Mixed in-situ rheology of viscoelastic surfactant solutions using a hyperbolic geometry

Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

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Mixed in-situ rheology of viscoelastic surfactant solutions using a hyperbolic geometry Brayan F. García, Soheil Saraji∗ Department of Petroleum Engineering, University of Wyoming, 1000 East University Avenue, Laramie, WY 82071, United States

a r t i c l e

i n f o

Keywords: Wormlike micelles Porous media Microfluidic devices Non-Newtonian rheological modeling Extensional rheometry

a b s t r a c t Microfluidic devices can be used to represent industrial and natural mixed extensional-shear flows and to measure, simultaneously, in-situ rheological properties of viscoelastic fluids. However, such measurements are challenging due to the nature of mixed flow kinematics. In this study, we present a new mathematical framework to quantify the extensional behavior of viscoelastic fluids in mixed flows through a hyperbolic contraction-expansion microfluidic device. This approach provides a better estimation of apparent extensional viscosity when the nonNewtonian fluid kinematic does not promote additional flow phenomena such as vortices, shear banding or high-velocity jet. We then apply this approach to analyze the mixed rheology of two different viscoelastic surfactant solutions. The results indicate that there is a critical flow rate where extensional properties of the studied viscoelastic fluids start losing importance possibly due to additional flow phenomena. We also found a viscoelastic transition regime from extensional tension-thickening to tension-thinning as flow rate increases.

1. Introduction Viscoelastic surfactant solutions (VES) have been the center of attention in different applications including drug delivery, bioengineering, cooling/heating systems, packed bed reactors, and others [1–3]. In upstream oil and gas industry, VES have been used as clean fracturing fluids [4], self-diverting acidizing fluids [5], and well stimulation fluids [6], because of their viscoelastic properties and low impact on petroleum reservoirs (compared with polymers) [7,8]. Likewise, VES have been considered for Enhanced Oil Recovery as mobility-control agents [9–12] to prohibit viscous fingering [13], and increase oil recovery [11]: advantages that are attributed to their inherent rheology [9,14]. The flow of VES through hydrocarbon reservoirs and, in general, porous media with complex microscopic geometries (e.g. contractionexpansion configurations in pore-throats) may induce large shear and extensional components [15], affecting wormlike micellar structures [16], their rheological behavior, and flow patterns. To study the effect of these complex porous geometries and other variables/conditions on VES flow behavior, microfluidic devices and porous media models have been utilized [2,16,17]. Different non-linear flow behavior, depending on flow conditions and configurations, have been reported: creation of entanglements in the local pore space [18,19], formation of long-lived gel-like structures (Flow Induced Strcutured Phase -FISP-) [20,21], shear banding [2,22], interfacial undulations [23], flow instabilities [24–28],

∗

and elastic instabilities [14,17,29]. Although these studies have demonstrated the existence of several interesting flow phenomena, their contributions to the quantitative estimation of in-situ rheological properties of VES in porous media are limited. The appropriate quantification of shear and extensional forces in such geometries is extremely important to accurately predict (1) the in-situ fluid rheology and ultimately (2) the flow behavior in porous media. However, measuring the extensional rheology under flow conditions is challenging because of the nature of flow dynamics with shear and extensional components [15,16,30]. To date, the theoretical aspect of complex extensional-shear rheology for VES flows in microfluidic devices is not well understood and is still an active area of research. In order to measure the extensional rheology, special microfabricated geometries like abrupt contraction [31] and hyperbolic contraction/abrupt expansion [32] have been investigated, with some drawbacks such as elastic vortex growth and non-uniform strain rates. A square-square contraction was also used to study the flow of Newtonian and Boger fluids [33,34] leading to vortex growth upstream of the contraction and a three-dimensional highly unstable flow. Subsequently, low viscosity Boger fluids were used [35] for evaluation of elasticity in a microfluidic hyperbolic contraction/abrupt expansion similar to that presented in a previous study [32]. In this investigation, the development of upstream vortices was useful to assess the relaxation time using the dimensionless vortex length. The flow instabilities found in these works can be reduced by using a contraction-expansion hyperbolic geometry that impose a more uniform elongational rate [30]. Ober et al.

Corresponding author. E-mail addresses: [email protected] (B.F. García), [email protected] (S. Saraji).

https://doi.org/10.1016/j.jnnfm.2019.07.003 Received 10 January 2019; Received in revised form 4 July 2019; Accepted 9 July 2019 Available online 11 July 2019 0377-0257/© 2019 Elsevier B.V. All rights reserved.

B.F. García and S. Saraji

Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Table 1 Salt/surfactant molar ratio for TDPS/SDS/NaCl and APA-TW/CaCl2 solutions.

