Influence of the electronic screening on charges in the ionic liquid . 

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Gibbs-Thompson effect

Looking at Fig. 2.8a, the drastic change in both elastic response and dissipation from D > S to D < S would actually rather suggest a complete confinement induced phase change, with a freezing of the confined ionic liquid inside the gap. A confinement induced phase transition is expected when the unfavorable bulk free energy is balanced by favorable (wetting) surface contributions, thus stabilizing the unfavored phase in confinement. This results in a shift for the phase transition, as observed for capillary condensation (the shifted liquid-gas phase transition) or capillary freezing (shifted crystallization) [2]. The balance of free energy leads to the so-called Gibbs-Thompson equation, which characterizes the critical confinement S at which the free energies of the liquid and solid phase become equal [2]: T = TC 􀀀 TB = 2 TB Lh S with = wl 􀀀 ws (2.5) where T = TC􀀀TB is the shift in transition temperature, in confinement TC as compared to the bulk transition occurring at TB. For the specific ionic liquid used here, TB = 􀀀71 C; wl and ws are the surface energy of the liquid and solid phase with respect to the wall/substrate, = 1:21 g/mL the density of the liquid phase and Lh = 47 kJ/kg the latent heat of melting [8] (see Table 2.1). Eq. (2.5) shows that if wetting of the solid-phase on the substrate is favored compared to that of the liquid ( ws < wl) the freezing temperature of the confined phase TC is larger than the bulk freezing temperature TB (TC > TB in Eq. 2.5).
Putting numbers, one gets TC 25 C for a an ionic liquid confined in a gap of 20 nm with 0:3 J/m2 (anticipating on the values below). In other words, the ionic liquid may freeze in nanoconfinement at room temperature. In this scenario, the distribution of confinement length measured experimentally, see inset if Fig. 2.10, can be understood as a signature of activation due to the first order character of the freezing transition, potentially facilitated by the prewetting phase on the substrate.

Dependence on the metallicity of the substrate

A delicate question though is to understand the variations with the metallic nature of the substrate. Following the argument above in terms of the shifted liquid-solid transition, this raises the question of the surface energy of the solid (crystal) phase with the substrate and how it is influenced by the metallic character of the substrate. Physically, one may propose a simple explanation in terms of image charges. To highlight the argument, let us consider a semi-infinite ionic crystal at the interface with a perfect metal, as sketched in Fig. 2.10b. The network of image charges builds a crystal structure with a (nearly) perfect symmetry with respect to the real upper half-lattice. Accordingly one expects the electrostatic contribution to the surface free energy to (nearly) vanish, as the system behaves as a single bulk lattice: elec ws ! 0. This requires of course a perfectly symmetric crystalline structure and this cancellation is not expected to occur for insulating substrates, or for disordered liquid phases. In other words, the (semi-infinite) ionic crystal has a lower surface energy at the interface with a metal wall as compared to an insulating substrate: insulating ws > metal ws . This shows that the crystal phase is favored on metallic surface as compared to an insulating one and the Gibbs-Thompson equation (Eq. 2.5) accordingly implies that the threshold confinement for the freezing transition should be larger with metal as compared to insulating confining surfaces.

Electronic screening by a Thomas-Fermi metal

In order to account for the non-ideal metallic nature of the confining walls, one should model the electronic screening inside the substrates. To this end, we use the simple Thomas- Fermi framework, based on a local density approximation for the free electrons gas [37].
This description provides a simple screening equation for the electric potential V (V) in the metal, where the screening length TF = 1=kTF [m] characterizes the typical length over which a defect charge is screened in the metal and is defined in terms of the density of states at the Fermi level, according to k2 TF = 4e2 @n @F ; nT the state occupation and F the Fermi level (see Section 2.2.3). The limit of large kTF (small TF) corresponds to the perfect metallic case for which V is uniformly zero.

Influence of the electronic screening on charges in the ionic liquid

The electronic screening therefore modifies the interactions of charge close to the liquid-wall interface. The Green function for the electrostatic interaction, replacing the Coulomb interaction, obeys equations: = 􀀀 Q (r 􀀀 r0) for z > 0 (2.6) 􀀀 k2 TF = 0 for z < 0 (2.7).
This allows us to calculate the energy of a semi-infinite ionic system in the presence of the metal wall, as U = 1 2 R drc(r) (r), with c the charge density. For the crystal phase cr(r) = Q P n(􀀀1)n(r 􀀀 Rn), with Rn the lattice sites, while for the liquid phase, the charge density liq(r) vanishes beyond a few molecular layer close to the wall. The calculation of this energy is a very challenging task, because both the one-body interaction of ions with their image charges, and the two-body interactions between ions are strongly modified by the presence of the confining metallic (TF) wall. Here, we develop a simplifying description which captures the main effects of wall metallicity on the surface electrostatic energies, with the objective to rationalize the experimental data. This framework is described in Appendix A.

