# Strong Viscosity Increase in Aqueous Solutions of Cationic C22-Tailed Surfactant Wormlike Micelles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}on the surfactant concentration C η

_{0}~C

^{n}with the exponents n ranging from 2 to 10, depending on the charge and branching of micelles. In most of the papers, relatively small power law exponents n have been observed: η

_{0}~C

^{2}and C

^{1}(for cetylpyridinium chlorate with NaClO

_{3}) [13]; η

_{0}~C

^{3.3−3.8}(for cetyltrimethylammonium bromide (CTAB) with sodium salicylate and NaCl or KBr) [12]; η

_{0}~C

^{2.42}(for CTAB with KBr) [14]; η

_{0}~C

^{3.5−5.3}(for potassium oleate with KCl) [15]; η

_{0}~C

^{3.5−5.6}(for erucyl bis(hydroxyethyl) methylammonium chloride (EHAC) with KCl) [9]. Nevertheless, more steep viscosity increase based on strong growth of WLMs in length have not been observed, although it has been predicted [2,16,17]. This result is important because, in such systems, a high viscosity can be obtained at rather low surfactant concentration, which is quite important for practical applications.

## 2. Materials and Methods

#### 2.1. Materials

^{1}H NMR spectroscopy. Heavy water D

_{2}O (Astrachem, 99.95%, purity) and potassium chloride (Helicon, >99.8% purity) were used as received. Double distilled water was purified on Milli-Q Millipore Waters equipment.

#### 2.2. Methods

#### 2.2.1. Rheology

#### 2.2.2. Small-Angle Neutron Scattering (SANS)

_{2}O (Astrachem, 99.95%, purity) was used as a solvent to get higher scattering contrast. The measurements were carried out at 25 °C. All data were treated according to standard procedures of small-angle isotopic scattering [25]. Primary SANS data were corrected for the sample transmission, sample thickness, and electronic noise by SAS program [25]. The details of the measurements are described elsewhere [26].

## 3. Results and Discussion

#### 3.1. Effect of Salt Concentration

_{min}is the value of the loss modulus at the high frequency minimum, and l

_{p}is the persistence length of WLMs (30 nm [27]). If the micelles are branched, an effective contour length L

_{c}should be used instead of L [19]. The effective length L

_{c}corresponds to the contour length of linear WLMs forming a network, which has the same rheological properties as the network of the branched micelles. Figure 2d shows the variation of the estimated average contour length of WLMs with salt concentration. It is seen that the average contour length increases up to 15 μm with increasing salt concentration up to 3 wt.% and then the L

_{c}diminishes. This result confirms the appearance of branching points at salt content above 3 wt.%. This salt concentration corresponds to the Debye screening lengths κ

^{−1}of 0.53 nm, indicating that at these conditions the electrostatic repulsion between the head groups is essentially shielded. Thus, the variation of the salt content allows for estimation of the screening effect of KCl on the rheological properties of EHAC solutions and reveals the linear-to-branched WLMs transition.

^{−1}of 0.75 nm). At this concentration, the WLMs of EHAC are linear, and their contour length is 3 times smaller than the maximum length attained at 3 wt.% KCl (Figure 2d). One can expect that, at 1.5 wt.% KCl, the EHAC micelles will also remain linear at lower EHAC content because the transition from linear to branched WLMs always occurs with increasing surfactant concentration due to entropy reasons [12].

#### 3.2. Surfactant Concentration Dependencies

^{−1}, which confirms the formation of WLMs in the solution [3]. From the middle part of the scattering curve, the local shape and size of the micelles can be obtained [21]. In Figure 4 (inset), ln(Iq) is plotted as a function of q

^{2}for the q-range 2π/l

_{p}≤ q ≤ 1/R

_{g}, where R

_{g}is the cross-sectional radius of gyration of WLMs of EHAC, and l

_{p}is their persistence length. The curve represents a straight line confirming local cylindrical shape of the WLMs. The gyration radius can be calculated from the slope as ${R}_{g}=\sqrt{2\cdot tg\alpha}$ [2]. It was obtained that the R

_{g}= 2.1 ± 0.2 nm, and the cross-section radius R

_{cs}= 3.0 ± 0.3 nm. The last value is close to the length of EHAC molecule comprising a hydrophobic tail of 2.4 nm [27] and a bulky quaternary ammonium head group [27]. On the scattering curve (Figure 4), there is no evidence of a structural peak, meaning low electrostatic interaction between WLMs in the network in the presence of 1.5 wt.% KCl salt.

