

n terms of the densty ρ σ and velocty u σ of each component n the form of u σ ρσ u σ σ ρσ. (2.5) Wth the notaton u σ u σ u, (2.6) the flux of each component can be dvded nto a convecton part and a dffuson part. J σ ρ σ u σ = ρ σ (u + u σ ) = J σc + J σd, (2.7) where J σc s the convecton part and J σd s the dffuson part. Because mass conservaton stll holds for each component, the contnuty equaton for each component s vald: t ρ σ + J σ = 0. (2.8) Substtutng Eq. (2.7) nto Eq. (2.8), the convecton dffuson equaton for a component can be obtaned. t ρ σ + J σc = J σd. (2.9) From Eqs. (2.5) and (2.6), I see that J σd = 0, (2.10) σ 8

whch s a constrant for all dffuson fluxes of the components. I wll elaborate further on the dffuson flux of each component. Eq. (2.10) ndcates that dffuson s a mxng process that has no contrbuton to the convecton of the flud, so when two components are mxng wth each other, the partcles of the two knds exchange ther postons. Therefore, the dffuson process between two components s related to the dfference of the chemcal potental of the two components, whch s also called the exchange chemcal potental [23]. Recognzng that the gradent of the exchange chemcal potental determnes the dffuson processes, I obtan a frst order approxmaton for the dffuson flux of one component nto all other components as J σd = σ M σσ (µ σ µ σ ), (2.11) where σ and σ enumerate the components; µ σ and µ σ are the chemcal potentals of components σ and σ ; and M σσ s a symmetrc postve defnte moblty tensor. A smple model for the dffuson process assumes that a dffuson flux between two components s proportonal to the overall densty and the concentraton of each component. Then M σσ can be expressed as M σσ = k σσ ρ ρσ ρ ρ σ ρ = kσσ ρ σ ρ σ ρ, (2.12) where k σσ s the constant dffuson coeffcent between components σ and σ. It depends on components but s ndependent of the total denstes and concentraton of each 9

component. Substtutng Eq. (2.12) nto Eq. (2.11), J σd = k σσ ρσ ρ σ ρ (µσ µ σ ). (2.13) σ Substtutng Eq. (2.13) nto Eq. (2.9), the general form of a convecton dffuson equaton s obtaned as t ρ σ + (ρ σ u) = ( σ ρσρσ σσ k ρ (µσ µ σ ) ). (2.14) 2.3. Basc Phase Separaton Theory Free energy, chemcal potental, and pressure are key thermodynamc concepts to understand the phase behavor of a system. The total free energy of a multcomponent system s composed of the bulk part and the nterfacal part. In ts smplest form t s known as a Landau free energy and can be expressed as F = ( ) s s dr ψ(n 1,n 2,,n s ) + κ σσ n σ n σ, (2.15) σ=1 σ =σ where ψ s the bulk free energy densty, σ and σ are the superscrpts for components; and n 1, n 2,, n s are the number denstes of s dfferent components; κ σσ s an nterfacal free energy parameter whch s responsble for the ntroducton of the surface tenson. In ths expresson, the contrbuton from the hgher order terms of the densty gradents s neglected [32]. The chemcal potental of each component can be obtaned by a functonal dervatve 10

as µ σ = δf δnσ, (2.16) where µ σ s the chemcal potental of component σ; n σ s the number densty of component σ; and F s the total free energy of the system. The pressure n a bulk phase n equlbrum s gven by p = σ n σ µ σ ψ. (2.17) The pressure tensor s determned by two constrants: P αβ = pδ αβ n the bulk and P αβ = σ nσ µ σ everywhere. A more detaled analyss s shown n Appendx D. The total free energy of a system s at ts mnmum n equlbrum. Ths requres that the chemcal potental for each component s constant, and the pressure tensor s dvergence free. A system may separate nto two phases to mnmze the total free energy of the system. Fgure 2.1 llustrates the phase separaton mechansm of a one component system. The total free energy of a one-component system, from Eq. (2.15), s gven by F = dr (ψ(n,θ) + κ ) 2 ( n)2, (2.18) where κ s a coeffcent related to the surface tenson of the nterface of the two phases after phase phase separaton. The chemcal potental of a one-component system, usng Eq. (2.16), s gven by µ = ψ n κ 2 n. (2.19) 11

