Recent studies have measured or predicted thickness-dependent shifts in density or specific volume of polymer films as a possible means of understanding changes in the glass transition temperature *T*_{g}(*h*) with decreasing film thickness with some experimental works claiming unrealistically large (25%-30%) increases in film density with decreasing thickness. Here we use ellipsometry to measure the temperature-dependent index of refraction of polystyrene (PS) films supported on silicon and investigate the validity of the commonly used Lorentz-Lorenz equation for inferring changes in density or specific volume from very thin films. We find that the density (specific volume) of these supported PS films does not vary by more than ±0.4% of the bulk value for film thicknesses above 30 nm, and that the small variations we do observe are uncorrelated with any free volume explanation for the *T*_{g}(*h*) decrease exhibited by these films. We conclude that the derivation of the Lorentz-Lorenz equation becomes invalid for very thin films as the film thickness approaches ∼20 nm, and that reports of large density changes greater than ±1% of bulk for films thinner than this likely suffer from breakdown in the validity of this equation or in the difficulties associated with accurately measuring the index of refraction of such thin films. For larger film thicknesses, we do observed small variations in the effective specific volume of the films of 0.4 ± 0.2%, outside of our experimental error. These shifts occur simultaneously in both the liquid and glassy regimes uniformly together starting at film thicknesses less than ∼120 nm but appear to be uncorrelated with *T*_{g}(*h*) decreases; possible causes for these variations are discussed.

## INTRODUCTION

The experimentally observed large changes in the glass transition temperature *T*_{g} of ultra-thin supported and free-standing polymer films with decreasing thickness *h* have puzzled the field for more than two decades.^{1–15} Studies of local dynamics near the free surface^{16–18} and local *T*_{g} within the film^{19–21} have led to the understanding that decreases or increases in the average *T*_{g} of the film^{2,6,22} are typically the result of competing dynamical perturbations from the interfaces. The free surface with its reduced number of intermolecular contacts tends to increase the local dynamics and reduce *T*_{g} while the polymer-substrate interface can lead to a decrease in mobility and increase in *T*_{g}, especially if specific interactions such as hydrogen bonding are present.

An open question at present is what other material property changes occur in these thin films, and in particular, which correspond to the large shifts in *T*_{g}(*h*). To date, studies have looked at physical aging,^{23–27} viscosity,^{17,28,29} modulus,^{30–33} and permeability.^{34} Here, we investigate changes in the temperature-dependent specific volume (inverse density) with decreasing film thickness of polystyrene (PS) films supported on silicon using ellipsometry. It has been extensively reported that such supported PS films show a large ∼30 K decrease in average film *T*_{g} as the thickness is decreased below ∼60 nm down towards ∼10 nm.^{1,5,19,35} Our work was motivated by recent theoretical efforts from White and Lipson^{36} that predict a small (∼0.5%) increase in the liquid-line specific volume (decrease in density) with decreasing film thickness that may account for the observed *T*_{g}(*h*) shifts. All previous experimental efforts measuring the density of thin films have only investigated the glassy regime with experimental errors of ±1%.^{37–41} Although the focus of the present study is to comment on these measurements of the average film density and theoretical efforts to explain *T*_{g}(*h*) shifts based on the predicted temperature dependence of the average film specific volume, we note that computer simulations have previously compared depth-dependent density profiles near free surface and substrate interfaces with local mobility and generally find these quantities to be uncorrelated.^{42–47} Less frequently commented on in these studies is whether the density of thin films is the same as bulk. One study by Baschnagel and co-workers^{43} has shown that in computer simulations, even though the local density near interfaces varies considerably, the local density at the interior of the film is the same as bulk provided that the film is sufficiently thick that bulk density can be recovered in the interior.^{42}

Changes in mass density (or specific volume) in thin polymer films have been little studied. There was some early interest in the topic during the 1990s when *T*_{g}(*h*) shifts were first observed. In 1998, Wallace *et al.*^{37} used neutron reflectivity to investigate if the density of ultrathin PS films supported on silicon was reduced, as a possible explanation for the observed *T*_{g} reductions. Such an observation would correlate with the ideas behind the original free volume models of *T*_{g},^{48–51} although they have since been disproven.^{52,53} Wallace *et al.* found no evidence of density changes to within ±1% for PS films supported on silicon measured at room temperature down to film thicknesses of 6.5 nm.^{37} This conclusion was corroborated by Forrest *et al.*^{38} using Brillouin light scattering to measure the acoustic phonon velocities, related to the film’s mass density and mechanical stiffness, finding them to be equivalent to the bulk glassy values within ±1% at room temperature for free-standing PS films down to film thicknesses of 29 nm. Given that these early studies saw no change in mass density to within ±1% with decreasing film thickness and the experimental challenge with measuring such small changes within ultrathin films with small sample sizes, this issue has remained relatively dormant.

However, more recently, a couple of studies have reported increases in density with decreasing film thickness below ∼40 nm by as much as 30% for ∼5 nm thick films as measured by x-ray reflectivity^{39,40} or corresponding increases in index of refraction from ellipsometry.^{39–41} Such large increases in density seem unrealistic given that so-called stable glasses that are slowly formed by physical vapor deposition to optimize molecular packing, obtaining material densities equivalent to glasses that have been aged for thousands or millions of years to reach equilibrium well below the typical glass transition temperature measured, still only show an increase in density of ∼1% relative to ordinarily cooled glasses.^{54–56} Even samples of amber glass that have been aged for millions of years, the density increases are only ∼2%.^{57,58} In particular, Vignaud *et al.*^{39} measured supported PS films (*M*_{w} = 136 kg/mol) reporting an increase in mass density of the film of 26% below 25 nm relative to bulk values that varied within ±1% between 30 and 140 nm thick films as measured by the electron density profile from x-ray reflectivity measurements. They also used spectroscopic ellipsometry (λ = 250-1700 nm) to measure the index of refraction of the PS films at 20 °C using three different angles of incidence (65°, 70°, and 75°) and determined the density of the PS films using the Lorentz-Lorenz equation, where values for the polarizability α were estimated by benchmarking the index values for bulk films to the known bulk density for PS. These ellipsometry data also reported an increase in density of 30% below 40 nm relative to bulk values that varied within ±1% between 40 and 150 nm thick films. In support of these results, they cited work of Ata *et al.*^{40} that reported similar measurements for supported PS films (*M*_{w} = 980 kg/mol) measured using x-ray reflectivity and ellipsometry, also analyzed using the Lorentz-Lorenz equation, as well as index of refraction values reported by Li *et al.*^{41} for supported PS films (152 kg/mol and 590 kg/mol) showing an increase in index with decreasing thickness. To within experimental error, these studies report the same index of refraction despite the varying molecular weights used while simultaneously suggesting that the film’s thickness relative to the polymer’s radius of gyration *R*_{g} and chain distortion (and hence molecular weight) may be important factors. We also note that all these studies report only on the density or index of refraction at room temperature in the non-equilibrium glassy state.

