Frictional Aging of Single‐Asperity Nanoindentation Contacts in Quartz and Calcite

The evolution of fault friction during the interseismic period affects the mechanics of a future earthquake on the same fault patch. Frictional aging has been previously tied to time‐dependent contact area growth through observations made on rock analogs. However, our understanding of the processes that control frictional aging is limited and is dependent on experiments that explore only numerous mechanisms. We conduct slide‐hold‐slide experiments with a dual‐axis nanoindenter on single‐crystal surfaces of quartz and calcite. Our results show that frictional aging in diamond‐quartz contacts is independent of time and contact area, in stark contradiction to past experiments done on quartz‐quartz contacts in rocks. Diamond‐calcite contacts show modest frictional aging, but still well below previous reported values from calcite‐calcite contacts. These results suggest that frictional aging of like‐on‐like minerals may be of chemical origin, as suggested in recent studies with atomic force microscopy and molecular dynamics simulations.


Introduction
Friction between solid bodies is governed by the physical interactions of their contacting asperities (Bowden & Tabor, 1950).The frictional force F is a product of two properties: the average shear strength of the contact τ c and the real area of contact A r (i.e., the total area of contacting asperities) through F = τ c A r (Bowden & Tabor, 1950).This physical framework distinguishes the two parameters and their respective contribution to the frictional force.The current discussion on the underlying physics of friction is about these two end members, contact 'quality' (τ c ): the shear strength of the contacts which is controlled by the molecular interactions of the two contacting bodies (Bhushan et al., 1995), and contact 'quantity' (A r ): the total real area of contact controlled by the number and size of asperities in contact (e.g., Archard, 1953;Bowden & Tabor, 1950;Dieterich & Kilgore, 1994;Greenwood & Williamson, 1966).While this division is useful for discussion, we note that some combination of both members might also be able to explain experimental data.Understanding the physical mechanisms that control friction is of great importance in the study of earthquakes.Our knowledge of friction and how it evolves in the earthquake cycle is limited to empirical laws with only partial understanding of the physics behind them.Different frictional evolution laws affect whether or not an earthquake can nucleate on a fault patch and the spatiotemporal behavior of a propagating earthquake rupture (e.g., Ampuero & Rubin, 2008).
Contact quantity has been studied extensively in the past through theoretical models and experiments.Calculations show that the real area of contact is proportional to the applied load (e.g., Archard, 1953;Bowden & Tabor, 1950).Dieterich and Kilgore (1994) showed that the real area of contact increases with load and time in PMMA (a transparent polymer) and soda-lime glass in stationary contact.The time-dependence of contact area was interpreted to be due to plastic creep of asperities where the contact stress is a large fraction of the material's yield strength (Dieterich & Kilgore, 1994).These results, coupled with experimental observations that static friction in rocks increases approximately linearly with the logarithm of time for rock surfaces in stationary (or quasi-stationary) contact (e.g., Dieterich, 1978;Beeler et al., 1994;Marone, 1998b,a), has led to the idea that frictional contacts 'age', meaning, that the friction force increases with time and load due to an increase in contact area.The contact quantity approach has been widely accepted by modelers and experimentalists as the underlying physical mechanism of frictional aging (Ampuero & Rubin, 2008).However, a growing number of studies have questioned this interpretation, as the models utilizing time-dependent frictional aging cannot explain the full mechanical response of sliding friction in laboratory experiments (Ampuero & Rubin, 2008;Bhattacharya et al., 2015Bhattacharya et al., , 2017Bhattacharya et al., , 2022)).The contact quality approach has been recently explored through atomic force microscopy (AFM) experiments and molecular dynamics simulations and suggests that interfacial chemical bonding may be another contributing mechanism of frictional aging.More specifically, recent studies show that the density of siloxane (Si-O-Si) bonds may control the frictional response of amorphous silica (A.Li et al., 2014;Q. Li et al., 2011;Liu & Szlufarska, 2012;Mo et al., 2009).Furthermore, Thom et al. (2018) shows that the indentation hardness and creep rate of quartz are independent of relative humidity, which was previously thought to control asperity creep in experiments on quartz rocks (Dieterich & Conrad, 1984;Frye & Marone, 2002).These new studies suggest that frictional aging in quartz is not due solely to asperity creep, contradicting the classical contact quantity model.
We conducted slide-hold-slide (SHS) experiments with a diamond indenter tip on single-crystals of quartz and calcite at the single-asperity scale.Friction tests were run with a nanoindenter capable of inducing lateral displacements along the tested surface while maintaining constant normal force.The contact area between the indenter tip and the tested substrate was constantly monitored throughout the experiment using the continuous stiffness measurement (CSM) method (Oliver & Pharr, 1992).These experiments allow us to probe the mechanical behavior of a sliding asperity junction in real time, thus shedding light on the physical processes that control sliding friction.

