Integral Equation Methods in Macromolecular

1
Integral Equation Methods in Macromolecular
Science Selected Applications
Avik P. Chatterjee Department of Chemistry State
University of New York College of Environmental
Science Forestry
2
Outline

• Introduction
• 1. Fiber-reinforced composites, wood and
  cellulose, and
• connectedness percolation
• 2. Liquid State Theory
• 3. Polymer Reference Interaction Site
  Model (PRISM)
• Connectedness Percolation in Macromolecular
  Systems
• 1. One-Component Systems of Flexible and
  Rigid Polymers
• 2. Two-Component Mixtures of Flexible and
  Rigid Polymers
• Shear-Induced Effects on Miscibility in Polymer
  Solutions
• Conclusions

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Reinforced Composites Where ?

• Natural Composites Composites e.g., bone
  (calcium hydroxyapatite collagen) and wood
  (complex polysaccharides lignin) exhibit a
  multi-level, hierarchical structure which
  achieves greater toughness and stiffness than the
  individual component materials
• Man-made Composites synergistic improvement of
  material properties, e.g., Si-C reinforced Al
  brake rotors combine high stiffness with low
  density, and reinforced concrete combines the
  compressive strength of stone (concrete) with the
  tensile strength of steel and nanoclay/montmorill
  onite reinforced thermoplastics particle board
  epoxy resin wood chips, etc
• Wood for the Trees wood remains one of the
  most successful fiber-reinforced composites, and
  cellulose the most widely-occurring biopolymer
  (annual global production of wood estimated at
  1.75 x 109 metric tons).

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Wood A natural, fiber-reinforced Composite

• Cell walls layered cellulose microfibrils
  (linear chains of glucose residues, degree of
  polymerization ? 5000 10000, ? 40-50 w/w of
  dry wood depending on species), bound to matrix
  of hemicellulose and lignin.

5
Wood Selected Mechanical Properties Anisotropic
by Nature
Tensile Modulus and Tensile Strength for Steel
200 GPa and 400 MPa, respectively Tensile
Modulus and Tensile Strength for Carbon
Nanotubes 103 and 102 GPa, respectively.
6
Cellulose Nanocrystals (I)

• Individual fibers have major dimensions 1-3
  mm, consisting of spirally wound layers of
  microfibrils bound to lignin-hemicellulose
  matrix microfibrils contain crystalline domains
  of parallel cellulose chains individual
  crystalline domains 10-20 nm in diameter, 1-2
  ?m in length.
• Nanocrystalline domains can be separated from
  surrounding amorphous regions by carefully
  controlled acid hydrolysis (amorphous regions are
  degraded more rapidly)
• Crystalline domain elastic modulus 150 GPa
  compare S-glass 85 GPa, Aramid 65 GPa, and
  Carbon fibers 230 GPa
• Suggests possible role for cellulose
  nanocrystals as a renewable, bio-based,
  low-density, reinforcing filler for polymer-based
  nanocomposites.

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Cellulose Nanocrystals (II)

• Cellulose microfibrils secreted by certain
  non-photosynthetic bacteria (e.g., Acetobacter
  xylinum), and form the mantle of
  sea-squirts/tunicates (e.g., Ciona
  intestinalis)
• These are highly pure, largely crystalline
  forms, free from lignin/hemicelluloses
  fermentation of glucose could provide a microbial
  route to large-scale cellulose production.

Culture of Cellulose-secreting colonies of A.
xylinum, W.-T. Wu, et. al., Biotechnol. Appl.
Biochem., 35, 125, (2002).
Adult sea-squirts
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Cellulose Nanocrystals (III)
Nanocrystalline cellulose whiskers, obtained by
acid hydrolysis of bacterial cellulose. Image
courtesy of Drs. W.T. Winter and M. Grunert,
Dept. of Chemistry, SUNY-ESF.
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Connectedness Percolation What ?
Percolation The formation of infinite, spanning
clusters of connected (defined by spatial
proximity) particles.
Non-percolated system
Percolated system
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Percolation Why? --- Dramatic Effects on
Material Properties
Z. Bartczak, A.S. Argon, R.E. Cohen, M. Weinberg
Polymer, 40, 2331, (1999).
Izod impact energy (J/m)
Average ligament thickness (mm)
Toughness of HDPE/Rubber Composites Rubber
particle size range 0.36 0.87 mm
t Matrix ligament thickness
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Percolation and Cellulose-based Nanocomposites

• Polypyrrole-coated cellulose whiskers
  investigated as filler particles providing (i)
  electrical conductivity and (ii) mechanical
  reinforcement/shear modulus enhancement (J.Y.
  Cavaille, et. al., Compos. Sci. Technol., 61,
  895, (2001) Macromolecules, 28, 6365, (1995)
  the matrix was a styrene-butyl acrylate
  copolymer Particle aspect ratio 100).

