Experimental testing of the Prandtl-Tomlinson model: Molecular origin of rotational friction

Structural superlubricity, one of the most important concepts in modern tribology, has attracted lots of interest in both fundamental research and practical applications. However, the underlying model, known as the Prandtl-Tomlinson (PT) model, is oversimplified and not for real processes, despite its prevalence in frictional and structural lubricant studies. Here, with a realistic system, cholesteric liquid crystals, confined between two atomically smooth surfaces, we measure both the surface torque during rotational friction and the molecular rotation from the commensurate to incommensurate configuration at the onset of structural lubricity. Furthermore, by changing the surface potential or the strain, the Aubry transition is confirmed. The results agree well with the description by a quasi-one-dimensional version of the PT model and provide molecular evidence for rupture nucleation during static friction. Our study bridges the gap between theories and experiments and reinforces the connection between friction and fracture.

One of the 125 questions 1 published in the Science journal in 2021 is how to measure interfacial phenomena at the microscale.For the interface between solids and solids, contact mechanics have been well-established.However, friction, which dates back to research five centuries ago, has not been well understood due to the multiple contacts at the nanoscale and the complicated dynamics involving physical interactions and chemical reactions 2,3 .Two significant signs of progress in the understanding of stick-slip motions have been made in the past decades.
Another prediction based on the FK model, i.e., the Aubry transition 11 , has also been observed in 1D cold ions 12 and 2D colloidal lattices 13 .However, these two models are oversimplified and the direct experimental testing has not been accomplished for several decades 14 .Second, the paradigm using brittle fracture theory 15 was developed and experimentally confirmed to describe the onset of kinetic friction from the static friction that had previously been described with the static friction coefficient.However, the mechanism of rupture nucleation during static friction at the microscopic level is still unclear 15,16 .The above puzzles are due to the difficulties of experimental characterization at the atomic interface.
Similarly, the interface plays a big role in the viscoelastic behavior of liquid crystals.Some historical puzzles, such as the discontinuous cholesteric-nematic transition 17 and permeative flows 18 , have been ascribed to the boundary conditions.Recently, we proposed 18 that the historical measurement of anomalous viscosity of cholesterics in the capillary is, in fact, the twist elasticity resulting from the confinement with a rather small Ericksen number.In other words, liquid crystals are elastic solids at small scales.Furthermore, our study 17 showed that the cholesteric slip occurs only when the critical surface torque is reached.However, it is still unclear how the slip occurs.In the present work with more experimental data, we interpret the slip behaviors at the critical surface torque under mechanical winding as the reduction of friction by rotating liquid crystal molecules from the commensurate, namely the easy axis, to the incommensurate configuration, forming structural lubricity [Fig.1(b)].In this real-space system, the corrugated surface potential on the muscovite mica works like a grooved surface 19,20 to align molecules.At the easy axis, molecules lie on the commensurate position of grooves to minimize the twist distortion, while under stress, molecules rotate gradually to an incommensurate position with a maximum angle π/2 to the easy axis.With the decay of anchoring strength, it is easier to reach the incommensurate configuration, undergoing the Aubry transition.On the other hand, the rotation of molecules is the sign of rupture nucleation resulting in the propagation of rupture fronts until the slip occurs.Therefore, we consolidate the paradigm that static friction is interfacial rupture 15 and is indeed dynamic 21,22 .Finally, the relationship between dislocations and cracks is discussed.The experiments were performed using a standard instrument 23,24 , i.e., the Surface Force Balance (SFB), to measure force responses of confined cholesterics during surface approach and retraction [Fig.1(a)].Cholesterics are chiral liquid crystals with a pitch of about 244 nm.The main experimental procedure is to compress and stretch the confined cholesterics and measure the forces that respond.
