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J. Phys. Chem. C 2008, 112, 7836–7840
Dipolar Alignment in an Electric Field: Effect of Lattice Arrangement† Andrew P. Chafin* and Geoffrey A. Lindsay US NaVy, NAVAIR, NAWCWD, Michelson Laboratory, Chemistry Branch, M.S. 6303, 1900 North Knox Road, China Lake, California 93555 ReceiVed: NoVember 29, 2007; ReVised Manuscript ReceiVed: January 29, 2008
Monte Carlo calculations were carried out to elucidate the dependence of dipolar alignment in a strong electric poling field on the choice of lattice. It has been found that the lattice that provides the best alignment for a prolate-shaped chromophore is an offset tetragonal lattice (same as face-centered tetragonal), whereas the lowest-energy lattice is a non-offset tetragonal (primitive tetragonal) lattice arrangement. 1. Introduction The field of nonlinear optical (NLO) materials has undergone a tremendous expansion in the past decade or so. New organic chromophores have been found1 that outperform current inorganic NLO materials2 in electro-optic (EO) modulators.3 This, along with their advantages in processing, make organic NLO materials very attractive. These new organic NLO chromophores have a high firstorder hyperpolarizability (β).4 This molecular property must be translated into the macroscopic electro-optic coefficient r33.5 The actual physical property used in EO modulators is the change in refractive index (∆nzz), which is proportional to the modulating field times r33 (∆nzz ) -1/2nzz3r33Ez). If we assume that the principal axis of the second-order hyperpolarizability tensor is aligned with the dipole vector, then r33 ) -2\[ f02fω2βzzz(-ω,ω,0)]/(nzz4)N〈cos3 θ〉, where f0 and fω are local field factors, N is the number density of chromophores, and 〈cos3 θ〉 is the average of the cubed cosine of the angle between the dipole vectors and the poling axis, sometimes called the order parameter. This paper is concerned with the calculation of the loading parameter; LP ) N〈cos3 θ〉, which is directly proportional to r33. Most organic EO devices are based on host-guest polymeric systems in which the alignment of the dipoles is achieved by poling with an external field at high temperatures. The loading parameter in the low-concentration case (where dipole-dipole interactions can be ignored) can be calculated based solely on the dipole-field interaction as N〈cos3 θ〉 ) NL3[(Epµ)/(kT)], where L3(p) is the third-order Langevin function of p.5 However, at moderate concentrations the dipole-dipole interactions become important and this precludes an exact analytical solution. Pereverzev and Prezhdo have developed solutions using mean-field theory and assuming a cubic or tetragonal lattice.6,7 These authors take the temperature into account via a statistical averaging of the dipoles. Analytical solutions were proffered for low concentration,s but higher concentrations, where Dalton et al. showed that chromophore-chromophore electrostatic interactions are important,8 had to be calculated numerically. Recently, Robinson, et al. tackled the calculation of polar alignment of point dipoles in an electric field via a Monte Carlo simulation9 on both cubic and tetragonal lattices with the Z axis shortened or lengthened relative to the X and Y axes.10 Their † Part of the “Larry Dalton Festschrift”. * Corresponding author. E-mail:
[email protected].
10.1021/jp7113048
results lead to a synthesis effort toward chromophores that are oblate in shape; that is, they are bigger around the middle than in the dipole direction.11 However, with this approach the ultimate concentration of active material is limited by the amount of the inert material attached to the dye. There is another lattice type besides the primitive cubic and primitive tetragonal that has the same density. That is an offset cubic or offset tetragonal lattice, also called a face-centered lattice. In this lattice, every other lattice site is offset one-half of the periodic distance in the Z direction. In crystallographic terms, these offset lattices are cF and tF Bravais lattices, while the non-offset cubic and tetragonal lattices are designated cP and tP. This paper will explore the ramifications of the offset lattice geometry. In their most recent paper, Robinson, et al. introduce a modified Lennard-Jones term and allow their dipoles to wander off-lattice; they also explored the face-centered lattice, however only in the cubic case.12 2. Monte Carlo Simulations The electrostatic potential energy (W) is defined here as the sum of the field-dipole and dipole-dipole interactions (eq 1).
