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[Hiroshi Sakuma](https://orcid.org/0000-0002-6522-0704), Diane E. Moore, David A. Lockner, Toshihiro Kogure

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[Velocity-Independent Dry Friction on Mica: Realization of Ideal Amontons-Coulomb Friction](https://mdr.nims.go.jp/datasets/3d56cefa-dc6a-425b-b983-d98f86a3ca18)

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Velocity-Independent Dry Friction on Mica: Realization of Ideal Amontons-Coulomb FrictionVelocity-Independent Dry Friction on Mica: Realization of IdealAmontons-Coulomb FrictionHiroshi Sakuma ,1,* Diane E. Moore,2 David A. Lockner ,2,† and Toshihiro Kogure31National Institute for Materials Science, Tsukuba, Japan2U.S. Geological Survey, Moffett Field, California 94035, USA3University of Tokyo, Hongo, Japan(Received 7 January 2026; revised 27 April 2026; accepted 17 June 2026; published 20 July 2026)The Amontons-Coulomb friction law assumes that the frictional force between materials is independentof sliding velocity. However, as Coulomb noted, this is a rough approximation, and a second-orderdependence of friction on the logarithm of sliding velocity is incorporated in a commonly used “rate- andstate-dependent” friction representation. Here, we conduct shear experiments on mica, a layer-structuredmineral, at temperatures ranging from 22 to 200 °C and under normal stress of 100 MPa. The frictioncoefficient clearly depends on the logarithmic sliding velocity at 22 °C, but rate sensitivity decreases withincreasing temperature until at 200 °C, the friction coefficient is independent of sliding velocity. Ourfindings could initiate the development of velocity-independent frictional materials, realizing the idealAmontons-Coulomb friction.DOI: 10.1103/y3jy-nqcdAccording to Amontons-Coulomb friction law, frictionbetween objects is independent of velocity [1–3]. However,in reality, friction between materials weakly depends on thesliding velocity. This velocity-dependent friction is essen-tial for understanding slip stability of materials and earth-quakes caused by the movement of natural faults, and hasled to the development of what is generally referred to asthe “rate- and state-dependent” friction (RSF) formulationin the field of geoscience [4–6]. In the RSF formulation, thefriction coefficient μ (¼ the shear stress divided by thenormal stress) of materials is typically described as follows:μ ¼ μ� þ a lnðV=V�Þ þ b lnðV�θ=DcÞ. Here, V is slidingvelocity, θ is a function of state of the sliding surface, Dc isthe characteristic slip distance for strength evolutionfollowing a change in sliding velocity, a and b are semi-empirical parameters, and superscript � indicates variablevalues at reference state. Parameter “a” quantifies theinstantaneous response of friction to a change in velocitywhile “b” quantifies the subsequent evolution of frictionwith continued slip. The values of these parameters varywith materials and test conditions [7]. While reportedvalues of b typically fall in the range of −0.02 < b <0.02 [7], a is always positive and, to our knowledge, neverzero, that is, velocity-independent friction characterized bya ¼ b ¼ 0, representing an ideal material described by thevelocity-independent Amontons-Coulomb friction law.The physical and chemical origins of these two parametersare noteworthy. The a parameter is considered to reflect athermal activation process, characterized by the Arrheniusequation, which occurs at asperity contacts [8–10]. Multiplemechanisms may be involved in this process, such asdislocation glide in layered minerals [11,12], and subcriticalcrack growth in granite [13].Within the frameworkof thermalactivation theory, the a value is expected to vary linearlywith temperature [8,9], and can be defined as a ¼ kBT=σcΩ,wherekB is theBoltzmannconstant,T is the temperature,σc isthe average normal stress at the contacts, and Ω is theactivation volume associated with the process [14]. The bparameter is considered to be related to the time-dependentincrease in the real area of contact. Its value becomesnegligible under dry and vacuum conditions [15,16], indicat-ing that awater-assisted chemical creep process contributes tothe increase in the real area of contact over time.The friction coefficient of layered minerals is muchlower