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Deposition of Biocolloids on a Charged Collector Surface: An Ion-Penetrable Membrane Model Yung-Chih Kuo* Department of Chemical Engineering, Eastern College of Technology and Commerce, 110 Tung-Fang Road, Hu-Nei, Kaohsiung, Taiwan 829, Republic of China
Jyh-Ping Hsu and De-Fang Chen Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received October 10, 2000. In Final Form: February 28, 2001 The deposition of biocolloids onto a negatively charged, planar collector surface is modeled theoretically. Here, a biocolloid is mimicked by a rigid core covered by a negatively charged, ion-penetrable membrane. The results of simulation reveal that the following conditions lead to a faster rate of deposition: (1) thicker membrane (constant total amount of fixed charge); (2) lower fixed charge density; (3) smaller valence of the cations in the liquid phase; (4) lower surface potential of the collector surface.
1. Introduction Knowledge about the behaviors of biological cells, or biocolloids, is a prerequisite to a detailed understanding of the performance of blood systems. For instance, the release of thromboplastin by platelets is involved in their adhesion onto the wounded blood vessel to yield a platelet plug. The deposition of platelets on the wall of the blood vessel does not occur under normal circumstances. For the case of an injured blood vessel, its surface potential is lowered, which reduces the electrical repulsion force between suspended platelets and the vessel surface, the probability for the platelets to overcome an energy barrier is increased,1 and the deposition of platelets onto the vessel surface occurs. Moreover, since the successive release of adenosine diphosphate (ADP) by the deposited platelets leads to a lower surface potential of themselves, the release of ADP has the effect of attracting more platelets to vessel surface. However, if the number of deposited platelets exceeds a certain level, the electrical repulsion between suspended platelets and the vessel surface is restored by the termination of ADP release, and further deposition of platelets is prohibited. The migration and the subsequent attachment of biocolloids to a target surface in a real system, where the suspension medium is in a flow condition, is profound. In practice, the analysis is usually simplified by assuming a stagnant suspension and considering a flat surface. In a study of the adhesion of mouse tumor cells to glass overslips, Weiss and Harlos2 found that the time for cells to deposit on the surface is much longer than that estimated based on gravitational sedimentation. Ruckenstein et al.3 investigated the deposition of platelets onto a horizontal glass slide; both the sedimentation velocity of platelets and the probability of platelets to overcome the potential barrier were measured. Although the * To whom correspondence should be addressed. Fax: 886-76933254. E-mail:
[email protected]. (1) Srinivasan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1981, 79, 390. (2) Weiss, L.; Harlos, J. P. J. Theor. Biol. 1972, 37, 169. (3) Ruckenstein, E.; Marmur, A.; Rakower, S. R. Thromb. Haemostasis 1976, 36, 334.
experimental studies on the cellular interaction with a surface are ample in the literature,4-16 the relevant theoretical studies are still limited. Ruckenstein and Chen17 examined the attachment of cells to a rigid surface taking the effect of self-retardation of attached cells into account. A cell is represented by a rigid entity with constant surface properties. In the present study the rigid entity model of Ruckenstein and Chen is extended to the case of a hard core covered by an ion-penetrable membrane. The cell model considered here simulates more closely to a biological cell than the corresponding rigid entity model. The interaction force and energy between a biocolloid and a planar collector surface and the rate of deposition of particles to the surface are evaluated. The analysis performed here is also applicable to the evaluation of the biocompatibility of biomaterials such as dialyzers, vascular prostheses, and blood collectors. 