Stick-Slip Mode of Boundary Friction as the First-Order Phase Transition
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b GENERAL PROBLEMS OF THEORETICAL PHYSICS
ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 I.A. LYASHENKO, A.M. ZASKOKA
Sumy State University (2, Rimskii-Korsakov Str., Sumy 40007, Ukraine; e-mail: [email protected], [email protected])
STICK-SLIP MODE OF BOUNDARY FRICTIONAS THE FIRST-ORDER PHASE TRANSITION
PACS 05.70.Ce; 05.70.Ln;47.15.gm; 62.20.Qp;64.60.-i; 68.35.Af; 68.60.-p
A tribological system consisting of two contacting blocks has been considered. One of themis arranged between two springs, the other is driven periodically. The kinetics of the systemhas been studied in the boundary friction mode, when an ultrathin lubricant film is containedbetween the atomically smooth surfaces. In order to describe the film state, the expression forthe free energy density is used in the form of an expansion in a power series in the orderparameter, the latter being reduced to the shear modulus of a lubricant. The stick-slip modeis shown to be realized in a wide range of parameters, being a result of the periodic first-orderphase transitions between kinetic friction regimes. The behavior of the system governed byinternal and external parameters has been predicted.K e y w o r d s: ultrathin lubricant film, boundary mode of friction, tribological system
1. Introduction
Owing to the rapid development of high-precisionexperimental techniques aimed at researching thenanosystems, the processes of friction in the casewhere the thickness of a lubricant between rubbingsurfaces equals several atomic diameters have beenintensively studied recently [1–3]. In tribology, thisfriction mode was called “boundary friction”. It isoften realized in ordinary mechanisms, because therubbing surfaces contact with each other by meansof surface irregularities or inhomogeneities [1, 4]. Theboundary friction differs essentially from the hydro-dynamic mode, when the friction force is proportionalto a power function of the velocity. Note that theultrathin film of a lubricant does not form conven-tional, thermodynamically equilibrium phases, solidand liquid ones. Instead, we have liquid- and solid-like states, which are kinetic friction modes, and therecan be several of them [5, 6]. This occurs because thesymmetry of a lubricant state is substantially affectedby friction surfaces, this fact being not of importancefor bulk lubricants. In the course of friction, the phasetransitions of both the first and second orders cantake place between the stationary states [7, 8]. Thesetransitions often comprise the origin of the stick-slipmotion mode for contacting surfaces [5, 6, 9]. c (cid:13) I.A. LYASHENKO, A.M. ZASKOKA, 2013
In order to describe the boundary mode of fric-tion and nano-contact phenomena, phenomenologicalmodels are widely used [7–10]. In particular, a modelwas developed [11], in which the lubricant melting isdriven by the thermodynamic and shear mechanisms.In the framework of this model, the influence of ad-ditive fluctuations of principal quantities was stud-ied [12], and their presence in the system was demon-strated to result in the emergence of new stationarystates and new kinetic friction modes [13, 14], whichare not essential for bulk systems. The origin of thehysteretic behavior, which was observed experimen-tally [3, 15, 16], was elucidated in work [17]. The in-dicated model also made it possible to describe theperiodic stick-slip mode of motion [18, 19].In works [7, 20], a thermodynamic scenario ofboundary friction was proposed, which is based onthe phase transition theory developed by Landau [21].This model takes into account that the ultrathinfilm of a lubricant can melt and stay in a liquid-like disordered state both owing to the ordinary ther-modynamic melting and as a result of overcomingthe fluidity threshold by the shear stress component(“shear melting”). The influence of those factors wasalso studied in work [8], in which the excess vol-ume [22, 23] arising owing to the lubricant stochas-tization at its melting was selected as the order pa-rameter. As the excess volume increases, the shear .A. Lyashenko, A.M. Zaskoka
M KK
Fig. 1.
Diagram of a tribological system modulus decreases [8], which results in the melting.In works [7, 20], the shear modulus itself was selectedas the order parameter, which acquires zero valuesin the liquid-like phase. However, in works [7, 20],the melting is described as a continuous phase tran-sition of the second order, whereas jump-like phasetransitions of the first order are often observed in theboundary friction mode [5,6,8], which are responsiblefor the stick-slip motion [5, 6].This work is aimed at describing the first-orderphase transition in the framework of the model de-veloped in works [7, 20] and at studying the behaviorof tribological systems on the basis of the indicatedmodification. The model proposed does not make al-lowance for specific types of lubricants, because itstask consists in describing the origins of phenomenathat take place at the boundary friction. For specifictypes of lubricants and friction surfaces, the modelshould be modified. To some extent, it can be madeby choosing the numerical values of coefficients inthe series expansions of the free energy, relaxationtimes, and so on. The model describes only homoge-neous lubricants composed of non-polar quasispher-ical molecules [5, 6]. One of the reasons is the factthat we study a situation where the elastic stressesacquire zero values in the liquid-like state, i.e. themelting gives rise to the total disordering of lubricantmolecules, which does not takes place in thin lubri-cant films consisting of polymer molecules. Anotherreason is the fact that the obtained time dependencesof the friction force and stresses are strictly periodic,which is also observed for quasispherical moleculesonly [5, 6].
