Fredy R. Zypman

Evidence of self-organized criticality in dry sliding friction

This letter presents experimental results on unlubricated friction, which suggests that stick–slip is described by self-organized criticality (SOC). The data, obtained with a pin-on-disc tribometer examines the variation of the friction force as a function of time—or sliding distance. This is the first time that standard tribological equipment has been used to examine the possibility of SOC. The materials were matching pins and discs of aluminium loaded with 250, 500 and 1000 g masses, and matching M50 steel couples loaded with a 1000 g mass. An analysis of the data shows that the probability distribution of slip sizes follows a power law. In addition, the frequency power spectrum follows a 1/fα pattern with α in the range 1.1–1.3. We perform a careful analysis of all the properties, beyond the two just mentioned, which are required to imply the presence of SOC. Our data strongly support the existence of SOC for stick–slip in dry sliding friction.

Dry friction avalanches: Experiment and theory

Experimental evidence and theoretical models are presented supporting the conjecture that dry friction stick-slip is described by self-organized criticality. We use the data, obtained with a pin-on-disk tribometer set to measure lateral force, to examine the variation of the friction force as a function of time. We study nominally flat surfaces of matching aluminum and steel. The probability distribution of force drops follows a negative power law with exponents mu in the range 3.2-3.5. The frequency power spectrum follows a 1/f alpha pattern with alpha in the range 1-1.8. We first compare these experimental results with the well-known Robin Hood model of self-organized criticality. We find good agreement between theory and experiment for the force-drop distribution but not for the power spectrum. We explain this on a physical basis and propose a model which takes explicitly into account the stiffness and inertia of the tribometer. Specifically, we numerically solve the equation of motion of a block on a friction surface pulled by a spring and show that for certain spring constants the motion is characterized by the same power law spectrum as in experiments. We propose a physical picture relating the fluctuations of the force drops to the microscopic geometry of the surface.

Negentropy Generation and Fractality in the Dry Friction of Polished Surfaces

We consider the Robin Hood model of dry friction to study entropy transfer during sliding. For the polished surface (steady state) we study the probability distribution of slips and find an exponential behavior for all the physically relevant asperity interaction-distance thresholds. In addition, we characterize the time evolution of the sample by its spatial fractal dimension and by its entropy content. Starting from an unpolished surface, the entropy decreases during the Robin Hood process, until it reaches a plateau; thereafter the system fluctuates above the critical height. This validates the notion that friction increases information in the neighborhood of the contacting surface at the expense of losing information in remote regions. We explain the practical relevance of these results for engineering surface processing such as honing.

Squeezing out hidden force information from scanning force microscopes

A method to measure force-separation curves with a scanning force microscope is presented. Forces within the “snap to contact” are obtained by high-speed (MHz) measurement of cantilever deflection signals analyzed using the generalized beam theory. Numerical simulation is used to demonstrate the effectiveness of the method. Experimental results show that the method yields complete continuous force-separation curves with flimsy cantilevers in fluids allowing for sensitive force measurements in nonvacuum environments.

Exact expressions for colloidal plane–particle interaction forces and energies with applications to atomic force microscopy

We begin by deriving a general useful theoretical relationship between the plane–particle interaction forces in solution, and the corresponding plane–plane interaction energies. This is the main result of the paper. It provides a simple tool to obtain closed-form particle–plane forces from knowledge of plane–plane interaction energies. To illustrate the simplicity of use of this general formalism, we apply it to find particle–plane interactions within the Derjaguin–Landau–Verwey–Overbeek (DLVO) framework. Specifically, we obtain analytical expressions for forces and interaction energies in the van der Waals and the electrical double layer cases. The van der Waals expression is calculated here for benchmarking purposes and is compared with well-established expressions from Hamaker theory. The interactions for the electric double layer situation are computed in two cases: the linear superposition approximation and the constant surface potential. In both cases, our closed-form expressions were compared with existent numerical results. We also use the main result of this paper to generate an analytical force-separation expression based on atomic force microscope experiments for a tip and surface immersed in an aqueous solution, and compare it with the corresponding numerical results. Finally, based on our main result, we generalize the Derjaguin approximation by calculating the next order of approximation, thus obtaining a formula valuable for colloidal interaction estimations.

