# Prediction of the Viscoelastic Properties of a Cetyl Pyridinium Chloride/Sodium Salicylate Micellar Solution: (II) Prediction of the Step Rate Experiments

^{1}

^{2}

## Abstract

**:**

_{1}) versus time curve in the start-up experiment, the shear stress (τ

_{12}) in the start-up experiment, τ

_{12}in the long-term start-up experiment, the stress relaxation upon cessation of steady shear flow, and the transient N

_{1}/τ

_{12}in the step strain experiment. The study findings clearly show an improvement in the predictions of the viscoelastic properties of the micellar solution compared with those predicted previously. For example, the experimental N

_{1}/τ

_{12}is 9 at the strain of 9 in the step strain experiment, and the corresponding previous and present predictions are 2.47 and 8.45, respectively.

## 1. Introduction

## 2. Materials and Methods

_{1}versus time curve in the start-up experiment; the shear stress τ

_{12}in the start-up experiment; the shear stress in the long-term start-up experiment; the stress relaxation upon cessation of steady shear flow; and the transient N

_{1}/τ

_{12}in the step strain experiment.

**C**

_{t}

^{−1}is the Finger strain tensor, i.e., the inverse of the Cauchy–Green strain tensor

**C**

_{t}. The memory function m is written as:

_{i}and g

_{i}are the relaxation times and relaxation modulus coefficients, respectively, at low shear rate or at rest; i is the number of relaxation spectra; and $\dot{\gamma}$ is the shear rate. The strain-dependent function h used is the Papanastasiou–Scriven–Macosko (PSM) function [9,13,14,15].

## 3. Results and Discussion

_{1})—were predicted in [9], which indicates some reasonable aspect of the viscoelastic theory studied here. As mentioned in the first part of the work [9], another five groups of transient shear experiments reported by Pipe et al. [3] can also be predicted to check the capability of the model. Below are the predictions.

#### 3.1. Start-Up Experiment

#### 3.1.1. N_{1}

_{1}in the step rate experiment is also shown in Figure 8, as reported in [3], which is shown here in Figure 1, along with the prediction using the Vasquez–Cook–McKinley (VCM) model and the calculated result from the modified RS-PSM model with both parameters, f and ζ. The calculated results from both models show consistency with those from the experiments performed under the shear rate of 5 s

^{−1}. At the shear rate of 150 s

^{−1}, the result of the VCM model features a small and sharp stress-overshoot regime, which exhibits a large deviation from the experiment. The results of the present calculations at 30 and 150 s

^{−1}also show deviations from the experimental data, but the results are improved appearance, indicating that both the present model and the parameters describing the viscoelastic properties of the CPyCl/NaSal solution in this study can reflect the N

_{1}property of the solution more effectively. When the shear time reached 2 s and the stress had a steady status, the transient N

_{1}in Figure 1 was equal to that calculated in the steady shear experiment in [9]. The reason for the inefficiency of the present model in predicting N

_{1}is unknown. Remarkably, Gaudino et al. [16] predicted the transient N

_{1}of the same CPyCl/NaSal solution with 50 mM of NaSal at least at a steady state using the parameters obtained in characterizing the shear stress growth data.

#### 3.1.2. Shear Stress

^{−1}) were obtained using a single step rate. The other three groups (the minimum shear stress at the second low shear rate, the steady shear stress after long-term shear, and a group of stress developing data at the shear rate of 5 s

^{−1}after shearing at 150 s

^{−1}) were obtained using the two-step rates mode with a decreasing shear rate. Calculation of the two-step rates mode with a decreasing shear rate in this study was hindered by both the complex strain history and the deficiency of experimental data; therefore, the experiments involving the single-step rate were predicted.

^{−1}are predictions, and those at 5 and 30 s

^{−1}are fits. The maximum calculated value at 70 s

^{−1}was approximately 33% lower than that obtained experimentally, which is attributed to the linear interpolation between the ζ values at 30 and 150 s

^{−1}and the large gap between those at the two shear rates.

^{−1}is shown in Figure 3. In the stress-overshoot regime, the calculation result is lower than the experimental result due to interpolation; however, in a different regime, the calculated result is consistent with the experiment result. From the ζ curve in Figure 5 in the first part of the work [9], we can see that the ζ values at both 30 and 150 s

^{−1}in the overshoot regime are apparently larger than 1, which can cause corresponding calculation deviation at 70 s

^{−1}, and those in the steady shear regime approach 1, which will produce a calculation result similar to the experiment.

