# The Influence of Mechanical Deformations on Surface Force Measurements

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Figure 1.**(

**a**) Chemical structure and sizes of [C${}_{4}$C${}_{1}$Pyrr][NTf${}_{2}$]. Ion sizes are estimated from geometry, bond lengths and covalent radii, associated with the most stable configuration found by energy minimization (Chem3D 16.0, PerkinElmer Informatics). (

**b**) Schematic of the SFB experiment to measure the surface interactions and to characterize, in situ, the geometry, when a liquid is confined between two mica surfaces. (

**c**) Composition of the layers forming each solid surface, with associated thickness e, Young’s modulus E, Poisson’s ratio $\nu $ and elastic modulus $K=\frac{2}{3}\frac{E}{1-{\nu}^{2}}$ (values from [7,21,22]). For the anisotropic mica, the given mechanical properties correspond to the c-axis, the direction of interest for this study. A wide range of Young’s moduli have been reported for mica (50–500 $\mathrm{GPa}$ in [6]); the quoted value has been consistently obtained by Brillouin scattering [21] and nanoindentation [7].

## 3. Results and Discussion

#### 3.1. Calibrations in a Dry Atmosphere

- Since the conception of the SFA, negative distances from $-0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ to $-1.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ have been reported when two mica surfaces separated by water jump-in to contact [11,28,38,39,40,41,42]. This is due to the washing of gas molecules and organic contaminants (carbon compounds) that are spontaneously deposited on the mica surfaces in air [43] and to the dissolution in water of the potassium ions initially present on the mica surfaces. As adhesion is typically 10 times smaller in water than in a dry atmosphere, the mica is expected to be less compressed after the jump-in across water than during the calibration after the jump-in across a dry atmosphere. As compression of mica was not considered in these studies, the thickness of the contaminant layer may be underestimated, albeit not by more than a few angstroms given the much thinner mica used. Nevertheless, the dependence of this effect on the mica thickness and spring constant may explain—at least in part—the strong variability in the reported values.
- In the case of molecular liquid giving rise to a structural force profile under confinement, a good accuracy on the distance D is needed in order to identify the absolute number of ordered layers composing the film (as illustrated in the subsection titled “Influence of surface deformations on structural force profile”).
- Dynamic measurements can be performed with the SFA to determine the slip length associated with a flow of liquid in vicinity of a solid surface. By definition, the slip length is the distance between the hydrodynamic origin and the mechanical origin. Being able to measure nanometric slip lengths therefore requires having a subnanometric resolution on the mechanical zero [18,44,45,46,47,48].

#### 3.2. Contact Mechanics Across an Ionic Liquid

- The jump-out force ${F}_{\mathrm{s}}$ obtained with force measurement techniques is routinely used to deduce the surface energy W. As the relationship between these two quantities depends on the model ($W=-\frac{2{F}_{\mathrm{s}}}{3\pi R}$ in the JKR model, $W=-\frac{{F}_{\mathrm{s}}}{2\pi R}$ in the DMT model), it is crucial to know the regime of contact in order to extract reliable values [53].
- When investigating friction in the boundary lubrication regime with smooth adhering surfaces, the applied load is generally large enough to flatten the sliding surfaces. These mechanical deformations have to be known in order to interpret the data, in particular to determine whether the friction is controlled by the area of contact or by the load, and to unravel the role of adhesion [10,39,53,54,55,56,57,58,59,60,61].

