Article pubs.acs.org/JPCB
Electrophoretic Mobility of a Dilute, Highly Charged “Soft” Spherical Particle in a Charged Hydrogel Stuart Allison,*,† Fei Li,‡ and Melinda Le† †
Department of Chemistry, Georgia State University, Atlanta, Georgia 30302-3965, United States Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Hong Kong, P. R. China
‡
S Supporting Information *
ABSTRACT: In this paper, numerical modeling studies are carried out on the electrophoretic mobility of a dilute, highly charged “soft” spherical particle in a hard hydrogel subjected to a weak, constant, external electric field. The particle contains a solid core with either a uniform charge density or “zeta” potential on its surface. Outside of this lies a charged gel layer of uniform thickness, composition, and charge density. The present work extends previous studies by accounting for the “relaxation effect”, or distortion of the charge distribution in the vicinity of the model particle due to the imposition of an external electric and/or flow field. The particle gel layer and ambient hydrogel are modeled as porous Brinkman media. The (steady state) electrodynamic problem is solved at the level of the Poisson equation. Applications emphasize the influence of the relaxation effect and hydrogel charge density on the electrophoretic mobility.
use to this day.15,16 The early theoretical work on “small ions” addressed the complex phenomena of the “electrophoretic effect” and the “relaxation effect”10 which also play an important role in the electrophoresis of large macroions and nanoparticles. The free solution (no gel present) electrophoresis of “hard” particles that are large compared to the size of the solvent had its origins in the work of Smoluchowski17 and Huckel.18 By “hard”, we mean that the particle surface is smooth, welldefined, and inpenetrable by the solvent and background electrolyte, BGE. Henry19 solved the problem of the free solution electrophoresis of a weakly charged “hard” spherical particle of arbitrary radius, a, with a centrosymmetric charge distribution. This treatment adequately treated the “electrophoretic effect”, but did not account for the “relaxation effect” (polarization of the particle’s ion atmosphere by the imposition of an external electric and/or flow field). For weakly charged particles, however, the relaxation effect can be ignored. Overbeek20 and Booth21 included lowest order corrections for the relaxation effect. As in the work of Henry, the mobility expressions of Overbeek and Booth could be put in closed analytical form. Subsequent work has extended the range of validity to “hard” spherical particles of arbitrary (centrosymmetric) charge.22,23 These procedures, however, require numerical solution of the various coupled field equations
1. INTRODUCTION Electrophoresis continues to be a simple, extraordinarily effective, and widely used tool in the separation and characterization of charged biomolecules and nanoparticles in the physical and biological sciences.1 It is easy to understand why electrophoresis is so effective when you consider what happens when a voltage difference is simply applied across an aqueous solution of a mixture of charged particles which could include salt/buffer ions, peptides, proteins, nucleic acids, etc. The electric field that results from the applied voltage will cause the various ions to translate with different velocities and direction depending primarily on their charge and size, and this causes different particles/molecules to separate. With the introduction of a gel such as polyacrylamide2 or agarose,3 the ability of electrophoresis to separate large molecules on the basis of their size is improved. A gel also reduces convection4 that can spoil a separation. There are many applications of electrophoresis to a wide variety of systems described in the literature that are too numerous to cite in the present work. Fine summaries and reviews relevant to peptide,5 nanoparticle,6 and DNA7 separations are cited here as representative examples. Closely associated with electrophoresis is the related electrokinetic transport property of electrical conductance of ionic solutions.8 The success of atomic scale continuum theory9 to account for the electrical conductance of simple salt solutions10 was a remarkable achievement given the fact that we are dealing with a dense system with strong interparticle interactions. This theory which initially treated the ions as point ions was later generalized to account, to lowest order, for the finite sizes of the ions.11−14 Extensions of this theory remain in © XXXX American Chemical Society
Special Issue: J. Andrew McCammon Festschrift Received: December 14, 2015 Revised: January 25, 2016
A
DOI: 10.1021/acs.jpcb.5b12224 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
field conditions. The objective of the present work is developing an algorithm applicable to dilute soft particles in charged gels within the framework of the linear continuum model. Once this is achieved, it is possible to compute electrophoretic mobilities.
