Composites Design and Manufacture (Plymouth University teaching support materials) Rheology.

Lecture PowerPoint

Review papers

Subject Index

First, a couple of definitions:

• rheology: the science of flow and deformation of matter
• viscosity: the ability of a fluid to resist a change in the arrangement of the molecules when under an applied strain or stress.  The standard symbol for viscosity is the Greek symbol eta: η.
• dynamic viscosity: the force required to overcome internal friction. The standard symbol for dynamic viscosity is the Greek symbol mu: μ. The SI units are Pascal-seconds (Pa·s: identical to 1 kg·m⁻¹·s⁻¹) although the composites industry often uses cgs units: centiPoise (cP).  There is a direct numerical equivalence between milliPascal-seconds and centiPoise (1 mPa·s = 1 cP).
• kinematic viscosity: the ratio of the viscous force to the inertial force where the latter is a function of the fluid density (ρ).  The standard symbol for kinematic viscosity is the Greek symbol nu: η and the SI units are m²·s⁻¹. although the parameter is often given in centiStokes (cgs units: 1 centiStoke is 1 mm²/s)  Hence η = μ/ρ.

Effect of temperature on viscosity

Second, as the temperature rises it causes increased motion of the atoms in a polymer chain and thus the polymer becomes more fluid.  For a thermoplastic polymer, there is a direct and reversible relationship between temperature and viscosity (assuming no change in the molecular weight of the polymer due to further polymerisation or to degradation).  For the individual components in a thermosetting resin there should also be a similar reversible relationship between temperature and viscosity, but this will not be true for the mixed materials.  A typical relationship for the variation of viscosity with temperature would be:

.................... (Equation 1)

where η and T are the instantaneous viscosity and temperature respectively and η_o is the viscosity at temperature T_o.  For a polyester resin, a typical value of a would be -0.04ºC^-1 which would result in:

• a 33% fall in viscosity for a 10ºC temperature rise,
• a 56% fall in viscosity for a 20ºC temperature rise, and
• a 70% fall in viscosity for a 30ºC temperature rise respectively [12].

Effect of cure on viscosity

Third, most thermosetting resins start out as a mixture of two low viscosity components which react with one another to form a 3-D cross-linked network.  In consequence, the degree-of-cure and the viscosity increase with time after the mixing of the components. This is normally accompanied by the generation of heat which will act to accelerate the reaction.  In the limit, this may result in smoke and/or fire.

The heat evolved during cross-linking (cure) can be assumed to be proportional to the degree-of-cure of the resin system with the equation to describe the reaction rate having the form: δα/δt = K(1-α)ⁿ where K is based upon the Arrhenius equation: K = K_o.exp(-E_a/RT) and where:

• K is the overall reaction rate,
• K_o is the pre-exponential constant (per second)
• E_a is the activation energy (J/mol)
• R is the universal gas constant (J/mol.s), and
• T is the absolute temperature (K).

For resin systems cured by an addition reaction (e.g. polyester and vinyl ester systems), the Kamal and Sourour [13] model is often used.  It incorporates both n-th order kinetics and an auto-catalytic term (Equation 2 [14]).  

.................... (Equation 2)

where:

• ΣΔh is the cumulative specific heat release (J/kg),
• K₁ and K₂ are temperature dependent kinetic constants (per second)
• B₁ and B₂ are energy constants (J/mol)
• R is the universal gas constant (J/kg.K)
• T is the absolute temperature (K), and
• H is the specific enthalpy of reaction (J/kg)

For epoxy resin, White [15] modelled the PR286 resin system with the following assumptions:

• the temperature of both the mould and the resin were equal,
• the problem could be simplified by considering it as a one-dimensional infinite plane problem,
• Equation 3 applies:

.................... (Equation 3)

where:

• η is the viscosity at temperature T and time t,
• η_o is the viscosity at time zero and is solely a function of temperature,
• log₁₀(η_o) = 3.85E3/T(ºK)-8.93 is the rate of viscosity rise calculated from the constant temperature experiments, and
• log₁₀(K(T)) = -3.93E3T(ºK)+8.66.

White further demonstrated that the profiles of the isothermal viscosity-time curves of an epoxy resin system were identical when plotted on logarithmic axes.  The effect of temperature was a change in the position of the curve with respect to those axes.  The application of an appropriate shift along the logarithmic time and logarithmic viscosity axes could bring any pair of curves into coincidence.  By implication, it follows that any isothermal viscosity-time curve may be generated from any other by an appropriate scaling of the time and viscosity axes.

Roller [16] presented experimental evidence for the applicability of Equation 4 for the determination of the viscosity of curing B-staged epoxy resins as a function of both time and temperature. 

.................... (Equation 4)

where:

• η is the time-dependent viscosity,
• η_o is the zero-time viscosity given by η_o = η_xe^ΔE_η/RT, although this expression may not hold near (i.e. within 50ºC of T_g) the glass transition temperature,
• η_x is the calculated viscosity of the material at T=∞,
• ΔE_η is the Arrhenius activation energy,
• k is the kinetic factor given by k = k_xe^ΔE_k/RT, where k_x and ΔE_k are the analogues of η and ΔEη and the expression will not correctly model the kinetics if the reaction mechanism changes within the temperature range of interest,
• R is the gas constant, and
• t is time

In-process monitoring of viscosity: see the Process Control webpage.