2. Tribological System
The fabrication of atomically smooth surfaces withlarge dimensions is associated with considerable tech-nological difficulties. Therefore, to measure the dy-namic parameters of ultrathin lubricant films betweensuch surfaces, the surfaces characterized by small di-mensions and pasted on spherical or cylindrical sur-faces that rub each other are used. This scheme wasapplied while designing the surface force apparatus (SFA) [3, 24, 25]. Two types of SFA—Mk II andMk III—were described in review [3]. In the latter,the system to control distances between rubbing sur-faces was improved. The device allows the shape ofsurfaces to be determined, as well as the distance be-tween them to within an accuracy of 1 ˚A. The contactarea between surfaces is measured with an accuracy of ± %, the normal and shear components of operatingforces to within ± %, and the magnitude of appliedloading to within ± %.One of the rubbing surfaces in the SFA is fixed,and the other is driven to move periodically. In thecourse of motion, the shear stresses and the effec-tive viscosity of a lubricant are measured, the lubri-cant structure is determined, and so forth. In thiswork, we consider a simplified mechanical analog ofthe SFA exhibited in Fig. 1. Two springs character-ized by the stiffness constant K are connected witha block of mass M mounted on rollers. The rollingfriction for the latter is neglected below. On the in-dicated block, another block is arranged, which isbrought into a periodic motion by applying an exter-nal force. Provided that the surfaces of two blocksinteract with each other, the motion of the upperblock stimulates the motion of the lower one. Thetrajectory of the lower block substantially dependson the friction mode established in the system. Asimilar tribological system was experimentally stud-ied in works [26, 27]. Note that, in contrast to theSFA design, now both blocks are mobile, which en-ables the time dependences of block coordinates andvelocities to be registered, and, by analyzing them,the rheological and tribological characteristics of thesystem to be determined.Let X and V = ˙ X be the coordinate and the veloc-ity, respectively, of the upper block, whereas x and v = ˙ x denote the corresponding quantities for thelower block. Let us consider the case where the up-per block moves according to the cyclic law, X = X m cos ωt, (1) V = − X m ω sin ωt, (2)where X m is the amplitude, and ω is the cyclic fre-quency. We write down the equation of motion forthe lower block in the form [26] M ¨ x + 2 Kx − F = 0 , (3)where F is the friction force that arises between theblocks at their relative motion. From the last expres-sion, it follows that the character of motion in the ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 tick-Slip Mode of Boundary Friction system essentially depends on the friction mode andthe lubricant properties, because they determine theforce F .The friction force is determined in a standard way, F = σA, (4)where σ is the shear stress that arises in the lubricant,and A is the contact area between rubbing surfaces.In the boundary friction mode, the elastic, σ el , andviscous (dissipative), σ v , stresses arise in the lubri-cant layer [7, 8, 16]. As a rule, the melting is accom-panied by a reduction of the elastic stress component,whereas the viscous one grows owing to an increaseof the relative shear velocity between rubbing sur-faces [16]. Hence, the total stress is determined bythe sum of indicated components, σ = σ el + σ v . (5)The viscous stresses in the lubricant layer are deter-mined by the empirical formula [28, 29] σ v = η eff ( V − v ) h , (6)where the effective viscosity of the lubricant, η eff (itdepends on plenty of factors and is determined exper-imentally) is introduced into consideration, as well asthe relative velocity of surface motion, V − v .As a result, in the case of boundary friction, poly-mer solutions or melts are applied as lubricants. Thenecessity of such an application is caused by the factthat the friction surfaces are small in dimensions, andthe lubricant film between them must not be squeezedout under the influence of large tribological loadings.Such lubricants are non-Newtonian fluids, the viscos-ity of which depends not only on the temperature, butalso on the velocity gradient. However, the applica-tion of the SFA allows the behavior of a wide classof lubricants to be examined in the boundary frictionmode, because the rubbing surfaces in these experi-ments are completely imbedded into a vessel with aliquid to study, so that the latter is not squeezed outfrom the gap between the surfaces during their mo-tion [3]. However, we should note that even ordinarywater, when being used as a boundary lubricant, canbehave as a non-Newtonian fluid, because, owing toits interaction with the surfaces, it can create spa-tially ordered structures in the course of motion.The non-Newtonian fluids are divided into twoclasses: pseudoplastic fluids, the viscosity of which decreases with the growth of the strain rate (e.g.,these are polymer solutions and melts) and dilatantones, the viscosity of which increases as ˙ ε grows (e.g.,suspensions of solid particles). For both situations tobe taken into account, let us use a simple power-lawapproximation [28, 29] η eff = k ( ˙ ε ) γ . (7)Here, we introduced the