Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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between them, r. The proportionality factor is 1/4 πεε

= 8.99 × 10

/C, where ε

is the permittivity

constant factor which has the value 8.85 × 10

–12

/Nm

, and ε is the relative permittivity for the medium

across which the force acts.

(5.4)

If, as an illustration, we set q

= q

= 1.6 × 10

–19

C, the charge of one electron, and solve for the force

when two electrons are an atomic distance 0.2 nm apart, we ﬁnd that the resulting force is about 6 nN.

This is a magnitude that can be detected with most AFMs. One must also remember that free charges

induce surface charge on nearby surfaces that acts as an image charge buried within the material. Image

charges always carry the opposite sign of the original charge. In this case, the force relationship becomes

(5.5)

with ε

and ε

representing the permittivities of the medium and sample, respectively. For metals, where

the permittivities are inﬁnite, the term in parentheses approaches one. Because of the high permittivity

of liquids, if the system is immersed in a liquid, the force is drastically reduced and can even be repulsive

(i.e., positive), depending on the relative values of ε

and ε

5.3.2.2 Dipoles

Molecules can have positive and negative ends to them, due to one atom or atomic group having a

stronger electronegativity than the others. Dipoles have associated electric ﬁelds, and these electric ﬁelds

interact with other charges and dipoles. The magnitude of a dipole moment of a molecule or an atomic

bond is p = ql, where charges ±q lie a distance l apart. Ubiquitous water’s dipole moment is 6.18 ×

–30

Cm, which is a modestly high value, exceeded in general only by strongly ionic pairs such as NaCl.

It is interesting that the interaction potential of a dipole with a charge, another dipole, or a polarizable

atom or molecule is related to whether the dipole is ﬁxed or free to rotate. The functional dependence

upon distance changes. Although the force between two dipoles can be quite weak, collective effects may

be large enough to be measurable using SPM techniques. Typically, one integrates the force or potential

over the volumes where the charges, molecules, or dipoles are located. Once again, this changes the

functional dependence of the force law. For example, the interaction potential between a ﬁxed dipole

and a polar molecule is proportional to 1/r

. If the ﬁxed dipole ﬁnds itself in front of a semi-inﬁnite half-

space of polar molecules (a ﬂat sample), the interaction potential is then a function of 1/r

. The derivations

may be found in Israelachvili (1992).

5.3.2.3 Polarizability

All atoms and molecules are polarizable. The effect originates from the charged nature of atoms. In an

electric ﬁeld, the positively charged nucleus moves slightly in the direction of the ﬁeld, and the electrons

against it, until the force exerted by the electric ﬁeld is balanced by the internal restoring forces of the atom

or molecule. This is similar to the dipole moment p = ql, but it is an induced, rather than permanent,

dipole moment. The relation among the induced dipole moment µ, the electric ﬁeld E, and the polarizability

α is simply µ = αE. Polarizabilities are of the order of 10

–40

/J. Because electric ﬁeld strengths and

functional dependencies on distance depend on whether the source of the ﬁeld is a dipole or charge, the

interaction potentials between two individual atoms or molecules exhibit either 1/r

or 1/r

proportionalities.

5.3.2.4 Applied Electrostatic Fields

An easy way in which the experimentalist can actively control an SPM measurement is to apply a voltage

between the tip and sample, forming a capacitor between them. The energy stored in such an electric

ﬁeld is equal to W = –½CV

, where C is the tip–sample capacitance and V the applied voltage. The

4 εε

−













4 εε

εε

capacitance of two parallel plates separated by a distance δ is ε

A/δ, where A is the area of the plates.

Therefore, the work per unit area for two planes is ϖ(δ)

planes

= –½ε

V/δ, and by using the Derjaquin

approximation above (Equation 5.3), we arrive at

(5.6)

The presence of a dielectric material with dielectric constant ε of thickness b/2 upon each electrode (tip

and sample) will modify the equation to

(5.7)

Try taking the limits b → 0 and b → δ in order to check this latter result with the former one. A typical

value for the force at contact (δ ≈ 0.2 nm) with an applied voltage of 10 V and tip radius of 10 nm is

–14 nN. The functional dependence on δ for a capacitive interaction is plotted in Figure 5.6.

