ELECTRICAL THEORY OF ADHESION (STICKING) OF FILMS TO SOLID SURFACES AND ITS EXPERIMENTAL JUSTIFICATION

B. Deryagin and N. Krotova

By the work of adhesion, or simply the adhesion \(A\), of liquid \(1\) to another liquid or to a solid body \(2\), one understands, as is known, the specific work (per unit area) of isothermal equilibrium separation of phases \(1\) and \(2\) in the presence of a third phase \(3\)—usually air. \(A\) can be determined by using the Dupré–Young equation:

\[ A=\sigma_{13}+\sigma_{23}-\sigma_{12}=\sigma_{13}(1+\cos\alpha), \tag{1} \]

where \(\sigma_{13}\), etc., are the surface tensions of the corresponding interfaces (equal to the specific work of formation of the corresponding surface), and \(\alpha\) is the contact angle. This method of measuring \(A\) is based, evidently, on the high mobility of the liquid particles, which ensures the formation of an equilibrium contact angle (in the absence of hysteresis of the latter). This method becomes, however, inapplicable if it is desired to measure the adhesion of a solid body or film, for example a high polymer, to the same solid substrate, which is of practical interest, for example for characterizing the strength of adhesion of protective films to metals. The problem of the strength of gluing is also connected with this same question. In the case of elastic, nonbrittle films, however, \(A\) can be determined directly by measuring the work of detachment of the film from unit surface of the substrate. A number of methods and instruments for this purpose were developed and used by one of us\(^1\). The scheme of operation of one of them is shown in Fig. 1. \(AA\) is a plate that can rotate about a horizontal axis \(O\). The lower end of the film \(pp\) deposited on the plate is pulled by a load \(P_0\), tending to detach the film from the plate. For a number of types of films, however, detachment practically does not occur until the angle \(\alpha\) between the plate and the vertical exceeds a certain value \(\alpha_0\), at which detachment does occur, although very slowly. Equating the work of the force of gravity acting on the load descending during detachment to the work of detachment, we obtain from elementary considerations:

\[ A_0=\frac{P_0}{b}\left(1+\cos\alpha_0\right), \tag{2} \]

where \(b\) is the width of the strip of film being peeled off. The similarity of this equation to Dupré’s equation (1) is not accidental and depends on the fact that both equations can be derived in the same way from the condition of equilibrium of the surface, in one case with tension \(\sigma\), in the other with \(\frac{P_0}{b}\), if in both cases, in order to find the condition of mechanical equilibrium, one applies to the process of reduction of the surface separating phases 1 and 2 the first principle of thermodynamics.

Fig. 1. Diagram of the method for measuring the adhesion of elastic films.

Experiments, however, showed1 that if peeling is carried out at an angle \(\alpha>\alpha_0\) or with a load \(P>P_0\), then peeling proceeds faster, but only an insignificant part of the excess work expended, \(A-A_0\), is converted into the kinetic energy of the load.

The question arises: why does faster peeling absorb a greater amount of work? Moreover, it was found that at \(\alpha<\alpha_0\) peeling also proceeds, although so slowly that its detection requires prolonged observation. A second question also arises: how is one to find the values \(\alpha_0\) and \(A_0\) corresponding to equilibrium, infinitely slow peeling, and is it at all possible in such experiments to realize a thermodynamically reversible peeling process?

In Fig. 2 we present, for several typical cases, adhesion diagrams showing \(A\)[^2] as a function of the peeling rate \(v=\frac{l}{\tau}\), where \(l\) is the length of the film being peeled off and \(\tau\) is the duration of peeling; in other words, \(v\) is measured by the rate of advance of the peeling boundary. The change in the peeling rate was achieved by changing the load \(P\) at a constant value \(\alpha_0=\frac{\pi}{2}\). The results obtained clearly show the thermodynamically irreversible character of the peeling process and argue in favor of a negative answer to the second question. At the same time, the numerical values of \(A\) observed experimentally prove to be so large, reaching at high \(v\) values of the order of \(10^5\) erg/cm\(^2\), that they rule out the possibility of reducing it to ordinary molecular interactions or forces of chemical affinity. Indeed, forces of the first kind, whose order of magnitude does not depend on the aggregate state, would lead to values of the order of \(10^2\)–\(10^3\) erg/cm\(^2\);

characteristic of the “equilibrium” adhesion of liquids. Moreover, it would be impossible to explain by the action of molecular forces the dependence of \(A\), and a very sharp one at that, on the rate of separation. It is likewise impossible to attribute the observed high values of \(A\) and their dependence on \(v\) to the energy of chemical bonding, quite apart from the fact that such an assumption is in most cases untenable on purely chemical grounds.

In view of the fact that the separation of a film is accompanied by a considerable deformation of the latter, the question arises whether the enormous difference in orders of magnitude between the work in rapid separation of a film and the adhesion of liquids is not explained exclusively by the fact that in the first case the work is expended mainly on deforming the film being separated. In other words, is not the “true” work of separation, obtained by subtracting the work of deformation from the value \(A\) calculated by formula (2), a small fraction of the latter quantity?

Fig. 2. Typical dependences of adhesion on the rate of film separation (adhesiograms). 1—gutta-percha–gelatin; 2—nitrocellulose–gelatin; 3—gutta-percha–zinc; 4—gutta-percha–nickel; 5—gutta-percha–steel.

