Rheology Test on Shear Viscosity of Surfactant Solution - American

5 May 2016 - 0 b m b m m. 0. 1. (3). According to eq 3, the time t has an inverse dependence ... increasingly intense which is in conflict with the on...
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Rheology Test on Shear Viscosity of Surfactant Solution: Characteristic Time, Hysteresis Phenomenon, and Fitting Equation Na Xu,† Jinjia Wei,*,† and Yasuo Kawaguchi‡ †

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan

‡

ABSTRACT: A series of rheology tests on surfactant solutions (cetyltrimethylammonium chloride and sodium salicylate system, hereafter referred to as CTAC/NaSal) were performed in the present study. First we investigated the characteristic times (induction time and “plateau” time) of the viscosity variation process to study the effects of shear and temperature on micellar transition on the time scale. It was proved that the characteristic time has a power law dependence on shear rate. The power exponent varies around −1 and increases with temperature. By contrast, concentration influences the characteristic time little. Second the temperature-induced viscosity variation hysteresis phenomenon is studied for the first time. The viscosity variation obviously lags behind the varying temperature, and a “viscosity variation hysteresis circle” (VVH circle) is formed between curves tested under increasing and decreasing temperature. The shear-induced viscosity variation hysteresis also was studied from the induction time viewpoint. It was found that the hysteresis shows apparent power law dependence on both temperature varying rate (r) and shear rate, and the higher r or shear rate leads to the severer hysteresis. Third an equation is established to describe the correlation relation between viscosity and the influencing parameters of shear rate, temperature, and concentration. The calculated viscosity values from the equation agree well with the measured ones.

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INTRODUCTION In the preceding article,1 we investigated the viscosity variation of a quite dilute surfactant solution (CTAC/NaSal) utilizing a series of rheological measurements. A theoretical equation was established to describe the relation between variation of micellar size and its influencing factors in the shear flow field. It can be found that both the effects of shear and temperature have two aspects (promoting effect and restraining effect) on micellar structural transitions. Therefore, only at a proper “ratio” of shear effect and temperature effect, can the biggest and strongest micellar structure be formed and can viscosity reach the highest value for one given solution. We found an interesting “two-peaks” phenomenon, namely, the viscosity experiences twice increase−decrease processes forming two shear-thickening peaks on the viscosity curve (Generally, there is only one shear-thickening peak on the viscosity curve.). We proved the two-peaks phenomenon happens under conditions of appropriate concentration, shear strength, and temperature increasing rate, and explained the phenomenon utilizing our theoretical equation for the variation of micellar size. The twopeaks phenomenon is also caused by the two aspects of both shear effect and temperature effect. Finally, the theory of total free energy change was introduced to further analyze the simultaneous effects of shear and temperature on micellar transitions from the energy viewpoint. In that article, we focused on the uninterrupted responses of surfactant aggregates under the simultaneous effects of shear and increasing temperature, particularly the “response speed”. © XXXX American Chemical Society

We aimed to measure and unify the different effects of shear and temperature from the viewpoint of energy, and build contacts between energy and micellar transition. In this article, also using rheological measurement, first we investigate the characteristic times of viscosity variation process under constant shear. Under shear, the viscosity shows nonlinear responses due to the continuous molecular reorganization, namely the transitions of micellar structures. One of the responses is the shear-thickening phenomenon, which is due to the formation of shear-induced micellar structures. Another special phenomenon is that the shear stress in surfactant solution could remain steady in a certain range of shear rate. It has been proven that this plateau stage of shear stress is related to the formation of shear banding.2−4 Besides, temporal oscillation will appear in the viscosity behaviors. Gentile et al. proved that the oscillatory behavior should be caused by the variation of the fraction of different micellar structures.5 All the micellar structure transitions are obviously time-dependent. Usually, two characteristic times can be used to describe the rate of the SIS (shear induced structures) growth under constant shear. One is the induction time tI, defined as the length of the period before the significant and sharp increase in Received: February 20, 2016 Revised: May 4, 2016 Accepted: May 5, 2016

A

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

temperature first increases from 10 to 60 °C then without any pause, decreases from 60 to 10 °C at the same rate under constant shear rate during each test procedure. The increase− decrease temperature experiments were performed for two aims. One is to verify whether the two-peaks phenomenon happens with decreasing temperature. Second is to verify whether temperature change can induce the hysteresis phenomenon (therefore the shear rate should not be too large or too small). For the above two aims, the shear rate was set to 5 s−1 for all increase−decrease temperature experiments. The other experimental operations are same as those in the previous article.

