Complex flows of viscoelastic wormlike micelle solutions
Introduction
A number of recent review articles have been written on the rheology and applications of wormlike micelle solutions [1], [2], [3], [4], [5], [6], [7], [8]. In this review, we intend to focus less on the linear viscoelasticity and/or simple shear deformation response of these fluids and more on the complex flows where a combination of shear deformation and extensional flows exists. Such complex flows are found in a number of industries and applications for which wormlike micelle solutions are commonly used.
Surfactants are amphiphilic molecules which have both a bulky hydrophilic head, which is often charged, and a relatively short and slender hydrophobic tail typically consisting of an 8–20 carbon atom chain. Above their critical micelle concentration (CMC), surfactant molecules in water will spontaneously self-assemble into large aggregates known as micelles to minimize the exposure of their tails to water [9], [10], [11]. In oil, reverse micelles are formed where instead the head-groups are shielded from the oil [12, 13]. These large aggregates can form into a number of different complex shapes including spherical, wormlike micelles, vesicles and lipid bilayers [14]. The morphology of the aggregates depends on the size of the surfactant head group, the length and number of tails, the charge on the surfactant, the salinity of the solution, temperature, and the flow conditions [9, 14]. Surfactants with a large head group and/or a single short tail tend to form spherical micelles while surfactants with small head groups and/or a single long tail tend to form wormlike micelles. Surfactants with two or more tails tend to form bilayers. The basis for these relatively simple distinctions are clearly described by Isrealachvili [15] using a packing argument based on the relative effective size of the surfactant head and tail groups.
The phase diagram of surfactant solutions that form wormlike micelle solutions can be quite complex [14, 16]. As the concentration of surfactant in solution is increased, a transition is observed from dilute individual micelles, to semi-dilute entangled micelles, to nematic, hexagonal, cubic or other ordered phases at the highest surfactant concentrations. Within the semi-dilute regime, increasing salt concentration can modify the wormlike micelles from linear, to branched and finally to an interconnected network. For a linear wormlike micelle, the shape and area per unit surfactant molecule is optimized at all positions along the backbone except at the endcaps [9]. When a linear micelle breaks, it must pay an energy penalty by forming two new end caps. In this regime, the electrostatic repulsion of the head groups is strong enough that the increased curvature of an endcap, which spreads the head groups apart, is favored over the concave curvature of a branch point which drives the charged head groups of the surfactants closer together. However, as the salt concentration is increased and the head group charges are sufficiently screened, the wormlike micelles can form three-point or four-point junctions. Evidence of the existence of these branched micelles can be seen in cryo-TEM images [17], [18], [19], [20], [21], [22].
This review will focus on surfactants that tend to form either linear or branched wormlike micelles like those shown in Fig. 1. At large surfactant concentrations these wormlike micelles can grow very long, become entangled, and make the solution viscoelastic. As suggested by their pseudonym ‘living polymers,’ wormlike micelles display many of the same viscoelastic properties of polymers. However, although both wormlike micelle solutions and polymer solutions can be viscoelastic, wormlike micelles are physically quite different from polymers. Whereas the backbone of a polymer is covalently bonded, wormlike micelles are held together by relatively weak physical attractions and as a result are continuously breaking and reforming with time. In an entangled network, both individual polymer chains and wormlike micelles can relieve stress through reptation driven by Brownian motion [10]. However, unlike polymeric fluids, wormlike micelle solutions have access to a number of stress relief mechanisms in addition to reptation. Wormlike micelles can relieve stress and eliminate entanglement points by either breaking and reforming in a lower stress state [11] or alternatively by creating a temporary branch point which allows two entangled micelles to pull right through each other thereby eliminating the entanglement point and relieving stress in what has become known as a ‘ghost-like’ crossing [23].
