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Langmuir 2008, 24, 4260-4264
A Single-Dipole Model of Surface Relief Grating Formation on Azobenzene Polymer Films Jae-Dong Lee,† Mi-Jeong Kim,*,‡ and Tomonobu Nakayama§ School of Materials Science, Japan AdVanced Institute of Science and Technology, Ishikawa 923-1292, Japan, and International Center for Young Scientists and Nano System Functionality Center, National Institute for Materials Science, Tsukuba 305-0044, Japan ReceiVed December 10, 2007. In Final Form: January 31, 2008 A new model that keeps track of the dynamics of single dipoles is suggested for the photoinduced formation of a surface relief grating on azobenzene polymer films by two optical fields. Interfering optical fields provide a single dipole of an azobenezene molecule with two dynamical resources: rotation of the molecule caused by the torque, leading to an induced dipole, and its subsequent migration because of the electric force. Explicit development of the induced dipole of the molecule and its real-time migration, which depend on the details of the optical fields such as polarization and wavelength, can be understood self-consistently within the model.
I. Introduction During the past decade, photoinduced mass migration leading to surface relief grating (SRG) formation on azobenzene polymer films has attracted much attention because of its potential applications, such as acting as a new tool to fabricate organic optical devices.1-9 The trans-cis photoisomerization cycle is one important feature of azobenzene polymers that enables photoresponsive motions, that is, photoinduced alignment10,11 and photoinduced polymeric mass migration to form SRGs.1,2 Mass migration occurs from the bright region to the dark region.6 The SRG is fabricated below the glass transition temperature (Tg) of the polymers and can be erased by irradiation or heating.1,2 The SRG formation process strongly depends on the polarization of the two optical fields. The p:p polarization gives appreciable diffraction efficiency, while the s:s polarization is ineffective.12,13 Several mechanisms for photoinduced mass migration of azobenzene molecules have been suggested: the gradient force of the optical electric field,5-8 isomerization-driven free volume expansion,9 the diffusion model,14 the mean field theory of anisotropic intermolecular interaction,15 a thermodynamic origin,16 the combined particle model of gradient force and surface * To whom correspondence should be addressed. E-mail:
[email protected]. † Japan Advanced Institute of Science and Technology. ‡ International Center for Young Scientists, National Institute for Materials Science. § Nano System Functionality Center, National Institute for Materials Science. (1) For a review, see Natansohn, A.; Rochon, P. Chem. ReV. 2002, 102, 4139. (2) Sekkat, Z., Knoll, W., Eds. PhotoreactiVe organic thin films: Academic Press: New York, 2002. (3) Kim, D. Y.; Li, L.; Kumar, J.; Tripathy, S. Appl. Phys. Lett. 1995, 66, 1166. (4) Rochon, P.; Ballata, E.; Natansohn, A. Appl. Phys. Lett. 1995, 66, 136. (5) Kumar, J.; Li, L.; Jiang, X. L.; Kim, D. Y.; Lee, T. S.; Tripathy, S. Appl. Phys. Lett. 1998, 72, 2096. (6) Bian, S.; Williams, J. W.; Kim, D. Y.; Li, L.; Balasubramanian, S.; Kumar, J.; Tripathy, S. J. Appl. Phys. 1999, 86, 4498. (7) Keyang, K.; Yang, S.; Wang, X.; Kumar, J. Appl. Phys. Lett. 2004, 84, 4517. (8) Keyang, K.; Yang, S.; Kumar, J. Phys. ReV. B 1996, 73, 165204. (9) Barrett, C. J.; Rochon, P.; Natansohn, A. J. Chem. Phys. 1998, 109, 1505. (10) Todorov, T.; Nikolova, N.; Tomova, T. Appl. Opt. 1984, 23, 4309. (11) Todorov, T.; Nikolova, N.; Tomova, T. Appl. Opt. 1984, 23, 4588. (12) Jiang, X. L.; Li, L.; Kumar, J.; Kim, D. Y.; Shivshankar, V.; Tripathy, S. Appl. Phys. Lett. 1996, 68, 2618. (13) Viswanathan, N. K.; Balasubramanian, S.; Li, L.; Tripathy, S.; Kumar, J. Jpn. J. Appl. Phys. 1999, 38, 5928. (14) Lefin, P.; Fiorini, C.; Nunzi, J.-M. Opt. Mater. 1998, 9, 323.
