Numerical simulations of phase separation dynamics in a water oil surfactant system

Transcription

1 Joural of Colloid ad Iterface Sciece wwwelsevierco/locate/jcis Nuerical siulatios of phase separatio dyaics i a water oil surfactat syste Juseo Ki Departet of Matheatics Doggu Uiversity Seoul Republic of Korea Received 10 May 006; accepted 1 July 006 Available olie 0 July 006 Abstract We have studied uerically the dyaics of the icrophase separatio of a water oil surfactat syste We developed a efficiet ad accurate uerical ethod for solvig the two-diesioal tie-depedet Gizburg Ladau odel with two order paraeters The uerical ethod is based o a coservative secod-order accurate ad iplicit fiite-differece schee The oliear discrete equatios were solved by usig a oliear ultigrid ethod There is at ost a first-order tie step costrait for stability We deostrated uerically the covergece of our schee ad preseted siulatios of phase separatio to show the efficiecy ad accuracy of the ew algorith 006 Elsevier Ic All rights reserved Keywords: Noliear ultigrid ethod; Surfactat; Phase separatio; Gizburg Ladau odel 1 Itroductio I a water oil surfactat syste oolayers of surfactat olecules for icroeulsios as a rado phase [1] Microeulsios show two types of orphology as follows I the syetric case of eve copositios of water ad oil the syste shows a cocotiuous etwor patter Fig 1a The droplet patter Fig 1b is foud i the asyetric case of ueve copositios of water ad oil The dyaics of icrophase separatio i a water oil surfactat syste has bee ivestigated uerically by usig the tie-depedet Gizburg Ladau TDGL odel [] I [3] the cell dyaical syste approach is used A Mote Carlo siulatio is used i [4] I[5] a hybrid odel is used which is a pheoeological sei-icroscopic odel where the biary ixture ad the surfactat are treated as a cotiuous field ad with discrete olecules respectively I this paper we preset a fiite-differece ethod for the solutio of the TDGL odel The Cra Nicolso ethod is applied to the teporal discretizatio The resultig fiitedifferece equatios are coservative secod-order accurate i E-ail address: cfdi@dogguedu URL: space ad tie ad solved by a efficiet ad accurate oliear ultigrid ethod A advatage of usig a oliear ultigrid ethod is that the schee is uch ore efficiet tha traditioal iterative solvers i solvig the oliear equatios at the iplicit tie step It is straightforward to exted a twodiesioal code to a three-diesioal oe ad to parallelize the serial code It is also atural to icorporate hydrodyaic effects such as surface tesio force gravity ad shear flow i this odel [6] The cotets of this paper are as follows: i Sectio we briefly review the goverig equatios I Sectio 3 we cosider a fully discrete sei-iplicit fiite-differece schee ad describe a oliear ultigrid V-cycle algorith for the TDGL syste Nuerical experiets such as a secod-order covergece test ad tests of the effects of paraeters o the phase separatio of the syste are perfored i Sectio 4 I Sectio 5 coclusios are give I additio we preset a future directio for this curret algorith The future pla is to icorporate hydrodyaic effects Goverig equatios The dyaics of icrophase separatio i icroeulsio systes ca be odeled by the followig TDGL odel with two order paraeters xt ad Φxt These paraeters /$ see frot atter 006 Elsevier Ic All rights reserved doi:101016/jjcis