Testing Apparatus
Tests were conducted using the Gemini dual-axis nanoindenter (KLA Inc.) at Texas A&M University (Figure 1a).The apparatus consists of two independent indenter heads capable of simultaneously applying loads on the indenter tip of up to 50 mN in the vertical and lateral directions.The indenter tip is coupled to the actuators by two glass plates that join at a tip mounting block.The actuators are mounted on an aluminum frame with a stiffness of approximately 10 6 N/m in the vertical direction and 10 5 N/m in the horizontal direction (Brazil et al., 2021).The lateral stiffness is calculated as a one-dimensional spring in series with the contact point to obtain the true displacement of the tip on the sample surface.The lateral load P x on the sample is related to the lateral force F x applied by the horizontal actuator by P x = F x k x (d x d 0 ), where k x is the lateral stiffness of the lateral loading column, which equals approximately 540 N/m, d x is the present lateral position of the tip as measured by the capacitance plate in the lateral force head (Figure 1a) and d 0 is the lateral position immediately prior to the application of F x .Each of the lateral and vertical actuators is equipped with a dedicated lock-in amplifier which allows the measurement of the contact stiffness throughout a test via CSM (Oliver & Pharr, 1992).However, we only utilize the vertical lock-in amplifier so that the frictional behavior of the contact is not affected by the lateral oscillatory motion of the tip during CSM measurements.

Experimental Procedure
Slide-hold-slide (SHS) tests are used to characterize the apparent time-dependent strengthening of rocks (e.g., Beeler et al., 1994;Dieterich, 1972;Marone, 1998b).We follow the same experimental protocol adapted to the nanoindenter: (a) the tip was indented into the sample to a depth of 200 nm at a constant rate of Ṗz /P z = 0.1 s 1 , where P z is the normal load; (b) the tip was slid laterally at a constant rate of 1 μm/s for a distance of 6 μm; (c) a hold segment was initiated by setting the lateral sliding rate to zero for a predetermined hold time; and (d) the tip was reloaded and translated at a constant rate of 1 μm/s for another 2 μm of slip (Figure 1b).A starting indenter depth of 200 nm for the SHS tests was chosen so that the local strain field underneath the indenter tip does not significantly change with depth (Brazil et al., 2021).In addition, the starting indentation depth of 200 nm ensures that the effects of surface topography are negligible, where the average root-mean-square ± one standard deviation is 56.3 ± 14.9 nm for quartz and 14.2 ± 0.8 nm for calcite (Figure S5 in Supporting Information S1).The normal load remains constant throughout a test (Figure S4 in Supporting Information S1).All experiments were run at ambient relative humidity of 30%.

Materials
The materials tested in this study were single-crystals of quartz and calcite of unknown orientation and origin.The tested surface of the crystals was polished until the surfaces appeared shiny, then adhered using Crystalbond 509 to a cylindrical aluminum puck that was mounted into the nanoindenter.We chose quartz and calcite for two reasons: (a) quartz and calcite are primary constituents of crystalline and sedimentary crustal rocks and as such are the focus of experimental friction studies related to earthquakes; (b) quartz and calcite differ in their mechanical behaviors, where quartz is a 'harder' material with a ratio of elastic modulus to hardness of E/H = 116.42/13.59≈ 9 compared to E/H = 81.99/2.73≈ 30 for the softer calcite (Figure S1 in Supporting Information S1).Past experiments that examined the relation between contact area and friction were performed only on soft materials, such as PMMA (e.g., Dieterich & Kilgore, 1994;Rubinstein et al., 2004).The physical processes observed in those studies were then assumed to apply also to geological materials.Here, the sample types were chosen to cover a wide range of material properties relevant to geological faults.The nanoindenter tip used in our experiments was of Berkovich type (a three-sided pyramid) and made of diamond.The supporting information includes results from SHS tests run with a cone terminated at its tip by a sphere of 1 μm radius (also made of diamond).The tip geometry dictates how the substrate deforms under the rigid tip, where the Berkovich tip induces a plastic deformation response and the sphere an elastic-plastic response (Brazil et al., 2021).