#### 3.3. Influence of Surface Deformations on Structural Force Profile

- For this SFB study with an ionic liquid ($R\sim $1 $\mathrm{cm}$, $k\sim $3000 $\mathrm{N}/\mathrm{m}$, $K\sim $50 $\mathrm{GPa}$, ${W}_{0}\sim $1 $\mathrm{mN}/\mathrm{m}$, $\lambda \sim $0.6 $\mathrm{nm}$), we obtain ${N}_{k}\sim $2$\xb7{10}^{2}$ and ${N}_{K}\sim $4, in agreement with the fact that we have spring instabilities and a strong effect of the surface deformations on the structural force profile.
- For previous AFM studies with ionic liquids [63,68,72] ($R\sim $20 $\mathrm{nm}$, $k\sim $0.1 $\mathrm{N}/\mathrm{m}$, $K\sim $50 $\mathrm{GPa}$, ${W}_{0}\sim $50 $\mathrm{mN}/\mathrm{m}$, $\lambda \sim $0.8 $\mathrm{nm}$), we obtain ${N}_{k}\sim $$5\xb7{10}^{2}$ and ${N}_{K}\sim $$2\xb7{10}^{-1}$, in agreement with the fact that they have spring instabilities but a small effect of the surface deformations on the structural force profile.
- For a previous SFA study with liquid crystals [84] ($R\sim $$0.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$, $k\sim $$2000\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}/\mathrm{m}$, $K\sim $$50\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, ${W}_{0}\sim $$0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$, $\lambda \sim $$6\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$), we obtain ${N}_{k}\sim $1 and ${N}_{K}\sim $$6\xb7{10}^{-3}$, in agreement with the fact that they have no spring instabilities and a small effect of the surface deformations on the structural force profile.
- For a previous AFM study with polyelectrolytes [73] ($R\sim $$2\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $k\sim $$0.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}/\mathrm{m}$, $K\sim $$50\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, ${W}_{0}\sim $$0.02\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$, $\lambda \sim $$50\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$), we obtain ${N}_{k}\sim $${10}^{-1}$ and ${N}_{K}\sim $${10}^{-6}$. Additionally, for a previous AFM study with colloidal suspensions [74] ($R\sim $$7\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $k\sim $$0.03\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}/\mathrm{m}$, $K\sim $$50\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, ${W}_{0}\sim $$0.005\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$, $\lambda \sim $$70\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$), we obtain ${N}_{k}\sim $$7\xb7{10}^{-1}$ and ${N}_{K}\sim $$4\xb7{10}^{-7}$. This is in agreement with the fact that both studies have no spring instabilities and a small effect of the surface deformations on the structural force profile.

## 4. Conclusions

- SFA experiments are not always in the JKR regime but can be in the DMT regime, typically for situations of moderate adhesion over a range of a few nanometers, as we have seen in the case of the ionic liquid. Using the correct model of contact mechanics is crucial, notably for quantitative investigations of adhesion or friction. The two regimes of contact are usually distinguished by calculating the value of the Maugis parameter from an estimate of the range of the attractive forces; it is in fact more accurate to look at the value of the contact radius before jump-out.
- In classical SFA experiments using mica sheets glued on glass lenses, the mica does not only bend but can also experience a compression, which we observed for relatively thick mica.
- This compression has to be taken into account for a proper calibration of the undeformed mica thickness in a dry atmosphere; for this purpose, we presented a method based on the fitting of the relation between the contact radius and the force with the JKR model. The usual procedure, which consists of taking the jump-in point as a reference, can lead to an underestimation of the mica thickness and an equivalent outward shift of the force profile measured after injecting the liquid. We found that this effect amounts to ∼$1\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ for a ∼$7\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-thick mica, and we expect it to decrease with the mica thickness.
- For any system showing a structural force profile with SFA or AFM, the exponentially decaying harmonic oscillation due to local variations of liquid density may be convoluted with the mechanical response of the confining solids. A correct interpretation of the detailed shape of the structural force profile is necessary to understand the behaviour of liquids in nanoconfinement.
- We have proposed a simple scaling criterion to estimate the degree of convolution. Typically, compression in the solids is dominant over compression in the liquid for simple liquids (large energies, small length-scales) and easily deformable solids (small elastic modulus, large radius of curvature). For SFA experiments with mica sheets glued on glass lenses, the influence of mica compression is interpreted to be more important and independent of the mica thickness at low loads and smaller and reduced for thinner mica at large loads. This effect is expected to be even more important at all loads when mica is replaced by a softer layer (such as EPON glue) [85,86,87] or at high loads if the distance measurement includes the indentation of the whole solid bodies (not only the top layers) [70].
- When the solid compression is dominant compared to liquid compression, a fit of the structural force profile with an exponentially decaying harmonic function is not appropriate. We have illustrated that it can be useful to consider heuristic formulations, which are based on extensions of contact mechanics models to situations where the solid surfaces confine a structured liquid film.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SFB | Surface Force Balance |

SFA | Surface Force Apparatus |

AFM | Atomic Force Microscope |

FECO | Fringes of Equal Chromatic Order |

JKR | Johnson–Kendall–Roberts |

DMT | Derjaguin–Muller–Toporov |

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**Figure 2.**(

**a**) Picture of the FECO when the two solid surfaces are in contact across N${}_{2}$, observed in (wavelength $\lambda $)- (lateral distance x) space. (