2 JS Ki / Joural of Colloid ad Iterface Sciece Fig 1 Sapshot pictures of the syste obtaied by the coputer siulatios for a cocotiuous etwor ad b droplet/atrix patter describe the differece i the local desities of water ad oil ad the local cocetratio of surfactat at site x ad tie t respectively [3]: [ E Φ = FΦ D 1 sφ Ω D Φ Φ ] dx 1 FΦ= g 4 [ β νφ Φ ave ] λφ Φ ave where s g β ν λ ad D Φ are positive pheoeological paraeters ad Ω is the syste doai I E Φ theter sd Φ eergetically prefers a relatively high value of Φ at the iterface I FΦtheterλΦ Φ ave prevets surfactats fro forig clusters The ter νφ Φ ave eas the local couplig iteractios Notatios ave ad Φ ave correspod to the ad Φ field values averaged i space respectively The TDGL equatios i the coserved syste are writte explicitly as t = M 3 Φ = M Φ t 4 = FΦ D [1 sφ ] 5 = FΦ D Φ Φ D s Φ 6 where M Φ are the positive diffusioal obilities The boudary coditios for the TDGL syste are the zero Neua boudary coditios = Φ = = = 0 o Ω 0T 7 where is the oral vector to Ω Usig these boudary coditios we ca derive the followig equatio [7]: de Φ = M dt M Φ dx 8 Ω which eas the total eergy of the syste decreases with respect to tie 3 Nuerical solutio We first discretize the TDGL syste 3 6 i space Ω = [ab] [cd] Let[ab] ad [cd] be partitioed by a = x 1/ <x 3/ < <x Nx 1/ = b ad c = y 1/ <y 3/ < <y Ny 1/ = d For siplicity we assue the above partitios are uifor i both directios that is x i1/ x i 1/ = y j1/ y j 1/ = h = b a N x for 1 i N x 1 j N y Therefore x i1/ ad y j1/ ca be represeted as x i1/ = a ih ad y j1/ = c jh We deote by Ω h ={x i y j : 1 i N x 1 j N y } the set of cell cetered poits x i y j = x i 1/ x i1/ / y j 1/ y j1/ / For Neua boudary value probles it is atural to copute uerical solutios at cell ceters Let be approxiatios of x i y j We ipleet the zero Neua boudary coditios 7 by requirig that for exaple 0j = 1j Nx 1j = Nx j i0 = i1 iny 1 = iny for all i j We the defie the discrete Laplacia by the stadard five-poit stecil h = i 1j i1j /h Ny ad the discrete l Nx -or by =h i=1 j=1 The TDGL equatios 3 6 are itegrated with respect to tie usig the Cra Nicolso ethod: 1 1/ = M h 9

3 74 JS Ki / Joural of Colloid ad Iterface Sciece Φ 1 Φ 1/ 1/ 1/ = M Φ h = F1 Φ 1 D h [ 1 sφ 1 F Φ D h [ 1 sφ = F1 Φ 1 Φ e h 1 ] e h ] F Φ Φ D Φ h Φ 1 Φ D s c 1 h 1 c h where h e ad c h are cell-edge ad cell-ceter based discrete gradiets respectively ad are described i Eqs 16 ad A oliear ultigrid V-cycle algorith I this sectio we develop a oliear full approxiatio storage FAS ultigrid ethod to solve the oliear discrete syste at the iplicit tie level The oliearity is treated usig oe step of Newto s iteratio ad a poitwise Gauss Seidel relaxatio schee is used as the soother i the ultigrid ethod See Ref [8] for additioal details ad the followig otatios Let us rewrite Eqs 9 1 as follows: N 1 Φ 1 = s 1 s s 3 s 4 1/ 1/ 13 where the oliear syste operator N the left-had side of Eq 13 is defied as 1 M h 1/ 1/ Φ1 F1 Φ 1 D h [ 1 sφ 1 e h 1 ] 1/ F1 Φ 1 Φ D Φ h Φ 1 D s h c 1 1/ M Φ h ad the source ter the right-had side of Eq 13 is Φ F Φ D h [ 1 sφ e h ] F Φ Φ D Φ h Φ D s h c I the followig descriptio of oe FAS cycle we assue that a sequece of grids Ω Ω 1 is coarser tha Ω by factor Give the uber η of pre- ad postsoothig relaxatio sweeps a iteratio step for the oliear ultigrid ethod usig the V-cycle is forally writte as follows: FAS ultigrid cycle 1/ 1 Φ 1 = FAScycle Φ N s 1 s s 3 s 4 η 1/ 1/ 1/ That is { Φ 1/ 1/ } ad { 1 Φ 1 1/ 1/ } are the approxiatios of { 1 Φ 1 1/ 1/ } before ad after a FAScycle We ow defie the FAScycle 1 Presoothig 1/ Φ = SMOOTH η Φ N s 1 s s 3 s 4 1/ 1/ 1/ which eas perforig η soothig steps with the iitial approxiatios ad source ters to get the approxiatios { Φ 1/ 1/ } Oe SMOOTH relaxatio operator step cosists of solvig the syste give below by 4 4 atrix iversio for each : Φ 4M h = s 1 M h 1/ [ i 1j 1/ 4M Φ h = s M Φ h 1 1/ [ 1/ 1/ 1/ 1 ] 1/ 1 i 1j 1/ 1 ] 1/ i1j 1/ i1j 14 15

