Simulated Glass Transition in Free-Standing Thin Polystyrene Films

A. R. C. BALJON,1 S. WILLIAMS,1 N. K. BALABAEV,2 F. PAANS,3 D. HUDZINSKYY,3,4 A. V. LYULIN3

1Department of Physics, San Diego State University, San Diego, California 92128

2Institute of Mathematical Problems of Biology—Pushchino 142290, Russia

3Group Theory of Polymers and Soft Matter, Eindhoven Polymer Laboratories, Technische Universiteit Eindhoven,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

4Dutch Polymer Institute, P.O. Box 902, 5600 AX Eindhoven, The Netherlands

Received 23 July 2009; revised 23 February 2010; accepted 23 February 2010

DOI: 10.1002/polb.22005

Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: In this article, we investigate the glass transition

in polystyrene melts and free-standing ultra-thin films by

means of large-scale computer simulations. The transition

temperatures are obtained from static (density) and dynamic

(diffusion and orientational relaxation) measurements. As it

turns out, the glass transition temperature of a 3 nm thin film

is �60 �K lower than that of the bulk. Local orientational mo-

bility of the phenyl bonds is studied with the help of Legen-

dre polynomials of the second-order P2(t). The a and b

relaxation times are obtained from the spectral density of

P2(t). Our simulations reveal that interfaces affect a and b-relaxation processes differently. The b relaxation rate is faster

in the center of the film than near a free surface; for the arelaxation rate, an opposite trend is observed. VC 2010 Wiley

Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1160–

1167, 2010

KEYWORDS: glass transition; relaxation; simulations; thin films

INTRODUCTION When polymers are near surfaces, the num-ber of possible configurations, and, hence, their entropy isstrongly reduced. As a result, the properties of a polymericsystem start to deviate from bulk values when one of thedimensions of the system approaches nanometer lengthscales. Such confinement effects are significant for polymersin a slit,1 a nanopore,2 or a cavity,3 instances of one-, two-,and three-dimensional confinement, respectively. The physicsof an uncharged linear polymer chain under various types ofconfinement is known reasonably well in terms of the effecton the configurational statistics of the chain.4 The shape ofthe confining volume, the nature, size, and architectureof the polymer, and the type of interaction of the polymerwith the confining wall all has an impact on the structure ofthat polymer. Less known is the influence of confinement onglassy dynamics, segmental mobility, and as a consequence,on glass transition in general.

The internal relaxation dynamics of polymeric chains seemto be strongly affected by confinement.5 Scientifically inter-esting anyway because of the all-important role of entropy, abetter understanding of the physics of polymers under con-finement is relevant to fields as diverse as lubrication, filtra-tion, enhanced oil recovery, and transport through mem-branes.2 A porous medium, for example, can be used as aprobe to gauge the dimension of a polymer chain under dif-ferent conditions.6 The study of wall-induced conformational

changes in polymers may help elucidate the activity of bio-logical polymers near cell surfaces.7 Thin polymeric filmswith thickness less then 100 nm also play an important rolein the microelectronics industry, where they are used asmasks in lithographic processes.

The glass transition behavior of thin polymer films in recentyears has been the subject of a large number of experimentalstudies.8 A comprehensive review was recently published byAlcoutlabi and McKenna9 Techniques such as ellipsometry, X-ray reflectivity, local thermal analysis, and dielectric spec-troscopy have been used to determine the glass transitiontemperature (Tg) of these films. They reveal that the glasstransition temperature decreases with film thickness forfree standing films. In the case of supported films or filmsconfined by surfaces, the glass transition temperaturedecreases10,11 or increases12 with increasing confinement,depending on the interaction of the polymer with thesubstrate.

