1. Swelling of Network Structures: The essentials of the equation-state approach Costas Panayiotou University of Thessaloniki, Greece

2. The salient features of the development: The equation-of-state contributions The elastic and interlinking terms The challenges of aqueous systems and supercritical fluids Concepts from rubber elasticity

3. Uncrosslinked and crosslinked networks

4. Definitions in crosslinked networks

5. Extension of a single chain

6. The key assumptions in Rubber Elasticity

7. Entropy change upon deformation

8. The generalized formulation Most often the starting point is an expression for free energy change of the type: For the elastic term, the most often used expression is the Flory-Rehner: Birshtein: Flory-Rehner modif.: For f=4 it becomes: Adding the 2 nd term from mixing entropy, we obtain

9. The generalized formulation Flory 1950 says: or: Which leads to the widely used swelling equation: where: For swelling applications recall that

10. Most often used expressions for the elasticity term Flory-Rehner (with mix entropy term): Flory-Rehner (without mix entropy term): Birshtein: Kuhn: Non-Gaussian:

11. They are practically identical! Comparison of Birshtein and Flory 1950 (f=4) elastic expressions (no mixing term)

12. Significant differences! Comparison of Birshtein, Flory 1950, mod.Flory-Rehner (f=4), and Kuhn elastic expressions

13. The differences are magnified in small swelling degrees. Comparison of Birshtein, Flory 1950, mod.Flory- Rehner (f=4), and Kuhn elastic expressions

14. Comparison of Birshtein, Flory-Rehner with and without mixing term (f=4), non-Gaussian, and Kuhn elasticity expressions Only the non-Gaussian is different at high swelling degrees.

15. The Partition Function Approach to Rubber Swelling The Flory-Rehner rationale will be combined with the simple LF approach, or: The Flory-Rehner term consists of two terms, the elastic and the chain interlinking terms, or: Most often f=4. The value of Δτ will not concern us

16. The Partition Function Approach to Rubber Swelling The LF partition function will be formulated for a system of N 1 solvent molecules and n 2 = ν e effective (between cross-links) polymer chains, or: If r 1 and R 2 are the numbers of the corresponding segments, then: The symbols have their usual meaning

17. The Partition Function Approach to Rubber Swelling The resulting key equations are: and From these we easily derive the equations for pure fluids

18. The Partition Function Approach to Rubber Swelling Applications Calculations with M X = 3000

19. The Partition Function Approach to Rubber Swelling

21. The Free-Energy Change Approach to Rubber Swelling As mentioned before, in this approach we must specify the mixing term and the elasticity term, or: For the elasticity term we will use Birshtein’s while for the mixing term we may use any NRHB version, or: And Where, here m is the number of charged segments per network chain

22. The Free-Energy Change Approach to Rubber Swelling The key equations where In general where For Q R we may use the LF (NR-LFHB), the Guggenheim (QCHB), or the Stavermann (NRHB) expressions

23. The Free-Energy Change Approach to Rubber Swelling The key equations By opting for NR-LFHB, we have In this case we have and For the physical part of the chem. potential we have where

24. The Free-Energy Change Approach to Rubber Swelling The key equations The contribution from hydrogen bonding is: The contribution from the elastic term is: and the contribution from the ionic term is In total:

25. The Free-Energy Change Approach to Rubber Swelling Applications The swelling capacities of PEO gels of different cross-link densities in chloroform as a function of temperature. Symbols are experimental data53. The two dashed lines are the predictions of the model for the lowest (upper curve) and the highest (lower curve) cross-link densities. Solid lines are calculated by slightly varying the ζ binary parameter Symbol Cross-link density (mol m -3 ) ζ 12 1HT6000.991 2HT10500.992 3HT14000.993 4HT20500.999

26. The Free-Energy Change Approach to Rubber Swelling Applications The swelling capacities of PEO gels of different cross-link densities in water as a function of temperature. Symbols are experimental data53. The dashed line is the predictions of the model for the lowest cross-link density assuming availability for hydrogen bonding of all oxygens of PEO chains. Solid lines are calculated by varying the number of available oxygen sites per PEO chain (numbers near each line).

27. The Free-Energy Change Approach to Rubber Swelling Applications The swelling ratio of water + NIPA gel as a function of temperature. Symbols are experimental data from Marchetti et al.55. The dashed line are the LFHB model calculations by Lele et al.53.

28. The rubber swelling phenomenon is rather well understood Key Conclusions The connection with EOS is rather well established NRHB + rubber swelling eqns: a powerful tool for many important applications - especially in hydrogels and in gel foaming with SCF