#### 3.1.3. Long-Term Start-Up Experiment

^{−1}. The present modified RS-PSM model with parameters f and ζ was employed to calculate the two experiments at 4 and 10 s

^{−1}, shown in Figure 4. Under long-term shearing, the experimental stress exhibited a slightly decreasing trend and other subtle phenomena, and the calculated stress at 4 s

^{−1}was 21 ± 0.5 Pa after the shear time reached 2 s. For the calculation at 10 s

^{−1}, reaching the steady state took slightly longer (approximately 2.5 s), and the stress was 21 ± 0.5 Pa. The calculated stress was constant at approximately 10 s and did not exhibit subtle variation after 10 s, despite the slight difference between the steady stresses at the two shear rates. The deviation between the calculated and experimental results was small, e.g., the deviation between the calculated steady stress and the minimum experimental shear stress was less than 7%. The maximum deviation in this case was approximately one-fifth of the deviation between the calculation result and the experiment result of the maximum overshoot stress in Figure 3.

#### 3.2. Stress Relaxation Experiment

^{−1}. The present study also presents three calculations in the same conditions as those used by Pipe et al. [3], which are shown in Figure 5. The predictions of the present model are similar to those of the VCM model at the three shear rates; however, certain deviations can be observed between the experimental and calculated results. Pipe et al. [3] also used a two-mode exponential relaxation process to adequately describe the relaxation experiment at 150 s

^{−1}; however, the structuralized model in the present study was not modified using this method to improve the calculation results.

#### 3.3. Step Strain Experiment

_{1}/τ

_{12}and time t and the relation between N

_{1}/τ

_{12}and strain γ in the step strain experiment, as well as the prediction of the VCM model. According to the calculation process of the VCM model [3], the strain history of the step strain experiment can be formed by applying a triangular-like shear rate, such that the strain increases and reaches a constant at the end of shearing. Therefore, the strain application process in the calculation is indeed an applying-rate process, where the shear rate is applied according to a designed rule. The complex application of the shear rate used in the study by Pipe et al. [3] was not possible in this study, and the step rate mode was used to generate a strain on the sample. This is why the step rate was added to the experimental mode of this group of experiment in Table 1 [9].

^{−1}was denoted by “Mode 1”, and the shear time used was calculated using the given strain. The shear time t

_{0}in Table 1 is obviously long as the strain is large. In terms of the N

_{1}/τ

_{12}versus t experiment in Figure 5 reported by Pipe et al. [3], the shear stress approached a steady state or the maximum level at approximately 0.1 s. Therefore, the second mode, denoted by “Mode 2”, involves the use of 0.1 s as the shear time. Therefore, the shear rate can be calculated by dividing the strain by the shear time. Finally, the shear time of 0.1 s is adjusted manually and slightly according to the calculations of Mode 2 and the N

_{1}/τ

_{12}experiments, which is denoted by “Mode 3” in Table 1.

_{1}/τ

_{12}versus t relation in the CPyCl/NaSal solution using the three shear rate modes in Table 1 and the experimental and calculated N

_{1}/τ

_{12}versus γ relation. The steady N

_{1}/τ

_{12}values of the three shear rate modes are almost identical and almost correspond to the experimental data in Figure 6a. The difference between the calculations of the three modes is the time taken to reach a steady state of N

_{1}/τ

_{12}, which is equal to the shear time t

_{0}. The overshoot of N

_{1}/τ

_{12}is also not observed in the calculation, and the N

_{1}/τ

_{12}calculated is almost constant after t

_{0}. However, the experimental N

_{1}/τ

_{12}exhibits an overshoot phenomenon at large strain and approaches a constant after 0.1 s, but it is not stable. Figure 6b shows the results of the VCM model, where the curve of the VCM model at γ = 1 is consistent with the experiment after 0.1 s, and the other two curves at γ = 6 and γ = 12, respectively, show large deviations from the corresponding experiments. The results of Mode 3 are consistent with those of the experiments, including both the steady N

_{1}/τ

_{12}value and the time reaching the steady state or the maximum stress ratio. The consistency of time between the experiment and the calculation is attributed to the artificial adjustment of shear time according to the experiment; however, this consistency also reflects the capability of the present modified RS-PSM model with parameters f and ζ to fairly express the viscoelastic property of the micellar solution.

_{1}/τ

_{12}= γ, under the strain of 9, which indicates some deficiency of the VCM model. The present calculation using Mode 3 approaches both the experiment and the Lodge–Meissner relation, which shows a certain reasonable aspect of the present model. In the analysis of the data on N

_{1}/τ

_{12}versus γ, Pipe et al. [3] proposed that the deviation between the calculation of the VCM model and the experimental result could be related to the inhomogeneous flow of the solution, and the present calculation shows another possible explanation of such a viscoelastic phenomenon, i.e., the homogeneous flow could produce more of the experimental phenomena of N

_{1}/τ

_{12}versus γ curve owing to the viscoelastic property of the solution. In addition, the influence of applying the real shear strain in the calculation remains unknown, despite the attempted application of three modes, because all three modes differ from the experimental process used by Pipe et al. [3]. This could complicate the present calculation.

## 4. Conclusions

_{1}in the stress growth experiment was improved.

