Bilayer Membrane in Confined Geometry: Interlayer Slide and Steric Repulsion.

Transcription

1 layr Mmbran n onfnd Gomtry: Intrlayr Sld and Strc Rpulson. S.V. aoukna, S.I. Mukn Tortcal Pyscs Dpartmnt, Mosco Insttut for Stl and Alloys, Lnnsky Pr.,, 909 Mosco, Russa PAs: 68.5, s, 87.6.Dg. Abstract W drvd fr nrgy functonal of a blayr lpd mmbran from t frst prncpls of lastcty tory. T modl plctly ncluds poston-dpndnt mutual sld of monolayrs and bndng dformaton. Our fr nrgy functonal of lud-crystalln mmbran allos for ncomprssblty of t mmbran and vansng of t n-plan sar modulus and obys rflctonal and rotatonal symmtrs of t flat blayr. Intrlayr sld at t md-plan of t mmbran rsults n local dffrnc of surfac dnsts of t monolayrs. T sld ampltud drctly ntrs fr nrgy va t stran tnsor. For small bndng dformatons t rato btn bndng modulus and ara comprsson coffcnt, b / A, s proportonal to t suar of monolayr tcknss,. Usng t functonal prformd slf-consstnt calculaton of strc potntal actng on blayr btn paralll confnng alls sparatd by dstanc d. W found tat tmpratur-dpndnt (T curvatur α:, at t mnmum of confnng potntal s nancd four tms for a blayr t sld as compard t a unt blayr. W also calculat vscous mods of blayr mmbran btn confnng alls. Pur bndng of t mmbran s nvstgatd, c s dcoupld from ara dlaton at small ampltuds. Tr sourcs of vscous dsspaton ar consdrd: atr and mmbran vscosts and ntrlayr drag. Dsprson rlaton gvs to brancs, ( α T / d b. onfnmnt btn t alls modfs t bndng mod ( t rspct to mmbran n bulk soluton. Four dpndncs ar obtand: ~ - α d / ; - b d 6 / ;- A /b s and - A / m for t conscutv ntrvals of av vctor : <<(α/ b / ; (α/ b / <<<</d ; /d<<<<(b s / m / and (b s / m / <<. Smultanously, ntrlayr slppng mod, dampd by vscous drag, rmans uncangd by confnmnt: ~ - A /b s and ~ - b / for <</d and /d<<, rspctvly. Introducton ll mmbran s caractrd by compl structural and dynamcal proprts [,,]. Tortcal modlng and dscrpton of lpd mmbrans s of grat fundamntal and practcal ntrst and as long noug story. Pnomnologcal modl ntroducd by Hlfrc [] tratd lpd mmbran as a sngl st t bndng rgdty and spontanous curvatur. Ts modl as latr usd for calculaton of fruncy spctrum of mmbran n atr soluton [5] and for nvstgaton of strc ntractons of mmbrans n multlayr systms [6]. layr structur of lpd mmbran as analyd by Evans and Yung [, 7], o consdrd dynamc couplng btn

2 t monolayrs and ntrlayr sld. Allong for t couplng btn local curvatur and local dnsts of lpds tn t monolayrs t fruncy spctrum of mmbran n t bulk atr as rcalculatd [8]. Aftrards, vscous mods of a blayr adrng to a substrat r found [9] usng dnsty-dffrnc modl [8], supplmntd t bndng potntal [0]. In ts papr drv n fr nrgy functonal of a blayr mmbran t ntrlayr sld. Intrlayr sld functon, mmbran strtcng and bndng ampltud ntr drctly t stran tnsor of t mmbran. Our functonal s dstnct from but can b rducd (as partcular cas to dnsty-dffrnc modl usd n [8, 9]. W study dynamcs of blayr mmbran n atr soluton confnd btn paralll alls as a stp toards undrstandng ntr-mmbran ntractons. T ffct of confnmnt s modld by strc potntal []. In Scton ntroduc an ansotropc lastc modul tnsor contanng ntally ndpndnt componnts. T rflcton and rotaton symmtrs of t flat blayr rduc t numbr of componnts to 5. Nt, mpos ro sar strss modulus and ncomprssblty constrant. W rstrct ourslvs only to t cas of small bndng dformatons and clud t corrspondng stran and lastc tnsor componnts. Tus, t numbr of ndpndnt componnts of lastc tnsor n t fr nrgy functonal s rducd to to. T drvd fr nrgy functonal of a blayr mmbran contans tr flds dscrbng ara dlaton and bndng dformaton coupld to ntrlayr sld. In Scton a parabolc strc potntal actng on t mmbran btn confnng alls s ntroducd. W calculat slf-consstntly t curvatur of t confnng potntal at ts mnmum. W valuat t curvaturs of t strc potntal for a blayr t sld and for a unt blayr. In Scton us t drvd functonal to study dynamcal proprts and dsspatv mcansms of t blayr mmbran n atr soluton confnd btn paralll alls. W nvstgat only pur bndng dformatons of t mmbran (ro total latral strtcng, c dcoupl from ara dlaton. Vlocty fld n t surroundng atr s found by solvng Stocks uatons for ncomprssbl flud. Flud vlocty vanss at t alls. Euatons of moton ar dtrmnd as boundary condtons on t mmbran surfacs by rurmnt of forc balanc nglctng nrtal ffcts. Tr sourcs of dsspaton ar ncludd nto dynamc uatons: atr and mmbran vscosts and ntrlayr drag. In t last Scton of t papr dscuss lmtatons and possbl mprovmnts of our modl and corrspondnc t arlr rsults [9]. In Appnd A statc bavor of mmbran n aal-symmtrc cas s studd. Analytcal solutons ar obtand for a crcular mmbran bnt by trnal prssur. Mmbran bndng, ntrlayr sld and latral strss dstrbuton ar found as functons of prssur across t mmbran. In Appnd rdrv t dsprson rlaton [8] for a mmbran n t bulk atr soluton usng our fr nrgy functonal.

