In this chapter we introduce the methods conventionally used to explore friction on the nanoscale. The leading position among the instrumental setups is held by the atomic force microscope. Here we will briefly illustrate the type of forces sensed by this instrument and its basic modes of application. Other experimental techniques in nanotribology are the surface force apparatus, the quartz crystal microbalance and also, to some extent, scanning tunneling microscopy and transmission electron microscopy. Virtual experiments rely on molecular dynamics simulations. A short introduction to this method will be followed by a series of numerical results reproducing friction and wear measurements at the atomic level.

Atomic force microscopy

In a typical atomic force microscope (AFM) [24] a sharp micro-fabricated tip is scanned over a surface. Standard AFM tips are made of silicon or silicon nitride, but tips can be also coated to allow a large variety of material combinations. The probing tip is attached to a cantilever force sensor, the sensitivity of which can be well below 1 nN. Images of the surface topography are recorded by controlling the tip–sample distance in order to maintain a constant (normal) force. This is made possible by using piezoresistive cantilevers, or, most commonly, by a light beam reflected from the back side of the cantilever into a photodetector, which allows one to monitor the cantilever bending (Fig. 17.1). The lateral force between tip and surface is responsible for the cantilever torsion and can be measured if the photodetector is equipped with four quadrants. If this is the case the AFM can be used as a friction force microscope (FFM), see Appendix A. The design of a home-built AFM, optimized for friction measurements in ultra-high vacuum (UHV), is shown in Fig. 17.2.

The tip–sample force can be related not only to the static bending or torsion of the cantilever.

Introduction

Following the pioneering experiments on the glass transition in PMMA, which gave a good approximation to monodisperse hard-sphere systems, Pusey and van Megen (1989) and coworkers started a series of experiments with the aim of characterising the transition. Most of the experiments were performed through laser light scattered by density fluctuations characterising the transition from liquid to supercooled liquid and eventually to an amorphous solid. An efficient method was also used to measure the time average in a non-ergodic system using averages over different scattering volumes and wave vectors. The comparison with the predictions of MCT was performed in an extended fashion, showing a relatively good agreement with the experimental findings, to a 20% level of accuracy. The most relevant result of this important set of measurements was the detection of the structural arrest point, a result that is not easy to obtain in normal liquids due to the existence of activated dynamics or hopping effects. The latter are supposedly responsible for the crossing of the barriers that confine the system in a potential well.

MCT was subsequently applied to potentials with an attractive tail following the short-range repulsion, and lead to the behaviour described in the previous chapter on the the theory of supercooled liquids. The most relevant finding was the evidence of the existence of higher-order singularities, which were already defined and studied within MCT, in systems with short-range attractive interactions. Shortly after the predictions of MCT on the consequences of an attractive interaction in hard-sphere systems obtained, many attempts were made to experimentally demonstrate their validity. In particular, first the re-entrant glass line was detected, then the effect of the A3 singularity was shown and finally the higher singularity of type A4 was identified. The experiments on these various aspects of the behaviour of supercooled liquids are illustrated in the following sections.

Introduction

Interactions between particles, both in molecular fluids and colloidal systems, are generally characterised by a strong short-range repulsion, which is responsible for excluded volume effects, followed by an attraction of variable strength. The latter is at the origin of cluster formation, a process that produces many different physical phenomena of great importance in the physics of simple and complex fluids. In atomic and molecular systems the most relevant effect of attraction is the appearance of critical points accompanying phase transitions, while in complex fluids, besides critical effects, peculiar phenomena develop such as aggregation, percolation, glass and sol–gel transitions. Recently the latter have been collectively named arrest phenomena, since their common feature is the pronounced slowing down of the dynamics. We first outline briefly the phenomenology and the approaches based on aggregation and percolation, which describe situations in which the attractive interaction is so strong that the colloidal particles adhere, leading to the formation of macroscopic clusters that eventually invade the physical sample. In the case of reversible aggregation the particles form the so-called physical gels. When the aggregation is irreversible, chemical gels are formed.

