1. Costas Panayiotou University of Thessaloniki, Greece
Statistical Thermodynamics of Polymers and their Mixtures: The equation-of-state approach
Costas Panayiotou
University of Thessaloniki, Greece

2. Why equation-of-state?
Typical systems that require it:
Foaming
Micronization
Impregnation
Polymer
Alloying
Retrograde
Vitrification
Interf. Tension
At high T, P

3. Course outline:
Revisiting idealities and regularities in Thermodynamics
Lattice, Off-lattice, Lattice-Fluid
The non-random distribution
Flory-Huggins theory
Free volume – the pressure effect
SL and PV theories
Hydrogen bonding
The NRHB and NRCOSMO approach
Applications from SC fluids to polymer glasses and gels

4. 2-D visualization of the Lattice
N1 molecules of component 1
N2 molecules of component 2
The molecules are assumed to be arranged on a quasi-lattice
Each lattice cell has a volume, υ*
Coordination number, z
Pair interaction energy, εij
Average interaction energy per molecule or segment, ε* = z/2 εij

5. Mixing in ideal Solutions
N1 molecules of 1
N2 molecules of 2

6. Mixing in ideal Solutions
N1 molecules of 1
N2 molecules of 2
The molecules have a comparable molar volume
Zero volume of mixing, ΔVM=0
This solution is ideal when, in addition, molecules exchange their position with no change in the total energy of the system or, ΔΗΜ=0

7. ΔVM=0
The solution is ideal, ΔΗΜ=0
N1 molecules 1
N2 molecules 2
N0 = N1 +N2
Eqn. Boltzmann: S = k ln(W)
Number of discrete arrangements :
Stirling approximation :

8. ΔVM=0
The solution is ideal, ΔΗΜ=0
N1 molecules 1
N2 molecules 2
N0 = N1 +N2
Mixing entropy:
Entropy of pure components:
xi=1, then Si=-k Ni ln(xi) = 0
The mixing process in ideal solutions is spontaneous
Since x<1 and, thus, ln(x)<0: ΔSM>0 και ΔGM<0

9. Mixing Ideal Solutions
N1 molecules 1
N2 molecules 2

10. The mixing process in ideal solutions is spontaneous
Since x<1 and thus, ln(x)<0: ΔSM>0 και ΔGM<0

11. Real Solutions:
Real Solutions:
Athermal: ΔΗΜ=0, ΔSM does not obey the equation for ideal solutions
Regular: ΔΗΜ≠0, ΔSM does obey the equation for ideal solutions
Irregular (non-ideal): ΔΗΜ≠0, ΔSM does not obey the equation for ideal solutions

12. Polymer Solutions: The theory of Flory - Huggins
1. Chain transfer from solid state to a disordered state in order to be arranged randomly in the quasi-lattice

13. Polymer Solutions: The theory of Flory - Huggins1.
Chain transfer from solid state to a disordered state in order to be arranged randomly in the quasi-lattice

14. Polymer Solutions: The theory of Flory - Huggins
1. Chain transfer from solid state to a disordered state in order to be arranged randomly in the quasi-lattice
2. Mixing chain molecules with solvent molecules
Molar volume of solvent: V1
Molar volume of polymer: V2
Number of chain segments: r = V2 / V1
Total number of lattice sites: Ν0 = Ν1 + rΝ2
W: Number of discrete arrangements
Boltzmann equation: S = k ln(W)

15. Polymer Solutions: The theory of Flory - Huggins
W: Number of discrete arrangements
Εqn. Boltzmann: S = k ln(W)
When Ν2=0: S=S1=0
When Ν1=0: φ2 φ1
φι=niVi/V

16. Polymer Solutions: The theory of Flory - Huggins
In regular or in irregular solutions: ΔΗΜ≠0
Energy of interaction 1-1: ε11
Energy of interaction 2-2: ε22
z (1-1) +z (2-2)  z(1-2) +z (2-1)
½ (1-1) + ½ (2-2)  (1-2)
Δε12 = ε12 – ½ (ε11+ε12)
ΔΗΜ = Ν12 Δε12 1 2