W)
∑ i
[
µi · Ep +
( (
1 1 4πε0 2
∑ k
j*i
∑ j
(
µi · µj 3
|sij|
µi · µk 3
|sik|
-
-
3(µi · sij)(sij · µj) |sij|5
3(µi · sik)(sik · µk) |sik|5
)
) )]
+
(1)
The symbol · denotes an inner (dot) product between two vectors, Ep is the poling field vector, µi, µj, and µk, are dipole vectors, and sij is the displacement vector between dipoles i and j. The summations are divided by 2 inside the box in order to not count an interaction twice. k denotes a virtual dipole due to the periodic boundary conditions. The sign of the dipole-field term is positive to reflect the fact that the poling field is in the negative Z direction. The calculations were performed on a 12 × 12 × 12 lattice with periodic boundary conditions. The calculation is begun with a random arrangement of point dipole alignments. The dipole vectors are stepped by modifying cos θ and φ with step times a random number between 0 and 1, and a new energy is calculated. If ∆W e 0, then the new vector is kept. If ∆W > 0, then the value e-(∆W)/(kT) is computed and compared to a uniform random number between 0 and 1. If the random number is less, then the new dipole is kept, otherwise
This article not subject to U.S. Copyright. Published 2008 by the American Chemical Society Published on Web 04/22/2008
Dipolar Alignment in an Electric Field
Figure 1. tP and tF Lattices.
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7837
Figure 3. Variation of the Loading Parameter (N〈cos3 θ〉) with concentration (#/cc) for a 12 × 12 × 12 lattice of dipoles 5 D strong in a field of 200 V/µm at 100 °C. This number is proportional to the expected r33. The solid line (with crosses) corresponds to a 2:1 (v/h) offset tetragonal lattice (tF). The dotted line (with stars) corresponds to a 1:1 (v/h) offset cubic lattice (cF). The line consisting of a dash followed by a dot (with solid squares) corresponds to a 1:1 (v/h) nonoffset cubic lattice (cP). The dashed line (with solid diamonds) corresponds to a 2:1 (v/h) non-offset tetragonal lattice (tP).
Figure 2. CLD Chromophore.
the old vector is returned. A Boltzmann distribution is assured by adjusting the step size so that one-half of the comparisons are rejected.13 Each dipole is adjusted once during each cycle. A pseudoequilibrium is attained after about 100 cycles, and the results from at least the next 200 cycles are averaged. As noted previously, the lattice chosen was a 12 × 12 × 12 lattice with periodic boundary conditions. Because of the boundary conditions, a total of 46 656 point dipoles are considered of which 1728 dipoles are explicitly considered.14 The poling field was set at 200 V/µm (approximately the maximum experimentally attainable15), and the poling temperature was set to 100 °C. It should be noted that both the allowable concentrations and lattice size are constrained by the size of the chromophore. However, in the present study only point dipoles are considered (size exclusion is not invoked). A typical chromophore such as CLD (Figure 2) has an end to end distance of approximately 26 Å and a volume of 750 Å3. Table 1 shows the cell sizes for lattices of three different eccentricities and chromophore concentrations (the sizes for the offset lattices are identical). As an example, a 30 wt % solution of CLD in a polymer with a total density of 1.2 g/cc has a molar concentration of 4.55 × 1020 molecules/cc. Although it is certainly possible to add material to the middle of a chromophore in order to make it more oblate in shape, it will be difficult to attain high concentrations. Although size exclusion is not considered in the following calculations (which is different than the dipole-dipole interactions), for a CLDsize dye, even the physical interactions of nearest-neighbor dyes will be quite strong at a dye concentration of 5 × 1020 molecules/cc. 3. Results 3.1. Loading Parameter. Figures 3-5 detail the variation of loading parameter (N〈cos3 θ〉) with concentration for dipoles
Figure 4. Variation of the Loading Parameter (N〈cos3 θ〉) with concentration (#/cc) for a 12 × 12 × 12 lattice of dipoles 10 D strong in a field of 200 V/µm at 100 °C. The solid line (with crosses) corresponds to a 2:1 (v/h) offset tetragonal lattice (tF). The dotted line (with stars) corresponds to a 1:1 (v/h) offset cubic lattice (cF). The line consisting of a dash followed by a dot (with solid squares) corresponds to a 1:1 (v/h) non-offset cubic lattice (cP). The dashed line (with solid diamonds) corresponds to a 2:1 (v/h) non-offset tetragonal lattice (tP).