Recently, White and Lipson^{36} have used a thermodynamic lattice model to predict the temperature-dependent specific volume for the liquid state of thin free-standing PS films. This equation-of-state model uses bulk pressure-volume- temperature (PVT) data for PS (literature data for a molecular weight of 110 kg/mol were used^{36}) to determine the relevant model parameters: nearest neighbor non-bonded segment interaction energy, lattice site volume, and number of segments per chain. In addition, data for the temperature dependence of surface tension are used to define an additional surface parameter *f* representing the fraction of missing contacts or interactions at the free surface. With these model parameters so defined, predictions for temperature-dependent material properties in thin films can be made with no further adjustable parameters. The core foundation of this thermodynamic model has been shown to work well describing the miscibility and phase behavior of polymer blends^{59–62} and bulk glass transition values across different polymers.^{63} The 2011 work on free-standing PS films uses this thermodynamic model to predict the temperature-dependent specific volume for the equilibrium liquid-line above *T*_{g}.^{36} The model’s equation-of-state predicts an increase in the specific volume liquid-line with decreasing thickness of about 0.3% for 30 nm and 0.8% for 10 nm thick films of free-standing PS relative to the bulk value. In the model, this film expansion with decreasing thickness arises from a reduction in the attractive energy between polymer segments due to missing contacts at the free surface. The study then goes on to estimate an anticipated *T*_{g}(*h*) for the films by determining the intersection point of this predicted specific volume liquid-line from the model with a glassy-line. As the equilibrium thermodynamic model is unable to make predictions about the non-equilibrium thermodynamic glassy state, the authors simply assumed that the glassy line remains fixed at the bulk value obtained from PVT data. Given the lack of any other information at the time, this was as reasonable an assumption as any. Thus, based on the small predicted increase in specific volume of the liquid-line, the intersection temperature with the assumed bulk glassy-line was identified as *T*_{g}(*h*) and found to decrease by ∼40 K with decreasing film thickness in remarkable agreement^{36} with experimental data for low molecular weight free-standing PS films from the literature,^{64} given that the theory had no adjustable parameters. More recently, they have also expanded this model to incorporate substrate interactions and treat supported polymer films.^{65} To clarify, the thermodynamic model, which has a strong theoretical foundation and proven record in blends and bulk systems, predicts a driving force for film expansion (increase in specific volume) with decreasing film thickness. However, the subsequent prediction of a corresponding decrease in *T*_{g}(*h*) is less theoretically sound because it relies on the untested assumption that the glassy-line specific volume remains the same as bulk.

Here, we test the predictions of this thermodynamic model by using ellipsometry to measure the temperature dependence of the index of refraction and calculate the Lorentz-Lorenz parameter as a proxy for film density in order to compare the relative specific volume between thin and thick films. Because of the experimental challenges with measuring ultrathin free-standing films, we focus here on PS films supported on silicon with a 1.25-nm native silicon oxide layer. It has been previously demonstrated that the PS/silicon-oxide interface is neutral when it comes to *T*_{g} perturbations; the local *T*_{g} of a thin PS layer next to a silicon-oxide interface reports the bulk value.^{19} In addition, the *T*_{g}(*h*) behavior of low molecular weight free-standing PS films has the same *T*_{g}(*h*) functional form as that for PS films supported on silicon but with the magnitude of the *T*_{g} reduction at a given film thickness *h* being twice as large.^{64} This is consistent with the free surface being the source of the enhanced mobility leading to the *T*_{g}(*h*) reduction with free-standing PS films (two free surfaces) having a *T*_{g}(*h*) value in agreement with supported PS films (one free surface) of half the thickness, *T*_{g}(*h*/2).^{66} Thus, supported PS films should exhibit the same phenomenon as free-standing PS films, if only weaker by a factor of two.

From our investigation, we find that both the liquid and glassy lines of specific volume show equivalent shifts with decreasing thickness in supported PS films, negating the assumption made by White and Lipson^{36} to explain the *T*_{g}(*h*) decrease that the glassy-line specific volume remains unshifted and the same as bulk. Yet, for film thicknesses of ∼120 to 65 nm, we do observe an increase in specific volume (decrease in density) of 0.4 ± 0.2% consistent in magnitude with the film expansion, shifted specific volume liquid-line prediction of White and Lipson based on their detailed thermodynamic model.^{36} But then, for film thicknesses below 65 nm, where the *T*_{g}(*h*) decrease is observed, we find that the specific volume decreases returning back to the bulk value at ∼40 nm, suggesting that if the film expansion prediction from the thermodynamic model is correct, some additional factor acts to counteract the effect for very thin films where the *T*_{g}(*h*) decrease is observed. Below ∼30 nm, we observe a large unrealistic increase in apparent film density (∼5% increase for a 10 nm thick film) consistent with some of the more recent reports.^{39–41} We believe such unrealistic values in apparent film density likely arise from difficulties in measuring the index of refraction of very thin films and breakdown in validity of the Lorentz-Lorenz formula (a continuum approach assuming an isotropic medium^{67}) as the film thickness is no longer “large” relative to the monomer size. Also of note, in agreement with previous reports,^{68–70} we find that the slope of the liquid-line (thermal expansion coefficient) remains constant upon confinement, and the *T*_{g}(*h*) decrease is accompanied by a broadening of the transition and a small increase in the glassy-line thermal expansion consistent with a larger fraction of the sample remaining liquid to lower temperatures.

