Microfriction

I. R. Konyakhin

Causes of Friction

Friction is always accompanied by the conversion of mechanical energy into other forms, chiefly into heat, which is what we perceive as resistance to motion. In all, there are four factors that determine friction: molecular adhesion of the rubbing surfaces, elastic deformations of asperities, their plastic deformations, and, finally, destruction of the outer layer of the surface by dispersion, that is, by abrasion, the transformation into dust of the material participating in friction. These factors are, as it were, separate converters of mechanical energy into heat and into its other forms.

Molecular adhesion manifests itself more strongly if the contacting surfaces are polished, and the better the polish, the stronger the molecular adhesion. According to Rebinder, there are three kinds of polishing¹.

1. The limit of grinding, as, for example, in Johansson blocks, when scratches still remain.
2. Removal of an adsorbed layer by stearic acid dissolved in kerosene.
3. Plastic flow of the material on the surface being polished. In the latter case, the polished surfaces may weld together; in this process, dissimilar metals at the place of welding form an alloy.

The smoothness of a surface asymptotically approaches a certain value, and the coefficient of friction tends toward its limiting value. As smoothness increases, molecular adhesion increases; therefore an increase in friction may occur.

Elastic deformation of asperities obviously cannot absorb appreciable amounts of energy during friction, since the energy of the deformed body is almost completely returned. If one asperity elastically deforms another, then the latter accumulates energy and then returns it, pushing forward the asperity that had previously deformed it. During plastic deformation of the asperities of the rubbing surfaces, the mechanical energy of motion is converted into heat.

And the last factor of friction is surface wear due to the tearing away of particles from the rubbing surfaces. Energy of motion is, of course, expended on this tearing away, and the more so the lower the speed of friction, since at lower speeds the material is more plastic.

In the present work we have set ourselves the aim of elucidating, at least to some extent, the question of friction at microscopically small velocities of motion. Such friction differs substantially from friction at ordinary velocities.

Some phenomena of this friction were studied by A. V. Verkhovskii on a specially constructed apparatus. In Verkhovskii’s apparatus (Fig. 1), the specimen has a lower annular surface with diameters \(D = 24\) mm and \(d = 20\) mm. The experimental results differed depending on whether the specimen was placed quietly from above or was turned in one direction or the other. If the specimen had first been turned in the direction of action of the couple of forces \(PP\), then the displacements increased according to a straight-line law; if, however, it was turned in the opposite direction, curves were obtained.

If, by gradually increasing the couple of forces, the displacement is brought to the point \(K\), and then the couple is reduced to zero, it turns out that the displacement changes along the straight line \(nk\). When the driving couple is increased again, the displacement follows the straight line \(nk\), and then the curve is continued. In testing different materials, different displacements were obtained. The dependence of these displacements on the magnitude of the driving couple is shown in the graph (Fig. 3). Brass is displaced most easily, and steel with the greatest difficulty.

Fig. 1

Thus, according to the experiments of A. V. Verkhovskii, visible motion under dry friction does not arise immediately after the driving force is applied; the body at first yields somewhat forward, until this force reaches a certain magnitude. Then a sharp jump in velocity occurs, and ordinary motion, visible to the naked eye, begins. During the preliminary displacement, the coefficient of friction increases from zero to the coefficient of static friction and then drops abruptly to the coefficient of kinetic friction. The magnitude of the entire preliminary displacement reaches the order of \(S = 2, 2.5\,\mu\), after which the displacement breaks away, i.e., ordinary motion begins.

Fig. 2

Fig. 3

The author of the present article set himself the goal of broadening the study of the phenomenon of starting from rest under dry friction. For this purpose an apparatus was newly designed and built, the scheme of which is shown in Fig. 1.

Having carried out a large number of experiments, we came to the conclusion that Verkhovskii’s experiments had only made the beginning of the study of this question, and that certain phenomena in starting from rest had not been studied by him.

The author, of course, does not believe that he has studied all the phenomena to a sufficient extent—further work in this direction still remains to be done—and everything that he has obtained so far is described below.

Microdisplacement

If, during the preliminary displacement, the velocity is kept constant, then no breakaway will occur, and the motion can then continue for an arbitrarily long time. An experiment on the apparatus showed that, at a velocity equal to \(6\,m\mu\) per second, a displacement occurred by an amount equal to \(60\,\mu\), which is 24 times greater than in the experiments of A. V. Verkhovskii. The experiment lasted more than 4 hours, and no breakaway—that is, no abrupt transition to visible motion—occurred.

