# A Khomenko - Stochastic models of ultrathin lubricant film melting - страница 1

ВІСНИК ЛЬВІВ. УН-ТУ

Серія фізична. 2006. Вип. 39. С. 23-35

VISNYK LVIVUNIV. Ser. Physic. 2006. N 39. P.23-35

УДК 539.2

PACS number(s): 64.60.-i, 05.10.Gg, 62.20.Qp

STOCHASTIC MODELS OF ULTRATHIN LUBRICANT FILM MELTING

A. Khomenko

Sumy State University, Rimskii-Korsakov Str., 2, 40007 Sumy, Ukraine e-mail: khom@phe.sumdu.edu.ua

The melting of ultrathin lubricant film by friction between atomically flat surfaces is studied. In the case of a second-order transition the spatial distributions of the elastic shear stress and stain, and the temperature are described on the basis of Ginzburg-Landau equation. The additive noises of above quantities are introduced for building a phase diagrams with the domains of sliding, stick-slip, and dry friction. It is shown that increase of the strain noise intensity causes the lubricant film melting even at low temperatures of the friction surfaces. Taking into account the deformational defect of the shear modulus the analogical phase diagrams are obtained in the case of a first-order transition.

Key words: viscoelastic medium, phase diagram, stick-slip friction.

The study of the noise influence on the friction process has an evident fundamental and practical importance because in some experimental situations the fluctuations can change the frictional behavior critically, for example, providing the conditions for low friction [1]-[3]. In particular, the thermal noise, acting in any experiments, can convert the ultrathin lubricant film from stable solidlike phase state to the liquidlike one and, thus, transform the dry friction into the sliding or the stick-slip (the interrupted) modes. Therefore, in recent years the considerable study has been given to the influence of disorder and random impurities in the interface on the static and the dynamic frictional phenomena [4]-[6]. These investigations show that a periodic surfaces are characterized by smaller friction coefficient during sliding than non-regular ones. Besides, the stick-slip dynamics, inherent in solid friction, attracts an increased attention on the atomic [7]-[9] and the macroscopic [10, 11] levels as well as for granular mediums [12]-[14]. In order to achieve the better understanding of the above phenomena, here an analytic approach is put forward, which describes the transitions between friction modes due to variation of fluctuations of elastic and thermal fields. Moreover, in the simplest case the proposed model gives insight into their spatial distribution in the lubricant film.

In the previous work [15] on the basis of rheological description of viscoelastic medium the system of kinetic equations has been obtained, which define the mutually coordinated evolution of the elastic shear components of the stress a and the strain є, and the temperature T in ultrathin lubricant film during friction between atomically flat mica surfaces. Let us write these equations using the measure units

© Khomenko A., 2006

—v 10 , = —!>- = —t- гусь

G0

1/2 ґ \1І2ґ \1/2

xT

n0

Tc (1)

for variables a, є, T, respectively, where p is the mass density, cv is the specific heat capacity, Tc is the critical temperature, n0 = n(T = 2Tc) is the typical value of shear viscosity n, xT =pl2cv / к is the time of heat conductivity, l is the scale of heat conductivity, к is the heat conductivity constant, тє is the relaxation time of matter strain, G0 = n0 / тє:

Xa<s = -a + g, ; (2) тєє = -є + (T -1)a; (3) tt T = (Te - T) + оє + o2. (4) Here the stress relaxation time xa, the temperature Te of atomically flat mica friction surfaces, and the constant g = G /G0 are introduced, where G is the lubricant shear

modulus. It can be seen [15, 16] that Eqs. (2) and (3) are the Maxwell-type and the Kelvin-Voigt equations for viscoelastic matter, correspondingly. The latter takes into account the dependence of the shear viscosity on the dimensionless temperature П = n0 /(T -1). Equation (4) represents the heat conductivity expression, which describes the heat transfer from the friction surfaces to the layer of lubricant, the effect of the dissipative heating of a viscous liquid flowing under the action of the stress, and the reversible mechanic-and-caloric effect in linear approximation. These equations coincide with the synergetic Lorenz system formally [17, 18], where the elastic shear stress acts as the order parameter, the conjugate field is reduced to the elastic shear strain, and the temperature is the control parameter. As is known this system can be used for description of the thermodynamic phase and the kinetic transitions.

