Origin of Rubber Elasticity

Abstract

Under suitable conditions, virtually all rubbery materials exhibit the ability to sustain deformations followed by complete recovery upon removal of the stress. This phenomenon holds significance beyond the narrow confines of the term “rubber elasticity” This elasticity theory is also of great importance in the deformation of any substances., e.g., in the deformation of amorphous or semi-crystalline polymers. Rubber elasticity is also essential to the functions of elastic proteins and muscles. Thus, the theory of rubber elasticity is centrally essential to much of polymer science. In this chapter, we have touched upon some of the basic concepts of thermodynamics of rubber elasticity and other factors affecting it.

1. Introduction

The capability of Rubber-like materials to extend to several-fold their original length is undoubtedly the most striking characteristic, which has been the subject of research interest for decades. Questions like what factor contributes to such substantial deformation and what effects temperature and pressure cast on the elasticity of rubber, all these have been pursued by several investigators. Having these questions in the back of our mind, we shall try to cement our concepts regarding how rubber elasticity is different from those of crystalline solid [1, 2] and glasses [3] which cannot normally be extended to more than a small fraction of their original length without undergoing failure, and ductile materials such as metals [4, 5, 6] which can undergo large deformation but cannot return to their original length upon removal of stress.

In this chapter, we will first get up to understanding the thermodynamics of rubber elasticity. It should be noted that the classical thermodynamic approach is only concerned with the macroscopic behavior of material under investigation and has very little thing to do with their molecular structure. The next section of this chapter presents the quantitative description of elasticity of the network of rubber chains based on the classical principles of statistical mechanics. Finally, we discuss various factors affecting the elasticity of rubber.

2. Effect of various factors on the elasticity of rubber

2.1 Effect of temperature on rubber elasticity

We need an experimental setup for investigating the effect of temperature on rubber elasticity. There are numerous ways to measure this experimentally. One such experimental design that is widely used to probe the fundamentals of rubber elasticity is mentioned in Table 1. This experiment is primarily based on the automatic stress-relaxometry setup as shown in Figure 1, which was used by M.C. Shen and D.A. McQuarrie [7].

Step 1. Extend the rubber sample to a desired fixed length	Step 2. Allow the sample to equilibrate, i.e., allowing the stress-relaxation to proceed until constant modulus is achieved.
Step 3. Under equilibrium conditions, measure the restoring force as a function of temperature at constant pressure.	Step 4. Repeat steps 1 to 3 on a fresh sample at different elongated lengths.

Table 1.

Procedure for measuring the effect of temperature on the elasticity of rubber.

Figure 1.

The automatic stress relaxometry. By permission of the American Institute of Physics [7].

An outcome of such experimental exercise on natural rubber samples is shown in Figure 2, which exhibits the effect of temperature on the restoring force at various extension ratios. The testing sample was prepared from NBS pale creep rubber, which was cured by 1.5 phr dicumyl peroxide at 145°C temperature for 40 min duration. The restoring force is seemed linearly varying over a wide range of temperatures. The slop of the force-temperature curve however changes depending upon the extension ratio. The shift from negative slope at low degree of elongation to the positive slope at a higher extension ratio is called the thermoelastic inversion. The thermoelastic inversion value may lies around 10% for rubbery materials. It should be noted that this behavior is not confined to natural rubber, but it is general for all rubbery materials.

Figure 2.

Stress-temperature curves of natural rubber [7].

Now is the right time to get fully involved with the constitutive relationship between the restoring force (f) and various thermodynamic state variables such as temperature (T), pressure (P), etc. Most thermodynamic theories in general textbooks are confined to the gaseous pressure-volume form of work ($\mathit{dW}=\mathit{PdV}$). However, in the case of the deformation of rubber, there is something more than just pressure-volume work. When a strip of rubber sample is elongated by a length $\mathit{dl}$, a restoring force is generated inside the rubber system (Figure 3). Therefore, the tiny amount of work involved in this process can be expressed as,

Figure 3.