In this article, we concentrate on free standing films of poly-styrene (PS). Forrest and coworkers13–15 studied these filmsexperimentally. In films that were between 20 and 80 nmthick they observed a systematic decrease of the dilatometricglass transition temperature with film thickness of up to60� K below the bulk value. Such a large decrease was alsoobserved by Wubbenhorst and coworkers.16

Correspondence to: A. R. C. Baljon (E-mail: abaljon@mail.sdsu.edu)

Journal of Polymer Science: Part B: Polymer Physics, Vol. 48, 1160–1167 (2010) VC 2010 Wiley Periodicals, Inc.

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Dielectric relaxation spectroscopy16 shows a speed-up of thea relaxation process and an asymmetric broadening of therelaxation peak in thin PS films. A decrease in the a relaxationtime implies a decrease in Tg. Hence, these observations con-firm the system-size dependence. The broadening is believedto be related to a spatial distribution of Tg’s in the directionnormal to the film. This effect has been studied in detail byTorkelson and coworkers.17,18 in supported PS films. Theymeasure Tg as a function of the distance to the free film sur-face and find variations of Tg and the a process over a rangeof up to 50 nm. This range is even larger for the smaller scalestructural relaxations (b process). Hence, it has been sug-gested that interfaces influence a and b relaxation dynamicsdifferently. It could even be that there is an additional relaxa-tion process unique to thin freely suspended films and notpresent in the bulk.19 Kremer and coworkers20 on the otherhand, believes that the dependence of Tg on film thickness isdue to chain cleaving caused by oxygen in the air.

In addition to experimental work, molecular dynamics simu-lation studies of glass transition behavior of thin polymericfilms have been reported.21,22 Most of these simulationsemploy course-grained bead-spring models. They all confirmthe decrease in glass transition temperature with film thick-ness for free standing films or films near repulsive sub-strates. A review was written by Bachnagel and Varnik.23

In this article, we use an atomistic model of PS polymersand compare the glass transition behavior in the bulk andin films of two different thicknesses (12 and 3 nm). Weconfirm the decrease in Tg with film thickness in these com-putational films using a variety of methods. We analyzetheir spectral densities and calculate the a and b relaxationtimes. The spatial dependence of these relaxation times andTg is investigated as well. We confirm that the a relaxationprocess is faster and that Tg is lower near the free surfaces.On the other hand, b relaxation is faster in the center of thefilm.

COMPUTATIONAL METHODS

Atactic-polystyrene polymers are modeled using unitedatoms. Each polymer contains 80 monomers and hence has amolecular weight Mw ¼ 8.6 � 103. Free standing films con-taining 8 or 32 PS polymers are simulated. The united atomforce field is identical to the one employed in previous stud-ies of the glass transition in bulk systems.24 They include (1)nonbonded interactions between all united atoms three ormore bonds apart; (2) bending potential; (3) torsion poten-tial for the backbone; (4) torsion potential for the phenylring torsions; (5) phenyl ring out-of-plane bending potential;(6) torsional potential around the Carom-Carom-bond; (7)improper torsion potential. Details on all terms can be foundin previous publications.14,25 The leapfrog variant of the ve-locity Verlet algorithm was used to integrate Newton’s equa-tion with a timestep of 4 fs. The temperature is controlledusing a collisional thermostat.26 The system was equilibratedat 540 �K. At the very beginning, all trans conformations ofparallel chain molecules were generated in a large simula-

tion box. The box was then allowed to relax under a fixedimposed external pressure of 1400 atm using a Berendsenbarostat. Periodic boundary conditions are employed in the xand y direction. After equilibration, the lateral dimensionwas 70 by 70 Å. Subsequently, the simulation box was fixedat this setting and the system further equilibrated at zeropressure. The dimension perpendicular to the plane dependson the number of molecules. It is 112 and 35 Å for filmswith 32 and 8 molecules, respectively. Subsequently, thesample is cooled at a rate of 0.01 �K/ps. Configurations arestored at a temperature range between 540 and 210 �K. Atthese temperatures, the trajectories are sampled. The first nsof this trajectory is disregarded to allow the system to adjustto the specific temperature, before statistical properties arecalculated.

