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Dynamic Charge Regulation Model for the Electrostatic Forces between Ionizable Materials P. Maarten Biesheuvel Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received March 15, 2002 The electrostatic repulsion between similar surfaces with ionizable surface groups interacting across aqueous solutions is calculated for plane parallel and curved surfaces for the case in which the solution phase is at thermodynamic equilibrium, but the charged surface is not in equilibrium with the solution. Simulations are made for approaching and retracting oxidic surfaces for which the (equilibrium) surface chemistry is described by a standard 2-pK model. Population balances describe the concentration of each surface species as a function of the rates of the different reactions with the ions in solution. Approaching surfaces experience a higher repulsion than predicted by thermodynamic equilibrium, with the upper limit being the repulsion at a constant surface charge (at very high approach velocities). For retracting surfaces the electrostatic repulsion is lower than that for equilibrium and may decrease to zero when the surfaces are sufficiently discharged at contact and pulled apart fast enough. In that case, and with the attractive van der Waals force still operating, a net attractive force is predicted, which might explain the pull-off force (adhesion force) often measured in the atomic force microscope and the surface force apparatus.
Introduction When colloidal particles move rapidly toward or away from each other, dynamic aspects must be considered in the description of the interaction behavior. Based on earlier experiments,1 Frens and Overbeek2 argued that during a collision of two particles there might not be enough time for equilibrium to be attained between the particle surface and the solution phase. Overbeek,3 Lyklema,4 and Weaver and Feke5 quantitatively compared several mechanisms operating during particle approach and argued that relaxation of the double layers is often fast enough for thermodynamic equilibrium to be established in the solution phase. However, the adjustment of the surface charge by adsorption and desorption of ions may be too slow for thermodynamic equilibrium between the surface and the solution. In the limiting case of a very fast approach, the concentrations of the different surface species do not adjust at all, and the surface charge remains constant during the approach. 2-6 In this paper, the forces between plane and curved surfaces are modeled for surfaces that approach and retract with a constant velocity. A dynamic charge regulation model is used based on the assumption that surface reactions are rate limiting.3-5,7-9 The charging reactions are based on a standard 2-pK model with all surface charges assumed to reside in a single charging plane. Dynamic population balances are used instead of the equilibrium relations that are generally used but are (1) Frens, G.; Engel, D. J. C.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1967, 63, 418. (2) Frens, G.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1972, 38, 376. (3) Overbeek, J. Th. G. J. Colloid Interface Sci. 1977, 58, 408. (4) Lyklema, J. Pure Appl. Chem. 1980, 52, 1221. (5) Weaver, D. W.; Feke, D. L. J. Colloid Interface Sci. 1985, 103, 267. (6) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (7) Kijlstra, J.; Van Leeuwen, H. P. J. Colloid Interface Sci. 1993, 160, 424. (8) Shulepov, S. Yu.; Dukhin, S. S.; Lyklema, J. J. Colloid Interface Sci. 1995, 171, 340. (9) Mandralis, Z. I.; Wernet, J. H.; Feke, D. L. J. Colloid Interface Sci. 1996, 182, 26.
