On the influence of the fourth order orientation tensor on the dynamics of the second order one

Christina Papenfuss

doi:10.1515/jnet-2023-0066

Article Open Access

On the influence of the fourth order orientation tensor on the dynamics of the second order one

Christina Papenfuss

Published/Copyright: December 18, 2023

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 49 Issue 2

Abstract

The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation.

Keywords: Second Law of Thermodynamics; 4th order orientation tensor; Folgar-Tucker equation; uniaxial orientation distribution; orientation tensor dynamics; Duffing equation

1 Introduction

Several modern materials with important applications have an internal structure with an orientational degree of freedom. Examples are fiber suspensions, where the distribution of fiber orientations determines the material properties of the resulting fiber composite or ferrofluids or liquid crystals, where the optical anisotropy used in liquid crystal displays is based on the anisotropic orientational order of the molecules. In liquid crystals as well as in fiber suspensions second order tensors are introduced as a macroscopic measure of anisotropy. In terms of the orientation distribution function f ( x , n , t) they are defined as the second moment, either the alignment tensor a used in liquid crystal theory [1, 2]

(1)

or the orientation tensor A used in the case of flowing fiber suspensions [3]

(2) A = ∫ S 2 f ( x , t , n ) n n d 2 n .

n is a unit vector indicating the long axis of the fiber or the elongated molecule, respectively. x is the position of the continuum element at time t. denotes the symmetric traceless part of a tensor. Analogously we define the higher order moments

(3)

In the sense of continuum thermodynamics the orientation tensors or the alignment tensors of successive order are internal variables.

From the definition it follows, that the orientation tensor A is a symmetric second order tensor, and the alignment tensor a is symmetric traceless. For the orientation tensor we have

(4) trace ( A ) = ∫ S 2 f ( x , t , n ) n ⋅ n d 2 n = ∫ S 2 f ( x , t , n ) d 2 n = 1 .

The two measures of anisotropy are equivalent and are related by a = A − (1/3)trace( A )1 = A − (1/3)1.

An expansion of the orientation distribution function (ODF) with respect to its variable n

(5)

is an approximation if truncated after the second order term

(6)

In the context of flowing fiber suspensions the Folgar-Tucker equation [3, 4]

(7) d A d t = 2 ∇ v ⋅ A ̄ − 2 ∇ v ̄ : A ( 4 ) + 2 D r ( 1 − 3 A ) ,

is widely applied to predict fiber orientations as they occur in the moldflow process of fiber composites, [see f.i. [5–8]]. d A d t is the material time derivative, the overline denotes the symmetric part of a tensor, D _r the orientation diffusion coefficient (a material parameter) and v the material velocity. The Folgar-Tucker equation is derived from a differential equation for the orientation distribution function, based on Jeffery’s equation for the rotation of a single fiber [9] and orientation diffusion in addition [3].

In the case that the second order orientation tensor is considered as the only internal variable, a closure relation expressing the fourth order tensor in terms of the second order one is needed. A widespread and simple closure relation is the quadratic closure

(8) A ( 4 ) = A A .

Advantages and disadvantages of different closure relations have been investigated extensively in the literature [5, 10–15].

From the differential equation for the ODF equations of motion for the orientation tensors of any order can be derived, f.i. for the fourth order orientation tensor [3]:

(9) d A ( 4 ) d t = A ( 4 ) ⋅ ∇ v + ( ∇ v ) T ⋅ A ( 4 ) − 2 ∇ v : A ( 6 ) + D r − 20 A ( 4 ) + 12 A δ ̄ ,

where a closure relation for the sixth order tensor is needed. A reasonable assumption is [16]

(10) A ( 6 ) = A A A .

Our aim here is a comparison of a model with the second order alignment or orientation tensor on one hand and a model with the second and fourth order tensors as independent variables on the other hand. The fourth order orientation tensor cannot be avoided in a theory based on an orientation distribution function. The viscosity tensor of a polymer melt or a liquid crystal depends on the fourth order tensor [17–19]. The second and fourth order orientation tensors evolve in time in the flow field and cause the back coupling of orientation and flow. Polymer melts with suspended glass or carbon fibers are the precursors of fiber composites. During the production process the flow field influences the fiber alignment. After hardening the fiber orientations are ‘frozen’ and determine the anisotropic material properties of the resulting composite. The most important property, the elasticity tensor or the stiffness tensor depend on the fourth order orientation tensor. In ref. [20] a study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffness predicted by orientation averaging mean field homogenization is given. In ref. [21] the variety of fourth-order orientation tensors is analyzed and specified by parametrizations and admissible parameter ranges.