Schematic representation of change in configuration of rubber chains under the applied stress.

$$\mathit{dW}=\mathit{PdV}-\mathit{fdl}$$E1

In the expression above, the $\mathit{dV}$ factor is the volume change that arises due to the elongation of the rubber sample. Usually, $\mathit{PdV}$ is so small relative to the $\mathit{fdl}$ that $\mathit{PdV}$ can be dropped out from Eq. (1). However, we are going to have $\mathit{PdV}$ for the sake of completeness. Now, for the reversible processes, we may combine the first and second law of thermodynamics and write it as,

$$\mathit{dU}=\mathit{TdS}-\mathit{dW}$$E2

From Eqs. (1) and (2)

$$\mathit{dU}=\mathit{TdS}-\mathit{PdV}+\mathit{fdl}$$E3

Since the experiment is usually conducted at a constant atmospheric pressure value (1 atm or 14.696 psi), the enthalpy transfer $\mathit{dH}$ accompanying the volume change due to the elongation of the rubber may be written as,

$$\mathit{dH}=\mathit{dU}+\mathit{PdV}$$E4

By combining Eqs. (3) and (4), we arrive at the expression,

$$\mathit{dH}=\mathit{TdS}+\mathit{fdl}$$E5

By partially differentiating Eq. (5) and treating temperature and pressure as constant, we get,

$$f={\left(\frac{\partial H}{\partial l}\right)}_{T,P}-T{\left(\frac{\partial S}{\partial l}\right)}_{T,P}$$E6

$$f={\left(\frac{\partial H}{\partial l}\right)}_{T,P}+T{\left(\frac{\partial f}{\partial T}\right)}_{l,P}$$E7

Eqs. (6) and (7) hold an essential piece of information regarding the origin of elastic force. Let us take a moment to behold this relationship between restoring force and temperature and what role enthalpy and entropy play in this Eq. (6). According to Eq. (6), the restoring force depends on two factors: enthalpy and entropy change that occur in rubber due to elongation (or deformation in general terms). Generally, rubber molecules are so long that almost every chain participates in crosslinking and entanglement processes. During deformation or stretching, some rubber chains are forced to become linearly oriented, which causes a decrease in entropy of the rubber system. This decrease in entropy gives rise to the elastic force in the network chains, (Figure 4).

Figure 4.

Graphical representation of the relationship between restoring force and temperature.

It is important to understand the limitation of Eq. (6) that it does not fully comply with the behavior of rubber at extremely high elongation. The ${\left(\frac{\partial H}{\partial l}\right)}_{T,P}$ part has a finite value and cannot be ignored. At sufficiently high elongation, most rubber crystalizes and thus ${\left(\frac{\partial H}{\partial l}\right)}_{T,P}$ factor may overweight $-T{\left(\frac{\partial S}{\partial l}\right)}_{T,P}$. The coefficient ${\left(\frac{\partial H}{\partial l}\right)}_{T,P}$ can be experimentally obtained from the force-temperature curve as the intercept on restoring force axis at zero temperature value. The coefficient ${\left(\frac{\partial H}{\partial l}\right)}_{T,P}$ has a fundamental relationship with other thermodynamic quantities, which may be expressed as

$${\left(\frac{\partial H}{\partial l}\right)}_{T,P}={\left(\frac{\partial U}{\partial l}\right)}_{T,V}+T\frac{\alpha }{\beta }{\left(\frac{\partial V}{\partial l}\right)}_{T,P}$$E8

where, $\alpha$ is the cubical coefficient of thermal expansion,

$$\alpha =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{P,l}$$E9

and $\beta$ is the coefficient of isothermal compressibility,

$$\beta =\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_{T,l}$$E10

The coefficient value $\alpha$ and $\beta$ can be measured experimentally, however the coefficient ${\left(\frac{\partial V}{\partial l}\right)}_{T,P}$ is usually exceedingly small for most of the rubbers. Therefore, a great deal of experimental accuracy is required. Eqs. (7) and (8) can be combined to one the more refined and insightful expressions about the relative contributions of enthalpy and entropy towards the rubber elasticity. It should be noted that Eq. (11), in practice, does not comply well with the real behavior of rubber-like materials due to a lack of accurate ${\left(\frac{\partial V}{\partial l}\right)}_{T,P}$ data [8, 9, 10]. Therefore, various approximations have been suggested, which is beyond the scope of this chapter.