Effect of metallicity on the freezing transition

Altogether we obtain = 0 (1 + F(kTFa)), where 0 [J/m2] is the surface tension difference between of the liquid-wall versus crystal-wall interfaces for an insulating substrate and including also the non-electrostatic contributions to the surface energy (van der Waals, …) and the possible (constant) contribution from the tungsten tip. The dimensionless parameter = e2(C 􀀀 L)=(16 0) quantifies the contribution of metallicity to the surface energies.
Now using the Gibbs-Thomson result, Eq. 2.5, one predicts that the increase in surface energy difference for better metals, i.e. for larger Thomas-Fermi wavevector kTF (Eq. 2.8) will lead to a shift in the critical confinement distance for the freezing transition according to S = 0 S(1 + F(kTFa)): (2.9) with 0 S the value for the perfectly insulating material defined in terms of 0 (here Mica).

Comparison with the experimental data

In Fig. 2.10, we compare the prediction for S with the experimental data for the various substrates investigated. Note that in doing this comparison, we estimated the values of Thomas-Fermi length based on the substrate conductivity and carrier density (Section 2.2.3). We also fixed the molecular length to the crystal lattice constant as 2a = 0:67 nm (as estimated from the molar volume of the ionic liquid). As shown in Fig. 2.10, a good agreement between the theoretical predictions and the experimental results is obtained, yielding 0 S = 15 nm and = 10:1. From the value for 0 and Eq. 2.5, one gets 0 0:2 J/m2. Using Eq. 2.9, and assuming L=C 0:8 as typical from such systems [15], this would predict a value of 100, in fair agreement with the one obtained from the fit of the experimental data, 10.

Effect of tension and bulk melting temperature

Finally, we might also expect the bias voltage U [V] applied between the two confining surfaces to affect the observed freezing transition. As shown in Figure 2.11, we sweep the applied voltage U between -1.8 V and +1.8 V (red curve), while simultaneously measuring the critical confinement distance S (blue points) for each bias voltage between the tungsten tip and a HOPG substrate. We observe hysteresis cycles in the critical confinement length, with a peak to peak amplitude of approximately 20 nm. Note that such hysteretic effects of the tension are not observed systematically (see for example the last bias cycle for which no effect on critical confinement length is observed). This is not surprising, considering the fact that hysteresis evidence the presence of metastable effects in the freezing transition.

Typical rheological curves and reversibility

We show in Fig. 3.4 additional examples of rheological curves for various contact conductance. The general trends of the rheological curves are maintained, with (1) a plastic transition at a critical oscillation amplitude aY, corresponding to a decrease in Z0 and an increase in Z00, and an increase in current fluctuations, (2) a plateau in the dissipative impedance at large oscillation amplitude, (3) a decrease of the conservative impedance to negative values Z0 < 0, corresponding to a capillary-like attraction at a critical oscillation amplitude aL.
Additionally, we show in Fig. 3.5 an example of a rheological curve obtained for increasing and decreasing oscillation amplitude. The plastic transition is found to be reversible, with here no visible hysteresis.

Table of contents :

