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Surface Roughness and Surface Force Measurement: A Comparison of Electrostatic Potentials Derived from Atomic Force Microscopy and Electrophoretic Mobility Measurements Robert F. Considine† and Calum J. Drummond* CSIRO Molecular Science, Bag 10, Clayton South, Victoria 3169, Australia, and Cooperative Research Centre for Water Quality and Treatment, Bag 3, Salisbury, South Australia 5108, Australia Received December 8, 2000. In Final Form: September 4, 2001 The force of interaction between pairs of silica colloids in aqueous electrolyte has been measured using an atomic force microscope . The measured force of interaction has been compared to predictions based on the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory, and only partial agreement has been obtained. Specifically, the force-separation data is entirely repulsive, failing to manifest the attraction predicted from the van der Waals interaction. The experimental decay length of the repulsive force was found to be in good agreement with the predicted Debye length, implicating an electrical double layer interaction. At low electrolyte concentration (5 nm. The error in ψfit has been reported in terms of the error associated with the two radii, R1 and R2, the spring constant, k, the standard deviation among at least five repeat force-separation curves, ∆δ, and the result of the boundary condition (constant potential or constant charge) chosen in the DLVO fit, ∆fit. The value of ∆fit corresponds to the difference in ψfit obtained for the boundary condition of constant charge compared to constant potential and was determined for each individual force-separation curve. It is common place in AFM force experiments to associate an accuracy of around 10% for values of R and k; accordingly the fractional error in ψfit due (9) Senden, T. J.; Drummond, C. J.; Kekicheff, P. Langmuir 1994, 10, 358. (10) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403. (11) Jaschke, M.; Butt, H.-J. Rev. Sci. Instrum. 1995, 66, 1258. (12) Radmacher, M.; Cleveland, J. P.; Fritz, M.; Hansma, H. G.; Hansma, P. K. Biophys. J. 1994, 66, 2159. (13) Considine, R. F.; Dixon, D. R.; Drummond, C. J. Langmuir 2000, 16, 1323.
Considine and Drummond to R and k can be represented as14
∆R,k ) ψfit
x( ) ( ) ( ) ∆R1 R1
2
+
2
∆R2 R2
+
2
∆k k
)
x(0.1)2 + (0.1)2 + (0.1)2
(1)
which yields ∆R,k ) 0.1732ψfit, and the total error in ψfit has been calculated according to
∆ψfit ) x(0.1732ψfit)2 + (∆fit)2 + (∆δ)2 ψfit
(2)
DLVO Calculations. In DLVO calculations, we have assumed that the van der Waals interaction can be described by
F/R )
-AH
(3)
6H2
where AH is the nonretarded Hamaker constant (ca. 0.8 × 10-20 J for a silica-silica interaction),15 and H is the intersurface separation. We have also assumed that the electrical double layer interaction can be computed as a function of separation by employing the nonlinearized Poisson-Boltzmann equation. Accordingly, the mean electrostatic potential, ψ(x), valid for the interaction between two charged planar surfaces in symmetric electrolyte located at a distance d apart, can be written in the nondimensional form:16
d2y(ξ) dξ2
) sin(h)y(ζ)
(4)
where y ) (evψ/kBT) is the potential scaled by the thermal potential, (kBT/ev), with kB being the Boltzmann constant, T being the absolute temperature, v being the valence of the symmetric electrolyte (v:v), and e being an electronvolt (1.602 × 10-19 J). The scaled variable, ξ ) κx, gives the range of the electrical interaction, where κ-1 is the Debye screening length:
κ-1 )
(
)
8πnv2e2 kBT
-1/2
(5)
Here is the dielectric constant of the solvent, and n is the number density of ions in the bulk electrolyte. An important result of the Debye screening length is that the range of the electrical double layer interaction varies as a function of the ionic strength. It should also be noted that the magnitude of the electrical double layer interaction can be expressed as a function of the sign and magnitude of the electrostatic potential of the surfaces.
