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# RHEOLOGY 1,2

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1. 4.1 AND IT IS AIMED TO … build up mathematical models describing how materials respond to any type of solicitation (forces or deformations). 1 … build up mathematical models able to establish a link between materials macroscopic behaviour and materials micro-nanoscopic structure. 2 RHEOLOGY1,2 .. is the science that deals with the way materials deform or flow when forces (stresses) are applied to them.

2. 4.2 A cross section area STRESS F h F S A cross section area F h NORMAL STRESS (N/M2 = Pa) SHEAR STRESS (N/M2 = Pa)

3. A cross section area F h L0 L F S h F LINEAR STRAIN SHEAR STRAIN HENCKY STRAIN DEFORMATION

4. 4.3 HOOKE’s Law (small deformations) Normal stress Shear stress Incompressible materials E = 3G [SOLID MATERIAL] E = Young modulus (Pa) G = shear modulus (Pa) RHEOLOGICAL PROPERTIES A - ELASTICITY “ A material is perfectly elastic if it returns to its original shape once the deforming stress is removed”

5. B - VISCOSITY “ This property expresses the flowing (continuous deformation) resistance of a material (liquid) ” Very often VISCOSITY and DENSITY are used as synonyms but this is WRONG! EXAMPLE: at T = 25°C and P = 1 atm HONEY is a fluid showing high viscosity (~ 19 Pa*s) and low density (~1400 Kg/M3) MERCURY is a fluid showing low viscosity (~ 0.002 Pa*s) and high density (13579 Kg/M3) WATER: viscosity 0.001 Pa*s, density 1000 Kg/m3

6. NEWTON Law LIQUID MATERIAL Shear rate h = viscosity or dynamic viscosity (Pa*s) n = kinematic viscosity = h/density(m2/s)

7. On the contrary it can be “SHEAR THINNING” … or “SHEAR THICKENING” (opposite behaviour) IF h does not depend on share rate, the fluid is said NEWTONIAN WATERis the typical Newtonian fluid.

8. M M M M M K(T) K(T) K(T) K(T) M K(T) M K(T) Idealised polymer chain zfriction coefficient Usually h reduces with temperature Why h depends on liquid structure, shear rate and temperature?

9. LIQUID VISCOEALSTIC SOLID VISCOEALSTIC deformation deformation t t stress stress t t C - VISCOELASTICITY “ A material that does not instantaneously react to a solicitation (stress or deformation) is said viscoelastic”

10. STRESS Material behaviour depends on: ELASTIC (instantaneous) REACTION OF MOLECULAR SPRINGS 1 VISCOUS FRICTION AMONG: - CHAINS-CHAINS - CHAINS-SOLVENT MOLECULES 2 SOLVENT MOLECULES POLYMERIC CHAINS

11. D – TIXOTROPY - ANTITIXOTROPY A material is said TIXOTROPIC when its viscosity decreases with time being temperature and shear rate constant. A material is said ANTITIXOTROPIC when its viscosity increases with time being temperature and shear rate constant. The reasons for this behaviour is found in the temporal modification of system structure

12. h AT REST: structure MOTION structure break up COAL PARTICLE EXAMPLE: Water-Coal suspensions In the case of viscoelastic systems, no structure break up occurs t

13. 4.4 .. consequently, linear viscoelasticty enables us to study the characteristics of material structure LINEAR VISCOELASTICITY THE LINEAR VISCOEALSTIC FIELD OCCURS FOR SMALL DEFORMATIONS / STRESSES THIS MEANS THAT MATERIAL STRUCTURE IS NOT ALTERED OR DAMAGED BY THE IMPOSED DEFORMATION / STRESS

14. Shear modulus G does not depend on the deformation extension g0 Shear stress Tensile modulus E does not depend on the deformation extension e0 Normal stress Incompressible materials MAIN RESULTS

15. G(t) or E(t) estimation g E(t) = 3 G(t) h solid g0 g0is instantaneously applied liquid 1) MAXWELL ELEMENT1,2

16. g1 g2 g3 g4 g5 h1 h2 h3 h4 h5 g0 g0is instantaneoulsy applied E(t) = 3 G(t) 2) GENERALISED MAXWELL MODEL1,2

