Differential composition distributions

The compositional distribution of ethylene copolymers represents relative contributions of macromolecules with different comonomer contents to a given resin. Compositional distributions of PE resins, however, are measured either by temperature-rising elution fractionation (tref) or, semiquantitatively, by differential scanning calorimetry (dsc). Table 2 shows some correlations between the commercially used PE characterization parameters and the stmctural properties of ethylene polymers used in polymer chemistry. 

The ratio on the left-hand side can be obtained as a function of M since fw,app(M) / Mw can be determined, as described above, from experimental information. Thus, Eq. (5.7) allows determination of wa(M), the chain composition distribution when all other parameters on its right-and side are measured by differential refractometry. Once wa(M) is known, we are ready to compute v(M) from Eq. (5.3), fw(M) from Eq. (5.6), and finally Mw [59]. 

The latexes were cleaned by ion exchange and serum replacement, and the number and type of surface groups were determined by conductometric titration. The molecular weight distributions of the polymers were determined by gel permeation chromatography. The stability of the latexes to added electrolyte was determined by spectrophotometry. The compositional distribution was determined by dynamic mechanical spectroscopy (Rheovibron) and differential scanning calorimetry, and the sequence distribution by C13 nuclear magnetic resonance. 

The conditions of the existence of such bimodal distributions are well-known for the binary copolymerization [169-171], since in this case the differential equation proposed by Skeist [12] for the determination of the composition distribution has an explicit solution. Indeed when m = 2, the only one independent equation of the two equations (5.2) has a simple solution [169, 170, 172] ... 

One of the unique characteristics of Th-FFF is that retention depends not only on the molar mass but also on the chemical composition of the polymer. This chemical differentiation is due to the dependence of the underlining thermal diffusion process on polymer (and solvent) composition [84]. This effect can likely be used to determine compositional distributions in copolymers and blends [111]. Figure 10 compares the resolving power of Th-FFF and SEC on two polymers of similar molecular weight but varying chemical composition. The polymers coelute in SEC because their sizes are similar whereas Th-FFF resolves the polymers because they differ in chemical composition. 

From equation (2), it is possible to derive expressions for the differential and cumulative overall composition distributions (10). 

The reflectivity is defined as the ratio of the intensities of the reflected and incident beams and should be differentiated from the reflectance which is the ratio of the amplitudes of the incident and reflected waves. The reflectance in general is a complex number because there is usually a change in phase of a wave on reflection whereas reflectivity is a real number varying from zero to unity. The specular reflection can provide information on the composition distribution normal to the surface. The reflectivity is a function of both the angle of incidence of the beam to the surface and the refractive index changes of the substrate. The reflectivity is a function of the length scale of interactions of... 

For ethylene/1-olefin copolymers, chain crystaUizabihty is mainly controlled by the fraction of noncrystalhzable comonomer imits in the chain. Consequently, the differential Crystaf profile shown in Fig. 1, together with an appropriate cahbration curve, can be used to estimate the copolymer chemical composition distribution (CCD), also called the short-chain branch distribution. The CCD of a copolymer describes the distribution of the... 

Some of the principles as well as problems involved in the melting of random copolymers are found in olefin type copolymers. The melting temperatures of a large number of random type ethylene copolymers, as determined by differential scanning calorimetry, are plotted as a function of the mole percent branch points in Fig. 5.11. The samples represented in this figure are either molecular weight and compositional fractions or those with a narrow composition distribution with a most probable molecular weight distribution.(74) These samples were crystallized and heated rapidly. In this set of data there are two different copolymers that contain ethyl... 

Quite recently, Karoglanian and Harrison [128] noted the similarity of compositional distribution information generated by differential scanning calorimetry (DSC) and TREF. They showed that DSC thermograms could be generated from TREF chromatograms, and vice versa. 

The changes in the degree of crystallinity and subsequent melting behaviour, upon addition of an ICP to a PE matrix, have been studied using differential scanning calorimetry (DSC) (Figure 4.5). Linear low density PE, due to its broad and multimodal chemical composition distribution. 

The SCB distribution (SCBD) has been extensively studied by fractionation based on compositional difference as well as molecular size. The analysis by cross fractionation, which involves stepwise separation of the molecules on the basis of composition and molecular size, has provided information of inter- and intramolecular SCBD in much detail. The temperature-rising elution fractionation (TREE) method, which separates polymer molecules according to their composition, has been used for HP LDPE it has been found that SCB composition is more or less uniform [24,25]. It can be observed from the appearance of only one melt endotherm peak in the analysis by differential scanning calorimetry (DSC) (Fig. 1) [26]. Wild et al. [27] reported that HP LDPE prepared by tubular reactor exhibits broader SCBD than that prepared by an autoclave reactor. The SCBD can also be varied by changing the polymerization conditions. From the cross fractionation of commercial HP LDPE samples, it has been found that low-MW species generally have more SCBs [13,24]. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

At a more molecular level, the influences of the composition of the membrane domains, which are characteristic of a polarized cell, on diffusion are not specifically defined. These compositional effects include the differential distribution of molecular charges in the membrane domains and between the leaflets of the membrane lipid bilayer (Fig. 3). The membrane domains often have physical differences in surface area, especially in the surface area that is accessible for participation in transport. For example, the surface area in some cells is increased by the presence of membrane folds such as microvilli (see Figs. 2 and 6). The membrane domains also have differences in metabolic selectivity and capacity as well as in active transport due to the asymmetrical distribution of receptors and transporters. 

---