Phase diagram of aggregation of oppositely charged colloids in salty water

R. Zhang; B. I. Shklovskii

doi:10.1103/PhysRevE.69.021909

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Phase diagram of aggregation of oppositely charged colloids in salty water

R. Zhang and B. I. Shklovskii

Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA

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Phys. Rev. E 69, 021909 – Published 26 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.021909

Abstract

Aggregation of two oppositely charged colloids in salty water is studied. We focus on the role of Coulomb interaction in strongly asymmetric systems in which the charge and size of one colloid is much larger than the other one. In the solution, each large colloid (macroion) attracts a certain number of oppositely charged small colloids $(Z$ -ion) to form a complex. If the concentration ratio of the two colloids is such that complexes are not strongly charged, they condense in a macroscopic aggregate. As a result, the phase diagram in a plane of concentrations of two colloids consists of an aggregation domain sandwiched between two domains of stable solutions of complexes. The aggregation domain has a central part of total aggregation and two wings corresponding to partial aggregation. A quantitative theory of the phase diagram in the presence of monovalent salt is developed. It is shown that as the Debye-Hückel screening radius $r_{s}$ decreases, the aggregation domain grows, but the relative size of the partial aggregation domains becomes much smaller. As an important application of the theory, we consider solutions of long double-helix DNA with strongly charged positive spheres (artificial chromatin). We also consider implications of our theory for in vitro experiments with the natural chromatin. Finally, the effect of different shapes of macroions on the phase diagram is discussed.

References (24)

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Galland, New York, 2002).
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T. T. Nguyen and B. I. Shklovskii, J. Chem. Phys. 115, 7298 (2001).
This behavior is different from the one resulting from the assumption that for a given $(p, s)$ all macroion– $Z$ -ion complexes have the same charge. Such an assumption would lead to a picture of aggregate as a uniformly charged liquid of complexes. According to the classical Rayleigh theory [13] of instability of a charged liquid, this liquid would form droplets with the size optimizing the sum of their surface and charging energy (see also Ref. [14], and references therein). The possibility of redistributing Z-ions between complexes makes the system more flexible and leads to a deeper minimum of the total free energy.
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I. M. Ruzin, S. Marianer, and B. I. Shklovskii, Phys. Rev. B 46, 3999 (1992).
The definition of $ε$ is different from Ref. [11]. There notation $ɛ$ is used to define the additional correlation energy per Z-ion in the aggregate. The definition used here is more convenient for systems shown in Figs. 22 and 33 in which not all Z-ions in the aggregate can gain the additional correlation energy.
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[c1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Galland, New York, 2002).

[c2] J. Widom, J. Mol. Biol. 190, 411 (1986).

[c3] V. A. Bloomfield, C. Ma, and P. G. Arscott, ACS Symp. Ser. 548, 185 (1994).

[c4] J. Pelta, F. Livolant, and J.-L. Sikorav, J. Biol. Chem. 217, 5656 (1996).

[c5] E. Raspaud, M. Olvera de la Cruz, J.-L. Sikorav, and F. Livolant, Biophys. J. 74, 381 (1998).

[c6] E. Raspaud, I. Chaperon, A. Leforestier, and F. Livolant, Biophys. J. 77, 1547 (1999).

[c7] M. Saminathan, T. Antony, A. Shirahata, L. H. Sigal, T. Thomas, and T. J. Thomas, Biochemistry 38, 3821 (1999).

[c8] E. Lai and J. H. van Zanten, Biophys. J. 80, 864 (2001).

[c9] E. Lai and J. H. van Zanten, J. Controlled Release 82, 149 (2002).

[c10] H. W. Walker and S. B. Grant, Colloids Surf., A 119, 229 (1996), and references therein.

[c11] T. T. Nguyen and B. I. Shklovskii, J. Chem. Phys. 115, 7298 (2001).

[c12] This behavior is different from the one resulting from the assumption that for a given $(p, s)$ all macroion– $Z$ -ion complexes have the same charge. Such an assumption would lead to a picture of aggregate as a uniformly charged liquid of complexes. According to the classical Rayleigh theory [13] of instability of a charged liquid, this liquid would form droplets with the size optimizing the sum of their surface and charging energy (see also Ref. [14], and references therein). The possibility of redistributing Z-ions between complexes makes the system more flexible and leads to a deeper minimum of the total free energy.

[c13] Lord Rayleigh, Philos. Mag. 14, 184 (1882).

[c14] A. V. Dobrynin, M. Rubinstein, and S. P. Obukhov, Macromolecules 29, 2974 (1996).

[c15] C. Tse, T. Sera, A. P. Wolffe, and J. C. Hansen, Mol. Cell. Biol. 18, 4629 (1998).

[c16] I. Rouzina and V. A. Bloomfield, J. Phys. Chem. 100, 9977 (1996).

[c17] I. M. Ruzin, S. Marianer, and B. I. Shklovskii, Phys. Rev. B 46, 3999 (1992).

[c18] The definition of $ε$ is different from Ref. [11]. There notation $ɛ$ is used to define the additional correlation energy per Z-ion in the aggregate. The definition used here is more convenient for systems shown in Figs. 22 and 33 in which not all Z-ions in the aggregate can gain the additional correlation energy.

[c19] B. I. Shklovskii, Phys. Rev. Lett. 82, 3268 (1999).

[c20] K. Keren, Y. Soen, G. Ben Yoseph, R. Gilad, E. Braun, U. Sivan, and Y. Talmon, Phys. Rev. Lett. 89, 088103 (2002).

[c21] A. Yu. Grosberg, T. T. Nguyen, and B. I. Shklovskii, Rev. Mod. Phys. 74, 329 (2002).

[c22] M. Gueron and G. Weisbuch, Biophymers 19, 353 (1980); S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, P. Pincus, and D. Hone, J. Chem. Phys. 80, 5776 (1984); S. A. Safran, P. A. Pincus, M. E. Cates, and F. C. MacKintosh, J. Phys. (France) 51, 503 (1990).

[c23] T. T. Nguyen, A. Yu. Grosberg, and B. I. Shklovskii, J. Chem. Phys. 113, 1110 (2000).

[c24] P. Linse and V. Lobaskin, Phys. Rev. Lett. 83, 4208 (1999); V. Lobaskin and K. Qamhieh, J. Phys. Chem. B 107, 8022 (2003).

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Phase diagram of aggregation of oppositely charged colloids in salty water

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