On a thermodynamic foundation of Eyring rate theory for plastic deformation of polymer solids

ABSTRACT

The plastic deformation of almost all solid polymers can be represented by a thermally activated rate process involving the motion of cooperative mobile elements over potential barriers, based on the Eyring activated rate theory. The present study shows that the governing equations for the Eyring rate theory can be completely derived based on the principle of microscopic reversibility in the framework of non-equilibrium dynamics.

KEYWORDS:

• Fluctuation theorem
• non-equilibrium thermodynamics
• continuum mechanics
• polymer
• Eyring rate theory

1. Introduction

Glassy and crystalline polymer solids are naturally viscoelastic, and their mechanical properties depend largely on the temperature and strain rate [1]. Linear viscoelastic behaviours appear at relatively lower strains and can be described through combinations of the Hookean spring and Newtonian dashpot [2]. Moreover, plastic deformations such as molecular slippage, dislocation movements, and local structural damage occur irreversibly at higher strains [1, 3]. The most generic theory used to describe plastic deformation in polymers was proposed by Eyring et al. [4–6]. The basic concept is that plastic flow is activated via the stress applied under thermal fluctuation in accordance with the kinetic theory of chemical reactions. In particular, the Eyring rate theory has been successfully used to account for tensile yielding and necking behaviours in almost all polymer solids, including amorphous and crystalline polymers [1, 3, 7, 8].

These nonlinear mechanical responses of polymers, which include elasto-viscoplasticity, are undoubtedly relevant to irreversible non-equilibrium processes under tension. Coleman, Noll, and Truesdell [9, 10] established an axiomatization of nonlinear continuum mechanics in the framework of rational thermodynamics, presenting constitutive equations for elasto-viscoplasticity. They incorporated the second law of thermodynamics into continuum mechanics with memory as a restriction on the constitutive relations. The Clausius–Duhem inequality is derived as a non-negative rate of entropy production resulting from irreversible deformation. Notions of equilibrium thermostatics, such as temperature and entropy, which cannot be defined in the non-equilibrium state, are used as a priori parameters. Alternatively, the introduction of internal state variables that reflect internal structural rearrangements allows various dissipative processes, such as viscoelastic and/or plastic deformations, to be represented using the temperature and entropy employed in the corresponding local accompanying equilibrium state [11–14].

The fluctuation theorem, based on the principle of microscopic reversibility, has received considerable attention in recent years to describe non-equilibrium thermodynamics that are far from equilibrium, wherein the internal entropy production rate is not governed through deterministic functions [15–18]. The replacement of the Clausius–Duhem inequality with the fluctuation theorem allows for the expression of real non-equilibrium states, in which violations of the second law of thermodynamics occur spontaneously and instantaneously on micro- and meso-scopic scales. Adopting the aforementioned replacement, Ostoja-Starzewski et al. [19, 20] extended the rational continuum axioms of Coleman, Noll, and Truesdell to represent actual irreversible plastic deformation behaviours.

The objective of the present work is to derive the Eyring rate equation from a thermodynamic point of view and to explore the non-equilibrium thermodynamic origin of plastic deformation in polymer solids. In particular, the thermodynamic approach of the Eyring rate theory is essential for understanding the nonlinear mechanical properties of almost all polymer materials.

2. Outline of Eyring rate theory [4, 5]

In the Eyring rate theory, the plastic flow process is treated as the movement of a cooperative mobile element, which is referred to as a flow unit, in the stretching direction over a free-energy barrier. The model is based on chemical kinetics and describes the changes in the reaction rate with temperature. In the absence of stress, the detailed balance requires that an equal number of flow units move forward and backward over the potential barrier at a frequency ω given by: (1) $\omega ={\omega }_0{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}$(1) where $\mathrm{\Delta }{F}^{†}$ denotes the free-energy barrier, β indicates the inverse temperature (Boltzmann’s constant is assumed to be unity for convenience), and ${\omega }_0$ represents the universal frequency, $1/\text{β}\hslash$ ($\hslash$ is the Dirac’s constant).

