Friction of Solid Surfaces

T. A. Kontorova, Leningrad

Chapter I. Static Friction of “Dry” Surfaces

§ 1. Amontons’ Law

The first laws of friction of solid bodies were formulated in 1699 by the French physicist Amontons¹ in the following form: the resistance to the relative motion of solid bodies is proportional to the normal load and does not depend on the area of contact between the bodies. Later (in 1781) Coulomb² also arrived at similar conclusions.

Mathematically, Amontons’ law (usually attributed to Coulomb) is formulated as follows:

\[ P = \mu F, \tag{1} \]

where \(P\) is the friction force, \(F\) is the normal load (pressure per unit area), and \(\mu\) is the proportionality coefficient, called the coefficient of friction.

A distinction is made between static friction (friction at rest) and kinetic friction (friction of motion). In the first case, in relation (1), \(P\) is understood as the force that must be applied in order to set one body in motion relative to another; the corresponding coefficient of friction will be denoted below by \(\mu_s\). In the case of kinetic friction, \(P\) is understood as the force required to maintain the relative motion of the rubbing bodies (sliding) at a given velocity; the coefficient of kinetic friction will be denoted by \(\mu_k\). Amontons’ law (1) is generally regarded as the fundamental law of static and kinetic friction both for “dry” (unlubricated) and for lubricated surfaces.

Numerous experimental investigations carried out since the formulation of this law have shown, however, that in a number of cases deviations from it are observed, and have also revealed new factors affecting the friction of solid surfaces.

In this article we shall attempt to elucidate the principal regularities to which the friction of solid bodies is subject—regularities that are chiefly qualitative, since, strange as it may seem, there is no theory at all satisfactory for so widespread a phenomenon in everyday practice as friction.

In Chapters I and II we shall dwell on the consideration of static and kinetic friction of “dry” surfaces; the third and fourth chapters will be devoted to an analysis of questions connected with static and kinetic friction in the presence of lubrication. We shall then try to give a brief account of existing theories of friction and to indicate possible ways of explaining certain experimental regularities that still remain not fully understood.

§ 2. Influence of the Degree of Cleanliness of the Rubbing Surfaces

The special role of the cleanliness of rubbing surfaces in determining the coefficients of static and kinetic friction is noted by all authors of works on friction. Only in the case of thorough cleaning of the surfaces is it possible to obtain constant, reproducible values of \(\mu_s\) and \(\mu_k\) ¹).

We shall touch on the question of the dependence of the coefficient of kinetic friction on the degree of cleanliness of the rubbing surfaces somewhat later; here we shall dwell on the consideration of data relating to the coefficient of static friction.

Already in one of his earliest works, Hardy ³ notes that thorough cleaning of rubbing surfaces leads to a considerable increase in the coefficient of static friction. The explanation of this fact presents no special difficulties. Any solid surface may be regarded as “lubricated” by films of various foreign substances adsorbed on it; in most cases we are apparently dealing with monomolecular films formed by molecules of oxygen, water, or some impurities present in the atmosphere and adsorbed from the surrounding medium.

Langmuir ⁴, in his article “Mechanical Properties of Monomolecular Films,” describes experiments that directly clarify the nature of the influence of monomolecular films deposited on solid surfaces on the value of the coefficient of static friction of these surfaces. He was able to obtain monomolecular films of fatty acids on glass by transferring them from the surface of water onto glass; by this method it was also possible to obtain a number of monomolecular layers deposited one upon another.

The coefficient of static friction was determined by means of the so-called “inclination” method, which consists in the fact that plates lying one upon another are tilted from the horizontal position until one of them begins to slide—

¹) The numerical values of \(\mu_s\) and \(\mu_k\) given by different authors therefore cannot be compared with one another.

by another. The coefficient of static friction is then equal to the tangent of the angle at which the relative motion of the surfaces begins. In the case of “clean” glass surfaces such sliding began at angles of inclination close to \(60^\circ\). If, however, a monomolecular film of fatty acid was deposited on one of the surfaces, the experiment led to different results depending on whether the film had been deposited on the sliding body (the “slider”) or on the surface along which sliding occurs. In the first case the angle corresponding to the onset of sliding decreased to \(40^\circ\); after a very slight relative displacement of the surfaces, however, further sliding stopped and resumed only when the angle of inclination was increased to \(60^\circ\). If, on the other hand, the monomolecular film was on the surface relative to which sliding occurs, then the latter was greatly facilitated, beginning at angles of inclination of \(6\)—\(8^\circ\) and not ceasing for a fairly considerable interval of time.

Langmuir explains this difference by the fact that from the “slider” (the sliding body) the monomolecular film is very quickly rubbed off during the experiment, whereas the wear of the film on the lower surface, relative to which sliding takes place, is very slight (because of the small dimensions of the “slider” in comparison with the dimensions of this surface). Experiment shows that the monomolecular film is rubbed off the slider as a result of its displacement by only \(2\ \mathrm{mm}\).

Not in all cases, however, does removal of the film prove so easy to accomplish. Of great interest are Schaefer’s experiments, cited by Langmuir,\(^4\) on measuring the coefficient of static friction of a glass sphere against a glass plane covered with a film of stearic acid. The relative motion of the surfaces in these experiments took place along a strictly definite trajectory; in the case of a monomolecular film of acid, an increase in the coefficient of friction could be observed after the experiment had been repeated 700 times. In the case of a stearic-acid film with a thickness of 7 monomolecular layers, the coefficient of friction began to increase appreciably only after the experiment had been repeated 4000 times.

By creating on one of the surfaces a thin film of lubricant invisible to the eye, obtained as a result of keeping the “clean” surface in an atmosphere saturated with lubricant vapor, or else as a result of applying a drop of lubricant to some part of this surface, Hardy\(^5\) was able to obtain sharply reduced values of the coefficient of static friction (for more detail on this see § 14).

All these experimental facts undoubtedly testify that the presence on solid surfaces of thin films of foreign substances leads to a lowering of the coefficient of static friction of the surfaces, while the removal or wear of these films entails a noticeable increase in the coefficient of friction.

The increase in the coefficient of static friction observed as a result of cleaning “dry” (unlubricated) solid surfaces is undoubtedly due to the removal from them of various adsorbed contaminating films.

In some cases, as a result of thorough cleaning of surfaces, it was possible to observe an especially sharp increase in the coefficient of static friction. Thus, Tomlinson\(^6\) indicates that, with very thorough polishing and cleaning of the surfaces, the sliding of a piece of lead relative to a glass plate began only when they were brought almost into a vertical position; the coefficient of static friction, determined in the usual way [by formula (1)], then assumed an infinitely large value.

At times it was even possible to invert the glass plate so that the piece of lead was hanging on its lower surface.

Analogous phenomena were also observed by Tomlinson in studying the static friction of thin glass and quartz threads.

§ 3. Study of “Dry” Friction in Vacuum

One can speak of more or less complete cleaning of the surface of a solid body from the adsorbed films covering it, of course, only after prolonged heating of the surface under conditions of high vacuum. The study of friction in vacuum therefore represents a very substantial interest in considering the question of the influence of adsorbed films on the magnitude of the coefficient of friction.

The first measurement of the coefficients of both static and kinetic friction in vacuum was carried out by Charlotte Jacob\(^7\) in 1912. The friction of brass and glass in vacuum was studied with the surfaces heated to \(450^\circ\). In this case a considerable increase in the coefficient of static friction was observed. In view of the poor reproducibility of the results obtained by Jacob, however, we shall not dwell on their detailed consideration.

In 1929 Dow\(^8\) studied the static friction of copper, zinc, and aluminum in a vacuum of \(0.1\) mm Hg. No changes in the values of the coefficients of static friction \(\mu_s\), as compared with their values in air, were found. This result is quite understandable if one takes into account the very imperfect vacuum and, chiefly, the absence of heating of the surfaces studied.

In 1930 Shaw and Leavey\(^9\) published the results of measurements of the coefficients of static friction of iron, nickel, copper, silver, aluminum, and glass, carried out in vacuum with preceding prolonged heating of the surfaces. In view of the great interest of the data obtained in this work, we shall dwell on their consideration in somewhat more detail.

The determination of the coefficients of friction was carried out by the “inclination” method (§ 2). A cylindrical rod of the material under study was cut into 5 parts; four of them formed two

X-shaped supports, on which at the beginning of the experiment the fifth one rested motionless (Fig. 1). The entire system was inside a freely rotating glass tube connected to vacuum pumps. The tube was placed in an electric furnace; the temperature inside the tube was registered by means of a thermocouple. After prolonged pumping and heating, the tube was deflected from the horizontal position until the rod began to slide along the supports. The beginning of sliding was noted automatically: as it fell, the rod closed an electric circuit containing recording instruments.

The experiments were carried out in a vacuum of 0.01 mm Hg with heating of the surfaces to 350°. They led to the following results:

1. If the surfaces were not heated beforehand, then the coefficient of static friction in vacuum was just as low as in air, and remained low after arbitrarily prolonged pumping. The numerical values of the coefficients of friction of all the substances studied lay, in this case, in the interval between 0.1 and 0.3.

Fig. 1.

Fig. 2.

1. When the surfaces are heated in vacuum, at first there is a slight decrease of the coefficient of friction; a further increase of temperature to 350°, however, is accompanied by a sharp rise of \(\mu_s\). This increase of \(\mu_s\) proves to be irreversible—when the surfaces are cooled, the coefficient of friction does not return to its original low value, but lies considerably above it. A second heating leads to a new irreversible increase of the coefficient of static friction. The character of the change of \(\mu_s\) with change of temperature is illustrated by the curves of Fig. 2, where temperature is plotted along the abscissa axis, and the coefficient of static friction \(\mu_s\) along the ordinate axis.

Table 1

Surface material	\(\mu_s\)
Iron	0.8
Nickel	1.1
Copper	2.3
Silver	2.6
Aluminum	3.1

As a result of a series of repeated heatings and coolings, the coefficient \(\mu_s\) reaches a constant high value, no longer changing with further change of temperature (for certain exceptions to this rule see § 4).

The final values of \(\mu_s\) (measured at room temperature) are given in Table 1. They indicate that

as a result of preliminary heating in vacuum, the coefficient of static friction increases very sharply (approximately 10-fold). The data given in the table refer to the case in which the surfaces were heated while in contact with one another. If the surfaces are heated without being in contact, the values of \(\mu_s\) lie somewhat (approximately 10%) higher. This circumstance is probably due to the fact that in the second case the conditions for degassing the surfaces prove more favorable.

