CHEM. ES. CHINESE UNIVESITIES 00, 6(), 7 Mesoscale Smulaton fo Polyme Mgaton n Confned Unfom Shea Flow HE Yan-dong, WANG Yong-le, LÜ Zhong-yuan * and LI Ze-sheng State Key Laboatoy of Theetcal and Computatonal Chemsty, Insttute of Theoetcal Chemsty, Jln Unvesty, Changchun 300, P.. Chna Abstact The stuctue and dynamcs of confned sngle polyme chan n a dlute soluton, ethe n equlbum o at dffeent shea ates n the unfom shea flow felds, wee nvestgated by means of dsspatve patcle dynamcs smulatons. The no-slp bounday condton wthout densty fluctuaton nea the wall was taken nto account to mmc the envonment of a nanochannel. The dependences of the adus of gyaton, especally n thee dffeent dectons, and the densty pofle of the chan mass cente on the stength of the confnement and the Wessenbeg numbe(w n ) was studed. The effect of the nteacton between polyme and solvent on the densty pofle was also nvestgated n the cases of modeate and stong W n. In the hgh shea flow, the polyme mgates to the cente of the channel wth nceasng W n. Thee s only one densty pofle peak n the channel cente n the unfom shea flow, whch s n ageement wth the esults of the expements and theoy. Keywods Confned polyme; sspatve patcle dynamcs; No-slp bounday condton; Unfom shea flow Atcle I 005-9040(00)-0--06 Intoducton Wth the need of developng technologes n mco- and nano-mete scales, the eseaches of the stuctue and the dynamcs of confned polymes n dlute solutons, especally n the exstence of the extenal foce to dve the flow, have attacted much moe attenton than eve befoe. The applcaton backgound ncludes not only the chomatogaphy, the lubcaton, and the ol ecovey, but also the new technologes elated to macomolecule tanspotng n nanochannels, such as the NA sequencng [], the NA delvey though mcocapllaes n gene theapy [ 7], and the lab-on-chp applcaton [8 ]. In these hghly confned spaces, the effects of the channel wall and the shea ate ae amplfed, and the hydodynamc popetes ae also dffeent fom those n fee soluton. Appaently, t s a complcated poblem whch s elated to the nteacton between the polyme and the wall, the vscosty of the solvent, the wall oughness, and the exstence of the extenal foce to the flow. Thee have been some nvestgatons about the stuaton of pessue dven flow, but few of unfom shea flow, especally n mesoscale. Theefoe, the knowledge about the stuctue and dynamcs of the polyme chans unde Couette flow n confned dlute solutons s of geat mpotance. In expements, t s now possble to develop mcofludc and nanofludc devces wth chaactestc dmensons n the scale of tens of nanometes o even smalle [8,,3]. Some of these expements have evealed that the polyme chans mgate away fom the walls [,4], and the thckness of the depleton laye nceases wth nceasng shea ate. Avalable theoetcal and numecal studes have been focused on the behavo of dlute polyme solutons confned n dffeent nano-channels. Fo example, Ma and Gaham [5], wth the ad of a bead-spng dumbbell model of the polyme molecules, pedcted a mgaton of the dumbbells away fom the wall towad the mdsecton of the channel, whch was manly attbuted to the hydodynamc nteacton between the polyme and the wall. Usta et al. [6,7], va a bead model wth a lattce Boltzmann descpton fo the solvent, obseved that the densty pofle exhbts a peak n the unfom shea flow and two symmetc maxma n the pessue dven flow. A ecent molecula dynamcs smulaton by Khae et al. [8], also showed that a local mnmum of the polyme densty pofle n the pessue dven flow and a peak n the unfom shea flow. They also *Coespondng autho. E-mal: luzhy@jlu.edu.cn eceved Januay 7, 009; accepted Mach 7, 009. Suppoted by the Natonal Natual Scence Foundaton of Chna(No.0774036) and Fok Yng Tung Educaton Foundaton (No.4008).
