Electrostatic Friction

Abstract

We consider the effect of an external bias voltage and the spatial variation of the surface potential on the damping of cantilever vibrations. The electrostatic friction is due to energy losses in the sample created by the electromagnetic field from the oscillating charges induced on the surface of the tip by the bias voltage and spatial variation of the surface potential.

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We consider the effect of an external bias voltage and the spatial variation of the surface potential on the damping of cantilever vibrations. The electrostatic friction is due to energy losses in the sample created by the electromagnetic field from the oscillating charges induced on the surface of the tip by the bias voltage and spatial variation of the surface potential. A similar effect arises when the tip is oscillating in the electrostatic field created by charged defects in a dielectric substrate. The electrostatic friction can be compared with Casimir friction originating from the fluctuating electromagnetic field due to quantum and thermal fluctuation of the current density inside the bodies. We show that electrostatic and Casimir friction can be greatly enhanced if, on the surfaces of the sample and the tip, there are 2D systems; for example, a 2D electron system or on incommensurate layers of adsorbed ions exhibiting acoustic vibrations. We show that the damping of the cantilever vibrations due to the electrostatic friction may be of similar magnitude as the damping observed in experiments. We also show that, at short separation, the Casimir friction may be large enough to be measured experimentally. We consider the contribution from contact electrification to the work necessary to separate two solid bodies. The variations of the surface potential resulting from contact electrification give the contribution to the work necessary to separate two solid bodies. For silicon rubber (polydimethylsiloxane, PDMS), we discuss in detail the relative importance of the different contributions to the observed work of adhesion.

14.1 Effect of a Bias Voltage and the Spatial Variation of the Surface Potential

The electrostatic potential at the surface of a metal relative to its interior depends on the magnitude of the surface dipole moment per unit area, which, in turn, depends on the separation of the lattice planes that are parallel to the surface [317]. Variations of the crystallographic directions at the surface of a clean polycrystalline metal results in a variation of the surface potential. This is referred to as the ‘patch effect’. Patch potentials are also generated and influenced by surface contamination and, in the case of alloys, by variation in the chemical composition. The surface potential can be easily changed by applying a voltage between an atomic force microscope tip and the sample. The electrostatic forces between conducting surfaces due to spatial variation of the surface potential were studied in [318].

Patch potential variation is specific to the particular sample and depends on environmental factors. Spatial variation of surface potential is expected to be related to the physical size of the surface crystallites, which, in the case of metal, is typically of the order of $1\,\upmu$m. Thin films deposited on substrates at temperatures much lower than the melting point of the film are often amorphous, with non-uniform thickness and crystallite size of the same order as the thickness of the film [319]. Annealing of the film can produce grain structures that are substantially larger than the film thickness. The patch-potential variations have been measured under various conditions using vibrating or rotating plate electrometers [320]. Notably, it was shown that large-scale variations in surface potential were caused by adsorption of contaminants, which was transient and found to reduce the variation of surface potential of the clean surfaces [50, 52].

a. General theory We begin by considering a model in which the tip of a metallic cantilever of length L is a section of a cylindrical surface with the radius of curvature R (Fig. 14.1). The cantilever is perpendicular to a flat sample surface, which occupies the xy plane, with the z-axis pointing away from the sample. The tip displacement $\mathbf {u}(t)=\hat{x}u_0e^{-i\omega t}$ is assumed to be parallel to the surface (along the x axis), which will be a good approximation when the oscillation amplitude $u_0$ is sufficiently small. The cantilever width w, i.e. the size in the direction perpendicular to the xz plane, is taken to be much larger than its thickness c ($w\gg c$), and d is the separation between the tip and the sample surface. It is straightforward to obtain the static electric field distribution when $d\ll R$. In this case, the electrostatic field of the entire cylinder is effectively the same as that due to its bottom part. The problem is then reduced to solving the 2D Laplace equation with the boundary conditions that the potential has the constant values V and $0\,$at the metallic surfaces of the tip and the sample, respectively. In this case, the electric field distribution outside the conductors is equal to the field due to two charged wires passing through points at $z=\pm d_1=\pm \sqrt{(d+R)^2-R^2}$ [191]. The wires have charges $\pm Q$ per unit length, $Q=CV$, where $C^{-1}=2\ln [(d+R+d_1)/R]$. The electric potential at a point $\mathbf {r}$ exterior to the tip and the sample is given by

$$\begin{aligned} \varphi _0(\mathbf {r)}= & {} -2Q\left[ \ln |\mathbf {r-r}_{+}|-\ln | \mathbf {r-r}_{-}|\right] =\nonumber \\= & {} Q\int _{-\infty }^\infty \frac{dq}{|q|}e^{iqx}\left[ e^{-|q||z-z_{+}|}-e^{-|q||z-z_{-}|}\right] , \end{aligned}$$

(14.1)

where $\mathbf {r}_{\pm }=\pm \hat{z}d_1$. The attractive cantilever-surface force can be calculated in a straightforward manner using (14.1) [169].

Fig. 14.1

Scheme of the tip–sample system. The tip shape is characterized by its length L and the cylindrical tip radius of curvature, R

A somewhat different picture applies in the case of an oscillating charged tip. The cantilever charge does not change when its tip moves parallel to the surface, while the sample charge varies in time at any fixed point. Thus, the electric field from the oscillating tip will be the same as from an oscillating wire located at $\, z=d_1$. The oscillating electric potential due to the tip, at a point $\mathbf {r}$ exterior to the tip and the sample is given by

$$\begin{aligned} \varphi _1(\mathbf {r}, t)=\varphi _1(\mathbf {r)e}^{-i\omega t}+c.c., \end{aligned}$$

(14.2)

where

$$\begin{aligned} \varphi _1(\mathbf {r)=}iQu_0\int _{-\infty }^\infty \frac{dqq}{|q|} e^{iqx}\left[ e^{-|q||z-z_{+}|}-e^{-|q||z-z_{-}|}R_p(q,\omega )\right] , \end{aligned}$$

(14.3)

and $R_p(q,\omega )$ is the reflection amplitude for the p polarized electromagnetic waves. The electric field is given by $\mathbf {E}(\mathbf {r} )=-\mathbf {\nabla }\varphi (\mathbf {r})$. The energy dissipation per unit time induced by the electromagnetic field inside of the metallic substrate is determined by integrating the Poynting vector over the surface of the metal, and is given by

$$\begin{aligned} P&=\frac{c}{4\pi }\int dS\hat{z}\cdot \left[ \mathbf {E(r})\mathbf {\times B}^{*} \mathbf {(r})\right] _{z=+0}+c.c.= \nonumber \\&=-\frac{i\omega }{4\pi }\int dS\left( \varphi _1( \mathbf {r)}\frac{d}{dz}\varphi _1^{*}(\mathbf {r)}\right) _{z=+0}+c.c. = \nonumber \\&=4\omega Q^2|u_0|^2w\int _0^\infty dqqe^{-2qd_1}{\mathrm Im}{R_p}(\omega , q). \end{aligned}$$

(14.4)

Taking into account that the energy dissipation per unit time must be equal to $2\omega ^2\Gamma \left| u_0\right| ^2$, using (14.4) gives the friction coefficient:

$$\begin{aligned} \Gamma =\lim _{\omega \rightarrow 0}2C^2V^2w\int _0^\infty dqqe^{-2qd_1}\frac{\mathrm { Im}R_p(\omega , bq)}{\omega }. \end{aligned}$$

(14.5)

