Recent MD Results on Supercooled Thin Polymer Films

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001Recent MD Results on Supercooled Thin Polymer Films

F.Varnik 1?,J.Baschnagel 2,K.Binder 1

1Institut f¨u r Physik,Johannes-Gutenberg Universit¨a t,D-55099Mainz,Germany 2Institut Charles Sadron,6rue Boussingault,F-67083,Strasbourg Cedex,France Abstract The dynamic and static properties of a supercooled (non-entangled)polymer melt are investigated via molecular dynamics (MD)simulations.The system is con?ned between two completely smooth and purely repulsive walls.The wall-to-wall separation (?lm thickness),D ,is varied from about 3to about 14times the bulk radius of gyration.Despite the geometric con?nement,the supercooled ?lms exhibit many qualitative features which were also ob-served in the bulk and could be analyzed in terms of mode-coupling theory (MCT).Examples are the two-step relaxation of the incoherent intermedi-ate scattering function,the time-temperature superposition property of the late time α-process and the space-time factorization of the scattering func-tion on the intermediate time scale of the MCT β-process.An analysis of the temperature dependence of the α-relaxation time suggests that the criti-cal temperature,T c ,of MCT decreases with D .If the con?nement is not too strong (D ≥10monomer diameter)the static structure factor of the ?lm coin-cides with that of the bulk when compared for the same distance,T ?T c (D ),to the critical temperature.This suggests that T ?T c (D )is an important temperature scale of our model both in the bulk and in the ?lms.PACS :61.20.Ja,61.25.Hq,64.70.Pf contribution to EPJE Special Issue on Properties of Thin Polymer Film (guest editor:James Forrest)

Typeset using REVT E X

I.INTRODUCTION

Glass forming materials have been used by mankind since the very early days of civiliza-tion.An example are early ceramics from the neolithic period dating back to5000B.C.In addition to conventional glasses like those used for windows,bottles,etc.,polymers represent a new class of glassy systems with a large variety of thermal and elastic properties.Due to their(in general)low thermal conductivity,polymers are utilized as protective coatings in (micro-)electronic devices,optical?bres and other thermally fragile materials[1–3].Further-more,the high dielectric constant of some polymeric compounds suggests their application as porter of electric circuits,a new?eld under evolution[4,5].In all these applications,the polymer is in contact with another material.This gives rise to an interface.It is therefore important to know whether and how the interface alters the properties of the system.In particular,one would like to understand to what extent the glass transition temperature is in?uenced by the polymer-substrate interactions.Hence,the study of the glass transition in thin polymer?lms?nds a strong motivation from the technological side.

In addition to the technological importance,the study of the glass transition in thin polymer?lms is also of fundamental interest.It could help to elucidate the nature of this phenomenon.According to Adam and Gibbs[6],structural relaxation near the glass transition can take place only if many particles move in a correlated way to allow a collective rearrangement.The average size of such a“cooperatively rearranging region”de?nes a dynamic correlation lengthξ.The largerξ,the smaller the probability of a cooperative motion and thus the longer the structural relaxation time are supposed to be.If one assumes thatξperges at some temperature(Vogel-Fulcher-Tammann temperature)below T g,the glass transition can be considered as a thermodynamic second order phase transition.

However,the cooperatively rearranging regions are not directly correlated with static density?uctuations.The structure of a glass former changes only slightly upon cooling contrary to the hypothesized strong increase ofξ.As a consequence,scattering experiments cannot be used to determineξ(T)[7].One has to resort to indirect investigations.A possibility consists in studying con?ned systems.If the system size,L,is?nite,one could expect that the cooperatively rearranging region cannot grow beyond any bound,its largest extension beingξ=L<∞.Therefore,the glass transition temperature should depend on L and decrease with system size.

In thin polymer?lms it is the?lm thickness,D,which is?nite and thus one expects T g=T g(D).Indeed,one?nds a dependence of T g on D in both experiments[8–13]and computer simulations[14].However,the observed change of T g with D strongly depends on the system under consideration.If the interaction between the polymers and the substrate is attractive,the glass transition temperature of the?lms becomes larger than the bulk value for small?lm thicknesses[10].Intuitively,this e?ect can be attributed to chains which are close enough to the substrate to‘feel’the attractive interaction.The motion of these chains should be slowed down with respect to the bulk.In a thin?lm almost all chains touch the attractive substrate.So,T g should increase.

On the other hand,measurements(by ellipsometry)of polystyrene(PS)?lms(of rather large molecular weights)on a silicon substrate showed a signi?cant decrease of T g from 375K down to345K for the smallest?lm thickness of10nm,i.e.,a relative change of10% in T g was observed[9].There have also been many experiments in recent years on freely

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standing polystyrene?lms(i.e.,no solid substrate,but two polymer-air interfaces)[8,11–13] exhibiting a dramatic decrease of T g by up to20%if the?lm thickness becomes much smaller

than the chain size.An interesting explanation of this observation in terms of an interplay between polymer-speci?c properties and free-volume concepts has been proposed[15].

This sensitive dependence of T g on the polymer-substrate interaction was also obtained from computer simulations[14].One?nds that a strongly attractive wall leads to an increase of T g,whereas a weaker attraction has the opposite e?ect.Therefore,for repulsive walls,a

decrease of T g can be expected,provided that other parameters of the system,in particular the average density,are una?ected by the con?nement.As the simulations of our system in the bulk were done at constant external pressure[16,17],we also carried out the simulations

of the?lms at constant normal pressure in order to obtain an average density that lies as close as possible to the bulk density at the same temperature.This allows to separate the

e?ect of con?nement from that of the density.

