^{1}, Philip Spicer

^{1}, Nitish Nair

^{1}and Arthi Jayaraman

^{1,a)}

### Abstract

Functionalizing nanoparticles with organic ligands, such as oligomers, polymers, DNA, and proteins, is an attractive way to manipulate the interfacial interactions between the nanoparticles and the medium the particles are placed in, and thus control the nanoparticle assembly. In this paper we have conducted a Monte Carlo simulation study on copolymer grafted spherical nanoparticles to show the tremendous potential of using monomersequence on the copolymers to tune the grafted chain conformation, and thus the effective interactions between copolymer grafted nanoparticles. We have studied AB copolymers with alternating, multiblock, or diblock sequences, where either A monomers or B monomers have monomer-monomer attractive interactions. Our focus has been to show the nontrivial effect of monomersequence on the conformations of the grafted copolymers at various particle diameters, grafting densities, copolymer chain lengths, and monomer-monomer interactions in an implicit small molecule solvent. We observe that the monomersequence, particle diameter, and grafting density dictate whether (a) the grafted chains aggregate to bring attractive monomers from multiple grafted chains together (interchain and intrachain monomeraggregation) if the enthalpy gained by doing so offsets the entropic loss caused by stretching of chains, or (b) each grafted chain folds onto itself to bring its attractive monomers together (only intrachain monomeraggregation) if the entropic loss from interchain aggregation cannot be overcome by the enthalpic gain. For six copolymers of chain length grafted on a spherical particle of diameter , interchain and intrachain monomeraggregation occurs, and the radius of gyration varies nonmonotonically with increasing blockiness of the monomersequence. At larger particle diameters the grafted chains transition to purely intrachain monomeraggregation. The radius of gyration varies monotonically with monomersequence for intrachain monomeraggregation because as the sequence becomes blockier (like monomers are grouped together), the copolymer chain has to fold less compactly to maximize the enthalpically favorable contacts while maintaining high conformational entropy. The radius of gyration of alternating and diblock copolymers scales with chain length N through a power law with the prefactor and scaling exponent , varying with monomersequence and monomer-monomer attraction strength.

This work was financially supported by the University of Colorado Innovative Seed Grant Award. A.J. would like to thank University of Colorado Undergraduate Research Opportunities Program (UROP) for offering an undergraduate summer research assistantship to P.S.

I. INTRODUCTION

II. METHOD

A. Model

B. Monte Carlo simulation

C. Analysis

D. Parameters

III. RESULTS

A. Effect of monomersequence and monomer-monomer interactions

B. Effect of particle size (curvature) and grafting density

C. Effect of grafted copolymer chain length

IV. DISCUSSION

### Key Topics

- Polymers
- 250.0
- Copolymers
- 111.0
- Nanoparticles
- 56.0
- Block copolymers
- 52.0
- Aggregation
- 41.0

## Figures

A schematic of model AB copolymer grafted particles with alternating and diblock sequence for grafted chain length . Also shown are the various monomer sequences—alternating or , multiblocks , , and diblock or —for the grafted chain length .

A schematic of model AB copolymer grafted particles with alternating and diblock sequence for grafted chain length . Also shown are the various monomer sequences—alternating or , multiblocks , , and diblock or —for the grafted chain length .

(a) Radius of gyration and (b) number of attractive monomer contacts as a function of monomer sequence for six AB copolymers of length grafted on a spherical nanoparticle of diameter at athermal interactions (no symbols solid line) and monomer-monomer attractive interaction (circles), 0.5 kT (down triangle), and 1 kT (up triangle). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive. Monomer sequence 1 refers to alternating, 2 refers to , 3 refers to , and 4 refers to diblock copolymer.

