# Scaling Behavior of Entanglement Dynamics in Polyelectrolyte Solutions: Insights from High-Frequency Rheometry

Yahya Rharbi

► To cite this version:

Yahya Rharbi. Scaling Behavior of Entanglement Dynamics in Polyelectrolyte Solutions: Insights from High-Frequency Rheometry. ACS Macro Letters, 2025, 14 (3), pp.259-264. 10.1021/acsmacro-lett.4c00722. hal-05076069

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# Scaling Behavior of Entanglement Dynamics in Polyelectrolyte Solutions: Insights from High-Frequency Rheometry

Yahya Rharbi*

Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, 38000 Grenoble, France

*Corresponding author: Yahya.Rharbi@univ-grenoble-alpes.fr;

Y. Rharbi [https://orcid.org/0000-0003-3091-2775]

**ABSTRACT:** Entanglement dynamics in polyelectrolyte solutions remain a challenging topic, particularly in capturing the entire dynamic spectrum, from single entanglement relaxation ($\tau_e$) to reptation time ($\tau_{rep}$), and aligning these observations with scaling predictions in the semidilute entangled (SE) and fully entangled (FE) neutral regimes. Using piezo compressional and classic rotational rheometry, we investigate the scaling behaviors of key viscoelastic properties over several decades of time scales in ten millions Mw cationic polyacrylamide solutions. Specific viscosity ($\eta_{sp}$) and $\tau_{rep}$ scale as predicted within Fuoss, SE, and FE regimes, defining crossover concentrations between these regimes ($C_e$ and $C_D$). More importantly, $\tau_e$, the rubbery plateau width ($\tau_{rep}/\tau_e$), and the high-frequency modulus ($G_e$) scale as $C^{-1.14\pm0.02}$, $C^{1.25\pm0.07}$, and $C^{1.32\pm0.05}$, aligning with SE prediction in the early SE regime before transitioning to neutral scaling of $C^{-2.7\pm0.14}$, $C^{3.1\pm0.15}$, and $C^{2.35\pm0.07}$ at an intermediate concentration between $C_e$ and $C_D$ labeled $C_D^e$. These results indicate that electrostatic interactions affect single entanglements and reptation differently, leading to a transition to neutral behavior at $C_D^e$ for the former, and at $C_D$ for the latter.
For Table of Contents use only

# Scaling Behavior of Entanglement Dynamics in Polyelectrolyte Solutions:
## Insights from High-Frequency Rheometry

Yahya Rharbi*

Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, 38000 Grenoble, France

*Corresponding author: Yahya.Rharbi@univ-grenoble-alpes.fr;

<figure><img src="image_2.png" /></figure>
The extensive application of polyelectrolytes as rheology modifiers in various industries has sustained interest in their dynamic properties for decades.¹⁻¹⁴ The primary challenge is to understand how electrostatic repulsions influence dynamic behaviors across different concentration regimes.⁵,⁹,¹¹ Scaling theories provide frameworks that describe key viscoelastic properties, including specific viscosity ($η_{sp}$), reptation time ($τ_{rep}$), Rouse time for single entanglements ($τ_e$), rubbery plateau width ($τ_{rep}/τ_e$), and plateau modulus ($G_e$).⁹,¹⁰,¹²,¹⁴⁻¹⁷ These scaling laws predict distinct behaviors across regimes: In the semidilute unentangled (SU) regime, $η_{sp} ∝ C^{0.5}$ according to Fuoss's law.¹⁴,¹⁵ The semidilute entangled (SE) regime, spanning from the onset of entanglement concentration ($C_e$) to the critical concentration ($C_D$), where the electrostatic screening length equals the size of the electrostatic blob, is characterized by $η_{sp} ∝ C^{1.5}$, $τ_{rep} ∝ C^0$, $τ_e ∝ C^{-1.5}$, $G_e ∝ C^{1.5}$, and $τ_{rep}/τ_e ∝ C^{1.5}$.⁹,¹² Above $C_D$, in the fully entangled (FE) regime, polyelectrolytes adopt behavior similar to neutral polymers, with viscoelasticity dominated by entanglements: $η_{sp} ∝ C^{3.9}$, $τ_{rep} ∝ C^{1.6}$, $τ_e ∝ C^{-2.3}$, $G_e ∝ C^{2.3}$, and $τ_{rep}/τ_e ∝ C^{3.9}$.¹²