Thanks to the possibility of forming reversible or irreversible reactive links, during aggregation clusters of particles tend to coalesce and form larger aggregates. In the case of reversible bond formation a fragmentation process is also present. Aggregation is an ubiquitous process that can be observed in disparate situations at various length and time scales. Examples are polymer chemistry, aerosol systems, cloud physics, clusters of galaxies in astrophysics, etc. Although aggregation in colloidal suspensions has long been studied, it has become a subject of renewed interest in recent years because it is a non-equilibrium phenomenon, the final stage of which may lead, among other things, to the formation of a gel. We briefly summarise various aspects of clustering by introducing the Smoluchowski aggregation formalism, which is used in many different physical approaches to aggregation, and a brief summary of the salient aspects of percolation that are important for the physical phenomena we describe.

Introduction

As we mentioned in previous chapters, structural arrest refers in general to various phenomena which have in common a marked slowing down of the dynamics, including aggregation and cluster formation, percolation and glass transition. In the last few years the study of the glass transition of colloidal systems in the supercooled region revealed a series of new phenomena which were interpreted as the possibility of including in a single framework glass and gel transition in liquids (Sciortino and Tartaglia, 2005).

Summarising what we described at length in previous chapters, the study of the glass transition in supercooled liquids was initiated in hard-sphere systems, where the cage effect is the relevant physical mechanism for the structural arrest. In the case of colloids an interaction potential with a hard core and an attractive tail brings in the usual phase separation with a critical point, but also a new mechanism of structural arrest, predicted theoretically. Besides the repulsive glass transition due to excluded volume effects, a line of attractive liquid-glass appears, the driving mechanism of which is the bonding between the particles. Since the attractive glass line extends to a wide range of volume fractions of the dispersed phase, the colloid could also give rise to a low-density gel, similar to the gelling due to cluster formation and aggregation at very low densities.

In order to clarify the mutual relation between the coexistence curve and the glass lines, a detailed study was performed (Tartaglia, 2007) in a model binary system with hard-core repulsion followed by a short-range square-well attractive potential. The ideal glass transition line has been studied with a simulation of the dynamics of the colloidal system and by evaluating the diffusion coefficient D for long times. The loci of constant D, the iso-diffusivity lines, are used and by extrapolating to the limit D → 0 one gets an indication of the position of the arrest line.

In this chapter, we describe the elements of liquid theory. Our intention is not to present all aspects of liquid theory in its complete form, but instead only those that will be useful and sufficient for discussing its different applications for presentations in subsequent chapters. It will be addressed in the context of understanding problems that arise in complex fluids and colloidal science discussed in subsequent chapters.

In Section 2.1 we introduce the concept of the pair correlation function and the structure factor which are fundamental quantities when discussing applications of liquid theories to the analysis of scattering data, including light scattering, X-ray scattering and neutron scattering. We would like to remind the reader here that the pair correlation function and thus the structure factor are intimately connected with the statistical thermodynamics of the system. In this sense the study of the structure factor in the liquid state using scattering techniques is to investigate some aspect of the thermodynamics of the liquid system.

Then we discuss the solution of Ornstein–Zernike equation in its different approximations. In particular, in Section 2.2.3, we illustrate the use of the Baxter method, an elegant analytical method for solving hard-sphere and adhesive hardsphere systems in the Percus–Yevick approximation. These two systems are the model systems for simple liquids as well as for the colloidal solutions. In Section 2.2.4 we present an analytical solution for the case of a narrow-squarewell potential that avoids some aspects of the unphysical features of the Baxter solution of the adhesive hard-sphere system. We shall show in a later chapter the applications of this analytical solution for studying the kinetic glass transition in a micellar system.

We then skip the background introduction to the liquid theories of ionic solutions such as the classical Debye–Hückel theory and the Poisson–Boltzmann theory of ionic solutions, as well as the mean spherical approximation solutions of the so-called primitive model of ionic solution already available in Blum (1980).