Nanotribology: tip–sample wear under adhesive contact

Tribology International 33 (2000) 443–452 www.elsevier.com/locate/triboint

Nanotribology: tip–sample wear under adhesive contact R. Bassani, M. D’Acunto

*

Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione, Universita` di Pisa, Via Diotisalvi 2, I-56126 Pisa, Italy Received 5 May 1999; received in revised form 10 February 2000; accepted 20 February 2000

Abstract In this report, the irreversible variation of mass of the probe tip of an atomic force microscope (AFM) is considered from theoretical and numerical points of view through statistical methods. The tip–sample interaction due to the intermittent-contact operating mode of an AFM is modelled as a double-well potential where the wear mechanism, which reveals itself as mass sticking to the probe tip, is described as a transition between the two potential wells. We evaluate the interaction of a silicon nitride AFM/FFM tip with gold in order to compare the results with those obtained from previous experimental and numerical studies.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Ornstein–Uhlenbeck process; Fokker–Planck equation; Probe tip mass

1. Introduction Friction and wear between two sliding solids are very common processes in nature, but they are also among the most complex and least understood. Considerable success has been achieved recently in the quantitative measurement of friction forces on the atomic scale and in understanding the underlying macroscopic mechanism in the case of sliding friction without wear. This result has been made possible by considerable improvements in the characterization of interfaces. Moreover, the rapid development of atomic force microscopes (AFMs) and the availability of large computer resources have made possible quantitative predictions for the tribological processes. Successes on both the experimental and theoretical sides have opened up a new research field called nanotribology [1]. From an experimental point of view, the AFM is a rather versatile instrument that can be used to investigate how surface materials are moved or removed on the micro and nano scale; for example, in scratching and wear and nanofabrication/nanomachining. By scanning the sample in two dimensions with the AFM, wear scars are generated on the surface. Primarily, wear depth is a function of normal load. At loads below a critical value,

* Corresponding author. E-mail address: [email protected] (M. D’Acunto).

elastic deformations are responsible for low wear. For higher normal loads, wear depth is expected to increase with normal load value. But wear is not only the prerogative of the sample: the tip of the AFM probe also experiences wear. This paper investigates the complex wear process of probe tips by taking into consideration the variation of tip mass as a consequence of adhesive interaction between the tip and sample. 2. The model It is well known that, in the non-contact operating mode (NC-AFM), for small oscillations the cantilever can be considered as a damped harmonic oscillator characterised by a spring stiffness k and effective mass m*. The resonance frequency of the system is n=k/m*. A variation of the effective tip mass leads to a shift in the resonance curve for the cantilever. Unfortunately, the shift in the resonance curve is not only due to variation of the tip mass but also due to external force gradients. In fact, if the tip is immersed in an external force field F(z), where z is the vertical position, the cantilever behaves like an oscillator with an effective spring constant given by ∂F(z) keff⫽k⫺ . ∂z

(1)

The main effect of the external force gradient is to shift the oscillator resonance curve by the following amount

0301-679X/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 0 0 ) 0 0 0 2 8 - 1

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Nomenclature a.u. atomic units D diffusion coefficient f(t) white stochastic force Feff effective potential j probability flow density k spring stiffness kB Boltzmann constant m* effective mass of tip M rate at which mass attaches to the tip (%) nA initial atomic population P(x, t兩x0, t0) density of probability t time T temperature U(x) double-well potential UA initial bounding value for potential UB final bounding value for potential top barrier potential UC Ueff effective potential v velocity V wear volume (nm3) w wear rate resulting from atomic transitions x position x0 jump-to-contact position y0 vertical distance between layers b velocity dissipation coefficient g noise variable d Thomas–Fermi screening length h(t) white stochastic force l noise dissipation coefficient s decay factor w resonance frequency wA bottom well frequency wC top barrier frequency

