Characterizing the rheology, slip, and velocity profiles of lamellar gel networks

Emulsions comprised of ionic or nonionic surfactants and fatty alcohol (FA) in excess water often containing salt are widely used as pharmaceutical creams, cosmetic lotions, hair care products like shampoos and conditioners and as carriers of intradermal or transdermal drug delivery in part because of their rheological properties [1–4] and their ability to form intricate colloidal structures [5,6]. These ternary mixtures can withstand elastic deformations and are often referred to as “lamellar gel networks” [7] because the lipids are prepared below the gel-transition temperature. Most research on these systems focuses on optimizing the formulation to attain a physically stable emulsion and colloidal properties suitable for the delivery of the drug to the body or the cosmetic effects to the skin.[8]

Lamellar gel networks have a multiphase structure that is responsible for their opaque creamy appearance and texture, as well as their highly viscous, shear-thinning rheology [9], and an oily tactile feel and slipperiness that are desirable consumer properties. The fatty alcohol and surfactant in excess water self-assemble into highly swollen, regularly spaced bilayers, with hexagonal packing of molecules in the bilayers [10]. This swelling is caused by repulsive effects, either steric or electrostatic, the latter caused by the ionicity of the surfactant head group, which produces an osmotic pressure [11] in the hydrophilic region of the lamellae [12], leading to swelling by water. The extent of swelling is governed by the concentration of the surfactant counterion and any salt present [13].

Stable lamellar gel networks can only be formed over a limited range of fatty alcohol and surfactant concentrations [14]. Varying the ratio of the surfactant to fatty alcohol within this range results in creams of various consistencies, homogeneities, and viscosities [15]. Long chain alcohols (C₁₆ and C₁₈) used for these systems are weak emulsifiers by themselves and phase separate into lipid-dense lamellar crystalline structures upon cooling [16,17], while the lamellae of the mixed surfactant/fatty alcohol emulsifier swell with water, producing a creamy, flowable material. The hexagonally packed hydrocarbon chains have in-plane spacing of around 4.2 Å [9]. In a similar mixed emulsifier system containing cetrimide, a homolog mixture of alkyl trimethyl ammonium bromides (dodecyl 20%, tetradecyl 68%, and hexadecyl 10%), and fatty alcohol in 93% water, the lamellar sheets are separated from each other by interlamellar water giving rise to a lamellar spacing of 50 nm with a bilayer thickness of only about 3.57 nm [13]. This lamellar spacing varies with the water content, ionicity of the cationic surfactant (CS), and the salt concentration [18]. Junginger [19] showed using thermogravimetric tests that the interlamellar bound water is in dynamic equilibrium with the bulk water phase and that both water domains are continuous.

Above a transition temperature, surfactant/fatty alcohol mixtures in water form a liquid crystalline L_α phase, with alkyl chains positionally disordered in the plane of the lamellae. The positional disorder can transform upon cooling, typically into a hexagonal order, forming a stiffer gel phase known as the L_β phase. Several other arrangements have been reported [20,21] for the gel phase, such as the L_β′ phase with surfactant tails tilted to the bilayer normal [22], but these have not been widely reported in cosmetic emulsions used in commercial formulations.

In this work, a thorough rheological characterization is performed on a pseudoternary system comprised of (1) a monovalent cationic surfactant with a saturated C₂₂ tail, namely, behentrimonium methosulfate (BTMS), (2) a mixture of cetyl and stearyl alcohol (C₁₆ and C₁₈), and (3) water. The chemical structures of the surfactant and fatty alcohols are given in Fig. 1. These main ingredients, along with additives such as salt, ethylenediamine tetra-acetic acid, and preservatives are the key ingredients in commercial hair conditioners [18,23,24]. Here, for simplicity, we do not include any additives in our samples. The positive charge on the quaternary ammonium surfactant head group promotes adhesion to the negatively charged wet hair and provides a richer, heavier, more conditioned feel to the consumer.

