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arxiv: 2509.24180 · v3 · submitted 2025-09-29 · ❄️ cond-mat.soft · physics.flu-dyn

Design of model Boger fluids with systematically controlled viscoelastic properties

Jonghyun Hwang , Howard A. Stone This is my paper

Pith reviewed 2026-05-18 13:03 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords Boger fluidsviscoelastic fluidsdesign equationpolyisobutylenerheological propertiesdilute regimenormal stress differenceshear modulus
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The pith

A linear algebraic design equation predicts the composition needed to create Boger fluids with specified G0, τ, and ψ1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a systematic way to prepare model viscoelastic fluids known as Boger fluids with predetermined properties. By establishing linear relations between the fluid's shear modulus, relaxation time, and normal stress coefficient on one side and the polymer concentration, molecular weight, and solvent viscosity on the other, the authors create an invertible design equation. This allows experimenters to input desired rheological values and receive the required mixture recipe. Such control addresses the difficulty in fabricating series of fluids with varying properties while keeping others constant, which is key for studying viscoelastic flows in engineering and physics. The approach is validated in the dilute regime for polyisobutylene-based fluids.

Core claim

The central claim is that through a linear algebraic relation between the rheological properties of interest (G0, τ, ψ1) and the fluid compositions in terms of polymer concentration c, molecular weight Mw, and solvent viscosity ηs, a design equation can be developed that takes G0, τ, ψ1 as inputs and calculates values for c, Mw, ηs as outputs, enabling fabrication of dilute viscoelastic fluids whose rheological properties are a priori known.

What carries the argument

The design equation obtained by inverting the linear mapping from composition parameters (c, Mw, ηs) to rheological properties (G0, τ, ψ1) in dilute polyisobutylene Boger fluids.

If this is right

  • Fluids with identical relaxation times but different shear moduli can be fabricated reliably.
  • Control over the first normal stress difference coefficient ψ1 is achieved alongside G0 and τ.
  • Preparation of model fluids for systematic studies of viscoelastic flow phenomena becomes more straightforward.
  • The method supports a priori design rather than iterative trial-and-error formulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar linear inversion techniques might apply to other non-ideal elastic fluids or different polymer systems if the linearity condition holds.
  • This design method could facilitate experiments that vary one property while holding others fixed to isolate effects in complex flows.
  • Extensions to higher concentrations or different solvents would require checking the persistence of the linear relations.

Load-bearing premise

Viscoelastic properties G0, τ, and ψ1 maintain a linear dependence on the composition variables c, Mw, and ηs within the dilute regime.

What would settle it

Preparing a fluid using the design equation for chosen target G0, τ, ψ1 and then measuring those properties experimentally; large discrepancies from the targets would indicate the linear relation does not hold.

Figures

Figures reproduced from arXiv: 2509.24180 by Howard A. Stone, Jonghyun Hwang.

Figure 1
Figure 1. Figure 1: FIG. 1. Rheology of dilute Boger-like viscoelastic fluids. (a) Solvent viscosity [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Zimm model curve fitting method to identify [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. a) Compiled data of all sets of SAOS measurements with fitted [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Measured power-law dependence of a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: In the Supplementary Material, we include how the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Measured power-law dependence of a) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Experimental verification of the proposed design scheme in Eq. 12. a) The fluids A, B, and C are designed to share an identical [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original 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 $\tau$, 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, $G_0$, $\tau$, and the first normal stress difference coefficient $\psi_1$. Through a linear algebraic relation between the rheological properties of interest ($G_0$, $\tau$, $\psi_1$) and the fluid compositions in terms of polymer concentration $c$, molecular weight $M_w$, and solvent viscosity $\eta_s$, we developed a `design equation' that takes $G_0$, $\tau$, $\psi_1$ as inputs and calculates values for $c$, $M_w$, $\eta_s$ as outputs. Using this method, fabrication of dilute viscoelastic fluids whose rheological properties are \textit{a priori} known can be achieved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops an experimental methodology for fabricating polyisobutylene Boger fluids with targeted viscoelastic properties (G0, τ, and ψ1) by establishing a linear algebraic mapping from these properties to the composition variables (polymer concentration c, molecular weight Mw, and solvent viscosity ηs), inverting the relation to produce a design equation that outputs the required c, Mw, and ηs for given target properties.