No. HE Yan-dong et al. 3 emphaszed the effects of the themal dffuson n nanoscale channels. Moe ecently, Mllan et al. [9] and Fedosov et al. [0], usng dsspatve patcle dynamcs, have nvestgated the effects of molecula weght, polyme concentaton, and flow ate on the densty pofles on the ntesecton of the channel of the flud n the pessue dven flow. The dsspatve patcle dynamcs(p) smulaton technque s vey sutable to studng the flud behavo n mesoscale [], whch s n accodance to the length scale of the nanofludc expements. Theefoe n ths eseach, we adopted P wth a no-slp bounday condton, especally wthout the densty fluctuaton nea the wall, to nvestgate the stuctue and the dynamcs of a sngle polyme chan n a confned dlute soluton dven by unfom shea flow. The no-slp bounday condton as well as elmnatng densty fluctuaton nea the wall s nontval snce these ae the chaactestcs of fluds n mesoscale. It should be noted that wall slp and densty fluctuaton nea the wall may be obseved f the length scale s n seveal Angstoms. We wll patculaly dscuss the effects of the Wessenbeg numbe, the shea ate, and the confnement of the nano-channel on the chan confomatonal stuctues, the polyme cente of mass dstbuton, and the chan mgaton n the channel. P Method and No-slp Model P, a mesoscale flud smulaton method, was fst ntoduced by Hoogebugge and Koelman [] n the ealy 990s, whch s based on the consdeatons of momentum consevaton and Galllean nvaance. Then t was successfully appled to the study of polyme solutons [3] and modfed to ts pesent fom [4,5]. In P, each patcle epesents a lage numbe of atoms o molecules and moves accodng to Newton s equatons of moton unde the nfluence of ts neghbos, d = v () dt dv = f () dt The pawse nteactve foce actng on a patcle by a patcle j s chaactezed by thee pats: the consevatve foce( F ), the dsspatve foce( F ), C and the andom foce( F ). The consevatve foce C F, whch s deved fom a soft nteacton potental wthn a cetan cutoff adus c, s gven by C F = α ( ) e (3) whee =, j =, e =. α s the maxmum epulson between patcle and patcle j. In ode to ensue the consevatve foce soft and epulsve that acts along the lne of patcle centes, the mass functon ω c ( ) s chosen as ω c ( )= fo < and ω c ( )=0 fo. The numbe densty(ρ) n ou smulaton s 3. In pncple, ρ should be lage enough to coectly descbe the behavo of lqud, but ρ=3 s a compomse between computaton effcency and the coectness of the model. The nteacton paamete between the polyme and solvent s chosen as fom 5 to 30. The dsspatve and the andom foces ae F = γω ( )( e v ) e (4) F = σω ( )θ e (5) whee v = v v. γ and σ ae both pefactos whch j epesent the stength of the dsspatve foce and the andom foce, espectvely. θ s a Gaussan dstbuted andom vaable wth zeo mean and unt vaance. Accodng to the fluctuaton dsspaton theoem, the dsspatve and andom foces ae coupled togethe to ensue a canoncal equlbum dstbuton va the followng elatons [4] ω ( ) = [ ω ( )] (6) σ =γk B T (7) The smple functon fom fo ω ( ) s chosen accodng to ( ) ( < ) ω ( ) = (8) 0( ) The polyme chan s constucted by connectng the adjacent patcles va an exta hamonc spng, F = S k (9) whee the constant k s 0.0 n ou smulatons. It should be noted that the patcle mass m, the patcle-patcle nteacton cutoff adus c, and the tempeatue k B T ae taken as the unts n ou smulatons. The advantage of P [5] s that t allows a lage ntegaton tme step due to ts soft potental. Howeve, the dsadvantage s also esulted fom the soft nteacton whch can not pevent the penetaton of smulated solvent o polyme patcles nto the wall. We
4 CHEM. ES. CHINESE UNIVESITIES Vol.6 appled a technque to mpose an mpenetable wall whch confoms to the no-slp bounday condton [6]. In ths technque, the wall s constucted by two layes of feozen P patcles dstbuted on a egula lattce. Moeove, based on equvalent foce between wall and flud patcles, t s possble to mpose the no-slp bounday condton n ths technque. The consevatve foce coeffcent of the flud-wall nteactons s theefoe adjusted to be 9.0 n thee dmensons to acheve the no-slp condton n ou smulatons. Howeve, ths method nduces the depleton of patcles and the unacceptable densty fluctuatons nea the wall whch ae unealstc n mesoscale. To ovecome ths poblem, a self-consstent algothm whch s called adaptve bounday condtons [7,8] s adopted hee to update the flud-wall nteactons based on the densty devaton fom the desed dstbuton nea to the sold wall. The foce F W ( b ) actng on the patcles n bn b s then updated accodng to b b W W F ( ) = F ( ) + C [ ρ ( ) / ρ ( ) ] (0) b b W s = a = a whee b s the bn ndex, C W s a postve constant, ρ s () s the local densty values aveaged up to n av bns [fom a to b, b =max(n av +;)], and ρ d () s desed densty values n the same bns. To