E11

2.2 Effect of crosslinking on rubber elasticity

As discussed in the introduction section, rubber materials have the outstanding ability to return to their initial position almost instantaneously upon the removal of deforming load with virtually zero permanent deformation within the network chain structure. This snapping of rubber primarily happens due to the presence of crosslinks. We can think of crosslinks as the knot holding two or more threads together. Crosslink inhibits the long-range mobility of rubber chains. Therefore, based on our general understanding, we can confidently say that the crosslinked rubber would require a lot more force to stretch for the same amount of strain as the uncrosslinked rubber [11, 12, 13]. Figure 5 schematically represents the crosslinking process of the linear polymer chains into an infinite network. In practice, crosslinking process is performed by incorporating the appropriate crosslinking agents like sulfur or peroxide into the rubber matrix and heating it at elevated temperature under some pressure. The crosslinking process enhances dimensional stability, abrasion resistance, and many other properties [14, 15, 16, 17].

Figure 5.

Schematic representation of crosslinks in an ideal chain network structure.

The extent to which a rubber chain network is crosslinked directly impacts the elasticity of rubber. According to the statistical mechanics, rubber elastic modulus under a uniaxial elongation is directly proportional to the number of crosslinks per unit volume ($N_o\text{or}\mathit{mol}/{\mathit{cm}}^3)$ of the rubber chain network. If the rubber density is given as d $\left(g/{\mathit{cm}}^3\right)$, and the average molecular weight of network chain is $M_c\left(g/\mathit{mol}\right)$, then

$$N_o=\frac{d}{M_c}$$E12

For a small uniaxial strain value, the relationship between the elastic modulus $E_o$ and crosslink concentration $N_o$ can be mathematically written as

$$E_o=3N_o\mathit{RT}\frac{{\overline{\overline{r}}}_o^2}{r_f^2}$$E13

or

$$E_o=\frac{3\mathit{dRT}}{M_c}\frac{{\overline{\overline{r}}}_o^2}{{\overline{r}}_f^2}$$E14

Here, ${\overline{\overline{r}}}_o^2$ represents the mean square end-to-end distance of the chains within the network, and ${\overline{r}}_f^2$ represents the mean end-to-end distance of the isolated chains [18, 19].

Since rubber is considered an incompressible material in relation to their shear deformation, that is poisson’s ratio of rubber is close to 0.5, i.e., elastic modulus $E_o$ is approximately equal to three times the shear modulus $G_o$. Therefore, Eq. (14) can be rewritten as,

$$G_o=\frac{\mathit{dRT}}{M_c}\frac{{\overline{\overline{r}}}_o^2}{r_f^2}$$E15

Eq. (15), however is based on the idealized image of a perfect rubber chain network in which all network chains contribute to the elasticity of rubber. It is assumed that each crosslink combines four chains, and two crosslinks terminate each such chain. But in reality, there are several imperfections present in the rubber chain network. As illustrated in Figure 5, the non-effective crosslinks like wasted crosslinked, chain loops, and terminal ends significantly affect the elastic stress in the strained network of rubber chains. For example, linear polymer chains of average molecular weight M would have 2d/M number of terminals. Since these terminals would not take part in the elasticity, they should be excluded from the number of effective chains.

$$G_o=\mathit{RT}\frac{{\overline{\overline{r}}}_o^2}{r_f^2}\left(\frac{d}{M_c}-\frac{2d}{M}\right)$$E16

or,

$$G_o=\left(\frac{\mathit{dRT}}{M_c}\right)\frac{{\overline{\overline{r}}}_o^2}{r_f^2}\left(1-\frac{2M_c}{M}\right)$$E17

