Effect of free surface roughness on the apparent glass transition temperature in thin polymer films measured by ellipsometry.

Effect of free surface roughness on the apparent glass transition temperature in thin polymer films measured by ellipsometry Mikhail Yu. Efremov Citation: Review of Scientific Instruments 85, 123901 (2014); doi: 10.1063/1.4902565 View online: http://dx.doi.org/10.1063/1.4902565 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of substrate roughness on the apparent surface free energy of sputter deposited superhydrophobic polytetrafluoroethylene thin films Appl. Phys. Lett. 95, 033116 (2009); 10.1063/1.3186079 Vacuum ellipsometry as a method for probing glass transition in thin polymer films Rev. Sci. Instrum. 79, 043903 (2008); 10.1063/1.2901601 A polymer gate dielectric for high-mobility polymer thin-film transistors and solvent effects Appl. Phys. Lett. 85, 3283 (2004); 10.1063/1.1805703 Effect of Polymer‐Substrate Interactions on the Glass Transition of Polymer Thin Films AIP Conf. Proc. 708, 598 (2004); 10.1063/1.1764232 Extraordinary elevation of the glass transition temperature of thin polymer films grafted to silicon oxide substrates J. Chem. Phys. 115, 9982 (2001); 10.1063/1.1415497

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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 123901 (2014)

Effect of free surface roughness on the apparent glass transition temperature in thin polymer films measured by ellipsometry Mikhail Yu. Efremova) Materials Science Center, University of Wisconsin – Madison, Madison, Wisconsin 53706, USA

(Received 21 July 2014; accepted 14 November 2014; published online 5 December 2014) Ellipsometry is one of the standard methods for observation of glass transition in thin polymer films. This work proposes that sensitivity of the method to surface morphology can complicate manifestation of the transition in a few nm thick samples. Two possible mechanisms of free surface roughening in the vicinity of glass transition are discussed: roughening due to lateral heterogeneity and roughening associated with thermal capillary waves. Both mechanisms imply an onset of surface roughness in the glass transition temperature range, which affects the experimental data in a way that shifts apparent glass transition temperature. Effective medium approximation models are used to introduce surface roughness into optical calculations. The results of the optical modeling for a 5 nm thick polystyrene film on silicon are presented. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902565] I. INTRODUCTION

Despite of a growing number of thin film applications, there are numerous challenges in the characterization of various properties of sub-100 nm layers of materials. Determination of glass transition temperature (Tg ) in thin glassy polymer films is a well-known example of unresolved problems in the field: the results obtained by different researchers and experimental techniques vary significantly.1, 2, 69 The question is still open; however, two decades of intense research in this particular area were exceptionally fruitful for the whole nanometrology field. A number of new thin-film characterization methods have been developed: differential AC-chip calorimetry,3 dielectric spectroscopy with air gap geometry,4 dye reorientation in thin films,5 nanobubble inflation,6 nanospheres embedding,7 ATRFTIR-nanoDSC,8 and others. More dependable experimental practices became widely accepted, including controlled atmosphere and vacuum,8, 9 using derivative of extensive property over temperature to localize the transition,10,1,11,12 and employing the limiting fictive temperature concept for Tg assignment.13,14,12,15 Experimentalists became aware of a variety of factors that were crucial for probing nano-scale objects while being negligible for bulk materials: oxidative degradation,16, 17 thin-film preparation artifacts,1,18,17,19 moisture adsorption,17,12,20 dewetting,21–23 variety of interfacial effects,24 film-substrate bonding,25, 26 solvent purity for spin-coating solutions,22 etc. Here a new factor is added to this list – free surface roughness. There is a vast amount of literature discussing experimental aspects of deviation of surfaces from the flat geometry. This work demonstrates how the surface roughness and its evolution with temperature can affect manifestation of glass transition in thin films observed by ellipsometry. The choice of the method is important: ellipsometry is de facto the standard technique in the field. It was used for the discovery of Tg depression in thin films.27