5.3.2.5 Innate Electrostatic Fields

If there is no voltage applied between two conductors that form a capacitor, there may still be an

electrostatic ﬁeld between them. If the work functions Φ

(nominally, the potential difference between

the Fermi level and the vacuum level) of the two materials are not equal, when electrically connected

the Fermi levels equilibrate, and the voltage difference is Φ

– Φ

. The value can be of the order of a volt,

giving rise to a nonnegligible force. An offset voltage may be applied to compensate this contact potential

difference.

Even when the contact potential difference has been removed, the system may continue to be affected

by innate electrostatic ﬁelds due to work function anisotrophies. The work function is very sensitive to

perturbations at a surface. Surface preparation, uneven distribution of adsorbates, crystallographic ori-

entation, and the presence of surface steps, hillocks, pits, or defects can all inﬂuence the work function

and make its value change with position on the surface.

Let us now perform a short gedanken experiment. We assume a metallic material, the surface of which

incorporates a patch of adsorbates having work function Φ

and the rest of the surface, bare, having

work function Φ

. We know that energy must be conserved, and that everywhere inside the metal, the

electron has a mean energy equal to the Fermi level. An electron taken out of the material via the patch

and put back into it through the bare surface will not conserve energy (Φ

– Φ

≠ 0) unless there exist

electric ﬁelds external to the material. The electric ﬁelds emanate from the adsorbate patches, which can

be modeled as dipole sheets. The ﬁelds are strong enough to affect electron trajectories in ﬁeld emission

microscopy, and they induce surface charge in the sample. The resulting tip–sample forces are known as

patch charge forces.

The calculation of patch charge forces is complex, ﬁrst, because of the nontrivial nature of the electric

ﬁeld associated with the dipole sheet, and second because the distribution of image charges induced in

nearby bodies is heavily dependent on their geometry. Nonetheless, the resulting force is notable not

only in that it can initially be repulsive, as in Figure 5.6, or attractive, but also because induced image

charges are always of the opposite sign, and therefore the force always turns attractive as the distance

between tip and sample is reduced (Section 5.3.2.1). The normal component of the electric ﬁeld E

along

the central axis of a dipole disk with dipole moment per unit area µ and diameter ρ is

(5.8)

()

=−

sphere–flat

εε

εδ ε

()

=−

+−

()

[]

sphere–flat

=±

−

()













In the limits of either an inﬁnitely large disk or at an inﬁnite distance from the patch, E

→ 0. The

magnitude of the electric ﬁeld is maximum at about ρ ≈ z. If we assume that there is one extra electron

on the tip, then the force felt by that electron 10 nm from a patch on the sample of diameter ρ = 10 nm

and dipole moment per unit area µ = 1.6 × 10

–8

C/m will be F

= qE

= ±5 nN, and its image force will

be –2 pN. As the tip is lowered onto the sample, the image force grows and the charge-patch ﬁeld force

will drop. There will also be image charges built up in the tip from the patch ﬁeld, which is attractive to

the patch.

5.3.3 Electrodynamic Forces

5.3.3.1 The Dispersion Force

Now we need to consider oscillating electrons. The position of an electron about the nucleus of an atom

is not ﬁxed with time; it oscillates, generating a ﬂuctuating dipole ﬁeld. The ﬁeld interacts with nearby

atoms, inducing the appropriate instantaneous dipole moments in them that are always attractive. This

is known as the dispersion force because the frequencies correspond to those of visible and ultraviolet

light, which the ﬂuctuations disperse. The London equation gives the dispersion potential W(r) between

two nonpolar molecules a distance r apart:

(5.9)

The electrons orbit the nuclei at a frequency ν, h is the Planck constant, and α is the polarizability of

the atom. The dispersion force acts between all materials, because all atoms have ﬂuctuating electrons.

For two atoms with α/4πε

= 10

–30

, hv = 10

–18

J, and interatomic distance of 2 × 10

–10

m, W = –1.2

× 10

–20

J. This is three times more energy than the thermal energy kT at 300 K, so the dispersion force

may not be ignored at room temperature. The London constant 3hνα

/4 (4πε

)

for Ar–Ar is approxi-

FIGURE 5.6 General behavior of electrostatic and electrodynamic forces as a function of tip–sample separation δ.

Capacitive forces approximately follow a 1/δ force, with van der Waals forces following 1/δ

. Patch charge forces can

ﬁrst be repulsive before becoming attractive.

()

=−

()

να

mately 50 × 10

–79

, whereas for CCl

–CCl

it is about 1500 × 10

–79

, the difference being largely

due to the sixfold change in polarizability of the molecular pairs.