It should first of all be noted that the value of \(A\), calculated by formula (2), does not include the work of stretching of the film by the load producing the separation, even in the case when the film is greatly elongated under the action of this load. This follows from the fact that, in deriving formula (2), the work of the falling load \(P\) is computed on the assumption that the film is inextensible.

Let us now consider the work of deformation in bending of the tape being separated. It is quite obvious that if its length is sufficiently great for the curvature of the lower end of the tape to be neglected, then further separation will not be accompanied by a change in potential energy, and if the bending deformation of the various portions of the tape, including the portion of maximum curvature, takes place thermodynamically reversibly (and consequently without elastic hysteresis and relaxation), then the total work of deformation is equal to zero. In order to estimate the influence of the irreversible component of the deformation work, let us note that each small portion of the separated tape, in the course of its separation, undergoes a single cyclic bending deformation—straightening. To estimate the energy loss \(W_1\) due to elastic hysteresis (including all kinds of irreversible expenditures

work during deformation), we shall proceed from the relation:

\[ W_1=\delta W_{\max}, \tag{3} \]

where \(W_{\max}\) is the potential energy of the maximum elastic deformation corresponding to the moment when the section under consideration acquires the maximum curvature, and \(\delta\) is the loss coefficient for irreversibility of deformation, which depends, generally speaking, on the deformation rate and increases with an increase in the magnitude of the maximum deformation itself. Both quantities \(W_{\max}\) and \(W_1\) are taken in the calculation per unit area of the strip being torn off, whose thickness is equal to \(2h\).

Fig. 3.

\(W_{\max}\) can be calculated quite exactly for the case of an ideally elastic film whose deformation does not go beyond the limits of applicability of the classical theory of linear elasticity, i.e., when, knowingly, \(W_1=0\). Although we are now considering the opposite case, nevertheless, having only the task of obtaining an approximate estimate of the magnitude of \(W_1\), we shall still apply the theory of bending of ideally elastic (thin) plates to determine the form of the strip being torn off and the value of \(W_{\max}\) (see the Appendix).

The theory leads to the following formula for the curvature \(\varepsilon=\frac{1}{R}\) of an elastic rectangular strip bent without twisting by a vertical load \(P\), suspended at its lower infinitely distant end, the tangent planes to the different sections of the strip being parallel to one and the same horizontal straight line (the boundary of separation), and the centerline lying in a vertical plane (the particular case of determining the form of an elastic line, or “elastica”):

\[ \varepsilon=\frac{d\Theta}{ds}=2\sqrt{\frac{P}{B}}\sin\frac{\Theta}{2}, \tag{4} \]

where (see Fig. 3) \(\Theta\) is the angle made by the given section of the strip with the vertical, \(s\) is the distance of the section under consideration along the arc of the strip from some fixed place, and \(B\) is the stiffness of the strip, equal to:

\[ B=\frac{\frac{2}{3}Ebh^3}{1-\chi^2}, \tag{5} \]

where \(E\) is Young’s modulus, and \(\chi\) is Poisson’s ratio.

The curvature, as is evident, increases continuously as one approaches the boundary of separation: only at very small distances from this boundary does the curvature begin to decrease as a result of the action of forces,

causing adhesion (for more detail see the addendum). In the case when detachment occurs from the lower surface of the horizontal plate \(\left(\theta=-\frac{\pi}{2}\right)\), the maximum curvature is equal to:

\[ \varepsilon=\sqrt{\frac{2P}{B}}=\sqrt{\frac{3P}{Ebh^3}}=\frac{1}{h}\sqrt{\frac{6\Delta l}{l}}, \tag{6} \]

where \(\frac{\Delta l}{l}\) is the relative elongation of the strip being detached under the action of the load \(P\).

This equality can also be derived in another way (see the addendum). To estimate the magnitude, let us calculate the maximum deformations of relative elongation and contraction, \(\pm\Delta\), which occur respectively near the upper and lower surfaces of the tape at the boundary of detachment. We obtain:

\[ \Delta=\frac{h}{R}=h\varepsilon=\sqrt{\frac{3P}{Ebh}}=\sqrt{\frac{6\Delta l}{l}}. \tag{7} \]

One may also find, for the maximum stresses associated with tension and compression due to bending, equal to \(\pm\Delta\), the value

\[ p_{\max}=\sqrt{\frac{3EP}{bh}}=E\sqrt{\frac{6\Delta l}{l}}, \tag{8} \]

or

\[ \frac{p_{\max}}{E}=\sqrt{\frac{6P_0}{E}}, \tag{9} \]

where \(P_0\) is the tensile stress under the action of the load.

In our experiments with gutta-percha, nitrocellulose, and other polymer films, \(\Delta\), calculated from (8), is usually considerably less than 1, which justifies the application of the theory of bending of “thin” plates.

At the same time, the values of \(p_{\max}\), calculated from (9), lie within the limits in which the corresponding materials deform without significant expenditure of work on hysteresis.* Thus, for \(\beta\)-gutta-percha \(2h\) was of the order of \(50\text{–}60\,\mu\), \(\frac{P}{b}\) did not exceed \(100\,\text{g}\), \(p_0 \cong 0.1\,\text{kg}/\text{mm}^2\).

Assuming \(E=2.5\), we obtain \(p_{\max}\cong 1.2\,\text{kg}/\text{mm}^2\), which corresponds to the region of reversible-elastic deformations (cf.²), when \(\delta\) should be small (at those rates of deformation which occurred during rapid detachment).