[30] reported the appraisal of a geometry that can potentially work as a microfluidic extensional rheometer. They used a 2-D flow approximation and plug-like flow assumption to estimate viscous and elastic pressure drops. They found, despite their effort, that there is a deviation of 66% between the calculated and true extensional rates (measured), producing an unsatisfactory estimate of rheological properties at even low flow rates. Recently, Lubansky and Matthews [36] analyzed the flow of a model dilute polymer in a planar contraction/abrupt expansion to compare the apparent extensional viscosity using an energy minimization calculation with the calculated “true extensional viscosity” from Batchelor’s Theory. In conclusion, the values of apparent extensional viscosity were not reliable using this geometry at low or high Weissenberg numbers (Wi). Zografos et al. [37], optimised the concept of the microfluidic converging/diverging channel to generate a constant strain-rate (or extensional rate) along the centerline of the flow. It was shown that a constant strain rate could not be produced for viscoelastic fluids, though the device could be used accurately up to a Wi = 0.02. Cromer and Cook [38] performed a numerical study of the flow of wormlike micellar solutions (synonym for VES) in the coverging/diverging channel from Ober et al. [30], to investigate the combined effects of shear and extensional components. The implemented model was successful in capturing signature behavior of the viscoelstic flow. Lee and Muller [39] presented a mathematical method for the design of a micro-channel that can impose a nearly constant extension rate, where the first normal stress difference reaches a steady-state. Later on, Kim et al. [40] used this method to construct the differential pressure extensional rheometer (DPER) for evaluating the steady-state extensional viscosity of a polymer at low and high extensional rates. They found the following polymer behaviors: strain thinning at low rates, strain thickening at intermediates rates, and an extensional viscosity plateau at high rates. We believe that there is room for a similar study on viscoelastic surfactants and also to improve the accuracy of existing microfluidic rheometers, making them useful tools for research on viscoleastic flow through porous media. In this study, we develop a theoretical framework to improve the predictions of in-situ extensional rheology during mixed extensional-shear flow of viscoelastic surfactant solutions in a hyperbolic contractionexpansion geometry (an ideal pore/throat configuration). The new mathematical derivation demonstrates the source of deviation factor, reported by Ober et al. [30] between the extensional rate measured by 𝜇-PIV systems and the calculated nominal extensional rate. We also experimentally evaluate the mixed extensional and shear flow behavior of two VES systems at different temperatures, salt/surfactant concentrations, and flow rates. The results reported here, show the plausible existence of a critical flow rate (or critical Re or Wi number) where apparent extensional properties become less significant possibly due to additional flow phenomena or elastic instabilities. The outcomes of this work are valuable for diverse industrial processes involving flow in complex geometries and for the Non-Newtonian fluid mechanics scientific community.

(A) Molar ratio (−)

[SDS] (Fixed [TDPS] = 1.7 wt.%)

[NaCl]

0.5 wt.%

0.55 wt.%

3 wt.% 3.5 wt.%

8.009 (1) 9.344 (3)

7.798 (2) 9.098 (4)

(B) Molar ratio (−)

[APA-TW]

[CaCl2 ]

1.5 wt.%

2.5 wt.%

5.0 wt.%

5 wt.% 10 wt.% 15 wt.%

11.173 (5) 22.346 (8) 33.520 (11)

6.704 (6) 13.408 (9) 20.112 (12)

3.352 (7) 6.704 (10) 10.056 (13)

surfactants, salt and deionized water (resistivity higher than 10 MΩcm) using a magnetic stirrer. 2.2. Fluid preparation Concentrations for zwitterionic/anionic solutions were selected in order to ensure the existence of entangled wormlike micelles (semidilute regime). The overlapping concentration for TDPS/SDS/NaCl systems at 0.5 M of NaCl (3.0 wt.%), is equal to [TDPS]: 7–8 mM (0.25–0.29 wt.%) for a minimum surfactant ratio (RAZ =[SDS]/[TDPS]) of 0.45 [41]. As [TDPS] increases, less surfactant ratio is needed to form long wormlike micelles [41], and thus in this study, for a base [TDPS] of 46.7 mM (1.7 wt.%) a minimum RAZ of 0.37 was used (i.e., [SDS] = 17.3 mM or 0.5 wt.%). For TDPS/SDS/NaCl solutions, four samples were prepared considering one base concentration of TDPS (1.7 wt.%), two concentrations of SDS (0.5, 0.55 wt.%), and two concentrations of NaCl (3, 3.5 wt.%). TDPS/SDS/NaCl mixtures were stirred for 24 h and also kept at rest for at least 2 days at room temperature before use. Table 1(A) summarizes the matrix of concentrations and their respective salt/totalsurfactant molar ratio (total surfactant moles = TDPS moles + SDS moles) with an identification number for the solution within parenthesis, to facilitate the discussion of results. To the best of our knowledge, the APA-TW/CaCl2 overlapping concentration has not been reported; hence, the concentrations of forming wormlike micelles were selected based on our previous rheological study [42]. For APA-TW/CaCl2 solutions, nine samples were prepared with APA-TW concentrations equal to 1.5, 2.5 and 5 wt.%, and CaCl2 concentrations equal to 5, 10 and 15 wt.%. APA-TW/ CaCl2 were mixed for at least 20 h and then used after 2 h of rest at room temperature. No significant change in rheology was observed after 2 h for this VES system. Table 1(B) contents the concentrations and molar ratio (salt/surfactant), with an identification number for the solution within parenthesis, to facilitate the discussion of results. 2.3. Experimental apparatus and procedure Extensional rheology measurements were conducted in an ‘extensional -Viscometer-Rheometer-On-a-Chip’ device (e-VROC by RheoSense, Inc.), which uses a converging/diverging micro-channel (hyperbolically-shaped) with Micro-Electro-Mechanical System (MEMS) technology to measure local pressures. The experiments were performed at different flow rates (100, 200, 500, 1000, 1500 and 2000 μL/min, just one solution was tested at 20 and 50 μL/min to widen the range of Weissenberg number) and at different temperatures (25, 45, and 65 ∘ C). A schematic representation of the e-VROC with dimensions and location of pressure sensors is presented in the Fig. 1. The micro-channel temperature was controlled by an internal thermal jacket using a Thermo-Cube system. To avoid fluctuations in measured pressure, the flow times were set to be large enough to assure steady-state flow conditions. Consequently, the equilibrium time considered for flow rates of 100, 200, 500, 1000, 1500 and 2000 μL/min

2. Materials and methods 2.1. Chemicals Zwitterionic/anionic surfactant solutions were composed of Ntetradecyl-N,N-dimethyl-3-ammonio-1-propanesulfonate (TDPS), Sodium Dodecyl Sulfate (SDS), and Sodium Chloride (NaCl), and non-ionic surfactant solutions were obtained using Aromox APA-TW commercial surfactant (family of tallow alkyl amido propyl-dimethylamine oxides) and Calcium Chloride (CaCl2 ). TDPS, and SDS were purchased from Sigma-Aldrich, and APA-TW from AkzoNobel Chemicals Co. All chemical products were directly used without further treatment. All solutions were prepared by mixing desired amounts of 57

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Fig. 2. (a) Shear and shear-extensional deformation regions inside e-VROC, (b) normal (𝜎 ii ) and shear stresses (𝜏 ij ) applied on the control volume, and (c) expected velocity profile in x-direction during flow through the constriction.

