# Time–Concentration Superposition for Linear Viscoelasticity of Polymer Solutions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{T}and a modulus shift b

_{T}, which has its equivalent, now, in a

_{c}and b

_{c}for TCS. We will determine the concentration shift factors a

_{c}(c/c*) and b

_{c}(c/c*) in search of an unambiguous way to estimate the concentration scaling exponents of non-model polymer solutions in comparison to the universal properties known for model polymer solutions. The shift factor of the complex viscosity, b

_{c}/a

_{c}, is also included. Equivalently to TTS, TCS is shown to expand the experimental frequency window of the SAOS data. This has been studied experimentally with a typical non-model polymer solution. The results were compared to data available from other laboratories that have not yet been analyzed in this way.

## 2. Materials and Methods

#### 2.1. Polymer Solution

_{w}) of 18 × 10

^{6}g/mol was purchased from Polysciences, Inc. (Catalogue No.: 18522-100), Warrington, PA, USA, and used as received (hereafter called 18M PAAm). A series of ternary solutions containing 18M PAAm, deionized water and 60 wt.% sucrose (hereafter called 18M PAAm–sucrose–water solution) with a concentration range from 0.173 wt.% to 0.921 wt.% were prepared by firstly dissolving 18M PAAm powder in deionized water, obtained from the Direct-Q UV-R pure water system after three stages of treatment at room temperature, and then adding an adequate amount of sucrose, purchased from Chemical Technology Co., Ltd. (Catalogue No.:M20224), Meryer (Shanghai), China to make up 60 wt.% sucrose and gently stirring at room temperature using an IKA C-MAG HS 7 magnetic stirrer at 12 rpm for 72 h. The ternary solutions were then put into a 4 °C refrigerator for further homogenization and storage.

_{w}= 5.7 × 10

^{6}g/mol and M

_{w}/M

_{n}= 34.4, supplied by the same source and characterized in our previous work [7].

#### 2.2. Rheometry

## 3. Results

_{ref}/c*), G″(ω, c

_{ref}/c*). They belong to the solution at reference composition c

_{ref}/c*, abbreviated as G′

_{ref}, G″

_{ref}in the following.

_{ref}= 17.3c* was chosen here to serve as a reference state providing the frequency-dependent G′

_{ref}and G″

_{ref}in the relatively narrow frequency window of the SAOS experiment. These values for c

_{ref}= 17.3c* are fixed during the shifting while the higher c/c* moduli become shifted to the left and down thereby widening the frequency window. Stepwise progress of the shifting is pictured as a sequence. When the shifting is complete, the rescaled moduli superimpose on G′

_{ref}and G″

_{ref}of the reference state and a set of master curves is generated. These master curves belong to c

_{ref}= 17.3c* but can now be shifted to other concentrations.

_{ref}< 1, b

_{c}is positive and a

_{c}is negative and for c/c

_{ref}> 1, the signs change and b

_{c}< 1 and a

_{c}> 1. Since TCS in Figure 1 is based on the lowest concentration, all shifting occurs to one side.

_{c}is equivalent to the concentration scaling of either the terminal relaxation time or the Rouse relaxation time, respectively. The time–concentration shift effectively amounts to a multiplication of the dynamic moduli G′(ω, c) and G″(ω, c) with the above modulus shift factor b

_{c}and a multiplication of the experimental frequency with the above a

_{c}as

_{c}and b

_{c}, and hence is larger than the shifting of G′ or G″ as

_{c}and the vertical shifting factor b

_{c}over the terminal entanglement dynamics and Rouse dynamics for 18M PAAm–sucrose–water ternary solutions is compared with the concentration scaling of the terminal entanglement dynamics for 18M PAAm–water binary solution [11] and monodisperse PB–PHO binary solution [4]. The concentration scaling of the 18M PAAm–water ternary solutions over the terminal entanglement dynamics is in excellent agreement with those of the 18M PAAm–water binary solutions. It shows that the solvent quality of 60 wt.% sucrose and water mixture is very similar to that of pure water. However, the power law concentration scaling exponent of the shifting factor a

_{c}over Rouse dynamics shows a negative sign. It reflects the fact that the characteristic relaxation time over Rouse dynamics is decreased with the increase in polymer concentration, likely due to the screening effect of segment–segment interactions in polymer solutions. The estimated concentration scaling exponents $\alpha $ and $\beta $ for PB–PHO, 18M PAAm binary and ternary solutions are listed in Table 1 along with the results of other binary polymer solutions obtained from the TCS procedure over the terminal entanglement dynamic regime. The concentration scaling of complex viscosity is equivalent to the scaling of the shifting factors a

_{c}/b

_{c}. As shown in Figure 4, they all exhibit a well-defined power law concentration scaling. Table 1 also lists the estimated concentration scaling exponent (α − β) for all the polymer solutions analyzed here. The convolution of shift factors in the viscosity shift obscures its meaning and makes it hard to draw conclusions about concentration effects on other viscoelastic material functions.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Time–concentration superposition (TCS) of SAOS data belonging to a 18M PAAm–sucrose–water solution at semidilute concentrations from 17.3c* to 92.1c*. TCS merges the linear viscoelastic data G′, G″, η*(ω, c/c*) thereby generating master curves at the reference concentration of 17.3c* as measured by SAOS. (

**a**) Original data before shifting; (

**b**) 25% shifting towards the reference state; (

**c**) 50% shifting the curves; (

**d**) 75% shifting the curves; (

**e**) 100% shifting the curves; (

**f**) a master curve of G′(ω, c/c*), G″(ω, c/c*), η*(ω, c/c*) after completion of the TCS procedure. The shifting and plotting were executed using IRIS Rheo-hub software [8].