Note however that the glass transition temperature is an empirical quantity.It is usually de?ned as the temperature at which the viscosity reaches a value of1013poise.So,the

choice of another number would give a di?erent value of T g.Furthermore,T g depends on the cooling rate so that it is not a temperature in a strict thermodynamic sense.Therefore,

other temperatures have been introduced to characterize the glass transition.One example is the so-called Vogel-Fulcher-Tammann(VFT)-temperature,T0,at which the system viscosity seems to perge.This quantity is obtained by?tting the viscosity,η(T),to the empirical

function,η(T)=η0exp[E/(T?T0)].An attempt to rationalize the VFT-law is the free volume theory[18–21].The main idea of this approach is that a tagged particle can leave its initial position only if it?nds a“free volume”of size v f≥v c in its neighborhood(v c being some critical volume of the order of the size of a molecule).It is further supposed that the average free volume vanishes at T0.Assuming statistical independence of the free volumes

and using a Taylor expansion of the average free volume around T0,one obtains the VFT-law for transport coe?cients like the viscosity or the(inverse)di?usion coe?cient.This brief description illustrates that the free volume theory has a phenomenological character.The precise meaning of the free volume is unclear and the existence of T0,where v f is supposed to vanish,is not derived,but postulated.

Contrary to the VFT-temperature,the critical temperature of the so-called mode-

coupling theory(MCT)results from a microscopic approach to the dynamics of supercooled (simple)liquids[22–24].Within the idealized version of the theory,the structural relaxation time perges at a critical temperature T c,while the static properties of the system remain liquid-like.This implies that the system vitri?es at T c.Thus,from the point of view of MCT,the glass transition is a purely dynamic phenomenon.However,comparisons between the theory and experiments[24,25]reveal that T c lies in the region of the supercooled liquid, where the glass former is only moderately viscous and no absolute freezing occurs.In real-ity,there are additional relaxation mechanisms which are not incorporated in the idealized MCT and begin to dominate close to and especially below T c.An attempt was made to approximately include these relaxation processes in an extended version of MCT[23],but the validity of this extension is currently unclear[26].Nevertheless,the idealized MCT derives several empirically known phenomena,such as the stretching of theα-relaxation, which is often described by the Kohlrausch-Williams-Watts(KWW)-law(∝exp[?(t/τ)β], whereβ<1),or the time-temperature superposition principle.Furthermore,it also makes

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new predictions,such as the space-time factorization property,which have been tested both in experiments[24,25]and computer simulations[27].

An application of MCT to our polymer model in the bulk[17,28–31]revealed that the theory represents a suitable framework to analyze the dynamics of the supercooled melt. Therefore,we attempt to test whether MCT can also be applied to the supercooled polymer ?lms.We?nd that even for an extremely thin?lm of three monomer layers,some features of the dynamics at low temperatures can be successfully described by the MCT.This analysis yields T c as a function of?lm thickness and shows that T c decreases with D.Within the error bars the Vogel-Fulcher-Tammann temperature exhibits the same D-dependence so that we also expect the glass transition temperature of our model to behave analogously due to T0(D)≤T g(D)≤T c(D).

The paper is organized as follows:After a presentation of the model in section II,sec-tion III focuses on the in?uence of the con?nement on static properties of the system.A discussion of the dynamics is the subject of section IV.In section V we determine the dependence of T c and T0on?lm thickness and the last section summarizes our conclusions.

II.MODEL

We study a Lennard-Jones(LJ)model for a dense polymer melt[16,32]of short chains (each consisting of10monomers)embedded between two completely smooth,impenetrable walls[33,34].Two potentials are used for the interaction between particles.The?rst one is a truncated and shifted LJ-potential which acts between all pairs of particles regardless of whether they are connected or not,

U LJ-ts(r)= U LJ(r)?U LJ(r c)if r

0otherwise,

where U LJ(r)=4? (σ/r)12?(σ/r)6 and r c=2×21/6σ.The connectivity between adjacent monomers of a chain is ensured by a FENE-potential[32],

U FENE(r)=?k

R0 2

,(1)

where k=30?/σ2is the strength factor and R0=1.5σthe maximum allowed length of a bond.The wall potential was chosen as

U W(z)=? σ

The left panel of Fig.1compares the bond potential,i.e.the sum of LJ-and FENE-potentials,with the LJ-potential.It shows that the bonded monomers prefer shorter dis-tances than the non-bonded ones.Thus,our model contains two intrinsic length scales(see right panel of Fig.1for a schematic illustration).Since these length scales are chosen to be incompatible with a(fcc or bcc)crystalline structure and since our chains are?exible(no bond angle or torsion potentials),one could expect that the system does not crystallize at low temperatures,but remains amorphous[35].

This expectation is well borne out,as Fig.2illustrates.The upper part of the?gure shows a snapshot of a?lm of thickness D=20and at temperature T=0.44which corresponds to the supercooled state.The visual inspection of this con?guration suggests that the structure is disordered.This is corroborated by an analysis of the static structure factor S(q).