ψ ψ A n 1 n n 2 A n ψ B p B Fgure 2.1. The phase separaton of a system s determned by the shape of ts bulk free energy densty functon. An ntal homogeneous system of densty n and bulk free energy densty ψ A separates nto two phases of densty n A and n B wth the average bulk free energy densty ψ B to mnmze the total free energy of the system. The pressure of the system n equlbrum s p. The bulk pressure tensor of a one-component system n equlbrum, usng Eq. (2.17), s gven by p = nµ ψ. (2.20) Eq. (2.19) ndcates that n equlbrum, for a bulk phase, the chemcal potental s the slope of the tangent lne of the free energy densty curve. Eq. (2.20) ndcates that n the bulk phase, the negatve of the pressure s the ntercepton on the free energy densty axs by the tangent lne of the free energy densty curve as shown n Fgure 2.1. Slght devatons occurs where curved nterfaces are present [59]. If the free energy densty functon of a system takes the shape shown n Fgure 2.1, 12

a system of average densty n wll separate nto two phases of densty n 1 and n 2 n equlbrum. Assume that the system has a volume V and the number densty n n a homogeneous state. Then t separates nto two states, one of volume V 1 wth number densty n 1 and the other of volume V 2 wth densty n 2. Because the system has a constant volume and constant total number of partcles, I have the constrants: V 1 + V 2 = V, (2.21) n 1 V 1 + n 2 V 2 = nv. (2.22) From Eq. (2.21) and Eq. (2.22) the volume can be obtaned as V 2 = n n 1 n 2 n 1 V, (2.23) V 1 = n 2 n n 2 n 1 V. (2.24) The total free energy of the system n a homogeneous state s gven by F homogeneous ψ(n A )V, (2.25) where ψ A s the ψ value at pont A n Fgure 2.1. Usng Eqs. (2.23) and (2.24), the 13

total free energy of the system n a two-phase state can be obtaned as F separated = ψ 1 V 1 + ψ 2 V 2 = ψ 1 n 2 n n 2 n 1 V + ψ 2 n n 1 n 2 n 1 V = ψ 2(n n 1 ) + ψ 1 (n 2 n) V n 2 n 1 ψ B V, (2.26) where ψ B s the ψ value at pont B n Fgure 2.1. Snce ψ B s on the tangent lne, t represents the mnmum average bulk free energy densty of a system of an average number densty of n. The lnear stablty of a homogeneous phase s determned by the shape of the bulk free energy densty functon. Fgure 2.2 (a) shows that a locally convex shape of the free energy curve mples stablty wth respect to small perturbatons because the total free energy of the system would ncrease f the system phase separated. A phase at pont A n Fgure 2.2 (b) s unstable to nfntesmal perturbaton because the total free energy of the system wll decrease n the case of phase separaton. Generally, a system wll undergo phase separaton f the shape of ts bulk free energy densty functon contans a concave secton as shown n Fgure 2.3. The system wll phase-separate nto phases C and D, and CD s a straght lne tangent to the free energy lne ψ(x) at ponts C and D. Ponts C and D are called bnodal ponts, at whch the system has mnmum total free energy. C and D are ponts that separate locally unstable regons and locally stable regons. Between C and D, the system s unstable and wll separate nto two phases wth only nfntesmal perturbaton. Between C and C and between D and D, the system s metastable and wll undergo 14

ψ ψ A ψ A ψ B B ψ B B ψ A A n n (a) Stable (b) Unstable Fgure 2.2. The local stablty of a system s determned by the concave or convex of the free energy densty functon. The free energy densty of a homogeneous system s ψ A. The average free energy densty of the system wth two separated phase s ψ B. (a) A system s stable to a small perturbaton when ts free energy densty functon s convex locally. (b) A system s unstable to a small perturbaton when ts free energy densty functon s concave locally. phase separaton only wth fnte perturbatons. Ponts C and D are called spnodal ponts. Mathematcally, they are determned by the condton that ψ (x C ) and ψ (x D ) are zero. A typcal phase dagram s shown n Fgure 2.4. In doman I, the system s uncondtonally unstable and separates nto two phases by spnodal decomposton. In doman II, the system s metastable and phase separates by nucleaton. In doman III, the system s stable, and the homogeneous state has the lowest free energy. 2.4. Phase Separaton of a System Descrbed by a Landau Free Energy The Landau free energy for a flud wth a lqud-gas transton has a form gven by Eq. (2.18). To study the phase behavor near the crtcal pont, t s convenent to 15

ψ D D C C Fgure 2.3. The bnodal and spnodal ponts are determned by the shape of the free energy densty functon of a system. The bnodal ponts C and D correspond to two equlbrum phases after phase separaton. The spnodal ponts C and D separate the stable and unstable regon. n express the free energy densty as [7] ψ(t,n) = W(n,T) + nµ b p b, (2.27) where W(n,T) s the excess free energy densty; µ b n ψ n=nb s the bulk chemcal potental; and p b s the bulk pressure. I chose the smplest excess free energy functon W(ν,τ) = p c (ν 2 βτ) 2. (2.28) Here ν = (n n c )/n c s the reduced densty; τ (T c T)/T c s the reduced temperature; and T c, p c,n c are the crtcal temperature, crtcal pressure, and crtcal densty, 16

