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Modeling Mesoscale Structure In Comb Polymer Materials for Anhydrous Proton Transport Applications Barry Husowitz Peter Monson

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Proton Exchange Membrane (polyelectrolyte) Proton Exchange Membrane (polyelectrolyte) Membranes need to be Membranes need to be Chemically stable Chemically stable Thermodynamically stable Thermodynamically stable Reasonable proton conductance Reasonable proton conductance Important for Operation of Fuel Cell (Proton Exchange Membrane) U.S. Department of Energy Hydrogen Program Why is this important?

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Water Assisted Proton Conductance Nafion (comb polymer) Nafion (comb polymer) The conductance of water containing nafion is highly dependent on the state of hydration of the membrane and structural characteristics The conductance of water containing nafion is highly dependent on the state of hydration of the membrane and structural characteristics Temperature range of operation highly dependent on solvent Temperature range of operation highly dependent on solvent P. J. Brookman, J. W. Nichoson, "Developments in Ionic Polymers, vol 2" Elsevier, London, pp. 269-283 (1986)

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Anhydrous proton conductance Conductance does not depend on physical properties of the solvent Conductance does not depend on physical properties of the solvent Investigating proton-conducting polymers which do not rely on a solvent is a revolutionary approach to new hydrogen fuel cells Investigating proton-conducting polymers which do not rely on a solvent is a revolutionary approach to new hydrogen fuel cells

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Center for Chemical Innovations (CCI) : Candidates for Anhydrous Proton Conductance Comb diblocks Comb diblocks Comb polymers Comb polymers Imidazole Benzotriazole

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Dendrimer-Linear Dendrimer-Linear PT Center for Chemical Innovations (CCI) : Candidates for Anhydrous Proton Conductance

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Modeling Polymer Architecture (chemistry/structure) Predict (model) Mesoscale Structure (ordering) What is the correlation between the nanostructure and proton conductance?

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Coarse Grained Models bond vibrations ~ 10 -15 s diffusion ~ 10 -9 -10 -6 s conformational rearrangements ~ 10 -12 - 10 -10 s Number of Atoms Lumped into Effective Segment (interaction center) MC, MD, DPD Want a coarse-grained model for polymers that captures relevant interactions, excluded volumes, repulsion between unlike atoms - Using softer potentials and reducing the degrees of freedom is an efficient technique for large dense systems (10 6 segments) 5 1 2 3 4 5 F. Muller-Plathe, CHEMPHYSCHEM 3, 754-769 (2002)

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Multiscale Modeling: Force Fields Quantum Calculations of energy surface Force Field - Functional forms and Parameters to describe the potential energy of atoms or groups of atoms Molecular Mechanics – Newtonian mechanics to model molecular system Quantum Mechanics Classical Physics conformational rearrangements ~ 10 -12 - 10 -10 s bond vibrations ~ 10 -15 s diffusion ~ 10 -9 -10 -6 s 1 1 2 3 4 5 H. Sun, Macromolecules 28, 701-712 (1995)

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Basic Model For Polymers H=H b + H nb Bonding interactions within the chains (Conformational states of individual polymers), connectivity Interactions between other chains and non-bonded sites Base Case Model Hamiltonian Minimal coarse graining model that captures relevant interactions, connectivity, excluded volumes Minimal coarse graining model that captures relevant interactions, connectivity, excluded volumes

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Bonding Hamiltonian (Ideal Chain Models) Connectivity (Molecular Architecture) Connectivity (Molecular Architecture) Freely joined chain Freely joined chain Key Parameter b Key Parameter b Discrete Gaussian chain Discrete Gaussian chain r i (s) r i (s+1) b Fixed r i (s) r i (s+1) G. R. Strobl, Chapter 6 "The Physics of Polymers, 2'nd Ed." Springer, NY, (1997) R eo =bN 1/2

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Non-bonding Hamiltonian { { LL LL Φ A,m K. Daoulas and M. Muller, J. Chem. Phys. 125, 184904 (2006) Essential Interactions described through simple parameters, chain extension R eo, Flory-Huggins ( N AB ), compressibility (N ), acts like excluded volume Essential Interactions described through simple parameters, chain extension R eo, Flory-Huggins ( N AB ), compressibility (N ), acts like excluded volume Symmetric Diblock 01 r-r c R eo No explicit volume exclusion, segments can overlap, enforce low compressibility on length scale of interest R eo L=0.17R eo

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Metropolis Algorithm For Monte Carlo Simulation The basic idea is that we assume each configuration of a system has a probability proportional to a Boltzmann factor or Consider two configurations A and B, each of which occurs with probability proportional to the Boltzmann factor. Then The nice thing about forming the ratio is that it converts relative probabilities involving an unknown proportionality constant (called the inverse of the partition function), into a pure number. We can achieve the relative probability of the last equation in a simulation by proceeding as follows:

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Metropolis Algorithm For Monte Carlo Simulation 1. Start from configuration A, with know energy E A, make a change in the configuration to obtain a new (nearby) configuration B. 2. Compute E B (typically a small change from E A, but not to small) 3. If E B < E A, assume the new configuration, since it has lower energy (desirable thing, according to the boltzmann factor). 4. If E B > E A, accept the new (higher energy) configuration with This means that when the temperature is high, we don’t care if we take a step in the “wrong” direction, but as the temperature is lowered, we settle into the lowest configuration we can find in our neighborhood.