tension,17,18 and so on. There is still no consensus on the governing mechanism of the phenomenon. This is because many controversies remain to be resolved. In particular, there are questions on (i) the dependency of migration on the laser parameters such as intensity and wavelength,19 (ii) whether it is a bulk process or a surface process,5,9 (iii) how it depends on the field polarization,5,13,15 and so on. Considering these, we note that a model that takes into account the full dynamical effect of applied optical fields on the molecular dipoles is still lacking. For instance, the gradient force model cannot lead to the desired time-averaged force without specifying the special direction of the induced dipole moment Pinduced. In other words, to reproduce the experimental observation, the ad hoc assignment of a negative polarizability (χ < 0) is required in the equation Pinduced ) χE by the applied field E.6-18 However, this will remain unclear until there is a reliable study and understanding of χ itself. In this paper, we suggest a new model that keeps track of the dynamics of single dipoles under interfering optical fields. Here, we try to obtain pinduced of a single dipole, rather than Pinduced averaged over a macroscopic collection of dipoles. In the model, the SRG formation by two optical fields develops in a two-step process. First, depending on the applied linear optical field, there should occur sufficient trans-cis photoisomerization cycles to align the dipoles of the azobenzene molecules in a direction perpendicular to the optical field.10,11 Second, the driving force for the dipole displacement should be induced by the following two coupled equations of motion:
F ) (p‚∇) E(r, τ)
(1)
N ) p × E(r, τ)
(2)
where F is the electric force, N is the torque on the dipole, and p is the dipole moment of a single azobenzene molecular group. Here, we exclude the motion of the side chain and the polymer (15) Pedersen, T. G.; Johansen, P. M.; Holme, N. C. R.; Ramanujam, P. S.; Hvilste, S. Phys. ReV. Lett. 1998, 80, 89. (16) Saphiannikova, M.; Neher, D. J. Phys. Chem. B 2005, 109, 19428. (17) Barada, D.; Fukuda, T.; Itoh, M.; Yatagai, T. Jpn. J. Appl. Phys. 2006, 45, 465. (18) Fukuda, T.; Barada, D. Jpn. J. Appl. Phys. 2006, 45, 470. (19) Kim, M. J.; Lee, J. D.; Chun, C.; Kim, D. Y.; Higuchi, S.; Nakayama, T. Macromol. Chem. Phys. 2007, 208, 1753.
10.1021/la703842r CCC: $40.75 © 2008 American Chemical Society Published on Web 03/12/2008
New Model for Photoinduced Migration of Azobenzene
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Figure 1. Photoinduced migration of the azobenzene molecular mass: (a and b) Geometry of the azobenzene film (blue planar patch) and two applied optical fields E1 and E2 (green arrows). (c) Snapshot of the dimensionless standing optical field E′x() 2 cos θ cos(ω′τ′) cos(2πx′); red line). Blue rods at positions A, B, C, and D denote the dipoles of the azobenzene molecules aligned along the yˆ -axis. (d) Migration of molecular dipoles initially positioned at A, B, C, and D. θ ) 15°, I′ ) 1 × 10-4, and ω′ ) 1 × 109 are used.
backbone, which cooperatively move following the covalently linked azobenzene molecular group in reality. Equation 1 gives the lateral displacement of the dipole. On the other hand, eq 2 leads to rotation of the dipole p and results in a dipole component along the optical field, which corresponds to the induced dipole of the molecule pinduced. Solving the above coupled set of equations, one can properly account for pinduced and its subsequent displacement under the optical field. Our new model has several obvious merits: (i) the observed geometry of SRG can be reproduced without any ad hoc approximation, (ii) a real-time calculation incorporating the field oscillation is possible, (iii) the polarization and wavelength dependency of the grating efficiency can be investigated, and (iv) by the natural extension of the model a general understanding of the role of the photoisomerization cycle for SRG formation could be possible.
