Reduced Dimensionality in Organic Electro-Optic

Materials: Theory and Defined Order

Stephanie J. Benight, Lewis E. Johnson, Robin Barnes, Benjamin C. Olbricht, Denise H. Bale,

Philip J. Reid, Bruce E. Eichinger, Larry R. Dalton, Philip A. Sullivan and Bruce H. Robinson*

Department of Chemistry, University of Washington, Seattle, WA 98195

robinson@chem.washington.edu

Supporting Information

SAXS results for C1:

Thin films of C1 appeared glassy and amorphous with no visible crystalline domains.

Further investigation employing small-angle X-ray scattering (SAXS), shown in Figure S-1,

confirmed this result.

Figure S-1. Small angle X-Ray scattering results for spin coated samples of C1 on glass

substrates. No sharp peaks are present that would indicate an ordered array of crystalline

domains; only an “amorphous halo” is present.

Structure of P3:

Figure S-2: P3 dendrimer used for comparison. Previously reported as PSLD-33.1

Characterization details:

Details regarding the Attenuated Total Reflection (ATR) for acquisition of EO

coefficients and the VAPRAS instrument are presented elsewhere.2-5

A schematic of the different layers of a poled substrate is shown in Figure S-3.

1

EO

EO

out EOEO EO

EO out

VApprox E

d

dE E

d

εε

′=

′ = ⋅ +

Figure S-3. A schematic of different layers of an electro-optic (EO) device is shown.

Utilizing a ~90 nm thick TiO2 layer causes negligible change in the applied poling field

as shown in the derivation below.

out out EO EO

out out EO EO

EOout EO EO

out

D E E

V d E d E

V d d E

ε ε

εε

= ⋅ = ⋅

= +

= +

The required parameters used for these materials are shown in Table S-1.

Table S-1. Values for dielectric constants and thickness for the material systems presented.

Parameters Values

εEO 4 - 5.5

εout ~406

dEO ~ 1.0 micron

dout ~ 0.090 micron

Substituting the values in Table S-1 into the equation for EO

E ′above and setting EEO = 50 V, we

see that the new applied field across the EO layer is 50.6 V, a negligible change in applied field.

Simulation methods:

Our group has developed custom RBMC code in C++, compiled with ICC 11.1,7 and run

on the “Stuart” ROCKS 5 cluster at the University of Washington. Simulations were run using

the Metropolis Monte Carlo technique, using self-consistent reaction field boundary conditions

and the Hamiltonian

U = 4εLJ

σ eff (i, j)

rij

12

+1

ε∞rij

3µµµµ i ⋅µµµµ j − 3 µµµµ i ⋅ r̂( ) r̂ ⋅µµµµ j( )( )− 1

ε∞RC

3

2 ε − ε∞( )2ε + ε∞( )

µµµµ i ⋅µµµµ j

j>i

∑ −3ε

2ε + ε∞

µµµµ i ⋅Ep

i

∑ (S1)

where σ eff is the effective Lennard-Jones contact distance, calculated for every pair of ellipsoids

using the method of Perram and Wertheim,8, 9 ε is the static dielectric constant, ε∞ is the optical

dielectric constant, rij is the distance between the centers of the ith and jth ellipsoids, µµµµ is the

dipole moment of an ellipsoid, and RC is the cutoff radius for electrostatic interactions. Only

repulsive inter-nuclear interactions were considered; the attractive component of the Lennard-

Jones potential was neglected.

The dielectric constant

ε =4πM z

VboxEp,z

+ ε∞ (S2)

was self-consistently calculated10-12 at the beginning of every MC cycle (every m trial moves)

based on the total dipole moment M of the simulation box, with volume 1box

V mN−= (here, m

denotes the number of particles and N, the number density), in response to the external field Ep,

which was defined along the z-axis. The local field E0 acting on each ellipsoid was then

calculated as the field in the center of a cavity of dielectric constant ε∞ in an infinite region of

dielectric constant e, using the modified Onsager cavity field13, 14

E0 =3ε

2ε + ε∞

Ep (S3)

The high-frequency dielectric constant ε∞ = 2.25 was assumed to be the square of the refractive

index of PMMA, and the polarizability of the ellipsoids themselves was neglected ( )gasµ µ= .

Electrostatic interactions, including the external field, were switched on 1000 cycles into each

simulation at the three lower densities. Simulations at N = 5.04 × 1020 molecules/cc used 2000

randomization cycles combined with simulated annealing during the initial portions of the

simulation, and simulations at N = 6.30 × 1020 molecules/cc used 5000 randomization cycles and

simulated annealing.12 Only Lennard-Jones interactions were considered during the

randomization phase in order to break up the simple cubic lattice that all simulations were started

from. Maximum move sizes were optimized to maximize the RMS translational and rotational

displacement per move. Total energy for the simulations was recorded every cycle (m steps) and

simulations periodically checked to see if the energy appeared stable. A typical graph of the

energy convergence is shown in Figure S-4.

