Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

Baumgaertel, M.; Winter, H. Henning

doi:10.1007/bf01332922

Cited by 510 publications

(271 citation statements)

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“…The dynamic moduli (G 0 and G 00 ) and the spectra of relaxation times τ i were determined from unsteady rheometry experiments, and the fitting parameters g i were obtained by fitting the Oldroyd-B equations Eq. 1 and 2 following the method used by Baumgaertel and Winter (17). Four well-defined discrete relaxation times were found from the rheometry spectra (τ i ¼ 0.005;0.13;0.50, and 2.13 sec.)…”

Section: Resultsmentioning

confidence: 99%

Force-free swimming of a model helical flagellum in viscoelastic fluids

Liu

Powers

Breuer

2011

Proc. Natl. Acad. Sci. U.S.A.

191

167

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We precisely measure the force-free swimming speed of a rotating helix in viscous and viscoelastic fluids. The fluids are highly viscous to replicate the low Reynolds number environment of microorganisms. The helix, a macroscopic scale model for the bacterial flagellar filament, is rigid and rotated at a constant rate while simultaneously translated along its axis. By adjusting the translation speed to make the net hydrodynamic force vanish, we measure the forcefree swimming speed as a function of helix rotation rate, helix geometry, and fluid properties. We compare our measurements of the force-free swimming speed of a helix in a high-molecular weight silicone oil with predictions for the swimming speed in a Newtonian fluid, calculated using slender-body theories and a boundary-element method. The excellent agreement between theory and experiment in the Newtonian case verifies the high accuracy of our experiments. For the viscoelastic fluid, we use a polymer solution of polyisobutylene dissolved in polybutene. This solution is a Boger fluid, a viscoselastic fluid with a shear-rateindependent viscosity. The elasticity is dominated by a single relaxation time. When the relaxation time is short compared to the rotation period, the viscoelastic swimming speed is close to the viscous swimming speed. As the relaxation time increases, the viscoelastic swimming speed increases relative to the viscous speed, reaching a peak when the relaxation time is comparable to the rotation period. As the relaxation time is further increased, the viscoelastic swimming speed decreases and eventually falls below the viscous swimming speed. motility | propulsion | rheology S mall motile organisms often swim in complex fluids. Mammalian spermatozoa beat their flagella to move through cervical fluid (1). The Lyme disease spirochete Borrelia burgdorferi flexes and rotates its body to move through the extracellular matrix of our skin (2). The nematode Caenorhabditis elegans undulates its body to move through soil saturated with water (3). While there is an extensive framework for understanding the mechanics of swimming of small organisms in purely viscous Newtonian liquids such as water, our understanding of the basic principles of swimming in non-Newtonian fluids is still in its infancy (4). The behavior of complex fluids is varied, and there are many possible nonNewtonian effects a swimmer could encounter, including elastic response of the fluid, shear-dependent viscosity, adhesion to suspended particles or fibers, or the permeability of a porous medium. In this article we focus on swimming in an elastic liquid, and our goal is to determine how the speed of a model bacterial swimmer is changed by elastic effects.It is known that helically shaped bacteria such as Leptospira or B. burgdorferi swim more rapidly in solutions with methylcellulose than in nonviscoelastic solutions of the same viscosity (2, 5). On the other hand, C. elegans, which moves using planar undulations of its body, swims more slowly in a viscoelastic fluid than in a vi...

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Section: Resultsmentioning

confidence: 99%

Force-free swimming of a model helical flagellum in viscoelastic fluids

Liu

Powers

Breuer

2011

Proc. Natl. Acad. Sci. U.S.A.

