Sedimentation Equilibrium of Mixtures of Charged Colloids

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0.1i=320a0.080.06η i0 5 10 15 20ρ salt(µM)15100.04i=25X (cm)0.02i=100 5 10 15 20X (cm)210 −6bm 3g/Z 3eE (mV/cm)1.510.5Q/2ρ ssinhφ(x)−10 −6 00 5 10 15 20X (cm)m 2g/Z 2em 1g/Z 1e00 5 10 15 20X (cm)Figure 4.5: Density profiles (a) and the electric field profile (b) of anequimolar ternary mixture of equally-sized colloids with charges Z i =(1000, 250, 125) and gravitational lengths L i = (1, 2, 1)mm with total packingfractions ¯η i = 0.005 , with monovalent ionic strength ρ s = 10µM in thereservoir. Almost complete segregation takes place into well-defined layers ofpure components, with a slope η i(x) ′ = πρ s σ 3 /3Zi 2 L i and a constant electricfield m i g/eZ i in the layer with component i, as denoted by the dashed linesin (a) and (b), respectively. The inset in (b) shows a dimensionless measurefor local charge neutrality.24
in the calculations is chosen to be un-physically low (ρ s < 1pM), to imitatethe coion-less situation of the simulations.0.030.02η i0.0100 50 100x (σ)Figure 4.6: An agreement with Monte Carlo simulation data (wiggly lines)by [22]. The diameter of the particles is taken to be σ = 150nm, the systemheight H is 100σ the gravitational length is L = (4, 6, 6)σ, and the colloidalcharge is Z = (50, 20, 10). The salt concentration in the calculations is chosento be un-physically low (ρ s < 1pM), to imitate the coion-less situation of thesimulations.4.5 Polydisperse systemsEven more exciting examples. . .Now we will compare two polydisperse systems, example 1 and example 2.The fixed system parameters can be found in table 4.1. The two examplescan be discerned by the mass per charge ratio Z i L i of the particles.Example 1, figure 4.7, shows a system of 21 components of equal mass, suchthat L i = 2mm. The colloidal charge number Z i however varies between 150and 350 according to a Gaussian function:¯η i = ¯η mean · exp[− (Z i − Z mean ) 22∆ 2 ] (4.15)¯η mean is a normalization constant that provides the total mean packing fractionto be 0.005, ∆ ≈ 50 is the standard deviation, and Z mean = 250 themean charge number.25
Page 1: Sedimentation Equilibrium of Mixtur
Page 4 and 5: 5 Charge Regulation in the Low Dens
Page 6 and 7: programming?Do I like programming?T
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 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
Page 58 and 59: Z i (x) that converges, at least in
Page 60 and 61: Other extensions are planned to inc
Page 62 and 63: their buoyant mass. The force that
Page 64 and 65: (π/6)ρ i (x)σi3 the local packin
Page 66 and 67: 0.010.008ρ +0.006 i=10.5hη i/σi
Page 68 and 69: species, and (B) L i = 2 × (250/Z
Page 70 and 71: A.7 AcknowledgementsIt is a pleasur
Page 72 and 73: length by Peter Debye. In dense col
Page 74 and 75: One quick check shows ∫ ∞dx Q(x
Page 77 and 78: Appendix CThe block modelThe model
Page 79: Index of frequently usedsymbols and
Page 83 and 84:
Bibliography[1] J. Perrin, (1913),
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Sedimentation Equilibrium of Mixtures of Charged Colloids

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