proportionality coefficient k (its dimension is Pa · s γ +1 ) and the dimensionless in-dex γ (for pseudoplastic fluids, γ < ; dilatant onesare characterized by the index γ > ; and γ = 0 inthe case of Newtonian fluids).The strain rate is determined through the relativevelocity of motion and the lubricant thickness h [28], ˙ ε = V − vh . (8)Taking Eqs. (7) and (8) into account, the expressionfor viscous stresses (Eq. (6)) looks like σ v = k (cid:18) V − vh (cid:19) γ +1 . (9)Note that, according to Eq. (9), viscous stresses areavailable in both the liquid- and solid-like states ofa lubricant. The presence of viscous (dissipative)stresses in both phases was indicated in the exper-imental work [16]. However, if the lubricant is in asolid-like state, viscous stresses are low, because, inaccordance with Eq. (9), they are proportional to therelative shear velocity, V − v , which is low in this case.Substituting Eqs. (5) and (9) into Eq. (4), we ob-tain the final expression for the friction force [30–33], F = " σ el + k sgn( V − v ) (cid:18) | V − v | h (cid:19) γ +1 A, (10)where the function sgn( V − v ) = ( , V ≥ v, − , V < v (11)takes into account the direction of force action. Thefirst term in Eq. (10) describes the elastic componentof the friction force, and the second is responsiblefor the viscous one, which grows with the velocity.Hence, the friction force depends on the velocity ofthe lower block, v , and the elastic stresses, σ el , thatarise in a lubricant. ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 .A. Lyashenko, A.M. Zaskoka
3. Thermodynamic Model
In the homogeneous case, the free energy density foran ultrathin lubricant layer looks like [7, 20, 30, 31] f = α ( T − T c ) ϕ + a ϕ ε − b ϕ + c ϕ , (12)where T is the lubricant temperature; T c is the crit-ical temperature; ε el is the shear component of theelastic strain; α , a , b , and c are positive constants,and ϕ is the order parameter (the amplitude of theperiodic component in the microscopic medium den-sity function [7, 20]). The parameter ϕ equals zero inthe liquid-like phase and acquires nonzero values inthe solid-like one. In comparison with works [7, 20],potential (12) takes additionally the third-order terminto account. This form of expansion is used to de-scribe phase transitions of the first order [21, 34]. Inthe second term in Eq. (12), we also introduced thefactor a , which allows us to vary the contribution ofthe elastic energy to the potential.The elastic stresses that arise in the lubricant layer,according to Eq. (12), are determined as the deriva-tive σ el = ∂f /∂ε el , so that σ el = aϕ ε el . (13)Therefore, after the coefficient a has been introducedinto expansion (12), the shear modulus is determinedas follows: µ = aϕ . (14) Fig. 2.
Dependences of the free energy density f (see Eq. (12))on the dimensionless order parameter ϕ for various temper-atures T = 265 , 286, and 310 K (curves to , respec-tively). The calculation parameters are α = 0 .
95 J / (K · m ) , T c = 290 K, a = 4 × Pa, b = 230 J/m , c = 850 J/m ,and the shear strain ε el = 2 . × − Depending on the value of parameter a , it can acquireboth small and large values at | ϕ | < . Note that, inthe boundary friction mode, the shear modulus canbe several orders of magnitude larger than that inthe hydrodynamic mode for the same lubricant. Asa result, if the critical temperature T or critical elas-tic shear stress σ el become exceeded in the courseof friction, the lubricant does not melt completely;instead, a domain structure with regions of liquid-assisted and dry friction is created. For this situationto be studied, Eq. (12) must include gradient terms,which considerably complicates the subsequent con-sideration. However, the examination of such spatialstructures comprises a separate problem, which is notthe purpose of this work. Therefore, the gradientterms are excluded from Eq. (12), which correspondsto the consideration of a lubricant behavior in theframework of the one-domain model with a homoge-neous structure.According to the principle of minimum energy, thesystem tends to occupy a stationary state, which cor-responds to the minimum of the free energy f ( ϕ ) (seeEq. (12)), irrespective of its initial conditions. Sincethe parameter ϕ is the amplitude of the periodic com-ponent in the microscopic medium density function,we consider below only the physical range of values ϕ ≥ . Let us introduce the function B ( ε el , T ) = aε + 2 α ( T − T c ) . (15)The analysis of expression (12) for the free energy al-lows the following situations to be distinguished. Pro-vided that the condition B ( ε el , T ) ≤ is obeyed, themaximum of potential (12) at ϕ = 0 and its min-imum at ϕ > are realized (curve in Fig. 2).In this case, the lubricant is solid-like, because theshear modulus µ > . In the intermediate interval < B ( ε el , T ) < b / (4 c ) , the maximum of the poten-tial at ϕ = 0 transforms into a minimum and, ad-ditionally, there emerges a maximum that separatesthe zero and nonzero minima (curve in Fig. 2; it isalso shown scaled-up in the inset). In this case, thestate of a lubricant depends on the initial conditions,and the lubricant can be in either solid- or liquid-likestate. In the latter case, B ( ε el , T ) ≥ b / (4 c ) , and asingle minimum of the potential at ϕ = 0 is realized(curve in Fig. 2), which, according