Mode-coupling behavior of a Lennard-Jones binary mixture upon increasing confinement

Transcription

1 PHYSICAL REVIEW E 8, 5 9 Mode-coupling behavio of a Lennad-Jones binay mixue upon inceasing confinemen P. Gallo,* A. Aili, and M. Rovee Dipaimeno di Fisica, Univesià Roma Te, Via della Vasca Navale 8, Roma, Ialy Received Decembe 8; evised manuscip eceived Ocobe 9; published Decembe 9 Molecula dynamics simulaions ae pefomed on a Lennad Jones binay mixue confined in off laice maices of sof sphees wih inceasing adius. We focus on dynamics upon supecooling and in paicula on esing he mode coupling heoy popeies of he confined mixue. Paamees of mode coupling heoy in going fom bulk o weak confinemen, and fom weak o song confinemen ae exaced fom simulaions and analyzed. We focus on he sudy of he behavio of he single paicle densiy coelaos. We find ha he mode coupling heoy eains is validiy also in he case of song confinemen, wih a educion of ange of validiy. The ole of hopping is discussed in elaion wih he diffeences beween he esuls obained fom he diffusion coefficiens and he mode coupling heoy pedicions. DOI:./PhysRevE.85 PACS numbe s :.7.P,..Ja,..Lc I. INTRODUCTION Duing he las yeas an inense debae has been developing on glass ansiion in confinemen. This issue is of lage inees in biology, geophysics, and fo echnological applicaions whee confined fluids ofen play fundamenal oles,. How he glass ansiion scenaio ansfes in going fom bulk o confined phases i is a quesion whose answe migh depend on he ype of confinemen. Noneheless geneal ends and univesal feaues ae emeging ou of he many divesified kinds of confinemen ha have been unde he scuiny of he scienific communiy involved. One of he impoan achievemens in his field has been ha he mode coupling heoy 7, MCT, of he evoluion of glassy dynamics, woks well also in seveal kinds of confinemen 8 7. The MCT is able o descibe he dynamics of bulk liquids in he supecooled egion on appoaching a cossove empeaue T C. Above T C egodiciy is aained hough sucual elaxaions while below his empeaue sucual elaxaions ae fozen and only acivaed pocesses pemi he exploaion of he configuaional space. When also hopping is fozen he sysem eaches he empeaue of glass ansiion. Above T C he elaxaion mechanism of he supecooled liquid can be descibed as maseed by he cage effec. Neaes neighbos suound and ap he agged paicle foming a cage aound i. When he cage elaxes, due o coopeaive moions, he paicle diffuses. The MCT descibes he dynamics fo he densiy coelao q inoducing a eaded memoy funcion. In he idealized vesion of he MCT, hopping pocesses ae negleced and he nonlinea se of inegodiffeenial equaions can be solved analyically o he leading ode in = T T C /T C, he small paamee of he heoy, deiving univesal esuls fo he behavio of he densiy coelao. Wihin hese appoximaions T C is he empeaue of sucual aes of he ideal sysem. The success of his heoy is due o he fac ha on appoaching T C fom *Auho o whom coespondence should be addessed; gallop@fis.unioma.i above he pedicions of he idealized vesion of he MCT ae veified in expeimens and compue simulaions 7. In many sysems hopping sas o appea above and close o T C inoducing small deviaions fom he idealized behavio. The ole of hopping, closely conneced o dynamical eheogeneiies 8, seems o be enhanced in confinemen o in mixue wih lage size dispaiy 9 close and above T C. Depending on degee and ype of confinemen novel feaues ecenly emeged 8 in he densiy coelaos chaaceizing highe ode MCT ansiions 9,, analogous o wha found in colloids. Unified mean field appoaches fo bulk liquids, colloids and confined liquids have been also invesigaed. In his famewok i is impoan o undesand if he validiy of MCT is peseved in siuaions of song confinemen. Sudies upon deceasing he fee volume accessible fo he liquid ae heefoe significaive o deeply undesand he ole of esiced geomeies 7. Liquids confined in newok of ineconneced poes wih a lage value of poosiy, as silica xeogels, can be appopiaely sudied wih models whee he confining solid is buil as a disodeed aay of fozen micosphees 8. We conside hee one such a sysem, a liquid Lennad-Jones binay mixue, LJBM, composed of 8% of paicles A and % of smalle paicles B embedded in an off-laice maix of sof sphees. In he bulk phase his LJBM has been poven o show upon supecooling a dynamical behavio in ageemen wih he pedicions of MCT. In pevious sudies we esed he MCT behavio of his confined mixue fo a single size of he sphees and we compaed i wih an equivalen bulk. The pesen sudy epesens an exension of he pevious wok in which we sudy he behavio of seveal sysems upon inceasing he size of he confining sphees upon supecooling by pefoming MCT ess on he sysems and compaing he esuls. The pape is sucued as follows, in he nex secion we descibe he simulaion deails. In he Sec. III we analyze he saic popeies of he sysems. In he Sec. IV we sudy he mean squae displacemen MSD behavio. Secions V and VI deal wih he van Hove self-coelaion funcions and is space Fouie ansfom, he inemediae scaeing funcion, especively. The las secion is devoed o conclusions /9/8 /5 5-9 The Ameican Physical Sociey

