Thickness Dependence of the Glass Transition Temperature in Thin

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Langmuir 2001, 17, 2703-2710

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Thickness Dependence of the Glass Transition Temperature in Thin Polymer Films Jae Hyun Kim,† Jyongsik Jang,*,† and Wang-Cheol Zin*,‡ School of Chemical Engineering and Hyperstructured Organic Materials Research Center, Seoul National University, Shinlimdong 56-1, Seoul 151-742, Korea, and Department of Materials Science & Engineering and Center for Advanced Functional Polymers, Pohang University of Science & Technology, Pohang 790-784, Kyungbuk, Korea Received August 5, 2000. In Final Form: January 23, 2001 In this paper the glass transition temperature (Tg) in thin polymer films has been studied. Ellipsometry has been used to measure Tg of poly(R-methylstyrene) (PAMS) thin films as a function of the film thickness for two molecular weights. When the films were thinner than a few hundred angstroms, substantial reduction in Tg was apparent. The Tg depression pattern did not show the difference between molecular weights. The Michaelis-Menten equation was adapted and used to fit the experimentally obtained Tg data. From this analogized Michaelis-Menten function fitting, the fitting parameters Tg,∞ and ξ could be obtained. The obtained Tg,∞ corresponded with the bulk Tg, and the parameter ξ was correlated with the statistical segment length. A continuous multilayer model was proposed and derived to describe the effect of surface on the observed Tg reduction in thin films, and the depth-dependent Tg profile was obtained. These results showed that Tg at the top surface was much lower than the bulk Tg and gradually approached the bulk Tg with increasing distance from the edge of the film. The model and equation were modified to apply for the polymer coated on the strongly favorable substrate and the freely standing film.

Introduction Recently, much focus is given to the study of the glass transition temperature (Tg) in thin polymer films due to its importance in technology and in science.1-15 The large surface-to-volume ratio in the thin film geometry enables us to explore free surface, the role of interface that has various physical or chemical affinities, and chain confinement effect. Study on Tg in thin polymer films was initiated with a thin polymer film coated on the substrate that has no interaction with polymer. Keddie et al. gave the first direct evidence for reduction in Tg with decreasing thickness for polystyrene (PS) thin film coated on a Si wafer.3 When the thickness of the film was below 400 Å, the reductions in Tg were apparent and the measured Tg * To whom correspondence should be addressed. † Seoul National University. ‡ Pohang University of Science & Technology. (1) Frank, C. W.; Rao, V.; Despotopoulou, M. M.; Pease, R. F. W.; Hinsberg, W. D.; Miller, R. D.; Rabolt, J. F. Science 1996, 273, 912. (2) Beaucage, G.; Composto, R.; Stein, R. S. J. Polym. Sci., Phys. Ed. 1993, 31, 319. (3) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Europhys. Lett. 1994, 27, 59. (4) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Faraday Discuss. 1994, 98, 219. (5) Keddie, J. L.; Jones, R. A. L. Isr. J. Chem. 1995, 35, 21. (6) Orts, W. J.; van Zanten, J. H.; Wu, W.; Satija, S. K. Phys. Rev. Lett. 1993, 71, 867. (7) Wu, W.; van Zanten, J. H.; Orts, W. J. Macromolecules 1995, 28, 771. (8) Wallace, W. E.; van Zanten, J. H.; Wu, W. Phys. Rev. E 1995, 52, R3329. (9) van Zanten, J. H.; Wallace, W. E.; Wu, W. Phys. Rev. E 1996, 53, R2053. (10) Forrest, J. A.; Dalnoki-Veress, K.; Stevens, J. R.; Dutcher, J. R. Phys. Rev. Lett. 1996, 77, 2002. (11) Forrest, J. A.; Dalnoki-Veress, K.; Dutcher, J. R. Phys. Rev. E 1997, 56, 5705. (12) Forrest, J. A.; Mattsson, J. Phys. Rev. E 2000, 61, R53. (13) Xie, L.; DeMaggio, G. B.; Frieze, W. E.; DeVries, J.; Gidley, D. W.; Hristov, H. A.; Yee, A. F. Phys. Rev. Lett. 1995, 74, 4947. (14) DeMaggio, G. B.; Frieze, W. E.; Gidley, D. W.; Zhu, M.; Hristov, H. A.; Yee, A. F. Phys. Rev. Lett. 1997, 78, 1524. (15) Grohens, Y.; Brogly, M.; Labbe, C.; David, M.-O.; Schultz, J. Langmuir 1998, 14, 2929.