**Figure 2.**Three–dimensional plots of G′ and G″ against concentration and frequency. (

**a**) 18M PAAm–sucrose–water ternary solution; (

**b**) 18M PAAm–water solution [11]; and (

**c**) PB–PHO solution [4], respectively. The first row and the third row show the curves before TCS shifting. The second row and the fourth row show the curves after TCS shifting. The color scheme indicates the scale of frequency. The master curves are projected in the G′– or G″–Freq plane.

**Figure 3.**(

**a**) A master curve of G′(ω, c/c*), G″(ω, c/c*) complex viscosity η*(ω, c/c*) with c

_{ref}= 49.9c* superimposed with respect to the terminal entanglement dynamics; (

**b**) plots of shifting factor a

_{c}and 1/b

_{c}against concentration c/c* and in comparison with PB–PHO [4] and 18M PAAm–water binary solutions, respectively.

**Figure 4.**Plots of (a

_{c}/b

_{c}) as a function of (c/c

_{ref}) in the logarithmic scales for the polymer solutions listed in Table 1 with respect to reference concentration as c

_{ref}= 56c* for PB–PHO [4], c

_{ref}= 44.3c* for PB–DOP [4], c

_{ref}= 8.0c* for PI–OB [13], c

_{ref}= 49.1c* for λ-DNA [9], c

_{ref}= 125c* for UHMWPE [12], c

_{ref}= 48c* for 18M PAAm aqueous solution [11], and c

_{ref}= 49.9c* for 18M PAAm–sucrose–water solution, respectively.

**Table 1.**The concentration scaling exponents α, −β and (α − β) of the shifting factors a

_{c}, 1/b

_{c}and a

_{c}/b

_{c}, respectively, are estimated by TCS for a range of polymer solutions. PB–PHO and PB–DOP are monodisperse polybutadiene (PB) in phenyloctane (PHO) and in dioctyl phthalate (DOP) [4], respectively; PI–OB is monodisperse cis-polyisoprene (PI) in oligobutadiene (OB), which is a marginal solvent between good and theta solvent [13]; λ-DNA is completely monodisperse ultrahigh molecular weight linear lambda (λ) DNA in Tris-EDTA buffer [9]; UHMWPE is ultrahigh molecular weight polyethylene dissolved in oligo-ethylene [12]; Welan gum solutions but no information on molecular characterization available [10]; 18M PAAm is highly polydisperse 18M PAAm in aqueous binary solution [11] and in 60 wt.% sucrose–water ternary solution, respectively. The 18M PAAm ternary solution was analyzed with respect to the terminal entanglement dynamics as well as Rouse dynamics in which the exponent α shows a negative value.

Sample Code | M_{w} (g/mol) | M_{w}/M_{n} | c/c* | Solvent Quality | α | −β | α − β |
---|---|---|---|---|---|---|---|

PB–PHO [4] | 9.25 × 10^{5} | <1.1 | 8.56~400 | Good | 2.46 ± 0.06 | 2.12 ± 0.02 | 4.58 ± 0.05 |

PB–DOP [4] | 9.25 × 10^{5} | <1.1 | 3.1~144.9 | θ | 1.8 ± 0.1 | 2.14 ± 0.03 | 3.9 ± 0.1 |

PI–OB [13] | 4.88 × 10^{4} | 1.05 | 4~27.1 | Marginal between good and θ solvent | 2.9 ± 0.4 | 1.19 ± 0.05 | 4.1 ± 0.4 |

λ-DNA [9] | 3.15 × 10^{7} | 1.0 | 19.5~89.7 | Good | 4.7 ± 0.3 | 2.0 ± 0.1 | 6.7 ± 0.3 |

UHMWPE [12] | 3.2 × 10^{6} | 9.7 | 41.7~125 | θ | 1.0 ± 0.2 | 2.2 ± 0.1 | 3.2 ± 0.3 |

Welan [10] | - | - | - | - | 3.6 ± 0.2 | 1.16 ± 0.01 | 4.8 ± 0.2 |

18M PAAm [11] | 18 × 10^{6} | >34.4 | 10~250 in water | Super-good | 1.40 ± 0.06 | 1.05 ± 0.03 | 2.48 ± 0.09 |

17.3~92.1 in 60 wt.% sucrose–water | 1.37 ± 0.09 | 1.00 ± 0.04 | 2.4 ± 0.1 | ||||

−1.92 ± 0.08 | 1.95 ± 0.07 | 0.03 ± 0.04 |

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**MDPI and ACS Style**

Li, C.-Q.; Winter, H.H.; Fan, Y.-Q.; Xu, G.-X.; Yuan, X.-F.
Time–Concentration Superposition for Linear Viscoelasticity of Polymer Solutions. *Polymers* **2023**, *15*, 1807.
https://doi.org/10.3390/polym15071807

**AMA Style**

Li C-Q, Winter HH, Fan Y-Q, Xu G-X, Yuan X-F.
Time–Concentration Superposition for Linear Viscoelasticity of Polymer Solutions. *Polymers*. 2023; 15(7):1807.
https://doi.org/10.3390/polym15071807

**Chicago/Turabian Style**

Li, Can-Qi, Horst Henning Winter, Yuan-Qi Fan, Geng-Xin Xu, and Xue-Feng Yuan.
2023. "Time–Concentration Superposition for Linear Viscoelasticity of Polymer Solutions" *Polymers* 15, no. 7: 1807.
https://doi.org/10.3390/polym15071807