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

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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 Ê 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 Ê = 0, the solute is totally excluded from the

pores; when Ê = 1, it totally penetrates the pores.

It is instructive to consider a simple model for the significance of

the constant Ê 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/ýòà21. This fraction gives Ê

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 Ê 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 Ê 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 Ê 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 Ê 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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