There is consensus on the dynamics of polyelectrolytes from Newtonian behavior up to the reptation time, often characterized by classical rheometry to determine $η_{sp}$ in Newtonian regions and $τ_{rep}$ from low- to mid-frequency oscillations. This approach has consistently supported the SU ($C^{0.5}$) and SE ($C^{1.5}$) predictions for $η_{sp}$ in several salt-free polyelectrolytes, alongside the $C^0$ scaling of $τ_{rep}$ observed in the SE regime.²⁻⁵,¹¹,¹⁸⁻²²

While these findings shed light on specific aspects of chain dynamics, understanding the complete dynamic spectrum, from $τ_{rep}$ to $τ_e$ across the rubbery plateau, including plateau width and high-frequency $G_e$, remains challenging, partly due to experimental constraints in accessing high-frequency data within classical rheometry.⁵,²³ To address these challenges Time-Temperature Superposition (TTS)²⁴ with low-evaporation solvents like glycerol was used to access the entire rubbery plateau of polystyrene sulfonate in the SE regime, revealing neutral-like properties, with $τ_{rep}/τ_e ∝ C^4$ and $G_e ∝ C^{2.3}$ within the SE regime as defined by $η_{sp} ∝ C^{1.5}$.⁵ Alternatively, microrheology showed SE-like dynamics: $τ_{rep}/τ_e ∝ C^{1.82}$ and $G_e ∝ C^{1.5}$, yet under conditions where $η_{sp} ∝ C^{2.56}$.²²,²³ These discrepancies highlight the need for further experiments to fully map the dynamic spectrum across all polyelectrolyte concentration regimes.

The appropriate approach should involve rheometry tools capable of spanning the entire rubbery plateau from µHz to kHz,²⁵⁻²⁷,²² broadening the SE regime by lowering $C_e$ and increasing $C_D$ through the use of high $M_w$ polyelectrolytes in low Bjerrum length solvents such
as water,²³,²⁸ and maintaining constant temperature to minimize conformation and regime variability.

Rotational rheometric measurement using a DHR-3 rheometer (TA Instruments) with Couette geometry allows determination of low-shear viscosity ($\eta_0$), specific viscosity ($\eta_{sp} = (\eta_0 - \eta_s)/\eta_s$), the G' and G'' over a frequency range from 100 µHz to 10 Hz, enabling the evaluation of $\tau_{rep}$. Extending the frequency range for probing $\tau_e$ is achieved through piezo compressional rheometry, allowing measurements up to a 7 kHz.²⁵,²⁶ The Tripav piezo compressional (TriJet Limited) operates in compression mode between 25 mm diameter plates with a gap range of 10 to 500 microns, measuring the compressibility response to determine G' and G''.²²,²⁵,²⁷,²⁹ This approach provides measurements across more than 7 decades of frequencies, ranging from 100 µHz to 7 kHz. Details of the technical aspects and the relevance of the measurement are provided in Section S1.

This strategy was applied to high molecular weight cationic polyelectrolytes across four concentration decades at a constant temperature of 20 °C. Cationic polyacrylamide (CPAM, superfloc XD 5500) from KEMIRA is an ideal candidate for this study due to its high Mw of approximately $10^7$ g/mol,³⁰ corresponding to around 45,000 molecular units per polymer chain, significantly larger than in previous studies.⁵,²³ Made up of acrylamide units and around 80% quaternary ammonium groups (acryloxyethyltrimethyl ammonium chloride), it maintains a stable positive charge in aqueous solutions across a broad pH range.³¹,³⁰,⁴,¹⁸,¹⁹ These properties make this CPAM class particularly suitable as a flocculant in wastewater treatment.³² Characterization of this CPAM is conducted using NMR, light scattering, and polyelectrolyte titration (Section S2).³⁰,³¹,³³ Sample preparation and measurement methods are detailed in Section S3.