3 Fr nrgy functonal Fr nrgy dnsty of ansotropc mdum can b rttn to t lost ordr n lastc stran tnsor as [,]: F λ klm u k u lm, ( r a summaton ovr t rpatd ndcs, k, l, m s prformd. Indcs, k,l, m acur valus,,, numratng t spac as, y, appropratly. Hr u s t stran tnsor, k s t lastc (modulus tnsor. y dfnton t lastc tnsor s symmtrc undr t cang k, l m and,k l,m: λ λ λ λ, klm klm kml lmk and as ndpndnt coffcnts. Allong for (, t (symmtrc strss tnsor k σk s dfnd as: F σ k λklm ulm. ( u λ klm In a symmtrc mdum tr s a corrlaton btn dffrnt componnts λ klm and t numbr of ndpndnt lmnts of t tnsor of lastc modulus s rducd. Lt us ntroduc artsan coordnat systm t t -as prpndcular to t unprturbd (flat mmbran plan, and t t monolayr ntrfac (.. blayr mdplan postond n t (,y-plan (at 0. T mmbran tcknss s ual to, and t flat mmbran s modld as a tn blayr plat bound by t - and plans t n-plan lnar dmnson R>>. T (,y plan s a plan of rflcton symmtry. Ts mpls tat undr a transformaton, y y, - t fr nrgy must b nvarant. Trfor all t componnts λ klm t odd numbr of -ndcs ar ual to ro [9]. Mmbran can b consdrd latrally sotropc. Tn -as s an as of rotatonal symmtry. Tus t prsson for lastc nrgy dnsty, F, rducs to t follong [0]: F λ λ ( u u yy λ u λ yy u u yy ( λ λ yy ( u u u u λ ( u u yy Assumng tat t mmbran s n lud stat, rur tat t n-plan sar modulus y u y ( (t coffcnt n front of u y vanss, and tus obtan: λ λ. Hnc, prsson ( yy fartr smplfs and acurs t form: F λ λ ( u u yy λ u λ ( u u u yy u ( u u y (

4 Lt an trnal forc appld prpndcular to t mmbran plan nduc a small bndng dformaton along t -as. Allong for a typcal prmntal stuaton, consdr a tn mmbran t t rato of ts tcknss to latral lnar dmnson (ffctv radus of t ordr of 0 -. Hnc nglct t appld trnal strsss on t top and bottom mmbran surfacs t rspct to t ntrnal latral strsss n t. Du to t smallnss of t mmbran tcknss, ro strsss on t surfac also vans n t bulk of t mmbran. So mpos t follong condton usually mpld for t tn plats []: σ ( r σ y ( r σ ( r 0 ; (5 r r spans ovr t mmbran s bulk. In accord t (, (, ts componnts of t strss tnsor ar rlatd to t stran tnsor componnts as follos: σ λ u, y λ yy u y σ, (6 σ λ ( u u λ u. (7 yy ombnng t rlatons (5 and (7, fnd: λ u ( u u yy λ. (8 It s ntrstng to mnton tat as follos from (6 fulfllmnt of t frst to condtons (5 rurs vansng of t stran tnsor componnts u and u. Vansng of ts componnts ould also corrspond to nfnt lastc modul y λ and λ yy. ondton (5 prmts to omt t trms contanng u and u n (. Also usng (8 and prssng u va u u n (, fnd t prsson for t fr nrgy dnsty: yy ( u u yy λ F λ. (9 λ In addton, mpos t "ncomprssblty" condton,.. t constancy of t bulk dnsty of t mmbran: u u u 0. (0 yy T condton (0 s satsfd smultanously t (8 f: Fnally, t fr nrgy dnsty s rttn as: ( y λ λ. F u u yy, ( r dnots a suprposton of ansotropc lastc modul: ( λ r In t lnar appromaton for t stran tnsor on as: u k u u k u k λ., ( s t -t componnt of t dstorton fld.

5 In ordr to ntroduc t ssntals of our modl n a smplst ay, lmt t follong dscusson to t small bndng ampltud cas,.. mpos condton: u <<, r u (r s t -componnt of dsplacmnt, dscrbng t dformd mmbran. Also, sall nglct t -dpndnc of t componnt u (r n t tn plat appromaton [], tus dfnng t "sap"-functon ξ(, y ( r ndpndnt of t dpt. Substtutng ξ(, y n t dfnton ( u and traftr nto t rlatons (6 and condtons (5, on obtans t follong partal dffrntal uatons: u, ξ u y. ( ξ y No, l ntgratng uatons (, lt us ntroduc to functons: (nomognous latral strtcng of t mmbran a(,y and n-plan sld ± f(,yof t lor (<0 and uppr (>0 monolayrs at t md-plan 0 of t mmbran. Tus, t n-plan dstortons of a blayr mmbran av t follong form: u u ξ(, y ( Θ( Θ( f (, y a (, y, ξ(, y ( Θ( Θ( fy (, y ay (, y, ( y y u and u r t stp-functon s dfnd as: Θ( > 0 and Θ( < 0 0, and t coc of t sgn of Θ and of ts argumnt s mad for t fartr convnnc. It s ort mpasng r t lmtatons of valdty of t rlatons (. T prssons ( ar clarly dstnct from t usual prssons for tn plats []. In t lattr cas t dsplacmnts u and u ar st to ro at 0, mplyng t prsnc of nutral (not strtcd y y surfac at t md-plan of t plat n t small bndng appromaton: ξ << []. It ll b son n Appnd A (s uaton A. tat t scond trm n ( s of t sam ordr as t frst on: f,y~ ξ /R, r R s ffctv radus of t mmbran. Small bndng appromaton s justfd n t uadratc n ξ trm s nglgbly small t rspct to lnar trms n t prssons for n-plan dstortons u and u y :O(ξ /R<< ξ /R. T lattr condton s fulflld as long as ξ <<. On t otr and, for strongly bnt tn plat t ξ trm domnats ovr ξ trm and tus gr ordr trms sould b addd on t rgt and sd of uatons (. Lt us no dscuss t pyscal manng of t prssons (. T mmbran strtcng a(,y dfns poston-dpndnt sft of t nutral surfac (along coordnat l sld functon f(,y multpld by t stp-functons lad to t splttng of ts nutral surfac nto to surfacs blongng to uppr and lor monolayrs. Ts surfacs ar dtrmnd from t condtons: u (, y, 0 and u (, y, 0. T functon f(,y provds addtonal dgr of y 5

