Holographic kinetics for mixed volume gratings in gold nanoparticles doped photopolymer

Chengmingyue Li; Liangcai Cao; Qingsheng He; Guofan Jin

doi:10.1364/OE.22.005017

1. Introduction

Photopolymer has been considered as one of the most promising holographic recording material due to its unique properties and low cost [1–3]. To further improve its performances for holography, a novel polymeric nanocomposite system, the nanoparticles (NPs) doped photopolymer, has been extensively explored [4–7]. Owing to the photopolymerization-driven mutual diffusion between the NPs and monomers, the significant improved refractive index modulation has been demonstrated in different oxide NPs doped photopolymer to enhance the diffraction efficiency of photopolymer for high-density data storage devices [8–10]. Generally, the contribution of absorption modulation is neglected due to the maximum increase of 3.7% to the diffraction efficiency [11, 12].

Different from the oxide NPs, metal NPs exhibit strongly enhanced linear and nonlinear optical properties due to the coherent oscillation of surface-bound conduction electrons known as localized surface plasmon resonance [13–15]. By dispersing gold NPs into photopolymer, a plasmon-induced holographic absorption grating has been proposed to be an effective method to improve the holographic performances of photopolymer in our previous work [16]. To obtain a thorough understanding of the influence of gold NPs to the photophysical and photochemical process during holographic exposure, a quantitative model for dynamic holographic mixed gratings formation could be much beneficial.

In this work, we describe the holographic kinetics for the recorded holographic grating with both refractive index modulation and absorption modulation in gold NPs doped bulk photopolymer. A multicomponent diffusion process is evolved in the photochemical diffusion equations with the consideration of nonlocal photopolymerization. The refractive index modulation and absorption modulation are the result of spatial distribution of photoproduct and NPs, and modeled by using Lorentz-lorenz relation and Maxwell-Garnett theory, respectively. Consequently, a dynamic quantitative model is established for analyzing the temporal photophysical and photochemical process in a gold NPs doped photopolymer with both refractive index and absorption modulation. The effect of doped gold NPs to the formation of spatial distributed photoproduct and its enhanced rate of production of radicals contributed to the diffraction efficiency are analyzed based on this model. The simulation results are compared and found to agree well with the experimental results of a bulk PQ-PMMA photopolymer doped with gold NPs. Then the temporal evolution of refractive index modulation and absorption modulation are extracted from the fitting curve of the diffraction efficiency through the above simulation model. The extracted temporal evolution of refractive index modulation and absorption modulation confirm the theoretical predictions about the active effect of the doped gold NPs during holographic exposure.

2. Holographic kinetics in gold NPs doped photopolymer

2.1. Mechanism of mixed gratings

Under the illumination of holographic interference pattern, the photoinitiator system in the bright regions absorbs the photons and initiates a chain reaction of photopolymerizable monomer molecules. This photopolymerization process lowers the chemical potential of monomers in the bright regions. The balance of the thermo-dynamical equilibrium between the bright and the dark regions is broken. The monomers experience the diffusion from the dark to the bright regions [17]. In the nanocomposite material, the NPs experience counter-diffusion from the bright to the dark regions to reduce the chemical potential difference since the NPs are not consumed in the bright regions [18, 19]. When the monomers are exhausted by the polymerization, the spatially periodic distributions of both photoproduct and NPs are formed.

In the polymeric matrix of our previous work, the photopolymer contains methyl methacrylate (MMA) as monomer, 2, 2-azobis (2-methlpropionitrile) (AIBN) as thermal initiator and phenanthrenquinone (PQ) as photosensitizer. It can be fabricated to be a thick coupon with the thermo-polymerization method. While most MMA molecules are polymerized to form the prepolymer poly(methyl methacrylate) (PMMA) as the host matrix, a small percentage (~10%) of MMA molecules remain unpolymerized [20] and are uniformly distributed in the material before the exposure. During the holographic exposure, the PQ molecules are activated and attached with PMMA polymer matrix or MMA monomers to become the product [21]. As the PQ and MMA are both small molecules, they experience the diffusion from the dark to the bright regions due to the chemical potential gradient, which also drive the gold NPs diffusing from the bright to the dark regions [22]. This multicomponent diffusion process driven by the photopolymerization process of PQ with MMA or PQ with prepolymer PMMA is shown in Fig. 1(a). It should be noted that the prepolymer PMMA could not diffuse during the exposure for the great molecular weight. Eventually, the refractive index difference between the photoproduct and polymeric matrix results in the holographic refractive index grating; the absorption distinction between the dark regions with assembled gold NPs and the bright regions with decreased concentration of gold NPs leads to the holographic absorption grating. Figure 1(b) shows the schematic graph of holographic mixed gratings formation in gold NPs doped photopolymer [16].