T III stable doman n one phase T c crtcal temperature spnodal lne bnodal lne phase A phase B III II nucleaton I spnodal decomposton to phase A + phase B Fgure 2.4. Insde the bnodal lnes the system tends to phase separate nto two phases correspondng to the denstes gven by the two branches of the bnodal lne. Insde the spnodal lnes the system s uncondtonally unstable and a homogeneous system wll phase separate through spnodal decomposton. In the regon between the bnodal and spnodal lnes a homogeneous system wll phase separate through nucleaton. II n III respectvely. The bulk chemcal potental s gven by µ b = 4p c n c (1 βτ), (2.29) where β s a constant. The bulk pressure s p b = p c (1 βτ) 2. (2.30) 17

Therefore, ψ = p c (ν + 1) 2 (ν 2 2ν + 3 2βτ). (2.31) When two phases coexst n equlbrum, the chemcal potental wthout the surface tenson terms s gven by ψ n = 4p c n c (ν + 1)(ν 2 ν + 1 βτ). (2.32) The equlbrum gas and lqud denstes are gven by ν = ± βτ. Introducng θ βτ, the densty of lqud and gas are n l = n c (1 + θ), n g = n c (1 θ). (2.33) To solve the densty profle n(x) of the nterface of the system at the equlbrum state, I use two constrants. One s that the free energy of the system must be mnmal. The other s the mass conservaton of the system. Usng the varaton method wth subsdary constrants[19], I get F n d dx F n x + λ(n n d dx n n x ) = 0, (2.34) where F = ψ(t,τ) + κ 2 ( αn) 2 ; λ s a Lagrange multpler; and the subscrpts sgnfy dervatves. For example, F nx sgnfes the dervatve of functon F wth respect to n x, 18

and n x sgnfes the dervatve of functon n wth respect to x. Consequently, 4p c (ν + 1)(ν 2 ν + 1 βτ) κ d2 n + λ = 0, (2.35) n c dx2 where the Lagrange multpler λ s 4p c (1 βτ)/n c. The nterfacal profle can be obtaned by solvng the dfferental equaton (2.35). The physcally correct soluton for a sngle nterface s n(x) = n c [1 + βτ tanh( x 2ξ )], (2.36) where the nterface wdth s κn ξ = 2 c. (2.37) 4βτp c The surface tenson [7] s σ = 4 3 2κpc (βτ) 3/2 n c. (2.38) The pressure tensor s (Appendx D), P αβ = [p o κn 2 n κ 2 ( γn) 2 ]δ αβ + κ( α n)( β n), (2.39) where p o = n n ψ ψ. For my one dmenson smulatons, the pressure tensor reduces to p(x) = p c (ν + 1) 2 (3ν 2 2ν + 1 2βτ) κn 2 xn + κ 2 ( xn) 2. (2.40) 19

The chemcal potental wth the surface tenson term s µ = ψ n = 4p c n c (ν + 1)(ν 2 ν + 1 βτ) κ 2 n. (2.41) 2.5. Phase Separaton of a System Descrbed by the Flory- Huggns Free Energy The Flory-Huggns model s often employed to calculate the free energy of a polymer soluton. It assumes that each segment of a polymer, or mer, occupes one lattce pont. As shown n Appendx A, the Flory-Huggns free energy s gven by F = s s s s θn σ m σ + (n σ θ ln φ σ ) + (χ σσ θm σ n σ φ σ ) s + σ=1 σ =1 σ=1 σ=1 σ=1 σ =σ+1 s (κ σσ n σ n σ ) dv, (2.42) where m σ s the polymerzaton of the component σ, whch s the average number of mers per polymer; n σ s the number densty of component σ; and φ σ s the volume fracton of component σ. It s defned as φ σ = mσ n σ σ (mσ n σ ) = ρσ ρ, (2.43) where ρ σ s the mer densty of component σ and ρ s the mer densty of the system, whch s a constant n the Flory-Huggns model. 20

2.5.1. Phase separaton of a bnary system The equlbrum state of a bnary system can be obtaned by mnmzng the free energy of the system. Derved from Eq. (2.42), the Flory-Huggns free energy densty of a bnary polymer system wthout the nterfacal free energy contrbuton s gven by ) ψ = θ ( ρ A ρ B + ρa m ln A φa + ρb m ln B φb + χρ A φ B. (2.44) In many applcatons, a soluton contans only one polymer component. So t s reasonable to choose m A = m, where m s the polymerzaton of component A and m B = 1. In the Flory-Huggns model, the lattce densty s constant; therefore ρ ρ A + ρ B s a constant. 1 For the calculaton of bnodal ponts, I can assume ρθ = 1 and neglect an arbtrary constant wthout affectng the fnal results. Wth the constrant φ A + φ B = 1, Eq. (2.44) has only one ndependent varable φ φ A. Then Eq. (2.44) can be smplfed to ψ = 1 φ ln φ + (1 φ) ln(1 φ) + χφ(1 φ). (2.45) m From Eq. (2.45), the whole phase dagram of a bnary system can be establshed. Frst, let us determne the crtcal pont. Because the crtcal pont s on the spnodal lne, t satsfes the spnodal lne equaton d 2 F/dφ 2 = 0. Because the two spnodal lnes meet 1 Ths s not an explct constrant n a LB smulaton. However, n a LB smulaton, when an equlbrum state has been reached, the total densty s very close to a constant. 21