The lower part of Fig.2shows S(q)for a?lm of thickness D=10at temperatures corresponding to the normal liquid state and to the supercooled state.S(q)is calculated parallel to the walls[i.e.q=|q|,q=(q x,q y)]by averaging over all monomers in the system. Figure2shows that the maximum value of the S(q)increases at lower temperature and that its position is slightly shifted towards larger q-values.Thus,the average interparticle distance decreases with decreasing temperature,since the density of the?lm increases in our simulations at constant pressure[34].However,these structural changes are rather small. Even at a very low temperature of T=0.42which is quite close to the critical temperature of the system at this?lm thickness(T c(D=10)?0.39),no qualitative di?erence is observed between the structure factors at low and high temperatures.The packing of the system thus remains liquid-like(i.e.amorphous)at all studied temperatures.

All simulations have been carried out at constant normal pressure P N,ext=p=1[34]. However,to adjust the normal pressure,we do not vary the wall-to-wall separation,D,but the surface area.For each temperature,the average surface area is calculated by an iterative approach[36].The system is then propagated until the instantaneous surface area reaches the computed average value.At this point the surface area(and thus the volume)is?xed and a production run is started in NV T-ensemble,where the temperature is adjusted using the Nos′e-Hoover thermostat[37,38].This thermostat slows down or accelerates all particles depending on the sign of the di?erence between the instantaneous kinetic energy of the system and the desired value imposed,i.e.3Nk B T/2(N is the number of particles)[37–42].

One may therefore ask how reliable the resulting dynamics is when compared to pure Newtonian dynamics in the microcanonical(NV E)ensemble.This question was already examined for the bulk system in Ref.[16]and for our con?ned model in Refs.[43,44].In both cases,the results obtained from constant energy(NV E)simulations and from the Nos′e-Hoover thermostat are identical.More details about the applied simulation techniques can be found in[34,36,44].

III.STATIC PROPERTIES

In this section,we want to discuss the in?uence of con?nement on both the chain confor-mation and the static properties of the melt.When computed within thin layers parallel to the wall,the structure of a chain varies with the distance from the wall.On the other hand, the chain structure averaged over the whole?lm does not depend much on?lm thickness.

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Contrary to this insensibility of the average chain conformation to the con?nement,the dense packing of the melt exhibits a pronounced dependence on D.

A.E?ects of the Con?nement on the Structure of a Chain

Let us?rst look at the T-dependence of the single chain structure factor S c(q).Simi-larly to S(q),S c(q)is also calculated for q-vectors in direction parallel to the walls and by averaging over all chains in the system.

The upper panel of Fig.3shows S c(q)for a?lm of thickness D=20at two representative temperatures:a relative high temperature corresponding to the normal liquid state(T= 1)and a low temperature representative of the supercooled state(T=0.44).Contrary to the structure factor of the melt(see Fig.2),S c(q)is practically independent of the temperature,not only on scales larger than the radius of gyration,but also on the local scale of the intermonomer distance.This may be rationalized as follows:The interactions of the monomers along the backbone of a chain do not include speci?c potentials for the bond or torsional angles,which could make the chain expand with decreasing temperature.This would lead to a much stronger temperature dependence than that resulting from potentials used.Since the bond potential is very steep around the minimum and is in general very large,the bond length is essentially independent of T(b=

b·b =0.961at T=0.46.Here,b is the bond vector).The main e?ect is that the overall size of a chain slightly shrinks with decreasing temperature because the density increases.This also leads to a weak increase of the peak of S c(q)at q=7.6,which is, however,not visible on the scale of Fig.3and negligible compared to that of S(q)in Fig.2.

Furthermore,Fig.3shows that,for q≤2π/R bulk

g ,the q-dependence of S c(q)can be

described well by a Debye-function,

S Debye(q)=

2N p

z>z w+2R bulk

g .Here,z w≈1denotes the physical position of the wall,i.e.,the smallest

distance between a monomer and z wall=D/2.As the chain’s center of mass approaches

the wall,R2

g, and R2

ee,

?rst develop a shallow minimum and then increase to about twice

the bulk value followed by a sharp decrease to zero in the very vicinity of the wall where practically no chain is present.On the other hand,the perpendicular components,R2g,⊥and R2ee,⊥,?rst pass through a maximum before decreasing to almost0at the wall.This behavior has been observed in several other simulations(see[45]and references therein), also for larger chain length than that used here[46,47].

These results suggest that a spatially resolved version of the chain structure factor, S c(q,z),should depend upon the position of the chain with respect to the wall.To test this idea we pide the system into layers of thickness1/4(in units ofσwhich roughly corresponds to the monomer diameter)and evaluate S c(q,z)by taking into account all monomer pairs within the layer which belong to the same chain.Here,z is the distance of the middle of such a layer from the wall.The lower panel of Fig.4shows S c(q,z)for an extremely thin?lm of thickness D=5at T=0.35and for two typical layers:a layer in the?lm center(z=2.375) and a layer close to the wall(z=1.125)[note that all layers with a distance smaller than z=1.125to the wall are practically empty.Note also that there is no layer whose middle lies exactly in the?lm center z=2.5.Only the boundary of the central layer“touches”the?lm center so that its middle is closer to the wall than D/2=2.5].In the center of the

?lm S c(q,z)can be described by a Debye function with R2Debye

g =1.93,whereas a larger

radius(R2Debye

g =2.4)must be chosen close to the walls.These values for R2Debye

g

are taken

from the pro?le of the radius of gyration for D=5at T=0.35,which yields R2g=2.4and R2g=1.93when averaging over the intervals1≤z≤1.25and2.25≤z≤2.5,respectively. Furthermore,Fig.4shows that S c(q,z)for the?lm center coincides with S c(q)obtained by averaging over the whole system.Thus,the local variations of S c(q,z)close to the wall disappear when the whole system is considered.Qualitatively,this point can be understood by noting that there are only few chains close to the wall,where R2

g,

(z)is larger than the

bulk value,while most of chains are in the inner part of the?lm,where R2

g,

(z)is close to

R2bulk

g .Averaging over the whole system thus tends to cancel the e?ects of the walls on the

chain conformation.