Fig. 1 (a) Holographic multicomponent diffusion process in gold NPs doped PQ-PMMA photopolymer; (b) Schematic graph for the formation of holographic mixed gratings.

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2.2. Kinetic model for photochemical and photophysical process

To represent the photochemical and photophysical process in the gold NPs doped PQ-PMMA, a model for the kinetics of photopolymerization and multicomponent diffusion process is required. In the gold NPs doped PQ-PMMA, the photopolymerization process is composed of excited PQ molecules reacted with residual MMA molecules or PMMA prepolymer. Both of the photoproducts contribute to the change of refractive index. In the previous research of PQ-PMMA photopolymer material, the kinetic parameters for these two photopolymerization processes are considered as irrelevant variables with each other [22]. Similarly, the multicomponent process in the gold NPs doped PQ-PMMA photopolymer consist of PQ, residual MMA and gold NPs can be considered as two independent mutual diffusion processes. The physical mechanism of the spatial transfer of multicomponent can start with the volume conservation law. As the shrinkage is small enough to be neglected in PQ-PMMA polymer [3, 12], the volume fraction normalization condition is written as:

M_{P Q} + M_{M M A} + M_{N P s} + M_{P} + M_{B} = 1,

where M_i is the volume fraction of the i component. The subscripts of PQ, MMA, NPs, P and B represent the PQ, the MMA molecules, the NPs, the photoproducts and the immobile background, respectively. To satisfy the law of volume conservation, the condition of the total matter flow for the diffusion system is required, i.e.

j^{P Q} + j^{N P s} = 0

for the diffusion system of PQ and gold NPs. Here j^PQ and j^NPs are the flow densities of PQ and gold NPs, respectively. Taking into account the Fick’s first law

j = - D C_{i} (\partial φ_{i} / \partial x)

[23], which describe the diffusive flux j to the concentration C_i with driving force of chemical potential

φ_{i}

at the steady state, the flow density of the PQ and gold NPs mutual diffusion system can be depicted as

j_{P Q, N P s} = D_{P Q - A u} [[P Q] (x, t) \nabla [A u] (x, t) - [A u] (x, t) \nabla [P Q] (x, t)],

where D_PQ-Au is the diffusion coefficient of the PQ-Au NPs mutual diffusion system. [PQ] and [Au] are the concentration of PQ molecules and gold NPs, respectively. It is obviously that the driving force of the mutual diffusion for one component is the concentration gradient of the other.

Generally, the diffusion equation can be derived in a straightforward way by substituting the above flux j of the diffusion system into the continuity equation $\frac{\partial C_{i}}{\partial t} = \nabla \cdot j_{i} + G$ , in which G is a source term due to chemical consumption or generation of the corresponding component. Then the diffusion equation governing the concentration of PQ molecules is derived as:

\frac{\partial [P Q] (x, t)}{\partial t} = D_{P Q - A u} ([A u] (x, t) \frac{\partial^{2} [P Q] (x, t)}{\partial x^{2}} - [P Q] (x, t) \frac{\partial^{2} [A u] (x, t)}{\partial x^{2}}) - R_{t} [P Q] (x, t),

where

R_{t} = κ_{i} I_{t} (x, t) (s^{- 1})

is the rate of production of excited state photosensitive PQ* radicals, in which

κ_{i} = ϕ ε d

represents the number of radicals initiated per photon absorbed and

I_{t} (x, t)

is the incident interference pattern with appropriate units (Einstein/cm³s) [24].

Similar with the derivation of the mutual diffusion between PQ and gold NPs, the MMA molecules experience the mutual diffusion with the gold NPs and the consumption of polymerization with excited PQ* radicals. The concentration evolution of MMA molecules is given by

\begin{array}{l} \frac{\partial [M] (x, t)}{\partial t} = D_{M - A u} ([A u] (x, t) \frac{\partial^{2} [M] (x, t)}{\partial x^{2}} - [M] (x, t) \frac{\partial^{2} [A u] (x, t)}{\partial x^{2}}), \\ - k_{m} [P Q^{*}] (x, t) [M] (x, t) \end{array}

where [M] is the concentration of MMA molecules; D_M-Au is the diffusion coefficient of the MMA-Au NPs mutual diffusion system and k_m is the rate constant of polymerization between PQ* radicals and MMA monomers.