6 frdom n comparson t a blayr tout sld (or a sngl monolayr. Undr t condton of ro total latral strtcng (.. pur bndng dformaton, a 0 t prsnc of t functon f mans tat t nutral surfac splts nto to suc surfacs locatd n ac monolayr symmtrcally t rspct to t md-plan 0. T total ampltud of mutual ntrlayr sld at ac pont,y of t md-plan s tn gvn by f(, y, c sgnfs dscontnuty of n-plan dstortons u and u y across t md-plan 0. In t oppost cas: f 0 t monolayrs ar coupld togtr (no ntrlayr sld and t dstorton fld s t sum of bndng and strtcng (for small dformatons, t lattr bng contnuous across t md-pan 0. In gnral, t dstorton fld ( ncluds bndng, strtcng and mutual ntrlayr sld. Substtutng ( nto (, procd to dtrmn t stran tnsor componnts: u u yy ξ(,y ξ(, y y ( Θ( Θ( ( Θ( Θ( f (, y a (, y, fy (, y a y (, y, (5 y y and u can b prssd va u and u yy usng (8. It s mportant to mnton tat (n accord t dfnton ( t addton of dscontnuous trms f, y Θ( ± n ( lads to non-vansng contrbutons to t u and u y componnts of t stran tnsor proportonal t Drac s dlta functon δ (. Allong for t nrgy cost-fr (statc ntr-monolayr sld, dscard ts contrbutons to and u, tus kpng t lattr ual to ro n accord t condtons (5, (6. T fr nrgy functonal of t ol mmbran Fv s obtand by an ntgraton of t fr nrgy dnsty F ovr t mmbran s volum stps: frst ovr t tcknss coordnat (-<<, and tn ovr t mmbran plan {, y}. Usng prssons ( and (5, fnally fnd: ~ Fv u u yy dv ξ ddy ~ ~ ~ ( ξ ( ~ ( ( f f ddy a ddy ( ( ~ Hr t tld rfrs to to-dmnsonal dffrntaton (, y. Actually, uaton (6 s ut rmarkabl. T man curvatur of t ntrlayr surfac H, s prssd as follos: ~ ξ ξ ξ H. (7 y Trfor t frst trm on t rgt and sd gvs ffctv bndng nrgy,.. trnsc curvatur-bndng nrgy functonal, F c, of t "standard" form [, ]: u y (6 6

7 F c b ( H c0 ds, (8 t ro spontanous curvatur c 0. Hr b s bndng rgdty (modulus. omparng (6 and (8, fnd: b. T last trm n (6 accounts for lastc nrgy of ara dlaton t ara comprsson coffcnt dfnd as A. In gnral, local rlatv ara dlaton S / S uals u u yy []. ~ Accordng to uaton (5, t rlatv ara dlaton s gvn by a, l t dffrnc of rlatv ara dlatons btn t monolayrs s gvn by ( ~ f. Hnc, t ( ~ a trm n (6 arss du to contnuous (across t monolayrs ntrfac 0 latral strtcng of t mmbran, c lads to t cang n avrag lpd dnsty. T ~ ( f trm rprsnts t nrgy of local ara dffrnc of t monolayrs (ara-dffrnc lastcty [], c s uvalnt to dffrnc of lpd dnsts n monolayrs (dnsty-dffrnc modl [8]. In prncpl, ts nrgy s not rlatd to t prsnc of nutral surfacs tn t monolayrs (at larg mmbran strtcng/comprsson tr ar no nutral surfacs, c ould oby u 0, s prsson(. As apparnt from uaton (6, t rlaton btn bndng and ara comprsson coffcnts (s [] b / ~ A coms out naturally n our drvaton. Nt, t scond trm on t rgt and sd of uaton (6 prsss couplng btn bndng dformaton and ntrlayr sld producng local ara dlaton dffrnc btn monolayrs. Not, tat bndng s dcoupld from (contnuous ara dlaton, causd by latral strtcng, n t lost ordr appromaton. Du to ydropobc ffct t monolayrs, l sldng, ar forcd to stck togtr and to follo t sam sap dfnd by ξ(,y on t monolayr ntrfac. Mutual ntrlayr sld along t ntrfac lads to rlaaton of strtcng/comprsson of t monolayrs causd by bndng dformaton, and tus prmts t fr nrgy dcras. Fnally, our fr nrgy functonal s nvarant to transvrsal sld of monolayrs, suc tat dv(f0. Hnc, t nrgy dos not cang undr a mutual rotaton of t monolayrs (as a ol or a poston ndpndnt sft of on of t monolayrs t rspct to t otr. W ll consdr pur bndng dformatons of t mmbran t no ovrall strtcng Trfor, rur t latral stran ntgratd ovr t tcknss to b ro n ac pont of t mmbran. Ts mposs rstrcton on t form of u and u : t functon a(,y sould b ual to ro n vry pont of t blayr. Hnc, ts functon s omttd vryr blo. Tn stran tnsor componnts can b rttn as: ξ f u ( Θ( Θ(, u yy ξ y ( Θ( Θ( y fy, (9 y 7 u y