The structure of the article is the following. Section 2 is devoted to construction of the Ginzburg-Landau scheme describing the spatial distributions of elastic and thermal fields in the lubricant film. In Section 3 the additive noises of the shear components of the elastic stress and strain, and the temperature are taken into account. The phase diagrams are calculated defining the domains of sliding, stick-slip, and dry friction in the planes temperature noise intensity - temperature of friction surfaces and noise intensity of shear elastic strain - temperature noise intensity. In these sections the solidlike lubricant is assumed to be amorphous (disordered). Therefore I study the glass transition represented in terms of a second-order transition. In Section 4 taking into consideration the deformational defect of the shear modulus the crystal-fluid transition is modeled as a first-order transition.

Approach described in Ref. [15] permits to take into account the nonhomogeneity along the perpendicular direction y to the confining walls (value y=0 corresponds to the boundary of moving wall). With this aim, one should assume that divergence of heat current vector q = -kVT is equal to - divq « (к /12)(Te - T) + kV2 T , where V stands

for a derivative with respect to y. Accordingly, measuring the coordinate y in units of l equation (4) gets the additional term describing the nonhomogeneous distribution of the temperature:

xTT = V2T + (Te - T) -ає + а2. (5)

Experimental data for organic lubricant [7] show that relaxation time of the stress xa at normal pressure is ~10-10 s, and it increases by several orders of magnitude at large

pressures. The corresponding time of the strain can be estimated by ze и a/c ~ 10 12 s,

where a ~ 1 nm is the lattice constant or the intermolecular distance and c ~ 103 m/s is the sound velocity. Since the ultrathin lubricant film consists of less than ten molecular layers the relaxation process of the temperature to the value Te occurs during time tt satisfying the condition

xe, Tt <<xa . (6)

These inequalities of hierarchical subordination are called adiabatic approximation in synergetics and mean that in the course of medium evolution the strain s(t) and the temperature T(t) follow the change of the stress a(t). Due to conditions (6) the left-hand sides of Eqs. (3) and (5), containing the small relaxation times, are set to be equal to zero and the analytic solution is possible. Besides, within the framework of the one-mode

approximation the operator V2 is replaced by ratio (l/L)2, where L is the maximal value of the heat conductivity length [15].[1] As a result, the strain є and the temperature T are expressed in terms of the stress value a:

[ 1 - (l/L)2 +a2 J

T = Te + T(l/L)2 + (22 _ . (8)

e 1 - (l / L)2 +a2

Insertion of Eq. (7) into (2) provided that a2 << 1 + (l/L)2 and reverse transition

from (l / L)2 to the operator V2 give in the domain gTz и 1

и -[1 + g(1 - Te)]a + (2g - 1)a3 + V2a - 2gV2a3 + 2gV4a . (9) Then, within the framework of automodel representation in accordance with that each derivative V adds the order of smallness, neglecting the powers of a larger than the third order, the system (2), (3), and (5) is reduced to the time-dependent Ginzburg-Landau equation:

= V2a-— ; (10) da

which form is defined by synergetic potential

E = [1 + g (1 - Te)] у + (1 - 2g) (11) If the temperature Te is smaller than the critical value

Tc = 1 + g_1; g - G /G0 < 1/2; G0 ^ r,0 / тє, (12) the potential (11) assumes a minimum corresponding to the stress a=0, so that the melting can not take place and the solidlike state is realized. At the opposite case Te > Tc the stationary shear stress has the nonzero value

є0 = a0

T„ = і^+й. (14)

This causes the melting of film and its transition into fluidlike state. In accordance with Eqs. (7) and (8) at (l/L)=0, the corresponding stationary values of strain and temperature are as follows:

\ Te (1 - g) + g - 2

° g (Te - 3)

In the steady state a = 0 equation (10) has the first integral

i(Va)2 = E + |Е0|, E0 ^ E(a0) ^-[^ "^+^ . (15)

Here it is taken into consideration that in the "ordered" phase meeting y=-oo the fulfillment of conditions a = a0, Va = 0 requires the equality of integration constant to

the absolute value of "ordering" potential E0.

