The dynamic viscosity of resins undergoing soft gelation shows a marked sigmoidal shape, suggesting that similarly shaped functions might be considered for model predictions.
By Brian Love
We have all encountered instances where failure to characterize the complete physical state of formulated polymer-resin mixtures limits our ability to describe rheological advancement (gaining new insights into deformation and the flow of matter) from a more fundamental perspective. The presence of co-monomers, fillers, compatibilizers, and orientation all affect the thermodynamics of the glass transition. As monomers and formulated resins form longer chain lengths, the transition from small- to larger-molecule behavior is often tracked and observed based on rheological analysis, combined with the corresponding parametric studies, to resolve how fillers, formulation, cure temperature, and pressure affect the dynamics of these processes. The analysis might lack a true interpretation of free volume, for example, but the parametric sensitivity can be resolved, thus helping process engineers define routine and robust cure schedules.
There is no shortage of models describing cure advancement, either from a rheological or a cure-kinetics perspective. Several practical reviews have summarized specific models that are based in part on both kinetics and thermodynamics.1, 2 They interpret how the glass transition is affected by growing chain lengths. Further complications arise for network formation as the kinetics of polymerization also affect the induction time and rheological advancement.3, 4 On the one hand, phenomenological relationships interpret the shapes of various rheological profiles.5–7 On the other, ab initio calculations from first principles, aimed at resolving the contributions of entanglements and size to the path length of single polymer chains, are used to study rheology.8 The power-law model has a particular allure to describe time-dependent viscosity because of its simplicity. Predictions have been made using power-law coefficients to control polymerization conditions and avoid thermal runaway associated with auto-accelerating reactions.9, 10
We have been investigating whether other mathematical models could represent a wider range of dynamic rheology data. The Austrian physicist Ludwig Boltzmann elegantly described the toggling between two states following some sort of sigmoidal curve. He resolved the mathematics to describe such a function, n(t), which yields four parameters, two time constants and two physical constants attributed to the initial and terminal physical states.
For a cure curve, these states are related to the viscosities of the resin in the uncured (no) and cured (n∞) states, respectively. One time constant corresponds to the induction time for the viscosity to traverse though the midpoint between the cured and uncured conditions (to) and the second relates to the time associated with the rate of viscosity rise (Δt) at the midpoint. Interpretations of the gel time similar to the power-law model are described as algebraic expressions of the two time constants.
We have evaluated rheological advancement using this sigmoidal model of pre-polymers measured in our own laboratory and reanalyzed other published data sets, including epoxies,11–13 acrylics,14, 15 polyurethanes,16 polyacrylamides,17 and even network gels based on denatured and misfolded proteins such as insulin. An example of the analysis for an acrylamide gel with its corresponding Boltzmann fit is shown in Figure 1. We use the model in logarithmic form given the large range of most rheological data, as it seems to capture a wider array of resin behavior.
The model is nimble in resolving dynamic viscosity from a phenomenological perspective. It is encouraging that the Boltzmann parameters have some physical significance. On one level, the allure of including one simple equation into process models such as Moldflow® to describe time-dependent materials conveyance in reaction-injection molding and polymerization vessels as a function of time is compelling. Unresolved issues remain to be addressed. For one, it is not fair to consider the upper limit of viscosity accurate in the Boltzmann model if it is extracted from data in which the torque limit of the rheometer is reached. A more rugged rheometer would yield a different dynamic curve, which would result in different Boltzmann constants. This is one motivation to redirect our efforts to focus on weaker gels such as the acrylamides, which have gel stiffnesses well below the torque limit of the rheometer. The sigmoidal model should be more accurate for foodstuffs and other protein mixtures that have lower terminal viscosities. Our work is just the first step of a more comprehensive effort to justify and compare the sigmoidal model in terms of what we already know of other fundamental gelation models from a thermodynamic and kinetic point of view, incorporating cure temperatures, exothermic evolution, dynamic conversion, and free volume within the growing polymer.
I would like to acknowledge the contributions of Thibaut Savart and Caroline Dove, who conducted the acrylamide-cure experiments and the corresponding rheometry.
Author Information
Brian Love, University of Michigan Ann Arbor, MI
Brian Love, a materials scientist, joined the University of Michigan in 2008, following nearly 15 years at Virginia Tech. His primary research interests are tied to photopolymerization of resins and reactive dispersions.
References
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11. F. Teyssandier, M. Ivankovic, and B. J. Love, Modeling the effect of cure conversion on dynamic viscosty of epoxy resin cured by an anhydride curing agent, J. Appl. Polymer Sci. 115, pp. 1671–1674, 2010.
12. F. Teyssandier, Y. Y. Sun, C. P. Wong, and B. J. Love, Cure versus flow in dispersed chip underfill materials, Macromol. Mater. Eng. 293, pp. 828–831, 2008.
13. B. J. Love, F. Teyssandier, Y. Y. Sun, and C.P. Wong, Sigmoidal chemorheological models of chip underfill materials offer alternative predictions of combined cure and flow, Macromol. Mater. Eng. 293, pp. 832–835, 2008.
14. B. J. Love, F. Piguet-Ruinet, and F. Teyssandier, Chemorheology of photopolymerized acrylates using a modified Boltzmann sigmoidal model, J. Polymer Sci. B: Polymer Phys. 46, pp. 2319–2325, 2008.
15. F. Piguet-Ruinet and B. J. Love, Dynamic photorheological analysis of photopolymerizable urethane dimethacrylate resins with varying diluent content and light fluence, J. Appl. Polymer Sci. 107 (3), pp. 1523–1529, 2008.
16. F. Teyssandier and B. J. Love, Cure advancement of urethane networks using a sigmoidal chemorheological model, Polymer Eng. Sci. In press., doi:10.1002/pen.21560
17. T. Savart, C. Dove, and B. J. Love, In situ dynamic rheological determinations of polyacrylamide during gelation coupled with mathematical models of viscosity advancement, Macromol. Mater. Eng. In press.
