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Abstract
Copolymerization offers a promising route to the accurate control of material properties, such as refractive index. For compatible polymers, an ability to produce copolymers with incrementally varying refractive indices is a basic requirement for fabricating gradient-index (GRIN) optics. To determine the accuracy of this approach, we examined the composition dependence of index in several optical copolyimides. We observed deviations from standard model predictions that exceed 5 × 10−3 at certain compositions, consistent with non-ideal copolymer formation. This implies that state-of-the-art GRIN optical design requires careful calibration of the composition dependence in these copolymers. We further found that birefringence is sensitive to the structures of the component polymers, and that the thermo-optic and thermal expansion coefficients vary linearly with composition, within the experimental uncertainty.
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1. Introduction
Polymers are useful in manufacturing components for optical and photonic applications, where their light weight, toughness, and resilience are critical [1,2]. They can have good optical transparency and clarity, and they can be shaped into a variety of sophisticated shapes and surface profiles. As a result, polymer optics are widely used in many current commercial applications. More recently, we have demonstrated that layered polymers enable the fabrication of GRIN optical elements, which can reduce the size, mass and complexity of optical systems [3]. Among optical polymers, polyimides are of particular interest because they retain their useful properties even at very high temperatures. Their glass transition temperatures can be above 250 C and their decomposition temperatures can be above 400 C. Polyimides with combinations of optical, thermal and mechanical properties appropriate for a variety of microelectronic and optoelectronic applications are commercially available [4]. Recently, we reported a study of the refractive index and thermo-optic coefficient for a set of optical polyimides which have relatively low temperature dependence, thus making them especially appropriate in high-precision optical applications [5].
Copolyimides and polyimide blends have been used in microelectronic and optoelectronic applications because they enable control of material properties. Generally, the emphasis in these studies [6–14] has been on the influence of composition on mechanical properties, glass transition temperature, linear thermal expansion coefficient and, more recently, dielectric permittivity [15]. Notably, the composition dependence of the refractive index in fluoro-copolyimides has been found to deviate significantly from linearity in some studies [16].
The GRIN polymer optics we demonstrated previously [3] consist of layers with precisely controlled refractive index. These layers conform to an index profile specified by the optical design, which serves as the foundation for the imaging properties. Copolymerization could facilitate the fabrication of individual layers with varying indices if it could result in well-controlled refractive indices. Furthermore, since the interfaces between layers with different indices would have similar composition, the interlayer stress should be reduced and adhesion should be improved. In particular, this copolymerization approach could enable the substitution of optical polyimides having reduced thermal dependence [5] into high-precision, layered GRIN optical elements [3].
In order to examine whether copolymerization is a practical approach for controlling the refractive index of optical polyimides, we investigated two sets of copolyimides. We evaluated the dependence of the refractive index on composition at a level of accuracy appropriate for state-of-the-art polymer GRIN optics. The materials, selected from Ref. 5, are in a class of polyimides that have been found to have good thermal stability and optical properties, as well as good miscibility [17]. They are copolymers of BPDA-TFMB with either BPDA-DADP or BPDA-mBAPS. Thus, each copolymer structure involves the same dianhydride, BPDA, but with two different diamines, as shown in Table 1. These polyimides are similar, but do differ somewhat in structure. In particular, BPDA-TFMB and BPDA-DADP have relatively rigid, linear structures compared with the angular, less rigid BPDA-mBAPS material. We consider these differences when interpreting the experimental results.
Table 1. Chemical structures of monomers investigated in this study
2. Sample preparation
Biphenyltetracarboxylic dianhydride (BPDA, 98%), bis[4-(3-aminophenoxy)phenyl] sulfone (mBAPS, 98%), and 4,4’-diaminodiphenyl ether (DADP, 98%) were purchased from TCI. 2,2’-bis(trifluoromethyl)benzidine (TFMB, 98.5%) was purchased from Akron Polymer Systems. The monomers were used without further purification. Anhydrous N,N’-dimethylacetamide (DMAc, 99.8%) was obtained from Alfa-Aesar and used as received.