17. g1 = 90 Pa g2 = 9 Pa g3 = 0.9 Pa g4 = 0.1 Pa

18. g1 g2 g3 g4 g5 h1 h2 h3 h4 h5 g(t) = g0sin(wt) w = 2pf f = solicitation frequency SMALL AMPLITUDE OSCILLATORY SHEAR

19. On the basis of the Boltzmann1 superposition principle, it can be demonstrated that the stress required to have a sinusoidal deformation g(t) is given by: t(t) = t0sin(wt+d) d(w) = phase shift t(t) = g0*[G’(w)*sin(wt) + G’’(w)*cos(wt)] g(t) = g0sin(wt) Gd= t0/g0=(G’2+G”2)0.5 G’(w) = Gd*cos(d) = storage modulus G’’(w) = Gd*sen(d) = loss modulus tg(d)=G”/G’

20. LIQUID G’≈ 0 G”≈ Gd SOLID G’≈ Gd G”≈ 0

21. g1 g2 g3 g4 g5 h1 h2 h3 h4 h5 According to the generalised Maxwell Model, G’ and G” can be expressed by: li = hi/gi g(t) = g0sin(wt)

22. In the linear viscoelastic field, oscillatory and relaxation tests lead to the same information: Oscillatory tests Relaxation tests

23. 4.5 EXPERIMENTAL1

24. Rotating plate Gel Fixed plate SHEAR DEFORMATION/STRESS SHEAR RATE CONTROLLED SHEAR STRESS CONTROLLED

25. Linear viscoelastic range STRESS SWEEP TEST: constant frequency (1 Hz) G’(Pa) (elastic or storage modulus) t(t) = t0sin(wt) G’’(Pa) (loss or viscous modulus) w = 2pf

26. FREQUENCY SWEEP TEST: constant stress or deformation t0 = constant; 0.01 Hz ≤ f ≤ 100 Hz G’ (Pa) G’’ (Pa)

27. gi G’ (Pa) Fitting parameters gi, hi, n hi Black lines: model best fitting G’’ (Pa) t(t)

28. li+1 =10* li Np = generalised Maxwell model fitting parameters 0th Maxwell element (spring) -------> 1 fitting parameter (ge) 1st Maxwell element -------> 2 fitting parameters (g1, l1) 2nd Maxwell element ------->1 fitting parameters (g2, l2) 3rd Maxwell element -------> 1 fitting parameters (g3, l3) 4th Maxwell element -------> 1 fitting parameters (g4, l4)

29. 4.6 Polymer Solvent Polymer Solvent ≈ Crosslinks FLORY THEORY3

30. SOLVENT mgH2O = msH2O D=mgH2O - msH2O = 0 D = DM + DE + DI = 0 Mixing Elastic Ions SWELLING EQUILIBRIUM

31. rx= crosslink density in the swollen state np= polymer volume fraction in the swollen state np0=polymer volume fraction in the crosslinking state T= absolute temperature R=universal gas constant gi = spring constant of the Maxwell ith element DE= -RTrx(np/np0)1/3

32. Comments The use of Flory theory for biopolymer gels, whose macromolecular characteristics, such as flexibility, are far from those exhibited by rubbers, has been repeatedly questioned. 1 However, recent results have shown that very stiff biopolymers might give rise to networks which are suitably described by a purely entropic approach. This holds when small deformations are considered, i.e. under linear stress-strain relationship (linear viscoelastic region)9. 2 G can be determined only inside the linear viscoelastic region. 3

33. x 4.7 x Polymeric chains x SAME CROSS-LINK DENSITY (rx) EQUIVALENT NETWORK TOPOLOGY EQUIVALENT NETWORK THEORY4 REAL NETWORK TOPOLOGY

34. REFERENCES • Lapasin R., Pricl S. Rheology of Industrial polysaccharides, Theory and Applications. Champan & Hall, London, 1995. • Grassi M., Grassi G. Lapasin R., Colombo I. Understanding drug release and absorption mechanisms: a physical and mathematical approach. CRC (Taylor & Francis Group), Boca Raton, 2007. • Flory P.J. Principles of polymer chemistry. Cornell University Press, Ithaca (NY), 1953. • Schurz J. Progress in Polymer Science, 1991, 16 (1),1991, 1.