only valid at thermodynamic equilibrium, e.g., for a slow enough approach/retraction. The dynamic charge regulation model will be used to suggest that the finite rate of surface charge adjustment is a possible explanation of the pull-off force (adhesion force) that has been observed in the surface force apparatus (SFA),10-12 in the atomic force microscope (AFM),13-18 in the interfacial gauge, and between crossed filaments19 when close (metallic, ceramic, or polymer) surfaces in water with dissolved ions are pulled apart sufficiently fast. These forces are also observed for dissimilar surfaces15,18 and for metallic surfaces kept at a constant electrostatic potential.13,14 Pull-off forces are a function of the contact time (the hold time at closest contactsbefore retraction) and the pull-off rate (retraction velocity).15,18,20 Dynamic aspects of colloidal interaction have been studied theoretically by the following authors. Weaver and Feke5 model the interaction of plates and spheres held at a constant distance, assuming thermodynamic equilibrium in the double layer and a constant equilibrium potential. They incorporate the ion exchange between the Stern layer and the solution phase, as well as surface diffusion within the Stern layer. Serayssol and Davis21 (10) Ederth, T.; Claesson, P. M. J. Colloid Interface Sci. 2000, 229, 123. (11) Korchowiec, B. M.; Baba, T.; Minamikawa, H.; Hato, M. Langmuir 2001, 17, 1853. (12) Shubin, V. E.; Ke´kicheff, P. J. Colloid Interface Sci. 1993, 155, 108. (13) Serafin, J. M.; Gewirth, A. A. J. Phys. Chem. B 1997, 101, 10833. (14) Campbell, S. D.; Hillier, A. C. Langmuir 1999, 15, 891. (15) Vakarelski, I. U.; Ishimura, K.; Higashitani, K. J. Colloid Interface Sci. 2000, 227, 111. (16) Zauscher, S.; Klingenberg, D. J. J. Colloid Interface Sci. 2000, 229, 497. (17) Adler, J. J.; Rabinovich, Y. I.; Moudgil, B. M. J. Colloid Interface Sci. 2001, 237, 249. (18) Vakarelski, I. U.; Higashitani, K. J. Colloid Interface Sci. 2001, 242, 110. (19) Yaminsky, V. V.; Ninham, B. W.; Pashley, R. M. Langmuir 1998, 14, 3223. (20) Vigil, G.; Xu, Z.; Steinberg, S.; Israelachvili, J. N. J. Colloid Interface Sci. 1994, 165, 367. (21) Serayssol, J.-M.; Davis, R. H. J. Colloid Interface Sci. 1986, 114, 54.
10.1021/la025739w CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002
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describe a head-on collision of two deformable spheres and incorporate van der Waals and lubrication forces. For the electrostatic repulsion, they use a simple expression between spheres operating at constant potential conditions. Dukhin and Lyklema22 present equations for the interaction between flat plates with lateral ion movement. Together with Shulepov,8 they perform simulations for charge regulating spheres with the solution phase at thermodynamic equilibrium. The surface charge is considered an exponential function of the surface potential and the departure from equilibrium. The latter is a function of the concentration of charge-determining ions at the surface, which is a dynamic function of adsorption and desorption. Kijlstra and Van Leeuwen7 also model the interaction of spheres, with the solution phase at equilibrium and the surface charge a function of the equilibrium potential and a kinetic rate constant. In contrast to these authors, Krozel23 assumes that the ratelimiting step is the transport of ions in the double layer, not the adjustment of the surface charge by adsorption and desorption, and models the interaction force between charge regulating spheres based on a surface structural model, also incorporating surface diffusion. Whereas most of the former authors implemented the influence of the force field on the relative velocity of the particles, Mandralis et al.9 describe the electrostatic repulsion between spheres and plates that approach with a constant velocity. The solution phase is assumed to be at equilibrium and the surface charge is assumed to be a function of the overpotential. Like Krozel,23 Hsu et al.24 assume the transport in the double layer to be rate limiting. They24 make calculations for a step change in surface charge as well as for particle approach with a constant surface potential or charge. Lustfeld and Pohlmeier25 make calculations for an isolated cylinder for the case that ion transport in the double layer is the rate-limiting step, as well as for the case that surface kinetics is rate-limiting. They