The electronic motion is correlated at visible and ultraviolet frequencies, which are of the order of

Hz. During one period of the ﬂuctuation frequency, the electromagnetic ﬁeld travels 300 nm.

Correlated electron motion does not occur if upon leaving one atom, the ﬁeld travels to another atom

and returns to the original one to ﬁnd that the motion is out of phase. Therefore, at distances greater

than approximately 100 nm, the dispersion potential for two atoms drops off at a rate of 1/r

, instead of

1/r

. This is known as the retardation effect. Integrated over the volume of a ﬂat surface and an approaching

spherical tip, the distance dependence for nonretarded dispersion forces become 1/δ

; that for retarded

dispersion forces is 1/δ

5.3.3.2 van der Waals Forces

There is not just one van der Waals force, but rather there are van der Waals forces. The terminology

van der Waals forces encompasses three forces of different origin. The dominant contribution is the

dispersion, or London force, due to the nonzero instantaneous dipole moments of all atoms and molecules,

as described in the previous section. The second contribution is the Keesom force, which originates from

the attraction between rotating permanent dipoles. The interaction between rotating permanent dipoles

and the polarizability of all atoms and molecules generates the third contribution, the Debye force. The

interaction potential between atoms or molecules of each force is a function of 1/r

. The dispersion force

is the most important component of van der Waals forces because all materials are polarizable, whereas

for Keesom and Debye forces, there must be permanent dipoles present.

The Hamaker constant, A, reﬂects the strength of the van der Waals interaction for two bodies 1 and

2 in medium 3, with permittivities ε

and indexes of refraction n

. The ﬁrst term includes Keesom and

Debye interactions, the second the London interaction.

(5.10)

Inspection of this equation reveals that for identical materials across any medium (ε

= ε

and n

), A is positive (attractive), whereas if ε

and n

are intermediate to ε

, n

and ε

, n

, A is negative

(repulsive). If ε

and n

equal the permittivity and index of refraction of either of the two bodies, A

vanishes. For ε

= n

= 1 (air or vacuum), A is always positive. In other words, van der Waals forces can

be attractive, repulsive, or zero. The judicious choice of the medium in which an SPM experiment is

carried out helps control the van der Waals forces between tip and sample.

The form of the van der Waals interaction potential for two ﬂat surfaces is W(δ)

planes

= –A/12πδ

. Using

Derjaguin’s equation from Section 5.3.1, we ﬁnd that the force between a sphere and a ﬂat is F(δ)

sphere–ﬂat

–AR/6δ

, as seen in Figure 5.6. A typical value for A is 10

–19

J; in air or vacuum a force at contact of the

order of –4 nN would be expected for a 10 nm tip radius. In water, A is drastically reduced because of

the high permittivity of water (≈80).

5.3.4 Electromagnetic Forces

There are three classes of magnetism — diamagnetism, paramagnetism, and ferromagnetism. In paramag-

netic and diamagnetic materials, electron spins are randomly oriented due to thermal ﬂuctuations,

yielding no net permanent magnetic moment for the material. However, in ferromagnetic materials, a

strong quantum effect called exchange coupling causes the spins to align, giving the ferromagnet a

permanent magnetic moment.

Spins will respond to an external magnetic ﬁeld. In all materials, the change in electron orbital moment

is opposite to the external ﬁeld, giving a repulsive force. (This is an atomic analog of Lenz’s law, which

AkT

nnnn

nn nn nn nn

−













−













−

()

−

()













εε

says that it takes work to move a magnet toward a current loop.) Materials in which this is the only effect

are diamagnets. The weak diamagnetic interaction can easily be overridden. In paramagnetism, the

electron spins in atoms when magnetic moments line up in the same direction as the external ﬁeld,

yielding a net attraction. Paramagnetism is stronger than diamagnetism, but weaker than ferromagnetism

by several orders of magnitude.

The force on a magnetic dipole with magnetic moment

→

m is F = ∇

→

m ·

→

B, where

→

B is the magnetic

ﬂux density. For constant magnetic moment

→

m, the force depends on the gradient of the ﬂux density.

In magnetic force microscopy, if the tip is ferromagnetic, and the sample is paramagnetic or diamagnetic,

then the diverging ﬁeld from the tip interacts with the induced magnetic moment of the sample. If the

sample is ferromagnetic, then a para- or diamagnetic tip would detect magnetic force only when the tip

is over part of the sample where the ﬁeld has a gradient, for example, at the edge of a magnetic domain.

