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Filled van der Waals networks

  • H. -G. Kilian
Filled Networks
Part of the Progress in Colloid & Polymer Science book series (PROGCOLLOID, volume 75)

Abstract

New possibilities of characterizing filler-loaded networks with the aid of a generalized van der Waals formulation are presented. By arriving at a full description of stress-strain cycles, different means of cooperation within the composites are identified in the investigation of model systems. The role of filler-to-matrix contacts is discussed, including the Einstein-Smallwood boundary value problem. An interpretation of the Mullins softening is achieved, based on the description of a set of cyclic quasi-static experiments.

Key words

Networks filler-loaded vulcanisates Mullins softening van der Waals theory of filled networks Einstein-Smallwood effect 

Symbols

a

van der Waals parameter of global interactions between the chains

ar

interaction parameter within the rubber network

af

interaction parameter within the filler network constituted by crosslinking filler particles only

C, Cλ

Einstein-Smallwood parameter

D=λ- λ−2

deformation function of the „Gaussian continuum“

Dmm−λm−2

maximum value of the deformation function

f

nominal force

fh

enthalpy component of the force

fs

entropy component of the force

G0

modulus

G00=ρRT/M0

Gf modulus accounting for the Einstein-Smallwood effect

gfv

modulus with prefactor correction

H

enthalpy

H(i)

enthalpy of the oscillatory freedom (i)

kB

Boltzmann constant

x

fraction of filler to matrix bond

λ

macroscopic strain

λmax

maximum macroscopic strain achieved with the first stretch

λr

strain in the rubber matrix

λrr

matrix strain of a filled rubber

λrf

matrix strain of a filler network

λi

matrix strain in the filler-loaded network of the type (i)

λf

average total strain of the filler

λp

plastic strain of the filler-particle's ensemble

λfe

elastic strain of the filler

λm

maximum macroscopic strain

λmr

maximum matrix strain of the filled rubber

λmf

maximum matrix strain of the filler network

λmrf

maximum matrix strain of the filler network rubber

λrfv

maximum matrix strain of the filler network rubber plus Einstein Smallwood

λm max

maximum strain parameter after the first stretch

λb

remnant strain after the first stress-strain cycle

λbb

ad hoc assumption of how the remnant strain is developed in dependence on the macroscopic strain λ

Mc

molecular weight of the network chains

Ms

molecular weight of the Kuhn segment

M0

molecular weight of the “stretching-invariant unit”

N

number of chains

n

volume density of chains

NL

Loschmidt number

p

pressure

Q

heat exchanged during deformation

ρ

mass density

Rg

gas constant

S

entropy

S(i)