If the fourth order orientation tensor is not modelled in the production process, it has to be approximated. Replacing the fourth order tensor by the second order one via a closure relation is an approximation. It would be favourable to calculate the fourth order tensor directly. In ref. [22] the variety of fiber orientation tensors is used to determine a maximum deviation of the direction-dependent Young’s modulus, which can arise if only second-order directional information is included in a specific meanfield homogenization. Consequences of closure approximations, i.e., restriction to second-order directional information, are demonstrated and motivate measurement of fourth-order fiber orientation tensors. These studies investigate the importance of the fourth order orientation tensor and the consequences of different approximations in the form of closure relations, but do not take into account an equation of motion for the fourth order tensor.

The present work is organized as follows: In the first part the balance equations of mass, of momentum, of angular momentum and of energy are exploited together with a differential equation for the second order orientation tensor (or alternatively, for the second and fourth order orientation tensors) together with the entropy inequality. The case of a relaxation equation (without a flux term) for the second order orientation tensor is treated as well as the case of a balance type equation with a flux of the orientation. The implications of the Second Law of Thermodynamics on constitutive functions, as well as the entropy production are derived by the method of Liu [23]. The state space, i.e. the domain of the constitutive mappings is chosen to include density, temperature, second (and fourth) order orientation tensor, temperature gradient, orientation tensor gradient and velocity gradient. The resulting entropy production may be interpreted in terms of thermodynamic fluxes and forces in the sense of Irreversible Thermodynamics.

In the second part the differential equations for the second and fourth order orientation tensors, which result from a mesoscopic background are considered in the special case of a rotation symmetric distribution function, resulting in differential equations for scalar order parameters. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. If the fourth order parameter is considered as an auxiliary variable it is eliminated from the set of equations. The resulting equation for the second order parameter is of the type of a Duffing equation.

2 Macroscopic theory

In this section we exploit the dissipation inequality with the balance equations as constraints. The case with the second order alignment tensor as the only orientation variable is discussed in detail. Because the case with the second and fourth order alignment tensors is analogous, we present the results on constitutive functions.

2.1 Balance equations

On the continuum level we have the balance of mass (mass density ϱ)

(11) d ϱ d t + ϱ ∇ ⋅ v = 0 ,

and the balance of momentum (stress tensor t , ^T denotes the transposed, specific volume force f )

(12) ϱ d v d t − ∇ ⋅ t T − ϱ f = 0 .

A rotation of fibers causes an internal angular momentum with specific spin s

(13) ϱ d s d t − ∇ ⋅ Π T − ϵ : t − ϱ p = 0 .

Π: couple stresses, p specific couple forces, ϵ Levi-Civita tensor.

The viscous stress tensor as well as the couple stresses contribute to the production of internal energy e. To the spin density there corresponds the angular velocity of particles ω _p

(14) Θ − 1 ⋅ s = ω p

with specific moment of inertia Θ of the particles.

In order to keep the balance of energy quite general, we introduce additional power terms in analogy to [24] corresponding to changes of the alignment tensor m : a ̇ and its gradient M : ( ∇ a ) ̇ , respectively:

(15) ϱ d e d t + ∇ ⋅ q − ∇ v : t − ∇ ( Θ − 1 ⋅ s ) : Π + ( Θ − 1 ⋅ s ) ⋅ ϵ : t + m : a ̇ + M : ( ∇ a ) ̇ = 0

with specific internal energy e and heat flux q .

In addition an equation of motion for the alignment tensor is postulated:

(16) d a d t + a ⋅ W − W ⋅ a = G + ∇ ⋅ H

with W = (∇ v )^antisym and alignment production G . H is a non-convective flux (transport of alignment). It vanishes in case of a pure relaxation equation.

In addition the entropy inequality expresses the Second Law of Thermodynamics

(17) ϱ d η d t + ∇ ⋅ Φ = σ ≥ 0

with specific entropy η, entropy flux Φ and entropy production σ.

2.2 Exploitation of the dissipation inequality

The state space, the domain of the constitutive functions, is chosen as a first order gradient one

(18) Z = { ϱ , T , ∇ T , ∇ v , a , ∇ a }

with temperature T.

