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Application of a Modified Gouy-Chapman Theory to the Stability of a Biocolloidal Suspension Yung-Chih Kuo† Department of Chemical Engineering, National Chung Cheng University, Chia-Yi, Taiwan 62102, Republic of China Received January 29, 2003. In Final Form: April 17, 2003 A theoretical study on the stability of biocolloidal particles immersed in an electrolyte solution is presented. The outer border of each biocolloid is covered with an ion-penetrable membrane carrying nonuniformly distributed positive and negative fixed charges. The results of numerical simulations reveal that the classical point-charge model underestimates the stability ratio of a biocolloidal suspension, W, for both constant total amount of positive fixed charges and that of negative ones in membrane phase. W predicted by the case of a nonlinear distribution of positive and negative fixed charges is larger than that predicted by the case of a corresponding linear one. A thick membrane causes a small W, and the larger the dielectric constant of a suspension, the larger the value of W. Increasing the sizes of mobile cations or positive fixed charges provokes a reduction in W. The reverse is true for mobile anions or negative fixed charges. A small value of W can result from (1) a large average concentration of positive fixed charges, (2) a small average concentration of negative fixed charges, (3) a large nonuniformity index for positive fixed charges, or (4) a small nonuniformity index for negative fixed charges.
1. Introduction One of the essential quantities characterizing a colloidal suspension is the stability ratio, W, which measures the effectiveness of potential energy barrier in precluding colloids from coagulation. From the kinetic point of view, W is defined by (frequency of collisions between colloids)/ (frequency of collisions leading to coagulation) or (rate of rapid coagulation)/(rate of slow coagulation).1 The numerator of the definition can be described by the Smoluchowski theory for Brownian coagulation in the absence of a potential barrier, while the expression for the denominator, which contains an interaction energy, can be achieved by the Fuchs theory for the stability ratio of a dispersion.1-3 For estimation of the influence of potential energy on the stability of lyophobic colloids, successful attempts, including two opposing contributions, the electrostatic repulsion of overlapping diffuse double layers (DDL) and the London-van der Waals attraction of a particulate pair, was established independently by Derjaguin and Landau and Verwey and Overbeek.1-4 Their distinguished works are referred to, in the literature, as the DLVO theory, which is a central canon for colloidal stability. Through the foundation of the DLVO theory, appreciable efforts on the improvement in the evaluation of stability ratio have been accomplished thereafter.5-8 To estimate the electrical contribution, the information about the electrostatic potential distribution, which is commonly governed by the Poisson-Boltzmann equation † Tel: 886-5-272-0411 ext. 33459. Fax: 886-5-272-1206. E-mail:
[email protected].
(1) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1989; Vol. I. (2) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; M. Dekker: New York, 1986. (3) Heicklen, J. Colloid Formation and Growth; Academic Press: New York, 1976. (4) Russel, B. R. The Dynamics of Colloidal Systems; The University of Wisconsin Press: Madison, WI, 1987. (5) Wiese, G. R.; Healy, T. W. Trans. Faraday Soc. 1970, 66, 490. (6) Overbeek, J. Th. G. Pure Appl. Chem. 1980, 52, 1151. (7) Metcalfe, I. M.; Healy, T. W. Faraday Discus. Chem. Soc. 1990, 90, 335. (8) Kuo, Y. C.; Hsu, J. P. Chem. Phys. 1999, 250, 285.
(PBE) based on the assumption of mean-field liquid state physics, is required. The solution to the general PBE is, however, nontrivial, in other than a few limited cases. The difficulties in resolution of the PBE arise mainly from the natural properties of ionogenic species in a dielectric medium and the sophisticated physiochemical conditions on colloidal interfaces. Historically, Gouy and Chapman originated the investigation on the analysis of the PBE by a model of a charged planar particle surrounded by point electrolyte ions in a uniform-dielectric continuum solvent. Namely, the primitive Gouy-Chapman theory (GCT) entirely neglects any ionic characteristics but its valence. In a microscopic approach to the DDL behaviors, the classic GCT assuming that the ionic sizes vanish is unrealistic. To modify the GCT, Stern first proposed the concept of the closest approach of ionic species adjoining an interfacial boundary to reflect the effects of finite size of ions. In Stern’s DDL model, conventionally termed the modified Gouy-Chapman theory (MGCT), the dielectric constant within the designated repulsive region, which pertains to the closest approach, is lower than that in the bulk liquid solution. The location of the closest approach of charged species from a rigid surface was named as the outer Helmholtz plane by Graham about 2 decades later. Although Stern’s DDL model assumes that the size of every ionic hard core is the same, many of the succeeding studies of electrical properties when considering the effects of ionic sizes employed a reasonable MGCT by applying an adequate expression for the excluded volumes of electrolyte ions.9,10 Valleau and Torrie11 proposed an MGCT for unequal radii of cations and anions. They concluded that asymmetric electrocapillarity and differential capacitance can arise naturally by assigning different closest approach distances for ionic species. Bhuiyan et al.12 presented an MGCT for 2:1 and 1:2 electrolytes. They observed that a nonzero potential can (9) Blum, L. J. Phys. Chem. 1977, 81, 136. (10) Henderson, D.; Blum, L. J. Chem. Phys. 1978, 69, 5441. (11) Valleau, J. P.; Torrie, G. M. J. Chem. Phys. 1982, 76, 4623. (12) Bhuiyan, L. B.; Blum, L.; Henderson, D. J. Chem. Phys. 1983, 78, 442.
