PHYSICAL CHEMISTRY

A. T. SANZHAROVSKII and G. I. EPIFANOV

INVESTIGATION OF THE PROCESS OF FORMATION OF THE MECHANICAL PROPERTIES OF POLYMER COATINGS AND OF INTERNAL STRESSES IN THEM

(Presented by Academician P. A. Rebinder on 13 VII 1961)

In works (1) it was shown that the magnitude of the internal stresses arising in polymer coatings applied to solid substrates does not depend on the nature of these substrates. This indicates that the principal cause of the occurrence of internal stresses in a coating is the shrinkage processes taking place in it during formation. The role of the substrate, adhesively bonded to the coating, consists in preventing the free contraction of the coating during its hardening. Therefore, by studying the kinetics of formation of the mechanical properties of polymer films and the shrinkage phenomena occurring in them during hardening, it is possible to calculate the internal stresses that arise in a coating applied to a solid substrate. We have carried out an attempt at such a calculation using gelatin films as an example.

Let us suppose that a polymer coating dried from moisture content \(U_0\) to moisture content \(U\) has an instantaneous deformation modulus \(E_1\) and an equilibrium modulus of high-elastic deformation \(E_2\). As a result of shrinkage phenomena that occurred in the coating during its drying time \(t\), internal normal stresses \(\sigma\) arose in it. Let us subject such a coating to further drying for a time \(\Delta t\). During this time the moisture content of the coating will decrease by \(\Delta U\), and the linear shrinkage will increase by \(\Delta \varepsilon_y\):

\[ \Delta \varepsilon_y = \alpha \Delta U, \tag{1} \]

where \(\alpha\) is the coefficient of linear shrinkage.

The increase in linear shrinkage by \(\Delta \varepsilon_y\) will cause an increase in internal stresses by \(\Delta \sigma\), determined by the relation

\[ \Delta \sigma = \left(\Delta \varepsilon_y - \Delta \varepsilon_v\right) E_k, \tag{2} \]

where \(\Delta \varepsilon_v\) is the magnitude of the residual deformation that arose in the film during the time \(\Delta t\) under the action of internal stresses, and \(E_k\) is the apparent modulus of elasticity of the coating.

Assuming that the coating at the stage of formation is a Bingham body, the development of high-elastic deformation in which is described by the Kelvin–Voigt model (2), \(E_k\) and \(\Delta \varepsilon_v\) can be expressed by the equations:

\[ E_k = \frac{1}{ \dfrac{1}{E_1} + \dfrac{1}{E_2}\left(1 - e^{-\Delta t/\theta}\right) }; \tag{3} \]

\[ \Delta \varepsilon = \frac{\sigma - \sigma_0}{2\eta}\,\Delta t, \tag{4} \]

where \(\theta\) is the stress-relaxation period; \(\sigma_0\) is the yield limit of the coating; \(\eta\) is the viscosity of the coating.

Combining (2), (3), and (4), we obtain an equation determining the increase in internal stresses in the polymer coating:

\[ \Delta \sigma = \left( \Delta \varepsilon_y - \frac{\sigma - \sigma_0}{2\eta}\,\Delta t \right) \frac{1}{ \dfrac{1}{E_1} + \dfrac{1}{E_2}\left(1 - e^{-\Delta t/\theta}\right) }. \tag{5} \]

With very slow drying of the coating (as \(\Delta t \to \infty\)), the entire shrinkage \(\Delta \varepsilon_y\) is realized in the form of viscous flow, as a result of which at any stage of coating hardening \(\Delta \sigma\) becomes zero. In the case of instantaneous drying (as \(\Delta t \to 0\)), \(\Delta \sigma\) reaches a limiting value equal to:

\[ \Delta \sigma_{\mathrm{p}}=\Delta \varepsilon_y E_1 . \tag{6} \]

Fig. 1. Change in the relative shrinkage \(\varepsilon_y\), the modulus of elasticity of instantaneous deformation \(E_1\), and the apparent modulus \(E_k\) as functions of the moisture content of gelatin

During the formation of real films, which proceeds at a finite rate, \(\Delta \sigma\) has an intermediate value between 0 and \(\sigma_{\mathrm{p}}\).

Integrating equation (6) from the initial moisture content to the final one, we obtain the limiting stresses that would arise in the film if viscous flow and relaxation were absent in it. Integration of equation (5), however, makes it possible to find the actual stresses in a film forming at a finite rate.

Integration of equations (5) and (6), which we carried out by a numerical method, requires knowledge of the dependence of the shrinkage \(\Delta \varepsilon_y\), the modulus of instantaneous deformation \(E_1\), and the apparent modulus \(E_k\) of gelatin jellies on their moisture content. To obtain these dependences, a study was made of the mechanical properties of jellies of various concentrations and of the shrinkage phenomena accompanying the process of their drying. Fig. 1 shows the dependence of the relative linear shrinkage \(\varepsilon_y\) of jellies on their moisture content \(U\). As \(U\) decreases, the shrinkage increases (especially at the initial stage of drying), and for air-dry specimens containing 14–13% water it reaches a value of about 2.

In parallel with the study of shrinkage, the deformation properties of the jellies were investigated. For this purpose, specimens dried to various moisture contents were subjected to stepwise loading (as in (3)). Fig. 2 shows the curves of change in deformation of the specimens during loading, holding under load, and after unloading. From the magnitude of the deformation \(\varepsilon_1\), the modulus of instantaneous deformation \(E_1\) was calculated and its dependence on the moisture content of the specimens was plotted. Fig. 1 shows such a dependence, from which it is seen that a decrease in moisture content from 90 to 30% does produce a considerable increase in \(E_1\) (by a factor of 180), but its value still remains low (about 70 kgf/cm\(^2\)). With a further decrease in moisture content, the modulus increases sharply, and for air-dry gelatin it reaches a value of \(4.2\text{–}4.5 \cdot 10^4\) kgf/cm\(^2\). Specimens dried at 70–80° to constant weight have a modulus of the order of \(9 \cdot 10^4\) kgf/cm\(^2\).