to Eq. (14), cor-responds to the zero value of lubricant shear modulusand its liquid-like structure.The stationary values of order parameter ϕ aredetermined as the roots of the equation ∂f /∂ϕ = 0 ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 tick-Slip Mode of Boundary Friction [30, 31], namely, ϕ ∓ = b c ∓ s(cid:18) b c (cid:19) − (cid:18) ac ε + 2 α ( T − T c ) c (cid:19) . (16)The root ϕ − is related to the unstable stationarystate, because it corresponds to the maximum of po-tential (12). The stable state, which corresponds tothe potential minimum, is given by the root ϕ + . Be-sides roots (16), the stationary solution ϕ = 0 alwaysexists, which corresponds to the extremum of poten-tial (12) with the order parameter equal to zero; it canbe either a maximum or minimum of the potential.According to Eq. (16), the lubricant melts if eitherthe temperature T elevates or the shear componentof the elastic strain, ε el , grows. Thus, the model con-cerned makes allowance for both thermodynamic andshear meltings.As was already indicated above, at small valuesof temperature T and strain ε el , when the function B ( ε el , T ) ≤ , the lubricant is solid-like, because, inaccordance with Eq. (16), a stationary value of pa-rameter ϕ different from zero is realized, and, accord-ing to Eq. (14), the shear modulus µ is also non-zero.In this case, the potential has a single minimum at ϕ ≥ . If the temperature T exceeds the critical value T c = T c − a α ε + b αc , (17)the order parameter vanishes in a jump-like manner,when the lubricant passes into the liquid-like state, inwhich the potential f ( ϕ ) has a single minimum at ϕ =0 [30, 31]. If, after this transition, the temperature T falls down further, the lubricant solidifies followingthe mechanism of first-order phase transformation ata lower temperature, T c = T c − a α ε , (18)and the parameter ϕ becomes non-zero again. In theintermediate temperature region, T c < T < T c , thepotential is characterized by two minima at positive ϕ . Hence, the dependence ϕ ( T ) has a hysteresis char-acter [30, 31] and corresponds to the phase transitionof the first order. Expression (18) elucidates the phys-ical meaning of the critical temperature T c ; namely,it is the temperature of lubricant solidification at zerostrains, when only the mechanism of thermodynamicmelting is active in the system.From expression (17), it follows that the lubricantmelts not only at the temperature elevation, but also if it is subjected to an external mechanical action,when the elastic strain component exceeds the criticalvalue ε el ,c = r α ( T c − T ) a + b ac . (19)Using formula (18), we can determine the elastic de-formation ε el , at which the lubricant solidifies, ε ,c = r α ( T c − T ) a . (20)Note that, according to relation (19), the melting canoccur even at the zero temperature, T = 0 , if thestrain exceeds the critical value. At the zero strain,i.e. at ε el = 0 , the lubricant melts, when its temper-ature exceeds the critical value T c (see Eq. (17)).As a rule, it is the relative shear velocity betweenfriction surfaces rather than the shear strain compo-nent ε el that is registered in experiments [5,6]. There-fore, for our research to go further, it is necessary toobtain a relation between those two quantities. Let ustake advantage of the Debye approximation, accord-ing to which the elastic strain component ε el arises inthe lubricant layer, when the latter flows plasticallyat the velocity [7] ˙ ε pl = ε el τ ε , (21)where τ ε is the Maxwell relaxation time for internalstresses. The total strain in the layer is determinedas the sum of elastic, ε el , and plastic, ε pl , compo-nents [7, 23] ε = ε el + ε pl . (22)Combining relations (8), (21), and (22), we obtainthe kinetic equation for the evolution of the elasticcomponent of the shear strain [8, 30, 31, 33]: τ ε ˙ ε el = − ε el + ( V − v ) τ ε h . (23)Boundary friction experiments testify that the re-laxation time of the elastic strain is very short, as arule. This quantity can be estimated from the relation τ ε ≈ a/c ∼ − s , where a ∼ nm is the lattice con-stant or the intermolecular distance, and c ∼ m / s is the sound velocity [11]. However, in the boundarymode, the strain relaxation time, τ ε , can differ by sev-eral orders of magnitude [5, 6]. Bearing in mind thatthe value of strain relaxation time τ ε is small, below ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 .A. Lyashenko, A.M. Zaskoka we use the adiabatic approximation τ ε ˙ ε el ≈ [35],which allows us to determine the strain by its sta-tionary value ε = ( V − v ) τ ε h , (24)rather than by Eq. (23).In the general case, the free energy (12) depends onthe lubricant layer thickness h [20]. Note that, in theframework of our model, the second term in expres-sion (12) is proportional to the square of the elasticstrain, ε . In accordance with relation (24), the sta-tionary elastic strain increases with a reduction of thelubricant thickness h . Therefore, in the limiting caseof a very thin layer ( h → ), the strain ε el → ∞ .In this case, the second term in expansion (12) dom-inates, and the stationary value of order parameterequals zero, so that the lubricant is liquid-like, as inwork [20]. A detailed study of the influence of the lu-bricant layer thickness on friction modes was carriedout in works [36, 37].