Holm and Kirschstein\(^{10}\) studied the static friction of nickel and platinum under more refined experimental conditions than those used in the work of Shaw and Leavey.

In a glass tube connected to vacuum pumps, a thin wire is stretched, positioned horizontally before the start of the experiment. A light cylinder is suspended on the wire, its internal diameter being somewhat greater than the diameter of the wire. The tube is evacuated to \(10^{-4}\) mm Hg; the entire apparatus is heated for an hour in a furnace at \(400^\circ\), and the wire and cylinder are annealed by passing an electric current at a temperature of \(1000^\circ\). After thorough evacuation, heating, and subsequent cooling of the apparatus, the wire is deflected from the horizontal by rotating the ground joint until, at a certain value of the angle of inclination, the cylinder begins to slide along it.

Already after a single annealing the coefficient of friction is very large. As a result of repeated annealing it increases until, finally, conditions are produced under which the wire can be turned vertically and the cylinder still does not begin to slide along it. It “sticks” to the wire so strongly that only shaking the whole apparatus makes it fall. The usual determination of the coefficient of static friction as the tangent of the angle at which sliding of one of the surfaces begins gives, in this case, an infinitely large value of \(\mu_s\). This result, as well as the experiments described in the preceding paragraph, indicates that in the absence of foreign adsorbed layers on solid surfaces, the phenomenon of sliding of surfaces in its pure form is absent; it is replaced by direct cohesion, by the sticking of solid bodies to one another.

Table 2

Gas	Surface material	Surface material
Gas	nickel	platinum
Vacuum \(10^{-4}\) mm	—	—
Nitrogen . . . . .	—	—
Argon . . . . .	—	—
Hydrogen . . . . .	1.66	1.88
Dry air . . .	1.66	2.25
Water vapor . . .	0.62	1.38
Benzine . . . . .	0.45	—
Toluene . . . . .	0.38	—
Benzol . . . . .	0.31	—

Holm and Kirschstein also studied the effect of the subsequent admission into the vacuum of various gases on the magnitude of the coefficient of static friction. It was found that chemically inert gases—argon and nitrogen—do not affect \(\mu_s\): it still remains equal to

...infinity: admission of air, water vapor, gasoline, etc.

T. A. KONTOROVA

Table 3

Coefficient of static friction and elastic constants of solid bodies

Surface material	$\mu_s$	$g$	Experimental conditions
Fe	0,8	8300	“Dry” friction in vacuum (data of Shaw and Leavey $^{9}$)
Ni	1,1	7300	“Dry” friction in vacuum (data of Shaw and Leavey $^{9}$)
Cu	2,3	4700	“Dry” friction in vacuum (data of Shaw and Leavey $^{9}$)
Ag	2,6	2900	“Dry” friction in vacuum (data of Shaw and Leavey $^{9}$)
Al	3,1	2700	“Dry” friction in vacuum (data of Shaw and Leavey $^{9}$)
Hard steel	0,393	8900	“Dry” friction in air (data of Tomlinson $^{6}$)
Soft steel	0,411	8100	“Dry” friction in air (data of Tomlinson $^{6}$)
Pt	0,445	6200	“Dry” friction in air (data of Tomlinson $^{6}$)
Cu	0,600	4700	“Dry” friction in air (data of Tomlinson $^{6}$)
Brass	0,634	3500	“Dry” friction in air (data of Tomlinson $^{6}$)
Al	0,937	2700	“Dry” friction in air (data of Tomlinson $^{6}$)
Glass	0,940	2900	“Dry” friction in air (data of Tomlinson $^{6}$)
Sn	1,110	1810	“Dry” friction in air (data of Tomlinson $^{6}$)
Pb	3,310	780	“Dry” friction in air (data of Tomlinson $^{6}$)
Soft steel	0,74	8100	“Dry” friction in air (data of Hardy $^{5}$)
Glass	0,94	2900	“Dry” friction in air (data of Hardy $^{5}$)
Ni	1,66	7300	“Dry” friction in dry air (data of Holm and Kirschstein $^{10}$)
Pt	2,25	6200	“Dry” friction in dry air (data of Holm and Kirschstein $^{10}$)

were given in 1929 by Tomlinson $^{6}$; § 25 of the present article is devoted to a consideration of his theory.

§ 6. Friction of dissimilar surfaces

According to the data of a number of authors who have studied the dry friction of dissimilar surfaces, under ordinary experimental conditions the coefficient of static friction of substance $A$ on substance $B$, $(\mu_s)_{AB}$, has a value intermediate between the values of the coefficients of friction of pairs of homogeneous surfaces, $[(\mu_s)_{AA}$ and $(\mu_s)_{BB}]$.

A particularly careful study of the friction of dissimilar surfaces was carried out by Tomlinson, who measured the coefficients of static friction for 55 pairs of different solid surfaces. The data he obtained are given in Table 4. In all cases the following inequality is satisfied:

Table 4

Values of \(\mu_s\) according to Tomlinson

Surface material	Hard steel	Soft steel	Platinum	Nickel	Copper	Brass	Aluminum	Glass	Tin	Lead
Hard steel	0.393	0.410	0.398	0.428	0.548	0.535	0.649	0.605	0.785	1.955
Soft steel	0.410	0.411	0.427	0.429	0.533	0.506	0.705	0.721	0.786	1.930
Platinum	0.398	0.427	0.445	0.386	0.592	0.560	0.796	0.569	0.855	2.070
Nickel	0.428	0.429	0.386	0.389	0.562	0.504	0.745	0.775	0.895	2.150
Copper	0.548	0.533	0.592	0.562	0.600	0.618	0.695	0.675	0.857	1.945
Brass	0.535	0.508	0.560	0.584	0.618	0.634	0.706	0.873	0.752	2.110
Aluminum	0.649	0.605	0.796	0.745	0.695	0.706	0.937	0.845	0.905	2.000
Glass	0.605	0.721	0.569	0.775	0.675	0.873	0.845	0.940	0.941	2.420
Tin	0.786	0.786	0.855	0.895	0.857	0.752	0.905	0.941	1.110	2.250
Lead	1.955	1.930	2.070	2.150	1.945	2.110	2.000	2.420	2.250	3.310

In the case of static friction in vacuum, however, this relation is not justified; in the overwhelming majority of cases \((\mu_s)_{AB}\) proves to be considerably smaller than both \((\mu_s)_{AA}\) and \((\mu_s)_{BB}\). At the same time, a new factor begins to play a role—the affinity between the substances \(A\) and \(B\).

In Table 5, pairs of dissimilar metallic surfaces are arranged in order of increasing degree of affinity between the metals \(^{9}\); the first four pairs of metals in the molten state form immiscible liquids, while the following pairs possess the property of forming solid solutions.

Table 5

Friction of dissimilar surfaces in vacuum

Surfaces	\(\mu_s\)
Ag—Fe	0.3
Cu—Fe	0.69
Ni—Fe	0.69
Ag—Ni	0.86
Al—Fe	1.28
Al—Cu	1.47
Ag—Cu	1.72
Cu—Ni	2.01
Al—Ag	2.20
Al—Ni	2.36

This arrangement of the metals in order of increasing affinity between them is, as we see, at the same time also an arrangement of them in order of increasing coefficients of static friction. This regularity is naturally connected with the circumstance, already noted above, that in vacuum there arise on the surfaces regions of direct “adhesion” of solid bodies to one another—regions of “chemical” contact.

The strength of such “chemical” contact must be determined by the degree of affinity between the bodies.

Chapter II. Kinetic Friction of “Dry” Surfaces

§ 7. Kinetic Friction and Speed. The Influence of the Degree of Cleanliness of the Surface

The data available in the literature on the dependence of the kinetic friction of “dry” (unlubricated) surfaces on the speed of their relative motion are highly contradictory. Coulomb notes that, on the kinetic friction of metals, the speed of their motion has very little influence.

According to the data of Honda and Yamada[^12], the coefficient of kinetic friction \(\mu_k\) remains constant when the speed of relative motion is varied in the interval from 15 to 40 m per minute.

Galton and Westinghouse[^13], as well as Smith[^14], indicate, however, that as the speed decreases the coefficient of kinetic friction increases.

The most careful investigation of the laws of kinetic friction, both for unlubricated and lubricated surfaces, belongs to Beare and Bowden[^15]; we shall dwell here in detail on the results obtained by them.

The apparatus used by these authors for studying kinetic friction is very simple in its construction; it consists of a metal flywheel, onto which one of the rubbing surfaces, in the form of a cylindrical ring, is fitted. At the beginning of the experiment the flywheel rotates freely; then small balls (5 mm in diameter) of the material under investigation, fixed immovably above it, are lowered onto it. As a result of the friction of the balls against the plane, the rotation of the wheel gradually slows down.

The angular velocity of the wheel is recorded by photographing, on a moving film, a beam of light reflected from mirrors fixed on the wheel. From the “retardation” curve thus obtained one can determine the angular deceleration \(\Phi\) due to friction; it is related to the coefficient of kinetic friction by the following relation:

\[ I\Phi = l\mu_k F, \]

where \(I\) is the moment of inertia of the rotating wheel, \(F\) is the total normal force, and \(l\) is the distance of the rubbing ball from the center of rotation.

In each given experiment the authors determined successive values of the coefficient \(\mu_k\) at gradually decreasing speeds of relative motion of the rubbing surfaces. The surfaces studied (of steel, nickel, and glass) were ground with fine emery, polished, washed with alcohol, and dried by heating to \(150^\circ\).

The experiments showed that the degree of cleanliness of the surface has an extremely strong influence on the magnitude of the coefficient of kinetic—

of kinetic friction. Namely: if the surfaces had been cleaned and dried insufficiently thoroughly (dried at room temperature), then with the passage of time, i.e. as the relative velocity of the rubbing surfaces decreased, the coefficient of friction \(\mu_k\) increased noticeably. Thus, for example, in the case of friction of steel surfaces, 3 min. after the beginning of the experiment the coefficient of friction increased from 0.27 to 0.40.

If, however, the surfaces had been thoroughly cleaned and just as thoroughly dried (by heating to \(150^\circ\)), then successive measurements made at various velocities in the interval from 600 to 60 cm/sec in all cases gave a constant value of the coefficient \(\mu_k\); for steel surfaces it was equal to 0.41.