## EXPERIMENTAL METHODS

Films were made by dissolving monodisperse, weight average molecular weight *M*_{w} = 650 kg/mol, *M*_{w}/*M*_{n} = 1.06 (Pressure Chemical) polystyrene in toluene and spin coating onto 2 cm × 2 cm silicon wafers with 1.25 nm native oxide layers (Wafernet). All samples were annealed under vacuum at 120 °C for at least 16 h to evaporate residual solvent and allow the chains to relax and then cooled to room temperature. Ellipsometry measurements of the temperature-dependent film thickness *h*(*T*) and index of refraction *n*(*T*) were initiated by increasing the temperature of the sample from room temperature to 130 °C over a span of 10 min and equilibrating the film at 130 °C for 30 min. A second alignment of the sample was then performed at 130 °C immediately before beginning the 1 °C/min cooling run.

Ellipsometry measurements were performed on a J.A. Woollam M-2000D rotating compensator instrument that measures the change in polarization state (all four Stokes parameters) of the light reflected off the sample. The raw data are expressed as Ψ(λ) and Δ(λ) representing the amplitude ratio and phase shift of the *p*- to *s*-polarized light. Although this was varied as described in detail below, typically measurements were taken at an angle of incidence of 65° every 10 s on cooling at 1 °C/min. The PS film thickness *h* and index of refraction *n*(λ) were determined by fitting the transparent PS film with a Cauchy layer over the wavelength range of 400-1000 nm atop a semi-infinite temperature-dependent silicon substrate containing a 1.25-nm native oxide layer. Unless otherwise noted, the index of refraction values reported are those at a wavelength of λ = 632.8 nm (corresponding to a HeNe laser) obtained by evaluating the best fit Cauchy parameters at *n*(λ = 632.8 nm).

## RESULTS AND DISCUSSION

### Temperature-dependence of film thickness and index of refraction

Measurements of the temperature-dependent film thickness *h*(*T*) and index of refraction *n*(*T*) were collected by ellipsometry on cooling at a rate of 1 °C/min for PS films of different thicknesses supported on silicon. Figure 1 graphs the PS film thickness as a function of temperature for four different thicknesses: 977, 330, 65, and 31 nm. In order to compare the shape of the *h*(*T*) data for the different film thicknesses, the data sets have been normalized by the film thickness at 110 °C evaluated based on a linear fit to the liquid-line data from 105 to 125 °C. From the data, we can clearly see that the liquid-line thermal expansion is the same for all film thicknesses, as has been reported previously by other studies,^{68–71} along with an overall broadening of the transition. As the glass transition temperature *T*_{g} decreases with decreasing film thickness, it is the glassy line that deviates. The 65-nm data show a slightly reduced *T*_{g} value, but the glassy-line slope is essentially parallel to the thick (bulk) films. In contrast, the 31-nm data show a larger slope in the glassy line, at least down to 30 °C, consistent with a significant fraction of this film remaining liquid-like to lower temperatures.

We have determined *T*_{g}(*h*) values from the *h*(*T*) data in the usual manner by determining the intersection of linear fits to the liquid and glassy regions of the data.^{1,5,70} The corresponding fit ranges used, 40-80 °C (glassy) and 105-125 °C (liquid), come from consideration of fitting the data away from the transition. For consistency, the film thickness values reported for the corresponding *T*_{g}(*h*) values were all evaluated at 30 °C from a linear fit to the glassy-line data to closely match the room temperature thickness values typically reported in such studies. Figure 2 plots our measured *T*_{g}(*h*) values from the present work, along with data from previous literature studies^{1,69,72–74} (data collated in the review by Roth and Dutcher^{5}), showing our data are in excellent agreement with existing studies. The *y*-axis has been graphed relative to $Tgbulk$, the *T*_{g} value reported for thick films in each study, in order to accommodate small differences in $Tgbulk$ values between different studies. Based on an average of films thicker than 200 nm, the $Tgbulk$ for our samples is 95.8 °C, typical for ellipsometry measurements. The data clearly show that below ∼60 nm in film thickness, the average glass transition temperature *T*_{g}(*h*) of the film begins to decrease quite substantially. The reported decrease in *T*_{g}(*h*) for 10-15 nm thick films varies considerably from −10 K to greater than −35 K, which likely reflects the broadening of the transition in thin films as shown in Fig. 1.

Figure 3 graphs the temperature-dependence of the index of refraction *n*(*T*) for the PS layer (at λ = 632.8 nm) for the same four films as the thicknesses shown in Fig. 1. Surprisingly, we find that the absolute values of the index of refraction at any given temperature in both the liquid and glassy regimes are not constant but vary with film thickness in a non-monotonic manner. Multiple measurements on different samples find this non-monotonic trend to be very reproducible with comparable film thicknesses giving similar absolute values in refractive index, as shown in Fig. 3. The *n*(*T*) slopes of the liquid-line are all similar as would be expected for an equilibrium liquid; in contrast, the slopes in the glassy region vary slightly with film thickness, indicating differences in how the films fall out of equilibrium on cooling.