Viscoelastic wormlike micelle solutions are currently being used extensively as rheological modifiers in consumer products such as paints, detergents, pharmaceuticals, lubricants and emulsifiers where careful control of the fluid properties is required. In addition, micelle solutions have also become important in a wide range of applications including agrochemical spraying, inkjet printing, turbulent drag reduction and enhanced oil recovery where they are often used as a polymer-free fracture fluid for stimulating oil production [1,2,24]. In these applications, wormlike micellar solutions experience a combination of shear and extensional deformation. A fundamental understanding of the behaviour of these complex fluids in different flow regimes is therefore extremely important to a host of industries. This has prompted a considerable interest in the complex fluid community to interrogate the flow behaviour of wormlike micelles in simple shear, uniaxial extensions and complex flows where a combination of shear deformation and uniaxial extension exists. The response of the wormlike micelles in shear flows has been extensively studied in the past three decades and multiple reviews have been complied on this topic [4, 8]. However, although, a number of studies of the response of wormlike micelles in uniaxial extensional flows and complex flows have been published over the last two decades, a critical review of such studies is needed. It is the aim of this review to summarize the results of those studies.
The outline of this review is as follows. In Section 2, we discuss extensional rheology measurements. In Section 3, we discuss flows of wormlike micelle solutions past a falling sphere. In Section 4, we discuss flows of wormlike micelle solutions past a stationary cylinder. In Section 5, we discuss flows of wormlike micelle solutions in a range of other complex flows like contraction-expansions, cross-slots and sharp bends. Note that another important example of complex flow is the flow in porous medium. The response of micellar solutions in porous medium has been reviewed in great details in a recent review by Zhao et al. [7], and therefore, readers are encouraged to read this recent publication [7]. Finally, in Section 6 we conclude and present our outlook for future research in this area.
Section snippets
Extensional rheology of wormlike micelle solutions
Over the past thirty years, extensional rheology has become a topic of great interest and importance to the complex fluids community. In particular, the extensional behavior of polymeric fluids has received a considerable amount of attention [25]. However, even though they are less well studied, the behavior of wormlike micelles solutions in extensional flows has been shown to be incredibly rich and complex. The extensional behavior of wormlike micelle solutions was initially studied using the
Flow around a falling sphere
Over the last 20 years, a number of research groups have examined the flow of the self-assembled surfactant solutions past a falling sphere [53, [85], [86], [87], [88], [89], [90], [91], [92], [93]]. Historically, the first study was reported in 2003 by Jayaraman and Belmonte [85] who examined sedimentation of a sphere in a shear-banding wormlike micellar fluid based on 9 mM/9 mM solution of CTAB/NaSal [85]. These authors demonstrated that beyond a critical shear Weissenberg number of Wi > 45,
Flow around a circular cylinder
Fluid flow around a circular cylinder in cross-flow of a non-Newtonian fluid is a canonical problem that has been the subject of number of both experimental [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [120] and numerical studies [119, [121], [122], [123]] primarily for the flow of polymer solutions. Only recently has the flow of wormlike micelle solutions past a circular cylinder been investigated [124], [125], [126], [127], [128], [129]. Moss and Rothstein [124, 125]
Flow through other complex geometries
Flow of wormlike micelles in complex geometries have focused primarily on cross-slot, contraction-expansion and 90° degree bend. The common feature of these geometries is that micellar solutions experience a combination of both extension, shear deformation and/or stream-line curvature.
Conclusions and outlook
We have reviewed the experimental and theoretical literature on complex flows of wormlike micelles, in which a combination of shear and extensional deformation exists. Although wormlike micelles have been studied for quite sometimes, their response in complex flows still remains enriched due to a strong positive feedback loop between the flow and the microstructure. Flow changes the microstructure of the wormlike micelles, and the microstructural changes affect the resulting stress field, and
Declaration of Competing Interest
No conflict of Interest.
Acknowledgements
The authors would like to express his sincere thanks to both their current and former students whose work constituted a large part of this review article. The authors would also like to thank the National Science Foundation (CBET CAREER 1942150 for HM and CBET-1705251 for JPR) for financial support.
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