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Statistical Mechanical Modeling of Chromophore Interactions at High Concentration

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### Statistical Mechanical Modeling of Chromophore Interactions at High Concentration

### Why is order important for NLO materials The Electro-Optic Effect

### Monte Carlo Simulations

Electrical, Optical and Magnetic Properties of Organic and Hybrid Materials

Bruce H. Robinson

Robert D. Nielsen

Harrison Rommel

Outline

- Fundamental Concepts: Matter organization is a many body problem
- Why is order important for NLO materials
- Many different patterns of arrangement are possible in Materials
- Treating molecules as classical objects.
- The principles of Statistical Mechanics to determine organization
- Monte Carlo Methods to determine overall order in frustrated systems
- Analytic Methods to compute average order
- Application to EO materials and the EO coefficient

Organization of Molecules in Condensed phases

- Put the NLO molecules (aka chromophores) into a host.
- Hosts: neat, solvent, polymer, block-copolymer
- Condensed phases means liquid or solid states
- Most chromophores in a polymer slurry are glassy and have a solid to glass transition temperature (Tg).
- Usually: No real lattice or lattice order (not crystals)

To generate a voltage across the material when shining light we need to have the molecules somewhat aligned or asymmetrically arranged.

We now proceed to explain

how and why this is so.

How do we get asymmetric arrangements:

- Must start with asymmetric molecules
- use lager scale organization (dendrimers host-polymers)
- Grow crystals
- Generate ferro-electric phases
- Generate ordered phases that give higher order than “normal" para-electric or anti-ferroelectric order
- Generate some order using a poling or ordering Electric (E) field (place sample in capacitor):

Why do we get an E.O. effect for Organic NLO materials

- The electric field of the light pushes and pulls electrons in the NLO chromophore.
- The movement of electrons sets up a field (as a voltage) around the material
- The distribution of charges can be detected by a capacitor type arrangement.
- The response of the material to light (or a DC voltage) is equivalent to a change in the index of refraction of the material

Electro Optic (EO) Coupling

Explaining the Phenomenon:

- Molecules (arrows in the box) are somewhat aligned on z pointing up.
- r is the EO coupling coefficient – coupling electric field effects to the index of refraction changes.
- The applied E field (along z) changes the index of refraction on either z or x depending on the nature of the coupling coefficient r.
- Coupling along z causes a change in the index of refraction that then causes a phase delay in the light traveling along the x direction (which has its E field in the z direction).

Measurement and Functionality

- Phase of the light in the two arms is changed by the driving E field.
- When the phases differ by 180, or pi, no light out on z.
- Low value of driving E field or
- Large dynamic range
- Low insertion loss (<1 db)

Large Angle, Fast Response Spatial Light Modulator (SLM)

Experimental Results

Schematic Diagram

12 mrad/

volt

Literature Citations

• Steier, et al., “Polymeric waveguide prism based electro-optic beam deflector,” Opt. Eng., 40, 1217-22 (2001)

• Steier, et al., “Beam deflection with electro-optic polymer waveguide prism array,” Proc. SPIE, 3950, 108-116 (2000)

• Steier, et al., “Polymeric waveguide beam deflector for electro-optic switching,” Proc. SPIE, 4279, 37-44 (2001)

Phased Array Radar with Photonic Phase Shifter

Steier, et al., IEEE mW & Guided Wave Lett. 9, 357 (1999)

Goals

- Engineering goal is to optimize macroscopic E.O. effect

1) Bandwidth limitation

2) Push-Pull modulation

- What do we optimize on molecular scale

to yield best effect?

Microscopic basis of E.O. effect for organic NLO materials

- Large molecular Hyperpolarizability
- Classical perturbation picture
- Quantum perturbation picture

How a single molecule responds to light

- E field of the light interacts with the electron distribution in the molecule to change charge distributions.
- Molecules have no net charge, but a very large dipole moment. For two charges (plus and minus q) separated by a distance r: the dipole moment is

Microscopic basis of E.O. effect

The E field causes polarization and hyperpolarization

For the most part, consider effect on z, in molecule

Microscopic to Macroscopic

Net acentric ordering to give macroscopic Hyperpolarizability

The EO coefficient is proportional to molecular hyperpolarizability and the net order, which, we will see, is related to the intrinsic dipole moment.

Summary: Organic NLO materials

- Couple applied electric fields to optical transmission
- Solid state, Guest/Host, Chromophore/polymer, matrix
- High molecular electric dipole moment ( >5 D)
- Large molecular Hyperpolarizability ( )
- Organic Chromophores aligned by “Poling” with electric field above glass transition temperature of Guest Polymer followed by cooling
- Experimental observation: At high Chromophore concentrations poling becomes inefficient

How does a poling field give order?

- The dipole of the molecule interacts with the E field of the poling field
- How a single molecule interacts with E:

Energy:

Probability of being at a particular orientation q, is given by Boltzmann’s formula

Modeling an NLO Chromophore

We need to model the structures with simpler, classical rules. When considering many molecular interactions Q. M. too time consuming.