Remerciements
Introduction
1 The Tuning Fork based Atomic Force Microscope 
1.1 Force measurements at the nanoscale
1.1.1 Static force measurements
1.1.2 Dynamic force measurements
1.1.3 The quartz tuning fork based AFM
1.2 The tuning fork as a mechanical resonator
1.2.1 The tuning fork and its resonant frequencies
1.2.2 Quartz-based sensing
1.2.3 Mechanical excitation
1.2.4 Resonance
1.2.5 Quality factor and force sensitivity
1.2.6 Parameters calibration
1.3 Dissipative and conservative response
1.3.1 Conservative and dissipative force field
1.3.2 Tuning fork in interaction
1.3.3 Ring-down experiments
1.4 Tuning-Fork based AFM set-up
1.4.1 Tuning Fork preparation
1.4.2 Integrated AFM Set-up
1.4.3 Signal acquisition
1.4.4 Signal processing and control
1.5 Limitation
1.5.1 Fundamental limitations
1.5.2 Experimental limitations
1.6 Conclusion
2 Capillary Freezing in Ionic Liquids 
2.1 General Context
2.2 Experimental Set-up
2.2.1 General Set-up
2.2.2 Ionic Liquids
2.2.3 Substrates
2.2.4 Tip
2.3 Solid-like response and prewetting
2.3.1 Dissipation of an AFM tip oscillating in a viscous fluid
2.3.2 Approach curve in the ionic liquid
2.3.3 Prewetting
2.4 Confinement-induced freezing transition
2.4.1 Gibbs-Thompson effect
2.4.2 Dependence on the metallicity of the substrate
2.5 Role of electronic screening
2.5.1 Electronic screening by a Thomas-Fermi metal
2.5.2 Influence of the electronic screening on charges in the ionic liquid .
2.5.3 Effect of metallicity on surface tensions
2.5.4 Effect of metallicity on the freezing transition
2.5.5 Comparison with the experimental data
2.6 Effect of tension and bulk melting temperature
2.6.1 Effect of tension
2.6.2 Effect of bulk melting temperature
2.7 Conclusion
3 Molecular Rheology of Atomic Gold Junctions 
3.1 General Context
3.2 Experimental set-up
3.2.1 Experimental set-up
3.2.2 Static mechanical properties of the junction
3.3 Rheology of a gold nanojunction
3.3.1 Viscoelastic junction properties
3.3.2 Typical rheological curves and reversibility
3.4 Yield stress and yield force for plastic flow
3.4.1 Yield force, yield stress and yield strain
3.4.2 Interpretation of the deformation mechanism
3.5 Dissipative response in the plastic regime
3.5.1 Friction coefficient
3.5.2 Liquid-like dissipative response
3.5.3 Frequency dependence of the plastic transition
3.5.4 Solid-like dissipation regime at large oscillation amplitude
3.6 Conservative force response and capillary attraction
3.6.1 Capillary attraction
3.6.2 Shear induced melting of the junction
3.6.3 Jump to contact at large oscillation amplitude
3.7 Prandtl-Tomlinson model
3.7.1 Equations and non-dimensionalization
3.7.2 Simulation procedure
3.7.3 Simulation results and limiting cases
3.7.4 Discussion
3.8 Conclusion
4 Non-Newtonian Rheology of Suspensions 
4.1 General context
4.1.1 Rheology of non-brownian suspensions
4.1.2 Shear thickening
4.1.3 Shear thinning
4.2 Experimental Set-up
4.2.1 Measuring normal and tangential force profiles between two approaching beads with the AFM
4.2.2 Particles, substrate and solvent
4.2.3 Rheology of macroscopic suspensions
4.3 Nanoscale force profile
4.3.1 Typical approach curve
4.3.2 Normal dissipative force
4.3.3 Normal force gradient
4.3.4 Tangential dissipative force
4.3.5 Approach in presence of a surface asperity
4.3.6 Approach between cornstarch particles
4.4 Frictional force profile
4.4.1 Characterization of the frictional regime
4.4.2 Distribution of friction coefficient and normal critical load
4.4.3 Ring-down and characterization of non-linearity
4.4.4 Measurements under moderate and high normal load
4.5 Results and Discussions
4.5.1 The shear thickening transition in PVC and Cornstarch
4.5.2 Shear thinning at low shear rate in PVC suspensions
4.5.3 Shear thinning at high shear rate in PVC suspensions
4.6 Conclusion
5 Conclusion and Perspectives 
5.1 General Conclusion and Perspectives
5.1.1 Nanoscale Capillary Freezing in Ionic Liquids
5.1.2 Molecular Rheology of Gold Nanojunctions
5.1.3 Non Newtonian Rheology of Suspensions
5.1.4 Instrumental Perspectives
5.2 On-Going Perspectives on Reactive Lubrication
5.2.1 The Tuning Fork based dynamic Surface Force Apparatus
5.2.2 Reactive Lubrication in Skiing
5.2.3 Reactive Lubrication in Ionic Liquids
A Interfacial energies with Thomas–Fermi boundary 
A.1 Surface energy of a crystal with a TF wall
A.2 Physical interpretation and an approximated scheme
A.3 Surface energy of a liquid with a TF wall
A.4 Relative wetting of the crystal versus the liquid at a TF wall
A.5 Molecular dynamics of a molten salt in confinement

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