8 and u agan can b prssd va u and u yy usng (8. T fr nrgy functonal of t mmbran acurs t form: F v ~ ( u u yy dv ( ξ ~ ~ ~ ( ξ ( f ddy ( f ddy} ddy (0 To study t proprts of t functonal (0 n dtal a smpl problm t cylndrcally symmtrc dformaton s dscussd n Appnd A. Eulbrum stat of t mmbran s dfnd by t Eulr-Lagrang uatons, c ar obtand by uatng to ro t frst varatonal drvatvs of t lastc nrgy functonal F ( ξ, f t rspct to t functons ξ(r and f(r. onfnng potntal for a blayr t sld Drct nflunc of confnd gomtry on t mmbran bavor manfsts tslf n t rducton of t manfold of accssbl mmbran conformatons. Strc ntractons of mmbran t confnng alls (s Fgur can b modld [] by ntroducton of an tra potntal nrgy W dpndnt on t bndng ampltud: W confnng potntal, W, acurs t form: F v α ξ ~ ( u u dv ( ξ yy ~ ~ ~ α ( ξ ( f ddy ( f ddy}. T fr nrgy functonal (0 appndd t ddy ξ ddy ( urvatur of t confnng potntal at ts mnmum d W α dξ ξ 0 s calculatd blo usng slf-consstncy procdur. In Fourr spac {, } F v ( f t fr nrgy functonal ( s rttn as: r. y * * ( ξ f ξ f y α ξ d (π (π d d y ( (π d y d d y W dagonal t uadratc form n ( t rspct to ξ and f by lnar transformaton: ~ R ξ ~ Im ξ R ξ Imξ ~ ~ Im f, R f, ( 8

9 r ~ α, b. b ~ In t varabls and t nrgy functonal ( taks t form: ξ f F v ~ d d y ξ ~ f. ( 00 (π Usng rlatons ( and functonal ( calculat t trmodynamcal avrag ξ : kt kt ξ (5 α α 6 α r k s oltmann s constant, T s tmpratur. In t absnc of ntrlayr sld only t frst trm n uaton (5 ould rman, as obtand n [,]. T scond trm n (5 sgnfs nancmnt of t bndng fluctuatons causd by ntrlayr sld. T lattr lads to rlaaton of t latral strsss (s Appnd A and Fgur and tus to a dcras of fr nrgy of t bnt mmbran. T man-suar fluctuatons of bndng ampltud ar found as: ξ ( r 0 ξ d π k T α. (6 In t confnd gomtry t avrag bndng ampltud s rstrctd to fnt spac d, avalabl btn t alls (nglctng volum occupd by t mmbran tslf,..: << d, tus provdng t slf-consstncy condton for dtrmnaton of t ffctv rgdty α : r µ. ξ µd, (7 Substtutng (6 nto (7 obtan a slf-consstncy soluton for α : ( kt α. (8 6 µ d b W also valuat r t curvatur of t confnng potntal, α 0, for a unt blayr (tout ntrlayr sld. In ts cas t scond trm on t rgt and sd of (5 s ro and nc: ( kt α 0. (9 6 µ d b Tus, ntrlayr sld rsults n consdrabl nancmnt (α/α 0 of t curvatur of confnng potntal. 9

10 layr dynamcs: vscous mods To study t dynamcal proprts of t ntroducd modl of blayr mmbran t ntrlayr sld dtrmn r uatons of moton and fnd t gnmods of t mmbran surroundd by atr soluton. W ar ntrstd n t bavor of mmbran confnd btn paralll alls (s Fgur. Lt a flat mmbran l n t (,y-plan t t normal pontd along -as. W trat ac monolayr consttutng t mmbran as a (unt to-dmnsonal condnsd structur. W rur t ulbrum btn vscous strsss rtd on t mmbran surfac by atr soluton and mmbran rstorng forc. W nglct nrtal ffcts and ntroduc tr sourcs of vscous dsspaton: atr and mmbran vscosts and ntrlayr drag T forc balanc uatons ar prssd as: δfs δξ ( 0 ( 0 0 (0 δf δf s m t ~ f ( f b ( 0 ( 0 0 s t ( ( 0 ( 0 0 ( Hr t flud strss tnsor s dfnd as: ydrostatc prssur, v - vlocty and k p δ k v k v k, r p dnots - vscosty of atr soluton. T flud strss tnsor s valuatd at uppr (0 and lor (-0 mmbran surfacs and carrs t sgn of t normal. T frst trm on t lft and sd of (0 s t lastc rstorng mmbran forc, c s balancd by normal to t mmbran surfac vscous strss of t flud. Euaton ( rprsnts forc balanc n latral drcton and contans t follong contrbutons [,8]: a tangntal tracton on ntr-layr surfac du to monolayrs dffrntal flo; b cornt surfac flo of t monolayrs as unt surfacs (t dynamc vscosty m,; c vscous drag btn monolayrs (caractrd by coffcnt b s c arss at fnt vlocty of tr mutual sld; d tracton of t surroundng flud. Euaton ( accounts for t absnc of total strtcng forcs rtd by atr on t mmbran snc dscuss r only pur bndng dformatons of mmbran,.. n total ara dlaton s ro. sds t balanc uatons (0-(, Navr Stocks uatons for atr solutons surroundng t mmbran sould b addd. In t small vlocty lmt, tratng flud as ncomprssbl and nglctng nrta, t crpng flo uatons ar rttn: p v, v 0. ( T non-slp boundary condtons at mmbran-atr ntrfac provd t contnuty of normal and tangntal vlocts of t flud and t mmbran: 0

11 ξ t f t v v (±0 ( f ( 0, v ( 0,, y. (5 t onfnmnt btn paralll alls at dstanc d mpls vansng of atr vlocty (normal and tangntal componnts at t alls surfacs: v j ( ± d 0, j, y,. (6 In ordr to fnd dsprson rlaton mak Fourr-transform of t fr nrgy functonal ( and of t forc balanc and crpng flo uatons. For ts purpos t vbraton s pandd n plan avs propagatng along drcton. Tn, fr nrgy dnsty, F s (, taks t follong form: F * * ( ξ f ξ f f α ξ s (, ξ and L y Fv F (, dd s, (8 (π 0 r L y s systm dmnson along y-as, and av omttd nd n t subscrpts of t Fourr componnts. Rstorng mmbran forcs ar gvn by functonal drvatvs of t fr nrgy: (7 δf s α ξ f * δξ (9 F δ s ξ * δf f (0 Fourr transforms of crpng flo uatons ( for t componnts of atr vlocty and t prssur v (, v ( and p p ( ar rttn as: 0, p (, p ( t t. ( W fnd t follong solutons of dffrntal uatons ( t normal vlocty contnuous at 0, obyng also condton of ro latral strtcng forc actng on t mmbran (uaton ( and condton v ( 0 v ( 0 rsultng from uaton (5: > 0 : ( p ( < 0 : (