Fig. 1. Hyperbolic planar contraction-expansion geometry e-VROC.

were 3.3, 2.5, 1.7, 0.75, 0.1, and 0.08 min, respectively (as the flow rate increases, the time to achieve steady-state decreases [30]). After each test, the e-VROC system was thoroughly washed with isopropanol and deionized water. A 1 mL Hamilton Gastight glass syringe was used for all the measurements. After filling the syringe with fluid, all bubbles were completely removed to circumvent problems with the pressure measurements inside the chip. For shear rheology measurements, an Anton-Paar MCR 502 rheometer was used to carry out steady-state flow curves, amplitude and frequency sweeps. The experimental procedure and shear oscillatory results are reported in our previous publication [42]. Density measurements were conducted with an Anton Paar density meter DMA 5000 M, and values are also reported in Supplementary Material†. Unfortunately, tests at 65 ∘ C for TDPS/SDS/NaCl solutions could not be completed due to some flow instabilities that caused repeatability issues and meaningless data. We attribute this problem to the fact that TDPS/SDS/NaCl solutions lose their viscoelasticity at 65 ∘ C as reported in our previous investigation [42]. Thus, we do not report density values for TDPS/SDS/NaCl solutions at 65 ∘ C.

the contraction-expansion zone which is mixed shear-extensional region (depicted by red dotted lines in Fig. 2(a)). In the mixed shear-extensional region a planar deformation takes place, where both shear and normal stresses are present as shown in Fig. 2(b). Normal stresses (𝜎 ii ) are responsible for the elongational deformation while shear stresses (𝜏 ij ) generate the shear deformation. These stresses are not constant throughout the hyperbolic constriction, generating a spatially-variable velocity profile during the contraction-expansion of fluids as shown in Fig. 2(c). The spatial variations in velocity produce different deformation rates, which is mainly overlooked in the literature. Our main objective is to enhance the velocity prediction and incorporate its spatial variations into the mathematical treatment of mixed flow in the hyperbolic converging-diverging geometry and hence improve the quantification of shear and extensional components of flow. Here, the velocity is described only by the components 𝜐𝑥 = 𝜐𝑥 (𝑥, 𝑦, 𝑧), and 𝜐𝑦 = 𝜐𝑦 (𝑥, 𝑦, 𝑧) (planar extension implies 𝜐𝑧 = 0). Since the predominant velocity component is along the x-direction, where elongational and shear deformations occur simultaneously, the analysis is carried out on the x-velocity component. The variations of 𝜐𝑥 = 𝜐𝑥 (𝑥, 𝑦, 𝑧) in x, y and z direction are given by:

2.4. Mathematical analysis to quantify extensional rheology in mixed flows

𝜕𝜐𝑥 = 𝜖̇ 𝑥𝑥 , 𝜕𝑥

The converging/diverging geometry in e-VROC was theoretically designed to generate planar extensionally-dominated flows. However, in converging micro-channels the generation of a fully developed extensional flow at a constant extension is complicated, and other considerations such as (a) methods for establishing a fully developed flow, and (b) the accurate estimation of viscous and elastic pressure drops have to be included [39]. Lee and Muller [39] worked on the item (a) and formulated a mathematical method for constructing a micro-channel with conditions under which the normal stresses reach a steady-state at a constant extension rate. In this manuscript, we worked on item (b) and attempted to improve the mathematical estimation of pressure contributions in the eVROC system. As discussed before, the current mathematical treatment for this microchip fails to capture the true extensional component of flow with acceptable accuracy (66% of deviation at low flow rates, Wi < 10, see Supplementary Material, Section S.2.). Then, we mathematically analyze the predominant stresses in flows across the converging/diverging micro-channel, considering a twodimensional steady-state incompressible flow, to better understand the corresponding extensional and shear components. Fig. 2 presents the regions of deformation inside the microchip, the acting forces over a fluid control volume, and the main velocity profile at the contractionexpansion section. There are two primary deformation regions: (1) upstream and downstream the contraction-expansion zone which is essentially shear dominated region (depicted by blue dotted lines in Fig. 2(a)) and (2) within

𝜕𝜐𝑥 = 𝛾̇ 𝑦𝑥 , 𝜕𝑦

𝜕𝜐𝑥 = 𝛾̇ 𝑧𝑥 𝜕𝑧

(1)

where 𝜖̇ 𝑥𝑥 stands for the extensional rate, and 𝛾̇ 𝑦𝑥 and 𝛾̇ 𝑧𝑥 stand for the shear rates in y and z direction, respectively. In fact, the deformation rates are spatial-dependent functions as well, i.e.: 𝜖̇ 𝑥𝑥 = 𝜕𝑥 𝜐𝑥 (𝑥, 𝑦, 𝑧), 𝛾̇ 𝑦𝑥 = 𝜕𝑦 𝜐𝑥 (𝑥, 𝑦, 𝑧), and 𝛾̇ 𝑧𝑥 = 𝜕𝑧 𝜐𝑥 (𝑥, 𝑦, 𝑧). As a drawback, extensional and shear rates may be spatially non-uniform, varying at each point. Even though e-VROC device was designed to impose a nominally constant extensional rate [30], some experimental results show that similar hyperbolic configurations can induce a non-uniform extensional rate [32]. To avoid the ambiguity of dealing with local values of shear and extension, one can pose average functions to calculate the characteristic shear and extensional deformation rates in the mixed flow region. Then, the proposed average functions are: 𝜖̇ 𝑥𝑥 =