_{0}~C

^{10.5}and η

_{0}~C

^{4.4}. Similar two power-law regions of the concentration dependence of viscosity in the semi-dilute region were previously observed in EHAC solutions at higher salt concentration (3 wt.% KCl), but the exponents were much lower: η

_{0}~C

^{5.6}and η

_{0}~C

^{3.6}. The first exponent indicated the presence of rather short WLMs, which do not break during the characteristic reptation time. The values of the exponent are comparable to the theoretical values (5.25) predicted for “unbreakable” flexible micelles [2]. The second exponent is inherent to long micelles, which break and recombine many times while reptating. The exponents obtained experimentally are comparable to the theoretical value predicted for such “living” micelles (3.5) [7,12,30,31]. The transition from one regime to another occurring at increasing surfactant concentration is related with the growth of WLMs in length [9]: the longer the micelles, the higher is the reptation time, and the larger is their probability of breaking. Therefore, the long micelles represent “living” chains that break and recombine many times during reptation.

_{0}~C

^{10.5±1.5}and η

_{0}~C

^{4.4±0.2}) are detected for the first time. They may be explained by a high charge of micelles poorly screened by salt. Indeed, the steep slope of the first regime (η

_{0}~C

^{5.25}) was predicted by Cates and co-authors [32] for solutions of “unbreakable” WLMs that are uncharged or fully screened by excess salt. MacKintosh and co-authors [2,16,17] showed that the WLMs of ionic surfactant at rather low ionic strength (usually at salt concentration below 0.3 M) start to grow in semi-dilute region much faster than the uncharged WLMs. Therefore, one can suppose that the unscreened electrostatic repulsion is one of the reasons for high power law. Viscosity grows with length of WLMs L and surfactant concentration C as $\eta ~{L}^{3}\cdot {C}^{3.75}$ [33]; hence, if L rises faster than C

^{0.6}(uncharged case), one can expect higher power law dependencies.

_{0}. It means the formation of network of entanglements with increasing number and length of WLMs [12].

_{0}(Figure 5b), we took the value of ${G}^{\prime}$ at a frequency of 5 rad/s, for clarity. From Figure 5b, one can see that the ${G}^{\prime}$ values for both semi-dilute regimes are located on the same dependence G

_{0}~C

^{2.8±0.1}. Its power law is higher than the theoretical prediction for uncharged WLMs (2.25 [12]) and obtained in literature for EHAC solutions at high salt concentration (2.23 [9]). This theoretical power law value considers an increase of entanglements due to WLM concentrations rise with surfactant content [12]. However, at a low surfactant concentration, the shortest WLMs did not take part in the network because they are shorter than a distance between the entanglements. Hence, the plateau modulus appears lower than it was expected. Further growth of WLMs in length induces higher increase of entanglements because all WLMs become included in the network. As a result, the increase of the entanglements number (i.e., plateau modulus) exceeds the predicted dependence on surfactant concentration. The similar slope, 3.0, has been observed for salt-free dimeric surfactant solutions [33], indicating a crucial role of a low ionic strength for the steep rise of plateau modulus.

_{br}(τ

_{br}~1/L [12]), which becomes much lower than the reptation time τ

_{rep}, increasing with the length as τ

_{rep}~L

^{3}[12]. Thus, in the network of long WLMs, the breaking mechanism is dominating. Such long micelles break many times during reptation, which results in the averaging of the relaxation processes that become described by a single relaxation time, according to the Maxwell model [2,12]. One can propose that such a short concentration range (0.2–0.5 wt.%) of “unbreakable” chains is due to fast growth of WLMs in length at the rather small salt concentration under study.

_{e}(Figure 7a) were calculated from the plateau modulus G

_{0}by using the following expressions [2]:

^{−0.97}, which is lower than theoretical expectation ξ~C

^{−0.75}for dense network [2] and results in steep concentration dependence of plateau modulus. A steep decrease is also obtained for contour length between the entanglements (Figure 7a): l

_{e}~C

^{−1.6}. The concentration dependence of the longest relaxation time (Figure 7b) has two slopes, according to the regimes of the viscosity growth (Figure 3b). In the first region of “unbreakable” chains, the networks have multiexponential relaxation function, and the obtained result indicates that the longest relaxation time does not change appreciably with concentration. In the second region of living chains demonstrating Maxwellian behavior, the longest relaxation time must be equal to single relaxation time ${\tau}_{R}$, which is linked to the zero-shear viscosity ${\mathrm{\eta}}_{0}$ and plateau modulus ${G}_{0}$, according following expression [12]:

^{1.2±0.1}, which is quite surprising. A moderate growth L~C

^{0.6}has been usually observed for the WLM solutions [2,12], according to Cates’s prediction. However, in our case, this result confirms the suggestion about a fast growth of contour length in semi-dilute regions for EHAC solutions at rather low salt content. To the best of our knowledge, this is the first experimental observation of such fast growth of WLMs in length.