2 GALLO, ATTILI, AND ROVERE PHYSICAL REVIEW E 8, 5 9 II. SIMULATION DETAILS FIG.. Snapshos of he simulaion cells fo he diffeen sysems. Top lef M = bulk, op igh M =, boom lef M =., boom igh M.. Ligh gay sphees ae A paicles, dak gay small sphees ae B paicles. The sof sphees ha fom he confining sysem ae epesened by hei isopoenial sufaces dak gay anspaen sufaces. We epo daa analysis obained fom MD simulaions ajecoies of a Lennad-Jones 8: binay mixue LJBM defined as in Refs.. In he following he 8% paicles will be efeed o as A and he % as B. We analyze he behavio of he binay mixue in confined saes by embedding he LJBM in a disodeed aay of sof sphees, labeled wih M in he following. The paicles ineac wih he poenial, V =, whee A,B,M. The cuoff ange of he ineacions is given by C =.5 and, in ode o avoid he disconinuiy a = C, he poenials ae shifed. The simulaions have been pefomed in a cubic cell whee peiodic bounday condiions have been applied. In he following Lennad-Jones unis ae used. The oal numbe of A and B paicles is N=. Boh ypes of paicles have he same mass m A =m B =. The paamees and of he LJBM poenial = ae given by AA =, AB =, BB =8, AA =, AB =, and BB =.5. The simulaion box of he confined sysems conains a igid disodeed aay of N M = sof sphees. To obain diffeen confining condiions he paamee M of he sof sphee poenial is changed while mainaining fixed he volume of he simulaion box. The paamees chosen fo he sof sphees ineacion poenial M = ae M =,. and. and M =.. The Loenz-Behelo mixing ules have been used o obain he values of he paamees of ineacions of he sof sphees wih A and B paicles. We simulaed also he bulk and we will in he following efe o he bulk as he M = case. Snapshos of he simulaed sysems ae epoed in Fig.. The ealizaion of each of he confined sysems is descibed in he following: we equilibaed a high empeaue, T=5., a bulk sysem done of 8 A paicles and B paicles, using a consan volume wih a seleced simulaion box lengh L=9.87. This coesponds o a numbe densiy of he bulk of =.. This value is he lowes densiy epoed as hemodynamically sable in he bulk 5. Then we andomly chose A paicles in a way ha hey ae fa apa a leas M bounday condiion included. The chosen paicles ae fixed in hei posiions and hei poenial is gadually swiched o he sof sphees one unil equilibaion is eached. Saing fom T=5. all sysem ae hen cooled and equilibaed via a velociy escaling pocedue. Fo each hemodynamical poin, MD simulaions o evaluae saisical and dynamical popeies have been pefomed in he micocanonical ensemble. The confined sysems wee sudied fo empeaues anging fom T=5 down o T=. fo M =, o T=.5 fo M =, o T=.5 fo M =. and o T= fo M.. These ae he lowes empeaues a which fo each sysem we wee able o equilibae. Fo each empeaue invesigaed an equilibaion ime longe han he elaxaion ime of he sysem has been simulaed. The oal poducion ime of he lowes empeaue invesigaed, T=., was of un = millions of ime seps. This value would coespond o a empeaue of 5.9 K and un =. s fo liquid Agon. We also checked ha he esuls of he simulaions ae no sensiive o a specific choice of he disodeed maix by doing hee diffeen ealizaions of he disodeed aay fo each size of he sphees. Ou poducion imes ae always much longe han he elaxaion imes and heefoe he coelaion funcions calculaed fo he diffeen ealizaions of he sysems coincide. The behavio of he oal enegy calculaed in he micocanonical ensemble is epoed in Fig. as funcion of he invese empeaue fo sysems fom M = o M.. The enegies of he sysems wih bigge sof sphee size lie above hose wih he smalles ones. This effec is due o he em of he fluid-maix poenial enegy which in fac inceases wih he size M. Fom he figue we fuhe noe ha upon cooling we do no obseve any abup change in he enegy. We also checked sysemaically duing ou MD uns ha hee was no any signaue of insabiliies in he hemodynamical quaniies sudied as funcion of he simulaion ime. In compaing he cuves of Fig. we noe ha in he sysem wih M. he LJBM is songly confined. In fac he lowes empeaue whee equilibaion could be eached fo he sysem lies in a egion in which he enegy is sill apidly deceasing. In his kind of egion he ohe sysems ae sill vey liquidlike and fo hem he lowes eachable empeaues ae in egions whee a flaening of he enegy cuves is obseved. 5-

3 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES PHYSICAL REVIEW E 8, = = 8 5 = = = g AB () E. = /T FIG.. Toal enegy pe paicle as funcion of invese empeaue fo bulk M = and confined LJBM. III. STATIC RESULTS In he following we pesen and discuss he main sucual popeies obained fo ou sysems compaed wih he bulk case M =. In Figs. 5 we epo he adial disibuion funcions RDF g, =A,B. The esuls ae ploed fo each size M a he diffeen empeaues invesigaed. The RDF ae veically shifed by a quaniy g = T. The fis peak of each g is appoximaely locaed a he disance =. The heigh of he fis peaks of he RDF s of he confined sysems inceases by inceasing he sof sphee size due o he coesponding incease of he effecive densiy. In each sysem when he empeaue is loweed he fis neighbo peak of he g AA and g AB RDF becomes gadually naowe and highe, see Figs. and. This effec is enhanced in he confined sysems in compaison wih he bulk and i indicaes an inceasing packing of g AA () g AA () (σ) = = = (σ) FIG.. Radial disibuion funcions g AA fo he diffeen sysems as indicaed in each panel. The cuves ae shifed veically by g= T. Highe cuves coespond o lowe empeaues. g AB ().. FIG.. Radial disibuion funcions g AB epoed as descibed in Fig. fo he g AA. he paicles in he sysems due o he pesence of he sof sphees. The boad second neighbo peak becomes moe sucued a he lowe empeaues and evolves in a double peak sucue in he g AA fo he cases M =. and M., in he g AB fo all M. The spliing of he second peak is ofen obseved in supecooled liquids and also coesponds o an incease of local packing of he paicles,. Fom Fig. 5 i is eviden ha he fis shell of he B paicles is less defined wih espec o he fis neaes neighbo shell of he A paicles. This is due o he ineacion poenial whee he A-A and he A-B aacions pevail on he B-B aacion. While in he bulk a deceasing empeaue he fis peak of he g BB becomes moe eviden, fo he confined LJBM he fis peak is educed and evolves in a small shoulde a inceasing size of he sof sphees. This behavio join o he incease of he fis peak of he g AB upon cooling, see Fig., suggess ha he confinemen induces a mixing effec a leas up o he lages size invesigaed, g BB () g BB () = = = FIG. 5. Radial disibuion funcions g BB epoed as descibed in Fig. fo he g AA