It is well-known fact that an outcome of an ellipsometric measurement, angles and , depends on the film geometry and on the flatness of the film’s surfaces, in particular.28 Surprisingly, the factor of roughness (more generally, the morphology of the free surface) has been ignored in the publications regarding glass transition observations in thin films by ellipsometry. There are several possible reasons for this simplification. First, due to the complexity of the ellipsometric theory, it is not always clear how the experimental data – ellipsometric angles – reflect deviation of free surface from the flat geometry. Then, it would be natural to assume that surface roughness would have no effect in the first approximation. Second, polymer coatings used for ellipsometry measurements are often exceptionally smooth and uniform. Polystyrene (PS) films fabricated by spin-coating on Si substrates with subsequent annealing above Tg typically demonstrate 0.2–0.3 nm root-mean-square (rms) roughness by AFM measurements.22, 29 Intuitively, such a small factor could be neglected. Third, introduction of surface morphology indeed changes some film characteristics (average density, reflectivity, etc.). However, if the morphology is either temperatureindependent or this dependence is smooth enough, it should not affect the location of features on the temperature dependencies of film’s characteristics. To consider the effect of roughness on Tg , a mechanism for additional surface roughening in the vicinity of the glass transition should be introduced. The following discussion, using a 5 nm thick PS film on a Si substrate as an example, illustrates: (a) the effect of roughness on ellipsometric data, (b) the magnitude of the roughness effect in comparison with the glass transition, and (c) possible mechanisms of roughness growth acceleration in a vicinity of Tg , accompanied with optical models and calculations of apparent Tg bias based on these mechanisms. II. RESULTS AND DISCUSSION A. Effect of roughness on the apparent thickness

a) E-mail: [email protected]

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On a qualitative level, ellipsometry as well as many other thin film optical techniques tends to measure optical 85, 123901-1

© 2014 AIP Publishing LLC


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as

FIG. 1. Examples of the films for the optical thickness calculation. (a) A model film with two flat surfaces. (b) A model film with one flat and one rough surfaces.

thickness nf d where d is thickness and nf is index of refraction of the film, rather than d and nf separately.30,31,10 A simple numerical example shown in Fig. 1 illustrates the effect of surface roughening that should be expected in a general case. A model film with sharp surfaces shown in Fig. 1(a) is characterized by d = 50 nm and nf = 1.6 and has optical thickness nf d = 80 nm. The same film with one surface roughened by imprinting a groove-and-ridge pattern with f = 0.5 duty cycle and dr = 20 nm depth is sketched in Fig. 1(b). The patterned part of the film has effective refraction index nr ≈ fnf + (1 − f)na = (0.5 × 1.6) + (0.5 × 1) = 1.3, where na = 1 is refractive index of ambient vacuum or air. Obviously, the average thickness of the film is the same 50 nm. However, the optical thickness is (d − fdr )nf + dr nr = (40 nm) × 1.6 + (20 nm) × 1.3 = 90 nm. Significant increase of the optical thickness in the second case reflects the fact that the light needs more time (proportional to the optical thickness) to pass through the film expanded by surface roughening. A simple but more quantitative estimation for the roughness effect in thin coatings can be made using the three-phase model.32 The configuration consists of a substrate with homogeneous, isotropic and mathematically sharp surface, uniform and isotropic coating, and ambient air/vacuum – see Fig. 2(a). Refractive index for Si substrate ns is a complex number tabulated for a wide range of temperatures and wavelengths λ.33 For visible light ellipsometry, a thin PS film is transparent and its index nf is a real number. For these materials (|ns | |nf |) and very thin coatings (d λ) where λ is the wavelength of the probing light, the changes of ellipsometric angles and with d can be approximated

FIG. 2. Optical models for thin supported films. (a) Three-phase model for a 5 nm thick film with flat surfaces. (b) The same film as in (a) but roughened. (c) A typical model for the films discussed in this work.

sin φ tan φ 1 δ = 4π d 1− 2 , λ nf

(1a)