viscosity being detected (the length of Stages I and II6). The other is the “plateau” time tP at which viscosity reaches the dynamically stable value in the plateau region (the length of Stages I−IV6). Through reviewing the published results in literature, it can be found that both tI and tP have very close relationships with the following parameters: geometry,7 concentration of surfactant solution,8−10 shear rate,7,8,11,12 and counterion salt.13−15 In addition, according to our experiments, temperature has a great effect on the two characteristic times also. In the present study, we are concerned only with the influences of shear rate and temperature on the characteristic times to study the effects of shear and temperature on the time scale, on the basis of the dynamic analyses of viscosity variation in the preceding article. Besides, it has been found repeatedly that the relation between characteristic time and shear rate is consistent with the power law dependence of the form tI,P ∝ γ̇nI,P7,8,11,12,14 in experiment. However, the relation has not been proven mathematically, which is another issue we try to work out. Second, the viscosity variation hysteresis with changing temperature is studied. The rheology hysteresis phenomenon is frequently seen in wormlike micellar solutions on increasing and decreasing shear rate.16−19 Essentially, the reported hysteresis is caused by the lagging variation of molecular motion behind shear, consequently, theoretically it will also happen on the varying temperature which directly determines the molecular thermal motion in the surfactant micellar solution. In this article, we investigate the temperature-induced hysteresis phenomenon and study the relation between r and hysteresis for the first time, as well as the relation between shear rate and the concomitant hysteresis from the tI viewpoint. Third, we try to find the correlation relation between viscosity and the influencing parameters of shear rate, temperature, and concentration. Most researches on relatively dilute surfactant turbulent flow adopt the solvent (usually water) viscosity as the solution viscosity to perform calculation. That introduces error into the research results. Thus, an equation for calculating more precise solution viscosity is necessary for numerical research of surfactant drag-reducing flow.

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RESULTS Characteristic Times. The theoretical equation for the variation of micellar size we established in the previous article1 is as follows. dR /dt = 4γR ̇ 2(αmρ − 4αbkR /π )

where R is the half length of micelle; t is the time; γ̇ is the shear rate; αm is the cross-linking ratio after collision between micelles; αb is the breaking probability at any point of a micelle; ρ is the number density of micelles ; and k is a constant. In eq 1, the parameters αm, αb, and ρ are mainly influenced by the temperature, shear rate, and concentration of surfactant solution. But the conditions of temperature, shear rate, and concentration are constant in one test procedure in the present study, therefore, αm, αb, and ρ can be considered as constants with time for the test procedure. Thereby, eq 1 can be integrated. Suppose that the micellar size changes from R0 to Rt over time t, then eq 1 can be integrated as eq 2, and the relationship between t and shear rate can be obtained as eq 3.

∫R

EXPERIMENTAL SECTION Materials and Sample Preparation. The preparation of the surfactant system investigated in this study is the same as that described in the previous article.1 The concentrations of tested samples in the present study are listed in Table 1.

NaSal

0.94 1.88 2.50 3.13

0.94 1.88 2.50 3.13

1 dR = 4αmργR ̇ − 16αbkγR ̇ 3/ π 2

0

+

∫0

t

dt (2)

⎤ 1 (1/R 0 − 1/R t )⎥γ −̇ 1 αmρ ⎦

(3)

According to eq 3, the time t has an inverse dependence on the shear rate. The characteristic times of tI and tP are just two special points in t, therefore, tI and tP should also be inversely dependent on shear rate, namely tI,P ∝ γ̇−1. However, the values of parameters αm, αb, and ρ in eq 2 slightly change with test conditions (for example, geometry of rheometer, shear rate, temperature, concentration, and so on), and the changes have influence on the integration process. Therefore, the proportionality of the inverse relationship between tI,P and shear rate varies around −1, namely, nI and nP vary around −1 in the power law relation tI,P ∝ γ̇nI,P. This mathematics consequence coincides with the experimental results.7,8,11,12,14 In addition, when the micellar size no longer changes with time under shear, that is, dR/dt equals zero in eq 1, the “plateau” micellar size RP at plateau time tP will be obtained as

