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Handbook of Solvents - George Wypych - ChemTech - Ventech!

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7.1 Introduction to diffusion, swelling, and drying 345<br />

The chemical potential is the sum <strong>of</strong> the terms <strong>of</strong> free energy <strong>of</strong> mixing and the elastic free<br />

0<br />

energy. At swelling equilibrium, μ 1 = μ 1 , and thus the left hand term <strong>of</strong> the equation<br />

becomes zero. The equation [7.1.5] takes the following form:<br />

0<br />

0<br />

2<br />

( μ − μ ) =−( μ − μ ) = RT[ln( 1−v)<br />

− v + χv<br />

]<br />

[7.1.6]<br />

1 1<br />

mix 1 1 el<br />

2 2 2<br />

where:<br />

v2 =n2V2/(n1V1 +n2V2) volume fraction <strong>of</strong> polymer<br />

n1,n2 moles <strong>of</strong> solvent and polymer, respectively<br />

V1,V2 molar volumes <strong>of</strong> solvent and polymer, respectively<br />

R gas constant<br />

T absolute temperature<br />

χ Flory-Huggins, polymer-solvent interaction parameter.<br />

The interaction between the solvent and solid matrix depends on the strength <strong>of</strong> such<br />

intermolecular bonds as polymer-polymer, solvent-solvent, and polymer-solvent. If interaction<br />

between these bonds is similar, the solvent will easily interact with polymer and a relatively<br />

small amount <strong>of</strong> energy will be needed to form a gel. 12 The Hildebrand and<br />

Scatchard hypothesis assumes that interaction occurs between solvent and a segment <strong>of</strong> the<br />

chain which has a molar volume similar to that <strong>of</strong> solvent. 12 Following this line <strong>of</strong> reasoning<br />

the solvent and polymer differ only in potential energy and this is responsible for their interaction<br />

and for the solubility <strong>of</strong> polymer in the solvent. If the potential energies <strong>of</strong> solvents<br />

and polymeric segments are similar they are readily miscible. In crosslinked polymers, it is<br />

assumed that the distance between crosslinks is proportional to the molecular volume <strong>of</strong> the<br />

polymer segments. This assumption is the basis for determining molecular mass between<br />

crosslinks from results <strong>of</strong> swelling studies.<br />

The result <strong>of</strong> swelling in a liquid solvent (water) is determined by equation: 13<br />

⎛∂T<br />

⎞<br />

⎜<br />

⎟<br />

⎝∂w<br />

⎟<br />

1⎠<br />

gl /<br />

P<br />

g ⎛∂μ<br />

⎞<br />

T⎜ 1 ⎟<br />

⎜<br />

⎝∂w<br />

⎟<br />

1⎠<br />

=<br />

gl /<br />

ΔH<br />

1<br />

T, P<br />

[7.1.7]<br />

where:<br />

T thermodynamic (absolute) temperature<br />

w1 mass fraction <strong>of</strong> solvent in gel at saturation concentration<br />

g phase symbol (for gel)<br />

l symbol for liquid<br />

P pressure<br />

g<br />

μ 1 chemical potential <strong>of</strong> solvent in gel phase dependent on temperature<br />

g/l<br />

g -l<br />

ΔH1 =H1 − H01<br />

is the difference between partial molar enthalpy <strong>of</strong> solvent (water) in gel and<br />

pure liquid solvent (water) in surrounding<br />

Contrast this with the equation for water in the solid state (ice):<br />

⎛∂T<br />

⎞<br />

⎜<br />

⎟<br />

⎝∂w<br />

⎟<br />

1⎠<br />

gcr /<br />

P<br />

g ⎛∂μ<br />

⎞<br />

T⎜ 1 ⎟<br />

⎜<br />

⎝∂w<br />

⎟<br />

⎠<br />

=<br />

ΔH<br />

1 T, P<br />

gcr /<br />

1<br />

where: cr phase symbol for crystalline solvent (ice)<br />

[7.1.8]

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