Thermodynamics of Elasticity

Stretching an elastomer reduces its entropy and changes its free energy. The retractive force in an elastomer is primarily the result of its tendency to increase the entropy towards the maximum value it had in the original deformed state Current explanations of rubber-like elasticity are based on several assumption [20]. The first one is that rubber-like elasticity is entirely intramolecular in origin. The second one is that the free energy of the network is separable into two parts, an elastic one and a liquid one. The liquid one is presumably not dependent on deformation. When an elastomer is stretched, the free energy is changed, because it is subjected to work. If we consider the stretching in one direction only, the work done Wel is equal to f∂l, where f is the retractive force and ∂l is the change in length. The retractive force is then [19–21].  where F is the free energy, H is the enthalpy, and Sis the entropy of the system. An ideal elastomer can be defined as one where (∂H/∂L) _T,p ¼ 0 and f ¼ T(∂S/∂L)_T,p. The negative sign is due to the fact that work has to be done to stretch and increase the length of the elastomer. This description of an ideal elastomer is based, therefore, on the understanding that its retractive force is due to a decrease in entropy upon extension. In other words, the entropy of elasticity is the distortion of the polymer chains from their most probable random conformations in the unstretched condition. The probability that one chain end in a unit volume of space coordinates, x, y, z is at a distance r from the other end is [21, 22]:  where b² ¼ 3/2xL². The number of links is x and the length is L. The entropy of the system is proportional to the logarithm of the number of configurations. Billmeyer expresses it as follows [7]:  where k is the Boltzmann’s constant. The retractive force for a single polymer chain, f0 stretched to a length dr at a temperature T is, therefore, 

It is generally assumed that the total retractive force of a given sample of an elastomer is the sum of all the f0 forces for all the polymeric chains that it consists of. This is claimed to be justified in most cases, though inaccurate in detail [22]. Tobol sky wrote the equation for the entropy change of an unstretched to a stretched elastomer as depending upon the number of configurations in the two states [12]:  where Ω and Ωu represent the number of configurations. The evaluation of these configurations by numerous methods allows one to write the equations for the change in entropy as: 

where N0 and L and Lu are the relative lengths of the unstretched and stretched elastomer. Tobol sky derived the tensile strength as being [12]: 

By dividing both sides of the equation by the cross-sectional area of the sample, one can obtain the stress–strain curve for an ideal rubber. The retractive force of an elastomer, as explained above, increases with the temperature. In other words, the temperature of elastomers increases with adiabatic stretching [21, 22]. The equation for the relationship was written by Kelvin back in 1857 [22]: 

In actual dealing with polymers, stretching rubber and other elastomers requires overcoming the energy barriers of the polymeric chains with the internal energy of the material depending slightly on elongation, because  ∂V is the change in volume. At normal pressures the second term on the right becomes negligible. It represents deviation from ideality. The contribution of the internal energy E to the force of retraction is  Bueche [16] expressed differently the work done on stretching an elastic polymeric body. It describes deforming an elastomer of x length, stretched to an increase in length a in a polymeric chain composed of N freely oriented segments. The other dimensions of this polymeric chain are y and z. These are coordinates that become reduced as a result of stretching to 1/(α)^0.5.

Work done per chain =  where the chain ends are initially r0 distance apart [16],  Bueche [16] also described the average energy per chain as Ave: energy/chain=  The modulus of elasticity, G, was shown to be related to strain in elongation of polymeric elastomers. For up to 300% elongation, or more, the following relationship [22] applies:  Where k is Boltzmann’s constant, m represents the number of polymeric chains in the sample, and g is the strain. The relationship of stress to strain is:  There are several molecular theories of rubber-like elasticity The simplest one is based on a Gaussian distribution function for the end to end separation of the network chains: [23] (the dimensions of the free chains as unperturbed by excluded volume effect are represented by (r2)0)  Within this Gaussian distribution function (r2)0 applies to the network chains both in the unstretched and stretched state. The free energy of such a chain is described by a Boltzmann relationship [23]:  C(T) is a constant at a specified absolute temperature T. The change in free energy for a stretched elastomer can be expressed as follows:  where a is the molecular deformation ratio of r components in x,y,z directions from the unstretched or elastomer to one that was stretched and deformed. Additional discussions of this theory and other theories of elasticity are not presented here because thorough discussions of this subject belong to books dedicated to physical properties of polymers. 

اخر الاخبار

اخبار العتبة العباسية المقدسة

مؤتمر السيد المجاهد يشهد عرض فيلم وثائقي عن سيرته العلمية ومشروع إحياء تراثه

بحث حوزوي يتناول التمييز بين المعجزة والسحر عند السيد المجاهد

بحث يناقش أثر القواعد العقلية في استنباط الأحكام الشرعية عند السيد محمد المجاهد

بحث علمي يناقش ردود السيد المجاهد على اللغويين والنحويين في كتابه (مفاتيح الأصول)

المزيد

الآخبار الصحية

هل توقيت نشاطك اليومي يؤثر على صحة دماغك؟.. العلماء يجيبون

"مشروب محبوب" يخفي فوائد لا يعرفها كثيرون

طبيب يكشف 5 أنواع للسعال.. بعضها خطير

نقص فيتامينات "خفي".. قد يفسّر التعب والتنميل وقلة التركيز

المزيد

الآخبار التكنلوجية

دراسة: تغيّرات في أسلوب القيادة قد تكشف مبكرا عن الخرف !

عمرها 773 ألف سنة.. اكتشاف أقدم أحافير بشرية في المغرب

"اختراق علمي".. اختبار دم في منزلك قد يكشف ألزهايمر

ماذا يحدث للجسم عند قطع الكافيين فجأة؟

المزيد

مواضيع ذات صلة

The Crystalline State

Rheology and Viscoelasticity of Polymeric Materials

Elasticity

The Glass Transition and the Glassy State

Induction Forces in Polymers

Effects of Dipole Interactions

Structure and Property Relationship in Organic Polymers

تقدير الكتل المولارية من إتزان الترسب

الأسموزية والديلزة Osmosis and dialysis

الحجم والشكل

الوزن الجزيئي للبوليمرات

البوليمرات المؤلفة من سلاسل رئيسة لا عضوية