At the same time it becomes clear that reducing the increase in the work of detachment with speed to the influence of the period of cyclic deformation on elastic hysteresis or relaxation is fundamentally impossible, since with shortening of the cycle (with more rapid detachment) the losses due to hysteresis should not increase but decrease, since relaxation will not have time to occur.

Therefore, even at the largest values of the loads \(P\), the coefficient \(\delta\) could not be large. In those cases where excessively high values of \(p_{\max}\) were observed, by simply increasing the thickness of the strip being detached, \(2h\), it was possible to reduce \(p_0\), and consequently also \(\delta\), and to verify the small influence of the irreversibility of deformation.

Fig. 4. Detachment of surgical plaster from glass. ○—formation of “tails” at the boundary of detachment and their rupture. △—formation of tails with their subsequent lagging behind the substrate. ●—even detachment boundary, peeling of the film from glass.

Thus, the values of \(A\) calculated by formula (2) correspond, in the main, to the true work of detachment, and the correction for the work of deformation is of no substantial significance.

Calculations of the curvature of the strip near the detachment boundary will also be useful later in considering the action of forces that ensure adhesion.

In conclusion to this discussion, let us briefly consider the influence, on the work of detachment, of deformation during the detachment of films of certain “heavy” polymers, which can be observed in the detachment of films of some polymers (for example, natural and synthetic rubber, various synthetic resins, etc.).

Experiments by V. N. Yashin in our laboratory showed that, for such polymers as well, this phenomenon is not of a general nature, disappearing when the detachment rate is increased.

As an example we present an adhesion diagram of detachment, where the circles and triangles refer to experiments accompanied by the formation of tails, and the points to experiments not so accompanied. We see that the transition from one type of detachment to another is not accompanied by a break in the curve expressing the dependence \(A(v)\), and, moreover, the largest values of \(A\) are obtained precisely in the absence of “tails.”

The increase of \(A\) with decreasing film thickness deserves attention, both because of its practical interest and because it indicates that elastic phenomena can, on the contrary, reduce the work of detachment*). This occurs in those cases where the process of film formation (the sol–gel transition) is accompanied by the appearance of elastic stresses that are eliminated during detachment, and liber—

*) This is true for high detachment rates. For slow detachment, as will be shown below, the increase of \(A\) with thinning of the film is explained, within the framework of the electrical theory of adhesion developed, by an increase in the curvature of the tape near the detachment boundary.

excitation of the energy of these stresses is capable of reducing the work of detachment. This effect must increase sharply with the thickness of the film and, at a certain thickness, causes its spontaneous separation from the surface being coated. Conversely, at small thicknesses the effect disappears. It also decreases with increasing temperature and with the introduction of plasticizers, which promote timely relaxation of stresses during the process of film formation. A striking example is provided by the absence of adhesion of nitrocellulose to glass and gelatin at ordinary film thicknesses (tens of microns) and its sharp increase to values of the order of \(A \simeq 10^5 \ \mathrm{erg}/\mathrm{cm}^2\) at thicknesses of the order of several microns.

Thus, taking into account the work of deformation of the films being detached is not capable of explaining the high values of the work of detachment \(A\) and its velocity dependence.

As the only solution of this problem, a solution consistent with all, including the experimental facts described below, we put forward the following conception. When films are detached, the newly formed surfaces prove to be electrified with opposite charges, which may be the result of the separation, upon detachment, of the two plates of a molecular double electric layer. The question of the mechanism by which such a double layer arises, or by which electrification appears in the case of separation of two dielectrics or of a dielectric and a metal, has been little investigated, although it is extremely interesting and important. However, we shall not touch upon it here, especially since the very fact that charges appear upon separation or detachment of two phases is quite indisputable and at the same time is not new. In particular, the works of Coehn\(^5\) and others on electrification upon disruption of the contact of a liquid with a solid are well known.

There is no doubt, however, that the charge of the surfaces usually registered in such experiments constitutes a very small fraction of the initial \(\sigma_0\), which has time to dissipate during detachment.

From this point of view, the experiments of Gess\(^4\) are of interest; he measured electrometrically the charge obtained after the separation of two dielectrics (one with a flat surface, the other with a convex surface) that had previously been pressed together. The experiments were carried out in vacuum at a pressure of \(10^{-3}\) mm of mercury. The use of the formulas of Hertz’s theory of contact deformation made it possible to calculate the contact area and to determine the density of electrification \(\sigma\). The experiments showed its independence of the magnitude of the contact area. The values of \(\sigma\) for various dielectrics varied from 100 to 150 absolute electrostatic units. There is no doubt, however, that these values of \(\sigma\) are underestimated, mainly because, in the contact of two solid bodies with inevitable microroughness, the area of “actual,” intimate contact constitutes a small part of the area of apparent contact calculated according to Hertz.

In this respect, considerably better conditions of contact are obtained in the process of formation on the solid surface of a film from the initial liquid state (for example, a sol). However, experiments by one of us have shown that in this case too the density of electrification \(\sigma\) after detachment of the film under atmospheric conditions is small. At the same time it was found that it increases with increasing rate of detachment. This leads to the supposition that, in this case, detachment is accompanied by a discharge of one type or another, the more completely eliminating the initial charges the more slowly detachment takes place.