4 GALLO, ATTILI, AND ROVERE PHYSICAL REVIEW E 8, = = = 5 g AM () g BM () S AA (Q) = =..5 g AM () g BM () S AA (Q).. = Q 5 5 Q.5 g AM () g BM () FIG. 7. Saic sucue faco S AA Q fo each sysem and a diffeen empeaues. The cuves ae shifed veically by S = T. Highe cuves coespond o lowe empeaues. FIG.. Radial disibuion funcions beween he sof sphees and he LJBM paicles fo each confined sysem and vaious empeaues: g AM on he lef column and g BM on he igh column. M =, fis ow; M =., second ow; M., hid ow. The cuves ae shifed veically by g= T. Highe cuves coespond o lowe empeaues. M.. A diffeen behavio has been obseved fo he case M =5. and L=. whee he pesence of he maix inceases he epulsion beween he B paicles wih a consequen effec of demixing. We conside now he g M epoed in Fig. whee he RDF evaluaed a diffeen empeaues ae displayed ogehe fo each of he hee M sizes. I is easy o idenify he main coodinaion shells of he liquid paicles aound he confining sof sphees. The posiions of he fis neighbo peaks of g M ae locaed close o = M bu hey slighly shif o lowe values a inceasing empeaue. We noe ha he second peak is spli a he lowe empeaues. The main peculia feaue in Fig. is he behavio of he g BM. We obseve fo he fis peak of his coelao upon supecooling no only a shif bu also a consisen and gadual educion in coespondence of an enhancemen of he second shell. This effec is he signaue of a depleion of B paicles fom he inefacial egion of he confining maix. The B paicles end o avoid he sof sphees as he empeaue is loweed. The saic sucue facos SSF S Q fo he A and B paicles ae calculaed via he Fouie ansfom of he RDF. The SSF have he usual feaues ha hese funcions pesen in simple liquids, in paicula hey depend weakly on empeaue. The heigh of he peaks and he deph of he valleys become slighly moe ponounced upon loweing he empeaue, while he posiions of he peaks do no change. The posiion of he fis diffacion peak Q MAX slighly difs o highe values wih inceasing M. This effec is paiculaly elevan fo he S AA Q funcions shown in Fig. 7. Fo he S AA Q Q MAX anges fom 7. fo M = o 7.58 fo M.. In he case of S AB Q no shown Q MAX shifs fom 7.55 o 8., while fo he S BB Q no shown i goes fom 5.7 o 5.9. Also his obseved shif of Q MAX is he signaue of he incemen of he close packing of he paicles when he confinemen becomes songe upon inceasing he size of he sof sphees. IV. MEAN SQUARE DISPLACEMENT AND DIFFUSION COEFFICIENT The esuls epoed in he pevious secion show ha hee is no signaue of a phase ansiion upon supecooling a leas in he saic popeies. Accoding o he MCT we expec ha he dynamical popeies can be songly modified in he supecooled liquid insead. We conside now he behavio of he MSD fo deceasing empeaues. In Fig. 8 we epo he MSD fo each sysem, fo A paicles and fo he empeaues T=, T =., and T=.. The B paicles MSD no shown shows a simila end. In a nomal liquid sae he MSD is popoional o fo sho ime, he ballisic egime, and a he onse of he diffusive, Bownian, egime becomes popoional o. A he empeaue T =. he MSD of all he diffeen sysems show his behavio, in all cases he onse of he Bownian egime is a =. A T =. and moe significanly a T = a diffeen behavio sas o appea especially fo he M. sysem. The MSD does no swich o he diffusive egime afe he ballisic one, like in he high empeaue ange. A plaeau appeas a inemediae imes due o he cage effec indicaing ha he sysem is appoaching a glass ansiion. The paicle ales in he cage of he neaes neighbos and only fo ime long enough fo he cage o elax 5-

5 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES PHYSICAL REVIEW E 8, 5 9 < A ()> < A ()> < A ()> he paicle is able o escape. Fo he sysems wih M he plaeau appeas fo empeaues lowe han T =. Fo M. and T= he heigh of he plaeau is aound and fom his heigh he cage size can be esimaed o be 7. This value is in ageemen wih pevious calculaions on he confined LJBM. By compaing he MSD of A paicles and B paicles no shown we obseve ha fo each empeaue he A paicles ae slowe han he B. A simila behavio has been found in he bulk LJBM. A fi of he MSD in he asympoic Bownian egime can be pefomed o exac he diffusion coefficien D fo each specie accoding o he Einsein elaion =D. Fom he T dependence of he diffusion coefficien we can es he pedicion of MCT D A T= T=. T=. D A, = D A, = D A, = D A, = = = - FIG. 8. Mean squae displacemen MSD of A paicles A fo he diffeen sysems, M =,,.,.. a T=, b T =., c T= T-T c FIG. 9. Powe-law fis, see Eq., of he diffusion coefficiens D fo A paicles in he diffeen sysems M =, M =, M =., M.. The values of he fiing paamees ae epoed in Table I. a) b) c) D B D B, = D B, = D B, = D B, - - T-T c FIG.. Powe-law fis, see Eq., of he diffusion coefficiens D fo B paicles in he diffeen sysems M =, M =, M =., M.. The values of he fiing paamees ae epoed in Table I. D T T C, whee T C is he cossove empeaue. I comes ou ha in all ou cases he diffusion coefficien goes asympoically o zeo wih a powe law. The values of D and he fiing cuves ae epoed in Figs. 9 and as funcion of T T C. MCT pedics ha he cossove empeaue does no depend on he diffeen species and moeove A = B.We fied D A and D B by assuming he same T C and elaxing he condiion on he exponens. The values obained ae pesened in Table I. The cossove empeaues show a monoonically inceasing behavio a inceasing M, wih T C going fom he value 9 fo M = o. fo M.. This is in ageemen wih wha found in a film wih nonineacing had walls, whee confinemen is song, and whee his mixue shows an enhancemen of he glass ansiion empeaue wih espec o bulk. The exponens A and B esul o be vey simila fo each sysem bu hey do no show any egula behavio as funcion of M and, mos impoanly, he values obained. 9 ae ouside of he ange of MCT which pedics fo a minimum value of.7 7. As discussed in he nex secion his phenomenon TABLE I. Powe-laws fi paamees, fo A and B paicles. In he uppe half of he able ae epoed values obained fom he analysis of diffusion consans D, while in he lowe half values obained fom he analysis of elaxaion imes. M = M = M =. M. Fom D T c A..7.5 B.. 9. Fom T c 8. 5 A B

