Langmuir 2006, 22, 3533-3546
3533
Electrophoresis of Diffuse Soft Particles Je´roˆme F. L. Duval*,†,‡ and Hiroyuki Ohshima§ Department of Physical Chemistry and Colloid Science, Wageningen UniVersiteit, Dreijenplein 6, 6703 HB Wageningen, The Netherlands, Laboratoire EnVironnement et Mine´ ralurgie, UMR INPL-CNRS 7569, Centre de Recherche Franc¸ ois Fiessinger, 15 aVenue du Charmois, B.P. 40, 54501 VandoeuVre-le` s-Nancy Cedex, France, and Faculty of Pharmaceutical Sciences, Tokyo UniVersity of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan ReceiVed October 20, 2005. In Final Form: February 8, 2006 A theory is presented for the electrophoresis of diffuse soft particles in a steady dc electric field. The particles investigated consist of an uncharged impenetrable core and a charged diffuse polyelectrolytic shell, which is to some extent permeable to ions and solvent molecules. The diffuse character of the shell is defined by a gradual distribution of the density of polymer segments in the interspatial region separating the core from the bulk electrolyte solution. The hydrodynamic impact of the polymer chains on the electrophoretic motion of the particle is accounted for by a distribution of Stokes resistance centers. The numerical treatment of the electrostatics includes the possibility of partial dissociation of the hydrodynamically immobile ionogenic groups distributed throughout the shell as well as specific interaction between those sites with ions from the background electrolyte other than charge-determining ions. Electrophoretic mobilities are computed on the basis of an original numerical scheme allowing rigorous evaluation of the governing transport and electrostatic equations derived following the strategy reported by Ohshima, albeit within the restricted context of a discontinuous chain distribution. Attention is particularly paid to the influence of the type of distribution adopted on the electrophoretic mobility of the particle as a function of its size, charge, degree of permeability, and solution composition. The results are systematically compared with those obtained with a discontinuous representation of the interface. The theory constitutes a basis for interpreting electrophoretic mobilities of heterogeneous systems such as environmental or biological colloids or swollen/deswollen microgel particles.
1. Introduction In the past decades, major efforts have been invested in the development of theories for the electrophoresis of soft particles (i.e., particles coated with charged polyelectrolyte layers1-15). These studies are motivated by the increasing number of experimental electrokinetic analyses aimed at the electrical and hydrodynamic characterizations of various colloidal/biocolloidal systems, such as adsorbed polyelectrolytes,16-21 bacteria,22-25 * Corresponding author. E-mail:
[email protected]. Tel: 33 (0) 3 83 59 62 65. Fax: 33 (0) 3 83 59 62 55. † Wageningen Universiteit. ‡ UMR INPL-CNRS. § Tokyo University of Science. (1) Ohshima, H. AdV. Colloid Interface Sci. 1995, 62, 189. (2) Ohshima, H. J. Colloid Interface Sci. 2000, 228, 190. (3) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (4) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258, 56. (5) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 263, 478. (6) Saville, D. A. J. Colloid Interface Sci. 2000, 222, 137. (7) Lopez-Garcia, J. J.; Grosse, C.; Horno, J. J. Colloid Interface Sci. 2003, 265, 327. (8) Lopez-Garcia, J. J.; Grosse, C.; Horno, J. J. Colloid Interface Sci. 2003, 265, 341. (9) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2004, 274, 309. (10) Wunderlich, R. W. J. Colloid Interface Sci. 1982, 88, 385. (11) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (12) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (13) Ohshima, H. Colloid Polym. Sci. 2005, 283, 819. (14) Hill, R. J.; Saville, D. A. Colloids Surf., A 2005, 267, 31-49. (15) Hill, R. J. Phys. ReV. E 2004, 70, 051406. (16) Starov, V.; Solomentsev, Y. E. J. Colloid Interface Sci. 1993, 158, 159. (17) Starov, V.; Solomentsev, Y. E. J. Colloid Interface Sci. 1993, 158, 166. (18) Donath, E.; Voigt, A. J. Colloid Interface Sci. 1986, 109, 122. (19) Donath, E.; Pastuschenko, V. Bioelectrochem. Bioenerg. 1980, 7, 31. (20) Duval, J. F. L.; van Leeuwen, H. P. Langmuir 2004, 20, 10324. (21) Duval, J. F. L. Langmuir 2005, 21, 3247.
and environmental colloids such as humic substances.26 The theories so far proposed are based on the Poisson-Boltzmann equation for the electric potential distribution across the polyelectrolyte layer and the Navier-Stokes equation for the liquid flow velocity inside and outside this permeable layer. More specifically, the hydrodynamics is classically tackled within the framework of the Debye-Bueche model,27 which considers the charged polymer segments to be resistance centers that exert frictional forces on the liquid flowing in the layer. To understand the peculiar electrophoretic behavior of coated particles as compared to that of their bare counterparts,28 analytical solutions of the governing transport and electrostatic equations were provided, mainly by Ohshima,1-3,13 for a variety of soft colloids under the conditions that polarization and relaxation are absent. However, a recent semianalytical study6 pointed out the significant importance of these phenomena for coated particles at high potentials. Clearly then, if experimental results obtained for a wide range of electrolyte concentrations, coating thicknesses, and polyelectrolyte charges are to be compared with theory, numerical solutions of the key electrokinetic equations are required. Recent works4,5,7,8,14,15 have integrated this consideration and provided numerical evaluations for the electrophoretic mobility over a wide range of particle charges, coatings, and double-layer thicknesses. (22) Van der Wal, A. Ph.D. Thesis, Wageningen Universiteit, The Netherlands, 1996. (23) Poortinga, A. T. Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands, 2001. (24) Bos, R.; van der Mei, H. C.; Busscher, H. J. Biophys. Chem. 1998, 74, 251. (25) de Kerchove, A. J.; Elimelech, M. Langmuir 2005, 21, 6462. (26) Duval, J. F. L.; Wilkinson, K.; van Leeuwen, H. P.; Buffle, J. EnViron. Sci. Technol. 2005, 39, 6435. (27) Debye, P.; Bueche, A. J. Chem. Phys. 1948, 16, 573. (28) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 77, 1607.
10.1021/la0528293 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/11/2006
3534 Langmuir, Vol. 22, No. 8, 2006
Within the framework of most aforementioned analytical1-3,13 and numerical theories7,8 for the electrophoresis of soft particles, with the exception of the work by Hill et al.4,5,14,15 discussed below, the density of the polyelectrolyte chains is assumed to be constant in the soft surface layer and drops sharply to zero at the very interface formed with the electrolyte side. Very recently, Duval et al. abandoned this necessarily approximate discontinuous description of the interface and developed the model of diffuse soft interface within the context of the electrokinetics (streaming potential/streaming current) of macroscopic charged gels.20,21 Their treatment was based on considering an interfacial zone where the properties gradually changed from bulk gel to bulk aqueous electrolyte. It was clearly demonstrated that the inhomogeneous distribution of the charged polymer segments substantially affects the overall electrokinetic response, especially in the low electrolyte concentration regime where the electrokinetic characteristics of the soft layer become sensitive to the spatial details of the chain distribution in the diffuse interface. Experimental data on polyacrylamide gels could be successfully explained on the basis of a continuum modeling for the hydrodynamic and electrostatic properties of the polymer films. In particular, the role of electrostatic swelling on the determination of the electrokinetic features of the charged gels could clearly be established and quantified.29 Recent small-angle neutron scattering30 and light-scattering measurements31 on microgel particles demonstrated that the polymer content therein was nonuniformly distributed and decreased from the center of the particle to its periphery. These experimental findings illustrate the necessity of developing advanced electrokinetic models for heterogeneous soft particles that go beyond the classical interfacial step-function representation, as adopted in refs 1-3, 7, 8, and 13. Therefore, we propose in the current article to extend the ideas put forward in refs 20, 21, and 29 to the electrophoresis of diffuse soft particles characterized by a continuous interfacial density profile distribution for the polyelectrolyte chains. It should be noted here that the idea of a gradual spatial distribution for the polymer segments is not new. In ref 32, Ohshima derived an approximate analytical expression for the mobility of colloidal particles coated with charged polymers with an exponential distribution for the segment density. In another context, Lopez-Garcia et al.33 solved the Poisson-Boltzmann equation numerically for a charged spherical hard particle surrounded by a permeable membrane immersed in an electrolyte solution and for which the fixed charges were distributed according to linear and exponential type-functions. In ref 4, Hill et al. integrated in their numerical analysis of the key electrokinetic equations a fuzzy outer boundary for a brushlike coating. The latter study, however, does not explicitly consider the impact of the inhomogeneous character of the coating (as subsumed in a characteristic length denoted δ in ref 4) on the electrophoretic features but rather focuses on the effects of size, charge, and ionic strength on the mobility for a given constant δ. In ref 4, two situations are exclusively analyzed: one in which the segment density decay is slow (δ ) L with L being the nominal coating thickness) and one where that decay is abrupt (δ ) 0.1L). No indication is given for the continuous dependence of the mobility on the decay length δ and on how this dependence (29) Yezek, L.; Duval, J. F. L.; van Leeuwen, H. P. Langmuir 2005, 21, 6220. (30) Mason, T. G.; Lin, M. Y. Phys. ReV. E 2005, 71, 040801. (31) Fernandez-Nievez, A.; de las Nieves, F. J.; Fernandez-Barbero, A. J. Chem. Phys. 2004, 120, 374. (32) Ohshima, H. J. Colloid Interface Sci. 1997, 185, 269. (33) Lopez-Garcia, J. J.; Horno, J.; Grosse, C. J. Colloid Interface Sci. 2003, 268, 371.
