Integral adhesion curves

In order to find the characteristics of the random distribution, it is necessary to resort to experimental data. For this purpose, let us examine (for example) six parallel experiments with particles 30 pm in diameter. The integral adhesion curves for these six experiments are shown in Fig. 1.4. In order to determine the mathematical expectation, variance, correlation function, and normalized correlation function, it is necessary to select an interval of values for the function log Fad. In connection with the fact that the function otp changes quite smoothly, we will select an interval of 0.334 for the values of the function log Fad. Then, from the data of Fig. 1.4, we can determine the random values of the function log ad For each fixed value of log Fad, we obtain corresponding values of otp. Summing these values and dividing by the number of experiments, we obtain, in accordance with Eq. (1.30), the following values of the mathematical expectation ... 

Fig. IV.12. Integral adhesion curves for particles with following diameters (in fim) (1) 20 (2) 100. (Adhesive forces are in dynes.)...

Values of the median force can be obtained directly from integral adhesion curves. For this purpose it is necessary to draw a straight line parallel to the axis of the abscissas, corresponding to an adhesion number of 50%. The abscissa of the point of intersection of this straight line with the integral adhesion curves corresponds to the median force. 

The position of the point of intersection of the integral curves will depend on the properties of the specific system (particle-surface-ambient medium). In some cases, the integral adhesion curves for a specific range of particles may not intersect. Such a case has been found for particles with diameters of 30-80 jum with values of ap from 12 to 85% (see Fig. 1.3). Under these conditions, for all values of ap, the dependence of the forces of adhesion (including the median force) on particle size will be one and the same. 

Thus we see that, in the case in which the integral adhesion curves do not intersect and the value of the standard deviation is constant, i.e., o = const, the median force can provide an unambiguous characterization of the relationship between adhesive force and particle size. 

Average Force of Adhesion for Particles of Different Sizes. The average force of adhesion takes adhesion interaction into account and in independent of the adhesion number ap and of the position of the point K on the integral adhesion curves. Hence, it is a more objective criterion for the evaluation of adhesive interaction. 

Fig. V.11. Integral adhesion curves for cylindrical particles with a diameter of 20 jum, on a steel surface with a Class 6 finish, in air. Particle length, jum (1) 100 (2) 200 (3) 300 (4) 400. Adhesive force is given in dynes.

Features of Adhesion of Particles with Different Sizes, in Liquid Media. As noted previously, we find a log-normal distribution of particles with respect to force of adhesion in liquid media. The integral adhesion curves for particles of different sizes may intersect. Such an intersection does occur in the adhesion of spherical glass particles to glass surfaces, as indicated by the point K in Fig. 

Using integral adhesion curves similar to those shown in Fig. 1.3, we can find from Eq. (X.41) the velocity at the level of the particle center Vp, This velocity gives detachment of a part of the adherent particles, corresponding to a specific adhesion number. This relationship can be illustrated by the following results ... 

On comparing the adhesive force of the particles Fad calculated from Eq. (1.8) with experimental data relating to the detachment of a monolayer, it is easy to see that Fad corresponds to the force exerted by the most weakly held particles of the monolayer, i.e., to the initial part of the integral adhesion curves. Hence,when... 

The centrifugal method of measuring the value of the detaching force is the principal method used in determining forces of adhesion. The advantages of this method lie in its simplicity and accessibility, and also in the reliability of the results and the rapidity of the measurements. In addition to this, a variety of conditions may be created in the centrifuge test tubes (humidity, temperature, pressure, etc.), which widens the experimental potentialities of the method. However, in order to obtain the integral adhesion curve several measurements must be made with different numbers of revolutions. 

A method of dimensional modeling [90] based on the equidistance of integral adhesion curves, has also been used The essence of this method is that, by studying the known forces of adhesion between large particles, one may predict how small particles will behave, according to the particular properties of the bodies in contact. This method is only suitable in special cases. 

