3. Interactions at a liquid-solid interface
Interactions between particles in fluids are complicated many-body affairs. We give the briefest of sketches here while leaving the meat to expert treatments (Hansen and McDonald 1986; Israelachvilli 1991). One goal is to give the reader a listing of factors to be considered in choosing tip materials and sample preparation methods. Another is to dispel the naïve view that height in an AFM image is the true height of an object. Assuming that the sample is not surrounded by a co-adosorbate, and assuming that the tip does not compress the sample, the height must still reflect changes in the interaction forces, F1,2 (or stiffnesses, S1,2) as the tip passes from the substrate (F1, S1) to the sample, (F2, S2) as illustrated in Figure 2.
Figure 2: Interaction forces (F) and interfacial stiffnesses (S) are a function of the composition of the surface so that the AFM trace will not, in general, faithfully record topography.
In this case smaller interaction forces (or interfacial stiffness in the case of DFM) over
the sample result in a smaller apparent height (h2<h1). Muller and Engel
illustrate this distortion using data and calculations for electrostatic interactions
(Muller and Engel 1997).
It is convenient to group the interactions between tip and sample into the following classes:
(a) Hydrodynamic.
(b) Van der Waals.
(c) Ionic.
(d) Chemical (e.g., hydrophilic and hydrophobic).
(e) Hydration forces which arise from the changes in the structure of the intervening fluid as the tip is approached.
(a) Hydrodynamic effect cause interactions between the cantilever and substrate over large distances (comparable to the cantilever dimensions). This is because the free-flow of fluid around the cantilever is disrupted by the presence of a nearby surface, giving rise to an extra damping term
where 
is the
‘squeeze’ damping coefficient, R is the radius of a sphere a distance D
from a flat surface and h is the viscosity of the fluid.
The expression is valid for R>>D. We have used this simple form for
comparison with the well known Stokes’ law for a free sphere, but more accurate
expressions are given by O’Shea and Welland (O'Shea and Welland 1998). This term can
dominate the overall damping but it contributes little to image contrast because the
cantilever is held at an almost fixed height (the tip height) above the surface. This is
on the order of 10mm so that the nm scale fluctuations
encountered on scanning a surface cause negligible changes in this damping.
(b) & (c) Ionic and van der Waals interactions are described semi-quantitatively by the Derjaguin, Landau, Verwey and Overbeek (DLVO) theory (Israelachvilli 1991). This ascribes the total force on a sphere of radius R and surface charge density st a distance D (D<<R) from a plane surface of charge density ss to the sum of Coulomb and van der Waals interactions (Muller and Engel 1997)
. (4)>
Here 
is the Debye screening length and Ha is the Hamaker constant (which
can be taken to have a value on the order of 10-20J). e
and e0 are the relative dielectric constant of
the medium and the dielectric constant of free space respectively. The Debye theory is
valid only at low salt concentrations and yields
nm for 1:1
electrolytes (e.g., NaCl) and
nm for 1:2
electrolytes (e.g., MgCl2) (5)
where C is the molar concentration of the electrolyte. Plots of interaction forces for various conditions are given by Muller and Engel (Muller and Engel 1997). Here we focus on the dynamic force microscope so we are more interested in the interfacial stiffness (S=dF/dD). It is convenient to express this in terms of the stiffness per unit tip radius, which, from Eqn. 4 is
. (6)
A highly charged biomembrane may have ss=-0.05C/m2 while a reasonable value for a silicon-nitride tip in neutral (pH 7) solutions is -0.032C/m2 (Muller and Engel 1997). Plots of values of S/R for these parameters are given in Fig. 3. A tip radius of 100nm results in an effective stiffness of on the order of -1N/m at nm distances from the surface. Similar positive stiffnesses can result from Coulomb repulsion at very low salt concentrations. In practice, values similar to this are observed but they are often remain positive (i.e., interactions are repulsive) all the way into the surface. This implies that other forces, most probably associated the structure of the liquid itself, play a role.
Figure 3: Interfacial stiffness (normalized to
tip radius) as a function of distance for charged surfaces in various 1:1 salt
concentrations as a function of tip-sample distance according to the DLVO theory. This
theory accounts for the order of magnitude of the observed interfacial stiffness but fails
to account for the fact that negative stiffness (attractive interactions) is rarely
observed in water.