Two sets of random copolyimides, BPDA-(TFMB/DADP) and BPDA-(TFMB/mBAPS), with various monomer content, were prepared in a two-step procedure as described previously [5]. Random copoly(amic acid)s (coPAA’s) were typically prepared by polycondensation of 5 mmol BPDA-(TFMB/DADP) or BPDA-(TFMB/mBAPS) using various diamine monomer feed ratios in DMAc at room temperature for 3-4 days. The concentrations of the resulting coPAA solutions in DMAc were 16-21 wt.%, and were maintained by constant stirring with a magnetic stir bar. Molar ratios between TFMB and DADP/mBAPS in the monomer feeds were 1/9, 3/7, 5/5, 7/3, and 9/1.
Copolyimide films were prepared by thermal imidization of coPAA solutions using a conventional two-step procedure summarized schematically in Fig. 1. Each coPAA solution was cast onto a glass substrate using a doctor blade technique. The coPAA film thickness was controlled with spacers, resulting in copolyimide films with typical thickness in the range of 30-40 µm. The substrate with cast coPAA solution was dried in an oven at 80 C for 3 hours, followed by thermal imidization in a dry nitrogen atmosphere, first at 200 C for 30 min and then at 280 C for 30 min. Free-standing copolyimide films were obtained by peeling from the glass substrates, after cutting away the edges of the films.
3. Characterization of sample composition
Since these copolyimides are poorly soluble in deuterated polar solvents such as DMSO and DMF, it is difficult to determine the mole fraction (mol%) of TFMB using 1H NMR measurements. Therefore, we determined the composition based on elemental analysis performed by Micro-Analysis, Inc. for BPDA-(TFMB/DADP), and by the Microanalysis Laboratory, University of Illinois for BPDA-(TFMB/mBAPS). All samples were dried at 120 C for 24 h in vacuum prior to analysis. Results for % carbon and % fluorine were reported as averages from duplicate runs. The resulting copolyimide compositions are summarized in Table 2. Since the fluorine atom is characteristic of the fluorinated monomer (TFMB) in these copolyimides [18,19], we used the fluorine analysis results to determine the mass percentages of fluorine (${W_F}$), and thus the mole fractions of TFMB in the BPDA-(TFMB/DADP) or BPDA-(TFMB/mBAPS) copolyimides according to [20]
where ${F_{TFMB}}$ is the mole fraction (mol%) of TFMB; ${M_2}$ is the molecular weight per repeat unit of either the BPDA-DADP or BPDA-mBAPS homopolyimide; ${W_F}$ is the content (wt.%) of fluorine in the copolyimides, as determined by the elemental analysis; $\Delta M = {M_2} - {M_1}$, with ${M_1}$ being the molecular weight per repeat unit of the BPDA-TFMB homopolyimide; and the factor 114 accounts for the six fluorine atoms per repeat unit of BPDA-TFMB.Table 2. Elemental analysis results for a series of copolyimides of BPDA-(TFMB/DADP) and BPDA-(TFMB/mBAPS)
As seen in Table 2, the experimentally determined mole fractions of TFMB in the copolyimides (${F_{TFMB}}$) are in reasonable agreement with the mole fractions in the monomer feed (${f_{TFMB}}$). The greatest discrepancies are found for BPDA-(TFMB/mBAPS)-73 and BPDA-(TFMB/mBAPS)-91. The typical measurement errors associated with the mole fractions (${F_{TFMB}}$) are ±0.3% of the values listed in Table 2. The results of carbon elemental analysis on the same copolyimides were lower than the theoretical %C values by 0.4-1%, which can imply higher TFMB mole fractions than theoretically expected. However, a more likely explanation for the latter results is either the presence of trapped water, incomplete imidization, and/or unknown contaminants in the monomers.
