Network modulus

Equation (40) shows that the small deformation shear modulus of an affine network increases indefinitely over the phantom network modulus as junction functionality approaches 2. 

According to the arguments based on the constrained-junction model, the term Gch should equate to the phantom network modulus onto which contributions from entanglements are added. 

The result is important for the discussion in Part 3. Multiplication of the v-values by RT gives the corresponding moduli. The effective modulus of the first network after removal of first network crosslinks, Gie, has been calculated for a first network modulus, G-j, of 0.75 MPa. In Figure 1, G. e is plotted against the modulus of the second network before removal of the first network cross-links, G2. It can be seen that the memory effect increases with increasing modulus or degree of cross-linking of the second network. Gx and G2 max are related to the experiment to be discussed in Part 3. 

Figure 1. Effective first network modulus, Gle, after complete removal of first network cross-links plotted against second network modulus, Gi. Calculated from the composite network theory of Flory (19J for G, — 0.75 MPa.

Equation (2.53) is stating that the network modulus is the product of the thermal energy and the number of springs trapped by the entanglements. This is the result that is predicted for covalently crosslinked elastomers from the theory of rubber elasticity that will be discussed in a little more detail below. However, what we should focus on here is that there is a range of frequencies over which a polymer melt behaves as a crosslinked three-dimensional mesh. At low frequencies entanglements... 

Figure 2.11 Schematic of the storage modulus as a function of the frequency of the applied strain for a polymer melt. The plateau gives the network modulus, GN. The plateau extends down to lower frequencies as the molecular weight is increased because the relaxation time is proportional to M3... 

Figure 2.12 The network modulus of a solution of an HMHEC in water, showing the curves calculated from Equation (2.58) and the experimental points taken from reference 24. The molecular weight was Mn = 105 and there were 4 hexadecyl hydrophobes per cellulose chain at synthesis...

Here the 2 in the denominator is to avoid double counting because it takes two sites to form one link. However closed loops and chain entanglements are both possibilities and Equation (2.57) must be modified for these effects. We can write the network modulus as... 

An example of the effectiveness of this equation is given by an aqueous HEUR gel made up of a polymer with Mn = 20 x 103 Daltons at a concentration of 30kgm-3 filled with a poly(styrene) latex with a particle diameter of 0.2 pm at q> = 0.2. The unfilled gel had a network modulus of 0.4 kPa, whilst the modulus of the filled gel was 0.7 kPa. Equation (2.68) predicts a value of 0.728 kPa. The poly(styrene) particles act as a non-interactive filler because the surface is strongly hydrophobic as it consists mainly of benzene rings and adsorbs a monolayer of HEUR via the hydrophobic groups, resulting in a poly(ethylene oxide) coating that does not interact with the HEUR network. This latter point was... 

The above examples show that we can describe the network modulus of polymer gels by using the concept of entropic springs making up the network. In some cases corrections to the network are required to... 

The chain overlap parameter has been very successful at superimposing the data from the systems without hydrophobic modification, producing the continuous curve. However, it is clear from Flynn s work that once the hydrophobes are introduced into the polymer the viscosity rapidly increases at lower values of the chain overlap parameter. Increasing the mole percentage of hydrophobes also increases the viscosity at lower values of the chain overlap parameter. The position and number of the hydrophobes on a chain are important in determining the structure that forms and the onset of the increase in viscosity. The addition of side chains to hydroxyethyl cellulose modifies the network modulus as a function of concentration. This is discussed further in Section 2.3.4. 

The main subject of the following discussion is the mechanical behaviour of networks in terms of the behaviour of the system of weakly coupled macromolecules. The network modulus of elasticity is small in comparison to the values of the elasticity modulus for low-molecular solids (Dusek and Prins 1969 Treloar 1958). Nevertheless, large (up to 1000%) recoverable deformations of the networks chains are possible. 

Cross-linking effects are more important in the plateau and terminal zones. For moderately cross-linked polymers, such as soft vulcanized rubbers, the equilibrium modulus is similar in magnitude to the entanglement network modulus before vulcanization. In some cases, however, the former modulus may be higher by as much as a factor of 2 than the latter one, thus... 

The number of network strands per unit volume (number density of strands) is p = njV. In the last equality, p is the network density (mass per unit volume), Mg is the number-average molar mass of a network strand, and IZ is the gas constant. The network modulus increases with temperat-iire hpnaiisp its origin is entropic, analogous to the pressure of an ideal gas p = nkTjV. The modulus also increases linearly with the number density of network strands u = n/V = pJ fEquation (7.31) states that the modulus of any network polymer is kT per strand. 

For any functionality /, the phantom network modulus is lower than the affine network modulus [Eq. (7.31)] because allowing the crosslinks to fluctuate in space makes the network softer. The phantom network has the... 

These predictions need to be modified because real networks have defects. As shown in Fig. 7.7, some of the network strands are only attached to the network at one end. These dangling ends cannot bear stress and hence do not contribute to the modulus. Similarly, other structures in the network (such as dangling loops) are also not elastically effective. The phantom network prediction can be recast in terms of the number density of elastically effective strands v and the number density of elastically effective crosslinks ii. For a perfect network without defects, the phantom network modulus is proportional to the difference of the number densities of network strands v and crosslinks // = since there are fjl network strands per crosslink ... 

In the 1940s, it was recognized that the classical predictions of network modulus were bounded. A real network could certainly not be expected to have lower modulus than the phantom prediction, since it is based on unrestricted fluctuations of ideal strands that are allowed to pass through each other. At the other extreme, the classical models have no means to attain a higher modulus than the affine prediction, based on junctions that... 

The fact that two chains cannot pass through one another creates topological interactions known as entanglements that raise the network modulus. 

Computer simulations of network modulus for networks with three different strand lengths (filled circles with number of monomers per strand N= 2, 26, and 44). The open squares are the same networks but the modulus is measured when the strands are... 

Therefore, it is well established that topological entanglements dominate and control the modulus of polymer networks with long network strands. The Edwards tube model explains the non-zero intercept in plots of network modulus against number density of strands (see Figs 7.11 and 7.12). The modulus of networks with very long strands between crosslinks approaches the plateau modulus of the linear polymer melt. The modulus of the entangled polymer network can be approximated as a simple sum. 

The phantom network model assumes there are no interactions between network strands other than their connectivity at the junction points. It has long been recognized that this is an oversimplification. Chains surrounding a given strand restrict its fluctuations, raising the network modulus. This is a very complicated effect involving interactions of many polymer chains, and hence, is most easily accounted for using a mean-field theory. In the... 

Originally derived by James and Guth, this weak concentration depend-ence of the network modulus comes from two competing effects. As concentration is lowered, the number density of strands naturally decreases. However, the strands also stretch as concentration is lowered, and this stretching raises the modulus somewhat through the proportionality to (see Problem 7.29). The net effect is the weak decrease of the gel modulus upon swelling, given by Eq. (7.74). 

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 > 0 chains are nearly ideal in concentrated solutions, whereas for 0 < 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q < 0o) there are three... 

The swelling increases steadily as the excluded volume increases. If the network is prepared in the bulk (0o = 1), the general relation between the dry network modulus and the equilibrium swelling depends upon whether... 

The Iwata model probably overestimates the entanglement contribution to the rubber elasticity. This conclusion results from the observation that the topological contribution to the network modulus and the contribution arising from chemical crosslinks are approximately of the same order of magnitude A statistical... 

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