By Paul Davis
Any parent rushing to organize dinner for tired children can speak knowledgably about the rheological vicissitudes of cooked spaghetti. Sitting motionless in the strainer, spaghetti forms a pile of slippery threads. But scoop up one large portion to split between dishes and the strands solidify into a single mass that slithers onto one plate or another, parental intentions notwithstanding.
At the 2018 SIAM Annual Meeting, which took place in Portland, Ore., this July, Pam Cook of the University of Delaware spoke about molecular-level modeling of materials that exhibit similarly schizoid behavior. She was simultaneously delivering the Past President’s Address and the Julian Cole Lectureship. The latter’s prize citation recognized Cook’s “comprehensive mathematical modeling of the structure and dynamics of wormlike micellar solutions.” These anomalous materials, positioned “at the boundary between solids and fluids,” as Cook put it, are commonly used in drag-reducing agents in oil fields and as thickeners and surfactants in home and personal care products.
Despite their unappetizing name, wormlike micellar solutions have attracted their share of mathematical modelers. These solutions display horrendously complex rheological properties and can exhibit highly variable viscosity, shear thinning and thickening—sometimes visible as elastic recoil or extensional fracture—and even self-assembly. Their behavior varies widely over diverse time scales. Such micelles are wormlike because each has a length of about 2 microns and a radius of approximately 0.001 microns — a 2,000:1 ratio (for context, the radius of a human hair is a relatively enormous 25 microns).
Micelles typically have a hydrophilic head and a hydrophobic tail. In an aqueous solution, they become “living polymers” — “worms” that continuously break and reform; true polymers do neither. Environmental factors like temperature, solvent salinity, and the nature of the flow field influence the particular properties that micelles manifest.
The associated modeling challenges are substantial. One approach involves modeling a three-dimensional dynamic assembly of individual “worms” that randomly break and reform over a continuous range of lengths. Regardless of the method, such models must yield to analysis—numerical or otherwise—to fit to experiments, account for observed phenomena, and test for behavior that violates observations.
Figure 1. The two sizes of micellar “worms” in the Vasquez-Cook-McKinley model. Two short units can combine to form one double-sized unit, as shown in the lower left. A linear spring connects the ends of the half-sized “worms.” Figure courtesy of [1].
Ignoring the appeal of modeling these “worms” in their full behavioral glory, Cook and her collaborators Paula Vasquez (University of South Carolina) and Gareth McKinley (Massachusetts Institute of Technology) created a model that permits only two sizes rather than allowing a continuous distribution of worm lengths. In the Vasquez-Cook-McKinley (VCM) model, just two lengths—\(L\) and \(L/2\) —define the two species of worms whose interactions carry the burden of explaining a considerable range of novel rheological behavior that is evident both experimentally and in practice (see Figure 1).
The short worms are linear springs connecting two equal masses; Cook calls them “elastic dumbbells.” Their longer cousins arise when two short worms connect; this reforming rate is modeled as constant. However, the rate at which one double-length worm ruptures into its two constituent halves depends upon the deformation rate and the local elongation rate. The newly-created halves can retract slightly as the solvent convects them back into the flow. The VCM model tracks the halves with a configuration distribution function that depends on time, location of the worm’s center of mass, and orientation of the vector that connects the two end-point masses (\(\bf{Q}\) in Figure 1).
It is perhaps surprising that the drastic simplification of permitting worms of only two lengths pays off with a theory that leads to improved understanding of micellar solutions’ unusual rheology. This success seems to arise from two features of the VCM model. It accounts for the splitting and reconnecting of the worms (scission and reforming in the jargon of this branch of non-Newtonian fluid mechanics), and the corresponding reduction in the model’s overall complexity facilitates its analysis. Furthermore, the model incorporates the aqueous solvent’s viscosity as well as its corresponding interactions with the two length species, \(L\) and \(L/2\).
Nonetheless, the simplicity of the two-species VCM model is relative. It requires 18 coupled partial differential equations, while shear flow modeling demands only half as many.
A classic rheological experiment studies viscosity variations in simple shear flow in a Couette cell; a micellar solution is confined between two concentric cylinders, the inner of which rotates while the outer remains stationary. When the inner cylinder rotates relatively slowly, the spinning fluid’s velocity decreases almost linearly (curvature accounts for most of the change) from the inner rotating wall to zero at the outer stationary wall. Thus, the velocity gradient is close to constant across the gap between cylinders. At higher velocities, wormlike micellar fluids exhibit so-called shear banding; a band of high shear rate (a large velocity gradient) exists near the inner wall and quickly transitions to a band of lower shear rate (lower velocity gradient) through the bulk of the fluid to the outer wall (see Figure 2).
Figure 2. At sufficiently high velocities of the inner cylinder, the micellar solution in this Couette cell exhibits shear banding, a region of steep velocity gradient near the inner cylinder, a kink, and finally a broader band of more gradual velocity reduction to the stationary outer cylinder. Figure courtesy of [2].
In work with Lin Zhou (New York City College of Technology), Cook and her collaborators show that the relative dominance of either the shorter or longer species of micellar worms explains shear banding in the VCM model. As shear rate increases, breakage of the double-length species overwhelms the ability to stretch, and scission dominates. However, most of the longer worms disappear at very large shear rates; in these cases, the shorter worms dominate.
Besides its success in illuminating shear band mechanics, the VCM model has helped explain the possibility of multiple shear bands, elastic recoil, and other rate- and time-scale-dependent phenomena. Indeed, much of the VCM model’s effectiveness across a variety of flow types stems from its coupling of the two species’ local number density to local stresses and velocities via the solvent. These two number densities are surrogates for a smooth distribution function in a more complex model that incorporates a continuum of worm lengths.
In framing their model for this important class of materials, Cook and her team have found and followed the narrow path that separates uninformative models from the intractable. Cook once described her late colleague Julian Cole as having had “the ability to simplify a problem down to the essential elements.” He would have been pleased by the path finding she described.
Cook’s presentation is available from SIAM either as slides with synchronized audio or a PDF of slides only.
The Julian Cole Lectureship was established in 1999 in memory of Julian D. Cole and his work in mathematical applications to aerodynamics. The prize is awarded every four years to an individual for an outstanding contribution to the mathematical characterization and solution of a challenging problem in the physical or biological sciences or in engineering, or for the development of mathematical methods for the solution of such problems.
References
[1] Bird, R.B., Curtiss, C.F., Armstrong, R.C., & Hassager, O. (1987). Dynamics of Polymeric Liquids: Kinetic Theory (Vol. 2) (2nd ed.). New York, NY: Wiley-Interscience.
[2] Hu, Y.T., & Lips, A. (2005). Kinetics and mechanism of shear banding in an entangled micellar solution. J. Rheol., 49(5), 1001-1027.
Paul Davis is professor emeritus of mathematical sciences at Worcester Polytechnic Institute.