1 ∂F(z) ⌬n⫽n⬘⫺n⫽⫺ ∗ , 2m n ∂z

(2)

where n⬘ is the new oscillator resonance frequency. If the oscillations are small compared with the typical length scale of the force field, the force gradient ∂F(z)/∂z can be measured by detecting this resonance shift. One may ask what happens if the shift of the resonance curve is due to a variation of the effective tip mass m*, owing to an interaction of instantaneous time contact between the tip and the surface sample. If the force gradients are known, it is possible to measure the effect of adhesion of the sample material to the tip probe. In the NC-AFM mode a shift overlap between the external force field and the variation of tip mass should not happen since there is no contact between the tip and the sample. However, there is another AFM technique in which shift overlap can lead to a measurable effect: the intermittent-contact

(IC-AFM) mode. Intermittent-contact atomic force microscopy is similar to NC-AFM, except that for ICAFM the vibrating cantilever tip is brought closer to the sample so that at the bottom of its travel it just barely hits, or taps, the sample (Figs. 1 and 2). As in NC-AFM, for IC-AFM the oscillation amplitude of the cantilever changes in response to tip–sample spacing. An image representing surface topography is obtained by monitoring these changes. Some samples are best handled using IC-AFM instead of contact or non-contact AFM. ICAFM is less likely to damage the sample than contact AFM because it eliminates lateral forces between the tip and the sample. The measure of shift as consequence of effective tip mass can result in new insights into micro/nanowear, nanoscratching and nanofabrication processes. The fundamental idea is that the interaction between a probe tip and sample surface can be dealt with in a mesoscopic

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Fig. 1.

Interatomic force versus distance curve.

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interaction under the adhesive force is assumed to be an Ornstein–Uhlenbeck process transition [5]. The double-well potential system has a long-standing history in physics, chemistry and engineering investigations. In recent times, it has been employed to study the quantum coherence in surface–tip transfer of adatoms in AFM/scanning tunnelling microscopy (STM) by Salmeron et al. [6]. In the present paper the shape of double well and the difference between double minima and the barrier height (Fig. 3) depend on the length of contact time, normal load and distance travelled. Working with the IC-AFM mode, short time of contact, low load and the transfer of material due to adhesive forces only are considered. It is well known that, starting from contact, the curve of force versus tip–sample distance exhibits a marked hysteresis when the surfaces are separated. It was shown also that separating the surfaces prior to contact results in no hysteresis. Hysteresis is a consequence of adhesive bonding between the two materials and separation is accompanied by inelastic processes in which the topmost layers of the sample surface adhere to the tip probe. Moreover, it was observed that the diameter of the contact area decreases during lifting of the tip, resulting in the formation of a thin adhesive neck as a consequence of ductile extension until breaking occurred at a distance of separation of about 1 nm [7]. We follow the mechanism just described to evaluate the wear volume as a function of the energy barrier of separation between the tip probe and the sample, i.e., the difference in energy between point C and point A as shown in Fig. 3.

3. Wear rate

Fig. 2. Amplitude of vibration versus frequency plot in IC-AFM, showing that a small mass variation of the tip probe leads to a shift in resonance frequency.

In this section a quantitative evaluation is given of the wear rate of material transfer when a probe tip is in adhesive contact with a sample surface in a mesoscopic approach. In order to have a general framework for the

picture [2,3]. From this point of view, irreversible processes take place when material is induced to pass from the sample surface to the tip, and an oscillating behaviour can be obtained with the inverse transition from tip to sample. A double-well potential can be used to simulate the interaction between the tip and the sample, and a Fokker–Planck equation describes the rate at which atoms from the sample are stimulated to cross the energy barrier. Moreover, fluctuating noise can be considered in order to give a more realistic picture to these calculations [4]. In this paper, we develop a quantitative evaluation for wear volume, i.e., the wear rate for a single atomic volume, when atomic mass is transferred from sample to tip in an irreversible way. Moreover, the tip–sample

Fig. 3. A double-well potential simulating the interaction during tip– sample contact. The upper well represents the sample surface and the lower well the tip probe, U(x) being the interaction potential. The shape and minimum values of the double well depend on the normal load, the time of contact and the distance travelled.