δ = 0,

(1b)

where ϕ is the angle of incidence.32 Here, the product do = d(1 − nf −2 ) is determined instead of thickness d. For roughened (imprinted) film shown in Fig. 2(b), nr 2 = fnf 2 + (1 − f)na 2 . Taking, for instance, d = 5 nm and nf = 1.6, for the flat film do = (5 nm) × (1 − 1.6−2 ) = 3.05 nm, while for the roughened film do = (10 nm) × (1 − (0.5 × 1.62 + 0.5)−1 ) = 4.38 nm, which is more by 44%. It illustrates the rule that roughening of a thin coating indeed looks like increasing of the film thickness, in agreement with the qualitative argument above. It is worth to note that only one independent film characteristics (do in this case) can be determined, since the only one experimental parameter (angle ) is informative. According to Eq. (1a), variation of both ϕ and λ gives practically no additional information about the film (see also Ref. 30). In other words, if ellipsometry is used as a sole experimental method, any change in thin film surface morphology is hardly distinguishable from the variation of film thickness. For practical purposes, more realistic filmstacks than the three phase model are used. The optical model used for the rest of the discussion is shown in Fig. 2(c). Due to complexity of optical calculations, commercial Film Wizard Software ver. 8.0.3 (SCI) is used to convert parameters of optical layers to ellipsometric angles and backwards. Importantly, the expression for the effective refractive index nr of the roughness layer depends on surface morphology. Several optical representations of the roughness (effective medium approximations, or EMA) are used in practice.28 A practical example of optimization of the optical model by fitting raw spectroscopic elipsometry data for a few nm thick PS coating on Si is given in the supplementary material.75 This example demonstrates insensitivity of the optimization procedure to surface roughness. B. Surface roughening and glass transition: Comparison of the magnitudes

For our discussion, it is interesting to compare magnitudes of the features associated with glass transition and surface roughening in an ellipsometric experiment. A typical glass transition feature is shown in Fig. 3(a). The sample is an annealed 5.4 nm thick PS film on a Si wafer with native oxide surface. Linear heating run is performed in vacuum. The data are averaged from 20 runs. The detailed description of the experiments can be found in Ref. 22. The feature reflects increase in the thermal expansion coefficient (TCE) upon the glass-to-liquid transformation. The change in due to the glass transition over the apparent transition range (from 70 to 100 ◦ C) is about 0.08◦ . For the flat film model, it is equivalent to increase of the film thickness by 0.05 nm. Addition of 0.13 nm thick roughness layer would cause the same optical effect. (The roughness effect is computed using Bruggeman EMA28 ). As it is mentioned in the Introduction,


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1. Roughness due to lateral heterogeneity

FIG. 3. Magnitudes of thermal processes observed by ellipsometry. (a) Glass transition in 5.4 nm thick PS film on Si (circles), in comparison with (b) dewetting of the 5.2 nm thick PS film from the HMDS-treated Si (diamonds).

typical AFM roughness of PS films is 0.2–0.3 nm rms value.22, 29 This roughness is measured for coatings annealed above Tg . Note that the thickness of the rough layer, as it is probed by ellipsometry, can significantly exceed the rms roughness by AFM.28, 29 It means that the magnitude of the glass transition effect in 5 nm thick PS film is equivalent to ∼50% (or less) increase of existing roughness. This example shows that any significant change in surface roughness during the glass transition in a few nm thick PS film would noticeably alter the manifestation of the transition. As another comparison, film roughening caused by dewetting is illustrated in Fig. 3(b). The sample is as-spun 5.2 nm thick PS film on a hexamethyldisilazane (HMDS) treated Si wafer. Linear heating run in vacuum is shown. Experimental details can be found in Ref. 23. Dewetting occurs in the 120–180 ◦ C temperature interval. Corresponding change of is 5.6◦ . The effect of dewetting on is almost two orders of magnitude stronger than the glass transition effect. As a conclusion, a change in film roughness can cause much more pronounced effects in ellipsometric measurements than the glass transition, so surface roughening is able to either alter or mask the manifestation of glass transition if it occurs in the same temperature interval.