To characterize the small vertical shifts in the *n*(*T*) data with film thickness, we have evaluated the index of refraction at 110 °C and 50 °C, in the liquid and glassy regimes, respectively, by taking a linear fit of the *n*(*T*) data between 105-125 °C and 40-80 °C and evaluating the best fit parameters at *n*(*T* = 110 °C) and *n*(*T* = 50 °C). These refractive index values are plotted in Figure 4 as a function of film thickness, in the liquid regime at *n*(*T* = 110 °C) and the glassy regime at *n*(*T* = 50 °C). We find the same small vertical shifts in the absolute value of the refractive index for both the liquid and glassy regimes. At film thicknesses above 200 nm, the film’s index of refraction remains constant at 1.5712 ± 0.0007 in the liquid regime and 1.5801 ± 0.0008 in the glassy regime. Below 200 nm, the refractive index decreases until a minimum is reached at 65 nm before increasing again. In both the liquid and glassy regimes, the minimum in refractive index at 65 nm, 1.5685 in the liquid state and 1.5772 in the glassy state, is more than three times bigger than the variability observed for bulk films (*h* > 200 nm). Below this minimum in refractive index at 65 nm, the data increase again returning to the bulk index value for thicknesses of ∼30 nm. Below 20 nm, the measured index of refraction is observed to apparently increase substantially consistent with recent reports in the literature.^{39–41}

### Ellipsometry data fitting

Before proceeding any further, we should consider the accuracy of the index of refraction values obtained from fitting of the ellipsometry Ψ(λ) and Δ(λ) data defined as

where *r _{p}* and

*r*are the reflection coefficients for the

_{s}*p*- and

*s*-polarizations.

^{75}Formally, there is no generally accepted method to quantify the measurement accuracy of ellipsometry, primarily because the sensitivity of modern ellipsometers has exceeded the accuracy with which known reference samples can be created.

^{76}Thus, the most reasonable measure of the error for our results, especially given we are primarily interested in relative differences between bulk and thin films, is the sample-to-sample variability in the index of refraction data for nominally identical bulk films, which we have found to be less than ±0.001 for films thicker than 200 nm. Even for the thinner films ∼65 nm where we observe the decrease in refractive index, the sample-to-sample variability remains less than ±0.001. For comparison, the variability in the refractive index data for repeated measurements of a single sample is much less (±0.0002), even for the thinnest films, indicating stability of the films over time. (Note, no evidence of dewetting was observed for these high molecular weight films, even after multiple temperature ramps of the thinnest films, as determined by atomic force microscopy.) However, this all assumes we are accurately fitting the Ψ(λ) and Δ(λ) data with a valid layer model, and that the fits remain robust as the film thickness is decreased.

The common method of fitting ellipsometry data for transparent polymer films is to model the wavelength dependence of the index of refraction of the polymer layer using a Cauchy model

where the absorption term of the index is taken to be zero. Eq. (2) represents an expansion of the Sellmeier model,

which can be derived from the Lorentz model, where the absorption frequencies $\nu i=c\lambda i$ of the material are approximated by a simple spring-bonded electron model.^{77} Far from the characteristic absorption wavelengths *λ _{i}* of the material, the Cauchy and Sellmeier models are identical. We have fit some of the data at representative thicknesses treating the index of the PS film with the Sellmeier model (taking only a single term in the sum of Eq. (3)) and found there to be no difference in fitted index values to within the reproducibility of the measurement, as expected for wavelengths λ = 400-1000 nm where PS has no characteristic absorptions.

The layer model we use is comprised of air, the PS film modeled as a Cauchy layer, and a silicon substrate with a 1.25 nm native oxide layer. The known index values for the silicon and native oxide layer, including the temperature dependence for the index of silicon $N\lambda \u2261n\lambda \u2212ik(\lambda )$, are taken from the literature^{78} and provided as part of the Woollam software. When using the Cauchy model for the PS layer, employing the standard three parameters (*A*, *B*, and *C*) provides the best fit. However, we have found that for film thicknesses less than 300 nm, the third parameter *C* is not well defined during fitting; when it is allowed to vary in thin films, additional noise is introduced into the *B* parameter to compensate random fluctuations in *C*. These wavelength-dependent fitting parameters become less robust for thinner films because, as the film thickness and hence path length of the light through the film decreases, more of the ellipsometry signal comes from the interface and less from the dispersion (wavelength dependence) in the material. To be consistent with all the data presented in this manuscript, we have chosen to fit only the two parameters, *A* and *B* in Eq. (2), while holding the parameter *C* = 0.000 38, a value determined from the average of bulk films with thicknesses greater than 800 nm.^{79} Other ellipsometry studies have also reduced the number fitting parameters when modeling very thin films.^{71} (We have verified that the small shifts in refractive index with film thickness discussed in Fig. 4 do not change if a different value of *C* is used.) For these very thick films, we have also included an additional non-uniformity parameter into the fitting to account for small variations in film thickness across the measurement spot size common in films greater than a micron in thickness. The silicon oxide layer was held fixed at 1.25 nm, a value determined from many measurements of bare silicon wafers. Analysis of the data using a different value for the SiOx layer thickness (e.g., 2.0 nm) does not change the results appreciably (by less than 0.3% even for the thinnest PS films of 10 nm in thickness), and that the same film-thickness dependent trends in PS index are observed.

Figure 5 graphs experimental data of Ψ(λ) and Δ(λ) collected for PS supported on silicon at 110 °C for representative film thicknesses (977, 330, 65, and 31 nm), where the experimental data are shown as symbols and the best fit values as curves. For all film thicknesses, excellent fits to the experimental data are obtained with the exception of the thinnest films as described below. Nominally we fit all of the Ψ(λ) and Δ(λ) data between λ = 400-1000 nm; however, we also varied this wavelength range to be 400-700 nm, 700-1000 nm, or 450-950 nm with no significant change in the observed results. Accuracy of Ψ(λ) and Δ(λ) data fits within the Woollam measurement and analysis software is characterized by the mean squared error (MSE), a biased estimator that is weighted by measurement error,^{80,81}

where $\sigma \Psi exp$ and $\sigma \Delta exp$ are the measurements errors for Ψ and Δ, *n* is the total number of (Ψ, Δ) data pairs being fit, and *m* is the number of fitting parameters (typically 3 = *h*_{PS}, *A*, and *B*). Values of MSE provide a quantitative measure of the difference between the experimental data (Ψ^{exp}, Δ^{exp}) and model fit (Ψ^{mod}, Δ^{mod}). Note formally within the Woollam CompleteEASE software for the M-2000 ellipsometer, the experimental parameters being measured^{82} are $N=cos2\Psi $, $C=sin2\Psi cos\Delta $, and $S=sin2\Psi sin(\Delta )$, such that the MSE is actually calculated as