The last column of Table 2 shows the volume fraction of BPDA-TFMB for each copolymer composition studied. This was calculated from the measured mole fractions and the homopolyimide densities, which are listed in Table 3. The densities were measured with a gas displacement pycnometer (Micromeritics AccuPyc II 1340). The accuracy of these measurements was estimated to be ±2% due to the limited sample sizes available. The volume fractions are included in order to facilitate a comparison of the measured copolymer indices to predictions, which often describe the average refractive index in terms of the volume fractions of the components [21–23].
Table 3. Measured densities and refractive indices of the homopolyimides used in this study
4. Refractive index measurements
We measured the refractive indices of free-standing films with a Metricon Model 2010 prism coupler. This instrument uses a piston to press the film samples against a high-index prism. All index measurements were done in a dry nitrogen atmosphere, after each sample was dried to remove absorbed water. In order to minimize stress-induced effects in the index results, the minimum pressure needed to achieve optical coupling was used. We modified the instrument by adding heaters and calibrated thermocouples to monitor and control the sample temperature. The refractive index results reported here were obtained at either 25 or 105 C and 633 nm, a wavelength at which all the polyimide films in this study were highly transparent. A fused silica standard was used for index calibration.
In a previous study [24], we carefully considered the uncertainties in this type of refractive index measurement on polymer films. Under ideal circumstances, we showed that the uncertainty in the measured refractive index of a highly uniform sample with good surface quality was ±5 × 10−5. However, additional sources of experimental uncertainty often exist in polymer films, including the surface and/or bulk morphology, as well as differences in molecular orientation between or within samples. For example, we found that the measured index of PMMA films, fabricated for optical applications in large batches in a well-controlled industrial setting, had an uncertainty of ±2 × 10−4. Since the current study was done on film samples from several small-batch syntheses of homopolyimides and copolyimides, it is expected that the uncertainty associated with the current index measurements is somewhat greater. To estimate the uncertainty in the refractive index results in this case, we measured the index of 3-4 separate batches of the homopolyimides listed in Table 3. This allowed us to conservatively estimate the index uncertainty as ±4 × 10−4, limited by sample characteristics such as uniformity and batch-to-batch reproducibility. For copolyimides of varying composition, this uncertainty value is consistent with the uncertainty of the elemental analysis presented in the previous section and summarized in Table 2.
Refractive index measurements were performed for both $TE$- and $TM$-polarized light. Within the experimental uncertainty, all in-plane polarization directions resulted in the same value of $TE$ refractive index, ${n_{TE}}$, which was always larger than the corresponding $TM$ refractive index, ${n_{TM}}$. The in-plane index uniformity was confirmed visually by observing the samples between crossed polarizers, thus verifying the uniaxial nature of these films. The average refractive index, $\bar{n}$, was obtained from the polarized light measurements as
For these uniaxial films, ${n_{TE}}$ corresponds to the ordinary refractive index, ${n_o}$, and ${n_{TM}}$ corresponds to the extraordinary refractive index, ${n_e}$. This corresponds to the molecular chains being predominantly oriented parallel to the film plane [25]. The convention in optics is to define birefringence as [26,27]
According to this definition, all of the optical polyimides studied here, as well as all related optical polyimides that we have come across in previously published works, have negative birefringence. However, several previous studies of polyimide films have chosen to define birefringence instead as [16,25,28]
which, for all optical polyimides of which we are aware, is always positive. This sign difference is due solely to the adoption of a different definition of birefringence than is standard in optics. For consistency with previous work in the field, we adopt Eq. (4) as the definition of birefringence in the present study.5. Average refractive index of copolyimides
The measured average refractive index for the two sets of copolyimides we have investigated is plotted as a function of BPDA-TFMB volume fraction in Fig. 2. The error bars associated with these measurements (±4 × 10−4) are too small to be displayed on the scale of these plots. As seen in Fig. 2, the composition dependence of the refractive index is approximately linear in volume fraction for both sets of copolyimides. However, we observe deviations from linearity exceeding the measurement uncertainty at specific compositions in Fig. 2, and these are discussed below.