17. Polymer Solutions: The theory of Flory - Huggins
Energy of interaction 1-1: ε11
Energy of interaction 2-2: ε22
½ (1-1) + ½ (2-2)  (1-2)
Δε12 = ε12 – ½ (ε11+ε22)
ΔΗΜ = Ν12 Δε12
Each chain segment has (z-2) external neighbors
Each chain end segment has z-1 external neighbors
Total number of external neighbors per chain:
Q = (r-2)(z-2) +2(z-1) = zr - 2r +2
Q ≈ zr ( Flory – Huggins approximation)
Total number of external neighbors of type1 per chain:
z r φ1
ΔΗΜ = Ν2 z r φ1 Δε12

18. Polymer Solutions: The theory of Flory - Huggins
ΔΗΜ = Ν2 z r φ1 Δε12
Ν2 z r φ1 = z Ν1φ2
ΔΗΜ = Ν2 z r φ1 Δε12 = Ν1 z φ2 Δε12
k T χ12 = z Δε12
ΔΗΜ = k T χ12 N1 φ2
χ12: Flory – Huggins parameter
χ12 = 0 athermal solutions
χ12>0 mixing endothermic
χ12<0 mixing exothermic

19. Polymer Solutions: The theory of Flory - Huggins
ΔΗΜ = k T χ N1 φ2
Ideal solutions
ΔGΜ = kT [N1ln(x1) + N2 ln(x2)]

21. Phase diagram for a binary polymer – solvent system – Binodals and Spinodals

22. Typical loopholes in L-L calculations

23. Types of binary L-L equilibria

24. The F-H χ parameter in real systems

25. Major drawbacks of F-H theory
It describes only UCST
It cannot describe volumetric behavior
It does not apply to the gaseous or supercritical state
χ is not independent of composition, or P

26. The dependence of χ on free-volume key to LCST behavior

27. The pressure affects phase equilibria of polymer mixtures

28. Sanchez – Lacombe Equation of State Model (Lattice Fluid Theory)
r1 segments per molecule
for component1
N0 empty cells
N1 molecules of component 1
Total cells: Nr = N0 + r1N1
Total external contacts per molecule:
Q = q z = r (z – 2) +2
Energy:
-Ε = Nii εii
-Ε = q N Θr z/2 εii ~ rN(rN/Nr)ε* (S-L LF)
Volume: V = (N0+r1N1) υ*
G = H – TS
= E + PV – TS
= E + PV – kT ln(Ω)

29. Sanchez – Lacombe Equation of State Model (Lattice Fluid Theory)
G = H – TS
= E + PV – TS
= E + PV – kT ln(Ω)
At constant pressure (P) and temperature (Τ), a system reaches the equilibrium when G  minimum
In pure fluid systems:
μ = G

30. Sanchez – Lacombe Equation of State Model (Lattice Fluid Theory)
G = H – TS
= E + PV – TS
= E + PV – kT ln(Ω)
In multicomponent mixtures the chemical potential is the partial molar Gibbs free energy,
G = Σ Νi μi :

32. Non random distribution of molecular segments in the quasi lattice
In the previous examples, it was assumed that:
Molecular segments and empty cells are distributed in the quasi lattice in a random way
Is this a reasonable assumption ?
Contact energy 1-1: ε11
Contact energy 2-2: ε22
Contact energy 1-2: ε12
Molecules prefer as neighbors other molecules of the same type if the 1-1 and 2-2 contact interactions are stronger than the 1-2 cross contact interactions.
As a general rule, the molecules (or the molecular segments) are not distributed in a random way
There is a random distribution if ε11 = ε22 = ε12

33. Energy of contact 1-1: ε11 Energy of contact 2-2: ε22
(1-1) + (2-2)  (1-2) + (2-1)
Δε12 = 2ε12 – (ε11+ε12) 1 2

34. (1-1) (2-2)  (1-2) (2-1)
Random distribution
ε11 = ε22 = ε12
[z N1] [zΝ2]
[z N ] [zΝ ]
Δε12 = 2ε12 –(ε11+ε12)=0
[Ν110] [Ν220]

35. Non-random distribution
(1-1) (2-2)  (1-2) (2-1)
Random distribution
ε11 = ε22 = ε12
[Ν110] [Ν220] [Ν120 /2] [Ν120 /2]
Δε12 = 2ε12 –(ε11+ε12)=0
(1-1) (2-2)  (1-2) (2-1)
Non-random distribution
ε11 ≠ ε22 ≠ ε12
[Ν11] [Ν22] [Ν12/2] [Ν12/2]
[Ν110Γ11] [Ν220 Γ22]
Δε12 = 2ε12 – (ε11+ε12)≠0