An alternative derivation of (14.5) is given in Appendix R. Now, assume that the electric potential at the surface of the tip is inhomogeneous and consists of the domains or ‘patches’. The cylinder with linear size w is ‘divided’ into cylinder segments with the linear size $w_i$: $w=\sum _iw_i\gg w_i\gg \sqrt{dR}$, and with the surface potential $V_{is}=V+V_i$, where V is the bias voltage and $V_i$ is the randomly fluctuating surface potential for the domain i. In the case of a cylindrical tip geometry, all domains give independent contributions to the friction, which can be obtained from (14.5) after replacement $V\rightarrow V+V_i$ and $w\rightarrow w_i$. The contribution to friction from all domains is given by

$$\begin{aligned} \Gamma= & {} \sum _i\Gamma _i=\sum _i\lim _{\omega \rightarrow 0}2C^2(V+V_i)^2w_i\int _0^\infty dqqe^{-2qd_1}\frac{\mathrm {Im}R_p(\omega , q)}{\omega }= \nonumber \\= & {} \lim _{\omega \rightarrow 0}2C^2\left( V^2+V_0^2\right) w\int _0^\infty dqqe^{-2qd_1}\frac{ \mathrm {Im}R_p(\omega , vq)}{\omega }, \end{aligned}$$

(14.6)

where we have used that the average value of the fluctuating surface potential $\langle V_i\rangle =\sum _iw_iV_i=0$ and $V_0^2=\sum _iw_iV_i^2/w,$ so that $V_0$ is the root mean square variation of the surface potential. According to (14.6), bias voltage and patch contributions to the friction have the same dependence on d.

Many experiments use thermally evaporated thin films of gold [21]. The work function of gold is 5.47, 5.37, and 5.31 eV for the $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ direction, respectively [321]. If the surfaces are clean and amorphous then we can assume that they consist of equal areas of these three crystallographic planes, and the root-mean-square $\langle \sigma _v^2\rangle ^{1/2}$ of the potential distribution becomes:

$$\begin{aligned} \left\langle \sigma _v^2\right\rangle ^{1/2}=\sqrt{\left\langle {(V_i-V_j)}^2\right\rangle }=\sqrt{2\left( \left\langle V_i^2\right\rangle -\big \langle V_i\big \rangle ^2\right) }\approx 90\text{ mV }. \end{aligned}$$

(14.7)

When annealed, thin gold films form mesa structures with the $\langle 111\rangle$ crystallographic planes exposed. In this case, variations of the surface potential are presumably generated by the material lying between the mesas. The size of the mesas depends on the temperature of the substrate during the formation of the film.

Sukenik et al. measured the root mean square variation of the surface potential due to thermally evaporated gold using the Stark effect in sodium atoms [322]. The films were partially optically transparent with a thickness of 42 nm and heated at $120\,^{\circ }$C for several hours in vacuum. They deduced that the magnitude of the fluctuating surface potential is $V_0=150$ mV, and showed that the scale of the lateral variation of the surface potential is of the order of the film thickness. The measurement of the non-contact friction between a gold tip and the gold sample gave $V_0\sim 200$ mV [21], thus confirming the prediction of the theory that this parameter is determined by the root mean square variation of the surface potential.

Now, let us consider a spherical tip (radius R) with (constant voltage) surface domains with the linear size $R_i$. If $R\gg R_i\gg \sqrt{dR}$, the domain on the apex of the tip will give the main contribution to the friction. In this case, we can neglect the spatial variation of the surface potential and the electric field induced by the bias voltage is approximately the same as that which would be produced in the vacuum region between two point charges $\pm Q_i=\pm C(V+V_i)$ located at [170, 171]

$$\begin{aligned} z=\pm d_1=\pm \sqrt{3Rd/2+\sqrt{(3Rd/2)^2+Rd^3+d^4}}, \end{aligned}$$

(14.8)

where

$$\begin{aligned} C=\frac{d_1^2-d^2}{2d}. \end{aligned}$$

(14.9)

It can be shown that the electrostatic force between the tip and the metal surface within this approximation agrees very well with the exact expression for a sphere above a metal surface [323]. The vibrations of the tip will produce an oscillating electromagnetic field, which, in the vacuum region, coincides with the electromagnetic field of an oscillating point charge. The friction coefficient for a point charge moving parallel to the surface due to the electromagnetic energy losses inside the sample is determined by [242] (see also Appendix R)

$$\begin{aligned} \Gamma _{\Vert }=\lim _{\omega \rightarrow 0}\frac{Q_i^2}{2}\int _0^\infty dqq^2e^{-2qd_1}\frac{\mathrm {Im}R_p(\omega , q)}{\omega }. \end{aligned}$$

(14.10)

For motion normal to the surface, $\Gamma _{\perp }=2\Gamma _{\Vert }$. Thus, just as for the cylindrical tip geometry, for a spherical tip, the friction depends quadratically on the bias voltage. However, for a spherical tip, the parabola begins from zero in contrast to the cylindrical tip where the parabola begins from a finite positive value.

b. Clean surfaces For clean flat surfaces, the reflection amplitude is determined by the Fresnel formula (O.2). In this case, for a cylindrical tip with radius $R\gg d$ and for a metal substrate, (14.6) gives:

$$\begin{aligned} \Gamma _{cl}^c=\frac{w \left( V^2+V_0^2\right) }{2^6\pi \sigma d^2}. \end{aligned}$$

(14.11)

With $w=7\times 10^{-6}$ m and $\sigma =4\times 10^{17}$ s$^{-1}$ (which corresponds to gold at 300 K), and with $d=20$ nm and $V=1$ V, (14.11) gives $\Gamma =2.4\times 10^{-20}$ kg/s, which is eight orders of magnitude smaller than the experimental value $3\times 10^{-12}$ kg/s [21].

Assuming $R>>d,$ using (14.10) and (O.2) gives the friction between a spherical tip and a clean sample surface

$$\begin{aligned} \Gamma _{cl}^s=\frac{3^{1/2}R^{1/2}V^2}{2^7d^{3/2}\pi \sigma }. \end{aligned}$$

(14.12)

This expression is only a factor of 1.6 smaller than the result obtained independently in [169]. For the same parameters as above and at $d=20$ nm, the friction for a spherical tip is two orders of magnitude smaller than for the cylindrical tip. The friction determined by (14.12) has the same distance dependence as in the experiment [21]. However, the magnitude of the friction is too small to explain the experimental data.

To get insight into the possible mechanisms of the enhancement of non-contact friction, it is instructive to note that (14.11) can be obtained qualitatively from the following simple geometrical arguments [324]. The vibrating tip will induce a current in the sample in a volume with the spatial dimensions $L_x$, $L_y$ and $L_z$. The instantaneous dissipated power in the sample is given by $P\sim I^2r$, where I is the current and r is the effective resistance. The current I is proportional to the tip velocity $v_x$, and can be written as $I\sim v_xQ_t/L_x$, where $Q_t$ is the charge of the tip. The effective resistance r can be approximated by the macroscopic relation $r=\rho L_x/L_yL_z,$ where $\rho$ is the resistivity. Using these simple expressions for the current I and the resistance, and using the relation $Q_t=C_tV_s$ (where $C_t$ is the tip–sample capacitance) for the induced charge, the instantaneous power dissipation is

$$\begin{aligned} P\sim \rho \frac{v_x^2C_t^2V_s^2}{L_xL_yL_z}. \end{aligned}$$

(14.13)

Comparing this expression with $P=\Gamma v_x^2$ we get

$$\begin{aligned} \Gamma \sim \rho \frac{C_t^2V_s^2}{L_xL_yL_z}. \end{aligned}$$

(14.14)

For cylindrical tip vibrating above the clean surface $L_y\sim w$ and $L_x\sim L_z\sim d_1$. For $d\ll R$ the tip–sample capacitance $C_t\sim w \sqrt{R/8d}$ and $d_1\sim \sqrt{2dR}$. Substituting these expressions in (14.14) gives (14.11) to within a numerical factor of order of unity. From (14.14), it follows that the friction will increase when the thickness $L_z$ of the ‘dissipation volume’ decreases. This is the reason that 2D systems may exhibit higher friction than 3D systems.

c. Film on top of a higher resistivity substrate From the qualitative arguments given above, it follows that, for a thin metal film on top of a higher resistivity substrate—for example. a dielectric or a high resistivity metal—the friction will be larger, than for a semi-infinite sample with the same bulk resistivity as for the film. In this case, the main part of the dissipation occurs within the film, and according to (14.14) this will give rise to a strong enhancement of the friction.

