Friction between a temperature dependent viscoelastic body and a rough surface

In this study, we investigate the friction between a one-dimensional elastomer and a one-dimensional rigid randomly rough surface. Special emphasis is laid on the temperature dependence of the elastomer and its effect on the frictional behavior of the contact. The elastomer is modeled as a Kelvin body in a one-dimensional substitute model in the spirit of the method of dimensionality reduction. The randomly rough surface is a self-affine one-dimensional fractal. We provide a short discussion of a conical indenter pressed in a displacement controlled process into an elastomer. These analytical considerations are taken as a basis for the treatment of the randomly rough counter surface in contact to an elastomer with and without temperature dependent viscosity. We identify dimensionless quantities describing this process, introduce a thermal length scale, and give estimates for the coefficient of friction as function of velocity, indentation and thermal quantities.


Introduction
The roughness of interfaces seems to be the main source of friction according to the groundbreaking work of Bowden and Tabor [1]. Greenwood and Tabor [2] explained the behavior of polymers in frictional processes as deformation losses in the material. Experiments by Grosch [3] supported these ideas looking at friction between elastomers and hard specimens with controlled roughness. In the following years, the aspects of rheology [4] and surface roughness [5,6] in rubber friction were investigated. The notion of the coefficient of friction is mostly used in studies in the field. Hereby, Amontons' laws are implicitly presumed to be valid: The force of friction is proportional to the normal force and hence the coefficient of friction is independent of the normal force [7,8]. This is a widely accepted relation, which is a rather simplified picture of real frictional systems. It is well established that both the static and the dynamic coefficient of friction can vary by a factor of four depending on geometrical and loading conditions of the tribological system under investigation [9]. Schallamach [10] conducted experiments in polymer friction. Recently, deviations from Amontons' law were investigated [11,12]. They may have their origin in macroscopic interfacial dynamics [13−15] or in the contact mechanics of rough surfaces. In this note, we explore the thermal behavior of elastomers due to frictional interaction under circumstances when the applicability of Amontons' laws is at stake. Some fundamental understanding of the frictional answer of a viscoelastic body is learned from a simplistic model: (i) the polymer is modeled as a Kelvin body, which is completely defined by a constant elastic modulus and a thermally activated viscosity, (ii) the undeformed elastomer surface is flat and experiences no friction on microscopic scale, (iii) the counter body is rigid, thermally insulated and has a randomly rough, selfaffine fractal surface, (iv) capillary and adhesive effects [16] are not considered, and (v) the features are investigated in a one-dimensional substitute model. Even though these requirements simplify the situation greatly we still observe a non-trivial frictional and thermodynamic behavior of the elastomer. We do not claim a direct one-to-one correspondence to a complete three-dimensional analysis but see a broad utilization of our results on one-dimensional grounds if the rules put forward in the method of dimensionality reduction (MDR) [17−21] are obeyed. The MDR maps threedimensional frictional problems onto one-dimensional ones. Li et al. [22] deal with friction of an elastomer surface with a rough surface at constant temperature. Dimaki and Popov [23] consider a smooth cone indenting into a polymer with temperature dependent viscosity. In Ref. [24], a randomly rough rigid surface is brought into contact with an elastomer of temperature dependent viscosity in a force controlled process.
This study is organized as follows. After a short introduction to rough surfaces in Section 2, the polymer model considered is presented and the contact criterion is established. Power dissipation and temperature dependence are then introduced. Section 3 provides a discussion of a single asperity contact, of the extension to rough multi-asperity surfaces, and of temperature dependence. A thermal length scale is identified. Finally, in Section 4 conclusions are drawn and an outlook is presented.

Self-affine isotropic fractal surfaces
For the exploration of frictional behavior in the concept of MDR, two different steps have to be taken. On the one hand, the geometry of the contacting surfaces has to be transferred into a one-dimensional substitute model. On the other hand, physical observables in the contact have to be calculated in the model. Let us introduce a specific class of surfaces that we use for our model building.
Certain statistical key values may describe an arbitrary surface. Here, we name the root mean square (RMS) roughness 2 h h    and the RMS gradient 2 ( ) h h      , where  denotes the ensemble average over several realizations of the system. For a certain class of surfaces, so called randomly rough surfaces, there exists a close relation of the above quantities to the autocorrelation function (power spectrum) C, which completely characterizes the surface. Many natural surfaces are known to exhibit the property of fractality, at least in some range of wave vectors. In a one-dimensional substitute model, the roughness and the slope, i.e., the moments of the autocorrelation function, can be held constant at the values of an original two-dimensional surface [20] where q denotes the wave number. The considered surfaces are fractal and self-affine surfaces meaning that the surface looks about the same as the resolution is increased or decreased. No natural length scale is to be found in this kind of surfaces. The Hurst exponent H is another quantity to characterize fractal surfaces. Under a rescaling of the spatial coordinate x x The slope interpolates between the values for longest respectively shortest wave vector for Hurst exponents in the range 0 to 1 according to a power law with H as exponent.