To our knowledge, surface roughening due to lateral inhomogeneity has not been introduced before. The effect is closely related to the broadening of the molecular dynamics under confinement conditions, and in particular, glass transition broadening in thin films. Broadening of dynamics in confinement is a widely accepted concept.34 It is found in many low-dimensional systems and demonstrated by numerous experimental techniques, including ellipsometry,10 calorimetry,35,15 X-ray reflectivity,36 and dynamic mechanical analysis.37 The phenomenon manifests itself in experiments as either increase of the glass transition temperature range or widening the relaxation time spectrum for the transition related modes. In general, broadening of glass transition in thin coatings is explained as an interfacial effect.38,10,34,35,36,37 The layer model assumes that parts of the film in contact to the substrate and at free surface have molecular mobilities different from the bulk values. The layer adjacent to the substrate supposedly has lower than bulk mobility due to the strong interaction with the substrate. It is associated with longer relaxation times and increased local Tg . Free surface has an opposite effect: increased mobility, shorter relaxation times, and decreased local Tg . In other words, the layer model implies that broadening of molecular dynamics is caused by spatial heterogeneity of thin films in the direction normal to the surface or interface.38 Interestingly, that spatial heterogeneity in lateral directions (illustrated in Fig. 4) as another origin of the glass transition broadening in thin films has been neglected in the literature. However, the significance of the lateral heterogeneity factor is worth to be considered. First, it should be noted, that the concept of dynamic spatial heterogeneity in the glassy state, regardless of the direction, is not new and have many arguments of both theoretical and experimental character. Extensive reviews on this topic can be found in Refs. 39–41. Second, although the lack of permanent structural inhomogeneities in supercooled liquids is generally accepted,41 the existence of relatively stable lateral domains of different structures in thin-film glasses can be expected by a number of reasons. There are evidences that lateral dynamic inhomogeneities in confinement can be much larger and more stable than in the bulk material.42 Note that dynamic

C. Mechanisms of roughness growth acceleration in the glass transition region

To complete the argumentation for the roughnessinduced apparent Tg shift, a plausible mechanism of roughening acceleration in the glass transition temperature range is required. Two such mechanisms are discussed here: roughening due to lateral inhomogeneity and roughening associated with thermally activated capillary waves. The former one is proposed for the first time and allows constructing welldefined optical models. The latter phenomenon is well-known and supported by experimental observations but requires more parameters to create quantitative optical models.

FIG. 4. A sketch of the lateral distribution of the domains with different dynamics in a glassy thin film.


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domains with relaxation time larger than characteristic time of experimental observation are equivalent to stable structural heterogeneities.40 Conceivably, the tendency of amorphous polymers to form structure with a short range order43–45 can be amplified in thin films by a strong orienting influence of the flat interfaces.46–48 It can be hypothesized that such ordering can lead to formation of a spherulite-like mosaic. Classic spherulitic morphologies contain well-ordered lamellae interdigitated with amorphous zones.49 In analog, the formation of more ordered lateral domains in glassy thin films may be accompanied with concentration of defects (i.e., polymer segments oriented in less-preferable directions) in the interlaced regions. Obviously, the degree of ordering even for the most ordered domains should be far from the crystalline phase. It should be mentioned that both the characteristic length of the short range order and the size of heterogeneities are expected to be less than 10 nm.43,44,45,39 If the lateral inhomogeneity is a substantial factor for glass transition broadening in thin films, then the ellipsometry-specific shift in glass transition is an immediate consequence of this supposition. The mechanism of the transition broadening due to the lateral inhomogeneity is illustrated in Fig. 5. At the starting point below the glass transition, the depicted film has flat free and film/substrate interfaces and all domains are glassy. As the temperature of the system increases being still below the transition range, the domains expand equally in the normal direction. At the onset of the glass transition, an increasing part

FIG. 5. Evolution of film roughness upon heating, assuming the lateral heterogeneity mechanism.