36. Non-random distribution ε11 ≠ ε22 ≠ ε12 [Ν11] [Ν22] [Ν12/2] [Ν12/2]
(1-1) (2-2)  (1-2) (2-1)
Non-random distribution
ε11 ≠ ε22 ≠ ε12
[Ν11] [Ν22] [Ν12/2] [Ν12/2]
[Ν110Γ11] [Ν220 Γ22]
Δε12 = 2ε12 – (ε11+ε12)≠0 1
local composition: xij = xiΓij

37. local composition: xij = xi Γji
(1-1) (2-2)  (1-2) (2-1)
Non-random distribution
ε11 ≠ ε22 ≠ ε12
[Ν11] [Ν22] [Ν12/2] [Ν12/2]
[Ν110Γ11] [Ν220 Γ22]
Δε12 = 2ε12 – (ε11+ε12)≠0
local composition: xij = xi Γji
(1) xyelow = 0.333
(2) xred = 0.667
x12 = 0.375
Γ12 = Γ21=1.125
x21 = 0.750

38. Non-random distribution of molecular segments
Non-random distribution of free-volume and mol. segments

39. The non-randomness factors Γ00, Γ11, and Γ10 for polyethylene as a function of temperature t various pressures.

40. The strength of intermolecular forces
Dispersion / London forces
Polar interactions (dipolar, quandrupolar,…)
Hydrogen bonding forces – specific forces
Ionic (ion-dipole, ion – ion) forces
Mirror forces
Hydrophobic effect

41. Specific interactions as quasi-reactions

42. Towards a Unified Description
of Fluids, Polymers, Glasses, Gels, and Interfaces
The NRHB Model

43. The Essentials of the Equation-of-State Model: The NRHB (Non-Randomness + Hydrogen-Bonding) Model
1. The Lattice-Fluid Picture: Accommodation of systems of molecules differing in size and shape (r, v*, ε*)
2. The Non-Randomness in Solutions: The local composition in the neighborhood of a molecule may be different from the bulk composition. Guggenheim’s Quasi-Chemical theory for calculation of non-randomness factors Γij
3. The Hydrogen-Bonding: The strong specific intermolecular interactions as separate Quasi-Chemical reactions in an inert solvent. Veytsman’s statististics + cooperativity.
(Panayiotou et al., J. Phys. Chem. A, 1998, 102, 3574)
The molecules are arranged on a quasi-lattice leaving of course some sites empty

44. The Structure of the Model
The Partition Function:
Q = QRQNRQHB = QPQHB
QR The configurational (random distribution) PF
(Staverman)
QNR The correction factor for the non-random
distribution in the system (Guggenheim QC)
QHB The hydrogen-bonding factor (Veytsman-Statistics)

45. The NRHB (Non Random Hydrogen Bonding) Model
where:
ΩR : Random combinatorial factor
ΩNR : Non-randomness factor
ΩHB : Hydrogen bonding factor
Panayiotou C. et al., Ind. Eng. Chem. Res. 43:6592 (2004)

46. The NRHB (Non Random Hydrogen Bonding) Model Mixing / combining rules
Scaling constants
εh* και εs*, [ε* = εh* + (T )εs* ]
υsp*
EH HB energy
SH HB entropy
VH HB volume
Mixing / combining rules

47. Applications to Subcritical and Supercritical Fluids

48. Experimental (symbols) and calculated (line) orthobaric and
supercritical (red) densities of Carbon Dioxide .
Panayiotou C. et al., Ind. Eng. Chem. Res. 2004, 43,

49. Experimental (symbols) and calculated (line) orthobaric(a) and
supercritical densities(b) of Water.

50. a)VLE and LLE for Methane(1) + Methanol(2) at 298.15 K. Symbols are
experimental data [Ishihara et al. 1998], (—) NRHB, (----) SAFT . b) Detail

51. Solubility of Pharmaceuticals in Liquid Solvents
Solid compounds with very complex HB behavior
Solubility of Pharmaceuticals in Liquid Solvents
Solubility of ketoprofen in acetone and water
Tsivintzelis et al. J. Phys. Chem. B, 113, 2009,

52. Solubility of Pharmaceuticals in Mixed Solvents
Very often in pharmaceutical processes a mixed solvent.
Solubility of Pharmaceuticals in Mixed Solvents
Solubility of ketoprofen in water-acetone mixture
Tsivintzelis et al. J. Phys. Chem. B, 113, 2009,