6 GALLO, ATTILI, AND ROVERE PHYSICAL REVIEW E 8, 5 9 π G A (,).5 = T =. =. = 5 = 5 a) = b) T =. =. = π G B (,).5 = T =. =. = 5 = 5 a) = b) T =. =. = π G A (,).5 =. T =. =. = 5 = 5 c) =. d) T =. =. = 5 = 5 π G B (,).5 =. T =. =. = 5 = 5 c) =. d) T =. =. = 5 = 5 FIG.. SVHCF fo A paicles in he diffeen sysems: a M =, b M =, c M =., d M., a T=. fo inceasing ime. Resuls fo some of he imes ae evidenced wih hicke lines. is conneced o he pesence of long ime hopping, which influences wih D behavio. V. VAN HOVE CORRELATION FUNCTIONS We conside now he self-pa of he van Hove coelaion funcions SVHCF. G s, = N N i i i= G s, is he pobabiliy densiy o find a paicle of ype a disance a ime. The SVHCF have been evaluaed fo diffeen empeaues and fo imes ha span fom he ballisic o he diffusive egime of he MSD, 5 wih a n geomeic pogession. The mos significaive esuls ae epoed in Figs. 5. In Figs. and we show he SVHCF as funcion of and diffeen imes of each sysem M =,,.,. a T=. fo A and B paicles, especively. Fom he figues i is eviden ha a sho imes he SVHCF has a single peak. This peak moves o lage disances as he ime inceases. A deailed analysis shows ha he posiion of he peak moves appoximaely as. This behavio is consisen wih he ballisic egime found in he MSD a he ealy imes. The shape of he funcions is well appoximaed by a Gaussian. This shape is peseved also fo imes longe han he ballisic egime. In his egion he posiion of he peak MAX changes accoding o he diffusive egime MAX. A high empeaue heefoe he SVHCF do no depend subsanially on he confinemen, apa ha he dynamics fo inceasing value of M becomes slighly slowe, as aleady found fo he MSD. As he empeaue deceases diffeences in he behavio of he sysems sa o appea. In Fig. he coelaos of A paicles ae epoed a T=, he lowes empeaue invesigaed fo he sysem wih M.. While he SVHCF fo M show a behavio simila o ha of he FIG.. SVHCF fo B paicles in he diffeen sysems: a M =, b M =, c M =., d M., a T=. fo inceasing ime. Resuls fo some of he imes ae evidenced wih hicke lines. high empeaues, quie diffeen feaues of he coelaos appea fo he M. case. One of hese diffeen feaues is he cluseing of he cuves see he boom-igh panel in Fig. in he ange of ime. This ange coesponds o he plaeau in he MSD and defines he -elaxaion egion of he same sysem. Sholy afe he ime he fis peak of SVHCF fo he sysem M. does no change posiion and deceases wih ime. A second peak aound sas o appea and becomes moe ponounced as he ime inceases. The heigh of his second peak becomes compaable wih he fis one fo imes long enough,. A hid peak also appeas aound a long imes. The pesence of muliple peaks in he coelaion funcions is conneced o hopping phenomena aking place upon supecooling. The posiions of hese wo hopping peaks π G A (,) π G A (,).5.5 = T = =. T = =.5x =.5x =.5x =.5x = x a) = b) T = =.5x =.5x = x c) =. d) T = =.5x =.5x = x = 5 FIG.. SVHCF fo A paicles in he diffeen sysems: a M =, b M =, c M =., d M., a T= fo inceasing ime. Resuls fo some of he imes ae evidenced wih hicke lines. 5-