The optics generated in the SFB were recorded and analyzed to obtain results, such as distance, velocity, force, and twist angle.Detailed experimental methods are shown in Sec.SIII of the Supplementary Materials.In the SFB, the surfaces are made of muscovite mica with atomic smoothness, and the interaction between the surfaces and cholesterics is not as strong as chemical bonds.As a result, the liquid crystal molecules rotate both in the bulk and on the surface to minimize the free energy under stress [Fig.1(b)].The free energy per unit area  as a sum of the anchoring energy (one surface) and twist elastic energy by ignoring the dislocation energy is written as 17 , where  is the deviated angle of molecules to the easy axis,  is the anchoring strength,  22 = 6 pN is the twist elastic energy, Φ is the total twist angle,  is the smallest surface separation between two crossed cylinders thus, The form in Eq. ( 1) is widely used in the liquid crystal community, sometimes with different expressions in the anchoring energy 25,26 .In fact, this form is a modified threedimensional (quasi-1D) version of the PT model [Fig.1(c)] below or the FK model, with a quasi-static motion, where  is the free energy, The rotational friction in cholesterics under confinement is a balance of the twist elastic torque Γ  and the frictional anchoring torque Γ  , by differentiating Eq. ( 1) with respect to the twist angle Φ, while ignoring the viscous torque with a strong anchoring limit, From the fitting line in Fig. 2(d), the critical torque Γ  ≈ 0.22 mN/m and anchoring strength  ≈ 0.27 mN/m can be calculated using Eq. ( 6) and the pitch  = 244 nm.
Meanwhile, the deviated angle  ≈ 0.52π is obtained using Eq. ( 4), which is consistent with the previous result 17 , i.e.,  ≈ 0.49π.The effect of Burgers vector and noninteger layer on the measurement is discussed in the Supplementary Materials (Fig. S1).The deviation of molecules by around π/2 from the easy axis is a sign of structural lubricity at the maximum incommensurate configuration.Therefore, the barrier to having a slip is the activation energy required to get structural lubricity.
Furthermore, the critical surface torque is almost independent of the layer thickness or the motor speed of the SFB, ranging from a few nanometers per second to 80 nm/s [Fig.2, (e) and (f)].In the PT model, a dimensionless parameter  is calculated to predict the stick-slip behaviors, which is the ratio between the stiffness of the surface potential and the spring 3,27 , By analogy, the parameter   based on Eq. ( 2) is also calculated, where  chl =  22  is the spring constant of the cholesterics and  = Φ/.
In a previous study 17 , we showed three distinct regimes of cholesterics under mechanical stress during the decay of surface anchoring (Fig. 3).First, with strong anchoring strength, cholesteric layers were collectively removed at about 65% strain, i.e., the constrained regime.Second, with intermediate anchoring strength, cholesteric layers were compressed up to about 30% strain, before being removed one by one, i.e., the stick-slip regime.Third, with weak anchoring strength, cholesteric layers were continuously removed one by one almost without strain.
Although explained by the decrease of surface torque, the regime transition will be further interpreted here.With time evolution, probably due to the adsorption of water that smoothens the corrugation of the surface potential, the anchoring strength decreases.As a result, the transition goes from the constrained regime (initially   > 1), stickslip, to the sliding-slip regime 17 (  < 1), which is the Aubry transition 14 by decreasing the surface potential (Fig. 3).In other words, the stress-induced incommensurate status changes from a pinned status to a more continuous transition (sliding-slip regime) layer by layer with a smaller corrugation of surface potential, with which molecules easier rotate to reach incommensurate status.
These three regimes are consistent with the results predicted by a simulation 28 .On the other hand, the jump process (i.e., kinetic friction) is continuous in the constrained regime while it is discontinuous in the stickslip regime (Fig. 3), which is another Aubry transition 14 by increasing the spring constant (i.e., strain-stiffening) of cholesterics with larger compression ratio 18 .Although in the constrained regime, the anchoring potential is large, after compression the stiffness of cholesterics outweighs that of the anchoring potential.With the stiffer spring constant of cholesterics, there is not enough time for the slower molecules to recover from the incommensurate configuration during slip to the commensurate configuration, such that the structural lubricity continues.