II. Theoretical Model An azobenzene polymer film is considered to be located on the xy-plane of Figure 1a. In the experimental setup shown in Figure 1b, the two applied optical fields can be expressed as E1(r, τ) ) E10ei(k1‚r-ωτ) and E2(r, τ) ) E20ei(k2‚r-ωτ) with k1 ) (-k sin θ, 0, -k cos θ) and k2 ) (k sin θ, 0, -k cos θ). θ is the angle between the zˆ-axis and the propagating direction of the optical field. k is the wave vector given by 2π/λ (λ ) wavelength). The total optical field is E(r, τ) ) E1(r, τ) + E2(r, τ). For the moment, for a preferred polarization, we consider the p:p polarization and have E10 ) E0(cos θ, 0, -sin θ) and E20 ) E0(cos θ, 0, sin θ). Now, we can express E(r, τ) as, taking only its real part,
E(r, τ) ) 2E0 cos θ cos(ωτ) cos(k sin θx)xˆ + 2E0 sin θ sin(ωτ) sin(k sin θx)zˆ It should be noted that E(r, τ) is the standing wave along the xˆ -axis on the film (i.e., z ) 0). As the first step of our model, E(r, τ) begins the trans-cis photoisomerization cycle and aligns the dipole at an initial position i on the film to be pi ≈ (piyˆ. This initial alignment was also assumed in the mean field treatment of the intermolecular interaction by Pedersen et al.15 In fact, however, within our model, such an assumption is not essential for realization of the
experimentally observed dipole displacement (but it influences the displacement efficiency). This will be discussed later. For the next step, the driving force for the dipole displacement is considered: the coupled dynamics of electric force and torque. Equations 1 and 2 can be rewritten in a more figurative form in our geometry:
d2xi d mi 2 ) pi sin φi Ex(x, τ)|x)xi dx dτ
(3)
d2φi Ii 2 ) pi Ex(xi, τ) cos φi dτ
(4)
where mi and Ii are the mass and the moment of inertia of the molecular dipole at an initial position i, respectively, and φi is the angle between the yˆ -axis and pi. If we assume a collection of identical dipoles, this simplifies to mi ) m, Ii ) I, and pi ) p. It is useful to introduce the dimensionless parameters x′i and τ′ defined by x′i ≡ xi sin θ/λ and τ′ ≡ τ(pE0/mλ2)1/2. This changes eqs 3 and 4 to
d2x′i dτ′2
) -2π sin θ sin 2θ sin φi cos(ω′τ′) sin(2πx′i) d2φi
I′
dτ′2
) 2 cos θ cos φi cos(ω′τ′) cos(2πx′i)
where both ω′ and I′ are also dimensionless, that is, ω′ ≡ ω(mλ2/pE0)1/2 and I′ ≡ I/mλ2. The above coupled differential equations can be solved with the initial conditions of x′i(0) ) x′i0 and φi(0) ) 0 or π. Defining a characteristic velocity Vc ≡ (pE0/ m)1/2, we have τ′ ) τVc/λ and ω′ ) 2πc/Vc (c ) speed of light). It now remains to estimate the values of ω′ and I′. Noting that x2Vc is the velocity of the molecule with m and p under the static electric field E0, we estimate that ω′ would be a huge number that amounts to O(108-9) or larger depending on E0. Note that ω′ does not depend on the wavelength of the optical field. The moment of inertia I is given by I ≈ ml2/12 if the azobenzene molecule is assumed to be a rod of length l, so I′ ≈ (1/12)(l/λ)2.
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Figure 2. Initial-stage temporal behavior of detailed forces on molecules located at A, B, C, and D under the situation of Figure 1. Green line, dimensionless optical field gradients dE′x/dx′ () -2π sin θ sin 2θ cos(ω′τ′) sin(2πx′i)); blue line, rotation angles φi; and red line, net dimensionless forces d2x′i/dτ′2 () sin φi dE′x/dx′).
Figure 3. Dipole number density F(x, τ) due to photoinduced migration caused by the two optical fields for p:p polarization: (a) For the grating pitch λ/(2 sin θ) ) 1 µm with λ ) 488 nm and θ ) 14.12°, the time-dependent behavior of F(x, τ) at τ ) 2, 4, 6, and 8 s is given (indicated by red arrows). (b) Fixing the grating pitch () 1 µm) in terms of angle control (θ ) 15.43° for λ ) 532 nm), the wavelength dependency of SRG formation at τ ) 8 s is illustrated. We have taken I/m ) 23.81 nm2 and ω′ ) 2.5 × 109.
With l ∼ O(1)-O(10) nm and λ ∼ O(102) nm, typical values of I′ would be O(10-4)-O(10-2).
III. Results Figure 1 illustrates how the azobenzene molecular dipoles initially positioned at A, B, C, and D (see Figure 1c) move with respect to time under two interfering optical waves. The dipoles accelerate in different directions depending on their positions and migrate characteristically from the bright side to the dark side, as shown in Figure 1d. Figure 2 provides, under the same situation as Figure 1, a detailed analysis of the electric forces acting on the dipoles. The net direction of the electric force (red) is determined by a combination (i.e., multiplication) of the field gradient (green) and the angle of rotation (blue). This makes it possible to understand the dipole migration in Figure 1d. In Figure 3, modulation of the surface structure is simulated in terms of dipole number density F(x, τ) along the xˆ -direction.