Fig. 2. Dependence of the measured average refractive index on BPDA-TFMB volume fraction for two sets of copolyimides, at 25 C (blue) and 105 C (red): (a) BPDA-(TFMB/DADP), and (b) BPDA-(TFMB/mBAPS). A linear approximation is included for each set of data.
The basic linear behavior seen in Fig. 2 can be described by the empirical relation due to Arago and Biot [21], which has often been used to predict approximately the refractive index of mixtures, namely
In the present case, A and B refer to the homopolyimides in a set of copolyimides denoted by AB, and $\mathrm{\Phi }$ refers to the corresponding volume fraction. The indices of the 3 homopolyimides used in this study are listed in Table 3. A further assumption in applying Eq. (5) to the data in Fig. 2 is volume additivity, namely that ${\mathrm{\Phi }_A} + {\mathrm{\Phi }_B} = 1$, which is associated with the concept of an ideal mixture in classical thermodynamics [29]. An ideal mixture is one in which there are no volume and enthalpy changes resulting from mixing of the components. This implies volume additivity, while the condition that the enthalpy of mixing is zero means that interactions between segments of the A and B, A and A, or B and B components are identical. Deviations of the observed copolyimide indices from the ideal mixture prediction are a measure of the validity of this approximation.
A more widely applicable approximation for the refractive index of mixtures is based on the Lorentz-Lorenz model [21]. In this model, the average refractive index, $\bar{n}$, is related to the average molecular polarizability, $\bar{\alpha }$, by
In the case of the copolyimides in this study, the Lorentz-Lorenz prediction of refractive index differs from the Arago-Biot (linear) prediction by <1 × 10−3 over the full composition range, as calculated using the parameters in Tables 2 and 3. This is shown in Fig. 3. Such differences between the two model predictions are comparable with the measurement uncertainty. We conclude that, at the present level of experimental accuracy, both models perform equally in reproducing the measurements.
Fig. 3. Difference between the Arago-Biot (linear) and Lorentz-Lorenz predictions for the refractive index of ideal mixtures, corresponding to the two sets of copolyimides studied here, as a function of BPDA-TFMB volume fraction, at 25 C. Solid line: BPDA-(TFMB/DADP), dashed line: BPDA-(TFMB/mBAPS).
To examine the composition-dependent deviations of the measured refractive index from model predictions in more detail, we compare the experimental results to the Lorentz-Lorenz model in Fig. 4. We choose the Lorentz-Lorenz model for this purpose because of its broad applicability, as discussed previously [21]. Furthermore, since this model can be derived from electromagnetic theory, it has the advantage of providing a useful framework for interpreting the observed deviations, as well as for a detailed examination of polarization effects.
Fig. 4. Differences between the measured and ideal average refractive index, $\Delta {\bar{n}_{meas}}$ (red, open symbols, left axis) and ratio of the observed to ideal density, ${\rho _{AB,meas}}/{\rho _{AB,ideal}}$ (blue, solid symbols, right axis), as a function of BPDA-TFMB volume fraction, at 25 C, for the two sets of copolyimides in this study: (a) BPDA-TFMB:BPDA-DADP, and (b) BPDA-TFMB:BPDA-mBAPS. The ideal values are based on the Lorentz-Lorenz model. The error bars correspond to the measurement uncertainty, and the lines are guides for the eye.
According to Fig. 4, the difference between the measured composition-dependent refractive index and the Lorentz-Lorenz prediction can exceed 5 × 10−3 for the copolyimides in this study. These index differences, which are approximately an order of magnitude higher than the cited measurement uncertainty, reflect departures of the copolyimide properties from ideal mixtures, at certain compositions. In the following section, we discuss some potential monomer interactions that can lead to density deviations from the ideal mixture prediction in copolyimides, thereby explaining the refractive index differences highlighted in Fig. 4.
