^{1,a)}, Robert K. Prud’homme

^{1}and Athanassios Z. Panagiotopoulos

^{1,2,b)}

### Abstract

The solution phase behavior of short, strictly alternating multiblock copolymers of type was studied using lattice Monte Carlo simulations. The polymer molecules were modeled as flexible chains in a monomeric solvent selective for block type . The degree of block polymerization and the number of diblock units per chain were treated as variables. We show that within the regime of parameters accessible to our study, the thermodynamic phase transition type is dependent on the ratio of . The simulations show microscopic phase separation into roughly spherical aggregates for ratios less than a critical value and first-order macroscopic precipitation otherwise. In general, increasing at fixed , or at fixed , promotes the tendency toward macroscopic phase precipitation. The enthalpic driving force of phase change is found to universally scale with chain length for all multiblock systems considered and is independent of the existence of a true phase transition. For aggregate forming systems at low amphiphile concentrations, multiblock chains are shown to self-assemble into intramolecular, multichain clusters. Predictions for microstructural dimensions, including critical micelle concentration, equilibrium size, shape, aggregation parameters, and density distributions, are provided. At increasing amphiphile density, interaggregate bridging is shown to result in the formation of networked structures, leading to an eventual solution-gel transition. The gel is swollen and consists of highly interconnected aggregates of approximately spherical morphology. Qualitative agreement is found between experimentally observed physical property changes and phase transitions predicted by simulations. Thus, a potential application of the simulations is the design of multiblock copolymersystems which can be optimized with regard to solution phase behavior and ultimately physical and mechanical properties.

Financial support of this work was provided by the National Science Foundation through a Graduate Research Fellowship to M.G. and by the NSF NIRT Center on Nanoparticle Formation. Additional support was provided by the Department of Energy, Office of Basic Energy Sciences (Grant No. DE-FG02-01ER15121). M.G. also gratefully acknowledges financial support from Merck and Company through a Doctoral Research Fellowship.

I. INTRODUCTION

II. MODEL AND SIMULATION METHOD

III. RESULTS AND DISCUSSION

A. Macro-versus microphase separation

B. Microscopic phase separation

C. Aggregation properties

D. Aggregate size, shape, and density distributions

E. Network formation

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 64.0
- Biomolecular aggregates
- 45.0
- Polymers
- 39.0
- Phase separation
- 24.0
- Solvents
- 24.0

## Figures

Critical micelle concentration as a function of reciprocal temperature for selected aggregate forming multiblocks as indicated in legend. Points are from simulations, with estimated statistical uncertainties in measured data represented by error bars. Lines are fitted to points according to Eq. (1).

Critical micelle concentration as a function of reciprocal temperature for selected aggregate forming multiblocks as indicated in legend. Points are from simulations, with estimated statistical uncertainties in measured data represented by error bars. Lines are fitted to points according to Eq. (1).

Parameter values as determined from Eq. (1) as a function of total solvophobic monomers per chain for multiblocks (◼, ◻) (this work) and diblocks (○) (Ref. 42). Points are from simulations, with estimated statistical uncertainties in measured data represented by error bars. Linear regression fit of data represented by solid line for and dashed line for and systems (for ).

Parameter values as determined from Eq. (1) as a function of total solvophobic monomers per chain for multiblocks (◼, ◻) (this work) and diblocks (○) (Ref. 42). Points are from simulations, with estimated statistical uncertainties in measured data represented by error bars. Linear regression fit of data represented by solid line for and dashed line for and systems (for ).

Representative aggregate size distributions for (A) and (B) systems at the conditions of Table II, calculated as a function of solvophobic monomer number and number of chains . All points are from simulations.

Representative aggregate size distributions for (A) and (B) systems at the conditions of Table II, calculated as a function of solvophobic monomer number and number of chains . All points are from simulations.

Root mean squared radius of gyration [see Eq. (4)] of (◼) and (◻) systems of aggregate forming multiblocks as a function of . Values are calculated for aggregates in the vicinity of and at the conditions of Table II. Estimated uncertainties in values are represented by error bars.

Root mean squared radius of gyration [see Eq. (4)] of (◼) and (◻) systems of aggregate forming multiblocks as a function of . Values are calculated for aggregates in the vicinity of and at the conditions of Table II. Estimated uncertainties in values are represented by error bars.

(A) Principal radii of gyration of multiblocks in the vicinity of (circles) and (triangles), calculated at the conditions of Table II. Corresponding root mean squared radii of gyration for each aggregate type are reported in legend, with estimated statistical uncertainties in the last digit reported in parentheses. (B) Representative configuration of aggregates in the vicinity of at the conditions of Table II and a simulation box size of .

(A) Principal radii of gyration of multiblocks in the vicinity of (circles) and (triangles), calculated at the conditions of Table II. Corresponding root mean squared radii of gyration for each aggregate type are reported in legend, with estimated statistical uncertainties in the last digit reported in parentheses. (B) Representative configuration of aggregates in the vicinity of at the conditions of Table II and a simulation box size of .

(A) Volume fraction distributions of total solvophobic monomers (open symbols) and total solvophilic monomers (filled symbols) as a function of the distance to the center of mass for aggregate forming systems at the conditions of Table II (B) Radially averaged probability distributions as a function of for systems at the conditions of (A). Only aggregates in the vicinity of are considered in these analyses.

(A) Volume fraction distributions of total solvophobic monomers (open symbols) and total solvophilic monomers (filled symbols) as a function of the distance to the center of mass for aggregate forming systems at the conditions of Table II (B) Radially averaged probability distributions as a function of for systems at the conditions of (A). Only aggregates in the vicinity of are considered in these analyses.

Representative snapshots and chain distributions of multiblock for simulations at the following conditions: (A) , , , and and (B) , , , and .

Representative snapshots and chain distributions of multiblock for simulations at the following conditions: (A) , , , and and (B) , , , and .

Representative snapshots and aggregate size distributions of multiblocks for simulations at (A) , , , and , (B) , , , and , and (C) , , , and . Points are from simulations, with lines drawn for visual clarity.

Representative snapshots and aggregate size distributions of multiblocks for simulations at (A) , , , and , (B) , , , and , and (C) , , , and . Points are from simulations, with lines drawn for visual clarity.

## Tables

Micellization vs phase separation for systems studied. Cells in bold type correspond to systems that from aggregates, while cells in nonbold type designate systems undergoing macroscopic phase separation.

Micellization vs phase separation for systems studied. Cells in bold type correspond to systems that from aggregates, while cells in nonbold type designate systems undergoing macroscopic phase separation.

Properties of aggregating multiblock systems. is the temperature and is the critical micelle concentration. Micellar aggregate size distributions were measured at and volume fraction of copolymer . is the number of solvophobic sites corresponding to the maximum of the monomer aggregate distribution, while is the number of chains corresponding to the maximum in the chain aggregate distribution. Estimated statistical uncertainties in units of the last digit reported are given in parentheses.

Properties of aggregating multiblock systems. is the temperature and is the critical micelle concentration. Micellar aggregate size distributions were measured at and volume fraction of copolymer . is the number of solvophobic sites corresponding to the maximum of the monomer aggregate distribution, while is the number of chains corresponding to the maximum in the chain aggregate distribution. Estimated statistical uncertainties in units of the last digit reported are given in parentheses.

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