1

Glass Transition Behavior of Polymer Films ofNanoscopic Dimensions

Arlette R.C. Baljon*, Maarten H.M. Van Weert, Regina Barber DeGraaff, andRajesh Khare**

Department of Physics, San Diego State University, San Diego, CA 92182, USA(Dated: November 1, 2004)

Glass transition behavior of nanoscopically thin polymer films is investigated by means ofmolecular dynamics simulations. We study thin polymer films composed of bead-spring modelchains and supported on an idealized FCC lattice substrate surface. The impact on the glasstransition temperature of the strength of polymer-surface interaction, and of chain grafting to thesurface is investigated. Three different methods - volumetric, energetic, and dynamic – are used todetermine the glass transition temperature of the films. Based on these, we are able to distinguishtwo different transition temperatures. When the temperature is lowered, a first transition occurswhen the beads order locally. This transition is characterized by an anomaly in the heat capacity.Upon decreasing the temperature further, the point is reached at which internal relaxation timesdiverge, as calculated, using for instance mode coupling theory. In qualitative agreement with theexperiments, the former temperature depends on the characteristics of the polymer-surfaceinteraction. By contrast, the latter temperature, is independent of these.

I. INTRODUCTION

The properties of polymers start deviating fromtheir bulk values when one of the dimensions ofthe system approaches nanometer length scales.Thin polymeric films with thickness less then100 nm play an important role in themicroelectronics industry, where they are used asmasks in lithographic processes. It is critical thatin such operations they retain their patterns.Therefore, they should be so designed as toexhibit a high glass transition temperature atprocessing conditions. Hence, glass transitionbehavior of thin polymer films in recent yearshas been the subject of many experimentalstudies1,2,3,4,,5,6,7,8.

Two approaches have been discussed in theliterature for raising the glass transitiontemperature of thin polymer films. The firstconsists of tuning the interfacial energy betweenthe polymer and the surface supporting the film.In an early study, Van Zanten et al.3 showed that100 nm thin films of poly-(2)-vinylpyridine,supported on a silicon oxide substrate, exhibit an

increase in Tg by 20-50 deg C compared to itsbulk value.

More recently, Fryer et al.4 presented asystematic experimental study of the effect ofinterfacial energy on the glass transition behaviorof thin polymeric films. They found that for agiven film thickness, the difference between theglass transition temperatures of the thin film andthe bulk polymer scaled linearly with theinterfacial energy. This work also demonstratedthat for lithographically relevant filmthicknesses, the effect of the interfacial energywould not suffice to cause the desired increase inthe glass transition temperature of the film.

A second approach for raising the glasstransition temperature of thin polymeric films,has been to graft (attach) some of the chains inthe film to the surface.5,6,7,8 Using opticalwaveguide spectroscopy, Prucker et al. comparedthe Tg of grafted films of PMMA on a siliconoxide surface with that for films supported on ahydrophobized silicon oxide surface. Theyfound that chain grafting had a negligible effect

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on Tg. Tsui et al.6 used X-ray reflectivity tostudy the glass transition behavior of thin filmsof polystyrene supported on a silicon oxidesurface. Films consisted of polystyrene chainsspin-coated onto approximately 3 nm thick brushlayer of chains end-grafted on the supportingsurface. They found that, for 33 nm thick films,the Tg of the films containing a brush layer wasonly slightly (less than 3 deg C) higher than theTg of the films that did not contain any graftedchains. Effect of chain grafting on the glasstransition behavior of polystyrene films was alsothe focus of a study by Tate et al.7. Theseauthors studied two types of films: In one typeof film, some polystyrene chains in the film (thatcontained terminal hydroxyl groups) were end-grafted to the supporting silicon oxide surface.The other type of film contained poly (4-hydroxystyrene) chains, some of which wereside-grafted to the supporting surface. It wasfound that for 50 nm thick films of the first type,Tg exceeds the bulk Tg by about 15-20 deg C.For 100 nm thick films of the second type, the Tgwas raised above the bulk value by as much as55 deg C.

