Modeling the Optical Behavior of Complex Organic Media: From

Departments of Chemistry, Physics, and Materials Science and Engineering, The University of Washington, Seattle, Washington 98195, and Department of C...
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J. Phys. Chem. B 2009, 113, 15581–15588

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Modeling the Optical Behavior of Complex Organic Media: From Molecules to Materials Philip A. Sullivan,† Harrison L. Rommel,† Yoshinari Takimoto,‡ Scott R. Hammond,† Denise H. Bale,† Benjamin C. Olbricht,† Yi Liao,†,⊥ John Rehr,‡ Bruce E. Eichinger,† Alex K.-Y. Jen,†,§ Philip J. Reid,† Larry R. Dalton,† and Bruce H. Robinson*,† Departments of Chemistry, Physics, and Materials Science and Engineering, The UniVersity of Washington, Seattle, Washington 98195, and Department of Chemistry, UniVersity of Central Florida, Orlando, Florida 32816 ReceiVed: August 20, 2009

For the past three decades, a full understanding of the electro-optic (EO) effect in amorphous organic media has remained elusive. Calculating a bulk material property from fundamental molecular properties, intermolecular electrostatic forces, and field-induced net acentric dipolar order has proven to be very challenging. Moreover, there has been a gap between ab initio quantum-mechanical (QM) predictions of molecular properties and their experimental verification at the level of bulk materials and devices. This report unifies QM-based estimates of molecular properties with the statistical mechanical interpretation of the order in solid phases of electric-field-poled, amorphous, organic dipolar chromophore-containing materials. By combining interdependent statistical and quantum mechanical methods, bulk material EO properties are predicted. Dipolar order in bulk, amorphous phases of EO materials can be understood in terms of simple coarse-grained force field models when the dielectric properties of the media are taken into account. Parameters used in the statistical mechanical modeling are not adjusted from the QM-based values, yet the agreement with the experimentally determined electro-optic coefficient is excellent. Introduction A nascent telecommunications revolution is being driven by the development of new organic nonlinear optical (ONLO) materials that promise low-cost, low-drive-voltage EO devices.1–3 These devices are designed to control optical-based information flow compatible with the voltage constraints of existing technologies.4–6 Organic materials are especially suited for this task because their EO properties are determined by electronic rather than nuclear motions, providing for a very rapid response. The interconversion of light and electrical signals requires molecules that both have large hyperpolarizabilites and can be arranged acentrically.7 These two aspects of the problem are being addressed by (1) synthesizing molecules with ever-larger hyperpolarizabilities and (2) exploring a variety of methods to enhance acentric order. There may be limits to the magnitudes of intrinsic molecular hyperpolarizabilities that can be achieved.8,9 Experience in our laboratory has shown that molecular modifications of the fundamental chromophore units that improve the hyperpolarizability generally lead to a red shift of the lowest energy optical transition. Moreover, when the concentration of chromophores becomes larger than an (material-dependent) optimum value, the electro-optic response decreases with increasing concentration.10–12 This report shows how the results of QM calculations of the electrostatic and electro-optic properties of individual chromophores to may be used with simple coarse-grained force field models to describe the interaction of many chromophores using statistical mechanical methods to compute averaged order parameters. Electro-optic properties can be quantitatively pre* Corresponding author. E-mail: [email protected]. † Department of Chemistry, University of Washington. ‡ Department of Physics, University of Washington. § Materials Science and Engineering, University of Washington. ⊥ University of Central Florida.

dicted without resort to any adjustable parameters. The fundamental relation between the (Pockels effect) EO response coefficient, r33, and molecular parameters is given by13

r33 ) 2N βzzz(-ω;0, ω) 〈cos3 θ〉 g(ω)/nω4

(1)

Here, N is the number of EO molecules per unit volume, nω is the refractive index of the material at frequency ω, βzzz(-ω; 0,ω) is the molecular hyperpolarizability,7,13 g(ω) is a LorentzLorenz-Onsager local field factor, and 〈cos3 θ〉 is the order parameter. This order parameter is the average of the third power of the cosine of the angle between the EO molecular dipole axis and the external poling field, θ. The dipole axis is defined as the molecular z-axis, which identifies the appropriate component of the β tensor. The theoretical evaluation of eq 1 appears straightforward: One computes the hyperpolarizability and dipole moment using standard QM methods. The order parameter can then be computed from simple Langevin theory, which requires only the dipole moment of the chromophore and the poling field strength. This approach yields a value of r33 that is reasonably close to experiment, which has justified its use. It seems that theory does give quantitative results. However, in previous work,14 we found that the theoretically determined hyperpolarizability was 3- to 4-fold smaller than the value estimated from the experimental results. The modeling of the order parameter required that we use an empirical dipole moment of 12 D, which was 2-fold smaller than the theoretically determined value. The order parameter, needed to explain the experimental results, was about 3-fold smaller than that predicted by simple, Langevin theory. These unresolved discrepancies leave one with the impression that theory is not capable of quantitatively predicting r33. These discrepancies have been addressed more broadly in the literature. First, the hyperpolarizability determined from