Results
Experimental results from SHS tests are reported for quartz and calcite for comparison.The main parameters of interest in these tests are: the coefficient of friction µ, calculated as the ratio of lateral load to normal load µ = P x /P z ; frictional aging ∆µ (Figure 1b), calculated as the peak friction after reload minus the prehold friction averaged over the last second before the hold starts; lateral displacement d x ; lateral sliding velocity v x , calculated as the time derivative of the lateral displacement v = ḋx ; vertical depth of the indenter tip (hereafter referred to as depth) d z ; and the contact stiffness S as measured by CSM.The contact area is proportional to the square of contact stiffness A ∝ S 2 (e.g., Oliver & Pharr, 1992), thus, we calculated the contact area at any given point in time relative to a reference contact area A 0 as: where the subscript 0 denotes the reference point, defined here as the beginning of the 'hold' segment.
Figure 2 illustrates the variation of friction, contact area and lateral displacement with time during a 10-s hold for quartz and calcite with the diamond Berkovich tip.During the hold, friction relaxes while the contact area grows and the contact slips at an ever-decreasing rate.This behavior is due to the lateral compliance of the indenter column (Figure 1) during the hold period.By the end of the longest hold (1000-s) the contacts experience a total slip (from the start of the hold) of 150 200 nm.The slip rates by the end of the longest holds, calculated over the latter quarter of the hold segment, reach sub-nanometric rates of 0.025 ± 0.005 nm/s for quartz and 0.005 ± 0.007 nm/s for calcite.These are average rates calculated from several tests.The calculated slip rate for calcite indicates that the tip was pushed backwards in some tests in the latter stages of the 1000-s hold.The contact area grows during holds by two distinct patterns, growing rapidly in the first 0.5 s, then linearly with the logarithm of time (Figure 3).Contact area increases with the logarithm of time as approximately 0.023log 10 (t) 1 in quartz, compared to 0.124log 10 (t) 1 in calcite.Close examination of the reloads (Figures 2b and 2d) reveals that peak friction coincides with a reduction of contact area, compared to the maximum A/A 0 observed at the end of the hold.For example, in the reload after the 10-s hold in quartz (Figure 2b), A/A 0 drops by approximately 3% of its value by the end of the hold.Similar reductions in contact area are observed in quartz at peak friction for all hold durations.In calcite, however, the reduction in contact area at peak reload friction is smaller, < 1%, over the various hold times.Nevertheless, once sliding is resumed at the baseline 1 μm/s rate in the reloads, contact area returns to approximately its pre-hold level.
Figure 3 shows representative curves for contact area and indenter tip depth versus the logarithm of time during a hold for quartz (Figures 3a and 3b) and calcite (Figures 3c and 3d).The difference in the growth rates of the normalized contact area ( Ȧ = d (A/A 0 )/d (log 10 [t])) between quartz and calcite suggests that it arises due to the different hardness of the tested materials.The ratio of the growth rate of contact area of calcite over that of quartz is approximately equal to the inverse of their hardness ratio, that is, The change in tip depth with time during a hold (Figures 3b and d) further illustrates the difference between the contact area growth rate in quartz and calcite.The tip sinks into the quartz by less than 10 nm for all hold durations, whereas the tip sinks into the calcite by more than 10 nm in the 1-s hold and up to 30 nm in the 100-s hold.The 1000-s hold tests in quartz and calcite exhibit riding out of the tip from the sample during the holds.This behavior is not consistent with the continued contact area growth (from the stiffness measurements) in these tests.This behavior is observed in most but not all tests in these samples.The apparent riding out of the tip is likely due to thermal drift of the indenter tip, which was observed in other materials over similar time durations in the Gemini system and at similar drift rates.
Figure 4 presents a compilation of all frictional aging results versus hold time and contact area (at peak friction) for quartz and calcite.Frictional aging results from Beeler et al. (1994) for quartz-on-quartz rock contacts are plotted for comparison.The holds in the quartz-diamond contacts clearly do not exhibit an aging effect, where ∆µ remains relatively independent of hold time and equals roughly 0.006 (Figure 4a).This result is in stark contrast to the aging effect observed in the Beeler et al. (1994) study.We note that the Beeler et al. (1994) results have been re-interpreted as due to slip-dependent frictional aging within the rate-and-state framework (Bhattacharya et al., 2017).In our experiments the indenter tip slips on the quartz surface continuously during the hold period (Figure 2a), though it does not have an apparent effect on aging in diamond-quartz contacts.The ∆µ results for calcite show more variability and also an aging trend with hold time, which is due to solely the longest (1000-s) holds.The observed aging in calcite is more pronounced in holds ≥100 s, though still less than observed in quartzquartz contacts (Beeler et al., 1994) and significantly lower than observed in calcite-calcite contacts (Carpenter et al., 2016).When ∆µ is plotted against contact area (Figure 4b) we find that aging in quartz-diamond contacts results in seemingly random noise, whereas the calcite data show an overall increase in friction with contact area.Finally, the difference in the growth of contact area with hold time in quartz and calcite is evident from the calculated slopes in Figure 4c.