A mixture of cetyl and stearyl alcohol (henceforth referred to as cetostearyl alcohol) is commonly used in creams and lotions because of its enhanced stability against phase separation over that of pure cetyl or stearyl alcohol [16,25,26]. Microscopy and x-ray diffraction [13] have shown that either pure cetyl or pure stearyl alcohol phase separates from water into a crystalline phase usually with an orthorhombic (β-form) or monoclinic crystal structure (γ-form), with an excess water phase. This is manifested in the appearance of puddles of water in the sample or evaporation of the water from the sample (subsequently depositing on the sides of the sealed container holding the sample), as can be seen in Fig. 2. Apparently, the chain-length mismatch of the two fatty alcohols in the cetostearyl mixture confers more conformational freedom to the alkyl end of the longer stearyl alcohol chain. The bulky trimethylammonium surfactant head group along with its methosulfate counterion “pushes away” the fatty alcohols and, therefore, also contributes to this conformational freedom. The resulting increased population of gauche conformations prevents the formation of orthorhombic or monoclinic crystals, allowing the less tightly packed liquid crystalline hexagonal phase of the lamellar gel to be maintained, along with freedom of the chains to rotate about their axes [27]. In the orthorhombic and monoclinic crystals, on the other hand, there is long-range bond orientational order of the alkyl tails, making these materials more solidlike.

The cationic surfactants used in these products have a gel-to-liquid crystalline transition temperature above room temperature (65–70 °C). The formulations are prepared by adding preheated water to the fatty alcohol and the cationic surfactant mixture that had also been preheated to a temperature above their melting points, i.e., 90 °C. This high temperature ensures that the L_α liquid crystalline phase is formed during mixing of CS and FA; these convert to a stable L_β gel phase when the system is cooled to shelf-storage at room temperature [5]. The formation of a stable gel network is dependent on the polymorphism of the fatty alcohol. DSC and x-ray scattering studies [28,29] have shown that stable emulsions are formed when the fatty alcohol is arranged in its hexagonally packed bilayer structure (i.e., the “α-form” of the L_β gel, not to be confused with the L_α phase, in which the fatty alcohol is disordered in the plane). These samples are left to rest overnight, after which the samples appear swollen. Kudla et al. [30] showed in their work with a ternary system containing a cationic surfactant that a ripening process takes place after the formation of these viscoelastic creams, during which lamellae gradually form over around 3 days.

In this paper, we characterize the rheological and deformation behavior of these formulations more thoroughly than heretofore and examine the dependencies of their rheology on their compositions. Such knowledge is of fundamental importance for developing hair conditioners with desirable rheological properties like creaminess, and slipperiness, in addition to having acceptable shelf-life (stability) and ability to accommodate additives like perfume and oils [31]. “Slipperiness,” which is a critical property that facilitates the untangling of coils in the hair and the effective scrubbing of the head, cannot readily be determined from simple rheological measurements for these systems. Even though an extensive literature on the structure and rheology of these lamellar gel networks is available, the slip behavior of these systems has not been documented before, although slip has been commonly observed in gelled suspensions, emulsions, entangled polymers, pastes, and waxy crude oils [32–36]. Here, we image the flow of these gel phases during rheological measurements by particle tracking velocimetry and find that they slip even at roughened rheometer surfaces, leading to plug the flow at low shear rates. At higher shear rates, sample fracture is also seen. In oscillatory experiments, while the material shows continuous deformation at low strain amplitudes, at high strain amplitudes, the sample ruptures and material slides along the rupture plane. These heterogeneities occur even in cone and plate geometries where the stress is nearly constant. A key novelty of the work lies in uncovering their inherent tendency to slip so that these fluids lack defined, traditional rheological properties but instead display a combination of rheological and tribological signatures.