DuVal and Ohshima
is affected by the charge, hydrodynamic permeability, and thickness of the coating or by the electrolyte concentration. This is the major purpose of the current article. Furthermore, the current study focuses on the electrophoresis of a particle with an uncharged core (which is stricto sensu impermeable to fluid flow) surrounded by a charged coating, and particular attention is also devoted to the analysis of the electrophoresis of a soft particle devoid of any hard core. Despite its practical interest,26 the latter situation is not considered in ref 4. The method by Hill et al. may deal with any polymer segment distribution, as does that presented in our study, and also offers the advantage that it enables one to solve the full electrokinetic model in both ac and dc fields. Within the scope of the current article, timeindependent electric fields are solely considered. Given the above considerations, none of the above-referenced papers analyzes along the lines set forth in this study the impact of the characteristic length and interfacial gradients associated with the polymer segment distribution on the electrophoretic behavior of the particle as a whole. As judged by numerous experimental works pointing out the influence of the temperature or ionic strength on the interspatial distribution of polyelectrolytetype chains,29,34-38 it is timely to examine in detail the effects of inhomogeneity for the chain distribution on the electrophoretic motion of soft particles. This is done in this article where an exact numerical analysis of the electrokinetic equations derived on the basis of the formalism proposed by Ohshima1 is presented for the first time. Those equations are solved without any restrictions on the particle charge, particle size, double-layer thickness, and radial function that describes the diffuse (heterogeneous) nature of the interface. The analysis also includes elements of prime importance that deserve attention in the modeling of the electrokinetics of soft particles,39 that is, the partial dissociation (pH effect) of the ionogenic sites distributed throughout the polymeric shell (site dissociation model) as well as the possibility of specific interactions between those sites and ions from the background electrolyte other than chargedetermining ions.9,18,40-43 The theory outlined in this article has been successful in addressing the electrohydrodynamic properties of various environmental soft particles26 and biocolloids.39 In view of the complex surface structures encountered for those types of systems, the classical approximations of uniform charge and uniform permeability within the soft layer need to be revisited, as it is done in the current article. The recent electrokinetic study performed on Escherichia coli mutants25 clearly demonstrates the shortcomings of the classical soft particle analysis as originally proposed by Ohshima. The presence of a nonuniform distribution of charged groups on the lipopolysaccharides (LPS) and the patchlike distribution of the LPS at the bacterial surface are responsible for the failure of Ohshima’s (34) Makino, K.; Yamamoto, S.; Fujimoto, K.; Kawaguchi, H.; Ohshima, H. J. Colloid Interface Sci. 1994, 166, 251. (35) Taylor, L. D.; Cerankowski, L. D. J. Polym. Sci., Part A: Polym. Chem. 1975, 13, 2551. (36) Kawaguchi, H.; Fujimoto, K.; Mizuhara, Y. Colloid Polym. Sci. 1992, 270, 53. (37) Gao, C.; Leporatti, S.; Moya, S.; Donath, E.; Mohwald, H. Chemistry 2003, 9, 915. (38) Garcia-Salinas, M. J.; Romero-Cano, M. S.; de Las Nieves, F. J. J. Colloid Interface Sci. 2001, 241, 280. (39) Duval, J. F. L.; Busscher, H. J.; van de Belt-Gritter, B.; van der Mei, H. C.; Norde, W. Langmuir 2005, 21, 11268. (40) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2005, 286, 761. (41) Keh, H. J.; Ding, J. M. J. Colloid Interface Sci. 2003, 263, 645. (42) Yates, D. E.; Levine, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1974, 1, 1807. (43) Davids, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 180.
Electrophoresis of Diffuse Soft Particles
Figure 1. Schematic representation of a soft particle, composed of a rigid hard core and a permeable charged polymeric layer, moving with a velocity U B in a nonbound electrolyte subjected to a dc electric field B E. The spherical coordinate system and corresponding unit vectors are also given. In the absence of a hard core (a ) 0), the model corresponds to that of a spherical polyelectrolyte.
analytical theory to account for the electrokinetic properties of the E. coli bacteria.
Langmuir, Vol. 22, No. 8, 2006 3535
the soft layer are equal to those in the bulk electrolyte solution. This approximation is essentially true for rather loose layer structures29 but becomes necessarily questionable for more dense brushes. In ref 44, Zimmerman et al. estimated from grafting density and surface conductivity data that the mobility of ions in a poly(acrylic acid) layer was only about 14% that of free ions. Therefore, in the general situation, one should depict the ionic mobility of the ions (as well as the dielectric permittivity31 and the solvent viscosity) as a function of the position, with that function being dependent on the spatial distribution of the density of the soft material within the layer (see next section). One could think of evaluating that function by means of independent Monte Carlo simulations or molecular dynamics computations. The formalism reported in the article is sufficiently flexible for including these complex features met for dense soft layers, provided that the spatial functionalities for the ion mobility, solvent viscosity, and dielectric permittivity are a priori known. 2.1. Modeling of the Diffuse Soft Interface. The spatial dependence of the polymer segment density, ns, is chosen to be dependent on r only (radial profile) and is written for the sake of generality as
ns(r) nos
2. Electrokinetic Model for a Diffuse Soft Particle Consider a spherical, core/polymeric shell particle that moves with a velocity U B in a nonbound electrolyte, in which a uniform dc electric field is applied (Figure 1). The origin of the spherical coordinate system (r, θ, φ) is placed at the center of the particle, and the polar axis (θ ) 0) is set parallel to B E. The radius of the impermeable particle core is denoted by a, and δ represents the nominal thickness of the polymeric shell that is permeable to ions and solvent molecules to some extent. In the limit a f 0, the particle core vanishes, and the model reduces to that of a spherical polyelectrolyte, whereas for δ f 0 the other limiting case of a hard sphere is approached. The electrolyte is composed of N types of ionic mobile species with valency zi, bulk concentrations c ∞i , and limiting ionic mobilities λoi (i ) 1,..., N). Though differences in ion valencies may be easily accounted for, we consider here monovalent fixed charges distributed throughout the shell with a volume density, denoted as Ffix. The surface of the particle is uniformly charged, and the electrostatic conditions there are given either by a surface potential, ys, or a surface charge density, σs, depending on the nature of the particle investigated. We adopt the model of Debye-Bueche27 in the framework of which the polymer segments are regarded as resistance centers of radius as that exert frictional forces on the liquid flowing in the charged layer. The corresponding friction coefficient is denoted as k. The hydrodynamic volume fraction of polymer segments, φ, is given by φ ) 4nsπas3/3 with ns being the polymer segment density. The main assumptions of the formalism presented in this article are those given in ref 1: (a) the Reynolds numbers of the liquid flow inside and outside the polyelectrolyte layer are small, so the liquid may be regarded as incompressible; (b) the electrophoretic velocity, U B , is proportional to the applied field B E, which is correct for low E as considered in electrophoresis of the first kind (terms in E of order higher than 1 may be neglected); (c) the relative permittivities, r, inside and outside the polyelectrolyte layer are the same, which is reasonable for polymeric shells with sufficiently high water content, a situation commonly encountered in practice; and (d) the slipping plane (at which the velocity of the fluid relative to the particle is zero) is located on the particle core (r ) a). Following the assumption c, we will consider that the mobilities or diffusion coefficients of electrolyte ions within
) f(r)
(1)
where nos is the nominal segment density of the layer with homogeneously distributed chains and f is a radial function satisfying
f(r f ∞) f 0
(2)
which expresses the vanishing of the polymeric shell for r f ∞. Using eq 1, the dependence of φ on r is simply given by φ(r)/φo ) f(r) with φo ) 4nos πa3s /3. To obtain the distribution of k, the coefficient of friction exerted by the polymer segments on the flow, the Brinkman equation is employed.45 It relates k to φ by considering the flux of fluid across the group of spheres of radius as as follows
k(r) ) ηw
{[
(
as2 8 4 3+ -3 -3 18 φ(r) φ(r)
) ]}
-1
1/2
(3)
with ηw being the dynamic viscosity of water and the radial dependencies of k and φ explicitly written.20,21 From comparison with rigorous numerical calculations involving averaging over numerous initial stochastic configurations, eq 3 is shown to be valid for φ < 0.3.46 Equation 3 may be rewritten as follows
k(r) )
18ηW as2
∞
∑σn[φ(r)]n/2 n)2
(4)
where σn represents the coefficients in the Taylor expansion of eq 3 with respect to the variable φ1/2. For very low volume fractions (φ f 0), only the first term is significant and eq 4 reduces to Stokes’ equation, which predicts a linear dependence of k on φ so that
k(r) ) f(r) ko
(5)
(44) Zimmermann, R.; Norde, W.; Cohen-Stuart, M. A.; Werner, C. Langmuir 2005, 21, 5108. (45) Brinkman, H. C. Research 1949, 2, 190. (46) Sangani, S. A.; Yao, C. Phys. Fluids 1988, 31, 2435.