than that of materials such as quartz and feldspar[17,18]. Such low friction coefficients have been explainedby their nondilatant brittle deformation [19,20] and can bepartly explained by low atomic-scale potential barriersbetween layers for shear [21–25]. These layered materialsmay also have anomalous friction properties with respect tothe sliding velocity. In this Letter, triaxial shear experimentswere performed on dry sheets of the layer-structuredmineral mica, KAl2½Si3AlO10�ðOHÞ2, and not using apowdered mica gouge, to measure the velocity dependenceof friction at temperatures of 22, 65, 100, and 200 °C and a*Contact author: sakuma.hiroshi@nims.go.jp†Contact author: dlockner@usgs.govPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW LETTERS 137, 046201 (2026)0031-9007=26=137(4)=046201(7) 046201-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-6522-0704https://orcid.org/0000-0001-8630-6833https://ror.org/026v1ze26https://ror.org/00dcq6866https://ror.org/057zh3y96https://crossmark.crossref.org/dialog/?doi=10.1103/y3jy-nqcd&domain=pdf&date_stamp=2026-07-20https://doi.org/10.1103/y3jy-nqcdhttps://doi.org/10.1103/y3jy-nqcdhttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/normal stress of 100 MPa. Because of its ability to formatomically flat surfaces via cleavage [26], mica is widelyemployed in nanotechnology research as a substrate for atomicforce microscopy [27] and surface forces apparatus [28,29].Natural single-crystal mica (muscovite) was used fortriaxial shear experiments. The chemical composition of themica was analyzed by inductively coupled plasma opticalemission spectrometry and alizarin complex one-extractionmethod as K0.92Na0.09ðAl1.78FeðIIIÞ0.12Mg0.09Ti0.02ÞðOHÞ1.96F0.04½Si3.06Al0.94O10�Þ. Mica sheets with a thick-ness of 0.51 mm were rough-cut and then sanded toconform to the elliptical area of driving blocks [Fig. S1(a)[30] ]. These sheets were then cleaved to get a clean surfaceand rinsed with deionized water. The rinsed sheets andnovaculite driving blocks were placed in a vacuum ovenand dried overnight at 200 °C. The presence of water vaporstrongly influences the frictional properties of quartz [15] andclay minerals [56,57]. Such a heating procedure is thereforeessential for removing surface water and preventing wateradsorption during sample assembly. In our previous study[58], heating at 120 °C for a few hours was found necessaryfor eliminating adsorbed water on mica, as confirmed byatomic force microscopy.The sample number and experimental conditions are listedin Table S1 [30]. A standard triaxial test configuration wasemployed in which a cleaved mica pair was sandwichedbetween cylindrical driving blocks of porous novaculite(1.905 cm in diameter) with 30° inclined sawcuts[Fig. S1(b) [30] ]. The inclined surface of driving blockswas roughened with No. 60 SiC grit to ensure a no-slipinterface with the mica samples. After taking novaculiteblocks and mica sheets out of the oven, the sample cell wasassembled promptly. After the sample assembly was placedin the pressure vessel, it was evacuated with a scroll pump toless than 1%of atmospheric pressure (< −0.099 MPa gaugepressure). A transducer attached to the sample confirmed thatthis pressure remained constant during each experiment.Mechanical data were corrected for elastic deformationof the loading system, jacket strength, the confining-pressure dependence of piston seal friction, and thereduction in contact area during deformation [59,60].Briefly, the confining pressure was applied using argongas as the confining medium. A soft lead jacket isolated thesample assembly from the argon confining medium. Thejacket strength was determined in calibration tests over arange of temperatures and was subtracted from the shearstrength measurements. Samples were sheared by advanc-ing a piston against the lower TiC end cap at slidingvelocities of 0.01, 0.1, 1.0, and 3.0 μm=s along the sheardirection. Axial force was measured using an external loadcell. A normal stress of 100 MPa was maintained undercomputer control by adjusting the confining pressure basedon the measured axial force. Axial displacement wasmeasured using an external direct current displacementtransducer. Samples were sheared for approximately3.7 mm of axial displacement. The reduction in contactarea at an axial displacement of 