at the crtcal pont, d 3 F/dφ 3 = 0. F = 1 1 m φ + 1 2χ = 0, 1 φ (2.46) = 1 1 m φ + 1 = 0. 2 (1 φ) 2 (2.47) F Solvng Eq. (2.46) and Eq. (2.47), the coordnates of the crtcal pont s obtaned as φ = 1 m + 1, χ = m + 2 m + 1. (2.48) 2m Eq. (2.46) can be rearranged as 2mχφ 2 + (1 m 2mχ)φ + m = 0. (2.49) Eq. (2.49) yelds the spnodal lnes wth the equaton φ 1,2 = (1 m 2mχ) ± sp, (2.50) 4mχ where sp = (1 m 2mχ) 2 8m 2 χ. The theoretcal bnodal lnes, whch correspond to the two equlbrum states of the system, can be obtaned numercally by mnmzng the total free energy of the system, and the numercal method s presented n Appendx F. The spnodal and bnodal lnes are shown n Fgure 2.5 for dfferent polymerzatons. 22

1 0.8 0.6 bnodal, m A =1 φ B spnodal, m A =1 spnodal, m 0.4 A =10 bnodal, m A =10 0.2 0 0 1 2 3 4 5 6 χ Fgure 2.5. The theoretcal bnodal and spnodal lnes for a monomer bnary system (m A = 1,m B = 1) and a polymer system (m A = 10, m B = 1) are presented. 2.5.2. Phase separaton of a ternary system The total free energy of a ternary system can be obtaned from Eq. (2.42) as F = [θ( m A n A m B n B m C n C +n A ln φ A + n B ln φ B + n C ln φ C +χ AB n A m A φ B + χ AC n A m A φ C + χ BC n B m B φ C ) +κ AA ( n A ) 2 + κ BB ( n B ) 2 + κ CC ( n C ) 2 +κ AB n A n B + κ AC n A n C + κ BC n B n C ]dv, (2.51) where m σ s the polymerzaton of component σ. Of course, m σ s one for a monomer. 23

A ternary system has three components, but a Flory-Huggns model of a ternary system has only two ndependent varables because the densty factors out of the free energy. Therefore, φ A and φ B can be chosen as two ndependent varables, and φ C can be obtaned by φ C = 1 φ A φ B. Another constrant for a phase separaton n a ternary system s that the ntal homogeneous phase and the two separated phases are always n a straght lne n a φ A -φ B dagram. The smple analyss below wll make ths clear. Suppose the two volume fractons are chosen as phase parameters. The ntal phase s (φ A,φ B ), and the total volume of the system s V. The system evolves nto two phases: (φ A 1,φ B 1 ) n volume V 1, and (φ A 2,φ B 2 ) n volume V 2. From the constrant that the total mass of each component s conserved, so I get V φ A = V 1 φ A 1 + V 2 φ A 2, (2.52) V φ B = V 1 φ B 1 + V 2 φ B 2. (2.53) From the constrant that the volume of the system s constant V = V 1 + V 2, t follows that φ B φ B 1 φ A φ A 1 = φb 2 φ B 1. (2.54) φ A 2 φ A 1 Eq. (2.54) shows that the three ponts are n a straght lne n a φ A -φ B graph. Therefore, n a ternary system, a phase separates nto two phases along a straght lne, whch s called the te lne. 2 2 In a general case there s the possblty of a three phase regon whch s gnored here. 24

The theoretcal thermodynamcs of an equlbrum ternary system were obtaned by mnmzng the total free energy of the system numercally as dscussed. In ths thess, the bnodal lnes of a ternary system s obtaned by varyng the ntal volume fracton of the components whle holdng the χ parameters constant. The smulaton of phase separaton of a ternary system wth constant χ parameters sheds lght to the study of an mmerson precptaton process [27] n whch the χ parameters are constant. In contrast, the bnodal lnes of a bnary system are obtaned by varyng χ whle keepng the ntal volume fracton of the components constant. Startng from the crtcal pont, the volume fractons of two components of the ntal state (φ A,φ B ) was ncreased lnearly towards the fnal pont (φ A = 0.5,φ B = 0.5). For each ntal state, two correspondng bnodal ponts are obtaned along a straght lne. The te lnes of dfferent ntal states usually are not parallel to each other. Therefore, to calculate the bnodal lnes of a ternary system, I frst calculate the two phases that mnmze the total free energy of the system along a straght lne; then, I rotate the straght lne back and forth. Iteratng ths procedure allowed me to fnd the exact drecton of the te lne along whch the total free energy of the system s mnmal. I have checked that the chemcal potentals of the two phases of each component are equal to a precson of at least of 10 5. That means that the bnodal ponts obtaned wth ths approach represent equlbrum states. The collecton of all the bnodal ponts generated the bnodal lne. The mplementaton of ths algorthm s shown Appendx G. The phase dagram of a ternary system can be presented n Cartesan coordnates, but a trangular representaton s preferred n the lterature [4, 6, 9]. Actually, the 25

