James Gayvert1

, Alexandros Chremos2, Jack Douglas2

1Le Moyne College, 2National Institute of Standards and Technology(NIST), U.S. Department of Commerce, Gaithersburg, Maryland, USA

Molecular Dynamics Study of the Conformational Properties of Polymers 

in an Explicit Solvent and the Identification of the θ-Temperature
POLYMERS

Large molecule/macromolecule composed of repeating subunits called monomers

Polystyrene(Styrofoam) Polyisoprene(rubber) Insulin Hexamer DNA

Image from: 

https://www.theodysseyonline.com/ne

w-year-no-more-styrofoam

Image from: 

https://www.tirerack.com/content/t

irerack/desktop/en/homepage.html

Image from: 

https://en.wikipedia.org/

wiki/Insulin

Image from: 

http://exchange.smarttech.com/search.

html?q=%22DNA%20structure%22
POLYMER MODELS

o Ideal Chain:

o Modeled after a random walk model (no 

attractions, no repulsions)

o “Real Chain”:

o Incorporates interactions between all 

segments

o Includes a repulsive excluded volume 

interaction and an attractive 

component that allows us to account 

for the quality of the solvent

A typical ideal chain

https://en.wikipedia.org/wiki/Ideal_chain
FLORY THEORY/SCALING LAW

Rg~N

ν Rg2 =

1

N

σk=1

n

rk − rmean

2

Rgis the radius of gyration of the polymer For a good solvent, ν≈3/5 

N is the degree of polymerization(# of segments) For the θ-solvent, ν ≈ 1/2

ν Is the Flory exponent For a poor solvent, ν ≈ 1/3

Solvent quality depends on chemistry and temperature
SOLVENT QUALITY

• “Good solvent”:

• Monomers prefer to interact 

with the solvent particles rather 

than with other monomers

• Repulsive interactions dominate

• Polymer swells

• “θ-solvent”:

• Attractive and repulsive 

interactions are equal

• Polymer behaves as if it’s an 

ideal chain 

• “Poor solvent”

• Monomers prefer to interact 

with other monomers rather 

than with solvent particles

• Attractive forces dominate

• Polymer contracts and behaves 

more as a hard sphere

Images from: http://rkt.chem.ox.ac.uk/lectures/pol.html
OBJECTIVES

• Develop a model that can identify the θ-temperature 

of a polymer solution

• Use molecular dynamics to simulate polymers of 

varying chemistries and molecular architectures in an 

explicit solvent
MOLECULAR DYNAMICS SIMULATIONS

Set initial conditions, particle 

positions/velocities, and select 

ensemble (NPT,NVE etc.)

Calculate forces from particle 

positions using chosen 

interaction model

Solve Newton’s equations of 

motion(integrate)

Write Data to Disk/Analyze 

Results

Loop over 

time-steps

Animation from

https://en.wikipedia.org/wi

ki/Molecular_dynamics
IMPLICIT SOLVATION WORK

• Implicit solvation work on this topic has been done by Steinhauser M.O. In J. Chem. Phys. 122(2005) on flexible linear polymer chains of 

varying molecular masses

• Used a coarse grained bead-spring model that introduced a dimensionless parameter λ which determines the depth of the intermolecular 

potential

• Larger λ=greater attractive forces between segments 

• The overall intermolecular potential has the form

Vinter r = ൞

VWCA r − λε, 0 < r < 2

1/6σ

λVcos r , 2

1/6 ≤ r < rcut

0 else

VWCA r = 4ε

σ

r

12

−

σ

r

6

+ ε

Vcos r =

1

2

cos αr

2 + β + 0.5 ε

Where α and β are parameters that fit the

cosine curve to smoothly approach 0 at the cutoff

Image from: 

Steinhauser M.O 

Int. J. Mol. 

Sci. 2009, 10(12)

Steinhauser M.O. J. 