These lamellar gels show rheological characteristics similar to those reported for “soft glassy materials” [37], which comprise a class of complex fluids such as foams, emulsions, pastes, gels, and concentrated suspensions. The widely used soft glassy rheology (SGR) model [38–40] predicts the rheological properties of these systems which include shear hysteresis in up- and down-ramps of the shear rate [41], oftentimes leading to the formation of heterogeneous shear bands [42], a logarithmic power-law creep response [43] at stresses below their yield stress and a gradual stress decay following sudden imposition of a small strain over wide range of time scales [44,45]. In the linear viscoelastic regime, these materials show a storage modulus G′(ω) that typically exceeds the loss modulus, G″(ω), a weak power law frequency dependence of its moduli, and a significant loss modulus at the lowest frequencies [46]. Our mixtures exhibit these above-mentioned signatures of soft glassy materials, thus leading to the possibility that the behavior of these systems can be theoretically predicted in future work. The comparison, in our work, of the rheology of lamellar gels to an existing class of fluid models, the SGR models, is another novelty of our work.

The remainder of the paper is organized as follows: First, we describe the materials and methods used and then the rheological characterization and particle imaging methods. Next, we present the results of the rheological characterization and then of the imaging, including quantification of slip. The results are then summarized and discussed.

The cationic surfactant, BTMS, and the fatty alcohols—stearyl alcohol (1-hexadecanol) and cetyl alcohol (1-octadecanol), were provided in the pellet form by the Procter & Gamble Company (Cincinnati, OH). This commercial grade “BTMS” is only 80% pure and contains 20% by weight isopropyl alcohol (IPA), which is used as a processing aid to improve its manufacturability. We account for the presence of IPA in our specifications of compositions of BTMS. These substances were used without further purification. The fatty alcohol in the lamellar gel network was cetostearyl alcohol, which is a 1:2.5 (weight basis) mixture of cetyl and stearyl alcohol, respectively. Formulations were prepared in-house while maintaining the desired ratios between the fatty alcohol, cationic surfactant, and de-ionized water (MilliQ Plus, resistivity 18.2 MΩ cm). The term “CS mole fraction” used henceforth is defined as $mol(CS)/(mol(CS)+mol(FA))$ which quantifies the mole fraction of the cationic surfactant within its mixture with fatty alcohol. The number of moles of the cationic surfactant, represented by $mol(CS)$⁠, is calculated after subtracting the weight of IPA from the weight of the commercial grade BTMS to ensure that the CS mole fraction represents the actual fraction of CS in the mixture of CS and FA.

The lamellar gel networks were prepared as follows. A 100 g chassis of the sample was made by melting together the measured weights of the CS and FA in an oven at 90 °C, which is above their melting points. The desired quantity of water that was preheated to 80 °C was added to this molten mixture and was continuously stirred until a homogenous gel-like white base mixture was obtained. These samples were then left to cool to room temperature and were then stored in clear, glass vials.

The rheological experiments were performed using the Anton Paar MCR702 rheometer at 20 °C, with a cone and plate geometry made of stainless steel. The temperature was maintained through a Peltier plate and hood covering the geometry. The top fixture was usually a cone 25 mm in diameter with a 2° cone angle, with a truncation gap of 105 μm. Some experiments were also done with a top cone geometry of 1° cone angle to evaluate if the rheological results were sensitive to the cone angle. Other tests were performed with the 2° cone used as the bottom fixture with the plate at the top, or with a plate-plate geometry, to test the effect of geometry on the results. To minimize the slippage of the sample due to shaft rotation, both the cone and plate had been sandblasted with roughness, R_a, of 4.19 μm as described by the manufacturer.

For the velocimetry measurements, fluorescent particles (0.1 vol. %, radius = 1.5 μm, Fluoresbrite-YG, Polysciences) were added into the preheated water used to make the formulation. The sample was viewed through the edge meniscus by a fluorescence microscope (Anton Paar Module 1, bandpass 450−490 nm). Videos of the edge meniscus during the rheological tests were acquired with a CCD camera (Lm165C; resolution, 1392 × 1040 pixels; made by Lumenera) attached to the fluorescent microscope, which records at a frame rate of 15 frames/s (or higher rates at lower resolutions). The focal plane of the objective lens was set to 0.1 mm inside the sample-air interface. Experiments were performed with and without the particles, and no difference was seen in the rheological data.