Solution of Eq. (10) at the stationary conditions shows that the shear stress distribution is presented by the kink

a = a0tanh g2 s 2 + , (16)

where the correlation length g is introduced diverging at the critical value of friction surfaces temperature. The integration constant y0>>g defines the width of a boundary domain in which the shear stress decreases from the steady state value (13) to the zero. Substitution of Eq. (16) into (7) gives the similar strain є vs y dependence. Plugging distribution (16), decreasing on the correlation length g, into the formula (8) it is seen that in the transition region the temperature monotonically increases from the value

T = T+ Te(l/L)2 + (2 - Te)a2tanh2(yjg) e 1 - (l/L)2 +а>гі1і2(у0/g)

aty=0 to the maximal magnitude T = Te /[1 - (l/L)2] aty=y0. If the value of l is equal to

film thickness then at y0<y<l the shear stress changes sign and increases in absolute value but the temperature decreases. Such a(y) and T(y) dependencies imply that in the vicinity of confining walls the key role is played by the shear melting and near the boundary y=y0 the thermodynamic melting prevails. However, it is worth noting that formal using of Eq. (15) at y=y0, corresponding to the a=0, leads to the finite gradient of shear stress

00 = gTe - (g +1) g [2(1 - 2 g )]1

In this section, as well as in Ref. [15], a melting of ultrathin lubricant film by friction between atomically flat mica surfaces has been represented as a result of action of spontaneously appearing elastic field of stress shear component caused by the heating of friction surfaces above the critical value Tc=1+g"\ Thus, according to such approach the studied solid-liquid transition of lubricant film occurs due to both thermodynamic and shear melting. The initial reason for this self-organization process is the positive feedback of T and a on є [see Eq. (3)] conditioned by the temperature dependence of the shear viscosity leading to its divergence. On the other hand, the negative feedback of a and є on T in Eq. (5) plays an important role since it ensures the system stability.

Va0 =sJe vs „2 . (18)

0 •€. rrs гл rs \-|1/2 4 7

According to this approach the lubricant represents a strongly viscous liquid that can behave itself similar to the solid - has a high effective viscosity and still exhibits a yield stress [7, 16]. Its liquidlike state corresponds to the elastic shear stress which relax to the zero value during larger times Ta than that of the solidlike state. Moreover, at a=0 Eq. (3) reduces to the Debye law describing the rapid relaxation of the elastic shear strain during the microscopic time тє ~10-12 s. At that the heat conductivity equation (4)

takes the form of simplest expression for temperature relaxation that does not contain the terms representing the dissipative heating and the mechanic-and-caloric effect of a viscous liquid. Also, it is assumed that the film becomes more liquidlike and the friction decreases with the temperature growth due to decreasing activation energy barrier to molecular hops.

In accordance with Ref. [11] in the absence of shear deformations the temperature

mean-square displacement is defined by equality (u2^J = T/Ga , where a is the lattice

constant or the intermolecular distance. The average shear displacement is found from

the relationship (u2^J = a2a2/G2. The total mean-square displacement represents the

sum of these expressions provided that the thermal fluctuations and the stress are independent. Above implies that the transition of lubricant from solidlike to fluidlike state is induced both by heating and under influence of stress generated by solid surfaces at friction. This agrees with examination of solid state instability within the framework of shear and dynamic disorder-driven melting representation in absence of thermal fluctuations. Thus, the strain fluctuations, related to the stress ones, and the thermal fluctuations will be considered independently.

Consider now the affect of additive noises of the elastic stress and strain shear components a, є, and the temperature T . With this aim, one should add to right-hand

sides of Eqs. (2)-(4) the stochastic terms Ia/2g, I\!2g lT'2g (here the noise intensities

Ia, Іє, IT are measured in units of є2тЄ2, (Tck/1)2, correspondingly, and g(t) is the 5 - correlated stochastic function) [19]. Then, within the adiabatic approximation та >> тє,tt , equations (3) and (4) are reduced to the time dependencies

) = є+є|01), T (t) = T + T g(t); &=a{re-1 + a2 )d (a), є = ^Ie + IT a2 d (a). (19)

T =(re + 2a2 )d (a), T = ^IT + Iєa2 d (a), d (a) = (1 + a2)-1. (20)

Here, deterministic components are reduced to obtained in Ref. [15], whereas fluctuational ones follow from the known property of variance additivity of independent Gaussian random quantities [19]. Thus, using the slaving principle inherent in synergetics [17, 18] transforms noises of both strain є and temperature T , which are adiabatic initially, to multiplicative form. As a result, a combination of Eqs. (2), (19), and (20) leads to the Langevin equation

xai = f (a) +VIcO)g(t); f = - Ц, (21)

da

where the force f is related to the synergetic potential [15]

V = \d- g )a 2 + g{\--У ln(l + a 2 ) (22)

2V *' X 2 ) and an expression for the effective noise intensity

I (a) = /a+( /є+ ^ a2 ) g 2 J 2(a) (23)

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