4. Kinetics of Melting
The changes in the lubricant temperature T and thestrain ε el induce variations of the order parameter ϕ , which governs the free energy f (see Eq. (12))in accordance with the power-law expansion of thelatter [21]. The stabilization time for a new sta-tionary value ϕ + (see Eq. (16)) is determined bythe generalized thermodynamic force − ∂f /∂ϕ . If ϕ ≈ ϕ + , this force is small, and the relaxation pro-cess is described by the Landau–Khalatnikov linearkinetic equation [38] ˙ ϕ = − δ ∂f∂ϕ , (25)where the kinetic coefficient δ characterizes the iner-tial properties of the system. After substituting en-ergy (12) into Eq. (25), we obtain the equation in theexplicit form, ˙ ϕ = − δ (cid:0) α ( T − T c ) ϕ + aϕε − bϕ + cϕ (cid:1) + ξ ( t ) . (26)Equation (26) includes a term responsible for ad-ditive fluctuations with a low intensity [30, 31]. Theirintensity is selected to be so low that they do notaffect the deterministic behavior of the system. How-ever, their introduction is necessary, because, at sub-sequent numerical calculations, the root ϕ = 0 of Eq. (26) corresponding to the maximum of the poten-tial f ( ϕ ) , i.e. to the unstable stationary state, turnsout stable according to the structure of the equation.In this situation, the introduction of ξ ( t ) stimulatesthe system to transit from the unstable state intoa stable one, which corresponds to the energy mini-mum. Hence, fluctuations are taken into considera-tion by means of the features in subsequent numericalcalculations.The dynamic characteristics of any tribological sys-tem are governed by its properties in whole. Forinstance, in the geometry illustrated in Fig. 1, thebehavior of the system substantially depends on thestiffness constant, K , of the spring and the mass ofthe lower block, M . In contrast to the case of motionwith constant elastic strains, this tribological systemcan reveal the stick-slip mode of motion in the courseof friction [5, 6, 9, 18]. The indicated mode is estab-lished because the lubricant periodically melts andsolidifies in the course of motion, which leads to theoscillatory character of friction force F . To calculatethe evolution of this system in time, we need to solvethe system of kinetic equations (3) and (26) numeri-cally, determining the friction force F from Eqs. (10)and (11), the elastic stresses σ el from Eq. (13), andthe strain ε el from relation (24). In so doing, we haveto take into account the relation ˙ x = v , as well asdefinitions (1) and (2).While solving the differential equations numeri-cally, we used the Euler–Cramer method with thetime increment ∆ t = 10 − s . The initial conditions ϕ = x = v = 0 were chosen. The result obtainedis shown in Fig. 3. The dashed curve in the upperpanel corresponds to the time dependence of the up-per block coordinate X ( t ) (see Eq. (1)), and the solidone to that of the lower block, x ( t ) , which is morecomplicated. The figure also exhibits the time depen-dences for the block velocities, elastic shear stresses σ el (see Eq. (13)) that arise in the lubricant, and totalfriction force F (see Eq. (10)). Let us examine thesedependences in more details.At the initial time moment, t = 0 , the blocks aremotionless, and the lubricant is in the solid-like state,because the dependences are plotted for the lubricanttemperature T lower than the critical one, T c (seeEq. (18)), and ε el = 0 at rest. At t = 0 , the upperblock starts to move, and, at t > , its velocity growsin accordance with Eq. (2). Since the lubricant is inthe solid-like state, the friction force F possesses boththe viscous and elastic components, and the lowerblock moves together with the upper one. However, ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 tick-Slip Mode of Boundary Friction
Fig. 3.
Dependences of the coordinates X and x , the veloc-ities V and v , the elastic stresses σ el (see Eq. (13)), and thefriction force F (see Eq. (10)) on the time t for the same pa-rameters as in Fig. 2 and h = 10 − m, τ ε = 10 − s, γ = − / , A = 0 . × − m , k = 5 × Pa · s / , δ = 100 m / (J · s) , T = 200 K, X m = 0 . × − m, ω = 10 rad/s, M = 0 . kg,and K = 3000 N/m. The dashed curves correspond to the co-ordinate X ( t ) and the velocity V ( t ) of the upper block, andthe solid ones to the coordinate x ( t ) and the velocity v ( t ) ofthe lower block in the course of motion, the absolute value of elasticforce Kx , which impedes the lower block to move,grows and, as a result, the velocity v does not increaseso sharply as the velocity V does. Hence, the relativeshear velocity between block surfaces, V − v , increasesin time and, in accordance with Eq. (24), the elasticstrain ε el also grows. At a certain time moment, thecondition ε el > ε el ,c (see Eq. (19)) becomes satisfied,and the lubricant begins to melt following the “shearmelting” mechanism. The friction force substantiallydecreases at that, because the stresses vanish, andthe lower block can slide for a considerable distance,being driven by the elastic force from the compressedand stretched springs. Therefore, the relative shear Fig. 4.