For a planar film with thickness $d_f$ and dielectric constant $\epsilon _2$ on top of a substrate with dielectric constant $\epsilon _3$, the reflection amplitude is determined by (7.103). For a metallic film on a dielectric substrate, or a metallic film on a metallic substrate with $\sigma _2\gg \sigma _3,$ for $d_1\gg d_f$ and $R\gg d$, (14.6) and (7.103) give

$$\begin{aligned} \Gamma _f^c=\frac{w \left( V^2+V_0^2\right) R^{1/2}}{2^{9/2}\pi \sigma _2 d_fd^{3/2}}. \end{aligned}$$

(14.15)

This is greater, by a factor of $2\sqrt{2dR}/d_f$, than the corresponding friction for a semi-infinite sample with the clean surface with the same bulk conductivity as for the film. For a thin film, the effective resistivity of the substrate is increased, giving rise to additional ohmic dissipation. In [169], (14.15) was obtained using a different approach, and neglecting the spatial variation of the surface potential.

d. 2D system on top of a dielectric or metal substrate Let us now consider a 2D system such as an electronic surface states or a quantum well, or an incommensurate layer of ions adsorbed on a metal surface. For example, for the Cs/Cu(100) system, experimental data suggest the existence of an acoustic film mode even for the dilute phase (coverage $\theta \approx 0.1$). This implies that the Cs/Cu(100) adsorbate layer experiences a negligible surface pinning potential. The reflection amplitude for a 2D system is given by (7.110) and

$$\begin{aligned} \mathrm {Im}R_p\approx \frac{2\omega \eta qa\omega _q^2}{\left( \omega ^2-\omega _q^2\right) ^2+\omega ^2\eta ^2}, \end{aligned}$$

(14.16)

where $\omega _q^2=4\pi n_ae^{*2}aq^2/M$. In the case of a 2D structure on top of a dielectric, the factor qa in (14.16) and in the expression for $\omega _q^2$ must be replaced by $1/\epsilon$, where $\epsilon$ is the dielectric function of the substrate. Using (14.16) in (14.6) for $R\gg d$ we get

$$\begin{aligned} \Gamma _{ad}^c=\frac{w\eta MR^{1/2}\left( V^2+V_0^2\right) }{2^{9/2}d^{3/2}\pi n_ae^{*2}}. \end{aligned}$$

(14.17)

This friction exhibits the same distance dependence as observed experimentally [21]. The same expression for the friction is valid for a 2D structure on top of a dielectric. Comparing (14.11) and (14.17), we find that a 2D structure on top of a substrate gives the same magnitude of friction as for a semi-infinite solid (with a clean surface) with the effective conductivity $\sigma _{eff}=n_ae^{*2}/M\eta 2d_1$. Agreement with experiment [21] for $d=20$ nm is obtained if $\sigma _{eff}\approx 4\times 10^9$ s$^{-1}$. In the case of a 2D electron system, for $R=1\,\upmu$m, such an effective conductivity is obtained if $\eta _{\parallel }=10^{14}$ s$^{-1}$ and $n_a=10^{15}$ m$^{-2}$. For Cs/Cu(100) and for $n_a=10^{18}$ m$^{-2}$ ($\theta \approx 0.1$), the electric charge of the Cs ions $e^{*}=0.28e$ (see [200]). Due to the similarities of Cu and Au surfaces, a similar effective charge can be expected for the Cs/Au surface. For such a 2D system, agreement with experimental data is obtained for $n_a=10^{18}$ m$^{-2}$ and $\eta _{\parallel }=10^{11}$ s$^{-1}$. In [170] we estimated the damping parameter associated with the covalent bond for Cs atom on Cu(100): $\eta _{\Vert cov}=3\times 10^9$ s$^{-1}$. However, the collisions between the ions, and between the ions and other surface defects, will also contribute to $\eta$. In this case, $\eta _{col}\sim v_T/l$ where $v_T\sim \sqrt{k_BT/M}$, and l is the ion mean free path. For $T=293$ K and $l\sim 1$ nm, we get $\eta _{col}=10^{11}$ s$^{-1}$.

For a spherical tip and a 2D system on top of the substrate, from (14.16) and (14.10) for $R\gg d$, we get the contribution to the friction from the 2D system

$$\begin{aligned} \Gamma _{ad}^s=\frac{3RM\eta V^2}{2^6d\pi n_ae^{*2}}. \end{aligned}$$

(14.18)

At $d=20$ nm, this friction is $\sim$ two orders of magnitude smaller than for the cylindrical tip.

14.2 Friction Due to Spatial Fluctuations of Static Charge in the Bulk of the Sample

In this section, we consider a dielectric substrate with a stationary, inhomogeneous distribution of charged defects. Such a situation was investigated experimentally [21] by employing a fused silica sample irradiated with $\gamma$ rays. In the course of irradiation, positively charged centers (Si dangling bonds) are generated. Randomly distributed positive charges are compensated for by randomly distributed negative charges; thus, on average, the sample is electrically neutral. We model the sample as though it consists of microscopically small volume elements $\Delta V_i$. Each volume element chosen is sufficiently small that, within it, not more than one charge center is present. Thus, the electric charge $q_i$ of each element is equal to $\pm e$ or 0, in such a way that the average $\langle q_i\rangle =0$. We will consider the fluctuations of charges in different volume elements i, j to be statistically independent, so that $\langle q_iq_j\rangle =0$ for $i\ne j$. The mean square of charge fluctuations within a given element $\langle q_iq_i\rangle \approx 2Ne^2$, N is the average number of positive charges in one volume element. In the absence of the cross terms the average tip-sample friction coefficient is determined by adding the friction coefficient from all the charges $q_i$. According to (14.10), the contribution to the friction coefficient from the charge $q_i$ in the element $\Delta V_i$ is given by

$$\begin{aligned} \Delta \Gamma _{i\Vert }=\lim _{\omega \rightarrow 0}Ne^2\int _0^\infty dqq^2e^{-2qd_i} \frac{\mathrm {Im}R_p(\omega , q)}{\omega }, \end{aligned}$$

(14.19)

where $d_i=D(x_i, y_i)-z_i$. Here the coordinates $x_i, y_i, z_i$ give the position of the ith volume element in the substrate, and $D(x_i, y_i)$ is the distance between the sample and points $x_i, y_i$ located on the surface of the tip. The total friction coefficient is obtained by summing all of the volume elements. Replacing the sum by an integral ($N\sum \rightarrow n\int d^3r$, where n is the number of the positive charge centers per unit volume), and integration over z gives

$$\begin{aligned} \Gamma _{\Vert }=\lim _{\omega \rightarrow 0}\frac{ne^2}{2}\int _0^\infty dqq\int dx\int dye^{-2qD(x, y)}\frac{\mathrm {Im}R_p(\omega , q)}{\omega }. \end{aligned}$$

(14.20)

For a cylindrical tip, $D(x, y)=d+x^2/2R$, and we get

$$\begin{aligned} \Gamma _{\Vert }^c=\lim _{\omega \rightarrow 0}\frac{\sqrt{\pi R}ne^2w}{2}\int _0^\infty dqq^{1/2}e^{-2qd}\frac{\mathrm {Im}R_p(\omega , q)}{\omega }. \end{aligned}$$

(14.21)

Using the same parameters as in Sect. 14.1, for a cylindrical gold tip separated by $d=10$ nm from a dielectric sample with $n=7\times 10^{17}$ cm$^{-3}$ we get $\Gamma _{\Vert }=4.4\times 10^{-20}$ kg s$^{-1}$.