entropy of the i-th internal freedom

T

absolute temperature

U

internal energy

ur

volume correction in the filled rubber

uf

volume correction in the filler network

urf

volume correction in the filler network rubber

ur max

the minimum value of ur after the first stretch up to λmax

uf max

the maximum value of Uf after the first stretch

y

number of stretching invariant units per chain

ys

number of Kuhn segments per chain

v

relative number of crosslinks

vr

relative number of matrix crosslinks

vf

relative number of filler-to-matrix crosslinks

ξ

prefactor correction

V

volume

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References

  1. 1.
    Treloar LRG (ed) (1975) The Physics of Rubber Elasticity, 3rd Ed, Clarendon Press, OxfordGoogle Scholar
  2. 2.
    Flory PJ (ed) (1953) Principles of Polymer Chemsitry, Cornell University Press, IthacaGoogle Scholar
  3. 3.
    de Gennes PG (ed) (1979) Scaling Concepts in Polymer Physics, Cornell University Press, IthacaGoogle Scholar
  4. 4.
    Kilian H-G (1981) Polymer 22:209CrossRefGoogle Scholar
  5. 5.
    Kilian H-G (1982) Coll & Polym Sci 260:895CrossRefGoogle Scholar
  6. 6.
    Kuhn W, Grün P (1942) Kolloid-2 101:248CrossRefGoogle Scholar
  7. 7.
    Kilian H-G (1981) Coll & Polym Sci 259:1084CrossRefGoogle Scholar
  8. 8.
    Kilian H-G, Vilgis Th (1984) Makromol Chem 185:193CrossRefGoogle Scholar
  9. 9.
    Kilian H-G (1979) Phys Bl 12:641CrossRefGoogle Scholar
  10. 10.
    Green AE, Adkins JE (eds) (1970) Earge Elastic Deformations, 2rd Ed, Clarendon Press OxfordGoogle Scholar
  11. 11.
    Graessley WW (1982) Adv Polym Sci 46:67Google Scholar
  12. 12.
    Dusek K (1984) Int Rubber Conf, MoscowGoogle Scholar
  13. 13.
    Oppermann W, Rehage G (1981) Coll & Polym Sci 259:117Google Scholar
  14. 14.
    Langley NR (1976) Macromolecules 1:348CrossRefGoogle Scholar
  15. 15.
    Kilian H-G, Enderle HF, Unseld K (1986) Coll & Polym Sci 264:866CrossRefGoogle Scholar
  16. 16.
    Pak H, Flory PJ (1979) J Polym Sci Phys Ed 17:1845CrossRefGoogle Scholar
  17. 17.
    Rivlin RS, Saunders DW (1951) Philos Trans R Soc London A 243:251CrossRefGoogle Scholar
  18. 18.
    Kilian H-G (1984) Int Rubber Conf, MoscowGoogle Scholar
  19. 19.
    Kilian H-G, Unseld K (1986) Coll & Polym Sci 264:9CrossRefGoogle Scholar
  20. 20.
    Kilian H-G, Pietralla M (1988) in preparationGoogle Scholar
  21. 21.
    Enderle F, Kilian H-G (1987) Progr Coll & Polym Sci, in pressGoogle Scholar
  22. 22.
    Vilgis Th, Kilian H-G (1986) Coll & Polym Sci 264:137CrossRefGoogle Scholar
  23. 23.
    Kilian H-G, Schenk H, Wolff S (1987) Coll & Polym Sci, in pressGoogle Scholar
  24. 24.
    Callen HB (ed) (1959) Thermodynamics, Wiley, Int Ed, New YorkGoogle Scholar
  25. 25.
    Godovsky YK (1977) Vysokomol Soed A 19:2359Google Scholar
  26. 26.
    Godovsky YK (1986) Adv Polym Sci 76:31Google Scholar
  27. 27.
    Göritz H (1982) Coll & Polym Sci 260:193CrossRefGoogle Scholar
  28. 28.
    Kilian H-G (1986) Gummi, Faser-Kunststoffe 10:548Google Scholar
  29. 29.
    Kilian H-G (1986) Kautsch Gummi Kunstst 39:689Google Scholar
  30. 30.
    Rigbi Z (1980) Adv Polym Sci 36:21CrossRefGoogle Scholar
  31. 31.
    Bueche F (1960) J Appl Polym Sci 4:107CrossRefGoogle Scholar
  32. 32.
    Mullins L (1956) J Polym Sci 19:225CrossRefGoogle Scholar
  33. 33.
    Mullins L (1956) J Polym Sci 19:237CrossRefGoogle Scholar
  34. 34.
    Mullins L, Tobin NR (1965) J Appl Polym Sci 9:2993CrossRefGoogle Scholar
  35. 35.
    Harwood JAC, Mullins E, Payne AH (1965) J Appl Polym Sci 9:3011CrossRefGoogle Scholar
  36. 36.
    Bueche F (1961) J Appl Polym Sci 15:271CrossRefGoogle Scholar
  37. 37.
    Kilian H-G, Schenk H, J Appl Polym Sci (1988) submitted publication, in pressGoogle Scholar
  38. 38.
    Einstein A (1906) Ann Phys 19:289CrossRefGoogle Scholar
  39. 39.
    Einstein A (1911) Ann Phys 34:581Google Scholar
  40. 40.
    Smallwood HM (1944) J Appl Polym Sci 15:758Google Scholar
  41. 41.
    Enderle HF, Kilian H-G, Vilgis Th (1984) CoU & Polym Sci 262:696CrossRefGoogle Scholar
  42. 42.
    Ambacher H, Kilian H-G (1988) to be publishedGoogle Scholar
  43. 43.
    Becker GW, Rademacher AJ (1962) J Polym Sci 58:621CrossRefGoogle Scholar
  44. 44.
    Mergenthaler, Kilian H-G, Pietralia M (1987) Progr & Coll Polym SciGoogle Scholar
  45. 46.
    Kilian H-G, Höhne GWH, Trögele P, Ambacher H (1984) J Polym Sci Polym Symp 77:221Google Scholar
  46. 47.
    Kilian H-G (ed) (1986) German-Chinese Polymer Symposium, PeckingGoogle Scholar
  47. 48.
    Soos I (1982) Canditate Thesis: Characterization of the Rubber Filler Interaction, BudapestGoogle Scholar
  48. 49.
    Enderle HF, Kilian H-G, Vilgis Th (1984) Coll & Polym Sci 262:696CrossRefGoogle Scholar
  49. 50.
    Enderle HF, Kilian H-G, Heise B, Mayer J, Hespe H (1986) Coll & Polym Sci 264:305CrossRefGoogle Scholar
  50. 51.
    Blythe AR (ed) (1979) Electrical Porperties of Polymers, University Press, CambridgeGoogle Scholar
  51. 52.
    Westlinning H, Wolff S (1966) Kautsch Gummi Kunstst 19:470Google Scholar
  52. 53.
    Wolff S (1974) Kautsch Gummi Kunstst 27:511Google Scholar
  53. 54.
    Wolff S, Pöhnisch H, Hoffmann P (1975) Kautsch Gummi Kunstst 28:379Google Scholar

Copyright information

© Dr. Dietrich Steinkopff Verlag GmbH & Co. KG 1987

Authors and Affiliations

  • H. -G. Kilian
    • 1
  1. 1.Abteilung Experimentelle PhysikUniversität UlmUlmF.R.G.

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