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Percolation and Elastic Moduli in Filled
Nanocomposites

• Halpin-Tsai effective medium model for Eeff ,
  Geff below percolation
• threshold
• Elastic fiber network model above threshold,
  includes deformation energies due to rod bending,
  stretching, and shearing
• Connect across threshold using a scaling
  function chosen to reproduce the correct critical
  exponents.
• (X.F. Wu and Y.A. Dzenis, J. Appl. Phys., 98,
  093501, (2005), and M.D. Rintoul and S. Torquato,
  J. Phys. A. Math. Gen., 30, L585, (1997)).

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Elastic Network Model for Filled Nanocomposites

• Rods of length L, radius R, tensile modulus E,
  Poisson ratio ?
• Stretching energy ? E R2 L (?XX2) /2
• Bending energy 3 ? E R4 (?XY2) /( 2 L)
• Shearing Energy ? E R2 L (?XY2) /( 4 (1 ?)),
• where X-direction is parallel to the rod axis.
• Assume (i) isotropic orientational distribution,
  (ii) linear variation of number of rod-rod
  contacts with volume fraction, and (iii) scaling
  behavior for percolation probability near the
  threshold
• P (?rod) ? (?rod - ?rodP)0.474, in three
  dimensions, for ?rod gt ?rodP ,
• P Probability a randomly selected rod belongs
  to the infinite cluster.
• Calculate energy of elastic deformation,
  transform strain tensor to laboratory frame,
  average over rod orientations and segment
  lengths, to obtain estimate for moduli.

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Elastic Moduli, Yield Strains, from Nanocomposite
Model

• Assumes elastic limit ? elastic deformation
  energy per rod equals interaction energy
  (H-Bonds/Van der Waals) at rod-rod contacts (L.A.
• Hough, et al., Phys. Rev. Lett., 93, 168102,
  (2004)).

Parameter choice Erod150 GPa, Ematrix15 MPa,
rod radius 8.5 nm, rod-rod contact energy 5 x
10-17 J/contact, matrix rod Poisson ratios
0.5, chosen to mimic cellulose whiskers
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Model Polymer Structures
Flexible Coil-like Polymer
Rigid Rod-like Polymer
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Liquid State Theory

• Originally developed for monatomic and small
• molecule fluids
• An integral equation method used to calculate
  the
• radial distribution function g(r), which
  describes
• the inter-particle packing and degree of
  correlation
• Thermodynamics of liquids can be described
  based
• on complete knowledge of the radial
  distribution
• function.

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Radial Distribution Functions
g(r) represents the conditional probability of
finding sites on a different molecule at a
distance r from a given site.
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Ornstein-Zernike (OZ) Equation ---For Atomic
Liquids

• h( r ) g( r ) 1 h( k )
  Total correlation function
• C( r ) C( k) Direct
  correlation function
• r --- Site number density (number of sites per
  unit volume)

Closure Relation Percus-Yevick (PY)
approximation g( r ) 0 r lt d
Appropriate for athermal, c( r ) 0
r gt d excluded-volume,
interactions
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Polymer Reference Interaction Site Model (PRISM)

• Individual molecules represented by several
  interaction sites
• An intramolecular correlation functionthe
  form factor ?(r) --- is
• introduced
• Form factor Depends upon molecular
  architecture

• For a multi-component system of homopolymers
  (labelled M, M,
• etc), equivalent-site approximation ? PRISM
  equations

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Form Factors

• Form factor Important Limiting behaviors
• As k ? 0, (k) ? N (1 k2R2g)
• For k Rg gtgt 1, (k) ? 1 / kn where
  n is the Flory
• exponent, defined through the relation Rg ?
  Nn , where
• N is the number of monomers/subunits per
  chain
• Suggests the simple form
• (k) ? (R g/d)1/n / 1 k2R2g 1/2n ,
• used in much of our work.

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Correlation Pathways
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Molecular Closure Approximations

• The repulsive hard core and the soft
  attractive branch of the
• inter-site potential are treated separately
• Reference/ athermal state ? only the
  repulsive excluded
• volume interactions are accounted for
• The direct correlation function c(r) with the
  attractive potential
• accounted for is modified perturbatively
  based on the athermal
• direct correlation function.
• Reference Molecular Percus-Yevick / High
  Temperature Approximation (R-MPY/HTA)

Asterisks represent convolutions in r-space, and
superscripts 0 indicate athermal conditions.
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Thread and String Approximations

• Thread Approximation
• d ? 0 limit, closure relation becomes
• g(r ? 0) 0
• String Approximation
• Monomers are excluded from the
• region r lt d in an average sense

d

• These approximations frequently permit analytic
  solutions to the OZ eqns., as C (r) is now
  simply a Dirac delta function.