2. Analysis A schematic representation of the problem under consideration is illustrated in Figure 1. We consider the interaction between a biocolloid, which comprises a rigid, (4) Grinnell, F.; Milam, M.; Srere, P. A. Biochem. Med. 1973, 7, 87. (5) van Wachem, P. B.; Beugeling, T.; Feijen, J.; Bantjes, A.; Detmers, J. P.; van Aken, W. G. Biomaterials 1985, 6, 403. (6) Sharefkin, J. B.; Watkins, M. T. In Techniques of Biocompatibility Testing; Williams, F. F., Ed.; CRC Press: Boca Raton, FL, 1986; p 96. (7) van der Valk, P.; van Pelt, A. W. J.; Busscher, H. J.; de Jong, H. P.; Wildevuur, Ch. R. H.; Arends, J. J. Biomed. Mater. Res. 1983, 17, 807. (8) Lydon, M. J. T.; Minett, W.; Tighe, B. J. Biomaterials 1985, 6, 396. (9) Takayama, H.; Tanigawa, T.; Takagi, A.; Hatada, K. Biomaterials 1986, 7, 11. (10) Reich, S.; Rosin, H.; Levy, M.; Karkash, R.; Raz, A. Exp. Cell Res. 1984, 153, 556. (11) Gerson, D. F. Biochim. Biophys. Acta 1980, 602, 269. (12) Corry, W. D.; Defendi, V. J. Biochem. Biophys. Methods 1981, 4, 29. (13) Hertl, W.; Ramsey, W. S.; Nowlan, E. D. In vitro Cell Dev. Biol. 1984, 20, 796. (14) Crouch, C. F.; Fowler, H. W.; Spare, R. E. J. Chem. Technol. Biotechnol. 1985, 35B, 273. (15) Kataoka, K. Hyomen 1983, 21, 385. (16) Weiss, L.; Blumenson, L. E. Cell Physiol. 1967, 70, 23. (17) Ruckenstein, E.; Chen, J. H. J. Colloid Interface Sci. 1989, 128, 592.
10.1021/la001424+ CCC: $20.00 © 2001 American Chemical Society Published on Web 04/26/2001
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they vary with the distance between a particle and the collector surface. The electrical interaction force between a particle and the collector, FR, can be calculated by19
{
∫0φ[exp(bφ) - exp(-aφ)] dφ}
FR a + b dφ 2 )+ CRT 2 dX
( )
(2)
where C ) ana0/NA and R is the gas constant. The scaled electrical potential, wel, can be evaluated by19
wel ) 2πRc3 Figure 1. Schematic representation of the system under consideration. X, D, and L denote the scaled distance, the scaled thickness of membrane, and the scaled distance between the membrane layer and the collector surface, respectively.
uncharged core and an ion-penetrable membrane layer of scaled thickness D, and a constant-potential, planar collector surface immersed in an a:b electrolyte solution. Without loss of generality, we assume that the membrane layer contains uniformly distributed negative fixed charge. Suppose that the spatial variation in the scaled electrical potential, φ, can be described by the Poisson-Boltzmann equation
d2φ [-exp(-aφ) + exp(bφ) - iN] ) a+b dX2
(1)
where φ ) eφ/kBT, N ) ZN0NA/ana0, κ2 ) e2a(a + b)na0/ 0rkBT, and X ) κr. In these expressions, φ, N0, and Z are the electrical potential, the density of fixed charge, and the valence of fixed charge, NA and e are the Avogadro number and the elementary charge, 0 and r are the permittivity of a vacuum and the relative permittivity, na0, kB, T, and r denote the number concentration of cations in the bulk liquid phase, the Boltzmann constant, the absolute temperature, and the distance, and i represents a region index (i ) 0, double-layer region, i ) 1, membrane phase). Since only the averaged effect resulted from the plane perpendicular to the r direction is considered, the fixed charge can assumed to be smoothly distributed from the viewpoint of r even in small regions. The boundary conditions associated with eq 1 are assumed as
φ f φ0 as X f D + L
(1a)
φ|XfD- ) φ|XfD+ ) φD
(1b)
|
|
dφ dφ ) dX XfD- dX XfD+
(1c)
φ f φc and (dφ/dX) f 0 as X f 0
(1d)