2 𝑙𝑡 ∫0

𝑙𝑡 2

𝜕𝜐𝑥 𝑑 𝑥, 𝜕𝑥

𝛾̇ 𝑦𝑥 =

2 𝑤𝑐 ∫0

𝑤𝑐 2

𝜕𝜐𝑥 𝑑 𝑦, 𝜕𝑦

𝛾̇ 𝑧𝑥 =

2 ℎ ∫0

ℎ 2

𝜕𝜐𝑥 𝑑 𝑧 (2) 𝜕𝑧

where, the limits lt /2 and wc /2 correspond to mixed flow region in the constriction (where a mixed flow regime is present) according to the coordinate system presented in the Fig. 1. Notice that, the average function for the extensional rate is performed only along the center-line since the shallow height of the channel generates a dominant velocity gradient in z-direction [30] across the chip and shear dominates the flow everywhere except the centerline where the extension is dominant. For shear rates, average functions are performed at the center of the contractionexpansion region because this location presents the maximum shear 58

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

deformation. Examining the set of Eqs. (2), reveals that accuracy of xvelocity estimation, directly affects prediction of characteristic extensional and shear rates. The current method used to calculate the extensional rate in this hyperbolic converging-diverging geometry is based on plug-like flow [30], which is not a good approximation for the x-velocity, as we discussed previously. Indeed, this approximation neglects any shearing flow, inducing an underestimation of the real extensional rate [30]. To overcome this disadvantage, we suggest an improvement of the velocity profile prediction by better approximation of the x-velocity at the centerline using the analytical solution for rectangular channels [43]: 𝜐𝑥 = 𝑘(𝑥)𝜐𝑥

in square ducts [44,45], contractions [31,46] and hyperbolic geometries [32] such as e-VROC. Reviewing these studies, an assumption of parabolic velocity profile seems to be suitable to calculate the characteristic shear rate. It can be demonstrated that average functions for the shear rates (Eq. (2)) considering a parabolic velocity profile, are: 𝛾̇ 𝑦𝑥

(3)

(5)

where x varies from the beginning of the contraction to the end of expansion as shown in the Fig. 1. The extensional rate from the plug-like ve−1 locity is a constant (𝜕 𝜐𝑥 ∕𝜕𝑥 = 𝐶 = 2𝑄(𝑤−1 𝑐 − 𝑤𝑢 )∕𝑙𝑡 ℎ) whose value only depends on the microchip dimensions and flow rate [30]. It has been found that this theoretical extensional rate (C), underestimates the true extensional rate calculated from centerline velocity measurements using micro-PIV by about 66% (i.e., true extensional rate = 1.66C) [30]. To improve this deviation, one can get the partial derivative of the Eq. (3) with respect to x as follows: 𝜖̇ 𝑥𝑥

𝜕𝜐 𝜕𝜐 = 𝑥 ≈ 𝑘(𝑥) 𝑥 𝜕𝑥 𝜕𝑥

2𝐶 𝑙𝑡 ∫0

𝑙𝑡 2

𝑘(𝑥)𝑑𝑥

𝜂𝐸,𝑎 (𝜖̇ 𝑥𝑥 ) =

(6)

⇒

𝜖̇ 𝑥𝑥 ≈

2 ℎ ∫0

ℎ 2

𝜕𝜐𝑥 2𝜐 𝑑𝑧 = 𝑥 𝜕𝑧 ℎ

(8)

𝑃𝐿 1 Δ𝑃𝑒 𝜖 𝐻 𝜖̇ 𝑥𝑥

(11)

Eq. (11) provides a better estimation of the extensional viscosity since it includes the modified expressions for Hencky strain, viscoelastic pressure drop, and characteristic extensional rate. Therefore, the proposed equation significantly improves the estimation of rheological parameters compared to the current approach used in these geometries [30]. 2.5. Dimensionless numbers relevant to extensional rheology in mixed flows Dimensionless numbers are necessary to analyze elastic, viscous and inertial effects of the mixed shear-extensional viscoelastic flow. In microfluidic devices, inertial effects characterized by Reynolds number (Re) can be negligible, while elastic effects characterized by Weissenberg (Wi) or Deborah (De) number might be significant [16]. Nonetheless, in this work we consider the variation of flow rates, and therefore inertial effects are as important as elastic effects for viscoleastic fluids. The Reynolds number is the ratio of inertial forces to viscous forces given by the following expression [16]:

𝑤

𝑢 𝐶 𝑘(𝑤)𝑑𝑤 𝑤𝑢 − 𝑤𝑐 ∫𝑤𝑐

𝛾̇ 𝑧𝑥 =

where the superscript PL denotes power-law fluid model, and 𝑤𝑑 = 𝑤𝑢 − 𝑤𝑐 . Measuring the total pressure drop from experiments and calculating the pressure drop associated with viscous effects utilizing Eq. (10), the viscoelastic pressure drop of a power-law fluid can be computed as: Δ𝑃𝑒𝑃 𝐿 = Δ𝑃𝑐𝑃 𝐿 − Δ𝑃𝜐𝑃 𝐿 . Finally, correcting the approach from Ober et al. [30], the new apparent extensional viscosity (𝜂 E,a ) for a hyperbolic contraction-expansion geometry is described by the following equation:

Notice that the term 𝜕 k(x)/𝜕 x is neglected due to its small variation in x-direction (in the order of ∼ 0.001, see Figure S2, in Supplementary Material). Incorporating Eq. (6) into the average function for extensional rate (Eq. (2)), using the actual dimensions of the micro-chip, and solving the integral, the characteristic extensional rate become ∼ 1.78C (for derivations and calculations, see Supplementary Material, Section S.1.). This means 12% overestimation of the real value of the extensional rate. We believe that this overestimation comes from the expression for k in terms of x, which does not consider the shape effects of the hyperbolic contraction. To include the effects of the hyperbolic-width on the extensional rate, the variable k is taken as a function of the hyperbolic width in a change of variables: 𝜖̇ 𝑥𝑥 ≈