consde a flud n Couette flow, a velocty s added to all wall patcles, whch can take the value fom 0 to 3.0 wthout volatng the stablty of the smulaton. The walls ae movng n the opposte dectons paallel to X axs. The patcle penetaton though the wall n P s fobdden by applyng bounce fowad bounday condton(as shown n Fg.) to ensue the velocty gadent between the walls lnea and no-slp adjacent to the walls [9]. Fg. Bounce fowad bounday condton The gay domans ae the walls modeled by feezng P patcles. The walls ae n the XY plane, the Z decton s pependcula to the wall. We have smulated a dlute polyme soluton wth a volume facton smalle than 0.0 to satsfy the full development of hydodynamc nteacton [30]. The smulaton box sze s chosen lage enough to avod the nteacton between the chan and each of ts mages d fo dffeent molecula weghts. The sze of a polyme chan s chaactezed by the mean squae adus of gyaton [3] N g = ( cm ) N = () whee s the coodnate of -th segment, and cm s the coodnate of the chan cente of mass. To consde the stength of the confnement, the channel heght s egulated between H=.5 g and H=4.5 g. The neutal wall model s adopted. The ntegaton tme δt=0.0. All ou smulatons wee un at least ove a tme peod of 6 0 6 tme steps. The polyme chans wth lengths N=0 to N=40 wee nvestgated n dffeent solvent condtons and confnements. The Wessenbeg numbe(w n ) was changed fom 0 to 95 n the Couette flow. 3 Popetes of Polyme Unde Couette Flow n Confned lute Soluton We fstly consdeed the nfluence of confnement stength on the polyme confguaton n the condtons wthout flow. Takng a polyme chan wth N=00(the adus of gyaton g =4.30 n fee dlute soluton) as an example, the channel wdths ae vaed fom H=.5 g, 3.5 g, to 4.5 g. We set the nteacton paametes between polyme and solvent α sp, between solvent and solvent α ss, and between polyme and polyme α pp all equal to 5. Thus the system s n athemal condton. The tme evolutons of g n thee dectons ae used to analyze the effect of the confnement. In the case of H=4.5 g, as shown n Fg.(A), the evolutons of g n thee dectons ae vey smla, and they all appoach to 6.0 so that g 4.3. Fg.(B) coesponds to the case of H=3.5 g. Appaently the polyme chan s affected by the confnement. The mean squaed adus of gyaton n Z decton s educed slghtly, whch s 3.9 n contast to 5.3 and 5.7 n X and Y dectons, espectvely. Ths ndcates that the polyme chan s subjected to the epulson fom the wall, whch s the man eason fo the depleton nsde the channel. It can be futhe demonstated n the case of H=.5 g, as shown n Fg.(C), gz tuns to be 3., but gx and gy ae both 5.9. Thus unde ths condton the polyme chan suffes fom a lage loss n fee enegy and s subject to stonge stec depleton foces. Although
No. HE Yan-dong et al. 5 Fedosov et al. [0] ndcated that beyond a dstance of about.5 g, both the local sze and the shape of the polyme wee unaffected by the wall, we found that thee exsted nfluences even fo H=3.5 g. The dffeence may be attbuted to the adaptve shea coecton that they used and the dffeent chan length whch we chose. Fg. Mean squaed adus of gyaton n thee dectons fo the chan wth N=00 X. otted lne, Y. dashed lne, Z. sold lne. (A) H=4.5 g ; (B) H=3.5 g ; (C) H=.5 g. The chan centes of mass densty pofles of the polyme n dffeent confnements wee calculated[see Fg.3(B)]. Appaently the heghts of the chan centes of mass densty pofles ncease and the wdths of the pofles tun to be naowe wth nceasng the stength of the confnement. It shows that the polyme chan pefes esdng n the cente of the channel unde hgh confnement. In all the thee confnement condtons, t s found that the depleton layes ae always less than 0.6 g and nealy the same. α wp =9.0) condtons. Fg.3(B) compaes the densty pofles of chans wth dffeent chan lengths. It can be seen that wth nceasng chan length, the peak heght and the depleton laye thckness nea the wall ae both nceased. Moeove, the shote chans have been much dstbuted close to the walls. These esults agee well wth those of the depleton laye thckness calculated fom Monte Calo and P smulatons [3,0]. We then consdeed the nfluences of Couette flow. The shea vscosty of the confned dlute soluton can be calculated accodng to the pessue tenso, whch s measued by P = m v v + C αβ, α, β, α, β V V j> F () whee V s the volume of the smulaton box. Shea vscosty n assocaton wth the only non-vanshng off-dagonal component P xz = P zx s gven by η= P xz /γ& (3) whee η s the shea vscosty and γ& s the shea ate. Fg.4 shows that the pessue tenso P xz s popotonal to the shea ate γ&, thus the vscosty η s nealy constant whch s an obvous esult fo dlute polyme soluton. Fg.3 ensty pofles of chans wth dffeent chan lengths[n=80( ), N=00( ) and N=40( )(A)] and chan centes of mass