Thin film simulations are compared with bulk simulationsthat are cooled at the same rate as the free-standing films.These simulations are performed in a cubic box with peri-odic boundaries in all three dimensions. Each cell containseight polymers.

RESULTS

Glass Transition TemperatureFigure 1 shows the density profile of the 32-chain polymerfilm for several temperatures. The z-direction is perpendicu-lar to the free surfaces. The scale is set such that z ¼ 0 isalways in the middle of the film. As can be seen, the densityreaches a plateau value in the middle of the film and dropsnear the free surfaces. The average value of the density inthis plateau regime as a function of temperature is shown inFigure 2. The plateau is smooth at high temperatures, atlower ones the density fluctuates. This is due to internalordering of the united atoms. The same effect has earlierbeen observed in simulations of bead-spring polymers nearsurfaces.27 Apparently, even when united atom models areemployed to model a free-standing film, a similar orderingeffect occurs. The density plotted in Figure 2 has beenobtained from averages between two peak values in the pla-teau regime. Figure 1 and 2 show that density decreaseswith increasing temperature. Although the decrease is linear,a different slope is observed at low and high temperature.This is a well known experimental phenomenon and servesas an indication of a glass transition.28 A similar effect canbe observed when film thickness is plotted versus tempera-ture. The point at which the two lines cross is identified asthe glass transition temperature. It is 362 �K for the filmwith 8 polymers and 392 �K for the one with 32 polymers.The density as a function of temperature for the bulk systemis shown as well. The crossover temperature is 447 �K. Thethermal expansion coefficient can be calculated from theslope of the fit. From the bulk data, we obtain for the glassaglass ¼ 2.0 � 10�4 1/�K and for the melt amelt ¼ 5.3� 10�4 1/�K. These values agree well with those obtainedexperimentally: aglass ¼ 1.7–2.4 � 10�4 1/�K and amelt ¼5.1–6.0 � 10�4 1/�K.29,30

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GLASS TRANSITION IN PS FILMS, BALJON ET AL. 1161

A second estimate of the glass transition temperature isobtained from the orientational mobility of the phenyl bonds.The orientation correlation function is given by the secondorder Legendre polynomial P2:

P2ðtÞ ¼ 3

2ð~bð0Þ �~bðtÞÞ2 � 1

2

� �(1)

where ~b is the vector representing the phenyl bond. Bracketsrepresent averages over phenyl bonds. At t ¼ 0 the correla-tion function equals 1 and it decays to zero once the bondorientation is independent of the original orientation. At in-termediate times a stretched exponential fits the data:P2ðtÞ ¼ ae�ðtsÞb , where s is the relaxation time. The quantity0 � b � 1 represents the average width of the relaxationspectrum. Two different correlation functions have been cal-culated. In the first case, only phenyl bonds at both ends ofthe chains are included, in the second, the average is takenover the 68 phenyl bonds in the middle of the molecules.

For the first case, Figure 3 shows the inverse relaxation timeas a function of inverse temperature. At low temperature,simple activated dependence of the relaxation time on tem-perature is observed:

s ¼ s1eUactkBT (2)

At high temperatures the relation is given by the Vogel-Fulcher law:

s ¼ s1eUact

kBðT�T0Þ (3)

The points at which the fits cross define the glass transitiontemperature Tg. They are found to be 352 �K for films with8 chains, 396 �K for films with 32 chains, and 402 �K forbulk samples. The Vogel Fulcher temperature T0 is approxi-mately 50 �K below Tg. That this difference is the same forbulk PS and thin films was recently found experimentally by

Tsui and coworkers.31 The glass transition temperatures arelisted in Table 1, along with the results obtained when thesame analysis is repeated for the 68 phenyl bonds in themiddle of each chain.