𝜕𝜐𝑥 2𝜐 𝑑𝑦 = 𝑥 , 𝜕𝑦 𝑤𝑐

The total pressure drop related to the contraction, ΔPc , is usually expressed as a sum of elastic pressure drop, ΔPe , and viscous pressure drop, Δ𝑃𝜐 (𝑖.𝑒., Δ𝑃𝑐 = Δ𝑃𝑒 + Δ𝑃𝜐 ). For a shear-thinning fluid, Ober et al. [30] proposed that the viscous pressure drop portion can be estimated by using the power-law model (𝜂 = 𝑚𝛾̇ 𝑛−1 ) with consistency index m, and flow behavior index n. Here, we modify the proposed equation to estimate the corresponding viscous pressure drop, including the new characteristic extensional rate as: ( ) ( ) [( ) ( ) ] 𝑤𝑢 𝑛+1 𝑤𝑐 𝑛+1 2𝑛+2 2𝑛 + 1 𝑛 𝑙𝑡 𝑛+1 𝑛 Δ𝑃𝜐𝑃 𝐿 = 𝑚𝜖̇ 𝑥𝑥 − (10) 𝑛+1 𝑛 2ℎ 𝑤𝑑 𝑤𝑑

The width w(x), is expressed as [30]: 𝑙𝑡 𝑤𝑢 𝑤𝑐 𝑙𝑡 𝑤𝑐 + 2𝑥(𝑤𝑢 − 𝑤𝑐 )

𝑤𝑐 2

Remember that 𝜐𝑥 must be evaluated at the center of the contraction, i.e., 𝑤(𝑥) = 𝑤𝑐 = 400 𝜇𝑚. Furthermore, the calculation of Hencky strain experienced by the fluid, is modified since 𝐾 affects the calculation of extensional rate. Because we are dealing with a characteristic extensional rate for the contraction, we called the new Hencky strain as characteristic Hencky strain with the following expression: ( ) 𝑤𝑢 𝜖 𝐻 = 𝐾 𝑙𝑛 (9) 𝑤𝑐

where 𝜐𝑥 is the plug-like velocity given by 𝜐𝑥 = 𝑄∕(ℎ𝑤(𝑥)) (Q being the flow rate, h the height and w(x) the width), and k(x) is a correction factor for the unrealistic plug-like velocity calculated from the following equation [30,43]: ] (𝑖−2) [ ∑∞ (−1) 2 (1 𝑖𝜋ℎ ) 1 − 3 𝑖=𝑜𝑑𝑑 𝑖 𝑐𝑜𝑠ℎ 2𝑤(𝑥) 48 𝑘(𝑥) = (4) ( )] [ 𝜋3 𝑡𝑎𝑛ℎ 2𝑖𝜋ℎ (𝑥) ∑∞ 𝑤(𝑥) 1 − 192𝜋 𝑤 5ℎ 𝑖=𝑜𝑑𝑑 𝑖5

𝑤(𝑥) =

2 = 𝑤 𝑐 ∫0

(7)

where k(w(x)) is a power law equation based on the values obtained ( ) from Eq. (4) given by: 𝑘 𝑤(𝑥) = 3.9134𝑤(𝑥)−0.117 with 𝑅2 = 0.964 (in our microchip, widths varies from 400 to 3115 μm). Finally, solving Eq. (7) leads to the expression 𝜖̇ 𝑥𝑥 ≈ 1.659𝐶 (for derivations and calculations, see Supplementary Material, Section S.1.). The new characteristic extensional rate can be written as the extensional rate from plug-like velocity prediction (C) times a correction coefficient (𝐾 ), i.e.: 𝜖̇ 𝑥𝑥 = 𝐾 𝐶. 𝐾 is determined for hyperbolic contractions by employing Eq. (7) and its value may vary depending on the chip-dimensions, but this variation is always around 1.66. Regarding to the variation of x-velocity with respect to y and z direction, different studies have shown that a fully developed velocity profile for Newtonian and non-Newtonian fluids can be achieved

𝑅𝑒 =

𝜌𝜐𝑥 𝐷ℎ 𝜂(𝛾̇ 𝑧𝑥 )

(12)

where 𝜌 is density at flow conditions, 𝜂 is the shear viscosity, whose value is taken from the power-law model fit at the characteristic shear rate (𝛾̇ 𝑧𝑥 ), and (Dh ) is the hydraulic diameter at the contraction, which is taken as the characteristic length scale for the e-VROC system [30] with the following equation and value: 𝐷ℎ =

59

2𝑤𝑐 ℎ = 260.48 μm 𝑤𝑐 + ℎ

(13)

B.F. García and S. Saraji

Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Table 3 Power-law parameters for TDPS/SDS/NaCl solutions at different temperatures. The units for m is Pa.s and n is dimensionless.