For the tip surface with a 2D structure on it, using (14.16) we get

$$\begin{aligned} \Gamma _{2D\Vert }^c=\frac{1}{2^{5/2}}\left( \frac{e}{e^{*}}\right) ^2\sqrt{ \frac{R}{d}}\frac{nw}{n_a}M\eta =\frac{e^2nw}{16\sigma _{eff}d}. \end{aligned}$$

(14.22)

With $\sigma _{eff}=n_ae^{*2}/2M\eta d_1=4\times 10^9$ s$^{-1}$, $n=7\times 10^{17}$ cm$^{-3}$, and with the other parameters the same as before, we get, for $d=10$ nm, $\Gamma _{2D\Vert }^c=3.5\times 10^{-12}$ kg s$^{-1}$, which is nearly the same as the experimental observations [21]. Thus, the theory of the friction between a gold tip and silica substrate with an inhomogeneous distribution of the charged defects is consistent with the theory of friction between a gold tip and gold substrate (see Sect. 14.1). In both theories, we have assumed that the gold surfaces are covered by a 2D structure.

The analysis above has ignored the screening of the electric field in the dielectric substrate. This can be justified in the case of very small tip–sample separations (substantially smaller than the screening length), since only defects in the surface layer of thickness d contribute to the integral in (14.20). When the screening is included, the effective electric field outside the sample will be reduced by the factor $(\varepsilon +1)/2$ [191], and the friction coefficient will be reduced by the factor $((\varepsilon +1)/2)^2$, which is equal to 6.25 in the case of silica. However, the heterogeneity of the tip surface can be larger than for the sample surface, so that the damping parameter $\eta$ can be larger for the 2D structure on the surface of the tip. This increase in $\eta$ and screening effect will compensate for each other.

14.3 Contact Electrification and the Work of Adhesion

When two solid objects are removed after adhesional or frictional contact, they will in general remain charged [325–329]. At the macroscopic level, charging usually manifests itself as spark discharging upon contact with a third (conducting) body, or as an adhesive force. The long-range electrostatic force resulting from charging is important in many technological processes such as photocopying, laser printing, electrostatic separation methods, and sliding-triboelectric nanogenerators based on in-plane charge separation [330]. Contact charging is also the origin of unwanted effects such as electric shocks, explosions or damage of electronic equipment.

Contact electrification is one of the oldest areas of scientific study, originating more than 2500 years ago when Thales of Miletus carried out experiments showing that rubbing amber against wool leads to electrostatic charging [331]. In spite of its historical nature and practical importance, there are many problems related to contact electrification that are not well understood, such as the role of surface roughness [332–334], surface migration [335] and contact de-electrification [336].

The influence of contact electrification on adhesion has been studied in pioneering work by Derjaguin et al. [337, 338] and by Roberts [339]. These studies, and most later studies, have assumed that removing the contact between two bodies results in the bodies having uniform surface charge distributions of opposite signs. However, a very recent work [340–342] has shown that the bodies in general have surface charge distributions that vary rapidly in space (on the sub-micrometer scale) between positive and negative values, and that the net charge on each object is much smaller (sometimes by a factor of ${\sim }1000$) than would result by integrating the absolute value of the fluctuating charge distribution over the surface area of a body.

Contact electrification occurs even between solids made from the same material [340]. This has been demonstrated for silicon rubber (PDMS). If two rubber sheets in adhesive contact (contact area A) are separated, they obtain net charges $\pm Q$ of opposite sign. However, as discussed above, each surface has surface charge distributions that fluctuate rapidly between positive and negative values, with magnitudes much higher than the average surface charge densities $\pm Q/A$. The net charge scales with the contact surface area $Q\sim A^{1/2}$, as expected based on a picture where the net charge results from randomly adding positively and negatively charged domains (with individual area $\Delta A$) on the surface area A: when $N=A/\Delta A>> 1$, we expect from statistical mechanics that the net charge on the surface A is proportional to $N^{1/2}$ as observed [340]. Note that in the thermodynamic limit, $A\rightarrow \infty$, the net surface charge density $Q/A = 0$.

In this Section, we will present an accurate calculation of the contribution from contact electrification to the work of adhesion to separate two solids. The same problem has been addressed in a less accurate approach by Brörmann et al. [343]. They assumed that the charged domains formed a mosaic pattern of squares, where each nearby square has a charge of the opposite sign but of equal magnitude. The authors applied an approximate procedure [325] (see also [344, 345]) to this problem, in order to obtain the contribution to the work of adhesion from charging. In this Section, we will present a general theory in which, the surface charge distribution $\sigma (\mathbf{x})$ is characterized by the density–density correlation function $\langle \sigma (\mathbf{x})\sigma (\mathbf{0})\rangle$, the power spectrum of which can be deduced directly from Kelvin Force Microscopy (KFM) potential maps. We find that, for polymers, the contact electrification may only contribute a small amount to the observed work of adhesion. However, more KFM measurements at smaller tip–substrate separations are necessary to confirm the conclusion presented below.

Fig. 14.2

After separation the bottom solid has the surface charge distribution $\sigma _0 (\mathbf{x})$ and the top solid has the surface charge distribution $-\sigma _0 (\mathbf{x})$, i.e., the charge distribution on one surface is the negative of that of the other surface

We will calculate the force between the two charged solids when the surfaces are separated by the distance d, see Fig. 14.2. The lower surface has the surface charge density $\sigma _0(\mathbf{x})$, where $\mathbf{x} = (x, y)$ is the in-plane coordinate, and the upper surface has the surface charge density $-\sigma _0 (\mathbf{x})$, i.e., the charge distribution on one surface is the negative of that of the other surface. We write the electric field as $\mathbf{E} = -\nabla \phi$ so that the electric potential $\phi$ satisfies $\nabla ^2 \phi = 0$ everywhere except for $z=0$ and $z=d$. We write

$$\sigma _0 (\mathbf{x}) = \int d^2q \ \sigma _0 (\mathbf{q}) e^{i\mathbf{q}\cdot \mathbf{x}}.$$

The electrostatic stress tensor

$$\sigma _{ij} = {1\over 4 \pi } \left( E_i E_j -{1\over 2} \mathbf{E}^2 \delta _{ij}\right) .$$

Here we are interested in the zz-component:

$$\begin{aligned} \sigma _{zz} = {1\over 8 \pi } \left( E_z^2-\mathbf{E}_\parallel ^2\right) . \end{aligned}$$

(14.23)