10.1021/jp908057d CCC: $40.75  2009 American Chemical Society Published on Web 10/16/2009

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Figure 1. (Panel a) The chemical structure of trichromophore dendrimer D3. The full name is given elsewhere.14 (Panel b) The molecular structure as presented as united atom ellipsoids, with connecting linkages (coincident with bond vectors) about which rotation can occur. This picture illustrates the model structure employed in united-atom Monte Carlo (MC) calculations. Figure adapted from previous work.14

either electric-field-induced second harmonic generation15,16 or hyper-Rayleigh scattering (HRS)14,17,18 experiments are usually 3- to 4-fold larger than those determined by QM.19,20 The QM calculations have been carried out in vacuum, and the importance of dielectric and environment on the hyperpolarizability of the EO chromophores has been discussed by several groups.21–24 Second, the experimental r33 increases less than linearly with concentration, indicating that the dipolar interactions must be strong enough to reduce order below that predicted by Langevin theory.11,25,26 Third, the dipole moments predicted by QM (for molecules in vacuo) are about 2-3 times larger than those estimated from experiments as needed to describe ordering.14,20,27 This discrepancy is further exacerbated by the simple principle that the dipole moment can only increase (in the case of a neutral ground-state type chromophore) when a molecule is transported from the vacuum to a medium with dielectric constant ε > 1.28 If one were to use the dipole moments given by QM19,29–32 in a classical dipolar interaction picture,27,33 the dipole-dipole interactions would overwhelm the poling field, resulting in a very small order parameter. Given these difficulties, it has been generally concluded that quantum mechanics gives only a qualitative understanding of the relevant molecular properties. In this report,we show that the discord between theory and experiment can be resolved. Theory can quantitatively predict experimental r33 values. We find that the QM-computed dipole moment and polarizability can be used directly in Monte Carlo force-field-based simulations to obtain quantitative predictions of the order parameters and EO coefficients. Careful consideration of the dielectric of the environment is required to achieve this agreement.23 Parameters used to model the core repulsive and attractive dispersion interactions between the coarse-grained (united atom) chromophores have negligible effect on the order parameter and need not be adjusted to fit optical experiments.14,34 The results provide insights into electrostatic interactions in highly polar media, which is essential information as one looks for ways to make improvements in this new technology. Modeling Strategy The material chosen for this study consists of a single molecular component, the trichromophore-containing dendrimer, D3 (Figure 1). The conjugated π-electron core chromophores (attached by ester linkages at the periphery of each arm) are typical of the newest generation of organic EO molecules. Molecules in this class have demonstrated r33 values more than an order of magnitude larger than the current industry standard,

lithium niobate.35 The synthesis of D3, performed through wellestablished methods, was previously described.14,36 Despite its molecular complexity, D3 represents an excellent candidate for comparison of experiment and theory. The D3 dendrimer possesses sufficient mechanical stability and conformational flexibility to allow direct solution casting of optical quality thin films without the need of a polymer host. This attribute allows for the fabrication of pure dendrimer samples that are highly homogeneous molecular ensembles. Quantum mechanics has been used to compute the electrostatic momentssthe dipole moment, µ; polarizability, R; and hyperpolarizability, βsof the individual chromophores. The dipole moment and hyperpolarizability at zero frequency for the isolated chromophores in the gas phase are found by several different QM approaches to give µo ) 24 ( 2D and βzzz(0; 0, 0 ) (640 ( 20) × 10-30 esu.19 The enhancement of 0) ) βzzz both µ and β due to the reaction field of the dielectric medium have been explored for model ONLO chromophores.23 Two different DFT codes demonstrate a similar dependence of µ and β on the dielectric of the environment: The PCM37 method in Gaussian03 with the B3LYP functional and 3-21 g* basis set and the COSMO38 option in DMol3 with the PBE functional and numerical DNP basis set give the results shown in Figure 2 (left). A least-squares fit to the results of the QM calculations to simple functions of the dielectric are well-described by the empirical equations

