Structure and Elasticity of Rubber

L. R. Treloar*)

§ 1. Introduction

In its capacity for very large reversible deformations, natural rubber may be regarded as a representative of a large and rapidly growing family of substances of highly diverse chemical structure, some of which are given in Table 1. Not all of these

Table 1

Some rubber-like materials

*) Reports on Progress in Physics, vol. IX, p. 113–136, 1943; translated from the English and edited by A. G. Pasynkov.

substances possess high-elastic properties under normal conditions; for example, polystyrene and polymethyl methacrylate become elastic only upon heating, whereas at ordinary temperatures they are hard glassy substances, and gelatin becomes elastic only after swelling in water. For all types of rubber there are temperature limits to the high-elastic properties. The lower limit is associated with the transition to a glassy or crystalline state; the upper limit is associated with the transition to the state of a viscous liquid (unless chemical degradation intervenes). In a number of cases the upper limit can be raised by vulcanization (for example, for natural rubber); in other cases (for methacrylate) a shift of the high-elastic region toward lower temperatures is possible by introducing plasticizers.

In the present review we wish to try to show how the elastic and other states of rubber are connected with the properties of the molecule and with the arrangement of molecules in the material.

§ 2. Elastic Molecules

Most theories proposed to explain the elasticity of rubber linked this property directly with the structure of the molecule, so that the molecule was regarded as the fundamental elastic unit. But whereas earlier theories attributed elasticity to attractive forces between certain atoms or groups of atoms along the molecular chain, later theories explained this phenomenon by the thermal motion of the atoms of the chain. This fundamental change in point of view occurred about ten years ago and led to the clarification of the essential distinction between the rubber-like type of elasticity and the elasticity of a “solid” body of crystalline or glassy character. The kinetic theory of elasticity was first clearly expressed by Meyer, Susich, and Valko¹ and was developed mathematically by Kuhn²˒³ and Guth and Mark⁴. In the same period analogous ideas were developed by other investigators as well. For example, Griffith⁵ proposed a theory according to which molecules rotate, like a skipping rope, between the points of their mutual bonds; similarly, Mack⁶, although he explained elasticity by attractive forces between hydrogen atoms in a randomly tangled molecule, knew that the assumption that certain groups of atoms are capable of rotating around a simple C—C bond must lead to an unusual increase in the number of molecular configurations.

If we look at the molecular formulas given in Table 1, we shall in all cases notice a chain structure. This type of structure is characteristic of all kinds of rubber. The molecular weight of such chain molecules is invariably very high. For example, natural rubber has a molecular weight of about 300,000 (Gee⁷), which corresponds

chains of 20,000 carbon atoms. In addition, in all types of rubber the chains contain simple valence bonds, permitting comparatively free rotation of groups of atoms that make up the chain. This characteristic—a long chain structure containing simple bonds—is the most general feature inherent in the chemical structure of rubbers; other features of the molecule may vary widely. For example, the chain may or may not have side groups or double bonds (which do not permit rotation), the chain may be formed only of carbon atoms or of other atoms (for example, sulfur), or by an alternation of different atoms, etc.

Fig. 1. Rotation about bonds in a paraffin molecule.

Kuhn’s kinetic theory² idealized the molecule, abstracting from all its chemical properties and considering it simply as a chain of freely rotating bonds. This can be explained from Fig. 1, which represents a paraffin chain. If \(C_1\) and \(C_2\) are the first two carbon atoms, then the third atom \(C_3\) lies on the circumference of the base of a cone formed by the rotation of \(C_2C_3\) about the axis \(C_1C_2\), with the angle \(C_1C_2C_3\) being the valence angle \(\left(\text{equal to }109\frac{1}{2}^{\circ}\right)\). The fourth atom \(C_4\) similarly lies on the circumference of the base of a cone, whose generator \(C_3C_4\) moves around the first cone, etc. Thus there results a randomly entangled form of the molecule, in which the average distance between the ends of the chain is only a small part of the full length of the chain. If such a molecule is stretched and then released, the thermal motion of the chain atoms will tend to return it to the statistically most probable length. Thus, elasticity is a function of the statistical form of the free molecule and is due to the thermal motion of the chain atoms.

Some data on the subsequent development of the mathematical theory of elasticity will be considered below; for the present we have given only the general scheme of the elastic rubber molecule that is outlined by this theory.

§ 3. Experimental evidence for the kinetic theory

The kinetic theory directly leads to the following results³: 1) the reversible deformation of rubber is not accompanied, at constant temperature, by changes in internal energy; the work performed on the rubber is converted into heat, and as a result the entropy decreases upon stretching; 2) at constant

under extension, the elastic stress in rubber is proportional to the absolute temperature.

In both respects the behavior of rubber is entirely different from that of ordinary elastic solids. Therefore a check of any of these results may serve as a test of the correctness of the theory. In a careful study of the dependence of stress on temperature in vulcanized rubber, Meyer and Ferri^9 showed that the conclusions of the theory are confirmed, provided, of course, that the effects of irreversible crystallization and plastic flow are eliminated.

Figure 2 shows the proportionality between stress and temperature over a very wide temperature interval. Other

Fig. 2. Change of stress at constant length with absolute temperature for vulcanized rubber (Meyer and Ferri^9). a) Elongation 350%, experimental curve; b) Elongation 350%, curve corrected for thermal expansion; c) Elongation 170%.

investigators also found an increase of stress with increasing temperature, but it turned out that direct proportionality is not always observed. Deviations from this simple relation indicate the presence of changes in internal energy superimposed on the simple entropic effect. The most complete experimental investigation of such changes was apparently carried out by Wiegand and Snyder^10, who covered the whole range of extension of vulcanized rubber. Table 2 gives the relative changes in internal energy and entropy,

\[ \left( \frac{\partial U}{\partial l} \right)_T \quad \text{and} \quad T \left( \frac{\partial S}{\partial l} \right)_T , \]

per unit extension, calculated by us^11 on the basis of their data.