In the space between the surfaces, the electric potential:

$$\phi = \int d^2q \left[ \phi _0(\mathbf{q}) e^{-qz}+ \phi _1(\mathbf{q}) e^{qz} \right] e^{i\mathbf{q}\cdot \mathbf{x}}$$

where $\mathbf{q} = (q_x, q_y)$ and $\mathbf{x} = (x, y)$ are 2D vectors. Thus for $z=0$:

$$\begin{aligned} E_z = \int d^2 q \ q \left[ \phi _0(\mathbf{q}) - \phi _1(\mathbf{q}) \right] e^{i\mathbf{q}\cdot \mathbf{x}}, \end{aligned}$$

(14.24)

and

$$\begin{aligned} \mathbf{E}_\parallel = \int d^2 q (-i\mathbf{q}) \left[ \phi _0(\mathbf{q}) + \phi _1(\mathbf{q}) \right] e^{i\mathbf{q}\cdot \mathbf{x}}. \end{aligned}$$

(14.25)

Using (14.23), (14.24) and (14.25) gives

$$\begin{aligned} \int d^2 x \ \sigma _{zz} = 2 \pi \mathrm{Re} \int d^2 q \ q^2 \phi _0(\mathbf{q}) \phi ^*_1(\mathbf{q}). \end{aligned}$$

(14.26)

We now calculate $\phi _0(\mathbf{q})$ and $\phi _1(\mathbf{q})$. We write the electric potential $\phi (\mathbf{q}, z)$ as:

$$\phi = \phi _0 e^{-qz}+ \phi _1 e^{qz} \ \ \ \ \ \mathrm{for} \ \ 0<z<d,$$

$$\phi = \phi _2 e^{qz} \ \ \ \ \ \mathrm{for} \ \ z<0,$$

$$\phi = \phi _3 e^{-q(z-d)} \ \ \ \ \ \mathrm{for} \ \ z>d.$$

Since $\phi$ must be continuous for $z=0$ and $z=d$, we get:

$$\begin{aligned} \phi _0+ \phi _1 = \phi _2, \end{aligned}$$

(14.27)

$$\begin{aligned} \phi _0 e^{-qd}+\phi _1 e^{qd} = \phi _3. \end{aligned}$$

(14.28)

Let $\epsilon _0$ and $\epsilon _1$ be the dielectric function of the region between the bodies ($0<z<d$) and in the bodies ($z<0$ and $z>d$), respectively. In our application, the space between the bodies is filled with non-polar gas and $\epsilon _0 \approx 1$. From the boundary conditions $\epsilon _0 E_z(0^+)-\epsilon _1 E_z(-0^+) = 4 \pi \sigma _0$ and $\epsilon _1 E_z(d+0^+)- \epsilon _0 E_z(d-0^+) = - 4 \pi \sigma _0$, and using (14.27) and (14.28), we get:

$$\phi _0 + g \phi _1 = {2\pi \over q} \sigma $$

$$g \phi _0 e^{-qd}+\phi _1 e^{qd} = - {2\pi \over q} \sigma $$

where $\sigma = \sigma _0 2 / (\epsilon _1+\epsilon _0)$ and $g= (\epsilon _1-\epsilon _0)/(\epsilon _1+\epsilon _0)$. Solving these equations gives:

$$\phi _0 = {2\pi \over q} { \sigma \over 1+ge^{-qd}}, \ \ \ \ \ \ \phi _1 = e^{-qd} \phi _0.$$

Using these equations in (14.26) gives

$$\begin{aligned} \langle F_z \rangle =\int d^2 x \ \langle \sigma _{zz} \rangle = (2 \pi )^3 \int d^2 q \ \left\langle | \sigma (\mathbf{q})|^2 \right\rangle {e^{-qd} \over (1+ge^{-qd})^2}, \end{aligned}$$

(14.29)

where we have performed an ensemble average denoted by $\langle .. \rangle$.

Consider the correlation function:

$$\left\langle | \sigma (\mathbf{q})|^2 \right\rangle ={1\over (2\pi )^4} \int d^2x d^2x' \left\langle \sigma (\mathbf{x}) \sigma (\mathbf{x}') \right\rangle e^{i\mathbf{q}\cdot (\mathbf{x}-\mathbf{x}')}.$$

Assuming that the statistical properties of the surface charge distribution are translational invariant, we get:

$$ \left\langle \sigma (\mathbf{x}) \sigma (\mathbf{x}') \right\rangle = \left\langle \sigma (\mathbf{x}-\mathbf{x}') \sigma (\mathbf{0}) \right\rangle $$

and

$$\left\langle | \sigma (\mathbf{q})|^2 \right\rangle ={A_0\over (2\pi )^4} \int d^2x \big \langle \sigma (\mathbf{x}) \sigma (\mathbf{0}) \big \rangle e^{i\mathbf{q}\cdot \mathbf{x}}$$

where $A_0$ is the surface area. If $\bar{\sigma }= \langle \sigma (\mathbf{x}) \rangle$ denotes the average surface charge density, then we define the charge density power spectrum:

$$\begin{aligned} C_{\sigma \sigma } (\mathbf{q}) = {1\over (2\pi )^2} \int d^2 x \big \langle [ \sigma (\mathbf{x})-\bar{\sigma }] [ \sigma (\mathbf{0})- \bar{\sigma }] \big \rangle e^{i\mathbf{q}\cdot \mathbf{x}}. \end{aligned}$$

(14.30)

Using this definition, we get:

$$\begin{aligned} \left\langle | \sigma (\mathbf{q})|^2 \right\rangle ={A_0\over (2\pi )^2} \left[ C_{\sigma \sigma } (\mathbf{q}) + \bar{\sigma }^2 \delta (\mathbf{q}) \right] . \end{aligned}$$

(14.31)

Substituting (14.31) in (14.29) gives

$$\langle F_z \rangle = 2 \pi A_0 \bar{\sigma }^2 + 2 \pi A_0 \int d^2 q \ C_{\sigma \sigma }(\mathbf{q}) {e^{-qd} \over (1+ge^{-qd})^2}.$$

We expect the statistical properties of the surface charge distribution to be isotropic, which implies that $C_{\sigma \sigma }(\mathbf{q})$ only depends on the magnitude $q=|\mathbf{q}|$. This gives:

$$\langle F_z (d) \rangle = 2 \pi A_0 \bar{\sigma }^2 + (2 \pi )^2 A_0 \int d q \ q C_{\sigma \sigma }(q) {e^{-qd} \over (1+ge^{-qd})^2}.$$

The first term in this expression is the attraction between the surfaces due to the (average) uniform component of the charge distribution, which, as expected, is independent of the separation between the surfaces (similar to a parallel condenser). The second term is the contribution from the fluctuating components of the surface charge distribution. The contribution to the work of adhesion from the surface charge is given by:

$$U = \int _0^d dz \ \langle F_z(z) \rangle = 2 \pi A_0 \bar{\sigma }^2 d$$

$$\begin{aligned} + (2 \pi )^2 A_0 \int _0^\infty d q \ q C_{\sigma \sigma }(q) \int _0^d dz {e^{-qz} \over \left( 1+ge^{-qz} \right) ^2}. \end{aligned}$$

(14.32)