53. Solubility of Enantiomers in Supercritical CO2
Here, the parameters for the two enantiomers are the same, only the fusion properties are different, since such compounds form different crystal structures.
Solubility of Enantiomers in Supercritical CO2
Solubility of Ibuprofen in Supercritical CO2 (kij = )
Tsivintzelis et al. J. Phys. Chem. B, 113, 2009,

54. Melting Point Depression of Enantiomers in Supercritical CO2
The kij that was optimized in the S-Gas Equilibrium was used to predict the three phase SLG Equilibrium
Melting Point Depression of Enantiomers in Supercritical CO2
Pressure–temperature projection of the S–L–G equilibrium line for the binary system Ibuprofen-CO2. Experimental data (symbols) and NRHB predictions (lines), (kij= optimized in Solid-Gas equilibrium data)
Tsivintzelis et al. J. Phys. Chem. B, 113, 2009,

55. Applications to Polymer Systems

56. HO – CH2 – CH2 – O – CH2 – CH2 – OH n – HO….HO – – HO….– O – H2O….HOH
HB polymer systems
HO – CH2 – CH2 – O – CH2 – CH2 – OH n
– HO….HO –
– HO….– O –
H2O….HOH
H2O….HO –
H2O…. – O –
Water – PEG 600 VLE. Experimental data (points), NRHB predictions (kij=0.0, line)

57. Hexanol – PE liquid – liquid equilibrium
Hexanol – PE liquid – liquid equilibrium. Experimental data (points) and NRHB correlations (kij≠0, solid lines).

58. a) PS-PVME. Experimental (symbols, Nishi-Kwei) and calculated (lines) binodals. b) Change in μ of PS with a slight change in ζ12.

59. 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 NRHB model calculations53 by incorporating the elastic term

60. Handling the Glassy State with NRHB
Density: An order parameter for the non-equilibrium glassy state. Thus:
Boudouris-Panayiotou (Macromolecules 1998, 31, 7915; IEC Res. 2004, 43, 6841)
The Gibbs-DiMarzio criterion:
S at Tg

61. Plasticization of glassy polymers with SCF
Calculations with gLF and NRHB
These diagrams are in fact calculated by NRHB
Pantoula M. et al., J. Supercritic. Fluids. 37:254 (2006)

62. Non Random Hydrogen Bonding model, ΝRΗΒ
Experimental (symbols) and calculated (lines) sorption isotherms for the PMMA/CO2 system.
Pantoula M. et al., J. Supercritic. Fluids. 37:254 (2006)

63. Non Random Hydrogen Bonding model, ΝRΗΒ
Figure: Swelling isotherms for the system PS/CO2
Pantoula M. et al., J. Supercritic. Fluids. 39:426 (2007)

64. Selection of solvents: The Partial Solubility Parameters
Definition of δ:
In Hydrogen-bonded Systems:
Polar Component:

65. Radius of solubility. The best solvent 1 for a solute 2 is the one with the lowest Ra.

66. Solubility parameter calculations
Ionic Liquids:
Solubility parameter calculations
Table : Partial electrostatic and total solubility parameters for ionic liquids
Ionic liquid
δel (MPa1/2)
δtotal (MPa1/2)
Literature
(experimental
or calculated)a,b
Estimated with NRHB
[C2mim+][Tf2N-]
17.8
17.7
27.1
[C4mim+][Tf2N-]
16.0
16.5
26.1 – 26.7
26.2
[C6mim+][Tf2N-]
12.7
15.5
25.5 – 25.6
25.4
[C7mim+][Tf2N-] -
15.1
24.7
[C8mim+][Tf2N-]
14.6
23.6
a(literature 45), b(literature 50)
[Cnmim+] [Tf2N-] :
1-alkyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide
Tsioptsias C. et al, Phys. Chem. Chem. Phys., 2010, 12,

67. Ionic Liquids: Liquid-Liquid Equilibrium Calculations (a) (b)
[C4mim+][Tf2N-] – Water (a) and [C8mim+][Tf2N-] – Water (b) LLE. Experimental data13 (points),
NRHB correlations (solid lines) and and COSMO-RS predictions36 (dashed lines).
Tsioptsias C. et al, Phys. Chem. Chem. Phys., 2010, 12,

68. Vapor-Liquid Equilibrium Predictions
Ionic Liquids:
Vapor-Liquid Equilibrium Predictions
(a)
(b)
[C4mim+][Tf2N-] – Water (a) and [C8mim+][Tf2N-] – Water (b) LLE. Experimental data (Freire et al.),
NRHB predictions (solid lines) and COSMO-RS predictions (dashed lines).
Tsioptsias C. et al, Phys. Chem. Chem. Phys., 2010, 12,

69. Inhomogeneous Systems / Interfaces (Langmuir, 2002, 18, 8841, IEC Res
Experimental (symbols) and calculated (lines) surface tensions
of pure fluids as a function of temperature.

