The pitfalls of nonlinearity

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The pitfalls of nonlinearity

• Do we really understand
• the low density phase
• of charged colloids?

Markus Deserno
Max-Planck-Institut für Polymerforschung,
Ackermannweg 10, 55128 Mainz, Germany
CECAM, Lyon, July 8, 2003
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Outline

• A few words on the experimental side
• Some theoretical preliminaries
• Analysis within the cell-model
• Conclusions

CECAM, Lyon, July 8, 2003
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Experimental observation
Suspensions of charged colloids appear to phase
separate at sufficiently low volume fraction and
sufficiently low ionic strength.
? 0.0011
?0.023
Qeff ?1600, a ? 90nm
Confocal scanning laser microscopy
B. V. R. Tata et al., Phys. Rev. Lett. 69, 3778
(1992)
See however T. Palberg and M. Würth, Phys. Rev.
Lett. 72, 786 (1994).
B. V. R. Tata et al., Phys. Rev. Lett. 78, 2660
(1997)
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Donnan equilibrium
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Interactions, correlations
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Relative importance of these terms
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Strategy for the following
Discuss the problem within a cell-model
Evaluate the approximations involved
Compare to simulations
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Reminder The cell-model approximation
factorization in macroion coordinates!
density functional
cell-model
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Poisson-Boltzmann theory Functional
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Poisson-Boltzmann theory Pressure
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Linearized Poisson-Boltzmann theory
Expand PB functional up to quadratic order in
densities
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Linearized Poisson-Boltzmann pressure
Same game derivative of the (linear) functional,
evaluated at equilibrium
M. Deserno and H.-H. von Grünberg, J. Chem.
Phys. 66, 011401 (2002)
yields positive excess pressure (for symmetric
electrolytes)
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Illustrative example (1)
Colloidal suspension with Q 3500, r0 133 nm,
ns 3.6 ?M, lB 7.14 Å
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Illustrative example (2)
Physiological solution of DNA ? 4.2, r0 1.2
nm, ns 100 mM, lB 7.14 Å
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There clearly is a problem!
Linearized PB theory predicts a gas-liquid phase
separation for these systems, which is rigorously
absent on the nonlinear level!
Here the transition is an artifact of
linearization.
A. Diehl, M. C. Barbosa, and Y. Levin, Europhys.
Lett. 53, 86 (2001) H.-H. von Grünberg, R. van
Roij, and G. Klein, Europhys. Lett. 55, 580
(2001) M. Deserno and H.-H. von Grünberg, J.
Chem. Phys. 66, 011401 (2002) M. Tamashiro, J.
Chem. Phys. 119, 1855 (2003)
CECAM, Lyon, July 8, 2003
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In other words
Within the well-controlled settings of a
cell-model linearization is seen to lead to a
qualitative difference!
This dilemma persists independent of
questions regarding the validity of the
cell-model itself.
Still how reliable is the cell-model?
CECAM, Lyon, July 8, 2003
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Comparison with simulation data
MC simulations by Per Linse, J. Chem. Phys. 113,
4359 (2000)
CECAM, Lyon, July 8, 2003
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Comparison with simulation data
MC simulations by Per Linse, J. Chem. Phys. 113,
4359 (2000)
Thanks to E. Trizac for pointing this out to me!
CECAM, Lyon, July 8, 2003
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Comparison with simulation data
MC simulations by Per Linse, J. Chem. Phys. 113,
4359 (2000)
cell model is very good
PB theory works well
linearization is dubious
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Some thoughts and conclusions
Macroionmacroion correlations beyond
the cell-model can be included at the price
of linearizing macroionmicroion correlations.
Why should one be willing to pay this price?
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Some thoughts and conclusions
The term in linear theory which yields
the negative excess pressure is of
quadratic order in the electrostatic potential.
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Some thoughts and conclusions
The cell-model appears to work well, even at very
low volume fractions, as compared to simulations.
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Final apology
I have no alternative explanation for
the experimentally observed phase behavior to
offer!
I rather believe that the question is still open.
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Acknowledgements

• Hans-Hennig von Grünberg
• Mario Tamashiro
• Emmanuel Trizac
• Yan Levin
• Vladimir Lobaskin
• Itamar Borukhov
• Håkan Wennerström

CECAM, Lyon, July 8, 2003