Design of model Boger fluids with systematically controlled viscoelastic properties

TL;DR

This work tackles the challenge of engineering Boger fluids with prescribed viscoelastic properties by linking rheological parameters $G_0$, $\tau$, and $\psi_1$ to design variables $c$, $M$, and $\eta_s$ through an empirical design equation grounded in the Zimm model. By carefully characterizing PIB-based Boger fluids and mapping the observed power-law relationships into a design matrix, the authors enable independent control of elasticity and relaxation dynamics, as well as the first normal-stress coefficient. They validate the approach by designing fluids with targeted $G_0$ and $\tau$ (and separately $\psi_1$) and demonstrate robust data collapse and predictable rheological behavior across multiple formulations. The methodology provides a practical route for tailoring viscoelastic environments for flow experiments and could extend to other dilute polymer systems, offering a route to infer material parameters from design variables when torque measurements are limited.

Abstract

The subject of viscoelastic flow phenomena is crucial to many areas of engineering and the physical sciences. Although much of our understanding of viscoelastic flow features stems from carefully designed experiments, preparation of model viscoelastic fluids remains a challenge; for example, fabricating a series of fluids with different fluid shear moduli $G_0$, but with an identical relaxation time $τ$, is nontrivial. In this work, we harness the non-ideality of nearly constant-viscosity elastic fluids, commonly known as `Boger fluids', made with polyisobutylene, to develop an experimental methodology that produces a set of fluids with desired viscoelastic properties, specifically, $G_0$, $τ$, and the first normal stress difference coefficient $ψ_1$. Through a linear algebraic relation between the rheological properties of interest ($G_0$, $τ$, $ψ_1$) and the fluid compositions in terms of polymer concentration $c$, molecular weight $M_w$, and solvent viscosity $η_s$, we developed a `design equation' that takes $G_0$, $τ$, $ψ_1$ as inputs and calculates values for $c$, $M_w$, $η_s$ as outputs. Using this method, fabrication of dilute viscoelastic fluids whose rheological properties are \textit{a priori} known can be achieved.

Figures (12)

• Figure 1: Rheology of dilute Boger-like viscoelastic fluids. (a) Solvent viscosity $\eta_\text{s}$ as a function of PB weight fraction, $\phi_{\text{PB}}$, in a PB-light mineral oil mixture. We found empirically that $\eta_\text{s}\approx 0.022\exp(10.3\phi_{\text{PB}})~\left[\text{Pa}\cdot\text{s}\right]$. (b) and (c) For a fixed $\phi_{\text{PB}}$, we measured the specific viscosity $\eta_{\text{sp}}=(\eta-\eta_\text{s})/\eta_\text{s}$ as a function of PIB concentration $c$ (wt%), where $\eta$ is the measured solution viscosity in the low shear rate limit ($\dot\gamma=0.1$ s$^{-1}$). For $c<c^*$, where $c^*$ denotes the critical overlap concentration where neighboring polymers start to interact, $\eta_{\text{sp}}\propto c$, as expected for a dilute viscoelastic fluid. $c^*$ depends on molecular weight $M$ of the polymer; in the range of different polymer molecular weights we investigated in this study, $c<0.6$ wt% could generally be considered dilute. The indicated line with slope 2 is the theoretical prediction of $\eta_{\text{sp}}$ versus $c$ in the semi-dilute unentangled regime for a polymer solution in the $\theta$-solvent condition ($\nu=1/2$). (d) Two shear viscosity measurements of 1.27 MDa PIB at two different concentrations, $c=0.2<c^*$ and $c=0.8\approx c^*$. When $c\approx c^*$, polymer shear thinning starts to become non-negligible. Horizontal error bars are standard 5% deviations and the vertical error bars were measured.
• Figure 2: Zimm model curve fitting method to identify $G_0, \tau$, and $\xi$. a, i) For demonstration purposes, a measurement of the storage ($G'$) and loss moduli ($G"-\eta_\text{s}\omega$) of $0.4$ wt% of $0.4$ MDa PIB dissolved in $40$ % mineral oil mixed with $60$ % PB solution is presented. a, ii) Using Eq. (\ref{['eqn:fitting chi']}), $\xi$ can be found by looking at the region where $\Delta\log(\xi(\omega))/\Delta\log(\omega)$ is the smallest. In this example presented, $\eta_\text{s}=10.3~\text{Pa}\cdot\text{s}, G_0 =2.2 ~\text{Pa}, \tau=1.05 ~\text{s},$ and $\xi=0.30$. b) Fitted $\xi$ in relation to the design parameters, $c, M$, and $\eta_s$. $\xi$ shows weak dependence on $c$ and $\eta_s$, understandably due to changes in the fluid composition, but no appreciable dependence on $M$; the mean value is $\xi_{\text{avg}}\approx0.30$. Different symbols and colors represent different $M$ and $\eta_\text{s}$. c) From the rotational shear measurements, we determined $\psi_1$ by identifying the region where the first normal stress difference $N_1$ increases as $N_1\propto\dot\gamma^2$.
• Figure 3: Measured rheological data normalized by the fitted parameters. a) Compiled data of all sets of SAOS measurements with fitted $G_0, \tau$ and $\xi$. All data points collapse onto the Zimm model normalized by $G_0$ in the vertical axis and by $1/\tau$ in the horizontal axis. Despite the polydispersity of the sample polymers used, considering only the longest relaxation mode can well capture the trends in dynamic moduli. b) Measurement of $\psi_1$ normalized by $2\delta\xi G_0\tau^2$ versus $\dot\gamma\tau$. Values of $G_0, \tau$, and $\xi$ are obtained from SAOS measurements, and $\psi_1$ and $\delta$ are separately obtained from the rotational shear tests (see (d) panels of Figs. \ref{['fig:c-dependence']}, \ref{['fig:M-dependence']}, and \ref{['fig:eta-dependence']}). Typically when $10<\dot\gamma\tau<30$, $N_1\propto \dot\gamma^2$ ($\psi_1$ is nearly constant). Although the measurements below $\dot\gamma\tau=10$ are unreliable due to the limitations in the normal force reading resolution of the rheometer, it is expected that $N_1\propto\dot\gamma^2$ in that regime too. However, when $\dot\gamma\tau>30$, $\psi_1$ starts to decrease, following a power-law with an exponent $\approx -2/3$.
• Figure 4: Measured power-law dependence of a) $G_0$, b) $\tau$, c) $\psi_1$, and d) $\delta\xi$ on $c$, for constant $M=[0.4,1.27]$ MDa and constant $\eta_s\approx 10$ Pa$\cdot$s. Refer to Tables \ref{['tab:summary table']} and \ref{['tab:summary table exp']} for theory/experiment comparison. All data points shown in this figure were measured with solutions whose polymer concentrations were considered dilute.
• Figure 5: Measured power-law dependence of a) $G_0$, b) $\tau$, c) $\psi_1$, and d) $\delta\xi$ on $M$, for constant $c=[0.2,0.63]$ wt% and constant $\eta_s\approx 10.3$ Pa$\cdot$s. Refer to Tables \ref{['tab:summary table']} and \ref{['tab:summary table exp']} for theory/experiment comparison.