Results and Discussion Silica Sphere Topography. A Born-contact image (obtained with an AFM tip) of a representative silica particle is shown in Figure 1. On inspection of the image, it is immediately apparent that the silica particles are quite smooth and that any surface features are small as compared to the radius of curvature. The root-mean-square (RMS) of an xy plane-fitted central region (1 × 1 µm2) of the sphere is around 1 nm with a maximum vertical range of around 5 nm. The surface topography of five different spheres has been assessed, and, in general, values of RMS were found to lie in the range 1-2 nm and values of the maximum vertical range were found to be less than 20 nm. The inset of Figure 1 shows a 0.4 × 0.4 µm2 deflection image of the sphere surface, and it is apparent that the (14) Pentz, M.; Shott, M. Handling Experimental Data; Open University Press: Philadelphia, 1992. (15) Senden, T. J.; Drummond, C. J. Colloids Surf., A 1995, 94, 29. (16) McCormack, D.; Carnie, S. L.; Chan, D. Y. C. Langmuir 1995, 11, 177-196.
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Figure 1. Born-contact image of a representative silica sphere (3 × 3 µm2). Inset: 0.4 × 0.4 µm2 deflection image of central region of the sphere shown.
Figure 3. Representative force map (4 × 4 µm2) of the interaction between a pair of silica spheres (R1 ≈ R2 ) 1.6 µm). Maps correspond to (a) the height and (b) the experimental decay length. Data obtained in 2 mM KNO3 and pH 6.89.
Figure 2. Representative force-separation curve obtained for the interaction between two silica spheres (R1 ) 3.7 µm, R2 ) 0.6 µm). Data obtained in 1 mM KNO3 and pH 8.25. Filled symbols correspond to the force data on particle approach, and unfilled symbols correspond to the force data on separation.
roughness is distributed as gentle undulations in the sphere surface. Sphere-Sphere Interaction and Force Map. A representative force-separation curve for the interaction between a pair of silica spheres in 1 mM KNO3 is presented in Figure 2. The force-separation curve is entirely repulsive, and the repulsion was detected in every forceseparation measurement, regardless of electrolyte concentration or pH. The absence of a short-range attraction (originating from van der Waals interactions) is consistent with the result of incompressible surface asperities (see below). The repulsive force was analyzed in terms of the decay length, defined as the gradient of the repulsive force on the natural log(F/R) versus separation plot. A decay length force map, along with the corresponding height map, has been presented in Figure 3. The figure corresponds to a 16 × 16 matrix of height and force data, across a 4 × 4 µm2 scan, for the interaction between two spheres
of approximately equal radii (1.6 µm) in 2 mM KNO3. The decay length values are in the range 6-8 nm which is in general agreement with the anticipated Debye length (6.8 nm, see below). The decay length contrast over the sphere is not generally large, except for the few peaks distributed around the bottom right-hand corner of the map. On the other hand, significant contrast in the decay length was recently reported for force maps of the interaction between a functionalized AFM tip and a spherical bio-colloid (R ≈ 2 µm).13 Specifically, decay lengths obtained from force curves located around the perimeter of the bio-colloid were found to be significantly larger than those located in the central region, and it was proposed that this was due to the non-normal approach of the pyramid-shaped tip at the edge of the sphere. However, in the case of the interaction of two spheres of similar radii, the apparent contrast in the decay length may be reduced due to the location of the sphere-sphere contact point being different from the line of approach (see Figure 4). In any event, hereafter only centrally aligned force-separation curves have been analyzed to make comparisons with DLVO calculations (which are analogous to the co-axial alignment situation). van der Waals Interaction. The absence of an attractive region in the force-separation data is inconsistent with the anticipated van der Waals attraction (expected to dominate over separations 5 mM. It is proposed that the plateau in ψfit is associated with the absence of near separation data in these solutions. Numerous calculations of the influence of surface roughness on DLVO surface forces are reported in the literature.23-27 However, the results of these calculations are not all applicable for the interaction between surfaces that possess incompressible asperities. For such calculations, the minimum separation on surface approach should be defined as asperity-asperity contact, not based on separation data referenced to core-core contact as is often
Considine and Drummond
Figure 10. DLVO fitting at 0.1 M KNO3. The theoretical curve has been shifted by -0.4 nm on the separation axis to obtain agreement between ψfit and ζ.