The $\eta_{sp}$, calculated from the zero-shear extrapolation of the viscosity vs. shear rate curve (Figure S1), is plotted against polymer concentration in Figure 1 and listed in table S1. Beyond a concentration of 0.026 g/L, $\eta_{sp}$ adheres to a scaling relation of $\eta_{sp} \propto C^{0.55 \pm 0.01}$, closely aligning with Fuoss's prediction of $C^{0.5}$ for the SU regime.¹⁴,¹⁵ The onset concentration of the Fuoss regime is used to estimate the residual salt concentration as $C_s = 2C/f$, where $f = 0.3$ is the fraction of charged monomers, resulting in $C_s \approx 2.6 \times 10^{-5}$ mol/L.¹¹,¹² Beyond 0.2 g/L, $\eta_{sp}$ shifts to a power-law behavior of $\eta_{sp} \propto C^{1.45 \pm 0.07}$, approaching the predicted $\eta_{sp} \propto C^{1.5}$ for polyelectrolytes in the SE regime.⁹,¹² This transition at 0.2 g/L ($9 \times 10^{-4}$ M) likely indicates $C_e$, marking the shift from the SU to SE regime, where chain entanglements begin to dominate the viscoelastic response. The estimated $C_e$ aligns with the trend $C_e \propto N^{-0.77}$ observed in studies on sodium carboxymethyl cellulose and polystyrene sulfonate.⁵,²¹ For the current CPAM, with $N \approx 45,000$, this results in an estimated $C_e$ of $1.4 \times 10^{-3}$ to $2 \times 10^{-3}$ mol/L, which is consistent with the observed range in this study.
The SE regime spans over more than a decade of concentrations, reaching approximately 6 g/L, before transitioning to η<sub>sp</sub> ∝ C<sup>3.2±0.2</sup>, approaching the predicted C<sup>3.9</sup> behavior for the FE regime.<sup>12</sup> This change in behavior marks a shift to a regime where chain entanglements dominate the viscoelastic response as electrostatic interactions diminish, resulting in neutral behavior. This shift is likely associated with the transition from the SE to FE neutral regime, with the corresponding transition concentration attributed to C<sub>D</sub>. These results are consistent with prior studies, which reported comparable behaviors in low-salt conditions for polyelectrolytes bearing sulfonated groups, as well as those with quaternized species, similar to the polymer examined here.<sup>2–5,11,18–22,34</sup>

<figure><img src="image_3.png" /></figure>

Figure 1. Specific viscosity η<sub>sp</sub> at 20°C as a function of CPAM concentration across four decades. The solid line are power-law fits for η<sub>sp</sub> in different regimes: the SU (η<sub>sp</sub> ∝ C<sup>0.55±0.01</sup>), the SE (η<sub>sp</sub> ∝ C<sup>1.44±0.07</sup>) and the concentrated (η<sub>sp</sub> ∝ C<sup>3.2±0.2</sup>). Arrows indicate the transition concentrations C<sub>e</sub> and C<sub>D</sub> . The specific viscosity is extrapolated at low shear rates from DHR3 rheometer measurements using Couette geometry. The data in the dilute regime (first two points) reflect an apparent η<sub>sp</sub>, as the Newtonian plateau has not yet been fully established (see Figure S1).
<figure><img src="image_4.png" /></figure>

Figure 2. Storage modulus G' and loss modulus G'' as a function of angular frequency (rad/s) for an aqueous CPAM solution at 10.4 g/L performed at 20°C. The low-frequency G' (○) and G'' (□) are measured in rotational rheometry (Couette geometry, DHR3) while the high-frequency G' (●) and G'' (■) are measured in piezo compressional rheometer (Tripav). Arrows indicate the frequencies used to calculate the relaxation times τ<sub>rep</sub> and τ<sub>e</sub>.