Table 1. Materials and Concentrations of Samples CTAC

Rt

⎡ kα 1/R 0 − 4kαb/(παmρ) b ln t=⎢ 2 2 1/R t − 4kαb/(παmρ) ⎣ παm ρ

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concn (mmol/L)

(1)

Rheology Tests. The same rheometer and geometry as in the previous article1 were used to perform the rheological tests. The rheological tests were carried out under two different modes, namely, the peak hold mode and the temperature ramp mode. Under the peak hold mode, both shear rate and temperature were set to be constant and the viscosity data were sampled at 1 s interval in the shear duration for 10000 s for each new sample. While under the temperature ramp mode,

RP = B

παmρ 4kαb

(4) DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research In eq 4, k is related to the type of surfactant and counterion, αm and αb with shear rate, temperature, and ρ while ρ is dependent on the concentration of surfactant solution. Consequently, the micellar size in solution that achieves the final equilibrium state under shear, namely the plateau viscosity (the bigger micellar size corresponds to the higher viscosity20), depends on the type of surfactant and counterion, shear rate, temperature, and concentration. To investigate further, under the peak hold mode we tested the two characteristic times of CTAC/NaSal solutions at concentrations of 0.94 mmol/L (300 ppm), 1.88 mmol/L (600 ppm), 2.50 mmol/L (800 ppm), and 3.13 mmol/L (1000 ppm), temperatures of 15, 20, and 25 °C, and shear rates of 1 s−1, 5 s−1, 10 s−1, and 30 s−1. Note that the shear viscosity of surfactant solution can never reach an absolutely constant value due to the continuous change of shear-induced micellar structures. The “plateau” state of surfactant solution, in fact, means that the viscosity achieved a dynamic equilibrium state and is oscillating around a “plateau” value (namely the so-called plateau viscosity). The time that viscosity reaches the dynamic equilibrium state is the “plateau” time. The same methods for selecting values of tI and tP are used as other researchers,7,12,13 as illustrated in Figure 1. Namely, tI is

Figure 2. Induction time (tI) and plateau time (tP) as functions of shear rate γ̇ [tI,P ∝ γ̇nI,P].

Figure 1. Selecting methods for the characteristic times on the viscosity curve of CTAC/NaSal solution [2.50 mmol/L (800 ppm)].

obtained from the intersection of the prestructure viscosity value and the early viscosity growth slope; tP is quantified by identifying the first intersection of the viscosity curve with the average plateau value. The results are shown in Figure 2, wherein the relation between characteristic time and shear rate is described using ΔF(ΔT) ∝ rn(n > 0), and the corresponding slopes (nI and nP) of fitting curves are shown in Figure 3. It should be emphasized that, considering the testing error and the selecting value error, all of the data shown in Figure 2 represent the average of three or more measurements with new samples from the same mother solution. From Figure 3 one can find that nI is about −1.3 at 15 °C, −1.2 at 20 °C, and −0.8 at 25 °C, roughly independent of concentration. Temperature has a very evident influence on nI, namely, the higher is the temperature, the larger is the nI. In other words, the influence magnitude of shear rate on induction time decreases with increasing temperature. Although nP also increases with temperature, the increase rate is much smaller compared with that of nI, and the difference among nP values at different concentrations is very small. The reasons for the changes in nI,P with temperature are as follows. As temperature rises, the random thermal motion of molecules becomes increasingly intense which is in conflict with the one-directiondriving effect of shear. The conflict impairs the shear effect on the formation and growth of SIS, therefore, both nI and nP

Figure 3. Slopes of fitting curves in Figure 2.

increase. The evolution of viscosity is essentially the cumulative result of micellar behavior, and at the early stage (shorter than tI) the cumulative effect of shear is very weak (until the obvious increase in viscosity appears, which stands for the visible result of cumulative shear-effect on the macroscale). Thus, the influence of the increasing temperature on molecular action appears to be relatively prominent. That is why nI increases more obviously than nP with temperature. The relationships among micellar size, time, shear rate, and temperature can be schematically shown in Figure 4. Temperature-Induced Hysteresis Phenomenon. Part of the experimental results tested under the temperature ramp mode are shown in Figure 5. The results show that the twopeaks phenomenon happens with increasing temperature; however, it does not happen with decreasing temperature. It has been proven that the appearance of the two peaks depends on concentration condition, namely the micellar condition.1 The only difference between the I-solution (solution tested with increasing temperature) and D-solution (solution tested with decreasing temperature) lies in the different original states. Because of the complete destruction at high temperature, there C

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 4. Schematic relationships among micellar size (Rt), time (t), shear rate (γ̇), and temperature (T).