The MSE values function as a χ^{2} parameter that can be used to verify that a well defined and robust global minimum is being found during fitting by plotting values of MSE as a function of a given fitting parameter to visualize the shape and depth of the minimum. Figure 6 graphs the shape of the MSE function at the best fit minimum for each of our three fitting parameters (*h*_{PS}, *A*, and *B*) for many different film thicknesses that were measured. As can be seen from the plots, film thickness values larger than 30 nm have sharp, well-defined minima for each of the fitting parameters, which include the 65 nm samples where the unusual minimum in index of refraction is observed in Fig. 4. For the thinnest films, 21, 13, and 10 nm, the shape of the MSE minima becomes progressively flatter making the fitting less robust. The 21 nm thick film shows MSE minima that should still be sufficiently well defined, but the 13 and 10 nm thick films have very flat and ill-defined minima, especially for the *B* parameter, that the resulting best fit values should be suspect. It is well known and described in ellipsometry texts^{75,83} that ellipsometry is unable to reliably measure the index of refraction of very thin films, ∼10 nm or less. This ambiguity in index arises because the Ψ(λ) and Δ(λ) trajectories (graphs such as Fig. 8) for films with different indices all merge to a single point at zero film thickness called the film-free point (*h*_{PS} = 0).^{75,83} Even data measured at different wavelengths merge to this single point at *h*_{PS} = 0 such that enlarging the spectroscopic range does not resolve the problem. Thus, we conclude that the unusually large values of the index of refraction for the thinnest films (10 and 13 nm) plotted in Fig. 4 are unreliable because of fitting uncertainty in the Ψ(λ) and Δ(λ) data.

Figure 7 graphs the best fit parameters *A* and *B* as a function of film thickness. As expected, the *A* parameter follows the same trend shown in Fig. 4 for the index of refraction at λ = 632.8 nm, demonstrating that the results do not depend on the specific choice of wavelength. (In addition, we also evaluated the index at λ = 900 nm (data not shown) and found the same trend.) Interestingly, the *B* parameter is primarily constant for films greater than ∼20 nm except for an unusual hump in the data between 70 and 200 nm, with a peak at ∼120 nm. The *B* parameter accounts for the wavelength-dependent dispersion of the index of refraction, suggesting some change in the material is occurring. Even if we let the third *C* parameter vary in the Cauchy model fit, the hump in the *B* parameter data is still present.

The robustness and uniqueness of fitting ellipsometry data can be visualized by plotting Ψ–Δ trajectories.^{75} For a given layer model, expected (Ψ, Δ) values can be calculated for increasing PS film thickness showing how the Δ vs. Ψ trajectory circles counter-clockwise as the optical path length cycles through *λ*/2. In Figure 8, we plot the expected Δ vs. Ψ data evaluated at λ = 632.8 nm for PS films supported on silicon calculated using values from our layer model: *n*(*T* = 110 °C) = 1.5712 for PS, *n* = 1.7338 for native SiOx, and *N* = 3.9091–0.001 308*i* for Si.^{78} Figure 8 graphs Ψ–Δ trajectories for two different angles of incidence, 65° and 58°, where data were collected. (Note the Ψ–Δ trajectory for 65° appears discontinuous because a value of Δ = 0 is equivalent to Δ = 360.) The measured experimental (Ψ, Δ) values are also plotted demonstrating excellent agreement with the calculated curves, regardless of where in the trajectories the data falls. In addition, we have highlighted in orange a 20 nm region about 65 nm where the dip in index of refraction occurs in Fig. 4 and about 120 nm where the hump in the *B* parameter occurs in Fig. 7. In ellipsometry fitting, the part of the trajectory that is particularly hard to fit unambiguously is the region near the film-free point (*h* = 0) and the period-point when the trajectory returns to the same point as *h* = 0 due to destructive interference (for λ = 632.8 nm in Fig. 8, the period points are *h* = 245 nm for 65° and *h* = 237 nm for 58°).^{83} Here, trajectories for different values of the index of refraction merge and this is the primary reason why it is hard to measure the refractive index of very thin films (∼10 nm) with ellipsometry as described above.^{75,83} In reality, we are simultaneously fitting the entire range of wavelengths λ = 400-1000 nm at a single angle of incidence (nominally 65°) such that the period points range between 155 and 387 nm. The purpose of Fig. 8 is to note that neither of the regions where we observe variations in the index of refraction in Fig. 4 or hump in the *B* parameter in Fig. 7 occurs near the period-point that is hard to fit. We conclude that the index of refraction data presented in Fig. 4 is a robust measure, independent of the fitting details, with the exception of the two thinnest films at 10 and 13 nm. Thus, we continue with our analysis below focusing only on the films with thicknesses of 20 nm and larger.

### Lorentz-Lorenz equation as a measure of density and specific volume

The Lorentz-Lorenz equation, Eq. (6), relates the index of refraction, a macroscopic (continuum) quantity, to the microscopic polarizability of the material. As this inherently depends on the number of molecular dipoles per unit volume, which can be written in terms of the material’s mass density, the Lorentz-Lorenz equation has been frequently used as a measure of density.^{54,84–87} For common solids and liquids, the Lorentz-Lorenz relation gives accurate values of the density to within a few percent (typically 1%-2%);^{84,85,88} the largest discrepancies are for dense gases with large density fluctuations^{88,89} or highly polar compounds.^{84} We consider here what limitations may exist in the validity of this equation for very thin films based on the assumptions made in its derivation.