If both sample and tip are ferromagnets, then a magnetic force of some magnitude and direction will be

detected over all of the sample.

The interpretation of magnetic force microscope images is not simple, and requires considerable

theoretical effort to understand the data. They depend on the magnetic structure of both tip and sample.

The orientation of the tip with respect to the sample determines its sensitivity to the in-plane or

perpendicular components of the gradient of the ﬁeld. Topography may contribute to the overall “mag-

netic” image. The functional dependence on tip–sample distance will vary with induction effects. (Para-

magnetism and diamagnetism are induced magnetic effects.) The ﬁeld from a ferromagnetic tip may

alter the domain structure of the sample. Yet another complication is that of magnetic surface charge

which gives rise to an apparent double image of thin domains.

5.3.5 Forces in and Due to Liquids

The high permittivities of liquids have a strong affect on the strength of the force interaction between

tip and sample. Equations 5.5, 5.7, and 5.10 all explicitly state the inﬂuence of the permittivity on the

force magnitude. Also, there are other effects due to the presence of liquids. They are the double-layer,

charge regulation, hydration, structural, and capillary forces.

5.3.5.1 Double-Layer, Charge Regulation, Hydration, and Structural Forces

When immersed in a polar liquid such as water, surface charge on the tip and sample may be induced

by the ﬂuid. This may occur by either ionization or dissociation of the surface species, or by adsorption

of ions from solution. To maintain electrical neutrality, ions of the opposite charge gather near the surface

of the tip and sample to form a diffuse electrical double layer. If the tip and sample are pushed together,

a strong, long-ranged repulsive force is observed because of the overlap of the electrical double layers

(Figure 5.7). This does not necessarily remain repulsive all the way to contact, however. Charge

regulation — the readsorption of counterions onto the surface, reducing the surface charge density —

diminishes the repulsive force in concert with attractive van der Waals forces. Acting to keep the force

repulsive is the hydration force, which comes from the repellent interaction between hydrated ions bound

to the tip and sample surfaces. Whether attractive or repulsive forces dominate near contact depends on

the speciﬁc materials and medium involved.

In nonpolar liquids, much more subtle effects due to molecular ordering or structure at the liquid–solid

interface can be detected because of the low surface tensions and thereby reduced tip–sample force

interactions when nonpolar ﬂuids are present. Speciﬁcally, when tip and sample are within ten or fewer

molecular diameters of each other, it is possible to observe oscillations between attractive and repulsive

forces that display a period equal to the molecular diameter. Almost a standard experiment using a surface

forces apparatus, it is less reliable in SPM setups, probably because of the small curvature radius at the

end of the cantilever tip.

5.3.5.2 The Capillary Force

If a thin wetting ﬁlm (that is, no droplets, but rather a uniform layer) of water or another liquid covers

the tip and sample, one would expect a force–distance relationship as presented in Figure 5.8. The process

by which the peculiar shape is formed in the following. The tip approaches the sample, and when the

liquids touch, the system is driven to minimize the ﬂuid surface area. The liquids suddenly draw the tip

toward the sample (marked “A” in the ﬁgure). Although the work of adhesion between the tip and sample

is generally modiﬁed by the presence of the ﬂuid, in the displayed case there remains enough tip–sample

attraction such that the force becomes more negative until tip–sample contact is achieved. Upon retrac-

tion, the force curve shows extreme hysteresis. This corresponds to the elongation of a liquid bridge, or

capillary, between the tip and sample. If the microscope were stable enough, the liquid bridge would thin

until the waist of the capillary were of atomic dimensions, but usually mechanical or thermal vibrations

cause it to break, generating the second instability labeled “C” in the ﬁgure.

The overall magnitude of the capillary force can be large enough to obscure van der Waals forces. The

liquid ﬁlms surely make a large contribution to any damping measurements that are performed. The

permittivities of the ﬁlms must be considered if any attempt is made to quantify capacitive or other

electrical forces. (See Equations 5.7 and 5.10.) Any SPM imaging mode — contact, noncontact, or

intermittent contact — is subject to the effect of the capillary force via the change in work of adhesion,

the modiﬁed contact stiffness due to the presence of the liquid ﬁlms, or the viscous properties of the

ﬁlms. Experiments where quantitative results are desired should be performed in dry nitrogen, in

liquids, or in vacuum to temper the strong capillary inﬂuence on the data.