[

µ(ε) = µ0 1 + 0.5 ×

ε-1 ε+1

]

and

[

0 1 + 4.5 × βzzz(ε) = βzzz

ε-1 ε + 2.5

]

(2)

where µ0 and β0zzz are the vacuum values of the respective dipole moment and hyperpolarizability component. Moreover, the approximation that βHRS(ε) = (1/2)βzzz(ε) is valid when the zzzcomponent of the hyperpolarizability tensor is much larger than any other component.39 Several QM methods were used to explore the effect of frequency on the hyperpolarizability, including real-time, time-dependent density functional theory (RT-TDDFT, SIESTA).32 The SIESTA code was slightly modified to facilitate input-output of quantities needed to calculate the evolution of the wave functions and polarization at each time step, but was otherwise unchanged. The results of computations were compared with linear response DFT (LR-DFT,

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Figure 2. (Lower left) Dipole moment ratio as a function of dielectric constant, relative to µ0 ) µ(ε ) 1): diamonds, Gaussian03 B3LYP/3-21 g*PCM; squares, Dmol3 using PBE/DNP/COSMO. The dotted line represents the dipole enhancement calculated with Onsager theory28 for n ) 1.9 (estimated for the pure chromophore). (Upper left) Hyperpolarizability as a function of dielectric constant relative to β0zzz, which is the hyperpolarizability 0 at ε ) 1. A dielectric constant of ε ) 5 corresponds to βzzz(ε) ) 3.3 · βzzz . The solid lines are best-fit functions. (Upper right) The ratio of βzzz(-ω; 0, ω) to the long wavelength limiting value, βzzz(0; 0, 0), as a function of the frequency of the incident light. The functional form of the two-state model (dashed line) is given for an effective absorption wavelength parameter of λmax ) 734 nm.13,14 Results from calculations are the following: black diamonds, linear response DFT (Dalton); squares, RT-TDDFT SIESTA; blue diamonds, Gaussian 03 Hartree-Fock. (Lower right) The ratio 0 of βHRS(-2ω; ω, ω) to βHRS(0; 0, 0) ) βHRS , along with the two-state model (dashed line) prediction.

Dalton), Hartree-Fock (Gaussian03), and the simple (twostate) model function,13

βzzz(-ω;0, ω) 3 - ω2 /ωmax2 ) βzzz(0;0, 0) 3(1 - ω2 /ωmax2)2

(3)

where ωmax ) 2πc/λmax and λmax ) 734 nm is the frequency at the maximum absorption of the D3 film.13 Data obtained using these various computational approaches are plotted in Figure 2. The Figure demonstrates that both the computationally intensive LR-DFT and RT-TDDFT agree very well with the simple two-state model near the experimental wavelength λexp ) 1.310 µm (ν˜ exp ) 7630 cm-1), where r33 was measured. Figure 2 also shows that the two-state model compares favorably with the more comprehensive LR-DFT and RT-TDDFT methods for frequencies below the lowestenergy electronic transition. The RT-TDDFT method also yields the imaginary part of the polarizability, R, which gives the electronic absorption spectrum with the lowest energy transition at λmax ) 721 nm, in excellent agreement with the experimental λmax ) 734 nm.14 We have not yet computed the frequency dependence of the hyperpolarizability in the presence of the dielectric medium using combined time-dependent and reaction field QM methods. However, we anticipate that the frequency dependence should be somewhat independent40 of the dielectric strength as long as solavtochromic shifts are considered.41 In the high-frequency limit (optical frequencies), the rapidly oscillating external electric field should not significantly alter the reaction field of the