Table 2

Change in internal energy and entropy upon stretching of vulcanized rubber

Vigand and Snyder established the presence of three regions in the stretching curve. In the first region of the curve (stretching 0—350%) there occurs a decrease in entropy, accompanied by a comparatively small increase in internal energy, in approximate agreement with the simple theory; in the second region (stretching 400—700%) the internal energy decreases sharply and, finally, in the third region (\(>700\%\)) both terms—energy and entropy—become positive. It is clear that in the middle region the distortion is caused by crystallization (see § 5), which leads to changes in internal energy that obscure the changes directly caused by elastic stretching. In the third region rubber already ceases to behave like ordinary rubber and acquires properties analogous to those of a solid crystalline body.

Meyer\(^{12}\) showed that in the so-called inorganic rubber \((\mathrm{PNCl}_2)_n\), the stress at constant stretching increases with temperature, as in natural rubber.

Direct measurements of the heat liberated during stretching involve considerable experimental difficulties, and therefore there are no quantitative data suitable for testing the theory. Under isothermal stretching of rubber, the heat liberated should be equal to the work performed on the rubber. This reversible liberation of heat during stretching is generally known, but the heat of crystallization here also obscures the primary effect, as a result of which interpretation of the data becomes difficult.

§ 4. Molecular structure of rubber

Up to now the elasticity of the molecule has been considered without connection with the arrangement of the molecules and the intermolecular forces in the real material. Passing to an account of these factors, it is immediately clear that it is necessary to satisfy certain additional conditions in order that

so that the entire material would be elastic. On the basis of the works of Busse^13 and Treloar^14, we can formulate the principal conditions of rubber-like elasticity as follows.

1. The presence of chain molecules possessing bonds with free rotation.
2. Weak secondary forces between the molecules.
3. Entanglement of the molecules at certain points along the chain, with the formation of a three-dimensional “network.”

The first of these conditions has already been discussed. The second condition is necessary if the molecules are to possess the freedom of motion required by the kinetic theory of elasticity. The third condition follows from the general consideration that molecules under tension must be more or less firmly connected with one another, forming a continuous structure; however, these connections must not be so frequent as to prevent the free motion of the atoms of the chain.

Thus an “ideal” rubber forms a disordered network of long elastic molecules connected together by chemical bonds. The portions of the molecules between points of linkage move quite freely relative to their neighbors, like the molecules of a liquid. Well-vulcanized natural rubber at ordinary temperature approaches such an “ideal.” In this case the transverse bonds, which in general are assumed to be chemical bonds, are formed by the combination of rubber and sulfur. In unvulcanized rubber the transverse bonds cannot be so precisely defined, but it is quite probable that, as a result of complex local entanglements of the hydrocarbon chains, bonds of varying strength are formed (Treloar^14). Therefore unvulcanized rubber exhibits a complex dependence of its mechanical properties on time, owing to the effects of rupture and restoration of such bonds under tension, and also to plastic or irreversible deformation at high temperatures, whereas in vulcanized rubber these effects are much less noticeable.

§ 5. Crystallization

Crystallization has already been mentioned above in connection with phenomena of elasticity. Many works have been devoted to elucidating the mechanism of the crystallization process, and the information obtained sheds considerable light on the molecular nature of rubber.

It has long been known that unvulcanized rubber becomes hard and inextensible if it is kept for several days at 0° or at a lower temperature. It is also known that raw rubber, after being stretched to the limit, may under suitable conditions remain stretched for an indefinitely long time, but one can observe contraction to the original length,

if it is heated to a certain strictly defined temperature. It was established by X-ray methods that these effects are due to crystallization. Some, but not all, synthetic rubbers crystallize upon stretching. Rubber vulcanized at \(90^\circ\mathrm{C}\) crystallizes only at elongations above \(500\%^{15}\). In this case crystallization, though important, must be regarded as a secondary effect, only indirectly connected with the phenomenon of elasticity.

The molecular structure of unstretched and stretched rubber is shown schematically in Fig. 3. A rather similar structure was proposed by Mark\(^{16}\) for partially oriented cellulose fibers, with which rubber has a number of features in common. In unstretched crystalline rubber the crystallites are arranged at random.

(a) unstretched; (b) stretched

Fig. 3. Molecular structure of crystalline rubber (schematic). Parallel bundles represent crystallites.

These crystallites do not occupy the entire volume; owing to the presence of complex coils and portions of molecules that cannot fit into suitable lattices, there is always a noticeable fraction of amorphous rubber. In stretched crystalline rubber, the strong extension of the molecules favors crystallization; all crystallites are arranged approximately parallel, and the amount of amorphous rubber present is less than in unstretched crystalline rubber. From the breadth of the X-ray interferences the crystallite dimensions are estimated at approximately \(300\text{--}600\,\text{\AA}^{17}\), and since the average chain length is about \(20\,000\,\text{\AA}\), it follows that a molecule on average passes through a number of crystalline and amorphous regions.

The process of crystallization of unstretched rubber bears very little resemblance to the crystallization of low-molecular compounds. According to Bekkedahl\(^{18}\), it proceeds most rapidly at temperatures between \(-35\) and \(-15^\circ\mathrm{C}\), within which range the crystallization of raw rubber is completed after several hours. At \(-50^\circ\mathrm{C}\) crystallization is not observed at all. Similar observations were repeated by Meyer and Ferri\(^{9}\) and others. Vulcanization lowers the rate

crystallization, as is seen from the curves in Fig. 4, obtained by Bekkedahl and Wood\(^ {19}\), who used changes in density to study the course of crystallization.

Fig. 4. Dependence of the rate of crystallization (measured by the change in density) of vulcanized rubber on the amount of combined sulfur. Temperature 2°C (Bekkedahl and Wood\(^ {19}\)).

1. 0.0 and 0.1% S.
2. 0.20% S.
3. 0.30% S.
4. 0.35% S.
5. 0.40% S.
6. 0.43% S.
7. 0.46% S.
8. 0.50% S.

to investigate the course of crystallization. It is obvious that cross-links appreciably retard the rate of crystallization, although they have only a slight effect on the final degree of crystallinity. Moreover, the melting temperature of the crystals (\(\sim 14^\circ\)C), apparently, does not depend on vulcanization.

Fig. 5. Rate of crystallization of unvulcanized rubber at various elongations, according to measurements of double refraction. Temperature 0°C (Treloar\(^ {21}\)).