References

1. Kaye and Laby Tables of Chemical and Physical Constants, http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_3.html, accessed 15 November 2007.
2. S Coll, A Murtagh and C Ó Brádaigh, Resin Film Infusion of cyclic PBT composites: consolidation analysis, Proceedings of the Eighth International Conference on Flow Processes in Composite Materials, Newark (USA), 7-9 July 2004, pp 101-106.
3. L Dupont, M Aguilar and A Poirier, An objective study of resin injection machines for resin transfer moulding of both low cost products and advanced composites, Proceedings of the 9th International Conference on Composite Materials (ICCM-9), Madrid, 12-16 July 1993. Woodhead Publishing, Cambridge UK, 1993, pp 489-496.
4. A Cocquyt, The VIP (Vacuum Infusion Process) primer, GRPguru website, accessed at 17:14 on Wednesday 28 August 2013.
5. DW Becker, Tooling for Resin Transfer Moulding, Wichita State University, Wichita – Kansas, no date.
6. NRL Pearce, FJ Guild and J Summerscales, An investigation into the effects of fabric architecture on the processing and properties of fibre reinforced composites produced by resin transfer moulding, Composites Part A: Applied Science and Manufacturing, 1998, A29(1), 19-27.
7. P Ouagne, L Bizet, C Baley and J Bréard, Analysis of the film-stacking processing parameters for PLLA/flax fiber biocomposites, Journal of Composite Materials, May 2010, 44(10), 1201-1215.
8. LG Stringer, Optimization of the wet lay-up/vacuum bag process for the fabrication of carbon fibre epoxy composites with high fibre fraction and low void content, Composites, 1989, 20(5), 441-452.
9. U Riedel, J Nickel and AS Herrmann, High performance applications of plant fibres in aerospace and related industries, Fibres Seminar, IENICA, Copenhagen, 27 – 28 May 1999.
10. FN Cogswell, Thermoplastic Aromatic Polymer Composites, Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-1986-7.  PU CSH Library.
11. W Denissen, JM Winne and FE Du Prez, Vitrimers: permanent organic networks with glass-like fluidity (minireview), Chemical Science, 2016, 7, 30-38.
12. K Potter, Resin Transfer Moulding, Chapman & Hall, London, 1997. ISBN 0-412-72570-3.  PU CSH Library.
13. MR Kamal and S Sourour, Kinetics and thermal characterization of thermoset cure, Polymer Engineering and Science, January 1973, 13(1), 59-64.
14. CD Rudd, AC Long, KN Kendall and CGE Mangin, Liquid Moulding Technologies, Woodhead Publishing, Cambridge, 1997.  ISBN 1-85573-242-4.  PU CSH Library.
15. RP White, Time-temperature superpositioning of viscosity-time profiles of three high temperature epoxy resins, Polymer Engineering and Science, January 1974, 14(1), 50-57.
16. MB Roller, Characterization of the time-temperature-viscosity behaviour of curing B-staged epoxy resin, Polymer Engineering and Science, June 1975, 15(6), 406-414.

On-line resources:

• Rheology and Fluid Dynamics at the University of Plymouth
• The Society of Rheology
• Neil Cunningham, Practical rheology testing can be difficult so here's a few tips to help you on your way
• Deepak Doraiswamy: The origins of rheology - a short historical excursion
• Roderic Lakes (University of Wisconsin): Viscoelastic Materials - Internal friction in solids. Viscoelasticity. Anelastic solids.
• Scott Bader - Resins: type - viscosity (in Poise) - geltime (in minutes) - HDT temperature (in degrees Celcius) - approvals - access to datasheets (subject to registration)

Recommended reading:

• RB Bird and O Hassager, Dynamics of Polymeric Liquids: Volume 1 - Fluid Mechanics (second edition), Wiley, Bognor Regis, 1987. ISBN 0-471-80245-x
  ["THE reference - highly recommend chapters 2-3 as a start" - William Hartt]
• RB Bird, CF Curtiss, RC Armstrong and O Hassager, Dynamics of Polymeric Liquids: Volume 2, Kinetic Theory (second edition), Wiley, Bognor Regis, 1987. ISBN 0-471-80244-1.
• AH Cottrell, The Mechanical Properties of Matter, Wiley, London, 1964. UoP Library shelfmark 620.105 COT [Tony Atkins]. Also published by Krieger Publishing, 1981. ISBN 0-89874-168-8.
• PG DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca NY, 1979. ISBN 0-8014-1203-x [Steve Eichhorn]
• DK Felbeck and AG Atkins, Strength and Fracture of Engineering Solids - second edition, Prentice Hall, 1996. ISBN: 0-13-856113-3. UoP Library shelmark 620.112 FEL. ["The obscure Deborah number is explained in simple terms" - Tony Atkins]
• RS Lakes, Viscoelastic Solids, CRC Press, Boca Raton FL, 1998. ISBN 0-84939-658-1 [William Hartt]
• RG Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 1998. ISBN 0-19-512197-x [William Hartt].
• CW Macosko, Rheometry, Wiley-VCH, Weinheim, October 1994. ISBN 0-471-18575-2. UoP Library shelfmark 664.07MAC [William Hartt]
• Patrick Ostwald on "rheology" ? [Yves Brechet]
• RI Tanner and K Walters, Rheology: An Historical Perspective, Elsevier Rheology Series v7, 1988.  ISBN 0-444-82945-8.
• IM Ward and J Sweeney, An Introduction to the Mechanical Properties of Solid Polymers - second edition, John Wiley & Sons, April 2004. ISBN 0-471-49626-x [Steve Eichhorn].