These experiments testify quite clearly that the coefficient of kinetic friction \(\mu_k\) does not depend directly on the velocity; the increase in the coefficient of friction \(\mu_k\) observed in the case of insufficiently clean surfaces is, of course, due not to a change in velocity, but to the gradual removal, in the process of friction, of various adsorbed films contaminating the surfaces.

One may think that the increase in \(\mu_k\) with decreasing velocity observed by some authors is due to an analogous cause.

It would be very interesting to have the possibility of comparing with one another the values of the coefficients of static and kinetic friction of various solid surfaces. We do not know, however, of works in which data for the coefficients \(\mu_s\) and \(\mu_k\) were given simultaneously. A comparison of the data of different authors, on the other hand, seems to us meaningless in view of the specific dependence of both \(\mu_s\) and \(\mu_k\) on the conditions of the experiment.

§ 8. Kinetic Friction and Amontons’ Law

Measurements of the coefficients of friction of thoroughly cleaned steel surfaces at various values of the normal load applied to the surfaces showed that the coefficient of kinetic friction \(\mu_k\) does not depend on the load; Amontons’ law thus proves to be valid.

The study of the kinetic friction of steel surfaces contaminated by prolonged exposure to air led, however, to a somewhat unexpected result; in this case, as the load increases, the coefficient of kinetic friction rises noticeably. Table 6^15 gives the corresponding values of the coefficient \(\mu_k\), measured at various values of the load.

Table 6

Load in g	\(\mu_k\)
10.8	0.277
22.3	0.295
38.5	0.369
63.8	0.431

The presence of contaminating coatings on the rubbing surfaces thus leads to apparent deviations from Amontons’ law.

It may be assumed that, when the load is increased, more intensive wear of the films contaminating the surfaces takes place, which leads to an increase in the coefficient of friction.

In favor of such an assumption is the fact that this increase in the coefficient \(\mu_k\) is not completely reversible.

Thus, for example, if after a load of 63.8 g has been applied, the initial load of 10.8 g is again applied, the value of the coefficient of friction will no longer have its original value, 0.277 (Table 6), but an increased one—0.371.

§ 9. Kinetic Friction in Various Gases

Bire and Bowden[^15] measured the coefficient of kinetic friction of steel surfaces in various gases: in air, oxygen, carbon dioxide, and nitrogen. It turned out that the magnitude \(\mu_k\), with an accuracy lying within the limits of experimental error, does not depend on the nature of the surrounding gaseous medium. The corresponding values of \(\mu_k\) are given in Table 7.

Table 7

Gas	\(\mu_k\)
Air	0.57
\(O_2\)	0.58
\(N_2\)	0.57
\(CO_2\)	0.57

In the process of friction, as has already been established above, gradual wear occurs of the adsorbed films covering the surfaces. Those parts of the surface from which the film has already been worn away should be especially quickly covered again by atoms adsorbed from the surrounding medium. The nature of the surrounding medium should therefore, it would seem, exert a very substantial influence on the magnitude of the coefficient of kinetic friction. The circumstance that in fact \(\mu_k\) does not depend on the nature of the gas can, it seems to us, be explained by the assertion that in the process of friction there occurs not only wear of the films adsorbed on the surface, but also very intensive wear, abrasion of the solid surface itself. One may think that, in contrast to the case of static friction (where the nature of the surrounding medium exerts a strong influence on the magnitude of the coefficient of friction, § 3), in kinetic friction this wear of the surface itself plays a very substantial role. The nature of the surrounding medium under such conditions should, of course, have no influence on the magnitude of the coefficient of friction. In favor of such a point of view there is a series of experimental facts, first of all—the phenomenon of wear, which in practice almost always accompanies kinetic friction, and also the data presented below on the dependence of the coefficient of kinetic friction on the degree of polishing of the rubbing surfaces.

§ 10. Kinetic friction and preliminary polishing of surfaces

Preliminary polishing of rubbing surfaces does not always have the same effect on the magnitude of the coefficient of kinetic friction. According to the data of Beer and Bowden^15, preliminary polishing of a plane surface lowers the coefficient of kinetic friction only in the case when the balls rubbing against it (the method is described in § 7) are made of a softer material than the surface itself.

Table 8

Surface	\(\psi_k\)
Ground steel	0.552
Polished steel	0.449
Ground glass	0.457
Polished glass	0.309

Table 9

Surface material	\(\psi_k\)
Glass	0.40
Nickel	0.53
Steel	0.57

Table 8 gives the values of the coefficient \(\psi_k\) for the friction of cadmium balls on polished and unpolished surfaces of steel and glass.

If, however, the balls are made of a harder material than the surface against which they rub, then the coefficient of friction does not depend on the initial polishing of this surface: for an unpolished surface it has the same value as for a polished surface.

In this case the polishing of the surface, apparently, takes place very rapidly in the very process of friction, whereas soft balls cannot polish it.

Preliminary polishing of surfaces thus affects the magnitude of the coefficient of kinetic friction \(\psi_k\) only when the friction itself is not accompanied by polishing.

It would be appropriate here to pause for a brief account of current ideas about the essence of the process of polishing solid surfaces. We shall, however, postpone consideration of this question somewhat (§ 12).

§ 11. Kinetic friction of homogeneous and heterogeneous surfaces

The numerical values of the coefficient of kinetic friction obtained by Beer and Bowden^15 for the friction of certain homogeneous surfaces are given in Table 9.

The sequence of coefficients $\mu_k$ for nickel and steel corresponds to their arrangement in decreasing order of elastic moduli; the coefficient $\mu_k$ for glass, however, should have exceeded both $\mu_k$ for nickel and $\mu_k$ for steel (§ 5). In view of the small amount of experimental data, the question of whether there is, between the coefficients of kinetic friction of “dry” surfaces and the elastic constants, a relationship of the same type as in the case of static friction remains open.

In the case of friction of dissimilar surfaces, the picture is greatly complicated by the circumstance that the coefficient of kinetic friction of a substance $A$ on a substance $B$ has a substantially different value depending on which material constitutes the smaller of the rubbing surfaces (the ball) and which the larger (the plane). Thus, for example, when steel balls are rubbed against a nickel surface, $\mu_k = 0.49$; whereas when nickel balls are rubbed against a steel surface, $\mu_k = 0.66$. Numerical data on the friction of dissimilar surfaces are given in Table 10.

Table 10

Material of ball	Flat-surface material	Flat-surface material	Flat-surface material
	steel	nickel	glass
	$\mu_k$	$\mu_k$	$\mu_k$
Steel	—	0.49	0.61
Nickel	0.66	—	0.56
Glass	0.51	0.50	—
Copper	0.36	0.49	0.53
Coal	0.21	0.24	0.18

Beer and Bowden$^{15}$ note that $\mu_k$ assumes larger values in the case when the smaller of the rubbing surfaces consists of the softer material. If the smaller of the surfaces (the ball) is of a soft material, then the coefficient of friction increases as the hardness of the second of the rubbing surfaces decreases. If, however, the ball is of a hard material, then the opposite regularity is observed—$\mu_k$ increases as the hardness of the second surface increases.

These facts can readily be explained on the assumption that the process of friction itself is accompanied by polishing of the softer of the rubbing surfaces. Indeed, in the case when the smaller of the surfaces is of a soft material, it itself is polished in the process of friction; as the hardness of the second surface decreases, it is polished to a lesser degree, which leads to an increase in the coefficient $\mu_k$. In the case when the smaller of the surfaces is of a hard material, it itself is not polished during friction, but polishes the second surface.

The coefficient of kinetic friction of substance \(A\) on substance \(B\) thus proves to be different from the coefficient of friction of substance \(B\) on substance \(A\). The simple regularities observed in static friction are, in this case, greatly complicated by the accompanying frictional polishing of the surfaces.

§ 12. Measurement of the temperature of rubbing surfaces; melting of the surface and polishing

The measurement of the temperature of rubbing surfaces was first carried out in 1936 by Bowden[^16] with the aid of an equally simple and ingenious method. It is based on the fact that if the rubbing surfaces belong to dissimilar metals and if, in the process of friction, a rise in temperature occurs at the interface of the surfaces, then the rubbing bodies themselves will constitute a thermocouple.

By closing the rubbing bodies into an electrical circuit containing suitably calibrated measuring instruments, Bowden was able to measure the temperature of the rubbing surfaces and to investigate its dependence on the velocity of the relative motion of the surfaces, the magnitude of the applied load, and other parameters.

Fig. 4.

The regularities discovered are in good agreement with the following approximate theoretical calculation, made by Bowden, of the temperature rise at the interface of the surfaces.

In the friction of a cylinder against a plane (this was precisely the case in Bowden’s experiments), the amount of heat received by an element \(\delta x\) of the cylinder (Fig. 4) as a result of heating of its lower base is equal to

\[ K\pi r^2 \frac{d^2 T}{d x^2}\,\delta x, \]

where \(r\) is the radius of the cylinder, \(K\) is the coefficient of thermal conductivity of the material of the cylinder.

The amount of heat emitted by the element \(\delta x\) into the surrounding medium is

\[ 2\pi r \sigma (T - T_0)\,\delta x, \]

where \(\sigma\) is a coefficient, \(T_0\) is the temperature of the surrounding medium.

Integrating the condition of stationarity of the heat flux

\[ K\pi r^2\delta x\,\frac{d^2T}{dx^2}-\sigma 2\pi r\delta x\,(T-T_0)=0, \]

Bowden obtains the rise in temperature inside the rubbing cylinder at a distance \(x\) from the sliding plane

\[ T-T_0=\Delta T=Ce^{-\sqrt{\frac{2\sigma}{Kr}}\,x}. \tag{1} \]

He determines the constant \(C\) from the consideration that all the heat radiated into the surrounding space is obtained at the expense of the work of the frictional force. If all the work of the frictional force is converted into heat, then the amount of heat is

\[ Q=\mu_k Pv, \]

where \(P\) is the normal load, \(v\) is the velocity of relative motion of the rubbing bodies, and \(\mu_k\) is the coefficient of kinetic friction.