It is therefore natural to assume that before the discharge \(\sigma\) is so large that the interaction of opposite charges absorbs almost all the work of detachment. At the same time it is clear that, with decreasing detachment rate, the “discharge” has time, in the main, to be completed at a smaller separation of the surfaces, having absorbed a smaller amount of work; this also explains the decrease of \(A\) with falling \(v\).

Observations of detachment further show that, with increasing detachment rate, the latter is accompanied by luminescence (visible in the dark) and crackling*), indicating that here an actual discharge is plainly taking place. This phenomenon (by no means new) agrees with our explanation of the work of detachment and at the same time indicates the necessity of taking into account the different character of the discharge at small and large detachment rates.

Let us first consider the case of very large detachment rates. In this case the separated portions, electrified with opposite signs and forming a microcapacitor, acquire, as a result of the fall of capacitance with the gap thickness \(h\), so rapidly a discharge potential that the discharge occurs before \(\sigma_0\) has time to decrease appreciably owing to other causes. It is therefore obvious that the specific work \(A_0\) expended in detachment will be equal to

\[ A_0=\frac{2\pi\sigma_0^2}{D}\,h, \tag{10} \]

where \(h\) is the distance at which the discharge occurs (on the average), and \(D\) is the dielectric permittivity of the medium (in most cases, air: \(D=1\)).

Thus, at large \(v\), \(A\) should approach the limiting value determined by relation (10). To determine \(h\) we shall use Paschen’s similarity law for a gas discharge, which establishes in general form that the discharge potential \(V\) is a function of the “reduced” thickness \((ph)\) of the gas gap, where \(p\) is the pres—

*) These phenomena are more widespread than is usually thought and, for example, apparently explain the crackling when various fabrics are torn.

tion of the gas (in mm Hg). With sufficient accuracy this dependence is expressed by the equation:

\[ V=\frac{Bph}{C+\ln (ph)}, \tag{11} \]

where \(B\) and \(C\) are constants depending on the kind of gas*).

In Fig. 5 the curve expressing Paschen’s law is shown in the coordinates \(\lg (ph), \lg(V)\). Experiments with rapid detachment of films make it possible, expressing \(4\pi\sigma\) through the potential gradient \(V/h\), to write equation (10) (putting \(D=1\)) in the form:

\[ V^2=8\pi \frac{A_0}{p}(ph). \tag{12} \]

In the coordinates \(\lg V\), \(\lg(ph)\) this equation is expressed by a straight line with angular coefficient \(\frac{1}{2}\), whose position depends on the value of the parameter

\[ \lg\left[8\pi \frac{A_0}{p_0}\right]. \]

Fig. 5. Dependence of the discharge potential on the reduced gap thickness.

Experiments with gutta-percha for air \((p_0=760\ \text{mm Hg})\) give \(A_0=1.2\cdot 10^5\ \text{erg}/\text{cm}^2\). Drawing on the graph in Fig. 5 the corresponding straight line, from its point of intersection with the Paschen curve we find the values**):

\[ h \simeq 1.0\cdot 10^{-4}\ \text{cm}, \qquad V \simeq 5\cdot 10^3\ \text{V}, \qquad \frac{V}{h}=5\cdot 10^7\ \frac{\text{V}}{\text{cm}}, \]

\[ \sigma \simeq 1.3\cdot 10^4\ \text{CGSE}. \tag{13} \]

Having determined the value of \(\sigma\), one can graphically determine the work of detachment \(A\) at any other pressure \(p\). For this it is sufficient to draw on the graph (Fig. 5) the straight line corresponding to the equation

\[ \lg V=\lg\left(\frac{4\pi\sigma}{p}\right)+\lg(ph) \tag{14} \]

* The form of the Paschen curve depends somewhat on the metal of the electrodes. In our case, when at least one of the electrodes was always a dielectric, this influence is still greater. A second source of errors is the influence of the microrelief of the electrodes at small distances between them.

** Let us note that, according to some data, in calendering rubber mixtures potential differences of up to 6000 volts are observed\(^5\).

and having angular coefficient 1. The intersection of this straight line with the Paschen curve will give new values of the discharge potential and the reduced discharge distance \(ph\), and consequently of the gap \(h\) itself.

It is easy to see that lowering the pressure shifts this straight line upward, thereby increasing the discharge potential and, consequently, for a given value of \(\sigma\), also the work of separation

\[ A=\frac{\sigma V}{2}. \tag{15} \]

At the same time the discharge gap \(h_0\) also increases. One may also show in an analogous way, with the aid of straight line (14), that with increasing \(\sigma\), \(V\) and \(A\) increase, while \(h\) decreases.

With a decrease in the separation velocity \(v\), \(A\) must begin to decrease, since by the time of discharge \(\sigma\) will be smaller—owing to leakage of charge, which gradually weakens the attraction of the charges. Another reason for the dependence of \(A\) on \(v\) may be the statistical character of the discharge, the scattering of discharge potentials and gaps. It follows from this that, in a more exact treatment, one should speak of the probability of discharge per unit time\(^6\) as a function of \(V\) and \(ph\); since the probability of occurrence of a discharge during an elementary interval of time is proportional to it, then for slower separation the discharge, on the average, should occur at smaller gaps than for more rapid separation, as though the discharge lagged by some interval of time. It is clear that when the discharge occurs at a larger distance \(h_0\), the work expended is correspondingly increased.