This point is further examined in Fig.5.The upper panel of the?gure shows1.5R2

g,

,

3R2g,⊥,averaged over the whole?lm,and R2g=R2

g, +R2g,⊥versus temperature for two?lm

thicknesses D=5and D=20.For all temperatures the parallel component is larger than the bulk value,whereas the perpendicular one is smaller.The disparity between the two components becomes more pronounced for small?lm thickness.Nevertheless,the sum of these quantities,R2g,is fairly close to the bulk value for both?lm thicknesses.It slightly shrinks with decreasing temperature,since the density of the?lms increases.Similar trends are observed if D is varied and T is kept constant.The lower panel of Fig.5depicts1.5R2

g,

, 3R2g,⊥and R2g as a function of?lm thickness at T=0.5.R2g depends only weakly on?lm thickness and is close to the bulk value.This explains why the chain structure factors of Fig.3are essentially independent of D.

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B.The Packing Structure of the Melt

Figure6compares the structure factor of the melt,S(q),in the bulk with that of?lms

of various thicknesses at T=1(normal liquid state;upper panel of the?gure)and at

T=0.46(supercooled state;lower panel of the?gure).Qualitatively,the behavior of the ?lms and the bulk is identical.The structure factor is small at small q-values,re?ecting

the low compressibility of the system.Then,it increases and develops a peak at q max which

corresponds to the local packing of monomers(2π/q max≈1)before it decreases again and begins oscillating around1,the large-q limit of S(q).This behavior is characteristic of dense

amorphous packing.

However,there are quantitative di?erences which become more pronounced at low tem-

perature:While S(q)of the?lm of thickness D=20(almost)coincides with the bulk data

at T=1,deviations are clearly visible at T=46.Quite generally,the most prominent di?er-ences between the bulk and the?lm are found for small q and for q max.The compressibility of the?lm is higher,the value of q max is shifted to slightly lower q and the magnitude of S(q max)is smaller than in the bulk.Keeping the?lm thickness?xed,one can observe similar changes of S(q)as the temperature increases(see Fig.2).Therefore,the local packing of the monomers in the?lms seems to resemble that of the bulk at some higher temperature. Since the local structure of the melt has an important in?uence on its dynamic behavior in the supercooled state[22],Fig.6suggests that the?lm relaxes more easily than the bulk at the same temperature.Indeed,we will see later that the dynamics of the system is much faster in the?lm than in the bulk when compared at the same temperature.

IV.DYNAMICS

This section discusses the dynamics of the?lms at low temperatures and for various thicknesses,D,ranging from about3to about14times the bulk radius of gyration.To this end,the incoherent intermediate scattering function and various mean-square displacements were calculated.We will see that,despite the geometric con?nement,the?lms exhibit several dynamic features which are in agreement with predictions of mode-coupling theory (MCT)[22–25].In this respect,the?lms behave as the bulk[17,28–31].However,the onset of MCT e?ects is shifted to lower temperatures compared to the bulk.The presence of the smooth walls accelerates the dynamics and this in?uence of con?nement is the stronger,the smaller D.

A.Con?nement leads to Faster Dynamics

An interesting dynamic correlation function is the incoherent intermediate scattering functionφs q(t).It measures density?uctuations on various length scales,which are caused by the displacement of inpidual particles.For a planar system,we de?neφs q(t)for q-vectors parallel to the wall,i.e.,

φs q(t)= 1

Here,N is the total number of monomers in the system,q =(q x,q y),q=|q |=

k B T[22].This corresponds to free particle motion.At later times,the relaxation ofφs q(t)is strongly protracted.There is an intermediate time window(β-relaxation regime of MCT[22]),where the correlator changes rather slowly with time before the?nal structural relaxation(α-relaxation)sets in at long times.This two-step decay is a characteristic feature ofφs q(t)for temperatures close to T c and re?ects the temporary“arrest”of a monomer in its local environment(“cage e?ect”[22]).It has been analyzed in detail in Refs.[16,17,48,49].

Compared to the bulk,the?lm data relax faster.If the?lm thickness decreases,this acceleration is enhanced and the two-step decay gradually disappears.At D=5,no interme-diateβ-relaxation is observed.The same changes also occur in the bulk if the temperature increases[17,29,48].This suggests that the inverse?lm thickness qualitatively plays a similar role as the temperature.