7 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES π G B (,) π G B (,).5.5 = T = =. T = =.5x =.5x =.5x =.5x = x a) = b) T = =.5x =.5x = x c) =. d) T = =.5x =.5x = x = 5 FIG.. SVHCF fo B paicles in he diffeen sysems: a M =, b M =, c M =., d M., a T= fo inceasing ime. Resuls fo some of he imes ae evidenced wih hicke lines. coespond o he posiions of he fis wo peaks of he g AA epoed in Fig.. In fac he single paicle below a ceain empeaue sas o diffuse hough hopping pocesses and moves o enegeically favoed posiions coesponding o he peaks of he saic pai coelaion funcion. The behavio of he SVHCF of B paicles, shown in Fig. a he same empeaue T=, is simila o ha found fo A paicles. The posiions of he peaks ae slighly shifed a highe disances wih espec o hose of he SVHCF of A paicles due o he highe mobiliy of he B specie. In he case M. a muliple peak sucue appeas fo long ime as fo A paicles, bu now he seconday peaks ae boade wih espec o he esuls epoed in Fig.. This indicaes π G A (,) π G A (,).5.5 =. T =.5 =. T = x x x = 5 =.5x =.5x = x = 5 =. T =.5 =. T = x x x = 5 =.5x =.5x x = 5 FIG. 5. SVHCF fo A and B paicles and fo inceasing ime in wo diffeen sysems M =. and M. a T=.5 and T=, especively. These wo empeaues ae he lowes empeaues o which he sysems wee equilibaed and hey coespond o simila values of he small paamee of MCT, =. and.7, especively. Resuls fo some of he imes ae evidenced wih hicke lines. π G B (,) π G B (,) PHYSICAL REVIEW E 8, 5 9 a moe significaive pesence of hopping effecs fo B paicles. A behavio of he SVHCF simila o ha of he sysem wih M. fo T= can be found fo he ohe sysems a he lowes empeaues simulaed. In Fig. 5 we compae he behavio of he SVHCF fo M =. and M. a he lowes empeaues invesigaed, namely, T =.5 and T =, especively. These empeaues can be compaed in he wo sysems as hey coespond o simila small paamees, = T T C /T C of he heoy see nex secion fo he deeminaion of T C. Boh SVHCF of A and B paicles ae epoed fo inceasing imes. We obseve ha hopping phenomena fo smalle B paicles do no seem o be vey diffeen beween he wo sysems, and hey ae also vey simila in he bulk. Hopping phenomena of A paicles appea moe sensiive o confinemen. They ae in fac pacically absen in he bulk. In confinemen fo M =. hey appea only fo he longes imes while hey ae moe maked fo he case M., he songes confinemen invesigaed. This esul indicaes ha hopping pocesses may inevene cucially in confinemen possibly hiding he MCT behavio while sill pesen. In fac we noe ha, especially fo A paicles and especially fo he songes confinemen, when hopping sas o appea he fis peak, elaed o sucual aangemens, is sill in he egion of he plaeau of he MSD. As we will see in Sec. VI B his will inoduce small deviaions fom MCT also in he behavio fo he case M.. VI. SELF-INTERMEDIATE SCATTERING FUNCTIONS Sucual elaxaion and slowing down of he dynamics close o he cossove empeaue T C can be convenienly chaaceized by analyzing he self-inemediae scaeing funcion SISF, F s Q, = N exp iq N i i. i= The ime dependence of hese coelaion funcions is analyzed fo he wave veco Q=Q MAX coesponding o he posiion of he main peak of he saic sucue facos S AA Q and S BB Q. As discussed above he posiions of he main diffacion peak ae weakly dependen on he empeaue. Thus we fix he values of Q MAX o he aveage posiions found a he lowes simulaed empeaue. The wave veco values used ae: Q MAX,A =7.,7.5,7.,7.58 and Q MAX,B =5.7,5.9,5.7,5.9 fo inceasing M values. In Fig. esuls ae epoed of he SISF calculaed a hee empeaues T =,.,. fo he diffeen confinemens. A high empeaue T=. each coelao fo boh A and B paicles shows a quadaic dependence on ime fo. in he ballisic egime. Fo. he SISF decay quickly o zeo wih an exponenial elaxaion behavio. A sligh depaue fom he exponenial decay is howeve obseved fo he confined M. sysem. This effec is moe enhanced a T=., whee a small shoulde appeas fo inemediae ime. A he lowes empeaue simulaed fo he sysem M., T=, he shoulde becomes moe ponounced and 5-7

8 GALLO, ATTILI, AND ROVERE PHYSICAL REVIEW E 8, 5 9 F (Q MAX,) F (Q MAX,) F (Q MAX,) T = T =. T =. A = = = B = = = - FIG.. SISF F s Q MAX, fo A black lines and B gay lines paicles confined in he diffeen sysem as indicaed fo T =., T=., and T=. spans ove a geae ime ange. The SISF exhibis he wo sep elaxaion phenomenon pediced by MCT. The inemediae egion whee he SISF decays vey slowly is descibed as he elaxaion egime and coesponds o he plaeau egion of he MSD. The final decay o zeo of he SISF coesponds o he elaxaion egion. We now addess ou analysis o his impoan elaxaion pocess in all sysems. A. elaxaion Accoding o he MCT we can fi he behavio of he SISF in he elaxaion egion wih he Kohlausch-Williams- Was KWW seched exponenial f = f c exp /,, whee he exponen mus be in he ange,, is he elaxaion ime and f c is he nonegodiciy paamee. In Fig. 7 and 8 fis o he Eq. 5 of he SISF ae epoed fo each diffeen value of M fo A and B paicles. The ange of empeaues invesigaed in his analysis is diffeen fo each sysem. In paicula fo M = he bulk he ange is:. T, fo M =:.5 T, fo M =.:.5 T, fo M.: T.. Fom he figues i is eviden ha he low empeaue cuves fi vey well he KWW funcions fo all sysems. We found ha he exponens depend on he empeaue slowly deceasing as empeaue is loweed. The values obained fo he fied cuves ae in he ange and slighly deceasing wih inceasing M. c The non egodiciy paamee f Q, shows only a weakly inceasing behavio as funcion of deceasing empeaue wih a vey sligh incease wih inceasing M. We found f c c Q.7 fo he bulk M = sysem,.7 f Q.77 fo he M = sysem,.7 f c Q.78 fo he M =. sysem and.7 f c Q.78 fo he M. sysem. 5 F A (Q MAX,) - 5 T = 5. = Q MAX = 7. T = 5. = Q MAX = 7.5 T = 5. =. Q MAX = 7. T = 5.. Q MAX = 7.58 KWW fi T =. KWW fi KWW fi T =.5 KWW fi KWW fi - 5 The mos elevan quaniy o invesigae in ode o define he MCT cossove empeaue is he elaxaion ime and is dependence on he empeaue. In Fig. 9 and he values of,a and,b ae epoed on a log-log plos as a funcion of = T T C /T C. The elaxaion imes incease damaically up o fou odes of magniude as he empeaue is loweed. We noe ha he elaxaion imes of songly confined sysems ae lage han hose found fo sysems wih smalle M, his indicaes he slowing down of he dynamics upon inceasing he confinemen. In ageemen wih MCT pedicions hee ae no significaive diffeences beween he values of fo A and B paicles. The elaxaion imes behavio is fied in Fig. 9 and wih he MCT pedicion, T T C. We expec o find an univesal cossove empeaue T C and exponens. In he fiing pocedue we assume he same T C fo A and B paicles while he exponen can be diffeen. The values obained fom he fiing pocedue ae epoed in Table I and compaed wih he paamees obained fom he fi of he diffusion coefficiens, Eq.. The cossove empeaue shows a monoonically inceasing behavio wih he confining paamee M, A B and boh he exponens incease wih inceasing M. Moeove hei value is in he ange pediced by MCT a) T =.5 T = FIG. 7. SISF F s Q MAX, fo A paicles fo diffeen empeaues and values of M. Fom above: a M =, b M =, c M =., d M.. The dashed lines ae fis o he KWW funcions, see Eq. 5. b) c) d) 5-8