In these expressions, φ0 and φD are the scaled electrical potential of collector surface and that on the membrane-liquid interface, and L is the scaled separation distance between the membrane layer and the collector surface. Equation 1d suggests that the electrical field vanishes at the core-membrane interface. Here, we consider the general case in which the thickness of the membrane layer and that of the double layer can be comparable. This implies that φc may not reach the Donnan potential.18 Also, neither φD nor φc are fixed since
∫X∞∫l∞ c
FR(λ) dλ dl kBT
(3)
where Xc ) r/Rc, Rc being the particle radius, and λ and l are dummy variables. The scaled van der Waals potential, wVDW, can be evaluated by20
wVDW )
[
)]
-A132 1 + 2H H + ln 6kBT 2H(1 + H) 1+H
(
(4)
where H ) (X - D)/[2(κRc + D)]. The scaled gravitational potential, wg, can be expressed as
wg )
4 πRc3∆Fgh 3kBT
(5)
where ∆F is the difference of mass density between a particle and the surrounding fluid, g denotes the gravity, and h represents the closest distance between the membrane layer and the collector surface. The total potential, wTOT, is the sum of the electrical potential, the van der Waals potential, and the gravitational potential, that is
wTOT ) wel + wVDW + wg
(6)
On the basis of the conservation of the number of particles at the secondary minimum of the total energy curve, we have21,22
(
)
n2 dn2 2Rc2∆FgB ) 1dt 9µ n2m
5
dn1 dt
-
(7)
where n1, n2, and n2m are the number of particles per unit area deposited on the charged surface, that of particles per unit area at the secondary minimum, and that of particles per unit area at the secondary minimum corresponding to the monolayer coverage, respectively, t is time, B is the initial number concentration of particles in the liquid suspension, and µ denotes the viscosity of the liquid phase. The temporal variation in the number of attached particles is described by17
[(
|
d2wTOT dn1 ) n2kBT dt dh2
|
d2wTOT hmax
dh2 hmax
hmin
]
12µπ2Rc2
)
0.5
×
exp(-∆wTOT) (8)
where hmax and hmin denote, respectively, the distance from the collector surface to the position of the primary maximum and that to the position of the secondary minimum of the total interaction potential curve, and (18) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1994, 166, 208. (19) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1997, 195, 388. (20) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 184. (21) Maude, A. D.; Whitemore, R. L. Br. J. Appl. Phys. 1958, 9, 477. (22) Marmur, A. J. Colloid Interface Sci. 1985, 106, 360.
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Figure 2. Variation of the scaled electrostatic interaction force between particle and collector surface as a function of the scaled separation distance. Key: a ) 1, b ) 1, I ) 1 × 10-3 M, φ0 ) -0.3, N0D ) 2 × 10-7 M, Z ) -1, r ) 78, T ) 298.15 K. Curve 1, D ) 1.00; curve 2, D ) 0.95; curve 3, D ) 1.05.
Figure 4. Temporal variation of the percentage of deposition of particles on collector surface for the case of A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 particles/m2, B ) 2 × 1011 particles/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) -70 kg/m3. Key: same as Figure 2. Table 1. Primary Maximum Energy wTOT,max, Secondary Minimum Energy wTOT,min, and Energy Barrier ∆wTOT of the Potential Energy Curves casea
wTOT,max
wTOT,min
∆wTOT
1 2 3 4 5 6 7 8 9
12.090 14.832 11.071 14.522 11.011 13.752 12.841 14.704 11.312
-2.782 -2.667 -2.785 -2.669 -2.785 -2.406 -2.262 -2.668 -2.785
14.872 17.499 13.856 17.191 13.796 16.158 15.103 17.372 14.097
a Case 1, curve 1 of Figures 3, 6, 9, and 12; case 2, curve 2 of Figure 3; case 3, curve 3 of Figure 3; case 4, curve 2 of Figure 6; case 5, curve 3 of Figure 6; case 6, curve 2 of Figure 9, case 7, curve 3 of Figure 9; case 8, curve 2 of Figure 12; case 9, curve 3 of Figure 12.
Figure 3. Variation of the scaled total potential energy between particle and collector surface as a function of the scaled separation distance for the case of A132 ) 9 × 10-22 J. Key: same as Figure 2.