The combination of rotational and piezo oscillations allows estimation of G' and G'' across seven decades of angular frequency, from 10<sup>-3</sup> to 10<sup>4</sup> rad/s (Figure 2 and Figure S2). The overlap between G' and G'' demonstrates compatibility between these methods, providing a comprehensive view of the rubbery plateau and enabling coherent estimations of τ<sub>rep</sub> and τ<sub>e</sub> from low- and high-frequency crossovers.<sup>35</sup> Since the rubbery modulus (G<sub>e</sub>) depends on frequency, it is quantified using two methods: G<sub>e</sub><sup>min</sup>, based on G' at the frequency of minimum tanδ, and G<sub>e</sub><sup>max</sup>, from the high-frequency crossover.<sup>5,23,36,37</sup> Measurements from C<sub>e</sub> to above C<sub>D</sub> show a shortening of the rubbery plateau with decreasing concentration, as indicated by an increase in τ<sub>rep</sub> and a decrease in τ<sub>e</sub>, with both G<sub>e</sub><sup>max</sup> and G<sub>e</sub><sup>min</sup> decreasing as concentration decreases. Figures 3a and 3b and table S1 illustrate the power-law dependence of τ<sub>e</sub> and τ<sub>rep</sub> on CPAM concentration, from C<sub>e</sub> to above C<sub>D</sub>. τ<sub>rep</sub> shows greater dispersion than τ<sub>e</sub>, likely due to the longer times required for accurate τ<sub>rep</sub> capture. In the SE regime, τ<sub>rep</sub> follows τ<sub>rep</sub> ∝ C<sup>0.29±0.04</sup> up to C<sub>D</sub>, transitioning to τ<sub>rep</sub> ∝ C<sup>3.3±0.4</sup> above. The 0.29 exponent is close to the predicted C<sup>0</sup> for the SE regime and aligns well with many studies on a broad range of polyelectrolytes, either sulfonated or quaternized.<sup>5,18</sup> Noting that acquiring this exponent omits the logarithmic contribution, which could slightly contribute to the estimated exponent.<sup>10</sup> Beyond C<sub>D</sub> τ<sub>rep</sub> ∝ C<sup>3.3±0.4</sup> exceeds the predicted τ<sub>rep</sub> ∝ C<sup>1.6</sup> for neutral behavior.

On the other hand, τ_e demonstrates two distinct behaviors within the SE regime. Initially, it scales as τ_e ∝ C<sup>-1.14±0.02</sup> from C_e up to a critical CPAM concentration of 1.67 g/L, denoted as C_D^e, before transitioning
to $\tau_e \propto C^{-2.7 \pm 0.2}$ between $C_D^e$ and $C_D$. Beyond $C_D$, $\tau_e$ follows a different scaling pattern, $\tau_e \propto C^{-1.05 \pm 0.06}$, indicating a further shift in the system's dynamics.

The observed $\tau_e \propto C^{-1.14}$ below $C_D^e$ is more likely aligned with the SE regime of polyelectrolytes, as it closely follows the predicted scaling (-1.5) and deviates significantly from the scaling associated with neutral behavior (-2.3).¹² This suggests that within this regime, the single entanglement dynamics is mainly dominated by electrostatic repulsion. Conversely, above the $C_D^e$, $\tau_e$ transitions to neutral-like behavior, following $\tau_e \propto C^{-2.7}$. Notably, this transition occurs at a concentration significantly lower than $C_D$, as determined from $\eta_{sp}$ measurements. ¹² However, the slower decay of $\tau_e \propto C^{-1.05}$ above $C_D$, though correlated with the sharp rise in $\tau_{rep}$, remains unclear and requires further investigation.