medium. The monotonic increase of hyperpolarizabiltiy with dielectric is not a universal phenomenon for ONLO chromophores. Classes of chromophores (e.g., zwitterionic) exist that show a solvent-dependent blue shift in the wavelength of the maximum absorption and show a more complicated relationship between hyperpolarizability and dielectric.21,42–44 In summary, our QM calculations suggest that the dipole moment is approximately 30 D in a dielectric medium of ε ≈ 4.8-5.0 and that the hyperpolarizability of the chromophore considered here is increased by more than 3-fold over the vacuum calculation. At higher dielectric strengths, the hyperpolarizability may increase more than 5-fold. Moreover, on the low-energy side of the lowest-energy electronic transition, the frequency dependence of β is remarkably well described by the two-state model when compared with QM methods. Thus, for typical frequencies and dielectric strengths, the hyperpolarizability of this class of chromophores can be enhanced 6-8 fold over that predicted by the calculations performed at zero frequency in a vacuum. Applying these factors to model the individual chromophoric constituents of dendrimer D3 in bulk, we now can estimate hyperpolarizability as a function of both the dielectric of the environment and operating wavelength/ 0 frequency. Starting from βzzz(0; 0, 0) ) βzzz ) (640 ( 20) × 10-30 esu and applying eqs 2 and 3 yields the value βzzz(ω, ε) ) (640 × 10-30)(3.3)(1.91) ) 4033 × 10-30 esu. There is no established experimental method to directly measure βzzz(-ω; 0, ω), that is, hyperpolarizability in the solid state (relevant to r33). Therefore, trends in hyperpolarizability are commonly evaluated experimentally using HRS. The following discussion is designed to demonstrate that the theory

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gives a reasonably accurate accounting of the βHRS(-2ω; ω, ω) hyperpolarizability, from which we infer that the same theory also accurately accounts for βzzz(-ω; 0, ω), which is needed to estimate r33. The usual experimental technique measures βHRS(-2ω; ω, ω)17 for the EO chromophore in dilute solution relative to a standard, generally taken to be the solvent, so that βHRS,relative ) βsolute/βsolvent.45 In Figure 2 (bottom right), βHRS(-2ω; ω, ω) values calculated using the RT-TDDFT and LR-DFT codes are plotted together with the simple two-state formula

|

|

βHRS(-2ω;ω, ω) βzzz(-2ω;ω, ω) ) = βHRS(0, 0, 0) βzzz(0, 0, 0) 1 (1 - ω2 /ωmax2)(1 - 4ω2 /ωmax2)

|

|

(4)

using the experimental ωmax given above. More complete forms of eq 4, including line width effects, are available,46 but are not needed for our purposes. The two-state model equation agrees with the two time-dependent DFT calculations on the lowenergy side: ω < 6810 cm-1, or λ > 2λmax ) 1.47 µm.13 This may be somewhat fortuitous, as none of the theories considered here include vibronic motions, which have been shown to make important contributions to the frequency dependence of the HRS determined hyperpolarizability.47 The dielectric constant, ε, of chloroform is about 4.8; the D3 film, as a solid material at room temperature, is found (see below) to have a similar dielectric constant. Therefore, the enhancement of the hyperpolarizability due to the dielectric of the material should be about the same in chloroform and in the bulk. Measurements of βHRS(-2ω; ω, ω) relative to the solvent, βHRS(CHCl3), were performed at λexp ) 1.0 µm (ν˜ exp ) 10 000 cm-1), giving βHRS(-2ω; ω, ω)/βHRS(CHCl3) ) 7800 ( 500 and also at λexp ) 1.9 µm (ν˜ exp ) 5263 cm-1), giving βHRS(-2ω;ω,ω)/βHRS(CHCl3) ) 6088 ( 1950. The calculated value of βHRS(0; 0, 0) ) 320 × 10-30 esu in the vacuum. The presence of a medium with a dielectric of ε ) 4.8 increases the calculated value by 3.3 fold. The resonance enhancement on going to λexperiment ) 1.0 µm light is ∼1.8 fold, estimated from the two-state model expression (eq 4), and for λexperiment ) 1.9 µm, the factor is ∼3.2 fold. Although it may be an oversimplification, we assume the dielectric and frequency multiplicative factors are independent and obtain estimates to the hyperpoTSM (-2ω; ω, ω)ε)4.8 ) 1900 × 10-30 esu and larizability of βHRS TSM βHRS (-2ω; ω, ω)ε)4.8 ) 3379 × 10-30 esu at λexp) 1.0 and 1.9 µm, respectively. The values of βHRS(CHCl3) that would make theory and experiment agree here are βHRS(CHCl3) ) (0.24 ( 0.2) × 10-30 esu and βHRS(CHCl3) ) (0.43 ( 0.15) × 10-30 esu, respectively. If one uses the results of the RT-TDDFT, the (-2ω; ω, ω)ε)4.8 ) 5841 × 10-30 calculated results are βRT-TDDFT HRS TSM esu and βHRS (-2ω; ω, ω)ε)4.8 ) 2950 × 10-30 esu at λexp ) 1.0 and 1.9 µm, respectively. Thus, using the RT-TDDFT results, βHRS(CHCl3) ) (0.73 ( 0.2) × 10-30 esu, and βHR-30 esu. This data approximately S(CHCl3) ) (0.48 ( 0.15) × 10 spans the reported values for βHRS(CHCl3).48,49 Additionally, the close agreement between the TSM and RT-TDDFT derived values at λexp ) 1.9 µm (ν˜ exp ) 5263 cm-1) suggests that the TSM may be even less applicable at optical energies greater than the lowest energy optical transition. The differences in the estimates to βHRS(CHCl3) using different frequencies are probably due to the inadequateness of the theory to accurately exp (-2ω; ω, ω)ε for describe the frequency dependence of βHRS 47 the chromophore.