Discussion
SHS tests on diamond-quartz (D-Qtz) contacts performed using the dual-axis nanoindenter show a lack of frictional aging with time and with contact area.These results are in stark contradiction to past experiments with quartz-quartz contacts in simulated quartz gouge and on bare quartz surfaces (e.g., Beeler et al., 1994;Marone, 1998a).Diamond-calcite (D-Cal) contacts show some frictional aging, but this strengthening is still significantly lower than what is observed in past experiments on rocks (Carpenter et al., 2016).Timedependent contact area growth in both quartz and calcite (Figure 4c and Figure S2 in Supporting Information S1) is not translated into a proportional increase in ∆µ.Furthermore, if a change in fault normal displacement (∆depth in current study) is a proxy for changes in contact area during a hold, then our experiments on quartz exhibit ratios of ∆depth to ∆lateral displacement (both in nm) of ∼ 10/200 = 0.05 (Figure 3b) that are comparable to those of bare-surface SHS tests in granite (Beeler & Tullis, 1997).This means that fault normal displacement rates (and therefore contact area growth rate) during a hold is similar between our experiments and cm-scale experiments in rocks.These results suggest that frictional aging in these materials is predominantly controlled by mechanisms other than time-dependent growth of contact area.This realization has significant effects on models that describe how earthquakes nucleate and grow along a fault patch (e.g., Ampuero & Rubin, 2008;Lapusta & Rice, 2003;Scholz, 2019).If, in fact, frictional aging and evolution are predominantly of chemical origin (Li et al., 2011(Li et al., , 2014) ) and not solely dependent on contact area (e.g., Dieterich & Kilgore, 1994), then the prevailing earthquake nucleation and rupture models (e.g., Lapusta & Rice, 2003) need to be revised and past experimental studies of frictional evolution (e.g., Beeler et al., 1994;Dieterich, 1978;Marone, 1998b) will need to be revisited in light of the new interpretation for frictional aging.
The SHS tests show that contact area at peak friction is reduced in comparison to the maximum contact area at the end of the hold (Figure 2).Peak friction has been previously shown by Dieterich and Kilgore (1994) to occur after some destruction of contact area, which puts the relation between peak friction and contact area into further question.Additional results from SHS tests on D-Qtz contacts with a spherical tip exhibit even more destruction of contact area at peak friction than is observed in the tests performed with the Berkovich tip (Figure S3 in Supporting Information S1).Our single-asperity experiments suggest that frictional aging in quartz may be predominantly controlled by other mechanisms, most likely of a chemical origin.Formation of strong interfacial siloxane bonds between contacting SiO 2 substrates has been shown to be important in AFM experiments (Li et al., 2011) and molecular dynamics simulations (e.g., Li et al., 2014;Liu & Szlufarska, 2012;Mo et al., 2009).The next step for asperity-scale friction experiments in the nanoindenter is thus to replicate these SHS tests with fabricated quartz tips.Exploring quartz-on-quartz contacts at these scales will provide direct examination of the effects of chemical bonding on frictional aging of quartz.The discrepancy in ∆µ between the D-Cal results and macro scale calcite-calcite contacts (Figure 4a) needs further investigation.Frictional aging in calcite is sensitive to water (e.g., Bhamra et al., 2021;Chen et al., 2015) whereas the contribution of asperity plowing to frictional aging in the 'softer' calcite also needs to be quantified.Additional mechanisms such as asperity adhesion (Tesei et al., 2017) imply that frictional aging in calcite may be more complex than in quartz and involves multiple mechanochemical contributions.
Interfacial bonding between the indenter tip and the sample in our experiments is expected to occur and contribute to the observed frictional response.Bhamra et al. (2021) investigated the origin of friction of D-Qtz and D-Cal contacts from tribometer experiments and molecular dynamics simulations.They find that C-O bonds form directly between diamond and quartz and diamond and calcite, along with C-O-Ca, C-H-O bridging bonds in the presence of humidity (25%-30%).The number of bonds increases over the first 0.2-2 nm of slip (depending on their simulated slip rate) after which the number of bonds remains approximately constant (Bhamra et al., 2021).
During the hold period in our experiments the contacts slip roughly 40 nm (in the 1-s holds) to 200 nm (in the 1000-s holds), which suggests that if the bond-formation rate is similar to the calculations by Bhamra et al. (2021) then interfacial C-O bonds will form at the same rate as the contact area growth rate ( Ȧ). Bhamra et al. (2021) also show that the number of bonds is greater for D-Cal contacts than for D-Qtz contacts in a humid environment compared to a completely dry environment.Our experiments were conducted at the same relative humidity of 30% which therefore cannot explain the modest aging in D-Cal contacts.
Plowing of the indenter tip into the sample also contributes to the observed frictional response in our experiments.Brazil et al. (2021) show through nanondentation experiments (using the identical instrument) on diamond-metal contacts and finite element modeling that friction is independent of the total contact area but depends on the contact area at the leading edge of the indenter tip, in the direction of sliding, during plowing.Their results show that softer materials with higher E/H values exhibit higher friction, more sinking in of the indenter tip, and loss of contact at the trailing edge of the indenter tip.Our SHS test results show that during a hold the indenter tip sinks into the sample (Figures 3b and d).The sinking in is much more significant in the softer calcite than in the quartz.
These results agree with the findings of Brazil et al. (2021) and also suggest that the frictional aging observed in calcite may also be affected by asperity plowing.