3536 Langmuir, Vol. 22, No. 8, 2006
DuVal and Ohshima
with ko () 9ηwφo/2as2) being the friction coefficient of the polymeric shell for a homogeneous distribution of segments. In ref 4, the limit given by eq 5 was derived from a semiempirical relationship that yields the Brinkman theory valid at low φ and reproduces the results of numerical computations at higher φ. In the following text, we shall adopt the expression (eq 5) that is justified for the usual water contents of soft charged gel layers commonly reported in the literature (φ < 0.05). It is emphasized that eq 5 neglects the hydrodynamic interactions between the polymer segments. It is therefore strictly valid for κas > 1 with κ being the reciprocal screening Debye length given by κ ) N (∑i)1 z2i F2c∞i /orRT)1/2. As a consequence, within the framework of the current study, the soft layer is free draining, and the electroosmotic flow in the polymer layer is hindered by the hydrodynamic drag of the segments, also called electroosmotic drag. Assuming also that the ionogenic sites are uniformly distributed along the polymer chains, the local density of fixed charges, Ffix, satisfies the relation
Ffix(r) ) f(r) Fo
(6)
where Fo is the nominal charge density within the polymeric layer. Equation 6 is valid within the condition of complete dissociation of the ionogenic groups located at position r. In the more general case, the degree of dissociation of these groups depends on the pH and on the local equilibrium potential, denoted as ψ(0)(r), inside the polyelectrolyte layer9,18,39-41 so that
Ffix(r) ) f(r) g(pH, ψ(0)(r)) Fo
(7)
where g is the isotherm describing the protolytic properties of the polyelectrolyte layer. One may easily include in g, if necessary, the possibility of specific interactions between ionogenic sites and ions from the background electrolyte. These interactions depend on the chemical affinity of the ions for the sites considered and on the local potential within the polymeric shell. For the sake of simplicity and conciseness, all calculations given within the framework of the current study are performed for g ≡ 1 (complete dissociation). In ref 39, a detailed analysis of the electrophoretic behavior of various oral streptococcal bacteria is performed on the basis of the current model: the pH effect and specific ion interactions are therein explicitly included and discussed at length. In particular, the complementary information provided by the combination of quantitative analyses of electrophoretic mobility and titration data is strengthened. 2.2. Nature of the Distribution of the Polymer Segment Density within the Diffuse Interface. A rigorous determination of the distribution f(r) requires knowledge of the type of interactions existing within the system {polymer segmentssolvent molecules-electrolyte ions} and knowledge of the details concerning the preparation of the soft structure and (possible) subsequent grafting on the boundary particle surface. It is recalled here that the polymer segments constituting the layer are flexible and that their characteristic size and spatial arrangements strongly depend on their neighboring environment and, in particular, are mediated by the local distributions of electrolyte ions. In polymer chemistry, self-consistent-field calculations as based on the Scheutjens-Fleer theory, and molecular dynamic simulations are commonly used to determine the inhomogeneous spatial
distribution of adsorbed polymers at a given surface.47-50 Whereas such information may be obtained for relatively well-defined and uncharged soft layers, such as those examined in ref 15, it is more difficult to obtain when dealing with charged biological or environmental systems such as bacteria, viruses, and humic particles. Therefore, in the current article, we adopt for the sake of illustration the following form for f(r):
f(r) )
{
(
)}
r - (a + δ) ω 1 - tanh 2 R
(8a)
The parameter R has the dimension of a length and determines the degree of inhomogeneity of the distribution f for the polymer segment density. The dimensionless parameter ω is determined in such a way that the total number of polymer segments and thus the total number of charges remain constant upon variation of R and/or δ. ω is therefore given by the expression
ω)
∫a
∞
2 {(a + δ)3 - a3} 3 r - (a + δ) 2 1 - tanh r dr R
{
[
]}
(8b)
The choice of the radial profile given by eq 8 is motivated by an early study29 where electrokinetic (streaming potential) and swelling properties of polyelectrolyte gel layers were successfully examined on the basis of eq 8. It is emphasized that the numerical formalism detailed in section 2.3 may account for any segment distribution (empirical, analytical, or computational) as long as eq 2 is satisfied. Whereas mean-field calculations are feasible for soft particles with a relatively simple and controlled structure,15 they rapidly become intricate for complicated soft systems such as bacteria25,39 or environmental particles26 in terms of rigorous model variables. The latter (such as the Flory interaction parameters) are furthermore rarely accessible. The diffuse soft layer approach (eq 8), as presented here, is particularly powerful for analyzing experimental data collected on systems with complicated (biological or environmental) surface appendages.26,39 The analysis, such as that based on the profile in eq 8, has the ability to reproduce many practical situations, evoked below, upon tuning a single adjustable parameter R. Few profiles f(r) are given in Figure 2 as a function of R. For R f 0 (and ω f 1), the distribution is homogeneous within the surface layer whereas for increasing values of R (and therefore decreasing ω) the diffuse character of the interface becomes more pronounced. Let us now discuss a few examples illustrating the relevance of the parameter R. For noncharged polymers, the chains adsorb to the particle surface via multiple anchoring points along the chain. This produces a high segment density close to the surface that slowly decays further out, which corresponds to the situation R/δ ≈ 1. In contrast, polyelectrolyte chains typically have a short end group that anchors the chain, thus producing a brushlike coating with a lower R/δ. In other words, the segment density distribution and related radial heterogeneity are strongly influenced by the charge of the chains and their adsorption onto the surface, as known in polymer chemistry. We now examine the case of a biocolloid (for example, a bacterium) (47) Van Male, J. Self-Consistent-Field Theory for Chain Molecules: Extensions, Computational Aspects, and Applications. Ph.D. Thesis, Wageningen University, The Netherlands, 2003. (48) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (49) Van der Gucht, J.; Besseling, N. A. M. Phys. ReV. E 2002, 56, 51801. (50) Zhulina, E. B.; Klein Wolterink, J.; Borisov, O. V. Macromolecules 2000, 33, 4945.
Electrophoresis of Diffuse Soft Particles
Langmuir, Vol. 22, No. 8, 2006 3537 2.3R/ ) to appreciate the extension of the diffuse layer as compared δ
to that of the double-layer thickness. For diffuse interfaces, δ might be regarded as the average thickness at which f(r ) a + δ) ) ω/2 with ω defined by eq 8b. 2.3. Fundamental Electrokinetic Equations. 2.3.1. Field Equations. The standard set of equations governing the dynamics of a soft particle with a discontinuous distribution of polymer chains has been extensively discussed in the literature.1,4,5,7,8 Ohshima demonstrated that this set of fundamental electrokinetic equations can be expressed in terms of the liquid velocity, b u(r b), b) at the position b r relative to the particle and the deviation δµi(r of the electrochemical potential, µi(r b), of the ith ion species due to the applied electric field. Making explicit the dependence of the friction coefficient k on the position (as determined by eq 5), these equations are written for a diffuse interface as follows N
Figure 2. Representation of the spatial modeling of the interface for a diffuse soft particle (eq 8). The function f pertains to the spatial distribution for the polymer segment density (eq 1) to the distribution of the friction coefficients (eq 5) and fixed charges (eq 6) within the polymeric shell (sections 2.1 and 2.2).
exhibiting charged fibrillae39,51 forming a soft surface layer. These fibrillae are generally polydisperse in size so that a gradient of fibrillae density and hence charge density is generated. Such a gradient can be modeled by the appropriate R. A similar situation occurs when the surface layer is composed of charged and uncharged polymers.52 For particles covered with gel-like (hydrogels) surface layers, changes in temperature and electrolyte concentration may result in swelling/shrinking processes.30,31,37,38 These are connected to the exact distribution of the electrostatic potential across the interface.29 As such, the degree of swelling/ shrinking for segments close to the anchoring surface is necessarily different from that of segments facing the first water molecules on the solution side. This again induces an inhomogeneous distribution for the segment density, which may be modeled by the adjustment of R when varying the temperature and ionic strength, as done in ref 29 within the framework of streaming potential measurements on polyacrylamide gels. The quantity δ is defined as the thickness of the soft layer when the latter is homogeneous (δ may thus be called the nominal thickness). For diffuse (heterogeneous) soft interfaces, the (radial) profile for the density of the soft material is calculated according to eqs 8a and 8b. The degree of heterogeneity of the layer is strongly dependent on the ratio R/δ (Figure 2). When increasing R/ , the shell extends further to positions r > a + δ so that, for δ diffuse interfaces, δ loses the physical meaning it has for uniform layers. On the basis of eq 8a, one can estimate that at the position r ) a + δ(1 + 2.3R/δ) the density of polymer segments is only 2% of the nominal value. This indicates that the polymeric shell basically vanishes for r > a + δ(1 + 2.3R/δ) and that the thickness of the diffuse layer can, for practical purposes, be defined as δ(1 + 2.3R/δ). The scalar factor 2.3 may somehow vary depending on the criterion chosen for marking the end of the diffuse layer. However, this is a matter of definition only because the basic electrokinetic features of the diffuse soft particle are not primarily governed by the asymptotic tail of the polymer segment distribution where f ≈ 0.32 Given the above precision, the classical dimensionless parameter κδ is, strictly speaking, not relevant for diffuse interfaces. One instead should use the quantity κδ(1 + (51) Van der Mei, H. C.; Meinders, J. M.; Busscher, H. J. Microbiology 1994, 140, 3413. (52) Fernandez-Nieves, A.; Fernandez-Barbero, A.; de las Nieves, F. J.; Vincent, B. J. Phys.: Condens. Matter 2000, 12, 3605.