3.7 mmwas estimated to be14%, and the normal stress was adjusted accordingly tomaintain a constant value of 100 MPa under computercontrol. The friction force shows a linear relationship withthe normal force (Fig. S2).The surface topographies of the shear planes wereobserved by secondary electron imaging using a field-emission scanning electron microscope (Hitachi Hi-TechS-4500) operated at 5 kV. Thin-film specimens for trans-mission electron microscopy (TEM) analysis were preparedfrom the surfaces of the shear planes using a focused ionbeam system (Hitachi Hi-Tech FB-2100) equipped with amicrosampling function. TEM lattice images were acquiredusing a JEOL JEM-F200 operated at 200 kV in combinationwith a Gatan OneView high-speed CMOS camera.Temperature-dependent dry friction of mica—The fric-tion coefficient between (001) mica surfaces is in the range0.15–0.25 at the tested temperature-pressure conditions[Figs. 1(a), 1(b), S3 [30] ], which is consistent with valuesreported in previous studies [21,58]. The friction coefficientof mica increases generally with mm of accumulated axialdisplacement but modulates over the microns of the criticalslip distanceDc as a function of the imposed sliding velocity(dμss=dlnV ¼ a − b ¼ þ0.004) at 22 °C [Fig. 1(c)]. Here,μss is the steady-state coefficient of friction at a givenvelocity. The difference in friction coefficients from thereference sliding velocity of 0.1 μm=s was in the range from−0.01 toþ0.03 [Fig. 1(c)]. The friction coefficient at 200 °CFIG. 1. Friction coefficients of mica and their velocity depend-ence. Friction coefficients between single crystal micas under thevelocity-step tests at 22 °C (a) and 200 °C (b). A difference in thesteady-state friction coefficient, ΔμSS, is shown in the inset of (a).(c) The change in ΔμSS as a function of sliding velocity and thelinear fits (dashed lines), with a reference sliding velocityV� ¼ 0.1 μm=s. (d) The temperature dependence of a valuesobtained as the slopes of the linear fits to the data in (c).PHYSICAL REVIEW LETTERS 137, 046201 (2026)046201-2was comparable to that at 22 °C, but the velocity dependencewas no longer observed (a − b ¼ 0) [Fig. 1(b)]. The responseto the change in the sliding velocity gradually decreased withincreasing temperature [Figs. 1(a), 1(b), S3 [30] ]. The∼1 MPa stress drop at 1.1 mm axial displacement in the200 °C experiment [Fig. 1(b)] was an errant instrumentadjustment, not the effect of a velocity step. The most likelycause is a realignment of the sample column in response to thelateral shear displacement required for slip on the inclinedfault surface.Which parameter, a or b, controls the variation in thea − b value? Negative b values have been observed in someshear experiments [16], and the variation in the a − b valuein our tests may be explained by the combined effects of apositive a and a negative b. However, negative b suggests adecrease in the real area of contact over time, and itsphysical and chemical origin remains unclear. Commonly,b is known to correlate with the rate of frictional strength-ening during stationary hold time in slide-hold-slide tests[7,20,61], allowing b to be measured independent of a. Theslide-hold-slide tests (Fig. S4 [30]) confirmed that thestrength recovery with time was negligible, that is, b ¼ 0.Therefore, the response to the change in sliding velocity isinterpreted as being caused by a change in a. The a valuecalculated by the slope (dμss=dlnV) demonstrates a cleardecrease with increasing temperature, reaching zero at200 °C [Figs. 1(c) and 1(d)]. This phenomenon indicatesthat mica becomes an ideal Amontons-Coulomb frictionmaterial with velocity independence at 200 °C and is a topicfor further investigation.As discussed above, the a value is expected to exhibit alinear relationship with temperature in the framework ofthermal activation theory. One representative result [31]showing that a varies linearly with temperature, consistentwith the thermal activation theory, was plotted for com-parison [Fig. 1(d)]. Although the a values of most mineralsremain constant between room temperature and 300 °C andincrease at higher temperatures as discussed recently [62](Fig. S5 [30]), mica shows a negative to neutral temperaturedependence of a, up to 200 °C, the highest temperaturetested in this Letter. To our knowledge, this