Gissen and Wittstadt\(^ {20}\) followed the crystallization changes in stretched vulcanized rubber by observing double refraction. This method, together with density measurements, was used by us\(^ {21}\) to investigate the rate of crystallization in raw rubber at 0°C and elongations from 34 to 870%. The optical data (Fig. 5) reveal a gradual change in the rate of crystallization, beginning with weakly stretched

of fully stretched rubber, and leave no doubt that the slow crystallization of unstretched rubber and the very rapid crystallization of strongly stretched rubber are, fundamentally, one and the same process. It turned out that the density changes in a manner quite analogous to the birefringence, and the final increase in density before freezing ranges from 2.24% for unstretched rubber to 3.06% for rubber stretched to 700%. For stretches above 100%, the maximum change in density is approximately proportional to the birefringence; whereas at stretches below 100% the birefringence is comparatively lower because of the imperfect orientation of the crystallites at small stretches. As a conclusion from these experiments (which, apparently, are confirmed by the X-ray experiments of Gehman and Field ^{22,43}), it follows that comparatively small elongations lead to a rather high degree of orientation of crystallites in stretched rubber. Extrapolation of the density data shows ^{21} that the change in density corresponding to the maximum stretch is 3.75%. Thus, the density of rubber crystals must be 4% higher than the density of amorphous rubber.

Fig. 6. X-ray determination of the content of crystallites in (a) unvulcanized and (b) vulcanized rubber (Field ^{15}).

The curves of Figs. 4 and 5 agree with the idea of the development of crystallites from nuclei, “which may simply be ordered neighboring sections of long chain molecules” ^{22}.

Field ^{15} determined from X-ray photographs the relative content of the crystalline and amorphous parts in stretched rubber. His data, partially presented in Fig. 6, show the presence of up to 80% crystalline substance at room temperature both for raw and for vulcanized rubber. This still does not characterize the highest attainable degree of crystallization, and comparison of the birefringence at 0 and 25°C ^{21} shows that, at the limiting stretch at 0°C, probably no less than 90% of the rubber is crystallized *).

) Complete crystallization is impossible because of steric hindrance: with a known degree of crystal growth along the amorphous portions of the chains, the crystallites hinder to such an extent the freedom of motion of the nearest amorphous segments that they cause their partial vitrification ^{44}. (Translator’s note.*)

In unstretched rubber the crystals melt, if not at a precisely definite temperature, then in any case in the region above \(10^\circ \mathrm{C}^{23}\). The melting temperature gradually rises with the time during which the rubber has been kept over a number of years. In this respect crystalline rubber also differs from crystalline low-molecular substances. The melting temperature of crystals in stretched raw rubber affects the temperature at which a given stretching produces a definite degree of crystallization \(^{14}\). In explaining effects of this kind, Olfrey and Mark \(^{24}\) indicate that the free energy of a crystallite, upon the successive attachment to it of segments of a disordered molecule, increases by a constant amount with each added segment, whereas the decrease in the free energy of the “amorphous,” or statistically entangled, part of the molecule becomes (as statistical theory shows) progressively smaller with each segment subtracted from its length. Crystallization continues up to the state in which the increase in the free energy of the crystalline component is compensated by the decrease in the free energy of the amorphous component. Therefore an increase in temperature does not lead to melting of the crystal as a whole, but merely shifts the point of equilibrium, helping a larger segment of the molecule to detach itself from the lattice.

Roughly, but more vividly, one may say that the uncrystallized part of the molecule always tends to pull the crystalline part of the same molecule away from the lattice. In the process of crystallization the amorphous part becomes more stretched, and finally a state is reached in which the resultant molecular stress due to the thermal motion of the amorphous part is compensated by the attractive forces in the lattice. It is assumed that, with time, some molecular coils unwind and the molecular stress relaxes, creating the possibility of further crystallization. To obtain a state of crystallization corresponding to the initial state, it is therefore necessary to raise the temperature.

It is possible that the high strength of vulcanized rubber is connected with crystallization under stretching, just as the strength of textile fibers is connected with the degree of orientation of their molecules \(^{25}\). Mark \(^{16}\) expresses the opinion that the crystalline elements determine the strength, while the amorphous elements determine the flexibility or extensibility. The influence of crystallization on the reduction of plastic flow in raw rubber, as is seen from Fig. 7 \(^{14}\), is very marked. Flow with stretching rapidly increases up to the beginning of crystallization, then, after reaching a maximum, falls. The influence of entanglement, which was considered as a factor of intermolecular linkage (§ 4), cannot, of course, be so strong as to eliminate flow at high extensions, but it is sufficiently great to preserve the structure until the onset of crystallization, which leads to the formation of new and stronger intermolecular bonds.

The opposite effect—an increase in the plasticity of a crystalline body owing to the presence of a high-molecular amorphous component—is manifested in materials of the polyethylene type. Polyethylene crystallizes almost immediately after cooling and has comparatively low extensibility, but it contains a fairly large amount of amorphous substance, which binds the crystallites to one another and gives the material flexibility and plasticity, in contrast to the chemically similar paraffin, which is characterized by brittleness and lack of flexibility.

Fig. 7. Plastic deformation in raw rubber over the course of 1 hour at the indicated extensions (Treloar 14).

A brief comment must be made about gutta-percha and balata. It is generally accepted that the molecules of these two kinds of rubber differ from the molecules of natural rubber, respectively, only as trans- and cis-isomers. Apparently, the trans-configuration promotes crystallization, since gutta-percha and balata crystallize readily at room temperature and melting of the crystals occurs only at about 60°C*).

Fig. 8. Change of volume with temperature for purified rubber, with a transition of the “second order” at −72°C and melting of crystals at 11°C (Bekkedahl 18).

It is also necessary to mention that the so-called β-anomaly in the heat-capacity curve of rubber 26 was recently explained by distortions caused by crystallization at −10—20°C 27.