A time-resolved particle image velocimetry (PIV) algorithm calculated the 1D velocity profile from a pair of frames selected at arbitrary times of t and $t+Δt$⁠, where $Δt$ is the time between two image frames. We set $Δt≈1/(5γ˙)$⁠, in our calculation, with $γ˙$ the apparent shear rate applied on the sample. The exact value of $Δt$ is measured through the timestamps between the two selected frames. The images were cropped to retain only the region between the cone and the plate. The contrast-limited adaptive histogram equalization (CLAHE) algorithm was used to process the images, and cross-correlations of these two frames were used to determine the displacement of particles. The details and validation of the PIV algorithm can be found elsewhere [47].

To explain the behavior of these samples, a thorough rheological characterization was done. However, as we will show later in this section, the gap-dependence of the “viscosity” of these samples renders the term “viscosity” as merely the ratio of the stress to apparent shear rate and not a true viscometric rheological function. While we do compare our results to key signatures of the SGR model, no effect of waiting time (seconds up to hours) after sample loading was seen, indicating that the initial waiting period on the rheometer prior to any controlled deformation plays no significant role in the rheological results reported here, in disagreement with one of the characteristic behaviors of the SGR model.

The composition of 93.3% water and 0.23 CS mole fraction was chosen for our reference samples since this closely represents the composition used in commercial hair conditioners. As described below, we also considered samples that varied in both water and CS composition, over ranges that retained the gel-like properties of the mixtures. These stable systems showed reproducible rheological data over multiple repetitions of the experiment on different days, as shown in the supplementary material [55]. Over a longer period of time (weeks or months) after the sample is formulated, we observe small changes in creep rheology, however, that we discuss later.

First, we explore the time evolution of the viscosity of these samples in response to various imposed shear rates using two different cone angles, 1° and 2°, as shown in Fig. 3. Measurements were taken every 2 s and the shear rates were held constant for 1000 s. The apparent viscosity curves shift downward with increasing shear rate, thus demonstrating the shear-thinning of these samples. Viscosities also gradually decrease with time over the entire 1000 s runs, indicating that these systems do not readily reach the steady state. We, therefore, below report approximate “steady state viscosity” values for the subsequent experiments, using a criterion that we give below for determining when an “approximate steady state” has been achieved. Note in Fig. 3 that at the lower shear rates at early times, the viscosities for the 2° cone are somewhat lower (5%−50% lower) than for the 1° cone, indicating that the larger velocities of the 2° cone lead to somewhat lower stresses.

Review articles [48,49] on thixotropic fluids suggest the use of sudden step-down shear rate tests to differentiate the response of a thixotropic fluid from that of a viscoelastic material. Figure 4 shows that in this test, our sample shows a sudden drop in the stress following a step-down in the shear rate, followed by a more gradual increase in the stress and then a slow decrease over longer time periods toward a plateau value. The sudden drop followed by a more gradual increase in stress is the response expected for a thixotropic, nearly inelastic fluid. However, the gradual decrease thereafter indicates additional, more complex dynamics. We shall later describe creep-recovery tests, which show that the samples are capable of storing elastic strains of a few percent, making them slightly elastic.

Given that Figs. 3 and 4 show a very slow approach to steady stress, we elected to obtain approximate flow curves by imposing a shear rate ramp of increasing rate from 0.01 to 1000 s⁻¹ with the following protocol. At each imposed shear rate, measurements were taken until the viscosity satisfied an approximate “steady-state” criterion empirically defined as

for at least three consecutive time intervals, with each time interval $Δt=1s$⁠. This algorithm results in longer times at lower shear rates and shorter times at higher shear rates so that the viscosity comes as close as possible to steady state at all rates for a given overall ramp time. A diagrammatic representation of the time spent at each shear rate during the logarithmic ramp following the above “steady-state” criteria is available in the supplementary material [55]. The resulting “flow curve” was compared to those in which each imposed shear rate was imposed for a fixed duration of either 30 or 60 s, and the difference in the reported viscosity was negligible.