4(f)
].This breakup is not the same as the break of a liquid jet due to the Plateau-Rayleigh instability 32 .By contrast, during the approach, defects move to a larger radius, such that they never become cracks, but the whole layers undergo a tearing mode fracture encountering large energy before yielding [Fig.4, (a) to (f)].Under such circumstances, cholesterics behave like ceramics that resist compression but not tension.
The adhesion between two crossed cylinders or two spheres at contact during retraction has been proven to be the opening mode of fracture for decades 33 .We may also speculate that rolling friction is also the crack opening on one side that decreases the dissipated energy compared to the shearing mode of the normal stick-slip friction.separates the approach and retraction regimes.The non-integer layer has been deducted in (f).(g) The force profile of cholesterics during the surface approach.The red line is the theoretical fit using Eq. ( 9).R = 1 cm is the radius of crossed cylinders.The blue line is the slope of the SFB spring with a spring constant k = 179 N/m.(h) The force profiles up to 70% strain (an arbitrary value) with different layers calculated by Eq. ( 9).The black line is the linear fit of the maximum forces.(i) The yield energy calculated by integrating the forces with respect to the distance in (h).The black line is the parabolic fit of the yield energy.
With the strong anchoring assumption, the anchoring energy is negligible compared to the twist elastic energy.
Thus, the elastic force with  layers is calculated by the second term on the right side of Eq. ( 2) with Derjaguin approximation, The measured forces can be well fitted by the force calculation using Eq. ( 9  gradually increases at large distances 35,36 .In a more complex geometry with bumps and hollows [Fig.5(c)], the dislocations automatically form an isoheight map.The dislocations are denser in bumps with larger heights compared to those in hollows.
At large distances, thicker dislocation lines may be observed due to the increase of the Burgers vector.The geometry of confinement may provide a method to design defect networks 37 with different topological structures.
When the anchoring strength is strong, the disordered dislocations will be repelled to the bisector of the confinement, while with weak anchoring strength, the dislocations will be attracted to the surface 38 .This behavior provides a method to engineer the distribution of dislocations in materials.Furthermore, the paths of disordered dislocations are analogous to the crack paths designed on curved surfaces 39,40 .In other words, the disordered dislocations visualize the region with higher stress intensity 39,40 , and the order parameter in liquid crystals serves as a single variable to predict how cracks propagate in a complex geometry.Meanwhile, the thickness of the dislocation line is associated with the levels of stress intensity.Rotational friction on surfaces at the molecular level has been studied with absorbing molecules.The rotation of a single molecule can be controlled by external stimuli, such as electrons 41 , temperature 42 , and mechanical stress 43 .By contrast, the dynamics of absorbing molecules can be probed collectively as surface viscosity 44 or solid friction torque 45 .However, previous research mainly focused on the molecule-substrate interaction, i.e., the term of corrugated surface potential in the PT model.Without studying the interaction within molecules or between molecules, i.e., the elastic term, the PT model cannot be tested.In this work, we show that the surface potential and elastic energy are coupled and, therefore, should be studied simultaneously.Benefiting from the well-defined cholesteric system, the rotational friction at the molecular level is unraveled.
If we compare the static rotational friction with the dry sliding friction ruled by the Coulomb-Amontons laws, it is apparent that, firstly, the surface torque is independent of the contact area [Eq.( 4)].Additionally, despite different interaction areas with various layers (Sec.SII in the Supplementary Materials), the critical torque is the same [Fig.2(c)].Secondly, the critical surface torque is independent of the approaching speed outside the viscous torque regime [Fig.2(e)].Thirdly, the surface torque is proportional to the deviated angle [or deviated molecular rotation rate, Eq. ( 4)] rather than the normal load.However, in the Coulomb-Amontons laws, the normal load also could be described by the elastic deformation of the contact area through Hooke's law, which, therefore, is proportional to the strain.In other words, the traditional sliding friction is proportional to the strain; by contrast, the rotational friction is proportional to the molecular deviated angle.Taken together, we conclude that static rotational friction is still governed by the Coulomb-Amontons laws.
Many concepts in liquid crystals are borrowed from the crystalline community, such as dislocation and piezoelectricity.Conversely, the concept of disclination, which was developed in liquid crystals, has been used later in crystals 46 .Despite the well-developed elastic energy theory, liquid crystals are considered as fluids 47 .Therefore, fracture mechanics has rarely been used to describe liquid crystal behaviors.Even the concept of surface torque is rarely used in the liquid crystal community 17 .Our previous study 18 highlighted the importance of elasticity with strong surface anchoring under confinement.This work further underlines the friction and fractures in liquid crystals, which promotes the understanding of structural lubricity and interfacial ruptures in crystals.
Although the exact form of the anchoring potential is not proven, it is reasonable to use the parabolic potential 25 , since the elastic energy of liquid crystals is based on Hooke's law.The twist interaction of molecules with the surface is similar to the elastic interaction between cholesteric molecules.In fact, the anchoring potential could be verified by measuring both the surface separation and the deviated angle, simultaneously.For example, the newly developed μSFA 48 can be mounted into the polarized microscope for this purpose.The further understanding of surface anchoring may be assisted by first-principles calculations 49,50 .
It has been shown 51 that the maximum dissipated energy during friction is equal to the corrugation of the potential energy surface, which seems to be the first term of Equation 1 if the second term did not dissipate as phonons.
For cholesterics, the rotation of molecules along the helical axis also causes piezoelectricity 52 , which needs to be taken into account as electronic dissipation.In the stick-slip regime, apart from the dissipated heat, residual energy is also stored in the rest layers until all the layers are squeezed out.
There has been a lasting debate 53,54 on the mechanism of stick-slip motions during molecular friction.Three scenarios, including one-layer or whole-film melting and interlayer slip, have been proposed 53 .It is difficult to distinguish the melting and the slip when the heat dissipation from the yield energy is high enough to melt molecular layers or cause wear during stick-slip.However, from the results of this work, it may be more favorable for the scenario of interlayer slip 54,55 , which is also a sign of interfacial ruptures at the molecular level.
In summary, we tested the PT and FK models with

