Size of a polymer

We began this section with an inquiry into how to define the size of a polymer molecule. In addition to the molecular weight or the degree of polymerization, some linear dimension which characterizes the molecule could also be used for this purpose. For purposes of orientation, let us again consider a hydrocarbon molecule stretched out to its full length but without any bond distortion. There are several features to note about this situation ... 

The size of a polymer molecule may be defined either by its mass (see Chapter 6) or by the number of repeat units in the molecule. This latter indicator of size is called the degree of polymerisation, DP. The relative molar mass of the polymer is thus the product of the relative molar mass of the repeat unit and the DP. 

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constraints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r)o Such an approach fails to take account of... 

The symbol x is used here and in following pages in a somewhat different sense than in earlier portions of the book, where it represents the number of structural units. The segment employed in mixing problems often is conveniently defined as that portion of a polymer molecule requiring the same space as a molecule of solvent it is unrelated to the size of the structural unit, which is of no interest here. The present x, like the previous one, defines the size of a polymer species, however. 

Intrinsic viscosity measurements were done with a large number of solvents varying in pH, ionic strength, etc., using Cannon-Ubbelohde semimicro dilution viscometers. This was done to provide information on the effect of mobile phase composition on the size of a polymer molecule in solution and thus to facilitate the interpretation of GPC behavior. 

Polymers are examples of organic compounds. However, the main difference between polymers and other organic compounds is the size of the polymer molecules. The molecular mass of most organic compounds is only a few hundred atomic mass units (for reference, atomic hydrogen has a mass of one atomic mass unit). The molecular masses of polymeric molecules range from thousands to millions of atomic mass units. Synthetic polymers include plastics and synthetic fibers, such as nylon and polyesters. Naturally occurring polymers include proteins, nucleic acids, polysaccharides, and rubber. The large size of a polymer molecule is attained by the repeated attachment of smaller molecules called monomers. 

Here Cp is the accessible pore volume fraction of the gel which is a fijnction of the pore size distribution as well as the size of a polymer molecule. The value of p is given by Equation 3 ... 

The size of a polymer molecule in solution is influenced by both the excluded volume effect and thermodynamic interactions between polymer segments and the solvent, so that in general =t= . The Flory (/S) expansion factor a is introduced to express this effect, by writing ... 

One such model is the ideal freely jointed segments chain (Fig. 6.14). In this model the polymer is considered to consist of a chain of n links. We call each chain link a subunit . Each subunit has a length l. This parameter / can correspond to the length of a monomer but it can also be shorter or longer. The angle between adjacent chain links is taken to be arbitrary. The chain forms a random coil. To characterize the size and volume of such a coil we use the mean square of the end-to-end distance R2. The square-root of this value — we call it the size of a polymer chain — is given by... 

Q q R i l, i 2 Rb Rd Rg RP Ro r rc S Electric charge (As), heat (J), quality factor of a resonator Heat per unit area (J m-2), integer coefficient Radius of a (usually) spherical object (m), gas constant Two principal radii of curvature (m) Radius of a spherical bubble (m) Radius of a spherical drop (m) Radius of gyration of a polymer (m) Radius of a spherical particle (m) Size of a polymer chain (m) Radius (m), radial coordinate in cylindrical or spherical coordinates Radius of a capillary (m) Entropy (J K-1), number of adsorption binding sites per unit area (mol m-2), spreading coefficient (Nm-1)... 

Explain why the crystallite size of a polymer varies with the degree of undercooling. 

Another measure of the size of a polymer coil is the rms radius of gyration, Rq, defined by... 

The most obvious effect of branching in a polymer is to change the overall molecular conformation. Branching will decrease the size of a polymer molecule over that of its linear homologue. Such a decrease in size can be characterized by the following relation ... 

Unambiguous determination of the conditions under which slippage occurs requires a technique able to measure the velocity of the fluid in the immediate vicinity of the solid wall over a thickness comparable to the size of a polymer chain, i.e. a few tens of nanometers. Classical laser Doppler velocimetry does not meet this requirement even if it allows for the determination of velocity profiles which clearly reveal a non-zero velocity within typically a few 10 pm from the wall. We have developed a new optical technique. Near Field Velocimetry (N.F.V.) [14], which combines Evanescent Wave Induced Fluorescence (E.WF.) [27] and Fringe Pattern Fluorescence Recovery After Photobleaching (F.P.F.R.A.P.) [28]. The former technique gives the spatial resolution normal to the solid wall, while the latter one enables the determination of the local velocity of the fluid. A major constraint of the technique is that it needs polymer molecules labelled with an easily photobleachable fluorescent probe. 

Regime (i). On scales up to the thermal blob size the chain is nearly ideal because excluded volume interactions are weaker than the thermal energy. The size of a polymer subsection grows as the square root of the number of monomers in it up to the thermal blob size j-. 

Compared to a linear chain of the same number of units, a branched chain is more compact. As a result, the impact of branching on the size of a polymer chain is to decrease the mean-square radius as branching increases. To assess the decrease in size because of branching, the mean-square radius of a branched polymer is compared to the size of a linear analog of identical molecular weight. Quantitatively, this was defined by Zimm and Stockmayer with the following branching or contraction factor ... 

Another useful parameter is the radius of gyration P, which is a measure of the effective size of a polymer molecule (it is the root mean-square distance of the elements of the chain from its centre of gravity). 

Molecular conformation is an important subject of concern in liquid crystalline polymers. It is also the basic subject of polymer science. The mean square end-to-end distance or the radius of gyration is often used to characterize the size of a polymer molecule. According to the definition, the end-to-end distance is the sum of each rod s vectors in sequence... 

Fundamentally, studying the behaviour of polymers in solution amounts to determining a statistical geometrical structure. For instance, we can interest ourselves in the size of a polymer in solution to see how it depends on the number of links (or on a quantity that is proportional to this number). 

Thus, we are led to express the size of a polymer as a function of its molecular mass, which can be considered as proportional to the number of links... 

The size of a polymer is defined experimentally by its radius of gyration, which can be determined by static scattering it is therefore an important quantity. 

Fig. 13.24. Size of a polymer in solution. First-order diagram contributing to (fc, — k 0,S S). The interaction line is made of small crosses in order to symbolize the screened interaction b q).

Before going to the asymptotic limit, we must re-express P(k) in terms of the real size of a polymer. Now, according to (10.7.6), we have... 

Equation (3.93) indicates that the size of a polymer coil increases with increasing molecular weight of the polymer. According to this equation, at a temperature equal to the Flory temperature (i.e., T = 0), - a = 0, i.e., a = 1. Thus,... 

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