191

167

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“…The modeling of the G′, G′′ data consists of four major steps: (1) calculation of the relaxation spectra of networks at different extents of reactions (using the method of Baumgä rtel and Winter 28 ), (2) determination of characteristic relaxation patterns, (3) abstraction into a simple model and evaluation of model parameters, and (4) application of this model to predict the transient relaxation behavior during the gelation process and certain other rheological observations at the gel point, thus checking the model's self-consistency. The result should be, if the modeling is successful, a unified and self-consistent picture of the gelation process.…”

Section: Relaxation Patterns and Model Spectrummentioning

confidence: 99%

Relaxation Patterns of Nearly Critical Gels

Mours¹,

Winter²

1996

Macromolecules

142

116

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We measure the complex rheological behavior of nearly critical gels and analyze the data by searching for characteristic patterns and abstracting those patterns into a self-consistent model. The sample is a linear, flexible, nearly monodisperse polybutadiene which gets cross-linked on its vinyl side groups. The dynamic mechanical storage and loss moduli of cross-linking polymers change smoothly during the liquid-solid transition, while equilibrium rheological properties (e.g., zero-shear viscosity and equilibrium compliance) diverge. During gelation, the relaxation occurs in a distinct pattern which can be described in a quantitative way with a minimum number of parameters. The pattern can be understood as a combination of the BSW spectrum (representing the precursor relaxation behavior) and the selfsimilar Chambon-Winter gel spectrum (modeling the terminal relaxation due to growing clusters). The spectrum is cut off at the material's longest relaxation time, λmax. Our model parameters, λmax and Ge (equilibrium modulus), exhibit characteristic scaling behavior with respect to the distance from the gel point, |p -pc|. The relaxation exponent, n, in the terminal zone is a function of the extent of reaction. Hence, dynamic scaling (requires constant n values) is not valid for our system. The proposed model passes the self-consistency test by predicting the mechanical behavior (at different frequencies) as a function of the extent of reaction and other rheological observations during the sol-gel transition.

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“…The relaxation spectrum may be approximated from the storage and loss moduli though various methods, including continuous relaxation spectrum calculation [Ferry and Williams, 1952, Ferry, 1980, Schwarzl and Staverman, 1952 or numerical calculation for discrete relaxation spectrum [Baumgaertel and Winter, 1989, Winter, 1997, Stadler and Bailly, 2009. The former method includes emperical assumptions and the latter involves relative complex programming to tackle the ill-posed problem of direct fitting from the frequency domain.…”

Section: Verification By Ultrasonic Wave Testmentioning

confidence: 99%

Experimentally-based relaxation modulus of polyurea and its composites

Jia

Amirkhizi

Nantasetphong

et al. 2016

Mech Time-Depend Mater

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Polyurea is a block copolymer that has been widely used in the coating industry as an abrasion-resistant and energy-dissipative material. Its mechanical properties can be tuned by choosing different variations of diamines and diisocyanates as well as by adding various nano and micro-inclusions to create polyurea-based composites. Our aim here is to provide the necessary experimentally-based viscoelastic constitutive relations for polyurea and its composites in a format convenient to support computational studies. The polyurea used in this research is synthesized by the reaction of Versalink P-1000 (Air Products) and Isonate 143L (Dow Chemicals). Samples of pure polyurea and polyurea composites are fabricated and then characterized using dynamic mechanical analysis (DMA). Based on the DMA data, master curves of storage and loss moduli are developed using time-temperature superposition. The quality of the master curves is carefully assessed by comparing with the ultrasonic wave measurements and by Kramers-Kronig relations. Based on the master curves, continuous relaxation spectra are calculated, then the time-domain relaxation moduli are approximated from the relaxation spectra. Prony series of desired number of terms for the frequency ranges of interest are extracted from the relaxation modulus. This method for developing cost efficient Prony series has been proven to be effective and efficient for numerous DMA test results of many polyurea/polyurea-based material systems, including pure polyurea with various stoichiometric ratios, polyurea with milled glass inclusions, polyurea with hybrid-nano particles and polyurea with phenolic microbubbles. The resulting viscoelastic models are customized for the frequency ranges of interest, reference temperature and desired number of Prony terms, achieving both computational accuracy and low cost. The method is not limited to polyurea-based systems. It can be applied to other similar polymers systems.

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Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

Cited by 510 publications

References 16 publications

Force-free swimming of a model helical flagellum in viscoelastic fluids

Force-free swimming of a model helical flagellum in viscoelastic fluids

Relaxation Patterns of Nearly Critical Gels

Experimentally-based relaxation modulus of polyurea and its composites

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