The acceleration of the dynamics in the?lm compared to the bulk is not limited to the main peak of S(q).It is also found for other q-values,for instance,for very small wave vectors.Since the time dependence ofφs q(t)is directly related to the monomer mean-square displacement(MSD)in the low-q limit,it is instructive to illustrate the acceleration of the dynamics by an investigation of the MSD.For a polymer system,various kinds of MSD’s may be de?ned.An important example is the mean-square displacement of the innermost monomer,

g1(t)=

3

2M

M

i=1 [R cm i, (t)?R cm i, ]2 ,(6)

where R cm i, is the projection of vector to the i-th chain’s center of mass onto a plane parallel

to the wall.Again,the factor3/2accounts for the di?erence in the number of independent components,as above.

Figure8depicts g1(t)for two typical temperatures:T=1(upper panel),which cor-responds to the normal liquid state,and T=0.46(lower panel),which belongs to the supercooled state in the bulk.Both panels compare g1(t)for the bulk and for?lms of var-ious thicknesses.The in?uence of the walls is rather small at T=1so that g1(t)of the bulk almost overlaps with that of the?lm if D≥10.However,the lower panel of Fig.8 shows that the e?ect of the walls on the mobility becomes signi?cant at all studied thick-nesses with progressive supercooling.Outside the initial ballistic regime(g1(t)=3T t2),the motion resembles that of the bulk,but is the faster,the smaller the?lm thickness.For the bulk and D 7,g1(t)exhibits several regimes.In agreement with the predictions of the mode-coupling theory[22–24],a plateau regime emerges after the ballistic motion.At low temperature,the tagged particle remains temporarily in the“cage”formed by its neighbors. However,contrary to simple(atomic)liquids,where a direct crossover from the plateau to the di?usive regime occurs,an intermediate subdi?usive regime emerges due to the connec-tivity of the monomers[50].In this regime,which is present for all D,the center of mass already crosses over to the asymptotic di?usive motion,g3(t)?t,whereas the motion of the innermost monomer is described by a power law g1(t)～t x with an e?ective exponent x?0.63.The innermost monomer reaches the center of mass only if g1(t)is larger than the end-to-end distance of a chain.In this limit,the di?usive motion is dominated by the motion of the chain’s center of mass and g1(t)coincides with g3(t).

In addition to the parallel displacements the analysis of the motion perpendicular to the wall is also interesting because the oscillations of the monomer density pro?le,which can propagate through the whole?lm for small D and low T,could possibly suppress perpendicular motion substantially.To check that,Fig.9shows the MSD of all monomers, g0(t),computed in direction parallel to wall,

g0(t)=

3

N

N

i=1 [z i(t)?z i(0)]2 .(8)

Here,N is again the total number of monomers and r i, =(x i,y i).Furthermore,z i denotes the z position of i-th monomer.The factors3/2for g0(t)and3for g0,⊥(t)account for the di?erence of independent components:two in parallel and one perpendicular direction compared to3in a bulk system.

The upper panel of Fig.9depicts g0(t)at T=0.46(supercooled state).It compares the bulk data with the displacements parallel and perpendicular to the walls in?lms of thicknesses D=5,7and20.The comparison of g0,⊥(t)for the?lms and of g0(t)for the bulk reveals that the con?nement does not only accelerate the dynamics in parallel,but also in perpendicular direction if g0,⊥(t,D)is su?ciently smaller than the?lm thickness.For all D, one then?nds g0,⊥(t,D)>g bulk

(t)and g0,⊥(t,D1)>g0,⊥(t,D2)if D1

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increase of the mobility is less pronounced than that of the dynamics in parallel direction, so that,for a given?lm thickness,g0, (t,D)>g0,⊥(t,D).

Of course,the perpendicular displacement cannot grow in?nitely.It must be limited by the?lm thickness.This is illustrated by g0,⊥(t,D=5)which crosses over to a constant of approximately5at late times.In fact,using the density pro?le,ρ(z),one can compute the large time limit of g0,⊥(t)by

lim t→∞g0,⊥(t)=3

+D/2

?D/2

d z +D/2?D/2d z′ρ(z)ρ(z′)(z?z′)2

B.Incoherent intermediate scattering function

The discussion of the previous sections suggested that mode-coupling theory(MCT) could also be a relevant theoretical framework to describe the dynamics of the supercooled polymer?lms.In this section we want to test this suggestion by an analysis of the incoherent intermediate scattering functionφs q(t).

A quantitative application of MCT to the simulation data requires an intricate?t proce-dure which must simultaneously optimize several parameters subject to various theoretical constraints(some of them have to be independent of temperature,others independent of the wave vector).Before attempting this analysis simple tests should be carried out to check whether the approach is worthwhile at all.Two such tests can be performed.

Mode-coupling theory predicts that there is an intermediate time window in which the scattering function remains close the time-and temperature independent non-ergodicity parameter f sc q.This time window is calledβ-relaxation regime[22,25].In this regime the scattering function can be written as

φs q(t)=f sc q+h s q G(t),(10) where G(t)and h s q are theβ-correlator and the critical amplitude,respectively[22,25]. Equation(10)shows that the time-dependent corrections to f sc q have an important property. The space-and the time dependences factorize from one another.

This“factorization theorem”[22,25]suggests a simple test[48,49,51,52]which uses the simulation data directly without invoking any?t procedure.If t′and t′′denote two times belonging to theβ-regime,then the ratio

R s q(t)=φs q(t)?φs q(t′)

G(t′′)?G(t′)

=R(t)(11)

should only depend on temperature and time,but not on q,provided Eq.(10)holds.