(Poisson, Navier−Stokes, and Nernst−Planck equations). The underlying models employed in all of the studies discussed here are “coarse grained” in the sense that the solvent and ion atmosphere surrounding the host particle are represented as structureless continua. This shall be referred to as the “continuum model” for short. When a gel is present, hydrodynamic and direct interactions alter the transport, including electrophoresis, of a host particle in an otherwise solvent environment. For large particles such as high molecular DNA, reptation theories have been effective in describing many but not all of the features of gel electrophoresis.3,7 For particles that are small compared to the pore spacing in a gel, the gel can be modeled as an effective “Brinkman” porous medium.24−26 For a “hard” spherical particle with a centrosymmetric, but arbitrary, charge distribution, numerical algorithms comparable to those used in free solution22,23 have been developed to determine electrophoretic mobilities in uncharged27 and charged28 gels. The surfaces of many nanoparticles are not “hard” but rather contain an outer “gel” or “hairy” layer that is permeable to solvent and ions of the BGE. This includes, for example, surface modified metal nanoparticles.29 These shall be termed “soft” particles. Ohshima30−32 has developed the theory of free solution electrophoresis of dilute, weakly charged “soft” spherical particles and obtained analytical expressions of the electrophoretic mobility. The “gel” or “hairy” layer is modeled as a “Brinkman” effective medium24−26 using procedures similar to those discussed in the previous paragraph. Incidentally, J. A. McCammon also worked on related effective medium problems early in his career.33 Hill, Saville, and Russell34 developed numerical procedures to the free solution electrophoretic mobility of highly charged “soft” particles by accounting for the relaxation effect. The general problem of the electrophoretic mobility of a “soft” particle in a charged or uncharged gel has been the subject of recent past study and is the subject of the present work. The problem of a weakly charged “soft” particle in a charged hydrogel has been treated in detail previously.35,36 In these previous studies, electrostatics were treated at the level of the linear Poisson−Boltzmann equation, and the relaxation effect was ignored. This is a valid approximation for weakly charged particles, but breaks down if |ζ|/kBT exceeds about 1.537 where ζ is the “zeta” potential at the core surface of the particle. For highly charged particles, it is necessary to include the relaxation effect, and this requires solving the coupled Poisson, Navier−Stokes−Brinkman, and Nernst−Planck equations. Numerical theories of the electrophoresis of highly charged spherical particles containing a centrosymmetric charge distribution have been developed independently by a number of different laboratories starting with Wiersema et al.22 (free solution, hard sphere), O’Brien and White23 (free solution, hard sphere), Hill et al.34 (free solution, soft sphere), Allison et al.27 (uncharged gel, hard sphere), and Hill28 (charged gel, hard sphere). In all of the above theories, the particles are dilute. However, Tsai and Lee38 have included concentration effects using a cell model. All of the above continuum theories22,23,27,28,34,38 are linearized with respect to the imposed electric or flow field, but not the equilibrium electrostatics. This shall be referred to as the “linear continuum theory”. Recently, Bhattacharyya et al.39−41 have extended these studies to include nonlinear effects of the imposed electric or flow field which has made it possible to study electrokinetic transport under high
2. MODEL AND METHODOLOGY 2.1. Continuum Model and Field Equations. The model used here is similar to that described previously35,36 and is depicted in Figure 1. The particle is modeled as a sphere with a
Figure 1. Schematic of the gel−sphere−gel model. The core is a sphere of radius a of relative permittivity εp with net charge zc uniformly distributed over its surface (r = a) or with uniform equilibrium electrostatic potential, ζ, specified. Surrounding this is the particle gel layer of outer radius b, uniform charge density ρf,1, and Brinkman screening length S 1. Outside the model particle, the fluid is modeled as an effective medium with uniform charge density ρf,2, and Brinkman screening length S 2. The relative permittivity of the particle gel layer and fluid is taken to be εs.