5.3.6 Overview

Figures 5.6 through 5.8 compare the distance dependencies of the forces discussed in this section. One

can see that the shapes of the curves can be quite different. The shape of the curve, plus the magnitude

of the force, is an aid to the interpretation of the data. But remember that the shape will be accurately

revealed only if a stiff cantilever is used. Of course, more than one surface force may be operative, in

which case the data present more of a challenge.

FIGURE 5.7 Typical shapes of force curves taken in liquids. The logarithmic dependence on distance is characteristic

of the double-layer force. The higher the concentration of ions in the solution, the steeper the curve. Charge regulation

and van der Waals forces can overcome the repulsive interactions at small separations, whereas the most ionic solution

remains repulsive all the way to contact because of the hydration force.

We remind the reader that the discussion above has been limited to the situation where the tip and

sample may be modeled as a sphere and a ﬂat. This may or may not apply well to a given experiment.

In Section 5.1, we stated that surface forces predominate at small enough scales. The order of magnitude

of the forces from the examples given in this section is approximately 10

–8

N. We know that this is enough

force to attract an AFM tip to a sample. What radius spherical particles would stick to a surface if they

had the density of water? Of plutonium?

5.4 Adhesive Forces

One of the ﬁrst questions that comes to mind upon viewing a hysteretic force curve is “What is the origin

of the hysteresis?” Or, in other words, “Why is adhesion greater than attraction?” We shall see in this

section that the chemical and mechanical properties of the surface and near-surface region of the tip and

sample determine the adhesive behavior. A compliant tip–sample system of large curvature radius and

large mutual attraction is more likely to be hysteretic than a stiff system with small curvature radius and

negligible mutual attraction. Also, materials are not elastic for pressures approaching inﬁnity; at high

enough loads all materials permanently deform, and hysteresis results.

We ﬁrst cover anelasticity, where energy is dissipated as a function of relative tip–sample velocity. There

is no permanent damage. We then treat elastic continuum contact mechanics, where no permanent

deformation occurs and the tip and sample are assumed to be continuous media. The origins of adhesion

hysteresis will be examined in this limit. Subsequently, the role of surface forces in generating permanent

deformation will be introduced. This becomes important at the nanoscale. At the macroscale, permanent

deformation occurs only when force is purposefully applied. Then the central ideas of molecular dynamics

simulations, where the atomistic nature of material is incorporated into the calculations, will be presented.

FIGURE 5.8 Capillary force as a function of separation or penetration depth. As the tip approaches the sample,

the ﬂuid layers touch, and the tip is suddenly accelerated into the ﬂuid by surface tension (A). The attractive force

increases until tip–sample contact is established at B. Upon withdrawal, there is a lot of hysteresis in the curve due

to the extension of a liquid bridge between the tip and sample. Typically, a mechanical vibration causes the ﬁlm to

break at a ﬁnite bridge diameter (C).

5.4.1 Anelasticity

Taking the resonant behavior of a cantilever as an example, we can ask ourselves why its resonant peak

is not inﬁnitely sharp. It is because of the presence of the dashpot in the model for the system, visible

in Figure 5.1, and represented in Equation 5.1 as β

. Energy is absorbed by the dashpot. The dashpot is

not easily related to a speciﬁc physical process, but a result is the generation of heat. As seen in

Equation 5.1, the dissipative force is a function of the effective mass m*, β

, and

d, the velocity. Anelasticity

is usually studied by means of modulation techniques, where the phase lag between excitation and

response is monitored as a function of frequency or temperature. For velocities (frequencies) approaching

zero, this term may be ignored. Because force curve acquisition should be performed under equilibrium

(quasi-static) conditions, we shall not focus on anelastic behavior.

5.4.2 Adhesion Hysteresis in Elastic Continuum Contact Mechanics

The work of adhesion must be distinguished from adhesion which must be distinguished from adhe-

sion hysteresis. The work of adhesion is the energy per unit area required to separate two ﬂat surfaces

in vacuum from contact to inﬁnity. Adhesion, as we use the word here, is the maximum force needed

to separate two bodies. If one of the bodies is curved and indented into the second, then extra force is

required to separate them. In force curve analysis, the maximum negative force upon separation of the

tip and sample is often referred to as the pull-off force. Unfortunately, measuring the pull-off force does

not directly give the work of adhesion. The pull-off force is a function of the local curvature radius, the

equilibrium interatomic distance, the reduced elastic modulus of the tip and sample, as well as the work

of adhesion. Details are forthcoming in Section 5.4.2.6.