Note thatφs q(t)varies slowly for times around the plateau.Therefore,the denominator of Eq.(11)is fairly small and the accuracy of the test is predicated upon an appropriate choice of the parameters t′′and t′.To obtain a satisfactory signal-to-noise ratio,one would like to make t′′and t′as di?erent as possible.However,one has to be careful not to take t′′and t′outside theβ-relaxation regime,where Eq.(11)is no longer valid.We?nd that t′′=1 and t′=50is a reasonable compromise for all studied?lm thicknesses D=5,10and D=20.

With this choice two observations can be made from Fig.5:First,there is indeed an intermediate time window for all D where the correlators,measured at di?erent q,collapse, whereas they splay out at both shorter and longer times.This is a qualitative evidence for the factorization theorem.However,the best agreement with Eq.(11)is obtained for D=10.In the other two cases the superposition of the scattering functions for t′′

A possible reason for this di?erence could be that,for D=10,Eq.(11)is tested at a distance of T?T c(D=10)=0.42?0.39=0.03from the critical temperature of this?lm thickness. The tests for D=20and D=5,however,correspond to T?T c(D=20)=0.46?0.415=0.045 and T?T c(D=5)=0.35?0.305=0.045,respectively.As Eq.(11)is an asymtotic relation which is expected to hold the better,the closer T is to T c,Fig.5still suggests that the factorization theorem is not only satis?ed in the bulk[17,48],but also in the polymer?lms.

12

The second observation concerns the order of the q-values before and after theβ-regime. This order is preserved.The top curve,φs q=1(t),at short times is also the top curve in the α-regime.Similarly,the bottom curve,φs q=18.5(t),before theβ-regime also remains below all other q-values when leaving theβ-regime again.This behavior re?ects the theoretical pre-diction that the short-and long-time corrections to Eq.(10)exhibit the same q-dependence. It was pointed out in simulations of a binary Lennard-Jones mixture[52]and also found for our model in the bulk[48].

The second test of the applicability of MCT deals with the late-time relaxation ofφs q(t). An important prediction of the theory for theα-process is the time-temperature super-position principle(TTSP)[22,25].This means thatφs q(t)is not a function of time and temperature separately,but only of the scaled time t/τq(T),whereτq(T)is theα-relaxation time.So,we have

φs q(t)=F q(t/τq).(12) The function F q(t/τq)can often be well approximated by a Kohlrausch-Williams-Watts (KWW)function

F q(t/τK q)=f K q exp (t/τK q)βK q .(13) For time-temperature superposition to hold the amplitude f K q and the stretching exponent βK q must be independent of temperature.Only the Kohlrausch relaxation timeτK q is a function of T.MCT predicts thatτK q is proportional to theα-relaxation time and increases upon cooling as[22,25]

τK q∝τq～(T?T c)?γ.(14) Equation(14)also implies that any time from the window of theα-process should exhibit the same temperature dependence and can thus be used to test the TTSP.This means that it should be possible to collapse the late time decay of the scattering functions,measured at di?erent T,onto a common T-independent master curve by plottingφs q(t)versus t/τq.A convenient de?nition ofτq is to simply read o?the time whenφs q(t)has decay to a certain value.Again,such a test has the advantage that no complicated?t procedure is involved.It works directly with the simulation data.A possible choice isφs q(τq)=0.1[49].This low value warrants that the scattering function has decayed su?ciently so that possible perturbations from theβ-relaxation are completely negligible.

The resulting master curves are shown in Fig.11for D=5,10,20.For all?lm thicknesses, even for the extreme case of D=5,which corresponds to three atomic layers only,the TTSP is borne out by the simulation data and extends to shorter rescaled times with decreasing temperature,as predicted by MCT for homogeneous systems.In this respect,the?lms behave identical to the bulk[17,49].However,the results for D=10suggest that there could also be a qualitative di?erence.They include T=0.4,which is very close to the estimated T c,i.e.T?T c=0.01.Contrary to the bulk[17,49],the?lm data at this T?T c exhibit no apparent violation of the TTSP.Whether this is a general property of the con?ned systems or just a special feature of D=10is not clear at present.

While the scattering functions exhibit time-temperature superposition for all thicknesses studied,they do not superimpose if di?erent thicknesses are compared.This is illustrated in

13

Fig.12.The?gure showsφs q versus t/τq for D=5,20and the bulk at comparable distances to the corresponding critical temperature(i.e.,T?T c?0.045).Obviously,the shape of the late-time relaxation depends on D.Note that,compared toβK q(D=20),the stretching exponent of the thinner?lm,i.e.,βK q(D=5)is closer to that of the bulk.A possible explanation could be as follows:Due to the presence of the walls,there is a distribution of the relaxation times along the transversal direction.Regions closer to the wall decay faster and thus exhibit a smaller relaxation time.On the other hand,as the temperature decreases, the in?uence of the walls is“felt”throughout the?lm for all thicknesses studied.There is practically no region of(bulk-like)constant relaxation time in the inner part of the?lm. Now,if we pide a?lm into layers in which the relaxation time is calculated,the number of layers is smaller in a thin than in a thick?lm.The simplest assumption then is that the larger number of layers in the thick?lm leads to a broader distribution of relaxation times. Of course,if D increases further,there will be a region of bulk-like behavior which grows in the middle of the?lm and starts to dominate properties of the system for very large D. The?lm thicknesses studied here,however,are far from this limit.