4 JS Ki / Joural of Colloid ad Iterface Sciece [ F Φ D h 4 s Φ i 1j Φi1j 4Φ Φ 1 ] Φ 1 F Φ Φ = s3 F Φ F Φ Φ Φ D h 1 s 1/ Φ [ 1 s Φ Φ i1j Φ i 1j Φ 1 s Φ Φ 1 1 s F Φ Φ Φ 1 Φ 1/ 4DΦ h F Φ i 1j 1 1 i1j ] F Φ Φ = s4 F Φ F Φ Φ Φ F Φ Φ Φ D Φ h Φ i1j Φ i 1j Φ 1 Φ 1 Φ 1 1 D s 8h i1j i 1j Copute the defect def 1 def def 3 def 4 = s 1 s s 3 s 4 1/ 1/ N Φ 3 Restrict the defect ad { Φ def 1 1 def 1 def 3 1 def 4 1 = I 1 def 1 def def 3 def 4 1/ 1 Φ 1 = I 1 1 1/ 1 1/ Φ / 1/ } 1/ The restrictio operator I 1 aps -level fuctios to the 1-level fuctios d 1 x i y j = I 1 d x i y j = 1 4[ d x i 1/ y j 1/ d x i 1/ y j1/ d x i1/ y j 1/ d x i1/ y j1/ ] Coarse grid values are obtaied by averagig the four earby fie grid values 4 Copute the right-had side s 1 1 s 1 s 3 1 s 4 1 = def1 1 def 1 def 3 1 def 4 1 1/ N 1 1 Φ 1 1 1/ 1 5 Copute a approxiate solutio { ˆ 1 ˆΦ 1 ˆ 1/ 1 ˆ 1/ 1 } of the coarse grid equatio o Ω 1 ie 1/ 1/ N 1 1 Φ 1 1 = s 1 1 s 1 s 3 1 s If = 1 we explicitly ivert a 4 4 atrix to obtai the solutio If >1 we solve Eq 18 by perforig a FAS -grid cycle usig { 1 Φ 1 } as a iitial approxiatio: ˆ 1 1 ˆΦ 1 1 = FAScycle ˆ 1/ 1 1/ 1 ˆ 1/ 1 1/ 1 1 Φ 1 1 1/ 1 1/ N 1 s1 1 s 1 s 3 1 s 4 1 η 1 6 Copute the coarse grid correctio CGC ˆv 1 1 = ˆ 1 1 ˆv 1/ 3 1 = ˆ 1/ 1 ˆv 1 = ˆΦ 1 Φ 1 ˆv 1/ 4 1 = ˆ 1/ 1 1/ 1 1/ 1 7 Iterpolate the correctio ˆv 1 ˆv ˆv 3 4 ˆv = I 1/ 1 ˆv 1 1 ˆv 1/ 1 ˆv 1/ 3 1 ˆv 1/ 4 1 The iterpolatio operator I 1 aps the 1-level fuctios to the -level fuctios Here the coarse values are siply trasferred to the four earby fie grid poits ie v x i y j = I 1 v 1x i y j = v 1 x i1/ y j1/ for i ad