Fig. 2. Deformation curves of gelatin jellies of different moisture content (\(a\)—18.9%, \(b\)—17.2%, \(v\)—13.6%)

Having the dependences \(\varepsilon_y=f(U)\) and \(E_1=f(U)\), one can calculate the limiting stresses \(\sigma_{\mathrm{p}}=f(U)\) arising in a gelatin film during its formation. Fig. 3 gives the dependence \(\sigma_{\mathrm{p}}=f(U)\). As can be seen, in the moisture-content interval from 90 to 30%, \(\sigma_{\mathrm{p}}\) remains low (about 7 kgf/cm\(^2\)),

which is quite natural, since \(E_1\) remains small in this humidity interval. A sharp increase in \(\sigma_{\mathrm{p}}\) occurs below a humidity of 20%, which is caused by the transition of the jelly from a highly elastic state to a glassy one. For air-dry specimens, \(\sigma_{\mathrm{p}}\) reaches values on the order of 500 kg/cm\(^2\).

From the deformation curves of gelatin jellies (Fig. 2), the apparent modulus \(E_{\mathrm{k}}\) was determined. Fig. 1 shows the dependence of \(E_{\mathrm{k}}\) on the humidity of the specimens \(U\). For jellies containing from 90 to 40% water, the highly elastic deformation is approximately 25–30% of the instantaneous deformation, as a result of which the difference between \(E_{\mathrm{k}}\) and \(E_1\) is comparatively small. At the same time, the value of the limiting stresses \(\sigma_{\mathrm{p}}\), which should arise in the coating at this stage of curing, proves to be higher than the yield point \(\sigma_0\). Therefore, flow develops intensively in the coating, leading in practice to the complete removal of internal stresses.

Fig. 3. Change in the limiting \(\sigma_{\mathrm{p}}\) and actual \(\sigma\) internal stresses as a function of coating humidity

The stresses \((\sigma)\) also remain low in the humidity interval from 40 to 20%, since in jellies of this concentration considerable highly elastic deformations develop, leading to a sharp decrease in the apparent modulus \(E_{\mathrm{k}}\) in comparison with \(E_1\). Finally, in the humidity interval of 20% and below, the jellies pass from the highly elastic to the glassy state. Viscous flow almost completely disappears, while the highly elastic deformation, developing mainly at the initial stage of load application, amounts to an insignificant fraction of the instantaneous deformation (not more than 20% at a specimen humidity of 17% and not more than 10% at a humidity of 13%).

In Fig. 3 the curve \(\sigma\) represents the dependence of internal stresses on humidity, calculated from the apparent modulus and shrinkage. These stresses should be close to the actual stresses arising in the coating during formation. For air-dry specimens these stresses are approximately 350–400 kg/cm\(^2\). Direct experiments on the study of internal stresses in gelatin films \((^1)\) showed that the magnitude of the internal stresses is 270–300 kg/cm\(^2\). If the planar stressed state is taken into account, the agreement of \(\sigma\) will be still better. The satisfactory agreement of the data presented shows that the mechanism of the occurrence of internal stresses at the stage of formation of polymer coatings is essentially exhausted by the development of shrinkage–relaxation processes.

Equation (5) and the experimental data make it possible to draw two important conclusions concerning the influence of the concentration of the solution from which the coating is obtained, and of the coating thickness, on the magnitude of the stresses in it. The lower the concentration of the solution from which the coating is applied, the greater the shrinkage it undergoes during drying. Therefore, with a decrease in the solution concentration, the stresses in the coating ought to increase. However, since over the entire concentration range up to the transition to the glassy state only very insignificant stresses can arise in the coatings, the concentration of the initial solution cannot exert any substantial influence on the value of \(\sigma\) in the formed coatings. This conclusion is well confirmed by direct experiments on the study of \(\sigma\) in gelatin films \((^1)\). The film thickness does not enter directly into equation (5). But with an increase in coating thickness, the time \(\Delta t\) during which the coating is at each stage increases.

stage of drying. Therefore, as the thickness of the coating increases, the internal stresses in it should generally decrease. Thus, for example, in coatings of plasticized gelatin, in which high-elastic deformation and viscous flow persist up to the air-dry state, the internal stresses do indeed decrease as the coating thickness increases (Table 1).

Table 1

In coatings of pure gelatin, the formation of internal stresses occurs almost completely in the process of gelation, as a result of which the role of viscous flow and relaxation is sharply reduced. Consequently, the value of $\sigma$ in such coatings does not change with changes in thickness (Table 1). A similar picture should also be observed for polymerizing coatings for which the polymerization time does not depend on the coating thickness. This is confirmed by experiments on the study of internal stresses in coatings of cold-curing polyester varnish, the results of which are given in Table 1.

In conclusion, we express our gratitude to Academician P. A. Rehbinder and to Professors G. M. Bartenev and P. I. Zubov for a number of valuable comments made during the discussion of this work.

Institute of Physical Chemistry
Academy of Sciences of the USSR

Received
26 VI 1961

REFERENCES CITED

1. A. T. Sanzharovskii, G. I. Epifanov, Vysokomolek. soed., 2, 1698 (1960).
2. S. A. Glikman, Introduction to the Physical Chemistry of High Polymers, Saratov, 1959.
3. P. A. Rehbinder, L. V. Ivanova, Kolloid. zhurn., 18, 682 (1956).