12 [ ] [ ] [ ] [ ] < > 0 : 0 : ( < > 0 : 0 : ( Ts soluton mantans t symmtry rlatons compatbl t t confnd gomtry:, (, ( π,, (, ( π (5 Pyscal manng of (5, accordng to dfntons gvn bfor (, s tat / componnt of atr vlocty around vbratng mmbran bavs symmtrcally/ antsymmtrcally undr smultanous translaton by alf-prod ( / π along t av propagaton drcton and mrror rflcton n t md-plan btn t confnng alls (. Fartr, clud unknon coffcnts, usng stck boundary condtons at t alls (6. Tn substtut solutons n t form (-( nto Fourr transformd forc balanc uatons (0, ( (plotng (9, (0 and nto non-slp condtons (, (5 at t atrmmbran ntrfac. Tus, fnally obtan algbrac systm of four lnar omognous uatons for unknons,, and f : ξ [ ] [ ] [ 0 ( α ξ d d d d d f ] (6 ( ( ( [ ] [ ] 0 ξ d d d s m d b f (7 0 ( ξ d d d d d (8 0 ( ( d d d d d d d d f (9 Dsprson rlaton ( s found by uatng to ro t dtrmnant of t systm (6- (9. T lattr gvs uadratc uaton for (, c rsults n to brancs and ( (, s Fgur. To vscous mods: ydrodynamcally dampd bndng mod and ntr-monolayr slppng mod - m and t por la ( cangs t avlngt of fluctuatons. For pur bndng dformaton of t mmbran tr st up to four ydrodynamc rgms (dpndng on t paramtrs of t systm, sparatd by tr crossovr av vctors.

13 W us t rsult (8 to stmat an uppr lmt, 0, of t smallst -ntrval r t gnmods ar modfd by t confnng potntal,.. r t nducd rgdty trm domnats ovr t bndng trm ~ n ( (and n t frst brackt n uaton (6: << 0 α b kt b b d µ. (50 For typcal valu of bndng rgdty at room tmpratur b ~ 5kT [] 0 ~ α 0. ~. T d scond crossovr av vctor, /d, bounds t long avlngt rgm r confnmnt of t surroundng atr btn t alls ffcts mmbran dynamcs. For >>/d mmbran bavs as n t bulk atr soluton. W assum tat dstanc btn confnng alls s muc gratr tan monolayr tcknss (d/~0. T crossovr av vctor for t bulk flud (s Appnd at gvn coc of paramtrs 0-7 cm, 0 - dyn sc/cm, b s 0 7 dyn sc/cm acurs t valu 5 - ~ 0 cm and trfor obys <</d. Tus, t dos not nflunc dynamc b s bavor of t mmbran n confnd gomtry. In t ntrval of stll sortr avlngts tr s on mor crossovr av vctor: bs 7 ~ 0 / cm - ( m dyn sc/cm, c obys m /d<<. Hnc, nvstgat four ntrvals of av vctor valus: << 0, 0 <<<</d, /d<<<< and <<. For long avlngts: <</d, confnmnt btn t alls modfs t bndng mod t rspct to mmbran n t bulk soluton (s Appnd : ~ b, (5 and rsults tr n - or n 6 -dpndncs of nstad of -dpndnc of t bulk mod. For << 0 t bndng mod bcoms: αd. (5 T mod ( s drvn by strc potntal, caractrd by curvatur α, and s dampd by vscous losss n t surroundng flud. For 0 <<<</d t ydrodynamcally dampd bndng mod s gvn by: 6 d ~ 6 bd. (5 In ts av vctor ntrval fnt tcknss, d, of atr layrs ffctvly nancs atr vscosty from to /(d >>. Rsult (7 concds (modulo numrcal coffcnt t t dampd vbraton mod of rytrocyt alls consstng of to mmbrans, c comprs lud btn tm [5].

14 Smultanously, ntr-monolayr slppng mod, (, dampd by vscous drag at t monolayr s mutual ntrfac, rmans uncangd by confnmnt (s Appnd : A ~ (5 bs bs For a mmbran n t bulk soluton t mng of bndng and slppng mods occurs at, dfnd n Appnd. T rlatv ordr of t paramtrs, /d, by ncrasng valu dpnds on t coc of caractrstc paramtrs of t systm. Undr our coc <</d and t mng of t mods s dlayd up to / d, s Fgur. W spculat tat ts appns bcaus confnmnt ndrs bndng fluctuatons and trfor bndng mod rmans slor tan slppng mod up to / d. In sort-avlngt lmt, >>/d, rcovr, as pctd, t rsult for a mmbran n t bulk atr. onfnmnt s not rvald n ts cas bcaus mmbran-nducd vbratons of atr dcay ponntally bfor racng t alls. Namly, for >>/d t branc ( corrsponds no to bndng mod dampd by vscous losss n t surroundng flud. ~ 6 b. (55 T rnormald bndng rgdty ~ arss for g fruncy fluctuatons (compar t numrcal coffcnts n (5 and n (55 bcaus bndng mod s fastr tan ntrlayr slppng mod [8,9]; ntr-layr sld ladng to rlaaton of latral strsss n monolayrs s rtardd. In t ntrval /d<<<<, t branc ( ara comprsson modulus (suprscrpt atr cas: ~ bs b A s bcoms ntr-layr slppng mod t rnormald blo ndcats tat soluton concds t t bulk, (56 Fnally, for >> t ( mod s drvn by (g fruncy ffctv rgdty and s dampd by monolayr surfac vscosty monolayrs ar dynamcally coupld: m m, c domnats ovr ntrlayr drag as t. (57 Vscous mods for a mmbran n confnd gomtry obtand n ts papr ar n ualtatv accord t t rsults for mmbran bound to substrat [9]. W av not ncludd t mmbran tnson nto our fr nrgy functonal, bcaus n t consdrd lmt of small bndng dformatons of t blayr, t trm proportonal to gradnt of bndng ampltud vanss []. Dsprson rlaton for blayr mmbran n t bulk atr basd on our fr nrgy functonal (0 s drvd n Appnd and s also n accord t arlr rsults, obtand usng dnsty-dffrnc modl [8].

