Introduction to Statistical Mechanics of Macromolecules

1
Introduction to Statistical Mechanics of
Macromolecules
Monica Olvera de la Cruz Northwestern University
Department of Materials Science and
Engineering 2220 Campus Drive Evanston, IL 60202
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Outline

• Introduction
• Magnetic system with two states (up and down)
• Polymers
• Polymer Conformation
• Polymer Solutions, Melts and Blends
• Copolymers
• Melts (SCMF)
• Micelles (Simulations)
• Polyelectrolytes
• bulk
• surface segregation

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Introduction

• Random Walk. Consider a magnetic system with
  non-interacting N spins with µ and -µ values.
  There are N2N equally probable arrangements.
  Each arrangement specifies the configuration of
  the 1rst, 2nd,.Nth spin
• ()()(-)(-)(-)()()(-)(-)(-)(-)()(-
  )...
• The total magnetization M can be Nµ, Nµ-2µ,
  ,.-Nµ some values of M occur more than others.

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Enumeration of States

• To obtain mean or observable values (ltMgt, ltM2gt,)
  we define the number of states with N() spins
  up and N(-) spins down
• g(N,m) with m N() - N(-) M/2µ
• g(N,m) N!/(N/2-m)!(N/2m)!
• from (xy)N total arrangements N
  ?g(N,m)2N
• Ensemble Average. We assume a group of similar
  systems called ensemble of systems. In a closed
  systems is equally probable to be in any
  accessible state.

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Magnetization

• In the limit of N and m large (Sterling
  Approximation, used in polymers), we obtain a
  Gaussian distribution
• g(N,m) g(N,0) exp(-2m2/N)
• where g(N,0) 2N(2/pN)1/2.
• Gaussian integrals are useful to take averages
• ltmgt? m P(N,m) ?dm m g(N,m)/2N, ltm2gt?dm m2
  g(N,m)/2N
• We can include an external filed H and find the
  energy per spin Us - µs ? H and the total energy
  
• U ? Us -H ? µs -H M -H 2 µ m

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Thermodynamics
Charged flexible chains high a) to low c) salt
concentration (screened Coulomb interactions)

• The entropy is defined by
• s/KB s log g(N,U)
• Entropy increases is we add particles (N
  increases).
• Polymer conformation is dominated by entropy but
  in different environments (in solutions or in
  blends or if monomers are charged, etc) the
  conformation is affected.

Netz Andelman 2002
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Entropy

• g(N,m) g(N,0) exp(-2m2/N)
• so s s(N,0) -2m2/N,
• F U T S or F/KT U/KT s
• In polymers mR is in one dimension the end to
  end distance of the chain
• Get F (R) in units of KT

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POLYMER CONFORMATION 3D

• Real flexible polymer molecules change
  conformations by rotation about single bonds
  (typically C-C bonds) in the backbone.
• Steric repulsions cause the rotation to be
  hindered, with three staggered rotational states
  (gauche (G), trans (T) and gauche' (G'))
  energetically favored. Calculation of the
  average shape of a flexible polymer chain,
  including all bonding details, is demanding.
• The oversimplified model can be modified to
  describe in statistical terms the shape of real
  polymer molecules.

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Random Walk

• The freely jointed chain
• The average value of the end-to-end vector r is
  ltrgt 0.
• The average of square of the end-to-end distance
  r2 r?r
• nl 2 0
• Hence ltr2gt nl 2
• Diffusion eq. with tn and Dl2/6

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Interactions Flory argument
v is the excluded volume vgt0 is good solvent
or SAW Minimized F with respect to R and get
Flory exponent in good solvent
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Collapsed Chains in solvents

• When v lt 0 the chain is in a bad solvent it
  collapses
• (Add three body term in free energy)
• When v 0 the chain is a random walk

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Polymer Solution what is v?
N f
Flory-Huggins interaction parameter c
Free energy of mixing
Expand when solvent concentration is small
1 phase to 2 phase
Degree of incompatibility v1- 2c c ?
energetic N ? entropic
? vs volume fraction f
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Semidilute Solutions
semi-dilute overlapping of blobs Inside the
blobs chains are SAW or swollen For lengths
larger than blobs chains are RW of blobs
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Polymer melt state Mean Field
Free energy of mixing
N1
N, f
Expand when solvent concentration is small
(virial expansion)
Degree of incompatibility v is reduced by N then
ideal chains if N1N Expansion of F(R) for RRo
(1 4z/3 ) z v N1/2/b3 (from Flory
at RRo) then if v 1/N1 then z N1/2/N1
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Polymer blend phase diagram
NB, fB
NA, fA
Free energy of mixing
Flory-Huggins interaction parameter c
2 phase
cN
Critical point
1 phase
Degree of incompatibility cN c ? energetic N ?
entropic
Volume fraction fA
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Patterns at micro- and nano-scales due to
competing interactions
Well studied in block copolymers
www.polymermicroscopy.com
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Block copolymer melts
NA, f
NB, 1-f
A and B segments are covalently linked
Phase separation can only occur
microscopically Balance between repulsion and
stretching
B
A
B
B
A
A
L
Spheres
Spheres
Cylinders
Cylinders
Lamellae
cN
1
Gyroid
fA(z)
10.5
0
z
Volume fraction fA
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Gradient copolymers
The average composition varies in a controlled
manner along the chain backbone

