Langmuir 1994,10, 3350-3356
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Reordering of Surface Phases during Atomic Force Microscopy: A Simulation Study M. Callaway,?D. J. Tildesley,*itand N. Quirkes Department of Chemistry, University of Southampton, Southampton SO9 5NH, U.K., and European Centre for Computational Science and Technology, Biosym Technologies Sarl, Parc Club Orsay Universiti, 20 rue Jean Rostand, 91893 Orsay Cedex, France Received December 20, 1993. In Final Form: May 20, 199@ Atomic force microscopy detects forces between a scanning tip and molecules in the surface with which it is in contact. It is now widely used to characterize surface structure. In many cases it has been possible to achieve atomic resolution, and atomic force microscopy can therefore provide a very direct probe of molecular behavior at interface. We have used the molecular dynamics simulationof a Langmuir-Blodgett film to study the effect of tip shape on the observed pattern and to explore whether the structure of the adsorbate is significantlychanged by the presence of the tip. Our results indicate that the presence of the tip close to the film causes compressionof the adsorbateand subsequent enhancementof the translational structure of the film by the scanning tip and that the scanning results are sensitive to the tip geometry. 1. Introduction
Atomic force microscopy (AFM) detects forces between a scanning tip and molecules in the surface with which it is in contact. It is now widely used to characterize surface structure1as well as the adsorbed states ofisolated molecules2 and of condensed phase^.^ In many cases it has been possible to achieve atomic resolution, and atomic force microscopy can therefore provide a very direct probe of molecular behavior at interface^.^ There are, in our opinion, some serious uncertainties in the interpretation of images obtained by using this technique. The most important of these is the effect of the scanning tip on the sample. This is particularly pertinent in the scanning of a deformable adsorbate, for example the chain molecules in Langmuir-Blodgett films and soft polymer^.^-^ There are two questions which we wish to address in the paper. Is the adsorbed surface structure the native state of the system or is it modified by the presence ofthe tip? Secondly how does the geometry of the tip affect the observed AFM pattern. This is part of a more basic question as to how the tip resolves atomic structure in the film. Molecular dynamics simulation is an excellent tool for addressing these questions; it is possible to build up a number of idealized models of the film and to calculate the force on the tip directly in the simulation. There have been a number of previous simulation studies of the AFM experiment, particularly of metallic surface and adsorbed t University of Southampton.
Parc Club Orsay Universite. Abstract published in Advance ACS Abstracts, July 1, 1994. (1) Gould, S.A. C.;Burke, K.; Hansma, P. K. Phys. Rev. B 1989,40, 5363. (2) Blackman, G. S.; Mate, C. M.; Philpott, M. R. Phys. Rev. Lett. 1990, 65, 2270. (3) Marti, 0.; Drake, B.; Hansma, P. K. Appl. Phys. Lett. 1987, 51, 484. (4) Frommer, J. Angew. Chem. Int. Ed. Engl. 1992,31, 1298. ( 5 ) Meyer, E.; Howald, L.; Overney, R. M.; Heinzalmann, H.; Frommer,J.;Guntherodt, H. J.;Wagner, T.; Schier, H.;Roth, S.Nature 1991, 349, 398. (6) Weisenhorn, A. L.; Egger, M.; Ohnesorge, F.; Gould, S.A. C.; Heyn, S.P.; Hansma, H. G.; Sinsheimer, R. L.; Gaub, H. E.; Hansma, P.K. Langmuir 1991, 7, 8 . (7) Schwartz, D. K.; Garnaes, J.; Viswanathan, R.; Zasadzinski, J. A. N. Science 1992,257, 508. (8)Hoover, W. G.; De Groot, A. J.; Hoover; C. G.; Stowers, I. F.; Kawai, T.; Holian, B. L.;Boku, T.; Ihara, S.;Belak,J. Phys. Reu. A 1990, 42, 5844. (9) Annis, B. K.; Noid, D. W.;Sumpter, B. G.;Refner, J.;Wunderlich, B. Mackromol. Chem., Rapid Commun. 1992,13, 169. @