The Dielectric of the Film. A computation of the electrostatic interactions in the context of statistical mechanical simulations requires the effective dielectric of the material. To computationally estimate the dielectric constant of the dendrimer, we assume additivity of the contributions from the three chromophores and the dendrimer core. The mean index of refraction, n, for any of the components may be estimated from28

n2 - 1 4π tr{R(-ω;ω)} ) F0 3 3 n2 + 2

(5)

where the polarizability tensor, R, is computed from RTTDDFT, and F0 is the density of the pure material. Using eq 5 and the R tensor computed from the DFT codes (Gaussian03), the chromophore has an index of refraction of n1 ) 1.95. In the D3 dendrimer, the chromophore comprises about 60% (φ1 ) 0.6) of the molecular volume. The index of refraction of the core is similarly estimated to be n2 = 1.5, typical of transparent organic materials. The average index of refraction,nj ) φ1n1 + φ2n2, is nj ) 1.8 at λ ) 1300 nm and nj ) 1.67 at λ ) 2000 nm. Experimental values for nj for D3, measured by variable angle spectroscopic ellipsometry,50 are nj ) 1.84 at λ ) 1330 nm and nj ) 1.76 at λ ) 2137 nm, which agree well with theory. The classical theories of dielectric media51,52 relate the molecular dipole, mj, of the jth species to the scalar dielectric constant ε when polarization is induced by an external field of strength Epol applied in the direction specified by the unit vector e. Kirkwood (and Onsager) suggested using a sphere of volume V containing N molecules with dipole moment m within an infinite material of dielectric ε. The density of dipoles is F ) N/V, and the total dipole moment is M ) ∑iN) 1 mi. For a single species of dipoles, one has (ε - 1)Epol ) 4π〈e · M〉/V, and for several different species of dipoles, the straightforward average is

(ε - 1)Epol ) 4π

∑ j

〈e · Mj〉 ) 4π Vj

∑ Fj〈e · mj〉

(6)

j

Here, the sum is over the different species present in the material, and Fj is the molecular density of the jth species. For the case that there are no permanent dipoles and only the molecular polarizabilities contribute to the dielectric constant, now called ε0, the Onsager and Debye28 theories give

ε0 - 1 )

2

3ε0(nk2 - 1)

k)1

2ε0 + nk2

∑

φk

(7)

with use of eq 5. Using the refractive indices given above, we find ε0 ) 3.3. This represents the minimum value for the dielectric constant because it does not include the contributions from reorientable permanent dipoles. As such, it is less than the measured dielectric constant of 5.0 ( 0.2, obtained from the solid thin film at room temperature using an oscillating field less than 1 kHz in modulation.53 The difference between the measured dielectric constant of 5.0 and the estimated 3.3 value represents the contribution to the dielectric constant from dipoles in the solid material. This discrepancy between the dielectric constant calculated solely from the QM polarizability and the experimental value is within our expectations for the dipolar contribution typically seen in polymers.54

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Monte Carlo Simulations We now proceed to determine the order parameter, 〈cos θ〉, required in eq 1. In the dilute limit, the dipoles are independent of one another, which yields the Langevin result for the order parameters.11 However, in condensed phases, the dipole-dipole interactions contribute to the ordering forces. This requires a modeling strategy to simulate the combined influences of both the strong electrostatic and weaker dispersion interactions among chromophores. Experience with molecular dynamics simulations with all-atom force fields suggests that the complex dendrimer of this study would be extremely difficult to bring to equilibrium.26,34,54 Instead, we have chosen to use a Monte Carlo procedure with a united atom force field.14 Computer simulations were run in the NVT ensemble using Monte Carlo (MC) methods.34,55 Theoretical treatment of the D3 dendrimer began by modeling the molecule as a series of ellipsoids with connecting linkages about which rotation may occur. Figure 1, panel b, illustrates the mapping of the underlying molecular structure into the corresponding ellipsoidal representation. This mapping transforms the system into a tractable, coarse-grained or united atom model, and accelerates the calculations several-fold. The rotation of one ellipsoid relative to another is constrained to occur about the linkage that is coincident with the backbone bond connecting the underlying pair of atoms. By using linkages coincident with bonds, we are able to map the ellipsoidal structure back to the underlying molecular structure. In the model used for D3 shown in Figure 1, each of the three large ellipsoids represents a single chromophore, and each of these contains a dipole placed at the center of the ellipsoid and directed along the major axis. This model is consistent with the QM calculation of the dipole moment of this chromophore (to within a few degrees of angle owing to some ambiguity in the orientation of the ellipsoid with the asymmetric molecule’s axes). The interior of the Kirkwood sphere is considered to have unit dielectric and field, E0, where E0 ) fLEpol, and the energy of the interaction of each dipole with the electric poling field is 3