(a) Radius of gyration and (b) number of attractive monomer contacts as a function of monomer sequence for six AB copolymers of length grafted on a spherical nanoparticle of diameter at athermal interactions (no symbols solid line) and monomer-monomer attractive interaction (circles), 0.5 kT (down triangle), and 1 kT (up triangle). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive. Monomer sequence 1 refers to alternating, 2 refers to , 3 refers to , and 4 refers to diblock copolymer.

Average number of monomers N(r) at increasing radial distance from the surface of the particle r for particle diameter with six (a) alternating copolymers and (b) diblock copolymer grafted chains of length at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Average number of monomers N(r) at increasing radial distance from the surface of the particle r for particle diameter with six (a) alternating copolymers and (b) diblock copolymer grafted chains of length at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Radius of gyration as a function of monomer sequence for AB copolymers of length grafted on a spherical nanoparticle of diameter (a) and (b) 12 at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). (c) Number of attractive monomer contacts for (circles), (triangles), and 12 (squares) at monomer attractive interaction . Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Radius of gyration as a function of monomer sequence for AB copolymers of length grafted on a spherical nanoparticle of diameter (a) and (b) 12 at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). (c) Number of attractive monomer contacts for (circles), (triangles), and 12 (squares) at monomer attractive interaction . Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Representative simulation snapshots (best seen in color) for alternating and diblock AB copolymers of length grafted on a spherical nanoparticle of diameter , 8, and 12 at or . For alternating sequence we show snapshots only for because the chain conformations are similar for both and . exhibits intra- and interchain monomer aggregation and exhibits mainly intrachain aggregation only. shows purely intrachain aggregation for diblock copolymer and A block attractive and a combination of intra- and interchain aggregation for alternating and diblock and B block attractive.

Representative simulation snapshots (best seen in color) for alternating and diblock AB copolymers of length grafted on a spherical nanoparticle of diameter , 8, and 12 at or . For alternating sequence we show snapshots only for because the chain conformations are similar for both and . exhibits intra- and interchain monomer aggregation and exhibits mainly intrachain aggregation only. shows purely intrachain aggregation for diblock copolymer and A block attractive and a combination of intra- and interchain aggregation for alternating and diblock and B block attractive.

Radius of gyration as a function of monomer sequence (x-axis) for AB copolymers of length grafted on spherical nanoparticles of size , 8, and 12 nm at grafting density of and at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Radius of gyration as a function of monomer sequence (x-axis) for AB copolymers of length grafted on spherical nanoparticles of size , 8, and 12 nm at grafting density of and at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Radius of gyration as a function of monomer sequence for AB copolymers of length grafted on a flat surface at decreasing grafting density at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

Radius of gyration as a function of monomer sequence for AB copolymers of length grafted on a flat surface at decreasing grafting density at athermal interactions (black solid line) and monomer-monomer attractive interaction (circles) and 1 kT (triangles). Open symbols-dashed lines correspond to systems where A monomers are attractive and filled symbols-solid lines correspond to systems where B monomers are attractive.

as a function of log(N), where N is the length of the grafted or free copolymer at (a) athermal conditions, (b) alternating copolymers at A-A monomer-monomer attraction strength are 1 kT, and (c) diblock copolymers at A-A monomer-monomer attraction strength are 1 kT.

as a function of log(N), where N is the length of the grafted or free copolymer at (a) athermal conditions, (b) alternating copolymers at A-A monomer-monomer attraction strength are 1 kT, and (c) diblock copolymers at A-A monomer-monomer attraction strength are 1 kT.

## Tables

The exponent and prefactor in , where N is the grafted chain length for systems: chain, chains grafted on particle, chains grafted on particles, chains grafted on a flat surface with grafting density same as II, and chains grafted on a flat surface with grafting density same as III. Empty cells indicate that the data did not fit the scaling relationship.

The exponent and prefactor in , where N is the grafted chain length for systems: chain, chains grafted on particle, chains grafted on particles, chains grafted on a flat surface with grafting density same as II, and chains grafted on a flat surface with grafting density same as III. Empty cells indicate that the data did not fit the scaling relationship.

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