II. Interaction areas with various layers
With n layers confined in the SFB, the original distance is  0  .In the region with more than n layers, for example, n+1 layers, the layers are half compressed and half stretched [Fig.1(a)], which neutralizes the interaction.In other words, the innermost layer region is the effective interaction area, although the whole sample is under confinement.When the surface separation is compressed to D, ignoring the dislocation energy, there is an equilibrium position 5 at the radius  = √2( 0  − ), where R is the radius of the cylinder and r is the distance away from the surface contact point.In other words, the region within radius r is the effective interaction area.Therefore, with various layers but the same compression ratio at the jumping events, the interaction area depends on  0  , which varies with layers.
The cholesterics produce a pitch of about 244 nm.
The cholesterics were dried in the Schlenk line at around 80℃ overnight for the measurement of the compression ratio.
Without the Schlenk line, the strong anchoring strength may be obtained [Fig.2, (a) and (b) and Fig. 4], but sometimes the trace amount of water in the sample may affect the experiments.To obtain a surface with atomic smoothness, muscovite mica was freshly cleaved and deposited with a silver layer on one face.During the experimental setup, the silver side of the mica was glued onto the glass lens with a radius of 1 cm.Subsequently, the lenses were mounted into the SFB with standard procedures 6 .Finally, the SFB chamber was dried with the nitrogen for at least 1 h before the injection of the cholesterics.
The SFB is a standard instrument invented decades ago 6,7 .Inside the SFB, the optics reflected between the silver layers generate fringes of equal chromatic order (FECO) 8,9 on the spectrometer.The FECO can be analyzed to obtain information about surface separation, according to multiple-beam interferometry 8,9 .For force measurements, a cylindrical lens is connected to a motor that moves with a constant speed of a few nanometers, while another lens sits on a spring with a known spring constant.When a repulsive force is encountered during surface approach, the surface separation moves slower than the motor speed.Therefore, the force can be calculated.

FIG. 1 . 2 𝑈 0
FIG. 1. Mechanical winding of cholesterics.(a) Schematic diagram of cholesterics under the confinement of crossed cylinders (front view), where  = 244 nm is the pitch.(b) The deviation of molecules from the easy axis under stress, where Φ 0 is the original twist angle at the distance without compression  0 , and δ is the deviated angle at the compressed distance .The dot-dash line is the easy axis on the potential energy surface.(c) The Prandtl-Tomlinson model, 1 2  0 and  are the amplitude and periodicity of the surface potential, respectively,  is the spring constant, and  is the velocity along the -axis.