9 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES PHYSICAL REVIEW E 8, 5 9 F B (Q MAX,) - 5 T = 5. = Q MAX = 5.7 T = 5. = Q MAX = 5.9 T = 5. =. Q MAX = 5.7 T = 5.. Q MAX = 5.9 KWW fi T =. T = The powe-law behavio pediced by MCT, Eqs. and ae valid in a ange defined as = T T C /T C. Fom he figues we can noe ha he ange of validiy of MCT fo he elaxaion ime sas fo all sysems a, ha is T=T C. The slope of he saigh line inceases upon inceasing sof sphees size leading o a educion of ange in a) T =.5 T = FIG. 8. SISF F s Q MAX, fo B paicles fo diffeen empeaues and values of M. Fom above: a M =, b M =, c M =., d M.. The dashed lines ae fis o he KWW funcions, see Eq. 5. τ A 5 = = = - ε=(t-t C )/T C FIG. 9. Powe-law fis of he elaxaion ime, as a funcion of, fo A paicles fo each sysem M =,,.,.. The elaxaion imes ae obained fom KWW fis o he egime see Figs. 7 and 8. Paamees of he fi, T c and exponen ae epoed in Table I. b) c) d) τ B 5 = = = - ε=(t-t C )/T C FIG.. Powe-law fis of he elaxaion ime, as a funcion of, fo B paicles fo each sysem M =,,.,.. The elaxaion imes ae obained fom KWW fis o he egime see Figs. 7 and 8. Paamees of he fi, T c and exponen ae epoed in Table I. when compaing simila elaxaion imes as a funcion of he gowing sof sphee size. This leads o a % educion fo he case M. in coespondence o he lowes empeaues o which we wee able o equilibae each sysem. This empeaue maks fo each sysem he cossove o a egime whee hopping dominaes. The cossove empeaues obained by fiing he, ae lowe han he equivalen T C deived fom he fis o he powe law of D and his diffeence is moe siking upon inceasing sof sphees adii, faming in he picue of a moe sevee hopping ha influences moe makedly he D behavio fo he sysems wih songe confinemen. We noe moeove ha he cossove empeaues obained fom he MSD analysis ae slighly highe han he lowes empeaue simulaed in each confined sysem. Hopping coexiss wih sucual elaxaions fo he lowes empeaues invesigaed and masks MCT behavio a long imes. Also he discepancies beween he paamees deived fom he wo powe-law behavios can be aibued o he pesence of hopping ha causes he diffusion coefficien no o be invesely popoional o he elaxaion ime 8. This has aleady been obseved in bulk LJBM. In Figs. and we epo he powe-law fis of he invese of diffusion coefficiens and of he elaxaion imes fo A paicles vesus /T. The behavio of hese quaniies fo B paicles is simila no shown. The effec of hopping is eviden fom he plos. In fac we see fom he figues ha deviaion fom powe-law behavio is eviden only fo D a low empeaues. This is due o he fac ha hopping inevenes only fo he lowes empeaues and on ime scales longe han, causing D o swich o an Ahenius behavio. B. TTSP es MCT pedics a ime-empeaue supeposiion pinciple TTSP in he asympoic limi of T T C. The TTSP saes ha he shape of he coelao cuves in he lae elaxaion and ealy elaxaion ime egimes does no depend on empeaue. So he inemediae scaeing funcions behave as 5-9

10 GALLO, ATTILI, AND ROVERE PHYSICAL REVIEW E 8, a) /D A 5 = = = /T FIG.. Powe-law fis of he invese of he diffusion coefficiens fo A paicles shown fo all sysems vesus /T o bee show he deviaion fom he powe-law behavio a low empeaues. F A (Q MAX,) = = =. b) c) F s Q, = ˆ Q / T, 7 whee is a ime scale associaed o he -elaxaion decay of he coelaion funcion and ˆ Q is a mase funcion. The mase funcion ˆ, accoding o MCT, has in he -elaxaion egime he funcional fom of he von Schweidle VS law, ˆ = fq c h Q / b. whee f Q c is he non egodiciy paamee, h Q is he ampliude faco and b is he VS exponen. We used in he scaling of he SISF he, T obained via he KWW fis of he pevious subsecion. In Figs. and we epo fo A and B paicles especively he SISF a Q=Q MAX ploed as funcion of /, T fo each sysem. The SISF ae evaluaed in he anges of empeaues. T fo M =,.5 T fo M =,.5 T fo M =., T. fo M.. In he figues ae included he bes fi o he VS law, Eq. 8. Fom he figues i is eviden ha in he long ime scale of he -elaxaion egime he escaled cuves fall on op of each ohe defining a mase cuve ˆ wih he expeced τ Α 5 = = = /T FIG.. Powe-law fis of he elaxaion ime fo A paicles shown fo all sysems vesus /T o bee show he ageemen wih MCT a low empeaues /τ α FIG.. Self-pa of he SISF fo he A paicles, F A s Q,, evaluaed a Q=Q MAX fo each sysem M =,,.,. and diffeen empeaues, scaled by obained fom a Kohlausch- Williams-Was KWW fi, see Fig. 7; he dashed cuve is he fi o he VS law, Eq. 8. In a ae pesened M = bulk coelaos fo empeaues. T ; b M = coelaos fo empeaues.5 T ; c M =. coelaos fo empeaues.5 T ; d M. coelaos fo empeaues T.. seched exponenial fom. The TTSP es is veified in he egime. The TTSP holds also in he elaxaion egion, whee he mase funcion can be fied o he VS law. The paamees obained fom he VS fi ae epoed in Table II. Saing fom one paamee exaced fom MD daa we can calculae all he elevan MCT paamees hough he elaions = a + b, = a a = +b +b, whee is he Eule funcion, he a is elaed o he decay of he coelaion funcions owad he plaeau and he exponen paamee chaaceizes he sysem. The paamees mus be in he anges /, a.95, and b. Since we exaced fom ou analysis boh fom he and b fom he VS, we calculaed all he ohe paamees fom each one of hese wo and compaed he esuling ses. d) 9 5-

