programming?Do I like programming?Then you should copy chapter 6. It <strong>of</strong>fers you an opportunity to check thenumerical results in this document and to play with the programme, or evento optimize it.Many other chapters can be used to refresh the mind, like an encyclopedia,to recheck definitions or to gain some more insight in the origin <strong>of</strong> the finalresults.The reader will repeatedly be warned against unimportant information, advisedto skip chapters, but the reader should feel free to defy my understanding<strong>of</strong> importance.At last, and again: the form <strong>of</strong> this document is like its content.Rectangular?Seeking balance between opposing tendencies, and finding equilibrium, althoughseveral tensions and strong deviations from well known examplesremain. The structure will be diverse, with the permission <strong>of</strong> entropy, andlayered in density <strong>of</strong> information, suspense, entertainment, and many otherproperties too many to name.Let’s start!I hope you can enjoy reading now.vi
SummaryThis thesis is about sedimentation equilibrium <strong>of</strong> polydisperse colloidal suspensionsin the gravity field <strong>of</strong> the earth. The first part <strong>of</strong> the documentfocusses on the physical background <strong>of</strong> the subject. From a quick overview<strong>of</strong> some elementary thermodynamics, classical statistical mechanics, densityfunctional theory, the Bjerrum length λ B , the Debye screening length κ −1 ,and the gravitational length L will be introduced. These length scales willprove useful in describing the following colloidal systems.The first colloidal systems under consideration are monodisperse. By usingan analytical method, and a more consistent Poisson-Boltzmann theory, thenumber densities and electrostatic potential <strong>of</strong> the systems are obtained numerically.In the limit <strong>of</strong> low salinity a phenomenon is observed known as theentropic lift [8], colloids forming highly non-barometric density pr<strong>of</strong>iles. Thetheory will be expanded to describe polydisperse systems <strong>of</strong> charge stabilizedcolloids. In the same low salinity limit the entropic lift is again observed, aswell as segregation between colloidal species. The so-called Brazil nut effectis predicted by the theory. The layering is found to be dependent on themass per charge <strong>of</strong> the colloidal particles, as well as the system height. Afterthe consideration <strong>of</strong> binary, ternary and polydisperse systems (N = 10,and N = 21), two analytical models follow, that approximate the theory formulti-component systems. Monte-Carlo simulation data show almost perfectagreement with the numerical calculations, based on the theory.The last colloidal systems to be discussed contain charge regulating colloids.Three different methods determine the equilibrium distributions, and theexpectation value <strong>of</strong> the colloidal charge as a function <strong>of</strong> height. Still, manyexamples <strong>of</strong> systems remain to be explained, systems that violate primaryassumptions. In the region <strong>of</strong> high packing fractions, <strong>of</strong> non-spherical particleshape, dipolar colloids, the theory lacks answers. Some possible answers andadditional ideas are discussed in the outlook.One <strong>of</strong> the last chapters describes the C++ program that was used for thenumerical calculations. The summarizing article concludes the thesis, givingan overview <strong>of</strong> the main results.vii
- Page 1: Sedimentation Equilibrium of Mixtur
- Page 4 and 5: 5 Charge Regulation in the Low Dens
- Page 8 and 9: Additional information, often writt
- Page 10 and 11: 1.2 Sedimentation equilibrium of mi
- Page 12 and 13: make them spark. . . Wait, wait! I
- Page 14: dG = −SdT − V dP + ∑ αµ α
- Page 17: with ν λ = ν 0 +λw, the free en
- Page 20 and 21: eservoir is:µ res± = k B T ln(ρ
- Page 22 and 23: x-axis, only τ xx = −(ɛ/8π)E 2
- Page 24 and 25: is denoted by ρ i . The salt ions
- Page 26 and 27: numbers for the parameters Z i , L
- Page 28 and 29: 2Case (a)4.5Case (a)ρ +E (mV/cm)1.
- Page 30 and 31: is plotted as a function of Z 2 /Z
- Page 32 and 33: 0.1i=320a0.080.06η i0 5 10 15 20ρ
- Page 34 and 35: Z 16=2993a0.6210 3 η iZ 11=2500.40
- Page 36 and 37: 0.02210.83E x(mV/cm)0.60.40.0140.25
- Page 38 and 39: 4.6.1 The influence of the diameter
- Page 40 and 41: 0.01ρ s=10 −3 Mρ s=10 −5 Mρ
- Page 42 and 43: 0.03108642η i 0.01500 5 10 150.031
- Page 44 and 45: The next constant that can be solve
- Page 46 and 47: 4.6.5 Donnan-method for polydispers
- Page 48 and 49: and try to fill the holes, or try t
- Page 51 and 52: Chapter 5Charge Regulation in the L
- Page 53 and 54: 5.1.2 Numerical resultsThe example
- Page 55: Discussion The mean height of the s
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Z i (x) that converges, at least in
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Other extensions are planned to inc
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their buoyant mass. The force that
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(π/6)ρ i (x)σi3 the local packin
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0.010.008ρ +0.006 i=10.5hη i/σi
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species, and (B) L i = 2 × (250/Z
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A.7 AcknowledgementsIt is a pleasur
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length by Peter Debye. In dense col
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One quick check shows ∫ ∞dx Q(x
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Appendix CThe block modelThe model
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Index of frequently usedsymbols and
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Bibliography[1] J. Perrin, (1913),
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