h y d r o c a r b ~ n s . l ~Many - ~ ~ previous studies of AFMI3-l5 have adopted tapered models for the tip; this model is a natural choice as it relates directly to the bulk structure ofthe material. However, the actual structure of the apex of the tip is unknown on an atomic scale and it is possible that sharper tips are mechanically unstable to the shear forces experienced in a typical scan. We have attempted to address this point by considering two different tip geometries in this paper: a trigonal pyramid tip and full planar surface with a protruding atom. (Krantzman et al. have already studied the effect of varying tip structure by using single atom and multiatom t i p s . 9 We have studied equilibrated systems of a model stearic acid monolayer a t temperatures up to 298 K using the molecular dynamics method described elsewhere.'' The density of the model film is chosen to be 20.6& molecule-', which is close to the experimentally observed density of Langmuir-Blodgett films of stearic acid and other carboxylic acids. Model atomic force microscope tips were used to probe the surface structure by measuring the force on the tip due to the stearic acid molecules in the monolayer. In section 2 we give some details of the model Langmuir-Blodgett film and of the model AFM tips. Section 3 outlines the molecular dynamics method used in this paper. Section 4 contains the results of our simulated scans, and in section 5 we present our conclusions. 2. The Model of the Intermolecular Forces 2.1. The Model of the Adsorbate. The model stearic acid molecule is composed of a methyl tail group, 16 methylene groups, and a carboxylic head group. All groups are represented with explicit hydrogens with the exception ofthe methyl tail group which is modeled as a single united atom. The bond lengths are constrained to their equilibrium values while the bond angles are controlled by (10) Luedtke, W. D.; Landman, U. Comput. Mater. Sei. 1992, 1 , 1. (11) Landman,U.;Luedtke, J. D.;Ribarsky,M.R.J.Vm.Sci.Technol. A 1989, 7,2829. (12) Landman, U.; Leudtke, W. D.; Burnham, N. A.; Colton, R. J. Science 1990,248, 454. (13) Sumpter, B. G.; Getino, C.;Noid, D. W. J.Chem.Phys.1992,96, 7072. (14) Tang, H.; Joachim, C.; Devillers, J. Surf. Sci. 1993, 291, 431. (15) Mi-Taber, H.;Kawazoe, Y. Jpn. J.Appl.Phys. 1993,32,1394. (16) Krantzman, K. D.; Rees, C. D.; Farelly, D. J . Phys. Chem. 1991, 95, 9039. (17)Kim, K. S.;Moller, M.; Tildesley, D. J.; Quirke, N. J. Chem. Phys. 1901,94, 8390.
0743-7463/94/2410-3350$04.50/00 1994 American Chemical Society
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harmonic potentials of the form
1 (e - eOI2 v(e)= 5kb which is approximated by
v(e)= k, [I - cos(e - e,,)] The conformation of the hydrocarbon chain is controlled by dihedral potentials of the form proposed by Ryckaert and Bellemans'*
r=O
There are four dihedral potentials describing the orientation of the atoms in the head group with respect to the chain. Dispersion interactions between atoms within the same chain are modeled using the Lennard-Jones (12-6) potential: note that the individual atoms in two methylene groups separated by more than three carbon atoms in a particular chain interact through nine separate LennardJones interactions. The dipole moment of the head group is represented by partial charges placed on the four atoms of the head group. These charges produce an overall molecular dipole moment of 1.79 D. Each atom in the molecule interacts with all other molecules through W interactions within the minimum image convention. The partial charges within the head h o u p interact with partial charges in other head groups within a cylindrical cut-off. Our previous study indicates that the application of the cut-off has a negligible effect on the translation and orientational structure of the film.lg All atoms in the film interact with a structureless surface through a n integrated U (9-3) potential
This surface acts as a supporting substrate for the amphiphilic molecules but does not interact directly with the AFM tip. The charges, qc,in the head group induce image charges, qi, in the supporting surface. The charge-image charge interaction is given by Qc4i u. = -1 le 2 (47C€,)2(Zi - Zp)
where zi is the height of the charge and zp is the height of the image plane above the surface. The size of the image charges is E - E)
qi = [ 7 Qc ] E + €
where E is the dielectric constant of the vacuum above the surface and E' is the dielectric constant ofthe surface which is set to the value 4, appropriate for quartz glass. To summarize, the model film is composed of fully flexible chain molecules with explicit dispersion and electrostatic interactions. The parameters for the force fields employed are described in full detail e1~ewhere.l~ 2.2. The Models of the Tip. The size and shape of the area of AFM tip which is actually responsible for producing an image are not known with any certainty. (18) Ryckaert, J. P.; Bellemans, A. Chen. Phys. Lett. 1976,30, 123. (19)Kim, K. S.;Moller, M.; Tildesley, D. J.; Quirke, N. Mol. Sim. 1994, 13, 77.
