4372
Langmuir 1997, 13, 4372-4376
Adsorption of a Charge-Regulated Particle to a Charged Surface Jyh-Ping Hsu* and Yung-Chih Kuo Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. Received October 21, 1996. In Final Form: April 17, 1997X The adsorption of a spherical particle coated with an ion-penetrable membrane to a charged surface immersed in an electrolyte solution is analyzed. The former is capable of regulating the degree of dissociation of the functional groups in the membrane as a response to the variation in the electrical condition of the surrounding medium. The membrane contains both acidic and basic functional groups, and a general form for each is assumed which allows multiple-proton transfer among functional groups. Two classes of nonuniform distribution for the functional groups are considered. We show that the more concentrated the functional groups are distributed near the outer boundary of the membrane, the greater the electrostatic repulsion force between the particle and the surface, and the slower the rate of adsorption. Compared with a rigid particle carrying the same numbers of acidic and basic functional groups, the existence of the membrane has the effect of increasing the rate of adsorption. The electrostatic repulsion force between a particle with a membrane and a rigid surface is smaller than that between a rigid particle and a rigid surface if the separation distance is greater than a critical value. The reverse is true if the separation distance is smaller than the critical value.
1. Introduction The adsorption of charged, dispersed entities to a charged surface is a phenomenon of fundamental importance in various natural and industrial processes. A typical example of the latter includes the attachment of particles to surfaces. Here, the electrostatic interaction between two charged surfaces plays a key role in the estimation of the rate of attachment. Early efforts for the relevant subjects were mainly devoted to the interactions between lifeless entities and surfaces. Often, two interacting surfaces were assumed to maintain at a constant electrical condition (potential or charge density) to allow a simpler mathematical manipulation. While this may be adequate for inorganic surfaces, it is unrealistic for biological surfaces in which the surface conditions are determined by the degree of dissociation of the functional groups it bears. The so-called charge-regulation phenomenon is mainly due to the fact that the surfaces tend to minimize the total interaction free energy.1 Ninham and Parsegian,2 for example, showed that if two identical surfaces approaching each other are in ionic equilibrium with the surrounding medium, neither the surface potential nor the surface charge density remains constant. The analysis was extended by Prieve and Ruckenstein3 to the case of two different rigid planar surfaces bearing multiple ionizable functional groups, each of which is capable of donating or accepting a single proton. Since functional groups with multiple protons are common in practice, a more general treatment is highly desirable. Chang and Hsu4 studied the adhesion of a chargeregulated particle to a rigid surface, both being negatively charged, with the latter maintained at a fixed electrical condition. By neglecting the effect of hydrodynamic retardation, the presence of multivalent cations in the suspension medium was found to decelerate the rate of particle adhesion. The analysis was extended by Hus et
al.5 to take the effect of the time-dependent dissociation of the ionogenic groups on a cellular surface into account. Again, it was concluded that the presence of multivalent cations is disadvantageous to particle adhesion. As pointed out by Prieve and Ruckenstein,3,6 the effect of the multivalent cations on the electrostatic interaction between a cellular surface bearing ionogenic groups and a charged surface can be profound due to the requirement of continuous reequilibration of the ionogenic groups on the cellular surface as the separation distance between two surfaces varies. On the basis of the above discussions, it is apparent that the charge-regulated behavior of a particle surface is significant to the prediction of its electrostatic interaction with a charged surface. Neglecting this behavior may lead to an appreciable deviation. In the present work, the adsorption of a charge-regulated particle onto a rigid, charged surface is investigated; the effect of hydrodynamic retardation is taken into account. The particle contains an ion-penetrable membrane bearing dissociable functional groups. This is a generalization of the previous rigid surface models,3,4 which can be recovered as a limiting case of the present one. Furthermore, both acidic and basic functional groups are considered, and a general form for each is assumed which allows multiple-proton transfer among functional groups. Two classes of nonuniform functions for the distribution of the functional groups in the membrane are simulated.
* To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, July 1, 1997.
2. Modeling By referring to Figure 1, we consider a particle comprised of a rigid, uncharged core and an ion-penetrable membrane and a rigid, charged surface, both immersed in an a:b electrolyte solution. Let the dimensionless sizes of the uncharged core and the membrane be, respectively, X0 and d, X0 ) κr0, r0 and κ being, respectively, the radius of the uncharged core and the reciprocal Debye length. The membrane contains fixed charges due to the dissociation of the functional groups it bears. We assume that both acidic and basic functional groups are present.