^{−1.4±0.1}. This is an interesting result because, according to Cates theory [12], the breaking time is inversely proportional to the average contour length ${\tau}_{br}={\left(kL\right)}^{-1}$, where k is the rate constant that shows the probability of breaking of a unit length of the micelle per second. Thus, this expression is applicable to the studied WLM solutions in region 2.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Table of Symbols

Symbol | Definition | Units |

η_{0} | zero-shear viscosity | Pa·s |

${G}^{\prime}$ | storage modulus | Pa |

${G}^{\u2033}$ | loss modulus | Pa |

${G}_{0}$ | plateau modulus | Pa |

${G}_{min}^{\u2033}$ | minimum of loss modulus | Pa |

${G}_{max}^{\u2033}$ | maximum of loss modulus | Pa |

l_{e} | average contour length | nm |

ξ | mesh size | nm |

${\tau}_{R}$ | relaxation time | s |

${\tau}_{br}$ | breaking time | s |

L | contour length between entanglements | nm |

l_{p} | persistence length | nm |

R_{g} | gyration radius | nm |

k | breaking rate constant | (nm·s)^{−1} |

C* | overlap concentration | wt.% |

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**Figure 1.**Zero-shear viscosity as a function of KCl concentration for 0.8 wt.% aqueous solutions of EHAC at 25 °C.

**Figure 2.**(

**a**) Frequency dependencies of storage ${G}^{\prime}$ (filled symbols) and loss ${G}^{\u2033}$ (open symbols) moduli for 0.8 wt.% aqueous solutions of EHAC surfactant in the presence of 1 wt.% (circles), 1.5 wt.% (squares), and 5 wt.% (triangles) KCl at 25 °C. (

**b**) Plateau modulus or storage modulus at high frequency (5 rad/s), (

**c**) relaxation time, and (

**d**) average contour length or effective contour length (for branched WLM) as a function of salt concentration for 0.8 wt.% aqueous solutions of EHAC surfactant.

**Figure 3.**(

**a**) Viscosity as a function of shear rate for solutions of different EHAC concentrations: 0.1 wt.% (squares), 0.2 wt.% (circles), 0.4 wt.% (triangles), 0.75 wt.% (diamonds). (

**b**) Zero-shear viscosity as a function of EHAC concentration in 1.5 wt.% KCl at 25 °C.

**Figure 4.**SANS profile for 0.6 wt.% aqueous solutions (D

_{2}O was used as a solvent) of EHAC cationic surfactant at 1.5 wt.% KCl at 25 °C. Solid line shows the slope of I~q

^{−1}dependence. The inset graph represents the dependence of ln(Iq) on q

^{2}revealing the cylindrical local shape of WLMs.

**Figure 5.**(

**a**) Frequency dependencies of storage ${G}^{\prime}$ (filled symbols) and loss ${G}^{\u2033}$ (open symbols) moduli for aqueous solutions containing 0.3 wt.% (squares), 0.45 wt.% (circles), 0.75 wt.% (triangles) EHAC in the presence of 1.5 wt.% KCl at 25 °C. (

**b**) Concentration dependence of plateau modulus and storage modulus at high frequency. The line indicates the overlap concentration C* of WLMs obtained from concentration dependence of zero-shear viscosity.

**Figure 6.**The normalized dependencies of the loss modulus ${G}^{\u2033}$ on the storage modulus ${G}^{\prime}$ (Cole-Cole plots) for EHAC solutions at different concentrations: 0.3 wt.% (squares), 0.45 wt.% (circles), and 0.6 wt.% (triangles) in the presence of 1.5 wt.% KCl at 25 °C. The line is the semicircle of the ideal Maxwellian viscoelastic fluid.

**Figure 7.**Concentration dependencies of (

**a**) mesh size and contour length between entanglements, (

**b**) relaxation time obtained from crossover point of ${G}^{\prime}$ and ${G}^{\u2033}$ moduli, (

**c**) the average contour length estimated from moduli ratio, and (

**d**) breaking time estimated from loss modulus minimum in EHAC solutions in 1.5 wt.% KCl.

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**MDPI and ACS Style**

Molchanov, V.S.; Rostovtsev, A.V.; Shishkhanova, K.B.; Kuklin, A.I.; Philippova, O.E.
Strong Viscosity Increase in Aqueous Solutions of Cationic C22-Tailed Surfactant Wormlike Micelles. *Fluids* **2022**, *7*, 8.
https://doi.org/10.3390/fluids7010008

**AMA Style**

Molchanov VS, Rostovtsev AV, Shishkhanova KB, Kuklin AI, Philippova OE.
Strong Viscosity Increase in Aqueous Solutions of Cationic C22-Tailed Surfactant Wormlike Micelles. *Fluids*. 2022; 7(1):8.
https://doi.org/10.3390/fluids7010008

**Chicago/Turabian Style**

Molchanov, Vyacheslav S., Andrei V. Rostovtsev, Kamilla B. Shishkhanova, Alexander I. Kuklin, and Olga E. Philippova.
2022. "Strong Viscosity Increase in Aqueous Solutions of Cationic C22-Tailed Surfactant Wormlike Micelles" *Fluids* 7, no. 1: 8.
https://doi.org/10.3390/fluids7010008