Figure 5. Viscosity curves at different temperature increasing (I)/decreasing (D) rates under shear rate of 5 s−1 [2.50 mmol/L (800 ppm)].

conversely, the viscosity is much higher at high rates than that at low rates of temperature decreasing. It can be attributed to the following reasons. It is known that micelles will form SIS gradually under shear, leading to the obvious increase in viscosity. When temperature increases rapidly during shear process, SIS could not reach its optimal state before being destroyed thoroughly due to the exceedingly high temperature. That is why the viscosity is much lower at high rates than that at low rates of temperature increasing. But on the contrary, with temperature decreasing, the thoroughly destroyed SIS begins to reform through the connection and entanglement among micelles, and viscosity begins to recover correspondingly. When the temperature is further decreased, the thermal motion of molecule decreases, resulting in the same decline in growing speed of SIS, consequently the viscosity increasing rate becomes slower. If temperature decreases quite rapidly, the decline in thermal motion of molecule will fail to follow, namely molecules will still keep relatively high moving speed to form SIS. Consequently the viscosity is much higher at high rates than that at low rates of temperature decreasing. Therefore, the VVH circle should be caused by the lagging variation of molecular thermal motion behind temperature. Furthermore, it can be found that the difference between Tc-I and Tc-D listed in Table 2 becomes increasingly smaller with the decline in r, which represents that the lag is decreasing. Even we cannot perform the experiment at lower r (than

is no evident micellar structure (otherwise the viscosity should be larger) in the original D-solution. Consequently, the micellar condition for forming two peaks does not exist for the Dsolution. Besides, it can be found that the I-curve (curve tested with increasing temperature) and D-curve (curve tested with decreasing temperature) have increasing overlaps with decreasing r. However, the critical temperature (where SIS starts to recover, Tc-D) of the D-curve is always lower than the temperature (where SIS is thoroughly destroyed, Tc-I) of the corresponding I-curve (as shown in Table 2). Obviously, temperature-induced hysteresis phenomenon appears. We call the enclosed part “viscosity variation hysteresis circle” (VVH circle) which is derived from the viscosity difference between Icurve and D-curve as shown in Figure 5d by dashed line. In Figure 5, on the whole, the viscosity is much lower at high rates than that at low rates of temperature increasing; Table 2. Critical Temperatures in I-curves (Tc-I) and Dcurves (Tc-D) in Figure 5 curve 0.0167 0.0067 0.0017 0.0011

°C/s °C/s °C/s °C/s

Tc-I (°C)

Tc-D (°C)

46 45 42 42

31 33 35 35 D

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research 0.0011 °C/s); presumably, the I-curve and D-curve will completely coincide if temperature varies at infinitesimal r. However, under actual conditions, the VVH circle will always exist. Considering that the hysteresis phenomenon essentially indicates the deviation degree of the actual state from the ideal equilibrium state of surfactant solution with the change of testing condition, the total free energy change ΔF is introduced to measure the viscosity variation hysteresis on the time scale. The higher ΔF in solution stands for the severer hysteresis. The I-curves and D-curves of surfactant solutions at 0.94 mmol/L (300 ppm), 1.88 mmol/L (600 ppm), and 2.50 mmol/L (800 ppm) under a shear rate of 5 s−1 were tested. For each test procedure, the values of Tc-I and Tc-D can be obtained, as well as the difference ΔT between Tc-I and Tc-D. During the process of obtaining ΔT the total free energy change caused by shear can be removed, namely ΔT reflects the temperature-induced total free energy difference of the two states. Then the time difference Δt can be obtained from ΔT devided by the corresponding r. It is important to note that, ideally, only when temperature changes at an infinitesimal rate can ΔF remains 0. Namely, SIS needs infinite time to shift from one state to another without any change in the total free energy when temperature varies. Thus, the shorter Δt corresponds to the bigger ΔF(ΔT) in solution caused by ΔT, namely ΔF(ΔT) ∝ 1/Δt. As shown in Figure 6, Δt exhibits an apparent decrease with r, consistent with the power law relationship of Δt = arb (a and

correspondingly, that when SIS begins to recover as R0,D and that at the end of the experiment as Rt,D with decreasing temperature. Thereby Δt above can be calculated as follows according to eq 3. Δt = t(R t ,I − R 0,I) − t(R t ,D− R 0,D) ⎡ ⎢ kα b =⎢ ln 2 2 ⎢ παm ρ ⎢⎣