Putting

\[ Q=2\pi r\sigma\int_0^\infty (T-T_0)\,dx \]

and using equation (1), we obtain

\[ C=\frac{1}{\pi r}\frac{1}{\sqrt{2\sigma Kr}}, \]

whence finally:

\[ \Delta T=\frac{\sigma Q}{\pi r\sqrt{2\sigma Kr}}. \tag{2} \]

The rise in temperature of the rubbing surfaces \((x=0)\) is

\[ \Delta T=\frac{\sigma Q}{\pi r\sqrt{2\sigma Kr}} =\frac{\sigma\mu_k Pv}{\pi r\sqrt{2\sigma Kr}}. \tag{3} \]

It thus proves to be directly proportional to the magnitude of the load and to the velocity of relative motion of the bodies, and inversely proportional to the square root of the coefficient of thermal conductivity of the material of the cylinder (it is assumed here that the mass of the cylinder is very small in comparison with the mass of the second body). In Figs. 5 and 6 are given graphs of the dependence of the temperature of the rubbing surfaces on the load \(P\) and on the velocity of relative motion \(v\), obtained by Bowden for the case of “dry” friction of a constantan cylinder against a steel plane; they show that relation (3) is excellently justified by experiment.

The interval of velocities investigated in this case was from \(4\) to \(5000\ \mathrm{cm/sec}\), and of loads up to \(120\ \mathrm{g}\).

As for the dependence of the temperature of rubbing surfaces on the coefficient of thermal conductivity, it is somewhat less well confirmed. The reason for this circumstance lies, it seems to us, in the fact that in the calculation cited above Bowden assumed that contact between the rubbing bodies takes place over the entire plane of the base of the cylinder. In reality this is, of course, not so; contact between the rubbing bodies takes place only in separate, very small regions of the surfaces in contact.

Fig. 5.

Fig. 6.

An attempt to take this circumstance into account1 led to the same dependence of \(\Delta T\) on \(P\) and \(v\) as in Bowden’s work, and to an inverse proportionality between \(\Delta T\) and the coefficient of thermal conductivity \(K\) (but not the square root of \(K\)!). As the recalculation of Bowden’s data carried out by us showed, the relation \(\Delta T \sim \frac{1}{K}\) is confirmed quite well.

Fig. 7.

Fig. 8.

The straight lines shown in Figs. 5 and 6 refer to the case of friction of two refractory metals—constantan and steel.

In the case of friction of low-melting metals—gallium, bismuth, lead, Wood’s alloy, etc.—against a steel surface, the graph of the dependence of the temperature of the rubbing surfaces on the load $f$ and on the speed of relative motion $v$ has an entirely different character (Figs. 7 and 8): at first the relation (3) found above is valid, but beginning with a certain temperature (different for different bodies), any further rise of temperature ceases; with further increase of the load or speed the temperature remains constant. This constant temperature, within the limits of experimental error, coincides with the melting temperature of the material of the rubbing cylinder. Thus, for example, in the case of gallium the further rise of temperature ceases at $32^\circ$, in the case of Wood’s alloy—at $72^\circ$, of lead—at $327^\circ$, and of bismuth—at $270^\circ$.

It has long been suggested that the heating of surfaces during friction should play a major role in the polishing process of solid bodies. Many authors[^17] believe that the polishing of solid bodies is connected with melting of their surface. Bowden[^18] notes that, if this point of view is correct, then the relation between the melting temperatures of the polishing substance and of the solid body being polished must play a large role in the possibility of polishing. If the polishing substance melts or softens at a lower temperature than the solid body being polished, then upon heating it will begin to melt and flow and will have very little effect on the body being polished. A comparison of experimental data on the melting temperatures of polishing substances and of bodies being polished shows that polishing (and the formation of an amorphous layer—the so-called “Beilby layer”) takes place only when the melting temperature of the polishing substance is higher than the melting temperature of the solid body being polished. The relative hardness of the bodies in this case does not play an essential role.

Thus, for example, camphor (melting temperature $178^\circ$) polishes Wood’s alloy (melting temperature $69^\circ$), but does not polish metals that melt at higher temperatures, for example tin (melting temperature $232^\circ$). Calcite (melting temperature $1333^\circ$) can be polished with zinc oxide (melting temperature $1800^\circ$), but cannot be polished with copper oxide (melting temperature $1235^\circ$). Very many such examples could be cited.

The data given above on measurements of the temperature of rubbing surfaces constitute a remarkable experimental confirmation of the hypothesis of melting of the surface of solid bodies during friction and give grounds to think that, in the process of polishing solid bodies, such melting must indeed play a very substantial role.

Chapter III. Static Friction of Lubricated Surfaces

§ 13. Two Types of Lubrication

It is customary to distinguish two types of lubrication: “full,” or “complete,” lubrication, and “boundary,” or “partial,” lubrication.

A lubricant is called “complete” if the solid surfaces are completely separated from one another by a sufficiently thick layer of lubricating substance. In this case, static friction as such, from a theoretical point of view, should not be observed at all; in practice, however, it is very slight—a very small tangential force is sufficient for the relative sliding of the surfaces to begin. The resistance to motion in this case is entirely independent of the nature of the solid surfaces, being determined wholly by the properties of the lubricating substance; it proves, moreover, to be proportional to the viscosity of the latter. In practice, efforts are usually made to realize precisely this type of lubrication.

In the process of friction of the various parts of mechanisms, however, there occurs a gradual thinning and wearing away of the lubricant layer, accompanied by a gradual transition from the conditions of “complete” lubrication to “boundary,” or “partial,” lubrication.

By “boundary” lubrication is meant the case in which the resistance to the relative motion of the surfaces is determined not only by the properties of the lubricant itself, but also depends on the nature of the solid surfaces. In this case no direct dependence of the coefficient of friction on the viscosity of the lubricant is observed; the magnitude of the coefficient of friction is determined by quite different parameters. The numerical value of the coefficient of friction in the case of “boundary” lubrication is considerably higher than in the case of “complete” lubrication.

For different lubricating substances the transition from “complete” lubrication to “boundary” lubrication takes place under different conditions. In studying the kinetic friction of steel surfaces, Beare and Bowden^15 were able to observe this transition directly. In Fig. 9 is shown the graph obtained by them of the change in the angular velocity of one of the rubbing surfaces with time (a detailed description of the apparatus and method is given in § 7).

Fig. 9.

At first the coefficient of kinetic friction increases (the curvilinear portion of the graph); after a certain interval of time from the start of the experiment, however, the angular deceleration acquires a constant value, which corresponds to the transition to “boundary” lubrication, characterized by a constant value of the coefficient of friction. The curve shown pertains to the case of friction of steel surfaces lubricated with octyl alcohol; the initial instant of time corresponds to a coefficient of kinetic friction \(\mu_k \simeq 0.02\); under conditions of “boundary” lubrication (the rectilinear part of the graph) \(\mu_k = 0.073\).

coefficient in Hardy's experiments is rather large—as an example, Table 11 gives the values of \(\psi_s\) obtained by Hardy at various values of the load. The interval of loads studied in this case extends from 20 to 534 g. Hardy's investigations thus make it possible to consider that, under these conditions, Amonton's law, mentioned at the very beginning of our article as the basic law of friction, does indeed prove to be valid.

Table 11

Load in g	\(\psi_s\)
21,63	0,6321
31,64	0,6354
41,63	0,6301
51,63	0,6298
61,63	0,6352

b. Friction of flat surfaces

In the case of friction of a flat surface on a flat surface, the dependence of the coefficient of friction on the load, as follows from Fig. 11, proves to be quite different from that in the preceding case, namely: in the region of small loads the coefficient of friction \(\psi_s\), as the load increases, decreases (the part \(AB\) of the curve in Fig. 11) until, at a certain load (point \(B\)), the value of \(\psi_s\) reaches a minimum value, which remains unchanged with further increase of the load1.

This minimum value of \(\psi_s\) lies somewhat above the constant value of \(\psi_s\) corresponding to the case of friction of a spherical surface on a flat one (for the same material and with the same lubricant) (the dotted straight lines in Fig. 11). Thus, for example, in the case of flat steel surfaces lubricated with nonadecane, when the pressure is increased from 96 to 12,000 g/cm², \(\psi_s\) decreases from 0.305 to 0.184, whereas if one of the steel surfaces has a spherical shape, the coefficient \(\psi_s\) remains constant at all loads and equal to 0.181.

Fig. 11.
Legend: × — octyl alcohol; □ — caprylic acid; ○ — nonadecane. Horizontal axis: load in grams.

The larger the radius of curvature of the surface, the greater the normal pressure (the load divided by the area of contact) between the rubbing surfaces; for a given load, in case b we are dealing with a greater normal pressure than in case a.

The data presented above indicate that, in the case of friction of a spherical surface against a plane, Amontons’ law is justified for all studied values of the load; whereas in the case of friction of two plane surfaces Amontons’ law is justified only beginning with a certain minimum value of the load (point \(B\) on the curve). This makes it possible to arrive at a somewhat more general conclusion: at small values of the normal pressure (the region of small loads in the case of friction of plane surfaces), Amontons’ law is not valid; it comes into force only beginning with a certain minimum value of the normal pressure (the region of large loads in the case of friction of plane surfaces, and the case of friction of a spherical surface against a plane). The transition from conditions under which Amontons’ law has no place to conditions corresponding to its fulfillment is thus determined exclusively by the magnitude of the normal pressure between the rubbing surfaces.

The question of the physical nature both of Amontons’ law itself and of the deviations from it that are observed remains, however, open. In what follows we shall attempt to outline possible ways of considering it.

§ 16. Latent Period

In studying the static friction of lubricated surfaces, Hardy\(^{20,19}\) discovered the following very curious circumstance: the coefficient of static friction assumes a constant value, which does not change subsequently, only after a certain interval of time has elapsed after the rubbing surfaces have been brought into contact with one another. This interval of time, during which the stationary value of the coefficient \(\mu_s\) is established, Hardy calls the latent period.

The duration of the latent period and the character of the change in the coefficient of static friction during this period prove to be very different depending on the magnitude of the pressure between the surfaces, the temperature, the chemical nature of the lubricant, and other experimental conditions.

In the case of friction of plane surfaces under small loads, during the latent period there is an increase in the coefficient of static friction\(^1\). For a given load, the duration of the latent period is different for different lubricating substances. Table 12 gives values of the coefficient of static friction of plane steel surfaces measured after different intervals of time following the bringing of the surfaces into contact with one another. The duration of the latent period in this case is very great—it reaches several hours.

Hardy\(^{20}\) regards the increase in the coefficient of static friction with time as the result of a gradual decrease

\(^1\) In Fig. 11 the final stationary values of \(\mu_s\) are given.