Dependences of the relative displacement, X − x ,and the velocity, V − v , of blocks on the time t for the sameparameters as in Fig. 3 velocity diminishes, and, when the condition ε el <ε ,c (see Eq. (20)) is obeyed, the lubricant solidifiesagain. The considered process repeats periodically.In addition, Fig. 4 demonstrates the time depen-dences of the relative block displacement and thevelocity. At the time moments, when the surfaces“stick” to each other, their relative displacement X − x remains constant, and the relative shear velocity V − v is close to zero (in this case, the dependences V ( t ) and v ( t ) in Fig. 3 visually coincide). Hence, the pe-riodic stick-slip mode of motion takes place, whichis also typical of dry friction, when no lubricant isused [1, 2, 39]. For the chosen parameter values, theblocks “stick” to each other four times during a com-plete period of parameter changes: two times in eachdirection of motion, with the obtained dependencesbeing symmetric with respect to the motion direc-tion. However, a number of different situations canbe realized depending on the system parameters.The phase portraits of the system calculated at thesame parameters as in Fig. 3 and various values ofcyclic frequency ω are depicted in Fig. 5. The ki-netic dependences in Fig. 3 completely correspond tothe phase portrait in Fig. 5, a , because they were cal-culated for the same frequency ω value. It is im-portant to emphasize the fact that the phase por-traits in Figs. 5, a and c are symmetric with respectto the coordinate origin, whereas the phase portraitsin Fig. 5, b and d illustrate the situation where themotion of the upper friction surface in one directionaffects differently the motion of the lower block incomparison with its motion in the opposite direction.Hence, the system reveals memory effects, which were ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 .A. Lyashenko, A.M. Zaskoka Fig. 5.
Phase portraits of the system at the same parameters as in Fig. 3 and various values of the cyclic frequency ω = 10 ( a ),15 ( b ), 32 ( c ), and 38 rad/s ( d ) observed experimentally [5]. In this case, the motionof the lower block is also periodic in time, but thetime dependences of the parameters, which are givenin Fig. 3, are not symmetric with respect to theirzero values [30]. The inset in Fig. 5, b demonstratesthe enlarged section conditionally marked by letter A,because this section has pronounced features, whichcannot be distinguished in the main plot. Thus, thefrequency ω affects the behavior of the tribologicalsystem in a non-trivial manner. By varying ω , it ispossible to select various modes of motion, which con-siderably differ from one another. Note that, at somefrequencies, the stationary behavior of the system,which is established as a result of the system evolu-tion, depends on the initial conditions or the systemprehistory. For instance, in Fig. 5, d , the initial value ϕ = 0 gives rise to a mode similar to that exhib-ited in Fig. 5, c . This circumstance also confirms thepresence of memory effects in the system, which wereobserved experimentally [5].Figure 6 elucidates the influence of lubricant tem-perature T on the melting kinetics. The plotted de-pendences are divided into four sections. The tem-perature for the first section is the lowest, and, forevery next section, the temperature increases, i.e. wehave the inequalities T < T < T < T . The de-pendence obtained in the first section, at T = T , reproduces the dependence shown in Fig. 3 in moredetails, because it was obtained at the same T -value.As the temperature is elevated to T = T , the stick-slip mode of motion is realized, as it was at T = T .However, the maximum value of elastic stresses σ el decreases at T = T . As a result, the friction force F in the solid-like lubricant also decreases, as the tem-perature grows. As the temperature is elevated to T = T , this tendency survives. Note that a reduc-tion of the sticking peak number with the tempera-ture growth is not a rule, and the opposite situationcan take place. At T = T , the lubricant is liquid-like all the time, and the elastic stresses equal zero.It is so, because, at this temperature, the condition T > T c (see Eq. (18)) is obeyed even if ε el = 0 , i.e.the melted lubricant cannot solidify due to a reduc-tion of the relative shear velocity between the rub-bing surfaces. We do not know of any experimentsdevoted to similar researches of the influence of thetemperature on the friction mode. Therefore, the de-pendences exhibited in Fig. 6 are a forecast.
5. Numerical Experiment
The dependences shown in Fig. 6 testify that thegrowth of the temperature T gives rise to a reduc-tion of the elastic stress amplitude σ el and a reduc-tion of the friction force F maximum. Let is ana- ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 tick-Slip Mode of Boundary Friction
Fig. 6.