Entanglements in rubber chain network also impose additional restriction, which leads to increment in the elastic stress. In a closed packed network of rubber chains, it is quite natural to expect several such entanglements between two consecutive crosslinks [20, 21]. Therefore, the contribution of such entanglements to elastic stress cannot be overlooked, especially chains that are long enough to permit multiple entanglements. The deviation from the “normal” crosslinked structure can be accounted for by adding the entanglement factor (a) in Eq. (18),

$$G_o=\left(\frac{\mathit{dRT}}{M_c}+a\right)\frac{{\overline{\overline{r}}}_o^2}{r_f^2}\left(1-\frac{2M_c}{M}\right)$$E18

2.3 Effect of filler on rubber elasticity

Raw rubber is a mechanically weak material. Thus, rubber needs some extra compounding ingredients to enhance its physical properties in addition to crosslinking. Therefore, filler is an indispensable ingredient in the rubber industry. Carbon black, zinc oxide, silica, clay are some commonly used filler examples. Filler can be of two types, reinforcing and non-reinforcing. Reinforcing fillers increase the rubber’s stiffness without impairing the strength and losing the rubbery characteristic.

The most common expression describing the effect of filler on rubber elasticity is popularly called as Guth-Smallwood equation

$$\frac{E_f}{E_o}=1+2.5\ast {\varnothing }_f+14.1\ast {\varnothing }_f^2$$E19

where, ${\varnothing }_f$ is the volume fraction of filler and subscripts f and 0 refer to the filled and unfilled rubber respectively. Eq. (19) is inspired by the Einstein’s equation that relate viscosity of fluid containing small solid suspension particles [22, 23]. The validation of Eq. (19) is to be found in its good agreement with the experimental data as shown in Figure 6.

Figure 6.

Effect of filler volume fraction on filled to non-filled rubber modulus ratio.

Stress softening is another exciting behavior that is commonly seen in the filled rubber system. It was first observed by the Mullins, after whom it is named [24]. According to the stress-softening effect, the stress–strain curve depends on the maximum loading previously encountered. The term “Mullins effect” is also common to all rubbers, including non-filled rubber. Figure 7 illustrates the softening of stress with each step.

Figure 7.

Stress–strain curves for a filled rubber showing progressive cyclic softening, also known as the Mullins effect.

Bueche gave the first molecular interpretation of the Mullin’s effect and the explanation for the Figure 7 [25]. As illustrated in Figure 8, two nearby filler particles in a reinforced rubber are connected via polymer chains. One of them is relaxed, but the others are relatively elongated. When the rubber sample is stretched, the “prestrained” chains first reach maximum extension, and then they either become detached or break. In the second cycle, the detached/broken chains no longer share the overall applied load, thus giving rise to the observed softening of stress behavior. The same thing happens when the sample is stretched a third time.

Figure 8.

Schematic representation of molecular mechanism for stress-softening effect. (a) Unstrained rubber chain-filler particle assembly, and (b) rubber chain-filler particle assembly after deformation.

2.4 Effect of stress-induced crystallization

In unstrained conditions, the rubber sample is assumed to hold isotropic properties. However, when the rubber sample is stretched, anisotropic change occurs at the microscopic level. Polymer chains tend to orient more in the direction of stretch than in the lateral directions. Therefore, a greater number of ordered chins favor the formation of crystallites. These crystallites combine numbers of nearby network chains, which then act as crosslinks. As the sample is stretched further, more crystallite is formed and get transformed into physical crosslinks. These crosslinks then, in turn, cause a rise in elastic stress. It is known that such crystallites have very high elastic moduli (∼10¹¹ Pa), which is usually five orders of magnitude higher than the elastic moduli of rubbery materials (∼10⁶ Pa). At higher elongation, such crystallites also play the role of reinforcing filler, which further increases the elastic stress of the rubber sample.