η w∇ × ∇ × b u (b) r + ∇p(b) r -
δµi(b) r ∇c(0) r + ∑ i (b) i)1
r ∇ψ(0)(b) r + ∇(F(0) r δψ(b)) r + k(r) b u (b) r )B 0 (9) F(0) el (b) el (b)
{
}
1 r b u (b) r - c(0) r ∇δµi(b) r )0 ∇ c(0) i (b) i (b) λhi (10)
i ) 1,..., N:
b) ) c(0) b) + δci(r b) is the local where p(r b) is the pressure, ci(r i (r concentration of the ith ionic species, Fel(r b) ) Fel(0)(r b) + δFel(r b) is the charge density stemming from the mobile ions, and ψ(r b) ) ψ(0)(r b) + δψ(r b) is the local electrostatic potential. The quantities indicated by the superscript (0) refer to those at equilibrium (i.e., in the absence of the electric field), and the variables preceded by the symbol δ indicate small variations in the order E (i.e., linear in E). In eq 10, λhi denotes the drag coefficient of ion i and is related to the limiting conductivity λoi by the expression λhi ) |zi|eF/λoi with e being the elementary charge and F being the Faraday constant. Equation 9 represents the Navier-Stokes equation after linearization of the pertaining electrokinetic variables with respect to the applied field. Equation 10 results from the linearization of the combined equations expressing the velocity b Vi(r b) of an individual ion, as governed by diffusion and convection processes, and the continuity condition for that b)V bi(r b)) ) 0. To solve eq 9, it is convenient particular ion (i.e., ∇‚(ci(r to take the curl so that terms of third, fourth, and fifth order in E vanish.1 The result reads as
u (b) r + ∇ × {k(r)u b(b)} r ) η w∇ × ∇ × ∇ × b N
∇δµi(b) r × ∇c(0) r ∑ i (b) i)1
(11)
On the basis of symmetry considerations,53 the liquid velocity b u(r b) at the position brelative r to the particle may be further written in vectorial form
(
2 1 d{rh(r)} b u (b) r ) - h(r) E cos θ, E sin θ, 0 r r dr
)
(12)
where h is a function of r. It is noted that eq 12 automatically satisfies the continuity equation ∇‚u b ) 0 because b u(r b) can be rewritten as b u(r b) ) ∇ × ∇ × {g(r)E B} with h(r) ≡ dg(r)/dr.1 Following the strategy adopted in ref 1, the quantity δµi(r b) is written (53) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon: London, 1966.
3538 Langmuir, Vol. 22, No. 8, 2006
i ) 1,..., N:
DuVal and Ohshima
δµi(b) r ) -zieφi(r)E B‚e br ) -zieφi(r)E cos θ (13)
where the functions φi(r) are radial and b er is the radial unit vector. Within the common assumptions that solvent-mediated (dispersive) forces may be neglected and that the different ionic species are point-like, the distribution of the electrolyte ions at equilibrium c(0) b) obeys a Boltzmann statistic and the equilibi (r b) obeys the Poisson-Boltzmann equation, rium potential ψ(0)(r with both being function of the radial position only
∇2ψ(0)(r) ) -
1 (0) {F (r) + Fo f(r) g(pH, ψ(0)(r))} or el
with N
F(0) el (r)
)F
∑ i)1
N
)F
zic(0) i (r)
∑ i)1
zic∞i
{
(14)
}
{
exp
kBT
-
(
ηwr dr
+
r
)}
dh(r) dr
(16)
b u (r f ∞) f -U B ) (-U cos θ, U sin θ, 0)
µ h(r f ∞) f r 2
where µ is the electrophoretic mobility defined by µ ) U/E. Furthermore, in the stationary state, the net force acting on the particle must be zero. Let us consider a sphere S of radius r that is sufficiently large to encompass the particle in such a way that the net electric charge within S is zero. Then, there is no electric force acting on S, and the only hydrodynamic force u B H needs to be considered. One therefore obtains
∫SσH‚nb dS ) B0 as r f ∞
(17)
1/2
Lrφi(r) )
{
λhi h(r) dy dφi(r) -2 zi dr dr e r
dr2
(18)
is the softness parameter of the permeable polymeric shell for a uniform distribution of the segments. The quantity 1/λo has the dimension of a length and typically represents the characteristic penetration length of the flow within the soft structure. y(r)() eψ(0)(r)/kBT) in eq 16 denotes the dimensionless equilibrium potential. For a segment distribution f defined by f(r) ) 1 for a e r e a + δ and f(r) ) 0 for r > a + δ (discontinuous modeling of the interface), the results presented in ref 1 are recovered. Combining eqs 10 and 13 yields the governing equations for the functions φi(r)
i ) 1,..., N:
|
d2h(r)
()
}
(19)
where the continuity condition ∇‚u b(r b) ) 0 has been used. 2.3.2. Boundary Conditions. As a general comment, the boundaries associated with the governing electrokinetic equations written for a diffuse interface are substantially simpler than those considered for a sharp interface, as in refs 7 and 8. The gradual transition from the polymeric shell to the electrolyte side avoids the necessity of introducing other boundaries than that between the hard and soft components of the particle (i.e., at r ) a) and that pertaining to the far-field domain (i.e., for r f ∞).
(25)
which refines the boundary (eq 23). One may easily show that eq 25 is verified when h satisfies the conditions
and
ko ηw
(24)
where σH is the hydrodynamic stress and b n is the unit normal vector outward from the particle surface and the integration is carried over the surface of S. After calculation, eq 24 provides1
()
2 d 2 d - 2 + 2 r dr dr r
(22)
(23)
1 µ h(r f ∞) f r + O 2 r
2
λo )
(21)
The liquid velocity b u(r) relative to the particle must comply with the far-field condition given by
u BH )
where Lr is the differential operator defined by
Lr ≡
|
dh(r) )0 dr r)a
(15)
)
zi2c∞i φi(r) exp(-ziy(r)) ∑ i)1
(20)
and
-zieψ(0)(r)
df(r) h(r)
dr F dy(r) N
h(r ) a) ) 0
Using eq 12, eq 22 immediately yields
where () or) is the dielectric permittivity of water, kB is the Boltzmann constant, and T is the absolute temperature. The development of the curl terms in eq 11 may be performed after b), and c(0) b), substituting eqs 12, 13, and 15 for b u(r b), δµi(r i (r respectively. The result reads as follows
LrLrh(r) - λo2 f(r)Lrh(r) +
The slipping plane at which b u)B 0 is located at the position r ) a so that, following eq 12,
r2
rf∞
)0
|
d{h(r)/r} )0 rf∞ dr
(26)
(27)
Equations 20, 21, 26, and 27 are the boundaries associated with the differential equation of fourth order in h (eq 16). The two boundary conditions verified by the functions φi are written1
i ) 1,..., N:
dφi(r) )0 dr r)a
|
(28)
i ) 1,..., N:
φi(r f ∞) f r
(29)
Equation 28 results from combining the condition b u(r b)‚n b|r)a ) Vi(r b)‚ 0 with that for the individual ionic flow velocity b Vi (i.e., b b n|r)a ) 0). Equation 29 is obtained after considering that far from the particle, perturbations of the local electric potential and local ion concentrations caused by the applied field become negligible. The boundaries pertaining to the (dimensionless) equilibrium potential distribution, y, express the bulk electroneutrality condition far away from the particle
y(r f ∞) ) 0 (reference for the potentials)
(30)
and the electrostatic conditions at the surface of the particle core. The latter may be written in terms of the surface potential ys or
Electrophoresis of Diffuse Soft Particles
Langmuir, Vol. 22, No. 8, 2006 3539
surface charge density σs
dY1,i(x) ) Y2,i(x) dx
i ) 1,..., N:
y(r ) a) ) y
s
(31)
(
|
dy(r) -Fσ ) dr r)a orRT s
(32)
where R denotes the gas constant. For the case of a spherical polyelectrolyte (a ) 0), conditions 31 and 32 are replaced by
|
dy(r) )0 dr r)0
dY3(x) 2 ) Y4(x) Y (x) dx x + κa 3
(36)
dY4(x) ) Y5(x) dx
(37)
dY5(x) ) Y6(x) dx
(38)
(33)
so that continuity at r ) 0 is maintained. It is mentioned here that in the refs 1 and 3 by Ohshima the electrokinetic equations for a soft particle have been derived (albeit within the restricted step-function interfacial modeling) under the conditions that the net force acting on the soft particle as a whole (the particle core plus the polyelectrolyte layer) must be zero (eq 24) and the electrical force acting on the polymer segment is balanced with a frictional force exerted on the liquid flow. In refs 2 and 13, the latter condition has been replaced by an alternative and more appropriate condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution. The subsequent solving of the electrokinetic equations leads to a general mobility expression for a spherical soft particle. It is found (see ref 13 for further details) that this expression reproduces all of those derived previously and that the pressure continuity condition yields the correct limiting behavior of the electrophoretic mobility in the case where the frictional coefficient within the polymer layer tends to zero. (This behavior cannot be derived from the force balance condition applied to the polyelectrolyte layer.) In other words, the pressure continuity condition, which is automatically verified in our study because a diffuse interface with a continuous segment density profile is considered (there is no surface boundary), is completely in line with the fact that viscous forces are taken into account for the polymer layer and the core particle itself. There is therefore no contradiction with the work by LopezGarcia et al.,7 who within the approximate step-function interfacial modeling explicitly pointed out the importance of the viscous term in both components of a soft particle. To compute the electrophoretic mobility µ, it is necessary to solve consistently the nonlinear system composed of the N + 2 coupled differential equations (eqs 14, 16, and 19 of order 2, 4, and 2 in y, h, and φi, respectively) obeying the 2(N + 3) boundary conditions (eqs 30-33; 20, 21, 26, and 27; and 28 and 29, respectively). The complexity of the problem requires a numerical solution obtained according to the methodology outlined in the next section. Elements of discussion are further given so that we can compare the methodology used in the current article and that employed by Hill et al. in ref 4. 2.3.3. Solution of the GoVerning Transport and Electrostatic Equations. The distributions of the equilibrium potential y(r) and equilibrium concentrations c(0) i (r), as expressed by eqs 14, 15, and 30-33, may be solved accurately using an iterative finite differences scheme, as briefly presented in the Appendix. After a judicious change of variables, the differential equations of orders 4 and 2 pertaining to the variables h and φi (eqs 6 and 19, respectively) may be written as a system, noted Ω, that consists of a coupled set of 2N + 4 linear differential equations of the first order. The mathematics is straightforward but tedious. The resulting system Ω obtained is