The constitutive equations guarantee that the entropy production is positive [25]. The inequality

(19) ϱ d η d t + ∇ ⋅ Φ + Λ ϱ d ϱ d t + ϱ ∇ ⋅ v + Λ v ⋅ ϱ d v d t − ∇ ⋅ t T − ϱ f + Λ s ⋅ ϱ d s d t − ∇ ⋅ Π ⊤ − ϵ : t − ϱ p + Λ e ϱ d e d t + ∇ ⋅ q − ∇ v : t − ∇ ( Θ − 1 ⋅ s ) : Π + ( Θ − 1 ⋅ s ) ⋅ ϵ : t + m : a ̇ + M : ( ∇ a ) ̇ + Λ a : d a d t + a ⋅ W − W ⋅ a − G − ∇ ⋅ H .

with constitutive functions Λ^ϱ, Λ ^v, Λ ^s, Λ^e and Λ ^a is linear in the following higher derivatives

(20) { ϱ ̇ , T ̇ , ( ∇ T ) ̇ , v ̇ , ( ∇ v ) ̇ , s ̇ , a ̇ , ( ∇ a ) ̇ , ∇ ϱ , ∇ ∇ T , ∇ ∇ v , ∇ ∇ a } ,

which lead to the Liu equations with the following results on constitutive functions

(21) Λ v = 0 , Λ s = 0 , Λ e = − ∂ η ∂ T ∂ e ∂ T = − 1 T , Λ ϱ = ϱ T ∂ f ∂ ϱ , Λ a = ϱ T ∂ f ∂ a + 1 T m ,

where we introduced the specific free energy f = e − Tη. The Liu equations result in

(22) ∂ f ∂ ∇ T = 0 , ∂ f ∂ ∇ v = 0 , ϱ T ∂ f ∂ ∇ a = − M T

For the extra entropy flux we introduce k = Φ − q T and find that it is related to the flux of the alignment tensor

(23) ∂ k ∂ z i = ϱ T ∂ f ∂ a + 1 T m ⋅ ∂ H ∂ z i , z i ∈ { ϱ , ∇ T , ∇ v , ∇ a }

Special cases

If there is no flux of the alignment tensor, i.e. the differential equation (16) is a pure relaxation equation of an internal variable, equation (23) shows, that then k depends on temperature T and the alignment tensor a only – a scalar and a symmetric second order tensor. Consequently k  (T, a ) = 0 and the constitutive relation for the entropy flux is the classical one Φ = q T . An extra entropy flux is caused only by a flux of the alignment tensor.
In the case that there are no extra power terms in the balance of internal energy (15), i.e. m = 0, M = 0 then ∂ f ∂ y i = 0 y i ∈ { ∇ T , ∇ v , ∇ a } , i.e. the free energy is a function of the equilibrium variables and the internal variable only f (ϱ, T, a ), as it is assumed in Thermodynamics of Irreversible Processes.

Entropy production

The entropy production from the residual inequality is of the form

(24)

The alignment production G is not restricted by the Liu equations, but assuming a linear force-flux relation it may be derived from the entropy production (24). If m = 0, i.e. the classical form of the energy balance, we have

(25)

An explicit differential equation for the alignment tensor results from a constitutive assumption for the free energy.

2.3 Inclusion of the fourth order alignment tensor

A motivation to take into account the fourth order alignment tensor as an additional variable may be the elastic modulus of the resulting fiber composite or a better approximation of the orientation distribution function. A first order gradient state space is chosen:

(26) Z = { ϱ , T , ∇ T , ∇ v , a , ∇ a , a ( 4 ) , ∇ a ( 4 ) } .

The results are shown here for the case, that there are no non-convective fluxes of the second and fourth order alignment tensors and no extra power terms in the balance of energy. The dissipation inequality restricts the extra entropy flux and the specific free energy: k  (T, a , a ⁽⁴⁾) is a function of a scalar and symmetric tensors. The only vector invariant on this domain is k = 0 and consequently Φ = q T . The free energy depends on the equilibrium variables and internal variables f (ϱ, T, a , a ⁽⁴⁾). For the entropy production one finds

(27)

In linear force-flux relations the second and fourth order alignment tensors are not coupled, as the contraction over any pair of indices of a ⁽⁴⁾ is zero.

(28)

(29)

A coupling of the differential equations of the different order tensors may arise due to coupling terms in the free energy density.