TABLE 1: Three Eccentricities at Three Concentrations (Vertical and Horizontal Sizes of Cells) v/h 1:2 (oblate) 1:1 (spherical) 2:1 (prolate)
N ) 1 × 1020/cc N ) 5 × 1020/cc N ) 10 × 1020/cc 14 Å by 27 Å 21.5 Å 34 Å by 17 Å
8 Å by 16 Å 12.5 Å 20 Å by 10 Å
6 Å by 13 Å 10 Å 16 Å by 8 Å
of 5, 10 and 15 Debye strength at our standard conditions of 100 °C and a poling field of 200 V/µm. At a concentration of 1.0 × 1020 dipoles/cc or less, all lattices give approximately the same order parameter, which is proportional to the dipole moment. The exception to this is when the dipole is 15 D or greater, at which point there is already considerable dipole-dipole interaction with the non-offset 2:1 tP lattice (see Figure 5). Looking at just the 2:1 non-offset tetragonal lattice (tP), the concentration for the maximum loading parameter goes from 9
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Chafin and Lindsay
Figure 7. FTC Chromophore.
Figure 5. Variation of the Loading Parameter (N〈cos3 θ〉) with concentration (#/cc) for a 12 × 12 × 12 lattice of dipoles 15 D strong in a field of 200 V/µm at 100 °C. The solid line (with crosses) corresponds to a 2:1 (v/h) offset tetragonal lattice (tF). The dotted line (with stars) corresponds to a 1:1 (v/h) offset cubic lattice (cF). The line consisting of a dash followed by a dot (with solid squares) corresponds to a 1:1 (v/h) non-offset cubic lattice (cP). The dashed line (with solid diamonds) corresponds to a 2:1 (v/h) non-offset tetragonal lattice (tP).
Figure 8. Variation of the energy per dipole (in joules) with concentration (#/cc) for a 12 × 12 × 12 lattice of dipoles 5 D strong in a field of 200 V/µm at 100 °C. The solid line (with crosses) corresponds to a 2:1 (v/h) offset tetragonal lattice (tF). The dotted line (with stars) corresponds to a 1:1 (v/h) offset cubic lattice (cF). The line consisting of a dash followed by a dot (with solid squares) corresponds to a 1:1 (v/h) non-offset cubic lattice (cP). The dashed line (with solid diamonds) corresponds to a 2:1 (v/h) non-offset tetragonal lattice (tP).
TABLE 2: Nearest-Neighbor Distances for Each Lattice Typea lattice type
Figure 6. Variation of order parameter (〈cos3 θ〉) with concentration (#/cc) for a 12 × 12 × 12 lattice of dipoles 10 D strong in a field of 200 V/µm at 100 °C. The solid line (with crosses) corresponds to a 2:1 (v/h) offset tetragonal lattice (tF). The dotted line (with stars) corresponds to a 1:1 (v/h) offset cubic lattice (cF). The line consisting of a dash followed by a dot (with solid squares) corresponds to a 1:1 (v/h) non-offset cubic lattice (cP). The dashed line (with solid diamonds) corresponds to a 2:1 (v/h) non-offset tetragonal lattice (tP).
× 1020/cc for a dipole of 5 D to 3 × 1020/cc for a dipole of 10 D and 2 × 1020/cc for a dipole of 15 D. This shows that in this lattice dipole-dipole interactions are much stronger than those in the other lattices. With a dipole of 5 D (Figure 3), the cubic lattices (cF and cP) and the offset tetragonal lattice (tF) show a nearly linear relationship of the loading parameter with concentration. This indicates little dipole-dipole interaction. The non-offset tetragonal lattice (tP) is already showing that dipole-dipole interactions are important. The dependence of dipole alignment on the concentration and lattice can be seen a little clearer in Figure 6. Here the order parameter (〈cos3 θ〉) is graphed versus concentration. With the offset tetragonal lattice (tF), there is very little degradation of the order parameter with concentration increase as shown in Figure 6. In fact, there is a synergistic effect until the concentration rises above 5 × 1020 dipoles/cc. The cubic lattices (cF and cP) show a nearly linear decrease with concentration
non-offset cubic (cP) 2:1 non-offset tetragonal (tP) offset cubic (cF) 2:1 offset tetragonal (tF) a
nearest-neighbor distance 3
√V 3 0.7937 × √V 3 1.118 × √V 3 1.1225 × √V
V is the volume per dipole.