behavior hasnot previously been reported. Nonlinear behavior of the aparameter of materials has been explained by temperature-dependent material properties of σc and Ω [14]. Forexample, the average normal stress at the contacts, σc,may decrease with temperature because of the enhancementof plastic deformation at the contact. This may explain thelarge increase in a at elevated temperatures (> 600 K) ofsome materials [63]. However, the decrease in a of micawith increasing temperature does not seem to be explainedby the changes in σc. When the decrease in σc occurs atelevated temperatures, the decrease in a requires a largeincrease in the activation volume, Ω. However, no theoryhas been established to account for such a large increase inΩ at elevated temperatures.A mechanism of the realization of Amontons-Coulombfriction—The shear planes of the recovered mica samplesexhibit two distinctive characteristics [Fig. 2(a)]. First,slickenlines are present parallel to the shear direction, acommon feature in the shear planes of numerous materials.Second, lines perpendicular to the shear direction areevident. The cross section parallel to the shear directionreveals that the lines perpendicular to the shear directionreflect the structure of the ripped off and overlapped surfacelayers [Fig. 2(b)]. A difference in these surficial structuresis not evident among the recovered samples tested from 22to 200 °C, but a difference in the structure correlated withtemperature was observed at 500–1000 nm below the shearsurfaces. Some white lines oriented nearly parallel to theshear plane were observed in the recovered sample at 65 °Cat 500 nm below the shear plane [Fig. 2(c)]. Most linescorrespond to the opening of layers as the bending of layers[Fig. 2(e)], and this would be ripplocations [64,65] (adislocation of layered materials). Linear features were alsoobserved in the sample tested at 200 °C [Fig. 2(d)], but thedepth extent from 100 to 300 nm seems to be shallowerthan those at lower temperature and bending of the layerwas not observed at 200 °C [Fig. 2(f)]. The white linearlines at 200 °C are considered to indicate that the region hasbecome amorphous due to electron beam damage duringthe initial stage of TEM observation.Based on observations of the recovered samples, wepostulate that mica’s instantaneous response to a change invelocity, characterized by the RSF a parameter, reflects theshear response of ripplocations. Nonequilibrium moleculardynamics (NEMD) simulations were conducted to evaluatethe a value for mica models without dislocations [Fig. 3(a)]and those containing ripplocations [Fig. 3(b)]. Force field ofmica used in this Letter is an interatomic potential model,and it has been developed to reproduce the mica structureand physical properties [32–34]. The normal stress of 6 GPawas chosen because the normal stress of real area of contactcan be approximated by the indentation hardness [21,66], byassuming the contacts are yielding normal to the interface.Shear strain was applied along the h100i sliding direction onthe (001) plane. The a value was calculated based on thedifferences in steady-state friction coefficients at differenttested sliding velocities. The sliding velocities at the toplayer of the supercell relative to the bottom layer can be onthe order of 1 m=s at a shear rate of 109 s−1 and 100 m=s at1011 s−1. These velocities are much higher than those usedin experimental sliding tests due to the computational cost(see simulation details in Supplemental Material [30]). Thestructures of mica were drawn using VESTA [67].At lower shear rates (109 and 1010 s−1), the slidingvelocity remains much higher than in shear experiments;however, the shear stress-strain behavior of dislocation-freemica [Fig. 3(c)] resembles that obtained from density-functional-theory calculations at nominally zero velocity[21], suggesting that the friction mechanism is similar toPHYSICAL REVIEW LETTERS 137, 046201 (2026)046201-3that at nominally zero velocity. Moreover, the frictioncoefficient of 0.15, which is comparable to experimentalobservations [21], supports the notion that the effect ofsliding velocity is negligible for simulations at shear rates≤ 1010 s−1. The friction coefficient of the dislocation-freemodel at shear rates of 109 and 1010 s−1 appears to beindependent of sliding