Since the excited state PQ* radials are generated by the PQ molecules and react with the polymeric matrix or MMA monomers as one-to-one attachment, the concentration evolution of PQ* radicals can be depicted as:

\begin{array}{l} \frac{\partial [P Q^{*}] (x, t)}{\partial t} = κ_{i} I (x, t) [P Q] (x, t) - k_{m} [P Q^{*}] (x, t) [M] (x, t) \\ - \int_{- \infty}^{+ \infty} R (x, x^{'}) k_{p} [P Q^{*}] (x^{'}, t) [P M M A] (x^{'}, t) d x^{'}, \end{array}

where k_p is the rate constant of polymerization between PQ* and PMMA polymeric matrix;

R (x, x^{'}) = \exp [- {(x - x^{'})}^{2} / 2 σ] / \sqrt{2 π σ}

is the nonlocal spatial response function with the nonlocal response length

\sqrt{σ}

[25]. As the attached MMA molecules do not experience a propagation process of polymer chain growth involving link to other MMA molecules, the spatial nonlocal effect of MMA can be neglected [22].

Based on the description in the Section 2.1, the counter-diffusion of gold NPs is dominated by the polymerization-driven diffusion process of PQ and MMA small molecules. Considering both mutual diffusion processes of the PQ-Au NPs and MMA-Au NPs system, the rate equation governing the concentration evolution of gold NPs is presented as follows:

\begin{array}{l} \frac{\partial [A u] (x, t)}{\partial t} = D_{P Q - A u} ([P Q] (x, t) \frac{\partial^{2} [A u] (x, t)}{\partial x^{2}} - [A u] (x, t) \frac{\partial^{2} [P Q] (x, t)}{\partial x^{2}}) \\ + D_{M - A u} ([M] (x, t) \frac{\partial^{2} [A u] (x, t)}{\partial x^{2}} - [A u] (x, t) \frac{\partial^{2} [M] (x, t)}{\partial x^{2}}) . \end{array}

According to the photochemical reactions in the nanocomposite, the photoproducts consist of the polymers generated from the reaction of excited state PQ* radicals with MMA or PMMA. Consequently, the spatial and temporal evolution of corresponding photoproducts can be derived as:

\frac{\partial [P] (x, t)}{\partial t} = k_{m} [P Q^{*}] (x, t) [M] (x, t) + \int_{- \infty}^{+ \infty} R (x, x^{'}) k_{p} [P Q^{*}] (x^{'}, t) [P M M A] (x^{'}, t) d x^{'} .

In summary, the photochemical and photophysical processes under the holographic illumination in the gold NPs doped photopolymer have been investigated. The partial differential equations governing the concentration evolution related to these processes in the nanocomposite have been derived and presented. In the next section, with the method of Fourier series expansion and suitable initial conditions, the derived model is simulated to predict the holographic behaviours of important components in the nanocomposite.

2.3. Simulation for multicomponent diffusion and the active effect of gold NPs

The Fourier series $[X (x, t)] = \sum_{i = 0}^{m} X_{i} (t) \cos (i K x)$ are employed to describe the species concentrations of [PQ], [PQ*], [MMA], [P] and [Au], in which m = 3 ensures the simulation rapid, stable and convergent [24]. By substituting the Fourier series coefficients to Eqs. (3)-(7), a set of first order coupled differential equations can be derived. In order to examine the multicomponent diffusion processes, these equations in terms of time varying spatial harmonic amplitudes are numerically solved with the practical initial conditions. By estimating from the weight ratio of PQ: MMA: AIBN = 0.7:98.3:0.1 and ~10% residual MMA monomers after the thermo-polymerization fabrication process, the initial concentrations of PQ, PMMA and MMA at time t = 0 are set as: [PQ(x, 0)] = 3.2 × 10⁻⁵ mol/cm³, [PMMA(x, 0)] = 4 × 10⁻⁴ mol/cm³, [MMA(x, 0)] = 9 × 10⁻³ mol/cm³. The initial concentrations of radial PQ* and photoproducts are [PQ*(x, 0)] = 0 and [P(x, 0)] = 0. The values of the associated photophysical and photochemical kinetic parameters are sensibly assigned to be D_PQ-Au = 2.51 × 10⁻¹⁸ cm²/s, D_M-Au = 5.25 × 10⁻¹⁷ cm²/s, k_p = 2.64 × 10⁻² cm³/mol·s, k_m = 1.35 × 10⁻⁴ cm³/mol·s. The nonlocal response length is chosen to have a value of $σ = 0.2$ , which has been demonstrated to be a suitable value for PQ-PMMA material [12].