Upol )

∑ mi · E0 ) fLEpole · M i

Here, fL ) 3ε/(2ε + 1) is the Lorentz-Lorenz cavity correction factor. For the case of a prolate spheroid with an axial ratio of about 2, the correction due to the elliptical shape of the chromophores is minimal.56 Although the polarizability can be treated in detail,25,26,51 we consider a simplifying approximation: The net effect of smallmoment dipoles spread throughout the material together with a uniform polarizability may be treated in terms of a continuum dielectric, ε0. When this approximation is made, the dipoles of the chromophores are considered to interact with the poling field and with each other within a medium of dielectric, ε0. The dipole-dipole interaction energy, Uij, is56

Uij ) µi · Tij · µj /ε0

where

Tij )

(1

- 3rˆijrˆij† ) rij3

(8) where the µi are the permanent dipole vectors modeled with a fixed magnitude, using the QM estimates. Here, rˆij is a unit vector joining the two dipoles that are separated by the distance, rij. The dielectric term in eq 8, ε0, is that part of the dielectric

contributed by the linear polarizability of the system, which many be computed from eq 7.51,56 The total dielectric, ε, given by the Kirkwood theory, eq 6 in the present treatment, requires an estimate of the dielectric constant. This ε0 arises from the small, rapidly reorientable dipoles of the matrix and from the polarizability of the entire medium. To estimate the contribution due to the carbonyl, CHx, etc., groups, we use the Onsager approximation for the mean polarization of a single species according to

(

)[

]

〈e · m〉 ε(n2 + 2) 〈e · µ〉 ) +R Epol E0 (2ε + n2)

(9)

Onsager’s theory takes into account both polarizability and dipolar reorientation and in practice gives results that are quite consistent with experimental observations.28,52 The term 〈e · µ〉 ) µ〈cos θ〉 shows the connection between the average dipole moment and the net dipolar order, 〈cos θ〉, relative to the poling field. Equations 6 and 9 are combined to give a single equation for the effective dielectric. The effective dipole moment of each carbonyl is estimated to be µs ) 1.5, as is typical of organic units. This leads to a dielectric constant ε0 ≈ 7, which is comparable to the dielectric of typical polymers: PMMA, PVA, and PVC.54 This dielectric comes from the contribution of the medium, omitting the chromophore dipoles, to the dielectric when the material is near the poling temperature and the arms of the dendrimers are considered free to rotate. The observation that the hardened material, containing the chromophore dipoles, is only 4.8 reflects only the restricted motion of the dipoles at room temperature.53 Using eq 8, the dielectric can be interpreted to scale the magnitude of the chromophore dipoles so that molecules with µ ) 32 D in a medium with ε0 ) 7 have the same average interaction energy as those with µ ) 12 D = (32/7j) D in a vacuum (ε0 ) 1). The dipole moments of many of the older chromophores are computed to be in the 12-15 D range;20 however, quantum mechanical calculations estimate the dipole moment for the choromophore within D3 to be in excess of 25 D.19 The presence of a polarizable medium may reduce the apparent, experimental dipole moment by around a factor of 2. The united atom force field also includes a pairwise additive Lennard-Jones interaction energy that is a function of the centerto-center distance, Rij, and the distance between centers of the two ellipsoids if the two were just touching, Roij.57 For the sake of simplicity, a single energy parameter, σ, is assigned to each and every ellipsoid, regardless of size. The van der Waals energy, ULJ, is modeled by a 6-12 Lennard-Jones (LJ) potential.