The width of the rubbery plateau $\tau_{rep}/\tau_e$ is calculated either by dividing $\tau_{rep}$ and $\tau_e$ measured on the same sample or, when not possible, using $\tau_{rep}$ or $\tau_e$ values extrapolated from their concentration relationships to match the corresponding measurement conditions. The scaling behavior of $\tau_{rep}/\tau_e$, shows two transitions: one at $C_D^e$ and another at $C_D$ (Figure 3C). Between $C_e$ and $C_D^e$, $\tau_{rep}/\tau_e$ scales as $\tau_{rep}/\tau_e \propto C^{1.25 \pm 0.07}$, shifting to $\tau_{rep}/\tau_e \propto C^{3.1 \pm 0.2}$ up to $C_D$. Above $C_D$, $\tau_{rep}/\tau_e$ follows a trend with a higher exponent, $\tau_{rep}/\tau_e \propto C^{4.3 \pm 0.1}$ aligning closely with the predicted neutral behavior for this regime. The observed exponent of 1.25 below $C_D^e$ is remarkably close to the predicted 1.5 for the SE regime, strongly suggests that electrostatic interactions dominate the dynamics of this polyelectrolyte in this regime. Notably, this exponent differs significantly from the value of 4 observed in TTS measurements,⁵ indicative of neutral behavior, despite $\eta_{sp} \propto C^{1.5}$ under the same conditions, reflecting the SE regime. It is also lower than the 1.82 exponent found in microrheology under conditions where $\eta_{sp} \propto C^{2.56}$.²³ The exponent ($\tau_{rep}/\tau_e \propto C^{3.1}$) between $C_D^e$ and $C_D$ suggests neutral-like behavior, aligning with the predicted value of 3.9. Interestingly at $C_e$, $\tau_{rep}/\tau_e$ is 300, far exceeding the value of 1 observed in neutral polymers. Notably, extrapolating $\tau_{rep}/\tau_e$ along the neutral trend to $C_e$ reduces it to a range of 3–8, much closer to the model's predicted unity (Figure S3). Using this extrapolation to estimate $C_e$ by setting $\tau_{rep}/\tau_e$ to 1 reveals a value of 0.12 g/L, consistent with the value determined from $\eta_{sp}$. Taking $\tau_{rep}/\tau_e = 300$ at $C_e$ suggests that the number of entanglements (N/N<sub>e</sub>) at $C_e$, derived from $\tau_{rep}/\tau_e = (N/N<sub>e</sub>)^3$, is 6.69. This raises questions about the process underlying the onset of entanglement formation occurring at $C_e$ in polyelectrolytes.

The high-frequency modulus, $G_e^{max}$, derived from the G' and G'' crossover,⁵,³⁶,³⁷ scales as $G_e^{max} \propto C^{1.3 \pm 0.1}$ between $C_e$ and $C_D^e$, then deviates from this behavior above $C_D^e$, following $G_e^{max} \propto C^{2.35 \pm 0.07}$ up to $C_D$ (Figure 4 and Figure S4 and table S2). Beyond $C_D$, $G_e^{max}$ shows a lower exponent. The 1.3 exponent provides a strong indication of SE behavior, as it closely approaches the predicted value of 1.5 for this regime.⁹ The transition to $C^{2.35}$ at $C_D^e$ signals a shift to neutral behavior, consistent with the expected scaling for this regime.⁵,³⁸
The G' at the minimum tanδ,²³,³⁶ also scales as Gₑ⁻min ∝ C¹.³, further supporting the SE behavior of the plateau modulus (Figure 4 and Figure S4).¹² Yet, unlike GₑURA, Gₑmin transitions to neutral-like behavior at a higher concentration, scaling as Gₑmin ∝ C².¹±⁰.¹ at C_D instead of C_Dᵉ.³⁸ This highlights differing behaviors between the modulus derived from single entanglement (GₑURA) and that from the minimum tanδ (Gₑmin). Furthermore, the observed C².³ scaling above C_Dᵉ supports Colby's findings,⁵ which also identified neutral behavior with Gₑ ∝ C².³ within the SE regime identified by ηₛₚ. Remarkably, extrapolating the terminal modulus in unentangled regime at Cₑ using Cₑ kT/N yields Gₑ(Cₑ) ≈ 0.09 Pa, consistent with Gₑ(Cₑ) values in Figure 4: 0.07–0.1 Pa from GₑURA and ~0.2 Pa from Gₑmin, further confirming the estimated Cₑ value from ηₛₚ.