( (

1 R 0,I

−

4kαb παmρ

1 R t ,I

−

4kαb παmρ

)( )(

1 R t ,D

−

4kαb παmρ

1 R 0,D

−

4kαb παmρ

) )

⎤ 1 ⎛ 1 1 1 1 ⎞⎥ − 1 ⎜⎜ ⎟⎟⎥γ ̇ + − − + 4αmρ ⎝ R 0,I R t ,I R 0,D R t ,D ⎠⎥ ⎥⎦

(5)

In eq 5, R0,I is constant in all test procedures for solutions with the same concentration, Rt,I and R0,D should be equal and also constant, and Rt,D varies with r. With decreasing temperature at different rates, the viscosity curves of solutions at 1.88 mmol/L and 2.50 mmol/L are shown in Figure 7. A comparison of the viscosity values when temperature is decreased to 10 °C at the end of the experimental procedure in each concentration group shows that higher r gives rise to the bigger terminal viscosity. Therefore, Rt,D, proportional to the terminal viscosity, increases with r. Additionally, although the parameters αm, αb, and ρ have a little changes with temperature, they have no relations with r, therefore Δt decreases with r. Accordingly, there is a power law correlation between ΔF(ΔT) and r as ΔF(ΔT) ∝ r−b, namely, the higher r will lead to higher ΔF in solution. In addition, one can find that the parameter b decreases with concentration as listed in Table 3, which means Table 3. Fitting Parameter Values of Fitting Curves in Figure 6

Figure 6. Time difference (Δt) as a function of temperature varying rate (r) [Δt = arb].

concn (mmol/L)

a

b

0.94 1.88 2.50

207.3276 119.7150 45.05887

−0.3648 −0.5146 −0.7212

the change in r leads to more obvious change in ΔF(ΔT) for the higher concentration solution. But the higher concentration solution does not necessarily show higher ΔF(ΔT) with r according to Figure 6. As to the relations between ΔF(γ̇) and shear rate γ̇, it can be measured with the plateau time tP. When a constant shear rate is applied to the surfactant solution, the start-up process is completed instantaneously, after that, the whole viscosity

b are constants). The reversely proportional relationship between Δt and r can be explained utilizing eq 3. For one experimental procedure with increasing−decreasing temperature at the rate of r, we define the micellar size at the beginning of the experiment as R0,I and that when SIS is thoroughly destroyed as Rt,I with increasing temperature,

Figure 7. Viscosity curves with decreasing temperature at different rates: (a) 1.88 mmol/L (600 ppm); (b) 2.50 mmol/L (800 ppm). E

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 8. Schematic relationships among total free energy change (ΔF), temperature varying rate (r), shear rate (γ̇), concentration (C), and temperature (T).

dissipation scales and increase it on the large scales. The micellar structures seem to act as a kind of buffer-stop so that turbulent kinetic energy cannot be transferred to smaller scaled vortices where it should be dissipated by viscosity, leading to the drag reduction effect in surfactant turbulent flow. Fitting Equation for Viscosity. The turbulent channel flow is fully developed in research on dilute surfactant solution, therefore it is the plateau viscosity in the rheology test that should be the focus. According to the experimental results under the peak hold mode, some fitting works were performed to study the relationships between plateau viscosity and its influencing factors. It is found that the plateau viscosity has a power-law dependence on shear rate as shown in Figure 9;