{( ) ( ) }

ULJ(Rij) ) 4σ

(

Rijo Rij

(

12

-

Rijo Rij

6

+W

Rij - 21/6Rijo - 2 1 W ) 1 - tanh 2 2

))

(10)

The additional factor, W, removes much of the attractive part of the LJ potential and is inserted to allow greater motion during MC moves. The presence of the LJ repulsive potential avoids ellipsoidal overlap. The order obtained in the MC simulations is insensitive to the strength of the LJ energy parameter, σ, as the LJ parameter is varied over a rather wide range: 0.1 e σ/kBT e 2.0.34 The MC model begins to “freeze up” when the LJ energy parameter,

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Figure 3. (Left) Experimentally determined electro-optic coefficient as a function of the poling field strength applied across the D3 dendrimer sample film near Tg. (Right) Order parameter computed from MC methods for D3 dendrimer as a function of the poling field strength. Dashed line represents the independent particle (or Langevin) limit. The error bars on the theoretical points are the standard error from the order parameters of the parent distribution obtained from 500 independent trajectory runs. Figure adapted from previous work.14

σ/kBT, is larger than ∼1.0. To simulate the molecular reorientation near the glass transition temperature, Tg, for the dendrimer, the LJ energy parameter was set to σ ) 0.1kBT, which is ∼1 order of magnitude smaller than typical LJ parameters used in all-atom simulations. The molecular order parameter, 〈cos3 θ〉, was evaluated from runs over multiple configurations at a temperature corresponding to the experimental poling temperature of 370 K.34 The NVT-MC computational model of the interacting dendrimer ensemble consists of 150 dendrimers in a box at a density of 1 g/cm3 and a number density of N ) 6.45 × 1020 molecules/ cm3. To simulate the bulk material, minimum image, periodic boundary conditions were used.55 Results are shown in Figure 3. Experimental Results The sample was spin-cast onto an ITO slide and kept near the glass transition (Tg ) 96°C) under a poling field, Epol, while it was allowed to come to equilibrium. The poling field was measured directly from the voltage, V, and the distance, d, between the two electrodes as Epol ) V/d. The experimentally determined r33 (left side of Figure 3) was measured using the typical reflection ellipsometry method after returning the sample to room temperature and removing the poling field.58 Although it has been suggested that simple treatment of data obtained using this technique is subject to significant experimental error, by maintaining sample parameters within the guidelines presented in the literature, the error can be limited to e20%.59 The slope of the best-fit line is r33/Epol ) 1.42 ( 0.04 (nm/V)2. The error is obtained from the least-squares fit. Figure 3 (right) shows the dependence of 〈cos3 θ〉 on the poling field strength for D3 computed from the NVT-MC simulations. The order is proportional to the poling field, which parallels the findings of the experimental system (unlike the independent particle model shown by the dotted line). The slope is much lower (about 4-fold) than that seen using an independent chromophore model. The resulting order parameter is 〈cos3 θ〉 ) 1.35 × 10-3 · Epol. Similar NVT-MC simulations of simple 1:2 prolate ellipsoids, at an equivalent chromophore density, gave 〈cos3 θ〉 ) 1.1 × 10-3 · Epol. The two quite different systems give remarkably similar order parameters.14,34 The odd-power (acentric) order parameters cannot be measured directly by any known spectroscopic method. However, the even-power order parameters, such as 〈P2〉 ) (1/2) (3〈cos2θ〉 - 1), can be measured using polarized absorption

Sullivan et al. spectroscopy. Thin films of D3 were measured before and after poling; no alignment was detected before the film was poled. After poling, the order parameter 〈P2〉 ) (0.75 ( 0.02) × 10-3 · Epol was measured.60,61 For comparison, simulations gave 〈P2〉 ) (0.5 ( 0.1) × 10-3 · Epol, with errors estimated from the variances among different runs.34 The MC simulations, at this poling strength, show that 〈P2〉 ≈ 0.4〈cos3θ〉. At such low order parameters, both the experimental and theoretical values are difficult to determine. Additionally, the simulations showed that the ratio of the two acentric order parameters remained nearly constant, 〈cos3θ〉/〈cosθ〉 ≈ 0.6, over a wide range of densities.14,34 The measured dielectric constant of the solid film at room temperature was about 4.8. However, the total polarization under poling is much higher because the dipoles of both the chromophores and the dendritic matrix medium are able to rotate more freely, as discussed above. A modifed OnsagerKirkwood theory gives

(

εtot - 1 ) 4π F

)

fL · µs2 µ 〈cos θ〉 + Fs + Epol 3kT 2

3εtot(nk2 - 1)

k)1

2εtot + nk2

∑

(11)