5 76 JS Ki / Joural of Colloid ad Iterface Sciece j odd-ubered itegers The values at the other ode poits are give by v x i1 y j = v x i y j1 = v x i1 y j1 = v 1 x i1/ y j1/ where i ad j are odd-ubered itegers 8 Copute the corrected approxiatio o Ω after CGC = ˆv 1 1/ after CGC after CGC 1/ after CGC = Φ = Φ ˆv = 9 Postsoothig 1 Φ 1 = SMOOTH η 1/ after CGC 1/ 1/ Φ ˆv 1/ 3 ˆv 1/ 4 1/ after CGC 1/ after CGC 1/ after CGC N s1 s s 3 s 4 This copletes the descriptio of a oliear FAS cycle 4 Nuerical experiets I this sectio we describe how we perfored a covergece test of the proposed schee ad preset several siulatios of phase separatio Above all we ivestigate the effects of s ad λ o the phase separatio We also describe the surfactat dyaics diffusig ito a droplet iterfacial regio fro the outside 41 Covergece test of the proposed schee To obtai a estiate of the covergece rate we perfored a uber of siulatios for a saple proble o a set of icreasigly fier grids The iitial coditio is give by xy = 01 cos3x 04 cosy Φxy = o a doai Ω =[0 π] [0 π] The uerical solutios are coputed o the uifor grids h = π/ for = ad 7 The uifor tie steps = 01h g = 1 β = ν = 01 λ = 05 s = 01 D Φ = 005 ad M Φ = 1 are used to establish the covergece rates For each case the calculatios are ru to tie T = 01 I our forulatio of the ethod for the TDGL syste sice a cell-cetered grid is used we defie the error to be the discrete l -or of the differece betwee Table 1 l -Nor of the errors ad covergece rates Case 3 64 Rate Rate E E E 3 Φ 6155E E E 4 Fig The tie-depedet total eergy E Φ of the uerical solutios with the iitial data 19 that grid ad the average of the ext fier grid cells coverig it: def e h/ h = h h h h ij i 1j ij 1 / h 4 i 1j 1 The rate of covergece is defied as the ratio of successive errors: log eh/ h / e h / h 4 The errors ad rates of covergece are give i Table 1 The results suggest that the schee is ideed secod-order accurate I Fig the tie evolutio of the eergy E Φ with sae iitial data 19 is show As expected fro Eq 8 thetotal eergy is oicreasig ad teds to a costat value 4 Spiodal decopositio with off-critical quech We begi the uerical experiets with a exaple of spiodal phase separatio of a terary ixture Here we cosider the effect of λ i the ter λφ Φ ave i Eq For the iitial coditio we tae radoly perturbed cocetratio fields: xy = ave 001radx y Φxy = Φ ave 001radx y 0 where the rado uber fuctio radx y isi[ 1 1] ad has zero ea The ave Φ ave sets are chose as 0 03 The coputatioal doai usig a spatial esh of

6 JS Ki / Joural of Colloid ad Iterface Sciece Fig 3 Colus a b ad colus c d show the tie evolutios of spatial patters of the ad Φ fields with λ = 05 ad λ = 005 respectively Dar area deotes the regio of positive values of i a ad higher values of Φ i b Ties are at t = ad is Ω =[0 π] [0 π] The uifor tie step = 01h g = 1 β = ν = 01 s = 05 D Φ = 005 ad M Φ = 1 are used I Fig 3 colus a b ad colus c d show the tie evolutio of the spatial patters of the ad Φ fields with λ = 05 ad λ = 005 respectively Dar area deotes the regio of positive values of i a ad higher values of Φ i b The ties are at t = ad 78 A oderate value of λ= 05 prevets the surfactats fro forig clusters as is show i Fig 3b However whe λ= 005 is too sall the surfactats cluster at soe regio ad chage the global dyaics as is show i Fig 3d Next we ivestigate the effect of s i the ter sd Φ i Eq 1 The iitial cofiguratios of the ad Φ fields are chose to be radoly distributed as i Eq 0 with the ave Φ ave = values All other paraeters are the sae as before except λ= 05IFig 4 colus a b ad colus c d show the tie evolutio of spatial patters of the ad Φ fields with s = 005 ad s = 05 respectively The ties are at t = ad 78 The ter sd Φ i Eq 1 eergetically prefers a relatively high value of Φ at the iterface Oe ca see this pheoeo fro Figs 4b ad 4d At the higher value of s= 05 ore surfactat accuulates at the iterface tha i the case of s= Quatitative result doai growth rate We ivestigate quatitatively the coarseig aer observed i uerical siulatios The growth of the ordered doais is easured through the average doai size calculated as the iverse of the first oet of the circularly averaged structure factor [5] Aother reliable easure of the characteristic legth is the average radius of gyratio of the droplets sice the droplets are foud to be circular i the siulatio [9]We use the weighted average radius Rt = i=1 R i i=1 R i R i = Si π where S i is the droplet area ad is the total uber of droplets at tie t The coputatioal doai is Ω =[0 40π] [0 40π] ad the esh size is with tie step = 01h All the other paraeters are the sae as i the previous cases except ave Φ ave = ad s= 05 To obtai a averaged behavior te siulatios are ru with idetical coditios except for the seed of the rado uber I Fig 5 we show the characteristic doai size Rt as a fuctio of tie t The otatio deotes a average over 10 differet iitial rado coditios O a log log plot the growth appears to be slower tha the Lifshitz Slyozov growth