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whole process, internal degrees of freedom are not considered. For the sake of simplicity, single components of material transferred are considered to have atomic mass, although in a more realistic approach a cluster of atoms should be considered [6]. As already mentioned, before getting in contact, the surface layers of atoms interested in adhesive contact exist in their potential well. When the tip–sample contact is made active, a double-well potential simulating the contact has to be considered as shown in Figs. 3 and 4. Then the wear mechanism for the probe tip arises as a transition of surface sample atoms or molecules between state A, describing the surface sample, and state B, describing the probe tip. Moreover, this contact process can be affected by environmental fluctuations or white noise of thermal nature on the potential barrier. For a mass existing in a well potential A (Fig. 3), the equations of motion are as follows x˙⫽v, v˙⫽⫺bv⫺

(3a) dU(x) ⫹f(t) dx

(3b)

and g˙ ⫽⫺lg⫹h(t),

(3c)

where f(t) and h(t) are white stochastic forces of thermal nature; x and v are atomic position and velocity, respectively; g is a random variable; b and l are dissipation coefficients; and U(x) is a potential. If a linear coupling between position and the random variable g is assumed, then for a deterministic double well of the type a b U(x)⫽ x2⫹ x4 2 4

(4)

where a⬍0 and b⬎0, we have to insert into Eq. (3b) an effective force related to an effective potential Ueff(x, t) of type dU Feff⫽gx⫺ . dx

(5)

It is well known that the model described by Eqs. (3) and (5) relates to Brownian motion in a force field of a fluctuating double-well potential U(x), where the fluctuation is driven by the random variable g which, along with stochastic forces, is responsible for noise-induced transitions, equivalent to catalytic reduction of the potential barrier [5]. Although they are related above all to catalytic reactions, Langevin equations [Eqs. (3)] describe a transition A→B for a single atom. However, in order to develop a statistical approach for a population of atoms, it is possible to start from Eqs. (3) to reach a Fokker–Planck equation for a probability density P(x ,t兩x0, t0), where (x0, t0) is an initial state denoting position and time. Moreover, it is possible to distinguish two separate regimes, high internal viscosity and low internal viscosity, where the viscosity is relative to an internal atomic state of motion. Under conditions of high internal viscosity, i.e., when the velocity of the cluster relaxes much more rapidly than the position x, substituting v˙=0 into Eq. (3b) gives the following form for Eq. (3a) x˙⫽⫺

1dUeff(x) 1 ⫹ f(t), b dx b

(6)

to which an equation for the transition density P(x ,t兩x0, t0=0) can be associated [7]:

冉

冊

∂ ∂ 1dU(x) D ∂2 P(x, t兩x0)⫹ ⫺ P ⫽ 2 2P. ∂t ∂x b dx b ∂x

(7)

It is necessary to evaluate the transition A→B as shown in Fig. 3, i.e., wear rate in the stationary state. The stationary state is obtained for ∂ P(x, t兩x0)⫽0 ∂t or rather

冉

(8)

冊

∂ 1dU(x) D ∂P ⫺ ⫽0, P⫺ 2 ∂x b dx b ∂x

(9)

so that j⫽⫺

Fig. 4. After contact, the tip leaves the sample (arrow) carrying some of the surface material.

1dU(x) D∂ P⫺ 2 P b dx b ∂x

(10)

can be identified with the probability flow density. In the case of the double-well potential, supposing that the potential barriers are not too large, the velocity of passing over the barrier will be little compared with the dynamics of the settling of particles at the bottom of

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the well. Following the classical Kramers picture, it is assumed that it is possible to approximate the shape of the potential around the bottom of well A and around the top of the energy barrier C by the use of quadratic expressions:

冑

(11a)

冑

(11b)

UA(x)⫽UA⫹( 2pwA)2(x⫺xA)2 around A and UC(x)⫽UC⫺( 2pwC)2(x⫺xC)2 around C,

so that the number of particles nA in the proximity of A is

冕

447

to treat the limiting case of low viscosity in a similar fashion, which results in the following expression [4]:

冉

冊

UC−UA UC−UA exp ⫺ w⫽b kBT k BT

(16)

where the wear rate is now directly proportional to the friction term b. Finally, we obtain the wear volume for unit time of adhesive tip–sample contact as a product between initial atomic population, the rate of atoms passing through the energy barrier and the individual atomic volume n: V⫽nAwn.