φk

which can be used as a guide to the expected total dielectric under poling conditions. The quantity 〈cos θ〉/Epol, referring specifically to the chromophore order determined by the MC simulations, is a modification of Onsager’s equation and makes this estimator for the dielectric similar to the Kirkwood theory, which uses the approximation that

〈Mz〉 Fµ〈cos θ〉 1 ) ≈ f 〈M · M〉0 Epol VEpol 3kTV L

(12)

The order parameters can be simulated by either directly applying a poling field or by using Kirkwood’s method.62 The total dielectric (calculated from eq 11) is εtot ) 25. The dielectric constant, when measured near the poling temperature, was found to be εexp ) 27 ( 2.53 This level of agreement supports the idea that the dipole moments and linear polarizability are correctly estimated from QM-DFT methods, and that the fieldinduced ordering as well as the order parameters are welldescribed by statistical mechanics. The final factor to be computed is the Lorenz-Lorentz-Onsager local field factor, g(ω), for the thin film of D3 at room temperature, which is given by13

()

fω g(ω) 0 l ) fl 2 nω4 nω

2

)

ε(n02 + 2) n02 + 2ε

·

( ) nω2 + 2 3nω2

2

) 0.52

(12a)

Here, we have used n0 ) 1.7, nω ) 1.85, and ε ) 4.8 (as determined by variable angle spectroscopic ellipsometry; see the Supporting Information). All terms required in eq 1 have been individually determined and can be used to estimate the EO coefficient where Epol ) 104 V/µm as

Optical Behavior of Complex Organic Media r33 ) 2 ·

J. Phys. Chem. B, Vol. 113, No. 47, 2009 15587

4π · 6.45 × 1020 · (0.14 ( 0.02) · 3 × 104 4033 × 10-30 · 0.52 × 1012 pm/V r33 ) 1.53 ( 0.2 (nm/V)2 Epol

tion. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

(13)

which is in remarkably good agreement with the experimental value of 1.42 ( 0.02 (nm/V)2. Conclusions The electro-optic coefficient, a measure of the performance of an organic nonlinear optical material, has been difficult to estimate theoretically because it is the product of the hyperpolarizability and the order parameter, both of which are difficult to determine in the solid state. Both terms are heavily dependent on the specifics of the molecular environment and are not independent of one another. In this work, we have estimated each quantity from quantum mechanical simulations and have avoided the use of adjustable parameters. Each theoretical estimate has been closely related to experimental measurement to support the hypothesis that the quantum mechanical values are, in fact, quantitatively accurate. The parameters needed for the statistical mechanical calculations of the order parameter (the dipole moment of the individual constituents of the molecule D3, the molecular size and shape, and the polarizability or high frequency dielectric constant) are all estimated from related quantum mechanical calculations. Accurately estimating the electro-optic coefficient requires a detailed understanding of the dependence of the hyperpolarizability on both the dielectric of the environment and the frequency of the light field. Although the detailed modeling of the effects of dielectric and frequency required extensive QM calculations, the parametrization of those calculations showed that a very simple form for the dielectric dependence accurately describes the dependence on dielectric, whereas the two state model (which takes into account the dispersion effects) adequately describes the dependence on frequency. The modeling of the poling-induced order requires proper treatment of the effective screening of the dipole-dipole interaction by the polarizability portion of the matrix dielectric. The close connection between the acentric order parameter, 〈cos3 θ〉, needed for the EO coefficient and the acentric order parameter 〈cos θ〉 needed to estimate the dielectric constant allowed us to validate the estimates of the order parameters from experimental measurements of the dielectric constant, and the centrosymetric order parameter, 〈P2(θ)〉. Two main conclusions of this work are that the poling field-induced acentric order in the material is much lower than anticipated from previous estimates,11,27 and that the dielectric of the entire material is a very important component in the design of the EO properties. Acknowledgment. The authors acknowledge the support of the STC-MDITR Program of the National Science Foundation (DMR0120967) and NSF-(DMR-0092380). Support from the Air Force Office of Scientific Research under AFOSR-(F4962003-1-0110-P000) as well as from the DARPA MORPH program Phase I ((N) 14-04-10094), is also gratefully acknowledged. The authors thank Professor Bart Kahr for many helpful discussions and also for help in analysis of the VAPAS data. Supporting Information Available: Complete details of computer modeling, experimental material characterization, and expanded discussions thereof, are given as Supporting Informa-

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