∆wTOT ) wTOT,max - wTOT,min, wTOT,max and wTOT,min, being the scaled total potential energy at the primary maximum and that at the secondary minimum, respectively. The initial conditions associated with eq 7 and 8 are assumed as
n1 ) 0, t ) 0
(8a)
n2 ) n2m ) 0.35nm, t ) 0
(8b)
where nm is the number of particles per unit area for the compact monolayer coverage on the collector surface. The condition specified in eq 8b is based on the result of Thakur et al.23 3. Results and Discussion The behaviors of the electrical repulsive force and the total interaction potential between a particle and the collector surface and the rate of deposition of particles onto the collector surface are investigated through numerical simulation. For illustration, the Hamaker constant (23) Thakur, S. C.; Brown, F. F.; Haller, G. L. Am. Inst. Chem. Eng. J. 1980, 26, 355.
and the temperature are fixed, and therefore, the van der Waals potential is a function of the separation distance between particle and collector surface only. As expressed in eq 5, the gravitational potential is also a function of the separation distance only. Thus, the effects of parameters on the system under consideration are mainly electrical. Here, other possible short-range repulsion forces are ignored for simplicity. In the following illustrations, the magnitude of the numerical error for the scaled electrical potential is 2 × 10-5. The electrical force is ignored for a large separation distance if the evaluated FR/CRT is smaller than 10-5. For the electrical energy, the magnitude of the numerical error in the evaluation of the double integrals in eq 3 is 10-6. The variation of the scaled electrical interaction force between a particle and the collector surface as a function of the scaled separation distance between them for various scaled membrane thickness D is presented in Figure 2. The corresponding variation in the total potential and the temporal variation of the percentage of deposition of particles are illustrated in Figures 3 and 4, respectively. In Figures 2 through 4 since the total amount of fixed charge in the membrane layer of a particle is constant, the thinner the membrane, the higher the fixed charge density. Figure 2 reveals that the thinner the membrane, the greater the electrical repulsion force. As mentioned above since both the van der Waals potential and the gravitational potential are constant at a fixed separation
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Figure 5. Variation of the scaled electrical interaction force between particle and collector surface as a function of the scaled separation distance. Key: a ) 1, b ) 1, I ) 1 × 10-3 M, φ0 ) -0.3, D ) 1, Z ) -1, r ) 78, T ) 298.15 K. Curve 1, N0 ) 2.0 × 10-7 M; curve 2, N0 ) 2.2 × 10-7 M; curve 3, N0 ) 1.8 × 10-7 M.
Figure 7. Temporal variation of the percentage of deposition of particles on collector surface for the case of A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 particles/m2, B ) 2 × 1011 particles/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) -70 kg/m3. Key: same as Figure 5.
Figure 6. Variation of the scaled total potential between particle and collector surface as a function of the scaled separation distance for the case of A132 ) 9 × 10-22 J. Key: same as Figure 5.
Figure 8. Variation of the scaled electrical interaction force between particle and collector surface as a function of the scaled separation distance. Key: b ) 1, I ) 1 × 10-3 M, φ0 ) -0.3, N0 ) 2 × 10-7 M, D ) 1, Z ) -1, r ) 78, T ) 298.15 K. Curve 1, a ) 1; curve 2, a ) 2; curve 3, a ) 3.