(1) Healy, T. W.; Chan, D.; White, L. R. Pure Appl. Chem. 1980, 52, 1207. (2) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (3) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (4) Chang, Y. I.; Hsu, J. P. J. Theor. Biol. 1990, 147, 509.
(5) Hsu, J. P.; Kuo, Y. C.; Chang, Y. I. Colloid Polym. Sci. 1994, 272, 946. (6) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1978, 63, 317.
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© 1997 American Chemical Society
Adsorption of a Charge-Regulated Particle
Langmuir, Vol. 13, No. 16, 1997 4373
Ni)1 ) Mj,vSj,v ) 1, 2, v ) a, b v
(2)
where
M1,v ) N0,v/{(πRv)[(X0 + d)4 - X04] + 4π[(1 - RvX0)/3] × [(X0 + d)3 - X03]}, v ) a, b (2a)
{
M2,v ) N0,v/ (4π/3)[(X0 + d)3 - X03] +
}
(-1)2-u2! 4π [(X0 + d)u exp(Rvd) - X0u] , 3-u u)0 R v u! v ) a, b (2b) 2
∑
Figure 1. Schematic representation of the system under consideration.
2.1. Electrical Potential Distribution. Assuming that the linear size of a particle is sufficiently large compared to the separation distance between particle and surface, the electrical potential distribution of the system under consideration can be approximated by the Poisson equation
X02 d2ψ ) (g + iN), i ) 0, 1 dXc2 a + b
(1)
In this expression, ψ ) eφ/kBT, g ) [exp(bψ) - exp(-aψ)], Xc ) r/r0, κ2 ) e2a(a + b)n0a/0rkBT, and N ) NA (Ni)1 - 0 Ni)1 )/an . Here, φ denotes the electrical potential, N is + a the distribution of the fixed charges in the membrane, Ni)1 and N)1 + are, respectively, the densities of the negative fixed charges and the positive fixed charges in the membrane, e is the elementary charge, NA represents the Avogadro number, r and 0 are the relative permittivity of solution and the permittivity of a vacuum, respectively, n0a is the number concentration of cation in the bulk liquid phase, kB is the Boltzmann constant, T is the absolute temperature, i is a region index (i ) 0 represents the double-layer region, i ) 1 denotes the membrane), and r is the distance measured from the center of the particle. The boundary conditions associated with eq 1 are assumed as
(dψ/dXc)xcf1 f 0
(1a)
(dψ/dXc)xcf(1+d*)- ) (dψ/dXc)xcf(1+d*)+
(1b)
ψXcf(1+d*)- ) ψXcf(1+d*)+
(1c)
ψ ) ψ0 at Xc ) 1 + d* + H*
(1d)
where ψ0 is the dimensionless potential at the rigid surface, d* ) d/X0, and H* ) H/X0, H being the dimensionless closest surface-to-surface distance between particle and surface. 2.2. Distribution of Functional Groups. Suppose that the fixed charges in the membrane arise from the dissociation of the acidic and basic functional groups it bears. We consider two classes of nonuniformly distributed functional groups; one is linear and the other is a nonlinear function of the position variable. These distributions can be summarized by
Sj,v )
{
1 + Rv (X - X0), j ) 1 1 + exp[Rv(X - X0)], j ) 2, v ) a, b
(2c)
In these expressions X ) κr, Ra and Rb are, respectively, the parameters characterizing the distributions of acidic and basic functional groups, N0,a and N0,b are, respectively, the average space densities of acidic and basic functional groups, and j is an index representing the type of functional group distribution. Mj,vSj,v ≡ Ni)1 v , v ) a, b, are the concentration distributions of acidic and basic functional groups in the membrane, respectively. As shown in Appendix A, the total number of acidic functional groups is independent of Ra and j, and that of basic functional groups is independent of Rb and j. Some significant properties of Ni)1 are summarized in Appendix A. v 2.3. Dissociation of Functional Groups. Suppose that the dissociation of the functional groups in the membrane can be summarized by
a AHZn-a-n + H+ AHZ(n-1)a-(n-1)
Ka,n, n ) 1, 2, ..., Za (3a)
-(n-1)]+ -n)+ a BHZ(Zbb-n + H+ BHZ[Zbb-(n-1)
