Polymer number fraction

Dividing both sides of Eq. (6.58) by [M-], the total radical concentration, gives the number fraction of n-mer radicals in the total radical population. This ratio is the same as the number of n-mers in the sample containing a total of N (no subscript) polymer molecules ... 

This expression gives the number fraction or mole fraction of n-mers in the polymer and is thus equivalent to Eq. (5.25) for step-growth polymerization. 

Run Number Fraction Dead Polymer, (j) Run Number Fraction Dead Polymer, (f)jj... 

Figure 1 Is a flow sheet showing some significant aspects of the Iterative analysis. The first step In the program Is to Input data for about 50 physical, chemical and kinetic properties of the reactants. Each loop of this analysis Is conducted at a specified solution temperature T K. Some of the variables computed In each loop are the monomer conversion, polymer concentration, monomer and polymer volume fractions, effective polymer molecular weight, cumulative number average molecular weight, cumulative weight average molecular weight, solution viscosity, polymerization rate, ratio of polymerization rates between the current and previous steps, the total pressure and the partial pressures of the monomer, the solvent, and the nitrogen.

Mass spectrometry can be used to measure the molar mass distribution (MMD) of a polymer sample by simply measuring the intensity, Nt, of each mass spectral peak with mass m . This is due to the fact that mass spectrometers are equipped with a detector that gives the same response if an ion with mass 1 kDa or 100 Da (actually any mass) strikes against it. In other words, the detector measures the number fraction and this implies that Nt also represents the number of chains with mass m,. Thus, the number-average molar mass, Mn, is given by ... 

Here Nn gives the number fraction of n-mer in the total number of polymer molecules present, N. The weight fraction Wn, similarly, is given by... 

If the polymer concentration increases so that the number of high order bead-bead interactions is significant, c>>c =p, (when c is expressed as the polymer volume fraction. Op), the fluctuations in the polymer density becomes small, the system can be treated by mean-field theory, and the ideal model is applicable at all distance ranges, independent of the solvent quaUty and concentration. These systems are denoted as concentrated solutions. A similar description appHes to a theta solvent, but in this case, the chains within the blobs remain pseudoideal so that =N (c/c ) and Rg=N, i.e., the global chain size is always in-... 

N is the total number of monomers, (p the polymer volume fraction and Pi and Pi/2 the form factors of the total copolymer and of the single blocks respectively. 12=Vd=Vh is the excluded volume interaction parameter which relates to the second virial coefficient A2=vN/ 2Mc). 

Since N x is synonymous with the mole or number fraction of molecules in the polymer mixture that are x-mers (i.e., that contain x structural units), then... 

Consider the situation where one polymer molecule is produced from each kinetic chain. This is the case for termination by disproportionation or chain transfer or a combination of the two, but without combination. The molecular weight distributions are derived in this case in exactly the same manner as for linear step polymerization (Sec. 2-7). Equations 2-86, 2-88, 2-89, 2-27, 2-96, and 2-97 describe the number-fraction, number, and weight-fraction... 

Nj. is synonymous with the mole- or number-fraction distribution. When x is an odd number, there are x pathways and the (x — 1) term in Eq. 3-190 should be replaced by x. For polymerizations yielding high polymer, the difference between x and (x + 1) is negligible and can be ignored. [Some derivations of size distributions show exponents y and z, respectively, in Eqs. 3-186 and 3-187 instead of (y — 1) and (z — 1), which results in an exponent of x instead of (x — 1) in Eqs. 3-189 and 3-190). The differences are, again, unimportant for systems that yield high polymer.]... 

Polymer Number of Weight fraction Tm(°C) Water absorption... 

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... 

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. 

Fig. 19 Polymer volume fraction (j) = h( iy //jsw in swollen films of two PS-h-PB diblock copolymers (.S //47 (circles) and SB10 (squares)) that have been equilibrated at p/po 50% of the partial chloroform (non-selective solvent) vapor pressure [114], and of SV films (triangles) equilibrated under p/po = 80% of toluene (selective solvent) [119] versus the number of layers (film thickness normalized by the respective structure dimension in bulk)...

As we will see, some anomalies in the isotopic composition of carbon, hydrogen and oxygen can be explained on the basis of this assumption, and we will start the discussion with the deuterium-rich matter in carbonaceous chondrites. This deuterium-rich matter is essentially present as complex macromolecules 70 73 96 97). The carbon in these samples is essentially normal 76,98). For some polymer-type fractions, the deuterium content is up to 32 times higher than the galactic value (D/H 2 x 10s in the number of atoms per cubic centimeter). High deuterium enrichments are known in interstellar molecules and the mechanism of this enrichment is fully understood. For an excellent review dealing with interstellar chemistry, see the paper by Winnewisser 99) and the previously mentioned book by Duley and Williams 13). 

From the picture presented in Fig. 7, one can expect that the sequential hydrophobization of a polymer coil should lead to a copolymer with a non-random sequence distribution. This is indeed the case. As an example, let us consider the average number fractions of blocks consisting of l neighboring amphiphilic monomers, /a( ), occurring in a copolymer chain. Some results are shown in Fig. 8 on a semilogarithmic scale. 

Fig. 29 Characteristic wave number q as a function of polymer volume fraction for the systems of protein-like copolymers with L = 63 and random-block copolymers with different block lengths. The domain spacing is defined as r = lir/q. Adapted from [153]...

Number fraction of polymer chains having length /... 

A schematic T-c diagram is shown in Fig. 3 for fixed values of n, /, and w. Note that the excluded volume v is related to T through Eq. (4), and the polymer volume fraction

mass concentration c and chain number density p/n through... 

Fig. 3. Typical TLC chromatograms of cellulose triacetate (CTA) fractions and whole polymer (Ac w = 61.0wt%) having various Mw 12) solid lines fractions broken line whole polymer numbers on curves represent 10 4 Mw. (Rf = rate of flow.)...

The average number of polymer chains per unit volume of solution Nv (pm3) can be obtained from the density p= 1.13 g/cm3 of PEO and the polymer volume fraction Vj inside the clay gel through the relation... 

Let N cylindrical rods be situated in volume V, their concentration being c = N/V. The polymer volume fraction in the solution is then - jrpcd V4. Let us introduce the orientational distribution function for the rods f(u) cf(u)df2 is the number of rods per unit volume, which have the orientations within the small spatial angle dQ around the unit vector u. It is dear that in the isotropic state f(u) = const = l/4a. In the liquid-crystalline state the function f(u) has two maxima along the anisotropy axis. 

Let us assume that the solution of semiflexible macromolecules occupies the volume V. Let i be the polymer volume fraction in the solution. Then, the average concentration of segments is c = 4 fibrpd3, the total number of segments N = Vc, and, finally, the average concentration of macromolecules is c//L, where L denotes the contour length of one macromolecule. 

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