Discrete realization of rough surface
For the generation of a randomly rough surface with the desired properties, we fall back on the Fourier transform: The one-dimensional Fourier coefficients of the surface are proportional to the square root of the power spectrum [20]: The randomness of the different realizations of the surface is assured by a uniformly distributed random phase . For a discrete realization, the maximal and minimal wave vectors depend on the spatial step dx and the system length L: The one-dimensional surface is generated as the inverse discrete Fourier transform (DFT). In the following we use a one-dimensional rigid surface z 0 expressed as

Elastomer
After having described the generation of the onedimensional substitute surface we now turn to the modeling of the elastomer and the contact itself. In the considered model, two one-dimensional substitute surfaces z and u move relative to each other with constant velocity 0 v . The surfaces are discretized and the individual sites are labeled by   1, , . i N In every site i, a spring damper combination models the elastomer. This combination is a Kelvin body as shown in Fig. 1.
It consists of a spring with stiffness [20]  4 d k G x (10) which is coupled in parallel with a damper with damping constant [20] 4 dx    (11) for incompressible media with Poisson number 1 / 2   . The material of the original polymer is described by an elastic modulus G and a viscosity η. The factors of four originate from the employment of a one-dimensional substitute model [20,25]. The onedimensional model, the elastomer surface is considered to be a chain of non-interacting Kelvin elements (Fig. 2).
For the viscoelastic model, the equation of motion is easily found for the Kelvin element at every site ext 0 for some reference coordinate u 0 . In terms of discretized variables, the force exerted at each site in contact is is the relaxation time of the Kelvin element. The spring experiences a force according to   the deviation from the undisturbed soft surface u 0i = 0 while the damper exerts a force that is proportional to the rate of change in the deformed surface. We identify as the ratio between the time scale of motion 0 d d / t x v  and the time scale of the viscoelastic material (relaxation) Between the elements of the surfaces, a force acts according to viscoelastic model Eq. (13) if it is in contact. The force is set to zero if surfaces are not in contact. Negative forces are not considered since this would correspond to adhesive effects which we exclude from our study.
Initially, the rigid surface described by coordinates i z is generated with a certain given RMS roughness h, system length L, spacing dx and Hurst exponent H. This fixes also the RMS gradient and the cutoff wave vectors q i and q f . Note that the mean of z 0 is zero by construction. The moving rigid surface is pressed into the deformable surface in such a way that a given indentation d is sustained. The soft deformed lower surface is now described by coordinate u i . The situation is viewed as a stationary system so that all transient features have disappeared. The side step dx is fixed as well as time step dt throughout this study. In particular, this means that spatial derivatives are linked to time derivatives via Imagine we move along with the rough surface at constant speed 0 v to the right. One point of the surface is transferred to a new position some distance ahead. At this new position, the coordinate of the relaxing interface u i is calculated according to Eq. (13) without external force. In a continuous description, this leads to a simple ordinary differential equation which is solved by after some time t for an undisturbed surface at 0 0 u  for some constant (0) u . In the discretized case, the solution relates one site to the coordinate one step further to the right 1 d exp This solution corresponds to a free evolution and relaxation of the elastomer.