Kb,n, n ) 1, 2, ..., Zb (3b) where Za is the number of dissociable protons of an acidic group AHZa, and Zb is the number of absorbable protons of a basic group B, Ka,n, n ) 1, 2, ..., Za, and Kb,n, n ) 1, 2, ..., Zb, are the corresponding equilibrium constants. It can be shown that (Appendix B) the densities of negative i)1 fixed charges, Ni)1 - , and positive fixed charges, N+ , are, respectively,
Ni)1 - )
Ni)1 a Qa 1 + Pa
(4)
and
(
i)1 Zb Ni)1 + ) Nb
)
Qb 1 + Pb
(5)
where Pv and Qv, v ) a, b, are defined in Appendix B. 2.4. Motion of a Particle. Suppose that the movement of a particle toward the surface is governed by
m(dvp/dt) ) FD + FH
(6)
where m and vp denote, respectively, the mass and the velocity of the particle, and FD and FH are, respectively, the DLVO force and the hydrodynamic force. The hydrodynamic force can be calculated by7
4374 Langmuir, Vol. 13, No. 16, 1997
Hsu and Kuo
FH ) -6πµacλvp
(7)
AH2+ ) AH + H+
where µ is the viscosity of liquid, ac ) (X0 + d)/κ is the radius of the particle, and
λ)
∞
4
∑
n(n + 1)
× [sinh(β)] 3 n)0(2n - 1)(2n + 3) 2 sinh[(2n + 1)β] + (2n + 1) sinh(2β)
[
4 sinh2[(n + 1/2)β] - (2n + 1)2 sinh2(β)
]
- 1 (7a)
β ) cosh-1(1 + Hc)
(7b)
Hc ) H/κac
(7c)
Let φD be the DLVO potential, which comprises the electrical potential, φel, and the van der Waals potential, φvdw, i.e.,
(8)
φD ) φel + φvdw The electrical potential can be estimated by8
∫H∞∫l∞
φel ) 2πac3
c
FR (l′) dl′dl RT
(9)
The electrostatic interaction force, FR, is calculated by
FR an0akBT
)
1 1 bψ (e - 1) + (e-aψ - 1) b a
( )( ) a + b dψ 2X02 dXc
2
+i
∫ψ0 N dψ
(10)
For the present case, the van der Waals potential is8
φvdw )
[ ( )
]
Hc + 2 A132 1 (Hc + 1) ln 3RT 2 Hc Hc(Hc + 2)
(11)
where A132 is the Hamaker constant. The DLVO force can be evaluated by
( )( )
FD ) -
RT ∂φD ac ∂Hc
(12)
Substituting eqs 7 and 12 into eq 6 yields 2
d Hc 2
dt
+
9µλ dHc 3RT ∂φD )0 + 2 2ac F dt 4πac5F ∂Hc
(13)
where F is the density of the particle. Solving this equation gives the temporal variation of the position of the particle. The solution procedure is summarized in Appendix C. 3. Results and Discussion In a study of the electrostatic interaction between two amphoteric surfaces, Chan and co-workers9,10 considered the following mechanism for proton transfer: (7) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: Boston, 1983. (8) Hunter, R. J. Foundations of Colloid Science, Vol. I; Oxford University Press: London, 1989. (9) Chan, D. Y. C.; Healy, T. W.; Perram, J. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1 1976, 71, 1046. (10) Chan, D.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2844.
AH ) A- + H+ This is an example of eq 3a. The following compounds all contain multiproton functional groups: H3AsO4, H3PO4, HOOC(OH)C(CH2COOH)2, H4Y, and NH2C2H4NH2. The first two compounds are capable of forming a coordinate bond with membrane materials, and the rest are capable of undergoing a dehydration reaction with membrane materials. The dissociation of the functional groups of the membrane thus obtained can be described by eqs 3a and 3b. The simulated temporal variation in the dimensionless distance between particle and surface for the case when d ) 1 and j ) 1 at various Rv, v ) a, b, is shown in Figure 2. The corresponding variation in the electrostatic repulsion force between the particle and the surface as a function of separation distance is illustrated in Figure 3. The parameter Rv characterizes the distribution of the functional groups in the membrane. The smaller its value, the more homogeneous the distribution, and if Rv ) 0, the functional groups are distributed uniformly (i.e., independent of position) in the membrane. On the other hand, for a positive Rv, the greater its value, the more concentrated the functional groups are distributed near the outer boundary of the membrane. This leads to a greater electrostatic repulsion force between the particle and the surface, as suggested by Figure 3, and results in a slower movement of the particle toward the surface. As revealed by Figure 2, assuming that the