11 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES F B (Q MAX,) a) = = = /τ α FIG.. Self-pa of he SISF fo he B paicles, F B s Q,, evaluaed a Q=Q MAX fo each sysem M =,,.,. and diffeen empeaues, scaled by obained fom a Kohlausch- Williams-Was fi, see Fig. 8; he dashed cuve is he fi o he VS law, Eq. 8. In a ae pesened M = bulk coelaos fo empeaues. T ; b M = coelaos fo empeaues.5 T ; c M =. coelaos fo empeaues.5 T ; d M. coelaos fo empeaues T.. The values obained ae epoed in Table III. We emind ha exponens obained fom diffusion coefficiens canno be used fo his es since hey ae ou of MCT ange and heefoe bing o nonconsisen b, a, and b) c) d) TABLE II. Von Schwleide fi paamees fo A and B paicles of each sysem. c f q,a c f q,b PHYSICAL REVIEW E 8, 5 9 M = M = M =. M h q,a h q,b b A.5.7 b B 8.8. values. This failue is elaed o he song pesence of hopping pocesses in ou sysems fo low empeaues and long imes. We look a he ends of he paamees as funcion of he inceasing size M. We noe ha all values ae simila fo A and B paicles, he exponens a and b decease and he exponen paamee inceases. I is impoan o sess ha he wo ses of paamees obained independenly fom and fom b ae in vey good ageemen as expeced fo consisency wih MCT equaions excep fo M. whee especially in he case of he diffeence beween he wo ses appeas moe consisen. As we have seen ha hopping is moe significaive fo he songe confinemen as i sas a highe empeaues and mos impoanly on shoe ime scales compaed o he ohe sysems invesigaed, heefoe some deviaion fom MCT migh sa o be visible also in he values. C. Wave-veco analysis In Figs. 5 and ae epoed he SISF fo A and B paicles especively, evaluaed a seveal values of he wave veco in he ineval.5 Q 5.. This is he egion whee he wo fis peaks of he S AA Q and S BB Q appea. The empeaues consideed ae T =.,.5,.5, fo he sysems M =,,.,., especively. These empeaues coespond o simila small paamees of MCT. TABLE III. Paamees of MCT fo bulk and confined mixues, A and B paicles. In he op half of he able he values ae calculaed wih Eqs. 9 and fom he b values obained fom he VS fi, see Eq. 8. In he boom half he values ae calculaed wih Eqs. 9 and fom he values obained fom he powe-law PL fi of he elaxaion imes o he Eq.. M = M = M =. M. A B A B A B A B b VS a b b b PL a b a

12 GALLO, ATTILI, AND ROVERE F A (Q,) F A (Q,) = T =. =. T =.5 a) b) = T =.5 c) d) =. T = FIG. 5. SISF fo he A paicles, F A s q, fo diffeen wave veco values,.5 q 5. Dashed lines ae KWW fi o seched exponenial. a Panel: M = bulk a T=.; b panel: M = bulk a T=.5; c panel: M =. a T=.5; d panel: M. a T=. The empeaue of each sysem coesponds o simila small paamees of MCT. F B (Q,) F B (Q,) = T =. =. T =.5 a) b) = T =.5 c) d) =. T = FIG.. SISF fo he B paicles, F B s q, fo diffeen wave veco values,.5 q 5. Dashed lines ae KWW fi o seched exponenial. a Panel: M = bulk a T=.; b panel: M = bulk a T=.5; c panel: M =. a T=.5; d panel: M. a T=. The empeaues coespond o simila small paamees of MCT. f c Q,A τ α,a β A = = = 5 5 Q PHYSICAL REVIEW E 8, Q FIG. 7. Wave-veco analysis fo he fiing paamee of KWW cuves shown in Figs. 5 and. Fis ow of panel: Nonegodiciy paamees f q ; second ow: elaxaion ime ; hid ow: seching paamee. The wo columns of panels coespond o diffeen species of paicles. The analysis is caied ou fo each sysem: M = a T=., M = a T=.5, M =. a T=.5, M. a T=. The empeaues coespond o simila small paamees of MCT. In he same figues we show he KWW bes fis dashed lines fo each SISF. The fiing paamees of he KWW cuves, he elaxaion ime, he seching exponen and he nonegodiciy paamee f c Q ae epoed in Fig. 7. The elaxaion imes show a deceasing end wih inceasing wave veco Q. The values fo M = and ae vey simila. The slope of he cuves fo all sysems changes above Q=5 indicaing a cossove fom diffusive low Q o ballisic egime lage Q. The nonegodiciy paamee shows a monoonic deceasing end. The cuves fo he diffeen sysems ae vey simila and almos coincide fo B paicles. The seching paamee Q shows a slowly deceasing end. The asympoic behavio expeced fo Q is b, he von Schweidle exponen. We see ha fo M = and M = he asympoic values appea o convege on he same values obained fom he VS b values. Fo M =. and M. he asympoic values ae highe han ha calculaed fom he VS b values, also his inconsisency can be ascibed o he pesence of hopping. VII. CONCLUSIONS We epoed esuls fom MD simulaion of a bulk LJBM and he same mixue embedded in off laice maices of sof sphees. We simulaed fou sysems ha diffe in he size M of he sixeen confining sof sphees anging fom M = bulk o M.. All he sysems conain LJ paicles and he box lengh is fixed fo all sysems o L=9.87. The densiy of he bulk coesponds o he lowes sable densiy fo he LJBM. By inceasing he M values he sysem expeiences an inceasing effec of confinemen. Fo M. he confinemen esuls o be exemely song as eviden fom he enegy behavio. The consisen incemen of packing upon inceasing he sof sphees adius induces a mixing effec especially fo he highes M. In spie of his he g AA, g BB and g AB appea all vey simila o he bulk. Analogous o wha found fo a confined LJBM wih M =5. and L=. peviously sudied, B paicles end o avoid he sof sphees upon supecooling fo all M f c Q,B τ α,b β B 5-