Contact criterion
For the deformable surface at a single site, four distinct possible scenarios exist: (i) The first possibility is a site that is already in contact and remains so. The old coordinate of the element is u j+1 = z j+1 . It evolves freely according to Eq. (18). Its new coordinate fulfills the requirement uj ≥ z j and hence stays in contact. The force at this site in contact is calculated according to Eq. (13). The coordinate of the deformable surface is set to the rigid one u j = z j .
(ii) The second possible result of the evolution of a site, which already has been in contact, is that it loses its contact u k ≤ z k . The force acting at this site is set to zero, f k = 0.
(iii) Another outcome of the evolution is that a former free site hits the rigid surface and thus gets into contact u j ≥ z j . Again the force f j in this newly established contact site is calculated in accordance with Eq. (13). Finally, the coordinate is set to the rigid one u j = z j .
(iv) The last possibility of evolution is a free contact that stays free. Its coordinate u k evolves according to the equation of motion Eq. (13) with external force set to zero and reference surface u 0k = 0. The solution of Eq. (18) u k gives the height of the surface at this site. There is no force acting between the surfaces at this site f k = 0.
In Fig. 3, a typical picture of the surfaces in contact can be found. We consider a displacement controlled process. In order to achieve a certain indention d, the coordinates of the rigid body is adjusted by an overall shift of this surface through this indentation Summing the local forces fi over all sites yields the total force exerted between the surfaces. This force equals the normal forc f For the computation of the frictional force F x , the tangential force for all sites is calculated. The coefficient of friction is defined as ratio between the total tangential and total normal force The contact length is computed as the sum over the number of contact site times the spatial step Another quantity of interest is the surface gradient in contact sites which is a measure for the roughness that is actually experienced by the elastomer surface.

Temperature dependence
The temperature dependence of the simulated tribo-logical contact is included through a viscosity, which is considered to be thermally activated where T is temperature, U 0 an activation potential, and k B the Boltzmann constant. This influences the thermal behavior of the contact. For small deviations from a certain configuration (η 0 , T 0 ), the temperature dependence of the viscosity is expressed as where 0 T T T    and . An increase of the temperature by 30 K typically halves the viscosity [23]. This corresponds to an activation energy U 0 of 11log2 = 7.62 in units of k B T 0 at T 0 = 300 K.

Power dissipation
The temperature change originates from the dissipated power in a viscoelastic material. The energy loss in the viscous part of the element leads to a rise in temperature in the contact. Simple considerations about the heat flow in the contact and bulk give rise to a temperature field in the contact. As shown in Ref. [20], the MDR concept gives a straightforward recipe to include heat transfer.
In the substitute MDR model [20,24], the heat flow in a single element of the Winkler foundation is given by for a thermal conductivity λ. Reversing this relation, we obtain for the temperature change in the onedimensional substitute model For the Kelvin model in a one-dimensional Winkler foundation, the power dissipated in the element amounts to Thus, for an element with thermal conductivity λ, For elements in contact the change of viscosity leads to a different force experienced by the surface.

Single asperity-conical indenter
Since a rough surface may be seen as a collection of single asperities we review shortly the indentation of a cone into an elastomer (see Fig. 4). This set up is far easier to analyze since the associated temperature is constant for the entire contact zone. In this section, we consider a rigid cone with slope c penetrating a viscoelastic medium with indentation depth d. The single conical indenter can be treated in a displacement controlled process: The indentation is fixed and the normal force F N for this given indentation is calculated. A similar treatment for the force controlled process is found in Ref. [23]. The shape of the indenter is Since the indenter is moving at constant speed v 0 and we follow with it, we denote the corresponding coordinate by The viscoelastic medium is characterized as above by a parallel spring damper combination [20] The start and end point of the contact region are called 2 a  and 1 a . The vertical displacements u in the contact can be found from the geometry The velocity of the elastomer surface in contact is then The surface exerts a force on a single element The first contact point is determined by the condition that the surface is undisturbed, i.e., The surfaces detach when the force in the element vanishes These requirements may be rewritten as We introduce a dimensionless velocity and consider two speed regimes. In the first range 1 c v  , the elastomer attaches to the indenter even after passing by the tip. In the second regime 1 c v  , the relaxation of the elastomer is so slow that it loses contact after the tip. In the first speed domain, the normal force between the indenter and the elastomer is The frictional force is given by the tangential force. Hence, The coefficient of friction thus becomes Looking at higher velocities 1 c v  , the contact region ends at the tip, 2 0 a  . Hence, the integration limits have to be adapted properly The coefficient of friction thus becomes Summarizing these results coefficient of friction in this picture is given in terms of the dimensionless velocity The coefficient of friction for a conical indenter is described by a rather simple rational function of the normalized velocity up to normalized speed one. Thereafter it is unity.
In the next step, we want to include heat generation in the contact and the corresponding temperature. The power dissipation and hence the heat production in each element take place in the damper according to Eq. (29). As long as there is contact between the indenter and the elastomer the surface u follows the rigid profile so that 0 u cv    for the conical indenter. Substituting this into Eq. (30) we learn that the temperature is shifted by This temperature is the same for all sites in the contact as long as the slope is constant for the indenter.
We introduce a number of short hand notations for frequently appearing combinations of quantities  provides a dimensionless temperature. In these variables, the consistency equation for the temperature has to be fulfilled   Since the derivation of coefficient of friction Eq. (46) goes unaltered through, the normalized velocity is replaced by its temperature dependent analogue Figure 7 provides a plot of the two different velocities. The corrected coefficient of friction is expressed as It should be emphasized that c v  should never exceed one for the coefficient of friction to remain at less than unity. This also leads to a critical value for the parameter  . From Eqs. (50) and (51), one can deduct that at the solution * 1   the maximum is reached. Simultaneously the speed at the extreme is given by Hence, the critical value of 1 / c e   divides the behavior of the coefficient of friction into two regions (Fig. 8). For value below the critical value the coefficient of friction may rise to the saturation level and hit unity. Above the critical value, the speed c v  never exceeds unity and the coefficient of friction never reaches unity. A comparison between the simulation of such a contact and the analytical formula Eq. (52) is displayed in Fig. 9. The agreement is good for a large range of dimensionless velocities. Numerical deviations are due the fact that for very small indentations the number of contacts is very small. In the other extreme, there might be full contact over the entire simulation range.
c v fulfills the consistency equation Eq. (50) Hence, Substituting this into Eq. (52), we find for the temperature dependence of the coefficient of friction for the rigid cone shift in the contact as a further inverted temperature. Note that  is independent of c, i.e., the slope is not involved here. Neither, there is a dependence on 0 v . Figure 10 shows a comparison between the numerical simulation of a cone and the analytical expression Eq. (54). The difference between the expression Eq. (52) and Eq. (54) lies in the fact that the first one describes the coefficient of friction in a system with speed and indentation. Given these two quantities the coefficient of friction and the temperature are calculated. In the latter parametrization, the frictional process is parametrized by the indentation and the temperature. Hence, we have given expressions for the coefficient of friction for a conical single asperity as a function of indentation, normalized velocity, or temperature.

