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Introduction to Depletion Interaction and Colloidal Phase Behaviour

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Soft Matter at Aqueous Interfaces

Part of the book series: Lecture Notes in Physics ((LNP,volume 917))

Abstract

Efforts to explain physical properties of colloidal suspensions in terms of the forces that act between the colloidal particles go back to the beginning of the 20th century. In the second half of the last century theoretical progress clarified that the stability of colloidal particles is also affected by non-adsorbing polymers in solution, as first explained by Asakura and Oosawa in Japan using the excluded and free volume concepts. Here an introduction to the depletion interaction and resulting phase behaviour in colloidal suspensions is provided. The theory for the phase behaviour of colloidal dispersions is developed here starting from the Van der Waals theory for the as-liquid phase transition. Subsequently, the hard sphere fluid-solid phase transition is explained. Next, an attractive Yukawa hard-core model is used to outline the effects of varying the range of attraction on the phase behaviour of a colloidal suspension of attractive particles. Finally, the phase states that can be found in a colloidal hard sphere dispersion plus depletants are explained.

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Notes

  1. 1.

    The quantity \(n_{b}^{*}\) is the bulk polymer number density at which the polymer coils overlap. In terms of the volume fraction of polymer segments \(\varphi\) (\(0 \le \varphi \le 1\)), one then uses \(\phi_{p} = \varphi /\varphi^{*}\), with \(\varphi^{*}\) the segment volume fraction at which the chains start to overlap: \(\varphi^{*} = N_{p} v_{s} /v_{p}\), where \(N_{p}\) is the number of segments per polymer chain, \(v_{s}\) is the monomer (segment) volume, and \(v_{p} = (4\pi /3)R_{g}^{3}\) the coil volume, so \(\varphi^{*} { \sim }N_{p} /R_{g}^{3}\). The overlap number density \(n_{b}^{*}\) hence follows as \(n_{b}^{*}\) = \(3/(4\pi R_{g}^{3} )\).

  2. 2.

    A smooth transition between these forms is:

    $$V_{\text{ov}} (r) = \frac{2\pi }{3}(R_{d} - r/2)^{2} (2R_{d} + r/2).$$

    .

  3. 3.

    \(\varLambda = h/\sqrt {2\pi m_{\text{c}} k_{B} T}\), with the colloid mass \(m_{\text{c}}\) and Planck’s constant \(h\).

  4. 4.

    the ‘0’ refers to hard spheres and the subscript ‘f’ indicates a fluid phase.

  5. 5.

    The term Yukawa potential originally stems stems from the quantum mechanical theory of nuclear interactions. In a more general context, it is often used for potentials with a distance profile of the type \(\exp \{ - \kappa r\} /r\).

  6. 6.

    Although here only Yukawa attractions are considered this description also holds for spheres interacting through a hard-core repulsive Yukawa interaction.

  7. 7.

    This can be regarded as an explicit definition for \(\gamma\) within the van der Waals model when the attraction is described as a long-ranged Yukawa attraction.

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Acknowledgments

This text is highly inspired by several parts of the book I wrote with Henk Lekkerkerker and I thank him for the wonderful collaborations. I also acknowledge Agienus Vrij, Alvaro Gonzalez Garcia and Maartje S. Feenstra for useful discussions.

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Authors and Affiliations

  1. Laboratory of Physical Chemistry, Department of Chemical Engineering and Chemistry and Institute for Complex Molecular Systems, Eindhoven University of Technology, 513, 5600 MB, Eindhoven, The Netherlands

    Remco Tuinier

  2. Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry, Debye Institute, Utrecht University, Utrecht, The Netherlands

    Remco Tuinier

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Editors and Affiliations

  1. Forschungszentrum Jülich, Juelich, Germany

    Peter Lang

  2. Forschungszentrum Juelich GmbH, Juelich, Germany

    Yi Liu

Appendix

Appendix

As was the original objective of SPT [71], the pressure \(\varPi^{0}\) of the hard sphere system can be obtained from the reversible work of inserting an identical sphere \((q = 1)\)

$$\frac{W}{{k_{B} T}} = - \ln [1 - \eta ] + \frac{6\eta }{1 - \eta } + \frac{{9\eta^{2} }}{{2(1 - \eta )^{2} }} + \frac{{4\pi R^{3} \varPi^{0} }}{{3k_{B} T}},$$
(3.88)

to obtain the chemical potential of the hard spheres

$$\mu_{c}^{0} = {\text{const}} + k_{B} T\ln \frac{{N_{c} }}{V} + W.$$
(3.89)

Applying the Gibbs-Duhem relation

$$\frac{{\partial \varPi^{0} }}{{\partial n_{c} }} = n_{c} \frac{{\partial \mu_{c}^{0} }}{{\partial n_{c} }}$$

one obtains

$$\frac{{\varPi^{0} v_{c} }}{{k_{B} T}} = \frac{{\eta + \eta^{2} + \eta^{3} }}{{(1 - \eta )^{3} }},$$
(3.90)

the SPT expression for the pressure of a hard sphere fluid [71], which preceded the slightly more accurate Carnahan-Starling equation Eq. (3.30), which contains an additional \(\eta^{4}\)-term.

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Tuinier, R. (2016). Introduction to Depletion Interaction and Colloidal Phase Behaviour. In: Lang, P., Liu, Y. (eds) Soft Matter at Aqueous Interfaces. Lecture Notes in Physics, vol 917. Springer, Cham. https://doi.org/10.1007/978-3-319-24502-7_3

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