Besides experimental work, moleculardynamics simulation studies of glass transitionbehavior of thin polymeric films have beenreported in the literature9,10. Torres et al. 9 usedhard-sphere molecular dynamics simulations toinvestigate the glass transition behavior ofultrathin films of short polymeric chains. Theyfound that freestanding films exhibit a reductionin Tg for small film thicknesses. For supportedfilms of thickness less than 30 σ (where σ is thewidth of the square well potential used fordescribing non-bonded interactions betweenvarious chain sites in their model), they foundthat the systems with weakly attractive polymer-substrate interactions showed a decrease in Tg,whereas the systems with strongly attractivepolymer-substrate interactions showed anincrease in Tg compared to the bulk value.Varnik et al.10 also have used moleculardynamics simulations to study the glass

transition behavior of thin films of a non-entangled polymer melt confined betweensmooth and repulsive walls. They found that theglass transition temperature decreases with filmthickness, in agreement with experimental results

In this work, we use a molecular model similarto that of Varnik et al.10 to investigate the effectsof chain grafting and polymer-substrateinteractions on the glass transition temperature ofthin polymer films. Our work is inspired by theaforementioned experiments. The mainobjective of the work, however, is not to make aquantitative comparison with these experiments;rather it is to obtain an understanding of themolecular level mechanisms that are responsiblefor the elevation of the glass transitiontemperature caused by chain grafting. To ourknowledge, none of the existing theories for theglass transition, such as those based on theconcept of free volume, can explain such aphenomenon. In this study, we compare resultsfor the glass transition temperature (Tg) obtainedusing three different techniques: measurement ofchanges in the film thickness (volumetric),calculations of the specific heat (energetic), andmeasurements of relaxation times from diffusiondata (dynamic). In each case, we investigate thedependence of Tg on polymer-substrateinteraction and grafting.

The rest of the paper is arranged as follows.First, we present the molecular model and thesimulation details employed in the work. Resultsfor the glass transition temperature of the thinpolymer films obtained using differentapproaches are discussed next. The paper endswith a discussion of our results and conclusions.

Molecular Model and Simulation Details

In this work, we study thin polymer filmssupported on a substrate. The other surface ofthe films is free. A bead spring model11 is usedfor the polymer chains. The system contains M= 40 linear polymer chains, each of whichconsists of N = 100 monomers. This chain

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length is believed to be about twice theentanglement length at the density simulated inthis work12. The particles that form the substrateare fixed to their positions in an FCC lattice.

Non-bonded interactions between polymerchain beads are modeled through a truncated andshifted Lennard-Jones potential:

+

−

= 008742.04

612

ijij

LJij rr

Uσσ

ε ,rij<2.2σ

(1)

0=LJijU , rij>2.2σ

The length and energy scale of this interactionsets the units. All quantities presented in the restof this paper are expressed in terms of σ, ε, and

εστ /2m= , where m is the monomer mass.Appropriate values for real materials are in therange of a fraction of a nanometer, a few tens ofmeV, and a few nanoseconds, respectively. Ourmodel for polymers is similar to that used byVarnik et al.10 These authors obtained the glasstransition temperature for a bulk sample byfitting relaxation times obtained from diffusiondata to predictions made by the mode couplingtheory. They determined the bulk criticaltemperature of this model system to be 0.45.

Interactions between polymeric beads andsurface particles are modeled through the samepotential with modified length scale σps =0.9 andenergy scale εps. Two values for the strength ofthe attraction between chain beads and surfacesites have been modeled: εps = ε and εps = 0.1 ε.