For practical purposes, and within the context of our previous demonstrations of layered polymer GRIN lenses [30], Fig. 4 implies that basing a GRIN lens design on varying the copolyimide composition must be approached with caution. For copolymers which follow the ideal mixture behavior, it is sufficient to measure the indices of the homopolymers and then calculate the index of any copolymer based on its composition, using (for example) the Lorentz-Lorenz model. However, Fig. 4 indicates that such index predictions are only accurate to approximately ±5 × 10−3 in the case of copolyimides, due to the departure of copolyimides from ideal mixture behavior, as discussed below. This level of accuracy in refractive index mapping is generally not sufficient for precision GRIN optics [31]. Instead, successful GRIN optics incorporating copolyimides are likely to require a more extensive set of measurements of index vs copolyimide composition.
6. Density and molecular packing in copolyimides
In the Lorentz-Lorenz model, as seen in Eq. (6), the average refractive index depends on the molar volume, ${V_m}$, and the average molecular polarizability, $\bar{\alpha }$. Deviations of the refractive index from this model prediction correspond to deviations of the density, $\rho $ (or, equivalently, molar volume) from that of an ideal mixture, resulting from a nonzero enthalpy of mixing. These density effects have been described previously in terms of a molecular packing coefficient [28], under the assumption that the copolyimide formation has a negligible effect on polarizability. Based on Eq. (6), we can thus relate the deviations in the observed refractive index from the ideal mixture prediction to deviations in the observed density as
By defining $\Delta {\bar{n}_{meas}} = {\bar{n}_{AB,\,meas}} - {\bar{n}_{AB,\,ideal}}$, and assuming that $\Delta {\bar{n}_{meas}} \ll 1$, Eq. (8) simplifies to
Because the factor multiplying $\Delta {\bar{n}_{meas}}$ in Eq. (9) is a monotonic function of BPDA-TFMB content, the ratio of the observed to ideal density has a qualitatively similar composition dependence to that of $\Delta {\bar{n}_{meas}}$. This is shown in Fig. 4. However, the same factor in Eq. (9) implies that, with increasing BPDA-TFMB content, the deviations from ideal predictions become more pronounced for density than for index. This explains the difference between the two sets of results plotted in Fig. 4.
As shown in Fig. 4, the maximum difference of the observed density from the ideal mixture prediction is more than 0.5%, with the peaks in deviation corresponding to dilute copolymers. Such density changes may arise because copolymer properties are sensitive to the sequence of the monomer components [32]. In particular, an occasional dissimilar monomer unit in a polymer chain allows the copolymer to pack more tightly. In dilute copolymers, even ∼10% of an additive breaks up a long, uniform chain sequence into shorter segments of monomer units which are linked by different bonds. This additional flexibility can enable closer packing. Furthermore, X-ray investigations of polyimides have found that the formation of a copolyimide can also increase the correlation length, which represents the average distance between entanglements [33]. This is consistent with a more efficient copolymer packing upon the initial introduction of a new monomer in the chain. At concentrations above ∼10%, the new monomer has reduced impact on the correlation length. Near the middle of the composition range, where all monomers contribute significantly, the density deviation from ideality is reduced.
We note that the observed density is always greater than the ideal prediction in the present study, implying a more efficient packing of the copolyimide chains than predicted. In the BPDA-(TFMB/DADP) copolyimides, both TFMB and DADP are relatively rigid, and an increase in density results from a small addition of either component. By contrast, in BPDA-(TFMB/mBAPS) copolyimides, the mBAPS structure is relatively flexible, and a small addition of the more rigid TFMB units does not impact the density as strongly.
The copolyimides studied here consist of two similar, essentially amorphous, polyimides. Greater deviations from ideal mixture behavior is expected for component polyimides with a degree of crystallinity. For example, the refractive index of (PMDA/6FDA)-TFDB copolyimides has been reported to exhibit an unambiguously nonlinear composition dependence [15,34]. The PMDA-TFDB homopolyimide typically has localized quasi-liquid-crystalline domains [35]. Within these domains, the PMDA-TFDB segments form relatively tightly-packed regions of higher density. In this case, the introduction of 6FDA-TFDB segments into PMDA-TFDB chains disrupts the packing of these close-packed domains [36], causing a significant reduction in density relative to the ideal mixture prediction.
