Determining surface potential of the bitumen-water interface at nanoscale resolution using atomic force microscopy.
When two surfaces/interfaces (of oil droplets, solids, bubbles or their combination) approach each other, colloidal forces become significant at separations of a few tens of nanometres. These forces arise from molecular interactions between charged and uncharged atoms or molecules of the interacting bodies and the surrounding medium. According to the DLVO model proposed by Derjaguin and Landau (1941) and Verwey and Overbeek (1948), the net interaction between two surfaces/interfaces in a liquid medium is the arithmetic sum of van der Waals and electrostatic (electric double layer) forces. The van der Waals forces arise from dipole-dipole (Keesom), dipole-induced dipole (Debye) and/or instantaneous dipole-dipole (London) interactions. In a polar solvent, like water, most interfaces become electrically charged due to either dissociation of ionizable groups present at the interface or preferential hydration of lattice ions or adsorption/reaction of ions at the interface, and they repel or attract each other through Coulombic interactions.
Measurements of colloidal forces have been carried out with surface force apparatus (Israelachvili and Adams, 1976; Israelachvili, 1992), atomic force microscope (Ducker et al., 1991, 1992), and other instruments (Craig, 1997; Froberg et al., 1999), providing justification for the DLVO model. In the quantification of van der Waals forces, Hamaker constants (1) for all three phases involved in the interactions are needed. For many solids and liquids, Hamaker constants have been determined (Israelachvili, 1992; Bergstrom, 1997) and as the result, the magnitude and range of van der Waals forces are usually predictable, at least roughly, for many systems without further experimentation. Long-range electrostatic forces can be calculated if both the solution chemistry of the intervening liquid and surface potentials (or surface charge densities) of the interacting surfaces are known. The composition of the intervening liquid is needed to calculate the thickness of the electric double layer (Debye length). (2) Since most applications/experiments involve aqueous electrolyte solutions of known ionic strength, the electric double layer thickness can be calculated based on the concentration of electrolytes (Israelachvili, 1992). Surface potentials, on the other hand, are unpredictable for most of the systems and must be measured experimentally.
The effective surface potential, called zeta potential, is commonly determined from electrophoretic mobility measurements for particles moving in the solution owing an electric field applied between two electrodes (Adamson and Gast, 1997; Masliyah and Bhattacharjee, 2006). The situation can be reversed and the solution can be forced to flow through a plug of packed particles (Adamson and Gast, 1997; Masliyah and Bhattacharjee, 2006); the zeta potential of the particles is calculated from the streaming potential measured by this technique. Measurements of streaming potential conducted for a solution flow between two macroscopic parallel plates had also been demonstrated (Ramachandran and Somasundaran, 1986; Scales et al., 1990).
A significant limitation of electrophoretic mobility and streaming potential measurements, both classified under electroosmosis techniques, is that only an average value of the zeta potential/streaming potential is detected--regardless of whether the surface charge distribution is homogeneous or otherwise. However, in real-world situations, nearly all solids (and liquids) of technological significance exhibit surface heterogeneity. (3) The system of particular interest to us is the interface between water and bitumen. (4) The surface charge distribution at the bitumen-water interface, as in many other systems, can very likely be heterogeneous.
To detect heterogeneities in surface charge, analytical tools which provide accurate and spatially resolved information about material surface potential--particularly at microscopic and submicroscopic resolutions--are needed. Atomic force microscopy (AFM) is capable of characterizing solid surfaces at these desired length scales by providing three-dimensional images of surfaces and probing materials properties such as adhesion, elasticity, friction, and magnetic properties (Wiesendanger, 1991; Drelich and Mittal, 2005). The AFM is also commonly used in the measurement of long-range and short-range surface forces for microscopic particles through the colloidal probe technique (Ducker et al., 1991; Butt et al., 2005; Liu et al., 2003, 2005). Nano-sized probes having desirable chemical functionality are also frequently used in surface force measurements; such a technique is known as chemical force microscopy (Vezenov et al., 2005). Analysis of AFM-measured forces, based on the DLVO theory, can be used to either compare model predictions with experimental results, or to calculate parameters such as the Hamaker constant, surface potentials, or surface charge densities. In this communication, we use the DLVO theory to calculate, from AFM-measured colloidal forces, surface charge densities and surface potentials for the bitumen-water interface. We will demonstrate that these charge densities/potentials have significant variations along the interfacial plane, thus providing evidence for sub-microscopic charge heterogeneities of the bitumen surface and pointing to the existence of nanometresized domains of differing surface potentials. This research is motivated by the need to understand the mechanism(s) that control the stability of bitumen suspensions in aqueous media (which, in turn, is vital to the recovery of crude oil from the Canadian oil sands (Masliyah et al., 2004)). As will be demonstrated in a related paper (Esmaeili et al., 2007), surface charge heterogeneities can have considerable influence on the probability of droplet-droplet coalescence. (5) In addition to the traditional theory of Fuchs (Overbeek, 1952), surface heterogeneities may indeed be another fundamental mechanism that contributes to the probabilistic nature of coalescence. In this paper, we will focus on demonstrating the existence of bitumen surface heterogeneities through direct AFM measurements.