§ 6. Transition to the glassy state

For our knowledge of the transition from the elastic to the glassy state, an important role was played by the work of Bekkedahl and his collaborators. As is seen from Fig. 8 18, this transition, which is expressed

*) Kuhn’s works 45 showed that, in the distinctions between rubber and gutta-percha, in addition to cis- and trans-isomerism, an important role is played by rotational isomerism, due to the presence of a rotational barrier. (Translator’s note.)

in the change of the coefficient of expansion, by approximately a factor of three, is common to amorphous and crystalline rubber. The transition temperature (\(-70^\circ\)C) is also manifested in changes in a number of other physical properties: the dielectric constant\(^{18}\), the heat capacity\(^{26,28}\), and the thermal conductivity\(^{29*}\).

This low-temperature transition, called a transition of the second kind, is associated with the transition of amorphous rubber from the elastic to the glassy state. It can hardly be doubted that the change in mechanical properties here is connected with the loss of freedom of molecular motion, when the relative freedom of rotation about valence bonds is lost and the atoms become, as in glass, relatively fixed in their positions. It is impossible to say precisely how this loss of freedom of molecular motion leads to changes in such physical properties as thermal expansion. The existence of a transition in crystalline rubber undoubtedly presents difficulties for explanation. O’Frey and Mark\(^{40}\) believe that the theory explaining the transition by a change in molecular rotation is confirmed by Moeller’s X-ray data\(^{30}\) on the rotation of molecules in paraffin crystals, but it should be noted that Moeller’s observations concern transitions near the melting point. Let us recall that crystalline rubber contains a considerable amount (\(\sim 30\%\)) of amorphous component, but if the transition effect is due simply to the residual amorphous component, it should be much smaller in the “crystalline” than in the amorphous material. In fact, the volume changes are almost identical, and although the break in the heat-capacity curve is sharper in amorphous than in “crystalline” rubber\(^{28}\), the difference is not very great.

*) Boyer and Spencer\(^{46}\), in their latest work, give the following table for some of the polymers they investigated:

where \(T_m\) is the transition temperature, and \(\beta_1\) and \(\beta_2\) are the coefficients of thermal expansion below and above \(T_m\); the last line is taken from the work of Robinson et al.\(^{47}\). Boyer and Spencer found that for two-component systems (polystyrene + polyisoprene) the curves \(T_m\) and \(\Delta\beta = \beta_2 - \beta_1\) lie between the corresponding values for the pure polymers; the copolymers show a monotonic change of \(T_m\) between the \(T_m\)’s of the pure polymers. The product \(\beta_2 T_m\) for almost all polymers (various rubbers, nylon, esters, and celluloses) lies within the range 0.12–0.20. (Translator’s note.)

STRUCTURE AND ELASTICITY OF RUBBER

A further difficulty is connected with the observation that the transition in thermal conductivity upon lowering the temperature sometimes occurs at a temperature lying 50°C below the equilibrium transition temperature[^29]. The glassy state is usually regarded as a nonequilibrium state in which the liquid structure is “fixed,” but it is difficult to admit that the transition to a supercooled liquid could be so delayed. This question requires further experimental investigation.

If one accepts the hypothesis that explains the transition into the glassy state by the cessation of rotation around a single bond, then the effect may be due to forces: 1) between groups of atoms within the molecule itself and 2) between the given molecule and its neighbors. It should be supposed that, in the case of readily crystallizing types of rubber (polyisoprene, polyethylene), the influence of intermolecular forces or “lattice” forces is predominant, whereas in the case of more complex molecules (polymethyl methacrylate) the energy barrier to internal rotation is rather high, and such internal “hindrance” may determine the low-temperature transition more strongly than the influence of intermolecular forces. This may explain the difference between polymethyl acrylate, which is elastic at room temperature, and polymethyl methacrylate, which becomes highly elastic at about 90°C[^31]. On the other hand, the transition temperature of many polymers, in the normal state of glassy character, can be lowered by tens of degrees by the introduction of “plasticizers” (or, more precisely, “elastifiers”). This is true, for example, for methyl methacrylate[^31], which raises doubts as to the correctness of the hypothesis of internal hindrance. The “elastifier” lowers the forces of intermolecular interaction, but it cannot affect the forces that restrict rotation in an individual molecule.

In other cases the transition temperature may be raised by the introduction of foreign molecules. This occurs when divinylbenzene is introduced into polystyrene and is explained by the formation of cross-links that reduce the mobility of the molecules[^32]. The increase in transition temperature during the vulcanization of rubber with sulfur is explained in the same way[^32], but in this case it is doubtful whether the formation of cross-links is the only, or even the most important, factor in the changes, since the simple addition of sulfur to the molecule, without the formation of bridges, should increase the intermolecular forces and thereby affect the transition temperature[^33].

Whatever the point of view regarding the relative importance of the various factors listed in the transition from the glassy to the elastic state, all agree that at the transition temperature the atoms of the molecular chain cease to be fixed, as in a solid body, and acquire a certain degree of mobility, comparable with the mobility of molecules in a liquid, but limited, of course, by their mutual connectedness as links of a single chain. As

and in the case of a liquid, one may consider that the atoms of the chain normally occupy potential “wells,” from which they jump out from time to time, receiving the necessary thermal energy.

Whether a substance will be highly elastic or not depends on whether, during the time of observation, the atoms receive an amount of energy sufficient for jumping out of the potential “wells.” It follows from this that a substance may be rubber-like under slow deformation and not possess this property under rapid deformation. This question was studied by Aleksandrov and Lazurkin^34, who investigated the relation between the amplitude of oscillations and the temperature in rubbers subjected to a variable compressive force whose frequency varied from 1 to 1000 oscillations per minute.

Fig. 9. Dependence of the deformation amplitude of vulcanized rubber (3% S) on temperature; the frequency of oscillations per minute is indicated beside the curves (Aleksandrov and Lazurkin^34).

In Fig. 9, which is typical of their results, it is seen that the transition temperature for vulcanized rubber rises by approximately 20°C when the oscillation frequency is increased from 1 to 1000 per min. These curves can be interpreted on the basis of the following equation:

\[ D = D_{\infty}\left(1 - e^{-t/\tau}\right), \]

which relates the deformation \(D\) to the time \(t\) of action of a constant force; \(D_{\infty}\) is the deformation at \(t=\infty\). In this equation \(\tau\) has the meaning of a relaxation time, or “orientation time” of the molecules, and is related to the potential barrier \(U\), which limits the change in the shape of the molecule, by the usual equation

\[ \tau = Ae^{U/kT}. \]

This relation represents the experimental data well. One may hope that determination of \(U\) for various materials will help to understand the processes of molecular orientation and viscous flow in polymers.