It is defined by F(x, τ) ) [∑i∆(x - xi(τ))]/F0, where ∆(x) is the Lorentzian density profile of a single dipole and F0 is the uniform density before irradiation with the optical field. Initially, |xi(0) - xi+1(0)| , γ is assumed (γ ) Lorentzian width). SRG formation is illustrated in Figure 3a for the p:p polarization. Migration of individual dipoles in this case has already been well described in Figures 1 and 2. The grating pitch is determined by λ/(2 sin θ), which is consistent with experiment.1,2 Now, it is meaningful to compare this model with the gradient force model. First, the present single-dipole model can reproduce the correct molecular mass migration, as shown in Figures 1-3. In the gradient force model, χ < 0 would give the correct direction of migration, while χ > 0 gives the opposite direction (i.e., from dark side to bright side). However, χ < 0 would not be fully justified. Second, the gradient force model incorporates the time-averaged (timeindependent) force so that it cannot provide a real-time calculation. Because of that, it is difficult to estimate the relevant time scales. In the present model, however, we could investigate the wavelength dependency of SRG formation by fully accounting for the real-time field oscillation (see Figure 3b). The long wavelength field is found to be more efficient for SRG formation for a fixed grating pitch, which is consistent with our recent experimental results.19 Another experimental geometry with the s:s polarization can be considered. The two optical fields are E1(r, τ) ) E0ei(k1‚r-ωτ)yˆ and E2(r, τ) ) E0ei(k2‚r-ωτ)yˆ, and the total field E(r, τ) (with z ) 0) is written as E(r, τ) ) 2E0 cos(ωτ) cos(k sin θx)yˆ. Under E(r, τ), eq 1 cannot lead to any force along the xˆ -direction. Therefore, the molecular dipoles do not undergo any displacement along the xˆ -direction, which is consistent with experimental observations.7,8,13 Instead, eq 1 gives the force along the yˆ -direction from m(d2yi/dτ2) ) p sin φi(d/dx) Ey(x, τ)|x)xi. However, Id2φi/ dτ2 ≈ 0 because pi ≈ (pxˆ (i.e., φi ) (π/2), and therefore, the force along the yˆ -direction simply oscillates without a preferred direction so that it vanishes in the time average. Therefore, under s:s polarization, molecular mass migration on the film (i.e., SRG) is suppressed for both the xˆ - and yˆ -directions. Moreover, it is naturally expected that SRG formation for the s:p polarization would show intermediate behavior. It is important to estimate the strength of the migrating force induced by the present single-dipole model. The gradient force
New Model for Photoinduced Migration of Azobenzene
Figure 4. Dipole number density F(x, τ) after 4 s for two initial alignments. All the parameters are identical to those in Figure 3a. Calculation for the random initial alignment was performed by averaging over 25 sets of calculation results starting from 25 different random configurations of dipoles.
model5-8 has been criticized because the induced force might be too weak to migrate azobenzene molecules. In fact, it is estimated to be ∼102 N/m3, which is smaller than the gravitational force (∼104 N/m3).20 If one explicitly compares the maximum force strengths between two models, one has f ) 2πpE0/λ and fg ) 2π0E02/λ, where f and fg are the forces induced by the present model and the gradient force model, respectively. Here, p is the dipole moment of a single molecule, 0 ) 8.85 × 10-12 C2/Nm2 is the vacuum permittivity, λ is the wavelength of the optical field, and E0 ) (2Ilaserz0)1/2 is the electric field strength of the used laser (Ilaser is the laser intensity, and z0 is the vacuum impedance). Taking Ilaser ) 100 mW/cm2 and z0 ) 377 Ω,20 one obtains f/fg ) p/(0E0) ) p/(2.6 × 1018 D/m3). For five azobenzene derivatives with different electron-withdrawing groups such as CF3, CN, CNCl, NO2, and (CN)2, their dipole moments (corresponding to a single molecule) are found to range around 6.6-9.1 D.19 Estimating the mass density of the polymer film as about 103 kg/m3 and the molar weight of the polymer as about O(10) kg/mol (around 30 monomers), one can finally find f/fg ∼ 107-108, that is, f ∼ 109-1010 N/m3. This might be enough to drive the photoinduced mass migration.