A typical response of the shear stress to this test protocol is represented in Fig. 5(a). The shear stress initially decreases up to an imposed shear rate of 0.1 s⁻¹ before it begins to increase up to the final shear rate of 1000 s⁻¹. There are indications of different structural regimes in the dependence of stress on shear rate in Fig. 5, including “bumps” in the curves, suggesting transitions to different regimes. Roux et al. [50] studied a simpler lamellar system formed from sodium dodecyl sulfate, pentanol, and dodecane in water, in which the rheological data also showed bumps and transitions. Since the geometry of the system (Mooney cell corresponding to a Couette cell ended by a cone/plate at the bottom) used by Roux et al. allowed them to image the fluid from the bottom by optical microscopy, they were able to attribute rheological changes to the transition from defect-ridden sheets at lower shear rates to multilamellar vesicles (MLVs) or “onions” observable at higher shear rates. (Our imaging of our sample at the edge of the cone-and-plate geometry does not permit viewing its microstructure over length scales needed to make comparable observations.) While their system was simpler and contained only small-molecule surfactants, we might infer that the bumpy transitions between different flow regimes in our data might also reflect underlying transitions between flow-induced structures. Figure 5(b) shows corresponding shear-thinning viscosity. The water content and cationic surfactant ratio in these formulations were then independently varied to investigate the differences in the flow curves for the same test protocol. Error bars are plotted for each point in Figs. 5(a) and 5(b) with a 95% confidence interval for seven independent repetitions. Figure 5(c) indicates that the samples exhibit almost identical flow curves for water contents ranging from 85% to 95% above which the samples exhibit somewhat lower viscosity. The similarity in results for different water contents may reflect the self-adjustment of the network between excess bulk water and water trapped between the lamellar sheets, so that the gel network remains similar at different water contents. On the other hand, Fig. 5(d) shows increasingly viscous samples with increasing cationic surfactant mole fraction.

Subjecting the sample to a high shear induces microstructural changes that are seen macroscopically through rheological tests. A hysteresis loop, shown in Fig. 6, is found upon decreasing the shear rate (represented by unfilled symbols) immediately after the completion of the increasing-shear-rate ramp (represented by filled symbols). The hysteretic difference between the flow curve for the increasing shear-rate ramp and that for the decreasing ramp increases when one increases the highest shear to which the sample is subjected.

The destruction of the microstructure at high shear rates, evidenced by reduced viscosity, can be reversed after sufficient rest time. Figure 7 shows the results of repeating the hysteresis loop after a rest time of fixed duration in the rheometer. Evaporation is minimized by the use of the hood over the sample. Moreover, the use of the cetostearyl alcohol mixture is known to provide stability and helps prevent the expulsion of water from the sample.

An immediate repetition of the shear hysteresis test shows essentially no hysteresis and gives a stress vs shear rate curve that is almost identical to that for the first decreasing shear-rate ramp. This indicates that the fluid remembers the highest shear imposed on it and that the structure created at this shear rate remains unchanged at shear rates lower than this maximum shear rate and thus determines the response at all these lower shear rates. After allowing the sample to rest on the rheometer for 3 and 6 h, respectively, the behavior remains similar to this, highlighting the fact that the sample retains the memory of its highest shear even up to 6 h. However, as seen in Fig. 7(d), a partial recovery is obtained if the sample is left to rest for 12 h after shearing ceases. This indicates that 12 h is long enough for the network to rebuild and recover partially its initial viscosity. This sets the time over which the viscosity of the sample remains lower than what it would have been if it had not been subjected to a shear treatment.