T. A. KONTOROVA

Table 12

Lamp oil	Lamp oil	Octyl alcohol	Octyl alcohol
time from the start of the experiment	\(\mu_s\)	time from the start of the experiment	\(\mu_s\)
5 sec.	0.02	3 sec.	0.13
1 min.	0.11	5 min.	0.38
1 hour	0.304	1 hour	0.469
3 hours	0.331	3 hours	0.484
8 hours	0.338	6 hours	0.485

of the upper surface through the layer of lubricant, i.e., as the result of the gradual squeezing out of the lubricant from the space between the solid surfaces. Such squeezing out leads to a decrease in the distance between the solid surfaces, i.e., to an increase in the forces of adhesion between them, which also accounts for the increase in the coefficient of friction.

The greater the pressure between the surfaces, the more rapidly the squeezing out of the lubricant should occur. Indeed, the latent period depends very sharply on the magnitude of the load, decreasing as the latter increases; thus, for example, in the case of friction of flat steel surfaces lubricated with nonadecane, when the load is increased from 120 to 820 g, the latent period decreases from 12 to 1 min.

In the case of friction of a spherical surface against a flat one, i.e., in the case of large pressures between the rubbing surfaces, the above-described latent period of “squeezing out” is absent, i.e., the time necessary for squeezing out the lubricant is reduced so much that it becomes practically unobservable.

But in this case as well, the coefficient of static friction does not immediately reach its stationary value, i.e., a latent period is also observed; during this latent period, however, the coefficient of static friction does not increase, as during the above-described period of squeezing out, but decreases.

Table 13

Lubricant	Initial value \(\mu_s\)	Final value \(\mu_s\)	Latent period in minutes
Caprylic acid	0.57	0.34	60
Enanthic acid	0.50	0.40	45
Octyl alcohol	0.59	0.52	15

In Table 13 ¹) data are given on the friction of a spherical steel surface against a flat one for several lubricating substances. The first column of the table contains the initial values of the coefficient of friction, measured immediately after bringing the lubricated surfaces into contact with one another; the second column contains the final stationary values of the coefficient of friction; the third contains the duration of the period during which these stationary values are established.

If the explanation of the increase in the coefficient of static friction that occurs during the latent period of “squeezing out” presents no special difficulties, then the explanation of such a decrease in the coefficient of friction with time is considerably more difficult.

Hardy ²⁰ assumes that this decrease in the coefficient of friction is connected with a gradual increase in the degree of orientation of the molecules in the layer of lubricating substance, and regards the latent period as the time required for the attainment of a certain equilibrium orientation.

In contrast to the latent period of “squeezing out” described above, we shall call this period the latent period of “orientation.” In favor of the correctness of this point of view the following extremely curious circumstance testifies: the latent period of “orientation” occurs in the case of lubrication of surfaces with fatty acids and alcohols, and is never observed when paraffins are used as the lubricant.

Paraffin molecules, as is known, possess no polarity; they are constructed symmetrically, having groups CH₃ at their ends; the molecules of fatty acids and alcohols, however, possess clearly expressed polarity. The nature of the arrangement of paraffin molecules in the lubricating layer should therefore play no role in determining the conditions of sliding of the surfaces; whereas in a layer containing polar molecules of acids or alcohols, with time there will occur the establishment of a certain equilibrium orientation, leading to an easier sliding of the solid surfaces relative to one another.

In the molecules of fatty acids and alcohols it is customary to distinguish the “head” (the COOH group in fatty acids and the OH group in alcohols) and the “tail” (the CH₃ group, both in acids and in alcohols).

According to Langmuir, in the case of a monomolecular equilibrium layer the energetically most advantageous orientation of these molecules is one in which they are arranged perpendicular to the solid surface, attaching themselves to it by their “heads.”

It may be assumed that precisely such an orientation is established in the first monomolecular layer of lubricant, immediately adjacent to the solid surface.

In the case under consideration, apparently, one should speak not only of a single orientation of the molecules of the lubricating substance relative to the solid surfaces, but of a volume orientation in

throughout the entire layer of lubricant enclosed between the solid surfaces. This is evidenced by the following experimental facts, cited by Hardy. If each of the solid surfaces, separately, had been kept for some time before the experiment in an atmosphere of saturated vapor of the lubricant, then the latent period of orientation subsequently proved to be greatly shortened. The data given in Table 14 refer to the case of a three-hour exposure

Table 14

(Latent period—5 min.)

Lubricant	Initial value $\mu_s$	Final value $\mu_s$
Caprylic acid	0.45	0.34
Enanthic acid	0.48	0.40
Octyl alcohol	0.54	0.52

of the surfaces in an atmosphere of saturated lubricant vapor; as a result, the latent period was shortened to 5 min. (for comparison see the preceding table). Under these conditions an oriented layer of lubricant must already have formed on each of the surfaces before the experiment; indeed, the initial values of the coefficient of friction in this case are lower than the initial values $\mu_s$ usually obtained (Table 13). After the surfaces are brought into contact with one another, however, a further decrease of the coefficient $\mu_s$ is observed, apparently connected with a further change in the degree of orientation already throughout the entire layer of lubricant.

Such a layer completely loses its liquid properties; its state has nothing in common with the state of a lubricating layer under conditions of “complete” or “perfect” lubrication.

The latent period of “orientation,” as was to be expected, depends sharply on the temperature of the experiment, decreasing as the latter rises. Thus, for example: raising the temperature of the experiment from 15 to 70° in the case of lubrication with octyl alcohol shortens the latent period from 30 to 3 min., and with caprylic acid—from 45 to 5 min. These data also testify in favor of Hardy’s proposition concerning the orientational nature of the latent period—an increase in temperature must lead to a more rapid establishment of equilibrium orientation in the lubricant layer, i.e., to a shortening of the latent period.

The question of the kinetics of the latent period of orientation is considered in the work of Deryagin\(^ {22}\) under the assumption of a bimolecular structure of the lubricating layer. Deryagin divides (in accordance with Hardy’s hypothesis) the oriented molecules into two classes: stably oriented (with surface concentration $x_1$) and unstably oriented (situated at an angle of 180° with respect to

molecules of the first class1 with surface concentration \(x_2\). The “degree of orientation,” defined by the ratio \(\frac{x_1}{x_2}\), he calculates by means of the equations for the monomolecular change in the concentrations \(x_1\) and \(x_2\). The results obtained in this way are in qualitative agreement with the experimental data.

The possibility of observing both latent periods, or either one of them, is connected with their relative duration. According to Hardy[^20], in the case of low pressures the latent period of “orientation” is difficult to detect because of the large magnitude of the “squeeze-out” period; in the case of high pressures the reverse picture is obtained.

But even at low pressures (in the case of friction of flat surfaces) it is possible to observe the latent period of “orientation” if each of the surfaces is lubricated by producing a layer of lubricant from saturated vapor and allowing it to reach a stable state before bringing the surfaces into contact with one another. In this case the layer of lubricating substance is apparently considerably thinner than in the case of ordinary direct lubrication of the surfaces, which considerably shortens the “squeeze-out” period; after the surfaces lubricated in this way are brought into contact, in the case of acids and alcohols the coefficient of friction falls with time, while in the case of paraffins it remains unchanged.

Of great interest are the data given by Hardy[^19] (Table 15) on the static friction of a spherical surface against a flat one for the case in which, before the experiment, a thick layer of lubricating substance had been applied to each of the surfaces. In this case, at first there is an increase in the coefficient of friction \(\mu_s\) up to a certain \(\mu_{s\max}\), and then a decrease of it to the usual stationary value, i.e., both latent periods are observed—both “squeeze-out” and “orientation.”

Table 15

Lubricating substance	Initial value \(\mu_s\)	Maximum value \(\mu_s\)	Final value \(\mu_s\)
Caprylic acid	0.26	0.44	0.34
Enanthic acid	0.31	0.46	0.40
Octyl alcohol	0.47	0.54	0.52

These data may perhaps be regarded as directly testifying to the correctness of the views set forth on the nature of latent periods.

§ 17. Static friction of homogeneous and heterogeneous lubricated surfaces

In the case of friction of homogeneous unlubricated surfaces, the arrangement of substances in order of increasing coefficients of static friction, as was already noted above (§ 5), is at the same time their arrangement in order of decreasing elastic moduli. This sequence of coefficients of static friction, as Hardy showed, is also preserved for lubricated surfaces when they are lubricated with normal paraffins, alcohols, and fatty acids. In all cases the coefficient of static friction of glass exceeds the coefficient of static friction of steel, and the latter exceeds the coefficient of friction of bismuth.

In the case of friction of heterogeneous lubricated surfaces, the coefficient of static friction of substance \(A\) on substance \(B\), as also in “dry” friction, has a value intermediate between the coefficients of friction of pairs of homogeneous surfaces \([(\mu_s)_{AA}<(\mu_s)_{AB}<(\mu_s)_{BB}]\). According to Hardy’s data, this value is very close to the arithmetic mean of the coefficients \((\mu_s)_{AA}\) and \((\mu_s)_{BB}\).

Table 16

Friction of homogeneous and heterogeneous lubricated surfaces

Material of the lower plane	Material of the spherical slider	Lubricating substance: butyl alcohol, \(\mu_s\)	Lubricating substance: butyl alcohol, \(\bar{\mu}_s\)	Lubricating substance: octyl alcohol, \(\mu_s\)	Lubricating substance: octyl alcohol, \(\bar{\mu}_s\)
Glass	Glass	0.606	—	0.5176	—
Steel	Steel	0.3924	—	0.2981	—
Bismuth	Bismuth	0.30	—	0.25	—
Steel	Glass	0.493	0.4992	0.41	0.4078
Glass	Bismuth	0.451	0.453	0.38	0.3837
Steel	Bismuth	0.348	0.3464	0.27	0.274

Table 16 gives the coefficients of static friction \(\mu_s\) for certain homogeneous and heterogeneous surfaces lubricated with butyl and octyl alcohols, as well as the arithmetic mean values \(\overline{(\mu_s)_{AB}}\), calculated by the formula

\[ (\overline{\mu_s})_{AB}=\frac{(\mu_s)_{AA}+(\mu_s)_{BB}}{2}. \]

The totality of these data allows us to come to the conclusion that, under conditions of “boundary” lubrication, the elastic properties of solids

surfaces play exactly the same role as in the case of “dry” friction.