According with the previous experimental conditions, an unslanted transmission volume holographic grating is recorded. Then the incident intensity is represented as $I_{t} (x, t) = I_{0} [1 + V \cos (K x)]$ , in which the exposing fringe visibility V is unity; the grating vector is $K = 2 π / Λ$ with the period of grating is Λ = 1.03 μm (corresponding to the incident angle of 15°); and the intensity I₀ is converted into the appropriate units (Einstein/cm³s) with the original value of 5 mW/cm². Substituting the initial concentrations of different components, the corresponding kinetics parameter values and optical setup conditions into the Eqs. (3)-(7), the predicted spatial distribution of all the components is obtained.

Figure 2 shows the 2D spatial and temporal distribution of different components during holographic exposure with the initial concentration of gold NPs [Au(x, 0)] = 1.78 × 10⁻⁸ mol/cm³. The consumption of PQ and MMA molecules leads to a lower concentration, which is presented as dark color in the bright regions of Fig. 2(a) and Fig. 2(c), respectively. On the contrary, as shown in Fig. 2(d), the photoproducts are generated as the holographic polymerization processes in the bright regions. As the exposure time increases, the multicomponent diffusion process progresses and contributes to the spatial distribution of gold NPs in Fig. 2(b). During the exposure, some of gold NPs diffusing from the bright areas to the dark areas could induce the assembly in the certain area between the brightest region and the darkest region. Then in the area with higher density of gold NPs, more PQ molecules are activated to initiate the polymerization process as a result of the greater absorption induced by plasmon resonance of gold NPs. The narrow region of gold NPs assembly and the broaden generation of the photoproducts can be observed in Fig. 2(b) and Fig. 2(d), which may be attributed to the enhanced polymerization process and the induced nonlinear response of the material by the doped gold NPs.

Fig. 2 Spatial and temporal evolution of PQ molecules (a), gold NPs (b), MMA molecules (c) and photoproducts (d) in the polymeric nanocomposite under holographic exposure. Bright and dark regions of the interference pattern are indicated in the corresponding area of each subgraph.

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To obtain a thorough understanding of the modified photochemical and photophysical process by the active gold NPs dopant dispersed into the photopolymer, the effects of gold NPs to the spatial distribution of other components are investigated. In principle, the excitation of PQ molecules dominates the production of PQ* radicals and the latter reactions in the material. It might be affected by the absorbance of the PQ molecules under different wavelength and the enhanced absorption effect of the doped gold NPs. Figure 3(a) shows the comparison of the spatial distribution of PQ molecules in the photopolymer without and with the initial concentration of gold NPs. All the other kinetic parameters for the simulations are the same. It can be immediately seen that more PQ molecules in the photopolymer with gold NPs are consumed in the bright regions, which also drive more PQ molecules diffusing from the dark regions to bright regions. The nonlinear phenomenon departure from the cosine form can also be observed in the spatial distribution of PQ molecules in the gold NPs doped photopolymer. Figure 3(b) presents the spatial distribution of PQ molecules in the photopolymer with gold NPs under different exposure energy. As the exposure time or intensity increases, the diffusing and gathering gold NPs near the boundary of the bright and dark regions enhance the photon absorption and excitation process, which promotes the photopolymerization consumption of PQ molecules. The enhanced absorption with the diffusion of gold NPs from the bright regions to the dark regions contributes to the nonlinear spatial distribution of PQ molecules.

Fig. 3 (a) Comparison of the spatial distribution of PQ molecules at t = 6000s in the photopolymer without and with gold NPs; (b) The temporal evolution of spatial distribution of PQ molecules in the photopolymer with gold NPs.