Dependences of the coordinates X and x , the velocities V and v , the elastic stresses σ el (see Eq. (13)), and the frictionforce F (see Eq. (10)) on the time t for the same parametersas in Fig. 3 and the temperatures T = 200 K, T = 220 K, T = 250 K, and T = 300 K . The dashed curves correspondto the X ( t ) and V ( t ) dependences, and the solid curves to the x ( t ) and v ( t ) ones lyze the dependences of the σ el and F amplitudes onthe temperature T at various modes of functioningof the system in more details. We define the stressamplitude as σ el , max := ( σ el , max − σ el , min ) / and thefriction force amplitude as F max := ( F max − F min ) / ,where σ el , max and F max are the maximum values ofelastic stresses and friction force, respectively, and σ el , min and F min are their minimum values, which aredetermined within the complete period of parameterchanges, T = 2 π/ω , after the stationary friction modehas been established.The dependences of the indicated quantities on thetemperature are depicted in Fig. 7 for three typesof lubricants: pseudoplastic ( γ < , Newtonian( γ = 0 ), and dilatant ( γ > ) fluids. The upper paneldemonstrates that, as the temperature increases, theelastic stresses σ el , max decrease for all three types offluids, i.e. the temperature elevation favors the lu- Fig. 7.
Dependences of the elastic stress, σ el , max , and frictionforce, F max , amplitudes on the temperature T for pseudoplastic( γ = − / ), Newtonian ( γ = 0 ), and dilatant ( γ = 2 / ) fluidsas a lubricant. The parameters are the same as in Fig. 3 bricant melting. Note that, for pseudoplastic fluids( γ = − / ), which are used most often as lubricantsin such systems, the stress amplitude attains maxi-mum values within almost the whole presented rangeof temperatures, but the melting occurs at lower T inthis case. The lower panel of the figure shows the de-pendences of the friction force amplitudes F max on thelubricant temperature T . It follows from the figurethat the friction force decreases with the temperaturegrowth only for pseudoplastic fluids, and it is mini-mal within the whole range of temperatures in com-parison with other types of fluids. For dilatant andNewtonian fluids and for the selected parameter val-ues, the maximum friction force does not change withthe temperature growth. Since the elastic stresses forthose fluids decrease as the temperature grows (theupper panel of the figure), this means that the growthof T gives rise to an increase of the viscous compo-nent of the friction force, to which the second term in ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 .A. Lyashenko, A.M. Zaskoka Fig. 8.
Dependences of the elastic stress, σ el , max , and frictionforce, F max , amplitudes on the proportionality coefficient k (seeEq. (7)) for pseudoplastic ( γ = − / ), Newtonian ( γ = 0 ), anddilatant ( γ = 2 / ) fluids as a lubricant. The parameters arethe same as in Fig. 3 formula (10) corresponds. In the situation concerned,this can happen only if the relative velocity of motion, V − v , increases. Note that, according to the figure,the amplitude of the friction force for the Newtonianand dilatant fluids remains constant when the tem-perature grows, even in the case σ el = 0 , i.e. when thefriction force has only the viscous component. Sincethe F -amplitudes in the cases γ = 0 and γ = 2 / coincide at all temperatures, it is not sufficient to ex-perimentally measure the total friction force in orderto determine the friction mode. That is why the be-havior of the elastic, σ el , and viscous, σ v , stresses areadditionally studied as a rule [16]. Note also that,according to the results demonstrated in Fig. 7, theapplication of pseudoplastic fluids is optimal to re-duce friction, because they favor the establishment ofa mode with minimum force F , despite that the elas-tic stresses for such lubricants are maximum withinalmost the whole range of temperatures.To determine the dependence of the viscosity on thevelocity gradient and the temperature, both real [28]and computer-assisted [29] experiments are carried Fig. 9.