(17)

⫹⬁

nA⫽

pA exp(⫺UA(x)/kT) dx,

(12)

⫺⬁

where pA dx is the number of particles between xA and xA+dxA, and extend to infinity the limits of integration, justified by the dominating contribution coming from a small region around A. Then nA can be calculated as a Gaussian integral and the results is

冪 2p exp(⫺U /k T).

pA nA⫽ wA

kBT

A

(13)

B

It is possible to define a wear rate w in one second by considering the ratio between the probability flow density j and the number nA j w⫽ ⫽ nA

冑

wA 2pkBT

exp(⫺UA/kBT),

冕 B

Eqs. (15) and (16) can be modified slightly if we consider additional fluctuating terms to the potential barriers due to noise that is not thermal in origin, which may be caused by the complexity of the dynamic interactions between the tip and the sample in a sliding process. The consideration of such fluctuations in the potential barrier is important since it may be hypothesised that, in certain catalytic reactions, the catalyst–environment interaction could give rise to effects that can be represented by an effective potential containing a noise supply coupled with a position variable. Fluctuating potential barriers are usually used in electrical conduction between metallic islands in a disordered material, for example. The resultant effect considering fluctuating potential barriers with respect to Eqs. (15) and (16) is a smooth variation in wear rate in the presence of sliding friction due to noise-induced transitions [8–10].

(14)

b exp(−U(x)/kBT) dx

4. Numerical results

A

where the integral that appears on the right-hand side is needed in order to normalize the wear rate, because the dominating contribution comes from around C. Therefore, substituting the approximate expressions [Eqs. (11a) and (11b)] for U(x) and extending the limits of the integration to infinity, it is also possible to evaluate this quantity as a Gaussian integral. Thus we obtain a constant wear rate as w⫽2p

冉

冊

wAwC UC−UA . exp ⫺ b kBT

(15)

As can be expected, the wear rate depends on an Arrhenius factor and it is interesting to note that the rate does not depend on the shape of the potential barrier, but only on the difference between the top barrier and the bound values A and C. The total number of particles passing the barrier in one second is equal to wnA. This is the case of high viscosity. It is also possible

In our simulation we consider a fixed number of atomic layers, about 200 atoms/layer exposing a [001] face, in which each layer is separated by y0, the vertical distance from the next neighbour. Moreover, s=1/ny0 is a decay factor for the probability of having a transition to the tip. Assuming the value of 0.2 nm for the layer spacing, it is reasonable to conclude that, for n⬇5÷6 layers, the decay factor s varies between 0.8 and 1 nm⫺1. This value is in good agreement with the result obtained in Ref. [6], in which was shown that attractive forces should decrease exponentially with a decay length of the order of s. The existence of a vertical decay imposes consideration of a weight factor for each single layer as exp(⫺n). This factor is needed to ensure consideration of individual Fokker–Planck equations for the probability density relative to different layers. In the spirit of the simulation results, we consider yielding steps involving a transformation zone consisting of a fixed number n of atomic layers, and which for simplicity is modelled as a cylindrical section. Only atoms within this zone are

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Fig. 5.

Wear rate w*=w/10⫺12 Hz as a function of energy barrier ⌬U* under the condition of high viscosity for the first three layers.

Fig. 6.

