A new interpretation for the dynamic behaviour of complex fluids at the sol - Gel transition using the fractional calculus

Rheologica Acta

Volumn 48, Issue 1, 2009, Pages 51-58

  Warlus, Stéphane ^a   Ponton, Alain ^a  

^a UNIVERSITÉ PARIS DIDEROT   (France)

Author keywords

Fractional calculus; Gel; Power law; Small amplitude oscillatory shear; Stress relaxation

Indexed keywords

DIFFERENTIATION (CALCULUS); ELASTIC MODULI; MATHEMATICAL MODELS; RHEOLOGY; SHEAR STRESS; SOL-GEL PROCESS; STRESS RELAXATION;

EID: 58149196403     PISSN: 00354511     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00397-008-0306-z     Document Type: Article

References (71)

1. Adam M, Delsanti M, Durand D (1985) Mechanical measurements in the reaction bath during the polycondensation reaction near the gelation threshold. Macromolecules 18:2285-2290
2. Adolf D, Martin JE, Wilcoxon JP (1990) Evolution of structure and viscoelasticity in an epoxy near the sol-gel transition. Macromolecules 23:527-531
3. Alcaraz J, Buscemi L, Grabulosa M, Trepart X, Fabry B, Farré R, Navajas D (2003) Microrheology of human lung epithelial cells measured by atomic microscopy. Biophys J 84:2071-2079
4. Alvarez F, Alegra A, Comenero J (1991) Relationship between the time-domain Kohlrausch-Williams-Watt and frequency-domain Havriliak-Negami relaxation functions. Phys Rev B 44:7306-7312
5. Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201-210
6. Bausch AR, Ziemann F, Boulbitch AA, Jacobson K, Sackmann E (1998) Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys J 75:2038-2049
7. Benny Lee KC, Siegel J, Webb SED, Lévêque-Fort S, Cole MJ, Jones R, Dowling K Lever MJ, French PMW (2001) Application of stretched exponential function of fluorescence lifetime imaging. Biophys J 81:1265-1274
8. Berry GC, Plazek DJ (1997) On the use of stretched-exponential functions for both viscoelastic creep and stress relaxation. Rheol Acta 36:320-329
9. Bu H, Kjoniksen AL, Knudsen KD, Nyström B (2004) Rheological and structural properties of aqueous alginate during gelation via the Ugi multicomponent condensation reaction. Biomacromolecules 5:1470-1479
10. Caputo M, Mainardi M (1966) A new dissipation model based on memory mechanism. Pure Appl Geophys 91:134-137
11. Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. Riv Nuovo Cim 1:161-198
12. Chambon F, Winter HH (1987) Linear viscoelasticity at the gel point of a crosslinking PDMS with imbalanced stochiometry. J Rheol 31:683-697
13. Chenite A, Buschmann M, Wang D, Chaput C, Kandari N (2001) Rheological characterization of thermogelling chitosan/glycerol phosphate solutions. Carbohydr Polym 46:39-47
14. Cocard S, Tassin JF, Nicolai T (2000) Dynamical mechanical properties of gelling colloidal disks. J Rheol 44:585-594
15. Cumbrera FL, Sanchez-Bajo F, Guibertau F, Solier JD, Munoz A (1993) The Williams-Watt dependence as a common phenomenological approach to relaxation process in condensed matter. J Math Sci 28:5387-5396
16. de Arcangelis L, Del Gado E, Coniglio A (2002) Complex dynamics in gelling systems. Eur Phys 9:277-282
17. de Gennes PG (1979) Scaling concept in polymer physics. Cornell University Press, Ithaca and London, pp 137-145
18. de Rosa ME, Winter HH (1994) The effect of entanglements on the rheological behaviour of polybutadiene critical gels. Rheol Acta 33:220-237
19. Del Gado E, Fiero A, de Arcangelis L, Coniglio A (2003) A unifying model for chemical and colloidal gels. Europhys Lett 63:1-7
20. English RJ, Raghavan SR, Jenkis RD, Khan SA (1999) Associative polymers bearing n-alkyl hydrophobes: Rheological evidence for microgel like behaviour. J Rheol 43:1175-1194
21. Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, Fredberg JJ (2001) Scaling the microrheology of living cells. Phys Rev Lett 8:148102-1-148102-4
22. Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, New York
23. Flory PJ (1942) Constitution of three-dimensional polymers and the theory of gelation. J Phys Chem 46:132-140
24. Gemant A (1936) A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7:311-317
25. Gemant A (1938) On fractional differentials. Philos Mag 25:540-549
26. Hess W, Vilgis TA, Winter HH (1988) Dynamical critical behavior during chemical gelation and vulcanization. Macromolecules 21:2536-2542
27. Heymans N, Bauwens J-C (1994) Fractal rheological and fractional differential equations for viscoelastic behavior. Rheol Acta 33:210-219
28. Hsu SH, Jamieson AM (1993) Viscoelastic behaviour at the thermal sol-gel transition of gelatin. Polymer 34:2602-2608
29. Isuka A, Winter HH, Hashimoto T (1997) Self-similar relaxation behaviour at the gel point of a blend of a crosss-linking poly(ε -caprolactone) diol with a poly(styrene-co-acrylonitrile). Macromolecules 30:6158-6165
30. Izuka A, Winter HH, Hashimoto T (1992) Molecular weight dependence of viscoelasticity of polyprolactone critical gels. Macromolecules 25:2422-2428
31. Kjoniksen AL, Nyström B (1996) Effects of polymer concentration and cross-linking density on rheology of chemically cross-linked poly(vinylalcohol) near the gelation threshold. Macromolecules 29:5215-5222
32. Koike A, Nemoto N, Watanabe Y, Osaki K (1996) Dynamic viscoelastic and FI-IR measurements of end-crosslinking α, ω dihydroxyl polybutadiene solutions near the gel point in the gelation process. Polymer 28:942-950
33. Lang P, Burchard W (1991) Dynamic light scattering at the gel point. Macromolecules 24:814-815
34. Larson RG (1985) Constitutive relationships for polymeric materials with power law distributions of relaxation times. Rheol Acta 24:327-334
35. Lin YG, Mallin DT, Chien JCW, Winter HH (1991) Dynamic mechanical measurement of crystallization-induced gelation in thermoplastic elastomeric poly(propylene). Macromolecules 24:850-854
36. Martin JE, Wilcoxon JP (1988) Critical dynamics of the sol-gel transition. Phys Rev Lett 61:373-376
37. Martin JE, Adolf D, Wilcoxon JP (1988) Viscoelasticity of near critical gels. Phys Rev Lett 61:2620-2623
38. Martin JE, Wilcoxon JP, Odinek J (1991) Decay of density fluctuations in gels. Phys Rev A 43:858-872
39. Mason TG, Gang H, Weitz DA (1997) Diffusing-wave-spectroscopy measurements of viscoelasticity of complex fluids. J Opt Soc Am A 14:139-149
40. Maxwell JC (1867) On the dynamical theory of gases. Philos Trans R Soc 157:49-88
41. Michon C, Cuvelier G, Launay B (1993) Concentration dependence of the critical viscoelastic properties of gelatin at the gel point. Rheol Acta 32:94-103
42. Ming JM, Jerome R, Teyssié P (1997) Triblock copolymer based thermoreversible gels: 1 self association of sPMMA end-blocks ino-xylene and viscoelasticity of the gels. Polymer 38:347-354
43. Mours MM, Winter HH (1996) Relaxation patterns of nearly critical gels. Macromolecules 29:7221-7229
44. Muthukumar M (1989) Screening effect viscoelasticity near the gel point. Macromolecules 22:4656-4658
45. Nijenhuis te N, Winter HH (1989) Mechanical properties at the gel point of a crystallizing poly(vinyl chloride) solution. Macromolecules 22:411-414
46. Nutting PG (1921) A new generalized law of deformation. J Franklin Inst 191:679-685
47. Palmer A, Mason TG, Xu J, Kuo SC, Wirtz D (1999) Diffusing wave spectroscopy microrheology of actin filament networks. Biophys J 76:1063-1071