The application of an external traction ${\sigma }_0$ raises the energy level of the original site of a flow unit and lowers the energy level of its adjacent empty site, as illustrated in Figure 1. The forward and backward flows along the direction of ${\sigma }_0$ are expressed respectively as (2) $\{\begin{array}{c}{\omega }_{+}={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\text{β}\left(\mathrm{\Delta }{F}^{†}-{\upsilon }^{†}{\sigma }_0/2\right)\right]\\ \\ {\omega }_{-}={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\text{β}\left(\mathrm{\Delta }{F}^{†}+{\upsilon }^{†}{\sigma }_0/2\right)\right]\end{array}$(2) where ${\upsilon }^{†}$ is a parameter referred as the activation volume. Equation (2(2) $\{\begin{array}{c}{\omega }_{+}={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\text{β}\left(\mathrm{\Delta }{F}^{†}-{\upsilon }^{†}{\sigma }_0/2\right)\right]\\ \\ {\omega }_{-}={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\text{β}\left(\mathrm{\Delta }{F}^{†}+{\upsilon }^{†}{\sigma }_0/2\right)\right]\end{array}$(2) ) is generally referred to as the Eyring equation in polymer science: the physical interpretation of ${\upsilon }^{†}$ remains unclear and controversial. In general, the value is determined as a fitting parameter derived from the experimental data on the temperature or strain rate dependence of plastic deformation behaviours, and this value lies on the mesoscopic scale [1, 7, 21–24]. The ${\upsilon }^{†}$ parameter is of the order of 10 nm³, which is close to the volume of a Kuhn’s segment consisting of several repeating monomer units [3,7]. ${\upsilon }^{†}{\sigma }_0$ has dimensions of work and is identical to the difference between the free energies of the two sites.

Figure 1. The energy profile of Eyring rate theory [5]: the equilibrium position (dotted line) and the position under plastic flow (red line).

The net flow in the external flow direction is given by $\omega ={\omega }_{+}-{\omega }_{-}$, which is expressed using a hyperbolic function as (3) $\omega =2{\omega }_0{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\text{β}{\upsilon }^{†}{\sigma }_0/2\right)$(3)

The flow rate ω is directly related to the strain rate $\dot{ϵ}$. If ${\upsilon }^{†}{\sigma }_0\ll {\text{β}}^{-1}$, the strain rate is proportional to the applied stress because $\sinh x\approx x$ when $x\ll 1$(see (4a)). This indicates an ideal Newtonian flow. If ${\upsilon }^{†}{\sigma }_0\gg {\text{β}}^{-1}$, the logarithm of the strain rate is proportional to the work owing to the applied stress because $2\sinh x\approx {\mathrm{e}}^x$ when $x\gg 1$ (see (4b)). The system exhibits the typical plastic flow behaviour of polymer solids. In addition, equation (4b) has been widely applied to fracture behaviours [25, 26]. (4a, b) $\dot{ϵ}\propto \{\begin{array}{c}{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}\text{β}{\upsilon }^{†}{\sigma }_0\left({\upsilon }^{†}{\sigma }_0\ll {\text{β}}^{-1}\right)\\ \\ {\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}{\mathrm{e}}^{\text{β}{\upsilon }^{†}{\sigma }_0/2}\left({\upsilon }^{†}{\sigma }_0\gg {\text{β}}^{-1}\right)\end{array}\left(4\mathrm{a},\mathrm{b}\right)$(4a, b)

3. Thermodynamics of plastic deformation

We considered the uniaxial deformation of a homogeneous elasto-viscoplastic body [12, 27–30] that is in thermal contact with a heat bath at $\theta \left(={\text{β}}^{-1}\right)$. Thermodynamics with internal variables was adopted in this work. The introduction of internal variables supposes that a fictive thermodynamic equilibrium state, known as the local accompanying state described by the current state variables, exists behind the non-equilibrium state of the deformed body. The local accompanying equilibrium state, onto which the local non-equilibrium state can be projected adiabatically, permits the introduction of the notions of temperature and entropy [31]. Ostoja-Starzewsli et al. [19] confirmed that the introduction of the internal variables $\mathit{\alpha }$ allows for the notions to be applied to any local non-equilibrium state in terms of the dissipation functions.