Wear rate w*=w/10⫺12 Hz as a function of energy barrier ⌬U* under the condition of low viscosity for the first three layers.

able to migrate when the neck deformation yields, whereas atoms outside this zone are assumed to be static [7]. First, we have plotted wear rate (in units of 1012 Hz) obtained from Eqs. (15) and (16) with a weight factor relative to layers for a population of atoms to pass through the barrier in the regimes of both high and low viscosity, see Figs. 5 and 6. Since the thermal energy at room temperature is about 10⫺14 erg, we have to con-

sider UC⫺UA=⌬U in units of 10⫺14, i.e., ⌬U* =⌬U/10⫺14 erg. For well frequencies and the dissipation coefficient b under high viscosity conditions we consider the relationship wA⬇wB⬇b⬇1012 s⫺1 [2]; under conditions of low viscosity b is chosen to be three orders of magnitude lower than under high viscosity conditions. As shown by Fig. 5, we note that for ⌬U*=0 a nonzero value for wear rate is expected. Summing up all six layers we get the total number of

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Fig. 7. Wear volume V (nm3) as a function of energy barrier for gold under the condition of high viscosity.

atoms crossing from A to B and, finally, the wear volume V is evaluated from Eq. (17) for gold populations, see Figs. 7 and 8. Fig. 9 shows a comparison between the numerical values obtained in this paper and the experimental values given in Refs. [2,3,7]. Note that now the wear volume is delineated as a function of the height of the energy barrier expressed in atomic units (a.u.), and the values for which the comparison is performed are inferred by the relation for energy

U(c)⫽⫺(1⫹c) exp(⫺c),

(18)

where c=(x⫺x0)/d is the dimensionless ratio between position and displacement. x0 is the jump-to-contact position and d is the Thomas–Fermi screening length, considered in our simulation as equal to unity in a.u. Fig. 10 shows the percentage rate difference (Err%) between the values considered in Fig. 9; we can see that the difference is less than 10%. Finally, Fig. 11 presents the percentage mass that remains attached to the silicon

Fig. 8. Wear volume V (nm3) as a function of energy barrier for gold under the condition of low viscosity.

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Fig. 9.

Wear volume for short contact time: comparison between simulated values (———) and experimental values (- - -).

Fig. 10. Percentage rate difference (Err%) between the values shown in Fig. 9.

nitride tip probe after a contact with a sample surface. To get a qualitative idea of the change in tip mass we can consider that, before the contact, the tip consists of a bottom layer of 72 Si3N4 molecules exposed on a [001] face; the next layer consists of 128 molecules; and the remaining five layers contain 200 atoms each. The real situation foresees also wear transfer from the tip or in a lighter form such as smearing of the tip, but more complexity arises from deformation of both the tip

and the sample during the sliding process, neck formation, variation of pressure contact, creep and general rearrangements of the atoms [7]. All of these features require sophisticated methods of analysis, either to throw new light on experimental results and/or on theoretical analysis (as in molecular dynamics simulations, for example) or to contribute further to understanding surface contact mechanics when the relative friction force involves adhesive terms as prescribed by Johnson-

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Fig. 11. Percentage mass M remaining attached to the silicon nitride tip probe after contact with the sample.

Kendall-Roberts and Derjaquin-Muller-Toporov theories [11–15].

5. Conclusions A general approach to the problem of wear in IC-AFM operating mode due to adhesive tip–sample interaction, under the form of the transfer of material from the sample to tip probe, has been developed. The objectives and results of the present paper can be summarized as follows: 1. Irreversible material transfer between a tip probe and a sample in adhesive contact is considered from theoretical and numerical points of view through statistical methods. A double-well potential transition simulates the tribological feature of tip–sample contact. Wear develops as the passage of atoms from well to well in the presence of thermal noise. A Fokker–Planck equation describes the passage rate and finally a wear volume is evaluated. 2. The results are independent of the shape of the potential barrier, depending only on the top and bound values of the barrier itself and on high and low internal states of the atomic friction. An increasing mass of the probe tip as consequence of wear is evaluated.

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