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String Model Solutions of Flexible Polymers (I)

• Accurately describes Mw-dependence of second and
  third virial coefficients under good solvent
  conditions (PS in toluene)

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String Model Solutions of Flexible Polymers (II)

• Accurately describes concentration-dependence of
  osmotic pressure in the semidilute regime
  (poly-?-methyl-styrene in toluene)

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Alternative Treatment of Ideal Swollen Coils (I)

• Use blobs, defined by the mesh size/screening
  length ?, as the elementary structural units
  (instead of entire chains)
• For semidilute solutions, treat a swollen
  polymer molecule as an ideal chain of swollen
  blobs

• Rg N n (dilute), and
• N n f n / (1-3 n ) , (semidilute)
• (k) k , for k ? gtgt 1
• Reproduces results from scaling theory for second
  virial coefficient, osmotic pressure

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Alternative Treatment of Ideal Swollen Coils
(II)

• Unified treatment consistent with (i) chain
  fractal dimensions, and (ii) screening of
  excluded volume interactions

Osmotic Compressibility N 105, 106 and 107,
swollen coils
Second Virial Coefficient Upper Curve Swollen
Coils Lower Curve Ideal Coils
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String Model Solutions of Rigid Rod-like
Polymers

• Analytic treatment uses approximate form factor
  ? (k) , which (i) reproduces Guiniers law for
  small wavevectors, and (ii) behave as ? (k) 1 /
  k for k gt 1 / L
• Correctly describes the dependence of rod-rod
  second virial coefficient on rod-aspect ratio
• Recovers result of Onsager theory that the
  critical volume fraction for onset of nematic
  ordering varies inversely with the rod aspect
  ratio.
• For both rod-like and flexible, coil-like,
  molecules, string closure predicts an upper limit
  to the monomer density at which the
  compressibility route pressure diverges
  logarithmically similar to the Sanchez-Lacombe
  lattice model.

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Connectedness Percolation in Macromolecular
Systems
Step 1 Solve OZ equations for the
athermal/reference correlation
functions Step 2 Direct correlation functions
with attractive branch of potential
accounted for are obtained via
molecular closure approximations Step 3 Solve
Connectedness Ornstein-Zernike (COZ) equation
to locate percolation thresholds.
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Connectedness Ornstein-Zernike (C-OZ) Equation
Subscript 1 for connectedness analogs of
correlation functions
Percus-Yevick (PY) approximation h1( r ) g ( r
) r lt R C1( r ) 0 r gt R Two sites
are directly connected if their separation is
less than R.
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Macromolecular Systems Investigated
One-Component Systems

• Athermal and thermal flexible coil-like
  polymers
• ( Square-well interaction is included via
  R-MPY/HTA)
• Athermal rigid rod-like polymers

Two-Component Systems

• Athermal and thermal mixtures of flexible
  and rigid polymers
• ( Rod-like particles dispersed in matrix of
  coil-like polymers )

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Categories of Model Studied

• Analytical thread/string models
• d ? 0 thread/string idealizations
  direct correlation
• function C( r ) approximated by
  d-function with
• amplitude C0 determined from closure
  relation.
• For C-OZ Equation use string-like
  approximation within
• the connectedness range R to determine
  C1
• Fully Numerical calculations on finite
  diameter chains
• Excluded volume interactions are
  treated explicitly
• within the Percus-Yevick closure,
  applied for all r lt R.

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Explicitly Two-Component Systems

• Analytical string model ( finite site
  diameters )
• 1. Zeroth-order Evaluate the direct and
  pair correlation
• functions in the limit rod density ? 0
  (Onsager-like model)
• 2. First-order Evaluate the direct and
  pair correlation
• functions through terms linear in the rod
  density
• 3. Full Calculation Determine Cij
  numerically at desired
• coil and rod site densities
• C-OZ Equation solved using string-like
  approx. for the rods.
• Fully Numerical calculations on finite
  diameter chains
• Similar to those for one component
  systems, excluded volume
• enforced for all r lt d.

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Percolation Thresholds in ModelOne-Component
Systems

• Relation to semidilute crossover concentration
• Set R Rg for flexible coils, and R L / 2 for
  rigid rods
• as the choice for connectedness distance
• ? gives ?p ? N-1/2 for Gaussian coils, ?p ? N-4
  / 5 for self-avoiding
• coils, and ?p ? L-2 for rods
• Consistent with usual picture of semidilute
  crossover concentration