trend was not dependent on the molecular weight (Mw) of samples. It was suggested that the reduction of Tg in thin films is caused by the presence of a liquidlike layer at the polymer-air interface, i.e., the polymer surface. The existence of the lower Tg layer at the polymer surface has been found in various experiments, so this suggestion seems to be much plausible.16-18 On the other hand, other experiments show rather contradictory results. These films are composed of the thin polymer film coated on a strongly favorable substrate, and Tg of these samples usually increases with decreasing film thickness. van Zanten et al. showed this phenomenon with poly(2vinylpyridine) (P(2)VP) coated on the oxide surface of a Si wafer.9 Another result by Keddie et al. clearly showed the importance of interface with two different substrates, and this result revealed that both surface and interface affect the Tg in thin films.4 Grohens et al. identified the importance of the substrate by contrasting an increase of Tg with decreasing thickness for isotactic PMMA on silicon and aluminum with a decrease for syndiotactic PMMA.15 These results have highlighted the importance of interactions with the substrate on Tg in thin films. Because the geometry of a thin supported film has inherent asymmetry in its structure, Forrest et al. investigated film which has symmetry, i.e., freely standing films, to determine the effects from the surface and the interface more clearly.10,11 According to their experimental results, much larger reductions in Tg were observed for the freely standing film than for the supported film, and this reduction was dependent on Mw of the sample. This finding of Mw dependence in Tg reductions for freely standing films provides strong evidence of the importance of chain confinement effects on Tg. The discrepancy in Mw dependence between the freely standing film and the film coated (16) Meyers, G. F.; DeKoven, B. M.; Seitz, J. T. Langmuir 1992, 8, 2330. (17) Kajiyama, T.; Tanaka, K.; Takahara, A. Macromolecules 1995, 28, 3482. (18) Jean, Y. C.; Cao, Z. H.; Yuan J.-P.; Huang, C.-M., Nielsen, B.; Asoka-Kumar, P. Phys. Rev. B 1997, 56, R8459.

10.1021/la001125k CCC: $20.00 © 2001 American Chemical Society Published on Web 04/05/2001

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on a substrate still remains a puzzling problem.A recent result obtained with the lower Mw freely standing film does not show any dependence of Mw on the reduction in Tg.12 Until now, there is no general equation or model to estimate Tg in thin polymer films. To construct a model that includes both surface and interface effects, a continuous multilayer model is proposed. On the basis of this model, the general equation to estimate thickness dependence on Tg in thin films is developed and this equation is applied to various cases of experimental results about Tg in thin polymer films. The origin for each experimental result will be considered and the factors that can affect Tg in thin polymer films will be discussed for various geometric cases. Experimental Section The investigated samples consist of a polymer thin film on the clean native-oxide surface of a Si wafer (100). Two different molecular weight poly(R-methyl styrene)s purchased from Pressure Chemicals have been used. These two poly(R-methyl styrene)s will be denoted as PAMS24k (Mw ) 24 000 g/mol, Tg ) 443 K) and PAMS450k (Mw ) 450 000 g/mol, Tg ) 447 K). Tg values of these samples were measured by differential scanning calorimetry (DSC). Thin films of different thicknesses were prepared by spin-coating toluene solutions of various concentrations at a speed of 2000 rpm about 40 s onto a Si wafer (100). To ensure that all experiments are done on samples with equivalent and well-defined thermal histories, samples were annealed at about 50 K above the bulk Tg before experiments. Before and after Tg measurement, optical microscopy and atomic force microscopy were used to observe the morphology of sample. The heating stage that was devised in our laboratory was used in conjunction with an ellipsometer. A single wavelength null type ellipsometer (Rudolf Research, AutoEL II) and a variable angle rotating analyzer type ellipsometer (J. A. Woollam Co., VASE) were used to complement each other and the ellipsometric angles (ψ and ∆) were continuously monitored while the sample was being heated or cooled at a constant rate of 2 K/min. Because the sensitivity of this Tg measurement technique decreases with decreasing thickness of the film, we limited the thinnest thickness of the investigated film to 200 Å or so.

Figure 1. Typical ellipsometric cooling scan for the PAMS450k thin film. The thickness of this film is 1177 Å and the vertical arrow indicates Tg for the film. The angle of incidence of radiation is 70°, and its wavelength is 6328 Å. These data points are obtained at every minute during the time that the sample is cooled at the rate of -2 K/min.

Results and Discussion The typical scan of the usual ellipsometric experiment is shown in Figure 1. The ellipsometric angle ψ is essentially linear with thickness in the studied thickness range; thus the point at which the two straight-line segments intersect marks the glass transition temperature of the film. Regardless of initial thickness of the sample film, the temperature range of each segment line for the estimation of glassy state and rubbery state was 330370 K and 460-485 K, respectively. This range was far from the glass transition temperatures of various samples, so the possibility of a wrong estimation of Tg due to the slope in the transition range between glassy and rubbery regions could be avoided. As the thickness of the film is reduced, the range whose data deviate from the linear line of the glassy region increases, as can be seen in Figure 2. In the ellipsometric experiment Tg is obtained by this plot, and thickness or index of refraction versus temperature plot also can be used to detect Tg of a thin film.2 Reproducible Tg values, between cooling and heating cycles, in various thickness samples were obtained by the same method as stated above, and the cooling scan was used to determine Tg. There was no remarkable change in sample morphology before and after the Tg measurement experiment, so it could be confirmed that the dewetting19 did not happen in this experimental condition. (19) Reiter, G. Macromolecules 1994, 27, 3046.