Figure 1. Asketch ofthe model systems: (A) 394 atom trigonal tip model; (B)the planar tip model. We have therefore considered two different models for the AFM tip. The first, a trigonal pyramid shown in Figure lA, is composed of 394 carbon atoms with an apex consisting of 4 carbon atoms. The rigid tip is constructed from a diamond lattice, using the equilibrium carboncarbon bond length appropriate to diamond. Atoms connected to the surface of the tip by less than two bonds are removed. The trigonal-pyramid tip provides a suitable means for investigating the force pattern due to a sharp tip. It is important to note that the model of the tip is rigid. We believe that this is a reasonable approximation when modeling a soft film, such as a hydrocarbon adsorbed in a direction normal to the surface. The bond-angle and bond-length forces in the diamond tip are an order of magnitude greater than the barriers to conformational change in the adsorbed hydrocarbon. The second model comprises a single carbon atom embedded in a structureless W(9-3) surface, Figure 1B. The protruding atom, which is fixed rigidly with respect to the supporting surface, behaves like a piece of grit in a shoe, sensing the local variations in force above the uniform background force registering on the supporting surface. In both models, the atoms forming the tip interact with those in the film through the appropriate LJ dispersion potentials. The Lennard-Jones parameters for the individual carbon atoms in the trigonal pyramid, the protruding atom, and the structureless su porting surface are taken to be d k g = 28 K, u = 3.4 A technical problem arises with the trigonal pyramid and the planartip due to the divergence in the dispersion interaction between the tip and the substrate supporting the Langmuir-Blodgett film. This problem is discussed in the Appendix and is avoided in the simulation by ignoring the tipiLB-substrate interaction which is a constant a t constant height. In nature, this divergence is avoided by the relativistic retardation of the dispersion force at large distances.
zZo
3. The Molecular Dynamics Simulations Model Langmuir-Blodgett monolayer films of stearic acid were constructed by creating an 8 by 8 hexagonally close packed array of model stearic acid molecules physisorbed on a structureless hydrophillic surface with periodic boundary conditions in the plane of the surface. The molecules were placed head-group down on the surface. The system was brought to equilibrium at room temperature, 298 K, after 30 000 time steps, or 60 ps, followed by a production run of 35 000 steps, or 70 ps, (20) Joshi, Y. P.; Tildesley, D. J.Mol. Phys. 1983, 55,999.
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Callaway et al. Table 1. Hexagonal Order Parameter for a Series of Systems"
3
the system
hexatic order parameter of the center of mass
hexatic order parameter of the tail group
isolated film trigonal tip, 22.5 A planar tip, 26.5 A, 1.87 A planar tip, 24.5 A, 0.935 A planar tip, 22.5 A, 0 A
0.81 f 0.01 0.781 f 0.001 0.698 f 0.001 0.919 f 0.001 0.831 f 0.001
0.56 f 0.01 0.475 f 0.001 0.530 f 0.001 0.730 f 0.001 0.323 f 0.001
2
51 \
L! 0
-1
'0
500
1000
1500
0
a All films are high density,Am = 20.6 A2,and at 298 K. Planar tips are characterized by two heights: the first value refers to the separation between the supporting plane of the tip and the supportingplane of LB film;the secondvalue is the distance between the supporting plane of the tip and the center of the protruding atom.
t /steps Figure 2. A typical plot of the variation of force with time during a simulation with the 394 atom tip. A, = 20.6A2, T = 298 K.
during which time the properties of the system were calculated. Before describing the method used to simulate full AFM scanning of the thermalized Lamgmuir-Blodgett film, it is useful to review the experimental and simulation time scales. A typical experiment involves scanning an area of sample of side approximately 3 nm, with a frequency ca. 50 Hz. During a molecular dynamics simulation with time step of 2 fs, 1013steps would be required to complete a scan. Clearly on the time scale of our simulations, lo3 steps, the tip will be motionless. Therefore in our simulations scanning is performed by averaging the forces on the static tip a t a series of positions within the simulation box. To avoid unphysical starting configurations, the AFM tip was inserted into position within the simulation box in a series of small steps. Starting from a height well above the surface of the film the tip was lowered in 0.5-A steps, each lowering of the tip was followed by a 50-fs equilibration. Once the tip was in position and the system fully equilibrated for a further 400 fs, production cycles of 3600 fs were performed, during this period the forces on the tip and average properties of the film were calculated. Mean tip forces did not vary significantly on doubling the length of the production phase or doubling the length of the relaxation phase. Figure 2 shows the force as a function of time after a typical insertion using the trigonal pyramid tip.
