1. Introduction

Optically driven diffusion in photopolymer media is an appealing material platform for holography [1], Bragg filters [2], waveguides and optical circuits [3, 4], gradient index lenses [5, 6], and diffractive optical elements [7]. It affords high refractive index contrasts and high sensitivity, in materials that are low cost, self-processing, widely tunable and environmentally robust [8].

Two general design strategies exist for these materials. In the single-chemistry strategy [9] that has been extensively studied by Sheridan and Gleeson [10], a liquid monomer is partially and uniformly converted to a polymer gel, followed by non-uniform illumination that records features in the remaining freely-diffusing monomer. In the two-chemistry strategy, a solid host matrix is formed by a first (typically thermoset) polymer, followed by recording into a second (typically radical) photopolymer [11]. This two-chemistry approach is increasingly preferred in commercial holographic media [12] due to its additional design freedoms: the matrix polymer that dominates passive properties (e.g. modulus or phase flatness) can be engineered independently from the writing polymer that dominates the recording properties (e.g. refractive index contrast or scatter).

Despite sustained academic [13] and commercial [14] interest in two-chemistry media, their kinetics have not been studied in depth. A quantitative understanding of reaction/diffusion kinetics is necessary in order to fully exploit the additional design freedoms of these materials. However, these kinetics are difficult to study directly. Conventional metrology such as FTIR is of limited utility here, since recorded features are typically weak (< 1% fractional conversion) and spatially small (submicron grating pitch).

Instead, the preferred measurement technique is Bragg grating diffraction, since this yields a strong signal even from weak, small features [15]. However, a limitation of this technique is that it probes only the fundamental spatial harmonic of the refractive index response, and not the spatially uniform or higher harmonic components. These recording errors arise from radical sub-linear response [16], overlapping time scales of reaction and diffusion [17], and/or local depletion of species. Bragg diffraction can only measure these effects indirectly, by simultaneously fitting all model parameters to a single measured quantity, the diffraction efficiency [18]. This approach thus provides only indirect insight into the reaction/diffusion dynamics, which in turn limits the ability to rationally design such materials.

We demonstrate an alternative “divide-and-conquer” strategy for quantifying such reaction/diffusion kinetics, in which the critical processes occur on separated timescales and are measured with independent experimental methods. We apply this strategy to a model material similar to commercial two-chemistry media. This enables prediction of a material’s potential refractive index contrast solely on the basis of its chemical components. It also reveals that refractive index formation depends crucially on matrix displacement, a mechanism which was neglected in the foundational work of Colburn and Haines [19] but which explains the result reported by Dhar et al [20] that achievable refractive index modulation depends on the matrix bulk index. Finally, this strategy produces a reaction/diffusion model that quantitatively predicts recorded index features, with no additional adjustable parameters. This model is experimentally validated over a range of exposure doses (~1 to ~10³ mJ cm⁻²) and feature sizes (0.35 to 500 μm). Insights from this reaction/diffusion model are shown to enable rational design of new materials.

2. Model material formulation

This strategy is demonstrated in a model material (Table 1) similar to commercial two-chemistry media [7]. The total usable peak-to-mean refractive index modulation is Δn = 1.5 × 10⁻³, via the standard technique of measuring the Bragg diffraction from many angle-multiplexed weak holograms [21]. The recording-induced volume shrinkage is 0.5%, via the standard slanted-grating technique [22]. The glass transition temperature T_G is ~20 °C via dynamic mechanical analysis; see supporting information.

Table 1. Model material formulation. Components are used as received, except TBPA which is purified by dissolving in methylene dichloride and filtering with a Millipore 0.5 micron pore membrane filter. Components 3-6 are mixed into the polyol at 60 °C, degassed, then mixed with isocyanate and cast between glass slides. For Fig. 1, initial concentration of writing monomer is varied from 0% to 6%, and writing monomer is later added or removed from the gelled matrix via bulk diffusion.