The FENE (finitely extendable nonlinearelastic) potential is used to model chainconnectivity and chain grafting to the surface.Thus interactions between neighboring beadsalong a chain and that between grafting sites onthe chains and wall atoms are modeled by meansof the following potential:

−−=

2

0

20 1ln

2

1

R

rkRU ijFENE

ij rij<R0

(2)

∞=FENEijU , rij>R0

Here R0 equals 1.5 and the distance between theparticles i and j in a chain at which Uij is at aminimum is approximately rij=0.96. The springconstant k = 30 is used to set the rigidity of abond. It needs to be large enough to assure thatbonds in the polymers do not break or cross, yetsmall enough to allow usage of a reasonableintegration time step Δt. The diameter of thesubstrate particles is set to 0.8σ, the nearest-neighbor distance to 0.946σ . To prevent thepolymers from positioning themselves along thelattice of the substrate, the distance betweensubstrate particles differs from that betweenmonomers along a chain

We carry out molecular dynamics simulationusing a thermostat to maintain a constanttemperature in the system. For beads on thepolymer chains the equation of motion is

( ) ( )tUUm iiij

FENEij

LJiji Wrr +Γ−+−∇= ∑

≠

&w (3)

Here the damping constant Γ=0.4 and Wi(t) is awhite noise source. The strength of the noise isrelated to Γ via the fluctuation-dissipationtheorem12. In the simulations, the equation ofmotion is integrated with a fifth-order Gear-predictor-corrector algorithm13 with Δt=0.005 τ.

Initial states are prepared at the highesttemperature. Subsequently, the film is cooled ata rate of 1/25,000 1/τ to a new temperaturesetting at which it is then equilibrated for at least40,000 τ. Finally, runs are performed for up to200,000 τ. Although the data obtained fromthese runs fluctuate, no systematic trends due toaging were detected. Moreover cooling rates upto five times slower, did not yield detectabledifferences in the results.

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The film has dimensions of 19 x 16 in theplane of the film. As is customary, periodicboundary conditions are used in the plane of thefilm (x and y directions). In some of thesimulations, 25 percent of the chains in the filmare side-grafted to the supporting surface. Thisyields the same ratio of thickness of the graftedlayer to the radius of gyration of individualchains as reported in experimental work.7

However, we note that we are unable to deducethe average number of grafting sites on eachchain from the experimental data. In thesimulations at hand, the grafted chains contain anaverage of 4.8 connections with the substrate.This number will be varied in future work. Inorder to prepare the grafted film, temporarycrosslinks will be created through an algorithmdescribed in an earlier publication14 by one of theauthors. In this algorithm, Monte Carlo movesare employed to form or break the FENE bondsbetween the polymer beads and the surface. Theaverage number of connected beads per graftedchain is controlled through a parameter thatmodels the associative attraction and controls thesuccess rate of the Monte Carlo moves. Afterthe system is equilibrated, the FENE bonds aremade permanent.

The glass transition temperature of the thinfilms in this work is determined using threedifferent approaches. The first approach consistsof monitoring the film thickness as a function oftemperature. Film thickness in this case isdetermined using the density distribution of thechain beads as a function of the distance from thesupporting substrate.

At a critical temperature Th the plot of filmthickness versus temperature changes slope. Themethod is essentially the same as that employedexperimentally using ellipsometry. The secondapproach for determining the glass transitionfollows earlier work by Perera et al.15 Theyobtained the specific heat-temperature curve fora binary mixture of soft disks, which is a modelglass forming system. In the isothermal-isobaricensemble at hand, the constant pressure heatcapacity Cp at zero pressure is given by

2

2

NT

ECp

Δ= (4)

where E is the total energy and N the number ofbeads. This approach is similar to thecalorimetric method used experimentally wherethe DSC experiments are used to determine theglass transition by monitoring the temperaturederivative of the enthalpy. For a system atequilibrium, expression (4) can be derived fromthe fluctuation theory of statistical mechanics. Interestingly, as Perera et al.15 show, thecorrespondence expressed by this expressionholds into the supercooled state, although onecould argue that the system was only “locally”equilibrated in all the basins of the glassy energylandscape that were visited during the run time. The heat capacity versus temperature datashow an asymmetric peak. Two transitiontemperatures can be deduced from them. First,the fictive temperature Tf, is the temperature atwhich the heat capacity begins to rise. Second,the temperature denoted by Tp, at which the heatcapacity peaks. In their simulations of liquidsilica, Saika-Vovoid et al.16 relate the observedpeak in the heat capacity to changes in thepotential energy hypersurface, whereas, ingeneral, Tf is related to the dynamic arrest ofparticles at the glass transition.