For an infinite system, the first term in U increases without limit as the surfaces are separated. For bodies of finite size, the expression given above for the contribution from the net charging is of course only valid for separations smaller than the linear size of the bodies (i.e. $d < L$, where $A_0=L^2$), and the interaction energy will decay like $\sim 1/d$ for large separation. Thus, for a finite-sized system, the contribution to the normalized work $U/A_0$ to separate the solids, from the first term in U, will be of order $\sim$ $\bar{\sigma }^2 L$, with a prefactor which depends on the actual shape of the bodies. Note that, in the thermodynamic limit $L\rightarrow \infty$, since $\bar{\sigma }\sim 1/L$ this contribution to $U/A_0$ will actually vanish. Roberts (see [339]) has argued that the first term in (14.32) gives a negligible contribution to the work of adhesion also for finite-sized objects. Here, we take a more pragmatic approach and we will not include this term in the work of adhesion, in particular since it depends on the shape of the bodies, and also because, experimentally, it is easy to measure the work to separate the solids such small distance that the first term in (14.32) is completely negligible, see [346]. The contribution to the work of adhesion from the second term in (14.32) (for $d \rightarrow \infty$) is:

$$\begin{aligned} w_\mathrm{ch} = {U \over A_0} = {(2 \pi )^2 \over 1+g} \int _0^\infty d q \ C_{\sigma \sigma }(q). \end{aligned}$$

(14.33)

Note that the integral

$$\begin{aligned} \int d^2q \ C_{\sigma \sigma }(\mathbf{q}) = \left\langle [ \sigma (\mathbf{x})-\bar{\sigma }]^2 \right\rangle = \left\langle \Delta \sigma ^2 \right\rangle \end{aligned}$$

(14.34)

is the mean of the square of the fluctuating surface charge distribution. Using this equation we can write:

$$\begin{aligned} w_\mathrm{ch} = {2 \pi \over 1+g} {\left\langle \Delta \sigma ^2\right\rangle \over \langle q \rangle } \end{aligned}$$

(14.35)

where

$$\begin{aligned} \langle q \rangle = {\int _0^\infty d q \ q C_{\sigma \sigma }(q) \over \int _0^\infty d q \ C_{\sigma \sigma }(q)}. \end{aligned}$$

(14.36)

The study above is for the limiting case where the surfaces separate so fast that no decay in the surface charge distribution takes place before the separation is so large as to give a negligible interaction force. Experiments [341] have shown that the charge distribution decays with increasing time as $\mathrm{exp} (-t/\tau )$, where the relaxation time $\tau \approx 10^3 \ \mathrm{s}$ depends on the atmospheric condition (e.g., the humidity and concentration of ions in the surrounding gas). Taking into account the decay in the surface charge distribution, and assuming $z=vt$ (where v is the normal separation velocity), we need to replace the integral over z in (14.32) with:

$$ f(q, v) = \int _0^\infty dz {e^{-qz} e^{-2t/\tau } \over \left( 1+ge^{-qz} \right) ^2} = \int _0^\infty dz {e^{-(qz +2z/v\tau )} \over \left( 1+ge^{-qz} \right) ^2}$$

and (14.32) becomes

$$\begin{aligned} w_\mathrm{ch} = (2 \pi )^2 \int _0^\infty d q \ q C_{\sigma \sigma }(q) f(q, v). \end{aligned}$$

(14.37)

In the limit $v \rightarrow \infty$, we have $f\rightarrow 1/[q(1+g)]$ and, in this limit, (14.37) reduces to (14.35). In the opposite limit of very small surface separation velocity, $f \rightarrow v \tau /[2(1+g)^2]$ and in this limit

$$\begin{aligned} w_\mathrm{ch} = {(2 \pi )^2 v \tau \over 2(1+g)^2} \int _0^\infty d q \ q C_{\sigma \sigma }(q) = {\pi v \tau \left\langle \Delta \sigma ^2\right\rangle \over (1+g)^2}. \end{aligned}$$

(14.38)

Note that this expression is of the form (14.35) with $1/\langle q \rangle$ replaced by $v \tau /[2(1+g)]$. Since, typically, $\tau \approx 10^3 \ \mathrm{s}$ and $(1+g) \approx 1$ and $\langle q \rangle \approx q_1 \approx 10^9 \ \mathrm{m}^{-1}$ (where $q_1$ is defined below), we get $v_\mathrm{c} = 2 (1+g)/(\langle q \rangle \tau ) \approx 10^{-12} \ \mathrm{m/s}$. In most applications, we expect the separation velocity to be in the vicinity of the crack tip $v>> v_\mathrm{c}$, and in this case the limiting equation (14.35) holds accurately. Note, however, that the separation velocity v may be much smaller than the crack tip velocity.

In the KFM measurement, the local potential at some fixed distance d above the surface is measured, rather than the surface charge density. From the measured data, the potential power spectrum

$$C_{\phi \phi } (\mathbf{q}) = {1\over (2\pi )^2} \int d^2 x \left\langle [\phi (\mathbf{x})-\bar{\phi }] [\phi (\mathbf{0})-\bar{\phi }] \right\rangle e^{i\mathbf{q}\cdot \mathbf{x}}$$

can be directly obtained. However, we can relate the potential to the charge density:

$$\phi (\mathbf{q}) = {2 \pi \over q} \sigma (\mathbf{q}) e^{-qd}.$$

Thus

$$\begin{aligned} C_{\sigma \sigma }(\mathbf{q}) = {q^2 \over (2 \pi )^2} C_{\phi \phi } (\mathbf{q}) e^{2qd}. \end{aligned}$$

(14.39)

The results presented above are in Gaussian units. To obtain (14.39) in SI units we must multiply the right-hand side with $(4\pi \epsilon _0)^2$, where $\epsilon _0 = 8.8542 \times 10^{-12} \ \mathrm{C V^{-1} m^{-1}}$. Thus:

$$\begin{aligned} C_{\sigma \sigma }(\mathbf{q}) = 4 \epsilon _0^2 q^2 C_{\phi \phi } (\mathbf{q}) e^{2qd}. \end{aligned}$$

(14.40)

To get (14.33) in SI units we must multiply the right-hand side by $(4 \pi \epsilon _0)^{-1}$:

$$\begin{aligned} w_\mathrm{ch} = {\pi \over 2\epsilon _0 (1+g)} \int _0^\infty d q \ C_{\sigma \sigma }(q). \end{aligned}$$

(14.41)

Fig. 14.3

a The voltage power spectrum $C_{\phi \phi }$ and b the surface charge density power spectrum $C_{\sigma \sigma }$ as a function of the wavevector. The results have been calculated from the measured (KFM) voltage maps for PDMS/PDMS (blue) and PDMS/polycarbonate (PC) (red) [341]

We now analyze experimental data involving elastically soft solids with smooth surfaces, where the initial contact between the solids is complete due to the adhesion between the solids. In [341], several such systems were studied and here we focus on PDMS rubber against PDMS. After breaking the adhesive contact between two sheets of PDMS (which involves interfacial crack propagation), the electrostatic potential a distance d above one of the surfaces was probed using KFM measurements. From the measured potential map, we have calculated the potential power spectrum $C_{\phi \phi }(q)$ and then, from (14.40), the charge density power spectrum $C_{\sigma \sigma }(q)$. The measurements were done at the tip–substrate separation $d\approx 10^{-7} \ \mathrm{m}$, and since the electric potential from a surface charge density distribution with the wavevector q decay as $\mathrm{exp}(-qd)$ with the distance d from the surface, the KFM is effectively limited to probing the surface charge distribution with wavevector $q < 1/d$. In Fig. 14.3 we show both power spectra for $q < 2\times 10^7 \ \mathrm{m^{-1}}$. Note that the charge density power spectrum appears to saturate for a large wavevector, say $q>q_0$, with $q_0 \approx 10^7 \ \mathrm{m}^{-1}$. This result follows if, as expected, the process of creating surface charges is uncorrelated in space at short length scales. In that case, $\langle \sigma (\mathbf{x}) \sigma (\mathbf{0}) \rangle \sim \delta (\mathbf{x})$ and using (14.30), this gives $C_{\sigma \sigma } (\mathbf{q}) = \mathrm{const}$. The fact that $C_{\sigma \sigma } (\mathbf{q})$ decays for decreasing q for $q < q_0 \approx 10^7\,\mathrm{m}^{-1}$ implies that at some length scales $\lambda _0 = 2 \pi /q_0 \approx 0.6\,\upmu \mathrm{m}$ the charge distribution becomes correlated. The physical reason for this may relate to inhomogeneities on the PDMS surface; for example. due to filler particles (see below).