48. Ponton A, Barboux Doeuff S, Sanchez C (1999) Rheology of titanium oxide of based gels: Determination of gelation time versus temperature. J Coll Int Sci A 162:177-192
49. Power DJ, Rodd AB, Paterson L, Boger DV (1998) Gel transition studies on non ideal polymer networks using small amplitude oscillatory rheometry. J Rheol 42:1021-1037
50. Puig-de-Morales M, Grabulosa M, Alcaraz J, Mullol J, Maksym GN, Fredberg JJ, Navajas D (2001) Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation. J Appl Physiol 91:1152-1159
51. Ren SZ, Shi WF, Zhang WB, Sorensen CM (1992) Anomalous diffusion in aqueous solutions of gelatin. Phys Rev A 45:2416-2422
52. Richtering HW, Gagnon KD, Lenz RW, Fuller RC, Winter HH (1992) Physical gelation of a bacterial thermoplastic elastomer. Macromolecules 25:2249-2433
53. Rodd AB, Cooper-White J, Dunstan DE, Boger DV (2001) Polymer concentration dependence of the gel point for chemically modified biopolymer networks using small amplitude oscillatory. Polymer 42:3923-3928
54. Romer S, Schefflod F, Schurtenberger P (2000) Sol-gel transition of concentrated colloidal suspensions. Phys Rev Lett 85:4980-4983
55. Rosa ME, Mours M, Winter HH (1997) The gel point as reference state: A simple kinetic model for crosslinking polybutadiene via hydrosilation. Polym Gels Netw 5:69-94
56. Rosa ME, Winter HH (1994) The effect of entanglements on the rheological behavior of polybutadiene critical gels. Rheol Acta 33:220-237
57. Sato T, Watanabe H, Osaki K (2000) Thermoreversible physical gelation of block copolymers in a selective solvents. Macromolecules 33:1686-1691
58. Scanlan JC, Winter HH (1991) Composition dependence of the viscoelasticity of end-linked poly(dimethylsiloxane) at the gel point. Macromolecules 24:47-54
59. Schaefer DW, Keefer KD (1984) Fractal geometry of silica condensation polymers. Phys Rev Lett 53:1383-1386
60. Schiessel H, Blumen A (1995) Mesoscopic pictures of the sol-gel transition: Ladder models and fractals networks. Macromolecules 28:4013-4019
61. Stockmayer WH (1943) Theory of molecular size distribution and gel formation in branched-chain polymers. J Phys Chem 11:45-55
62. Takenata M, Kobayashi T, Hashimoto T, Takahashi M (2002) Time evolution of dynamic shear moduli in a physical gelation process of 1,3:2,4-bis-O-(p-methylbenzylidene)-D-sorbitol in polystyrene melt: critical exponent and gel strength. Phys Rev, E 65:041401-01-041401-07
63. Tanaka T (1999) In: Tanaka T (ed) Experimental methods in polymer science: Modern methods in polymer research and technology, chap 9. Academic, New York, pp 347-362
64. Tempel M, Isenberg G, Sackmann E (1996) Temperature-induced sol-gel transition and microgel formation in α-actin crosslinked actin networks: A rheological study. Phys Rev E 54:1802-1810
65. Tixier T, Tordjeman Ph (2003) Dynamic exponent of PDMS networks at the sol-gel transition. Polymer 44:6937-6942
66. Venkataramann SK, Winter HH (1990) Finite shear strain behavior of a crosslinking polydimethylsiloxane near its gel point. Rheol Acta 29:423-432
67. Vlassopoulos D, Chira I, Lapinet B, McGrail PT (1998) Gelation kinetics in elastomer thermoset polymer blends. Rheol Acta 37:614-623
68. Warlus S, Ponton A, Leslous A (2003) Dynamic viscoelastic properties of silica alkoxide during the sol-gel transition. Eur Phys E 12:275-282
69. Williams G, Watts DC (1970) Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans Faraday Soc 66:80-85
70. Winter HH, Chambon F (1986) Analysis of linear viscoelasticity of a crosslinking polymer at gel point. J Rheol 30:367-382
71. Winter HH, Morganelli P, Chambon F (1988) Stoichiometry effects on rheology of model polyurethanes at the gel point. Macromolecules 21:532-535