use population balances for the surface species and relate the surface charge to the surface concentrations of charged species. In this paper we will set up a dynamic model for the interaction of similar, parallel, charge regulating surfaces, plane as well as curved, that approach and retract with a constant velocity. The adsorption and desorption rates at the surface are considered rate-limiting, with the solution phase at thermodynamic equilibrium. Van der Waals forces are included; lubrication forces and surface diffusion are neglected. A single charging plane (also called Stern layer, outer Helmholtz plane, or adsorbed layer surface26) is considered at which the charging reactions take place (and the surface charge is defined). This is a simplified model which neglects that charge-determining ions typically reside in a plane closer to the surface than the (possibly hydrated) indifferent ions; as a consequence, the point of zero charge and the isoelectric point are not distinguished.27 When the van der Waals force is included, it is assumed that it emanates from a plane beyond the charging plane.2,11,12,20,26,28-31 The distance between the planes is on the order of a few tenths of a nanometer (0.20.3 nm,20 0.3 nm,30,31 0.35 nm,12 0.5 nm,11,29 0.6-1.2 nm,26,28 (22) Dukhin, S. S.; Lyklema, J. Langmuir 1987, 3, 94. (23) Krozel, J. W. J. Colloid Interface Sci. 1994, 163, 437. (24) Hsu, J.-P.; Kuo, Y.-C.; Tseng, S. J. Colloid Interface Sci. 1997, 195, 388. (25) Lustfeld, H.; Pohlmeier, A. J. Colloid Interface Sci. 2001, 239, 113. (26) Claesson, P.; Horn, R. G.; Pashley, R. M. J. Colloid Interface Sci. 1984, 100, 250. (27) Lyklema, J. Fundamentals of Interface and Colloid Science. Volume II: Solid-Liquid Interfaces; Academic Press: New York, 1995.
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Figure 1. Electrostatic repulsion between retracting surfaces for the equilibrium charge regulation (CR) model (upper line) and the dynamic CR model for five initial separations (d ) 0.05, 0.1, 0.2, 0.4, and 0.6 nm) and a retraction velocity of 200 nm/s. Data in Table 1.
0.4-1.0 nm2) and may be due to adsorbed ions,2,12,26,29 hydroxyl groups,30,31 a polymer gel layer,20 or surface roughness.32 Besides electrostatic and van der Waals forces, one additional force will always be observed in dynamic systems. The lubrication force3,16,33 (hydrodynamic force, or Spielman correction) is a direct consequence of the viscosity of the liquid that is pushed out from between the surfaces (on approach), or flows back (on retreat). The lubrication force can be calculated by solving the NavierStokes equation for the creeping flow regime and can be readily implemented in the analysis of forces between moving surfaces.16 Any remaining deviation of measured forces from that calculated for the equilibrium case must have a nonhydrodynamic cause. Because lubrication forces are independent of the type of surface and type and concentration of ions in solution (the viscosity of a liquid is mainly a function of temperature), dynamic aspects (e.g., adhesion forces) that turn out to be a function of these chemical aspects must also have a nonhydrodynamic cause. In this paper, the possibility will be investigated numerically that one such nonhydrodynamic effect, namely the limited rate of surface adsorption and desorption, is the cause of the hysteresis observed when surfaces in water are subsequently pushed together and pulled apart. Indeed, two similar and amphoteric (e.g., oxidic) surfaces will discharge when pushed together.6,31,34-38 In the limit where the charging planes come into full contact, the surface charge has become zero, because no (diffuse) ions in the (now vanished) solution phase are left to balance (28) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992; p 252. (29) Israelachvili, J. N.; Wennerstro¨m, H. Nature 1996, 379, 219. (30) Biesheuvel, P. M. Langmuir 2001, 17, 3553. (31) Biesheuvel, P. M.; Lange, F. F. Langmuir 2001, 17, 3557. (32) Considine, R. F.; Drummond, C. J. Langmuir 2001, 17, 7777. (33) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97. (34) Chan, D.; White, L. R. J. Theor. Biol. 1974, 48, 253. (35) Ettelaie, R.; Buscall, R. Adv. Colloid Interface Sci. 1995, 61, 131. (36) Behrens, S. V.; Borkovec, M. J. Phys. Chem. B 1999, 103, 2918. (37) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (38) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier, New York, 1948; p 74.