15 5 onclusons A novl fr nrgy functonal of blayr flud mmbran drvd n ts papr rflcts mportant pyscal proprts of t mmbran dfnng ts dynamc bavor. T functonal allos for to-dmnsonal lud-crystalln structur of t mmbran and ak adrnc btn t consttutng t monolayrs c rsults n tr mutual sld undr (bndng dformatons. Our fr nrgy functonal contans tr coupld flds paramtrng dgrs of frdom rlatd t bndng of mmbran, ntrlayr mutual sld and ara dlaton. Usng ts functonal av calculatd slf-consstntly t curvatur of ffctv strc potntal actng on t mmbran btn to paralll confnng alls. W found tat t curvatur at t potntal s mnmum (locatd at t mddl btn t alls s nancd four tms for a blayr t ntrlayr sld n comparson t a unt mmbran (t forbddn sld of t sam tcknss. Ts ncras can b ascrbd to (partal dcras of latral strss n t bnt mmbran va ntrlayr sld. T rlaaton of strsss ffctvly lors t nrgtc cost of mmbran bndng and ncrass trmodynamc probablty for conformatons t gratr bndng ampltuds. Ts n turn amplfs strc rpulson. W av also calculatd t dsprson rlatons for a mmbran confnd btn paralll alls. Our rsults ar n ualtatv accord t tos for mmbran bound to a substrat [9]. onfnmnt modfs vscous mods ( at long avlngts compard to t bulk atr cas. W av found four av vctor ntrvals sparatd by tr caractrstc av vctor valus: 0<</d<<, dfnd n Scton. T nvrs of t alf-dstanc d btn confnng alls dvds -as nto to ntrvals t confnd (<</d and bulk (>>/d bavor, rspctvly. Wav vctor 0 dlmts t ntrval of -valus, n c strc potntal modfs spctrum of bndng mods. In t ntrval 0 <<<</d found ( 6 dpndnc of bndng mod, smlar to prstaltc mods of a soap flm [5]. Unlk n [9], do not obtan ( dpndnc, bcaus ovrall mmbran tnson s not ncludd nto our fr nrgy functonal. Snc consdr t lmt of small bndng dformatons of a flat blayr t trm proportonal to gradnt of bndng ampltud vanss []. In t ntrval >>/d confnmnt s not mportant snc mmbran-nducd vbratons of atr dcay ponntally bfor racng t alls. As n t bulk cas, at > t monolayr surfac vscosty domnats ovr ntrlayr drag and t monolayrs bcom dynamcally strongly coupld. Fnally, mnton som lmtatons and possbl mprovmnts of our approac. Our functonal rspcts rflctonal symmtry of a flat blayr and trfor mpls tat spontanous curvatur s ro. W assumd a tn-plat appromaton for ac monolayr t constant lastc modul. In otr ords, dvlopd pnomnologcal ffctv mda modl. Hnc, only fluctuatons t avlngt largr tan ntr-molcular dstanc n lpd monolayr ar consdrd. W av plotd smallnss of bndng-to-tcknss rato usng lnar appromaton for t strss tnsor. In t small bndng appromaton ara dlaton s dcoupld from bndng. In ts papr 5 m

16 dscussd only pur bndng dformatons, nvrtlss, ara dlaton dynamcs can also b studd usng our functonal. W found only dampd gnmods of t mmbran n confnd atr soluton. T propagatng mods ll b consdrd lsr. T autors ar gratful to prof. Robjn runsma for t formulaton of t problm and to prof. Yu. A. madv and s coorkrs for usful commnts. 6

17 Rfrncs. A. n-saul, n Structur and Dynamcs of Mmbrans (Elsvr Scnc, U. Sfrt, R. Lposky, n Handbook of ologcal Pyscs, dtd by R. Lposky, E. Sackmann (Elsvr Scnc. V., A. Yung, E. Evans, J. Pys. Franc II, 5, 50 (995.. W. Hlfrc, Z. Naturforsc, 8 (, 69 ( F. rocard, J. F. Lnnon, J. Pys. 6 (, 05 ( W. Hlfrc, Z. Naturforsc, (a, 05 ( E. Evans, A. Yung, Lpds m. Pys. 7, 9 ( U. Sfrt, S. A. Langr, Europys. Ltt. (, 7 ( M. raus, U. Sfrt, J. Pys. II Franc, 7 ( U. Sfrt, Pys. Rv. E 9 (, (99.. F.. Macntos, Pys. Rv. E 50 (, 89 (99.. L.D. Landau, E.M. Lfst, Tory of Elastcty. (Tortcal Pyscs, V.7, Prgamon Prss, N York, H. lnrt, Gaug Flds n ondnsd Mattr (Strsss and Dfcts, V., World Scntfc Publsng o. Pt. Ltd., Sngapor, Safran S.A. Statstcal Trmodynamcs of Surfacs, Intrfacs and Mmbrans. (Frontrs n Pyscs, Prsus Pr. Publsr, 99. 7