The third method used to obtain the glass transition temper-ature is based on the translational diffusion of the unitedatoms. Again two different calculations are performed. In thefirst only the united atoms in the phenyl rings at the ends ofthe chains are included in the average; the second considersall united atoms in the 68 phenyl rings in the middle of thechains. The glass transition temperature can also be calcu-lated from the mean squared displacement of molecules.Below the glass transition, atoms are more or less frozen.They become more mobile as temperature is increased. Inprevious work,23 we have calculated the relaxation timefrom the diffusion constant. The glass transition occurswhen the relaxation time becomes infinite. These calcula-tions required extremely long runs. Recently, it was pointedout by Doi and coworkers32 that the glass transition temper-ature can be calculated from a plot of the mean squared dif-fusion over a short time interval t* versus temperature,where t* is in the subdiffusive regime. We choose t* ¼ 400ps, but verified that similar results are obtained for 200 or600 ps. Figure 4 shows the mean squared displacement aftert* as a function of temperature for united atoms in the phe-nyl rings at both chain ends. The characteristic temperatureTg is obtained by the intersections of the fits at low and hightemperature. It turns out to be 363, 403, and 440 �K forfilms with 8 chains and 32 chains and bulk samples, respec-tively. The calculations are repeated using only the 68 phenylrings in the middle of the chains. All results are summarizedin Table 1.

Next, we investigate the glass transition temperature at thesurface region of the film and compare it with that in the cen-ter region. To this end, we split the film with 32 polymers in a5 nm-thick layer in the center and the remainder near the

FIGURE 1 Density of a free-standing thin film with 32 polymers

as a function of the direction perpendicular to the film (z). The

film is centered around z ¼ 0. [Color figure can be viewed in

the online issue, which is available at www.interscience.

wiley.com.]

FIGURE 2 Density in the middle of the films as a function of

temperature for two different thicknesses. The error in the data

is comparable to the symbol size. [Color figure can be viewed

in the online issue, which is available at www.interscience.

wiley.com.]

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free surfaces. Figure 5 shows how the diffusion of the phenylrings at both ends of the molecules depends on where theyare located in the film. As expected, diffusion near the freefilm surfaces is faster than in the middle of the film. A sepa-rate glass transition can be defined for different layers of thefilm. In the case at hand, the glass transition for the centerlayer equals 432 �K; that for the boundary layers 389 �K.

Relaxation ProcessesFrom the orientational correlation function given in eq 1 thedistribution of relaxation times (spectral density) g(ln s) wascalculated using the CONTIN method:

P2ðtÞ ¼Z

gðln sÞe�t=sd ln s (4)

We have employed this method in previous studies.33 Theanalysis has been performed with the data obtained whenonly the phenyl bonds at both chain ends are considered.Figure 6a shows the results for the bulk sample, Figure 6bthose for the 32-chain film, and Figure 6c for the 8-chainfilm. Three distinct relaxation processes can be extracted.The transient time due to ballistic motion, the b relaxationtime related to relaxation within a cage formed by neighbor-ing chemical groups, and the a relaxation time related to therelaxation of the entire molecules. At high temperature, the aand b process merge. In Figure 7, we plot the spectral den-sities for the three different systems in the same figures, sothey can be more easily compared. Two different tempera-tures: 330 and 480 �K are shown. Moreover, a 5 nm thicklayer in the middle of the 32 chain film is split of from therest. P2(t) is then calculated for both parts separately. Thespectra obtained from these calculations are compared withthat of the total film in Figure 8.

DISCUSSION AND CONCLUSION

In Table 1, the measured glass transition temperatures arelisted. The temperature obtained from the static density data isaveraged over all united atoms. In the dynamic measurementsonly the phenyl bond is included. These calculations are eitherperformed for phenyl rings at both ends (end) or the 68 in themiddle (middle). All transition temperatures are obtained fromfits to the data at high and low temperature and hence dependon which data points are included in the fits. It has been foundthat the glass transition temperature also broadens withdecreasing film thickness. As a result for the ultrathin filmsthat we study the measured crossover temperature is sensitiveto the temperature range used and method employed. We esti-mate that all of this introduces an error in the order of 10 �K.

