Sedimentation Equilibrium of Mixtures of Charged Colloids

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0.03108642η i 0.01500 5 10 150.03108642η i 0.01500 5 10 15η 2/η 1η 1=20η 20 5 10 15 20η 2/η 1η 1=η 2h ih i00 5 10 15 20x (cm)0x (cm)Figure 4.15: The addition of a second component. Charge numbers areZ 1 = 1Z 2 2 = 250, gravitational lengths are L 1 = 1L 2 2 = 1mm. The meanpacking fraction of component 1 is fixed, ¯η 1 = 0.005, while the mean packingfraction of component 2 varies between 0.1¯η 1 < ¯η 2 < 10¯η 1 . The dotted lineis the packing fraction of component 1 in the absence of component 2. Theinsets show the mean heights of both components and the total mean height(see also eq. A.5).0.0223E x(mV/cm)10.80.60.4η 0.2i 0.01 4050 20 40 60 80 100x (cm)67891000 20 40 60 80 100x (cm)0.040.03123E x(mV/cm)0.80.60.440.2η i 0.02500 5 10 15 206x (cm)70.018109100 5 10 15 20x (cm)Figure 4.16: Packing fractions of a suspension with 10 different types ofcolloids. The charge number Z n = n · 250, while the gravitational lengthL n = 1mm. The other parameters of the system can be found in tab. 4.1,except for the left figure, height H = 100cm, instead of 20cm. The insetshows the electric field strength.stant.• Overlap of regimes can be neglected, segregation is strong. That is,34
2ρ s ≪ Z 2 i ρ i , the suspension height H is larger than the layer width ξ,and the charge per mass differs strongly per component Z i L i ≪ Z j L jfor all i < j < n, with n the number of components.For simplicity, consider an equimolar mixture, with mean packing fraction¯η i = ¯η. Call x i the height at which two components i and i + 1 form aninterface. Number the components from highest to lowest mass per charge,i.e. Z 1 L 1 < Z 2 L 2 < . . . < Z n L n . The form of the packing fractions is, byassumption,η i (x) = b i − c i x (4.18)where the slope c i = ρ s πσ 3 /3Z 2 i L i of η i is calculated from equation (3.8).Force balance (see equation (3.10)) dictatesβΠ(x i ) = limx↑xiρ i (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2= limx↓xiρ i+1 (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2 . (4.19)Now we use that the potential is continuous, whereas the electric field makesa jump (use equation (4.12)):φ ′ (x) ≈ 1Z i L i, for i ∈ I i , (4.20)where the approximation is justified by the assumptions, and I i is the intervalon which η i is non-zero. From equation (4.19) we now obtain:ρ i (x i ) − 1 ( 1) 2= ρi+1 (x i ) − 1 (1) 2(4.21)8πλ B Z i L i 8πλ B Z i+1 L i+1The jumps in the packing fraction are therefore((η i (x i ) − η i+1 (x i ) =σ3 1) 2 (1) 2 )− ≡ t i . (4.22)48λ B Z i L i Z i+1 L i+1Where t n is just a convenient abbreviation, called the jump constant. If oneη i (x) can be found, all packing fractions can be found by recursion. Thecomponent with the lowest mass per charge is expected on top, labelled byn. For b n the following relation applies:1H∫ xn +ξ nx ndx()b n − c n (x − x n ) = ¯η, (4.23)where ξ n is the layer width of component n. The constant b n is now foundto be:b n = √ 2H ¯ηc n . (4.24)35
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 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 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),

Sedimentation Equilibrium of Mixtures of Charged Colloids

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