18 Appnd A: Analytcal solutons for aally symmtrc cas W can obtan analytcal rsults dscrbng t ulbrum sap of and mutual monolayr sld n t blayr lpd mmbran undr constant trnal prssur for t cylndrcally symmtrc cas. onsdr a flat (unprturbd crcular mmbran n t plan (,y of t radus R. W sarc for an ulbrum soluton ndpndnt of t polar angl φ : ξ ξ(r, (A. r r s t radal coordnat n t rfrnc systm t t orgn stuatd at t cntr of t unprturbd mmbran's md-plan, and -as drctd along t mmbran s normal. Hnc, t sld-functons tak t form: f (, y f ( r cos φ, f y (, y f ( r snφ, (A. c tn lads to t follong prsson for t radal componnt of t dstorton fld: u r ξ( r ( r, ( Θ( Θ( f ( r. (A. r Snc t dformaton s purly radal, t angular componnt of t dstorton s ro: u 0. T symmtry of t dstorton flds (A., (A. prmts to prss t fr nrgy dnsty ( n t cylndrcal coordnats as follos: r ( u uφφ F rr, (A. u rr u r u u r φ ur, u φφ. (A.5 r r r φ r Eulbrum stat of t mmbran undr prssur s dfnd by t Eulr-Lagrang uatons, c ar obtand by uatng to ro t frst varatonal drvatvs of t lastc nrgy functonal F ( ξ, f t rspct to t functons ξ (r and f(r ntrng u and u n accord t (A.5 and (A.: δ F sr π r P 0, δξ(r δf sr 0, (A.6 δf(r r F v R 0 F dr, P s -componnt of an trnal prssur dffrnc appld to t oppost sr sds of t mmbran. Euatons (A.6 can b dcoupld by t ntroducton of t n unknon functons p(r and g(r nstad of functons ξ and f: ξ ξ p f, g f, (A.7 r r 8 rr φφ φ

19 In t n basc st of functons { p,g} uatons (A.6 ar transformd accordngly nto t follong form: r p r p r p p P r, r g r g g 0. (A.8 P r P. ot uatons n (A.8 blong to t Eulr s class of uatons and can b solvd analytcally usng t transformaton of t varabl: r varabl. T follong boundary condtons ar mposd:, r ( p ( r r p ( r p( r r 0 - t bndng ampltud ξ (r s arbtrary at r0; r 0 ξ( R 0 - mmbran s fd at t dg (no vrtcal dsplacmnt; ( p ( r r p( r 0 - ro toru at t mmbran s dg; r R ξ r 0 t slop at t cntr s ro ; r 0 5 f ( no ntr-monolayr sld at t cntr (aal symmtry; < < s t n 6 g ( r r g( r 0 - t ntr-monolayr sld at t dg s arbtrary. (A.9 r R Ts condtons av transparnt pyscal manng. T condtons and n (A.9 orgnat from t prsson for t varatonal drvatv t varatonal drvatv [ 0 r R δξ( r ] δ F sr δf sr δξ, and t condton 6 arss n δf ; bot drvatvs nclud ntgraton by parts n t sgmnt. In partcular, condton s obtand by uatng to ro t prfactor n front of 0. ondton s drvd by uatng to ro t prfactor n front of ξ r r R, c n turn corrsponds to ro toru, M, at t mmbran s dg (nc, mmbran s slop at t dg s arbtrary: ξ ξ f M π r r f (A.0 r r r ondton modls t faton of t mmbran at t prpry. ondton mpls a smoot sap at t cntr of t curvd mmbran. T rsultng solutons ar: ( r R r P ξ ( r R, ( r R r P f ( r. (A. 6 T bndng ampltud ξ( r u ( r s dfnd at t ntrfac (md-plan of mmbran and s -ndpndnt (for t consdrd r small bndng of mmbran. T functon f(r caractrs t ampltud of mutual sld of t monolayrs at t ntrfac of mmbran (0 9

20 (t total ampltud s gvn by f. As a rsult of ts sld t bottom surfac of t uppr monolayr s comprssd, and t top surfac of t lor monolayr s pandd. In t prsnt appromat approac f s constant along t tcknss (along -as of t monolayrs and dpnds on t poston n t plan of t mmbran. It s apparnt from (A. tat f~ ξ/r. Substtutng (A. n t prsson (A. for t radal dstorton u r P 6 Θ( Θ( r ( R r, fnd: P σ rr ( r, Θ( Θ( r R. (A. ( r d ~ urr ( r uφφ( r d 0. (A. u r. (A. T radal strss componnt corrspondng to t dstorton gvn by (A. s radly found: ( It s mportant to mnton r tat t latral strss componnt σ rr n (A. provs to b ndpndnt of t lastc modulus ( n our ak bndng appromaton. On t otr and, t dstorton and sld flds and t stran tnsor componnts dpnd on lastc modulus. In t consdrd cas of small bndng ampltud tr s no ovrall strtc of t dformd mmbran (.. t pur bndng taks plac and tus at any r: σ rr Fulfllmnt of ts ualty s guarantd by t -dpndnt factor n uaton (A.. T condton (A. s kpt by t ualty of t factors n front of Θ ( and Θ ( (.. t f-functon s takn to b t sam n bot monolayrs. Smultanously, strtcng dformaton of t monolayrs uals ro: n t dfntons of t dstorton fld componnts (s prsson ( n Scton. In gnral, f on dos not rstrct t problm to t ak bndng dformaton and/or f tr ar addtonal forcs actng n t latral drcton (strtcng t mmbran, on may ntroduc a 0 a ( r 0 or us to functons f f n front of Θ ( and Θ ( rspctvly. Rsults of t analytcal soluton of statc uatons n t cylndrcally symmtrc cas ar prsntd n Fgur. Latral strss σ ( r, s son for svral valus of -coordnat for a rr blayr t mutual ntrlayr sld (sold lns and dasd lns and for a unt blayr t forbddn sld, but of t sam tcknss (dottd lns. Rlaaton of latral strsss n bot monolayrs s nducd by mutual ntrlayr sld. T nutral (not strtcd surfac at t ntrfac of t mmbran splts nto to ons. onsuntly, a nutral surfac (t vansng latral strss appars n t mddl of ac monolayr: at (uppr monolayr and at (lor monolayr, s dasd ln. T monolayrs ar dformd as f ty r dsconnctd, ndpndnt layrs, but stll adjustd to t sam sap dfnd at tr mutual ntrfac nsd t mmbran. Trfor, t strss profls along -as concd t ac otr n bot monolayrs. As a rsult t strsss at t top and bottom trnal surfacs of t mmbran ( ±, sold lns dcras to tms t rspct to t cas tout sld ( ±, dottd lns. Smultanously, as t follos from (A., t latral strsss at t boundary rr 0

