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APS March Meeting 2012
Padden Award Symposium
February 28, 2012
Influence of Charge and Network Inhomogeneities
on the Swollen-Collapsed Transition in
Polyelectrolyte Nanogels
Prateek Jha (Northwestern University)
Jos Zwanikken (Northwestern University)
François Detcheverry (University de Lyon)
Juan de Pablo (University of Wisconsin-Madison)
Monica Olvera de la Cruz (Northwestern University)
This material is based upon work supported by NSF under
Award numbers DMR-0907781 and DMR-0520513.
Polymer Nanogels
• Finite-sized, solvent-permeated polymer networks
• Highly responsive materials
– High volume changes by absorbing/releasing solvent
– Small size (10-1000 nm) enables rapid kinetic response
– Ionic (Polyelectrolyte) nanogels superior than neutral nanogels
Kabanov and Vinogradov, Angew Chem Int Ed Engl. 2009 ; 48(30): 5418–5429
Polyelectrolyte Nanogels as Drug Delivery Carriers and in
Anti-Cancer Therapy
• Can easily incorporate oppositely
charged drugs/biomacromolecules
e.g. oligonucleotides, siRNA, DNA,
proteins, …
Kabanov and Vinogradov, Angew Chem Int Ed
Engl. 2009 ; 48(30): 5418–5429
• High swelling of nanogel used in
killing cancer cells
Park et al., Journal of Controlled Release 135 (2009) 89–95
Classical Description of Gel Swelling
Rubber Elasticity
Flory-Huggins Theory
Donnan Membrane Equilibrium
• Homogeneous deformation  uniform volume fraction,   Nv / V
• Charge neutrality in gel and solvent bath  No Coulomb interactions
• Electrostatic contribution from mobile ion entropy (Donnan)
– Mobile ion concentrations from association  dissociation equilibrium of salt
• Free Energy of gel: F  Felastic  FFH  FDonnan
• Condition of Mechanical Equilibrium:
Osmotic Pressure :   F / V  0
Solve for 
Flory, Principles of Polymer Chemistry
Charge and Network Inhomogeneities in Nanogels
  8lB cs N Av 
1
1 / 2
e2
where lB 
4k BT
 1  10  100 nm for cs  0.001 0.00001M
for wat erat room temperature
Influence on Swelling behavior ?
Polymer Volume Fraction
• Nanogels are not homogeneous
– Crosslinking inhomogeneities
(during preparation)
– Excess Charge due to presence of surfaces
• Excess Charge Coulomb Interactions
(Neglected in the Donnan picture)
• Important when nanogel size is comparable to
screening length
Distance from center [nm]
J Ramos et al; Soft Matter, 2011, 7, 5067-5082
Poisson-Boltzmann Description
Network charge fixed and homogeneous
f  Charge fractionof polymer
F  r , c r , c r 
  c r ln c r   c r  ln c r dr
k BT
1
 r r dr
2
PoissonEquation: 2  r   4lB  r 

• Mobile ion concentrations by mean-field approximation
F / c  0  c  cs exp 
• Dielectric mismatch between solvent and polymer can be incorporated by a
solvation term:
c  cs exp ESolvation   
 cs exp   
Gel
Solvent Bath
PK Jha et al, Current Opinion in Solid State and Materials Science, 15(6), 271-276 (2011)
• Smaller, collapsed, and gels at low salt concentration or in high dielectric
solvent have larger excess charge Donnan theory fails
PK Jha et al, Current Opinion in Solid State and Materials Science, 15(6), 271-276 (2011)
• Excess charge may be used to self-assemble nanogels to crystalline
structures
Gottwald et al, Phys. Rev. Lett. 92, 068301 (2004)
• Dielectric mismatch can give rise to re-entrant behavior: re-swelling at high
salt concentration
Salt Concentration
Theoretically Informed Coarse-Grained Simulations (TICG)






Detailed swelling behavior
Crosslink Inhomogeneities
Fluctuations
Few physical invariants
Free of discretization effects
Computationally Efficient
• Construct the free energy functional:
H ri ,(r), c (r)  Ubonded ri   Ηnon-bonded (r), c (r)
• Bonded (Elastic) Contribution from Gaussian Chain Approximation
• Non-Bonded Contribution = Solvent + Entropy + Coulomb
• Monte Carlo using the free energy functional
Detcheverry et al, Macromolecules, 2008, 41 (13), pp 4989–5001; Soft Matter, 2009, 5, 4858-4865; Faraday Discuss.,
2010, 144, 111-125
Effect of Polymer Charge
GOOD SOLVENT
POOR SOLVENT
• Very high swelling for ionic nanogels (Coulomb repulsions)
• Discontinuous volume transition for ionic nanogels
PK Jha et al, Soft Matter, 7, 5965-5975 (2011)
Effect of Salt Concentration
Swelling decreases with increase in salt concentration (screening)
PK Jha et al, Current Opinion in Solid State and Materials Science, 15(6), 271-276 (2011)
Summary
• Density and Charge Inhomogeneities strongly
contributes to the nanoscale gel behavior
• New Simulation Scheme for swelling behavior of
polyelectrolyte nanogels
• Discontinuous volume transition for ionic nanogels
SWOLLEN
COLLAPSED
Good Solvent
Bad Solvent
Highly Charged
Neutral/Weakly Charged
Low salt
High salt
Loosely crosslinked
Tightly Crosslinked
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