The elasticity of these formulations was quantified through creep and recovery tests in Fig. 8 that shows the strain as a function of time after imposition of a fixed shear stress, σ, for 1000 s, after which the imposed stress was removed, and the time-dependent recoil of the shear strain was measured for another 1000 s. Upon application of stress, there is a very nearly instantaneous jump in strain to a value that is roughly proportional to the imposed stress, corresponding to an instantaneous modulus of around 500 Pa. For stresses up to 16 Pa, the material then shows a solidlike creep response, following a power-law time-dependence with an exponent of 0.25, i.e., $γ∝t0.25$⁠. Upon removing the imposed stress, partial strain recovery occurs for these lower stresses. At 18 Pa, the creep response is initially similar to that at lower stresses, but at around 200 s, or a strain of around 10%, it transitions to an apparent “yielding” behavior of some kind, wherein the power-law relation at long times becomes much steeper, namely, $γ∝t1.35$⁠. Thus, the material seems to show an apparent “yield-stress” at 18 Pa above which it behaves more like a fluid and below which it has a more “solidlike” creep response. Furthermore, on imposition of a stress below this “yield-stress” for long time intervals (3 h), the strain exceeds 10%, but the response remains roughly power-law with slope corresponding to $γ∝t0.25$⁠, until a mild upturn appears at around an hour after start of creep, as shown in the supplementary material [55]. This indicates that for times up to an hour, the fluid response is not governed by a yield strain but instead by a “yield-stress.” However, samples at stresses below this “yield stress” do not stop flowing even up to 3 h but continue to deform with strain slowly accumulating approximately following the power law $γ∝t0.25$⁠. The “yield stress” is thus not a stress below which flow eventually halts but rather is the stress above which a transition occurs from a weak power-law creep to a much faster creep satisfying $γ∝t1.35$⁠.

The creep response of these samples is somewhat dependent on the long-term age of the sample since its preparation from the raw ingredients. Over a period of weeks, the material softens somewhat, and the stresses at which the transition behavior occurs drops by 1–2 Pa. Images of the flow field in the samples at stresses above 18 Pa show that the velocity profile becomes that of a plug flow, with the sample completely slipping. Additionally, at these stresses, no strain recovery is seen when the stress is removed. Ribeiro et al. [51] have shown for a similar ternary system forming a lamellar gel network that the initial elastic component of creep is phenomenologically well modeled by a spring and a dashpot in parallel, i.e., a Voigt model. However, at least for our material, the bulk of the sample does not become fluidlike at high stress, but instead the transition to fast creep at stresses of 18 Pa and above corresponds to a transition from viscoelastic to slip behavior.

After subjecting a previously undeformed sample to a step strain at time t = 0, we see a rather sluggish power-law relaxation of the shear modulus over the measured time scale of 1000 s, as shown in Fig. 9. The similarity of the moduli for strains of 1% and less indicate a nearly linear viscoelastic regime up to this strain, above which the gels show strain softening. In the linear regime, the relaxation follows a power law, with an exponent of −0.25, i.e., $G∝t−0.25$⁠, which is consistent with the time-dependence of the creep data. Ramos and Cipelletti [45] have shown similar stress relaxation response after step strain in a gel of closely packed MLVs.

The viscoelastic character of the gels is further explored through oscillatory shear experiments. Amplitude sweeps at 1 rad/s over the strain range of 0.1%–100% as in Fig. 10(a) show that the linear regime in this deformation history extends to 10% strain, which is, surprisingly, a decade higher than in the step-strain results presented in Fig. 9. The onset of nonlinear rheology coincides with a plunge in the storage modulus, which drops below the loss modulus above a critical strain amplitude of 10%. This collapse of stress at a critical strain was seen in other lamellar systems as well [17,52,53]. Based on this, frequency sweeps with a logarithmic ramp in angular frequency were performed at 1% amplitude, which is well within the linear viscoelastic regime. Figure 10(b) shows that in the linear regime these samples are more elastic than viscous over the frequency range, since the storage modulus, G′, is much higher than the loss modulus, G″. This is also a signature of soft, glassy fluids. The storage and loss moduli do not cross over even at angular frequencies as low as 10⁻³rad/s, indicating again that these samples behave as soft solid gels over a wide frequency range and do not have a readily accessible terminal region. Note that, oddly, the linear rheological data in Fig. 10(b) do not follow a power law and therefore appear inconsistent with what one might infer from a Kramers–Kronig analysis of the power-law relaxation in the linear regime of the step-strain data of Fig. 9. The magnitudes of the moduli in the two experiments are, however, roughly of the same order of magnitude (i.e., between 100 and 1000 Pa). We believe that the dissimilar linear regimes of the step-strain and oscillatory shearing, and the seeming inconsistency between their time and frequency dependencies, likely reflect the presence of slip and inhomogeneous velocity profiles in these materials, as will be discussed shortly. Finally, we note that Korhonen et al. [8] ascertained that a high storage modulus is a key for structural stability and resistance of these creams to external forces for long durations.