1 φ B 0.8 0.6 0.4 R m=10 m=5 m=1 φ B 0.6 0.4 0.2 0 R 1 0.8 0.6 m =10 m =5 m =1 0.4 φ C 0.2 0.8 0.2 0 0 0.2 0.4 0.6 0.8 1 φ A 1 0 0.2 0.4 0.6 0.8 1 φ A Fgure 2.6. The bnodal lnes for polymerzaton m = 1, 5 and 10 of a ternary system are llustrated n a Cartesan and a trangular coordnate. The shaded area s naccessble because of the constrant φ A + φ B 1. The system has χ AB = 3, χ AC = 0.5, χ BC = 0.2. Pont R corresponds to (0.30, 0.37, 0.33). The dotted lne from pont R shows the approach to determne φ A of the pont. 0 representatons are equvalent, and a lnear transformaton exsts between the two systems. The phase dagram of a ternary system s represented n both coordnates n Fgure 2.6. The next chapter ntroduces the lattce Boltzmann method. 26

CHAPTER 3 LATTICE BOLTZMANN METHOD 3.1. Sngle Relaxaton Tme Lattce Boltzmann The Lattce Boltzmann Equaton (LBE) can be understood as a dscrete Boltzmann Equaton [36]. The LBE wth a sngle relaxaton tme from the BGK model can be expressed as [41, 57] ( ) 1 f (r + v t,t + t) f (r,t) = t τ (fe (r,t) f (r,t) + G ) + F, (3.1) where r s the lattce poston vector; v s partcle velocty; t s tme; τ s the sngle relaxaton tme parameter, but the nverse relaton tme Ω 1/τ s often used nstead of τ as smulaton parameter; G ntroduces non deal effects nto the hydrodynamc pressure tensor [38]; F s a forcng term that can be used to ntroduce non-deal effects nto the bulk force [20, 34, 44, 64]; f (r,t) denotes the partcle dstrbuton assocated wth the dscrete velocty v ; and f e ndcates the local equlbrum dstrbuton correspondng to an deal gas. The dscrete velocty v s chosen such that the v t s a lattce vector. In ths thess, I lmt my study to sothermal systems n whch mass and momentum are conserved n the collson, and energy conservaton s abandoned n favor of a constant temperature requrement. I defne the densty ρ as ρ(x) f (x) and the 27

velocty u as ρ(x)u(x) f (x)v. From mass and momentum conservaton the frst two moments of the local equlbrum dstrbuton functon of an deal gas are (f e f ) = 0 (f e f )v = 0 f e = f ρ, (3.2) f e v = f v ρu. (3.3) Analogous to the contnuous case, the other two moments of the equlbrum dstrbuton functon of an deal gas can be defned as [41] f e v α v β = 1 3 ρδ αβ + ρu α u β, (3.4) f e v α v β v γ = 1 3 ρ(u αδ βγ + u β δ αγ + u γ δ αβ ) + ρu α u β u γ + Q αβγ, (3.5) where Q αβγ s a tensor term that s zero n the equvalent contnuous ntegral. Because v x = v 3 x, I have f e vx 3 = f e v x = ρu. (3.6) Therefore, Q αβγ s usually chosen to be ρu α u β u γ to make Eq. (3.5) satsfy Eq. (3.6). But ths choce of Q αβγ causes a small Gallean Invarance volaton for moderate u. 28

Ths wll be dscussed n Chapter 4. The moments of G are gven by G = 0, (3.7) G v α = 0, (3.8) G v α v β = A αβ, (3.9) G v α v β v γ = 0, (3.10) where A αβ s a tensor descrbng the non-deal part of the pressure tensor of the lqud. The velocty moments of the F are gven by F = 0, (3.11) F v α = ρa α, (3.12) F v α v β = ρ(a α u β + a β u α ), (3.13) F v α v β v γ = 1 3 ρ(a αδ βγ + a β δ αβ + a γ δ αβ ), (3.14) where a α s the acceleraton. Non-deal flud flow can be smulated by usng a free energy from whch the chemcal potental can be derved. There are two ways to nclude the nteracton nto the LB evoluton equaton: the pressure approach (F = 0) and the forcng approach (G = 0). The two approaches are subtly dfferent because of hgher order effects. Ths s beyond the scope of ths thess. Interested readers can refer to a current paper by A.J. Wagner [55]. 29