evolution process should be attributed to the release of the total free energy change and part of the energy change ΔF(γ̇) remains in solution after tP. Likewise, the relation between ΔF(γ̇) and tP coincides with the reversely proportional law of ΔF(γ̇) ∝ 1/tP. As one of the characteristic times, the relationship between tP and shear rate was studied above as shown in Figure 2b. Consequently, the relation between ΔF(γ̇) and γ̇ coincides with the power low of ΔF(γ̇) ∝ γ̇−np. According to Figure 2b, ΔF(γ̇) roughly decreases with temperature and does not exhibit constant dependence on concentration. Either of the ΔF(γ̇) variation with concentration or temperature is much smaller than that with shear rate. In summary, consequently, the viscosity variation hysteresis shows apparent power-law dependence on both r and shear rate, and the higher r or shear rate leads to the severer hysteresis, as schematically shown in Figure 8. Essentially, the viscosity variation hysteresis is a kind of dynamic viscoelastic action. When the surfactant solution obtains some kind of energy (either the thermal energy from the increasing temperature or the kinetic energy from shear), the microinternal structures in solution will change to adapt to the new solution conditions. In the adaptation process, some changes in the microstructure (the formation of SIS, for example) will lead to the obvious variation in viscosity, but part of the energy is hidden and stored as hysteresis. The storage and release of energy by the surfactant microstructures in turbulent flow can modulate the turbulence and thus produce drag reduction phenomenon. In turbulent flow, the principal potion of flow resistance stems from turbulent vortices with random sizes and random motions. The vortex scale may have a very wide range of variations, and different leveled turbulent kinetic energy distributes locally among different scaled vortices; the turbulent kinetic energy then successively transfers from larger scaled vortices to smaller ones, and is finally dissipated at the smallest scaled (namely, at the dissipation scale or Kolmogorov scale) vortex due to viscosity. In surfactant drag reducing flow, the micellar network structures will deform at a high-shear-rate region, meanwhile, absorbing the turbulent kinetic energy of small scaled vortices and storing it. Specially, the energy of the smallest scaled vortices will be absorbed thoroughly and the vortices vanish. Because of the hysteresis effect, the value of the absorbed kinetic energy always will be larger than the theoretical energy-containing value of the micellar structures. When being diffused or convected to a low-shear-rate region, the deformed micellar structures will relax and the stored energy will be released to the large scaled vortices (in the form of elastic stress waves, for example). The overall effect is that the micellar network structures decrease the vorticity on the

Figure 9. Plateau viscosity as a function of shear rate (γ̇) [η = Aγ̇B].

however, both the fitting parameters A and B change with temperature and concentration (shown in Table 4). Meanwhile, the plateau viscosity has a linear dependence on concentration according to our experimental results as shown in Figure 10. Table 4. Fitting Parameter Values of Fitting Curves in Figure 9 temp (°C)

concn (mmol/L)

A

B

15

0.94 1.88 2.50 3.13 0.94 1.88 2.50 3.13 0.94 1.88 2.50 3.13

0.06571 0.09952 0.1014 0.1173 0.05393 0.08169 0.09043 0.09754 0.04666 0.07008 0.07660 0.08182

−0.2931 −0.2783 −0.2625 −0.2549 −0.3949 −0.3752 −0.3655 −0.3457 −0.4851 −0.4720 −0.4626 −0.4370

20

25

F

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

the varying temperature. A “viscosity variation hysteresis circle” (VVH circle) is formed between curves tested under increasing and decreasing temperature. The hysteresis shows apparent power law dependence on both the temperature varying rate and shear rate, and the higher temperature varying rate or shear rate leads to the more severe hysteresis. (3) An equation is established to describe the correlation relation between viscosity and the influencing parameters of shear rate, temperature, and concentration. The calculated viscosity values from the equation agree well with the measured ones.

These correlations are determined by the essence of the viscosity variation.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 29 82664462. Fax: +86 29 82669033. E-mail: jjwei@ mail.xjtu.edu.cn.

Figure 10. Plateau viscosity as a function of concentration.

Notes

The authors declare no competing financial interest.

The increase in viscosity of surfactant solution is caused by the formation of SIS and SIS is derived from the collisions and aggregations of surfactant molecules and/or micelles. When the applied shear rate is increased, the plateau viscosity will decrease in spite of the increase in the collision frequency, because the microlevel intermolecular force is much smaller than the macroshear force, and thus the increasing shear rate prevents the molecules from aggregation. But the characteristic times will decrease with shear rate because of the increasing collision frequency. As concentration increases, there are more molecules per unit volume which increases the collision and aggregation chance for molecules, consequently the plateau viscosity increases. While temperature increases, the plateau viscosity will decrease according to the Andrade equation. That is, the increases in 1/γ̇ (the reciprocal of shear rate), C (concentration), and 1/T (the reciprocal of temperature) have essentially equivalent impacts on the increase in plateau viscosity. Consequently, it is reasonable to assume that the correlation relation between plateau viscosity and the above three influencing factors coincides with the equation of η(γ ̇, T , C) = (α1/T + β1C + γ1)γ −̇ (α2 / T + β2C + γ2)

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support from the project of National Natural Science Foundation of China (No. 51225601) and the Fundamental Research Funds for the Central Universities.