Figure S-4. The convergence data for a simulation run with Epol = 50 V/µM at N = 2.5 × 1020

molecules/cc is shown as an example of the typical energy convergence observed in these

simulations. U represents the potential energy; µE is the interaction of the dipoles with the poling

field, µµ represents the dipole-dipole interactions, and LJ is the Lennard-Jones potential.

While runs at the lower three densities converged easily, with 40000 cycles sufficient to

obtain convergence and a reasonable averaging range, the higher density simulations required

longer runs and use of simulated annealing12 to obtain convergence. Runs at N = 5.04 × 1020

molecules/cc were run for 80000 total cycles, averaging over the last 20000, and runs at N = 6.30

× 1020 molecules/cc were run for 300000 cycles, averaging over the last 50000. Convergence

was frustrated by the slow formation of the lower-energy centrosymmetrically ordered phase;

shorter runs showed higher energies and lower

P2

values.

DFT Confirmation of ( 2 , , )HRS

β ω ω ω− :

DFT calculations were run for the core chromophore of C1 and F2. The structures for

both C1 (see Figure S-5) and F2 (see Figure S-6) were optimized in chloroform and computed

with Gaussian0915 using the B3LYP level of theory with the 6-31G* basis set. The solvent

reaction field was calculated with the PCM method using default parameters for the solvent. The

wavelength for frequency dependent properties was set to 1906nm, the frequency used for HRS

experimental measurements. All values are given in the Taylor Series convention.

The calculated value for ( 2 , , )HRS

β ω ω ω− of C1 compared quite well to the

experimentally ascertained value (Figure S-5).

Calculated [units = ( × 10-30 esu)]

Experimental [units = ( × 10-30 esu)]

( ;0, )zzz

β ω ω− 2830 3000 ± 300

Figure S-5. Structure of the optimized chromophore core (left). Comparison of DFT and

experimental HRS hyperpolarizabilities (right). These values were corrected to 1310nm, the

frequency of the ATR measurements of EO activity.

The structure of F2 as used in hyperpolarizability calculations is shown in Figure S-6.

The TDBMS protecting groups were omitted.

Figure S-6. Optimized structure of F2 for hyperpolarizability calculations.

Hyperpolarizabilities for F2 were calculated at two wavelengths, 1906nm (corresponding

to HRS experiments) and 1310nm (corresponding to ATR experiments). The values are given in

Table S-2.

Table S-2. Hyperpolarizabilities [units = ( × 10-30 esu)] for the optimized structure of F2.

Freq. dep. β 1310nm 1906nm

βHRS(0;0,0) 1080 1080

βzzz(0;0,0) 2438 2438

βHRS(-ω;0,ω) 1563 1068

βzzz(-ω;0,ω) 3540 2409

βHRS(-2ω,ω,ω) 7518 2027

The calculated value of βzzz(-ω;0,ω) = 3540 at 1310nm is nearly identical to the experimentally

measured value of βzzz(-ω;0,ω) = 3500.

Calculation of 3cos θ :

The equation used to calculate 3cos θ from experimental parameters with proper conversion to

SI units:

( )3 20 12 3033 4 4

( )4cos 2 ( 10 ) 10 ( ( ;0, ) 10 ) /

3 10p p zzz

gE r E N pm V

ω

ωπθ β ω ω

η−= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅

⋅ (S4)

Further derivations of Reduced Dimensionality Theory:

For the three-dimensional case, in the specific case of n=1, the first acentric order parameter

is:

( )113

1 1cos coth( )

f f

f fD

e eL f f

e e f fθ

−

−

+= = − = −

− [S5]

For the two dimensional case, dipolar rotation is limited to a single plane containing the z-axis,

the direction of the poling axis. The integral can be reduced to

2cos

022

cos

0

cos

cos

n f

n

Df

e d

e d

πθ

θπ

θ

θ

θ θθ

θ

=

=

=∫

∫ [S6]

The integral above may be written in terms of Bessel functions, ( )nJ z . In the specific case of

n=1, the result is

( )( )

1 cos

11

cos20

cos

cos

f

fD

e di J i f

J i fe d

θ

θθ

θ

θ θθ

θ

⋅ − ⋅= =

− ⋅

∫

∫

[S7]

In the one-dimensional case, the dipole vectors can point only in the direction parallel or

antiparallel to the field, so the probability distribution is the sum of two Dirac delta functions

at cos 1θ = ± , giving

( ) ( ) ( )1

1 1 tanhcos

1{

n nf f

n

f fD

e e f odd n

e e even nθ

+ −

+ −

+ + −= =

+ [S8]

At low values of f the order parameters for 1n = are linear in f for all dimensionalities, with

the slopes being 1/1, 1/2, and 1/3 for the three different dimensions. In the limit of low f:

1 1 1

1 2 3

1

0

1 1 11 2 3

1

cos cos cos

cos

D D D

M Df M

f f f

Lim f

θ θ θ

θ−→

= = =

=

For the 2D case, we can quantify the order parameter 3

2cos

Dθ in terms of Bessel functions

using the identity:

( )3 14cos cos3 3cosθ θ θ= +

( )

( ) ( )( )

( )( )

2 23 cos cos

3 0 02 22

cos cos

0 0

3 1 3 1

20 0

14

31 14 4 4

cos cos3 3cos

cos

3cos

f f

Df f

D

e d e d

e d e d

J if J if iJ ifi

J if J if

π πθ θ

θ θπ π

θ θ

θ θ

θ θ θ θ θθ

θ θ

θ

= =

= =

+

= =

− − + − − −= = +

− −

∫ ∫

∫ ∫

In the low f limit the third-power order parameter is proportional to the first-power order

parameter. The ratio depends on the dimensionality:

3 1 3 1 3 1

1 1 2 2 3 3

3 1 3 1

0

3 3

10

3 3 353 4

32

cos cos cos cos cos cos

cos cos cos cos

cos cos3lim 1

cos 2 cos

D D D D D D

M D M D M D M Df f

M D M D

f fM D M D

MLim Lim

or

LimM

θ θ θ θ θ θ

θ θ θ θ

θ θ

θ θ

− − − −→ →∞

− −

→ →∞− −

+

= = =

= =

= =+

For the 2D case, we want 2

2cos

Dθ . To get this order parameter in terms of Bessel functions we

use the identity:

( )2

2

2

12cos cos 2 1

3cos 1 3cos 2 1

2 4P

θ θ

θ θ

= +

− += =

Then the second moment is related to the appropriate Bessel Functions:

( )( )

2cos

202 22

cos 0

0

14

3cos 2 1

341

f

Df

e dJ if

PJ if

e d

πθ

θπ

θ

θ

θθ

θ

=

=

+ −

= = − −

∫

∫

For the one-dimensional case the identity holds for all values of f . For the two- and three-

dimensional cases, the ratio increases and goes to 1 for all dimensionalities as f goes to infinity.

(All order parameters also go to one as f goes to infinity.)

One can use the generators of the order parameters to relate order parameters directly to one

another. For example:

33

cos

cos cos

n nn n

n n

G d Gy whereG

G df

G Gy and y

G G

θ

θ θ

′′= = =

′ ′′′= = = =

We can now relate the third and first moment order parameters using the generating function:

1

2

1

2

1

n

n n

n n

n n n

nn n n

Gd

G G GG

df G G

G Gd d

G GG G G GG

G df G df G G

dy dy y y y y

df df

+

+

+

′ ′ ′ ′= −

′ ′ ′ ′ ′ ′ ′= + = +

= + ⋅ = + ⋅

We just apply this generator of the order parameters twice to get

( )

22

2 33 2 3

dy y y y y

df

d dy y y y y y y yy y

df df

′= + ⋅ = +

′ ′ ′′= + ⋅ = + + = + +

This is a general method to relate the first and third order parameters for all dimensionality

problems. For example, for the one-dimensional case

22

3

tanhtanh 1

d fy f

df

y y

= + =

=

For the three-dimensional case one finds

2

2

22

21

21

3 1 31

2

yy y

f

yy

f

y yP

f

′ = − −

= −

−= = −

Then

3 2 2

2 2

2 2 2 21 1

2 2 2 2 21 1 3 1 1

23 1

d d y y y yy y y y y

df df f f f f

y y y y yy y y y

f f f f f f f

yy

f f

′ = + ⋅ = + ⋅ − = − + −

= − − − + − = + − + −

= − +

From here one can directly obtain the ratio:

( )( )

32 2

2 tanh6 2 61 1

tanh

fy

y f fy f f f= + − = + −

−

which limits to 0.6 as 0f → and 1 as f →∞ .

As an example, we can show that the ratio limits to 0.6 by writing:

( ) ( ) 2 4

32 2 2 2

2 4

32 2 2

1 23 15

13

1 23 15

1 1 13 3 3

tanhtanh (1 ) 1

6 2 (1 ) 6 (1 ) 6 2 1 11 1 2 3 3 2

(1 )

6 33 2 3 2 3 2

5 5

ff f f f

f

y f

y f f f f f f

f fy

y f f f

ζ ζ

ζ ζζ ζ ζ ζ

ζ

ζ

= − = − ≈ −

− −= + − = + − = + − = + − − −

− −= + = + = − ⋅ = ⋅

As f →∞ 1ζ → so

2 2

32 2

1 13 3

1 13 3

3 2 3 2 3 2 1f fy

y f f

ζ

ζ

− −= + → + = − =

From this, we get both limits from the same form for tanh(f).