Figure 2. Typical ellipsometric cooling scan for the PAMS450k thin film. The thickness of this film is 208 Å and the vertical arrow indicates Tg for the film. The angle of incidence of radiation is 75° and its wavelength is 6308 Å. These data points are obtained every 15 s during the time that the sample is cooled at the rate of -2 K/min.

Figure 3 shows the measured glass transition temperatures for two different molecular weights of poly(Rmethylstyrene), plotted against each film thickness. When the films are thinner than a few hundred angstroms, substantial reduction in Tg is apparent. On the other hand Tg values of thick films approach Tg values of bulk samples (PAMS450k, 447 K; PAMS23k, 443 K), measured by DSC. For the molecular weights used, Figure 3 does not exhibit any difference in Tg depression pattern between the two molecular weight samples. 1. Data Fitting. As the measured Tg pattern with thickness seemed to be a form of growth with saturation,

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Figure 3. Measured glass transition temperature (Tg) vs film thickness for the PAMS thin film. The solid circles and the open squares indicate Tg for the samples of PAMS450k and PAMS23k, respectively.

the Michaelis-Menten (M-M) function is used to fit the data. The Michaelis-Menten model is widely used in enzyme kinetics and also known as the Mono function in population dynamics. By analogy with this M-M function, eq 1 is obtained.

Tg(t) )

Tg(t) Tg,∞t t or ) ξ+t ξ Tg,∞ - Tg(t)

(1)

In eq 1, t is the thickness of polymer film and Tg(t) is the thickness-dependent glass transition temperature of the polymer film. Tg,∞ defines the saturation level with increasing t, and ξ decides the saturation rate of growth in this function. As t decreases to 0 Å, Tg(t) converges on 0 K, and as t increases, Tg(t) approaches Tg,∞. This limitation for the eq 1 is physically reasonable since Tg cannot be lower than 0 K for the thinnest film, and the thickness dependence of the film will disappear and Tg should be a bulk value for the thick film. In the eq 1 Tg,∞ is the bulk Tg, which could be detected by various normal techniques for Tg measurement. For a certain thickness t (0 Å < t < ∞ Å), Tg(t) is somewhere between 0 K and Tg,∞ K due to thickness dependence of Tg in thin polymer films. When t is ξ, the half-saturation value ()Tg,∞/2) is reached. Recently, Fukao et al. also used a model that is similar to eq 1 for their thickness-dependent Tg data fitting.20 By inverting eq 1, the fitting parameters (Tg,∞ and ξ) can be directly obtained from the y-intercept ()1/Tg,∞) and slope ()ξ/Tg,∞), as can be seen in eq 2.

ξ 1 1 1 ) + Tg(t) Tg,∞ t Tg,∞

(2)

Thickness-dependent Tg data have been fitted with this function, and the fitted results are summarized in Figure 4. It can be found that the measured thickness-dependent Tg is much agreeable with eq 2, and the best fit parameter values are Tg,∞ ) 446.8 ( 0.4 K and ξ ) 6.3 ( 0.6 Å for PAMS450k. (20) Fukao, K.; Miyamoto, Y. Europhys. Lett. 1999, 46, 649.

Figure 4. The modified Michaelis-Menten model linear fitting for the PAMS450k and PAMS23k thin films. The solid line is a fit of the measured Tg by eq 2 in the text. Fitting the modified Michaelis-Menten model to the data yields the y intercept (1/ Tg,∞) and slope (ξ/Tg,∞).

2. Continuous Multilayer Model. To explain the experimental Tg depression phenomena with decreasing film thickness, a continuous multilayer model will be proposed. The background of this model comes from various experimental and theoretical results about polymer surfaces and thin films. The molecular dynamics simulation of a glass polymer has shown that the decaying density region, whose density drops from its bulk value to practically zero, exists at the surface region.21,22 Mayes has suggested that, due to chain end segregation to the surface, the larger free volume compared with that in the bulk sample is located at the surface.23 From the free volume viewpoint this decreased surface density, and hence increased free volume, implies that Tg of the polymer surface is much lower than that of bulk. This could lead to an overall decrease in Tg of the thin film because the relative fraction of the reduced Tg region at the surface to total volume only increases with decreasing film thickness. Additionally, because the surface region has sigmoidal density gradient,21,22 its glass transition temperature could be very low at the top surface and converge to a bulk value with increasing distance from the edge of the film. As was seen in Figure 2, the data of ψ vs temperature contained curvature over a wide temperature range around Tg in the investigated thinnest films. This curvature is thought to be indirect evidence for the existence of many different Tg layers. Kajiyama et al. experimentally observed the depth dependence of the surface glass transition temperature of a poly(styreneblock-methyl methacrylate) diblock copolymer film, and the greater decrease in Tg was found for the depth closer to the outermost surface.17 To construct a model for describing the thermal behavior of the thin film that satisfys the research results obtained for the polymer surface, we assume that Tg at the top surface layer is very low and Tg increases to the bulk value with increasing distance from the edge of the film. (21) Mansfield, K. F.; Theodorou, D. N. Macromolecules 1991, 24, 6283. (22) Theodorou, D. N. Macromolecules 1989, 22, 4578. (23) Mayes, A. M. Macromolecules 1994, 27, 3114.