If, from the start of the displacement, the driving force is increased, while its increment is taken to decrease asymptotically to zero, then continuous motion with constant velocity is obtained. Such motion, occurring at microscopic velocities, differs essentially from rapid motion in that it occurs only under the action of an increasing force. Thus, on the basis of the foregoing, friction should be divided into 4 types:

1) static friction,
2) elastic displacement,
3) plastic friction,
4) sliding friction.

Fig. 4

Elastic displacement is characterized by the fact that, if the driving force does not exceed a certain limit, there is no plastic friction, and after removal of this force the specimen returns to its former position; in this case, an elastic aftereffect is observed for 2–3 minutes.

Static Friction

A driving force applied to a body resting on some surface gives rise to a frictional force between the contacting surfaces. First elastic and then plastic displacement begins. Static friction force is usually taken to mean that resistance to motion at which the transition to sliding occurs.

Static friction is substantially influenced by the duration of stationary contact. Experiments were carried out with durations of stationary contact \(t = 4\) min; \(t = 10\) min and \(t = 25\) min. Displacement of the specimen was produced by a continuously increasing force, created by means of a water jet with a capacity of 110 g/min, which on our apparatus corresponds to an increment of the force applied to the specimen,

\[ \frac{\Delta P}{\Delta t}=1650\,\text{g/min}. \]

Fig. 5

The results of the experiments are given in Table 1. Time is given in minutes, displacement \(S\)—in microns.

Table 1

If the displacement is in all cases taken equal to \(4\,\mu\), then the value of the force \(P\) is expressed by Table 2.

Table 2

The duration of stationary contact \(t_s\), and the resistance to displacement \(P\).

The duration of stationary contact has a more substantial significance in the first minutes. In our experiment the shortest duration of standing was \(4\) min. It would be interesting to take this time as possibly smaller, but, unfortunately, this is difficult to accomplish because the apparatus for setting up the experiment requires a certain amount of time during which the specimen is under the action of the vertical load. During prolonged standing, the specimen under the action of the vertical load descends by a retarded motion\(^1\), probably according to the same law by which the displacing force increases as a function of the duration of standing.

Fig. 6

As the specimen descends, plastic deformation of the asperities occurs in the vertical direction. As a result, the true contact area increases, and together with it the molecular adhesion. As the specimen descends, the speed of its downward motion decreases. The asperities, while decreasing in height, increase in thickness. In addition, the plastically deformed material becomes ever harder, and therefore more work is required for cutting and deforming the asperities in the horizontal direction. Likewise

\(^1\) Kuznetsov V. D. Physics of the Solid State, vol. IV. Tomsk, 1947.

more work is required to overcome molecular cohesion, since the number of molecules brought close together increases with the duration of immobile contact.

During horizontal displacement of the specimen, its asperities, located in depressions, must climb out of the depressions and ascend onto the protrusions, which must undergo elastic and plastic settling; this also requires expenditure of work.

Elastic Displacement

Elastic displacement is characterized by the fact that, under the action of a horizontal force \(P\), the specimen is displaced somewhat forward; after this force is removed, it returns to its former position. The experiment was carried out by applying and removing a driving force \(P\), equal to 15 kg, at a vertical pressure on the specimen of \(N = 70\) kg. The elastic displacement occurred by an amount \(S = 0.36\,\mu\). When the force \(P\) was removed, the specimen returned back. The complete elastic displacement was obtained in this way: the entire path \(S = 0.36\,\mu\) was traversed by the specimen as though in two stages; it traversed \(0.27\,\mu\) at once, and the remaining \(0.09\,\mu\) over the course of 3 min. It returned back in the same sequence, that is, three quarters of the path were traversed together with the removal of the force \(P\), while the remaining quarter moved slowly. Here, evidently, the elastic aftereffect of the rubbing surfaces was manifested.

Fig. 7

Plastic Friction

Plastic deformations of the asperities of rubbing surfaces absorb the lion’s share of all the mechanical energy of motion, converting it into heat.

When a roller rolls over a hard surface, the latter is continuously crushed, so that as a result a track remains, or, when a wheel rolls, a rut remains. Mechanical work is expended on the formation of this rut or track.