distance between particle and collector surface, the total potential is influenced only by the electrical interaction (or by the membrane thickness). As can be seen in Figure 3, the variation of the total potential as a function of the separation distance exhibits both a primary minimum and a primary maximum. The former arises from the fact that the van der Waals potential is much greater than the electrical potential as the separation distance becomes small. The positions of the primary maximum and that of the secondary minimum in Figure 3 appear at D values about 0.2-0.3 and 2.2-2.3, respectively. From Figures 2 and 3, one concluded that the larger the electrical force, the higher the primary maximum. This conclusion is also true for all the cases considered in the present study. Note that the x axis of Figure 2, L, excludes the membrane thickness, but the x axis of Figure 3, L + D, includes the membrane thickness. Note that the greater the repulsion force between a particle and the collector surface, the more difficult it is for the former to overcome the energy barrier. For the time evolution of the deposition process predicted by eq 8, the quantities (d2wTOT/dh2) at hmax, (d2wTOT/dh2) at hmin and the hmax are much insensitive than the value of exp(-∆wTOT) in the present study. The primary maximum energies, the secondary minimum energies, and
the energy barriers of the potential energy curves are listed in Table 1. As presented in Figure 4, where n0 denotes the initial number of particles per unit area in the suspension fluid, the deposition process is rapid in the initial stage, but the rate of deposition decreases with time due to the shielding retardation from the deposited particles. The fractional deposition reaches 35% after 6-30 h. Figure 4 also shows that the thinner the membrane layer of a particle, the slower the rate of deposition. Figures 3 and 4 also reveal that the greater the electrical repulsion, the slower the rate of deposition as expected. The variation of the scaled electrical interaction force between a particle and the collector surface as a function of the scaled separation distance between them for various N0 is shown in Figure 5. The corresponding total potential and the temporal variation of the percentage of deposition of particles on collector surface are illustrated in Figures 6 and 7, respectively. Figure 5 shows that the greater the N0, the greater the electrical repulsion force. This is expected since the greater the N0, the stronger the electrical field. Also, the greater the N0, the higher the primary maximum of the total potential curve as can be seen in Figure 6. Figure 7 suggests that the greater the
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Figure 9. Variation of the scaled total potential between particle and collector surface as a function of the scaled separation distance for the case of A132 ) 9 × 10-22 J. Key: same as Figure 8.
Figure 11. Variation of the scaled electrical interaction force between particle and collector surface as a function of the scaled separation distance. Key: a ) 1, b ) 1, I ) 1 × 10-3 M, N0 ) 2 × 10-7 M, D ) 1, Z ) -1, r ) 78, T ) 298.15 K. Curve 1, φ0 ) -0.300; curve 2, φ0 ) -0.305; curve 3, φ0 ) -0.295.
Figure 10. Temporal variation of the percentage of deposition of particles on collector surface for the case of A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 particles/m2, B ) 2 × 1011 particles/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) -70 kg/m3. Key: same as Figure 8.
Figure 12. Variation of the scaled total potential between particle and collector surface as a function of the scaled separation distance for the case of A132 ) 9 × 10-22 J. Key: same as Figure 11.
N0, the slower the rate of deposition of particles, as can be inferred from Figures 5 and 6. The variation of the scaled electrical interaction force between a particle and the collector surface as a function of the scaled separation distance between them for various cationic valences is presented in Figure 8. The corresponding total potential and the temporal variation of percentage of deposition of particles on collector surface are illustrated in Figures 9 and 10, respectively. Since both the membrane layer of a particle and the collector surface are negatively charged, more cations appear near and/or inside the membrane than anions. Note that for monovalent anions if ionic strength is fixed, the concentration ratio of cations for a cationic valence ratio 1:2:3 is 1:1/3:1/6. This implies that a cation with a higher valence has a lower shielding effect. Therefore, an electrolyte with a higher cationic valence leads to a higher electrical repulsion at a small separation distance between particle and collector surface. Since the electrical potential of the collector surface remains constant, the relaxation of charges from the collector surface will compensate the cationic effect on the electrical interaction force. As the separation distance increases, the order of the rate of decrease in the electrical interaction force for various
electrolytes is 3:1 > 2:1 > 1:1. Figure 8 corresponds to the behaviors of the electrical interaction force, elaborated in the above descriptions for various cationic valences. As can be seen in Figures 9 and 10, the smaller the cationic valence, the lower the energy barrier and the faster the rate of deposition. The variation of the scaled electrical interaction force between a particle and the collector surface as a function of the scaled separation distance between them for various collector surface potentials φ0 is shown in Figure 11. The corresponding total potential and the temporal variation of the percentage of deposition of particles on the collector surface are illustrated in Figures 12 and 13, respectively. We conclude from Figures 11 and 12 that the greater the absolute value of φ0, the greater the electrical repulsion and the higher the primary maximum. As shown in Figure 13, the greater the absolute value of φ0, the slower the rate of deposition of particles. As shown in eqs 4 and 5, the van der Waals potential and the gravitational potential have different distance dependence. The shorter the separation distance between two surfaces the greater the van der Waals attraction. For the above cases, if the separation distance is 7 times that of the double layer thickness, the gravitational
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Langmuir, Vol. 17, No. 11, 2001 3471 Table 3. Characteristic Times (h) for Various Scaled Membrane Thicknesss, D, and Fixed Charge Densities, N0 (M)a N0 (M)
D ) 0.95
D ) 1.00
D ) 1.05
1.8 × 10-7 2.0 × 10-7 2.2 × 10-7
2.8 6.9 13.3
6.7 16.8 32.3
11.8 29.7 54.6
a Parameters used: φ ) -0.3, I ) 1 × 10-3 M, Z ) -1, ) 78, 0 r T ) 298.15 K, a ) 1, b ) 1, A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 cells/m2, B ) 2 × 1011 cells/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) 70 kg/m3.