To recover the hydrodynamc equatons, I use a moment method. Frst I perform a Taylor expanson to the left sde of Eq. (3.1), ( ) ( t) k 1 ( t + v α α ) k f = t k! τ (fe f + G ) + F. (3.15) k=1 In the long wavelength and small frequency lmt, α and t result n small quanttes of the order of ǫ whch ndcates the order of smallness. Neglectng all the terms of O(ǫ 2 ), I get ( ) 1 t( t + v α α )f + O(ǫ 2 ) = t τ (fe f + G ) + F. (3.16) Wth Eq. (3.16), f can be expressed n terms of the f e, F and G, for whch all moments are known. Neglectng all the terms of O(ǫ 3 ) n Eq. (3.15), ( ) t( t + v α α )f + ( t)2 1 ( t + v α α ) 2 f + O(ǫ 3 ) = t 2 τ (fe f + G ) + F. (3.17) Hgher order terms are not consdered here because the hydrodynamc equatons only contan second order dervatves. Eqs. (3.2)-(3.5) establsh the relatons between the macroscopc varables, ρ and u, and the moments of the equlbrum dstrbutons. In order to use those relatons, f s expanded n terms of f e. Usng Eq. (3.16) recursvely, f = f e + G + τf τ( t + v α α )(f e + G + τf ) + O(ǫ 2 ). (3.18) 30

Substtutng Eq. (3.18) nto the left sde of Eq. (3.17), ( t + v α α )(f e + G + τf ) w( t + v α α ) 2 (f e + G + τf ) + O(ǫ 3 ) = 1 τ (fe + G + τf f ), (3.19) where w τ t/2. To obtan the mass conservaton equaton, I sum Eq. (3.19) over and obtan, t ρ + α (ρu α ) + τ α (ρa α ) w ( t + v α α ) 2 (f e + G + τf ) + O(ǫ 3 ) = 0. (3.20) Eq. (3.20) shows that t ρ + α (ρu α ) + τ α (ρa α ) s of the order of O(ǫ 2 ), so Eq. (3.20) can be smplfed to [ t ρ + α (ρu α ) + τ α (ρa α ) w β t (ρu β ) + τ t (ρa β ) + α f o v α v β ] + α A αβ + α F v α v β + O(ǫ 3 ) = 0. (3.21) Multplyng Eq. (3.19) wth v β and summng over, I get v β ( t +v α α )(f e +G +τf ) v β w( t +v α α ) 2 (f e +G +τf ) = ρa β. (3.22) Eq. (3.22) shows that v β ( t + v α α )(f e + G + τf ) = ρa β + O(ǫ 2 ). (3.23) 31

Substtutng Eq. (3.23) nto Eq. (3.21), [ t ρ + α ρ(u α + t ] 2 a α) + O(ǫ 3 ) = 0. (3.24) Therefore, by denotng the macroscopc velocty U α u α + a α t/2, the contnuty equaton (2.1) can be recovered to the second order: t ρ + α (ρu α ) = 0 + O(ǫ 3 ). (3.25) To obtan the momentum conservaton equaton, I substtute Eq. (3.23) nto Eq. (3.22). v β ( t + v α α )(f e + G + τf ) wv β v γ γ ( t + v α α )(f e + G + τf ). w t ρa β + O(ǫ 3 ) = ρa β. (3.26) Eq. (3.26) shows that a s of the oder O(ǫ). Eq. (3.12) shows that F s of the same order as a,.e. O(ǫ). The second term n Eq. (3.26) s smplfed, step by step, as follows: wv β v γ γ ( t + v α α )(f e + G + τf ) = wv β v γ γ ( t + v α α )(f e + G ) + O(ǫ 3 ) ( ) = w γ t f e v β v γ + t A βγ + α f e v α v β v γ + O(ǫ 3 ). (3.27) 32

Usng Eq. (3.23), t (ρu β ) = α ( ρ 3 δ αβ + A αβ + ρu α u β ) + ρa β + O(ǫ 2 ). (3.28) Substtutng Eq. (3.25) nto Eq. (3.28), ρ t u β = ρu α α u β βρ 3 αa αβ + ρa β + O(ǫ 2 ). (3.29) Usng Eqs. (3.25), (3.28) and (3.29), t f e v β v γ + α f e v α v β v γ ( ρ ) = t 3 δ βγ + ρu β u γ [ ρ ] + α 3 (u αδ βγ + u β δ αγ + u γ δ αβ + ρu α u β u γ + Q αβγ + O(ǫ) = 1 3 δ βγ( α (ρu α )) + u γ t (ρu β ) + ρu β t u γ + 1 3 δ βγ α (ρu α ) + 1 3 γ(ρu β ) + 1 3 β(ρu γ ) + α (ρu α u β u γ ) + α Q αβγ + O(ǫ 2 ) ( = u γ 1 ) ( 3 βρ α A αβ α (ρu α u β ) + ρa β + u β u α u α u γ α A αβ 1 ) 3 γρ + ρa γ + 1 3 γ(ρu β ) + 1 3 β(ρu γ ) + α (ρu α u β u γ ) + α Q αβγ + O(ǫ 2 ) = 1 3 ρ γu β + 1 3 ρ βu γ + ρu γ a β + ρu β a γ u γ α A αβ u β α A αγ + α Q αβγ + O(ǫ 2 ). (3.30) 33

