HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

Transcript

1 HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying the chain rule: P zz = ψ λ z = ψ λ z = ψ = ψ λ. The elastic energy per unit volume can be written as: ψ = [ λ z + ] λ z. Hence: ψ = λ z λ z.4 Finally, the nominal stress P zz can be written as: P zz = λ z λ z = λ z λ z.5 In order to plot the stress versus the stretch we define a normalized stretch: P zz = P zz = λz λ z.6 In the figure bellow we can see how as we increment the compression on the hydrogel, the stress also increments.

2 Figure : Normalized nominal stress P zz versus λ z b In the slow loading case, we know that the elastic energy per unit volume can be expressed as: f gel = λ z + λ z + f mix.7 Because the loading is very slow, the system is always in an equilibrium state. Therefore: f gel =.8 f gel = λ z f mix + f mix =.9 And rearranging the terms we obtain: λ z λ z = f mix + f mix = f mix + f mix. Remind that the osmotic pressure can be computed as: Π = f f + f.

3 And for long chains N the mixing energy per unit volume is: k B T ν [ ln + χ ]. From the previous expression it s easy to see that f =, so we can conclude: λ z = Π. c First of all we need to derive an expression for Π: Π = f + f = k BT [ ln + χ ] + k BT [ ln + χ χ ] ν ν = k BT [ ln + + χ ].4 ν Which means that the relationship λ z can be expressed as: Or = k BT [ ln + + χ ].5 λ z ν ν + [ ln + + χ ] =.6 λ z For a given value of χ, using any solver we can find the pairs λ z where λ z = λz ν. As expected, when λ z tends to zero, the volume fraction tends to one, because the polymer is so compressed that there is almost no solvent in it. Then, λ x is directly found thanks to the following relation: λ x = λ z = λ z ν.7 Since we know the values of the pairs λ z and we can directly plot the pairs λ x λ z where: λ x = λ x ν.8

4 Figure : versus the normalized λ z for three different values of χ Figure : Normalized λ x versus versus normalized λ z, computed for a value of χ = d As before we know that P zz = ψ.9 4

5 For slow loading the elastic free energy density can be written as: ψ = λ z + λ x = λ z + λ z. And the derivative then becomes: ψ = λ z λ z. So the nominal stress becomes: P zz = λ z λ z. The term <. Hence comparing this equation to.6 we can clearly see that the stress is much less in this case. The more solvent that leaves, the less stress we have. Finally, in order to plot the stress we define: P zz = λ z λ z. Figure 4: Normalized nominal stress P zz versus λ z 5

6 Problem In order to prove that the gel shrinks, we first compute the volume fraction of the polymer that we added. Considering that we added a total amount of polymer V p we get: sol = V p V V = V p V Vg = sol V sol V p. And if we take the derivative of sol respect to we obtain: = V sol V p V sol sol V sol V p = V p V. Since this derivative is negative, it means that the slope of the curve sol is negative and therefore, whenever sol decreases, the gel block must shrink increases. Which is due to the fact that when we add polymer the mixing energy on the gel is reduced some solvent wants to mix with the new polymer instead., sol V g, V g, f el = trc = [ ] / V Vg f mix sol = sol f mix sol sol Figure 5: Scheme of the full process carried out in this experiment Next we want to find the equilibrium state after introducing the new polymer. First we write the total energy in the two states. On the first state, since there is no polymer in the solution, the total energy is the elastic energy of the gel plus its mixing energy: U tot = V g f sol + f el. On the second state we added some polymer to the solution which affected the gel, that changed its volume to V g. This polymer, however, does not have elastic energy because the chains are not cross-linked. Hence, we can write the total energy on state as the elastic energy on the gel plus the mixing energy on both gel and solution: U tot = V g f sol + f el + V V g f sol sol.4 Which means that the difference in energy between the two states is: U tot = f el f el + V g f sol V g f sol + V V g f sol sol.5 6

7 Then, we consider the total energy per unit reference volume of gel: f tot = U tot = f el f el + V g f sol V g f sol + V V g f sol sol.6 Because the amount of polymer in the block is the same during all the process we can establish the following relationship: V g = V g =.7 So we obtain: f tot = f el f el + f sol + V f sol + f sol sol.8 The system will find the minimum energy state, which means that the variation of energy will be minimum too, hence: f tot = = f el + f sol V f sol sol.9 The elastic energy of a system which shrinks λ in all directions can be written as: f el = trc = λ = [ ]. Which means that: f el =. Then, going back to.9 we obtain: = f sol + f sol + V f sol sol + f sol sol. And rearranging the terms we can write: = f sol + f sol } {{ } Π V +f sol f sol sol sol +. For the term in parenthesis, we can write: V Then, using the chain rule we can express: = V.4 7

8 f sol sol = f sol sol sol sol.5 And from equation. it s easy to show that sol = sol V sol = sol V V.6 Therefore, we obtain = Π + f sol sol V sol f sol sol V.7 sol = Π + f sol sol sol f sol sol sol }{{} Π sol.8 So we finally obtain: = Π Πsol.9 8