7 78 JS Ki / Joural of Colloid ad Iterface Sciece Fig 4 Colus a b ad colus c d show the tie evolutios of spatial patters of the ad Φ fields with s = 005 ad s = 05 respectively The ave Φ ave sets are chose as Ties are at t = ad Diffusio of surfactat ito the droplet iterfacial regio Fig 5 Tie evolutio of the ea droplet size Rt is show o a logarithic-scale plot The straight lie has slope 1/3 ad correspods to the Lifshitz Slyozov growth law The dash dot lie has slope 1/35 law Rt t 1/3 We observe a slight decrease i the expoet Rt t 1/35 This fidig is cosistet with previous results [1011] For exaple previous results usig a lattice gas odel have show that the surfactat slows dow the growth [11] Fially we discuss the diffusio of the surfactat ito the droplet iterfacial regio We have a iitial coditio as follows: xy = tah 1 x π y π 05 D Φxy = 05 1 tah 1 x y 05 D Φ The coputatioal doai usig a spatial esh of is Ω =[0 π] [0 π] The uifor tie step = 03h g = 1 β = ν = 001 s = 0 λ = 5 D Φ = 001 ad M Φ = 1 are used Fig 6 shows the tie evolutio of the surfactat diffusio ito the droplet iterfacial regio The ties are at t = ad 469 fro left to right ad top to botto order Iitially there are a droplet i the ceter of the doai ad oe quarter of dis of surfactat at the left botto corer As the tie goes o the surfactat cocetratio diffuses to the bul regio ad accuulates at the iterfacial regio of the droplet 5 Coclusios We have preseted a uerical ethod for solvig the TDGL odel The uerical schee is fiite differece ad is

8 JS Ki / Joural of Colloid ad Iterface Sciece Fig 6 Surfactat diffusio ito the droplet iterfacial regio Dotted circles are iterfacial regio ad filled cotour lies are surfactat cocetratio The higher value of cocetratio is the darer color is The ties are at t = ad 469 fro left to right ad top to botto order solved by a efficiet ad accurate oliear ultigrid ethod Nuerical experiets showed that the schee is secod-order i both space ad tie ad has oly a first-order tie step restrictio We have studied the effects of paraeters λ ad s o the dyaics of phase separatio for a water oil surfactat syste A oderate value of λ prevets the surfactats fro forig clusters Whe λ is too sall the surfactats cluster at soe regio This clusterig chages the global dyaics At a higher value of s ore surfactat accuulates at the iterface tha at a lower value of s We also described diffusio of surfactat ito the statioary droplet iterfacial regio We view the wor preseted here as preparatory for a study of three copoet liquids such as two iiscible fluids ad oe surfactat syste with hydrodyaics I a copaio paper [1] we will describe couplig the terary TDGL odel to the equatios of fluid flow Navier Stoes equatios to siulate the hydrodyaics of flows cosistig of three copoets Such a syste is govered by u = 0 ρu t u u = p [η u u T ] αd σφ ρg u = M t Φ t = FΦ D [1 sφ ] u Φ = M Φ = FΦ D Φ Φ D s Φ where u is the velocity ρ is the desity η is the viscosity p is the pressure α is a costat σφ is the surface tesio coefficiet ad g is the gravity vector The ter αd σφ accouts for the iterfacial capillary force Acowledget This wor is supported by the Doggu Uiversity Research Fud Refereces [1] S Koura H Seto T Taeda M Nagao Y Ito M Iai J Che Phys [] T Teraoto F Yoezawa J Colloid Iterface Sci [3] S Koura H Kodaa Phys Rev E [4] T Kawaatsu K Kawasai J Colloid Iterface Sci [5] T Kawaatsu K Kawasai M Furusaa H Oabayashi T Kaaya J Che Phys [6] JS Ki JS Lowegrub Iterfaces Free Boud [7] Q Du RA Nicolaides SIAM J Nuer Aal [8] U Trotteberg C Oosterlee A Schüller MULTIGRID Acadeic Press 001 [9] A Charabarti R Toral JD Guto Phys Rev E [10] AN Eerto PV Coveey BM Boghosia Phys Rev E [11] FWJ Weig PV Coveey BM Boghosia Phys Rev E [1] JS Ki i preparatio