Concentration of junctions

The amount of branching introduced into a polymer is an additional variable that must be specified for the molecule to be fully characterized. When only a slight degree of branching is present, the concentration of junction points is sufficiently low that these may be simply related to the number of chain ends. For example, two separate linear molecules have a total of four ends. If the end of one of these linear molecules attaches itself to the middle of the other to form a T, the resulting molecule has three ends. It is easy to generalize this result. If a molecule has v branches, it has v 2 chain ends if the branching is relatively low. Branched molecules are sometimes described as either combs or... 

If the concentration of junction points is high enough, even branches will contain branches. Eventually a point is reached at which the amount of branching is so extensive that the polymer molecule becomes a giant three-dimensional network. When this condition is achieved, the molecule is said to be cross-linked. In this case, an entire macroscopic object may be considered to consist of essentially one molecule. The forces which give cohesiveness to such a body are covalent bonds, not intermolecular forces. Accordingly, the mechanical behavior of cross-linked bodies is much different from those without cross-linking. 

Let us now assume momentarily that the elastomers are perfect, in the sense that all functional groups (originally present in equivalent amounts) have reacted in such a manner that no sol exists. Under such idealized conditions, each molecule of monohydroxy PPO will in effect produce one inactive junction. (Strictly speaking, i molecules of monohydroxy PPO can produce somewhat more than i inactive junctions.) Thus, the concentration of junctions in the network becomes about 0.073 mole/kg instead of 0.100 mole/kg (Table II). As each junction gives 3/2 chains, the modulus, after equating h to zero in eq 1, is given by... 

Fig. 46 Schematic diagram of elemental process during transition from Hex cylinder to bcc sphere (i) undulation of interface (a, b), (ii) break-up of cylinders into ellipsoids (b, c), (iii) relaxation of domains from ellipsoids into spheres (c, d), and (iv) relaxation in junction distribution to attain uniform distribution (d, e). Pole where concentration of junction points is low may work as memory of grain conservation upon reverse transition from bcc sphere to Hex cylinder. Small arrows in part (b) indicate diffusion of chemical junctions along interface in process (ii). From [136], Copyright 2000 American Chemical Society...

Figure 10.34, curve 1 shows an example of such a relationship. The equation was developed based on the assumption that the glass transition depends on concentration of active groups in polymer, capable of forming labile junctions of space network. The increase in polymer modulus on cooling can be interpreted as an increase in density of labile physical bonds network. The attaimnent of certain concentration of junctions in this... 

Since at long times pendant chains do not contribute to permanent elastic properties, the elastic equilibrium behavior of networks containing these chains should not differ substantially from that of regular networks. The elastic modulus from a network with pendant chains can then be obtained from the molecular theories of rubber elasticity provided that the concentration of elastically active network chains (v) can be calculated accurately. Depending on the different approaches that can be used for the rubber elasticity theory, the calculation of some other parameters, like the concentration of junctions points (p) and trapped entanglements (Te), also may be needed. 

Liquid Junction Potentials A liquid junction potential develops at the interface between any two ionic solutions that differ in composition and for which the mobility of the ions differs. Consider, for example, solutions of 0.1 M ITCl and 0.01 M ITCl separated by a porous membrane (Figure 11.6a). Since the concentration of ITCl on the left side of the membrane is greater than that on the right side of the membrane, there is a net diffusion of IT " and Ck in the direction of the arrows. The mobility of IT ", however, is greater than that for Ck, as shown by the difference in the... 

Electrochemical methods covered in this chapter include poten-tiometry, coulometry, and voltammetry. Potentiometric methods are based on the measurement of an electrochemical cell s potential when only a negligible current is allowed to flow, fn principle the Nernst equation can be used to calculate the concentration of species in the electrochemical cell by measuring its potential and solving the Nernst equation the presence of liquid junction potentials, however, necessitates the use of an external standardization or the use of standard additions. 

The concentration of crosslink junctions in the network is also important if too low, flow will be possible if too high, the maximum attainable elongation will be decreased. From the point of view of theoretical analysis, the length of chain between crosslink points must be long enough to be described by random flight statistics. 

In general, in a planar process, — junctions are formed just below the surface of a siUcon wafer by the implantation of donor ions into a type region or acceptor ions into an n-ty e region. Thus, the general concern is with -p or -n junctions. As the initial wafer concentration of acceptors or donors in sihcon increases from 10 to 10 cm increases from about 0.81 to 1.04 V for a p n junction and is about 10 mV higher for an -p... 

An electrode potential varies with the concentration of the ions in the solution. Hence two electrodes of the same metal, but immersed in solutions containing different concentrations of its ions, may form a cell. Such a cell is termed a concentration cell. The e.m.f. of the cell will be the algebraic difference of the two potentials, if a salt bridge be inserted to eliminate the liquid-liquid junction potential. It may be calculated as follows. At 25 °C ... 

Standard potentials Ee are evaluated with full regard to activity effects and with all ions present in simple form they are really limiting or ideal values and are rarely observed in a potentiometric measurement. In practice, the solutions may be quite concentrated and frequently contain other electrolytes under these conditions the activities of the pertinent species are much smaller than the concentrations, and consequently the use of the latter may lead to unreliable conclusions. Also, the actual active species present (see example below) may differ from those to which the ideal standard potentials apply. For these reasons formal potentials have been proposed to supplement standard potentials. The formal potential is the potential observed experimentally in a solution containing one mole each of the oxidised and reduced substances together with other specified substances at specified concentrations. It is found that formal potentials vary appreciably, for example, with the nature and concentration of the acid that is present. The formal potential incorporates in one value the effects resulting from variation of activity coefficients with ionic strength, acid-base dissociation, complexation, liquid-junction potentials, etc., and thus has a real practical value. Formal potentials do not have the theoretical significance of standard potentials, but they are observed values in actual potentiometric measurements. In dilute solutions they usually obey the Nernst equation fairly closely in the form ... 

An element of uncertainty is introduced into the e.m.f. measurement by the liquid junction potential which is established at the interface between the two solutions, one pertaining to the reference electrode and the other to the indicator electrode. This liquid junction potential can be largely eliminated, however, if one solution contains a high concentration of potassium chloride or of ammonium nitrate, electrolytes in which the ionic conductivities of the cation and the anion have very similar values. 

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