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Figure 5. Schematic representation of the continuous multilayer model. x1, x2, x3, . . . indicate the depth from the edge of the film and Sg(x1), Sg(x2), Sg(x3), . . . denote the Tg at that layer. The thickness of the film is t, and Tg(t) is the average Tg of the film whose thickness is t.

Figure 5 displays the schematic representation of the continuous multilayer model. This model film is composed of many continuous layers and each layer has a different Tg. In other words, Tg of the polymer at the top surface layer is lower than that of its inner layer and Tg of each layer increases to the bulk Tg with increasing distance from the extreme edge of the film. Every layer in the film is designated as x1, x2, x3, . . ., and the Tg at that layer is assigned as Sg(x1), Sg(x2), Sg(x3), . . .. Generally, the distance from the edge of the film is x and Sg(x) indicates Tg at that layer. That is to say, Tg of each layer is depth-dependent. The thickness of the film is t, and Tg(t) means average Tg of the film, which is composed of many different Tg layers whose Tg values are Sg(x) values. Because Tg of the thin polymer film is measured by ellipsometry in this study, the obtained Tg is not the information from the one of any specific layers. Rather, it is average thermal behavior from the each layer. As the measured thickness of the film by ellipsometer is just the sum of all the layers which consist the polymer film and the obtained Tg of the film is the average thermal behavior from each layer, it is possible to elicit Tg of each layer by using the mean value theorem. Mathematically, because both the functions Sg(x) and Tg(t) can be treated as continuous and differentiable functions, the mean value theorem can be used to draw the function Sg(x), as can be seen in following equations. From the mean value theorem, the Sg(x) and Tg(t) have a relationship like eq 3.

∫0tSg(x) dx ) Tg(t) ∫0t dx

or

∫0tSg(x) dx ) Tg(t) ∫0t dx

(3)

By differentiating eq 3, the function Sg(t) can be expressed as the known function Tg(t).

Sg(t) ) tT ′g(t) + Tg(t)

(4)

T ′g(t) can be obtained from differentiating eq 1, and finally, Sg(t) can be derived by inserting T ′g(t) and Tg(t) in eq 4.

Sg(t) )

Tg,∞‚t(2ξ + t) (ξ + t)2

(5)

In eq 5, the parameters Tg,∞ and ξ were obtained from linear fitting of the experimental Tg vs t data with eq 2. Using eq 5 we can draw the depth-dependent Tg profile, i.e., Sg(t), from the top surface to its inner layers. In this mathematical derivation, Sg(x) is converted to Sg(t), in other words, the variable is changed from x to t. Sg(t) denotes the depth-dependent Tg of the layer whose distance

Figure 6. Measured glass transition temperature (Tg) vs film thickness for the PAMS450k thin film (open circle). The dashed line is the thickness-dependent Tg, i.e., Tg(t), which is drawn by eq 1 at the condition of Tg,∞ ()446.8 K) and ξ ()6.3 Å). The solid line is the depth-dependent Tg, Sg(t), which is obtained by eq 5 at the condition of Tg,∞ ()446.8 K) and ξ ()6.3 Å). Inset: The enlarged picture of the main diagram at the initial range of the graph.