We note that for our materials, the moduli do not significantly change on varying the amount of cationic surfactant over the range of 0.23 mole fraction to 0.6 mole fraction CS or the total water content over the range of 87%–95%. However, for samples with water content above 95%, G′ and the G″ and the steady shear viscosity are significantly lower, as are seen for the latter in Fig. 5(c).

The slip, fracture, and banding in lamellar gel networks are characterized for the first time in this work through PIV. To obtain the velocity profiles, we imaged the edge of the gap in a shear start up test where the sample was sheared at a fixed rate for a fixed time. The fluorescent tracer particles added to these systems to image them for velocimetry data did not have an effect on the rheological data, as determined by measuring rheology without the particles. An in-house PIV algorithm [47] for the 1D shear flow was used to compute the velocity profile between a pair of frames at successive intervals of time, with one such frame shown in Fig. 11. The imaging shows that under shear the material becomes inhomogeneous macroscopically, for example, by shear banding, wall-slip, and possibly inertial effects, which may influence the time-dependent stresses on the rheometer surfaces.

The velocity profiles for these formulations at different imposed shear rates were extracted by calculating the displacement of tracer particles between a pair of frames. The shear was imposed by rotating either the bottom 25 mm sandblasted plate or the top sandblasted cone. Figure 12(a) shows that with the bottom plate moving, at the lowest shear rate of 0.001 s⁻¹, the velocity profile is initially linear until about 10 s, after which inhomogeneities in the velocity profile are seen. By 45 s, the profile developed into two nearly linear regions, in addition to the jump in velocity near the bottom surface. This indicates the existence of shear banding in the sample at this shear rate [54]. At a shear rate of 0.01 s⁻¹ or higher in Figs. 12(b)–12(d), the velocity profile almost immediately (within 1 s) transitions to a uniform-velocity, plug flow in which the tracer particles move together albeit at a lower velocity than that imposed by the rotating bottom plate, implying that there must be slip along both the top and bottom surfaces. The time evolution of the stress for an imposed shear rate is shown in Fig. 12(b). Corresponding stress values for the other imposed shear rates follow a similar trend and can be derived from Fig. 2.

When the top cone rotates with the bottom plate fixed, this transition to a plug flow does not take place until a much higher shear rate of about 1 s⁻¹ is reached instead of 0.01 s⁻¹. At shear rates below 1 s⁻¹, we see a similar case of shear banding or local regions of a nonlinear velocity profile. This suggests that the fluid adheres better to the top cone than to the bottom plate, thus making it more difficult to slip or flow as a plug at low shear rates when the cone is rotating. Repeating this exercise but instead having the plate at the top and the cone at the bottom, with the cone rotating, showed a similar plug flow only at 1 s⁻¹ and not below this shear rate, validating that the higher shear rate required for the plug flow is due to the rotation of the cone, not its location at the top or the bottom surface. A similar exercise was performed with two parallel plates and regardless of whether the top or the bottom plate was rotating, with the other fixed, the transition to a plug flow from a linear velocity profile took place at 0.01 s⁻¹. Thus, the geometry of the rotating surface seems to affect the onset of slip and raises the question of whether the behavior of these samples is to some extent tribological, i.e., dominated by a friction coefficient of solidlike surfaces sliding over one another. In oscillatory shear experiments with a fixed top cone and a rotating bottom plate, at high strain amplitudes above 25%, we observe that a fracture plane develops, usually in the bulk away from the surfaces, with the material sliding along this fracture plane.