70. Experimental (symbols) and calculated (solid lines)
surface tensions of pure polymers.

71. Experimental (symbols) and predicted (lines) surface tensions of
Cyclohexane(1) + n-Hexane(2) mixture at K as a function of composition.

72. Methanol(1) – Toluene(2) at 308.15 K: The evolution of composition
profiles across the interface, as predicted by the present model.

73. Solvation Thermodynamics
NRHB PV
Panayiotou, C., J. Chem. Eng. Data 2010, in press / sp. Issue H. V. Kehiaian

74. Solvation Thermodynamics
This peculiar change as crit point is approached could shed light to supercritical enhancement of solvation
Ethanol
Panayiotou, C., J. Chem. Eng. Data 2010 / sp. Issue H. V. Kehiaian

75. From NRHB to NRCosmo The GC/QC Partition Function
and the COSMO Partition F.
NRHB
NRCosmo
The full PF can be used (as in 2003) to get a complete EOS Cosmo model but very slow calculations. If you have the sigma profiles from quantum mechanics calc you have the hb-cosmo term

76. From NRHB to NRCosmo Boundary Conditions: At zero densities (IG state)
In the absence of empty sites (compact state)
A little diversion for g
Both conditions are satisfied with:

77. From NRHB to NRCosmo The approximation Analytical expression
In direct analogy with NRHB g(rho) is directly prop to rho. Applying Γ the Z becomes this analyt expression. Γm-hb is the SAC from cosmo perspective or the NR factor from QC persp
Analytical expression

78. From NRHB to NRCosmo The equivalence Ai : surface area of molecule i
αeff : standard surf area
pi(σm): probability of group
m in molecule i
Γm : segment (m)
activity coefficient
With this dictionary you may transform the qc lattice statistics to cosmo statistics

79. Calculations with NRCosmo
Two scaling constants per molecule: ε* and ρ*=1/vsp*
As in NRHB, v* =9.75 cm3mol and
HB parameters are no longer needed !!!

80. Calculations with NRCosmo
1-Alkanols: E* = Acosmo (R=0.9994)

81. Calculations with NRCosmo
1-Alkanols: M/ρ* = Vcosmo (R=0.9999)

82. Calculations with NRCosmo
NRHB
1-Alkanols: The free-energy change upon hydrogen bond formation as calculated by the COSMO approach at K.

83. Calculations with NRCosmo
1-Propanol: Experimental and calculated surface tension.
Solid line: NRCosmo, Dashed line: NRHB

84. Calculations with NRCosmo
Experimental and predicted vapor liquid equilibria for the system CO2 - Ethanol at K (circles + solid lines) and at K (squares + dashed lines) by the NRCosmo model

85. Calculations with NRCosmo
Polymer solutions: VLE of Ethanol –PVAc. Experimental data and predictions by the NRCosmo model (mcos file for polymer) and NRHB

86. Calculations with NRCosmo
Polymer solutions: VLE of Methanol –PVAc. Experimental data and predictions by the NRCosmo model (mcos file for polymer)

87. Calculations with NRCosmo
NRHB
Solid-Liquid Equilibria: Prediction of solubility of paracetamol in ethanol as a function of temperature). Symbols are experimental data

88. NRCosmo vs. NRHB In Paracetamol + Ethanol
NRHB requires interaction energies for:
OH ---OH OH ---OH
OH ---O=C OH ---O=C
OH ---NH OH ---NH
OH ---OH With the assumption OH = OH :
OH ---OH
OH ---O=C
OH ---NH
NH ---NH NH ---NH
NH ---O=C NH ---O=C

89. Process Design and Development:
Processing Polymeric Materials with Supercritical Fluids

90. Pressure Quench: A Process for Porous-Structure Formation

91. Figure : Porous structures of polystyrene,
Here is the effect of the starting point (the pressure)
Figure : Porous structures of polystyrene,
80 οC, (α) 180 bar, (β) 230 bar, (γ) 280 bar, (δ) 330 bar, (ε) 380 bar