The salt dependence of the electrostatic interaction is the basis of one
method of using the AFM to map the surface charge distribution on bioploymers as described
by Heinz and Hoh (Heinz and Hoh 1999) and Czajkowsky et al. (Czajkowsky, Allen et al.
1998).
(d) Chemical Bonds. Giessibl (Giessibl 1997) has ascribed atomic resolution in DFM images to the rapid variations in the atomic-scale interaction force between atoms on the surface and a single atom asperity on a tip. He estimated the very local fluctuations owing to such forces using a Lennard-Jones potential. By taking the derivative of this potential, we can estimate the interfacial stiffness as follows:>
(7)
where Eb is a binding energy which we will take to be characteristic of a 1eV bond (1.6x10-19J) and s is a an atomic diameter which we will take to be 0.2nm. Here the subscripts refer to forces between the ith atom in the surface and the jth in the tip. The assumption here is that one dangling atom can dominate fluctuations in force (or stiffness) at small distances. The stiffness calculated with the parameters given above is shown in Fig. 4. It is only positive (repulsive force) at very small (angstrom) distances, suggesting that the observation of positive stiffness at much larger distances is a consequence of the structure of the intervening liquid.
Figure 4: Interaction stiffness for an atomic bond described by a
Lennard Jones potential. The interaction is significant only at sub nm distances.
Eqn. 7 describes the formation of a chemical bond between tip and sample,
and, at very short distances (e.g., when true contact is established) the tip chemistry
will play an important role in adhesion. This is the basis of one method of chemical
identification using the AFM and it will be discussed later in this chapter.
(e) Hydration forces.One important ‘chemical’ interaction in fluids is the hydrophobic/hydrophilic interaction. It this not a simple pair-wise interaction at all, arising from the degree to which various groups are miscible/soluble in water. Oily (hydrophobic) moeties are not wetted and thus water tends to push them together, giving rise to an apparent (and very strong) attractive interaction. Similarly charged or polar species tend to dissolve well and are effectively pulled out of the interior of a structure so that they may be more fully hydrated. Hydrophillic groups tend to interact strongly with each other (though the effect can be repulsive if they are similarly charged). Hydrophobic groups are nearly always repelled from hydrophilic groups. Biomolecules are almost invariably hydrated (the exception being the intramembrane region of membrane proteins) and thus expose hydrophillic groups on their outer surface. It would clearly be advantageous to use hydrophobic tips for imaging them since such tips would not stick to the surface on contact. Such tips may be made using electron-beam deposition of carbon (Keller and Chou 1992) or chemical vapor deposition of fluorocarbons (Knapp, Guckenberger et al. 1999) but they are not, as yet, commercially available.
Figure 5: Direct experimental observation of solvation forces in an organic solvent at a graphite surface. (a) shows that amplitude oscillations as a DFM tip approaches the graphite. The period is equal to the minor axis of the molecule. (b) shows the Young’s modulus deduced from such measurements for two different molecular fluids of differing size. Data is taken from Han et al. (Han and Lindsay 1998) with permission
Directly-measured solvation forces are illustrated in Fig. 5. This shows how the oscillation amplitude of a DFM changes as a graphite surface is approached in an organic solvent (octamethylcyclotetrasiloxane, OMCTS) (Han and Lindsay 1998). The large fluctuations in amplitude are spaced by 0.82nm, the diameter of the minor axis of this molecule, and they are clearly influenced by structuring of the liquid near the graphite surface (Fig. 5a). Fig. 5b shows the value derived for Young’s modulus from these data (along with data for another fluid, mesitylene). At nm distances from the surface, the Young’s modulus becomes significant (it is, of course, zero in the bulk). The stiffness of the nth layer (which is what is actually calculated from the amplitude decrease, see below) is related to the effective Young’s modulus of the nth layer (Pethica and Oliver 1987) by (Overney, Meyer et al. 1994
(8)
where R is the radius of the tip and Ln is the load on the
nth layer. Ln is calculated from 
where 
is the amplitude decrease for the nth layer. With R taken as 30nm, the
interfacial stiffness is on the order of 1N/m at a distance of about 3nm from the surface
purely as a consequence of the change in fluid properties near the surface. This
discussion is based on the low frequency data of Han and Lindsay (Han and Lindsay 1998).
In general a more complicated approach is needed to separate the effects of stiffening
from those of viscosity (O'Shea and Welland 1998). Increased viscous losses owing to
structuring of the fluid near an interface can also play an important role in DFM (O'Shea,
Lantz et al. 1998).