V.DETERMINATION OF T c(D)

The analysis of the preceeding section showed that the relaxation of the?lms is qual-itatively compatible with predictions of mode-coupling theory and that it speeds up with increasing con?nement.To some extent,the dynamics of the?lms corresponds to the be-havior of the bulk obtained at a higher temperature.This suggests that con?nement reduces the characteristic temperatures of the?lm compared to those of the bulk.In this section, we want to quantify this reduction by determining the critical temperature T c(D)and the Vogel-Fulcher-Tammann temperature T0(D).

Mode-coupling theory[22–24]predicts that the relaxation times of any correlation func-tion,which couples to density?uctuations,should exhibit the power-law temperature de-pendence of Eq.(14)if T is close to T c.Indeed,theα-relaxation times of the incoherent and coherent scattering functions in the bulk may be described by Eq.(14)in some q-dependent temperature interval[17,49].Deviations are found both for temperatures too close and to far

away from T bulk

c .These deviations are theoretically expected.Since Eq.(14)is an asymtotic

result of the idealized MCT,it can be violated if T is very close to T c due to relaxation chan-nels which are not treated by the idealized theory,and very far from T c where the asymtotic formula is no longer applicable.Similarly,the q-dependence of the temperature interval, where Eq.(14)is valid,has been rationalized by calculating the leading-order corrections to the asymptotic behavior within the framework of idealized MCT[53,54].Due to our?ndings in the bulk and due to the results of the previous section we expect to obtain similar results when analyzing the?lms.

To examine this expectation,we de?ne relaxation times as the time needed by a given mean-square displacement,like g1,g3or g0[see Eqs.(5),(6)and(7)],to reach the monomer size

g i(t=τ):=1(de?ning equation forτ).(15) This is a reasonable choice because the bulk analysis revealed that a monomer,which has moved across its own diameter,contributes to theα-relaxation[48].Using Eq.(15)we

14

computed τ(g i =1)as a function of temperature for various ?lm thicknesses,where,in addition to g 0,g 1and g 3,the MSD of the end monomers,

g 4(t )=3

T ?T 0(D ) ,(17)

where c is a constant which can depend on ?lm thickness.The possibility of describing the same data by both a power law (MCT)and a VFT-?t has also been observed for the bulk (see Fig.10in

[16]).We therefore use the VFT-formula as an independent approach to determine the variation of T 0with ?lm thickness.Table I contains the results.A plot of T c (D )and T 0(D )is shown in Fig.14.Since we expect T 0

The critical temperatures,T c (D ),were determined,for instance,from the mean-square displacement of all monomers [see Eq.(15)]and thus from a quantitiy which corresponds to the low-q limit of the incoherent scattering function.Figure 15shows that the same T c (D )can also be used to linearize the relaxation times τq at maximum of the static structure factor.Here,τq was de?ned by φs q (t =τq )=0.3.The upper panel depicts a log-log plot of τq versus T ?T c for D =5,7,10,15,20and for the bulk.It illustrates that a shift of the temperature scale by T c (D )leads to a very good superposition of the ?lm data for all T studied and also of the ?lm and the bulk if T ?T c >0.1.For smaller T ?T c deviations occur.

15

The relaxation time of the bulk increases less steeply than that of the?lms.This implies that the exponentγφof Eq.(14)is smaller in the bulk than in the?lms.If one?ts Eq.(14)to that T-interval,where the?lm data are linear,one obtainsγφ-values which range between γφ(D=10)=2.68andγφ(D=5)=3.24and which do not increase monotonically with decreasing D[see table I].The lower panel of Fig.15illustrates the same data in a slightly di?erent way.Here,τ?1/γφ

q

is depicted versus T?T c(D).The exponents,γφ,used for this plot are obtained from?ts to Eq.(14)usingτq as input data and keeping T c(D)constant.As expected,curves at all?lm thicknesses converge towards the origin of coordinates.In other words,the computed critical temperatures are consistent with a temperature dependence ofτq according to Eq.(14).On the other hand,the curves splay out at large T?T c(D)

indicating that the prefactor,B,inτ?1/γφ

q

=B|T?T c(D)|depends on the?lm thickness,D.

As mentioned above,although not monotonically,the exponentγφseems to increase with decreasing D[see also table I].To some extent,this?nding is unexpected,given the variation of the Kohlrausch exponentβK q in Fig.12.For the bulk,mode-coupling theory predicts lim q→∞βK q=b[56].Here,b is the von-Schweidler exponent which is related to γφbyγφ=1/2a+1/2b[22–24].The exponents a(exponent of the“critical decay”)and b are correlated with one another.A decrease or an increase of b entails a decrease or an increase of a.When applying these bulk predictions to the?lms the D-dependence ofβK q in

Fig.12[βK q(D=20)<βK q(D=5)<βK bulk

q ]suggests b(D=20)

γφ(D=20)>γφ(D=5)>γφ,bulk.Whileγφ(D)>γφ,bulk for all D,the results of table I do

not con?rm the expected order between the di?erent?lm thicknesses.

However,one should not conclude from the preceding discussion that a consistent ap-

plication of MCT might not be possible.On the one hand,it is not clear whether the

D-dependence of the?nite-q result in Fig.12(βK q at q=6.9)is the same as in the limit q→∞(i.e.,as for b).The corrections to the asymptotic behavior could depend on both q

and D.On the other hand,the?tted value forγφat D=5is the least certain because the lowest temperature simulated corresponds to a rather large T?T c(D=5)?0.045contrary

to T?T c?0.025for D=20.Therefore,the simulations have to be extended to lower temperature to test whether there are clear deviations from the superposition ofτq at low T in Fig.15.The present quality of the?ts would rather suggest that theγφ-values for the

di?erent?lm thicknesses are very close to each other.