while the non-offset tetragonal lattice (tP) shows a strong dependence on concentration reflecting the close approach of neighboring dipoles in the horizontal plane. 3.2. Lowest-Energy Lattice. The following question arises: What lattice does a solution of dipoles in a polymer naturally adopt? This is a question of energy and not order. As shown in ref 9 for chromophores CLD and FTC (Figures 2 and 7, respectively) in poly(methyl methacrylate) the electro-optic coefficient (r33) typically peaks at concentrations of between 1 and 4 × 1020 molecules/cc. The lattice that comes closest to this behavior is the non-offset 2:1 tetragonal lattice (tP). We have examined the variation in energy with concentration for the various lattices and have found that the nonoffset 2:1 tP lattice stands out as the lowest-energy lattice even though this lattice has the least vertical ordering (Figure 8). The other lattices, 1:1 cP, 1:1 cF, and 2:1 tF, have approximately the same energy profiles. This can be rationalized by the fact that the dipoles in the 2:1 tP lattice are closer together in the horizontal plane than the other lattices (see Table 2 and the Appendix). A large increase in electro-optic coefficient could be gained if a way were found to constrain the chromophores to an offset tetragonal lattice (tF). Clearly, anything that keeps the chro-
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J. Phys. Chem. C, Vol. 112, No. 21, 2008 7839
Figure 9. Staggered FTC trimer that may facilitate offset lattice packing.
Figure 10. Cubic non-offset lattice cP.
mophores apart horizontally will be beneficial. A padding of inert material around the chromophore has been proposed and demonstrated.1,11 The disadvantage, of course, is that the chromophore is being diluted with inert material. Therefore, it would seem to be more appropriate to maintain the prolate shape for higher chromophore packing density and devise methods of forcing the offset lattice and simultaneously preventing aggregation. In that vein, Figure 9 proposes a chevron-shaped trimer. The relevant geometrical guidelines here are to have the three chromophores connected middle to end to middle. In this way, tF packing would be encouraged. Dendrimer trimers, essentially three-arm stars, have been explored extensively by the Dalton group,12,16 but these materials are different from the chevron structure in that they were connected by longer spacers that did not force alignment. A trichromophore bundle design was also explored, and although it forced parallel alignment of the three dyes, the β was reduced because of detrimental chargetransfer band interactions.17 4. Conclusions In glassy solutions, a system of prolate dipoles has been found to adopt a non-offset tetragonal (tP) lattice as the lowest-energy configuration. A large increase in electro-optic coefficient could be gained if a way were found to constrain the chromophores to an offset tetragonal lattice (tF). A new type of dendrimeric attachment has been proposed, which maintains the individual chromophores’ prolate shape and yet encourages the adoption of an offset tetragonal lattice packing. Acknowledgment. We thank DARPA MTO for supporting this work. Appendix The nearest-neighbor distance can be calculated in the following manner. For each lattice, the volume per dipole is equal to the inverse of the concentration (dipoles/volume).
Figure 11. 2:1 tetragonal non-offset lattice tP.
Figure 12. Cubic offset lattice cf.
For the cubic non-offset case (cP, Figure 10), the volume V ) a × b × c a)b)c 3 Therefore, b ) √V The nearest-neighbor distance is equal to b, which is equal 3 to √V . For the 2:1 tetragonal non-offset case (tP, Figure 11), the volume V ) a × b × c b)c a ) 2b V ) 2b × b × b 3 3 3 3 Therefore, b ) √ V / 2 ) 1 / √2 × √V ) √V 3 The nearest-neighbor distance is equal to b ) 0.7937 × √V . For the cubic offset case (cF, Figure 12), the volume V ) a × b × c a)b)c 3 Therefore, b ) √V
The nearest-neighbor distance is equal to √b2 + ( a / 2 )2 ) b 3 × √b2 + b2 / 4 ) √1.25 , which is equal to 1.118 × √V For the 2:1 tetragonal offset case (tF, Figure 13), the volume V ) a × b × c
7840 J. Phys. Chem. C, Vol. 112, No. 21, 2008
Figure 13. 2:1 Tetragonal offset lattice tF.