The transition temperature calculated using only the phenylbonds at the ends is within errors the same when calculatedeither from the second order autocorrelation function P2 or

FIGURE 3 Orientational relaxation time of the phenyl bonds at

both ends of the PS molecules versus temperature. The glass

transition temperatures are determined from fits to the data at

high and low temperature. Dashed lines represent fits to eq. 2

and solid lines fits to eq. 3. [Color figure can be viewed in the

online issue, which is available at www.interscience. wiley.com.]

TABLE 1 Glass Transition Temperatures Obtained from the

Measurements Described in the Text for the Bulk, 32 Polymer,

and 8 Polymer PS Film

Bulk 32-Chain 8-Chain

Density 447 392 362

P2-end 402 396 357

P2-middle 449 412 385

Diffusion-end 420 403 363

Diffusion-middle 463 423 400

The first row of data is obtained from the change in density with tem-

perature. The second and third row are obtained from the decay of the

second order autocorrelation function P2 for the phenyl groups. Either

the 2 phenyl rings at the end of each polymer (end) or the 68 in the

middle (middle) were included in the calculation. Data in the last two

rows are obtained from the diffusion of the united atoms in the phenyl

rings at either the ends or in the middle of the film.

FIGURE 4 Mean squared displacement after 400 ps of united

atoms in the phenyl groups at chain ends. The data are

obtained by averaging over all six atoms in each phenyl ring.

[Color figure can be viewed in the online issue, which is avail-

able at www.interscience.wiley.com.]

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GLASS TRANSITION IN PS FILMS, BALJON ET AL. 1163

from diffusion data. The transition temperature is alwayslower for the phenyl bonds at the ends than when bonds inthe middle are included in the calculation. This is to beexpected, since chain ends are more mobile than the middleparts. The transition temperature calculated from the densityseems to be closer to that obtained for the middle phenylrings for the bulk and closer to that obtained for the end ringsfor the films. Most importantly, the simulation data show that,independent of the method employed, the glass transition tem-perature decreases with decreasing film thickness. Tg for the3 nm film is much lower than the bulk Tg. The reduction ofabout 60 to 80 �K is comparable to that observed in experi-ments.10 The absolute values are higher than those obtainedexperimentally. As we have pointed out in previous work,34

this is due to the high cooling rate employed. Experimentally,it has been found that the relation between the glass transitiontemperature Tg and film thickness h depends on molecularweight.15 Free-standing films of low molecular weight poly-mers obey the following empirical relation:

TgðhÞ ¼ Tbulkg 1� a

h

h i1:8(5)

According to ref. 15 our chain molecules are in this low mo-lecular weight regime. However eq. (5) does not fit our data.Hence, we fit their empirical relation that describes thereduction in Tg in experiments on high molecular weightpolymers (Mw > 575 103).

Tgðh < h0Þ ¼ Tbulkg 1� h0 � h

f

� �� �

Tgðh � h0Þ ¼ Tbulkg

(6)

This yields h0 % 21.5 nm and f % 150 nm. The value of h0is close to the end-to-end distance REE as is the case inexperiments.15 REE is of the order of 20 nm for our simu-lated polymers, which is similar to the value of ho we obtain

from a fit to eq. (6). Mw ¼ 8.6 � 103 is much lower for thesimulated chains, though, than the threshold value abovewhich eq. (6) was found to be true experimentally. We think

FIGURE 5 Averaged mean squared displacement after 400 ps of

all united atoms in both phenyl groups at the chain ends. Data

are for the 32 polymer film. Data for a 5 nm thick layer in the

center of the film as well as those for the remainder (boundary)

are shown and compared to those of the entire film.

FIGURE 6 Normalized distribution function of the relaxation

times for the P2(t). Only the phenyl bonds at chain ends are

included into the analysis. Data are shown for bulk (top), 32

polymer film (middle) and 8 polymer film (bottom).