functional groups are homogeneously distributed may lead to a deviation in the adsorption time on the order of 20%. The effect of the existence of a membrane on the movement of a particle is illustrated in Figure 4, and the corresponding electrostatic interaction force is shown in Figure 5. Note that if the thickness of the membrane approaches zero, the problem reduces to the interaction of two rigid surfaces. In this case the surface charge i)0 i)0 density of the particle is Ni)0 + - N- , where N+ can be i)1 calculated by eq B6 with Nb replaced by Ni)0 b , and i)1 Ni)0 evaluated by eq B5 with N replaced by Ni)0 a a , i)0 Nv , v ) a, b, being the surface concentrations of acidic and basic functional groups, respectively. We have
Ni)d ) v
2 ∫XX +d Ni)1 v (X)X dX,
1 κ(X0 + d)2
0
0
v ) a, b (14)
This expression means that all the functional groups are distributed uniformly over the particle surface. In this case, eq 1 with i ) 0 needs to be solved subject to eq 1d and
( ) dψ dXc
i)0 r0e(Ni)0 - - N+ ) ) 0rkBT Xcf1
(15)
As can be seen from Figures 4 and 5, for fixed Rv and d, a particle with type 2 functional group distribution (j ) 2) will experience a greater electrostatic repulsion force than a particle with type 1 functional group distribution (j ) 1). This is because the functional groups for the former are more concentrated near the outer boundary of the membrane than the latter. Figure 4 suggests that, for constant total numbers of acidic and basic functional groups, the rate of adsorption of a particle with a membrane to a rigid surface is faster than that of a rigid particle to a rigid surface. The difference in the adsorption time is on the order of 15%. As can be seen from Figure
Adsorption of a Charge-Regulated Particle
Figure 2. Temporal variation in the dimensionless separation distance between particle and surface for the case d ) 1 and j ) 1: (s) Ra ) Rb ) 0; (- - -) Ra ) Rb ) 0.5; (- - -) Ra ) Rb ) 1. Key: r ) 78, T ) 298.15 K, ψ0 ) -1, a ) 2, b ) 1, pH ) 7, ionic strength ) 10-3 M, N0,a ) N0,b ) 10-3 M, Za ) Zb ) 4, pKa,n ) (2 + n), pKb,n ) (4 + n), n ) 1, 2, 3, 4, r0 ) 10-6 m, A132 ) 10-20 J, F ) 103 kg/m3, µ ) 10-3 Pa s, initial H/κ ) 10-8 m, and vp ) 0 m/s.
Figure 3. Variation in the electrostatic repulsion force between particle and surface as a function of the dimensionless separation distance between particle and surface for the case shown in Figure 2.
5, the electrostatic repulsion force between a particle covered by a membrane and a rigid surface is smaller than that between a rigid particle and a rigid surface if the separation distance is greater than a critical value. The reverse is true if the separation distance is less than the critical value. In summary, the adsorption of a charge-regulated particle to a rigid surface is modeled. The former is covered by an ion-penetrable membrane carrying both acidic and basic functional groups, and a general form for each is assumed. The space distribution of these functional groups can assume essentially any continuous distribution, and the dynamic characteristic of the system under consideration is investigated. This is by far the most general model discussed in the literature. It simulates, for example, the behavior of a biological cell and a particle covered by an artificial membrane. Typical application for the former includes the aggregation between biological
Langmuir, Vol. 13, No. 16, 1997 4375
Figure 4. Temporal variation in the dimensionless separation distance between particle and surface for the same total numbers of acidic and basic functional groups. Ra ) Rb ) 1. Curve 1: d ) 1. Curve 2: d ) 0. Solid line: j ) 1. Dashed line: j ) 2. Key: same as that of Figure 2.
Figure 5. Variation in the electrostatic repulsion force between particle and surface as a function of the dimensionless separation distance between particle and surface for the case shown in Figure 4.