dY6(x) dx
)
dY2,i(x) dy(x) 2 + ) -Y2,i(x) - zi dx x + κa dx 2λhi dy(x) 1 2 Y (x) (35) Y (x) 2 1,i eβ dx x + κa 3 (x + κa)
i ) 1,..., N:
or
(34)
)-
2
() (
dy(x)
1 N
(x + κa)
{
() } λo
2
+ f(x) Y5(x) + κ (x + κa)2 Y3(x) λo 2df(x) Y4(x) κ dx x + κa
x + κa
Y6(x) +
2
∑ i)1
dx
c∞i zi2
)
N
c∞i zi2 exp{- ziy(x)}Y1,i(x) ∑ i)1
(39)
where x is the scaled space variable defined by
x ) κ(r - a)
(40)
The functions (Y1,i)i)1,...,N, Y3, and Y4, of which the definitions are required for the completeness of the system Ω, are given by
i ) 1,..., N:
Y1,i ) κφi
Y3(x) ) κβh(r)
(41) (42)
and
Y4(x) )
dY3(x) 2 + Y (x) dx x + κa 3
(43)
respectively. In eqs 35 and 42, the constant β is the factor eηw/ orkBT, which has the dimension of inverse electrophoretic mobility. The 2N + 4 boundary conditions associated with the system Ω are inferred from the boundaries (eqs 20, 21, and 26-29). After some rearrangements, they may be expressed in terms of algebraic combinations of the functions Y as given below:
i ) 1,..., N:
i ) 1,..., N: Y5(x) -
Y2,i(x ) 0) ) 0
(44)
Y3(x ) 0) ) 0
(45)
Y4(x ) 0) ) 0
(46)
Y1,i(x f ∞) f x + κa
2 6 Y3(x)|xf∞ ) 0 Y (x) + x + κa 4 (x + κa)2
-3Y3(x) + (x + κa)Y4(x)|xf∞ ) 0
(47) (48) (49)
3540 Langmuir, Vol. 22, No. 8, 2006
DuVal and Ohshima
The multidimensional root problem that consists of the 4(N + 2) equations (eqs 34-39 and 44-49) was solved by numerical shooting54 from x f ∞ to the position x ) 0 using an adaptive step size Runge-Kutta method of fifth-order implemented with a Newton-Raphson-type scheme. Further information is given in the Appendix. Once the spatial distributions of the various functions Y have been calculated, the dimensionless electrophoretic mobility, denoted as µ j and defined by
µ j ) µβ
(50)
is simply evaluated from the relationship
{
µ j ) 2 Y4(x) -
2 Y (x) x + κa 3
}
xf∞
(51)
Precision of three to four significant figures was easily obtained. We note that the exact numerical evaluation of the electrophoretic mobility for an infinitely long soft cylinder in a transverse dc field55 may be easily carried out following the strategy exposed here for the spherical geometry. This will be the subject of a forthcoming communication. In the work of Hill et al.,4 the solution to the field equations is obtained by the linear superposition of two situations: (i) the fluid velocity is prescribed a given value at r f ∞ in the absence of an applied electric field, and the particle is at rest and (ii) the electric field is applied in the absence of a far-field flow. The corresponding methodology improves the stability issues as expected for the resolution of stiff systems of linear ordinary differential equations. Within the scope of our article, the field equations submitted to numerical analysis are those derived following the strategy by Ohshima, who previously considered step-function interfacial modeling.1-3,13 It is based on the linearization of the key governing equations, thus decomposing all pertaining variables as the summation of an equilibrium term and a induced-field perturbation. The underlying strategy is therefore similar to that of Hill et al. or O’Brien and White.28 The differences are in the very numerical method employed. In our article, we derive, by means of judicious changes of variables explicitly and directly (eqs 34-39 and 44-49 and the Appendix), the final system of ordinary differential equations of first order to be solved for the evaluation of the electrophoretic mobility. Our method does not require the solution of the two independent problems described above in (i) and (ii), as done by Hill et al. We recognize that a major advantage of the methodology used by Hill et al. is that it may deal with the electrophoresis in ac and dc fields whereas we have developed our computational method for the dc case only, which is the purpose of the article. In Hill et al.’s paper,4 the perturbed electrostatic potential and perturbed ionic concentrations must be solved by two coupled equations whereas in our formalism both of these perturbed variables are included in a single quantitysthe perturbed electrochemical potential. Whereas our methodology is rather similar to that of Hill et al. regarding the manipulation of the key starting equations (by linearization of the variables with respect to the applied field), the final system to be solved is different because we consider the perturbed electrochemical potential to be a key quantity in the problem. Furthermore, attention is paid in the current article to the derivation of the Ohshima-like equations for diffuse soft particles because so far these were derived for homogeneous polymer segment distributions only. Besides, to the best of our knowledge, the formalism by Ohshima (54) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1986. (55) Ohshima, H. Colloid Polym. Sci. 2001, 279, 88.
has so far been used only for the derivation of approximate mobility expressions but was never subjected to rigorous numerical evaluation. In our analysis, the number of grid points used for the computation of the hydrodynamics (Navier-Stokes equation) is not fixed by the user because the system of ordinary differential equations is solved with an adaptive step size control algorithm implemented in a Runge-Kutta method of fifth order (Appendix). The major advantage of this quality-control Runge-Kutta method is the achievement of some predetermined accuracy in the solution with minimum computational effort. Many small steps tiptoe through a delicate and treacherous spatial region (close to the particle surface) whereas larger step sizes are employed for the computation of smooth uninteresting (far-field) domains. As such, this method is very valuable and efficient for solving stiff systems for which numerical instability is expected if using more conventional schemes. In the paper by Hill et al.,4 the perturbed variables are solved using a standard algorithm that exploits the banded structure of finite-difference coefficients. The grid points therein are controlled by an iterative mesh-generating algorithm, the number of grid points and the spatial extent of the domain being specified by the user. (See ref 4 for further details.) As such, both methodologies do integrate the necessity of an inhomogeneous step size distribution to cope with stability issues, but our method is probably easier to program because the quality control Runge-Kutta subroutine is intrinsically designed to get rid of instabilities during the resolution process itself.
3. Results and Discussion. 3.1. Validation of the Numerical Method (Results Available in Supporting Information). The dependencies of the electrophoretic mobility on the size, charge, and electrolyte concentration for a step function modeling of the interface (R f 0) have been extensively studied in the literature.4,7,8 In Supporting Information, the computational method was systematically and successfully tested by comparing the numerical results obtained for limiting cases of size (for example, flat plate geometry, i.e., κa . 1) or charge (for example, within the Debye-Hu¨ckel regime, i.e., y < 1) with the corresponding available analytical expressions.1 High electrolyte concentration regimes (κa . 1), where polarization and relaxation effects are insignificant, could be very well reproduced by our methodology. The basic features observed for the mobilities at low to intermediate electrolyte concentrations (κa ≈ 1) are further in line with those obtained by others.4 These analyses were carried out for the case when the particle core is uncharged, that is, ys ) 0 or equivalently σs ) 0, and special attention was also devoted to spherical polyelectrolytic colloids (a ) 0). These choices are mainly motivated by (i) the fact that the aforementioned situations have been so far barely explicitly mentioned and by (ii) the large number of soft biocolloidal25,39,56,57 and environmental systems26,58 for which the existence of a surface charge σs beneath the volume charge density Ffix is an abstraction from reality. It is noted here that even for hard surfaces the positioning of the charges on a well-defined plane is questionable, as correctly recognized in refs 59 and 60. (56) Hammond, S. M.; Lambert, P. A.; Rycroft, A. N. The Bacterial Cell Surface; Croom Helm: London, 1984. (57) Wicken, A. J. Bacterial Cell Walls and Surfaces. In Bacterial Adhesion: Mechanisms and Physiological Significance; Savage, D. C., Fletcher, M., Eds.; Plenum Press: New York, 1985; pp 45-70. (58) Buffle, J. Complexation Reactions in Aquatic Systems: An Analytical Approach. Ellis Horwood: Chichester, West Sussex, England, 1988. (59) Lyklema J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1995; Vol. 2, Chapter 3. (60) Lyklema J.; Rovillard, S.; De Coninck, J. L. Langmuir 1998, 14, 5659.
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Langmuir, Vol. 22, No. 8, 2006 3541
(Figure 2). The Donnan potential yD is effectively reached in the bulk of the soft layer when the conditions κδ . 1 and R , δ are fulfilled, as expected from the profiles given in Figure 2. yD depends on the ratio R/δ according to the relationships
yD ) sinh-1
{
( )]} ≈
Foω δ 1 + tanh 4Fc∞ R
[
sinh-1
Figure 3. Spatial distribution of the equilibrium electrostatic potential across a diffuse interface for various values of R/δ (indicated). Model parameters: a ) 100 nm, δ ) 50 nm, Fo ) 5 mM, c∞ ) 10 mM, NaCl electrolyte at 298 K, (panel A, κδ ≈ 17), and c∞ ) 10-2 mM (panel B, κδ ≈ 0.5).