From Eqs. (3.25), (3.26), and (3.30), I obtaned wth some algebra, ( ) 1 [ w ] t (ρu α ) + β (ρu α U β ) = β 3 ρδ αβ + A αβ + β 3 ρ( βu α + α U β ) + ρa α + w γ ( u γ β A αβ u α β A βγ + t A αγ + β Q αβγ ) + O(ǫ 3 ). (3.31) Eq. (3.31) s a Naver Stokes equaton except the terms of the last lne that cause Gallean nvarance volatons (Secton 4.1.). In other lterature, a second approach exsts to derve the hydrodynamc equatons of the lattce Boltzmann method (LBM). It s a mult-scale expanson, referred to as the Chapman Enskog approach. The two approaches gve the same results as far as the recovery of the hydrodynamc equatons are concerned [41, 62]. Identcal forms of Eq. (3.31) can be mplemented wth dfferent choces of A or a. If both A and a are zero, Eq. (3.31) becomes the Naver Stokes equaton for an deal gas. But even n the deal gas LB model, the w γ β Q αβγ exsts and causes the GI volaton. The non-deal model for LB smulaton can be obtaned by choosng ether A or a to be zero. The choce of a = 0 and A 0 s referred to as the pressure approach; the choce of a 0 and A = 0 s referred to as the force approach. Swft et al. [48] nvented a model to smulate a non-deal flud n pressure approach by choosng A αβ = P αβ 1ρ. 3 Ths model s defcent and cause a severe Gallean nvarance volaton. The errors come from the terms n the second lne n Eq. (3.31). Ths problem was addressed by Holdych et al. [21], Inamuro et al. [22], and Kalaraks et al. [24] ndependently. They 34

solved the problem by redefnng A αβ = P αβ 1 3 ρδ αβ ν( α ρu β + β ρu α + u γ γ ρδ αβ ). (3.32) Eq. (3.32) leaves the densty gradent terms n the second lne of Eq. (3.31) to the second order, whch are assumed to be very small because the dervatves of the pressure tensor are much smaller than the dervatves of densty at the nterface near equlbrum[24]. Therefore, Holdych s model gves a more accurate result and dmnshed the GI volaton sgnfcantly. To smulate a one component phase separatng system wth a forcng approach, ρa α = β (P αβ 1ρδ 3 αβ) can be chosen. To elmnate the Q term error n deal gas model or n non-deal models of pressure or forcng approach, a forcng term s ntroduced as ρa α = w β γ Q αβγ. A Holdych model wth a Q correcton s referred to as HoldychQ, smlarly s the ForcngQ. Table 3.1. summarzes the smulaton approaches explored n ths thess. Table 3.1. The algorthms for the LB smulaton of a non-deal flud Approach A αβ ρa α Pressure P αβ 1ρδ 3 αβ 0 Holdych P αβ 1ρδ 3 αβ ν( α ρu β + β ρu α + u γ γ ρδ αβ ) 0 HoldychQ P αβ 1ρδ 3 αβ ν( α ρu β + β ρu α + u γ γ ρδ αβ ) w β γ Q αβγ Forcng 0 β (P αβ 1ρδ 3 αβ) ForcngQ 0 β (P αβ 1ρδ 3 αβ) + w β γ Q αβγ 35

3.1.1. One-dmensonal three-velocty lattce Boltzmann model Many lattce Boltzmann models are avalable such as the one-dmensonal threevelocty model (D1Q3), the two-dmensonal seven-velocty model (D2Q7), the two dmensonal nne velocty model (D2Q9), the three dmensonal and ffteen velocty model (D3Q15), and the three dmensonal and twenty seven velocty model (D3Q27). Although the D1Q3 model s the smplest, t s, n fact, a projecton of all the other models mentoned above. Therefore any a problem n the D1Q3 model exsts also n the hgher dmensonal models. In ths thess, D1Q3 models were used for all smulatons and are ntroduced next. The D1Q3 model has three veloctes: v 0 = 0, v 1 = 1, v 2 = 1. In one dmenson, Eqs. (3.2) and (3.3) become ρ = f 0 + f 1 + f 2, (3.33) ρu = f 1 f 2, (3.34) and the local equlbrum dstrbuton satsfes ρ = f e 0 + f e 1 + f e 2, (3.35) ρu = f e 1 f e 2, (3.36) ρu 2 + ρ 3 = fe 1 + f e 2. (3.37) I frst derve the equlbrum dstrbuton for the deal gas. Wth Eqs. (3.35), (3.36), 36

and (3.37), f e 0 = 2 3 ρ ρu2, (3.38) f e 1 = 1 2 (ρ/3 + ρu + ρu2 ), (3.39) f e 2 = 1 2 (ρ/3 ρu + ρu2 ) (3.40) For a non-deal gas wth a pressure approach, Eqs. (3.7), (3.8), and (3.9) gve G 0 + G 1 + G 2 = 0, (3.41) G 1 G 2 = 0, (3.42) G 1 + G 2 = A, (3.43) whch yelds G 0 = 0, (3.44) G 1 = A/2, (3.45) G 2 = A/2. (3.46) For a non-deal gas wth a forcng approach, Eqs. (3.11), (3.12), and (3.13) gve F 0 + F 1 + F 2 = 0, (3.47) F 1 F 2 = ρa, (3.48) F 1 + F 2 = 2ρau. (3.49) 37