Vacuum-distillation-feed bitumen made of 83.1 wt.% carbon, 10.6 wt.% hydrogen, 0.4 wt.% nitrogen, 1.1 wt.% oxygen and 4.8 wt.% sulphur was received from Syncrude Canada Ltd. Bitumen substrate was prepared by spin-coating a thin layer of bitumen on a cleaned silicon wafer with a P6700 spincoater (Specialty Coating Systems Inc.) at 6000 rpm using 1-3 wt.% bitumen-in-toluene solution.
Topographical and phase images of the bitumen surface were obtained using the intermittent contact mode on a Digital Instruments Nanoscope E AFM. Before imaging, the bitumen substrate was immersed in water for 1 h in order to facilitate migration of polar organic compounds to the bitumen surface, and then residual water was removed with a stream of compressed nitrogen. Images with 512 x 512 resolutions were captured using Veeco RTESP10 cantilevers at a scan rate of 1 Hz. The cantilevers have a resonance frequency of about 300 kHz, spring constant of 20-80 N/m and radius of tip curvature of about 10 nm. The driving frequency of a cantilever during imaging was kept at about 85% of the cantilever's resonant frequency. (6) Example of the recorded AFM images is shown in Figure 1.
[FIGURE 1 OMITTED]
Colloidal force measurements between an AFM cantilever tip and bitumen were performed using a Nanoscope E AFM (Digital Instruments Inc.) in a fluid cell. This commonly used surface force measurement technique and the interpretation of recorded results are well described in the literature (Butt et al., 2005; Drelich and Mittal, 2005) and will not be repeated here. Briefly, the forces exerted on a tip by the bitumen surface were measured through changes in the deflection of a flexible cantilever. These measurements were carried out as a function of normal tip-bitumen separation. Sharp tips were used instead of colloidal probes to improve the lateral resolution in recording a variation of the surface forces. Triangular shaped contact-mode cantilevers (NP, Veeco) with pyramidal silicon nitride tips were used and the tips with a spring constant of 0.12 N/m were chosen for the force measurements. The cantilevers were cleaned by UV irradiation for 30 min prior to the experiments.
The deflection of the AFM cantilever was monitored by a laser-photodiode system. The cantilever is treated as a simple Hookean spring with its deflection proportional to the force acting on the tip. The value of the spring constant provided by the manufacturer was used in this study. About 10% variation in the spring constant value was reported for similar triangular cantilevers taken from the batch of cantilevers (Veeramasuneni et al., 1996). Measurements of surface forces between AFM tips and substrates were carried out in water (pH 6.0-6.5) and 1 mM KCl solution (pH 9.0) in the period of 0.5-1.5 h after the fluid cell was filled with water/electrolyte solution.
In colloidal force measurements, the silicon nitride AFM tip was moved stepwise to various locations of the substrate and then along x-axis at 10 nm per step using an operator-controlled offset adjustment. Surface forces between the AFM tip and bitumen were measured at each offset location. A schematic of this approach is shown in Figure 2. The force curves were analyzed with the SPIP software (Image Metrology, Lyngby, Denmark), which translates the cantilever deflection-piezo extension/retraction data to force-separation profiles. The resulting force-separation curves were obtained after the baseline correction. The force-separation graphs in this paper present individual probe-substrate approaching curves and do not represent averaging of the data points from different tests.