The derivation of the Lorentz-Lorenz relation is typically done by considering separately the electric field contribution from inside and outside an arbitrary small spherical cavity.^{67,90} The size of the cavity must be large with respect to the molecular dipoles such that the local microscopic electric field at the center of the cavity can be replaced by the macroscopic field contribution from those charges residing outside the cavity, $E\u20d1outsidemacro(r\u20d1)$. This macroscopic field outside the cavity can then be written in terms of the total macroscopic electric field, $E\u20d1totalmacro(r\u20d1)\u2261E\u20d1(r\u20d1)$, minus the macroscopic field inside the cavity: $E\u20d1outsidemacro(r\u20d1)=E\u20d1(r\u20d1)\u2212E\u20d1insidemacro(r\u20d1)$. For a spherical cavity with uniform polarization, i.e., negligible spatial variation in the molecular dipoles, this macroscopic field inside the cavity can be easily calculated^{90,91} giving $E\u20d1insidemacro(r\u20d1)=\u221213\u03f50P\u20d1$, where *P⃑* is the polarization density equal to the number density of molecular dipoles inside the cavity, $P\u20d1=NvP\u20d1$. The remaining quantity needed is the microscopic local electric field inside the cavity $E\u20d1insidelocal(r\u20d1)$, which has been shown to explicitly sum to zero for a uniform spatial distribution of dipoles having either a cubic lattice or random distribution.^{67,90} Combining all of these, we have for the total local (or effective) electric field

which is the standard Lorentz relation. At the microscopic level, the induced molecular dipole moment *p⃑* at a given location is the molecular polarizability *α* times this total local electric field

At the macroscopic level, the polarization density *P⃑* is defined in terms of the total macroscopic electric field *E⃑*(*r⃑*) and the dielectric constant $\kappa =\u03f5\u03f50$ of the material as *P⃑* = *ϵ*_{0}(*κ* − 1) *E⃑*(*r⃑*), where *ϵ*_{0} is the permittivity of free space. By merging the microscopic and macroscopic definitions of *P⃑* one arrives at the classic Clausius-Mossotti relation

For optical frequencies, the dielectric constant is usually written in terms of the index of refraction $n=\kappa $. In addition, the number density of molecular dipoles $Nv$ is often replaced with the material’s mass density *ρ* by using Avogadro’s number *N*_{A} and the molar mass of the molecular dipole unit, which for polymers is typically taken to be the molar mass of the monomer *M*_{0}.^{40,86} Thus, $Nv=\rho NAM0$ leads us to the commonly used form of the Lorentz-Lorenz equation^{39–41,84–86}

where the quantity $n2\u22121n2+2$ is often defined as *L*. (Note this has been written in SI units, whereas much of the literature is historically in Gaussian units, where $P\u20d1=(\u03f5\u22121)4\pi E\u20d1(r\u20d1)$ and *ϵ*_{0} = 1.)

We can see that two key assumptions were made in the derivation of Eq. (6) that are relevant and potentially a concern when applying the Lorentz-Lorenz relation to very thin films: (1) an arbitrarily sized cavity must fit inside the film but still be large relative to the molecular dipole unit, the monomer and (2) these molecular dipoles need to have a uniform spatial distribution inside the cavity for their contribution to the microscopic local electric field to sum to zero. First we consider what the minimum size for this arbitrary cavity is in the derivation of the Lorentz-Lorenz relation. For PS, although the main polarizing unit is the large phenyl ring, the relevant size is that of the entire molecular dipole unit, the monomer. The PS monomer size has been estimated at ∼0.7 nm.^{92} Within the derivation of the Lorentz-Lorenz relation, the radius of the cavity should be large (at least an order of magnitude) relative to the interparticle spacing of the molecular dipoles, i.e., at least *r* ≈ 7 nm or larger.^{90} Meaning a cavity of at least ∼14 nm in diameter or larger should fit inside the film. Second, we should consider if the molecular dipoles are uniformly randomly oriented across this arbitrary cavity. Experimental studies primarily with second harmonic generation indicate alignment of the phenyl rings locally at the free surface and substrate interface of PS films typically to a depth ∼1 nm.^{93–96} As the film thickness is decreased and the cavity size starts to span the entire thickness of the film, the non-random spatial orientation of the molecular dipoles at the interfaces may also invalidate assumptions in the derivation. Thus, we can certainly see that there will come a point in decreasing the film thickness where the Lorentz-Lorenz relation must fail because the film can no longer be accurately approximated as a continuum material. Based on the above discussion, we conclude that a cavity of at least ∼14 nm in diameter must fit inside the film, excluding a couple of nanometers at either interface where molecular orientation occurs, meaning that we anticipate the Lorentz-Lorenz relation, Eq. (6), to only be valid for film thicknesses larger than ∼20 nm.

The temperature dependence of the density and index of refraction have often been used as a stringent test of the validity of the Lorentz-Lorenz formula.^{54,84,85} The reason being that the molecular polarizability *α* at optical wavelengths is primarily determined by the electronic quantum states, which have a very weak temperature dependence.^{85} Thus, the refractive index *n* should only vary with temperature through density, implying that, according to Eq. (6),

should be a constant, independent of temperature. For a polymer film supported on silicon, the thermal expansion and hence volume change will occur entirely in the thickness direction.^{97} As such, the film thickness *h* can be used to represent the change in density = sample mass/volume with temperature. We find that the quantity $hn2\u22121n2+2$ for PS films on silicon is independent of temperature to within ±0.1% for film thicknesses greater than 150 nm, within ±0.2% for *h* = 50-150 nm, and within ±0.3% for *h* = 30-50 nm. Below 30 nm, the deviations in the temperature dependence of $hn2\u22121n2+2$ begin to increase substantially, varying by ±0.4% for films 12-30 nm thick and exceeding 0.9% for 10 nm thick films. This is consistent with our conclusion above that the Lorentz-Lorenz equation should become invalid for thin films as the thickness approaches ∼20 nm.