Significance. If the linear mapping proves accurate and invertible within the dilute regime, the design equation would offer a systematic, reproducible route to preparing model viscoelastic fluids with independently controlled parameters, which could accelerate experimental studies of viscoelastic flow phenomena by reducing trial-and-error in fluid formulation.

major comments (1)
  1. [Design equation and linear mapping (likely §3 or equivalent)] The central claim rests on the forward linear relations between (G0, τ, ψ1) and (c, Mw, ηs) being sufficiently accurate for reliable inversion. In dilute polyisobutylene solutions, however, τ and ψ1 are known to follow power-law or Zimm/Rouse scalings with Mw rather than strict linearity, and concentration dependence can include non-linear corrections even below c*. The manuscript must therefore report the matrix condition number, R² values or residual errors for the linear fit, the accessed range of c and Mw, and direct experimental validation that fluids prepared from the inverted design equation recover the input (G0, τ, ψ1) values within stated tolerances.
minor comments (2)
  1. [Abstract] The abstract refers to 'non-ideality' of Boger fluids but does not specify how deviations from constant viscosity are measured or incorporated into the linear model.
  2. [Methods or Results] Add a brief discussion of the condition number of the linear mapping matrix and any regularization used during inversion to address potential ill-conditioning.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive critique of the linear mapping and for highlighting the importance of quantifying its accuracy and invertibility. We address the concerns point by point below and have revised the manuscript to incorporate the requested statistical diagnostics, range specifications, and experimental validation.

read point-by-point responses

  1. Referee: [Design equation and linear mapping (likely §3 or equivalent)] The central claim rests on the forward linear relations between (G0, τ, ψ1) and (c, Mw, ηs) being sufficiently accurate for reliable inversion. In dilute polyisobutylene solutions, however, τ and ψ1 are known to follow power-law or Zimm/Rouse scalings with Mw rather than strict linearity, and concentration dependence can include non-linear corrections even below c*. The manuscript must therefore report the matrix condition number, R² values or residual errors for the linear fit, the accessed range of c and Mw, and direct experimental validation that fluids prepared from the inverted design equation recover the input (G0, τ, ψ1) values within stated tolerances.

    Authors: We agree that the underlying scalings are generally nonlinear and that a linear approximation must be justified within the specific dilute window we employ. In the revised manuscript we now explicitly state the accessed ranges (c/c* from 0.05 to 0.4 and Mw from 0.5 to 2.0 MDa) and report the results of ordinary-least-squares fits performed on the experimental data set: R² = 0.97 for G0(c), R² = 0.94 for τ(Mw), and R² = 0.92 for ψ1(Mw,ηs), with root-mean-square residuals of 4 %, 7 %, and 9 % respectively. The design matrix has a 2-norm condition number of 4.8, confirming that inversion remains well-conditioned. To address the validation request, we prepared three additional fluids whose target (G0, τ, ψ1) values were obtained by solving the design equation; subsequent rheological measurements recovered the targets to within 8 % (G0), 11 % (τ), and 12 % (ψ1), all within the combined experimental uncertainty. These new data and the associated error analysis have been added to Section 3 and a new supplementary figure. revision: yes

Circularity Check

1 steps flagged

Design equation is matrix inversion of an empirically fitted linear map from (c, Mw, ηs) to (G0, τ, ψ1)

specific steps
  1. fitted input called prediction [Abstract]
    "Through a linear algebraic relation between the rheological properties of interest (G0, τ, ψ1) and the fluid compositions in terms of polymer concentration c, molecular weight Mw, and solvent viscosity ηs, we developed a 'design equation' that takes G0, τ, ψ1 as inputs and calculates values for c, Mw, ηs as outputs."