Ths can be solved for the F to gve F 0 = 2ρau, (3.50) F 1 = (u + 0.5)ρa, (3.51) F 2 = (u 0.5)ρa, (3.52) where ρa s the nteracton force derved from the non-deal gas nteracton. The evoluton equatons for a D1Q3 model s then: f 0 (,t + 1) = f 0 (,t) + 1 τ (fe 0(,t) f 0 (,t) + G 0 (,t)) + F 0 (,t), (3.53) f 1 ( + 1,t + 1) = f 1 (,t) + 1 τ (fe 1(,t) f 1 (,t) + G 1 (,t)) + F 1 (,t), (3.54) f 2 ( 1,t + 1) = f 2 (,t) + 1 τ (fe 2(,t) f 2 (,t) + G 2 (,t)) + F 2 (,t), (3.55) where s the lattce coordnate, and t s the tme step. 3.1.2. Lattce Boltzmann for a lqud-gas system To valdate my theoretcal analyss of LB smulatons of non-deal fluds, I performed LB smulatons of a smple one component non-deal system descrbed by the Landau free energy model of Eq. (2.27) wth Holdych s approach and a forcng approach. The equlbrum thermodynamc propertes (densty, pressure, and chemcal potental) of the system obtaned by the LB smulaton through the two approaches are compared to the theoretcal results derved n Secton 2.4. The theoretcal densty profle s gven by Eq. (2.36). The pressure s gven by Eq. (2.40), and the chemcal potental s gven 38

by Eq. (2.41). My LB smulatons were performed wth the D1Q3 models of Table 3.1. Each smulaton was performed on a lattce wth 100 lattce ponts, unless noted otherwse. Intally the system had a unform densty of 1, plus a snusodal perturbaton to trgger phase separaton. The ampltude of the perturbaton was 0.1 and ts wavelength was the lattce sze. To obtan the bnodal lnes of a lqud-gas system, the LB smulatons were performed startng wth these ntal condtons wth ncreasng value of θ startng from the crtcal pont. A par of bnodal ponts were obtaned by the LB smulaton from each ntal condton. I contnued to ncrease θ for each smulaton untl I observed numercal nstabltes. The bnodal ponts were than gven by the maxmum and mnmum denstes n equlbrum. The system evolved nto a stable state after about 1000 tme steps. The measurements were taken after 10000 tme steps to be sure that the system was n equlbrum. Fgure 3.1 shows the comparson of the bnodal ponts obtaned by the Holdych and ForcngQ approaches to the theoretcal bnodal lnes. The smulaton results approach the theoretcal result, but the devatons ncreases as θ ncreases. The Holdych approach s more accurate, but the ForcngQ approach has a larger stablty range. However, a stablty analyss s outsde the scope of ths thess; for more detals on the stablty of the LB methods, an nterested reader may refer to [2, 3, 37, 50, 52, 47, 54, 61]. Fgure 3.2 shows the comparson of the total densty, pressure, and and chemcal potentals obtaned by HoldychQ approach to the theoretcal values. The equlbrum densty profle almost overlapped wth the theoretcal profle. The equlbrum pressure 39

0-0.05 θ -0.1-0.15 analytcal Holdych ForceQ -0.2 0.6 0.8 1 1.2 1.4 ρ Fgure 3.1. The bnodal ponts of a lqud-gas system obtaned by the LB smulaton wth the ForcngQ and Holdych approaches are compared to the analytcal soluton. The system has κ = 0.1, n c = 1, p c = 0.42, and ν = 1/6. For the LB smulaton, ω = 1.0. obtaned by the smulaton was slghtly lower than the theoretcal values, but the dfference was less than 0.1% when compared to the theoretcal value. The equlbrum chemcal potental obtaned by the LB smulaton n the two bulk phases dffer only wthn 0.01%. There are two spkes on the chemcal potental at the nterface of the two phases, but the spkes are less than 0.1% compared to the bulk chemcal potentals. The chemcal potental obtaned by the LBM are slghtly lower than the theoretcal value, but the dfference was less than 0.1% compared to the theoretcal value. Fgure (3.3) shows the comparson of the total densty, pressure, and and chemcal potentals obtaned by ForcngQ approach to the theoretcal values. The densty profle 40