Symmetrical polymers

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Phase Inversion (Solution Precipitation). Phase inversion, also known as solution precipitation or polymer precipitation, is the most important asymmetric membrane preparation method. In this process, a clear polymer solution is precipitated into two phases a soHd polymer-rich phase that forms the matrix of the membrane, and a Hquid polymer-poor phase that forms the membrane pores. If precipitation is rapid, the pore-forming Hquid droplets tend to be small and the membranes formed are markedly asymmetric. If precipitation is slow, the pore-forming Hquid droplets tend to agglomerate while the casting solution is stiU fluid, so that the final pores are relatively large and the membrane stmcture is more symmetrical. Polymer precipitation from a solution can be achieved in several ways, such as cooling, solvent evaporation, precipitation by immersion in water, or imbibition of... 

In fact, the variable x /Gi controls the "crossover" from one "universality class" " to the other. I.e., there exists a crossover scaling description where data for various Gi (i.e., various N) can be collapsed on a master curve Evidence for this crossover scaling has been seen both in experiments and in Monte Carlo simulations for the bond fluctuation model of symmetric polymer mixtures, e.g Fig. 1. One expects a scaling of the form... 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

A third factor influencing the value of Tg is backbone symmetry, which affects the shape of the potential wells for bond rotations. This effect is illustrated by the pairs of polymers polypropylene (Tg=10 C) and polyisobutylene (Tg = -70 C), and poly(vinyi chloride) (Tg=87 C) and poly(vinylidene chloride) (Tg =- 19°C). The symmetrical polymers have lower glass transition temperatures than the unsymmetrical polymers despite the extra side group, although polystyrene (100 C) and poly(a-meth-ylstyrene) are illustrative exceptions. However, tacticity plays a very important role (54) in unsymmetrical polymers. Thus syndiotactic and isoitactic poly( methyl methacrylate) have Tg values of 115 and 45 C respectively. 

The melting point is the temperature range in which total or whole polymer chain mobility occurs. The melting point (T ) is called a first-order transition temperature, and Tg is sometimes referred to as a second-order transition. The values for T are usually 33%-100% greater than for Tg. Symmetrical polymers like HDPE exhibit the greatest difference between T and Tg. The Tg values are low for elastomers and flexible polymers such as PE... 

The value of the Tm is almost always at least 33% higher than the Tt expressed in degrees Kelvin. Symmetrical polymers, like linear polyethylene, exhibit the greatest percentage difference between the Tm and the Tt and these values are usually well-known in symmetrical polymers. 

Polymers with flexible chains, such as natural rubber (NR), have low Tg values. The Tg is always less than the Tm> and the ratio of Tg to Tm is lower for symmetrical polymers like polyvinylidene fluoride (PVDF) than for those with unsymmetrical repeating units, such as polychlorotrifluoroethylene. Raymond Boyer has proposed a relationship of Tm — KTg, where the constant K — 2 for symmetrical and 1,4-asymmetrical chains. 

Note that since o is equal to unity for all flexible molecules and ( ) is equal to unity for all rigid molecules, one of the terms in Equation (18) will always equal zero. The two extremes of Equation (18) describe small, spherical (non-flexible) molecules and large, flexible (non-symmetrical) polymers. 

At short separation, where both the polyelectrolytes and the ions can distribute more or less freely between the walls, the interaction agrees with that in a symmetric polymer system. With increasing separation the interaction turns back into a repulsion in a symmetric case as the bridging disappears, while in an asymmetric system it remains attractive due to the electrostatic interaction between two oppositely net charged half cells at intermediate separations. In the presence of salt this effect may be strong enough to remain... 

Fig. 18. Log-log plot of the parameter co = l7rwoG j > that controls the behavior of critical wetting [11,220,277,278], vs %/%crit -1, for symmetrical polymer mixtues (NA=NB=N) with chain lengths ranging from N=128 to N=1024, showing the predictions Eqs. (124)-(126). From Werner et al. [266]...

Fig.20. Order parameter profiles m(z)=([pA(z)-pB(z)])/([pA(z)+pB(z)]), where pA(z), pB(z) are densities of A-monomers or B-monomers at distance z from the left wall, for LxLx20 films confining a symmetric polymer mixture, polymers being described by the bond fluctuation model with N=32, ab=- aa=- bb=8 and interaction range 6. Four inverse temperatures are shown as indicated. In each case two choices of the linear dimension L parallel to the film are included. While for e/kBT>0.02 differences between L=48 and L=80 are small and only due to statistical errors (which typically are estimated to be of the size of the symbols), data for e/kBT=0.018 clearly suffer from finite size effects. Broken straight lines indicate the values of the bulk order parameters mb in each case [280]. Arrows show the gyration radius and its smallest component in the eigencoordinate system of the gyration tensor [215]. Average volume fraction of occupied sites was chosen as 0.5. From Rouault et al. [56].

Fig. 21. Log-log plot of Tc(oo)-Tc(D) versus D, for the bond-fluctuation model of a symmetric polymer mixture with NA=NB=N=32. For small D, the straight line corresponds to a shift Tc(°o)-Tc(D)oc1/d, while the second straight line for larger D shows the result Tc(°o)-Tc(D)ocD"1/v, with v=0.63 being the critical exponent of the three-dimensional Ising model correlation length [229,230]. From Rouault et al. [55]...

Fig. 22. Phase diagrams of the confined polymer mixtures for thin films of various thicknesses D, using the bond fluctuation model for symmetric polymer mixtures for NA=NB= N=32. The symbols refer to different film thicknesses D=8,10,12,14,16,20,24,28,36 and 48 (from the bottom to the top). From Rouault et al. [55]...

The PRISM theory may also be applied to study polymer blends. An interesting point of discussion is a publication by Deutsch and Binder, who studied symmetric polymer mixtures by using the bond fluctuation variation of the MC technique.The results are reportedly in disagreement with the predictions of PRISM theory. This area is obviously very fertile for further research, and significant developments are around the corner. 

There exists a fairly good correlation between the Tm and Tg for a large number of polymers. A useful rule of thumb is that the ratio Tg/Tm is 1 /2 for symmetrical polymers, i.e. those containing a main-chain atom having two identical substituents, and 2/3 for unsymmetrical polymers. 

To help the design of optimized polymeric materials for BHJ solar cells, several models have been recently proposed [87-89]. The combination of these models and DFT calculations has recently led to the synthesis of several other poly(2,7-carbazole) derivatives (P17, P19-P22). Symmetric polymers (P17-P19) show better structural organization than asymmetric polymers (P20-P22), resulting in higher hole mobilities and power conversion efficiencies. Moreover, their low HOMO energy levels (ca. (- 5.6)—(— 5.4)eV) provide an excellent air stability and relatively high Voc values (between 0.71-0.96 V). 

For the simple example of a symmetric polymer blend with Aa — Ab = A . 

For a symmetric polymer blend Nfi, = NB = N), the whole phase diagram is symmetric (see Fig. 4.8) with the critical composition... 

Estimate the size of the critical region near the critical point in a symmetric polymer blend by comparing the mean-square composition fluctuations with the square of the difference in volume fractions of the... 

Detailed (mean field) analytical calculations were performed by Tang et al. [219] to evaluate the shift in the critical temperature TCD of a thin film as a function of D. They considered a symmetric polymer blend confined by neutral... 

For symmetrical polymers, the chemical nature of the backbone chain is the important factor determining the chain flexibility and hence Tg. Chains made up of bond sequences which are able to rotate easily are flexible, and hence polymers containing -(-CH2-CH2-)-, -(-CH2-0- H2-)-, or -(-Si-O-Si-)- links will have correspondingly low values of Tg. For example, poly(dimethyl siloxane) has one of the lowest Tg values known (-123°C) presumably because the Si-0 bonds have considerable torsional mobility. 

While Tm is a first order transition, Tg is a second order transition and this precludes the possibility of a simple relation between them. There is, however, a crude relation between Tm and Tg. Boyer [25] and Beamen [265] inspected data for a large number of semicrystalline polymers, some of which are shown in Table 2.4. They found that the ratio TgjTm ranged from 0.5 to 0.75 when the temperatures are expressed in degrees Kelvin. The ratio is closer to 0.5 for symmetrical polymers such as polyethylene and polybutadiene, but closer to 0.75 for unsymmterical polymers, such as polystyrene and polyisoprene. The difference in these values may be related to the fact that in unsymmterical chains with repeat units of the type -(CH2-CHX-)- an additional restriction to rotation is imposed by steric effects causing Tg to increase, and conversely, an increase in symmtery lowers Tg. 

Figure 7.3 Morphology study of polymeric mixtures of PE/PEP/PE-PEP by TEM. Samples were annealed at 119°C and then frozen, sectioned and stained volume fraction of the symmetric polymer PE-PEP (a) 0.86, (b) 0.90, (c) 0.91 and (d) 0.92. (From Ref. [5], reprinted with permission of the American Physical...

Membrane symmetric or asymmetric microporous. Ceramic, sintered metals, or polymers with pores 0.2 to 1 pm. Symmetric polymers have a porosity of 60 to 85% asymmetric ceramic membranes, porosity 30 to 40%, are used for high pressure and higher temperature <200°C. 

The first of these difficulties can be avoided for symmetrical polymer mixtures (Na = Nb = N) by working in the semigrandcanonical ensemble of the polymer mixture [107] rather than keeping the volume fractions < )A, B and hence the numbers of chains nA, nB individually fixed, as one would do in experiment and in the canonical ensemble of statistical thermodynamics, we keep the chemical potential difference Ap = pA — pB between the two types of monomers fixed as the given independent variable. While the total volume fraction 1 — < )v taken by monomers is held constant, the volume fractions < )A, B of each species fluctuate and are not known beforehand, but rather are an output of the simulation. Thus in addition to the moves necessary to equilibrate the coil configuration (Fig. 16, upper part), one allows for moves where an A-chain is taken out of the system and replaced by a B-chain or vice versa. Note that for the symmetrical polymer mixture the term representing the contributions of the chemical potentials pA, pB to the grand-canonical partition function Z... 

Fig. 17. Distribution function P M) of the order parameter M plotted over a range of temperatures for N = 128, L = 80, J> = 0.5, using the bond fluctuation model of symmetric polymer mixtures and using the extrapolation formula Eq. (145) and data from a single temperature run at kBT/e = 266.4.The number of (statistically independent) samples was N — 16800. From Deutsch and Binder [92]...

Similarly, for the symmetrical polymers A and B whose M —> °o, the interfacial thickness, Al , and the interfacial tension coefficient, V , were derived as ... 

Correlation of Dt with intrinsic viscosity [rf] also permits a determination of M for spherically symmetric polymers through the Mandel-kern-Flory equation ... 

Geometric factors, such as the symmetry of the backbone and the presence of double bonds on the main chain, affect Tg. Polymers that have symmetrical structure have lower Tg than those with asymmetric structures. This is illustrated by two pairs of polymers polypropylene vs. polyisobutylene and poly(vinyl chloride) vs. poly(vinylidene chloride) in Table 4.4. Given our discussion above on chain stiffness, one would have expected that additional groups near the backbone for the symmetrical polymer would enhance steric hindrance and consequently raise Tg. This, however, is not the case. This discrepancy is due to conformational requirements. The additional groups can only be accommodated in a conformation with a loose structure. The increased free volume results in a lower Tg. 

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