the case.23 Elimelech and O’Melia27 assumed a separation corresponding to a zero separation at asperity-asperity contact and showed that the presence of surface asperities substantially reduced the height of the primary energy barrier to the deposition of particles onto a flat plate. The results of Elimelech and O’Melia27 are consistent with the reduction in ψfit (compared to ζ) observed at high ionic strength, where the Debye length approaches the roughness range. On the other hand, very few reports are available in the literature regarding the influence of surface roughness on the ζ-potential. Some calculations have been performed on the result of a charge layer penetrable to hydrodynamic flow,28,29 although these have been primarily based on the concept of a “hairy layer” extending from the surface. The calculations have generally focused on describing the anticipated dependence of the electrophoretic mobility on the ionic strength, since reducing the Debye length is expected to reveal the various charges located at different depths of the charged surface layer. The present study reveals that the electrostatic potentials derived by electrophoresis are consistently greater (in terms of absolute magnitude) than the corresponding ψfit potentials at high ionic strength. However, in the present situation it is unclear as to how close the measured ζ-potential is to the actual diffuse layer potential (which corresponds to the electrostatic potential at the average plane of charge for rough surfaces). In light of this, more theoretical investigations are required regarding the influence of surface roughness (in terms of incompressible surface asperities) on the ζ-potential. Summary and Conclusions The AFM has been used to measure surface topography and force-separation curves between pairs of silica colloids in aqueous electrolyte. The spheres have been shown to (23) Czarnecki, J.; Dabros, T. J. Colloid Interface Sci. 1980, 78, 25. (24) Kostoglou, M.; Karabelas, A. J. J. Colloid Interface Sci. 1995, 171, 187. (25) Suresh, L.; Walz, J. L. J. Colloid Interface Sci. 1996, 183, 199. (26) Walz, J. Y. Adv. Colloid Interface Sci. 1998, 74, 119. (27) Elimelech, M.; O’Melia, C. R. Langmuir 1990, 6, 1153. (28) Chow, R. S.; Takamura, K. J. Colloid Interface Sci. 1988, 125, 226. (29) Donath, E.; Budde, A.; Knippel, E.; Ba¨umler, H. Langmuir 1996, 12, 4832.
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be smooth relative to the radius of curvature, but possess nanometer-scale surface roughness. The force curves have been analyzed in terms of the experimental decay length, and maps of the decay length reveal less contrast than corresponding AFM tip versus sphere decay length maps, and this has been attributed to the differing geometry of the force probes. Comparison of the dual silica colloid force-separation data with DLVO theory reveals an absence of the van der Waals attraction, and it is proposed that surface roughness attenuates such short-range forces. The short-range steric and hydration forces proposed previously in early studies in systems of ideal geometry are also proposed to be attenuated by surface roughness. The experimental decay lengths have been shown to be in good agreement with the corresponding Debye lengths, implicating the origin of the repulsive force to be an electrical double layer interaction. Comparison of the fitted diffuse layer potential with the ζ-potential reveals good agreement below 5 mM KNO3, with a divergence being
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measured at higher ionic strengths. The divergence has been rationalized in terms of the heightened importance of surface roughness as the Debye length shortens. The results of the present study highlight the importance of the concept of apparent separation in the interaction between nonsmooth surfaces and also that conclusions drawn from ideal geometry experiments may not always be applicable to practical systems such as those involving the interaction of nonideally shaped colloids. Acknowledgment. R.C. gratefully acknowledges the support and friendship of S. Gillies. R.C. is the recipient of an Australian post-graduate research award. We also thank the University of South Australia and the Cooperative Research Centre for Water Quality and Treatment for support. LA0017227