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(1) Xu, N.; Wei, J. J.; Kawaguchi, Y. Dynamic and energy analysis on the viscosity transitions with increasing temperature under shear for dilute CTAC surfactant solutions. Ind. Eng. Chem. Res. 2016, 55, 2279−2286. (2) Makhloufi, R.; Decruppe, J. P.; Aït-Ali, A.; Cressely, R. Rheooptical study of worm-like micelles undergoing a shear banding flow. Europhys. Lett. 1995, 32, 253−258. (3) Decruppe, J. P.; Ponton, A. Flow birefringence, stress optical rule and rheology of four micellar solutions with the same low shear viscosity. Eur. Phys. J. E: Soft Matter Biol. Phys. 2003, 10, 201−207. (4) Weiss, R. G.; Terech, P. Molecular Gels: Materials with SelfAssembled Fibrillar Networks; Springer: The Netherlands, 2006; p 699. (5) Gentile, L.; Silva, B. F. B.; Lages, S.; Mortensen, K.; Kohlbrecher, J.; Olsson, U. Rheochaos and flow instability phenomena in a nonionic lamellar phase. Soft Matter 2013, 9, 1133−1140. (6) Xu, N.; Wei, J. J. Time-dependent shear induced non-linear viscosity effects in dilute CTAC/NaSal solutions: mechanism analyses. Adv. Mech. Eng. 2014, 6, 179394. (7) Hu, Y. T.; Matthys, E. F. Characterization of micellar structure dynamics for a drag-reducing surfactant solution under shear: normal stress studies and flow geometry effects. Rheol. Acta 1995, 34, 450− 460. (8) Hu, Y. T.; Wang, S. Q.; Jamieson, A. M. Rheological and flow birefringence studies of a shear-thickening complex fluid-a surfactant model system. J. Rheol. 1993, 37, 531−546. (9) Hu, Y. T.; Matthys, E. F. Rheological and rheo-optical characterization of shear-induced structure formation in a nonionic drag-reducing surfactant solution. J. Rheol. 1997, 41, 151−166. (10) Macias, E. R.; Gonzalez, A.; Manero, O.; Gonzales-Nunez, R.; Soltero, J. F. A.; Attane, P. Flow regimes of dilute surfactant solutions. J. Non-Newtonian Fluid Mech. 2001, 101, 149−171. (11) Hu, Y. T.; Wang, S. Q.; Jamieon, A. M. Kinetic studies of a shear thickening micellar solution. J. Colloid Interface Sci. 1993, 156, 31−37. (12) Boltenhagen, P.; Hu, Y. T.; Matthys, E. F.; Pine, D. J. Inhomogeneous structure formation and shear-thickening in wormlike micellar solutions. Europhys. Lett. 1997, 38, 389−394. (13) Hu, Y. T.; Matthys, E. F. The effects of salts on the rheological characteristics of a drag-reducing cationic surfactant solution with shear-induced micellar structures. Rheol. Acta 1996, 35, 470−480.

(6)

where η(γ̇,T,C) is the plateau viscosity of surfactant solution at shear rate γ̇, temperature T, and concentration C; and α1,2, β1,2, and γ1,2 are the constants. On the basisof the experimental results, the constants in eq 6 can be determined. Substituting the determined constants into eq 6, we can get η(γ ̇, T , C) = (0.4391/T + 0.01857C + 0.02236) T + 0.02111C ‐ 0.8051) γ (7.5749/ ̇

REFERENCES

(7)

Comparing the plateau viscosity values between experiment and calculation from eq 7, we can see that the calculation values agree well with the measured ones. The standard deviation is 9.5% and the maximum error does not exceed 20%.

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CONCLUSIONS (1) All the three factors of concentration, temperature, and shear rate have influences on the characteristic time. The characteristic time has a power law dependence on shear rate. The power exponent varies around −1 and increases with temperature. By contrast, concentration influences the characteristic time little. (2) Temperature-induced viscosity variation hysteresis is obvious, namely, the viscosity variation obviously lags behind G

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

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H

DOI: 10.1021/acs.iecr.6b00703 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX