Tube Linear polymers

Fig. 23.4. Each molecule in o linear polymer con be thought of as being contained in a tube made up by its surroundings. When the polymer is loaded at or above Tg, each molecule can move (reptate) in its tube, giving strain.

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

The observations above can be rapidly turned into a semi-quantitative theory for star-polymer stress-relaxation [24] which is amenable to more quantitative refinement [25]. The key observation is that the diffusion equation for stress-re-lease, which arises in linear polymers via the passage of free ends out of deformed tube segment, is now modified in star polymers by the potential of Eq. (16). Apart from small displacements of the end, the diffusion to any position s along the arm will now need to be activated and so is exponentially suppressed. Each position along the arm, s, will possess its own characteristic stress relaxation time T(s) given approximately by... 

Unless the molecular weight of the star arm is rather low, this period of relaxation by fluctuation is generally interrupted by the reptation of the linear polymers. This happens when the centre of mass diffusion of the linear chains is able to renew the tube of all (central) portions of the chain not relaxed by the star-like modes. So the reptation time is given by... 

The challenge within our programme is to follow up the consequences of the tube model for the non-linear rheology of branched polymers - would such a theoretical framework lead to any understanding of the special behaviour of, for example, LDPE in complex flows We build up our tools as before in the context of linear polymers. 

Fig. 17. Damping functions in shear from the tube model for linear polymers (lowest curve) and various branched architectures. In the cases of comb and tree, the lower curves give the case of the structure with four levels of branching, the upper the limit of large structures hatched area covers published results on LDPE...

In CGE the capillary is filled with a gel containing cross-linked or linear polymers. The gel thereby acts as molecular sieve. Traditionally, cross-linked polyacrylamides and agarose have been utilized in the slab and tube format. Polyacrylamides when cross-linked have smaller pore sizes and are used for protein separations. The larger pore sizes of agarose gels are more suitable for DNA separation. Polyacrylamides yield very viscous gels. Therefore, polymerization is usually done on column, which has a lot of practical problems. 

Toms BA (1948) Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds-numbers Proc 1st Intern Congr on Rheology North Holland Amsterdam II 135... 

In the MSF theory, the function,/, in addition to simple reptation, is also related to both the elastic effects of tube diameter reduction, through the Helmholtz free energy, and to dissipative, convective molecular-constraint mechanisms. Wagner et al. arrive at two differential equations for the molecular stress function/ one for linear polymers and one for branched. Both require only two trial-and-error determined parameters. 

In thel980s, Leger and coworkers [9,10] studied the aspiration of linear polystyrene in pores of polycarbonates. A theoretical picture was also constructed for this [11] and is summarised in Fig. 2. The main result is that linear chains are sucked in when the pore current J (the solvent volume flowing per unit time) is beyond a certain threshold value Jc.This Jc turns out to be independent of tube diameter and of the molecular weight of the linear polymer... 

H and Si NMR spectra were recorded on a Brnker Avance 300 spectrometer respectively at 300.13 and 75.6MHz at room temperatnre. CDClj and toluene were used as solvents. For Si NMR, Cr(acac)3 (0.03 M) was added in the tube and a delay between pulses of 20 s was set. IR spectra were recorded on a Perkin-Ehner IR ET 1760-X. The average molecnlar weight of the linear polymers was determined by size exclusion chromatography in TFIF (flow rate l.OmL.min" ) on an apparatus equipped with a Waters refractive index detector, a Waters column pack (Ultrastyra-gel 10, 10 100 A) and a Minidawn Wyatt light scattering detector. 

Figure 1 (a) Reptation of a linear polymer molecule in a tube, (b) Arm retraction mechanism in the tube model for a star polymer... 

Complex Architectures. Perhaps the most significant recent advances in molecular understanding of polymer melts have emerged from the study of branched polymer architectures. We have noted above how a tube theory for star-polymers provided the means to treat fluctuations in entangled path length in linear polymers (see Figure lb). This is simply due to the complete suppression of reptation in star polymers without fluctuation there is no stress-relaxation at all ... 

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. 

Linear polymers move a distance of order of their own size during their relaxation time, leading to a diffusion coefficient D R /r [Eq. (9.12)]. However, the diffusion of entangled stars is different because at the time scale of successful arm retraction, the branch point can only randomly hop between neighbouring entanglement cells by a distance of order one tube diameter a. For this reason, diffusion of an entangled star is much slower than diffusion of a linear polymer with the same number of monomers ... 

There are four different regimes of monomer displacement in entangled linear polymer melts, shown in Fig. 9.20. The subdiffusive regime for the mean-square monomer displacement is a unique characteristic of Rouse motion of a chain confined to a tube, which has been found in both NMR experiments and computer simulations. 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

To understand this viscosity enhancement, it is easier to start with the theory for linear polymers. The behavior of linear polymers can be described by the reptation model.For a linear polymer of high molecular weight in the melt, chains can be modeled as a confined tube where the diffusion of the chain is restricted along the tube contour. Entanglements are formed between chains where the reptation of a chain along its contour becomes the dominant mode of movement. The addition of a branch point prevents reptation and other forms of movement must occur for the chain to change its configuration. In the case... 

Preparation of linear polymers. Into a 5 L, three-necked flask equipped with an overhead stirrer, an argon inlet tube and a condenser are placed 89.57 g (0.621 mole) frans-eyelohexanedimethanol, 39.52 g (0.334 mol) 1,6-hexanediol, and 1.8 L distilled tetrahydrofuran. The mixture is stirred until all solids have dissolved then, 200 g (0.942 mole) 3,9-bis (ethylidene 2,4,8,10-tetraoxaspiro [5,5] undecane) is added. The polymerization is initiated by the addition of 2 mL of a solution of p-toluenesulfonic acid (20 mg/mL) in tetrahydrofuran. 

Linear polymers have been observed by electron microscopy (113,114)-The linear tubes depicted in Fig. 1 consist of four polymer chains arranged in helical array. The two-dimensional network visible in the electron micrographs of Munn (114) probably represents an initial stage in formation of crystals. 

B. A. Toms, Some Observations on the Flow of Linear Polymer Solutions through Straight TUbes at Large Reynolds Numbers, Proa 1st Int. Cong. Rheol., North-Holland, Amsterdam, vol. II, p. 135, 1949. 

Brush polymers. A linear component, such as polystyrene, is grafted onto a polyethylene film or tubing. The polymer chains extend from the rigid surface in a brushlike fashion. Functionalization can be achieved along the extending chains, and the functional groups can be further modified with spacers prior to solid-phase synthesis. 

Comparative examination of Rv and Rh values show that Rvzwitterionic polymers, meaning that the aggregates dessociate to some extent in the capillary tube, due to the shear forces app ed therein. It seems that the increased steric repulsions introduced by the unfimctionatized arms lead to the formation of not so strong associates as in the case of linear polymers. 

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