3 Equations of motion for the orientation tensors from the orientation distribution function

Because the Folgar-Tucker equation has been formulated originally [4] in terms of the orientation tensor with trace 1, we will use the orientation tensors as variables, although the equations may be reformulated in terms of alignment tensors. For the second order orientation tensor we have the differential equation (7), and for the fourth order orientation tensor equation (9).

We will consider here only the case of a rotation symmetric distribution function. For symmetry reasons the orientation tensors can be expressed in terms of scalar order parameters S ⁽²⁾, S ⁽⁴⁾, S ⁽⁶⁾, … and a unit vector d pointing in the direction of the symmetry axis:

(30) A = S ( 2 ) d d , A ( 4 ) = S ( 4 ) d d d d , A ( 6 ) = S ( 6 ) d d d d d d

3.1 The second order orientation tensor as variable

In the Folgar-Tucker equation (7) together with the quadratic closure A ⁽⁴⁾ = AA the uniaxial form of the orientation tensors A = S ⁽²⁾ dd , A ⁽⁴⁾ = S ⁽⁴⁾ dddd , with the closure relation S ⁽⁴⁾ = S ⁽²⁾² is inserted. Taking the double scalar product with dd ( d ⋅ d ̇ = 0 ) we end up with the following differential equation for the second order scalar order parameter

(31) S ( 2 ) ̇ = S ( 2 ) 2 ∇ v : d d − 6 D r − 2 ∇ v : d d S ( 2 ) 2 + 2 D r .

This differential equation is a Riccati-equation. In the case that ∇ v : dd is constant, f.i. in a stationary flow with symmetry axis of the orientation distribution prescribed by the symmetry of the flow geometry, equation (31) has the analytical solution

(32) S ( 2 ) ( t ) = − C 1 ( y + x ) e x t + C 2 ( y − x ) e − x t 2 ∇ v : d d C 1 e x t + C 2 e − x t

with the abbreviations

(33) x = ( ∇ v : d d ) 2 − 2 ∇ v : d d D r + 9 D r 2 = ( ∇ v : d d − D r ) 2 + 8 D r 2 > 0 ,

(34) y = ∇ v : d d − 3 D r

and

(35) C 1 = 1 2 x x − y − 2 S ( 2 ) ( 0 ) ∇ v : d d , C 2 = 1 − C 1 = 1 2 x x + y + 2 S ( 2 ) ( 0 ) ∇ v : d d

The solution (32) is composed of real exponential functions. It has a constant long term limit

(36) lim t → ∞ S ( 2 ) ( t ) = 1 2 ∇ v : d d ( y + x ) = 1 2 ∇ v : d d ∇ v : d d − 3 D r + ( ∇ v : d d ) 2 − 2 ∇ v : d d D r + 9 D r 2

For very large values of ∇ v : dd this long term solution tends to 1, which means perfect alignment in the direction d .

3.2 The second order and fourth order orientation tensors as variables

Equations (7) and (9) are the equations of motion for the two internal variables. In the uniaxial case differential equations for the scalar order parameters S ⁽²⁾ and S ⁽⁴⁾ are derived by scalar multiplication with dd and dddd , respectively

(37) S ( 2 ) ̇ = S ( 2 ) 2 ∇ v : d d − 6 D r − 2 ∇ v : d d S ( 4 ) + 2 D r

(38) S ( 4 ) ̇ = S ( 4 ) 2 ∇ v : d d − 20 D r + 12 D r S ( 2 ) − 2 S ( 6 ) ∇ v : d d

where S ⁽⁶⁾ is eliminated by a closure relation S ⁽⁶⁾ = S ⁽²⁾³ (which is the lowest order series expansion of the maximum entropy closure [16]).

If now finally one is not interested in the fourth order parameter, it is possible to eliminate S ⁽⁴⁾ from equations (37) and (38). In the case that ∇ v : dd is constant (see also Section 3.1) we solve equation (37) for S ⁽⁴⁾, insert S ⁽⁴⁾ and S ( 4 ) ̇ into equation (38) and end up with a second order differential equation for the second order parameter

(39) S ̈ ( 2 ) = S ̇ ( 2 ) 4 ∇ v : d d − 26 D r + S ( 2 ) − 4 ( ∇ v : d d ) 2 + 28 D r ∇ v : d d − 120 D r 2 + 4 ( ∇ v : d d ) 2 S ( 2 ) 3 − 4 D r ∇ v : d d + 40 D r 2

It is of the form of the differential equation of the Duffing oscillator with a constant driving force. Analytical solutions of this equation exist only in the ‘undamped’ case, i.e. if the coefficient of S ̇ ( 2 ) is zero, 4∇ v : dd − 26D _r = 0. Analytical approximations and a comparison to numerical solutions is given for example in ref. [26]. If 4∇ v : dd − 26D _r is not too large, an oscillating behavior of the solution is observed in contrast to the solution of the Riccati equation (32). Taking into account the second and fourth order orientation tensors as independent variables and eliminating the fourth order tensor later qualitatively changes the behaviour of the solution compared to the case of the second order alignment tensor as the only variable.