The poor man's scanning force microscope

The Macroscope (Zypman F R and Guerra-Vela C 2001 Eur. J. Phys. 22 17-30), an educational large-scale version of a scanning force microscopes cantilever-tip system, is used in the presence of nonlinear forces. This paper presents quantitative experimental evidence confirming the validity of the beam model (BM) (Eppel S J, Todd B A and Zypman F R 2000 Materials Issues and Modeling for Device Nanofabrication ed L Merhari et al (Pittsburgh, PA: Materials Research Society) pp 584, 189) as a proper reconstruction algorithm. As a teaching laboratory experiment, the force measurements are first done directly with a variety of dynamometer-like setups. Subsequently, the measurements are performed indirectly with the Macroscope from the cantilever resonant frequency shifts and the BM algorithm. Two central results of this work lie in its ability to compare forces obtained by traditional algorithms with known forces, and to illustrate in a hands-on fashion the principles behind the working of a scanning force microscope.

Improved analysis of the time domain response of scanning force microscope cantilevers

The snap-to-contact instability encountered in scanning force microscopy-force spectroscopy (SFMFS) limits the range of forces measurable in SFM force–distance experiments. We have generalized the flexural beam theory for SFM cantilevers to include tip interactions that are present in the snap-to-contact region. We compare solutions for the beam theory with the simple harmonic oscillator (SHO) that is often used to approximate SFM cantilevers. The limitations of the SHO model are encountered when large force gradients are present. This causes the beam shape to change leading to error when the SHO is used to reconstruct force curves collected in the snap-to-contact region. We quantify the error introduced into a force–separation curve reconstructed using the SHO approximation by numerical simulation. The force–separation curve reconstructed by the SHO was significantly inaccurate and had distorted separation dependence. This makes physical interpretation of force curves reconstructed using the SHO approxim...

The macroscopic scanning force `microscope'

A homemade, macroscopic version of the scanning force microscope is described. It consists of a cantilever under the influence of external forces, which mimic the tip-sample interactions. The use of this piece of equipment is twofold. First, it serves as a direct way to understand the parts and functions of the scanning force microscope, and thus it is effectively used as an instructional tool. Second, due to its large size, it allows for simple measurements of applied forces and parameters that define the state of motion of the system. This information, in turn, serves to compare the interaction forces with the reconstructed ones. This comparison is in itself extremely relevant, as it cannot be done directly with the standard microscopic set-up.

Uncertainties of coherent states for a generalized supersymmetric annihilation operator

This study presents supersymmetric coherent states that are eigenstates of a general four-parameter family of annihilation operators. The elements of this family are defined as operators in Fock space that transform a subspace of a definite number of particles into a subspace with one particle removed. The emphasis is on classifying parameter space in various regions according to the uncertainty bounds of the corresponding coherent states. Specifically, the uncertainty in position-momentum is analyzed, with specific focus on characterizing regions of minimum uncertainty states, regions where the uncertainties are bounded from above, and where they grow unbound.

Intrinsic dissipation in atomic force microscopy cantilevers.

In this paper we build a practical modification to the standard Euler-Bernoulli equation for flexural modes of cantilever vibrations most relevant for operation of AFM in high vacuum conditions. This is done by the study of a new internal dissipation term into the Euler-Bernoulli equation. This term remains valid in ultra-high vacuum, and becomes particularly relevant when viscous dissipation with the fluid environment becomes negligible. We derive a compact explicit equation for the quality factor versus pressure for all the flexural modes. This expression is used to compare with corresponding extant high vacuum experiments. We demonstrate that a single internal dissipation parameter and a single viscosity parameter provide enough information to reproduce the first three experimental flexural resonances at all pressures. The new term introduced here has a mesoscopic origin in the relative motion between adjacent layers in the cantilever.