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of the domains becomes liquid characterized by an increased thermal expansion coefficient (TCE) α L . Liquid domains expand faster than their still-glassy neighbors, thus creating surface roughness. The onset of the transition corresponds to the onset of roughness. The onset of roughness exaggerates the onset of glass transition in the ellipsometry data, since both phenomena appear as an accelerated thickness growth. As a result, Tg shifts towards the onset of the transition at a lower temperature. It is worth noting that this mechanism can shift apparent Tg within the transition temperature interval only. To estimate the magnitude of Tg depression due to the heterogeneity-induced roughness, the optical modeling of 5 nm thick Si-supported PS film is performed in the 0–200 ◦ C temperature interval. The filmstack shown in Fig. 2(c) is used, λ = 500 nm, ϕ = 70◦ . Temperature dependence of ns is taken from Ref. 33. Parameters of the native oxide layer are considered as being temperature-independent: thickness dox = 1.6 nm and refractive index nox = 1.462 (software library value). Experimental PS refractive index nf and its temperature coefficients both for glass and liquid states are taken from Ref. 12. Thickness of the rough layer dr is equal to the difference between the heights of the tallest and the shortest domains. A combination of serial and parallel EMA models is used to compute nr .23 It should be noted that EMA models are applicable if the size of the domains are less than a tenth of the probing light wavelength.23 This condition is met since the size of domains is expected to be in the sub 10 nm range – see above. Temperature dependence of df and nf for each domain in the vicinity of Tg follows the experimental curve for thick (117 nm) PS film obtained by ellipsometry upon cooling.12 The distribution of local Tg among the domains is the free parameter of the model. For the further discussion, the separate symbol for the local glass transition temperature, τ g , is used to avoid confusion with the “global” Tg of the film as a whole. The main comparisons are made for box-distributed τ g centered at 100 ◦ C and singular τ g = 100 ◦ C (shown in Fig. 6(b)). Also, the standard layer model is implemented, where the film consists of layers with different τ g instead of lateral mosaic of domains. The mean temperature for all distributions, 100 ◦ C, is chosen to be close to Tg for medium and high molecular weight PS measured by calorimetry.50,12 The effect of Tg depression is calculated as a difference between Tg computed for the lateral inhomogeneity model and Tg obtained for the layer model with the same parameters. The results of modeling are presented in the integral form (T). Limiting fictive temperature (Tf ’) concept is used to assign Tg .51,50,12 The modeling results for the films with (1) singular τ g , (2) ±30 K box-distributed τ g and layered structure, and (3) ±30 K box-distributed τ g and lateral heterogeneity, are shown in Fig. 6. Chosen ±30 K width of the τ g distribution, although being arbitrary, reasonably reflects sparse literature data on the breadth of glass transition in thin films. Examples of glass transition broadness reported: 340–410 K for 4.5 and 5.7 nm PS films on Si, X-ray reflectivity,36 340– 410 K for 37 nm PS film on polyimide, dynamic mechanical analysis,37 100–150 ◦ C for 7 nm PS film on silicon nitride, nanocalorimetry,15 60–120 ◦ C for 16 nm PS film on


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distribution52 is shifted to 100 ◦ C for this model. The result can be described by the equation, 1 P = 2 50K 2 −1 TD TD dTD 1+ T 1+ T BR

−50K

BR

for −50 K ≤ TD ≤ 50 K, and P = 0 otherwise,

FIG. 6. (a) Calculated (T) curves for the roughness caused by lateral heterogeneity (shaded symbols). Data for comparable layer models are shown by open symbols. (b) Distributions of τ g used for modeling. 1 – Box distribution Tg ± 30 K, 2 – truncated bell-shape distribution (see text), and 3 – single τ g . (c) The sketch of the layer model.

Si, ellipsometry.10 More detailed data on τ g distribution presented by Fukao and Miyamoto52 are used in a special model below. The model with single τ g is used as a control. The backcomputed Tg for this model is 100 ◦ C. The layered film with ±30 K box τ g distribution demonstrates back-computed Tg = 99 ◦ C and significantly broadened transition. In contrast, the laterally heterogeneous film shows depressed Tg = 89 ◦ C. Importantly, this shift in Tg is not due to the altering the curve shape in the vicinity of Tg because the assignment of Tg as Tf ’ is based on the glass and liquid line segments outside of the transition temperature range. Instead, it is caused by the shift of the liquid line segment to lower . As it was stated above, both increase of roughness and expansion of the film thickness lead to lower . In this particular case, the roughness induced by the lateral heterogeneity is equal to 0.11 nm peak-to-peak value. While the arbitrary box distribution of τ g is used to demonstrate the concept, modeling of real systems requires experimental τ g distributions for sub-10 nm thick films. The information in this field is very limited and often indirect. Here the normalized temperature dependence of dielectric loss for 9 nm thick PS film52 is used as a representation of τ g distribution. This bell-shape distribution is generated by formula given in Ref. 52, then truncated to Tg ± 50 K interval and finally normalized. Note that the center of the original