4 Conclusions

The exploitation of the Second Law of Thermodynamics has shown, that a non-convective flux of the alignment tensor may result in an extra entropy flux. If there is no flux of the alignment tensor, the entropy flux equals heat flux divided by temperature. If the balance of internal energy is assumed in the classical form, the free energy depends on the equilibrium variables density and temperature and on the internal variable alignment tensor only. With extra power terms related to the time derivatives of the alignment tensor and its gradient the restrictions from the dissipation inequality are weaker, and the free energy may depend on gradients.
The production of the alignment tensor is not restricted by the Liu equations.
The entropy production from the residual inequality has the classical contributions due to heat conduction, shear and volume viscosity, an additional contribution due to rotation of the elongated particles relatively to the surrounding fluid, rotation viscosity and production of the alignment tensor. An exploitation of this entropy production in the sense of Irreversible Thermodynamics results in a differential equation for the alignment tensor. Production of alignment is caused by the flow field and by the derivative of the free energy with respect to the alignment tensor.
The inclusion of the fourth order alignment tensor and its gradient in the state space leads to analogous restrictions on constitutive functions. In the entropy production there arises an additional term due to production of the fourth order alignment tensor. Linear relations between thermodynamic forces and fluxes lead to differential equations for the fourth and second order alignment tensors. There is no coupling between the different tensor orders, but an indirect coupling of the equations of motion for the second and fourth order alignment tensors may result from coupling terms in the free energy.
In the second part explicit differential equations for the second and fourth order orientation tensor, resulting from a differential equation for the orientation distribution function, have been investigated in the special case of a rotation symmetric distribution function. Firstly the second order orientation tensor is considered as the only internal variable. Its differential equation is supplemented with a quadratic closure relation, leading to a Riccati type differential equation for the scalar order parameter. The analytical solution is not oscillating in time.

Secondly the differential equations for the second and fourth order orientation tensors are considered together with a closure relation for the sixth order orientation tensor in the uniaxial case. They form a set of two first order non-linear differential equations for the scalar order parameters. In the case of constant coefficients the fourth order parameter may be eliminated from the equations. The resulting differential equation for the second order parameter is a second order in time non-linear differential equation of the form of the Duffing-equation. It cannot be solved analytically, but an oscillating behavior of numerical solutions is known.

A comparison of these two examples shows that including the fourth order orientation tensor as an independent variable and eliminating it later may have considerable influence on the dynamics of the second order orientation tensor. The differential equation for the orientation distribution function as well as the system of coupled differential equations for the orientation tensors of successive order are of first order in time. Eliminating the fourth order parameter, we ended up with a differential equation for the second order parameter, which is of second order in time.
Analogously the sixth order parameter with its differential equation could be taken into account and eliminated later together with the fourth order parameter in the case of a rotation symmetric orientation distribution. This will be left for future research as well as the treatment of distribution functions with lower symmetry, for example planar orientation distributions.

Corresponding author: Christina Papenfuss, Faculty 2, School of Engineering – Technology and Life, Hochschule für Technik und Wirtschaft Berlin, University of Applied Sciences, Berlin, Germany, E-mail: Christina.Papenfuss@htw-berlin.de.

Funding source: HTW Berlin, Forschungsfonds

Research ethics: No studies on human beings or animals have been performed.
Author contributions: The manuscript was written by Christina Papenfuss.
Competing interests: The author has no competing interests to declare.
Research funding: Open Access publication was enabled by funding of the HTW Berlin (Forschungsfonds).
Data availability: There are no data available.

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Received: 2023-08-16

Accepted: 2023-11-29

Published Online: 2023-12-18

Published in Print: 2024-04-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/jnet-2023-0066

Keywords for this article

Second Law of Thermodynamics; 4th order orientation tensor; Folgar-Tucker equation; uniaxial orientation distribution; orientation tensor dynamics; Duffing equation

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BY 4.0