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Electrophoresis of Charged Macromolecules

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Electrophoresis of Charged Macromolecules Electrophoresis of Charged Macromolecules Christian Holm Institut für Computerphysik, Universität Stuttgart Stuttgart, Germany 1! Charge stabilized Colloids! The analytical description of charged colloidal suspensions is problematic:! n " Long ranged interactions: electrostatics/ hydrodynamics! n " Inhomogeneous/asymmetrical systems! n " Many-body interactions! Alternative! : the relevant microscopic degrees of freedom are simulated! via Molecular Dynamics! ●Explicit" particles (ions) with charges ε ●Implicit" solvent approach, but hydrodynamic interactions of the solvent are included via a Lattice-Boltzmann algorithm Test of LB implementation for Poiseuille! Simulation box of size 80x40x10. Velocity profile for a Poiseuille flow in a channel, which is tilted by 45◦ relative to the Lattice-Boltzmann node mesh. Computed using ESPResSo. Profile of the absolute fluid velocity of the Poiseuille flow in the 45◦ tilted channel. Red crosses represent simulation data, the blue line is the theoretical result and the dashed blue line represents the theoretical result, using the channel width as a fit parameter 3! EOF in a Slit Pore! Simulation results for a water system. Solid lines denote simulation results, the dotted lines show the analytical results for comparison. Red stands for ion density in particles per nm3, blue stands for the fluid velocity in x-direction, green denotes the particle velocity. All quantities in simulation units. 4! Colloidal Electrophoresis! local force balance FE = FDrag leads to stationary state ν FE F = Z E − Zeff E eff Electric field E v Zeff Observable: mobilty µ µ = = E γ eff µred = 6πη Bµ e2 viscosity η Bjerrum length B = 4πεkBT SEM (O‘Brien, White 1978)! Dimensional Analysis ˜ Z = ZlB /R Di: collective diffusion coefficient 2 2 κ = 4πlB ∑zi ci € # & 2 fi = ci /% ∑z jc j ( $ i ' € € View Online 1622 ELECTROPHORETIC MOBILITY OF A COLLOIDAL PARTICLE as a function of Ka for specific values of the dimensionless zeta potential y=- 4 kT‘ (9.2) This is not the most convenient way of plotting p(r, h-a) if the graphs are to be used for converting experimental mobility measurements to zeta potentials, since these experiments are usually carried out at fixed electrolyte concentration, i.e., fixed Ka. I E ”E E: reduced mobility µred 4 E 3 View Online 2 Downloaded by Universitat Stuttgart on 25 March 2011 Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 1 I I Electrophoretic Mobility of a Spherical Colloidal Particle BY RICHARDw. O’BRIENt AND LEE R. WHITEf* Department of Applied Mathematics, Institute of Advanced Studies, I,, ll*,, 0 5 Research School of Physical Sciences, The Australian National University, y: reducedCanberra, zeta-potentoial A.C.T. 2600, Australia Y ζred FIG. 3.-Reduced mobility E of a spherical colloidal particle in a KCl solutionReceived as13th a February,function 1978 of reduced zeta potential y, for KU < 2.75. In this regime the mobility appears to increase monotoni- The equations which govern the ion distributions and velocities, the electrostatic potential and cally with zetathe potential. hydrodynamic flow field around a solid colloidal particle in an applied electric field are re- examined. By using the linearity of the equations which determine the electrophoretic mobility, We have adopted the alternative procedure forwe show plotting that for a Ecolloidal as a particle function of any shapeof y the for mobility various is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface. The mobility Ka values. This has the additional advantagedepends ofonly mimicking on the particle size the and formshape, the of properties a typical of the electrolyte solution in which it is suspended, and the charge inside, or electrostatic potential on, the hydrodynamic shear plane in the experiment where mobility is changed, atabsence fixed of anIC, applied by fieldsystematically or any macroscopic varyingmotion. the concentration of a potential determining ion. NewThis expressions is due for to the the forces monotonic acting on the particlevariation are derived and a novel substitution is developed which leads to a significant decoupling of the governing equations. These analytic of surface potential with the p.d.i. concentration.developments allowSince for the the construction chemist of a rapid,usually robust numericalhas scheme for the solution of the governing equations which we have applied to the case of a spherical colloidal particle in a general control over the value of Ica, the experiment electrolytecan be solution. performed We describe with a computer a Ka valueprogram corres-for the conversion of mobility measurements to zeta potential for a spherical colloidal particle which is far more flexible than the Wiersema graphs ponding to a computed curve. which have traditionally been used for the interpretation of mobility data. Furthermore it is free of the high