Atomic Scale Friction: What can be Deduced from the Response to a Harmonic Drive?

V. Zaloj, M. Urbakh, J. Klafter
1998 Physical Review Letters  
In this Letter we investigate the response of a confined chain to a harmonic driving force. A model is introduced which mimics recent measurements on friction using surface forces apparatus. The model predicts a critical driving amplitude below which the response is linear. For higher amplitudes the system exhibits a nonlinear behavior and shear thinning. A novel origin for the thinning is proposed which stems from energy dissipation due to stick-slip motion and the transition to smooth
more » ... We establish relationships between the microscopic parameters of the system and phenomena observed in rheology and tribology. [S0031-9007(98) There has been a growing number of attempts to understand the relationship between frictional forces and the microscopic properties of nanosystems. Recent studies on friction [1-9] have exposed a broad range of phenomena and new behaviors which help shed light on some "old" concepts which are already considered textbook material. These include the static and kinetic friction forces, transition to sliding, and thinning, which have been widely discussed but whose microscopic meaning is still lacking. There have been, generally, two approaches used to investigate shear forces of confined liquids: rheological (oscillatory external drive) and tribological (constant driving velocity). In the bulk the two approaches lead to similar results, but less is known about the relationship between rheology and tribology in nanoscale confined systems. Establishing a relationship between these approaches is essential for creating a unifying description of the response to shear and for further progress of related fields. In this Letter we concentrate on the rheological side of the problem and its relationship to tribology. Our proposed predictions can be tested experimentally by simultaneously analyzing the time series of the spring forces and the shear moduli. We suggest an interpretation to the observed dramatic enhancement in the effective viscosity [3, 6] and to the effect of shear thinning in thin confined systems [3, [10] [11] [12] . In order to mimic the commonly used experimental configuration [13] we introduce a model of a chain embedded between two plates, one of which is externally driven, as depicted in Fig. 1a . The top plate of mass M is connected to a spring, of spring constant K 1 , which is harmonically driven, and to a spring K 2 , which is a response spring. The chain consists of N identical particles each of mass m 0 , which interact harmonically. The dynamical behavior of the system (chain 1 plates) follows the equations of motion: Here the coordinate of the harmonically driven stage is X d ͑t͒ X dm sin͑v d t͒. The microscopic parameter h 0 is responsible for the dissipation of the kinetic energy of each particle. U͑x͒ 2U 0 cos͑2px͞b͒ represents the particle-plate interaction. We assume that the interparticle interaction is harmonic V ͑x i 2 x i61 ͒ k͑x i 2 x i61 7 a͒ 2 ͞2; a and b are the periodicities of the undisturbed chain and of the potential U͑x͒, respectively. Let us introduce the following dimensionless variables and parameters: the coordinates of the top plate Y X͞b and the chain particles y i x i ͞b; the time t v 0 t, where the v 0 ͑2p͞b͒ p NU 0 ͞M is the frequency of the top plate oscillations in the minima of the potential U͑x͒ for the case of noninteracting particles; g Nh 0 ͞Mv 0 , which is the dimensionless friction constant,´ Nm 0 ͞M, the ratio of the chain and top plate masses, a V͞v 0 , the ratio of frequencies of the mechanical free oscillations of the top plate, V p ͑K 1 1 K 2 ͒͞M, and v 0 , k K 1 ͑͞K 1 1 K 2 ͒ the mechanical factor, D 1 2 ͑a͞b͒ the misfit of the substrate and chain periods, and r ͑v c ͞v͒ 2 , the ratio of the frequencies related to interparticle v c p k͞m 0 and particle-plate v ͑2p͞b͒ 3 p U 0 ͞m 0 interactions. In rheological experiments the behavior of the system is governed by two dimensionless parameters: The amplitude A d X dm ͞b and frequency d v d ͞v 0 of the driving stage. Another relevant quantity is the driving velocity V d A d d, which is used in rheology to analyze data, and which is relevant when making comparison to tribological results. 0031-9007͞98͞81(6)͞1227(4)$15.00
doi:10.1103/physrevlett.81.1227 fatcat:6tdswp3xynbb3pdd57f2wa2mke