Dependences of the elastic stress, σ el , max , and frictionforce, F max , amplitudes on the spring stiffness constant K (seeEq. (3)) for pseudoplastic ( γ = − / ), Newtonian ( γ = 0 ), anddilatant ( γ = 2 / ) fluids as a lubricant. The parameters arethe same as in Fig. 3 out. The problem urgency is connected with thefact that the dependences of the viscosity on the in-dicated quantities are anomalous in the boundaryfriction mode in the case of nano-sized tribologicalsystems. There can even be a mode, when the fric-tion force almost vanishes at cryogenic temperatures,which corresponds to a low viscosity of a lubricantand, accordingly, a very weak energy dissipation. Inthe English-language scientific literature, this modewas coined as “superlubricity” [40, 41]. Let us ex-amine the dependences of the friction force and thestresses for three types of lubricants – however, noton the temperature (as in Fig. 7), but on the propor-tionality coefficient k between the viscosity and thevelocity gradient (see Eq. (7)). The correspondingplots are depicted in Fig. 8. Note that, in contrastto Fig. 7, different k -values correspond to differentlubricants, friction surfaces, or experimental geome-tries. This means that every point in the dependencesexhibited in Fig. 8 corresponds to tribological sys-tems different by their properties. As one can see, forpseudoplastic fluids ( γ = − / ), the elastic stresses ISSN 2071-0186. Ukr. J. Phys. 2013. Vol. 58, No. 1 tick-Slip Mode of Boundary Friction remain constant with increase of the coefficient k . ForNewtonian and dilatant fluids, the maximum stressesmonotonously decrease with the increase of k . Thefriction force amplitude F max grows with the coeffi-cient k in the case of pseudoplastic fluid ( γ = − / ).At the same time, for the indices γ = 0 and γ = 2 / ,the friction force behaves identically as in Fig. 7, i.e.it remains constant. However, within the whole pre-sented range of k -values, F max is minimal just for thepseudoplastic fluid; therefore, the latter is optimal forcreating the conditions to reduce the friction in thiscase as well.In Fig. 9, the behavior of the examined quantitiesis illustrated, as the spring stiffness constant K in-creases. For the dilatant and Newtonian fluids, theelastic stresses σ el , max monotonously and slowly grow.In the case of the pseudoplastic fluid ( γ = − / ), thestresses drastically increase firstly, and afterward re-main almost constant. The friction force in this case( γ = − / ) also grows to a certain value and, then,does not almost change. For the indices γ = 0 and γ = 2 / , the amplitudes of friction force F max lin-early increase with the spring stiffness constant K ,and their magnitudes are equal as in the previous twofigures. Hence, in this case, the pseudoplastic fluidalso provides the minimum friction force in the sys-tem. Thus, a general conclusion can be drawn thatthe pseudoplastic fluids provide an optimal frictionmode in the tribological system exhibited in Fig. 1,because the maximum friction force F max is the low-est for them.
6. Conclusions
In this work, a thermodynamic model was developedto describe the behavior of a tribological system func-tioning in the boundary friction mode. The model al-lowed a number of effects observed experimentally tobe explained. It was shown that the stick-slip modeof motion is a result of the phase transition of thefirst order between the liquid- and solid-like states ofa lubricant. The influence of the lubricant tempera-ture, the spring stiffness constant, and the coefficientof proportionality between the viscosity and the ve-locity gradient on the system behavior was analyzed.For pseudoplastic fluids, the elastic stresses and thefriction force were found to decrease with the tem-perature growth. The increase of the spring stiffnessconstant induces the growth of the friction force andstresses for all types of lubricants. When the coef-ficient of proportionality k increases, the maximum stresses do not change substantially in the case ofpseudoplastic fluids, whereas the friction force grows.For the sake of comparison, the results of calcula-tions obtained for the dilatant and Newtonian fluidswere also reported. Modes, in which the displacementbetween the friction surfaces does not correspond tothe direction of motion of the upper block, were re-vealed, which evidences the presence of memory ef-fects in the system. While developing the model, thethermodynamic potential with two stable stationarystates was used, in which the zero and nonzero min-ima were separated by a maximum. However, it wasfound experimentally that the lubricant is character-ized by more than one type of transition and it canexist in a few (solid- or liquid-like) metastable states.For such a situation to be described, the additionalterms of higher orders in the free energy expansionare sufficient to be taken into account.I.A. Lyashenko is grateful to Prof. B.N.J. Pers-son for his invitation to make a research visit tothe Forschungszentrum (J ¨u lich, Germany), with thiswork being partially fulfilled there. He also thanksthe organizers of the Joint ICTP-FANAS Confer-ence on Trends in Nanotribology (September 12–16,2011, Miramare, Trieste, Italy) for their invitationand financial support, as well as to A.E. Filippov andV.N. Samoilov for the discussion of this work at theindicated conference.The work was supported by the the Ministry ofEducation and Science, Youth and Sport of Ukrainein the framework of the project “Modeling of fric-tion for metal nanoparticles and boundary liquidfilms interacting with atomically smooth surfaces”(N 0112U001380).
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Я.О. Ляшенко, А.М. Заскока
ПЕРЕРИВЧАСТИЙ РЕЖИМ МЕЖОВОГО ТЕРТЯЯК ФАЗОВИЙ ПЕРЕХIД ПЕРШОГО РОДУР е з ю м еРозглянуто трибологiчну систему, що складається з двохконтактуючих блокiв, один з яких закрiплений мiж дво-ма пружинами, а iнший приведений в неперервний перiоди-чний рух. Дослiджено кiнетику системи в режимi межово-го тертя, коли мiж атомарно-гладкими поверхнями блокiвзнаходиться ультратонка плiвка мастила. Для опису станумастила записано вираз для густини вiльної енергiї у ви-глядi розкладання в ряд за степенями параметра порядку,який зводиться до модуля зсуву. Показано, що в широко-му дiапазонi параметрiв реалiзується переривчастий режимруху, до якого приводять перiодичнi фазовi переходи пер-шого роду мiж кiнетичними режимами тертя. Спрогнозо-вано поведiнку системи при змiнi зовнiшнiх та внутрiшнiхпараметрiв.102