IV. Discussion We discuss how to understand a more realistic SRG formation on the azobenzene polymer film surface based on the present single-dipole model. It is true that the model describes a simplified version of the complicated real system. Nevertheless, we claim that the model could provide a seminal clue for understanding the actual situation. First, in reality, the azobenzene molecules are not aligned completely along the yˆ -axis under trans-cis photoisomerization cycling but are distributed around the yˆ -axis. That is, one may think of two molecules with the initial angles φ and -φ to the yˆ -axis. The opposite forces from the initial angles cannot produce an actual displacement because of the time oscillation, which implies that the permanent dipole cannot generate the displacing force under the oscillating field. Instead, only the force induced by the subsequent rotation dynamics of the molecules (i.e., by the induced dipoles) can produce an actual displacement, whose direction is the same for φ and -φ. This is clear from the calculation for the random initial alignment of Figure 4. The calculation was performed in the following way: (i) 25 different random initial configurations of dipoles are generated. (ii) A calculation from each random configuration is carried. We repeat this calculation 25 times. (iii) Finally, 25 sets of calculation results are averaged. As shown in the figure, the same SRG pattern (but with lower efficiency) can be obtained (20) Saphiannikova, M.; Geue, T. M.; Henneberg, O.; Morawetz, K.; Pietsch, U. J. Chem. Phys. 2004, 120, 4039.
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even from the random initial alignment. Therefore, at least within the present model, the SRG formation does not depend on the initial dipole alignment. This is an important fact in understanding how the natural extension of the present model could work for the real situation. Second, now one may ask “Why is SRG formation obserVed mainly in azobenzene polymers and not in general polymers?”. To answer this question, we should consider the photoisomerization cycle of the azobenzene molecule and its effect on the intermolecular force and the displacing force. In a general sense, the total force f acting on the dipole should be f ) fext + fint, where fext is the force from the external optical field and fint is the intermolecular force between the molecules. fint is not included in the present model. It is clear that the irrelevance of the dipole distribution to SRG formation (as in Figure 4) results from the fact that the present model incorporates only fext. As fint increases, the molecule becomes difficult to move, which is the opposite limit from the present model. SRG formation can then be understood based on the competition between fext and fint. Here, we point out that the photoisomerization cycle would have two nontrivial effects on the competition: (i) the photoisomerization cycle would reduce the intermolecular force fint through the dynamical transformation of the molecular structures and (ii) the photoisomerization cycle would increase fext by aligning the dipole perpendicular to the optical field. Therefore, it is noted that the present single-dipole model can be better applied to azobenzene polymers undergoing efficient photoisomerization cycles. Furthermore, these are the reasons why the azobenzene polymers are special for SRG formation. Another point worth mentioning for the further extension of the present calculation is to take into account the thickness (along zˆ) of the film. This is directly related to the grating formation under the optical fields with the circular polarizations. For the circularly polarized fields, the direction of polarization is not constant but rotates through the thickness of the film. Experimental observation shows that the polarization of RCP:LCP gives much higher efficiency for the grating formation than the p:p polarization, where RCP (LCP) is the right (left) circular polarization.13 However, according to the present model using a single layer of molecular dipoles, the RCP:LCP polarization gives the efficiency just similar to or slightly larger than the p:p polarization. One could obtain more realistic results using multilayers of molecular dipoles. However, it is not simple to set the aligned direction of azobenzene along the zˆ-direction. It has been reported that azobenzenes are aligned helically along the zˆ-direction under irradiation with circularly or elliptically polarized light.21-23 Further, nontrivial helical grating formation was reported by using elliptical polarization.24 In addition to the present migration mechanism on the film surface, therefore, we believe that the helically aligned azobenzene molecules along the zˆ-direction and the interaction between those could play a role in the grating formation under the circularly or elliptically polarized optical field.
V. Conclusions We have suggested a single-dipole model for photoinduced mass migration leading to SRG formation on azobenzene polymer films. In the model, we combine two dynamical resources on the azobenzene molecular dipoles under the optical fields: an electric (21) Iftime, G.; Labarthet, F. L.; Natansohn, A.; Rochon, P. J. Am. Chem. Soc. 2000, 122, 12646. (22) Kim, M. J.; Shin, B. G.; Kim, J. J.; Kim, D. Y. J. Am. Chem. Soc. 2002, 124, 3504. (23) Hore, D. K.; Natansohn, A.; Rochon, P. J. Phys. Chem. B 2003, 107, 2506. (24) Kim, M. J.; Kumar, J.; Kim, D. Y. AdV. Mater. 2003, 15, 2005.
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force and a rotating torque on the dipoles. With the model, the experimental observation of SRG can be reproduced without any ad hoc approximation. In particular, SRG formation depending on the polarization and wavelength of the optical field is self-consistently understood. Finally, we discuss how to understand SRG formation in a more realistic situation based on photoisomerization and its effects on the competition between
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the displacing force and the intermolecular force depending on the polarization geometry of the optical fields. Acknowledgment. This work was supported by Special Coordination Funds for Promoting Science and Technology from MEXT, Japan. LA703842R