Let us now consider the case of extremely small separation velocities. In this case the predominant importance will be gradual mutual neutralization of the charges due to autoelectronic emission, which should have quite appreciable magnitudes for the values of the potential gradient that we have found. The discharge current will be equal to

\[ -\frac{d\sigma}{dt}=f\left(\frac{V}{h}\right)=f(4\pi\sigma). \tag{16} \]

Equation (16) shows that \(\sigma\) decreases with time according to a definite law. Therefore \(\sigma\) decreases as the boundary of separation is receded from (see Fig. 6). The attractive force per unit area of the surfaces being separated during detachment is equal to

\[ F=2\pi\sigma^2 \tag{17} \]

and, consequently, is likewise some function \(F(\tau)\), independent of the separation velocity \(v\), of the time \(\tau\) elapsed from the beginning of separation. This remains true until \(v\) is so small that the surface has time to discharge by autoelectronic emission before the drop in capacitance, which causes an increase in the potential difference \(V\), can produce a gas discharge or (see below) cause appreciable leakage of charge along the surface (around the line of separation).

For the case when \(F\) is a function of time \(\tau\), using a special method of calculation (see the appendix), formula \((20')\) was found, which, after expressing \(F\) in terms of \(\sigma\), takes the form:

\[ A=\frac{3\pi^2}{2}\frac{(1-\chi^2)}{E}\frac{v^4}{h^3} \left[\int_0^\infty \sigma^2(t)\,t\,dt\right]^2 . \tag{18} \]

Thus, \(A\) becomes dependent on the “rigidity” of the film, being inversely proportional to it and, at the same time, increasing rapidly with \(v^4\). However, as \(v\) increases, the discharge of charges along the surface (around the boundary of rupture, as shown by the arrow in Fig. 6) may acquire greater significance, since with increasing \(h\), \(V\) increases faster than the surface resistance. If (for mean values of \(v\)) only this discharge mechanism is taken into account, then instead of (16) we obtain:

\[ -\frac{d\sigma}{dt}=L\sigma h, \tag{19} \]

where the constant \(L\) depends on the conductivity of both phases.

Fig. 6.

In this case \(F\) is not expressed directly either as a function of \(h\) or as a function of \(t\), as a result of which the calculation of the quantity \(A\) becomes somewhat more complicated mathematically and will not be considered by us here.

We see, therefore, that the theory leads to the following characteristic course of the curves of the dependence of \(A\) on \(v\), coinciding with experiment, namely: at small \(v\) the curve rises rapidly, being concave toward the ordinate axis. At mean values of \(v\), a bend of the curve is observed; at large \(v\) the curve approaches an asymptote parallel to the abscissa axis. In addition, in order to confirm the theory, experiments were carried out measuring the rate of detachment of a gutta-percha film under the action of various loads at different pressures of dry air. For this purpose an apparatus was used consisting of two glass drums: one (\(A\))—with divisions—onto which a gutta-percha film was applied, and a second (\(B\))—a guide drum—rotating on ball bearings and placed under the bell jar of an air pump (see Fig. 7).

The film applied to drum \(A\) was undercut and fastened to a tape passed over drum \(B\), and a detaching load was suspended from the lower end of the tape. With such an arrangement the boundary of detachment did not move in space. The length of the detached film was determined from the divisions of the first drum, and the time was record-

was measured with a stopwatch. The adhesion strength, characterized by the rate of detachment, was determined as a function of the load at a pressure of \(10^{-6}\) mm of mercury and at atmospheric pressure, and as a function of pressure at a specified load.

Fig. 7. Vacuum adhesiometer. \(A\)—upper drum, \(B\)—guide drum, \(N\)—ball bearings, \(S\)—film, \(C\)—trigger mechanism, \(P\)—load, \(L\)—handle.

The experiment (see Fig. 8) reveals, first of all, a sharp dependence of the adhesion strength on pressure: in vacuum (\(10^{-6}\) mm Hg) the rate of detachment at \(A=\mathrm{const.}\) falls by three orders of magnitude in comparison with the values obtained at atmospheric pressure (the air was dried with phosphorus anhydride).

Furthermore, the form of the curve expressing the dependence of the work of detachment on the rate, in a high vacuum of \(10^{-6}\), differs sharply from the curve obtained at atmospheric pressure. The former (Fig. 9) is characterized by concavity with respect to the ordinate axis over the whole range up to values of \(A\) corresponding to \(10^5\) erg/cm\(^2\), whereas the form of the second curve (as indicated above, see Fig. 2) at these values approaches a straight line parallel to the abscissa. All these experimental facts are consistent with the theory developed and undoubtedly indicate the electrostatic character of the detachment process. Electrical phenomena indicating high potentials had already earlier been noted by I. V. Obreimov\(^7\) in experiments on the splitting of mica along the cleavage plane. He also observed an increase in the work of splitting mica in high vacuum as compared with normal pressure.

In-figure labels: \(V\) cm/sec; gutta-percha—glass; \(P\), mm.

Fig. 8. Dependence of the rate of detachment of a film (at constant load) on the air pressure.

A number of other experimental facts are also consistent with the theory developed in the present communication. A characteristic feature of rapid detachment under constant load is the alternation of halts and “jumps,” while in detachment at constant speed on Polanyi’s apparatus one can record

strate quasi-periodic oscillations of the stress (see Fig. 10). The average amplitude of the oscillations depends on the speed, increasing together with it. The average amplitudes and frequencies of the oscillations also depend on the nature of the adhesive, the substrate, and the external medium. Thus, replacing air by water and by other (organic) liquids in which the given polymer does not swell almost eliminates the stress oscillations during peeling (and also reduces their average magnitude) (see Fig. 11).