zeta potential convergence difficulties which limited Wiersema’s calculations to moderate values of 5. Some sample computations in typical 1 : 1 and 2 : 1 electrolytes are exhibited which illustrate the existence of a maximum in the mobility at high zeta potentials. The physical explanation of this effect is given. The importance of the mobility maximum in testing the validity of the governing equations of electrophoresis and its implications for the colloid chemist’s picture of the Stern layer are briefly discussed. Downloaded by Universitat Stuttgart on 25 March 2011 Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 1. INTRODUCTION The velocity at which a charged colloidal particle moves in the presence of a weak electric field is linearly related to E, the strength of the applied field. For a spherical particle this relation takes the form 5 U = pE, where U is the velocity, and p is termed the “electrophoretic mobility” of the particle. This paper deals with the problem of calculating the quantity p. The electrophoretic velocity U is determined by the balance between the electrical and viscous forces which act on the particle, a balance which is strongly affected by the concentrations of the electrolyte ions in the solvent. In order to understand the way in which the ions affect the forces which act on the particle, we must consider the motion of ions around a spherical particle moving under the action of a steady electric field. t Rothmans Fellow ; Queen Elizabeth I1 Fellow. 5 For a non-spherical particle the velocity U will depend on the particle orientation ; there is no reason to expect that the velocity and field strength E will in general be in the same direction, 1607 View Online R. w. O’BRIEN AND L. R. WHITE 1623 The point is academic however since the method, presented in the previous sections of this paper, enables a rapid, exact numerical solution to the general mobility problem.* Thus the chemist, armed with this computer program can generate mobility as a function of zeta potential for the Ka value, the particular electrolyte solution, and temperature of interest to himself. In view of this, the utility of displaying graphs of p(5, K-a) for restricted electrolyte systems, as a conversion aid for the experimentalist, seems marginal. r 7- 6- 5- 4- E 3- 2- Downloaded by Universitat Stuttgart on 25 March 2011 Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 Y FIG.4.-Variation of mobility in KCI with zeta potential for KLZ > 3. In this case the mobility curves have a maximum. However, we have solved the mobility equations for a typical 1 : 1 electrolyte, viz. KCl, in order to display some of the striking features of the function &, ica). In fig. 3, we have plotted the reduced mobility E in KCl solution as a function of reduced zeta potential y (on the positive side where Cl- is the counterion) for values of rca 5 2.75. The following important features of fig. 3 should be noted: (i) in * A copy of the program constructed as a self-contained Fortran subroutine is available from the authors. View Online 1624 ELECTROPHORETIC MOBILITY OF A COLLOIDAL PARTICLE the region where Wiersema could obtain solutions to the equations [i.e., y < 5 ( N 125 mV)] our results are in agreement. (ii) For all rca values 52.75, the mobility is a monotonic increasing function of the zeta potential [at least in the experimentally accessible range y < lO(w-250 mV)]. (iii) At low values of zeta potential, all curves asymptote to the Huckel form E=y (9.3) irrespective of ica value. (iv) For values of Ka - 2.75 the mobility E plateaus at -3.17 For y 2 5. All these features are also present in Wiersema's work but are obscured by his method of displaying &, rca) and the restriction on the range of [ values that his method of solution imposes. I I I I *,I 21 ' ' """'5 10 50 100 ua FIG.5.-Maximum mobility of a spherical particle in KCl as a function of ua. a=R in lecture notes notation It is this restriction which prevented Wiersema from displaying, in a definitive way, Downloaded by Universitat Stuttgart on 25 March 2011 the most important feature of the p([, rca) function, viz. the maximum in the mobility as a function of zeta potential for all values of rca 2 2.75. In fig. 4, we display E Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 in KCI solution as a function of (positive) y for Ka 3 3. The following features should be noted: (i) a maximum in the mobility exists for all values of rca 2 3; (ii) the value of the maximum mobility increases with rca for rca > 4 (the very shallow maximum for rca = 3 is, however, slightly higher than the rca = 4 value); (iii) the reduced zeta potential corresponding to the maximum mobility is - 5 (125 mV) and increases very slowly as rca increases (the maximum for rca = 3 is at - 7, however) ; (iv) E(y) curves tend to the Smoluchowski form E = 3y (9.4) as rca is increased although, unlike the small Ka curves, there is no regime where all curves reduce to the Smoluchowski form and become independent of Ka.
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