Figure 9 illustrates the stress–strain behavior of natural rubber carried at two temperatures, i.e., 30 and 60°C. The graph shows how stress steeply rises above a certain ratio, i.e., λ = 0.3. However, it should be noted that the stress-induced behavior also depends on the temperature at which experiment is conducted. At 60°C natural rubber sample exhibit slight upturn in the elastic stress towards the extension ratio, which is primarily caused by the finite extensibility of the polymer chain in the network [27].

Figure 9.

Natural rubber stress–strain curve measured at two different temperatures [26].

2.5 Time-temperature superposition principle

So far, we have done a quantitative and qualitative discussion on the influence of various factors like temperature, extension ratio, crosslink density, fillers, and crystallinity affecting the elastic stress of modulus of the rubber sample. These considerations present a fair picture of physical behavior of a rubber material. However, the constitutive relations those we have seen till now are based on the static experimental data that is the effect of an immediate change to a system is calculated without regard to the longer-term response of the system to that change.

Rubber is regarded as a highly viscoelastic material that is rubber resembles characteristics of both elastic and viscous material. There are three main characteristics of viscoelastic materials: creep, stress relaxation, and hysteresis. These viscoelastic phenomena arise due to the long-term response of the material to a constant load or strain. Clearly, there is a time factor involved in these observations, which leads to the question that how one can relate viscoelastic responses with time scale mathematically. There are quite some constitutive models that quantitatively express viscoelastic response as a function of time. Maxwell, Kelvin–Voigt, Generalized Maxwell models are some well-known examples. Detailed discussion on these models is however beyond the scope of this chapter. Rather, we shall focus on understanding the time-dependent variation of elastic modulus by using time–temperature superposition principle.

To better understand time–temperature superposition principle let us us take an example of a stress relaxation experiment conducted at different temperature. At a given temperature T₁, a polymer sample of unit cross section area is subjected to an instantaneous strain that is maintained constant throughout the whole experiment. Then the stress as function of time is measured and stress relaxation modulus is obtained according to the Eq. (20). Where t can be experimentally accessible time period. Polymer sample is then set free from stress and allowed to undergo relaxation. Next, temperature is changes to T₂, and same procedure is repeated yielding $E\left(t\right)$tensile stress relaxation modulus at new T₂ temperature. Theis process is repeated at several different temperature and “t second stress-relaxation modulus” is obtained as a function of temperature.

$$E\left(t\right)=\frac{\sigma \left(t\right)}{{ϵ}_0}$$E20

where, $\sigma \left(t\right)$ is stress as a function of time, and ${ϵ}_0$ is constant strain value.

Time–temperature superposition principle states that the change in temperature from T₁ to T₂ is equivalent to multiplying the time scale by a constant factor $a_T$ that is only a function of the two temperatures T₁ to T₂ according to the Eq. (21).

$$E\left(t_1,T_1\right)=E\left(\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$a_T$}\right.,T_2\right)$$E21

$$\mathit{log}{a}_T=\mathit{log}\frac{t_1}{t_2}$$E22

where t₂ is the time required to reach $E\left(t_2,T_2\right)$ stress relaxation modulus measured at T₂ temperature. Figure 10(left). exhibits data obtained from one such experiment (i.e., stress relaxation) conducted on bis-phenol-A-polycarbonate ($M_w$ = 40,000 g/mol), and Figure 10(right) represents the transformed modulus-time curve for a referenced temperature of 141°C [26].

Figure 10.

Transformation of stress relaxation modulus-time obtained at different temperatures into single master curve for 140°C reference temperature.

3. Conclusions

Overall, we have discussed the basic concepts of thermodynamics of rubber elasticity. The quantitative effect of temperature on the elasticity of rubber has been scribbled down. Then the qualitative discussion on the role of crosslink density, filler concentration and strain induced crystallization on the elastic modulus of rubber-like materials provide supporting explanations backed by mechanism from statistical to molecular scale standpoint. Additionally, we discussed what role time scale plays in elastic modulus of general polymer at a given temperature using time–temperature superposition principle. The contents discussed in this chapter is simple and sufficient to develop an interest in readers’ mind to take next step towards studying more complex rubbery materials.