The diffusion coefficient (Dα) of the chainbeads is monitored in the third method. Thecharacteristic time for translational α-relaxation(τtr=Dα

-1) is then obtained for a range oftemperatures above the glass transition.According to the idealized mode coupling theory(MCT), this characteristic time diverges at a

critical temperature Tc as ( ) γτ −−∝ cTT .

Alternatively, the Vogel-Fulcher-Tammann

(VFT) equation ( )0/ TTce −∝τ can also be

employed to analyze the data and to obtain aVFT temperature T0. In our simulations, as wellas those by others10,17,18, simulated relaxationtimes cover at the most one or two decades. As a

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result, both power-law (MCT) and exponential(VFT) fits can represent the data equally wellover this limited range. For this reason, weconsider an analysis based on diffusion to be lessaccurate than the other two approaches.

III. RESULTS

A. Film thickness

For an ungrafted film with polymer-substrateinteraction εps=1, Fig 1a shows the bead densityas a function of distance from the first layer ofatoms in the substrate. The temperature equals0.55. As has been reported before19,20, layeringis observed near the surface. It is morepronounced and extends further inward at lowertemperature. At the lowest temperatures studied,layering extends throughout the film. At hightemperatures (T=1.4), only one layer is observed.A bulk-like region of constant bead densityoccurs farther away from the surface.Ultimately, the bead density reduces to zero nearthe free surface of the film. The film thickness isdetermined as the midpoint of the region wherethe bead density falls from its bulk-like value tothe free surface value. We find that the filmthickness equals 13.3 ± 0.2 at T = 0.55. Fig. 1bshows a plot of film thickness for a range oftemperatures. Linear sections of different slopescan be clearly distinguished for the liquid andglassy regions. The point of intersection of linesrepresenting the liquid and glassy regions yieldsa transition temperature value Th of 0.51. Wewould like to point out that there is some degreeof subjectivity involved in determining the valueof the glass transition using this way. It dependson the temperature range chosen for representingthe linear portions of the film thickness in theliquid and glassy regions.

Fig 2a compares the density profile of theungrafted film with that of the grafted one. Thecurves for the two films essentially lie on top ofeach other. For the grafted film, the first peak in

the density profile, is closer to the substrate. Weattribute this to the FENE interactions betweenthe grafted beads and atoms in the substrate.Temperature dependence of the thickness of thegrafted film (shown in Fig. 2b) yields a transitiontemperature of 0.54, which is slightly higher thanthat obtained for the ungrafted film. The shift isdue to a small difference in film thickness valuesat higher temperatures.

FIG 1. (a) Bead density as a function of distance (z) from thesubstrate for an ungrafted film with εps = 1 at a reducedtemperature T = 0.55. (b) Film thickness as a function oftemperature for the same film. Fits to the data at low and hightemperature are shown. Glass transition temperature is obtainedfrom fits to the data at low and high temperature. The crossovertemperature Th = 0.51 .

Fig. 3 displays the same data for simulationsin which the LJ interaction strength betweenpolymer and substrate is decreased to εps=0.1.Data were obtained for both ungrafted and

a

b

6

grafted films. For ungrafted films, however, allthe simulated temperatures are less than 0.6,since the films separate from the substrate athigher temperatures. Density profiles at T = 0.55are shown in Fig 3a. Comparison to that of theungrafted film at εps=1 reveals the expectedchange in layering near the substrate. For theungrafted film, at the lower polymer-substrateattraction, layering is more or less absent,

FIG 2. (a) A comparison of the bead density of the ungrafted(solid ) and grafted (long dashed) films with εps = 1 as a functionof the distance (z) from the substrate. (b) Film thickness as afunction of temperature for grafted film, εps = 1. The glasstransition temperature obtained from fits equals 0.54.

