Phantom model

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

A measurement of S(x) requires that the SANS spectrometer be calibrated by some absolute standard, a process which is often difficult to achieve with precision. An easier measurement in the ratio of scattering intensities of an anisotropic sample in two different directions. Figure 5 shows a graph of Sj (x)/S (x) versus x for the phantom model and the fixed junction case. 

In the affine model of network deformation, the cross-links are viewed as firmly embedded in the elastomeric matrix, and thus as moving linearly with the imposed macroscopic strain [1-4, 20]. In the alternative phantom model, the chains are treated as having zero cross-sectional areas, with the ability to move through one another as phantoms [2-4]. The cross-links in this model undergo consider-... 

Experimental estimates of p and // usually must rely on a model for the crosslinking chemistry, making quantitative tests of the phantom model difficult. Network defects preclude the use of Eqs (7.31) and (7.41), written in terms of the molar mass of a network strand M. Indeed, since Ms is not known for real networks and the affine and phantom models predict the same classical form of the stress-elongation curve [Eqs (7.32) and (7.33)] there is no practical means of determining which (if either) model is correct for small deformations of unentangled networks. For these reasons, we henceforth describe the modulus of all classical models as a network of strands with apparent molar mass M ... 

When the junctions (crosslinks) are allowed to fluctuate (phantom model), the Hory-Rhener equation takes form shown in Equation 9.9 ... 

Wonnell, T. L., Stauffer, P. R., and Langberg, J. J., 1992, Evaluation of Microwave and Radio Frequency Catheter Ablation in a Myocardium-equivalent Phantom Model, IEEE Trans. Biomed. Engineering, 39 1086-1095. 

The affine and the phantom models derive the behavior of the network from the statistical properties of the individual molecules (single chain models). In the more advanced constrained junction fluctuation model the properties of these two classical models are bridged and interchain interactions are taken into account. We remark for completeness that other molecular models for rubber networks have been proposed [32,57,75-87], however, these are not nearly as widely used and remain the subject of much debate. Here we briefly summarize the basic concepts of the affine, phantom, constrained junction fluctuation, diffused constraint, tube and slip-tube models. 

The Phantom Model. In this model polymer chains are allowed to move freely through one another and the network junctions fluctuate around their mean positions [3,91-93], The conformation of each chain depends only on the position of its ends and is independent of the conformations of the surrounding chains with which they share the same region of space. The junctions in the network are free to fluctuate around their mean positions and the magnitude of the fluctuations is strain invariant. The positions of the junctions and of the domains of fluctuations deform affinely with macroscopic strain. The result is that the deformation of the mean positions of the end-to-end vectors is not affine in the strain. This is because it is the convolution of the distribution of the mean positions (which is affine) with the distribution of the fluctuations (which is strain invariant, i.e., nonaffine). The elastic free energy of deformation is given by... 

The Constrained Junction Fluctuation Model. The affine and phantom models are two limiting cases on the network properties and real network behavior is not perfectly described by them (recall Fig. 29.2). Intermolecular entanglements and other steric constraints on the fluctuations of junctions have been postulated as contributing to the elastic free energy. One widely used model proposed to explain deviations from ideal elastic behavior is that of Ronca and Allegra [34] and Hory [36]. They introduced the assumption of constrained fluctuations and of affine deformation of fluctuation domains. 

FIGURE 29.4. Effect of constraints on the fluctuations of network junctions, (a) Phantom model and (b) constrained junction fluctuation model. Note that the domain boundaries (circles in the figures) are diffuse rather than rigid. The action of domain constraint is assumed to be a Gaussian function of the distance of the junction from B similar to the action of the phantom network being a Gaussian function of AR from the mean position A. 

The classical affinity model assumes that the doss-links are immobile with respect to the whole network. The fluctuations of the positions of cross-links induced by thermal motion are taken into account in the phantom model proposed by James and Guth. It suggests that the fluctuations of a given CTOss-link proceed independently of the presence of subchains linked to it, and during such fluctuations the subchains can pass freely through each other like phantoms. The classic phantom theory predicts the shear modulus G as ... 

To make the phantom model more realistic, the corrections for the effects of entanglements resulting from the exclusion of volume of a subchain to others were introduced. They lead to the reduction of the number of configurations available to each chain. According to the constrained jtmction model proposed by Flory, the deviation of a real network from the phantom network model results from constraints affecting the flurtua-tions of junctions, that is, the jimctions are chosen as structural... 

Several in-vitro evaluations were reported. Although phantom models have inherent limitations represented by the ideal conditions of the study design (a true colon is a moving organ, with peri-... 

Beaulieu CF, Napel S, Daniel BL et al. (1998) Detection of colonic polyps in a phantom model implications for virtual colonoscopy data acquisition. J Comput Assist Tomogr 22 656-663... 

Fernandez A, et al. First prize a phantom model as a teaching modality for laparoscopic partial nephrectomy. J Endourol 2012 26(l) l-5. 

For example, the phantom network model of James and Guth (1,2) gave a recipe for predicting the deformation of a polymer network by an applied stress, and allowed predictions of the change in chain dimensions as a function of network expansion or distortion. In an effort to make the phantom model more realistic, and to fit the model to a variety of experimental results, P.J. Flory and collaborators (3,4,5) proposed that the fluctuation of crosslink junction points calculated by the James-Guth method should be very much restricted by chain entanglements. 

It has been shown that for a uniaxial distribution the determination of is sufficient-in most cases. Previous treatments of segmental orien-e also based on the affine or phantom model molecular... 

In view of the intricate involvements of junctions and their pendant chains in a real network, full compliance with the characteristics of the phantom model network can scarcely be expected. Fluctuations of the junctions about their mean positions may be severely impeded. More important, relocation, under strain, of the neighbors about a given junction may be difficult cooperative rearrangement of chains obviously is required. These concerns lend support to the view of W. Kuhn that the junctions of a network are firmly embedded in the matrix provided by neighbors. Fluctuations deduced for a phantom network are frozen in, and the junctions must undergo displacements that are affine in the macroscopic strain, according to this view. 

Hence, the network may be expected to approach conformity with the phantom model as its elongation is increased, and especially when dilated by swelling. 

V is the total volume of the system and 0 < /i < 1, is a constant which interpolates between the two extreme cases. h = I for the phantom model and /j = 0 for the affine model. For short strands, < Ne, this equation is expected to reasonably describe the behavior in the linear stress regime. It is important to note here that there is no spring-spring interaction. This picture can be extended to take into account a distribution of strand lengths, but this does not give significant differences. ... 

The classical models of rubber elasticity reduce the elastic properties to a study of entropic springs. In the phantom model the EV interaction only gives the volume conservation, while in the affine deformation model it also is responsible for the affine position transformation of the crosslinks. Otherwise the entropic forces and the excluded volume interactions and consequently the entanglements, as they are a result of the EV interaction, completely decouple from the chain elasticity. The elasticity is entirely determined by the strand entropy. It is obvious that this is a... 

Taking this value for CmocL in account reduces the difference with the standard model somewhat. The actual value of the modulus then can reasonably well be described by the affine network model for the LJ system. Including Cmod = 1.5 one gets G% 0.027. Using p = 0.37cr (the density of the elastically active part) one gets within the affine model an effective strand length of Neff 13-14, while the cluster analysis yields Ng/f 11. The soft core model is somewhere in between the affine and the phantom model. 

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