§ 18. The Role of the Chemical Nature of the Lubricating Substance

The dependence of the coefficient of static friction on the nature of the lubricating substance was studied in detail by Hardy and Doubleday5, 20, 21 for various solid surfaces (glass, bismuth, and different grades of steel) under conditions of “boundary” lubrication with normal paraffins, alcohols, and fatty acids.

The regularities found in this case proved to be extremely simple. They may be formulated as follows:

1. For members of a given homologous series (paraffins, alcohols, acids), the coefficient of static friction decreases as the molecular weight of the lubricating substance \(M\) increases. In this case the relation

\[ \mu_s=\mathrm{const}-aM, \]

is justified with sufficient accuracy, where the coefficient of proportionality \(a\) does not depend on the nature of the solid surface and is a property of the given homologous series. Numerical values of the coefficients of friction are given in Table 17, where the members of each of the homologous series are arranged in order of increasing molecular weight \(M\). According to Hardy, the decrease in the coefficient of friction with increasing molecular weight of the lubricating substance is a consequence of the lengthening of the hydrocarbon chain of the lubricant molecule, which leads to an increase in the distance between the solid surfaces.

As an analogy one might consider the friction of two hair brushes against one another. The longer the bristles of the brushes, the more easily they slide over one another.

1. For members of a given homologous series (paraffins, alcohols, fatty acids), the straight lines

\[ \mu_s=\mathrm{const}-aM, \]

constructed for different solid surfaces (steel, glass, bismuth), prove to be parallel to one another. Hardy assumes

\[ \mu_s=b-aM, \]

where \(b\), in the general case, depends both on the nature of the solid surface and on the chemical nature of the lubricating substance (its belonging to one or another homologous series). For a given homologous series and a given solid surface, \(b\) is constant.

Figures 12, 13, and 14 show the corresponding straight lines for steel and glass surfaces lubricated with paraffins, alcohols, and fatty acids; the abscissa axis gives the molecular weight of the lubricant \(M\), and the ordinate axis the coefficient of static friction.

Table 17

Lubricating substance	Chemical formula	Molecular weight	Glass	Steel	Bismuth
Paraffins
Pentane	C₅H₁₂	72	0,7102	0,4763	—
Hexane	C₆H₁₄	86	0,6908	0,4528	0,37
Heptane	C₇H₁₆	100	0,6751	—	0,346
Octane	C₈H₁₈	114	0,6552	0,3421	0,32
Undecane	C₁₁H₂₄	156	0,5903	0,1785	—
Nonadecane	C₁₉H₄₀	268	0,4119	—	—
Tetracosane	C₂₄H₅₀	338	0,3251	—	—
Alcohols
Methyl	CH₃OH	32	0,6772	0,4610	0,29
Ethyl	C₂H₅OH	46	0,6512	0,4408	0,32
Propyl	C₃H₇OH	60	0,6301	0,4173	0,34
Butyl	C₄H₉OH	74	0,6061	0,3924	0,30
Amyl	C₅H₁₁OH	88	0,5854	0,3752	0,27
Octyl	C₈H₁₇OH	130	0,5176	0,2981	0,25
Undecyl	C₁₁H₂₃OH	172	0,4455	0,2298	—
Cetyl	C₁₆H₃₃OH	242	0,3253	0,1143	0,17
Fatty acids
Formic	HCOOH	46	0,6823	—	0,45
Acetic	CH₃COOH	60	0,6003	—	0,40
Propionic	C₂H₅COOH	74	0,6387	—	0,31
Butyric	C₃H₇COOH	88	0,5721	—	—
Valeric	C₄H₉COOH	102	0,5259	—	0,28
Caproic	C₅H₁₁COOH	116	0,4654	0,3108	—
Enanthic	C₆H₁₃COOH	130	0,4051	0,2556	—
Caprylic	C₇H₁₅COOH	144	0,3417	0,2003	0,19
Capric	C₉H₁₉COOH	172	0,2006	0,0742	—
Lauric	C₁₁H₂₃COOH	200	0,0983	—	—

1. For a given solid surface and at one and the same molecular weight of the lubricating substance, the greatest value of the coefficient of static friction is given by paraffins, a smaller value by alcohols, and the smallest by acids (see Fig. 15, relating to the case of friction of glass surfaces). This circumstance should evidently be connected with the nonpolarity of the paraffin molecules and the polarity of alcohols and fatty acids. The numerical data given in Table 17 relate to the case of friction of a spherical surface against a plane surface.

In the friction of two plane surfaces, the coefficient of static friction usually has a significantly greater value; this ...

...the regularities noted above remain, however, valid in this case as well. In this case too, for a given load, in coordinates coef-

Fig. 12.

Visible plot labels: \(\mu_s\); “glass”; “steel”; \(M\).

Fig. 13.

Visible plot labels: \(\mu_s\); “glass”; “steel”; \(M\); “Molecular weight of alcohols.”

Fig. 14.

Visible plot labels: \(\mu_s\); “glass”; “steel”; \(M\).

ficient of friction—molecular weight, straight lines are obtained; they are arranged in the same order.

As the load increases, the coefficient of friction decreases; in the limit, for flat surfaces, we obtain the same straight lines as for the friction of a spherical surface against a flat one.

Fig. 15.

Chapter IV. Kinetic Friction of Lubricated Surfaces

§ 19. Kinetic friction under conditions of “boundary” lubrication

The kinetic friction of certain solid surfaces under conditions of “boundary” lubrication by paraffins, alcohols, and fatty acids was studied by Beare and Bowden\(^{15}\) with the aid of the apparatus and method described by us in detail in § 7. At the same time, as was already noted above (§ 13), it was possible to observe the direct transition from conditions of “full” lubrication to conditions of “boundary” lubrication.

At the beginning of the experiment, when there is a thick layer of lubricating substance on the rubbing surfaces, the coefficient of kinetic friction proves to be a function of the velocity of the relative motion of the surfaces and of the viscosity of the lubricant; it decreases as the speed of motion decreases and increases when more viscous lubricating substances are used.

In the course of the experiment, however, a transition takes place to conditions of “boundary” lubrication, characterized by a constant value of the coefficient of kinetic friction \(\mu_k\), independent both of the viscosity of the lubricating substance and of the velocity of the relative motion of the bodies.

For different lubricating substances this transition occurs at different values of the pressure between the surfaces and different velocities of their relative motion. For some lubricating substances, conditions of “full” lubrication can be achieved only at velocities of motion lying above a certain threshold.

minimum; for other (very viscous) lubricating substances the conditions of “complete” lubrication are, on the contrary, preserved down to very small velocities of relative motion.

As special investigations have shown, with “boundary” lubrication the coefficient of kinetic friction \(\mu_k\), throughout the investigated range of velocities—from 600 to 5 cm/sec—does not depend on the velocity.

The “deceleration” curves of the rubbing surfaces recorded in this case prove to be strictly rectilinear (for comparison see the deceleration curve in Fig. 9, corresponding to the case of transition from the conditions of “complete” lubrication to “boundary” lubrication).

§ 20. The coefficient of kinetic friction and the load

In contrast to the case of static friction, in kinetic friction of solid surfaces the character of the dependence of the coefficient of friction on the magnitude of the load proves to be different depending on the nature of the lubricating substance.

Beer and Buden \(^{15}\) divide lubricating substances into two classes. For one of them the coefficient of kinetic friction, when the load is varied, remains unchanged, i.e., Amontons’ law is justified. In the case of friction of steel surfaces, lubricants of this type include, for example, octane, tetradecane, cetyl iodide, ethyl palmitate, etc.

For lubricants of the second type, however, the coefficient of kinetic friction \(\mu_k\) decreases as the load increases, tending thereby toward a certain minimum value; Amontons’ law thus proves invalid. Lubricants of the second type include, for the most part, alcohols and fatty acids.

Table 18

Load, g	Lubricating substance	Lubricating substance	Lubricating substance	Lubricating substance
	octyl alcohol	caprylic acid	oleic acid	rapeseed oil
	\(\mu_k\)	\(\mu_k\)	\(\mu_k\)	\(\mu_k\)
10.8	0.105	0.080	0.124	0.140
14.0	0.090	—	—	—
22.0	0.083	—	—	—
38.5	0.064	0.072	0.093	0.110
41.7	0.061	—	—	—
95.7	0.053	0.064	0.071	0.091

In Table 18 and in Fig. 16 are given values of the coefficient of kinetic friction \(\mu_k\) of steel surfaces at various

loads under conditions of “boundary” lubrication by certain lubricating substances of the second type. The smallest value of \(\mu_k\), i.e. the best lubrication conditions, is obtained in the case of octyl alcohol (\(\mu_k = 0.053\)).

§ 21. The Role of the Nature of the Solid Surface

The study of the kinetic friction of various solid surfaces has shown that, depending on the nature of the solid surface, one and the same lubricating substance may belong either to the first or to the second type of lubricant; i.e., the character of the dependence of \(\mu_k\) on the load is determined not only by the nature of the lubricating substance, but also by the nature of the rubbing surfaces.

Fig. 16.

Thus, for example, in the friction of glass surfaces, octyl alcohol and caprylic acid, in contrast to the case of friction of steel surfaces, give constant values of \(\mu_k\) for all values of the load (for octyl alcohol \(\mu_k = 0.259\), for caprylic acid \(0.216\)); for oleic acid, however, when the load is increased from \(10.8\) to \(95.7\) g, \(\mu_k\) decreases from \(0.110\) to \(0.084\); in the case of friction of nickel surfaces lubricated with octyl alcohol, \(\mu_k\) decreases with increasing load.

For one class of solid surfaces and lubricating substances, Amontons’ law is thus justified; for another class of surfaces and lubricants, the coefficient of kinetic friction decreases as the load increases.

In the chapter on the static friction of lubricated surfaces (§ 15), we arrived at the conclusion that the fulfillment or non-fulfillment of Amontons’ law is connected only with the magnitude of the normal pressure between the rubbing surfaces. In the case of kinetic friction the picture proves more complex. In considering it, one should apparently take into account both the change in the state of the films of lubricating substances and possible changes in the state of the solid surfaces themselves during the process of friction.

As regards the dependence of the coefficient of friction on the elastic constants, noted above for the case of static friction both of “dry” and of lubricated surfaces (§§ 5, 17), the data reported by Bidrom and Bowden testify that, in the case of kinetic friction of lubricated surfaces, this dependence remains in force.