from the edge of the film is t. This Sg(t) will be used as the general form of the depth-dependent Tg profile. By inserting the obtained values Tg,∞ ()446.8 K) and ξ ()6.3 Å) in eq 5, the depth-dependent Tg profile, i.e., Sg(t), for the PAMS450k can be drawn as Figure 6. For comparison, the thickness-dependent Tg profile, i.e., Tg(t), is also displayed in the same figure. Both Tg(t) and Sg(t) profile patterns are the growth with a similar saturation value, but the Sg(t) profile pattern shows the higher growth rate. The Sg(t) changes in a continuous manner, not in a discrete type. On the basis of this model, it can be found that regardless of the film thickness, all samples have this surface Tg profile, i.e., Sg(t), and the Tg reduction phenomena with decreasing film thickness originated from the increased effect of this much reduced Tg region at the top polymer surface. And the increased free volume at the surface layer could be attributed to the origin of this reduced Tg profile at that region, since the decreased density will induce the reduction in Tg. According to the molecular dynamics simulation result, the decaying density profile thickness is roughly 10 Å and the dynamical thickness of the free surface is indeed higher than that of the density profile.21,22 Though the reduction of Tg is dominant in that decaying density region, the obtained Sg(t) shows that the decreased Tg region extends over 100 Å. Thus, there must be some mechanism by which the effect of the reduced density at the surface in increasing the mobility of nearby segments can be transmitted to segments deeper in the film where the density is essentially at the bulk value. The reason for this extended effect on the glass transition temperature by the decaying density region can be attributed to the correlation in the motion between different parts of the chain and correlation in the motion between different chains. If a chain passes through the low-density surface region, its segments lying there will have an enhanced mobility and this enhanced mobility may inevitably be transmitted to the deeper region due to these correlations. The length scale for these dynamic correlations may not be very sensitive to the

Glass Transition Temperature in Polymer Films

molecular weight and it may rather be commensurate with the size of cooperatively rearranging domains, which is associated with the glass transition.24 As the temperature approached from below, the top surface would be already in a rubbery state, due to the lower density, and then the Tg at the deeper boundary layer may decrease through the correlated motions. Accordingly, it could be inferred that the surface which has the decaying density region and the dynamically correlated motions between chains plays a dominant role in the pattern of the Sg(t) profile. 3. Origin of Tg Depression. The experimental results of this study show that Tg is much reduced with decreasing film thickness. To explain this reduction of Tg in thin polymer films, various possible factors that have been known to affect the properties of polymer films will be considered in this part. The chain mobility at the solid surface has been investigated by many researchers, and they concluded that, according to the degree of the interaction between polymer chain and solid surface, the Tg at the interface region can be different from the bulk value.4,7-9,14,15 While an increase of Tg with decreasing film thickness was found for other polymers for which the interaction is known to be strongly attractive,4,9 this study showed only a decrease of Tg with decreasing film thickness. It has been believed that for the combination of the polymer and substrate which have no specific interaction, the influence on Tg by the substrate is trivial and this sample shows only decrease of Tg with decreasing film thickness.3,8 According to the simulation work in the polymer and solid interface, the variation in density is weaker than that in the surface region and the spatial distribution of chain ends also does not depart from uniformity.25 As the Si wafer whose surface is covered with a native-oxide layer was used and this solid surface does not have any specific interaction with the used polymer, the possible influence on Tg of the thin film from the substrate could be excluded. Thus, in this study only the surface effect was considered for describing Tg of the thin film, but for the case where a specific interaction exists, the restricted mobility at the interface region also should be taken into consideration in the model. The measured Tg values of the PAMS23k thin films by ellipsometry were also linearly fitted with eq 2 and the fitting parameters Tg,∞ ()443.2 ( 0.8 K) and ξ ()6.7 ( 0.6 Å) were acquired. By the same method with the case of PAMS450k, the thickness-dependent glass transition temperature, Tg(t), and the depth-dependent glass transition temperature, Sg(t), for the PAMS23k samples were obtained. The modified Michaelis-Menten model linear fitting, the obtained Tg(t), and the Sg(t) for the PAMS23k samples are all depicted in Figure 4 and Figure 7. The thickness dependence on Tg of these different molecular weight samples showed a similar pattern. They only exhibited the well-known molecular weight dependence on Tg26 between PAMS23k and PAMS450k samples and DSC also detected this Tg discrepancy (∼4 K) between the two molecular weight samples. Finally, the main concern is focused on the fitting parameter ξ since it shows similar values (∼7 Å) for PAMS450k and PAMS23k. In other words, the parameter ξ is not dependent on the molecular weight of the sample. As previously mentioned, the parameter ξ defines the saturation rate of growth in our model; namely, it determines the degree of thickness dependence on the (24) Theodorou, D. N., University of Patras, personal communications. (25) Theodorou, D. N. Macromolecules 1989, 22, 4589. (26) Fox, T. G.; Flory P. J. J. Appl. Phys. 1950, 21, 581.

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Figure 7. Measured glass transition temperature (Tg) vs film thickness for the PAMS23k thin film (open circle). The dashed line is the thickness-dependent Tg, i.e., Tg(t), which is drawn by eq 1 at the condition of Tg,∞ ()443.2 K) and ξ ()6.8 Å). The solid line is the depth-dependent Tg, Sg(t), which is obtained by eq 5 at the condition of Tg,∞ ()443.2 K) and ξ ()6.8 Å). Inset: The enlarged picture of the main diagram at the initial range of the graph.