• l the fractional length of the gradient
  along the chain backbone
• g(n) a function that describes the average
  composition of the chain along the backbone

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copolymer phase separation

• Numerical
• Self-consistent mean field theory
• Lamellar phase separated region
• Analytical
• Random phase approximation
• Scattering function
• Critical microphase separation transition

Equilibrium spacing Density profiles
(cN)c
f
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Self-consistent mean field theory
L
cN
(cN)c
f(z)
0.5
0
1
z
f
Goals

• Find dependence of the equilibrium repeat
  distance (L) on cN
• Find unit cell composition profiles

Inputs Copolymer profile g(s) Interaction
parameter c Length of chain N
Outputs Composition profiles f(z) Free energy per
chain
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SCF Method

• Polymer chains are on a simple cubic lattice
• i lattice layer in the z direction
• j segment along a polymer chain
• a lattice width statistical segment length
• The probability that a segment j is in layer i is
  the probability distribution function q(i,j)

i
1
2
3
4
5
6
j 1
j N
z
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Distribution function

• There is a recursive relationship for q(i,j) due
  to chain connectivity
• If segment j is in layer i, then segment j-1 must
  be in layer i-1, i, or i1

i
i-1
i1
1 neighbor in i-1
1 neighbor in i1
j
4 neighbors in i

• w(i,j) is the local mean field due to the
  presence of all surrounding molecules (an average)

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Mean field

• The mean field w(i,j) depends on
• The surrounding composition, volume fraction
  fa(i)
• The local composition of the polymer chain, g(j)
• c is modified for copolymer melt
• Incompressibility is enforced

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Diffusion equation

• Remember the diffusion equation
• The distribution function q also obeys a
  diffusion equation
• Concentration ? probability q
• Time ? number of monomer steps
• External field k ? mean field w(z,j)
• This equation can be solved for q
• Discretize derivatives
• Distance z ia
• c is small so w/kBT ltlt 1

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Circular equations
q2
j
jN
N-j
q1
j0

• To get q, need w
• To get w, need fa, fb
• To get f, need q

(1)
(2)
(3)
All three equations must be obeyed simultaneously
for f
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Composition profiles Block copolymer

• Transition to ordered phase at cN10.5
• At cN11
• Weak segregation
• Sinusoidal profile
• At cN140
• Strong segregation
• No longer sinusoidal

cN11
fa(z)
Weak segregation
z/Rg
cN140
cN20
fa(z)
Strong segregation
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Composition profiles Gradient copolymers
l0.5
cN140
cN15
fa(z)
z/Rg
cN35
cN140
fa(z)
z/Rg
Profile sinusoidal at high cN
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Free energy

• In each layer, an energy Dw is added to enforce
  incompressibility
• The interfacial free energy is calculated by
  summing Dw over all L layers
• The minimum of the free energy vs. L/Rg curve is
  the equilibrium repeat period for that cN

cN100
cN80
cN60
cN40
cN30
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Equilibrium repeat period

• L/Rg found for each cN from minimum in free
  energy
• For a given cN
• L/Rg decreases with increasing gradient
• For a given L/Rg
• cN increases with increasing gradient
• Decrease cN ? critical transition
• Block copolymer critical point by Leibler
• Find gradient critical points in a similar way

increasing l
Critical transition
(cN)c 10.495 L/Rg 3.23
Leibler, L. Macromolecules 13, 1602 (1980)
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Scattering function
The scattering function S(k) measures the
strength of the scattering at different
wavevectors k2p/l for fluctuations in the
disordered phase
Scattering increases with cN Transformation to
ordered phase at (cN)c, S(k)??
S(k)
cN
Scattering peak at lD Disordered periodic
fluctuations scatter light
No scattering at l? One component system A and B
are connected and cant separate
No scattering at small l Single chain decay Small
regions of all A and all B exist
k2p/l
k2p/D
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Microphase separation transition
L
The scattering function S(k) diverges at ODT at
wavelengths l2p/k equal to the ordered periodic
spacing L
Derive S(k) analytically using the random phase
approximation (RPA)
S-1(k)
Increase cN
Find l2p/k and cN where S-1(k)0
This is the critical point (cN)c, L
k2p/l
k
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Weak segregation

• Use order parameter expansion for the free energy
• Minimize with respect to one harmonic amplitude
• Minimize with respect to k