For a given lubricating substance, the numerical value of the coefficient of kinetic friction of glass is greater than the numerical value

coefficient \(\mu_k\) for nickel; the latter exceeds the coefficient of kinetic friction of steel surfaces.

In the case of kinetic friction of dissimilar lubricated surfaces, the coefficient of kinetic friction of substance \(A\) against substance \(B\), as in the case of static friction, has a value intermediate between the friction coefficients of pairs of homogeneous surfaces:

\[ (\mu_k)_{AA} < (\mu_k)_{AB} < (\mu_k)_{BB}. \]

At the same time, however, as in the case of kinetic friction of “dry” surfaces, the magnitude of the coefficient \((\mu_k)_{AB}\) proves to be different depending on which material the smaller of the rubbing surfaces is made of and which material the larger one is made of. Thus, for example, in the case of lubrication with octyl alcohol and hexadecane, the coefficient of kinetic friction of glass balls against a steel surface is greater than the coefficient of kinetic friction of steel balls against a glass surface; friction of nickel balls against a steel surface gives a larger value of \(\mu_k\) than friction of steel balls against a nickel surface, etc.

This gives us grounds to suppose that in this case as well the process of friction is accompanied by polishing of one of the rubbing surfaces.

§ 22. The Role of the Chemical Nature of the Lubricating Substance

In Figs. 17, 18, and 19 are given curves showing the dependence of the coefficient of kinetic friction \(\mu_k\) on the molecular weight of the lubricating substance \(M\) for the case of friction of steel and glass surfaces lubricated with paraffins, alcohols, and fatty acids\({}^{15}\). They show that, as in the case of static friction (§ 18), the coefficient \(\mu_k\) decreases as the molecular weight of the lubricating substance increases. The character of the dependence of \(\mu_k\) on \(M\), however, proves in this case to be considerably more complex,

Fig. 17.

Fig. 18.

Table 19

Lubricant	Length of carbon chain corresponding to \((\mu_k)_{\min}\): glass surface	Length of carbon chain corresponding to \((\mu_k)_{\min}\): steel surface	\((\mu_k)_{\min}\): glass surface	\((\mu_k)_{\min}\): steel surface
Paraffins	\(C_{22}\)	\(C_{14}\)	0.099	0.098
Alcohols	\(C_{16}\)	\(C_{8}\)	0.079	0.053
Acids	\(C_{11}\)	\(C_{6}\)	0.104	0.063

In the case of each of the homologous series (paraffins, alcohols, fatty acids), as \(M\) increases a certain minimum value of \(\mu_k\) is reached, after which, with further increase of \(M\), \(\mu_k\) no longer depends on \(M\). As follows from Table 19, the length of the carbon chain at which \(\mu_k\) reaches its minimum value is different for different lubricants and different solid surfaces. This table also gives the minimum values of \(\mu_k\). For a given solid surface, the smallest value of \(\mu_k\) is given by the alcohols.

Fig. 19.

Chapter V. Theories of Friction

§ 23. Theories of Friction

One of the first quantitative theories of friction of lubricated surfaces was Reynolds’ theory \(^{23}\), subsequently supplemented by Sommerfeld \(^{24}\). This theory is based on purely hydrodynamic conceptions of the layer of lubricating substance, treating it as a viscous medium. The nature of the solid surfaces themselves is ignored. Without dwelling on a detailed exposition and critique of such a point of view, we shall note only that it is of practical interest only in the case when the thickness of the lubricating layer is so great that it can be regarded as liquid, i.e. only under conditions of “complete” (“perfect”) lubrication.

Under the conditions of “boundary” lubrication (and likewise under “dry” friction), as is shown by the whole body of experimental facts cited above, the principal role is played by the physical processes occurring at the interface of the solid surfaces.

The coefficient of friction in this case does not depend on the viscosity of the lubricating substance, and the very concept of viscosity loses its usual meaning; the state of a thin oriented layer of lubricant could be classified as “solid.”1

The hydrodynamic Sommerfeld—Reynolds theory is completely inapplicable in this case.

The theory of “dry” friction and of friction under conditions of “boundary” lubrication must above all be based on taking into account the forces of interaction between the molecules of the solid surfaces and the molecules of the films of foreign substances adsorbed on them, and must also take into account the effect of orientation of the molecules of the lubricating layer.

There are, however, very few attempts to construct such a theory. Among them the works of Derjaguin[^25] and Tomlinson[^6] are of the greatest interest. Both of them, however, are not free from a number of shortcomings and require further improvement.

§ 24. Derjaguin’s Theory

In an article published in 1934, “Molecular Theory of Friction and Sliding,” Derjaguin[^25] sets as his principal task the proof of Amontons’ empirical law.

To this end he considers the relative sliding of two crystalline planes, basing himself on the mechanical model of a “saw.” In doing so he deliberately neglects the forces of attraction between the molecules of the sliding surfaces, regarding their absence as a necessary prerequisite for the fulfillment of Amontons’ law. The presence of repulsive forces, however, Derjaguin takes into account by imposing rigid constraints on the coordinates that determine the configurations of the molecules of the rubbing surfaces. As a result of considering the conditions of mechanical equilibrium of the sliding planes, Derjaguin does indeed succeed in obtaining Amontons’ law

\[ P=\mu F. \]

The theory proposed by him is, however, of a very formal character, not revealing the physical nature of the entire complex aggregate of phenomena accompanying the friction of solid surfaces.

Its principal shortcoming, as it seems to us, consists in the refusal to take the forces of attraction into account. Amontons’ law is indeed, as Derjaguin assumes, fulfilled only when there are some foreign coatings on the solid surfaces—lubricating substances or adsorbed gas films. This, however, by no means signifies a “withdrawal from the game” of the forces of molecular attraction.

If this were so, then the nature of the solid surfaces themselves would have no influence on the magnitude of the coefficient of friction; the latter would be completely determined by the properties of the lubricant. We know, however, that such conditions occur only in so-called “complete” or “perfect” lubrication, to which Deryagin’s theory again cannot be applied.

The whole totality of the facts set forth above—in particular the essential role of films adsorbed on the surfaces in the case of “dry” friction, the influence of the degree of orientation of the molecules of the lubricating layer in friction under conditions of “boundary” lubrication, and the dependence of the coefficient of static friction on the elastic constants of the rubbing bodies, a dependence which is also preserved in the case of friction of lubricated surfaces—all this undoubtedly testifies to the significant role of the forces of adhesion between the solid surfaces themselves and to the necessity of taking them into account in constructing a consistent molecular theory of friction.

Applying the results obtained by him to the phenomena of sliding in single crystals, Deryagin arrives at the conclusion that there is a relation between the plasticity of solids and the coefficients of static friction; in his opinion, the most plastic are those crystals for which the coefficient of static friction \(\mu_s\) has the smallest value.

A comparison of experimental data on the plasticity and static friction of certain solids shows that a relation between the plastic properties of solids and the coefficients of static friction does indeed exist; however, it proves to be the inverse of the relation whose existence Deryagin assumes1.

§ 25. Tomlinson’s Theory

a. Sliding Friction

Published in 1929, Tomlinson’s theory of friction[^6] remains to this day almost the only “molecular” theory of friction. Despite a number of shortcomings, it leads to the establishment of certain very substantial relations that are confirmed by experiment. We shall therefore dwell in detail on its exposition.

It should be noted that at the time this theory was created, physicists possessed very scanty data on the nature of the forces of molecular interaction; this circumstance affected, however, only the outward form of the theory, without being reflected in its fundamental propositions.

Tomlinson treats the contact between rubbing bodies as contact between individual pairs of molecules of these bodies. The applied

for bodies, the normal load \(F\) is balanced by the forces of molecular repulsion

\[ F = np, \tag{1} \]

where \(n\) is the number of pairs of molecules in contact, \(p\) is the mean value of the repulsive force of such a pair. The latter is here assumed to be independent of \(F\).

According to Tomlinson, the process of friction consists of the successive separation, from one another, of molecules in contact and the formation of new molecular contacts. In this, the work of the external force \(P\) over the path \(x\)

\[ Px = NW, \tag{2} \]

where \(W\) is the mean energy of separation from one another of a pair of molecules forming a contact, and \(N\) is the number of such “separations” over the path \(x\).

In the presence of \(n\) molecular contacts, \(N\) is equal to \(\dfrac{nx}{e}\), where \(e\) is the lattice constant; or, on the basis of relations (1) and (2),

\[ N = \frac{F}{pe}x \]

and

\[ P = \frac{W}{pe}F. \tag{3} \]

According to Amontons’ law, the external force is

\[ P = \mu F, \tag{4} \]

whence the coefficient of friction is

\[ \mu = \frac{W}{pe}. \tag{5} \]

All the quantities entering into this relation—\(W\), \(p\), and \(e\)—are molecular constants; molecular constants, Tomlinson concludes, are connected with the elastic constants of the given body, whence it follows that the coefficient of friction must be connected with the elastic constants of the body.

This conclusion of Tomlinson, as we know, is confirmed by experiment (§ 5).