10.1021/la034156z CCC: $25.00 © 2003 American Chemical Society Published on Web 06/06/2003
Stability of a Biocolloidal Suspension
Langmuir, Vol. 19, No. 14, 2003 5943
Figure 1. Schematic representation of the system under consideration.
occur at zero charge on an electrode wall, even in the absence of specific adsorption. For nonrigid biocolloids such as mammalian cells and collagen molecules, a particle possesses normally an ionpenetrable membrane layer bearing ionogenic groups, which are fastened in a three-dimensional volume rather than scattered over a two-dimensional solid surface.13 A typical example is that of a charged glycoprotein layer roughly 15 nm thick forming the peripheral zone of the surface lipid layer on human erythrocyte.14 Hence, for describing the interaction among biocolloidal particles, a model of hard particles becomes inappropriate and needs modification. To reflect more rational aspects of biological entities, Ohshima and Ohki15 proposed a theoretical model for uniform charged membranes, which was later improved to predict the behaviors of membranes with nonuniformly distributed fixed charges.16-18 By employing this model, Hsu and Kuo19 were able to obtain the stability ratio of a dispersion of charged membranes. Recently, we examined a membrane version of MGCT by incorporating the size effects of both freely mobile ions and fixed charges in the membrane phase.20-23 The validation and applicability of a membrane MGCT were further demonstrated by estimation of biocolloidal mobility,24 critical coagulation concentration,25 flocculation,26 and adsorption.27 At the present stage, it seems to be inevitable to achieve reliable (13) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1961. (14) Kawahata, S.; Ohshima, H.; Muramatsu, N.; Kondo, T. J. Colloid Interface Sci. 1990, 138, 182. (15) Ohshima, H.; Ohki, S. Biophys. J. 1985, 47, 673. (16) Kuo, Y. C.; Hsu, J. P. J. Chem. Phys. 1995, 102, 1806. (17) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1995, 103, 465. (18) Hsu, J. P.; Kuo, Y. C. J. Membr. Sci. 1995, 108, 107. (19) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 184. (20) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1999, 111, 4807. (21) Kuo, Y. C.; Hsu, J. P. J. Phys. Chem. B 1999, 103, 9743. (22) Kuo, Y. C.; Hsu, J. P. Langmuir 2000, 16, 6233. (23) Kuo, Y. C.; Hsieh, M. Y.; Hsu, J. P. Langmuir 2002, 18, 2789. (24) Huang, S. W.; Hsu, J. P.; Kuo, Y. C.; Tseng, S. J. Phys. Chem. B 2002, 106, 2117.
estimations of the stability ratio of a biocolloidal dispersion through a membrane MGCT. This is carried out in the present study. It emphasizes particularly two unequal sizes for positive and negative fixed charges in the membrane and two nonuniform distribution types for positive and negative fixed charges. 2. Analysis A. Interfacial Interaction System. Figure 1 depicts a schematic illustration of the system under consideration for the interactions between two identical biocolloidal particles. Here, a biocolloid comprises a rigid core and an ion-penetrable charged membrane of scaled linear thickness D ) κd in an arbitrary a:b electrolyte solution. Here, d is the membrane thickness and κ is the reciprocal of the Debye parameter, which is defined by κ2 ) e2a(a + b)na0/0rkBT, where e is the elementary charge, na0 means the number concentration of cations in the bulk liquid phase, 0 and r are the permittivity of a vacuum and the relative permittivity, respectively, and kB and T denote, respectively, the Boltzmann constant and the absolute temperature. Since a biocolloidal particle is relatively large compared to the thickness of the electrical double layer under normal physiological conditions,1,2 the curvature of the surface can be neglected. XL denotes the scaled half closest separation distance between two membrane-solution interfaces. Let rf,n, rf,p, ran, and rca be the effective diameters of negative fixed charges, positive fixed charges, anions, and cations, respectively. Without loss of generality, we assume that rf,n > rf,p > ran > rca. The origin of the fixed charges may be in the dissociation of functional groups which the membrane bears or from the absorption of electrolyte ions by those functional groups. (25) Hsu, J. P.; Huang, S. W.; Kuo, Y. C.; Tseng, S. J. Phys. Chem. B 2002, 106, 4269. (26) Kuo, Y. C. J. Chem. Phys. 2003, 118, 398. (27) Kuo, Y. C. J. Chem. Phys. 2003, 118, 8023.