whereas for the grafted film the tendency to layeris strongly reduced. Naturally, the changes inbead packing near the surface result indifferences in film thickness. Film thicknessincreases to 13.85 for the ungrafted film at

εps=0.1 and to 13.55 for the grafted film atεps=0.1. Similar changes occur at othertemperatures. Film thicknesses as a function oftemperature at εps=0.1 for both films are shownin Fig 3b. In the glassy region, data for graftedand ungrafted films are the same. By contrast, inthe liquid region the ungrafted film is thicker- aswas also observed in Fig 3a. The glass transitiontemperature for the ungrafted film, obtained fromthe fits shown, equals 0.49. Since we wereunable to obtain data above T=0.6 for theungrafted film, we could not obtain a fit to thefilm thickness data in the liquid region. Nor

FIG. 3. (a) Bead density as a function of the distance (z) fromthe substrate for an ungrafted film with polymer-substrateinteraction εps = 1.0 (solid) and with εps = 0.1 (long dashed) aswell as a grafted film with εps = 0.1 (dotted). (b) Film thicknessas a function of temperature for an ungrafted (x) and grafted (o)film at εps = 0.1. Th = 0.49 for the grafted film. The ungraftedfilm separates from the substrates for temperatures above 0.6.

a

b

a

b

7

could we calculate the crossover temperature. Attemperatures between T=0.4 and T=0.6, theungrafted films are thicker than the grafted films,which suggests that the glass transitiontemperature of the ungrafted film must be belowthat of the grafted film.

B. Heat capacity

In Fig 4 a-d, the constant pressure heat capacityCp is plotted as a function of temperature for allfour films. Different methods have been followed in theliterature to infer the glass transition temperaturefrom calorimetric data. Glass transitiontemperature can be defined as the temperature atwhich the heat capacity as a function of

FIG. 4. Temperature dependence of constant pressure heatcapacity (Cp) for (a) ungrafted film, εps = 1 (b) grafted film, εps =1 (c), ungrafted film, , εps = 0.1 (d) grafted film, εps = 0.1. Glasstransition temperature resulting from shown fits to data equalTf=0.32 (a), Tf=0.33 (b), Tf=0.32 (c), Tf=0.33 (d).

temperature peaks, or as the temperature at theonset of a rise in the heat capacity.15,16,21,22,23 Wewill call the temperature obtained using the lattermethod the fictive temperature Tf and thatobtained using the former Tp . Data for graftedand ungrafted film at εps= 1 are shown in Figs 4aand 4b. Both types of films show an asymmetricpeak in the specific heat data. The specific heatfor the ungrafted film peaks at Tp = 0.5. For thegrafted film the peak is less pronounced, butappears to be shifted slightly to the right, to Tp =0.55. The temperature Tf at which the heatcapacity starts to rise is obtained from thecrossover of fits to the data just before the onsetof the rise and those during the rise. This yieldsTf = 0.32 for the ungrafted and Tf = 0.33 for the

a

b

c

d

8

grafted film. In Figs 4c and 4d we plot thespecific heat versus temperature for films withthe less attractive polymer-substrate interaction. These data peak at Tp = 0.45 (ungrafted) and Tp

= 0.49 (grafted). The onset of the rise of Cp is Tf

= 0.32 (ungrafted) and Tf is 0.33 (grafted). It isinstructive to display the data for all four films inthe same figure. From such a display, shown inFig 5, we arrive at the following conclusions.

FIG. 5. Comparison of specific heat data for all four films.

First, the specific heat versus temperatureshows a sharp peak for ungrafted films, whereasfor grafted ones the peak is much broader.Second, if the polymer-substrate attraction islowered from εps=1 to εps=0.1, the position inthe peak for ungrafted films shows a clear shiftto lower temperatures. Third, the onset of thepeak occurs at more or less the same temperaturefor all four films. Fourth, the slope in the datajust beyond the onset of the peak appears todepend on the polymer-substrate interaction,since it is lower for higher attraction. It isinsensitive to grafting, though.