Table 2 Equivalences between flow rate (μL/min), extensional and shear rates (𝑠−1 ). Flow rate (μL/min)

Extensional rate (𝑠−1 )

Shear rate (𝑠−1 )

20 50 100 200 500 1000 1500 2000

16 39 78 156 390 780 1170 1560

45 112 223 447 1117 2234 3351 4469

[SDS] (Fixed [TDPS] = 1.7 wt.%) 0.5 wt.% NaCl 3 wt.% 3.5 wt.%

2𝜐𝑥 ℎ

𝜆0 (s) ∘

(14)

where 𝜆0 is the conventional relaxation time calculated as the inverse of the frequency at which the cross point between elastic and viscous moduli occurs. Table 2 presents the corresponding extensional and shear rates at each flow rate, calculated using Eqs. (8) (𝛾̇ 𝑧𝑥 = 2𝜐𝑥 ∕ℎ) and (7) (𝜖̇ 𝑥𝑥 = 𝐾 𝐶). From these calculations, a relationship between shear rate and extensional rate is found to obey the equation 𝛾̇ 𝑧𝑥 = 2.865𝜖̇ 𝑥𝑥 . Since the selected flow rates provide Reynolds numbers larger than 1 (please see Fig. 6 in Results and discussion and Figure S4 in Supplementary Material), we consider that inertial forces cannot be neglected and there is a need to analyze their effect on flow behavior. Interestingly, this aspect has not been considered before, because microfluidic devices are designed to drastically minimize the inertia during flow, but, in real applications such as viscous dominated flow in porous media (e.g. in Enhanced Oil Recovery applications), inertial forces might have a nonnegligible impact on in-situ rheology. The Trouton ratio has been conventionally defined as the ratio of the extensional viscosity to √ both evaluated at an equiva√ the shear viscosity, lent shear rate (𝛾̇ 𝑒𝑞 = 3𝜖̇ [48] or 𝛾̇ 𝑒𝑞 = 4𝜖̇ for planar flows) to adopt a convention between extensional and shear rates. However, in the mixed flow region, extensional and shear viscosity are simultaneously present, which leads to what we call dynamic Trouton ratio (Tr,d ), expressed as: ( ) ( ) 𝜂𝐸,𝑎 𝜖̇ 𝑥𝑥 𝑇𝑟,𝑑 𝜖̇ 𝑥𝑥 , 𝛾̇ 𝑧𝑥 = ( ) 𝜂 𝛾̇ 𝑧𝑥

C

25 45 25 45

0.55 wt.%

m

n

m

n

0.373 2.601 0.242 6.719

0.502 0.291 0.542 0.207

1.220 21.26 0.215 2.760

0.368 0.137 0.605 0.273

Table 4 Relaxation time for TDPS/SDS/NaCl solutions at different temperatures in s.

The Weissenberg number, which represents the ratio of elastic forces to viscous forces [47], is defined at the contraction based on an akin geometry [31] using the dominant characteristic shear rate (in the zdirection since thickness is less than the width, h < wc ) as follows: 𝑊 𝑖 = 𝜆0 𝛾̇ 𝑧𝑥 = 𝜆0

∘

[SDS] (Fixed [TDPS] = 1.7 wt.%)

25 C: [NaCl]

0.5 wt.%

0.55 wt.%

3 wt.% 3.5 wt.%

0.00613 (1) 0.00489 (3)

2.35032 (2) 0.45975 (4)

45 ∘ C: [NaCl]

0.5 wt.%

0.55 wt.%

3 wt.% 3.5 wt.%

0.01318 (1) 0.08822 (3)

0.38312 (2) 0.33841 (4)

lines in Fig. 3 represent the shear rates corresponding to the selected flow rates reported in Table 2, which are within the shear-thinning region. This indicates that as flow rate increases, shear viscosity decreases. To approximate the viscous pressure drop in Eq. (10), the power-law model was fitted to shear-thinning flow regimes for all samples (the fitting parameters for TDPS/SDS/NaCl solutions are shown in Table 3 and for APA-TW/CaCl2 solutions in Table S1, Supplementary Material). Steady-state flow curves are presented in Fig. 4 with the same vertical dashed lines for the shear rates given by the flow rates (reported in Table 2). Samples at 25 ∘ C do not show a constant value of the shear stress in the range of experimental shear rates (there is no stress plateau at the selected flow rates), therefore, a shear banding phenomenon is unlikely at this temperature. Meanwhile, TDPS/SDS/NaCl solutions 2, 3 and 4 at 45 ∘ C, are prone to develop shear banding at the first four flow rates (20, 50, 100 and 200 μL/min) since the shear stress tends to have a constant value in this range of flow rates [30,49]. Frequency sweeps for the representative solutions are presented in Fig. 5, where it can be observed that solutions 1 and 3 have a liquidlike behavior in the whole range of frequency (Loss Modulus larger than Storage Modulus) and solutions 2 and 4 have a liquid-like behavior at low frequencies and a solid-like behavior at high frequencies (Storage Modulus larger than Loss Modulus). Relaxation times for TDPS/SDS/NaCl fluids are reported in Table 4, and for APA-TW/CaCl2 fluids in Table S3, Supplementary Material (the Tables contain identification numbers for the solutions within parenthesis). As for representative solutions 1 and 3, there is no crossover point between moduli and hence, relaxation times were defined by fitting the viscosity curve to the Carreau-Yasuda model (consult [42]). Solutions 2 and 4, exhibit higher relaxation times than solutions 1 and 3. Comparing the storage modulus between samples in Fig. 5 (from 1 rad/s to 100 rad/s), the storage moduli for solutions 2 and 4 are higher than those of 1 and 3, which means that the elastic portions in fluids 2 and 4 are larger than those of 1 and 3. This is correlated with the relaxation times, where those of fluids 2 and 4 are higher than those of 1 and 3.

(15)

where 𝜂 and 𝜂 E,a are the shear and apparent extensional viscosity present during the flow, respectively. The dynamic Trouton ratio implies a new manner to analyze the relative importance of extensional viscosity over shear viscosity in real applications with mixed flow conditions. 3. Results and discussion We used four solutions from TDPS/SDS/NaCl mixtures with IDs 1– 4 and nine solutions from APA-TW/CaCl2 mixtures with IDs 5–13 for experimental measurements. To clearly present the results derived from the experimental measurements, we only present data from mixtures 1– 4 in this section. The remaining 9 set of data are documented in the Supplementary Material as additional evidence for our observations. 3.1. Shear rheology

3.2. Mixed in-situ rheology analysis Viscosity curves for TDPS/SDS/NaCl solutions are illustrated in Fig. 3 (for APA-TW/CaCl2 mixtures, viscosity curves are reported in [42]). The curves show a shear-thinning behavior for all viscoelastic fluids regardless of temperature and concentration. The vertical dashed

Microfluidic tests were performed at the flow rates and temperatures specified in Section 2. For TDPS/SDS/NaCl viscoelastic fluids, Fig. 6 presents the behavior of total pressure drop across the 60

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Fig. 3. Viscosity curves for TDPS/SDS/NaCl mixtures at different temperatures and concentrations. Internal dashed lines represent the characteristic shear rates of the flow rates investigated.