All these phenomena indicate the role of electrical discharges and may be interpreted on this basis, though we shall not dwell on this here.

If peeling is carried out under irradiation by X-rays or by radioactive thorium \(D\), then in some cases the adhesion strength, characterized by the value reciprocal to the peeling rate, increases greatly; in other cases, on the contrary, it decreases.

Fig. 9. Dependence of film adhesion on the peeling rate in high vacuum.

In the plot: \(A\ \mathrm{erg/cm^2}\cdot 10^{-3}\); \(V\ \mathrm{cm/sec}\); gutta-percha—glass; vacuum \(10^{-6}\ \mathrm{mm\ Hg}\).

If the specimens are subjected to preliminary irradiation (with X-rays), but peeling is carried out under normal conditions, then the adhesion strength passes through a maximum as a function of the irradiation time (Fig. 12).

The effect persists for a short time; it disappears completely only after 15 hours of rest. In explaining the action of X-rays and gamma rays, it should be borne in mind that these ionizers may act in two ways. Acting on the gas gap, they reduce the time of de-

Fig. 10. Load oscillations during peeling at constant speed; lower curve—nitrocellulose—glass; middle curve—rubber—glass; upper curve—gutta-percha—glass.

In the plot: ordinate—“peeling load in grams”; abscissa—“length of peeled strip in mm.”

delaying the discharge (increasing its probability), which should lead to a decrease in the work of separation. At the same time they are a powerful factor causing electron emission at the surface of separation of two dielectrics.^8 An increase in the strength of adhesion under irradiation by X-rays may be explained by the fact that the double electric layer at the high-polymer—substrate boundary arises during the period of film formation, while the X-rays facilitate its completion.

Fig. 11. Character of the voltage oscillation during separation of rubber from glass in various media (on the Polanyi apparatus); \(a_1\) and \(a_2\) are the points corresponding to a change in the medium in which separation takes place. 1—separation in air, and after \(a_1\)—in ethyl alcohol; 2—separation in air, and after \(a_2\)—in water.

The reason for the decrease in adhesion strength under prolonged irradiation by X-rays and gamma-rays is still unclear.

In any case, the reversibility of the effect gives grounds to assert that the action of the radiation is not chemical in nature.

Let us note, in conclusion, that phenomena of electrification and discharge are also observed when a liquid is separated from a solid surface, for example in cavitation phenomena;^9 in the latter case the discharges may be the cause of destruction of the surface of propeller screws.

MATHEMATICAL APPENDIX

On the Shape of the Strip Being Peeled Off

Let us consider the conditions of equilibrium of an elastic strip being bent during peeling, of width \(b\), whose thickness is regarded as infinitely small, under the action of a load \(P\). Let (Fig. 3) \(ABCD\) be the “axis line,” or, in other words, the “elastic line” of the strip, situated in a vertical plane perpendicular to the plate \(ABA'\) from which peeling takes place. We shall denote by \(\theta_0\) the inclination of this plate to the vertical. Let \(s\) be the length of the “elastic line” from its left end to some arbitrary point \(C\). Let \(\theta\) be the angle made by the tangent to the elastic line at the point \(C\) with the vertical. For a sufficiently long peeled-off portion its lower end hangs vertically, i.e.,

\[ \text{as } s \to \infty,\qquad \theta \to 0. \]

At the point of peeling \(B\), for \(s=s_0\), \(\theta=\theta_0\).

Let us divide the peeled-off part of the strip into two sections: the first, throughout which the strip is still within the sphere of action of the adhesion forces, and the second—outside this sphere. Assuming (as will be justified below) that the radius of action of these forces is considerably smaller than the radius of curvature of the strip at its most curved point, we may suppose that for all points of the first section \(\theta\) is very close to \(\theta_0\), and that at the beginning of the second section, near the point of peeling \(B\), \(\theta=\theta_0\).

Fig. 12. Change in the adhesion strength (gutta-percha-glass) as a function of the time of irradiation by X-rays.

Let us consider the conditions of equilibrium of the segment of strip \(CD\), situated below the cross-section of the strip made at an arbitrary point \(C\) of the second of the above-considered sections. In doing so, we shall assume that the point \(D\) is located where the tangent to the elastic line is already almost vertical. Equating to zero the vertical projections of the forces acting on the segment \(CD\), we find that the interaction of the parts of the strip through the surface of the imaginary cut (apart from a certain bending moment \(M\), perpendicular to the plane of the drawing) reduces to a vertical resultant equal to the load \(P\).

Let us now consider the conditions of equilibrium of an element of length of the strip \(ds\) (between the neighboring points \(C\) and \(C'\)) under the influence of:

1) the bending moment \((-M)\), acting on the element of length under consideration from the side of the upper cross-section;

2) of the bending moment

\[ M+\frac{dM}{ds}\,ds, \]

acting (at the point \(C'\)) from the side of the lower section;

3) of the couple of forces with moment \(P\sin\theta\,ds\), formed by the forces \(P\) and \(-P\), acting in the vertical direction at the points \(C\) and \(C'\).