RESULTS AND DISCUSSION
Heterogeneous Nature of the Bitumen Surface
The bitumen surface, after its exposure to water for about 1 h, was imaged using the AFM Tapping mode, also known as intermittent contact mode. In this mode, the cantilever is forced to oscillate at a large amplitude and a frequency that is close to the cantilever's resonant frequency, gently tapping the substrate surface (Magonov and Reneker, 1997; Cleveland et al., 1998). The cantilever motion is characterized by its amplitude and phase relative to the driving oscillator. During tapping the substrate the amplitude of cantilever oscillation changes above asperities and valleys of rough surfaces. AFM uses this change in amplitude to track the surface topography. A phase in oscillation of cantilever changes when different amount of energy is dissipated by the cantilever during substrate tapping, and can be used to track the surface regions of different composition. A variation in amount of energy dissipated can be caused by local changes in either substrate viscoelastic properties or probe-substrate interactions or both. Distinction of these effects is currently unreachable for any ill-defined substrate such as bitumen, for which the surface composition is not well defined.
Figure 1 shows the topographic (height) and phase images of the Athabasca bitumen sample used in this study. The surface roughness of spin-coated bitumen, defined by the root-mean-square (RMS) roughness parameter, was RMS = 1-3 nm, depending on the area that was scanned, with asperities usually shorter than 2-3 nm.
[FIGURE 2 OMITTED]
Height and phase images were compared in order to judge whether a phase contrast is due to topographic effect or due to a difference in materials. Topographic slopes and edges are often highlighted in phase images and the contrast can be a consequence of imperfect tracking of the surface. As shown in Figure 1, the height image is in fact different from the phase image, indicating little, if any, effect of surface roughness on the contrast in the phase image. The contrast in the phase image points to coexistence of surface domains of different viscoelastic and/or adhesive (towards the [Si.sub.3] [N.sub.4] tip) properties in the bitumen surface structure. Soft and sticky/adhesive domains are more dissipative and cost more energy, have lower phase and are displayed as dark relative to stiff, less adhesive domains, which have higher phase and appear bright in images rendered by the NanoScope software.
The domains of seemingly stiffer, less adhesive components represented by the brighter disk-shaped domains in the phase image of Figure 1 have a size of about 20 to 30 nm. A similarity in the morphology and size of these domains to surface structure of compressed asphaletenes imaged by Zhang et al. (2003) suggests that they might represent asphaltene aggregates, likely low-molecular weight asphaltenes.
Colloidal Forces and Calculated Surface Charges
The silicon nitride AFM tip was placed over the bitumen surface in water or 1 mM KCl solution and then moved stepwise along x-axis at 10 nm per step using an operator-controlled offset adjustment. Colloidal forces between the AFM tip and bitumen were measured at each offset location. A schematic of this approach and an example of the recorded force-separation curve are shown in Figure 2.
Figure 3 shows examples of colloidal force-separation curves recorded at different locations in both water and 1 mM KCl solution. All long-range forces recorded were repulsive due to similar sign charges of interacting surfaces. The bitumen is negatively charged at pH 6.0-6.5 and pH 9.0 (Liu et al., 2004). The isoelectric point for [Si.sub.3][N.sub.4] is located at pH 6 to 7, depending on the treatment of the surface (Zhmud et al., 1999), and therefore, the charge of the tip is probably neutral or close-to-neutral at pH 6.0-6.5 and negative at pH 9.0. Weak attractions between the AFM tip and bitumen were sometimes observed at separations less than 5 nm as a result of attractive van der Waals forces; the Hamaker constant for the bitumen-water-silicon nitride system is [A.sub.123] = 2.7 x [10.sup.-20] J.