Fz /nN 50
40 30 20 10 0
0
0
xih
Figure 3. Force ma ped images of a n ideal frozen, 0 K, high density, Am = 20.6 film: (a) 394 atom tip, tip-surface separation 29 A. (b) planar tip model, tip-atom substrate separation 29 A,protruding atodsurface separation 1.87 A.
i2,
4. Results
At 20.6 A2 molecule-' the molecules are essentially upright, the centers of mass having hexagonal ordering, and the dipoles of the head groups forming a one sublattice structure with the dipoles arranged head to tail.lg At this density the structure of the film is dominated by the packing of the backbone hydrogen atoms. The translational order of the film can be characterized by the radial distribution function in the plane of the surface and the hexatic order parameter 1
N 6
where there are N molecules in the film and #j is the angle between vectors from molecule i to two of its nearest neighbors,j. There are six such angles associated with a molecule i, the value #j in the perfect triangular lattice is z/6and in this case 0 6 = 1.0. In the thermalized film
(298 K) the centers of mass of the molecules show a solidlike ordering with 0 6 = 0.81, Table 1. The methyl tail groups show a reduced translational ordering with 0 6 = 0.56. There is no net diffusion of the tail groups. To test our model AFM tips, scans were performed at constant height ( z ) on a frozen, 0 K, film using both the trigonal pyramid and the planar tips. The structure of the film was determined by energy minimization. Clearly in this case no dynamics of the film is required and there is no real scan. The force between the tip and the LB film is measured at fixed z for a variety of x and y values and the results are contoured. As expected, both tips produced sharp two-dimensional images of the surface. A typical image, a t a force which is comparable to those observed in an experiment: is shown for the trigonal pyramid in Figure 3. Peaks signifying a repulsive force correspond to positions of the molecular tails within the idealized film.
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8 3-
B, 10
15
x /A Figure 4. Tip force variation along a lattice vector in a thermalized film, 298 K, A, = 20.6 A.z The 394-atom trigonal
pyramid was used as a “ghost tip”, i.e. with tip film forces set to zero for the purposes of the dynamics of the film. As a further test the film was thermalized at 298 K using the molecular dynamics method and the trigonal pyramid tip was inserted into the film for fixed z a t various values of x and y. The tip-film interactions were switched off. In this case, the tip behaves as a set of ghost force-sites which sense the force from the film without disturbing the film. In other words, the distance from the molecule to the atoms of the tip is calculated and used to estimate the force for each configuration of the film. Figure 4shows a one-dimensional section across the surface using this approach. The force peaks in Figure 4 correspond to molecular sites, and the troughs correspond to 2-fold minima in the surface, i.e. the saddle points between molecules. This is the expected pattern and the figure demonstrates that the thermal motion of the film does not destroy the pattern. In the next set of simulations, the tip is lowered into a fully thermalized film, in this case the tip acts as a rigid external field and influences the dynamics of the atoms in the Langmuir-Blodgett monolayer. It is possible to obtain a section through the force surface by inserting the tip at one position to a fixed height and measuring the force. The tip is then removed, translated laterally, and reinserted into the film at the same height. Although these sections can be calculated, the computing time involved in simulating each point in the model AFM insertion makes it impractical to generate enough points to produce a clear surface plot or contour plot of the force above a significant portion of the surface. Each point requires approximately 28 h on a four processor cluster on the Alliant FX80. From observation of the average structure of the surface film on the simulations, it is clear that different regions of local structure can be identified, corresponding to the surface above the tail groups, the surface above the 2-fold hollows, and the surface above the %fold hollows. In the case of anisotropic trigonal tip, the class of 2-fold hollows splits to give two subclasses as shown in Figure 5. An approximate contour can be produced using the following method. A set of between 10 and 20 simulations are performed for each class at representative positions across the whole surface. The mean tip force for each position type is then calculated and used within the unit cell shown in Figure 5 to generate a force contour plot over the whole surface.
Q
Figure 5. Simulation point types on the ideal surface lattice used for the construction of the contour plots. M corresponds to the average position of the tail group of the amphiphile in the ideal structure.