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3. Volume holographic mixed grating analysis: experimental verification

3.1. Evolution of holographic diffraction efficiency with mixed gratings

Based on the investigation of photochemical and photophysical process in Section 2.1, the growth of the diffraction efficiency is determined by the diffracted evolution of refractive index grating and the absorption grating. For the experimental measurement, the diffraction efficiency of a holographic grating is calculated as the intensity ratio of the diffracted beam to the incident beam, which is the mixed diffracted result of refractive index grating and absorption grating. Theoretically, given a first-order harmonic of grating refractive index modulation n₁(t) and the temporal absorption modulation α₁(t), the evolution of the diffraction efficiency can be estimated using the coupled wave theory:

η = \exp (- \frac{2 α_{0} d}{\cos θ_{B}}) {\sin^{2} [\frac{π d \cdot n_{1} (t)}{λ \cos θ_{B}}] {+sh}^{2} (\frac{α_{1} (t) \cdot d}{2 \cos θ_{B}})},

Where α₀ is the average absorption constant; θ_B is the Bragg angle; d is the effective thickness of the grating and λ_B is the wavelength for recording the grating.

According to the Lorentz-lorenz relation [26], the average relative index of the composite is dependent upon the refractive index of the individual components and their concentrations or volume fractions. For gold NPs doped PQ-PMMA, since the refractive index of gold NPs at the wavelength of 532 nm is much smaller than the organic components, it can be neglected with such a small doped concentration. Thus the average refractive index n of a given nanocomposite is expressed using

\frac{n^{2} - 1}{n^{2} + 2} = φ_{P M M A} \frac{n_{P M M A}^{2} - 1}{n_{P M M A}^{2} + 2} + φ_{M} \frac{n_{M}^{2} - 1}{n_{M}^{2} + 2} + φ_{P} \frac{n_{P}^{2} - 1}{n_{P}^{2} + 2} + φ_{B} \frac{n_{B}^{2} - 1}{n_{B}^{2} + 2},

where φ_PMMA, φ_M, φ_P and φ_B are the volume fractions of the prepolymer macromolecule, monomer, photoproduct and background (including the gold NPs and the other remaining components), respectively. Considering the relation between the individual components and the composition, the volume fraction can be estimated by

φ_{i} = N_{i} W_{i} / (ρ_{i} N_{A})

[26], where N_i is the total number of molecules per unit volume; N_A is Avogadro’s number; W_i and ρ_i are average molar weight and the density of i^th component, respectively. Then relating Eq. (1) with the negligible gold NPs concentration and Eq. (9), the temporal evolution of the first-order harmonic of relative index modulation is expressed as

\begin{array}{l} n_{1} (t) = \frac{{(n_{D a r k}^{2} + 2)}^{2}}{6 n_{D a r k}} [φ_{P M M A 1} (t) (\frac{n_{P M M A}^{2} - 1}{n_{P M M A}^{2} + 2} - \frac{n_{B}^{2} - 1}{n_{B}^{2} + 2}) + φ_{M 1} (t) (\frac{n_{M}^{2} - 1}{n_{M}^{2} + 2} - \frac{n_{B}^{2} - 1}{n_{B}^{2} + 2}), \\ + φ_{P 1} (t) (\frac{n_{P}^{2} - 1}{n_{P}^{2} + 2} - \frac{n_{B}^{2} - 1}{n_{B}^{2} + 2})] \end{array}

where n_Dark, n_PMMA, n_M, n_P and n_B are refractive index value of the material before exposure, polymeric matrix, residual monomer, photoproduct and background, respectively. φ_PMMA₁(t), φ_M₁(t) and φ_P₁(t) are the time varying first harmonic volume fractions of polymeric matrix, monomer and photoproducts, which can be obtained by solving the partial differential equations of concentration in Section 2.2. Although the concentration or refractive index of gold NPs is not included in Eq. (10), it should be noted that the gold NPs affect the refractive index modulation by changing the spatial distribution of other components as mentioned in Section 2, in here, i.e. φ_PMMA₁(t), φ_M₁(t) and φ_P₁(t). The spatial distribution differences of PQ molecules between the polymer with and without gold NPs in Fig. 3(a) also confirm this effect. In fact, either mutual diffusion constants or the polymerization rates are changed as the doped gold NPs, which will be discussed in the following section.