Figure 3 also shows the fitting theoretical curves plotted based on the equations presented in the Appendix. In the fitting practice, we found that for the system studied, the constant potential condition resulted in attractive forces, which did not agree with the experimental results. All the fittings were therefore performed using the constant charge density condition. For the results obtained in water, the surface charge density of the AFM tip was assumed to be much smaller than that of the bitumen surface. For the results in 1 mM KCl solution at pH 9.0, the tip surface charge density of -0.012 C/[m.sup.2] was used as one of the fitting parameters. Although the tip's surface charge density value appears acceptable in view of the results presented by Zhmud et al. (1999), future research should rely on tips with experimentally defined surface charge characteristics. We also assumed homogeneity of the Hamaker constant in our theoretical analysis of the experimental force curves. Only a little variation in the Hamaker constant is expected among bitumen surface domains of different interfacial characteristics. This variation however, should have a negligible effect on the shape of the force curves at distances exceeding 5-10 nm; the tip-bitumen interactions at long-range separations are controlled by electric double layer forces.
The calculated surface charge density and surface potential values for the bitumen-water interface are shown in Figures 4 and 5. The surface charge density of the bitumen-water interface was used as one of the fitting parameters in describing the tip-water-bitumen colloidal forces, and the surface potential was calculated from the surface charge density value using the Graham equation (see Appendix).
As shown in Figure 4, the bitumen surface charge density in water varied from about -0.002 to -0.004 C/[m.sup.2], and the corresponding surface potential was -90 to -130 mV. The surface charge density of bitumen in 1 mM KCl at pH 9.0 changed from -0.005 to -0.022 C/[m.sup.2], and the calculated bitumen surface potential varied from about -45 to -110 mV. Liu et al. (2004) conducted electrophoretic mobility studies for the same bitumen in 1 mM KCl solution and reported zeta potential values ranging from -70 to -75 mV at pH 6.0 to 6.5 and about -80 mV at pH 9.0. The difference of 30 to 40 mV between the AFM-determined surface potential and zeta potential calculated from the electrophoretic mobility measurements for bitumen in water at pH 6.0-6.5 can be partially attributed to differences in ionic strength of aqueous phase used in both studies and partially due to a crude approach used in our study. The agreement between the zeta potential value and surface potential determined with the AFM in 1 mM KCl solution at pH 9.0 is remarkable, though this result should be viewed with caution as too many fitting parameters were employed in our approach. Specifically, the spring constant of cantilever was not confirmed by independent measurements and the magnitude of measured colloidal forces can be in error by at least 5-10% (Veeramasuneni et al., 1996). Also, the radius of curvature of the tip was not measured but assumed to be as 30 and 40 nm in studies with water and 1 mM KCl, respectively. Although most similar AFM cantilever tips examined in our laboratory (using scanning electron microscope and atomic force microscope) had radii of curvature between 30 and 50 nm, the accuracy of the surface charge density determination could be enhanced if a more accurate value of the tip end curvature is known. Finally, the theoretical model used in this study should be used with caution because the Derjaguin approximation might not be accurate on account of the geometry and dimension of the sharp tip used in this study. Future research will address all of the above drawbacks of the approach adopted in this exploratory study.
[FIGURE 3 OMITTED]
The analysis of the experimental data in Figures 4 and 5 indicates that the surface charge density/surface potential of the bitumen changes every 20 to 40 nm. It is unlikely that this variation could be explained by irreproducibility of measured colloidal forces. A 10-20% variation in the magnitude of colloidal forces is commonly observed in the AFM studies when measurements are repeated on the same substrate location. During the experiments with bitumen this variation sometimes can be larger due to instability of the (semi-solid) bitumen surface, its softness, and dynamic character. For example, the solid bar shown in Figure 5A represents a maximum range of variation of the surface charge density value among two sets of five consecutive measurements conducted on the same bitumen locations; this variation, though significant in this case, is about 30% of that for all results shown in Figure 5A. Therefore, >100% variation in the surface charge density noted in Figures 4 and 5 can only be explained by the presence of nano-domains in the bitumen surface of distinctly different surface charge characteristics.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
In a crude approximation, the size of the domain can be estimated through measurements of distances in Figures 4 and 5 with constant, or close-to-constant, surface charge density/ surface potential. The regions of constant surface charge density/surface potential can be easily identified in Figure 4. Most of the domains appear to be 20 to 40 nm. Nevertheless, taking into account the possible scatter of the colloidal force values for the bitumen domain with the same surface charge density/surface potential, the size of the domain could be as large as 70 nm.
More frequent fluctuations in the surface charge density/ surface potential in Figure 5 makes the measurements of domains more difficult. But even for these more scattered data, domains appear to be of similar dimensions as measured in water.