§ 7. Kinetic theory of the molecular network

Let us now return to the development of the kinetic theory of elasticity, already briefly considered in § 2. The distribution formula introduced by Guth and Mark^4 and Kuhn^2 gives the distribution of the lengths of a paraffin chain

\[ p(x,y,z)\,dx\,dy\,dz = \frac{\beta^3}{\pi^{3/2}} e^{-\beta^2(x^2+y^2+z^2)}\,dx\,dy\,dz, \tag{1} \]

where \(p(x,y,z)\) is the probability of a chain having components of length \(x, y\), and \(z\) along the three coordinate axes. In this equation

\[ \frac{1}{\beta^2}=\frac{2}{3}l_c^2 z \frac{1+\cos\theta}{1-\cos\theta}, \tag{1a} \]

where \(l_c\) is the length of the \(C—C\) bond, \(z\) is the number of bonds in the chain, and \((180^\circ-\theta)\) is the valence angle. In order to make the distribution function expressed by equation (1) clearer, imagine one end of the chain \(A\) of each molecule fixed at the origin \(O\), and consider the distribution of the coordinates of the other end \(B\). According to equation (1), the most probable position of \(B\) will be at \(x=y=z=0\), i.e. it is most probable that the end of the chain \(B\) coincides with its beginning \(A\).

The probability density decreases with distance from \(O\) along the curve in Fig. 10a, expressed by the function

\[ p(x)=\frac{\beta}{\sqrt{\pi}}e^{-\beta^2x^2}. \]

This does not mean that the most probable chain length, when all directions are considered, is zero. To verify this, denote the chain length by \(r=\sqrt{x^2+y^2+z^2}\) and determine the number of molecules for which the end \(B\) lies within the volume bounded by spheres of radii \(r\) and \(r+dr\). From equation (1) it follows \(^{2,53}\):

\[ p(r)\,dr=\frac{\beta^3}{\pi^{3/2}}\cdot 4\pi r^2 e^{-\beta^2 r^2}\,dr, \tag{2} \]

Fig. 10. Theoretical distribution of molecular lengths.

a) Component of length in a specified direction

\[ p(x)=\frac{\beta}{\sqrt{\pi}}e^{-\beta^2x^2}, \]

b) Total length in any direction

\[ p(r)=\frac{4\beta^3}{\sqrt{\pi}}r^2e^{-\beta^2r^2}. \]

where \(p(r)\) is equal to zero at \(r=0\) and reaches a maximum at \(r=\frac{1}{\beta}\) (Fig. 10b). Equations (1) and (2) correspond to those equations that occur in the kinetic theory of gases: 1) for the distribution of velocity components in a given direction and 2) for the distribution of the total velocity (taking all directions into account \(^{35}\)).

§ 8. Kuhn’s Work

To simplify the problem of the elasticity of a molecular network, Kuhn made the following assumptions:

1) all molecules have the same chain length;
2) the distribution of the components of the molecular length is expressed by equation (1);
3) under stretching, the increase in the components of the length of each molecule occurs in the same ratio as the components of the length of the macroscopic rubber;
4) the volume of the rubber remains constant under deformation.

Kuhn first finds an expression relating the entropy \(S\) of an individual molecule to the components of its length; on the basis of Boltzmann’s equation \(S = k \ln p\) and equation (1), one obtains

\[ S = c_1 - k\beta^2(x^2 + y^2 + z^2), \tag{3} \]

where \(c_1\) is a constant. Multiplying \(S\) by the number of molecules having length components \((x,y,z)\) according to equation (1), and integrating, one can find an expression for the total entropy \(S\) of an aggregate of \(N\) molecules in the unstretched state. If the specimen is stretched in the \(x\)-direction by a fraction \(\gamma\), then the distribution of the molecular lengths in the stretched state [taking assumptions (3) and (4) into account] will be as follows:

\[ p'(x,y,z)\,dx\,dy\,dz = \frac{\beta^3}{\pi^{3/2}} e^{-\beta^2[x^2(1+\gamma)^2+(y^2+z^2)(1+\gamma)]} \,dx\,dy\,dz. \tag{4} \]

The total entropy \(S'\), corresponding to the stretched state, is obtained from equation (4) by substituting it into Boltzmann’s equation in place of equation (1). Neglecting powers of \(\gamma\) higher than the second, we obtain the following approximate expression for the change in entropy upon stretching:

\[ S' - S = -\frac{3}{2}Nk\gamma^2. \tag{5} \]

According to Kuhn, this is not yet the full change in entropy upon stretching. Equation (5) represents only a partial change in entropy caused by the change in \(r_1\)—the distance between the ends of the molecule. But since the molecule has not only a “length” \(r_1\), but also a “width” \(r_2\) and a “thickness” \(r_3\), the fractions of the total entropy determined by the values of \(r_2\) and \(r_3\) must be added to equation (5). This additional change in entropy upon stretching is equal to \(-Nk\gamma^2\) for each of the values \(r_2\) and \(r_3\), whence

\[ S' - S = -\frac{7}{2}Nk\gamma^2. \tag{5a} \]

Applying the thermodynamic relation between the force \(F\) and the entropy \(S\),

\[ F=\left(\frac{dw}{dl}\right)_T=-T\left(\frac{dS}{dl}\right)_T, \tag{6} \]

Kuhn obtained a linear relation for the deformation curve:

\[ F=7NkT\gamma. \tag{7} \]

In this form, however, Kuhn’s work is subject to criticism. From Wall’s work (see § 9) it follows\({}^{36}\) that the introduction of the values \(r_2\) and \(r_3\) into the expression for the change of entropy upon stretching is incorrect. In considering the stretching of a molecule, one of its ends is assumed to be fixed, and the other to be located in the volume element \(dx\,dy\,dz\). The entropy is connected with the probability that the end of the molecule is located in this volume element, which, in turn, is determined by the number of possible configurations of the molecular chain compatible with the given values of \(x\), \(y\), and \(z\). In counting these configurations, all values of \(r_2\) and \(r_3\) are automatically taken into account, and therefore the calculation of their independent contribution to the total entropy is superfluous. Therefore the approximate expression for the change of entropy should be taken to be equation (5), and not equation (5a)*).