densty pofles fo chan wth N=00 as a functon of Z/ g n dffeent confnements wth H=.8 g ( ), H=.3 g ( ), and H=3 g ( )(B) To study the nfluence of chan length on the depleton laye n ou systems, we smulated the chan wth dffeent lengths N=80, 00, and 40 n a channel of H=3 g, n athemal solvent and neutal wall(α ws = Fg.4 Pessue tenso P xz vesus shea ate γ& fo chan length wth N=00 The dashed lne s the fttng to the data ponts whch gves the vscosty η=0.759. We thdly studed the dependence of the densty pofles of the polyme on the nteacton stengths between polyme and solvent. As shown n Fg.5(A), when the nteacton α sp =5(ndcatng that the chan s solvophlc), the polyme pefes to mgate to the cente of the channel. In ths case t can be attbuted to the enhanced hydodynamc nteacton n dlute soluton. When α sp =0, the polyme chan has smla mgatng behavo. When α sp nceases futhe to 30,.e., the chan s solvophobc, the densty pofle s
6 CHEM. ES. CHINESE UNIVESITIES Vol.6 Fg.5 Chan cente densty pofles as a functon of Z/ g wth N=00 (A) At dffeent nteacton stengths between polyme and solvent wth W n =6.5 and H=.5 g ; (B) fo dffeent W n at H=.5 g ; (C) fo dffeent α sp wth W n =30 and H=.5 g. (A) α sp =0; α sp =5; α sp =30; α sp =35. (B) W n =45; W n =65; W n =30; W n =95. (C) α sp =5; α sp =0; α sp =30; α sp =35. obvously flattened and chaactezed by two low peaks nea the channel walls. Fo α sp =35, the polyme chan bascally esdes n a egon vey nea to the wall. In these cases, the polyme confguaton s compact and the chan has the tendency to mgate nea to the walls. The effect of Bownan dft must be taken nto account whch makes the polyme mgatng to the wall and s efeed n pessue dven flow [33]. As s expected, the polyme n bad solvent pefes the compact confguaton so that the effect of the wall-nduced hydodynamc nteacton s educed to balance the polyme dft. We can clealy see that the mgaton of polyme n confned dlute soluton s stongly dependent on the solvent qualty. The stength of the shea flow can be specfed by the dmensonless Wessenbeg numbe(w n ), whch s the poduct of the shea ate, γ&, and the polyme elaxaton tme, λ [34]. The adus of the gyaton of the chan wth dffeent chan lengths and the coespondng elaxaton tme n equlbum ae shown n Table. We take the chan length N=00 as an example. The evaluaton of W n s W n =λγ&, n whch γ& = V x /L whee L s the sde length of the smulaton box and V x s the velocty of wall patcles. Table Equlbum adus of gyaton g and elaxaton tme fo dffeent chan lengths N 0 60 00 40 g /nm.60 3.05 4.30 5.4 λ/nm.78 96.55 30.66 373.5 Fg.5(B) shows the densty pofle of the polyme wth chan length N=00 and dffeent W n. In all these cases, the densty pofles have a maxmum n the channel cente. When W n =45, ths maxmum s the lowest but the densty pofle s boade. Wth W n nceasng fom 45 to 95, the densty pofle becomes to be naowe and ts maxmum s enhanced. These pofles show that the shea flow causes the chans to mgate towads the cente of the channel [8]. Moeove, on keepng Wessenbeg numbe constant, fo example W n =30, t s found that the densty pofles of the polyme ae lagely dffeent n vaous solvent condtons, as shown n Fg.5(C). The densty pofles stll have the maxma n the cente of the channel, but when the chan changes fom solvophlcty(α sp =0) to solvophobcty(α sp =35), the densty pofle becomes boade and flatte. It s evdent that the hydodynamc nteacton s moe developed fo the polyme bette dspesed n the solvent, so the polyme can mgate easly to the cente of the channel. Wth nceasng chan solvophobcty, the polyme pefes a compact confguaton and the wall-nduced hydodynamc nteacton s educed, thus the polyme chan tends to mgate towads the channel walls. Smla esults ae also obtaned wth a lage Wessenbeg numbe W n = 95. 4 Conclusons We have used P, n whch the solvent patcles ae smulated explctly to allow fo the wall-nduced hydodynamc nteacton, to study the stuctue and dynamcs of confned polyme chan n dlute soluton, n dffeent shea flow felds. The confnement s taken nto account unde no-slp bounday condton and no densty fluctuaton nea the wall, whch s well used n mesoscale to mmc nanochannel. In zeo-shea systems, the polyme chan s gadually stetched along the unconfned dectons wth nceasng confnement stength, and the polyme densty pofle changes fom boad to naow. In slow and modeate Couette flows, thee exsts a constant shea vscosty. The polyme, n both modeate and stong Couette flows, mgates to the wall when the solvent changes fom good one to bad one. As the Bownan dffuson and the wall-nduced hydodynamcs has a balance, only one peak s found n the mddle of the densty pofle fo hgh W n and the polyme mgates to the cente wth nceasng
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