^{−1}was low for the maximum shear stress in the stress-overshoot regime owing to the deficiencies of both ζ data and the linear interpolation method. During the long-term shearing in the start-up experiment, the model yielded a steady state value, and the experiment showed a slight decrease or variation, but the deviation between the steady calculated stress and the minimum experimental stress was less than 7%.

_{1}/τ

_{12}versus γ and the Lodge–Meissner relation under the strain of 9 in step strain can be adequately expressed by the model.

_{1}/τ

_{12}was 9 at the strain of 9 in the step strain experiment in Figure 6c, and the corresponding previous and present predictions were 2.47 and 8.45, respectively. These results indicate that the model and parameters in the present study are relatively more suitable for describing the viscoelastic properties of the CPyCl/NaSal solution at 22 °C.

_{2}in the RS equation [14,23] was not included in the present model; therefore, the N

_{2}experiment was not used. Moreover, another group of data—i.e., Figures 11 and 12 reported by Pipe et al. [3]—that were obtained by applying designed stress were not predicted. In the present model, stress is a function of deformation or shear rate; therefore, the model cannot be used to calculate the shear rate by inputting the shear stress. The step stress experiment should be calculated in future studies.

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Transient N

_{1}of the CPyCl/NaSal solution in the start-up experiment. Symbols are the experimental data in Figure 8 in the study conducted by Pipe et al. [3], and lines are the calculations. The calculated values for the VCM model are at 0.8, 5, and 150 s

^{−1}[3], and “PSM with f and ζ” is the present calculation.

**Figure 2.**Maximum shear stress in step rate experiment and the steady shear stress of the CPyCl/NaSal solution. The square symbol is the steady experiment in Figure 6 of Pipe et al. [3], the circle is the transient experiment in Figure 13 of Pipe et al. [3], and the solid circle is the calculation here.

**Figure 3.**Prediction of the stress growth at 70 s

^{−1}. The symbol is the experiment in Figure 13 of Pipe et al. [3], and the line is the calculation here.

**Figure 4.**Long-term stress growth. Both the solid and the dashed lines are the experiments in Figure 10 of Pipe et al. [3], and the bold solid line is the calculation here.

**Figure 5.**Stress relaxation property of the CPyCl/NaSal solution after steady shear. The symbols represent the experimental data in Figure 9 reported by Pipe et al. [3], and lines represent the calculated results. The solid line is the prediction of Pipe et al. [3], and the dashed line is the present calculation.

**Figure 6.**(

**a**) Relation of N

_{1}/τ

_{12}versus t obtained using three shear rate modes, (

**b**) N

_{1}/τ

_{12}of both Mode 3 and VCM model, (

**c**) the correlation between N

_{1}/τ

_{12}and γ. The symbols represent the experimental data in Figure 5 of Pipe et al. [3], and the lines are calculations, in which “VCM” and “Lodge-Meissner rule” are from Pipe et al. [3], and “Mode 1”, “Mode 2”, “Mode 3”, and “present prediction” are the present calculations.

γ | Mode 1 | Mode 2 | Mode 3 | |||
---|---|---|---|---|---|---|

$\dot{\mathit{\gamma}}$ (s^{−1})
| t_{0} (s) | $\dot{\mathit{\gamma}}$ (s^{−1})
| t_{0} (s) | $\dot{\mathit{\gamma}}$ (s^{−1})
| t_{0} (s) | |

1 | 0.006667 | 10 | 16.67 | 0.06 | ||

2 | 0.01333 | 20 | 33.33 | 0.06 | ||

4 | 0.02667 | 40 | 61.54 | 0.065 | ||

6 | 150 | 0.04000 | 60 | 0.1 | 75 | 0.08 |

8 | 0.05333 | 80 | 100 | 0.08 | ||

10 | 0.06667 | 100 | 125 | 0.08 | ||

12 | 0.08000 | 120 | 133.3 | 0.09 |

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**MDPI and ACS Style**

Huang, S.
Prediction of the Viscoelastic Properties of a Cetyl Pyridinium Chloride/Sodium Salicylate Micellar Solution: (II) Prediction of the Step Rate Experiments. *Polymers* **2022**, *14*, 5561.
https://doi.org/10.3390/polym14245561

**AMA Style**

Huang S.
Prediction of the Viscoelastic Properties of a Cetyl Pyridinium Chloride/Sodium Salicylate Micellar Solution: (II) Prediction of the Step Rate Experiments. *Polymers*. 2022; 14(24):5561.
https://doi.org/10.3390/polym14245561

**Chicago/Turabian Style**

Huang, Shuxin.
2022. "Prediction of the Viscoelastic Properties of a Cetyl Pyridinium Chloride/Sodium Salicylate Micellar Solution: (II) Prediction of the Step Rate Experiments" *Polymers* 14, no. 24: 5561.
https://doi.org/10.3390/polym14245561