7. Birefringence in copolyimides
As mentioned earlier, polyimide films are generally uniaxial. Although the composition dependence of the average refractive index is of primary concern in GRIN optical design, birefringence can also affect the phase of the transmitted wavefront in optical systems. Therefore, it is of interest to consider the dependence of birefringence on the copolyimide composition. In particular, variations of birefringence with composition can provide insights into the effects of copolymerization on the molecular orientation.
An extension of the Lorentz-Lorenz equation (Eq. (6)) due to Vuks [37] has been shown to describe the refractive indices corresponding to the principal axes of an anisotropic medium. According to the Vuks description, each principal refractive index, denoted by $\xi $, is related to the associated polarizability component by
Fig. 5. Birefringence (${n_{TE}} - {n_{TM}}$) as a function of BPDA-TFMB volume fraction, at 25 C. Symbols represent measurements and solid lines are the ideal mixture prediction. Solid symbols (blue): BPDA-(TFMB/DADP). Open symbols (red): BPDA-(TFMB/mBAPS). The dotted lines are guides for the eye.
We further compare the observed principal polarizability components of the copolyimides to the ideal mixture predictions, based on Eq. (10), by calculating
Using Eq. (8) for ${\rho _{AB,meas}}/{\rho _{AB,ideal}}$ in Eq. (12), this simplifies to
The ratio of the observed principal polarizability components to the ideal mixture values is shown in Fig. 6. We note that the effect of copolymerization on polarizability is distinct from the effect on density, shown in Fig. 4.
Fig. 6. Ratio of observed to ideal polarizability, ${\alpha _{meas}}/{\alpha _{ideal}}$, as a function of BPDA-TFMB volume fraction, at 25 C, for (a) BPDA-(TFMB/DADP), and (b) BPDA-(TFMB/mBAPS). Solid symbols (blue): $TE$ polarization. Open symbols (red): $TM$ polarization. The lines are guides for the eye.
We note that by combining the Vuks and Lorentz-Lorenz equations (Eqs. (6) and (10)), the ratio of each principal polarizability component, ${\alpha _\xi }$, to the average polarizability, $\bar{\alpha }$, can be written as
Under the earlier assumption that the average polarizability is unchanged during copolymerization, Eq. (14) is clearly consistent with Eq. (13).
Figure 6 reveals changes in the orientation of the average polarizability as a result of copolymerization, representing additional deviations from ideal mixture behavior. When ${\alpha _{meas}}/{\alpha _{ideal}} = 1$, the polarizability maintains the orientation predicted for the ideal mixture. While the $TE$ and $TM$ polarizability components always have opposite deviations relative to the ideal mixture behavior, as expected in order to maintain the magnitude of the average polarizability, the sign of the deviations is different for the two sets of copolyimides studied here. For BPDA-(TFMB/DADP), the increase in the $TE$ component corresponds to greater ordering in the in-plane direction as a result of copolymerization. This is associated with the relatively rigid TFMB and DADP components. By contrast, the more flexible mBAPS component in BPDA-(TFMB/mBAPS) causes additional disorder, which decreases the in-plane polarizability. These polarizability deviations are consistent with the results in Fig. 5, where BPDA-(TFMB/DADP) copolyimides show increased birefringence, while BPDA-(TFMB/mBAPS) copolyimides show decreased birefringence, relative to an ideal mixture.
Interestingly, the composition dependence associated with the birefringence and polarizability, in Figs. 5 and 6, is qualitatively different from the composition dependence of density in Fig. 4. In particular, while the deviations of density from the ideal mixture behavior are greatest in dilute copolyimides, the effects on birefringence and polarizability tend to be higher at more balanced compositions, especially for copolyimides that include the relatively flexible mBAPS component. This suggests that different interaction mechanisms are responsible for the effects on birefringence and polarizability, as compared to density, during copolymerization.