C. Diffusion

The local translational mobility of themonomers in the films has been studied bymeans of the mean squared displacement (MSD)of the beads. For this purpose, the MSD is

calculated by averaging the squareddisplacement in the xy-direction over all particlesin the system. Displacement of the beads in thethird direction (perpendicular to the plane of thefilm) has not been included, since the motion ofthe particles in this direction is influenced by thesubstrate and the free surface. The MSD iscorrected for the center-of-mass diffusionintroduced by the thermostat. For filmscontaining grafted chains, contribution frombeads belonging to the grafted chains is excludedfrom the MSD. Figure 6 shows the results of theMSD for different temperatures of the ungraftedfilm at εps=1.

Several groups10,17,18 have used MSD data toobtain the glass transition temperature frommolecular dynamics simulations using modecoupling theory (MCT). Some of thesesimulations use models that include chemicaldetail17, others, like ours, employ somewhatmore coarse-grained bead-spring models ofpolymers10,18 In all these studies the MSD dataare employed to calculate relaxation times as afunction of temperature. A critical temperatureTc is defined as the temperature at which therelaxation time diverges. Although thesimulations can only access a small range ofrelaxation times, excellent agreement withexperimental results is obtained if the simulationmodel includes the detail necessary to make sucha comparison.17

For the simulations at hand, the mean squaredisplacement over time can be seen in Fig 6. Aplateau regime occurs at low temperature(T=0.3), a result of the beads getting trapped inthe cages formed by their neighbors. For shorttimes, one would expect the motion to beballistic. Although observed in our simulations,this regime is not shown. As temperatureincreases, the horizontal line representing aplateau regime gets shorter and is followed by asubdiffusive or α-relaxation regime. Beadmotion in this regime can be represented using apower law expression:

αα )()(2 tDtR >=< (5)

9

FIG. 6. Log-log plot of mean squared displacement of chain beads as a function of time in the ungrafted polymer film, εps = 1.

where Dα is the diffusion constant. Fitting thesimulation data to this expression in the timeinterval t = 500 to t = 4000 at differenttemperatures gives an average value of α = 0.46± 0.04.

According to MCT, the characteristic time ofthe translational α-relaxation in the sub-diffusiveregime, τtr=Dα

-1, algebraically diverges at thecritical temperature as17

( )γτ

τc

trTT −

= 0 (6)

The value of Dα and hence the relaxation time ateach temperature was determined by fitting theMSD data to equation (5) and using a value of α= 0.46. The relaxation time τtr in Fig 7 isplotted against the temperature T . Fitting thedata in this plot with equation (6) gives: Tc =0.36 and γ = 2.9. This value of the exponent γcompares favorably with a value of 2.1 obtainedby Varnik et al.10 for thin films and a value of

2.85 obtained by Zon and Leeuw18 for bulkpolymers. Fig 7 also show data for the graftedchains at the same polymer-substrate attraction.The results of a fit to equation (6) (not shown)are very similar: α = 0.49, Tc= 0.36 and γ= 3.0.

FIG. 7. Relaxation time as a function of temperature forungrafted and grafted polymer films, εps = 1. The longdashed line corresponds to a fit to equation (6), which ismotivated by the idealized mode coupling theory. Thesolid line corresponds to a fit to equation (7), the VFTequation.

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One may wonder if the critical temperaturewould have been different if the relaxation timeshad been obtained from the diffusion data for theentire polymer chains (rather than those for theindividual chain beads). Varnik et al.10 haveanswered this question and shown that, whereasthe values of the relaxation times and τ0 obtainedusing the two different methods differ, the valuesof γ and Tc do not.Fig 7 includes a fit of the data of the ungrafted film with εps=1 to the Vogel-Fulcher-Tammann (VFT) equation

( )0/ TTce −∝τ (7)

This fit yields T0=0.24. The same value is foundfor the ungrafted film.