In Figure 6 we present the results of such a calculation using the 394-atom trigonal tip. The force map of the surface shows a hexagonal ordering a t a mean tip force per unit cell of 0.6 nN, which is close to the typical experimental forces used. However the maxima and minima along the direction (marked as X on the figure) correspond, counterintuitively, to intermolecular and molecular sites, respectively. This can be seen most clearly in a section along the box-fixedx-direction for the trigonal tip shown in Figure 7. The average positions of the hydrocarbon tails are marked as Ms and the most repulsive forces occur at the tip positions between the tails and not directly above the terminal methyl groups. This inverse force mapping along the x-direction was observed both with the trigonal pyramid tip model and with a single-atom tip model but not with the planar tip. A detailed analysis of the force acting on the tip shows that high repulsive force occurs when the tip is wedged between two neighboring molecules forming the hollow. When the trigonal tip is directly above a tail, the molecule below it can collapse by changing its conformational structure and the tip experiences a weaker repulsive force. This conformational distortion is shown in Figure 8, which is a snapshot from the molecular dynamics simulation. When the temperature of the system is lowered from 150 to 75 K the monolayer becomes less flexible and the molecule below cannot collapse. In addition the methyl tails of the model Langmuir-Blodgett monolayer become progressively more solidlike and an inversion occurs; i.e. the most repulsive force in all of the models corresponds to the tail positions as observed for the static film at 0 K. Although the scanning forces are small, over the area over which the tip acts the localized pressure is very high. For example a tip force of 1 nN acting over an area of 1 nm2 would apply a pressure of 1 GPa. We have already noted the conformational change under this pressure, but there is also a change to the translational structure of the tip below the film. The radial distribution function (rdf) of the tail groups is shown in Figure 9. The solid line is the rdf averaged over the complete layer. The dashed line is the rdf for the tail group directly below the tip. The rdf for the native film, without the tip (not shown here), is almost the same as the solid line in Figure 9. The disruption of the whole monolayer by the tip is slight but the change in the structure around the single molecule directly beneath the tip is significant. In Figure 10 we present the results of a series of scans using the planar tip model. The force map shows the
3354 Langmuir, Vol. 10,No. 9,1994
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8 8 7
6- 7 5- 6 5- 5 4- 5 3- 4 3- 3 2- 3 1- 2 1
BELOW
X
Figure 6. A contour plot of forces experienced by a 394-atom tip at 22.5 A above a thermalized, 298 K, high density, A, = 20.6 The scale point 3 corresponds to a force of 0.42 nN and the scale point 9 corresponds t o a force of 0.95 nN.
A2,film. 1.2 1*2
I-
1f
0.8
-
31
\
0.4 0.2
f
f
M
M
M
-
t
M
l
M 0 15
20
A
25
30
35
40
-0
45
Figure 7. Tip force variation along a lattice vector in the x-direction above a thermalized film, T = 298 K, Am = 20.6 A2 for the 394-atom trigonal tip.
expected hexagonal ordering. Sections through the contour (not shown here) confirm the peaks in the surface force correspond to molecular sites. While the overall tip force is higher, reflecting the repulsive contribution of the Lennard-Jones (9-3) supporting-surface,the variation in tip repulsion over the surface, measured by the protruding single atom, is of the same magnitude as that of the 394 atom tip at 22.5 A. Under the influence of the planar tip the disruption of a molecule beneath the protruding tip atom is reduced, as can be seen by comparing Figure 9 and Figure 11. The protruding atom in the planar tip is
10
15
Figure 9. Two-dimensional radial distribution function for tail groups within a film at T = 298 K, Am = 20.6 A2,in the presence of the 394-atom tip. The tip height is 22.5 A. The solid line represents the rdf of all 64 moleculeswithin the box. The dashed line represents the rdf for the amphiphile directly beneath the bottom atom of the tip.
at a height of 24.6 A above the supporting substrate while the bottom atom in the trigonal tip is at a height of 22.5 Interestingly, the differences in the forces during the scan of the surface are comparable for the two models. The planar geometry allows the scan to take place without disrupting the film. These results are also reflected in the hexatic order parameter shown in Table 1. In lowering the trigonal probe to a height of 22.5 A,0 6 for the center of mass of the amphilphiles is reduced to 0.78 and 0 6 for the tail groups is reduced by 15%to 0.48. This fall demonstrates the disruption of the film by the probe. We observe different results in lowering the planar tip to a height of 26.5 A with a probe atom supporting plane separation of 1.87 A. In this case 0 6 for the center of mass is reduced to 0.7 but 0 6 for the tail groups is reduced to 0.53, a reduction of only 5% from that of the native film. This confirms the stabilizing effect of the planar-probe on the tail groups. As the planar probe is lowered to 24.5 A the solid structure of the film is very significantly enhanced over that of the free film; the tail groups of the amphiphile are adsorbed in an ordered structure on the upper surface. The forces in this case are larger than those normally
A.