velocity [Fig. 3(e)], and the calcu-lated a values are nearly zero for the dislocation-free model[Fig. 3(f)]. The increased friction coefficients at 1011 s−1can be attributed to a change in the friction mechanism. Atsuch extremely high sliding velocities, interlayer ionscannot reach the potential energy minimum configurationbecause the sliding velocity exceeds the velocity of phononvibrations. The interionic potential force acting on the ionsbecomes the source of friction. Therefore, the results at1011 s−1 cannot be compared with our experiments.In contrast, the ripplocation-bearing model exhibited lowshear stress [Fig. 3(d)] and strong velocity dependence atthese sliding velocities of 109 and 1010 s−1 [Fig. 3(e)]. Thecalculated a values for ripplocations, derived from velocitychanges between 109 and 1010 s−1, were positive andconstant across the temperature range of 27 to 200 °C[Fig. 3(f)]. The presence of ripplocations in low-temperaturesamples and their absence in high-temperature samples, asFIG. 3. Friction coefficients of dislocation-free and ripploca-tion-bearing mica obtained by NEMD simulations. (a) Disloca-tion-free mica model and (b) ripplocation-bearing mica model forNEMD simulations, (c) shear stress versus strain of the dis-location-free model at 200 °C, (d) shear stress versus strain of theripplocation-bearing model at 200 °C, (e) calculated frictioncoefficients, and (f) a values in the RSF formulation. Highnormal and shear stresses on the order of GPa are required tosimulate the real area of contact. The experimental shear stress ata normal stress of 100 MPa can be estimated by multiplying theratio of real area of contact to the apparent contact area.FIG. 2. Structures of the sheared surface and cross sections.(a) Scanning electron microscope (SEM) image of the recoveredshear plane tested at 65 °C, (b) transmission electron microscopy(TEM) image of the cross section near the surface, and TEMimages of the cross section from surface to depth tested at 65 °C[(c),(e)] and 200 °C [(d),(f)]. White lines are observed in bothcross sections at low magnification [(c),(d)], whereas high-magnification images reveal that the white lines observed at65 °C correspond to bending of the layers [(e)], and thoseobserved at 200 °C may correspond to amorphous regions causedby electron-beam damage [(f)]. The shear direction is indicatedby arrows [(a),(b)]. The black material on the surface is the golddeposited for the SEM observation. The thickness of 1 nmcorresponds to the thickness of single mica layer.PHYSICAL REVIEW LETTERS 137, 046201 (2026)046201-4confirmed by cross-sectional observations of sheared spec-imens, suggests that the absence of ripplocations is asso-ciated with the velocity-independent frictional behavior ofmica at 200 °C.Our findings indicate that the physics underlying the avalues of mica are connected to the presence of ripploca-tions. TEM observations did not detect ripplocations infresh mica, suggesting that applied shear and normal stresscaused the layers to buckle and form ripplocations. Theformation and stability of ripplocations are influenced bythree main components: the elastic energy associated without-of-plane deformation, the adhesion energy related tointerlayer separation, and interlayer friction that facilitateslayer sliding [68]. Thermal effects can alter these compo-nents in both directions. Under our tested conditions,thermal expansion of mica may increase the elastic energyassociated with out-of-plane deformation, resulting in thedecrease of ripplocations at elevated temperatures.The velocity-independent friction confirmed in mica atT ¼ 200 °C is a realization of ideal Coulomb’s velocity-independent friction. If ideal Amontons-Coulomb frictionmaterial is realized at various temperature conditions, suchmaterial can be useful as a lubricant among mechanicalparts because it requires no precise control of frictiondepending on the sliding velocity. The case where a ¼b ¼ 0 removes all rate dependence in the RSF model,making the use of this formalism unnecessary. Thisvelocity-independent friction of mica must be realizeddue to the layered structure. In our analysis, velocitydependence of dislocations (ripplocations) in the layeredmaterials is the origin of the a value in the RSF model. 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