Find L scaling

• Block copolymer
• Approximation breaks down
• Not sinusoidal
• Amplitude increases quickly
• Gradient copolymer
• Will probably be a better approximation
• Remains sinusoidal
• Amplitudes grow slowly

Weak segregation
? 0.5
Leibler. Macromolecules (1980) Mayes and M.
Olvera de la Cruz Macromolecules (1990s)
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Strong segregation power law
0.17 ?
L
L
L
Minimize
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Conclusion on copolymer blends
Copolymer
Amplitudes are much larger
Expansion doesnt work
Narrow interphase approximation
The critical point for block to gradient
copolymers is from cN10.49 to 29.5
Helfand, E. Wasserman, Z.R. Macromolecules 9, 879
(1976)
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Micelles of A-B copolymers in C-D blend or
solution (simulations in solvent SCMF in blends)
PS-PPHS diblock

• For drug delivery
• For emulsification process
• Can we deliver materials inside a micelles?
• Generate swollen micelle.

PS PS-PPHS
Au
PVP
Au
PVP
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Amphiphiles in their corresponding matrix
D2
CnDm
C2
CnDm (Mgtn) amphiphile in its corresponding
short homopolymer matrix C2 and D2.
C2
C2 block
No solubilization of C2 homopolymer in the
micelle cores!!!
D2
?C2D80.0588
?C2D80.1498
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A typical way of solubilization of homopolymer
C2 into the micelle core
AnBm (m/n gt1, i. e. A2B6, A2B8,, A2B12 )
amphiphiles in the matrix of short homopolymer C2
and D2. C2 good solvent for A but bad solvent
for B D2 T solvent for B but bad solvent for A
Solubilization of C2 homopolymer in the micelle
cores-short An block in the intermediate regime
Strong attraction interaction between C unit and
A unit
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A possible efficient way of solubilization of
homopolymer C2 into the crew-cut micelle core
AnBm (m/n lt1, i e. A4B2) amphiphiles in the
matrix C2 and D2. C2 T solvent for A but bad
solvent for B. D2 good solvent for B but bad
solvent for A..
Strong attraction interaction between B unit and
D unit
Solubilization of C2 homopolymer in the crew-cut
micelle cores- long An block in the intermediate
regime
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1. Under what conditions do micelles,
and/or swollen micelles/swollen crew-cut micelles
form in the presence of monolayers?2. Can they
be in equilibrium with each other?3. What is
the dynamic process leading to their formation?
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Model
Coarse-Grained Bead-Spring Model
? the depth of the attractive well,
characteristic of the effective attraction
between the like monomer pair, plays the role as
the Florys interaction parameter ? .
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Formation of Swollen Micelles
A2B8 FAA FBB FCC FDD FBD 0.5 FAC 2.0
cABNA2B8 30 ?A2A89.29
C2
A2 C2 core B 8 corona
D2
C2
A2B8 (A unit white bead B unit blue bead)
solubilizes C2 homopolymers in D2 matrix by the
formation of swollen micelles under condition ?
ACgt  ? AA ? BB? CC ? DD for A2B8 C2D2  
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A2B8 FAA FBB FCC FDD FBD 0.8 FAC 3.2
cABNA2B8 56 ?A2A811.11
D2
C2
A2B8 (A unit white bead B unit blue bead)
cannot solubilize C2 in D2 matrix because of the
formation of micelles in the strong segregation
  ? AA ? BB? CC ? DD ? 0.7 for A2B8 C2D2
  in the strong segregation limit ?ABNA2B8 gt 50
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Formation of Swollen Crew-cut Micelles
A4 C2 core B 2 corona
A4B2 FAA FBB FCC FDD FAC 0.5 FBD 1.5
cABNA2B8 20 ?A4A27.49
D2
C2
A4B2 (A unit white bead B unit blue bead)
solubilizes C2 homopolymers in D2 matrix by the
formation of swollen crew-cut micelles under
condition ? BDgt  ? AA ? BB? CC ? DD for
A4B2 C2D2  
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B2 core A4 corona
A4 C2 core B 2 corona
In the strong segregation limit , cABNA4B2
gt40 Micelles form in the C2 matrix
In the intermediate segregation limit , cABNA4B2
2040 Swollen Crew-cut Micelles solubilize C2
hompolymers in D2 matrix
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Conclusions on micelles

• in AnBm( nltm)C2D2, with a strong affinity of A
  to C at the intermediate segregation regime, AnBm
  can solubilize C2 homopolymers in D2 matrix by
  the formation of swollen micelles.
• in AnBm( ngtm)C2D2, with a strong affinity of B
  block with D at the intermediate segregation
  regime, AnBm can solubilize C2 homopolymers in
  D2 matrix by the formation of swollen crew-cut
  micelles.
• Tuning the amphiphilic molecule architecture and
  the affinity of amphiphile with its miscible
  matrix constituents provide an optimal way to
  emulsifying homopolymers into a useful array.