The 20-40 nm size of surface domains that can be concluded from the results in Figures 4 and 5 coincides well with the dimensions of the phases recorded in Figure 1. Unfortunately, the size of the domains appears to be comparable or even smaller than the effective dimension of the AFM tip used in this study. (7) As a result, the size of the bitumen surface domains determined in this study should be treated only as a rough estimate. Also, likely, the variation in surface charge density/ surface potential of the bitumen-water interface could be enlarged if experiments are conducted with sharper tips. The tip area has a significant contribution to the tip-bitumen interactions and determines the shape of the recorded tip-bitumen force curves. As the effective area of the tip used in this study exceeded the diameter of one domain, the results in Figure 4 reflect averaged colloidal forces for the AFM tip placed over two or more domains.
Despite the above-mentioned uncertainties in measurement--which are secondary and can be improved in future studies--it is clear from Figures 4 and 5 that there exist significant surface charge heterogeneities at the bitumen-water interface. Based on averaged surface potentials (obtained, for example, from electrophoretic measurements), the DLVO theory invariably predicts the impossibility of any coalescence between bitumen droplets in aqueous media. From experience, however, it is known that such coalescence do occur, albeit randomly. (8) In a subsequent paper (Esmaeili et al., 2007), we will show that the existence of surface heterogeneities, as demonstrated clearly in this study, can account for: (1) the occurrence of coalescence between charged droplets, despite the apparently insurmountable energy barriers created by the average surface potentials; and (2) the stochastic nature of the coalescence process.
We used atomic force microscopy in studying surface charge and surface potential of heterogeneous substrates at a sub-microscopic spatial resolution. In this exploratory investigation, the colloidal forces were measured between the [Si.sub.3][N.sub.4] AFM pyramidal-shape tips and the spin-coated Athabasca bitumen in water at pH 6.0-6.5 and in 1 mM KCl solution at pH 9.0. The AFM tip was moved stepwise at 10 nm per step using an operator-controlled offset and the colloidal forces were measured at each step across the bitumen surface. The fitting of the experimental force data was done using a theoretical model combining both electrostatic and van der Waals forces for a conical tip-flat substrate system, and this fitting allowed the determination of the surface charge density of the bitumen-water interface. The fitted surface charge density values varied from -0.002 to -0.004 C/[m.sup.2] in water (pH 6.0-6.5) and from -0.005 to -0.022 C/[m.sup.2] in 1 mM KCl solution (pH 9.0), respectively. Next, the bitumen surface potentials calculated from these surface charge density values using the Graham equation are from -90 to -130 mV in water and from -45 to -110 mV in 1 mM KCl solution, respectively.
Variation in the fitted surface charge density suggests a heterogeneous structure of the bitumen surface and the presence of sub-microscopic domains of 20 to 40 nm in diameter, although smaller and larger domains cannot be ruled out at this stage of our research. The exact size of domains could not be determined in this study due to a limited resolution dictated by the size of the commercial cantilevers equipped with pyramidal tips; the effective interaction area of the tips was comparable or larger than the size of the bitumen surface domains.
In this study, the measurements of bitumen surface charge density were carried out stepwise in one lateral direction, but the technique can also be used to map two-dimensional (x-y) surface charge distribution. Because soft and adhesive bitumen often contaminates the AFM tips during adhesive contacts, two-dimensional mapping of the bitumen surface charge were too difficult to build; a probability of tip contamination with the bitumen increases with the increasing number of measurements. Due to the same reason only individual force curves were recorded for each tip location, which enlarges the scatter of the data and increases probability of error. Future work with tip-inert materials should involve multiple measurements of colloidal forces at one tip location and thus averaged results can be obtained.
Future research should also make use of sharper tips to improve the lateral resolution of colloidal force measurements. The tips or tips' surfaces could rather be made of other than silicon nitride material and have chemical functionality that is of homogeneous nature and well defined surface charge characteristics. This way the accuracy of calculation of surface charge density and surface potential will improve.