The roller must constantly be pulled up onto a protrusion, which in turn settles downward as a result of plastic deformation.

The same occurs when one body slides over another: the asperities of one body are pulled onto the asperities of the other, their settling occurs and, consequently, work is performed.

Fig. 8

In microfriction, there is no wear of the rubbing surfaces as a result of tearing off or shearing of asperities. Plastic deformations occur here completely; that is, in other words, pure plastic friction is obtained.

In our experiment the motion was at a speed of \(5\ \text{m}\mu/\text{sec}\); at such a speed one cannot speak of any brittleness of the material, and therefore it may be said that there was no dispersion.

Influence of various factors on microfriction

Material of the rubbing surfaces

For the experiments the following contacts were taken: copper on copper, steel on steel, and copper on steel. The roughness of the surfaces was the same in all these cases.

Fig. 9

Fig. 10

Copper on steel is displaced with the greatest difficulty, and steel on steel the most easily. The breakaway of motion in dissimilar materials occurs more quickly.

Roughness of the rubbing surfaces

For comparison, two kinds of experiments were carried out with one and the same specimen: in some cases the surfaces were treated with emery paper; in others, they were treated with finer emery paper and then polished with chalk to a shine. The polished specimen is displaced with greater difficulty, and the breakaway of motion occurs earlier.

Rigidity of friction

By the rigidity of a surface in friction we mean its capacity to resist displacement. With rigid surfaces the breakaway of microfriction occurs more quickly. Molecular adhesion is strong. The rigidity of friction depends on the hardness of the materials, the magnitude of the irregularities, and the presence of foreign solid substances. In this, the magnitude of the irregularities affects the rigidity in the opposite way: greater irregularities mean less rigidity.

At the beginning of displacement the surface is more rigid than later; that is, from an increment of force \(\Delta P\), the surfaces are displaced by a smaller amount. Subsequently the motion becomes smoother.

Fig. 11

The graph shows the change in the velocity of motion with time, for different rigidities (Fig. 11). The change in velocity occurs under the action of a continuously increasing force. The most rigid surfaces are polished surfaces contaminated with foreign substances.

Pressure between Rubbing Surfaces

To determine the effect of pressure on the magnitude of displacement, the moving force was taken to be \(P = 7500\,i\), while the pressure was varied.

As the pressure increases, the displacement under the action of one and the same force decreases, apparently according to an exponential law; that is, ultimately an increment of pressure ceases to have any influence on the magnitude of the displacement of the specimen.

The results of the experiments are given in Table 3.

Table 3

The area of the rubbing annular surface of the specimen was \(F = 1.4\ \text{cm}^2\). When the specimen is in motion, the duration of stationary contact is equal to zero. The motion may take place at different velocities, and therefore the sinking-in of the moving body will occur by different amounts; it will be smaller at higher velocity. In very rapid sliding it may be assumed that sinking-in does not occur and the adhesion remains constant, while also being small in magnitude. At the beginning of the motion the specimen rises over the irregularities, and the molecular adhesion decreases—the motion becomes accelerated. When the velocity decreases, the irregularities are crushed, the specimen descends, the adhesion increases, and the velocity decreases still further.

Fig. 12

Influence of the Presence of Foreign Substances

Foreign substances may always be present between rubbing surfaces in the form of contaminants or residues resulting from grinding, or simply from abrasion due to friction. The presence of such substances between the planes of friction greatly increases the resistance to motion.

We carried out an experiment with one and the same specimen. The tested surfaces of the specimen were first ground with fine emery paper, and then with clean paper until blackening ceased to appear on it. The second time the surfaces of the specimen were lightly dusted with a fine powder of chalk; the presence of chalk was not noticeable to the eye. The slight presence of chalk greatly increased the resistance to displacement; in this case the force of disruption of the micro-motion, that is, of the abrupt transition to ordinary motion, was approximately twice as large as for the clean specimen, while the displacement before disruption was twice as small. In other words, the stiffness of friction increased.

¹) \(S\)—displacement in microns.

Motion under the action of a constant force

The phenomenon of displacement of the specimen under the action of a constant force occurs as follows. The motion is at first accelerated; the velocity reaches a certain maximum. After this, retarded motion begins and the specimen stops. To cause further motion, the driving force must be increased; then the described process is repeated. This can be repeated until breakaway occurs, i.e., until the micromotion is replaced by ordinary motion.