glass transition temperature. From this similar ξ value, it may be learned that the thickness dependence on Tg of the thin polymer film is not the function of its molecular weight, and therefore, the parameter ξ is not expected to be dependent on chain dimension. Because at the depth of ξ, Sg(ξ) is invariably equal to three-quarters of the Tg,∞ value, the parameter ξ determines the degree of depth dependence of Tg. To know the molecular weight dependence on the Tg reduction phenomena in thin film, Sg(t) is compared between PAMS23k and PAMS450k at the radius of gyration of each sample. The radius of gyration of each polymer is obtained in an assumption that the characteristic ratio of this polymer is similar to that of PS (∼10). The radius of gyrations, 〈Rg〉, of PAMS450k and PAMS23k are 174 and 39 Å, respectively. The difference, S g,∞ Sg(〈Rg〉), can be used as a barometer for the size dependence on Sg(t). If this difference is similar between molecular weights, Tg reduction phenomena in thin polymer films could be associated with the overall chain dimension of polymer. Sg,∞ is a asymptotic value of the function Sg(t) with increasing t and can be regarded as a same value as Tg,∞, since Sg(t) approaches Tg,∞ with increasing t. So the difference Sg,∞ - Sg(〈Rg〉) can be rewritten as Tg,∞ - Sg(〈Rg〉). For the PAMS450k, Tg,∞ (446.8 K) - Sg (174 Å; 〈Rg〉 of PAMS450k) is 0.6 K and for the PAMS23k, Tg,∞ (443.2 K) - Sg (39 Å; 〈Rg〉 of PAMS23k) is 9.6 K. While these two values show a large difference, for the PAMS23k, Tg,∞ (443.2 K) - Sg (174 Å; 〈Rg〉 of PAMS450k) is 0.7 K, which is almost the same value as the obtained difference for PAMS450k. So it can be inferred that the possible influence from the whole chain confinement in dimension is not large enough to affect the depth-dependent Tg in the polymer film. Rather, the surface that has the decaying density region and the dynamically correlated motion between chains play a dominant role in a reduction of Tg in thin polymer films, and regardless of molecular weights of polymers, there exist the same depth-dependent Tg profile layers at the surface region.

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It has been known that according to the calculation with a variety of chain lengths, the density profile at the surface of a polymer glass or melt falls from the bulk value to zero over a region of thickness on the order of 10 Å. It is remarkably insensitive to chain length and is governed by a length scale commensurate with the segment diameter rather than with overall chain size.27 As we used the same samples which have only the difference in their molecular weights, they are believed to have the same surface density profile and this same surface density profile is thought as the reason for the similar fitting parameter ξ for the different molecular weights. Therefore, it can be concluded that the fitting parameter ξ is the material parameter and the value can be fixed for the kinds of polymer molecules, but it is not different with its molecular weights. 4. Estimation of Surface and Interface Effects on the Thickness-Dependent Tg. In section 2, the continuous multilayer model which consists of different Tg layers at the surface region was suggested and the depthdependent glass transition temperature, i.e., Sg(x), was derived to explain thickness dependence of the glass transition temperature in thin polymer films. According to the Sg(x) profile, Tg at the top surface layer was very low and it gradually increases to the bulk value with increasing distance from the edge of the film, as can be estimated from eq 6.

Sg(x) ) Tg,∞

x(2ξ + x) (ξ + x)

2

(6)

In eq 6, the variable x is the distance from the extreme edge of the film surface and the magnitude of the constant Tg,∞ is the bulk Tg. The fitting parameter ξ can be replaced by the statistical segment length (l) of the polymer.28 For a given polymer, this statistical length (l) can be calculated from an assumption that two rotatable parts in the chain are a unit segment. It has also been found that there exists a universality in the Tg behavior of polymers and the depthdependent Tg profile at the surface region regardless of the kind of polymer.28 Accordingly, the Sg(x) profile, which naturally exists at the surface region, can be obtained without any experiment. The validity of our model and equation can be tested by the estimation of the surface Tg of PS. Jean et al. reported a surface Tg of 317K for the 50 Å region nearest the free surface of a supported PS film.18 On the basis of our continuous multilayer model and equations, the surface Tg for the 50 Å region nearest the free surface of PS can be estimated. The value of the estimated surface Tg for the 50 Å region of PS is 329 K, which is in excellent agreement with the experimental data by Jean et al. Therefore we can confirm that the model and the Sg(t) provide a reasonable value for the estimation of Tg in thin films and at the surface region. For the case where a specific interaction exists, the restricted mobility at the interface region should also be taken into consideration in describing the Tg behavior of thin polymer films. When there is a specific interaction between the polymer and the substrate, Tg at the interface region may be much higher than the bulk value. Therefore, there can be another Tg profile at the interface region and the form of this interface Tg profile function would be similar to that of the surface. The depth-dependent glass transition temperature at the interface region, i.e., Ig(x), may be expressed as (27) Sanchez, I. C., Ed. Physics of Polymer Surfaces and Interfaces; Butterworth-Heinemann: Boston, MA, 1992; Chapter 7. (28) Kim, J. H.; Jang, J.; Zin, W.-C. Langmuir 2000, 16, 4064.

