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Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
Download (direct link): polymerchemistry1984.djvu
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volume of this solvent can be subdivided into two categories: the
stagnant solvent in the pores (subscript i for internal) and the
interstitial liquid in the voids (subscript v) between the packing
particles:
Vso.vent = Vv+Vj
(9.104)
The entire interstitial volume must pass through the column before any
polymer emerges. Then the first polymer that does appear is the one with
the highest
Gel Permeation Chromatography
647
molecular weight. This solute has spent all its time in the voids-not the
pores- of the packing and passes through the column with the velocity of
the solvent.
Progressively smaller molecules have access to successively larger
fractions of the internal volume. Therefore, as Vj emerges, consecutive
fractions of the polymer come with it. Thus we can write the retention
volume for a particular molecule weight fraction as
where is a function of both the pore size and the molecular size and
indicates what fraction of the internal volume is accessible to that
particular solute. The relationships between VR, Vv, Vj, and KVj are also
shown in Fig. 9.14. When = 0, the solute is totally excluded from the
pores; when = 1, it totally penetrates the pores.
It is instructive to consider a simple model for the significance of
the constant in Eq. (9.105). For simplicity, we assume a spherical
solute molecule of radius R and a cylindrical pore of radius a and length
1. As seen in Fig. 9.15a, an excluded volume effect prevents the center
of the spherical solute molecule from approaching any closer than a
distance R from the walls of the pore. This effectively decreases the
volume accessible to the solute to a smaller cylinder of radius a - R. In
this accessible cylinder the concentration of the solute is the same as
in the interstitial fluid outside the pore. The excluded volume-that
shell of thickness R around the walls of the pore-is devoid of solute.
Hence the average concentration of solute in the pore as a whole is less
than that outside the pore. The fraction of the external concentration in
the pore is given by the ratio of the accessible volume to the actual
volume of the cylindrical pore: n(a - R)21/21. This fraction gives
for the case of spherical solute molecules in cylindrical cavities. If we
assume that the pore is long enough to neglect end effects, we have
Note that the fraction is zero when R = a, and unity when R = 0.
This simple model illustrates how the fraction and, through it, VR
are influenced by the dimensions of both the solute molecules and the
pores. For solute particles of other shapes in pores of different
geometry, theoretical expressions for are quantitatively different, but
typically involve the ratio of solute to pore dimensions.
The extension of these ideas to random coils can proceed along two
lines. In one analysis the coil domain is visualized as a sphere, as in
the case above, with rg taking the place of R. Alternatively, statistical
methods can be employed

(9.105)

(9.106)
Figure 9.15 Schematic illustration of size exclusion in a cylindrical
pore: (a) for spherical particles of radius R and (b) for a flexible
chain, showing allowed (solid) and forbidden (broken) conformations of
polymer.
Frictional Properties of Polymers in Solution
Gel Permeation Chromatography
649
to consider those conformations of a random chain which are excluded for
a coil confined to a pore. This latter situation is illustrated in Fig.
9.15b. Figure 9.15b represents by solid and broken lines two
conformations of the same chain, with the filled-in repeat unit being
held in a fixed position. If the molecule were in bulk solution, both
conformations would be possible. In a pore, represented by the enclosing
circle in Fig. 9.15b, the broken line conformation is impossible. This is
equivalent to a decrease in entropy for the coil in the pore, and the
effect can be translated into an equilibrium constant between the solute
in the pore and in the bulk solution. The factor in Eq. (9.105) is just
such a constant-the distribution coefficient-and can be evaluated by this
approach for pores of different shape.
Figure 9.16 shows the theoretical predictions for versus rg/a
compared with experimental findings. The solid line is drawn according to
the statistical theory. The experimental points correspond to the same
porous beads used as the stationary phase with their pore size analyzed
by two different experimental procedures: mercury penetration (circles in
Fig. 9.16, a = 21 nm)and gas adsorption (squares in Fig. 9.16, a = 41
nm). We can draw several conclusions from an examination of Fig. 9.16:
1. The characterization of the solid is also a source of discrepancy:
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