As an interesting historical note, in 1965, Looyenga proposed a different quantity $n2/3\u22121$ to replace the $n2\u22121n2+2$ term in the Lorentz-Lorenz equation.^{98} Looyenga’s expression was based on a previous formula he had derived to describe the dielectric constant of heterogeneous media incorporating the volume fraction of each component.^{99} Looyenga argued that the same formula could be adapted to homogeneous media by treating it as a mixture where one component was vacuum,^{98} leading to Looyenga’s replacement for the Lorentz-Lorenz formula

Although Looyenga’s expression for heterogeneous mixtures is routinely used, the formula $n2/3\u22121$ replacing that of Lorentz-Lorenz’s has received little attention, despite both Looyenga^{98} and others^{85} demonstrating that the temperature dependence of Looyenga’s expression is superior. For gases, Looyenga’s $n2/3\u22121$ expression is nearly identical to Lorentz-Lorenz’s $n2\u22121n2+2$, as they have the same expansion in $n\u22121$ to second order.^{85,100} However, for organic liquids with *n* ≈ 1.3-1.5, the $n\u221213$ cubic and higher order terms give rise to small differences of 2%-4%.^{85} For completeness and out of curiosity, we evaluated the quantity $hn2/3\u22121$ as a function of temperature for our PS films on silicon. Consistent with previous observations,^{85,98} we find Looyenga’s expression, Eq. (8), to show smaller deviations with temperature then that of Lorentz-Lorenz, Eq. (6). The temperature dependence of $hn2/3\u22121$ is constant to within ±0.1% for film thicknesses down to 50 nm and shows deviations to within only ±0.2% for *h* = 30-50 nm before increasing for thinner films below 30 nm. Below we continue our discussion by using the more commonly accepted Lorentz-Lorenz expression, Eq. (6), to comment on the relative change in specific volume and density of PS films as a function of film thickness; however, we note that the same deviations from bulk as that shown in Fig. 10 are observed to within experimental error if Looyenga’s expression, Eq. (8), is used instead.

### Changes in specific volume and density with decreasing film thickness

Here we proceed in our data analysis assuming the Lorentz-Lorenz equation is valid for sufficiently thick films with the understanding that the continuum approximation will breakdown for thin films as the thickness approaches ∼20 nm. Using the Lorentz-Lorenz equation, Eq. (6), we can define an effective specific volume v_{sp}, equivalent to the inverse density, from the Lorentz-Lorenz parameter $L=n2\u22121n2+2$, recognizing that the remaining terms $\alpha NA3\u03f50M0$ are constants independent of temperature and film thickness

Following White and Lipson,^{36} we plot curves of effective specific volume v_{sp}(*T*) in Figure 9 for several different film thicknesses. We find supported PS films of bulk thickness (977 and 330 nm) trace out the same $vspT$ curve while thinner films (65 nm) are shifted to larger $vspT$ values by 0.4% relative to bulk. This magnitude for the increase in specific volume is comparable to that predicted by the White and Lipson model, suggesting it may result from a natural driving force for film expansion as missing contacts at the interface lead to a reduction in the attractive energy between polymer segments.^{36,65} However, the thinnest films (31 nm in Fig. 9) show a shift in $vspT$ in the opposite direction to smaller values relative to bulk. In Fig. 9, for reference, the data sets have been normalized to the bulk $vspT$ value at *T* = 110 °C, as measured for film thicknesses greater than 200 nm. It is also clear from the $vspT$ data in Fig. 9 that both the liquid and glassy lines shift uniformly with film thickness. As the films are cooled from the equilibrium liquid state along a shifted $vspT$ curve, the film falls out of equilibrium into a glass along a similarly shifted $vspT$ curve. The glass transition does not occur at the same specific volume (i.e., total free volume) for different film thicknesses in contradiction with ideas behind the original free volume models for the glass transition.^{48–51}

To study the film thickness dependent shifts in $vspT$ in more detail, in Figure 10, we plot the effective specific volume $vspT$ values in the liquid (*T* = 110 °C) and glassy (*T* = 50 °C) regimes as a function of film thickness. The data clearly show that these small shifts in the effective specific volume of the film are consistently the same for the liquid and glassy regimes. Fig. 10(a) with a logarithmic thickness scale shows that for bulk films between 200 and 3100 nm, the liquid and glassy $vspT$ values are independent of film thickness to within a standard deviation of ±0.11%. In Fig. 10(b), we focus on the unusual non-monotonic behavior of the data for films below ∼200 nm that varies outside this range, where we have highlighted the variation in the bulk values with horizontal lines. Below ∼120 nm, the effective specific volume increases to a peak value of 0.4 ± 0.2% at a film thickness of 65 nm, an increase in $vspT$ of more than three times larger than the variation in the bulk data. However below this peak at 65 nm, conspicuously around the thickness where the *T*_{g}(*h*) decrease begin in Fig. 2, the effective specific volume decreases again back to the bulk value at ∼40 nm before continuing to decrease further for much thinner films. We do not show data below 20 nm because as we argued above, the ellipsometry fitting is unreliable for very thin films ∼10-13 nm. However, we do note that if values for film thicknesses less than 20 nm were plotted, the values would be off the scale showing an ∼5% increase in density for a 10 nm thick film consistent with recent literature reports.^{39,40} Also as described above, the approximations made to derive the Lorentz-Lorenz equation breakdown as the film thickness approaches ∼20 nm, making the trends in the data shown in Fig. 10 for the very thinnest films somewhat suspect. Note that all the variations in $vspT$ plotted in Fig. 10 are within the ±1% experimental error of previous studies reporting no changes in density with film thickness.^{37,38}

To further demonstrate that these data in Figure 10 are representative of the properties of the polymer film and not some systematic artifact of the data collection and analysis procedure, we have made a handful of additional measurements of PS films on silicon substrates that contain an added 23.5 ± 0.8 nm aluminum oxide (AlOx) layer (nominal index of 1.64) creating a strong uniform index contrast with PS. Bare silicon wafers were sputtered with AlOx and then subsequently characterized by ellipsometry fitting the sputtered AlOx film thickness and index characteristics *n*(*λ*) with a Cauchy layer. These characterized AlOx substrates were then spin-coated with PS and further measurements of the PS specific volume $vspT$ were done. Figure 10(a) plots as gray data these additional measurements of $vspT$ for PS on the AlOx coated substrates at 50 and 110 °C. To within reasonable accuracy, these measurements agree with those collected on bare silicon wafers clearly showing the same film thickness dependent features. Thus, we conclude that the data shown in Figure 10 are accurately reflecting the properties of the thin PS films.