We assume that the charge density power spectrum saturate for $q>q_0$ at $C^0_{\sigma \sigma } \approx 2.2\times 10^{-23} \ \mathrm{C^2/m^2}$ (see Fig. 14.3b). The assumption that the process of creating surface charges is uncorrelated in space at short length scales gives $C^0_{\sigma \sigma } = (2\pi )^{-2} e^2 n$, where $n= 1/\lambda _1^2$ is the number of elementary charges ($\pm e$) per unit surface area. Thus, we obtain $n=3.4 \times 10^{16} \ \mathrm{m}^{-2}$ and $\lambda _1 \approx 6\,\mathrm{nm}$ and $q_1 = 2 \pi /\lambda _1 \approx 10^9\,\mathrm{m}^{-1}$. The charge density $\langle |\sigma |\rangle = ne \approx 0.5\,\upmu \mathrm{C/cm}^2$ is similar to what was estimated by Baytekin et al. [341]. Using (14.34) we get the mean square charge fluctuation $\langle \Delta \sigma ^2\rangle \approx \pi q_1^2 C^0_{\sigma \sigma } \approx 7\times 10^{-5}\,\mathrm{C}^2/\mathrm{m}^4$ or the rms charge fluctuation $\approx$ $1\upmu \mathrm{C/cm}^2$, which, as expected, is similar to ne.

From (14.41) we get $w_\mathrm{ch} \approx (q_1-q_0)C^0_{\sigma \sigma } /\epsilon _0$, where we have used that $\pi /[2(1+g)] \approx 1$. The large wavevector cut-off $q_1$ is of order $2\pi /\lambda _1$, where $\lambda _1$ is of the order of the average separation between the surface charges (which we assume to be point charges of magnitude $\pm e$, where e is the electron charge). Since $q_0<< q_1\approx 10^9 \ \mathrm{m}^{-1}$ we get $w_\mathrm{ch} \approx q_1 C^0_{\sigma \sigma } /\epsilon _0 \approx 0.002\,\mathrm{J/m}^2$. This value is smaller than the measured work of adhesion during adiabatic (very slow) separation of the surfaces where [346] $w \approx 0.05\,\mathrm{J/m}^2$.

The calculation above does not include the interaction between the charges when the surface separation is smaller than $\sim$1 nm. However, this contribution cannot be accurately estimated without an accurate knowledge of the exact location and spatial extent of the localized charges, and probably also require knowledge about how the charge separation processes occur; for example, whether it involves electron tunneling at some finite surface separation.

The analysis above is based on the assumption that the surface charge density power spectrum saturates at a value $C^0_{\sigma \sigma } \approx 2.2\times 10^{-23}\,\mathrm{C}^2/\mathrm{m}^2$ for large wavevectors. This hypothesis should be tested by performing KFM measurements to smaller tip–substrate separations. The number of surface charges per unit area, n, which determines the cut-off $q_1$ in the study above, may also be probed by surface reaction experiments, such as the bleaching experiments reported on in [342].

Sylgard 184, which was used in [341], is intrinsically a heterogeneous polymer with siliceous fillers [347]. Even though the filler is partially modified by organic groups, it imparts non-negligible polarity of the polymer as evidenced from the high contact angle hysteresis (${\sim }20^\circ$–$40^\circ$) of water on this polymer as compared with that (${\sim }5^\circ$) on a pure PDMS matrix. X-ray photoelectron spectroscopy [348] also shows that the Silicon (Si2p) peak of the silica is 1 eV higher than that of the surrounding matrix, thus suggesting that the electron affinity of the silica rich region is probably different from that of the surrounding matrix. So, when two surfaces of sylgard 184 are brought close to each other, electrons may be transferred from one type of domain to another, which may show up as heterogeneous patches when the surfaces are separated. If the binding energy of the Si2p peak is an indicator, the PDMS matrix is more electron-rich than the silica-rich region. The breaking of the siloxane bond requires a very large force and is unlikely to contribute to charging [348–350]. Silica almost always has silanol (SiOH) groups. The silanol groups may form very weak hydrogen bond with the oxygen of polydimethylsiloxane. If that happens, some charge transfer may occur during the separation of the surfaces, which will show up as heterogeneous charge after the two surfaces are separated. This idea may be tested experimentally by performing KFM experiments using a clean PDMS network that does not have silica fillers.

At low crack-tip velocities, where the viscoelastic energy dissipation at the crack tip, and other non-equilibrium effects, are negligible (see [346]), the work of adhesion is usually assumed to result from the van der Waals interaction between the surfaces at the interface. The study above indeed indicated that the contributions from contact electrification gives only a small fraction ($\sim$4%) of the observed work of adhesion.

To summarize, we have derived a general expression for the contribution to the work of adhesion from contact electrification, and we have shown that, for PDMS (and probably for polymers in general), the contact electrification gives only a small fraction of the observed work of adhesion. More KFM measurements at smaller tip–substrate separations are necessary to confirm this conclusion.

Fig. 14.4

Transverse displacement X(z, t) of the cantilever centerline. The attractive force T acts along the z axis. A restoring force $X^{\prime }T$ acts on the tip when $\alpha \ne 0$. From [169]

14.4 Influence of Attractive Force on Cantilever Eigenfrequencies

Besides the damping of vibrations, interaction of the tip with a sample leads to a change in the cantilever eigenfrequency. When the cantilever geometry satisfies the conditions $L\gg w\gg c$, where the $L,\, w$ and c are the length, the width, and the thickness of the cantilever, respectively, we can employ the equation of elasticity [351] to describe the dynamics. In the case of an isotropic material, the transverse displacement X(z, t), of the beam centerline (along the x axis) (see Fig. 14.4), satisfies the differential equation:

$$\begin{aligned} \rho _cS\frac{\partial ^2}{\partial t^2}X(z, t) = -EI_i\frac{\partial ^4}{\partial z^4}X(z, t) + T\frac{\partial ^2}{\partial z^2}X(z, t), \end{aligned}$$

(14.42)

where $\rho _c$ is the density of the cantilever material, $S=wc$ is the cross-section area, E is Young’s modulus, $I_i=c^3w/12$ is the bending moment of inertia, and T is a stretching force. The clamped end, at $z=0$, imposes the boundary conditions $X(z=0)=X^{\prime }(z=0)=0$. The free end at $z=L$ imposes the boundary conditions $M_y\equiv EI_iX^{\prime \prime }=0$ and $F_x\equiv -EI_iX^{\prime \prime \prime }+TX^{\prime }=0$, where $M_y$ and $F_x$ are the moment of the elastic force and the elastic force in the y and x directions, respectively. The term $TX^{\prime }$ describes the effect of the restoring force when there is a non-vanishing angle between T and the centerline direction at $z=L$ (see Fig. 14.4). This expression for the restoring force is valid in the case of small angles $\alpha$ between vertical axis and the beam baseline, i.e. when $\sin \alpha \approx \tan \alpha =-X^{\prime }(z=L)$. The fundamental vibration eigenfunction is given by $X(z, t)=X_0(z)\cos (\omega t)$, where

$$\begin{aligned} X_0(z) = A \left[ (\Lambda ^2 \cos \lambda L + \lambda ^2\cosh \Lambda L) (\cos \lambda z -\cosh \Lambda z) \right. + \nonumber \\ + (\Lambda \sin \lambda L - \lambda \sinh \Lambda L)(\Lambda \sin \lambda z - \lambda \sinh \Lambda z)\big ]. \end{aligned}$$