92. The Nucleation Theory
According to the nucleation theory, in a closed isothermal system in chemical equilibrium the difference of the free energy per unit volume related to the formation of new phase cluster is given by the following equation :

93. Modeling the foaming of polymers with scCO2:
Nucleation Theory
Activation energy for homogeneous nucleation:
Nucleation rate:
According to nucleation theory the energy barrier for the formation of a stable nucleus is a function of :
1 The interfacial tension between the metastable polymeric phase and a gas bubble
2 The supersaturation of the system
The rate of nucleation (the rate of the production of stable nuclei) is a function of:
1 The amount of the dissolved fluid inside the polymer matrix
2 One characteristic frequency factor, fo (the rate that nuclei with critical radius are transformed in stable pores)
3 The energy barrier (ΔGhom)
Finally, the total number of nuclei that are formed in the system is found by the integration of the nucleation rate
In order to apply the nucleation theory one must know:
1 the pressure at which the growth of pores stops (transition from rubbery to glassy state occurs)
2 the solubility of the supercritical fluid into the polymer matrix
3 the interfacial between and the metastable polymer phase and a gas bubble
These properties were correlated with NRHB model.
Total number of nuclei:

94. Figure: Sorption of CO2 in polystyrene. Experimental data:
NRHB calculations for sorption and Tg are presented (one kij temp independent was used)
Figure: Sorption of CO2 in polystyrene. Experimental data:
(o) 100 οC, () 120 οC, (—) NRHB
Figure: Glass transition temperature for the system polystyrene-CO2, () experimental data, (----) CO2 vapor pressure, (—) NRHB

95. System CO2 – polystyrene:
γmixr = (1-wCO2) γpolr
NRHB calculations for surface tension of the pure polymer are presented
The surface tension of the mixture is calculated using the empirical equation of the slide (r=13, was calculated by fitting the predictions of the equation to experimental data at high temperatures around 210 C, if I remember correctly)
Surface tension of polystyrene versus temperature, () experimental data, (—) NRHB

96. Critical radius for nucleus formation in PS-CO2
Critical nuclei radius vs Pressure for the three experimental temperatures

97. ΝRΗΒ combined with nucleation theory
Energy barrier for nucleation.
At constant temperature, as the pressure increases, the energy barrier exponentially decreases.
In other words as pressure increases nuclei are formed more easily in the system.
This result is in agreement with the increase of cell density that was experimentally observed with increase of pressure.
On the other hand, increase of temperature leads to an increase of the energy barrier for nucleation. Consequently at higher temperatures the generation of nuclei becomes more difficult and fewer cells are observed in the final porous structure. This can explain the decrease of the cell population density with increase of the foaming temperature, which was experimentally observed.
The main reason for this is the reduction of the surface tension. It is reasonable to expect that as the temperature rises the surface tension should be decreased. Nevertheless, this happens only at low pressures, in which the sorption of CO2 is not very high. At higher pressures, the increase of temperature decreases the CO2 solubility into the polymer matrix and consequently causes a decrease of the surface tension
Activation energy for homogeneous nucleation (polystyrene-CO2 )

98. ΝRΗΒ combined with nucleation theory
The nucleation theory, in combination with NRHB model, was used to correlate the experimentally observed cell population density as a function of pressure and temperature.
Figure: Observed cell density, (), and calculated nuclei density, (—), versus pressure for the system polystyrene-CO2 at 80 oC

99. ΝRΗΒ combined with nucleation theory
Thus, this is the tool we need for rational design of foaming or pore formation processes as in TE scaffolding
Observed cell density, (), and calculated nuclei density, (—), versus temperature for the system polystyrene-CO2 at 330 bar

100. Conclusions
Thermodynamics and Equation-of-State models, are valuable tools in understanding and designing numerous processes, including modern technology applications, such as TE scaffolding.
The NRHB is an approximate but versatile equation-of state model applicable to the gas, liquid, and glassy states and, thus, it could be a useful tool for understan- ding polymer – SCF systems and processes.
The NRCosmo enhances drastically the capacity of the plain NRHB model to describe complex thermodynamic systems with a minimum of parameters and, hopefully, to predict the phase behavior of mixtures from pure component information.