We have also shown\({}^{36}\) that if Kuhn’s simplification of neglecting powers of \(\gamma\) higher than \(\gamma^2\) is not introduced into the calculation of the change in entropy upon stretching, then the resulting exact form of the deformation curve will no longer be linear, but will be determined by the relation:

\[ F=NkT\left(a-\frac{1}{a^2}\right), \tag{8} \]

where \(a=(1+\gamma)\) expresses the ratio of the lengths in the \(x\)-direction after and before stretching.

This correction leads to a result that agrees with Wall’s conclusions, obtained by another method. Let us now consider Wall’s method.

§ 9. Wall’s Work

Wall\({}^{37,50}\), in his theory, uses the same four basic assumptions from which Kuhn also proceeds. The distribution of lengths (i.e., distances between the ends) of a system of \(N_0\) identical molecules in the undeformed state is assumed to be given by Kuhn’s equation (1). It is also assumed that elongation (or compression), in which the length changes in the ratio \(a:1\), changes the length components of all molecules in the same ratio in which the \(x\)-, \(y\)-, and \(z\)-dimensions of the rubber specimen change, so that the distribution of molecular lengths in the deformed state is described by equation (4), if \(a\)

*) On the role of the energy barrier in rotation, see\({}^{48,49}\).

substituted instead of \(1+\gamma\). Wall calculates the probability \(P\) of the distribution of molecular lengths from equation (4), if the probability that a given molecule has components of length \((x,y,z)\) is determined by equation (1); in the same way he determines the probability \(P_0\) of the most probable (equilibrium) distribution. For the relative probability in the stretched and unstretched states he obtains the expression

\[ \ln \frac{P}{P_0}=-\frac{N_0}{2}\left(a^2+\frac{2}{a}-3\right). \]

The change in entropy accompanying the deformation is easily obtained from this relation, if one takes into account that \(S=k\ln P\). For the stretching curve of a specimen with an initial cross-sectional area of \(1\ \mathrm{cm}^2\), this gives

\[ F=NkT\left(a-\frac{1}{a^2}\right)= -\frac{\rho RT}{M}\left(a-\frac{1}{a^2}\right). \tag{9} \]

In this equation, which is identical with Kuhn’s corrected equation, \(F\) is the force per \(1\ \mathrm{cm}^2\) (referred to the unstretched cross-section), \(\rho\) is the density, and \(M\) is the molecular weight, which here denotes the weight between the junction points of the network. It should be noted that equation (9) is equally applicable both to stretching and to uniaxial compression. In Fig. 11 the curve represented by equation (9) is shown.

Fig. 11. Theoretical force–deformation curve for a molecular network at \(M=15\,000\) and \(T=298^\circ\mathrm{K}\).
(Axis labels: force \((\mathrm{kg}/\mathrm{cm}^2)\); \(a(=L/L_0)\). Curve regions: “compression,” “extension.”)

Shear deformation was also considered by Wall,³⁷ with the method differing only in details from the preceding one.

If \(\sigma\) is the magnitude of the shear, then for the elastic energy the following result is obtained:

\[ W=\frac{1}{2}NkT\sigma^2=\frac{1}{2}G\sigma^2, \tag{10} \]

where \(G\) is the modulus of elasticity. Under shear, therefore, the dependence of deformation on stress is linear, whereas under extension or uniaxial compression it is nonlinear. Equation (10) can also be derived by means of Kuhn’s basic method.³⁶

The Wall approach surpasses Kuhn’s method in that it does not contain poorly substantiated assumptions about the value of the entropy of an individual molecule [see equation (3)].

It is curious that equations (9) and (10) contain only one molecular constant \(M\)—the molecular weight between junction points. Therefore the formulas should be applicable not only to a network of paraffin chains, but also to any rubber in which the distribution of molecular lengths is expressed by a formula of the Kuhn type, for example, to natural rubber, in which not all C—C bonds permit rotation. It is important, however, to recall the initial assumptions. First of all, the distribution function (1) according to Kuhn is applicable only as long as the molecule has not been stretched to its full length. Equations (9) and (10), based on equation (1), cannot be applicable if the deformation (extension, compression, or shear) is so large that the principal parts of the molecule are completely, or almost completely, stretched out.

Secondly, it is obvious that the assumption of equal length for all molecules is not satisfied even approximately. When there is a distribution of chain lengths, the shorter chains will be completely extended already at comparatively small extensions, which must further restrict the range of applicability of Wall’s equation. Nevertheless, despite these difficulties, equation (9) gives a sufficiently satisfactory expression for the relation between stress and deformation in vulcanized rubber (see below).

§ 10. Other attempts to solve the problem of the molecular network

The methods of Kuhn and Wall are not the only methods that have been applied to the solution of the problem of the elastic network. They were described first, and in somewhat greater detail, only because in them the general principles of the approach and the nature of the difficulties that arise are expressed most clearly.

Pelzer1 noted that if the entropy of an individual molecule is expressed as a function only of the \(x\)-coordinate (which corresponds to stretching in the \(x\)-direction), applying the formula \(S = k \ln p\) to equation (1), then the corresponding stress \(f\) can be obtained:

\[ f = 2kT\beta^{2}x, \]

which vanishes only when \(x = 0\). This is, obviously, inapplicable to a piece of rubber. Therefore Pelzer proposes that one should start from the distribution function for the \(r\)-values [equation (2)]. This gives for \(f\)

\[ f = -kT \frac{d}{dr}[\ln p(r)] = kT\left(2\beta^{2}r - \frac{2}{r}\right). \]

The stress in this case vanishes at \(r_0=\frac{1}{\beta}\). Denoting \(r/r_0=\alpha\), we obtain for the deformation of a single molecule

\[ f=2\beta kT\left(\alpha-\frac{1}{\alpha}\right). \]

The magnitude of the stress under deformation of rubber as a whole is assumed to be simply equal to \(Nf\), where \(N\) is the number of molecules.