These results reaffirm, once again, that the Dobrynin *et al.* model¹² is an appropriate framework for describing the global chain dynamics of entangled polyelectrolytes, effectively capturing τ<sub>rep</sub>, η<sub>sp</sub> and G<sub>e</sub><sup>min</sup>, across both electrostatically dominated and neutral-like regimes.

The primary finding reveals that small-length-scale dynamics, reflected in the single-entanglement
dynamics identified through τ<sub>e</sub>, τ<sub>rep</sub>/τ<sub>e</sub> and G<sub>e</sub><sup>max</sup>, adopt all the predicted features of semidilute entangled
polyelectrolyte properties, supporting the validity of the Dobrynin et al. model<sup>12</sup> as a reliable framework
for capturing these aspects of polyelectrolyte behavior. This contrasts with Han and Colby's findings,
which report neutral behavior for such dynamics throughout all concentrations regimes.<sup>5</sup>

Interestingly, the model-experiment agreement is restricted to the early part of the semidilute entangled regime, between C<sub>e</sub> and C<sub>D</sub><sup>e</sup>, with C<sub>D</sub><sup>e</sup> being manifestly lower than the nominal C<sub>D</sub> defined from μ<sub>sp</sub>. Between C<sub>D</sub><sup>e</sup> and C<sub>D</sub>, a clear contrast emerges: the single-entanglement dynamics transition to neutral behavior, while the chain reptation dynamics retain electrostatically dominated characteristics. This divergence supports Han and Colby's observations<sup>5</sup> but challenges the Dobrynin et al. framework, particularly the approach used to derive τ<sub>rep</sub>, defined as τ<sub>rep</sub> = τ<sub>e</sub> (N/N<sub>e</sub>)<sup>3</sup>.<sup>12</sup> Beyond C<sub>D</sub>, however, both dynamics converge to neutral behavior, coming back into alignment with the model's predictions.<sup>12</sup>

While these results align few aspects of the polyelectrolyte model with experimental observation, they
also raise other questions, such as whether the transition to neutral behavior corresponds to C<sub>D</sub><sup>e</sup> or C<sub>D</sub>
and whether this transition is length-scale dependence.

At first glance, these results can be interpreted as indicating that each investigated length scale has its
own SE-neutral transition concentration. Specifically, C<sub>D</sub><sup>e</sup> can be viewed as marking the onset of the
transition from SE to neutral behavior for single entanglements, while C<sub>D</sub> represents the corresponding
transition for chain reptation dynamics. This interpretation may provide insight into Colby's observation
of neutral behavior in entanglement dynamic within the SE regime ($\tau_{rep}/\tau_e \propto C^4$).⁵ If the range between $C_e$ and $C_D$ is narrow, the SE regime for $\tau_e$ may be overlooked, as it briefly emerges after $C_e$ before rapidly shifting to neutral behavior. In this study SE behavior for $\tau_e$ is successfully captured, likely due to the broad range between $C_e$ and $C_D$, which spans nearly two decades of concentration. Such interpretation warrants further investigation, particularly through a refinement of the model's morphological description of chain conformations. For example, the coexistence of neutral behavior at short-length scales and electrostatic dominance at larger scales could be explored by examining more specific counterion distributions along the chain and further detailing necklace-like morphologies. On the other hand attributing single-entanglement dynamics solely to the G' and G'' crossover may oversimplify the process. Alternatively, a comprehensive analysis of the full dynamic spectrum across length scales could offer insights for enhancing the understanding of polyelectrolyte behavior.

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>
**Figure 3.** Dependence of viscoelastic characteristics on polymer concentration for aqueous CPAM solutions, spanning from the semidilute entangled (SE) to fully entangled (FE) regimes at 20°C. (a) Reptation time ($\tau_{rep}$), (b) Rouse time of a single entanglement strand ($\tau_e$), and (c) the width of the rubbery plateau ($\tau_{rep}/\tau_e$). Dashed lines represent the best fit curves to the power law in each regime. Power law exponents are reported on each curve. For solutions where $\tau_{rep}$ and $\tau_e$ were measured on different samples, the ratio $\tau_{rep}/\tau_e$ is calculated using extrapolated $\tau_{rep}$ values to match the concentration where $\tau_e$ was measured, and vice versa.