For the 3D case in the low f limit, y goes as:

( ) ( ) ( )

( )

( ) ( ) ( )

2 2

2 2 2

1 253

1 2 1 2 1 15 53 3 3 15

tanhtanh (1 ) 1 1

1 1 1 1 1

tanh (1 ) (1 )

1 (1 ) 1 1

ff f f f

f

yf f f f f

y f f f f f f

ζ ζ

ζζ ζ

ζ ζ

= − = − ≈ −

= − = − =− −

≈ − + = − + = −

From the relation of y and 2P above, we get the leading term for 2P

( )2 22

1 115 15

31 1 1

yP f f

f= − ≈ − − =

In the low f limit we can relate y, y3, and 2P

22 3

3 2 2

1 1 153 15

315 515

y f P f y f

y P P

= = =

= =

This low f limit is extremely good, even to much larger f values than one should have any right to

use. An empirical functional form that works very well for any value of f is:

( ) ( )33

3 2 2 2 23 3 325 5 59y P P P P≈ + ≈ +

With less than a 0.8% error (which occurs around f=12) over the entire range of f. For 2P less

than ½, the extra term gives less than a 5% enhancement to y3, so for most practical cases, the

low f limit estimate is quite good.

References:

1. Sullivan, P. A.; Rommel, H.; Liao, Y.; Olbricht, B. C.; Akelaitis, A. J. P.; Firestone, K. A.; Kang, J.-W.; Luo, J.; Choi, D. H.; Eichinger, B. E.; Reid, P.; Chen, A.; Robinson, B. H.; Dalton, L. R., Theory-Guided Design and Synthesis of Multichromophore Dendrimers: An Analysis of the Electro-Opitc Effect. J. Am. Chem. Soc. 2007, 129, 7523-7530.

2. Sullivan, P. A.; Rommel, H. L.; Takimoto, Y.; Hammond, S. R.; Bale, D. H.; Olbricht, B. C.; Liao, Y.; Rehr, J.; Eichinger, B. E.; Jen, A. K.-Y.; Reid, P. J.; Dalton, L. R.; Robinson, B. H.,

Modeling the Optical Behavior of Complex Organic Media: From Molecules to Materials. J.

Phys. Chem. B 2009, 113, 15581-15588.

3. Sullivan, P.; Dalton, L. R., Theory-Inspired Development of Organic Electro-optic Materials. Acc. Chem. Res. 2009, Online August 7; http://pubs.acs.org, doi:10.1021/ar800264w.

4. Dalton, L. R.; Sullivan, P. A.; Bale, D. H., Electric Field Poled Organic Electro-optic Materials: State of the Art and Future Prospects. Chem. Rev. 2010, 110, (1), 25-55.

5. Davies, J. A.; Elangovan, A.; Sullivan, P. A.; Olbricht, B. C.; Bale, D. H.; Ewy, T. R.; Isborn, C. M.; Eichinger, B. E.; Robinson, B. H.; Reid, P. J.; Li, X.; Dalton, L. R., J. Am. Chem.

Soc. 2008, 130, (32), 10565-10575.

6. Kim, S. K.; Kim, W.-D.; Kim, K.-M.; Hwang, C. S.; Jeong, J., High dielectric constant TiO2 thin films on a Ru electrode grown at 250 °C by atomic-layer deposition. Appl. Phys. Lett.

2004, 85, (18), 4112.

7. Intel C++ Compiler, 11.1; Intel, Inc.: 2009.

8. Perram, J. W.; Wertheim, M. S., Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function. Journal of Computational Physics 1985, 58, (3), 409-16.

9. Rommel, H. L.; Robinson, B. H., Orientation of Electro-optic Chromophores under Poling Conditions: A Spheroidal Model. J. Phys. Chem. C 2007, 111, (50), 18765-18777.

10. Jia, R.; Hentschke, R., Dipolar particles in an external field: Molecular dynamics simulation and mean field theory. . Physical Review E: Statistical, Nonlinear, and Soft Matter

Physics 2009, 80, (5-1), 051502/1-051502/9.

11. Bartke, J.; Hentschke, R., Dielectric properties and the ferroelectric transition of the Stockmayer-fluid via computer simulation. Mol. Phys. 2006, 104, (19), 3057-3068.

12. Leach, A. R., Molecular Modeling: Principles and Applications. 2nd ed.; Pearson Education: Harlow (UK), 2001.

13. Onsager, L., Electric moments of molecules in liquids. J. Am. Chem. Soc. 1936, 58, 1486-93.

14. March, N. H.; Tosi, M. P., Atomic Dynamics in Liquids. Dover Publications, Inc.: New York, NY, 1976.

15. Gaussian 09, R. A., Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski,

J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian, Inc., Wallingford CT, 2009.