The computational method was further tested by comparing the results for particles with an inhomogeneous distribution of polymer segments (R * 0) with those obtained by Hill et al.4 For that purpose, calculations were performed for the case where the particle core is uncharged, (ys ) 0 or σs ) 0) and the segment density distribution was set to
(
)
r - (a + δ) 1 f(r) ) erfc 2 R
(52)
which is that adopted in ref 4. Results are given in Supporting Information, and very good agreement was obtained. 3.2. Impact of the Segment Distribution on the Electrophoretic Mobility. 3.2.1. Electrostatic Potential Distributions at Equilibrium. Before examining in detail the impact of the interfacial gradients subsumed in the function f (eq 8) on the electrophoretic motion of a diffuse soft particle, it is helpful first to analyze the dependence of the equilibrium potential distribution on the decay length R, which governs the diffuse character of the interface. Typical examples are represented in Figure 3. For sufficiently large electrolyte concentrations (that is, κδ . 1; Figure 3A), starting from the situation pertaining to a discontinuous segment distribution across the interface (R ) 0), the potential y decreases (increases) for x < κδ (x > κδ) upon increasing R. This is caused by the corresponding decrease (increase) of the local charge density of fixed charges, Ffix(x)
{
( )]} (53)
F oω 2δ 1 - exp 2Fc∞ R
[
where the second equality is valid for R , δ. For sufficiently low values of y(x), the potential distribution basically matches that for the segment density f. This is in agreement with the results derived for planar geometry within the Debye-Hu¨ckel approximation for a macroscopic charged gel-like layer (ω ) 1).20 Upon increases in the nominal charge density Fo, the potentials y(x) are shifted upward (not shown), and the symmetry features for the potential distribution around the coordinates (κδ,yD/2) are governed by those of f and those induced by the nonlinear coupling between Ffix(r) and F(0) el (r) (introduced in eq 14), as explained in ref 20. We recall that the spatial functionality for the density of fixed charges is given by eq 6 with f defined by eq 8a (Figure 2), whereas the distribution for the density of charges that stems from the mobile ions is given by eq 15 with the potential distribution determined by the Poisson-Boltzmann equation (Figure 3). It is clear that the distributions for F(0) el (r) and Ffix(r) are nonlinearly related, which results in different symmetry properties with respect to the position r ) a + δ when varying the parameter R/δ. These different symmetry properties affect those of the potential distribution,20,21 as can be judged from the comparison of Figure 2 with Figure 3. When decreasing the electrolyte concentration, that is, upon decreasing κδ (Figure 3B), the potential profiles across the diffuse interface show a pronounced asymmetry with respect to the position x ) κδ even at low values of the ratio R/δ. This is a consequence of (i) the nonlinear coupling (evoked above) between the spatial functionalities for Ffix and Fel, which exhibit very different symmetry properties with respect to the position x ) κδ (see ref 21 for further details) and (ii) curvature effects that arise for sufficiently thin coatings (deviation from the flat-plate representation). In the limiting case κδ , 1, the extension of the electric double layer goes well beyond that of the coating so that the potential distribution barely depends on the spatial details of the polymer segment distribution (i.e., it is independent of R/δ). 3.2.2. Electrophoretic Mobility of a Diffuse Soft Particle. In the following text, the mobilities of diffuse soft particles are given for various values of R/δ and R/δ varying from 0, which corresponds to a discontinuous modeling of the interface (corresponding results in Supporting Information), to 1, which pertains to a polyelectrolyte brush-like coating (section 2.2). Increasing values of R/δ may be attributed to the swelling of the soft particle, as mentioned in section 2.2. We recall that the total number of polymer segments (and charges) is maintained at a constant value when varying R/δ (eq 8b). In ref 14, Hill et al. underlined the very close connection between the hydrodynamic layer thickness, denoted as Lh (which is experimentally accessible), and the mobility for a soft particle with an uncharged polymer layer. They showed that constraining a uniform layer (R f 0) to have the same Lh as its respective nonuniform layer (R * 0) leads to similar mobilities. Within the current study, computations are performed with nonfixed Lh, the values of which may be a posteriori evaluated from the flow-field profiles for possible comparison with experience.39 This choice is motivated by the fact that charged coatings (which are exclusively examined
3542 Langmuir, Vol. 22, No. 8, 2006
Figure 4. Influence of the diffuse character of the interface (as indicated by the decay length R) on the electrophoretic mobility for various values of Fo (NaCl electrolyte at 298 K). Model parameters: a ) 10 nm, δ ) 40 nm, λo-1 ) 3 nm, and c∞ ) 1 mM (κ-1 ) 9.6 nm). (Inset) Variations of the dimensionless mobility with space charge density Fo for various values of R/δ (indicated). Model parameters are the same as for the main panel.
here) have hydrodynamic properties (in particular, Lh) that are necessarily dependent on the solution composition (pH, ionic strength, and nature of the electrolyte ions), the space charge of the polymeric layer (e.g., electrostatic stiffening), and the polymer segment distribution (f and R). The mandatory prerequisite for the physical modeling of this complex dependence is a knowledge of the interactions within the system {polymer segments-solvent molecules-electrolyte ions}, which except for very well defined systems is rarely accessible in practice, especially when dealing with bacteria or environmental systems (section 2.2). Modeling the electrokinetic data collected for a given system on the basis of the theory of section 2 may be achieved by varying f and/or R until a consistent picture is achieved for the modulations of the mobility and of the (independently determined) titratable charge and hydrodynamic radii when changing the pH and/or the electrolyte concentration.39 The electrophoretic mobilities of a diffuse soft particle are shown in Figure 4 as a function of the ratio R/δ and the nominal charge density Fo. To appreciate quantitatively the impact of the gradual distribution of the polymer segments on the mobility, the quantity µ/µ(R ) 0) is reported on the ordinate axis. Let us first analyze the situation met for low Fo. Upon increasing R, the mobility decreases significantly. The rate of this decrease, as given by the derivative dµ/dR, is relatively large for R/δ < 0.4 and tends to zero for R/δ f 1. On the basis of Figure 3 only, these results might seem surprising. Indeed, increasing R increases the potential y(x) in the region x > κδ where flow penetration within the coating is significant. However, extending the thickness of the diffuse interface also results in a local increase in the friction forces (eq 5, Figure 2) within the region where there is an excess of electrokinetically active counterions. This hydrodynamic effect overwhelms the electrostatic effect and is responsible for the overall decrease in the mobility with increasing R at sufficiently low Fo. The low value of the space charge density prevents the increase in electrostatic potential from being strong enough to counteract the increase in electroosmotic drag at r > a + δ. In other words, increasing R/δ freezes ions that are potentially
DuVal and Ohshima
electrokinetically active, thus correspondingly diminishing the mobility. Going further in the analysis, dµ/dR gradually tends to zero with increasing R because the corresponding increase in the potentials at x > κδ is now large enough to compensate (partially) for the increasing electroosmotic drag. Comparable results were obtained in refs 20 and 21 when examining the incidence of R on the streaming current/streaming potential of macroscopic charged gels. The analogy between the two situations may possibly be inferred from the use of the reciprocal Onsager relationships. In the other limit of large charge density (g75 mM in the example given in Figure 4), the mobility increases for growing values of R. The latter situation is the mirror image of that mentioned for low Fo. The dependence (increase) of the equilibrium potential distribution on R (at x > κδ) now predominantly determines the increase in the mobility. This electrostatic feature wins over the increase in the electroosmotic drag at r > a + δ that would result in a decrease in the mobility. For intermediate values of Fo, the balance between the effects of R on the hydrodynamics and the electrostatics of the diffuse interface is eventually reflected by the presence of a minimum in the mobility. Upon increasing Fo, that minimum is shifted to lower values of the ratio R/δ and disappears for sufficiently large Fo. From Figure 4, it is shown that the screening of the electroosmotig drag exerted by the polymer segments on the flow is strongly dependent on the magnitude of the space charge density. The larger Fo, the more important that screening, the larger the resulting “effective” permeability of the particle, and thus the larger the corresponding electrophoretic mobility. The increase or decrease in µ/µ(R ) 0) with R/δ is basically governed by the magnitude of the aforementioned screening, as determined by the extent of diffuseness of the interface. The increase in µ with increasing values of R/δ, as mentioned before for large Fo, is in line with electrophoretic data on cationic microgel particles that show a larger mobility of swollen particles as compared to that for deswollen particles.61 In the inset of Figure 4, we report the mobility µ as a function of Fo for various values of the “swelling degree” R/δ. The results clearly show that the mobility-charge dependence changes with the degree of heterogeneity of the interface (as subsumed in R/δ). For R/δ ) 0, µ rapidly increases with Fo before leveling off and eventually decreasing (not shown) because of relaxation of the double layer by the applied electric field (this is the mobility maximum as predicted in the semianalytical study by Saville6). With increasing R/δ, the mobility basically decreases for Fo lower than 30-40 mM and then increases for larger Fo. These trends are consistent with the larger screening of the electroosmotic drag upon increasing Fo, as mentioned before. In other words, the maximum mobility gradually disappears upon increasing the swelling of the particle. This is consistent with the experimental results of Fernandez-Nieves et al.61 who reported mobilityconcentration curves for swollen and deswollen microgel particles. In their analysis, polarization arises when decreasing the electrolyte concentration whereas in ours it originates from the increase in Fo, with both situations being equivalent because they ultimately result in an increase in the electrostatic potentials for which polarization/relaxation of the double layer may come into play. As in the studies20,21 where the number of polymer segments and charges is always conserved (macroscopic gel-like layers are considered therein), the dependence of the electrokinetic quantity µ/µ(R ) 0) on the ratio R/δ can be explained on the basis of the R dependence of the distributions for the equilibrium potential and the friction coefficient. (61) Fernandez-Nieves, A.; Marquez, M. J. Chem. Phys. 2005, 122, 084702.
