Hayley Boigenzahn, Worcester Polytechnic Institute

Advisor: Dr. Debra Audus

Using Molecular Dynamics to 

Model the Structure of 

Polyelectrolyte Micelles
Complex Coacervation

In water

Polymer-Rich Phase 

(Complex Coacervate)

Water-Rich Phase

• Oppositely charged 

polymers attract

• Form a coacervate

• Liquid-liquid phase 

separation

• Hydrophilic neutral 

polymer goes to water 

phase + Neutral Polymer
Complex Coacervation

• Hydrophilic neutral polymers 

bonded to a charged polymer 

• Form spherical micelles at low 

concentrations

• Micelles are responsive to a 

variety of stimuli

• pH, salt concentration, among 

others

• Potential as drug delivery 

agents

Diblock Copolyelectrolytes

Coacervate

(Core)

Water + Neutral 

Polymers (Corona)
• Other molecules can be 

encapsulated in the core

• Helpful to know size

• R – Radius of the core

• D – Size of the corona

• C – Concentration in the core

• n – The number polymers in the 

micelle, or the aggregation 

number

Micelle Structural Properties
Molecular Dynamics

• Method of computer simulation

• Uses Newton’s equations of motion to 

determine positions of atoms

• LAMMPS

• Coarse-grain simulations – ‘necklace beads’

• Charge of ±1 on charged beads (Coulomb 

interactions)

• Repulsive interactions models a good solvent

• Why not just throw a bunch of polymers in a 

box, wait for it to equilibrate, and take the 

average micelle size?

• Many molecules in a very large box → Extremely 

long computational time
Finding the Aggregation Number

• Lowest free energy per chain → most favorable energy state → most 

common/accurate aggregation number

• Free energy cannot be directly measured from a single simulation

n = 20 n = 10 n = 20 n = 30

• Model with different numbers of 

polymers

• Find free energy per chain minimum
• Free energy is the area 

under the curve

• Free energy is found with thermodynamic integration

• Relate free energy to coulomb potential energy (energy due to 

charged beads)

• Plot from known state (neutral, q=0) to desired state (micelle, q=1)

Thermodynamic Integration

Free 

energy

ΔF = Ffinal − Fref = 2 න

0

1

dq

Ucoul

q

Charged 

system

Neutral 

system

Change in 

free energy
• Generate thermodynamic integration curves using coulomb energy

• Simulations record coulomb potential energy through time

• Once the system equilibrates, find the mean → one point on TI curve

Equilibrium Coulomb Energy

Equilibration Data collection

Average 

value
• Not all polymers make micelles

• Too many charged beads → 

macro-phase separation

• Too few charged beads → paired 

polymers/unstable micelles

• Significantly too many or too few 

polymers also tend to break 

apart

Micelle Conditions = Micelle = No 

Micelle

Nc

↓

Nn → 40 35 30 25 20 10

10

15

20

25

30

40
• Polymer with 20 charged 

and 30 neutral beads

• n ≅ 40

• Additional points between 

30 and 50 can be tested to 

improve accuracy

n Results

Nc=

20 charged beads

Nn=

30 neutral beads
• Concentration does not depend on 

knowing the aggregation number

• Controlled by interactions

• Agrees with a theoretical prediction

• Since C is constant, we know the 

scaling of R

Structural Results & Relations

D

R

R ∝

n

C

1

3
• Polymer with 20 charged 

beads and 20 neutral beads

• n ≅ 52

n Results

Nc=

20 charged beads

Nn=

20 neutral beads
• Polymer with 15 charged 

beads and 30 neutral beads

• n ≅ 26

• Additional points needed to 

improve accuracy

n Results

Nc=

15 charged beads

Nn=

30 neutral beads
Structural Results & Relations

D

R

D

R

20/20

15/30
Comparison

20/20 20/30 15/30

n 52 40 26

C 0.318 0.318 0.279

R 9.17 8.52 7.03

D 9.14 12.81 12.40

Accuracy expressed in significant figures

20/20 20/30 15/30

• n is more dependent on Nc

• C only depends on Nc

• R scales with n

• D is more dependent on Nn
Comparison

20/20 20/30 15/30

n 52 40 26

C 0.318 0.318 0.279

R 9.17 8.52 7.03

D 9.14 12.81 12.40

Accuracy expressed in significant figures

20/20 20/30 15/30

• n is more dependent on Nc

• C only depends on Nc

• R scales with n

• D is more dependent on Nn
Comparison

20/20 20/30 15/30

n 52 40 26

C 0.318 0.318 0.279

R 9.17 8.52 7.03

D 9.14 12.81 12.40

Accuracy expressed in significant figures

20/20 20/30 15/30

• n is more dependent on Nc

• C only depends on Nc

• R scales with n

• D is more dependent on Nn
Comparison

20/20 20/30 15/30

n 52 40 26

C 0.318 0.318 0.279

R 9.17 8.52 7.03

D 9.14 12.81 12.40

Accuracy expressed in significant figures

20/20 20/30 15/30

• n is more dependent on Nc

• C only depends on Nc

• R scales with n

• D is more dependent on Nn
Comparison

20/20 20/30 15/30

n 52 40 26

C 0.318 0.318 0.279

R 9.17 8.52 7.03

D 9.14 12.81 12.40

Accuracy expressed in significant figures

20/20 20/30 15/30

• n is more dependent on Nc

• C only depends on Nc

• R scales with n

• D is more dependent on Nn
• Coacervation

• Use Molecular Dynamics and Thermodynamic Integration

• Established sizes and explored stability of polymer micelles

• Questions?

Conclusion