13 MODE-COUPLING BEHAVIOR OF A LENNARD-JONES values. Also his sysem specific behavio does no appea o inefee wih MCT. The onse empeaue of he plaeau chaaceisic of supecooled liquids on he MSD depends on M and appeas a highe empeaues fo he lage M. The shape he MSD and he SISF is in ageemen wih MCT pedicions. In paicula all he SISF could be fied wih he KWW funcion. The exponens depend weakly on T and slighly decease wih deceasing empeaue and inceasing M. Powe-law fi values exaced fom SISF show ha T C inceases fo inceasing packing anging fom 8 o 5. The values of he exponens ae simila fo A and B paicles and in he ange pediced by MCT and incease upon inceasing M. We obseve a pogessive educion of he ange of validiy of MCT ha is educed of % fo M.. We find discepancies in he paamees exaced fom he powe-law fi o D values. The T C values obained fom D, ae sysemaically highe wih espec o he ones exaced fom he. Moeove he values obained fom D ae ouside MCT pediced ange. The eason of he obseved discepancies beween he wo ses of is he pesence of PHYSICAL REVIEW E 8, 5 9 long ime hopping in he SVHCF fo all sysems. Hopping inevenes o mask MCT behavio fo he paamees exaced fom D, as also found in he bulk. Impoanly he Von Schweidle law holds fo all sysems. VS paamees can be exaced and ogehe wih he exaced fom he allow o deemine independenly he es of he MCT exponens. The wo independen deeminaions show simila esuls confiming he MCT behavio of he confined mixue. I is impoan o sess ha MCT behavio is sill pesen also fo exemely song confinemen alhough some deviaions can be seen in he values calculaed fom MCT paamee equaions and in he asympoic value of Q. These findings ae of inees fo all he cases whee he micoscopic behavio of supecooled confined liquids is elevan such as echnological and biological applicaions. In his espec ou esuls can help o build a unified eamen of confined liquids appoaching he glass ansiion having idenified some feaues ha ae likely geneal popeies of confined liquids such as he enhancemen of hopping above T C. P. G. Debenedei, Measable Liquids: Conceps and Pinciples Pinceon Univesiy Pess, Pinceon, 997. Poceedings of he Inenaional Wokshop on Dynamics in Confinemen, edied by B. Fick, R. Zon, and H. Büne J. Phys. IV, ; Eu. Phys. J. E,, Special Issue fo he Inenaional Wokshop on Dynamics in Confinemen, edied by B. Fick, M. Koza, and R. Zon; Eu. Phys. J. Spec. Top., p. 7, Special Topic fo he Inenaional Wokshop on Dynamics in Confinemen, edied by M. Koza, B. Fick, and R. Zon. M. Alcoulabi and G. B. McKenna, J. Phys.: Condens. Mae 7, R 5. W. Göze, Complex Dynamics of Glass Foming Liquids: A Mode Coupling Theoy Oxfod Univesiy Pess, New Yok, 9. 5 W. Göze and L. Sjögen, Rep. Pog. Phys. 55, 99. W. Göze, Liquids, Feezing and Glass Tansiion, edied by J. P. Hansen, D. Levesque, and J. Zinn-Jusin, Noh-Holland, Amsedam, W. Göze, J. Phys.: Condens. Mae, A P. Gallo, M. Rovee, and E. Spoh, Phys. Rev. Le. 85, 7. 9 P. Gallo, M. Rovee, and E. Spoh, J. Chem. Phys.,. A. Aili, P. Gallo, and M. Rovee, J. Chem. Phys., P. Gallo, R. Pellain, and M. Rovee, Euophys. Le. 57,. P. Gallo, R. Pellain, and M. Rovee, Phys. Rev. E 7,. P. Gallo, R. Pellain, and M. Rovee, Phys. Rev. E 8, 9. F. Vanik, J. Baschnagel, and K. Binde, Phys. Rev. E 5, J. Baschnagel and F. Vanik, J. Phys.: Condens. Mae 7, R85 5. P. Scheidle, W. Kob, and K. Binde, Euophys. Le. 5, P. Scheidle, W. Kob, and K. Binde, J. Phys. Chem. B 8, 7. 8 C. Donai, S. C. Gloze, P. H. Poole, W. Kob, and S. J. Plimpon, Phys. Rev. E, A. J. Moeno and J. Colmeneo, J. Phys.: Condens. Mae 9, 7. A. J. Moeno and J. Colmeneo, Phys. Rev. E 7, 9. A. J. Moeno and J. Colmeneo, J. Chem. Phys. 5, 57. A. J. Moeno and J. Colmeneo, J. Chem. Phys., V. Kakoviack, Phys. Rev. E 79, 5 9. V. Kakoviack, Phys. Rev. E 75, V. Kakoviack, Phys. Rev. Le. 9, V. Kakoviack, J. Phys.: Condens. Mae 7, S J. Kuzidim, D. Coslovich, and G. Kahl, Phys. Rev. Le., W. Fenz, I. M. Myglod, O. Pyula, and R. Folk, Phys. Rev. E 8, 9. 9 W. Göze and M. Spel, Phys. Rev. Le. 9, 57. W. Göze and M. Spel, Phys. Rev. E, 5. F. Mallamace, P. Gambadauo, N. Micali, P. Taaglia, C. Liao, and S.-H. Chen, Phys. Rev. Le. 8, 5. F. Scioino, P. Taaglia, and E. Zaccaelli, Phys. Rev. Le. 9, 8. S. H. Chen, W. R. Chen, and F. Mallamace, Science, 9. M. Tokuyama, Physica A,. 5 M. Tokuyama, Physica A 78, M. Tokuyama, T. Naumi, and E. Kohia, Physica A 85,

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