ABSTRACT

The change in free energy (∂G) of a system at constant pressure, volume and temperature equals the change in internal energy (∂U) minus the product of the change in entropy (∂S) times temperature (T):

∂ ∂ ∂G U T S= − (8.1)

The change in free energy of a †occulated colloidal network as a function of changes in extensional strain (∂ε), at constant temperature, pressure (P), volume (V) and composition (μ), where deformation takes place without any rupture of bonds equals the product of stress (σ) times volume (Sonntag and Strenge, 1987):

∂ ∂

∂ ∂

∂ ∂

G U T

S V

ε ε ε σ

= ⎛

= ⋅

(8.2)

The main assumption in this model is that the change in free energy of the network upon deformation arises due to changes in elastic energy (∂Eelastic), namely,

∂ ∂

∂ ∂

G E

ε ε =

(8.3)

The elastic energy of the network can be expressed in terms of deformation (ΔL), or strain (ε = ΔL/L),

E k L L kelastic = =

1 2

1 2

2 2 2( )Δ ε

(8.4)

where k is the elastic constant of the network. The elastic constant k can be substituted by A · E/L, where E corresponds to the Young’s modulus, A is the area over which the force is applied, and L is the size of the system. The product of the area times the length corresponds to the volume of the network, while the product of the Young’s modulus times the strain corresponds to stress. Thus, the change in elastic energy as a function of strain can be expressed as

∂ ∂

∂ ∂

E LAE VE Velastic

ε ε ε ε σ=

= =

1 2

(8.5)

Sonntag and Strenge (1987) have shown that the entropic contribution to the change in free energy of a †occulated colloidal network upon a small elastic deformation is much smaller than the contribution from changes in the internal energy of that network. Thus, for the case where ∂U ≫ T∂S,

∂ ∂

∂ ∂

G U VE

ε ε ε≈ =

(8.6)

Upon integration and rearrangement, an expression for the Young’s modulus of the network can be obtained,

E

U

V =

2 2

Δ

ε (8.7)

where ΔU corresponds to the change in the internal energy of the network upon deformation, which equals the total interaction energy per †oc.