We have labeled the *y*-axes of Fig. 10 as the effective specific volume $vspT$ because according to Eq. (6), density $\rho =1vsp$ seems like the most reasonable parameter to vary with film thickness within the Lorentz-Lorenz expression. However, we do note there is another potential variable within the Lorentz-Lorenz expression, Eq. (6), the molecular polarizability α. It seems unlikely that such a local property would vary with film thickness at such large length scales, as α is not even expected to vary much with temperature. However, within the Lorentz-Lorenz derivation, the polarizability is assumed to be uniformly isotropic. One could imagine that as the film thickness is reduced and polymer chain segments must orient more within the plane of the film that the polarizability could become slightly anisotropic leading to deviations from the expected Lorentz-Lorenz expression. Perhaps the non-monotonic behavior in the *y*-axes of Fig. 10, formally $1L=n2+2n2\u22121$, occurs because of some film thickness change in density coupled with some film thickness change in the isotropic polarizability. Future work will investigate such a possibility.

Although we can only comment on the validity of ellipsometry measurements, we do note that Wallace *et al.*^{37} previously questioned the reliability of reflectivity measurements on very thin films. To avoid some of these concerns for their own neutron reflectivity measurements, Wallace *et al.*^{37} used a “twin” reflectivity technique that collected data of the critical angle for reflection from both the free surface side and substrate side of the PS film on silicon samples. A data reduction scheme was then used to identify the crossover point in the momentum transfer vector to account for the unknown tilt of the sample. This allowed for reliable determination of the film’s density to within ±1%. Wallace *et al.*^{37} called into question previous neutron and x-ray reflectivity works^{101,102} on thin polymer films that had claimed density changes of ∼5% with decreasing film thickness because they had not accounted for interfacial roughness or other confounding factors inherent in fitting reflectivity curves. In their careful study, Wallace *et al.*^{37} found the mass density of PS films to be consistent with the bulk value within ±1% down to film thicknesses of 6.5 nm. Perhaps it is also worth considering how unrealistic ∼5% increase in density really is. One can use PVT data for PS from handbooks^{103} to estimate what equivalent pressure increase would be required for such an isothermal increase in density. For PS in the melt state at 115 °C (the lowest temperature for which such data are available), pressures in excess of 100 MPa would be needed for ∼5% density increase, and in excess of 4 GPa of pressure would be required for ∼30% increase in density (glassy materials with larger bulk moduli values would require even greater pressures). Also as noted in the Introduction, stable glasses carefully formed by physical vapor deposition to optimize molecular packing only show increases in material density of ∼1% relative to ordinary glasses.^{54,55} Thus, given the unrealistic value of 5% or 30% increases in density, it seems likely that the recent x-ray reflectivity results^{39,40} may suffer from similar uncertainties in fitting of reflectivity curves as expressed by Wallace *et al.*^{37}

## CONCLUSIONS

We have used ellipsometry to measure the temperature dependence of the index of refraction *n*(*T*) for PS films supported on silicon and explored the validity of the Lorentz-Lorenz relation in very thin films. We find that the specific volume (density) of supported PS films does not vary by more than ±0.4% of the bulk value for film thicknesses above 30 nm, and that the small variations we do observe are uncorrelated with the *T*_{g}(*h*) reductions exhibited by these films. Based on the assumptions made in its derivation, we conclude that the Lorentz-Lorenz equation is not valid for very thin films, breaking down as the thickness approaches ∼20 nm. We believe that the large increase in index of refraction and apparent density (∼5% for a 10 nm thick film) we observe, consistent with recent experimental reports,^{39–41} are an experimental artifact because ellipsometry is known to be unreliable in accurately measuring the index of refraction of very thin films, ∼10 nm or less,^{75,83} and that such density increases would be unrealistic given stable glasses with optimum, equilibrium packing only show increases in density of ∼1% relative to ordinary glasses.^{54,55}

We have also tested recent theoretical predictions and assumptions made by White and Lipson.^{36,65} Their thermodynamic model predicts a driving force for film expansion with decreasing film thickness in thin films, caused by the missing interactions at the free surface, that should result in a small increase of the liquid-line specific volume $vspT$ of less than 1%. Based on this prediction, they suppose that such a shift in $vspT$ of the liquid line could explain the observed *T*_{g}(*h*) decreases exhibited by these films, assuming that the glassy-line specific volume remains the same as bulk. Experimentally we observed that both the liquid and glassy specific volume $vspT$ shift consistently together such that the glass transition does not occur at the same specific volume for different film thicknesses. In the thickness regime where we believe the Lorentz-Lorenz equation is valid (*h* > ∼ 20 nm), we do observe a small increase in specific volume $vspT$ of 0.4% ± 0.2% relative to bulk for film thicknesses between ∼120 and 65 nm. However, below ∼60 nm where the *T*_{g}(*h*) reductions begin, the effective specific volume is observed to decrease again returning the bulk value at ∼40 nm. We speculate that the non-monotonic changes in index of refraction with film thickness we observe for film thicknesses greater than 20 nm may result from small competing changes in film density (specific volume) and uniform polarizability.

## Acknowledgments

The authors wish to thank Jane Lipson, Justin Pye, Roman Baglay, and scientists at J.A. Woollam Co. for useful discussions. Support from the National Science Foundation CAREER program (Grant No. DMR-1151646) and Emory University is gratefully acknowledged.

## REFERENCES

Note that the values of *A*, *B*, and *C* for the Cauchy equation are typically quoted with the wavelength λ in Eq. (2) calculated in microns, and we have followed this common convention used within the Woollam software.