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Langmuir, Vol. 22, No. 8, 2006 3543
Figure 5. Influence of the diffuse character of the interface (as indicated by the decay length R) on the electrophoretic mobility for various values of λo-1 (NaCl electrolyte at 298 K). Model parameters: a ) 10 nm, δ ) 40 nm, and c∞ ) 1 mM (κ-1 ) 9.6 nm). (A) Fo ) 1 mM. (B) Fo ) 35 mM.
Figure 6. Influence of the diffuse character of the interface (as indicated by the decay length R) on the electrophoretic mobility for various coating thicknesses δ (NaCl electrolyte at 298 K). Model parameters: a ) 10 nm, c∞ ) 1 mM (κ-1 ) 9.6 nm), and λo - 1 ) 3 nm. (A) Fo ) 10 mM. (B) Fo ) 50 mM.
Given the previous comments, the dependence of µ on R for various nominal penetration lengths λo-1 (Figure 5) is according to expectation. Increasing λo-1 at constant R leads to an overall decrease in the electroosmotic drag by the polymeric layer and thus to an increase in the electrophoretic mobility. For sufficiently large λo-1, µ increases with R because a larger fraction of the double layer expansion for x g κδ is effectively probed by the electroosmotic flow. The mobility minimum as eventually observed for low values of λo-1 (panel B) is, as discussed in Figure 4, gradually shifted to lower R/δ and finally disappears when λo-1 is large enough. We note that the higher the charge density Fo (i.e., the larger the number of counterions that are potentially electrokinetically active), the more the mobility will depend on the permeability λo-1, as intuitively anticipated. To summarize, Figure 5 illustrates the screening of the electroosmotic drag upon the increasing permeability of the polymeric shell. It constitutes the pendant of Figure 4 where the screening was analyzed as a function of the charge density Fo. In Figure 6A, mobilities are examined as a function of R and the coating thickness δ. When δ decreases at constant R/δ, the following features occur. (i) The coating recedes further within the double-layer region, (ii) the potential distribution becomes strongly asymmetric because of curvature effects (section 3.2.1), and finally (iii) the local friction exerted by the polymer segments on the flow decreases (eq 8). Feature iii is responsible for the observed increase in the electrophoretic mobility when δ decreases at constant R/δ. For very thin coatings (κδ , 1), µ becomes
practically independent of R, which is line with feature (i) mentioned above and thus the mobility becomes determined by the layer heterogeneity to a lesser extent. For thicker coatings and low to moderate Fo (as in panel A), the mobility decreases continuously upon increasing R as a result of the increasing electroosmotic drag at x g κδ. This hydrodynamic feature wins over the increase in the electrostatic potentials at x g κδ that would result in an increase in µ/µ(R ) 0) with R/δ (Figure 3). The low value of the nominal space charge density prevents this increase in electrostatic potential from being strong enough to counteract the increase in electroosmotic drag at x g κδ. In Figure 6B, the mobility is examined under the same conditions used in Figure 6A except that now Fo is 5 times larger than in panel A. The explanations advanced for explaining Figure 6A still hold for R/δ < 0.2 (hydrodynamic effect). However, for R/ > 0.2, µ/µ(R ) 0) increases with R/ for δ ) 10, 20, and 40 δ δ nm. This increase is now determined by the increase in electrostatic potentials in the region of the soft layer where the electroosmotic flow is the larger (i.e., for x g κδ). The potential distribution is shifted upward namely because curvature effects disappear (increasing κδ) so that the potential in the bulk of the soft layer gradually reaches a maximum value given by the Donnan potential (eq 53). The large value of the charge density chosen for the calculations in Figure 6B allows the electrostatic effect to overwhelm the increase in the electroosmotic drag at x g κδ. In other words, for R/δ > 0.2, the local increase in potentials or for that matter the local increase in the charge density
3544 Langmuir, Vol. 22, No. 8, 2006
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because an increasing fraction of mobile ions distributed within/ outside the coating becomes hydrodynamically frozen. Equivalent results were obtained in ref 21 for the electrokinetics of macroscopically flat charged gels. For large κa, µ/µ(R ) 0) f 1, meaning that the expression µ(κa . 1) ) Fo/ηλo2 as elaborated for a sharp interface remains valid for a diffuse soft polymeric shell. This counterintuitive result may be demonstrated as follows. For sufficiently large c∞, the mobility is solely determined by that of the resistance centers constituting the permeable shell, as first recognized by Ohshima.1 The electrical body force Fe acting on the diffuse particle is given by
Fe )
3
∫aa+δr2f(r) dr
FoE
[(a + δ)3 - a3]
(54)
whereas the drag (viscous) force Fh acting on the fluid is
Fh )
Figure 7. Influence of the diffuse character of the interface (as indicated by the decay length R) on the electrophoretic mobility for various electrolyte concentrations (NaCl electrolyte at 298 K). Model parameters: a ) 100 nm, δ ) 50 nm, and λo-1 ) 5 nm. (A) Fo ) 5 mM. (B) Fo ) 20 mM.
Fel(r > a + δ) (eq 14) that accompanies the increase in the number of active mobile counterions governs the increase in mobility. Therefore, the screening of the electroosmotic drag is more efficient upon increasing the space charge density and the thickness of the diffuse layer. This implies that the particle becomes more “permeable” with a higher electrophoretic mobility, which conforms to ref 61. Finally, for the sake of completeness, the effect of the electrolyte concentration on the electrophoretic mobility of a diffuse soft particle is presented in Figure 7. Within the framework of a discontinuous modeling for the interface, the mobility decreases upon increasing κa as the result of the increase in the electroosmotic drag.4 For sufficiently large electrolyte concentrations, µ reaches a constant finite value (Supporting Information), as recognized by Ohshima1 and others.4 This value is given by the quantity µ ) Fo/ηλo2 (valid for λoδ . 1). In the κa , 1 regime, the dimensionless electrophoretic mobility µ j is proportional to the potential at the surface of the particle core. This result is expected by analogy with Hu¨ckel’s theory for hard particles, µ j ) 2eζ/3kBT, which is valid for low values of the electrokinetic potential ζ and small κa. This was recognized by Hill et al. in ref 4 for particles with charged hard cores. When increasing the parameter R at constant κa, the mobility decreases because of the ongoing increase in the electroosmotic drag as governed by the increase in the friction coefficients at x g κδ. The lower the electrolyte concentration, the larger the decrease
∫aa+δr2f(r) dr
3
k oU
[(a + δ)3 - a3]
(55)
Equating Fe and Fh yields µ ) Fo/ηλo2, where expression 18 has been used. We note that for dense polyelectrolyte layers (i.e., layers with low water content), high-order terms in eq 4 must be considered so that the linear relationship between k(r) and Ffix(r) as obtained for low polymer volume density φ is lost, thus rendering the limiting expression µ ) Fo/ηλo2 inappropriate for diffuse particles.4 The classical mobility limit at high electrolyte concentrations would also be necessarily incorrect for charged microgel particles synthesized with cross linkers used in a controlled way so as to fix the spatial distribution of k. In a recent study,62 Lietor-Santos et al. experimentally found that in the high-concentration regime where the mobility reaches a finite nonzero value the mobility µ of spherical uncharged colloidal particles covered with a polyelectrolyte shell decreases with electrolyte concentration c following the power law µ ≈ c-1/3. The authors successfully justified this scaling law by combining the general expression µ(c f ∞) ) Fo/ηλo2{1 sech(λoδ)} (valid at any λoδ, see Supporting Information) with the dependence of the shell thickness on the electrolyte concentration (swelling process), as described by Pincus.63 This power law cannot be verified on the basis of the results in Figure 7A. The reason for this is that the simulations are performed as a function of the electrolyte concentration with fixed thickness for the soft layer. In other words, the very dependence of that thickness on the electrolyte concentration is not taken into account. Therefore, the experimental data reported in ref 62 as a function of c correspond to various values of R/δ with the dependence of R/ on c as given by Pincus’ law. However, if we include this δ specific dependence in the calculations, our theory would reproduce the aforementioned scaling law because we show in Supporting Information that the mobility limit at high concentrations, as predicted by Ohshima, is recovered by our numerical scheme. In panel B of Figure 7, the nominal charge density Fo is increased. This leads to the occurrence of a mobility minimum at intermediate κa for R ) 0, as extensively discussed in Supporting Information and in other studies.4 For low electrolyte concentrations, the mobility decreases with R for the reasons set forth in panel A. At high κa, all mobilities approach the same limit, as shown previously. In the intermediate κa regime, the (62) Lietor-Santos, J. J.; Fernandez-Nieves, A.; Marquez, M. Phys. ReV. E 2005, 71, 042401. (63) Pincus, P. Macromolecules 1991, 24, 2912.
Electrophoresis of Diffuse Soft Particles
ratio µ/µ(R ) 0) presents a minimum for which the position is shifted to lower values of R upon increasing the electrolyte concentration. This is essentially an electrostatic feature: when the double layer sufficiently extends inside the coating and for values of the concentration that are not too large, the increase in the electrostatic potentials for x g κδ (Figure 3B) with increasing R significantly counteracts the drag exerted by the diffuse shell on the electroosmotic flow. This is manifested in a shift of the mobility minimum to lower κa with increasing R and an increase in the minimum for sufficiently large values of R. This result is the pendant of that discussed for Figure 4: the screening of the electroosmotic drag is larger in panel B as compared to that in panel A because the charge density is much larger. As a consequence, the mobility minimum is gradually shifted and removed for increasing values of R/δ.
