# Friction

Friction is a general interfacial property of surfaces in contact. Friction manifests itself as a force that resists the displacement of one surface relative to the other. The friction between two paper surfaces is often more important than the friction of paper against other materials such as metals. The requirements of paper-to-paper friction vary. Sometimes one needs high friction and sometimes low friction. Slippery paper with low friction probably causes more problems than high friction. Friction is a factor in many converting processes such as winding, calendering, printing, corrugating, etc.

In this article, we consider the general physical and chemical aspects of friction. As we shall see below, the friction properties of paper arise mainly from the chemical characteristics of the heterogeneous surface. Roughness properties would not seem to play a major role, since paper is for the purposes of friction quite compressible. This means that the variations in the effective contact area, a very important concept in the general understanding of the physics of friction, can be neglected. In some cases, it may be of importance that the sliding of a paper surface against another surface will make the surface fibres deform on time scales that may have relevance, for example, by giving rise to hysteresis 1. Measurement of paper friction is discussed more in detail in Surface properties.

## General physical and chemical effects

The following equation defines the frictional force, F:

(1)

where μ is the coefficient of friction, and
FN the normal force to the interface.

The body in Figure 1 remains at rest if the external force, P, is less than F. If P gradually increases, the body will eventually start sliding. The force, P, at the onset of sliding defines the coefficient of static friction: μ =μs = P/FN. The value of P required to keep the steady body in motion gives the coefficient of kinetic friction, μ = μk. The latter is almost invariably smaller than the former, μkμs. Figure 1. Forces acting in friction (a) and (b) definition of the friction angle, θf .

A friction angle, θf, which is the largest inclination angle of a surface before a body on the surface starts to slide, can also describe static friction, as shown in Figure 2. The coefficient of static friction obeys

(2)

Defining the coefficient of friction through normal force and not pressure is important. The apparent area of contact and the shape of the body therefore do not influence the friction force. In addition, the classical laws of dry or unlubricated friction state that μ is independent of the normal force, F, surface roughness, and sliding velocity 2. Actually, all these factors affect the friction of paper or of any material for that matter, but the classical laws are still useful as a first approximation.

In modern theory, friction results from two factors: the adhesion forces between the surfaces and the deformations of the interface during sliding. Adhesion is probably more important. Interfacial deformations are important in the transition from static to kinetic friction, although this is poorly understood. They affect friction because of energy dissipation. The effect is obvious if a hard, rough surface slides against a soft one and ploughs into the soft material. Energy is consumed in permanent deformations, when the surfaces become scratched and grooved, and even in elastic deformations. In the latter case, energy is lost in hysteresis particularly with visco-elastic materials like paper. It is unlikely that the recovered elastic energy could be fully used in overcoming friction. Based on recent advances in friction theory 3, the non-local deformations of the underlying fibre network below the rough surface and the visco-elastic response should play a role which has been hitherto neglected.

Forces between the molecules of the two surfaces cause adhesion. Since the range of these forces is very short, adhesive bonds are possible only in areas that come into molecular contact. At that scale, all surfaces are rough, and real contact occurs only at the surface asperities. These represent only a small fraction of the apparent contact area, and the local pressure may be very high. Assuming that τ is the shear strength of adhesive bonding and Ar is the area of real contact, the friction force, F, caused by adhesion is:

(3)

The bond shear strength, τ, depends on the materials. The contact area, Ar, depends on the topology and compressibility of the surfaces and on the normal force. For paper, it appears that the coefficient of friction does not depend on the apparent pressure 2,4-6.

If one assumes that the material is perfectly plastic at the asperities, the area of real contact is

(4)

where p0 is the yield pressure of the material that cannot be exceeded. Then

(5)

In plastic materials, the coefficient of friction should depend only on the shear strength of adhesion and yield pressure and not on the surface properties or apparent pressure. This result is in agreement with the classical laws of friction.

## Friction of paper surfaces

The coefficient of static or kinetic friction describes the friction of paper. As in most materials, paper μs is usually larger than μk. Only in some coated papers is μk>μs. The friction coefficients of paper-to-paper typically vary from 0.25 to 0.70, depending on the paper grade and its surface properties.

Measurement conditions have a large influence on the values of the paper-to-paper friction coefficient 7. The measurement system should be very rigid. Special care is necessary to avoid surface contamination. The result generally changes if the paper surfaces slide against one another before the actual measurement.

The friction of a paper-to-metal interface does not differ drastically from that of a paper-to-paper interface. The chemical condition and roughness of the metal surface are the key factors. When metal surfaces oxidise or gather dirt, the friction properties change accordingly. Usually, rough metal surfaces give higher friction than polished surfaces. This could result from the ploughing effect mentioned earlier. Paper-to-metal friction clearly decreases if the metal surface is hot. An increase in temperature from 20 °C to 100 °C halves the coefficient of friction 2.

Experiments suggest that adhesion is the main cause of paper friction. External variables, such as the contact pressure, sliding speed and surface roughness, are much less important. The strength of adhesion depends on the chemical nature of fibre surfaces. Wood extractives that occur naturally on fibre surfaces dramatically increase or decrease paper friction 6. The net effect is that friction increases markedly when the total amount of wood extractives decreases, as shown in Figure 2.

The roughness of paper has little effect on friction, as indicated by Figure 2 2,4,8. Calendering may increase or decrease friction by a small amount 2,6. For particularly rough papers, such as brown linerboards, calendering increases friction 2,7. The paper-to-paper friction sometimes also clearly depends on the relative orientation of the surfaces in contact, i.e., MD vs. CD. A difference may also exist between the top and wire side 4. High humidity increases paper friction. Possible explanations are that moisture increases surface compressibility 2 or increases the adhesion of cellulosic surfaces. Figure 2. Static coefficients of friction of various paper grades vs. PPS roughness before and after removing extractives 2.

Sizing can modify the surface properties of fibres. Sizing usually reduces friction, but friction can also be increased by choosing the right sizing agent 2,9. The coefficient of friction can also increase due to higher surface energy imparted through corona or flame treatment 2,6.

Coated papers usually have a coefficient of friction different from that of uncoated papers. The visco-elastic properties of coating are important for friction3. Colloidal silica coating increases friction. Increased surface energy, a stiffer paper surface and the ploughing effect are possible explanations for this 2,8.

Fillers may either reduce or increase μ. This depends on the morphology, surface area and porosity of the particles. Plate-like particles with a low surface area and porosity, such as kaolin and talc, reduce friction. More spherical particles with a high surface area and porosity, such as synthetic silicates, increase friction 6.

In some cases, surface strength may become a dominant factor for friction. If fibres and other particles break loose from the surface, friction usually decreases. Wear has a similar effect of reducing 2,5 μ but in some coated papers, μ increases with increasing wear 10.