Within the framework of MCT,quantities,such as the exponents a,b orγand the critical

temperature,are determined by the thermodynamic properties of the glass former,in par-

ticular by the static structure factor[22–24].The discussion of Fig.6already suggested that S(q)changes with decreasing D in a quite similar fashion as the bulk S(q)if the temperature is increased.Therefore,the question arises of whether it is possible to superimpose the bulk and?lm results for S(q)by comparing the data for the same T?T c(D).This would imply that the reduction of T c in the?lms is closely related to the fact that the development of the local packing,characteristic of the supercooled bulk,is shifted to lower temperature by the presence of the smooth walls.Figure16shows that such as superpostions for the same T?T c(D)is possible if the con?nement is not too strong(here,for D≥10).The upper panel compares S(q)of the bulk with that of a?lm of thickness D=10for T?T c=0.01. With the exception of the(slightly)di?erent amplitude of the?rst maximum,both structure factors are identical over the whole q-range.The lower panel shows the same comparison for the bulk and?lms of thicknesses D=5,10and20at a larger distance from T c,i.e.,for

16

T?T c=0.05.While S(q)of the bulk and?lm still coincide for D≥10,this is no longer the case for the thinnest?lm studied(D=5).The in?uence of the con?nement on the packing structure of the system at this?lm thickness cannot be explained by a mere shift of the temperature axis.

VI.SUMMARY

Results of extensive MD simulations of thin(non-entangled)polymer?lms are presented, which focus on the in?uence of con?nement on the sluggish dynamics of the system and in particular on the glass transition temperature.The?lm geometry is realized by introducing two perfectly smooth and purely repulsive walls.All simulations are carried out at constant normal pressure P N,ext=1.

The static properties of the system show that chains close to the walls prefer a parallel alignment.However,when averaged over the whole?lm and all directions,the in?uence of the walls on the chain conformations becomes very weak.In particular,we?nd that the chains’radius of gyration,R2g,does not depend much on?lm thickness.Even for the extreme case of D=5,R2g lies only by10%below the corresponding bulk value.

On the level of the overall packing structure of the melt,we observe that the structure factor,S(q),of a?lm of thickness D resembles that of the bulk at a higher temperature. If the con?nement is not too strong(D≥10),S(q),measured for a particular D at some T′,almost coincides with the bulk result for that temperature T′′which lies at the same

distance to T bulk

c as T′to T c(D)[i.e.T′′?T bulk c=T′?T c(D)].This indicates that T?T c(D)

is a relevant parameter for our con?ned system.

This static property of our model?nds a counterpart in the dynamic behavior.Our main ?ndings for the dynamics can be summarized as follows:(1)The relaxation of the super-

cooled?lms is accelerated compared to the bulk so that characteristic temperatures,such as

the mode-coupling critical temperature,T c(D),or the Vogel-Fulcher-Tammann temperature, T0(D),decrease with decreasing?lm thickness.As we expect T0≤T g≤T c,our results sug-gest that also T g(D)should decrease with decreasing?lm thickness.(2)The?lms exhibit

several features predicted by mode-coupling theory,such as the space-time factorization property in the intermediate time window of theβ-process,time-temperature superposition of theα-relaxation,and a power-law increase of theα-relaxation time in a T-interval that is close,but not too close to T c.This implies that the implications of the cage e?ect,whose ap-proximate mathematical treatment leads to these predictions,also seems to be an important factor to understand the dynamics of the con?ned system in the supercooled state above T c.

(3)A comparison of the mean-square displacements in direction parallel and perpendicular

to the walls shows that not only the parallel motion,but also the perpendicular motion is

accelerated compared to the bulk if the displacement is su?ciently smaller than the?lm thickness.However,for a given thickness,the parallel motion is always faster than that in transverse direction.In other words,the enhancement of the dynamics in more pronounced when relaxation processes parallel to the walls are considered.Due to the?lm geometry,it is clear that the long-time limit of the perpendicular mean-square displacements must be ?nite.We gave an expression which allows a computation of this limitting value from the density pro?le of the particle that corresponds to the MSD under consideration(i.e.,inner monomer,center of mass,etc.).

17

ACKNOWLEDGEMENT

We thank J.Horbach for helpful discussions on various aspects of this work.We grate-fully acknowledge the?nancial support by the“Deutsche Forschungsgemeinschaft”(DFG) under the project number SFB262and by BMBF under the project number03N6015.We are also indebted to the European Science Foundation for?nancial support by the ESF Pro-gramme on“Experimental and Theoretical Investigations of Complex Polymer Structures”(SUPERNET).Generous grants of simulation time by the computer center at the university of Mainz(ZDV),the NIC in J¨u lich and the RHRK in Kaiserslautern are also acknowledged.

18

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20

TABLES

715bulk

0.253±0.0130.297±0.0070.328±0.008 0.365±0.0070.405±0.0080.450±0.005

2.4±0.2 2.2±0.1 1.84±0.1

3.15±0.1 2.76±0.1 2.09±0.07

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