We also investigated how the polymer-substrate interaction affects the bead mobility atT=.55. Figs 8a and 8b show mobility for a 4 σlayer of the film, either close to the free interface(a) or close to the substrate (b). All beads havebeen included in the calculations. We concludethat, near the free interface, bead diffusion isindependent of polymer-substrate attraction,whereas at the substrate it depends on theinteraction in a predictable way.

IV. DISCUSSION

The glass transition is a complex, time-dependent kinetic process, hard to characterize interms of one single transition temperature23. Afree volume concept is often employed todescribe the glass transition. As the free volumeor the average unoccupied volume available forbead motion decreases, the relaxation timeincreases. Mode coupling theory predicts thatthe relaxation time diverges at the criticaltemperature Tc. Unoccupied volume by itself isnot, however, an adequate measure for the stateof a glassy film. Films with the sameunoccupied volume, but different prepared, hasbeen shown24 to possess different mechanicalproperties. Hence, an additional mechanismmust exist by which the material remembers itshistory. It has been hypothesized that thefluctuations in the state of local packing mayplay a role.24,25 This would cause a structural

change, manifested in changes in the potentialenergy hypersurface. Such a change in the

FIG. 8. Mean squared displacement (MSD) of chain beads ofthe polymer films as a function of time for (a) layer near the freesurface (b) layer near the substrate.

energy landscape with decreasing temperaturehas recently been investigated by Salka-Voivodet al.16 in simulations of liquid silica. Theyassociate it with a dynamical transition from afragile liquid state at high temperature to a strongliquid one at low temperature. At the transitiontemperature Tp the specific heat peaks. Hencetwo distinct transitions at different temperaturesare of importance, one is due to volumetriceffects, the other to structural ones.

As shown in Table 1, which summarizes ourresults, in our data two transition temperaturescan indeed clearly be distinguished. For all fourtypes of model films studied, the values of thecrossover temperature Th, obtained from the film

a

b

11

Th Tf Tp Tc T0

ungrafted, εps = 1 0.51 0.32 0.5 0.36 0.24grafted, εps = 1 0.54 0.33 0.55 0.36 0.24ungrafted, εps= 0.1 – 0.32 0.45 – –grafted, εps = 0.1 0.49 0.33 0.49 – –

TABLE 1. Glass transition temperature values obtained using different methods. The table shows values determined fromthe changes in film thickness (Th), as the temperature (Tf) at which the specific heat starts to rise, as the temperature (Tp)where the specific heat peaks. Critical temperatures determined from fits of relaxation times obtained from diffusion dataare included as well. The values denoted by Tc result from a power-law fit, whereas the values denoted by T0 result from anexponential fit.

thickness data, match with the Tp values. On theother hand, the transition temperature Tf, definedby the onset of the rise in Cp versus T data, isslightly lower than Tc, which was obtained froma fit of the relaxation time data to equation (6).Hence we hypothesize that at Th≈Tp, a transitionoccurs due to changes in the energy landscape,whereas the temperature Tf characterizes theglass transition. Note that this implies that thechange in slope in the film thickness versustemperature at Th is due to differences in localpacking at high and low temperature and not dueto dynamic arrest. Our data for Tc (MCT) and T0

(VFT) are consistent with Tf, especially the valueof MCT critical temperature is approximately10% higher than the glass transition temperatureTf.

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Fig 5 and Table 1 further indicate thatgrafting some of the chains to the substrateincreases Tp, in agreement with experiments.Although intuitively one might have predictedthe observed trend of grafting on the structuraltransition temperature Tp, the underlyingmechanism is far from understood. Itsexplanation will most likely not be obtainedwithout the development of a good theoreticalunderstanding of the mechanisms responsible forthe dynamical fragile-to-strong transition16 andthe accompanying specific heat anomaly.