The absorption modulation in the pure PQ-PMMA is neglected as the slight absorption coefficient of organic dye. However, with the plasmon resonance effect of doped gold NPs, even a small concentration can affect the dielectric constant of the host medium, either core-shell structure or dielectric matrix. Derived from the Maxwell-Garnett theory [27], the effective dielectric constant ε of composite material containing f_m volume fraction of doped NPs uniformly dispersed in the host medium can be expressed by

ε_{e f f} = \frac{(2 A + 1)}{(1 - A)} ε_{e}, A = f_{m} \frac{ε_{m} - ε_{e}}{ε_{m} + 2 ε_{e}},

where ε₀ and ε_m are the dielectric constants of the metal NPs and the host medium, respectively. Then the corresponding absorption effect can be estimated by the effective complex dielectric constant from

α = - k_{0} Im (ε) / \sqrt{Re (ε)}

, in which k₀ is the wave vector.

3.2. Evolution of the refractive index modulation and absorption modulation

To verify the established model for mixed gratings in gold NPs doped photopolymer, the smoothed experimentally obtained diffraction efficiency curves for samples without gold NPs, with 0.05 vol.% and with 0.24 vol.% concentration of gold NPs are used to fitting with the simulation equations of diffraction efficiency. These samples were made with the two-step thermo-polymerization technique [21]. The photopolymer syrup with weight ratio of PQ: MMA: AIBN = 0.7:98.3:0.1 and different volume concentration of gold NPs with the diameter of 6-8 nm were mixed [16]. The mixture were sonicated for 5 minutes to make them uniform and stirred until it became viscous. The viscous liquid contained in a glass mold with a 1 mm spacer was solidified by baking at 50°C for 24 h. Then the solid samples were achieved after removing the molds and cutting the raw coupon into 1 × 1 inch squares for holographic exposure. The diffraction efficiency of prepared samples was measured by using two-beam interference technology. The recording wavelength is 532 nm near the plasmon resonance peak of the gold NPs. The intensity of each split beam was 5 mW/cm² with the beam diameter of 6 mm. The unslanted gratings were recorded into the prepared samples with the incident angle of 15°. The absolute diffraction efficiency of volume gratings were obtained by η = I_d/I_inc, which I_d and I_inc are the intensity of diffracted beam and incident beam, respectively [16].

To simulate the evolution of refractive index grating and absorption grating, the initial concentration of gold NPs are set to be 1.78 × 10⁻⁸ mol/cm³ for 0.05 vol.% and 8.53 × 10⁻⁸ mol/cm³ for 0.24 vol.%. From Fig. 4, there is good agreement between experimental results and theoretical predictions to the temporal diffraction efficiency. It is indicated that the above proposed models are suitable for analyzing the holographic kinetics of mixed gratings in gold NPs doped PQ-PMMA photopolymer. The small differences between simulation and experimental results for the 0.05 vol.% concentration of gold NPs may be attributed to the constant diffusion coefficient of the PQ-Au and MMA-Au mutual diffusion systems.

Fig. 4 Comparisons of experimental results and simulation fitting results for the temporal diffraction efficiency for photopolymer without gold NPs (hollow circles for the experiment and solid green line for the theory), with 0.05 vol.% gold NPs (hollow triangles for the experiment and solid red line for the theory) and with 0.24 vol.% gold NPs (hollow squares for the experiment and solid blue line for the theory).

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By combining the kinetic model for the nanocomposite and the temporal description of refractive index modulation and absorption modulation, the kinetic parameter values related to the photochemical and photophysical process are obtained. The results are summarized in Table 1. It can be seen that the rate of production of the excited state photosensitizer κ_i increases as the increasing doped concentration of gold NPs, which might induce stronger nonlinear effect during the holographic exposure. These results are also according with the analysis of the active effect of gold NPs to the enhanced polymerization process during the holographic exposure in Section 2. The enhanced diffusion rate of PQ also confirms the active function of gold NPs in the photopolymer for the mutual diffusion effect. However, the polymerization rate of PQ molecules with PMMA macromolecules or MMA molecules is decelerated as the doped concentration of gold NPs. It might be the results of capturing ability of gold NPs to the initiator radicals or some active sites on the surface of gold NPs terminate the free radicals via the electron transfer reactions [28]. Since the MMA small molecules act as movable component, the chemical influence of gold NPs to decrease the polymerization rate is not as strong as that on the PMMA matrix. Correspondingly, the fitting results also show that the average weight of photoproduct decreases as the increasing concentration of gold NPs, which could also be the result of a strongly decreased polymerization rate of PQ with PMMA.