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3. Refractive index formation mechanism

Existing reaction/diffusion models typically infer the writing monomer concentration profile indirectly, from the holographically measured refractive index modulation [6]. This provides no direct understanding of the mechanism of index formation in these materials. In particular, it does not reveal whether the host matrix undergoes perfect 1:1 volume displacement by in-diffusing writing monomer, as a binder would in a single-chemistry system [16], or undergoes less than 1:1 volume displacement resulting in net densification, as observed in some two-chemistry systems [24].

In order to measure this displacement non-holographically, we construct a series of samples with varying concentrations of writing monomer, which has been introduced in one of two ways: either via mixing it with the liquid matrix components before thermoset, in which case nearly perfect volume displacement is guaranteed; or via bulk diffusion into an already thermoset matrix, in which case some densification is possible. This bulk diffusion is induced independently of any photo-process, simply by laminating together two thin media samples with different initial writing monomer concentrations, so that writing monomer diffuses from one into the other, causing some change in sample volume.

This strategy is demonstrated in a model material (Table 1), that the measured relationship between bulk refractive index and writing polymer concentration is the same in both cases, and thus the covalently cross-linked gelled matrix must be exhibiting the same 1:1 volume displacement as the liquid matrix components.

 

Fig. 1 Bulk refractive index as a function of concentration of polymerized writing monomer, TBPA. Writing monomer is either dissolved into the liquid resin, or diffused into the thermoset matrix, with the in-diffusion process monitored gravimetrically. In either case, the sample is then flood-exposed and allowed to rest for >3 days to ensure complete polymerization; then bulk index is measured via Metricon 2010 prism coupler. The same refractive index relation holds in both cases: dashed line is a good (R² = 0.992) linear fit to all data points. Therefore the solid matrix must be exhibiting nearly perfect volume displacement, just as the liquid resin is.

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Figure 1 not only demonstrates 1:1 volume displacement of matrix by writing polymer, it also establishes an analytical relationship between polymer concentration and refractive index. This relationship can be rewritten in terms of the component refractive indices as follows. Given 1:1 volume displacement, and in the limit where writing polymer is dilute (here <10 wt%) and low index contrast with the matrix (here <10% fractional contrast), the Lorentz-Lorenz relation reduces to [23]:

In the preceding experiment, 1:1 matrix displacement is inferred indirectly, from measurements of mass transport and refractive index. Next, this result is corroborated by direct measurement of matrix and writing polymer concentrations, as follows. A large (500 μm) feature is exposed with a dose sufficient to polymerize 100% of the writing monomer at the peak. After sufficient diffusion time and a final flood exposure, the concentrations of both writing polymer and urethane matrix polymer are measured via scanning confocal Raman spectroscopy [24]. Figures 2(a) and 2(b) show that the crosslinked urethane matrix is displaced by in-diffusing writing monomer even at this macroscopic scale. The small relative decrease in urethane concentration is challenging to accurately quantify, explaining why the curves in (b) are not mirror images (see supplementary information). However, the directly measured writing polymer concentration agrees with the concentration inferred from refractive index (c) on the assumption of 1:1 matrix exclusion. The agreement between these two independent measurements of concentration profile constitutes a rigorous validation of the analytical relationship obtained in Eq. (1).

 

Fig. 2 (a) Confocal Raman microspectroscopy yields concentration profiles of TBPA writing polymer, top, and urethane matrix polymer, bottom, within a gradient-index feature. The local densification of writing polymer causes a corresponding displacement of the host matrix, even over large scales (scale bars = 200 μm). (b) Spatial averages of those concentration profiles. (c) Writing polymer profile measured as a refractive index via phase-sensitive imaging and converted via Fig. 1 to concentration. This is compared to the confocal Raman result from (b), and to the prediction of the reaction/diffusion model discussed below.

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4. “Formula limit” on dynamic range

This analytical relationship between refractive index and polymer concentration enables precise measurement of multiple classes of oligomer through standard refractive index measurements. To begin, Eq. (1) allows us to calculate the “formula limit”, defined as the maximum achievable holographic refractive index modulation Δn, only reached if 100% of the writing monomer is polymerized with perfect spatial fidelity. For the standard case [19] of sufficiently many weak superposed angle-multiplexed gratings to consume all writing monomer, the total Δn_formula is given by (n_{writing polymer} – n_matrix) φ_{writing polymer} = 5.1 × 10⁻³, whereas the measured Δn is only a fraction of this limit, as shown in Fig. 3.The wasted remainder must correspond to the fraction of polymerized writing monomer that is not patterned at the holographic spatial frequency. The possibility of patterning at higher spatial frequencies, due to local depletion of monomer, can be ruled out in the case of weak gratings [15]. Therefore the wasted remainder must be uniformly distributed.

 

Fig. 3 (a) Holographically usable peak-to-mean refractive index modulation Δn, as a function of grating pitch Λ, compared to the ideal “formula limit” calculated from Fig. 1. For multiple weak exposures, this formula limit is simply Δn = n(6% TBPA) – n(0% TBPA), where 6% is the initial uniform TBPA concentration. (b) Distribution of polymerized TBPA at Λ = 0.70 μm and Λ = 0.35 μm, drawn to match the Δn scale in (a). For any exposure dose, three categories of writing oligomer are generated, in a ratio determined by Λ: (i) attached oligomer that is holographically patterned, (ii) attached oligomer that is spatially uniform, and (iii) mobile oligomer, which diffuses to become spatially uniform.

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The magnitude and resolution of this usable refractive index response will be shown to depend crucially on the processes of immobilization and re-mobilization of radicals. Immobilization is typically attributed to reduced diffusivity of long chains [11]; however, this is inconsistent with theory. The writing monomer forms non-branching chains, since typical monomers including TBPA are mono-functional to minimize shrinkage-inducing functional groups per high-refractive-index group [18]. Diffusivity of linear chains in a matrix is predicted to decrease, at the most extreme, only as the square of chain length [25], resulting in a predicted hologram lifetime of only hours for chains up to 100 monomer units long. We observe lifetimes of at least 6 months, and decades have been reported [7]; thus, immobility must be due to bonding to the host matrix and not to the reduced diffusivity of long-chained oligomers.

However, even though the final hologram is composed entirely of matrix-attached species, this must be preceded by significant diffusion of mobile growing radical chains before they attach to the matrix, since the usable holographic Δn (Fig. 3, quantity i) exhibits the 1/Λ² scaling that is characteristic of diffusional blurring. This blurring leads to a spatially uniform component in the attached oligomer profile (quantity ii). This is surprising, because reaction/diffusion models typically [26] assume that mobile radicals become immobilized so quickly that they can be neglected (although exceptions exist [27]). (It should be noted that reaction-diffusion can, in general, produce the same 1/Λ² scaling, but is not a plausible mechanism here since the average chain length (<10 units from stoichiometric arguments) is small relative to the scale of the blurring (characteristic 1/e distance of 0.12 μm).

This blurring causes a fundamental media resolution limit, readily found by extrapolating the 1/Λ² scaling to the point at which the holographic component of the material response goes to zero. At this scale, diffusion of radicals results in a uniform polymer distribution and thus no measurable holographic response. This is noteworthy because no consensus exists in the literature on what causes media resolution limits in general (compare, for instance, references [16] and [28]). In the model material, this extrapolation yields a resolution limit of Λ = ~300 nm, consistent with our observation that the material does not support reflection holograms at wavelengths up to 532 nm, corresponding to Λ ~180 nm .

Extrapolating instead to large feature size (1/Λ² → 0) reveals that mobile terminated oligomer must have a large effect on Δn. Figure 3 shows that holographically patterned Δn corresponds to at best only ~30% of the polymerized TBPA even when feature size is so large that polymerization must terminate before significant diffusion. Thus the remaining ~70% (quantity iii) must remain as mobile unreactive oligomer, diffusing into a uniform distribution invisible to Bragg diffraction but visible to the prism coupler used in Fig. 1. This large fraction of mobile oligomer is consistent with the existence of attached oligomer (Fig. 3, quantities i and ii), for the following reason. The attachment of radicals to the host matrix most plausibly occurs via hydrogen abstraction, a process which terminates the short live oligomer chain but leaves it still mobile. Accounting for this mobile dead oligomer allows us to correctly predict the usable Δn, and in particular the previous phase imaging result at Λ = 500 μm, shown in Fig. 2(c).

Finally, re-mobilization and spatial equilibration of attached radicals causes the eventual cessation of hologram growth. This cessation is typically ascribed instead to radical termination [14]; however, we find in this material that unconsumed monomer (measured as above) continues to be depleted according to the polymerization rate constant, long after the growth of the sinusoidal refractive index structure revealed by Bragg diffraction has ceased. We conclude that radicals are still present after cessation of growth, but are blurred by diffusion into a uniform distribution so that the continued polymerization is invisible to Bragg detection. Since this is observed even for Λ as large as 5 μm, the blurring mechanism cannot be reaction diffusion. Most plausibly, a chain transfer reaction terminates a propagating oligomer chain, resulting in an immobile terminated oligomer and a new mobile radical. Thus, reaction/diffusion kinetics not only cause the usable Δn to fall short of the formula limit, but the fraction of monomer not used in reaching this limit undergoes unwanted reactions that restrict both resolution (attached oligomer) and environmental stability (mobile oligomer).

5. Reaction/diffusion model

As shown in the preceding section, the set of reaction paths (Fig. 4) must include not only polymerization, but also immobilization and re-mobilization of radicals. These most plausibly occur via hydrogen abstraction and chain transfer respectively, but for the sake of generality we give only a phenomenological description in terms of species balance.

 

Fig. 4 Species balance description of the set of reaction paths. As shown above, this set must be extended to separately track all of the following species: unconsumed photoinitiator PI; unreacted monomer M; mobile radicals R_mob and matrix-attached radicals R_fix (attached to chains of any length); and polymerized chain units P_mob and P_fix, belonging respectively to mobile or matrix-attached chains of any length.

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In order to describe this set of reaction paths with a tractably small set of coupled reaction/diffusion equations (Table 2), the following simplifying assumptions are made. First, we assume that local polymerization of writing monomer has no effect on any rate constants, including diffusivity (that is, the diffusional decay parameter α = 0, in the Sheridan and Gleeson notation). This assumption is reasonable due to the low concentration of writing monomer and the fact that it forms short, non-branching chains, in striking contrast to single-chemistry systems in which photo-polymerization significantly increases the crosslink density [29]. Furthermore, this assumption is experimentally validated by dynamical mechanical analysis showing that even complete polymerization of writing monomer via floodcure only changes T_G by ~4°C. (See supplementary information.) This assumption also rules out the possibility of residual unreacted monomer due to reduced local diffusivity.

Table 2. Reaction/diffusion equations based on the reaction paths in Fig. 4. I is the exposure intensity and 2ε is the radical quantum yield of the photoinitiator. For generality, both hydrogen abstraction and chain transfer are treated as first-order processes, rather than explicitly tracking the concentrations of matrix abstraction sites and of chain transfer agents.

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Second, we assume that oligomers of all chain lengths can be described by a single set of molecular-weight-averaged rate constants. This neglects, for instance, chain-length-dependent diffusivity, which is reasonable since the average chain length is stoichiometrically expected to be <10 units, consistent with gel permeation chromatography results (see supporting information). Third, we assume that, both primary initiation and chain propagation can be described by a single rate constant, K_p. The K_p, as measured below, is five orders of magnitude smaller than the reaction-limited polymerization rate constant for typical acrylate polymerizations [30]. Therefore, the polymerization rate must be diffusion-limited and can be modeled with a single rate constant because the mobility of the short propagating chains is similar to primary radicals.

Finally, we explicitly neglect bimolecular termination, since it is ruled out by the observed linearity of material response with exposure dose and intensity, discussed below when measuring K_p. Physically, this is attributed to the fact that mobile radicals quickly become matrix-attached, so that bimolecular termination can only occur via the much slower mechanism of reaction diffusion. The observed linear response is, however, consistent with the presence of a unimolecular termination mechanism, such as less-than-unity efficiency of chain transfer re-initiation. But this unimolecular termination, if present, is sufficiently weak that its effect is not measurable over the order 10³ sec timescale of these experiments. Under these assumptions, the model is characterized by five rate constants, each independently measured and verified as described next.

The rate coefficient for photolysis of photoinitiator is calculated ab initio from the reported absorption cross-section [31] and quantum yield [32]. For verification, the calculated value K_d = 1.1 × 10⁻³ cm² mJ⁻¹ is used to predict the exposure dose needed to overcome oxygen inhibition and found to be within 10% of the measured value, based on an assumed equilibrium O₂ concentration of 1 × 10⁻³ M [33].

The diffusivity of monomer and small oligomers, D, is measured to be 1.6 × 10⁻³ μm²/sec by recording a large (>500 μm) feature and fitting the in-diffusion of new TBPA monomer (measured with phase-sensitive imaging [34]) to a simple Fickian diffusion model. This value is corroborated by measuring fluorescence recovery after photobleaching (FRAP) [35] using rhodamine 101, a fluorophore with similar molecular weight to TBPA monomer (491 and 385, respectively), which yields a diffusivity of D = 1.5 × 10⁻³ μm²/sec.

The polymerization rate coefficient, K_p = 0.21 M⁻¹ sec⁻¹, is measured by monitoring the diffraction of a Bragg hologram in the dark after a brief initial exposure. Diffraction efficiency is converted to refractive index modulation [36] and then, via Eq. (1), to concentration of polymerized TBPA. To ensure independent measurement of K_p, the time scales of diffusion, polymerization and apparent termination are separated in this material: the characteristic diffusion time is only ~10 sec, whereas the grating continues to grow at a nearly constant rate for ~300 sec before decelerating.

K_p is expected to be independent of exposure intensity or total dose, since the proposed reactions structure (Table 2) does not include bimolecular termination. As a validation of Table 2, measured values of K_p are found to be nearly constant (to within R² = 0.994) for exposure doses ranging over three orders of magnitude, ~1 to ~10³ mJ cm⁻², reached by varying exposure intensity from 5 mW cm⁻² to 100 mW cm⁻², and concurrently varying exposure time from 200 msec to 10 sec.

The rate of matrix attachment, K_a = 3.5 × 10⁻³ sec⁻¹, is found from the slope of the 1/Λ² line in Fig. 3. Assuming Fickian diffusion, we find that radicals must diffuse a characteristic (1/e) distance r = 0.12 μm before attaching to the matrix, in order to account for the observed 1/Λ² slope. This is then converted, via the measured diffusivity D, to an attachment rate coefficient K_a = D/r².

The chain transfer rate, K_ct = 3.5 × 10⁻³ sec⁻¹, is found as a single fit parameter to the long-term shape of grating growth curves, because, as discussed above, chain transfer is responsible for the deceleration of grating growth. This occurs as radicals equilibrate to a uniform distribution, via a process of chain-transfer to a mobile state, diffusion, and then reattachment. Deceleration occurs on a timescale of order 10³ sec, long enough to be well separated from both diffusion and polymerization. One consequence of this diffusion-mediated grating deceleration mechanism is the consumption of monomer even after cessation of grating growth, mentioned previously. Another unusual consequence is that the deceleration of grating growth is slower for larger Λ. As shown next, the model accurately predicts this slowing with the single set of rate constants derived through the independent experiments described above.

6. Model validation

To validate the model, we first show in Fig. 5(a) that the rate constants found above accurately predict material response without additional adjustable parameters, over a wide range of conditions, including varying grating pitch, exposure intensity, and exposure time. These predictions continue to be accurate even at spatial scales ~10³ times larger, as shown in Fig. 2(c). Finally, a more rigorous test of such a model is to predict the loss of recording fidelity that arises when reaction/diffusion coupling becomes strong [15]. This coupling occurs at large grating pitch (Λ = 5 μm) so that the timescales of reaction and diffusion are mixed, and large exposure dose (600 mJ cm⁻²) so there is significant depletion of writing monomer. To quantify fidelity, we monitor the second-order diffracted beam, which is visible for sufficiently thin gratings (here 100 μm) and which reveals the weak second harmonic of the grating caused by material nonlinearity [37]. The model successfully predicts the growth curves of both the fundamental and second-harmonic components, as demonstrated by Fig. 5(b), over a range of 10² in refractive index and thus 10⁴ in diffraction efficiency. This accurate prediction of nonlinear material response due to coupled dynamics serves as a validation of both the proposed reactions structure (Table 2) and the process of extracting individual rate constants in uncoupled experiments.

 

Fig. 5 a) Grating growth curves measured by Bragg diffraction, compared to predictions from the reaction/diffusion model without introduction of any additional fit parameters. Note that the apparent termination rate varies with Λ, since grating termination is actually governed by diffusional mechanisms that are successfully captured by the model. (i) Λ = 0.7 μm, exposure intensity I = 100 mW cm⁻², exposure time t = 3 sec, ii) Λ = 5 μm, I = 100 mW cm⁻², t = 1 sec, (iii) Λ = 0.7 μm, I = 10 mW cm⁻², t = 3 sec. b) Growth curves of a grating’s fundamental and second spatial harmonics, when reaction and diffusion are strongly coupled. (Λ = 5 μm, I = 60 mW cm⁻², t = 10 sec).

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7. Implications for the design of holographic media

The “divide and conquer” approach proposed here can validate and extend existing simultaneous fitting approaches by revealing the dominant reaction paths and by independently measuring critical rate constants. Although this approach of independent measurements is particularly facile in the high T_G model material where the timescales of reaction and diffusion are well separated, it leads to fundamental physical insights that are applicable even to holographic media with overlapping timescales. These insights enable rational material design, as we now discuss.

First, it is critical that the coupled material dynamics translate optical intensity into refractive index distribution with high fidelity. The model material exhibits an index response that is linear in exposure dose ranging from ~1 to ~10³ mJ cm⁻², and similar results have been reported in commercial media [11]. This is unexpected since radical photopolymerizations generally exhibit sublinear index response due to bimolecular termination. The model quantitatively predicts this linear response and explains the underlying physics: the only species that contributes to the steady-state holographic pattern is matrix-tethered oligomer for which bimolecular termination is strongly suppressed. The resulting linear response is highly advantageous for recording, since it enables faster and brighter exposures with no sensitivity penalty and reduced coupling effects between successive exposures.

Second, the model provides guidance for the requirements on the matrix polymer. To enable a large formula limit, the refractive index of the matrix must be significantly different (typically lower) than that of the writing monomer. Refractive index is typically reduced for low T_G. Additionally, elasticity of the matrix is revealed to be an important parameter, since writing monomer will only diffuse if matrix can adequately counter-diffuse [10]. These two facts, along with improved sensitivity [12], reveal why typical matrices have T_G well below room temperature.

Third, the potential refractive index modulation Δn of a material is of utmost importance in all applications. The formula limit introduced here predicts the potential Δn of candidate media formulations based only on chemical composition. The degree to which a material does not reach this limit indicates the fraction of monomer that has participated in unwanted reactions.

This unwanted diffusion of reactive and unreactive mobile oligomers into the unexposed regions is revealed to also be the limiting factor on spatial resolution. The degree of this unwanted diffusion is governed by the ratio of a mobile radical’s characteristic lifetime versus its characteristic time to diffuse across the recorded feature. A set of design strategies are immediately suggested by this ratio. For instance, increasing the concentration of matrix attachment sites should shorten the lifetime of mobile radicals, which will favorably shift this ratio, consequently increasing usable Δn, improving the maximum spatial resolution and minimizing mobile species that could impact lifetime. Such a guided strategy has clear advantages over blind exploration in the large space of potential formulas.

Appendix

1. Dynamic mechanical analysis of model material

The glass transition temperature (T_g) of the photopolymer formulation with 0 and 100% writing monomer conversion to polymer was determined by dynamic mechanical analysis (DMA). Both samples were made by the synthesis procedure as described in the Model Material section in the body of the manuscript. The 0% writing monomer conversion sample was measured by the DMA immediately after the material synthesis procedure. To create a sample with 100% writing monomer conversion, a sample was exposed with a mercury arc lamp with sufficient optical dose to completely consume all the TPO and polymerize all monomer in the sample. After the photocure and subsequent polymerization the sample was measured by the DMA. The T_g of photopolymer with 0 and 100% monomer conversion is shown in Fig. 6.

 

Fig. 6 The dependence of glass transition temperature (T_g) on monomer conversion. a. The measured T_g of the photopolymer formulation with no writing monomer conversion. b. The measured T_g with 100% writing monomer conversion. All measurements are at 1 Hz, and T_g is defined as the tan(δ) peak.

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The T_g only increases 4°C after 100% conversion of the writing monomer. This supports the assumption that all rate constants are independent of the degree of writing monomer conversion.

2. Calculation of n_{writing polymer}

Equation (2) gives the refractive index n as a function of the weight fraction of polymerized writing monomer p_{writing polymer}:

$$\begin{array}{l}n(p_{writingpolymer})=n_{matrix}+(n_{writingpolymer}-n_{matrix})({\rho }_{total}/{\rho }_{writingpolymer})p_{writingpolymer}\\ n(p_{writingpolymer})=n_{matrix}+\kappa p_{writingpolymer}\end{array}$$

Here κ = 0.085 is determined phenomenologically from the linear fit shown in Fig. 1. However, it is also of interest to extract a value for the more physically fundamental quantity $n_{writingpolymer}$, the refractive index of solute polymerized writing monomer. This quantity is not readily measurable by other means, and is distinct from the index of solute unreacted writing monomer, or dry crystalline writing monomer.

All the other terms in κ are measured as follows. The refractive index of pure matrix $n_{matrix}$is found from the linear fit in Fig. 1. The density of solute writing polymer ${\rho }_{writingpolymer}$ = 2.5 g/mL is estimated by dissolving writing monomer in cyclohexanone and observing the volume displacement; a more precise measurement is outside the scope of this work. The density of the resin as a whole ${\rho }_{total}$ = 1.1 g/mL is then calculated using the manufacturer’s reported densities for the matrix components. This set of values fully determines $n_{writingpolymer}$ = 1.696 ± 0.002.

3. Confocal Raman spectroscopy

The refractive index formation mechanism of the photopolymer formulation is determined by chemical imaging of a large (500 μm) index features in the photopolymer, using a scanning confocal Raman microscope. The scanning Raman microscope provides a direct, quantitative measurement of the spatial modulation in the photopolymer chemical components and the mechanics of the refractive index modulation.

The large index features are written in 1 mm thick photopolymer between two glass microscope slides. The writing dose is 4 J cm⁻², which is empirically found to be sufficient to reach maximal peak index change of the structure. After 10 days of development time at 60 degrees Celsius, the samples are stabilized against further reaction by complete photocure of the sample. Images of the polymeric components of the index structure, shown in Fig. 2 of the manuscript, are built by collecting a 29x27 Raman spectra grid, spaced 26 and 21 µm respectively, where each spectrum is integrated for 60 seconds. To resolve and quantify the modulation of the two chemicals of interest, the matrix and writing polymer, distinctive Raman peaks are identified and calibrated to accurately quantify them.

To create an accurate calibration of the chemical components, uniform bulk samples of the photopolymer formulation consisting of 0, 2, 4, 6, 8, 10 and 12 percent by weight writing polymer were synthesized and cast one millimeter thick between two glass microscope slides. The samples were then stabilized by complete photocure of the sample and then measured with the scanning confocal Raman microscope. The spectra and the resulting calibration for the writing monomer of the samples is shown in Fig. 7.

 

Fig. 7 (a) Raman spectra centered at 237 (cm⁻¹) of unstructured sample with varying amounts of writing monomer. (b) The relationship between the writing monomer concentration and the integrated normalized spectra of the 237 (cm⁻¹) spectral line.

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The small relative decrease in the matrix concentration in the calibration samples makes an accurate calibration challenging to create. Therefore, to accurately image the matrix polymer, it must have a clearly distinguishable Raman peak that does not overlap with a writing polymer peak. As shown in Fig. 8, there are no peaks writing monomer spectrum between 2800 and 3000 cm⁻¹, where there is a strong matrix polymer Raman signal. The chemical image of the matrix monomer is, therefore, produced by integrating the peak from between 2810 and 3020 cm⁻¹ at each point on the spectra grid and then normalizing those spectra to spectra taken from an unstructured area.

 

Fig. 8 Confocal Raman spectrum of the matrix and writing monomer independently. (a) Measured Raman spectrum of the matrix material. (b) Raman spectrum of tribromophenol, as reported by manufacturer Sigma-Aldrich.

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4. Gel permeation chromatography measurements

To determine if propagating oligomers can be described by a single polymerization rate constant, gel permeation chromatography (GPC) was used to characterize the molecular weight distribution of unattached oligomer in a photocured sample. Samples for the GPC were prepared by soaking a 500 µm thick sample of photocured polymer in tetrahydrofluorine for 1 week at 50°C to extract all unattached oligomer chains. The molecular weight distribution as measured by the GPC is shown in Fig. 9.

 

Fig. 9 The molecular weight distribution of unattached oligomers extracted from a photocured sample of the polymer, normalized to the peak value. Dashed lines show multiples of the molecular weight of the writing monomer.

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The molecular weight distribution of unattached oligomer suggests that the oligomer distribution primarily consists of monomer. This supports our use of a single set of molecular-weight-averaged rate constants to describe all unattached oligomers.

5. Monomer recovery after photopolymerization

The diffusivity of monomer, D, is measured by recording a large (>500 μm) feature and fitting the in-diffusion of new monomer, to a simple Fickian diffusion model. The in-diffusion of new monomer is determined by measuring the refractive index profile of the sample with phase-sensitive imaging [33]. The refractive index profile of an undeveloped structure after 30 days at room temperature, long enough for significant, but not complete diffusion, is measured and compared to refractive index profiles of structures with an array of diffusion coefficients that use the previously found initial concentrations of both monomer and polymer.

Figure 10 shows that the diffusion coefficient of the model that provides best fit to the measured monomer profile is 1.6 × 10⁻³ μm²/sec. This results was corroborated by measuring the diffusivity with fluorescence recovery after photobleaching (FRAP) [34], which yields a diffusivity of D = 1.5 × 10⁻³ μm²/sec.

 

Fig. 10 Finding the diffusivity of monomer by fitting the in-diffusion of new monomer to a simple Fickian diffusion model. (a) The refractive index profile of the measured and simulated data. The green line is the simulated profile and the red line is the measured refractive index profile. (b) The dependence on the diffusion coefficient on the quality of the fit between the measure and simulated data.

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Acknowledgments

The authors acknowledge the use of experimental facilities at the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, and at the Nanoscale Characterization Facility at the University of Colorado at Boulder. This research was supported by an Intelligence Community (IC) Postdoctoral Research Fellowship and by the following grants: NSF CAREER ECCS 0954202, NSF STTR IIP-0822695, NSF EFRI 1240374, NSF IPAS 1307918, AF MURI FA9550-09-1-0677, NSF IGERT 0801680, DoED GAANN P200A120063, NSF EPAS 1307918.

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