Tomlinson establishes the quantitative relation between \(\mu\) and the elastic constants with the aid of the theory of elastic contact developed by Hertz. According to Hertz, the area of elastic contact \(a\) between bodies \(A\) and \(B\) is proportional to the quantity

\[ F^{\frac{2}{3}}\left(\vartheta_A + \vartheta_B\right)^{\frac{2}{3}}, \]

where \(F\) is the normal load, and \(\vartheta\) is a function of the elastic constants,

associated with the compressibility modulus \(K'\) and the shear modulus \(g\) by the relation

\[ \vartheta=\frac{3K'+4g}{g(3K'+g)}. \tag{6} \]

Tomlinson assumes that the number \(n\) of pairs of molecules in contact must be some function of the magnitude of the area of elastic contact,

\[ n=f(\alpha), \]

whence, according to relations (1) and (5),

\[ \mu=\frac{W}{eF}n=\frac{W}{eF}f(\alpha). \tag{7} \]

Further, Tomlinson regards \(F\) as a constant quantity and determines the character of the dependence of \(\mu\) on \((\vartheta_A+\vartheta_B)\) on the basis of experimental data. For the static-friction coefficients measured by him for 55 different pairs of solid surfaces, the ratio

\[ \frac{(\vartheta_A+\vartheta_B)^{\frac{2}{3}}}{\mu} \]

proved to be a constant quantity, i.e.

\[ \mu \sim (\vartheta_A+\vartheta_B)^{\frac{2}{3}}. \tag{8} \]

Table 20

Values of \(\left(\dfrac{\vartheta_A+\vartheta_B}{\mu}\right)^{\frac{2}{3}}\cdot 10^8\) according to Tomlinson

Surface material	Hard steel	Mild steel	Platinum	Nickel	Copper	Brass	Aluminum	Glass	Tin	Lead
Hard steel	5.55	5.47	5.97	5.32	5.20	5.71	5.38	6.12	5.19	5.23
Mild steel	5.47	5.64	5.69	5.44	5.43	6.14	5.80	5.20	5.20	5.36
Platinum	5.97	5.69	5.70	6.29	5.09	5.73	4.58	5.56	4.94	5.02
Nickel	5.32	5.44	6.29	6.03	5.18	6.19	4.81	4.85	4.63	4.81
Copper	5.20	5.43	5.09	5.18	5.70	5.88	5.80	6.24	5.35	5.44
Brass	5.71	6.14	5.73	6.19	5.88	6.02	5.98	5.03	6.02	5.10
Aluminum	5.38	5.89	4.58	4.81	5.80	5.98	4.94	5.65	5.65	5.51
Glass	6.12	5.20	5.55	4.85	6.24	5.03	5.65	5.24	5.61	4.60
Tin	4.19	5.29	4.94	4.63	5.35	6.02	5.65	5.61	5.06	5.10
Lead	5.23	5.36	5.02	4.81	5.44	5.10	5.51	4.60	5.10	4.78

Table 20 gives the values of

\[ \frac{(\vartheta_A+\vartheta_B)^{\frac{2}{3}}}{\mu}, \]

calculated by Tomlinson. The mean value of this ratio is \(5.47\cdot 10^8\); forty-two of the figures presented in the table deviate from the mean-

of its value within 10%. These small deviations are quite natural, since the coefficient of proportionality between \(\mu\) and \((\vartheta_A+\vartheta_B)^{\frac{2}{3}}\) depends on the ratio \(\dfrac{W}{e}\), which is different for different pairs of substances.

Relation (8) is in complete agreement with the connection noted above between the coefficient of friction of homogeneous surfaces and the elastic constants of the given body: in the case of friction of homogeneous surfaces [formula (6)]:

\[ \mu \sim \left[\frac{3K+4g}{g(3K+g)}\right]^{\frac{2}{3}}, \tag{9} \]

i.e., the coefficient of friction must decrease as the shear modulus \(g\) increases.

Relation (8) also explains why, in the case of friction of heterogeneous surfaces, the coefficient of friction has a value intermediate between the coefficients of friction of pairs of homogeneous surfaces.

The theory of Tomlinson deserves great credit for explaining these experimentally observed regularities.

Tomlinson, however, overlooked the following extremely important circumstance. Relation (8) means that the coefficient of friction \(\mu\) is directly proportional to the magnitude of the contact area \(a\), but the latter contains the factor \(F^{\frac{2}{3}}\), omitted by Tomlinson in his calculations.

Substituting the value of \(a\) into formula (7), we obtain

\[ \mu \sim \frac{W}{e}\frac{F^{\frac{2}{3}}}{F}(\vartheta_A+\vartheta_B)^{\frac{2}{3}}, \]

i.e.

\[ \mu \sim \frac{1}{\sqrt[3]{F}}. \tag{10} \]

This relation we shall henceforth call Tomlinson’s law.

Thus the application of the theory of elastic contact leads to a contradiction with Amontons’ law (4): the coefficient of friction \(\mu\), usually assumed to be constant, itself turns out to be a function of the normal load \(F\).

We shall dwell in detail on a possible interpretation of this, at first sight paradoxical, result in one of the following paragraphs.

6. Relation between the coefficients of sliding and rolling friction

Tomlinson also establishes a relation between the coefficient of sliding friction \(\mu\) and the coefficient of rolling friction \(\lambda\), and subjects it to experimental verification.

Rolling friction, just like sliding friction, is described by Tomlinson as a successive contact of pairs of molecules of the rubbing bodies, followed by their separation from one another.

If a cylinder of radius \(r\) and length \(b\) rolls on a plane under the action of an external force, then the work of this force over the path \(x\) is determined by the relation (2) given above,

\[ Px = NW, \]

where \(N\) and \(W\) have their former meaning. The force \(P\) is related to the load \(F\), per unit length of the cylinder, by the relation

\[ P = \lambda F, \tag{11} \]

where \(\lambda\) is the coefficient of rolling friction, while the number of successively separating pairs of molecules over the path \(x\) in this case, as is easily shown, is equal to

\[ N = \frac{3}{4}\frac{nx}{a} = \frac{3}{4}\frac{F}{pa}x, \tag{12} \]

where \(a\) is half the width of the area of contact between the cylinder and the plane. Relations (2), (11), and (12) give

\[ \lambda = \frac{3}{4}\frac{W}{ap}. \tag{13} \]

With the aid of formula (5) we find the relation between the coefficient of rolling friction \(\lambda\) and the coefficient of sliding friction \(\mu\):

\[ \frac{\lambda}{\mu} = \frac{3}{4}\frac{e}{a}. \tag{14} \]

The quantity \(a\) can be determined with the aid of Hertz’s formula for elastic contact between a cylinder and a plane:

\[ a^{2} = \frac{1}{\pi}Fr(\vartheta_A+\vartheta_B), \tag{15} \]

where \(r\) is the radius of the cylinder.

Tomlinson subjects relation (14) to experimental verification, determining \(\lambda\) experimentally for the case of a steel cylinder rolling on a steel plane.

Equation (15) in this case gives \(a = 4 \cdot 10^{-4}\ \text{cm}\); for \(e\), Tomlinson takes the lattice constant of iron, \(2.8 \cdot 10^{-8}\ \text{cm}\). Под-

Substitution into formula (14) of the value \(\mu_s\) for steel, quoted by Hardy, \(\mu_s = 0.79\), gives

\[ \lambda = 0.000041. \]

Taking \(\mu = 0.39\) (according to Tomlinson), we obtain

\[ \lambda = 0.000020. \]

The experimental value of \(\lambda\) is

\[ \lambda = 0.000044. \]

In view of the large scatter of the values of \(\mu_s\) determined for different grades of steel under different experimental conditions, this agreement may be regarded as quite satisfactory.

§ 26. On the question of the mechanism of friction

We have established that, both in the case of lubricated and in the case of “dry” surfaces, at small values of the normal pressure the coefficient of static friction decreases as the pressure increases,\(^1\) i.e. deviations from Amontons’ law occur. Starting from a certain value of the load, any further decrease in the coefficient of friction ceases, and Amontons’ law comes into force.

Up to the present time there has been no theoretical explanation of these basic regularities; the causes determining the validity of Amontons’ law remain just as unclear as the causes producing deviations from this law.

We shall attempt here to indicate possible ways of explaining the observed regularities. First of all, it seems to us, clarity should be brought to the question of the nature of the contact between the rubbing surfaces.

One may suppose that at small values of the pressure between the surfaces, the contact between them is elastic. It is precisely to this case that the Tomlinson theory set forth above should apply. This theory, as has already been mentioned, leads to contradictions with Amontons’ law, establishing an inverse proportionality between the coefficient of friction and the cube root of the normal pressure [formula (10), § 25]:

\[ \mu \sim \frac{1}{\sqrt[3]{F}}. \]

As our calculation showed for the curve of the dependence of \(\mu\) on \(F\), shown in Fig. 16, the product \(\mu \sqrt[3]{F}\) indeed remains constant.

The regularities observed at small pressures may thus be explained by the elastic character of the contact between the rubbing surfaces, which determines the validity of Tomlinson’s law (10).

\(^1\) See § 15 and the note to it, as well as §§ 20 and 21.

At high pressures, however, Amontons’ law comes into force. Its interpretation encounters considerably greater difficulties than the explanation of deviations from it. It is natural to suppose that, with increasing pressure, the character of the contact between solid surfaces changes. One may think that this change consists in the fact that, at high pressures, the contact between solid surfaces acquires a plastic character. The friction of bodies against one another in this case may be accompanied by flow of the surface layers of the substance itself.

For verifying the possibility of such a point of view, the following considerations are very important. The dependence of the coefficients of static friction on the elastic constants of solids, noted above, is observed both at low and at high pressures. The assumption of the plastic character of the contact can be reconciled with this circumstance only on condition that there is a definite correspondence between the elastic and plastic properties of solids.

A comparison of the plasticity of a number of solids with the elastic constants of these bodies shows that such a correspondence does in fact exist. Arranging the metals in order of increasing plasticity ¹) (see the hardening curves of Boas and Schmid ²⁶)

\[ \mathrm{Fe,\ Ni,\ Cu,\ Ag,\ Al,\ Au,\ Mg,\ Zn,\ Sn,\ Cd,} \]

we come to the conclusion that they are thereby arranged in the order of decreasing elastic constants and, consequently, in the order of increasing coefficients of static friction (for comparison see Tables 3 and 5).

It follows hence that the more plastic a body is, the greater its coefficient of static friction.

The question of the character of the contact between solid bodies is also closely connected with the question of the location of the plane of sliding of the bodies relative to one another. In the case of elastic contact it apparently lies at the boundary separating the solid bodies, between the films adsorbed on the surfaces. In the case of plastic contact it may also be located inside one of the solid bodies, namely, inside the more plastic body. In this case, as a result of friction, a layer of the second substance must remain on the less plastic surface. We encounter such cases at every step in everyday practice—the traces of pencil on paper, of chalk on a blackboard, etc.

A very substantial role must also be played by the circumstance that, as a result of the squeezing out of the lubricating substance or adsorbed films (in “dry” friction) from the space between solid surfaces, regions of direct “adhesion” (“chemical” contact) of the solid bodies to one another may appear on the latter.

¹) These data were kindly communicated to me by M. V. Klassen-Neklyudova.

The existence of such “adhesion” is indicated by the experiments described above on the study of “dry” friction in vacuum (§ 3), in particular by the fact that in vacuum the coefficient of friction of dissimilar surfaces depends on the affinity between the solid bodies.

It has not been proved, of course, that the existence of plastic contact could account for the fulfillment of Amontons’ law; nor is the role of “chemical” contact and of those changes in the relief of the surfaces that may occur during friction entirely clear.

The elucidation of these questions must be the subject of special investigations.

References

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1. For more detail on this see § 26 of this article. ↩↩↩↩↩