Fig. 4. Flow curves for TDPS/SDS/NaCl solutions. Internal dashed lines represent the characteristic shear rates of the flow rates investigated.

contraction, ΔPc, (left-column) and the ratio between the elastic pressure drop, ΔPe , and the total pressure drop, ΔPc, (right-column) as a functions of Reynolds number (Re). The total pressure drop increased by Reynold’s number (or flow rate) monotonically for both 25 and 45 ∘ C (Figs. 6(a) and 6(b)). The elastic/total pressure ratio, however, reached a quasi-plateau around 1 for all solutions. The presence of this plateau means that the total pressure drop is mainly due to elastic pressure contribution. For solutions 2, and 4 this happened from a low Reynold’s number at 25 ∘ C (Re ∼ 0.2, Fig. 6(c)), while at 45 ∘ C this threshold was at a higher Reynold’s number (Re ∼ 1, Fig. 6(d)). For APA-TW/CaCl2 solutions, Figure S4 (in Supplementary Material) present similar trends for total pressure drop, while the elastic/total pressure ratio for some samples exhibits a perfect plateau at 45 and 65 ∘ C. Fig. 7 depicts the behavior of the same parameters as a functions of Weissenberg number (Wi). As observed in this figure, the total pressure drop increased as Weissenberg number (or flow rate) increased (Fig. 7(a) and (b)). From Fig. 7(c) and (d), the behavior of the elastic/total pres-

sure ratio shows that the elastic pressure drop becomes dominant over the viscous pressure drop as Weissenberg number increases, up to some value where a mild slope or quasi-plateau (increasing value but approximately constant) are reached. This behavior is more pronounced in solutions with higher elastic behavior (fluids 2, 4). This is similar to what we found for Re number plots and suggests that the more elastic fluid has more elastic contribution to the total pressure drop, as expected. Figure S5 (in Supplementary Material) shows the same trend in pressure drops for APA-TW/CaCl2 solutions. The apparent extensional viscosity as a function of Re and Wi numbers are illustrated in Fig. 8. From this figure, the apparent extensional viscosity of TDPS/SDS/NaCl solutions show an increasing-decreasing behavior as Re and Wi increase for all temperatures. Some mixtures of APA-TW/CaCl2 exhibit the same tendency in the apparent extensional viscosity (see Figure S6 in Supplementary Material). This indicates the existence of a critical point in Re or Wi, after which the elongational resistance (extensional viscosity) starts degrading. Similar behavior for the extensional viscosity has been reported previously, including pure 61

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Fig. 5. Frequency sweeps for TDPS/SDS/NaCl mixtures at different temperatures and concentrations. In the legend, LM indicates Loss Modulus (open symbols) and SM Storage Modulus (filled symbols).

Fig. 6. Pressure-drop, ΔPc , and elastic/total pressure ratio, ΔPe /ΔPc , through the contraction-expansion region for TDPS/SDS/NaCl solutions versus Reynolds number. Legends contain the respective ID number for each sample (see Table 1 to read concentrations).

Fig. 7. Pressure-drop, ΔPc , and elastic/total pressure ratio, ΔPe /ΔPc , through the contraction-expansion region for TDPS/SDS/NaCl solutions versus Weissenberg number. Legends contain the respective ID number for each sample (see Table 1 to read concentrations).

62

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

Fig. 8. Apparent extensional viscosity for TDPS/SDS/NaCl solutions versus Reynolds and Weissenberg numbers. In the legend, the respective ID number for each sample (see Table 1 to find concentrations).

Fig. 9. Dynamic Trouton ratio for TDPS/SDS/NaCl solutions versus Reynolds and Weissenberg numbers, and apparent extensional viscosity versus extensional rate. In the legend, the respective ID number for each sample (see Table 1 to read concentrations).

extensional experiments [50–53]. In those reports, the apparent extensional viscosity initially shows a plateau at low deformation rates, similar to that of Newtonian plateau for shear-viscosity, called the zero extensional viscosity. By increasing the deformation rate, the extensional viscosity rises in the so-called tension-thickening region, but with further increase in rate, the extensional viscosity reaches a maximum value [50,51]. If the rate further increases, the extensional viscosity may decrease in the so-called tension-thinning region. The tension-thickening and tension-thinning regions are observed in Fig. 8. Dynamic Trouton ratio as a function of Re and Wi numbers, as well as the apparent extensional viscosity as a function of extensional rate are depicted in Fig. 9. Dynamic Trouton ratios vary from 0.1 to 500 for TDPS/SDS/NaCl solutions (Fig. 9). Those values are comparable with conventional Trouton ratio values for viscoelastic shear-thinning fluids, which are usually found to be as large as 1000 [54,55]. The high values of the Dynamic Trouton ratio can be explained by the high elastic

pressure drop contribution (as flow rate increases), that increases the resistance to elongation of the viscoelastic fluids (extensional viscosity). Trouton ratio also evidenced an increasing-decreasing behavior for some samples with Re and Wi, implying the presence of a critical Re or Wi value at which the apparent extensional viscosity starts reducing and loosing importance. The apparent extensional viscosity versus extensional rate plots also exhibit tension-thinning, tension-thickening or transition behavior of the extensional viscosity on the rate at all temperatures for some samples (as shown by Fig. 9(c) and (f)). The main hypothesis that we suggest to physically explain these experimental results, is that secondary flow phenomena are occurring, dramatically affecting the flow kinematics. The quasi-plateau of elastic/total pressure ratio (presented in Figs. 6 and 7, in solutions with higher storage modulus) and predominant elastic pressure drop at Re numbers larger than 1 and Wi numbers larger than 100, may be associated with some phenomena in flow of viscoelastic solutions through