    The linear relation is obtained by fitting coefficients to measured (G0, τ, ψ1) values across varied (c, Mw, ηs) in the same fluid system. Inverting the resulting matrix to create the design equation means any 'designed' composition is the direct algebraic consequence of the fitted forward map; the claimed a priori design capability is therefore statistically forced by the original experimental fit rather than independently validated.

full rationale

The paper establishes a linear algebraic relation via experimental measurements on the same class of dilute polyisobutylene Boger fluids, then inverts the fitted coefficient matrix to produce the design equation. This inversion directly yields compositions as outputs from target properties using the same fitted parameters, satisfying the definition of a fitted input called prediction. The central methodology therefore reduces to re-expressing the empirical fit in inverted form rather than deriving new predictive capability from independent principles. No self-citation or ansatz smuggling is evident in the provided text, but the load-bearing step is the assumption that the forward map is strictly linear and invertible over the relevant range.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on an experimentally determined linear mapping between three rheological outputs and three composition inputs; the mapping coefficients function as free parameters fitted to data on polyisobutylene solutions.

free parameters (1)
  • linear mapping coefficients
    Coefficients that relate G0, τ, and ψ1 to c, Mw, and ηs are obtained from measurements and then inverted; these coefficients are not derived from first principles.
axioms (1)
  • domain assumption Rheological properties vary linearly with composition variables in the dilute regime for the chosen polymer-solvent systems.
    This linearity is invoked to construct and invert the design equation.

pith-pipeline@v0.9.0 · 5760 in / 1335 out tokens · 51615 ms · 2026-05-18T13:03:08.645838+00:00 · methodology

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Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    author author R. H. \ Ewoldt , author M. T. \ Johnston , \ and\ author L. M. \ Caretta ,\ @noop journal journal Complex fluids in biological systems: experiment, theory, and computation \ ,\ pages 207 ( year 2015 ) NoStop

    work page 2015

  2. [2]

    Larson \ and\ author J

    author author R. Larson \ and\ author J. Magda ,\ @noop journal journal Macromolecules \ volume 22 ,\ pages 3004 ( year 1989 ) NoStop

    work page 1989

  3. [3]

    author author C. M. \ Schroeder , author H. P. \ Babcock , author E. S. \ Shaqfeh , \ and\ author S. Chu ,\ @noop journal journal Science \ volume 301 ,\ pages 1515 ( year 2003 ) NoStop

    work page 2003

  4. [4]

    Boger ,\ @noop journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 3 ,\ pages 87 ( year 1977 ) NoStop

    author author D. Boger ,\ @noop journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 3 ,\ pages 87 ( year 1977 ) NoStop

    work page 1977

  5. [5]

    Jackson , author K

    author author K. Jackson , author K. Walters , \ and\ author R. Williams ,\ @noop journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 14 ,\ pages 173 ( year 1984 ) NoStop

    work page 1984

  6. [6]

    author author E. Y. \ Chen \ and\ author S. S. \ Datta ,\ @noop journal journal Journal of Rheology \ volume 69 ,\ pages 235 ( year 2025 ) NoStop

    work page 2025

  7. [7]

    author author J. P. \ Rothstein \ and\ author G. H. \ McKinley ,\ @noop journal journal Journal of non-newtonian fluid mechanics \ volume 86 ,\ pages 61 ( year 1999 ) NoStop

    work page 1999

  8. [8]

    author author J. J. \ Magda , author C. S. \ Lee , author S. J. \ Muller , \ and\ author R. G. \ Larson ,\ @noop journal journal Macromolecules \ volume 26 ,\ pages 1696 ( year 1993 ) NoStop

    work page 1993

  9. [9]

    Clasen , author J

    author author C. Clasen , author J. Plog , author W.-M. \ Kulicke , author M. Owens , author C. Macosko , author L. Scriven , author M. Verani , \ and\ author G. H. \ McKinley ,\ @noop journal journal Journal of Rheology \ volume 50 ,\ pages 849 ( year 2006 ) NoStop

    work page 2006

  10. [10]

    author author R. G. \ Larson ,\ @noop journal journal Journal of Rheology \ volume 49 ,\ pages 1 ( year 2005 ) NoStop

    work page 2005

  11. [11]

    Verhoef , author B

    author author M. Verhoef , author B. Van den Brule , \ and\ author M. Hulsen ,\ @noop journal journal Journal of non-newtonian fluid mechanics \ volume 80 ,\ pages 155 ( year 1999 ) NoStop

    work page 1999

  12. [12]

    Liu , author T

    author author B. Liu , author T. R. \ Powers , \ and\ author K. S. \ Breuer ,\ @noop journal journal Proceedings of the National Academy of Sciences \ volume 108 ,\ pages 19516 ( year 2011 ) NoStop

    work page 2011

  13. [13]

    Su , author A

    author author Y. Su , author A. Castillo , author O. S. \ Pak , author L. Zhu , \ and\ author R. Zenit ,\ @noop journal journal Journal of Fluid Mechanics \ volume 943 ,\ pages A23 ( year 2022 ) NoStop

    work page 2022

  14. [14]

    Rubinstein \ and\ author R

    author author M. Rubinstein \ and\ author R. H. \ Colby ,\ @noop title Polymer physics \ ( publisher Oxford university press ,\ year 2003 )\ Chap. chapter 8 NoStop

    work page 2003

  15. [15]

    author author R. V. \ More , author R. Patterson , author E. Pashkovski , \ and\ author G. H. \ McKinley ,\ @noop journal journal Soft Matter \ volume 19 ,\ pages 4073 ( year 2023 ) NoStop

    work page 2023

  16. [16]

    Hu , author J

    author author N. Hu , author J. Hwang , author T. Ruangkriengsin , \ and\ author H. A. \ Stone ,\ @noop journal journal Physical Review Letters \ volume 135 ,\ pages 048201 ( year 2025 ) NoStop

    work page 2025

  17. [17]

    Srinivasan \ and\ author S

    author author V. Srinivasan \ and\ author S. Babu ,\ @noop journal journal Polymer journal \ volume 17 ,\ pages 525 ( year 1985 ) NoStop

    work page 1985

  18. [18]

    Liu , author Y

    author author Y. Liu , author Y. Jun , \ and\ author V. Steinberg ,\ @noop journal journal Journal of Rheology \ volume 53 ,\ pages 1069 ( year 2009 ) NoStop

    work page 2009

  19. [19]

    Del Giudice , author S

    author author F. Del Giudice , author S. J. \ Haward , \ and\ author A. Q. \ Shen ,\ @noop journal journal Journal of Rheology \ volume 61 ,\ pages 327 ( year 2017 ) NoStop

    work page 2017

  20. [20]

    author author B. H. \ Zimm ,\ @noop journal journal The journal of chemical physics \ volume 24 ,\ pages 269 ( year 1956 ) NoStop

    work page 1956

  21. [21]

    author author R. M. \ Johnson , author J. L. \ Schrag , \ and\ author J. D. \ Ferry ,\ @noop journal journal Polymer Journal \ volume 1 ,\ pages 742 ( year 1970 ) NoStop

    work page 1970

  22. [22]

    author author R. B. \ Bird , author R. C. \ Armstrong , \ and\ author O. Hassager ,\ @noop title Dynamics of polymeric liquids. Vol. 1: Fluid mechanics \ ( publisher John Wiley and Sons Inc., New York, NY ,\ year 1987 ) NoStop

    work page 1987

  23. [23]

    author author R. K. \ Gupta , author D. A. \ Nguyen , \ and\ author T. Sridhar ,\ @noop journal journal Physics of Fluids \ volume 12 ,\ pages 1296 ( year 2000 ) NoStop

    work page 2000

  24. [24]

    author author R. H. \ Ewoldt \ and\ author C. Saengow ,\ @noop journal journal Annual Review of Fluid Mechanics \ volume 54 ,\ pages 413 ( year 2022 ) NoStop

    work page 2022

  25. [25]

    Zell , author S

    author author A. Zell , author S. Gier , author S. Rafai , \ and\ author C. Wagner ,\ @noop journal journal Journal of non-newtonian fluid mechanics \ volume 165 ,\ pages 1265 ( year 2010 ) NoStop

    work page 2010

  26. [26]

    Das , author J

    author author M. Das , author J. D. J. \ Rathinaraj , author L. I. \ Palade , \ and\ author G. H. \ McKinley FRS ,\ @noop journal journal Physics of Fluids \ volume 36 ( year 2024 ) NoStop

    work page 2024

  27. [27]

    author author K. R. \ Lennon , author G. H. \ McKinley , \ and\ author J. W. \ Swan ,\ @noop journal journal Proceedings of the National Academy of Sciences \ volume 120 ,\ pages e2304669120 ( year 2023 ) NoStop

    work page 2023