8. Thermo-optic coefficient of copolyimides
The refractive index measurements in Fig. 2 also enable calculations of the thermo-optic coefficient, $d\bar{n}/dT$ (corresponding to the average refractive index, $\bar{n}$, defined in Eq. (2)) and hence of the volume coefficient of thermal expansion, $\beta = ({1/V} )({dV/dT} )$. Both of these thermal effects play an important role in optical applications, as discussed previously [5]. Minimizing these coefficients increases the temperature range of acceptable optical performance for devices fabricated from the corresponding materials. A low $\beta $ can also minimize reliability concerns due to thermal mismatch among components in a typical system or device comprising multiple materials.
The measured thermo-optic coefficients for the two sets of copolyimides investigated here, obtained from the data in Fig. 2, are shown in Fig. 7. The $d\bar{n}/dT$ values of these copolyimides generally vary linearly with composition, within the experimental uncertainty. We note that the Lorentz-Lorenz model implies a relation between $d\bar{n}/dT$ and $\beta $, namely
Fig. 7. Thermo-optic coefficients corresponding to the data in Fig. 2, for the two sets of copolyimides in this study: (a) BPDA-(TFMB/DADP), and (b) BPDA-(TFMB/mBAPS). The lines indicate linear fits to the results.
This equation assumes negligible contributions from a temperature-dependent polarizability, although such effects can be included [38].
9. Summary
In order to examine whether copolymerization is a practical approach for controlling the refractive index of optical polyimides in GRIN optics, we have investigated two sets of copolyimides, BPDA-(TFMB/DADP) and BPDA-(TFMB/mBAPS). We have measured the composition dependence of the refractive index in these copolyimides with an accuracy of ±4 × 10−4, a level appropriate for state-of-the-art layered polymer GRIN optics. The measured indices deviated from the predictions for an ideal mixture, based on either a linear or Lorentz-Lorenz model, by about ±5 × 10−3 at certain compositions, consistent with non-ideal copolymer formation. These deviations are an order of magnitude higher than the measurement uncertainty, and imply that predicted index values are generally not sufficiently accurate for GRIN optical designs based on these copolyimides. This is somewhat surprising since these copolyimides consist of two similar, essentially amorphous, polyimide components. Even under these favorable conditions, we find that controlling the refractive index at the 10−4 level, as required for many GRIN optical designs, must rely on a specific calibration of index vs composition, rather than on model predictions. Furthermore, in cases where the components are more susceptible to non-ideal mixing, as for example when one component is quasi-crystalline, such a calibration will be indispensable.
Based on the Lorentz-Lorenz model, we interpreted the deviations of the refractive index from the ideal mixture prediction in terms of density changes with composition, which we found to be more pronounced in dilute copolyimides. This behavior is typical of slightly non-ideal mixtures. In both sets of copolyimides investigated here, interactions resulted in closer packing between the components, thereby increasing the density. Greater deviations from the ideal mixture prediction were observed for the relatively rigid components, BPDA-TFMB and BPDA-DADP, than for the relatively flexible BPDA-mBAPS component.
We have also identified composition-dependent variations in the birefringence of these anisotropic materials, which we interpreted within the context of an extension of the Lorentz-Lorenz model due to Vuks. In the case of copolyimides consisting of relatively rigid components, namely BPDA-TFMB and BPDA-DADP, interactions between the components increase the birefringence by enhancing the in-plane order. By contrast, the inclusion of a relatively flexible component, BPDA-mBAPS, decreases the birefringence by leading to increased disorder. We note that the composition dependence of the birefringence deviations from the ideal mixture model differs from that of the density deviations, implying that more than one interaction mechanism is responsible for these effects.
Funding
Office of Naval Research.
Acknowledgments
We acknowledge the assistance of Drs. Kevin Anderson and James Wollmershauser with the density measurements.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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