Table 1. Parameters values extracted from fits to experimental growth curves of diffraction efficiency

View Table

Based on the proposed model and the fitting parameter values from the experimental results, the refractive index modulation and absorption modulation for the samples with different concentration of gold NPs can be extracted, as shown in Fig. 5. Comparing the Fig. 4 to Fig. 5, the profiles of the diffraction efficiency is mainly dependent on the shape of temporal refractive index modulation curves. It confirms the influence of gold NPs on the refractive index modulation by largely changing the distribution of other components in the nanocomposite. The decelerated growth of refractive index modulation in the photopolymer with gold NPs may contribute to the retarded initial step of polymerization process. And it may also cause a greater decrease to the diffraction efficiency than the pure photopolymer as the exposure time increases. Different from the evolution of refractive index modulation, the absorption modulation keep increasing with decreasing growth rate as the polymerization-driven counter diffusion processes of gold NPs. At the beginning of the holographic exposure, the absorption modulation in the sample with 0.24 vol.% doped concentration of gold NPs grows faster than that with 0.05 vol.% doped concentration of gold NPs due to the higher diffusion rate of PQ-Au NPs system. As the exposure time increases to a certain value, the growth rate of absorption modulation decreases as a result of the reduced mutual diffusion due to the eliminating concentration gradient. The amplitude of decrease in the sample with 0.24 vol.% doped concentration of gold NPs is greater than that in the sample with 0.05 vol.%, which may be the result of more gold NPs remained in the bright regions.

Fig. 5 Temporal evolution of refractive index modulation (a) and absorption modulation (b) for photopolymer without gold NPs (dashed green line), with 0.05 vol.% gold NPs (solid red line) and with 0.24 vol.% gold NPs (dotted blue line).

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4. Conclusions

A holographic kinetics model for mixed volume gratings is proposed and employed to analyze the formation of refractive index grating and absorption grating in a gold NPs doped photopolymer. The spatial and temporal evolution of the concentration for different components confirm the polymerization driven multicomponent mutual diffusion process between gold NPs and small molecules of the photopolymer. The enhanced absorption effect of gold NPs and the induced nonlinear influence to the photophysical and photochemical process have also been presented. The model for holographic dynamics is then incorporated with the equations depicting the refractive index change and absorption modulation. A good agreement between the simulation results and the experimental measurements for the temporal diffraction efficiency of the prepared samples without and with gold NPs is presented. The fitting kinetic values of the corresponding parameters are consistent with the previous analysis of the active effect of the gold NPs dopant. The proposed model has been demonstrated to be an effective way to analyze the influence of the gold NPs to the diffraction efficiency of volume mixed gratings. More accurate theoretical prediction will be further investigated with the consideration of nonlinear effect of gold NPs and the time varying viscosity effects to the mutual diffusion process in the nanocomposite.

Acknowledgments

The work is supported by National Basic Research Program of China (2013CB328803), Nature Science Foundation of China (61361160418, 61275013), and Tsinghua University Initiative Scientific Research Program.

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Samples	κ_i (s⁻¹)	D_PQ/D_PQ-Au (cm²/s)	D_M/D_M-Au (cm²/s)	k_p (cm³/mol· s)	k_m (cm³/mol·s)	M_p^a (g/mol)
0vol.%	1.16 × 10³	1.43 × 10⁻¹⁵	1.41 × 10⁻¹⁵	5.44 × 10⁻²	5.44 × 10⁻⁴	19995.38
0.05vol.%	1.53 × 10³	4.20 × 10⁻¹⁵	4.19 × 10⁻¹⁶	1.91 × 10⁻³	3.82 × 10⁻⁴	16011.45
0.24vol.%	1.60 × 10³	8.98 × 10⁻¹⁵	5.68 × 10⁻¹⁵	1.79 × 10⁻³	3.25 × 10⁻⁴	15500.02

Holographic kinetics for mixed volume gratings in gold nanoparticles doped photopolymer

Abstract

1. Introduction

2. Holographic kinetics in gold NPs doped photopolymer

2.1. Mechanism of mixed gratings

2.2. Kinetic model for photochemical and photophysical process

2.3. Simulation for multicomponent diffusion and the active effect of gold NPs

3. Volume holographic mixed grating analysis: experimental verification

3.1. Evolution of holographic diffraction efficiency with mixed gratings

3.2. Evolution of the refractive index modulation and absorption modulation

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (11)

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