Equating to zero the algebraic sum of these three moments (with axes perpendicular to the plane of the drawing), we obtain the condition:

\[ \frac{dM}{ds}+P\sin\theta=0. \tag{1} \]

But from the elementary theory of bending it is known that

\[ M=-B\frac{d\theta}{ds}, \tag{2} \]

where \(B\) is the flexural rigidity, equal to

\[ B=\frac{2}{3}\frac{Eh^3}{(1-\varkappa^2)}, \tag{3} \]

where \(E\) is Young’s modulus, \(\varkappa\) is Poisson’s ratio, and \(2h\) is the thickness of the tape.

From (1) and (2) we obtain the equation determining the shape of the tape being torn off:

\[ B\frac{d^2\theta}{ds^2}-P\sin\theta=0, \tag{4} \]

the first integral of which, under the condition that \(\theta=\dfrac{d\theta}{ds}=0\) as \(s\to\infty\), is:

\[ \frac{B}{2}\left(\frac{d\theta}{ds}\right)^2+P(\cos\theta-1)=0. \tag{5} \]

From (5) we obtain for the curvature of the tape the formula

\[ \varepsilon=\frac{d\theta}{ds}=2\sqrt{\frac{P}{B}}\sin\frac{\theta}{2}, \tag{6} \]

which coincides with formula (4) of the main text.

Let us now consider the shape of the first segment of the strip being torn off, lying in the zone of action of the adhesive forces, which decrease to the right from the point of separation \(B\) according to a definite law:

\[ F=F(s), \tag{7} \]

where \(F\) is the adhesive force referred to unit area.

Let us formulate the equilibrium conditions for an element of length \(ds\) under the influence of the forces and moments acting on its ends and the adhesion forces acting on its upper surface. Projecting the forces onto the direction perpendicular to the plate \(AA'\), we obtain:

\[ \frac{dR}{ds}\,ds-F(s)\,ds=0, \tag{8} \]

where \(R\) is the corresponding projection of the force of interaction of the portions of the strip being detached, mentally separated by its transverse section. Integrating equation (8) from \(s\) to \(\infty\) and taking into account that, as

\[ s \to \infty \qquad F(s)\to 0,\qquad R\to P\sin\theta, \]

we obtain

\[ R=P\sin\theta_0-\int_s^\infty F(s)\,ds . \tag{9} \]

Equating to zero the sum of the moments acting on an element of length \(ds\), we now find instead of (1):

\[ \frac{dM}{ds}+P\sin\theta-\int_s^\infty F(s)\,ds=0 . \tag{10} \]

Instead of (2), for the portion of the strip under consideration we may write (in view of the smallness of \(\dfrac{dh}{ds}\)):

\[ M=B\frac{d^2h}{ds^2}. \tag{11} \]

From (10) and (11) there follows the equation:

\[ B\frac{d^4h}{ds^4}=-F(s). \tag{12} \]

To find the initial conditions necessary for integrating it, let us first consider a generalization of formula (5), which makes it possible to take into account the action of the adhesion force \(F(h)\). For this purpose we apply to the portion \(AB'\) the law of conservation of energy for a process in which the length of the detached strip increases by the amount \(\delta s\) owing to the peeling of the strip from the portion \(B_0B\) of the plate \(AA'\):

\[ \delta U=-M\delta\theta+Rdh-\delta A, \tag{13} \]

where \(\delta U\) denotes the increase, in this process, of the elastic bending energy of the portion \(AB'\) of the strip; \(\delta\theta\) is the change of the angle \(\theta\) at the point \(B'\) (as it passes into the new position \(B'_1\)); \(\delta h\) is the change of the distance \(h\), equal to the displacement \(B'-B'_1\); \(\delta A\) is the work expended against the adhesion forces acting to the left of the point \(B'\).

The first term on the right-hand side of equation (13) expresses the work of the bending moment, the second that of the force \(R\).

It is easy to see that if the length of the detached part of the tape is sufficiently large, and the tangent to it at the point \(D\) attains an asymptotic vertical position, then further detachment, lengthening only the vertical portion of the tape, cannot change the shape of the remaining curved part. The same result follows from the fact that, by virtue of the boundary conditions, the shape of the elastic line is determined [from equation (5)] on the assumption that \(s=\infty\).

In view of this, the calculations of the variations denoted in equation (13) by the symbol \(\delta\) may be carried out by displacing the point \(B'\) to the point \(B''\), distant from \(B'\) by \(\delta s\). In this case \(\delta U\) reduces to the elastic energy of the arc \(\delta s\), \(\delta A\) to the work against the adhesion forces expended in detaching this segment, starting from the initial position of contact with the plate \(AA'\) up to the present position, distant from the latter by \(h\), \(\delta \theta\) to the difference of the angles \(\theta\) at the ends of the segment \(B'B''\), equal in magnitude to \(\delta s\); \(\delta h\) may be represented in the form:

\[ \delta h=\frac{dh}{ds}\,\delta s. \tag{14} \]