If the maximum observed force upon tip–sample approach (the pull-on force) in the absence of

cantilever instabilities is the same as the pull-off force, there is no adhesion hysteresis. Adhesion hysteresis

is the energy difference between loading and unloading, and is proportional to the difference between

pull-on and pull-off forces. The energy lost is presumed to be transformed into heat via the generation

of phonons. Recent publications indicate that friction and adhesion hysteresis are strongly linked, much

more so than friction and adhesion.

Although there are many different sources of adhesion hysteresis including plasticity, chemical bonding,

and molecular entanglement, its fundamental behavior can be understood from classical contact mechan-

ical theories. In classical continuum mechanics, no atomistic structure to the materials is evident. Five

theories are summarized below. The ﬁrst three behave nonhysteretically; that is, the loading curve is the

same as the unloading curve, but the second two may. Before proceeding, we deﬁne some notation

common to all ﬁve. The ﬁve approaches are discussed in increasing order of sophistication.

5.4.2.1 Notation

The Poisson ratio ν and the Young’s modulus E are needed to write about the reduced elastic modulus K

for the tip–sample system

Other variables used in this section are ξ

the interatomic equilibrium distance at the tip–sample interface,

a the radius of the contact area between tip and sample, ϖ the work of adhesion evaluated at contact,

1/R = 1/R

tip

+ 1/R

sample

the reduced tip–sample curvature, P the load, and δ the penetration depth of the

tip into the sample. The load P for various contact mechanical theories is plotted as a function of δ in

Figures 5.9 and 5.10.

KE E

−













−

























νν

tip

sample

FIGURE 5.9 Normalized force plotted as a function of normalized penetration depth for Hertz, DMT, BCP, and

JKRS. The normalization factors are given in Equations 5.18. Hertz, DMT, BCP are not hysteretic, whereas JKRS is.

Even a hypothetical inﬁnitely stiff cantilever, indicated by the vertical lines with arrows, would detect adhesion

hysteresis for the JKRS case.

FIGURE 5.10 Maugis contact mechanics, where normalized force is displayed as a function of normalized pene-

tration depth, using the notation of Equations 5.18. Low λ values approach DMT mechanics, and high λ values

JKRS. Adhesion hysteresis appears for λ > 0.94.

5.4.2.2 Hertzian Mechanics

The Hertzian theory goes back over a hundred years to 1888. Neither surface forces nor adhesion hysteresis

are assumed (ϖ → 0). The relationship between the load P and the system parameters is

(5.11)

and between the contact radius a and the indentation depth δ,

(5.12)

One can see from the above two equations that P is a function of δ

3/2

. In the limit of high loads or

low surface forces, Hertzian mechanics can be applied to SPM experiments. But since an SPM experiment

is typically run under conditions where the cantilever is near its free equilibrium position in the presence

of nonnegligible attractive forces, one of the four following approaches should be more applicable.

5.4.2.3 DMT Mechanics

DMT (Derjaguin, Muller, Toporov) equations date back to 1975. They apply to rigid systems, low

adhesion, and small radii of curvature, but may underestimate the true contact area. They account for

long-ranged attraction around the periphery of the contact area, but constrain the tip–sample geometry

to remain Hertzian. The equations take the following form:

(5.13)

In other words, DMT is Hertz with an offset due to surface forces. There is no hysteresis between loading

and unloading. At δ = a = 0, P = –2πRϖ. When one tries to match attractive forces that exist up to the

point of contact at δ = 0, the slope at zero is not continuous. The next theory attempts to rectify this

unphysical behavior.

5.4.2.4 BCP Mechanics

The BCP (Burnham, Colton, Pollock) semiempirical approach was formulated in 1991, after several years

of force curve study revealed that existing possibilities (Hertz, DMT, and JKRS) did not match the

experimental SPM data. No surface forces are incorporated into Hertzian mechanics, DMT mechanics

predict a sharp discontinuity in slope at contact, and JKRS mechanics predict no detectable attractive

forces before contact. Experimentally, with stiff cantilevers, one observes long-ranged attractive forces

well before contact, and then a gradual transition between negative and positive slopes in the curves.

BCP behavior is representative of many combinations of tip–sample materials.

(5.14)

δ=

=−π

2 ϖ

=−

−π

=−













222