Determining surface potential of the bitumen-water interface at nanoscale resolution using atomic force microscopy.
Drelich, Jaroslaw ; Long, Jun ; Yeung, Anthony 等
INTRODUCTION
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.
EXPERIMENTAL
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.
SUMMARY
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.
APPENDIX
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.
Electrostatic Force
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)
and thus:
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]
or
r = L - D - R (1 - cos [alpha])/tan [alpha] + R sin [alpha] (A6)
and thus:
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]
Thus:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10)
where
[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)
and thus:
[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)
and then:
[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)
Total Force
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)
where:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A23)
and
[[rho].sub.[infinity]] = [[kappa].sup.2]
[epsilon][[epsilon].sub.0]kT/ [e.sup.2][z.sup.2] (A24)
NOMENCLATURE
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, [1] * Jun Long [2] and Anthony Yeung [2]
[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:
jwdrelic@mtu.edu