Ig(x) ) Tg,∞

(t - x)(2ξ + t - x) + 2kξ (ξ + t - x)2

(7)

This Ig(x) is a symmetrical function with respect to the thickness of the film (t). The adjustable parameter k determines the degree of the interaction between the polymer and the surface of the substrate. For the case in which the strong specific interaction exists, Tg of the polymer next to the substrate would be the most highly influenced value in the Ig(x) profile and such an effect from the substrate will naturally decrease with increasing distance from the substrate. The measured Tg of polymer films coated on substrates should be determined depending on the total effects from the surface Tg profile and the interface Tg profile. To obtain a novel equation including effects from the surface and the interface, the previously proposed continuous multilayer model was generalized. This model film is composed of many continuous layers, and each layer has a different Tg. The location of a layer, i.e., x, is expressed by the distance from the edge of the film surface, and Lg(x) indicates Tg at that layer. That is to say, Tg of each layer is depth-dependent and this depth-dependent glass transition temperature of the film is Lg(x). This Lg(x) profile can be derived from Sg(x) and Ig(x), as can be seen in eq 8.

Lg(x) ) Sg(x) + Ig(x) - {Sg(0) + Ig(0)}

(8)

The initial function value (at x ) 0) in eq 8 is assumed to be 0 K. Mathematically, because both the functions Lg(x) and Tg(t) can be treated as continuous and differentiable functions, the mean value theorem can be used to elicit the function Tg(t), as can be seen in the following relationship.

∫0tLg(x) dx ) Tg(t) ∫0t dx

(9)

The thickness of the film is t, and Tg(t) means the average Tg of the film composed of many different Tg layers whose Tg values are Lg(x) values. To extract Tg(t), this method assumes a linear averaging of Lg(x). With Lg(x) inserted in eq 9, the thickness-dependent Tg function including the effects from the surface and the interface can be finally derived in eq 10.

Tg(t) ) Tg,∞

t(2k + t) (ξ + t)2

(10)

As mentioned previously, the magnitudes of Tg,∞ and ξ can be known from the bulk Tg and the statistical segment length of a given polymer; thus, there is only one adjustable parameter (k) in eq 10. This parameter k can be obtained from fitting this equation with the measured Tg versus the thickness (t) data. 5. Tg of the Thin Film Coated on the Strongly Favorable Substrate. Equation 10 was applied to fit the measured thickness-dependent Tg data of poly(2vinylpyridine) (P(2)VP). These experimental data were obtained by van Zanten et al. with X-ray reflectivity and representative results concerning the polymer coated on a strongly favorable substrate.9 The value ξ was replaced by the statistical segment length (l). The simple relationship 〈Ro2〉 ) nl2 was used to obtain this segment length

Glass Transition Temperature in Polymer Films

Figure 8. Comparison between the estimated Tg and the measured Tg of poly(2-vinylpyridine) and PMMA thin films. The solid line is the thickness-dependent Tg of P(2)VP, which is drawn by eq 10 at the condition of Tg,∞ ()371 K), ξ ()6.8 Å), and k ()15 Å). The dot line is the thickness-dependent Tg of PMMA which is drawn by eq 10 at the condition of Tg,∞ ()390 K), ξ ()6.6 Å),29 and k ()8 Å).

and 〈Ro〉 of P(2)VP was obtained from the literature.29 The statistical segment length of P(2)VP is 6.8 Å. Figure 8 displays the measured Tg versus the thickness data and the fitted curve drawn by eq 10. Both the experimental data and the curve are in good agreement at the fitting parameter k ) 15. Thickness-dependent Tg data of PMMA coated on a native-oxide surface4 and the fitting curve are also added for comparison. The experimental data of PMMA are also in a good agreement with the eq 9 at the k ) 8. Due to its chemistry, interactions between P(2)VP and the native oxide surface of a Si wafer are more favorable than those between PMMA and the same substrate; thus a stronger interaction should be exhibited.9 As already mentioned, the fitting parameter k determines the degree of the interaction between the polymer and the surface of the substrate. Thus, this fitting parameter k could be used as a measure of interaction between the polymer sample and the substrate pair. The reason for this thickness dependence in Tg of P(2)VP, thus, can be attributed to the combination effect from the surface and the interface. Figure 9 shows the depthdependent Tg profiles (Lg(x)) of P(2)VP for two samples (the thicknesses of the films are 77 and 413 Å). The thickness of these two films was selected since van Zanten et al. already measured the Tg of these thickness films. These curves can be drawn by eq 8 at k ) 15. At surface and interface regions, there exist Tg different regions with the bulk value, respectively, and Tg of the thin film can be determined by both contributions. In other words, the free surface acts to lower Tg, while the substrate raises Tg depending on the interaction between the polymer and the surface of the substrate. In the case of P(2)VP, the effect of the interface is much higher than that of the surface, so the measured Tg shows an increasing tendency with decreasing film thickness.9 Both effects from the surface and interface will increase with decreasing film thickness, since the fraction of the surface and interface (29) Brandrup J., Immergut, E. H., Eds. Polymer Handbook, 3rd ed.; John Wiley: New York, 1989; Chapter 7.

