Theory of

Supramolecular Polymer Systems

Edward H. Feng and Glenn H. Fredrickson

Department of Chemical Engineering

University of California, Santa Barbara

Reversible Intermolecular Bonding

Meijer and coworkers, Science. 278, 1601, 1997.

2−ureido 4−pyrimidone bonding group

forms linear and network structures.

Potential Technological Applications

use temperature to control the number of bonds and hence

the physical properties and processability of the material.

Higher Temperature

Inhomogeneous Supramolecular Polymers

J. Ruokolainen et. al., Science, Vol. 280, 557-560. ’98.

A

C

B

3 component graft copolymer, C conducts electricity

χAB small, χAC and χBC large

Inhomogeneous Supramolecular Polymers

melting of "inner" lamellae

breaking of hydrogen bonds

higher T

Electrical Conductivity

use temperature to control the properties of the material.

Supramolecular Diblock Copolymer

• consider the most simple system that will form inhomo-

geneous phases.

• the energy of bonding will compete with the immiscibility

of the two types of polymers.

• started a collaboration with experimentalists at UCSB to

study this model system

Model for Supramolecular Diblock

• use a continuous Gaussian chain model

• assume an energy change for the reversible reaction of

two different homopolymers forming a diblock: ε

• an incompressible melt in the grand canonical ensemble

Parameters for Supramolecular Diblock

• zA/zB: ratio of activities of the two polymer species

• g = NB/NA: ratio of length of B to length of A polymer

• χAB: Flory-Huggins parameter that captures chemical

immiscibility of two species

• ε: energy of bonding

Model for Supramolecular Diblock

• for ε → −∞, this system is a binary blend

• for ε → ∞, this system has only diblock copolymers

• for intermediate values of ε, this system contains both

homopolymers and diblock copolymer

Theoretical Resultsparameters: zA, g, χAB, and ε.

Ξ(zA, V, T) =

∫DW+

∫DW−e−H[W±]

H[W±] =C

χABNA

∫dxW2

−(x) − iC∫

dxW+(x)

−V̄ (zAeεQAB[W±] + zAQA[W±] + QB[W±])

• for each choice of parameters, there is a corresponding

ternary blend system

P.K. Jannert, M. Schick, Macro. 30:137 , ’97. 30:3916, ’97.

Mean Field Equations

Ξ(µA, V, T) =∫

DW+

∫DW−e−H[W±]

δH[W±]

δW+(x)= φA(x; [W±]) + φB(x; [W±]) − 1 = 0

δH[W±]

δW−(x)=

2

χABNAW−(x) − φA(x; [W±]) + φB(x; [W±]) = 0

• find the mean field solution computationally by relaxing

the W± fields from random initial conditions

• calculate φA and φB density fields with pseudospectral

algorithm

Density Profile Results

parameters are zA = 1, ε = 0, χABNA = 4.0 and g = 1.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 0

2

4

6

8

10

12

14

16

lamellar structures for this system with equal parts A and B

Order Disorder Transition, zA = 1, g = 1.

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0 0.2 0.4 0.6 0.8

tem

pera

ture

bonding energy

Lamellar

Disorder

Equilibrium Polymers

• system with annealed disorder in the polymer length

distribution.

• this is a model for giant micelles, which can break and

recombine at any point along the micelle.

• we will study this model in confined environments, such

as between two parallel plates

Equilibrium Polymer Model

• use energy of bonding idea and formulate model in grand

canonical ensemble

• parameters of model:

– z, monomer activity

– u0, excluded volume parameter

– ε, bonding energy

• study this system confined between two parallel plates

separated by distance L.

Equilibrium Polymer Model

• effective Hamiltonian:

H[w] =1

2u0

∫w(r)2dr − V̄ e−2ε

∫ ∞

0zNQ(N ; [iw])dN

• the polymer length distribution ∼ zNQ(N ; [iw])

• mean field equation:

δH[w]

δw=

1

u0w(r) + iρ(r; [iw]) = 0 (1)

where density involves integral over all polymer lengths

Homogeneous Limit

H[w] =1

2u0

∫w(r)2dr − V̄ e−2ε

∫ ∞

0zNQ(N ; [iw])dN

• u0 → 0 implies w(r) = 0

• zNQ(N ; [iw]) = zN = e(ln z)N

• for z < 1, polymer length distribution is exponential with

characteristic length 〈N〉 = −(ln z)−1

Confined Equilibrium Polymer

Polymer Length Distribution

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

prob

abili

ty

polymer length

homoL=1L=2L=4L=6

L=10

Confined Equilibrium Polymer

Density Within Slit

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

rho

r

Conclusions

• formulated a field theoretic model for a supramolecular

polymer systems with reversible intermolecular bonding

• used computational methods to find saddle point solu-

tions of the model

• future work will involve graft copolymer systems

Effect of Temperature

• in original formulation, we scale chemical and bonding

energy by kT

χNA =eNA

kT, ε =

b

kT

• now we scale the temperature and bonding energy by eNA

by kT

Θ =kT

eNA, E =

b

eNA