The experiment was carried out on copper-on-copper specimens of matte roughness. The observation was made at the end of the displacement, before breakaway, since in this case the displacement proceeds more gently and the character of the motion is manifested more clearly. An additional load of the driving force of 1500 g was applied over 20 sec, or, in other words, the increase in force proceeded with an intensity of

\[ \frac{\Delta P}{\Delta t}=75\ \text{g/sec}, \]

while the initial velocity at which the additional loading began was \(V=0.03\ \text{m/min}\).

The force applied earlier, by means of which the specimen had been brought close to breakaway, was 31500 g. Thus, in this experiment a constant force \(P=33\ \text{kg}\) was involved, with a vertical pressure on the specimen \(N=80\ \text{kg}\).

The results of the experiment are shown in Table 4.

Table 4

Time is given in minutes, displacement in microns.

Fig. 13

These data are presented in the form of a velocity graph (Fig. 13). Point \(A\) corresponds to the end of the increase in force; from this point onward the motion proceeds under a constant force.

The Significance of the Initial Velocity of Micromotion

As we saw above (the velocity graph), micromotion under the action of a constant force dies out. A new portion of load may be used to produce the corresponding increment of such a displacement; moreover, this new displacement may begin when the preceding motion begins at an initial velocity equal to zero. The new displacement may also be begun when the preceding motion has not yet ceased, that is, at some initial velocity not equal to zero.

This initial velocity will have a substantial influence on the subsequent motion; its character will be different than when \(V_0 = 0\). The magnitude of the displacement caused by the action of the new constant force will be greater.

It must be said that in the experiments of A. V. Verkhovsky the initial velocity was not taken into account; therefore the results obtained were different: copper was displaced more easily than steel, whereas in our experiments the opposite was obtained.

Breakdown of Micromotion

The force at which breakdown of micromotion occurs and, consequently, transition to a visible velocity, may in general have different magnitudes depending on the intensity with which it has increased. The greater its increment, the smaller the breaking force. In the graph shown it is seen that it depends on \(\Delta P\) according to the same law as the velocity changes under the action of a constant force. Obviously, it depends on the velocity.

Fig. 14

The path traversed by the specimen before breakdown depends not only on the driving force but, consequently, also on the velocity of motion; therefore it is impossible to compare different materials in terms of their displacement without taking velocity into account.

Displacement of a Specimen by a Force Increasing in Separate Portions

As was described above, motion under the action of a constant force gradually dies out with time. To produce further displacement, it is necessary to increase the driving force; then the motion will be repeated.

In our experiments, with each subsequent portion of load, the displacement becomes greater and greater. The total displacement as a function of the increment of force is shown in the graph as a curve of wavy character (Fig. 15).

Fig. 15

The velocity from subsequent loads increases and, finally, a rapid transition of micromotion into ordinary motion occurs within \(2\text{–}3\) sec; the velocity in this case increases by approximately 30,000 times.

Displacement under a Continuously Increasing Force

In these experiments, the continuously increasing force was produced by a stream of water. In each experiment the intensity of the stream was varied as follows:

\[ \frac{\Delta P}{\Delta t}=720\ \text{g/min};\qquad \frac{\Delta P}{\Delta t}=914\ \text{g/min}\quad \text{and}\quad \frac{\Delta P}{\Delta t}=1140\ \text{g/min}. \]

The dependence of displacement on force is shown in the form of curves depicted in the graph (Fig. 16). It is clear that the greater the intensity of the stream, the faster the motion. In the coordinates shown in Fig. 17 these curves straighten out.

Fig. 16

Fig. 17

In examining the straight lines obtained, the following relation was observed between the increments of the driving force and the angular coefficients of the straight lines:

\[ \left(\frac{\Delta P_2}{K_2}\right)^2 = \frac{\Delta P}{K_1}\cdot \frac{\Delta P_3}{K_3} \]

All three straight lines converge at one point \(C\). The height of the position of this point should depend only on the character of the rubbing surfaces, that is, on the material, roughness, and cleanliness. The height of the position of point \(C\) should play the same role as the coefficient of friction in ordinary sliding.

1. Kuznetsov V. D. Physics of the Solid State, Vol. IV. Tomsk, 1947. ↩