Figure 8. A snapshot of the monolayer film under the 394atom tip, T = 298 K, Am = 20.6 A2.
5
r /A
x /A
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9 9 8
ABOVE 88-
7- 8 7- 7 6- 7 5- 6 5- 5 4- 5 3- 4 3- 3 BELOW 3 X
Figure 10. A contour plot of forces ex erienced by the planar tip model a t a plane-LB substrate separation of 26.5A and a tip atom-plane separation of, 1.87 (26.5X70.5a).The scale point 3 corresponds to a force of 6.66nNand the scale point 9 corresponds to a force of 6.84nN.
from the trigonal tip invert and become like those observed for the planar tip. It is clear that the sharp trigonal pyramid model distorts the local structure that it probes and if real experimental tips behave in this way they would not scan the native structure of the adsorbate. The planar tip, which is likely to be more mechanically stable than the trigonal tip, gives the more intuitive results with the most repulsive forces above the adsorbate molecules. Most of this probe is a large flat surface which helps to stabilize the solid structure of the adsorbate tails, while the protrusion measures the force variation.
4
3
CI
5
0)
2
1
n "0
5
10
15
r /A
Figure 11. Two-dimensional radial distribution function for tail groups within a film at T = 298 K, in the presence of the planar tip (plane height 26.5 protruding atodplane separation of 1.87 The solid line represents the rdf of all 64 molecules. The dashed line represents only that of the molecule directly beneath the protruding atom of the tip.
A).
A,
foundin the real experiment. Finally a t a height of 22.5 A the plane probe actually buckles the film and the tail groups are much less highly ordered on the upper surface.
5. Conclusions It is possible to obtain sensible AFM patterns of a deformable adsorbate using a molecular dynamics simulation of the adsorbate with the tip represented as a static field. However, the precise force pattern obtained depends on the shape of the model tip. A fine tip bores into the film and produces the most repulsive force when it is wedged between adsorbate molecules. The plane tip registers the most repulsive force when the protruding atom is directly above the average methyl tip position. The sharp tip of the tapered probes have to be lowered a significantdistance into the film to register a measurable force and as a result these tips change the local translational and conformationalstructure of the film, but they do not change the global structure on the scale of 40 A. The local distortion is a function of the film stiffness and is temperature dependent. As the temperature is lowered and the adsorbate film becomes less flexible, the results
Acknowledgment. Martin Callaway thanks BP and the SERC for a CASE award. He thanks Sally Rogers of BP Sunbury on Thames for help in performing a number of experiments using the AFM tip. We acknowledge a useful discussion with Ian McDonald, Ruth Lynden-Bell, and Trevor Rayment in the Chemistry Department in Cambridge. The Computer used in this work was supported by the Science and Materials Computing Committee of the SERC under Grant GWJ74459. Appendix 1. Nonconvergence of Trigonal Pyramid Substrate Forces Consider, for simplicity, a conical tip of height 2 with a conical half angle 8. If the number density of the tip, ,o, is constant, then in a thin layer of width dz a t a height z above the base of the tip there are N(z) atoms where
Each atom in this layer interacts with a solid surface through a potential given by
where 6 is the height of the base of the tip above the surface and n is the number density of the surface. The total interaction between the tip and the surface is
Callaway et al.
3356 Langmuir, Vol. 10, No. 9, 1994 We can rewrite this into two parts
AJZ~~[$&~
-
(&r]
&
at large separations the dispersion interaction would fall offmore sharply than T~because of relativistic e f f e c t ~ . ~ l - ~ ~ Therefore when using a tip such as the trigonal pyramid model or *e planar tip, the tiphubstrate interaction cannot be directly included within the simulation and is most conveniently ignored.
If z’ >> 6 the second integral approximates to
therefore V&) diverges as tends to infinity. In nature
(21)Caeimir, H.B. G.; Polder, D. Nature 1946,158,787. (22)Caeimir, H. B. G.; Polder, D. Phys. Rev. 1848, 73, 360. (23) LiRehitz, E. M.; Berestetakii,V. B.; F’itaevskii, L. D. Quantum Electrodynemics; Pergamon Press: Elmsford, NY,1982;p 347.