Theoretical Model for Surface Forces in a Conical Tip-Flat Substrate System
The pyramidal-shaped AFM tips can be reasonably approximated as conical with a spherical cap at their apex. These tips interact with a substrate surface. Geometry of the system and the parameters used in the modelling are shown in Figure A1. The following set of equations describing electrostatic and van der Waals forces were derived for such a system. In the derivation, we used the Derjaguin approximation (Derjaguin, 1934), which breaks down for small [kappa]R values. Although the use of the Derjaguin approximation may reduce the accuracy of the surface charge density/surface potential analysis, we note that the work of Stankovich and Carnie (1996) indicates that such approximations do not necessarily invalidate analyses of systems with the pyramidal-shaped AFM tips. Future work should focus on using sharper tips and the theoretical analysis of the measured colloidal forces using more sophisticated models than the one presented in this communication.
The electrostatic force per unit area between two planar, semi-infinite surfaces separated by a distance D was approximated with an expression derived by Parsegian and Gingell (1972). For the constant potential case the electrostatic force per unit area is:
f = 2[[epsilon].sub.0] [epsilon] [[kappa].sup.2] [[PSI].sub.1] [[PSI].sub.2] [e.sup.-[kappa]D] - ([[PSI].sup.2.sub.1] + [[PSI].sup.2.sub.2]) [e.sup.-2[kappa]D] (A1)
where [PSI] is the surface potential, [epsilon] the dielectric constant of the medium separating surfaces, [[epsilon].sub.0] the permittivity of vacuum, [[kappa].sup.-1] the Debye length, D the surface-surface separation, and subscripts 1 and 2 refer to two surfaces. For the constant charge density case the force per unit area is:
f = 2/[[epsilon].sub.0][epsilon] [[sigma].sub.1][[sigma].sub.2] [e.sup.-[kappa]D] + ([[sigma].sup.2.sub.1] + [[sigma].sup.2.sub.2]) [e.sup.-2[kappa]D]] (A2)
where [sigma] is the surface charge density.
[FIGURE A1 OMITTED]
For the geometry depicted schematically in Figure A1, the electrostatic force between the tip (denoted by subscript T) and the substrate (denoted by subscript S), [F.sub.TS], can be obtained by integration using Equation (A1) or (A2) as follows:
[F.sub.TS] = [[integral].sub.[infinity].sub.0] f x 2[pi]rdr (A3)
In the spherical region of the tip end (0 < r < R sin [alpha]):
R - [square root of [R.sup.2] - [r.sup.2] + D = L] (A4)
rdr = [square root of [R.sup.2] - [r.sup.2] x dL = (R + D - L) x dL] (A5)
In the conical region (r > R sin [alpha]):
L = D + R (1 - cos [alpha]) + (r - R sin [alpha]) x tan [alpha]
r = L - D - R (1 - cos [alpha])/tan [alpha] + R sin [alpha] (A6)
dr = 1/ tan [alpha] x dL (A7)
The case of constant potential
Substituting Equations (A1), (A3) and (A5) gives the electrostatic force between the spherical portion (0 < r < R sin [alpha]) of the tip and substrate by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8)
This integration produces:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)
where [L.sub.1] = D + R(1 - cos [alpha]), [a.sub.0] = [kappa]R - 1, [a.sub.1] = [kappa]R cos [alpha] - 1, [a.sub.2] = [a.sub.0] + 0.5, and [a.sub.3] = [a.sub.1] + 0.5
Using Equations (A1), (A7) and (A3), the electrostatic force between the conical portion of the tip (r > R sin [alpha]) and the substrate is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The total electrostatic force ([F.sup.e]) between the tip and the substrate with an attached nanoparticle can be obtained by combining all the forces obtained from Equations (A9) and (A10) as follows:
[F.sup.e] = [F.sup.s.sub.TS] + [F.sup.C.sub.TS] (A11)
The case of constant charge
The electrostatic force between the tip and substrate can be obtained using Equation (A2) as follows:
In the spherical region of the tip (0 < r < R sin [alpha]), Equation (A8) can be changed as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A13)
In the conical region (r > R sin [alpha]), Equation (A10) can be replaced by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A14)
Consequently, for the case of constant charge, the total electrostatic force ([F.sub.e]) of the system can be obtained by combining all forces obtained from Equations (A13) and (A14) using Equation (A11).