hard inner core of radius, a, surrounded by a gel layer of uniform composition that extends out to a radius, b. It is characterized by a Brinkman screening parameter, S 1, that has units of length, and uniform charge density, ρf,1. The particle core either contains a centrosymmetric charge distribution of uniform density, σ, located on its outer surface, or else the equilibrium electrostatic potential, ζ, is specified. When the charge density is specified, let eZc = 4πa2σ denote the net charge of the core (e is the protonic charge). It shall be assumed that when a weak external electric field, e0, is applied or the particle is subjected to a weak flow field of uniform velocity, u0, that the core charge density remains at its equilibrium value if the equilibrium core charge density is specified. Similarly, when the ζ-potential is specified, it is assumed that there is no electric or flow induced change in the ζ-potential on the core of the particle. It is also assumed that the particle concentration is sufficiently low so that particle− particle interactions can be ignored. This is what we mean by a “dilute” solution. The particle is immersed in a continuum Newtonian fluid with viscosity η, and relative permittivity εs, at temperature T. We shall assume that the relative permittivities of the core and gel layer are εp and εs, respectively. In addition to the particle gel layer, a stationary gel matrix of semi-infinite extent is also included, characterized by a Brinkman screening parameter, S 2. The gel matrix may also bear a fixed charge B
DOI: 10.1021/acs.jpcb.5b12224 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B density, ρf,2. Let zg denote the valence charge of these fixed sites. The mobile ions in the fluid are also modeled as a continuum that obeys, in the absence of an external field, the Poisson−Boltzmann equation
For highly charged particles, however, it is necessary to solve eqs 1 and 2 numerically. In addition to the necessity of solving the Poisson− Boltzmann and Poisson equations, it is also necessary to solve the Navier−Stokes−Brinkman and solvent incompressibility equations which can be written36
ε0εs∇2 Λ 0(r ) = −ρt (r ) = −(ρf ,1w1(r ) + ρf ,2 w3(r ) + F ∑ cα 0zαw2(r )e−βezα Λ 0(r))
η∇2 v̲ ( r̲ ) − ∇̲ p( r̲ )
(1)
α
= s ̲ e( r̲ ) + η S1−2w1( r̲ ) v̲ ( r̲ ) + η S −2 2w3( r̲ )( v̲ ( r̲ ) + u̲ 0)
In eq 1, ε0 is the permittivity of free space, ∇2 is the Laplacian operator which, for a function that depends only on the radial variable, equals r−2d(r2d/dr)/dr in spherical coordinates, Λ0 is the equilibrium electrostatic potential in the absence of an external electric or flow field, ρt is the local total charge density at a distance r from the particle, F is the Faraday, the sum over α extends over all mobile ions present in solution, zα and cα0 are the valence charge and ambient concentration of ion α (in mol/m3), β equals 1/kBT (kB is the Boltzmann constant), and the wj(r) terms are step functions defined in Table 1. No restriction is placed on the number of
(5)
∇̲ · v̲ ( r̲ ) = 0
Above, v̲ is the local fluid velocity, p is the local pressure, s̲e(r) is the local external force/vol on the fluid (due to interaction of mobile ion charges with the electric field of the particle and external field), and u̲0 is the velocity of the particle in the laboratory frame of reference. Equations 5 and 6 must be dealt with whether the particle is weakly charged or not. For highly charged particles, where the relaxation effect must be accounted for, it is also necessary to solve a Nernst−Planck equation for each mobile ion present in the BGE. Let jα̲ denote the local current density of ion α. Under steady state conditions and assuming no creation or destruction of ions, the equation of continuity gives
Table 1. Step Functions J 1
r-range
wj(r)
r r r r r r r r r
0 1 0 0 1 1 0 δol 1
0 a b 0 a b 0 a b
2
3
< < < < < < < <