(2)

where P is the probability density for a domain to have τ g at the temperature (Tg + TD ), and TBR = 22 K is the broadening parameter.52 The distribution and the results of modeling are shown in Fig. 6. Back-computed Tg = 87 ◦ C. The Tg depressions for both bell-shape and ±30 K box distributions are 11.4 and 10.5 K, respectively. Summarizing, the surface roughening due to the lateral heterogeneity can be responsible to significant, on the order of 10 K, shift of apparent Tg to the lower temperatures. Within the proposed optical model, the thickness dependence of the roughness is weak. Apparently, the taller domains depicted in Fig. 5 correspond with the larger height distribution under the same temperature. Calculations show 12.8 K, 11. 4 K, and 10.2 K Tg depressions for 2 nm, 5 nm, and 10 nm thickness, respectively (for the bell-shape distribution). However, the model requires a “monolayer” of the domains. This supposition is meaningful if the film thickness is less or comparable with the size of heterogeneities (> 1/a the factor of gravity can be neglected. Here |q| = 2π /l, √ where l is the wavelength, and a is the capillary length a = γ /gρ, where γ is surface tension, ρ is the mass density of the liquid, and g is the gravitational acceleration. For medium and high molecular weight PS, γ ≈ 0.04 N/m54 and a ≈ 2 mm. If the gravity is neglected, a set of capillary waves with |q| in the range between |qmin | and |qmax | has rms amplitude A determined by l kB T (3) ln max , A (T ) = 2π γ (T ) lmin where kB is the Boltzmann constant, lmax = 2π /qmax , and lmin = 2π /qmin . Amplitude A is the measure of the surface roughness due to the thermal capillary waves observed by a particular technique. Values of lmax and lmin depend on the experimental and some general limitations. For ellipsometry, λ is a natural choice for lmax and the reasonable value for lmin is 0.5 nm.53 Accurate evaluation of lmax and lmin is not necessary due to the weak dependence of A on these values. As the viscosity of the liquid increases while temperature approaches the glass transition, at some point the capillary waves become frozen-in.55 The freezing temperature TF is identified with Tg .55 Then the further decrease of temperature does not affect the roughness associated with the capillary waves. It should be noted that elastic surface waves, namely Rayleigh waves and bulk elastic shear waves, also contribute to the glass surface roughness.55 Elastic modes are temperature dependent and can cause positive slope of A(T) below TF .56 The contribution of the elastic waves to surface height fluctuations is negligible if |q|γ /2G∞ 1, where G∞ is high-frequency shear modulus.55 This expression is equivalent to l π γ /G∞ = l0 . For PS at Tg = 100 ◦ C, extrapolated γ ≈ 0.04 N/m54 and G∞ is about 1 GPa or more.57 Then l0 ≈ 0.1 nm, which is significantly less than l even for the smallest wavelengths approaching lmin . Consequently, the contribution of elastic modes can be neglected so the amplitude A is constant below TF . Theoretical dependence A(T) for PS is shown in Fig. 7. This example assumes lmax = λ = 500 nm; surface tension is experimentally found for monodispersed PS (Mn = 107.2 kg/mol, Mw /Mn = 1.07) to be γ (T) = b − cT, where b = 52.02 mN/m, c = 0.0885 mN/(m × K), and T expressed in ◦ C.54 The average slope computed for the 100–200 ◦ C temperature range is dA/dT|av = 1.0 pm/K. It agrees with the experimental value 1.0 ± 0.2 pm/K found by surface-diffuse X-ray scatter-

FIG. 7. Computed dependence of roughness caused by capillary waves using parameters characteristic for PS. Uncertainty in TF is shown by the set of dashed glass lines.