<figure><img src="image_8.png" /></figure>

**Figure 4.** Dependence of elastic modulus ($G_e$) on polymer concentration for aqueous polymer solutions, spanning from the semidilute entangled (SE) to fully entangled (FE) regimes at 20°C. $G_e^{max}$ is the modulus value taken from the higher frequency $G'$ and $G''$ crossover, and $G_e^{min}$ is the $G_e$ value from $G'$ at the minimum tan delta. Dashed lines represent the best fit curves to the power law in each regime, with power law exponents reported on each curve.

In conclusion, the entanglement dynamics of cationic polyacrylamide (CPAM) solutions with a molecular weight of several million were investigated across the semidilute entangled (SE) and fully entangled (FE) regimes using piezo compressional and rotational rheometry. The $\eta_{sp}$ followed the
expected scaling law in the Fuoss ($C^{0.55\pm0.01}$), SE ($C^{1.44\pm0.07}$), and FE ($C^{3.2\pm0.2}$) regimes, allowing the recovery of $C_e$ and $C_D$. The $\tau_{rep}$ in the SE regime scaled as $C^{0.29\pm0.04}$, close to the predicted $C^0$, with notable transition to neutral behavior at $C_D$. The $\tau_e$, the $\tau_{rep}/\tau_e$, and $G_e$ scaled as $C^{-1.14\pm0.02}$, $C^{1.25\pm0.07}$, and $C^{1.3\pm0.1}$, respectively, values that align with the predicted behavior in the SE regime of $C^{-1.5}$, $C^{1.5}$, and $C^{1.5}$. A shift towards FE neutral behavior was observed at a concentration $C_D^e$, well below $C_D$, with the scaling of $C^{-2.7\pm0.14}$, $C^{3.1\pm0.2}$, and $C^{2.35\pm0.07}$. At $C_D$, $\tau_{rep}/\tau_e$ followed a $C^{4.3\pm0.1}$ scaling, suggesting entry into the FE neutral regime. These results strongly establish the Dobrynin et al. model as a framework for predicting single-entanglement dynamics ($\tau_e$, $\tau_{rep}/\tau_e$, $G_e^{max}$) in the early semidilute entangled regime (below $C_D^e$), while also effectively describing reptation dynamics ($\tau_{rep}$, $\eta_{sp}$, $G_e^{min}$) across both electrostatically dominated and neutral-like regimes. These findings highlight the crucial role of electrostatic interactions in polyelectrolyte dynamics and offer valuable perspectives for future studies across varying concentrations and environmental conditions.

**Supporting Information**

* Section 1: Details of rheological measurement methods

* Section 2: Characterization of the polyelectrolyte

* Section 3: Sample preparation and measurement methods

* Figure S1: Viscosity vs. shear rate

* Figure S2: Storage modulus ($G'$) and loss modulus ($G''$) vs. frequency

* Figure S3: Width of the rubbery plateau as a function of concentration

* Figure S4: Normalized elastic modulus vs. concentration

* Table S1: Numerical data of $\eta_{sp}$, $\tau_{rep}$, $\tau_e$, and $\tau_{rep}/\tau_e$.

* Table S2 = Numerical data of $G_e^{max}$ and $G_e^{min}$.

**ACKNOWLEDGEMENTS**

The Rhéologie et Procédés Laboratory is affiliated with the LabEx Tec 21 (Investissements d'Avenir - Grant agreement no ANR-11-LABX-0030). Additionally, the laboratory is associated with the PolyNat Carnot Institute (Investissements d'Avenir - Grant agreement no ANR-11-CARN-030-01). The author expresses gratitude to Mohamed Karrouch and Vincent Verdoot for their invaluable assistance in conducting the experiments, and to Frédéric Hugenell for the insightful discussions.
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