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Percolation Thresholds in Rod-Coil Mixtures
A variety of models (numerical and analytical,
with several representations for the form
factors) find the same aspect ratio dependence of
rod percolation thresholds for one and
two-component systems ?p ? L-1 in all cases,
when R ltlt L.
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Percolation Thresholds --- Hard Rods in a
Matrix of Coils
Finite diameter models, rods/coils composed of
overlapping spherical beads
Overlapping bead coils dcoil 2 lcoil drod
10dcoil R 1.1 drod rmatrix 1.375
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Percolation Thresholds --- Hard Rods in a
Matrix of Coils
Analytical results from the string-like model for
rods
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Percolation ThresholdsEffects of Rod-Matrix,
Rod-Rod, Interactions

• Thread-like, simple, approximations for the form
  factors
• Yukawa potential for rod-coil, rod-rod,
  interactions
• R-MPY/HTA closure approximation C-OZ equations
  determine
• percolation thresholds as functions of (i)
  effective rod-rod
• second virial coefficient, (ii) rod-coil diameter
  ratio, and
• (iii) interaction range
• Fully Numerical calculations on finite diameter
  chains in an
• explicitly two-component model.

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Percolation Thresholds Effects of Rod-Matrix,
Rod-Rod, Interactions (I)
Rod-rod, rod-matrix, specific interactions do not
affect dependence of percolation threshold on
aspect ratio
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Percolation Thresholds Effects of Rod-Matrix,
Rod-Rod, Interactions (II)
Percolation thresholds correlate closely with
Rod-Rod Second virial coefficient
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Depletion Interactions Induced byFlexible Coils
(I)

• Excluded volume interactions create an imbalance
  in osmotic pressure when depletion zones
  overlap ? effective medium-induced attraction
  between solute particles

42
Depletion Interactions Induced byFlexible Coils
(II)

• Competition between (i) increasing osmotic
  pressure, and (ii) decreasing mesh size as
  concentration of flexible depletants increases
  leads to minimum in the solute-solute second
  virial coefficient near the semidilute crossover
  for both spherical and rod-like solute particles

Rod-like Solute Solid L 100 d, Ncoil 102,
103, 104, 105 Broken L 100 d, Ncoil 102,
103, 104, 105
Spherical Solute R/Rg 100, 1, 0.1, 0.025, 0.01,
top to bottom
43
Chain Deformation in Shear Flow

• Rouse and Rouse-FE
• Zimm and Zimm-FE

Finite Extensibility
Dimensionless shear rate
44
Shear-Induced Effects on Miscibility in Polymer
Solutions
Polymer Solutions

• Effect of accounting for finite chain
  extensibility on predictions of flow-induced
  shifts in miscibility ?
• Rouse, Rouse-FE, Zimm and Zimm-FE models used to
  describe chain conformational deformation under
  flow
• Thread closure approximation models the excluded
  volume effects
• Square-well interaction is included via
  R-MPY/HTA.
• Can the finite extensibility of polymer molecules
  be responsible for observed non-monotonicities in
  the dependence of critical temperature on shear
  rate ?

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Determining Spinodal Boundaries
Compressibility route is used, as this is less
sensitive to local, large wavevector, features of
the model.

• Two Free Energy Contributions
• Free energy of a stationary polymer solution
  in
• which coil conformations have been
  deformed, i.e., change
• in cohesive energy due to coil deformation
• Free energy explicitly due to coil
  extension/deformation under flow.

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Critical Temperatures--- Polymer Solutions
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Comparison With Experiments on Flowing Polymer
Solutions
C.C. Han, et. al., J. Chem. Phys., 100, 3224,
(1994).
Crosses Exptl. results for PS of Mw 1.8 ?106 in
dioctyl phthalate
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Conclusions --- Percolation in Macromolecular
Systems

• Percolation thresholds depend strongly on
  macromolecular architecture and dimensions
• Attractive segmental interactions strongly affect
  the percolation threshold
• The dependence of critical volume fractions on
  rod aspect ratio found in two-component model is
  similar to that found for the analogous
  one-component model, and is consistent with MC
  simulation results on fluid of phantom ellipsoids
  (J. Douglas et. al., 1995)
• Results from fully numerical calculations are
  consistent with
• those from analytical approximations
• A comparison of results from different methods
  suggests the analytical treatment could be of use
  in understanding percolation in more complex,
  multicomponent, macromolecular systems.

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• Future Directions
• Influence of
• (i) rod length polydispersity, and
• (ii) anisotropic rod orientational
  distributions, on percolation thresholds and
  elastic moduli
• Treatment of non-equilibrium situations within
  the Replica-OZ model, which considers the rods
  (say) as reaching an equilibrium distribution
  with respect to a quenched, previously
  equilibrated, matrix of flexible coils.

50
Acknowledgements
Dr. Xiaoling Wang Ms. Darya Prokhorova Prof.
William T. Winter, Department of Chemistry,
SUNY-ESF Donors of the Petroleum Research Fund,
administered by the American Chemical
Society The USDA CSREES National Research
Initiative Competitive Grants Program The USDA
CSREES McIntire-Stennis Program.