The observed shift in the transitiontemperature due to grafting is quite small. Thebead-spring model polymers can be mapped ontoreal polymers11. In this mapping, the value of

ε/kB ranges between 300 and 500 K, dependingon the experimental polymer system that ismapped. Hence, the increase in the value of theglass transition temperature as a result of chaingrafting in our simulations compares to a 15-25K increase in the value of Tg for a real polymericsystem. This increase in Tg is much smaller thanthat observed in the experimental work of Tate etal.7, who found that grafting leads to an increasein Tg by as much as 55 K above it bulk value.

Since for a film thickness comparable to thatused in our simulations the glass transitiontemperature of the film is about 30 degrees lessthan the bulk value, the shift in transitiontemperature due to grafting observedexperimentally is at least 85 degrees. Onepossible reason for this discrepancy could be thatthe experimental system contained more graftedsites per chain than studied in this work (seeearlier text). It is also possible that crosslinking(either temporary or permanent) within the filmhas played a role in further raising the transitiontemperature of the film in the experiments. Adetailed investigation of the effects of the variousgrafting characteristics (number of grafted sitesin the film, distribution of the grafting sites onthe chains, fraction of grafted chains etc.) on theglass transition temperature will be the subject ofa future publication.

Future work will also investigate molecularlevel structure of the films and hopefully willgive an explanation for the observed dependenceof Tp of the films on the interaction between the

12

polymer and the substrate, including the effectsof chain grafting. Changes in molecular leveldynamical27 properties of these films across theglass transition range have recently beeninvestigated in our laboratory as well.28

Conclusions

This paper presents a detailed investigationof the glass transition behavior of nanoscopicallythin polymer films. The work is motivated bythe experimental observation that grafting someof the polymer chains to the substrate can lead toa significant increase in the glass transitiontemperature. Such an elevation in grafting someof the polymer chains to the substrate is desirablefor manufacturing processes in themicroelectronics industry, such as lithographyand millipede data storage technology.29

Although the phenomenon of glass transitionhas been studied for decades, developing afundamental understanding of the molecularmechanisms underlying glass transition is still anactive area of research. A strong dependence ofTg on chain grafting is definitely not predicted byany of the existing theories. Simulation resultssuch as those obtained in this work can be usedto refine the theories of glass transition.

In this work, we have employed threedifferent methods for determining the glasstransition temperature. Two distincttemperatures, both of which characterizedifferent aspects of the glass transition, can beextracted from our data. First, the structuraltransition temperature, either defined as thetemperature Tp at which the heat capacity peaksor the temperature Th at which the slope in a plotof film thickness as a function of temperaturechanges. In qualitative agreement with theexperimental data in the literature, thistemperature is found to show a dependence on

both chain grafting on the substrate surface, andon the strength of the substrate-polymerinteraction. Second, a critical temperature Tf,defined as the fictive temperature at which thespecific heat as a function of temperature startsto rise. Diffusion data indicate that below thistemperature, bead motion is frozen over the timescale of our computer experiments. Thistemperature is slightly lower than the criticaltemperature (Tc) defined by the mode couplingtheory which is obtained by monitoring thetemperature dependence of the relaxation timededuced from the diffusion of the beads. Thismethod is less accurate, though, given thelimited range of relaxation times that can beaddressed in computer simulations. Both chaingrafting and the strength of the polymer-substrate interaction are found to have noinfluence on the values of Tf and Tc.

The three different methods used fordetermining the glass transition temperatureprobe different aspects of this process. A moredetailed analysis that differentiates between themolecular mechanisms underlying thesedefinitions of glass transition is currentlyunderway in our laboratory.

ACKNOWLEDGMENT

This research is supported by a grant from theDonors of the Petroleum Research Fund,administrated by the American ChemicalSociety. Moreover, Maarten v. Weertcontributed to this work during a three monthlong student internship. He acknowledgessupport from the Dept. of Applied Physics of theTechnical University Eindhoven, TheNetherlands. We acknowledge P. F. Nealey, J. J.de Pablo, A.V. Lyulin, and M.A.J. Michels formany insightful discussions.

13

* Electronic address: abaljon@mail.sdsu.edu.** Work on this project was begun and partially completed when RK held a position at Accelrys Inc.

References and Notes

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