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Metropolis Algorithm For Monte Carlo Simulation If we follow these rules, then we will sample points in the space of all possible configurations with probability proportional to the Boltzmann factor, consistent with the theory of equilibrium statistical mechanics. We can compute average properties by summing them along the path we follow through possible configurations. The hardest part about implementing the Metropolis algorithm is the first step: how to generate “useful” new configurations. Canonical Partition function – Sum of all the different possible energy states of the system (However, maybe one of these configurations is very very low in energy and dominates the sum)

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Monte Carlo Moves Incorporated 1. Moved single beads in the chain 2. Translated the whole chain 3. Reptation

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Simulation Methods Single Chain in Mean Field (SCMF) Single Chain in Mean Field (SCMF) Chains move in a field created by other chains via model Hamiltonian Chains move in a field created by other chains via model Hamiltonian Update the fields periodically based on segment distribution Update the fields periodically based on segment distribution Direct Monte Carlo (MC) simulation of the model Hamiltonian Direct Monte Carlo (MC) simulation of the model Hamiltonian Consider new and old fields for every MC move Consider new and old fields for every MC move Update fields after every accepted move Update fields after every accepted move

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Advantage of Model Retains computational advantage of Self-consistent Field theory (SCF), but includes fluctuations. Retains computational advantage of Self-consistent Field theory (SCF), but includes fluctuations. The evaluation of interactions via a grid and densities speeds up the energy calculation by about two order of magnitude compared to explicit pairwise interactions. The evaluation of interactions via a grid and densities speeds up the energy calculation by about two order of magnitude compared to explicit pairwise interactions. Can easily implement different polymeric structures (branched polymers, dendrimeric polymers etc.) or architectures with this model. Can easily implement different polymeric structures (branched polymers, dendrimeric polymers etc.) or architectures with this model.

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Experimental Results: Morphology of Comb Polymers by X-ray Scattering q1 q2 q2/q1= √ 3 X-ray scattering clearly reveals nano-scale phase- segregation induced by alkyl chains Chen et al, Nature Chemistry, 2010, 2, 503-508

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X-ray scattering clearly reveals nano-scale phase- segregation induced by alkyl chains Experimental Results: Morphology of Comb Polymers by X-ray Scattering q1 q2 q2/q1= 4 Chen et al, Nature Chemistry, 2010, 2, 503-508

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Center for Chemical Innovations (CCI) Comb Polymers - Coarse Graining

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Self- Consistent Field Theory (SCF) Study of Comb Polymers Comparison of SCMF with Previous Self- Consistent Field Theory (SCF) Study of Comb Polymers Liangshun Zhang et al, J. Phys. Chem. B 2007, 111, 351-357 Changing the position of the graft points and number of branch points provides a route to a cylinder morphology

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Alkyl side chains gives rise to cylinder morphology ΑΒ = 52.2 Monte Carlo simulations of coarse grained model q2/q1= q2 q1 Experimental Our Calculated Structure Factor

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Disordered structure in this case But, if we lower the interaction parameter to 40.6 then … ΑΒ = 52.2 q2/q1= 4 q1 q2

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MC simulations of coarse grained model ΑΒ = 52.2 Alkyl side chains give rise to lamellar morphology q2 q1 q2/q1= 4 Experimental Our Calculated Structure Factor

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Disordered structure in this case ΑΒ = 52.2 But, as before if we lower the interaction parameter To 40.6 then … q2/q1= 4 q1 q2

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Summary and Conclusions Coarse grained models of CCI polymers show lamellar, cylindrical and disordered structures found in experiments Coarse grained models of CCI polymers show lamellar, cylindrical and disordered structures found in experiments Simulations may explain origins of disordered structures seen in some experiments Simulations may explain origins of disordered structures seen in some experiments Addition of alkyl groups increases value of required for ordering in the system (effect of chain branching on order- disorder transitions) Addition of alkyl groups increases value of required for ordering in the system (effect of chain branching on order- disorder transitions) Simulations of comb polymer systems for large starting from a disordered state can lead to quenched disorder Simulations of comb polymer systems for large starting from a disordered state can lead to quenched disorder Thus simulations of alkylated polymers can show ordered structures when those of nonalkylated polymers Thus simulations of alkylated polymers can show ordered structures when those of nonalkylated polymers at the same value of do not

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Thank You Peter Monson Peter Monson NSF NSF Center for Chemical Innovation (CCI) at the University of Massachusetts, Amherst Center for Chemical Innovation (CCI) at the University of Massachusetts, Amherst Everyone here today Everyone here today

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