Taking all this into account and using, in addition, the expression for the bending energy of a thin plate, equal per unit area to

\[ \frac{B}{2}\left(\frac{d\theta}{ds}\right)^2, \]

we obtain, instead of (13):

\[ \frac{1}{2}B\left(\frac{d\theta}{ds}\right)^2\delta s = \]

\[ = B\left(\frac{d\theta}{ds}\right)^2\delta s +\left[P\sin\theta_0-\int_s^\infty F(s)\,ds\right]\delta h -\int_0^h F(h)\,dh\,\delta s, \tag{15} \]

or, after reduction and taking account of (14),

\[ \frac{1}{2}B\left(\frac{d^2h}{ds^2}\right)^2 +\left[P\sin\theta_0-\int_s^\infty F(s)\,ds\right]\frac{dh}{ds} -\int_{s_0}^{s}F(s)\frac{dh}{ds}\,ds=0. \tag{15′} \]

Taking the point \(B'\) of the strip being detached at the junction of both portions of the tape considered above \((s=s')\), for which both \(F(s)\) and \(\left(\dfrac{dh}{ds}\right)\) are simultaneously very small, we obtain*) from (15′):

\[ \frac{1}{2}B\left(\frac{d^2h}{ds^2}\right)_{s'}^{2} =\int_{s'}^\infty F(s)\frac{dh}{ds}\,ds=A, \tag{16} \]

which follows from formula (5) and coincides with formula (6) of the main text for the special case \(\theta=\dfrac{\pi}{2}\).

Applying (15′) to the boundary of detachment (at the point \(B\)), where:

\[ \begin{aligned} s&=s_0,\\ h&=0,\\ \frac{dh}{ds}&=0, \end{aligned} \left. \begin{array}{l} \\ \\ \end{array} \right\} \tag{17} \]

*) For a special case, a similar relation was derived earlier by I. V. Obreimov\(^7\).

we obtain that in this case also:

\[ \frac{d^{2}h}{ds^{2}}=0, \tag{17'} \]

and from (10) and (11) it follows:

\[ B\left(\frac{d^{3}h}{ds^{3}}\right)_{s=s_{0}} = - P\sin\theta+\int_{s_{0}}^{\infty} F(s)\,ds. \tag{17''} \]

These conditions are sufficient for determining the shape of the strip in the zone of action of the adhesion forces.

Taking into account that, under the assumption made, \(\dfrac{dh}{ds}\ll 1\), the term with \(P\sin\theta_{0}\) is small in comparison with the term \(\int_{s_{0}}^{\infty} F(s)\,ds\), from (17'') and (12) we obtain:

\[ B\frac{d^{2}h}{ds^{2}} = (s-s_{0})\int_{s}^{\infty}F(s)\,ds + \int_{s_{0}}^{s}(s-s_{0})F(s)\,ds = \]

\[ = \int_{s_{0}}^{\infty}F(s)(s-s_{0})\,ds - \int_{s}^{\infty}F(y)(y-s)\,dy. \tag{18} \]

Hence it is clear that the curvature of the strip being detached, equal to zero at the boundary of detachment (at \(s=s_{0}\)), increases monotonically with increasing \(s\), reaching a maximum value equal to

\[ \max\left(\frac{d^{2}h}{ds^{2}}\right)_{s=s'} = \frac{1}{B}\int_{s_{0}}^{\infty}F(s)(s-s_{0})\,ds \tag{19} \]

at the value \(s=s'\), at which \(F(s)\) becomes vanishingly small, i.e., at the junction of the two portions of the strip being detached.

It follows from this that the maximum values of the curvature and elastic energy can be determined by formulas pertaining to the second portion of the tape, free from the direct action of the adhesion forces and applied to its extreme left point, where \(\theta \simeq \theta_{0}\).

In those cases when (as, for example, in slow detachment, according to the electrical theory of adhesion) \(F\) is a function of time and, consequently, may be regarded as a function of \(s-s_{0}=vt\) (where \(v\) is the detachment velocity, explicitly independent of \(h\)), the work of detachment should be calculated by formula (16) using relation (19). We then obtain:

\[ A= \frac{1}{2B} \left[ \int_{s_{0}}^{\infty}F(s)(s-s_{0})\,ds \right]^{2} = \frac{v^{4}}{2b} \left[ \int_{0}^{\infty}F(t)t\,dt \right]^{2} \tag{20} \]

or, substituting the value of \(B\),

\[ A= \frac{3v^{4}(1-\varkappa^{2})}{4Eh^{3}} \left[ \int_{0}^{\infty}F(t)t\,dt \right]^{2}. \tag{20'} \]

References Cited

1. B. V. Deryagin and S. M. Sorokin, Physicochemical Foundations of Printing Processes, Proceedings of the Research Institute of OGIZ, 1937, part II, p. 207.
2. Mark, Physics and Chemistry of Cellulose. Leningrad, ONTI, Khimteoretizdat, 1935, vol. I.
3. Coehn und Lotz, Phys. Zeits. 21, 327 (1920); Coehn und Curs, Zeits. f. Phys. 29, 186 (1924).
4. Hess, Zeits. f. Phys. 78, 117 (1932).
5. See B. A. Dogadkin, Physics and Chemistry of Rubber, p. 286, Goskhimizdat, 1947.
6. Zingerman, Zh. É.T.F. 15, 507 (1945).
7. Cf. the work of I. V. Obreimov, Proceed. Roy. Soc. (A), 127, 290 (1930).
8. Ya. Frenkel, Phil. Mag. 33, 297 (1917); Zeits. f. Physik, 51, 232 (1928).
9. V. A. Konstantinov, DAN, 56, 259 (1947).

1. Since the kinetic energy developed, even in comparatively rapid peeling, is very small, in all cases the work \(A\) can be calculated by formula (2). ↩