Figure 13. Temporal variation of the percentage of deposition of particles on collector surface for the case of A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 particles/m2, B ) 2 × 1011 particles/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) -70 kg/m3. Key: same as Figure 11. Table 2. Characteristic Times (h) for Various Ionic Strengths I (M) and Cationic Valencesa a:b
I ) 9 × 10-4 M
I ) 1 × 10-3 M
I ) 1.1 × 10-3 M
1:1 2:1 3:1
31.1 52.7 87.6
16.8 32.5 60.3
7.0 13.5 36.1
Parameters used: φ0 ) -0.3, D ) 1, N0 ) 2 × 10-7 M, Z ) -1, r ) 78, T ) 298.15 K, A132 ) 9 × 10-22 J, Rc ) 1 × 10-6 m, n0 ) nm ) 2 × 109 cells/m2, B ) 2 × 1011 cells/m3, µ ) 0.73 × 10-3 kg/m‚s, ∆F ) 70 kg/m3. a
potential is 10.7% of the van der Waals potential. If the separation distance is 50 times that of the double layer thickness, the van der Waals potential becomes 9.3% of the gravitational potential. This implies that the gravitational force dominates the deposition of colloidal particles toward a vessel surface if the separation distance is larger than 50 times that of the double layer thickness. Figures 4, 7, 10, and 13 suggest that the surface coverage of deposited particles on the collector surface approaches saturation after a certain period of time. For convenience, we define tc to be a characteristic time, the time required for 35% of saturation deposition. Although the curves in Figures 4, 7, 10, and 13 look rather similar, they are not likely to collapse into a master curve by the scale of tc. The variations of tc for various ionic strength and cationic valences are summarized in Table 2. Table 2 reveals that,
for a fixed ionic strength, the larger the cationic valence the larger the tc. This is because the larger the cationic valence the higher the energy barrier. Table 2 also shows that, for fixed electrolyte valences, the larger the ionic strength, the smaller the tc. This is because the larger the ionic strength the thinner the electric double layer, the shorter the range of the electrical repulsion, and the lower the energy barrier. The effects of the thickness of the membrane layer of a particle and that of fixed charge density on the characteristic time tc are summarized in Table 3. This table shows that a thicker membrane or a higher fixed charge density leads to a longer tc. This is because both of these conditions yield a higher energy barrier. 4. Conclusion The deposition of biocolloids onto a collector surface is studied theoretically. Here, a biocolloid is regarded as a rigid core covered by an ion-penetrable, charged membrane layer. The applicability of the model proposed is justified through numerical simulation. We show that a lower fixed charge density in the membrane layer, a lower surface potential of collector surface, a smaller cationic valence, and a higher ionic strength are advantageous to particle deposition. If the total amount of fixed charge in the membrane layer is constant, a particle with a thicker membrane layer leads to a faster rate of deposition; for constant fixed charge density, a particle with a thinner membrane yields a faster rate of deposition. A characteristic time, the time required to reach roughly the saturation of surface coverage, is defined for the assessment of the effects of relevant parameters on the rate of particle deposition. Acknowledgment. This work is financially supported by the National Science Council of the Republic of China. LA001424+