**b**) Corresponding profile of the distance z between the surfaces along the lateral coordinate x (in red). A parabolic fit at small scale close to the apex (in green) allows one to measure the apical distance D, while a fit with the JKR profile (Equation (8) of the Supplementary Materials) at all measured scales (in blue) is used to extract the contact radius a. (

**c**) Picture of the FECO when the two solid surfaces are in contact across [C${}_{4}$C${}_{1}$Pyrr][NTf${}_{2}$], observed in (wavelength $\lambda $)-(lateral distance x) space. (

**d**) Corresponding profile of the distance z between the surfaces along the lateral coordinate x (in red). A parabolic fit at small scale close to the apex (in green) allows one to measure the apical distance D, while a parabolic fit at large scale (in blue) is used to extract the contact radius a. In both cases, the FECO images were recorded with a black and white camera, then recolored using the calibration of the wavelength axis with a mercury lamp. The two particular cases shown here correspond to the points of maximum load reached in Figure 3.

**Figure 3.**(

**a**) Force F as a function of distance D and (

**b**) contact radius a as a function of force F when approaching (in red) and retracting (in blue) the top surface across N${}_{2}$ with the stepper motor at $v=13.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}/\mathrm{s}$. The green curve is a JKR fit of $a\left(F\right)$ (Equation (4) of the Supplementary Materials) with $K=47.0\pm 0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, $W=131.79\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$ and ${D}_{\mathrm{ref}}=\delta +D=0$; the black curve is a JKR fit of $F\left(\delta \right)$ with $K=900\pm 200\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, $W=131.79\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$ and ${D}_{\mathrm{ref}}=\delta +D=0$. (

**c**) Force F as a function of distance D and (

**d**) contact radius a as a function of force F when approaching (in red) and retracting (in blue) the top surface across [C${}_{4}$C${}_{1}$Pyrr][NTf${}_{2}$] with the stepper motor at $v=10.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}/\mathrm{s}$. The green curve is a DMT fit of $a\left(F\right)$ (Equation (3) of the Supplementary Materials) with $K=16.7\pm 0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, $W=8.18\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$ and ${D}_{\mathrm{ref}}=\delta +D=1.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$; the black curve is a DMT fit of $F\left(\delta \right)$ with $K=600\pm 200\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$, $W=8.18\phantom{\rule{3.33333pt}{0ex}}\mathrm{mN}/\mathrm{m}$ and ${D}_{\mathrm{ref}}=\delta +D=1.4\pm 0.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$.

**Figure 4.**(

**a**) Force profile measured with [C${}_{4}$C${}_{1}$Pyrr][NTf${}_{2}$] between mica surfaces when approaching or retracting the top surface with the piezoelectric tube at $v=0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}/\mathrm{s}$, showing structuring with five distinguishable layers labeled by i. For clarity, only the full approach is shown (in red), together with retractions from layers $i=1$ (in blue), $i=2$ (in green), $i=3$ (in orange), $i=4$ (in purple), $i=5$ (in yellow). The black lines are the fit with Equation (5). Inset: proposed structure of alternating layers of cations and anions. (

**b**) Distances ${D}_{i}$ measured at the points of maximum and minimum force (respectively, in red and blue), as a function of the layer index i. Straight lines are the corresponding linear fits (Equations (3)). (

**c**) Forces $\left|{F}_{i}\right|$ measured at the points of maximum and minimum forces (respectively, in red and blue), as a function of the layer index i, in log-lin representation. Straight lines are the corresponding exponential fits (Equation (4)).

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Lhermerout, R.
The Influence of Mechanical Deformations on Surface Force Measurements. *Lubricants* **2021**, *9*, 69.
https://doi.org/10.3390/lubricants9070069

**AMA Style**

Lhermerout R.
The Influence of Mechanical Deformations on Surface Force Measurements. *Lubricants*. 2021; 9(7):69.
https://doi.org/10.3390/lubricants9070069

**Chicago/Turabian Style**

Lhermerout, Romain.
2021. "The Influence of Mechanical Deformations on Surface Force Measurements" *Lubricants* 9, no. 7: 69.
https://doi.org/10.3390/lubricants9070069