21 turn to ro troug t ol dpt of t mmbran: σ ( R, 0, corrspondng to t absnc of t appld trnal strtcng forcs. rr

22 Appnd : layr mods n t bulk atr In ordr to tst t rlvanc of our approac for dscrpton of dynamcal proprts of a blayr, rdrv r t dsprson rlaton for a mmbran n t bulk atr soluton usng our fr nrgy functonal (0, ntroducd n Scton. Our rsults ar n accord t arlr ons, obtand for a mmbran n t bulk flud usng curvatur lastc modl [5] and dnsty-dffrnc modl [8]. For surroundng bulk flud sarc for t soluton of crpng flo uatons ( (Scton satsfyng t non-slp condtons at mmbran-atr ntrfac (-(5. In addton, mpos t follong boundary condtons for flud vlocty componnts v : 0 ( ± v j, j, y, ; (. c rur t flud vlocty fld vans at larg dstancs from t mmbran. As n Scton, pand vbratons n plan avs propagatng along -as. W mak Fourr transform of t fr nrgy functonal (0. T fr nrgy dnsty s rttn as:, ( F s ( ξ ξ ξ * *, ( f f f F s (. T componnts of atr vlocty and prssur n t form : v and ar substtutd nto Fourr transformd crpng flo uatons ( (s Scton. T solutons of dffrntal uatons ( satsfyng boundary condtons (., t normal vlocty contnuous at 0, and obyng also condton of ro latral strtcng forc actng on t mmbran (uaton ( n Scton ar t follong: t (, t v ( t p p ( < > p 0 : 0 : (. [ ] [ ] < > 0 : 0 :. (. < > 0 : 0 :. (.5 Hr constants,, c ar prsnt n t uatons (-( n Scton, turn to ro du to boundary condtons (.. Unknon coffcnts, ar dtrmnd from non-slp condtons (, (5. Tn, substtut solutons (.-(.5 nto Fourr transforms of forc balanc uatons (0, ( and obtan an algbrac systm of to lnar omognous uatons for componnts and f : ξ [ ] 0 ξ f (.6 ( ( 0 ( ξ s m b f (.7

23 Euatng to ro t dtrmnant of t systm (.6-(.7 obtan uadratc uaton for (, c rsults n to brancs ( and (. Tr ar tr ydrodynamc rgms: <<, <<<< and <<, sparatd by crossovr av vctors and [8]: s, b bs. (.8 m For long avlngts, <<, t dsprson rlatons ar gvn by: ~ b. (.9 slppng mod, A ~, (.0 bs bs c dscrb rspctvly ydrodynamcally dampd bndng mod, (, and ntr-monolayr (, dampd by vscous drag at t mmbran md-plan. Hr suprscrpt s ntroducd to labl mmbran mods n t bulk flud. For av vctors n t ntrval <<<< t bndng and slppng mods m [8]: A ~, (. b s bs b ~. (. 6 T branc ( corrsponds no to bndng mod dampd by vscous losss n t surroundng flud, and t branc ( dscrbs t dampng of t slppng mod. T lastc modul n (. and (. dffr n gnral from tat n (.9 and (.0, bcaus g-fruncy (bndng fluctuatons occur at non-rlad monolayr surfac dnsts [8]. In sort-avlngt lmt, >>, obtan:, (. m ~ 6 b. (. T ( mod s drvn by (g fruncy ffctv rgdty and dampd by monolayr surfac vscosty. Effctv rgdty s nducd by dynamc couplng of monolayrs []. m Monolayr surfac vscosty ovrlms ntrlayr drag and bcoms t man sourc of dsspaton.

24 Fgur captons Fgur. Mmbran n t confnd gomtry. A blayr mmbran (ac monolayr of tcknss s placd n atr soluton btn paralll alls sparatd by dstanc d. T bndng ampltud ξ u s dfnd at t md-plan and s ndpndnt of t dpt n t mmbran. Intr-layr sld functon monolayrs at tr ntrfac. f paramtrs poston-dpndnt mutual sld of t Fgur. Vscous mods of a blayr mmbran n atr soluton confnd btn paralll alls for t cas of pur bndng dformatons. Dampng rats (/sc ar plottd as functons of dmnsonlss paramtr (d, r s av vctor, d s dstanc btn t alls. To brancs and orgnat from bndng and ntrlayr sld. T follong valus of paramtrs ar usd: d0-6 cm, 0-7 cm, 0 - dyn sc/cm, m dyn sc/cm, b s 0 7 dyn sc/cm, 0 8 rg/cm. Fgur. T latral strss σ rr normald by P R for varous -postons nsd t mmbran (>0 for t uppr monolayr, <0 for t lor s plottd as t functon of radal coordnat r (n dmnsonlss unts. T sold lns so strsss n t uppr and lor monolayrs (t strss profls along -as n bot monolayrs concd du to ntrlayr sld, s tt. T dasd ln rprsnts to nutral surfacs (at ± n t lor und uppr monolayrs. T dottd lns (, - caractr t strsss n t mmbran of t sam tcknss t forbddn sld.

25 Fgur Fgur log( log(d 5

26 Fgur.,5,5 σrr ; -0 /; -/ 0; - - 0, , 0, 0,6 0,8 r/r -0,5 - -,5 - -,5 6