ing. This value is estimated from the linear fitting of the data in Figure 6 of Ref. 58. The exact behavior of A(T) function in the vicinity of Tg is not clear. Experimental works for various glass-forming liquids report TF > Tg by 10–40 K,59, 60 TF < Tg ,56 or both cases.61 More complicated effect is reported for glycerol: a sharp increase of surface roughness takes place at about 100 K above Tg ; however, TF defined as a temperature of intersection between glass and liquid roughness-temperature dependencies is well below (by ∼100 K) Tg .60 Thin film effects may add another layer of complexity: sensitivity of free surface dynamics to the thickness of coating and to the mechanical properties of the substrate is reported.71,70,62 Due to the uncertainty in the experimental data, general considerations are used to construct a reasonable optical model to demonstrate the effect of roughness due to capillary waves in ellipsometric measurements. Optical properties of the roughness layer are calculated using Bruggeman EMA model with dr = cA. Coefficient c = 2 is introduced here because A reflects deviation from the mean position, while dr corresponds to the peak-to-peak distance. Here the layer model is used, where τ g varies for the layers within the film. The model employs the same ±30 K box distribution of τ g as was used above for the models based on lateral heterogeneity. The lowest local τ g , τ g - = Tg – 30 K is associated with free surface layer in accord with the widely accepted opinion.40, 71 Then the natural supposition TF = τ g - finalizes the construction of the model. Fig. 8 demonstrates the integral ellipsometric curve (T) for the model. The curve for the film where capillary waves remain frozen through the whole temperature interval is shown for comparison. Tg depression due to the roughness is 7 K. Note that the choice of the coefficient c = 2 is rather conservative. Tg depression increases up to 14 K for c = 6 suggested by Persson.63 Since the capillary waves are essentially the surface effect, it should be expected that the corresponding Tg


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FIG. 8. Effect of roughness due to the thermal capillary waves on (T) curve. (Shaded symbols) Capillary waves un-freeze at TF = 70 ◦ C. For comparison, (T) for the film where the waves remain frozen is shown by open symbols. Modeled sample is 5 nm thick PS film on Si.

depression decreases (roughly proportionally) with increase of film thickness. The model shown in Fig. 8 yields 7 K, 4 K, 2 K, and 0.7 K Tg depression for 5 nm, 10 nm, 20 nm, and 40 nm thicknesses, respectively. It should be emphasized that this result demonstrates the order of the effect only. Besides abovementioned ambiguity of the temperature interval where the roughness onset takes place, there are much uncertainty in the applicability of the Eq. (3) to the real systems of thin supported films. For instance, significant reduction of surface tension at large q, anticipated in Refs. 64 and 65, would lead to increase the surface roughening. In opposite, the effect of damping of thermal capillary waves in the vicinity of a solid substrate due to the strong repulsive forces decreases the roughness.72 However, this suppression becomes large for films typically much thinner than 5 nm (adsorption layers, a few monolayer thick films, etc.) and for materials with low surface energy.73, 74 It can be speculated that the effect of capillary waves interferes with the effect of lateral heterogeneity on the apparent Tg . The domain boundaries can influence wave’s propagation. The lateral heterogeneity model does not assume the enhanced surface mobility, while the part of the apparent Tg shift caused by the capillary waves is due to the decreased local tg at the film surface. Thus, if both mechanisms do exist, their superposition may have smaller effect on the apparent Tg than the sum of Tg shifts calculated independently. D. Additional comments

It is worth to note that ellipsometry is not the only thinfilm technique sensitive to the surface roughness. A good example is grazing angle X-ray diffraction (XRD) measurements that have been used extensively to determine Tg in thin coatings.2 Modeling of roughness for XRD experiments is still under development.66 In light of remarkably close results obtained by both ellipsometry22 and XRD9 for glass transi-