The resulting curve does not differ appreciably in form from Wall’s curve [equation (9)], but Peltzer’s reasoning is, evidently, unsatisfactory and arbitrary. It leads to the incorrect conclusion that a molecule with a length smaller than the most probable length \(\frac{1}{\beta}\) must exert a compressive force in the direction of its length, whose magnitude must become infinite at \(r=0\). Since, according to equation (1), when the end \(A\) of the molecule is at the origin, the probability density for the other end \(B\) is maximal when \(B\) coincides with \(A\), it is clear that the only force which may legitimately be regarded as existing between the ends of a simple molecule is an elastic force that vanishes at \(r=0\). If at a given moment the end \(B\) is at a distance \(r\) smaller than the most probable distance \(r_0\), then at the next moment, although it is more probable that the distance from the origin will increase, there will be no preference in direction, and therefore the motion of the end \(B\) cannot be regarded as radial, as it would be under the action of a repulsive force from the center. Therefore equation (2) cannot serve as a basis for calculating the entropy of a single molecule. In general, the concept of the entropy of a single molecule contains so many ambiguities that it is better to avoid it.

Guth and James\(^{39,51}\) also approached the problem of a network of molecular chains by using the assumption of constancy of volume under stretching. Denoting by \(L_x\), \(L_y\), and \(L_z\) the side lengths of a specimen initially having the form of a cube with edge \(1\ \mathrm{cm}\), one may therefore write

\[ L_y=L_z=\sqrt{\frac{1}{L_x}}. \]

The probability that a piece of rubber has dimensions \(L_x\), \(L_y\), and \(L_z\) is obtained as follows (slightly changing the notation):

\[ p\cdot dL_x\,dL_y\,dL_z=Ae^{-\beta^2\left(L_x^2+L_y^2+L_z^2\right)}dL_x\,dL_y\,dL_z, \tag{11} \]

which, after substituting the relation between \(L_x\), \(L_y\), and \(L_z\) and multiplying by the number of molecules \(N\), leads to the following form of the deformation curve:

\[ F=2NkT\beta^2\left(L_x^2-\frac{1}{L_x}\right). \tag{12} \]

Guth and James thus anticipate the general form of Wall’s equation (9). Their method, however, is unsatisfactory in that equation (11) represents the probability of a molecule having components of length \(L_x\), \(L_y\), and \(L_z\), and there is no justification for the possibility of using it to express the probability of these lengths for an entire rubber specimen*).

§ 11. Comparison of the theory with experiment

Guth and James\(^{39}\) showed that a theoretical equation of type (9) agrees well with the experimental data of Meyer and Ferri\(^{9}\) on the elongation of rubber and with the data of Shepard and Clapson\(^{41}\) on the uniaxial compression of rubber, if, however, different \(M\)’s are assumed for the two branches of the curve (extension and compression). This was to be expected, since different rubber specimens were used. However, in the work of Shepard and Clapson\(^{41}\) data are also given on the extension of the same rubber specimens that were tested in the compression experiments, and it would seem more correct to compare the data from one group of experiments with the theoretical formula. Suitable data for such a comparison are presented in Fig. 12, on which \(\lg F\) (compressive or tensile force in \(\mathrm{kg}/\mathrm{cm}^2\) based on the original cross section) is plotted against \(\lg \alpha\) (a logarithmic scale is necessary in order to represent the very wide range of \(F\)). For large extensions or compressions, equation (9) approximately has the form

\[ F = \frac{\rho R T}{M}\alpha \]

and

\[ -F = \frac{\rho R T}{M}\cdot \frac{1}{\alpha^2} \]

(shown in Fig. 12 by straight lines with slopes 1 and 2). The intersections \(A\) and \(A'\) of these lines with the \(\lg F\) axis give the values of \(\frac{\rho R T}{M}\). As can be seen, Wall’s equation represents the experimental data very well, especially on the compression branch, but it is still necessary to adopt different values of \(M\) (13,000 and 8,400) to represent both branches. The values of \(M\)

*) Flory\(^{52}\) points out that in the Guth–James formula (12) the dimensions are incorrect, since \(N\) denotes in equation (12) the number of chains crossing a unit area of cross section, whereas \(N\) is the number of chains per unit volume.

Flory\(^{52}\) himself considered the elastic deformation of a network on the basis of the deformation of an “average cell” of the network, formed by four segments of chains between a given node and four nearest nodes. Flory arrived at the same equation for elastic deformation as was obtained by Wall and by Kuhn–Treloar. In numerical calculations of the elastic deformation of vulcanized rubber, besides the nodes specified by the chemical bonds formed during vulcanization, it proved necessary to take into account, to some extent, the nodes of chain entanglement and the nodes between parts of the same chain. Flory came to the interesting conclusion that the number of nodes determining mechanical deformation coincides with the number of nodes determining the restricted swelling of vulcanized rubber, while the modulus of elasticity of the swelling rubber proves to be approximately inversely proportional to the \(5/3\) power of the relative magnitude of swelling in the given solvent. (Translator’s note.)

have the expected order of magnitude, but it is difficult to understand why the effective molecular weight under extension should be 50% greater than under compression. Still, to obtain such an order of agreement when $\alpha$ changes by up to 300 times is not so bad, especially considering the very general character of the theoretical assumptions.

It should be noted that Sheppard and Clapson’s data on compression were obtained not by direct compression of rubber, but by biaxial extension through the inflation of a balloon. The necessity for such an indirect

Visible labels in the figure: $\lg \alpha$; $\lg(-f)$ (kg/cm$^2$); $\lg F$ (kg/cm$^2$); Compression; Extension; Experiment (Sh–K); Wall ($M=8{,}400$); Wall ($M=13{,}000$).

Fig. 12. Comparison of the theoretical force–deformation curve with the experimental data of Sheppard and Clapson on the extension and compression of vulcanized rubber.

method is due to the fact that a piece of rubber can be compressed only to a small extent (for example, up to 50%) in order to avoid serious complications from bulging of the specimen, which assumes a barrel-like shape. Taking the volume during deformation to be constant, biaxial extension in the $yz$ plane may be regarded as equivalent to uniaxial compression in the $x$-direction, and in this way a much greater decrease in the $x$-dimension can be obtained (down to $1/40$ of the initial value).