For a viscous fluid, at a fixed shear rate, the shear stress is independent of the gap between the measuring plates, and thus, the measured viscosity is independent of the gap. On the contrary, for a sample whose response is primarily governed by its tribological properties, the shear stress at a fixed plate velocity will be independent of the gap, while the stress at the fixed shear rate will depend on the gap. Figure 13 shows that the apparent viscosity and the apparent friction coefficient (the latter given by the ratio of the shear stress to the plate velocity) are both gap-dependent. Thus, these samples cannot be characterized adequately by either a viscosity or a friction coefficient but display behavior intermediate between rheological and tribological.

This slip behavior implies that the measured stresses result from a combination of rheological and tribological properties. This makes these lamellar gel networks a very interesting type of complex fluid, which, to characterize properly, requires imaging the flow field as well as measuring the stress. In oscillatory amplitude sweeps using a parallel-plate geometry, Fig. 14 shows a monotonic increase in G′ and G″ with an increasing gap, except in G′ at high strains. Since at a particular fixed strain amplitude the plate velocity is larger for larger gaps, this indicates some degree of sample slip either at the wall or in the interior of the sample.

The rheology and velocimetry of cosmetic multiphase L_β gel networks made from a cationic surfactant behentrimonium methosulfate, a fatty alcohol mixture of cetyl and stearyl alcohol, and water, displays a combination of slight elasticity and pronounced thixotropic hysteresis in up- and down-ramps of shear rate, as well as slip and banding. These viscous, shear-thinning fluids remember the highest shear-rate that they have been subjected to for up to around 12 h and then lose this memory. Oscillatory and creep testing confirm the gel-like viscoelastic nature of these formulations with higher elastic than viscous modulus and lack of a crossover frequency even at an angular frequency of 10⁻³rad/s.

Tracking the velocity profile with particle image velocimetry, we find that if the moving surface is a flat plate, the sample transitions from a linear velocity profile to a plug flow at shear rates above around 0.01 s⁻¹. If the rotating surface is a cone, this transition occurs at a higher shear rate around 1 s⁻¹. At shear rates higher than this transition rate, the sample slips along both the top and bottom surfaces (even though they are roughened), with all the tracer particles moving together in a plug. Although the flow develops inhomogeneities macroscopically, including local shear bands, fracture planes, and slip at the wall, the apparent viscosity and stress data are uncannily reproducible upon multiple repetitions. In a purely rheological domain, shear stress (or viscosity) is gap independent for the same shear rate, whereas in a pure tribological domain, the shear stress (or friction coefficient) should remain gap independent when the velocity of the moving surface is the same. However, the lamellar fluids studied here do not show either of the above-mentioned behaviors but rather a combination of the two since the shear stress was found to be gap-dependent for either fixed velocity or fixed shear rate, suggesting that these samples are perhaps best termed triborheological. Oscillatory shear experiments showed that the elastic and viscous modulus, G′ and G″, respectively, are somewhat gap-sensitive even at low strains.

These measurements, especially of the velocity profile, contribute uniquely to the extensive body of literature characterizing these lamellar gel networks. We note that several of the properties of these fluids are similar to those predicted by the SGR model. A comparison is made to the criterion listed out by Fielding [46] for soft glassy materials in Table I.

The similarity with the characteristics of soft, glassy materials that we find in our rheological results for these samples opens up the possibility of using an SGR model, or some variant of it, to predict the behavior of these cosmetic emulsions in the future. Our first efforts to quantify the slipperiness of these cosmetic emulsions should facilitate an improved understanding of the formulation space for product developers. Several parameters not within the scope of this introductory work are also worth pursuing in the future, including the effect of salt on the swelling of the lamellar gel networks and on their rheology and slipperiness. It would also be valuable to develop a thixotropic constitutive model and slip and fracture models of these systems correlating these behaviors to the water content and the cationic surfactant content.

The authors gratefully acknowledge Procter & Gamble Company for supporting this work financially. They acknowledge helpful discussions with Mike Weaver and Fred Wireko of P&G, and with Cyrus Safinya and Emily Wonder of University of California, Santa Barbara. R.G.L. also acknowledges support from the National Science Foundation (NSF) (under Grant No. CBET-1907517). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.