Fig. 1.4. Integral distribution curves for 30-jLtm diameter spherical particles on a glass surface (1-6), results of individual experiments (7), average value of adhesive force (in dynes).

As the surface quality worsens to class 10, the adhesion number falls subsequently, it rises again. The roughness of the substrate has practically no effect on the adhesion numbers of small particles for a small detaching force (curve 1) or of large particles (70 M in diameter) for a considerable detaching force (curve 3 ). In the first case, almost all the particles are held on the surface, and in the second almost all are removed. Thus, the roughness of the surface has no effect on adhesion for the two extreme points on the integral adhesive-force curves. 

In the second section the adhesive force exceeds the weight of the particles, and hence the number of particles remaining changes very little for a considerable change in the detaching force. With falling particle size the difference between the two sections of the integral adhesive-force curves becomes less marked. 

The constant of integration in Equation 4 was adjusted by using the average value, Tgxp, obtained from the receding and advancing adhesion tensions at c lne = 10 M. Then the calculated curve for the adhesion tension silica-aqueous solutions of LNBr is obtained (Figure 5,a). This curve verifies the experimental results satisfactorily. 

The measured shear stress-shear slip (v-v) curve of the adhesive layer as shown in Fig. 7 above was too cumbersome to be directly integrated into the simple calculation programme. It was first approximated by a 5th order polynomial as shown in equation (6). 

Evaluation of Adhesive Interaction by Means of Force. The dependence of adhesion number on the forces holding the particles of a monodisperse dust on a surface is commonly characterized by integral curves of adhesive force [14] (Fig. 1.2). The force of adhesion can be expressed in integral curves either in absolute quantities or in -units.f The force of adhesion ing -units, i.e., the ratio of adhesive force to the particle weight, has been termed the coefficient of adherence by G. I. Fuks [12]. 

The results of studies on the distribution of particles with respect to adhesive force are presented in the form of integral curves. These curves can also be represented on logarithmic probability coordinates (Fig. 1.3), with values of the... 

Thus we see that the parameters F and a enable us to determine the character of the particle distribution, to establish the form of the integral curves, and to show how the adhesive force varies with particle diameter also, values of these parameters are needed in determining F. ... 

By a comparison of the forces of adhesion of particles Fad as calculated from Eq. (1.42) with experimental data on the detachment of a monolayer, it is easy to establish that ad corresponds to the force of the most weakly held particles of the monolayer, i.e., the initial section of the integral curves for adhesive force (see Fig. 1.2). Consequently, in the detachment of a powder layer by tilting a dust-covered surface, we measure the average force of adhesion of the readily removable particles. As they sUde, these particles produce an avalanchelike removal of the remaining particles. If the force of adhesion of the layer to the substrate is greater than the autohesion in the layer, the detachment will take place across the weakest autohesive bonds. 

By the use of this method, the distribution of the dust particles with respect to adhesive force can be determined in a single experiment i.e., such an experiment will give an integral curve of adhesive forces. 

With certain values of ap that may vary from zero to a finite value, in most cases below 50%, and also in the case of large particles, we find a direct variation of adhesive force with particle diameter. This corresponds to Zone A of the integral curve for the distribution of adherent particles with respect to adhesive force (see Fig. IV. 12). 

In Fig. V. 13 we show integral curves of adhesion for spherical glass particles with a diameter of 20-40 [xm and for irregularly shaped particles of equivalent size. The distribution of irregularly shaped particles with respect to adhesive force, the same as the spherical particles, follows a log-normal law. 

VI. 14. In contrast to the situation in air (see Fig. IV. 12), the point of intersection of the integral curves for adhesion in liquid media lies at values of ap above 50%. Such a position of the point K indicates that for the majority of values of ap or adhesion number yp, the adhesive force varies directly with particle size. 

Fig. VI.14. Integral curves characterizing distribution of spherical glass particles with respect to adhesive force (in dyn), in aqueous medium. Particle diameter, jum (1) 20 (2) 100.

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