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FIGURE 7 Comparison of relaxation processes in the bulk and the films with 32 and 8 polymers resp. Figure 7a shows the spectral

densities at T ¼ 330 K and Figure 7b at T ¼ 480 K. [Color figure can be viewed in the online issue, which is available at

www.interscience.wiley.com.]

FIGURE 8 Comparison of the relaxation processes in the entire 32-polymer film (black), the 5 nm thick layer in the center (blue),

and the boundaries outside of the 5 nm layer (red). Data are shown for four different temperatures. The arrows at the lower tem-

peratures indicate the peaks in the a and b relaxation processes. They indicate that the b-process is faster in the center of the film,

while the a-process is faster near free surfaces. [Color figure can be viewed in the online issue, which is available at

www.interscience.wiley.com.]

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GLASS TRANSITION IN PS FILMS, BALJON ET AL. 1165

that eq. (6) still holds, because our films are much thinnerthan those used in the experiments. Apparently eq. (6) holdswhen the size of the polymers is larger than the film thick-ness. We realize that for a careful comparison with experi-ments more data, especially for thicker films, are required.This in turn requires faster computers than currently avail-able. However, we hope that the rough estimate given aboveconvinces the reader that the glass transition temperature ofour simulated PS chains depends on film thickness in a simi-lar way as it does in experimental systems. The main goal ofthis study is the determination of spectral densities, whichare not easily obtained experimentally.

These are shown in Figure 6. The distribution of the relaxa-tion times of P2 shows three distinct peaks. They are causedby the large-scale motions of cooperative segment (a relaxa-tion), smaller scale structural dynamics (b relaxation), and atransient processes. At high temperatures, the ballistic and brelaxation peaks merge and only two peaks are visible. If thetemperature approaches the fluid regime, only one peakremains. The temperature at which the peaks merge dependson the distance to the glass transition one. Hence, peaks tendto merge at lower temperatures in thin films than in the bulk.In Figure 7 data, for the two films and bulk are compared.The ballistic relaxation time is a little higher for films thanbulk at T ¼ 330 �K. This is due to the higher density in thebulk at this temperature (Fig. 2), which decreases the freepath length. It can also be seen in Figure 7 that the a relaxa-tion time decreases with decreasing film thickness. A similarstrong decrease of the a relaxation time of thin films wasreported recently by Wubbenhorst and coworkers.11 It isbelieved to be responsible for the reduction in the glass tran-sition temperature of the films, which is mainly dictated bythe long-time a relaxation of entire molecules. On the otherhand, Figure 7 shows that the local structural b relaxationrate is more or less identical for the bulk and films. Our simu-lations (Fig. 8c and 8d) also reveal that interfaces affect a andb-relaxation processes differently. The a relaxation rate isfaster near the free surface. A possible theoretical explanationfor this was given by DeGennes,35 who argued that, when thethickness of the film is smaller than the size of the individualpolymers, free surfaces can facilitate collective sliding motionof large chain segments. A similar effect could cause thedecrease in glass transition temperature in the layer near thesurface observed in diffusion measurements (Fig. 5). Onewould intuitively expect that in the center of the film the brelaxation rate is slower as well. Interestingly, the oppositeeffect is observed. As can be seen in Figure 8, the small scalestructural relaxation is faster in the center of the film thannear a free surface. A similar effect has been observed inexperiments on PMMA by Torkelson and coworkers.36 It is asyet not understood. Experiments have found that the differ-ence in b relaxation time between the surface layer and cen-ter of the film is even more significant when the film is sup-ported by an attractive surface.36 In future simulations, weplan to investigate this topology as well. It will be interestingto investigate what causes the increased local mobility in thecenter of these thin films and how it depends on the chemis-try of the polymer under investigation.

A.B. thanks support from the NSF (DMR 0517201). We thank G.McKenna andM. A. J. Michels for insightful discussions.

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