Langmuir, Vol. 17, No. 9, 2001 2709

Figure 9. Depth-dependent Tg, Lg(x), which is obtained by eq 8 at the condition of Tg,∞ ()371 K), ξ ()6.8 Å), and k ()15 Å): (a) for the case where the P(2)VP film thickness is 77 Å; (b) for the case in which the P(2)VP film thickness is 413 Å.

regions only increases with decreasing thickness. This phenomenon can also be known from the difference in Lg profiles between t ) 77 Å and t ) 413 Å in Figure 9. Since the t ) 413 Å film has a larger portion of the region whose characteristics are similar to the bulk, the measured Tg of this film approaches a similar value with that of the bulk. In addition, these depth-dependent Tg profiles estimate extremely high values for Tg at the interface region. DeMaggio et al. suggested that there may be a dead layer (∼50 Å) whose expansion coefficient is almost zero at the interface region.14 Since Tg of the thin polymer film is measured by the volumetric method, the Tg of this dead layer can be thought to be very high. Perhaps it is absolutely impossible to measure Tg of the polymers that are located at the surface of a strongly favorable substrate, since the molecules are highly constrained to the substrate; thus, no volumetric change would be detected at the interface region. This could be attributed to the reason for the unrealistically high value for the Tg at the interface. Additionally the assumption that the initial function value (at x ) 0) in eq 8 is 0 K is another reason for this overestimation. However the effect from the strongly favorable substrate will continuously decrease with increasing distance from the substrate and finally the bulk behavior will be recovered, as can be seen in Figure 9. A drastic change at the surface and the interface is not natural in itself. There should be some range for a change. Therefore, the continuous multilayer model can play a good role as a method to describe the behavior of thin polymer films. 6. Tg of the Freely Standing Thin Film. Until now, there have been rather conflicting results concerning the Tg behavior in thin polymer films. For the kinds of polymer and substrate pairs, there should be different interfacial interactions between the coated polymer and substrate. The effects from the substrates on the Tg of polymers are strongly dependent on the degree of interfacial interaction between them, while the effect from the free surface is

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Figure 10. Depth-dependent Tg of PS, Lg(x), which is obtained by eq 8 at the condition of Tg,∞ ()374 K) and ξ ()7 Å). The solid line is the freely standing film (k ) 0) and the dotted line represents the film coated on a substrate that does not affect the Tg of the film (k ) 3.5).

similar between polymers. The reason for this discrepancy in Tg behavior is mainly thought to come from the difference in interaction between the polymer and the surface of the substrate. Because the geometry of a thin supported film has inherent asymmetry in its structure, it is very difficult to differentiate the surface effect from the interface effect. Forrest et al. investigated the films which have symmetry, i.e., freely standing films, to determine the effects from the free surface and the interface more clearly.10-12 According to their experimental results, much larger reductions in Tg were observed for the freely standing films than for the supported film and this reduction was molecular weight dependent.11 Equations 8 and 10 also can be applied to the freely standing film. The Lg profile for the PS freely standing film whose thickness is 200 Å is constructed in Figure 10. Because the freely standing film can be regarded as the film that has two surfaces, the k value for the freely standing film is 0. For comparison the Lg profile for the PS coated on the substrate that does not affect the Tg of the film (k) ξ/2) is added. Because there is the additional lower Tg region in the freely standing film, the Tg of the entire film will be much lower than that of the film coated on the substrate that does not affect the Tg of the film. The measured thickness-dependent Tg data of the PS freely standing film are compared with the estimated Tg by eq 10 at k ) 0 and ξ ) 7, as can be seen in Figure 11. The REE denotes the average end-to-end distance of the unperturbed polymer molecules. The Tg of these films seems to have a close correlation with the molecular dimension, since the reduction in Tg is accelerated below their REE. In the case of the thin supported film, the

Kim et al.

Figure 11. Comparison between the estimated Tg and the measured Tg of the PS freely standing film. The solid line is the thickness-dependent Tg, i.e., Tg(t), which is drawn by eq 10 at the condition of Tg,∞ ()374 K), ξ ()7 Å), and k ()0 Å). The vertical arrows indicate the REE values for the two higher Mw samples.

measured Tg data did not show any molecular weight dependence.3,11,14 Therefore, it can be inferred that this discrepancy arises from the difference in geometry between the freely standing film and the supported film. There might be another factor in thin freely standing films in addition to the two surface effects. The measured Tg of PS freely standing film with the lower molecular weight (