Conclusions
We conducted SHS tests on single-asperity contacts between a rigid diamond indenter and surfaces of singlecrystal quartz and calcite.Frictional aging is independent of time and contact area in diamond-quartz contacts.Diamond-calcite contacts show modest frictional aging with time and contact area that is significantly lower than what is observed in tests on calcite-calcite contacts in rocks.These results suggest that aging mechanisms other than time-dependent contact area growth are expected to control the evolution of friction in quartz and calcite.Our

Geophysical Research Letters
10.1029/2023GL105471 analysis suggests that frictional aging in our experiments is due to interfacial bonding between the diamond indenter tip and the samples, and the sinking-in of the indenter tip into the sample.Our results pave the way for the exploration of the effects of contact chemistry on frictional aging for quartz-quartz and other mated mineral contacts and suggest that contact quality may be an important factor in rock friction.

Figure 1 .
Figure 1.(a) Main components of the nanoindenter, including the normal and lateral force actuators, coupling plates, tip and sample.Displacements are measured in each of the heads separately using a three plate capacitance gauge (green moving plates sandwiched between the fixed blue capacitor plates).Forces are applied by electromagnets in each head (violet components), and each head is mounted on the aluminum frame (hatched gray).(b) Results from a typical SHS test on quartz with the hold segment denoted.The inset shows a blowup of the portion of the friction curve enclosed by the upper rectangle.Frictional aging ∆µ is the difference between the pre-hold steady-state friction and the peak friction after reloading.

Figure 2 .
Figure 2. Variation of friction (black), contact area (green) and lateral displacement (blue) with elapsed time around a 10-s hold in SHS tests on (a) quartz and (c) calcite.The vertical dashed red lines depict the start and end of the hold segment.Blowups of the reload (including the end of the hold segment) are shown in (b) and (d) for quartz and calcite, respectively.Note that both contacts are sliding throughout the hold segments.Lateral slip rates calculated over the last 2.5 s of the hold segment are given in (a) and (c).

Figure 3 .
Figure 3. Contact area versus the logarithm of time for different hold durations in (a) quartz and (c) calcite and the change of indenter depth (∆depth, from the beginning of the hold) versus the logarithm of time in (b) quartz and (d) calcite.Negative ∆depth means the indenter is riding out of the substrate and positive means the indenter is sinking into the substrate.The apparent decrease in contact area observed in some of the data in the very early stages of the hold (<0.5 s) is affected by the slowing down of the indenter tip and the instrument response to the transient, and does not represent the true mechanical response of the sample during hold.The dashed black line represents the calculated contact area growth rate (A˙) for the 1000-s hold, fitted only for the linear portion of the curve.

Figure 4 .
Figure 4. Frictional aging versus hold time (a) and contact area (b) and contact area versus hold time (c).Frictional aging from SHS experiments of like-on-like contacts are given for calcite and quartz rocks are added for comparison in (a).Contact area in (b) and (c) are determined at peak friction.