Effect of Inertia on Linear Viscoelasticity of Harmonic Dumbbell Model

Uneyama, Takashi; Nakai, Fumiaki; Masubuchi, Yuichi

doi:10.1678/rheology.47.143

Cited by 9 publications

(10 citation statements)

References 19 publications

(50 reference statements)

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“…(The superscript "(UD)" means the underdamped system.) The single dumbbell stress tensor can be decomposed into the contributions of the center of mass and the bond vector as [3]:…”

Section: Virtual Work Methodsmentioning

confidence: 99%

“…The dynamic equations ( 4) and ( 5) can be decomposed into two set of statistically independent equations. We introduce the center of mass position R(t) ≡ [R 1 (t) + R 2 (t)]/2 and the bond vector r(t) ≡ R 2 (t) − R 1 (t) [3]. We also introduce the momenta for the center of mass and the bond vector, P (t) ≡ P 1 (t) + P 2 (t) and p(t) ≡ [P 2 (t) − P 1 (t)]/2.…”

Section: Flexible Dumbbell With Tethering Potentialmentioning

confidence: 99%

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Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

Uneyama¹,

Oishi²,

Ishida³

et al. 2022

Preprint

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We derive the stress tensor of a rigid dumbbell by using the virtual work method. In the virtual work method, we virtually apply a small deformation to the system, and relate the change of the energy to the work done by the stress tensor. A rigid dumbbell consists of two particles connected by a rigid bond of which length is constant (the rigid constraint). The energy of the rigid dumbbell consists only on the kinetic energy. Also, only the deformations which do not violate the rigid constraint are allowed. Thus we need the dynamic equations which is consistent with the rigid constraint to apply the virtual deformation. We rewrite the dynamic equations for the underdamped SLLOD-type dynamic equations into the forms which are consistent with the rigid constraint. Then we apply the virtual deformation to a rigid dumbbell based on the obtained dynamic equations. We derive the stress tensor for the rigid dumbbell model from the change of the kinetic energy. Finally, we take the overdamped limit and derive the stress tensor and the dynamic equation for the overdamped rigid dumbbell. We show that the Green-Kubo type linear response formula can be reproduced by combining the stress tensor and the dynamic equation at the overdamped limit.

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“…(The superscript "(UD)" means the underdamped system.) The single dumbbell stress tensor can be decomposed into the contributions of the center of mass and the bond vector as [3]:…”

Section: Virtual Work Methodsmentioning

confidence: 99%

Section: Flexible Dumbbell With Tethering Potentialmentioning

confidence: 99%

Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

Uneyama¹,

Oishi²,

Ishida³

et al. 2022

Preprint

Self Cite

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show abstract

“…The contribution of the kinetic part stress per one particle reduces to −k B T1. 13) Thus the kinetic part stress tensor field should be defined as…”

Section: Theory 21 Microscopic Strain Tensor and Microscopic Stress T...mentioning

confidence: 99%

Linear Response Theory of Scale-Dependent Viscoelasticity for Overdamped Brownian Particle Systems

Uneyama

2022

J. Soc. Rheol., Jpn

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We show the linear response theory of spatial-scale-dependent relaxation moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood stress tensor field as the microscopic stress tensor field. We show that the scale-dependent relaxation modulus tensor, which characterizes the response of the stress tensor field to the applied velocity gradient field, can be expressed by using the correlation function of the Irving-Kirkwood stress tensor field. The spatial Fourier transform of the relaxation modulus tensor gives the wavenumber-dependent relaxation modulus. For isotropic and homogeneous systems, the relaxation modulus tensor has only two independent components. The transverse and longitudinal deformation modes give the wavenumber-dependent shear relaxation modulus and the wavenumber-dependent bulk relaxation modulus. As simple examples, we derive the explicit expressions for the relaxation moduli for two simple models the non-interacting Brownian particles and the harmonic dumbbell model.

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“…However, the memory kernel is difficult to handle with, and the memoryless Langevin equation under the Markovian approximation is widely employed as an approximate dynamic equation. (A recent work [39] showed that the generalized Langevin equation can be approximated by a Langevin equation with an effective mobility tensor, if the memory effect is weak.) Our derivation of the coarse-grained Langevin equation does not use the projection operator and simple (although the validity and accuracy of approximations are not fully justified).…”

Section: Coarse-grainingmentioning

confidence: 99%

Dissipation in Langevin Equation and Construction of Mobility Tensor from Dissipative Heat Flow

Uneyama

2019

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The rheological behavior of a material is strongly related to the energy dissipation, and the understanding and modeling of dissipation is important from the view point of rheology. To study rheological properties with some mesoscopic and macroscopic dynamics models, the modeling of dynamic equation which appropriately incorporates the dissipation is important. Although there are several methods to construct mesoscopic and macroscopic dynamic equations, such as the Onsager's method, their validity is not fully clear. In this work, we theoretically analyze the dissipation in a mesoscopic Langevin equation in detail, from the view point of stochastic energetics. We show that the dissipative heat flow from the heat bath to the system plays an important role in the mesoscopic dynamics. The dissipative heat flow is unchanged under the variable transform, and thus it is a covariant quantity. We show that we can construct the Langevin equation and also perform a coarse-graining based on the dissipative heat flow. We can derive the mobility tensor from the dissipative heat flow, and construct the Langevin equation by combining it and the free energy. Our method can be applied to various systems, such as the dumbbell model and the diffusion type equation for the density field, to give the coarse-grained dynamic equations for them.

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Effect of Inertia on Linear Viscoelasticity of Harmonic Dumbbell Model

Cited by 9 publications

References 19 publications

Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

Linear Response Theory of Scale-Dependent Viscoelasticity for Overdamped Brownian Particle Systems

Dissipation in Langevin Equation and Construction of Mobility Tensor from Dissipative Heat Flow

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