The elastic restoring force

Examining the two molecular conformations pictured above, it is possible to identify the source of the restoring force. 

the coiled shape can adopt many different conformations, all giving the circular contour. However, when the chain section is pulled completely out into the zigzag constrained by the covalent bond angles, there is only one possible 

Now consider Boltzmann s definition of the disorder of a system, using the thermodynamic measure of entropy. Boltzmann s equation relates entropy to the number of possible arrangements of a system by  

This equation shows us two things. Firstly, the process of stretching a polymeric material gives rise to a negative entropy change, since is less than Secondly, in principle we can count the number of possible arrangements that the chain can adopt and so calculate the entropy of stretching. 

The consequence of the negative entropy change can be found in the familiar rules of thermodynamics. Nature prefers disordered systems of high entropy to ordered systems of low entropy. So the direction of spontaneous change for a system is towards the state of higher entropy. To be more precise, the direction of spontaneous change is towards the state of lower free energy  

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When an elastic polymer network is stretched, the polymer chains are deformed. The verification of the theory has been largely based on measurements of the elastic restoring forces... 

At equilibrium the electrical force causing compression is balanced by the elastic restoring force, as expressed by the equation ... 

The elastic term takes into account the elastic restoring force, tending to establish a spatially uniform LC ordering. It is expressed as... 

The elasticity of rubbers is very different from that of materials such as metals or even glassy or semicrystalline polymers. Young s moduli for metals are typically of the order of 10 MPa (see table 6.1) and the maximum elastic extension is usually of order 1% for higher extensions fracture or permanent deformation occurs. The elastic restoring force in the metal is due to interatomic forces, which fall off extremely rapidly with distance, so that even moderate extension results in fracture or in the slipping of layers of atoms past each other, leading to non-elastic, i.e. non-recoverable, deformation. 

Figure 9.3 Schematic illustration of the elastic restoring force for a polymer molecule after stretching. (MacRitchie, F. 1990. Chemistry at interfaces. San Diego, CA Academic Press.)...

The elastic restoring force is due to internal energy variation under deformation. Atoms are slightly displaced from their equilibrium position, causing an energy increase (cf. Fig. 7.18). When the force is removed, the system returns to zero deformation because this minimizes the displacement energy of the atoms relative to their equilibrium positions. 

In directly measuring vacuum gauges, the elastic restoring force of a spring element (diaphragm) or of gravitation (liquid column) is used for pressure measurement. 

The associated spring constant that describes the elastic restoring force in response to a weak vertical displacement of the CL is ... 

Hence, it is the decrease in entropy as a chain is stretched that yields the elastic restoring force. Typically, a polymer chain can be stretched to many times its most entropically relaxed coil state. This is in strong contrast to the origin of elasticity in hard materials, such as metals, where it is the minute interatomic displacement in steep potential wells that provides the restoring force, and the entropic portion of G plays no role. 

Thus, we have indeed found an expression for Copt which is constant for a given polymer. We learn that the stress-optical coefficient includes three microscopic parameters, the size ao of a monomer, expressing the chain stiffness, the optical anisotropy per monomer, A/3, and the coefficient which relates to the elastic restoring forces. 

For a thin nematic film (a few to several tens of pm thick) with n strongly anchored at the boundaries, the reorienting field (electric or magnetic, F = E or H) must be sufficiently strong, F > Fc to overcome the elastic restoring force of the material and deform the equilibrium pattern of n =ti K,... 

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