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Figure 2. Total interaction force (ESR + van der Walls) between surfaces retracting from an initial separation of d ) 0.05 nm. Equilibrium charge regulation model (upper line), equilibrium constant charge model (“CC equilibrium” with the surface charge based on the equilibrium CR model at d ) 0.05 nm, namely σ ) -0.408 mC/m2), and dynamic charge regulation model (“CR dynamic”) for three retraction velocities (v ) 60, 120, and 200 nm/s, top to bottom). Data in Table 1.
the surface charge. Though the surface charge decreases on approach, the electrostatic repulsion force increases continuously when the surfaces are forced together, having a maximum at contact.6,31 When the now discharged surfaces are pulled apart without the surfaces being able to respond to this change, the electrostatic repulsion decreases dramatically. With the attractive van der Waals force present, a net attractive force (pull-off force, or adhesion force) is now possible. When the charging planes do not come into full contact (always the case for curved surfaces), the pull-off effect will be less extreme but can still be quite significant; see Figure 2 and Figure 3b. For surfaces that are sufficiently far apart, the electrostatic surface potential can be assumed constant (independent of separation),which is the constant potential (CP) model. Also, the surface charge can be assumed constant, which is the constant charge (CC) model. However, in reality both the surface charge and the surface potential change as function of separation. This is the charge regulation model (CR), which relates surface charge and surface potential by assuming thermodynamic equilibrium between the surface groups and the ions in the solution next to the surface. The electrostatic repulsion predicted by the CR model is always intermediate between that predicted by the CC model (highest) and the CP model (lowest, see Figure 3a). The charge regulation model implements the reactions between surface species and ions in the solution. Current versions of the charge regulation model assume thermodynamic equilibrium between surface and solution; thus, they assume that these reactions are infinitely fast. However, the charge regulation model can be modified to describe the nonequilibrium case in which the adsorption and desorption rates are too slow for equilibrium to be established, as suggested by Hsu and Liu.39 The use of finite rates of adsorption and desorption implies that after an imposed change (e.g., due to approaching surfaces) it (39) Hsu, J.-P.; Liu, B.-T. J. Colloid Interface Sci. 1999, 217, 219.
Figure 3. Forces in plate-sphere geometry. (a) Electrostatic repulsion (ESR) for constant potential (CP) model, constant charge (CC) model and charge regulation (CR) models, at equilibrium and for approach and retraction at v ) (200 nm/s. (b) Total interaction force (ESR plus van der Waals) for the three CR models. Equilibrium CR model (T); dynamic approach (r, v ) -200 nm/s); dynamic retraction (f, top to bottom: v ) 60, 200, and 600 nm/s, initial condition based on CR equilibrium for d ) 0.05 nm). Data in Table 1.
will take time for the surface charge to attain the equilibrium value. For finite adsorption and desorption rates (as in the dynamic version of the CR model), the surface charge will be higher than predicted by the CR equilibrium model when the surfaces approach, resulting in a higher electrostatic repulsion (with the repulsion for a constant charge as the upper limit). However, whensafter equilibrationsthe two surfaces are pulled apart, the surface charge will be lower than at equilibrium, resulting in a lower electrostatic repulsion. As a consequence, hysteresis will be observed in the (AFM or SFA) force curves, counterclockwise in the typical diagrams in which separation is plotted on the positive x-axis and repulsion is plotted on the positive y-axis; see Figure 3b.
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Theory In this paper the dynamic interaction of plane parallel and curved ionizable surfaces interacting across water with added salt, acid, and/or base is described. The solution phase is at thermodynamic equilibrium, with the adsorption and desorption of ions at the surface being the ratelimiting step. For the solution phase the Debye-Hu¨ckel equation relates the electrostatic repulsion of plane parallel similar surfaces, PESR [Pa], to the surface charge, σ [C/m2] (eq 21 in ref 40), by
( (
PESR ) 2RTc∞ cosh
) )
F σ -1 r0RT κ sinh(κx)
(1)
with R the gas constant [8.3144 J/(mol‚K)], T temperature (here, 298.15 K), c∞ the ionic strength of the bulk solution [mol/m3], F Faraday’s constant [96 485 C/mol], r the relative permittivity of water [for field strengths