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Langmuir, Vol. 19, No. 14, 2003
Kuo
Typical reactions include
R-COOH S R-COO- + H+
(1a)
R′-NH2 + H+ S R′-NH3+
(1b)
Equations 1a and 1b exemplify the formation of negative (basic) fixed charges and positive (acidic) fixed charges, respectively. The two classes of fixed charges are arranged in the membrane phase so that the margin of the leftmost one is coincident with the core-membrane interface and that of the rightmost one is coincident with the membrane-liquid interface. Referring to Figure 1, the interacting system is divided into seven regions: I, X < Xca, which comprises the charge-free domain in the internal core of a biocolloid (-Xc < X < 0) and the inner uncharged membrane (0 < X < Xca), where X is the scaled distance; II, Xca < X < Xan, which contains only cationic charges in the inner membrane layer; III, Xan < X < Xf,p, which contains both cationic and anionic charges in the inner membrane layer; IV, Xf,p < X < Xf,n, which contains cationic, anionic, and positive fixed charges in the membrane layer close to the uncharged core; V, Xf,n < X < X′f,n, which contains all the four kinds of charges; VI, X′f,n < X < X′f,p, which contains cationic, anionic, and positive fixed charges in the membrane layer near the membrane-solution interface; VII, X′f,p < X < XL, which includes the outer membrane that is free from both positive and negative fixed charges (X′f,p < X < D) and the liquid solution of electrical double layer containing both cationic and anionic charges (D < X < XL). Xf,p and X′f,p are the location of the inner and outer plane of positive fixed charges, respectively. Xf,n and X′f,n are the location of the inner and outer plane of negative fixed charges, respectively. The scaled symbols are defined by X ) κr, Xc ) κr0, Xca ) κrca/2, Xan ) κran/2, Xf,p ) κrf,p/2, Xf,n ) κrf,n/2, X′f,p ) D - Xf,p, and X′f,n ) D - Xf,n, where r and r0 are the distance measured from the core-membrane interface and the linear radius of rigid core of a biocolloid, respectively. Note that the positive and negative fixed charges are present, respectively, in the range Xf,p < X < X′f,p and in the range Xf,n < X < X′f,n. B. Nonuniform Fixed Charge Distribution. The nonuniform distribution of fixed charges in the membrane phase can be formulated by
{
2NAZvNv[1 + Rv(X - Xf,v)]
Nj,v )
ana0(Av + 2) NAZvNvAv{1 + exp[Rv(X - Xf,v)]} ana0[exp(Av) + Av - 1]
j)1 ,
(2)
j)2
where subscript v represents the sign of the fixed charges (v ) n for negative or basic charges and v ) p for positive or acidic charges), Av ) Rv(X′f,v - Xf,v), Rv symbolizes a nonuniformity index for species v, which is a parameter characterizing the extent of nonuniform distribution of fixed charges (species v) in the radial direction of the membrane phase, Nj,v and j denote, respectively, the dimensionless concentration of fixed charges for species v and a distribution type index, which is a parameter characterizing the distribution category of fixed charges (species v) in the membrane (j ) 1, linear type; j ) 2, exponential type), Nv and Zv mean, respectively, the average concentration and the valence of fixed charges for species v in the membrane phas,e and NA represents the Avogadro number. eNA(ZnNn + ZpNp) becomes the average density of fixed charges in the membrane phase. The scaled concentration distribution of fixed charges in
region V becomes Nj,p + Nj,n. It is worth notice that the distribution of fixed charges described in eq 2 ensures that the space-averaged concentration of fixed charges is constant in the membrane phase no matter the values of Rv and j, i.e.
aena0
∫XX′
f,v
f,v
Nj,v dX ) eNAZvNv(X′f,v - Xf,v)
(2a)
Under the condition of Rv f 0 (Av f 0), eq 2 becomes
Nj,v ) NAZvNv/ana0
(3a)
This means that as Rv f 0, the fixed charges reduce to the typical uniform distribution for both the cases of linear (j ) 1) and exponential (j ) 2) types with the concentration of fixed charges Nv.16-18 On the other hand, as Rv approaches infinity, the asymptotes of Nj,v need to be
N1,v(X)Xf,v) ) 0 N1,v(X)X′f,v) )
2NAZvNv ana0
(3b)
N2,v(Xf,veX