(14.43)

Here A is a constant,

$$\begin{aligned} \Lambda = \big [\tau /2 + (\kappa ^4 + \tau ^2/4)^{1/2}\big ]^{1/2}, \quad \lambda = \big [-\tau /2 + (\kappa ^4 + \nonumber \\ +\tau ^2/4)^{1/2}\big ]^{1/2}, \quad \tau = T/EI_i, \quad \kappa ^4 = \omega ^2\rho _cS/EI_i. \end{aligned}$$

(14.44)

The value of $\kappa$ in (14.44) obeys the following eigenvalue equation:

$$\begin{aligned} 2\kappa ^4 + (\tau ^2 + 2\kappa ^4)\cos \lambda L \cosh \Lambda L + \kappa ^2\tau \sin \lambda L \sinh \Lambda L = 0. \end{aligned}$$

(14.45)

The minimum value of $\kappa$ ($\kappa _{min}$) that obeys (14.45) determines the fundamental eigenfrequency and the explicit form of the fundamental eigenfunction X(z, t). In the absence of the stretching force, ($T=0$), we get $\kappa ^2 = 3.516/L^2$ and

$$\begin{aligned} \omega = (3.516/L^2)(EI_i/\rho _iS)^{1/2}. \end{aligned}$$

(14.46)

To estimate $\omega$ we set $c = 250$ nm, $w= 7$ $\upmu$m, $L = 250$ $\upmu$m, and use the material constants of Si: $\rho _c = 2.33\times 10^3$ kg$\cdot$m$^{-3}$, $E = 9\times 10 ^{10}$ kg $\cdot$ m$^{-1}\cdot$ s$^{-2}$. Then, we have $\omega /2\pi \approx 4\times 10^3$ s$^{-1}$, which is approximately the same as the value quoted in [21].

In the general case, the fundamental frequency is a function of the tip–sample separation, d, bias voltage V, tip geometry, and concentration of charge centers via the attraction force, T(V). In the case of attraction due only to an external bias voltage, there is a quadratic dependence of T on V. The proportionality coefficient depends on the geometry of the tip–sample system. The results of numerical calculations of the $\omega (V)$ dependence are shown in Fig. 14.5 for a cylindrical geometry showing that the bias voltage can have a considerable effect on the frequency shift.

Fig. 14.5

Numerical results for the cantilever frequency $\omega$ and effective mass $m_{ef}$ as a function of the bias voltage V in the case of a cylindrical tip and $d=20$ nm (see text). From [169]

In the limit of very large V (when $TL^2/(EI_i)\gg 1$), the first term on the right-hand-side of (14.42) can be omitted, and (14.42) is transformed into the equation for the vibration of a string. In this case, a linear dependence of $\omega$ on V ($\omega \propto V/L$) should occur.

Along with the resonant frequency, the cantilever effective mass also depends on V. The effective mass $m_{eff}$ is the coefficient entering the oscillator equation,

$$\begin{aligned} m_{eff}\frac{\partial ^2X(t)}{\partial t^2} +\Gamma \frac{\partial X(t)}{\partial t} + kX(t) = 0, \end{aligned}$$

(14.47)

as a parameter that determines the inertial force ($\Gamma$ is a friction coefficient, k a spring constant, and X(t), as stated previously, is the coordinate of the tip). The mass $m_{eff}$ depends on the coefficients in (14.42). This dependence can be obtained from the requirement that the kinetic energy of the flexible beam equals that of the oscillator, the two being of one and the same physical quantity. This condition defines $m_{eff}$ as

$$\begin{aligned} m_{eff}=\rho _cS\int _0^L dz X_0^2(z)/X_0^2(L), \end{aligned}$$

(14.48)

where $X_0(z)$ is given by (14.43).

Thus, $m_{eff}$ depends on the variation of $X_0$ with z, which in turn depends on the stretching force T and the other parameters in (14.42). In the simplest case, $T=0$, we have $m_{eff}=\rho _cLS/4$, i.e., a quarter of the beam’s actual mass. In general, the possible variation of $m_{eff}$ should be taken into account when the friction coefficient $\Gamma$ is obtained experimentally from the relation $\Gamma = m_{eff}\omega /Q$, where Q is the quality factor. In other words, not only the quality factor and the eigenfrequency, but also the value of $m_{eff}$ is required to obtain $\Gamma$. The dependence $m_{eff}(V)$ is shown in Fig. 14.5. It is seen to be similar to that of $\omega (V)$ at small values of V.

Figure 14.6 shows the effect of the Casimir–Lifshitz force on the fundamental frequency in the case of a spherical tip. The three curves illustrate the tendency of $\omega (d, R)$ to increase with the attractive force, which varies with d and R as $T\sim R/d^2$ (see (S.4) in Appendix S1). The frequency shift observed in [21] at $d=2$ nm for a gold sample corresponds to the curve $R =0.5$ $\upmu$m in Fig. 14.6. When we use the experimental value 1 $\upmu$m for the radius of curvature in the y direction, and set the radius of curvature in the x direction equal to the cantilever thickness $c=0.25$ $\upmu$m, then, according to Appendix S2, we obtain radius of equivalent spherical tip $R_{exp}=\sqrt{R_xR_y}=0.5$ $\upmu$m, i.e., $R_{exp}=R$. Hence the frequency shift observed by Stipe et al. [21] might be attributable to the Casimir–Lifshitz effect.

Fig. 14.6

Effect of Casimir–Lifshitz force on the frequency $\omega$. Solid line, $R=0.5$ $\upmu$m; dashed line, $R=1.0$ $\upmu$m; dotted line, $R=2$ $\upmu$m. From [169]

There are other physical mechanisms that may contribute to the frequency shift. Spatial variation of the surface potential, whose role in damping of vibrations was discussed in Sect. 14.1, will also contribute to the attraction force between the tip and the sample. In the presence of a bias voltage V, the attraction force of the tip to the sample is given by (see (S.8) from Appendix S2)

$$\begin{aligned} T^s(d)=\frac{RV^2}{4d} . \end{aligned}$$

(14.49)

In the case of variation of the surface potential bias voltage, V in (14.49), should be replaced on $V_0$, where $V_0$ is the root-mean-square variation of the surface potential. As discussed in Sect. 14.1, in the experiment in [21] $V_0\approx 0.2$V. Thus, the ratio of the Casimir–Lifshitz force (see (S.4) in Appendix S1) to the force induced by the work function anisotropy is of the order

$$\begin{aligned} \frac{R\hbar \omega _p}{32\sqrt{2}d^2}\Bigg /\frac{RV_0^2}{4d} \approx (12 \text{ nm }/d). \end{aligned}$$

(14.50)

For $d=2$ nm, this ratio is equal to 6. Hence, the anisotropy of the work function has a negligible effect on the frequency shift obtained in [21].