Chromophore: DMC3-97

- Model Chromophores as Ellipsoids
- Each ellipsoid has a dipole moment at its center along the major axis.
- Energy due of E-field induced polarization will be neglected.
- Leonard Jones type ellipsoidal repulsions will be included.

Bolzmann’s Rule

- G is the Gibbs-Distribution or the Generating Function.
- Probability of being at a particular orientation q:
- Need the integral at the bottom to be sure probabilities sum to 1.

The strength of the interaction is relative to temperature, as a single parameter, x

How much order can we get

- Problem: At low T a molecule would align but it could not move to align
- At T>Tg molecules can rotate but will not perfectly align.

- Example:
- One electron displaced by 1 Angstrom from a proton has a dipole of 4.8 Debye
- For a 10 Debye dipole: F=0.7
- So you get some, but not a lot of order

Boltzmann Probability Distribution

- Probability as a function of the cosine of the angle shows most probable orientation is to be aligned but but it is not a lot.
- For F=0.7

and 2.0

Area under the two curves are the same.

F=2.0

F=0.7

Computing Average Quantities

- We want to know how much, on average, is aligned.
- With the generating function, G, one can compute any quantity.
- Example, the mean cos angle raised to the nth power.

Averages: How much order can you get?

- Order as a function of dipole strength:
- Maximum possible average would be 1
- In practice won’t get above 0.6

Low Strength limit: F/3

L1(F)

The influence of Number Density

- Dipoles form anti-ferroelectric state at high density.
- Local Dipole-Dipole interaction fights poling-field induced order
- Local steric forces restrict order
- We investigated Number Density effect by:

Analytic mean field theory

MC simulations of dipole orientations

Analytic Theory of Piekara

Piekara, A. (1939) Proc. R. Soc. London A 172, 360-381

- The general Statistical Mechanical Problem is a complicated (N-body) interaction.
- Probability distribution of each chromophore, G(1), in a field of many other chromophores, and poling field.

Analytic Theory of Piekara (cont’d)

- Treat poling field E as interacting with dipoles:
- Have an effective interaction with all other chromophores, within a “crystallite”
- Average over all “crystallites” uniformly.

Comparison with Low Chromophore Density LangevinTheory

- Langevin Theory has the partition function:
- Piekara has crystallite partition function:

Piekara’s W

- F and W directly compete
- Problem: W is not known, nor is it self-consistent
- Hypothesis: W is related to dipole-dipole energy of interaction:

Piekara’s limit

Maximizing E.O. Effect

There is always a maximum loading, which is very simply related to the dipole moment strength, and nearly independent of the poling field at typical fields

Ratio of E.O. Coeffcients

Propagation along controlling field

Summary

- Increasing dipole moment alone does not give larger E.O. effect
- Steric forces indicate spherical shape is optimal
- is not constant, but is field dependent
- Aggregation of Chromophores not necessary to explain E.O. roll-off
- Can the choice of W be justified?

- A straightforward way to compute averages
- A Justification for Piekara’s W

Monte Carlo Calculations

- Use Monte Carlo methods to determine the effect of dipolar interactions between chromophores.

- Place dipoles on a grid (simple cubic lattice and body centered cubic lattice)
- M by M by M array with r as nearest neighbor distance.
- Periodic boundary condition

A 5 by 5 two dimensional array

Randomly oriented dipoles

How Monte Carlo Works

- Choose a dipole
- Rotate dipole by: a rotation axis and angle, selected randomly
- Compare the energy before and after rotation.

If the energy is higher, compare Boltzmann Probability with a [0,1] random number, and keep if larger.

If the energy is lower, keep the move

Equivalent MC runs

Along Z

Lattice StretchingX-Y Compression, No Field

X projection

Lattice StretchingX-Y Dilation, No Field

Z projection

Summary of MC Calculation Results

- Planes Statistically Oppose One Another for cubic lattice, and represent long range order.
- The dominant angular distribution depends on dipole moment to the Fourth Power
- The dominant angular distribution depends on chromophore density to the Second Power.
- MC predictions not sensitive to Cubic vs. BCC
- MC statistics depend upon lattice distortion and are a first step to understanding steric effects
- Scaling of W is needed to compare with analytic theory

Conclusions

- Strongly anti-ferro-electric interactions resist poling field in condensed matter. This is the source of roll-off of EO response with chromophore number density.
- Aggregation model not necessary
- Piekara’s crystallite interaction, W, can be interpreted as a field and competes with poling in a simple geometric way.
- As the dipole moment increases, the maximum loading decreases.
- For a given dipole moment there is an optimum number density that is easily calculated from the analytic theory.
- Spherical shaped chromophores are the optimum geometry (better than either prolate or oblate ellipsoids)
- MC shows the expansion of N body interactions has a cosine term as the dominant distribution. You will never get this from a small number of chromophores.
- MC results reproduce W dependence on ,N,T.

Future

- Include steric effect in MC calculations
- Generalize from point Dipoles to charges
- Explore extended multi-chromophore structures

Microscopic basis of E.O. effect for organic NLO materials

- Large molecular Hyperpolarizability
- Net acentric ordering to give macroscopic Hyperpolarizability

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