Conclusions In this article, the concept of diffuse soft interfaces as introduced in refs 20 and 21 is extended to the electrophoresis of soft particles characterized by a continuous distribution of polymer segments within the permeable coating. An analysis of the electrophoretic mobility of those so-called diffuse soft particles is performed by rigorous numerical evaluation of the relevant fundamental electrostatic and hydrodynamic equations. The numerical scheme is based on the analytical formalism originally developed by Ohshima,1 albeit within the restricted framework of discontinuous modeling of the interface between the soft particle and the neighboring electrolyte solution. After an examination of the mobility of a soft particle with a uniform coating as a function of its size, charge, and double-layer thickness (results available in Supporting Information), the impact of the diffuse character of the interface on the mobility is explicitly analyzed. As expected from the conclusions listed in refs 20 and 21, the inhomogeneous distribution of the polymer segments greatly affects the electrophoretic properties as derived from the commonly adopted discontinuous interfacial representation. The mobility is determined by a subtle balance between the electrical force and electroosmotic drag, both being significantly perturbed by the local finite gradients associated with the polymer segment distribution. Few illustrative situations are given and commented on the basis of the underlying equilibrium potential and friction coefficient distributions within the diffuse interface. In particular, it is shown that the screening of the electroosmotic drag increases upon increasing the charge density and hydrodynamic permeability, thus resulting in higher electrophoretic mobilities. For low to moderate charge densities, the mobility decreases with increasing diffuseness (heterogeneity) of the soft layer because the associated increase in electrostatic potentials in the region of large electroosmotic flow velocity is not strong enough to counterbalance the corresponding increase in the friction coefficient (and vice versa for larger charge densities). The analysis of the dependencies’ mobility charge and mobility electrolyte concentration revealed that the minima and maxima in mobility eventually observed as a result of double-layer relaxation gradually disappear upon increasing the heterogeneity of the layer. These findings, explained in terms of the screening of the electroosmotic drag, are in line with recent experimental work.61 An accurate numerical solution of the governing electrokinetic equations allows the experiment to be compared with theory over a wide range of particle size, charge, and external conditions such as electrolyte concentration, temperature, and pH. These conditions are known to possibly affect the arrangement and conformation of the polymer segments29,34-38 or the thickness of the soft coating as measured by dynamic light scattering or
Langmuir, Vol. 22, No. 8, 2006 3545
diverse imaging techniques (AFM, SEM, TEM, etc.).30,31,51 The model presented here is very useful for quantitatively appreciating the effect of these structural changes on the electrophoretic mobility. Its flexibility further permits (i) the analysis of electrophoretic data collected for various types of soft coatings (polyelectrolyte brush, neutral homopolymers, etc.), (ii) the taking into account of the peculiar surface structures of certain biocolloids such as bacteria, of which the cell wall (which may be assimilated to a soft coating) is surrounded by charged surface appendages (which constitute another coating with different hydrodynamic and electrostatic properties),25,39 (iii) the implementation of the dissociation characteristics (number of sites, pK values) of the charged groups, as evaluated by titration experiments or chemical elemental analysis, and (iv) the inclusion of possible specific interactions between ionogenic charged groups and ions from the background electrolyte other than charge-determining ions.39 A further improvement of our model would be the implementation of (i) spatial heterogeneity of the surface layer not only in the r direction (as done in this article) but also in the θ and/or φ directions and (ii) the effects of ionic correlations, which may play an important role in the electrophoresis of particles in the presence of multivalent counterions.64 This will be the subject of forthcoming studies.
Appendix 1. Computation of the Equilibrium Potential Profile y(r). Discretization of eqs 14, 15, and 30-33 leads to
j ) 2,..., m - 1:
[
yj - 1 1 -
(κRo∆x˜ )2 N
c∞i zi2 ∑ i)1
[
∆x˜
yj+1 - 2yj 1 -
{∑ N
]
x˜ j + (κa/κRo)
2∆x˜ x˜ j + (κa/κRo)
c∞i zi exp(-ziyj) +
i)1
]
+
)
Fo f(x˜ j) g(pH, yj) F
y1 ) ys
}
(A1)
(A2)
or
y 2 - y1 )
-(κRo)∆x˜ σsκ N
∑ i)1
(A3) (zi Fc∞i ) 2
ym ) 0
(A4)
where the space variable x˜ is defined by x˜ ) x/(κRo) and Ro is the radial position at which the potential y is zero. (x˜ j)j)1,...,m(∈[0, 1]) is the discretized equivalent of x˜ and is given by
j ) 1,..., m:
x˜ j ) j∆x˜
(A5)
with ∆x˜ j ) 1/m being the spatial step on the uniform distribution of grid points. The nonlinear tridiagonal system consists of m equations (A1-A4) with the unknown (yj)j)1,...,m solved using a globally convergent Newton-Raphson method.54 Ro was iteratively updated until the far-field condition (eq A4) was (64) Fernandez-Nieves, A.; Fernandez-Barbero, A.; de las Nieves, F. J.; Vincent, B. J. Chem. Phys. 2005, 123, 054905.
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DuVal and Ohshima
satisfied. Initial guesses for Ro and (yj)j)1,...,m were obtained from the analytical solution of the linearized Poisson-Boltzmann equation (Debye-Hu¨ckel approximation) written as
exp(-x) x (x + κa)χ(x) exp(x) dx 2(x + κa) 0 exp(x) exp(-x) x (x + κa)χ(x) exp(-x) dx + A + 0 x + κa 2(x + κa) exp(x) (A6) B x + κa
∫
y(x) )
∫
where χ is the radial function defined by
1
χ(x) )
N
∑ i)1
{∑
Fo c∞i zi + f(x) g(pH) F i)1 N
c∞i zi2
}
and at x2 ) R . Ro, the boundaries (eqs 47-49) are
For n2 ) 3, l ) 1,..., n2, and i ) 1,..., N: H2,l(x2, Y1,i, Y3, Y4, Y5) ) 0 (A14) The two-boundary problem (eqs A11-A14) was solved by numerical back shooting from position x2 to position x1.54 For that purpose, values for all of the independent variables Y are chosen at the boundary x2. These values are arranged to depend on the arbitrary parameters (ξ)p)1,2,3 as indicated below:
i ) 1,..., N:
∫0∞(x + κa)χ(x) exp(-x) dx
1 2
(A7)
A ) (κa)y - B
(A9)
κa - 1 FaσS κa B + orRT κa + 1 κa + 1
(
)
(A10)
for a constant surface charge boundary condition. For the sake of simplicity, the impact of the local potential distribution on the protolytic properties of the surface layer is omitted in eq A7, which means that the function g is dependent on the pH only. Following the aforementioned method, the potentials (yj)j)1,...,m could be obtained with fast convergence and a precision on the order of O(∆x˜ 2), with typical values for m being in the range of 500-1000. 2. Computation of the Electrophoretic Mobility µ j: Numerical Shooting. The set of differential equations (eqs 3439) may be written in the following concise form
For j ) 1, 2 and i ) 1,..., N: dYj,i(x) ) Gj(x, y, Y1,i, Y2,i, Y3) (A11) dx For j ) 3,..., 6:
(A16)
dYj(x) ) Gj(x, y, Y3,..., Y6) (A12) dx
with (Gj)j)1,...,6 being the operators defined by the right-hand sides of eqs 34-39, respectively. At x1 ) 0, the solution in Y satisfies eqs 44-46 rewritten as the following
For n1 ) 3, k ) 1,..., n1, and i ) 1,..., N: H1,k(x1, Y2,i, Y3, Y4) ) 0 (A13)
(A17)
3Y3(x2) R + κa
(A18)
2Y4(x2) 6Y3(x2) R + κa (R + κa)2
(A19)
Y6(x2) ) ξ3
(A20)
Y4(x2) ) Y5(x2) )
(A8)
for a constant surface potential boundary condition and
A)
Y2,i(x2) ) ξ1 ξ2(R + κa) 2
Y3(x2) )
The second constant A is given by s
(A15)
i ) 1,..., N:
and the constant B is obtained with the following expression
B)
Y1,i(x2) ) R + κa
Equations A15 and A16 are inferred from the boundary (eq 47), and eqs A17-A20 are inferred from the combination of the boundaries (eqs 48 and 49) and the asymptotic behavior of Y3 as given by eqs 25 and 42. The ordinary differential equations (eqs A11 and A12) are integrated for starting values of (ξ)p)1,2,3, arriving at the boundary x1 using an adaptive step size RungeKutta method of fifth order. In general, discrepancies from the desired boundaries (eq A13) are found. The parameters (ξ)p)1,2,3 are then iteratively updated with a globally convergent NewtonRaphson scheme until they zero the relationships (eq A13). Rapid convergence of the results is obtained for a judicious choice of the initial guesses pertaining to (ξ)p)1,2,3. From the asymptotic forms of Y, one can easily show that the desired (ξ)p)1,3 values must be ξ1 ) 1 and ξ3 ) 0. ξ2 actually corresponds to the searched dimensionless electrophoretic mobility µ j . As the initial guess for ξ2, the predictions from the flat-plate theory of Ohshima1 or of Hermans et al.64 were used. The scaled equilibrium potential y and the electric field dy/dr were evaluated on the self-controlled nonuniform grid by cubic spline interpolation of the results obtained from the finite difference algorithm with a uniform step size (previous section). Supporting Information Available: Discontinuous modeling of the soft interface R f 0: comments on the dependencies of the electrophoretic mobility on κa, Fo, κλo-1, and κδ. Testing the validity of the numerical scheme adopted: comparison of the numerical results with known approximate analytical expressions1,65 and with those obtained by Hill et al. in ref 4. This material is available free of charge via the Internet at http://pubs.acs.org. LA0528293 (65) Hermans, J. J.; Fujita, H. Koninkl. Ned. Akad. Wetenschap. Proc. 1955, B58, 182.