van der Waals Forces
The van der Waals force per unit area between two planar semi-infinite media separated by a distance D can be approximated by (Israelachvili, 1992):
[f.sub.vdw] = - A/6[pi][D.sup.3] (A15)
where A is the non-retarded Hamaker constant. The van der Waals force between the tip and the substrate is:
[F.sup.vdw.sub.TS] = [[integral].sup.[infinity].sub.0] [f.sub.vdw] x 2[pi]rdr (A16)
In the spherical region of the tip (r > R sin [alpha]), using Equations (A5) and (A16), the van der Waals force can be given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A17)
In the conical region (r > R sin [alpha]), combining Equations (A7), (A15) and (A16) gives:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A18)
[F.sup.vdw-C.sub.TS] = - A/3[tan.sup.2][alpha] (1.0/[L.sub.1] + R sin [alpha] tan [alpha] - D - R(1 - cos [alpha])/[Lsup.2.sub.1]) (A19)
The total van der Waals force ([F.sup.vdw]) between the tip and substrate can be obtained by adding the forces from Equations (A17) and (A19):
[F.sup.vdw] = [F.sup.vdw-S.sub.TS] + [F.sup.vdw-C.sub.TS] (A20)
The total force of the system, including the electrostatic force and van der Waals force, is given by:
F = [F.sup.e] + [F.sup.vdw] (A21)
Bitumen Surface Potential
The surface potential of bitumen was calculated based on fitted surface charge density values using the Graham equation (Israelachvili, 1992):
[[rho].sub.0] - [[rho].sub.[infinity]] = [[sigma].sup.2]/2[epsilon] [[epsilon].sub.0]kT (A22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A23)
[[rho].sub.[infinity]] = [[kappa].sup.2] [epsilon][[epsilon].sub.0]kT/ [e.sup.2][z.sup.2] (A24)
A non-retarded Hamaker constant, J D distance, surface-to-surface separation, m e electronic charge, = 1.602 x [10.sup.-19] C f force per unit area, N/[m.sup.2] F force, N k Boltzmann's constant, = 1.381 x [10.sup.-23] J/K L distance between a differential surface section of the tip and the substrate (Figure A1), m r radius of the circle of the tip at a given vertical position (Figure A1), m R radius of the tip apex, m T temperature, K Greek Symbols [alpha] geometric angle for the spherical cap at the tip end (Figure A1) [beta] half of the angle of the conical AFM tip [rho] concentration of ions, 1/[m.sup.3] [sigma] surface charge density, C/[m.sup.2] [epsilon] dielectric constant of the medium separating surfaces [[epsilon].sub.0] permittivity of vacuum, = 8.854 x [10.sup.-12] [C.sup.2]/Jm [kappa] reciprocal of the Debye length, 1/m [PHI] surface potential, V Subscripts 0 at the surface (D = 0) 1 surface one 2 surface two [infinity] at infinity from the surface (D = [infinity]) S substrate T AFM tip vdw van der Waals Superscripts C conical portion of the tip e entire tip (S+C) S spherical portion of the tip vdw van der Waals
Manuscript received November 8, 2006; revised manuscript received February 26, 2007; accepted for publication March 18, 2007.
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(1) Constants that measure the attraction between two surfaces of the same phase in vacuum.
(2) A surface-liquid region where the intensity of the electric field is larger than zero due to excess of charged species such as ions and oriented dipoles.
(3) Here, the surface is considered heterogeneous if it is composed of at least two distinct regions of different surface potentials, and these regions have dimensions larger than the size of a single atom or a functional group.
(4) An extra heavy form of crude oil.
(5) Commonly expressed as the "stability ratio," which is the reciprocal of this probability.
(6) Often referred as the damping ratio, [r.sub.sp] = [A.sub.sp]/[A.sub.o] = 0.85.
(7) Approximately two times the radius of curvature: between 60 and 80 nm.
(8) Under identical conditions, it is not possible to predict whether two droplets would coalesce in a given trial.
Jaroslaw Drelich,  * Jun Long  and Anthony Yeung 
[1.] Department of Materials Science and Engineering, Michigan Technological University, Houghton, MI, U.S.A. 49931
[2.] Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6
* Author to whom correspondence may be addressed. E-mail address: firstname.lastname@example.org
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|Author:||Drelich, Jaroslaw; Long, Jun; Yeung, Anthony|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Oct 1, 2007|
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