Rev. Sci. Instrum. 85, 123901 (2014)

tion in sub-10 nm thick PS films, the analysis of possible shift in XRD-determined Tg due to the surface roughness can be a valuable addition to the discussion presented here. Taking the free surface roughness into account does not exhaust the list of instrumental factors affecting Tg assignment by ellipsometry. Comparison between the effects predicted by this work (apparent Tg shift is about 10 K or less) and experimental Tg depression by ellipsometry (tens of kelvins for comparable thickness)1, 10 suggests existence of other significant factor(s) affecting the manifestation of glass transition in thin films. Experimental separation of these factors can be extremely challenging especially in the situation when the nature of other factors is either unknown or debatable. Another factor worth of further investigation is anisotropy of the film. Uniaxial symmetry should be expected in general for thin coatings even in amorphous state.67 Birefringence is the manifestation of anisotropy in the relevant case of the refractive index. It is reasonable to expect that temperature dependence of the birefringence has a feature at Tg . Increased molecular mobility due to glass transition should change (converge) the ordinary and extraordinary indices, thus affecting the optical thickness of the sample. Rotational randomization of the molecules, preferentially oriented in the glassy coating, can cause significant expansion of the film.68 Using methodology of this work, it can be expected that these processes shift apparent Tg as well. The detailed discussion on this subject will be given in a future publication.

III. CONCLUSION

Thermal changes of free surface roughness in an ultrathin polymer film undergoing glass transition can alter the determination of the transition temperature Tg by ellipsometry. A combination of several factors makes this effect to occur. Ellipsometric angles are sensitive to the amplitude and other characteristics of roughness. Increase of surface roughness and thermal expansion are hard to distinguish for a nanometer thick coating. Ellipsometric effect of the roughening can be significantly more pronounced than the manifestation of glass transition. Then, any acceleration of the roughness growth in the transition region leads to the shift of apparent Tg , even if Tg is assigned as a limiting fictive temperature. Two mechanisms of this accelerated growth are discussed. The first one relies on the lateral heterogeneity in thin amorphous films. The other mechanism is associated with freezing and unfreezing of thermal capillary waves during the transition. Both mechanisms are capable of causing Tg depression on the order of ten K for the 5 nm thick polystyrene coating on a silicon surface used as an example.

ACKNOWLEDGMENTS

The author would like to thank Paul Nealey and Mark Ediger for many useful discussions. This research is supported by the National Science Foundation through the University of Wisconsin Nanoscale Science and Engineering


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Mikhail Yu. Efremov

Center (DMR-0832760) and Materials Research Science and Engineering Center (DMR-1121288). 1 M.

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Molecular Weight Effects on the Glass Transition and Confinement Behavior of Polymer Thin Films.

The glass transition of polymers with different side-chain stiffness confined in free-standing thin films.

Influence of the glass transition on rotational dynamics of dyes in thin polymer films: single-molecule and ensemble experiments.

Densification and depression in glass transition temperature in polystyrene thin films.

The peculiar behavior of the glass transition temperature of amorphous drug-polymer films coated on inert sugar spheres.

Measurement of surface roughness of thin films by a hybrid interference microscope with different phase algorithms.

Breakdown of time-temperature superposition in a bead-spring polymer melt near the glass transition temperature.

Systematic Surface Phase Transition of Ag Thin Films by Iodine Functionalization at Room Temperature: Evolution of Optoelectronic and Texture Properties.

Influence of casting solvent on phenyl ordering at the surface of spin cast polymer thin films.

Thickness dispersion of surface plasmon of Ag nano-thin films: determination by ellipsometry iterated with transmittance method.

Polymer brushes: a controllable system with adjustable glass transition temperature of fragile glass formers.

Stable Freestanding Thin Films of Copolymer Melts Far from the Glass Transition.

Domain wall roughness in stripe phase BiFeO3 thin films.

Fabrication of Single Crystal Gallium Phosphide Thin Films on Glass.

Impact of surface roughness on liquid-liquid transition.

Measuring electro-mechanical properties of thin films on polymer substrates.

Sub-nanometer expansions of redox responsive polymer films monitored by imaging ellipsometry.

Effects of annealing temperature on optical, morphological, and electrical characteristics of polyfluorene-derivative thin films on ITO glass substrate.

Electron anions and the glass transition temperature.

Effect of surface treatment on enamel surface roughness.

Surface tension and disjoining pressure of free-standing smectic films above the bulk smectic-A-isotropic transition temperature.

Convective polymer assembly for the deposition of nanostructures and polymer thin films on immobilized particles.

Monitoring plasma treatment of thin films by surface plasmon resonance.

The elastic mechanical response of supported thin polymer films.