63

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Journal of Non-Newtonian Fluid Mechanics 270 (2019) 56–65

microfluidic devices: vortices and high-velocity jet. We suspect that a transition from no vortices to the generation of vortices at corners as Wi increases may be happening (as concluded in [56]). This phenomenon has been observed in similar configurations [30,31] and other microfluidic channels [56] with viscoelastic surfactant solutions. Those vortices can cause an additional constriction to flow, and contribute to a localized high-velocity jet in the middle of the channel [30,57]. Other localized high-velocity jets have been observed even in rectangular microchannels, where the fluid can be almost stagnant in some regions [58]. Fluid stagnation regions may also be occurring in the rectangular section channel before the contraction, which would significantly affect the flow. Since e-VROC is considered a microfluidic device with smooth wall surfaces, generation of microbubbles is not expected as seen in other experiments [59]. If this high-velocity jet is happening in the middle of the channel, it may be causing a severe stretch of the wormlike micelle structures and hence an enormous dissipation of elastic energy. This would explain, why as Re or Wi numbers increase (an increase in flow rate) the contribution of elastic pressure drop increases (Figs. 6 and 7). In fact, this stretching might be inducing a breakage of micelles. Based on these conceivable events at intermediate/high flow rates, where additional flow phenomena may appear generating non-linear effects, we conclude that our approach has deviations in the quantification of the extensional rate at intermediate/high flow rates (Wi >> 10, as supported in Supplementary Material, Section S.2.). Therefore, we use a qualitative analysis for explaining the results. Following this hypothesis, the elongational viscosity reduction (in Fig. 8) might be linked to the presumable micelles breakage due to a severe stretch when micelles pass through the high-velocity jet. The reduction in extensional viscosity with increase in stretching has been reported in other elongational experiments [52,53,60–62]). Likewise, the increasing-decreasing behavior for the Dynamic Trouton ratio in some viscoelastic samples (subtle but not less important as presented in Fig. 9) could be explained by this hypothesis. After the critical Re or Wi number, the Dynamic Trouton ratio decreases by the decrease of extensional viscosity due to the possible micelles breakage originated by the severe stretch in the middle of the channel [60–62]. Another plausible hypothesis, that might explain the experimental results is an increase in elastic instabilities after the critical flow rate or equivalently critical Wi number [14,17,29,58]. These instabilities can promote the large elastic energy dissipation in the flow (rather than a simple highvelocity jet in the middle of the channel), which generates a rupture, degradation or simply microstructural modification on the micelles at large Wi. To the best of our acknowledge, this is the first work that experimentally shows the increasing-decreasing behavior in apparent extensional viscosity in mixed flows through hyperbolic contraction-expansion microfluidic devices (previously reported in pure extensional flows). However, no physical explanation can be certainly provided since no visual evidence is available and all experimental results are physically explained by hypothetical phenomena that have been seen in other similar experiments. Hence, further investigation with visualization tools is absolutely required to confirm the presence of these phenomena. Although the approach may not provide accurate measurements at high flow rates (Wi > > 10) where additional phenomena are present, it can capture general trends and behaviors that could be of interest for the microfluidic research community.

1. Analysis of the deformation process during the viscoelastic flow across the contraction-expansion region led to a modification of the fundamental equations to calculate axial velocity, characteristic extensional rate, Hencky Strain, elastic and viscous pressure drop, and most importantly extensional viscosity. 2. As disadvantage, this new framework of equations is unable to capture additional flow phenomena such as vortices, shear banding, high-velocity jet, and others that strongly generate non-linear effects on flow at intermediate or high flow rates. Under these conditions the factor 1.66 loses its importance and only qualitative analyzes can be carried out. However, this approach provides the theoreticalmathematical evidence for the deviation factor of the extensional rate found by Ober et al. [30] when comparing the real extensional rate (from 𝜇-PIV measurements) with the one calculated at low flow rates. 3. This mathematical framework could be incorporated in new calculations of microfluidic extensional rheology (solely at low flow rates, i.e., Wi < 10, as supported in Supplementary Material, Section S.2.) for future research since fluid mechanics and soft-matter scientific areas are looking for new technologies to estimate the mixed flow behavior in real applications. 4. Elasticity is the main source of pressure drop in most flow rates for solutions with considerable storage modulus, while solutions with relatively low storage modulus only present a predominant elastic pressure drop contribution at high flow rates. 5. Experimental results can be physically explained by flow phenomena that have been identified in similar studies, such as vortices, shear banding or high-velocity jet. Although one or many of these phenomena might be occurring across the contraction-expansion geometry at intermediate and high flow rates, visualization tools are required to verify their existence. For future research, we suggest reproducing these experimental flows with these solutions and any visualization tool to conclude which phenomena are actually taking place at intermediate and high Re or Wi numbers. 6. Apparent extensional viscosity and Dynamic Trouton ratio exhibit an increasing-decreasing behavior for some solutions with respect to Reynolds or Weissenberg number. This indicates a critical point of flow rate where the severe stretching (due to probable additional flow phenomena) might be causing a micelles breakage and the extensional viscosity loses its importance in comparison with shear viscosity. Conflicts of interest There are no conflicts to declare. Acknowledgements The authors would like to thank the generous financial support from the School of Energy Resources at the University of Wyoming. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jnnfm.2019.07.003.

4. Conclusions

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