Chem. Phys 

122(2005)

λ=0 purely 

repulsive 

interactions
IMPLICIT SOLVATION WORK

Image from: Steinhauser M.O Int. J. Mol. Sci. 2009, 10(12) Steinhauser M.O. J. Chem. Phys 122(2005)

θ-temperature

λ ≈ 0. 65
WHY USE AN EXPLICIT SOLVENT?

Explicit Solvent Implicit Solvent

• Provides a more realistic and 

detailed full-atom description of 

the system

• Essential to reproduce certain 

properties of solute molecules 

• Reaction kinetics

• Dynamics

• Computationally expensive... 

but worth it

Image from: http://csb.stanford.edu/~koehl/ProShape/born.php
SIMULATION DETAILS

• Used a coarse grained bead model with an explicit solvent to try and 

determine the θ-temperature of various polymers of different molecular 

architectures and chemistries

• NPT Ensemble

• The Lennard Jones Potential was used with a cutoff of 2.5σ

• Used ZENO numerical path integration software to obtain conformational 

properties of the generated polymer chains (radius of gyration, 

hydrodynamic radius, etc)

VLJ r = 4ε

σ

r

12

−

σ

r

6

Polymer chain with 41 monomers 72000 solvent particles + polymer Image from: http://www.sklogwiki.org/SklogWiki/index.php/Lennard-Jones_model
SYSTEMS STUDIED

Linear Chain Ring Polymer

Molecular Mass: 41,81,161,321

Temperature=0.5,0.7,0.9

εpp for polymer-polymer interactions: 1.0

εss for solvent-solvent interactions: 1.0

εsp cross interactions: Modified Lorentz Berthelot 

mixing rule 

εsp= εpp ∗ εss 1 − α

α=.0, α=.02, α=.04, α=.06, α=.08, α=.1, α=.12, α=.14, α=.16,
WHAT SHOULD OUR MIXING RULE BE?

α ≥ 0.06

Collapsed Region

Always poor solvent 

conditions

α=0

Swollen Region

Good solvent 

conditions

0.02<α<0.04

Intermediate Region

Rgis sensitive to 

changes in temperature
SWOLLEN REGION-LINEAR

• α=0

• No clear trend between <Rg> and 

temperature, difficult to identify 

θ-temperature 

• Chains are more swollen, cross 

interactions too attractive (good 

solvent conditions)

• Cannot naively choose mixing rule 
α

≥0.06

Collapsed Region

Always poor solvent 

conditions

α=0

Swollen Region

Good solvent 

conditions

0.02<α<0.04

Intermediate Region <Rg> is sensitive to 

changes in temperature
COLLAPSED REGION-LINEAR

• α ≥ 0.06

• Chain is always collapsed, cross 

interactions too repulsive (bad 

solvent conditions)

• Desired behavior was not achieved 

for these simulations, no crossover 

occurs
α

≥0.06

Collapsed Region

Always poor solvent 

conditions

α=0

Swollen Region

Good solvent 

conditions

0.02<α<0.04

Intermediate Region Rg

is sensitive to 

changes in temperature
INTERMEDIATE REGION-LINEAR

α = 0.02 α = 0.04
RING POLYMERS

Swollen Ring

Collapsed Ring

Collapsed Region

Intermediate 

Region

Swollen Region
INTERMEDIATE REGION-RINGS

• α=0.04

• θ-temperature in range

0.8< θ<0.9
FUTURE WORK

• Increase resolution of search for θ-temperature

• Explore 0 < α < 0.06

• Explore more temperatures within range 

0.5<T<0.9

• Explore different types of molecular architectures

• Find θ-temperature for branched polymers

Branched/Star Polymer
ACKNOWLEDGMENTS

• Dr. Alexandros Chremos

• Dr. Jack Douglas

• Computational Soft Materials Working Group 

(COMSOFT)

• NIST Material Measurement Laboratory (MML)

• NIST Summer Undergraduate Research Fellowship 

(SURF) Program

• Le Moyne College