cells (microorganisms and plant cells, for example) and the attachment of cells such as leukocyte and cancer cells to specified sites, and that for the latter includes polymerinduced flocculation, a conventional process in wastewater treatments. We show that the presence of the membrane and the distribution of the functional groups inside can have a significant effect on the adsorption of a particle. For example, the deviation in the adsorption time due to the former can be on the order of 15%, and that due to the latter can be on the order of 20%. Acknowledgment. This work is supported by the National Science Council of the Republic of China under Grant NSC84-2214-E002-030. Appendix A For a spherical particle, eqs 2-2c suggest that Ni)1 v (X) increases with the dimensionless distance X for a positive Rv. This can easily be extended to the case that Ni)1 v (X) decreases with X through replacing (X - X0) in Sj,v
4376 Langmuir, Vol. 13, No. 16, 1997
Hsu and Kuo
by (X0 - X). The variation in the distribution of fixed charges is characterized by Rv. The total number of functional groups can be evaluated by 2 ∫XX +d Ni)1 v X dX ) N0,v 0
4π
0
(A1)
This indicates that the total number of functional groups is independent of Rv and j. If Rv f 0, the distribution of functional groups in the membrane approaches a uniform distribution. In this case, the asymptotic values of Mj,v are
M1,v f
M2,v f
3N0,v
, v ) a, b (A2a)
{4π[(X0 + d)3 - X03]}
, v ) a, b [8π(X02d + X0d2 + d3/3)]
[
3(X - X0,n)
un )
Ni)1 v
{
Ni)1 -
v ) a and u ) x, or
∑ um)
CH+ ) C0H+ exp(-ψ)
∑ nxn ) 1 + P
)
(B5)
a
Zb - 1
Zb
Ni)1 +
)
Zb(Ni)1 b
∑ yn) + n)1 ∑ [(Zb - n)yn]
-
n)1 Zb
) ZbNi)1 b
∑ (nyn)
n)1
(
) Ni)1 Zb b
)
Qb 1 + Pb
(B6)
where
where CH+ is the concentration of H+, which is determined by
(B1a)
0 + CH + is the bulk concentration of H , and
un-1Kv,n , n ) 2, 3, ..., Zv, CH+ v ) a and u ) x, or v ) b and u ) y (B2)
These expressions yield
(Kv,1Ni)1 v /CH+) , v ) a and u ) x, or (1 + Pv) v ) b and u ) y (B3)
where
Ni)1 a Qa
and
(A3b)
v ) b and u ) y (B1)
u1 )
H+
n)1
m)1
un )
Kv,w , n ) 2, 3, ..., Zv,
The densities of negative fixed charges, Ni)1 - , and positive fixed charges, Ni)1 , are, respectively, +
Zv
Qv )
Appendix B
(Ni)1 v
n
(A2b)
Denote xk, k ) 1, 2, ..., Za, and yk, k ) 1, 2, ..., Zb, as the -1)+ concentrations of AHZ , ..., AZa- and BHZ(Zbb-1 , ..., B, a-1 respectively. We have
Zy
(B3a)
v ) a and u ) x, or v ) b and u ) y (B4)
][ ]
0, X < X < X + d f ∞, X0 ) X + d0 0
u1CH+
, v ) a,b
H+
∏ w)1 C
1 + Pv
for j ) 2, v ) a, b. These expressions suggest that Ni)1 (j ) 1) increases linearly with X, and Ni)1 (j ) 2) v v approaches a delta function, δ(X0 + d), at the outer boundary of a particle.
Kv,1 )
Kv,w
∑∏ m)1 w)1 C
Substituting eq B3 into eq B2, we obtain
N0,v , j ) 1, 2 2 d(6X0 + 8X0d + 3d ) π v ) a, b (A3a) Ni)1 v
[ ] m
Za
N0,v
These expressions reveal that M1,v ) 2M2,v. Also, from eq 2c we have 2S1,v ) S2,v. Thus, Mj,vSj,v is constant; that is, both the linear and the nonlinear distributions of functional groups reduce to the same uniform distribution. If Rv f ∞,
f Ni)1 v
Zv
Pv )
[
n
Kv,j
∑ n∏ n)1 j)1 C
H+
]
, v ) a, b
(B6a)
Appendix C The numerical procedure for the resolution of the equations governing the movement of a particle toward a surface is summarized below. Step 1. At the kth stage, the time scale is tk. Assume that the dimensionless closest surface-to-surface distance between particle and rigid surface is Hk. Solving eq 1 with i ) 0 subject to eq 1d and a guessed value for the dimensionless differential potential at the rigid surface, ψs,g′, the potential distribution in the double layer is obtained. Step 2. Solving eq 1 with i ) 1 subject to eqs 1b,c, the potential distribution in the membrane phase is obtained. We check whether eq 1a is satisfied. If it is not satisfied, return to step 1 with a newly guessed ψs,g′. Step 3. FH and FR are evaluated by eqs 7 and 10, respectively, and φel and φvdw calculated by eqs 9 and 11, respectively. Step 4. Solving eq 13 subject to the condition H ) Hk-1 at t ) tk-1 yields the value H ) Hk,c. Here, we assume H ) 10-2r0 and vp,0 ) 0 at t0 ) 0; vp,k is the velocity of the particle at the kth stage. If Hk and Hk,c are inconsistent, return to step 1 with a newly guessed ψs,g′. The procedure is repeated until Hk and Hk,c are sufficiently close. Step 5. The time scale is advanced by tk+1 ) tk + ∆t (∆t is a prescribed time interval), and return to step 1. The numerical procedure for the case d ) 0 can be found in Hsu et al. (1994) with the number of functional groups calculated by eq 14. LA961018J