b)c a ) 2b V ) 2b × b × b 3 3 Therefore, b ) √ V / 2 ) 0.7937 × √V
The nearest-neighbor distance is equal to √b2 + ( a / 2 )2 ) √b2 + b2 ) b × √2 , which is equal to √2 × 0.7937 × √3 V or 3 1.1225 × √V . References and Notes (1) Kim, T.-D.; Kang, J.-W.; Luo, J.; Jang, S.-H.; Ka, J.-W.; Tucker, N.; Benedict, J. B.; Dalton, L. R.; Gray, T.; Overney, R. M.; Park, D. H.; Herman, W. N.; Jen, A. K.-Y. J. Am. Chem. Soc. 2007, 129, 488–489. (2) Wooten, E. L.; Kissa, K. M.; Yi-Yan, A.; Murphy, E. J.; Lafaw, D. A.; Hallenmeier, P. F.; Maack, D.; Attanasio, D. V.; Fritz, D. J.; McBrien, G. J.; Bossi, D. E. IEEE J. Sel. Top. Quantum Electron. 2000, 6, 69–82.
Chafin and Lindsay (3) Bortnik, B.; Hung, Y. C.; Tazawa, H.; Seo, B. J.; Luo, J.; Jen, A. K.-Y.; Steier, W. H.; Fetterman, H. R. IEEE J. Sel. Top. Quantum Electron. 2007, 13, 104–110. (4) Isborn, C. M.; Leclercq, A.; Vila, F. D.; Dalton, L. R.; Bre´das, J. L.; Eichinger, B. E.; Robinson, B. H. J. Phys. Chem. A 2007, 111, 1319– 1327. (5) Singer, K. D.; Kuzyk, M. G. J. Opt. Soc. Am. B 1987, 4, 968–976. (6) Pereverzev, Y. V. Phys. ReV. E 2000, 62, 8324–8334. (7) Pereverzev, Y. V.; Prezhdo, O. V. Chem. Phys. Lett. 2001, 340, 328–335. (8) Dalton, L. R.; Harper, A. W. Proc Natl. Acad. Sci. USA 1997, 94, 4842–4847. (9) Robinson, B. H. J. Phys. Chem. A 2000, 104, 4785–4795. (10) Nielsen, R. D.; Rommel, H. L. J. Phys. Chem. B 2004, 108, 8659– 8667. (11) Sinness, J.; Clot, O.; Hammond, S. R.; Bhatambrekar, N.; Rommel, H. L.; Robinson, B.; Jen, A. K-Y.; Dalton, L. Mater. Res. Soc. Symp. Proc. 2005, 846, 121. (12) Sullivan, P. A.; Rommel, H.; Liao, Y.; Olbricht, B. C.; Akelaitis, A. J. P.; Firestone, K. A.; Kang, J.; Luo, J.; Davies, J. A.; Choi, D. H.; Eichinger, B. E.; Reid, P. J.; Chen, A.; Jen, A. K.; Robinson, B. H. J. Am. Chem. Soc. 2007, 129, 7523–7530. (13) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987; Chapter 4. (14) The main 12 × 12 × 12 lattice is surrounded by 26 identical 12 × 12 × 12 lattices to give the boundary conditions. (15) Spave, M.; Blum, R. Appl. Phys. Lett. 1996, 6920, 2962–2964. (16) Luo, J.; Haller, M.; Ma, H.; Liu, S.; Kim, T.; Tian, Y.; Chem, B.; Jang, S.; Dalton, L. R. J. Phys. Chem. B 2004, 108, 8523–8530. (17) Liao, Y,; Firestone, K.; Bhattacharjee, S.; Luo, J.; Haller, M.; Hau, S.; Anderson, C.; Lao, D.; Eichinger, B.; Robinson, B.; Reid, P.; Jen, A.; Dalton, L. J. Phys. Chem. B 2006, 110, 5434–5438.
JP7113048