[Fig. 1 (
b)], and  0 is the molecular rotation rate of cholesterics at relaxation.The detailed derivations of the cholesteric equations in the following text can be found in a previous study 17 .With strong anchoring strength, twist angle Φ = Φ 0 − 2 ≈ Φ 0 =  0  0 keeps the relaxed twist angle Φ 0 at the original distance  0 [Fig.1(b)],

1 2
0 and  are the amplitude and periodicity of the surface potential respectively,  is the spring constant,  is the velocity along the  axis, and  is the time.

Figure 2 (
Figure 2(a) shows a typical experiment of cholesteric slippage under mechanical winding.During surface compression at a speed smaller than 4 nm/s, cholesterics underwent an elastic response and then yielded when reaching the maximum surface torque, resulting in a surface jumping event [Fig.2, (a) and (b)].Figure 2(c) shows that indeed the compression ratio at the critical jumping distance remains almost constant at about 0.2-0.4,with different half-pitch layers (i.e., layer thickness = halfpitch = π rotation) of cholesterics.If the threshold of the critical surface torque Γ  at the critical jumping distance   is assumed 17 ,

FIG. 2 .
FIG. 2. Slippage of cholesterics at the incommensurate position.(a) The elastic response and yield of cholesterics during the surface approach experiment.(b) Corresponding surface velocity (grey).The black curve is the smoothed data.(c) The compression ratio of cholesterics with various layers in the jumping events (i.e., yield events).(d) The critical yield distance as a function of the original distance.The line is the linear fit.(e) The compression ratio of cholesterics in the jumping events with various motor speeds of the SFB.The experiments were performed with 11 (circle) and 12 (triangle) integer layers.Some data (black) were reused from a previous study17 .(f) Twist elastic torque with various cholesteric layers under compression up to 80% strain [an arbitrary value, i.e., relative deformation (D0-D)/D0], calculated using Eq.(5).

FIG. 3 .
FIG. 3.Aubry transition by changing the anchoring strength or the stiffness of cholesterics.Three distinct regimes of force responses under compression were observed with the decay of the anchoring strength (experimental data reused)17 .

FIG. 4 .
FIG. 4. Fractures in cholesterics.(a) Dislocations of cholesterics in the SFB.(b) A rod with two cracks.(c-f) The differences in the distance profile, surface velocity, force profile, and twist transition during approach and retraction experiments.The red line in (c-e) ), confirming an elastic deformation [Fig.4(g)].Figure 4(h) shows that the forces generated by different layers with the same strain are different with a linear increase proportional to the number of layers, while the yield energy calculated from these force profiles increases parabolically [Fig.4(i)], which is different from the linear increase of elastic energy in normal springs with different lengths.The increase in yield energy is consistent with a previous study34 showing that lubricants increase the fracture energy since, in dry friction, the stiffness of the surface is quite large decreasing the stored energy.With fracture mechanics, it is easier to understand the formation of dislocation defects in the SFB and Grandjean-Cano wedge (Fig.5).At a height equal to  integer layers, the cholesterics are relaxed.By contrast, integer cholesterics are compressed on the left side and stretched on the right side, i.e.,  0 ± ∆ [Fig.5(b)], which is more significant at small heights with relatively strong deformation.By increasing the heights, the layers are further stretched until a new layer is added to minimize the free energy.Therefore, with less intense deformation, the Burgers vector  = 1 2

FIG. 5 .
FIG. 5. Dislocations in different geometries.Front-view and top-view of cholesteric dislocations in (a) the SFB, (b) the Grandjean-Cano wedge, and (c) a geometry with bumps and hollows.The dashed lines represent dislocations, the dot-dashed lines are guidelines, and the thick dashed lines in (c) are thick dislocations.
FIG. S1.The critical yield distance as a function of the original distance in the sample after 24 h.(a) The yield events with all sample layers.(b) The yield events with sample layers no larger than 12. (c) The yield events with thick samples.
Both the distance and the surface velocity profiles [for example, Fig 2, (a) and (b)) were directly measured by the SFB.