The considerable rise of $F$ above the theoretical curve under extensions or compressions near the breaking point can be explained by the ordering of molecules near their full extension, which is not taken into account in the theoretical treatment. From the extremely high crystallinity observed in rubber under linear extension, it follows that in fully stretched rubber the molecules lie almost entirely in the direction of extension. Similarly, under compression or biaxial extension, it is reasonable to suppose that near the breaking point almost

all the molecules lie completely in the plane of stretching. Therefore, if rupture does not occur, each of the two branches of the deformation curve must asymptotically approach a vertical line representing the limiting extension of the network. Although this condition is not attained, the approach to it is beyond doubt, and, assuming that at the point of rupture the network is fully stretched, this extension may be compared with the theoretical maximum extension.

§ 12. Maximum extensibility of the molecular network

a) Linear extension. Assuming that in a fully stretched network each molecule has its greatest possible length \(l_m\), one can easily calculate the extensibility. If, in the unstretched network, \(x_1, x_2, x_3,\ldots\) are the numerical values of the \(x\)-component of the length of an individual molecule, then after stretching they become equal to \(l_m\). The maximum extensibility in the \(x\)-direction may therefore be taken as equal to \(l_m/\overline{x}\), where \(\overline{x}\) is the mean value of \(x_1, x_2\). From equation (1)

\[ \overline{x}=\frac{1}{\beta\sqrt{\pi}}. \]

For a chain of \(z\) links of length \(l_c\), connected without restrictions created by valence angles, the value \(l_m = zl_c\), whence the relation is obtained

\[ \frac{1}{\beta^2}=\frac{2}{3}l_c^2 z, \]

and the maximum extension is equal to

\[ \left(\frac{3}{2}\pi z\right)^{\frac{1}{2}}=2.17\sqrt{z}. \]

Let us now consider the meaning attributed to the \(z\)-effective number of freely rotating bonds in a rubber chain.

Fig. 13 shows a part of a rubber chain containing two isoprene residues, where \(C_1—C_8\) are the carbon atoms of the chain. The system \(C_1C_2C_3C_4\) is rigid; it is equivalent to a bond of length 3.09 Å (taking the usual bond lengths and angles), freely rotating about the axis \(C_3—C_4\). The nearest long bond \(C_5C_8\) has the same motion relative to \(C_4C_5\) as does \(C_1C_4\). The comparatively short bond \(C_4C_5\) therefore acts like a hinge between the bonds \(C_1C_4\) and \(C_5C_8\), so that the rubber chain as a result becomes similar to a system of hingedly connected isoprene residues. The isoprene residue has molecular weight 68, whence for \(M=13\,000\) the approximate value is \(z=13\,000/68\), or 191, and the maximum linear extension is determined as \(a=30\); this is almost four times greater than the actual extensibility (\(a=7.6\)).

b) Biaxial extension. The maximum biaxial extension can be calculated in an analogous way. If \(r_1, r_2, \ldots\) are the projections of the molecular lengths in the \(yz\) plane, perpendicular to the direction of compression, then upon complete extension of the network all these values approach \(l_m\). The maximum extension (defined by the change in the length of a line in the \(yz\) plane) is therefore equal to \(\dfrac{l_m}{\bar r}\), where \(\bar r\) is the mean value of \(r_1, r_2, \ldots\). The distribution of the \(r\)-values is approximately given by a function analogous to equation (2):

\[ p(r)\,dr=\frac{\beta^2}{\pi}\cdot 2\pi r e^{-\beta^2 r^2}\,dr, \]

whence \(\bar r=\dfrac{\sqrt{\pi}}{2\beta}\), and the maximum linear extension is equal to

\[ \left(\frac{6z}{\pi}\right)^{\frac12}=1.38. \]

If \(a\) is the length, in the \(x\)-direction, of a line whose initial length is unity, then the linear extension in the \(yz\) plane is equal to \(\dfrac{1}{\sqrt a}\) (since the volume is constant). The linear extension corresponding to the smallest value of \(a\) reported by Sheppard and Clapson (\(a=0.0257\)) is therefore \(6.25\). Taking \(M=8400\), we find \(z=123\), and the maximum theoretical extension is \(15.3\), which is 2.5 times greater than the experimental data.

The discrepancy between the experimental values of extensibility in both cases and the extensibility calculated on the basis of molecular weights satisfying Wall’s equation proves to be considerable. Various causes may be put forward to explain this discrepancy. First, it is obvious that rupture may occur before complete extension of the network, as a result of which the true extensibility must always be less than the theoretical one, although such a large discrepancy can hardly be expected. Secondly, the theory assumes that all molecules have the same chain length, whereas in practice there must be a broad distribution of chain lengths; however, it is difficult to estimate the influence of this circumstance on extensibility. Thirdly, and probably most importantly, a source of errors may lie in the application of Kuhn’s formula to the distribution of molecular lengths, which forms the basis of Wall’s theory. The derivation of Kuhn’s formula contains assumptions that cannot be applied exactly to real molecules. These include the assumptions that: 1) each bond of the chain has complete freedom of rotation, so that all configurations are considered to have the same energy, and 2) the volume of the chain may be neglected, so that the motion of one part of the chain encounters no obstacles from another. With regard to (1), it should be noted[^42] that the energy barrier restricting rotation about a single bond is usually such that certain positions near an angle of \(120^\circ\)

are occupied most readily. If the three energy barriers have the same height, then the probability of occupation of each position will be the same, and the resulting distribution of configurations will be almost the same as in free rotation (provided only that the height of the barriers does not stop rotation altogether). But if the potential barriers have different magnitudes, then some positions will be more probable than others, and a considerable deviation from a random distribution may result. Each of the indicated effects probably causes an increase in the average chain length, i.e., lowers the extensibility. The analysis of experimental data given above shows that the actual average length of the rubber chain must be 2.5–4 times greater than the average length calculated from the statistical computation.

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1. Pelzer. ↩