Rough surface
Turning back to our original problem, we want to explore the frictional behavior of an elastomer in contact with a rough surface. The rough surface is characterized by Eq. (5). Thus, by construction the RMS slope is The rigid surface before indentation z 0 is normally distributed with vanishing mean and standard deviation h   by construction. Its derivative is a normally distributed random variable as well. Its mean is again vanishing but its standard deviation z  depends on H as one can see in Eq. (55). This rigid surface is brought into contact with an elastomer in a displacement controlled process. First the properties of the elastomer are considered to be temperature independent, later on the viscosity will be thermally activated.

Numerical values
Since

Displacement controlled process-constant viscosity
In a displacement controlled system, the contact configuration depends on the robust external quantity indentation depth d. Figure 11 shows a typical picture of the coefficient of friction as a function of indentation and speed. There exists a saturation plateau for sufficiently large indentations and velocities. Besides in a region of small velocities there is a linear dependence. Up to a value of about the roughness h the indentation does not play a role but for larger indentations the coefficient of friction declines with larger indentation. We want to find an estimate to express this behavior. The normal force and the coefficient of friction The empirical values give best fit to the numerical simulations. In Fig. 13, a comparison between the numerical simulation and the estimate Eq. (64) is shown. The estimate is a very good approximation over a large range of values of the dimensionless velocity H v . Especially, the most interesting plateau region but even the small velocity range is in good agreement. For very small velocities, the numerical results show a tendency to a constant coefficient of friction. This is rather a numerical than a physical effect.

Temperature dependence
So far we considered a temperature independent viscosity. As in the case of the rigid cone we introduce a thermally activated viscosity according to Eq. (25). For the rough surface, there is yet another difference to the geometrically well-behaved cone. We cannot claim a constant and equal temperature in the elements of contact any longer. Rather, there is a certain temperature associated with every element in contact. In Fig. 14, we display the change in the behavior of the normalized coefficient of friction as a reaction to temperature dependent viscosity. First, the plateau at sufficiently high velocities is reached as in the temperature independent case. For even higher velocities, more power is available at the contact sites and temperature rises in contacting elements. This in turn leads to a lower viscosity. The elastomer gets more liquid and fills non-contact regions better. The dimensionless velocities drop exponentially as well.

Thermal length
Starting from the expression in the consistency equation  with temperature dependent viscosity. The coefficient of friction drops from its maximal value for further rising velocities and indentations larger than the thermal length compared to Fig. 11 since a more power in the contact leads to a rising temperature and an exponentially falling viscosity.
qualitative understanding of the thermal behavior of elastomer surfaces in contact with rigid surfaces. The employment of master curves in this context has been addressed. Further investigations are required.