Polymer segment fraction

The vapor pressure of polyisobutylene (PIB) with a number-average molar mass of JVfpie = 45 kg/mol dissolved in benzene (B) (Mfl = 78.11 g/mol) was measured as function of polymer concentration at 313.15 K. The vapor pressure of benzene at this temperature is 24.3 kPa. For a polymer segment fraction of 0.4 the vapor pressure was 22.6kPa. 

Using Eqs. (37) and (38), vapor-liquid and liquid-liquid equilibria can be calculated as discussed above. Reference 5 gives details of liquid-liquid equilibrium calculations in polydisperse polymer-solvent systems. In Figure 2.11 the experimentally determined phase behavior of three PEG-water systems is compared with the calculated phase behavior of these systems using Eq. (37c) to represent g,p as a function of temperature and polymer segment fraction. The parameters were fitted to the data [29]. 

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

Each lattice site is defined to have z nearest neighbors, and 0i and 02 > respectively, can be used to describe the fraction of sites which are occupied by solvent molecules and polymer segments. The following inventory of interactions can now be made for the mixture ... 

Next we use the Flory-Huggins theory to evalute AG by Eq. (8.44). As noted above, the volume fraction occupied by polymer segments within the coi domain is small, so the logarithms in Eq. (8.44) can be approximated by the leading terms of a series expansion. Within the coil N2 = 1 and Nj = (1 - 0 VuNa/Vi, where is the volume of the coil domain. When all of these considertions are taken into account, Eq. (8.108) becomes... 

FIG. 12 Segment density profile as function of the distance from the wall Z for flexible (empty symbols) and semi-rigid (full symbols) living polymer chains at T = 0.4 [28]. The fractional occupancy of lattice sites by polymer segments is shown for the layers in the left half of the box. Dashed lines are guides for the eyes. 

In practice, it is difficult to assign the number of repeating segments in solvent or polymer unambiguously. For this reason, it is usual in using Flory-Huggins theory to replace segment fraction (f) in equation (3) by volume fraction ( >). 

Fraction of surface sites which have adsorbed polymer segments... 

Figure 11.10 Volume fraction of interior polymer segments in solution. PPI dendrimer in methanol, PAMAM dendrimer in methanol, random hyperbran-ched polyol in methanol, and PS dendrigraft in cyclohexane...

The fraction of the domain that is not occupied by polymer is (1 - ) = (1 - n ul). For large n this fraction is close to unity, which shows that there is plenty of space within the domain for additional polymer segments. 

Fig. 2. Adsorption isotherms for different molecular weights42 . The adsorbance nNavj/Sd is expressed in multiples of the amount that would fill the first layer. The polymer volume fraction in solution is given by nNfv,/V where n is the number of segments...

The thickness of the adsorbed region B which consists of a number of layers, each with Ms sites, is assumed to be <5 Pf where 6 and P are constants. Silberberg47) assumes that <5 = 1/2 and P = 1. Beyond this region is there a homogeneous bulk solution of polymer volume fraction < > from which adsorption has taken place. Except for solvent, all that is found in the region B are loop segments of the adsorbed polymer chains. Thus, the fraction of adsorbed polymer segments is expressed by... 

The infrared band of a particular group in a polymer shifts when some groups of the polymer are adsorbed onto active sites, e.g. silanol groups on a silica surface23. This phenomenon has been used to measure the fraction of adsorbed polymer segments. The fraction of die surface sites occupied by adsorbed polymer segments can also be determined from the frequency shift of 1R band caused by the interaction between functional groups and an active site. 

To determine the conformation of adsorbed polymers the fraction of adsorbed polymer segments (p) and the fraction of the occupied surface sites (6) are often measured. Fontana and Thomas2 were the first to measure p and 6 by IR spectroscopy. At present, the application of IR spectroscopy is limited to finely divided substrates, e.g. nonporous silica, and requires that the surface area and the number of surface sites (e.g. the silanol groups) per unit area are accurately known in advance. The adsorbed amount T of polymer per surface site can be determined from adsorbance A(g/cm2) and the total area of the adsorbent. However, it can also be evaluated from the ratio 6/p. 

If the swelling behavior of carboxylated latexes could be characterized only by the increase in the volume fraction, i.e., in particle size, Eq. 1 would still hold but this is not the case in carboxylated latexes. The latex particles are swelled to a great extent, and the polymer segments are dissolved into the aqueous phase and interact between the particles by the hydrogen bonds and chain entanglements. Thus, Eq. 1 is not expected to hold at all and the interparticle interactions are expected to be dominant in the viscosity development of the latex. 

Xv being a phenomenological interaction parameter for the noncombinatorial part of the solute (polymer) chemical potential, defined on a volume fraction basis. Equations similar to equation (1.9) and (1.12) serve to define x on a segment fraction basis, x -... 

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