According to the local accompanying state [32], the thermodynamic state depends only on instantaneous values of the state variables and not on their past histories. The local energy balance (or first law of thermodynamics) of the deformed continuum body is (5) $\rho \dot{\mathit{e}}=-\mathrm{d}\mathrm{i}\mathrm{v}\mathbf{h}+{\mathbf{T}}^T:\mathbf{L}$(5) where e denotes the specific internal energy, ρ indicates the current mass density, ${\mathbf{T}}^T$ represents the transpose of the Cauchy stress tensor for which the momentum balance ${\mathbf{T}}^T=\mathbf{T}$ holds, L symbolises the velocity gradient tensor, and h depicts the heat influx vector. The superscript dot denotes the material time derivative.

The vector $\mathbf{h}/\theta$ represents the heat flux entropy leaving the body through the surface. In the absence of a heat supply, the specific entropy rate $\dot{s}$ of the system is expressed as (6) $\rho \dot{s}=-\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathbf{h}/\theta \right)+\rho {\dot{s}}_i$(6) where ${\dot{s}}_i$ denotes the specific entropy irreversibly produced within the system. Equations (5(5) $\rho \dot{\mathit{e}}=-\mathrm{d}\mathrm{i}\mathrm{v}\mathbf{h}+{\mathbf{T}}^T:\mathbf{L}$(5) ) and (6(6) $\rho \dot{s}=-\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathbf{h}/\theta \right)+\rho {\dot{s}}_i$(6) ) are combined, and the specific free energy $\psi \left(\triangleq e-\theta s\right)$ is introduced, thereby yielding (7) $\rho \theta {\dot{s}}_i=-\rho \dot{\theta }s+{\mathbf{T}}^T:\mathbf{L}-\rho \dot{\psi }-{\displaystyle \frac{\mathbf{h}\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\theta }{\theta }}$(7) The second law of thermodynamics requires that the entropy production rate, ${\dot{s}}_i$, always be non-negative, which leads to the Clausius–Duhem inequality.

We assume that no temperature gradient exists within the body, i.e. $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\theta =\mathbf{0}$. Subsequently, the material description of (7) is obtained by multiplying both sides with Jacobian J in accordance with the relations $$\begin{gathered} J{\mathbf{T}}^T:\mathbf{L}={ \\ bfK}_{\mathrm{I}\mathrm{I}}:\dot{ \\ bfE} \end{gathered}$$ and mass balance ${\rho }_0=J\rho$, as follows: (8) ${\rho }_0\theta {\dot{s}}_i=-{\rho }_0\theta \dot{s}+{\mathbf{K}}_{\mathrm{I}\mathrm{I}}:\dot{\mathbf{E}}-{\rho }_0\dot{\psi }$(8) where ${\rho }_0$ indicates the initial density, K_II denotes the second Poila–Kirchhoff stress tensor, and E is the Lagrange–Green strain tensor.

In accordance with thermodynamics with an internal variable [13, 14, 33], the specific free-energy density $\psi$ is a function of E, θ, and $\mathit{\alpha }$. Therefore, the free-energy rate has the following form: (9) $\dot{\psi }={\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathbf{E}}:\dot{\mathbf{E}}+{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta }\dot{\theta }+{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathit{\alpha }}:\dot{\mathit{\alpha }}}}}$(9) Substituting (9) into (8), the entropy production can be expressed as (10) ${\rho }_0\theta {\dot{s}}_i=\left({\mathbf{K}}_{\mathrm{I}\mathrm{I}}-{\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathbf{E}}}\right):\dot{\mathbf{E}}-{\rho }_0\left(s+{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta }}\right)\dot{\theta }-{\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathit{\alpha }}:\dot{\mathit{\alpha }}}$(10) For any choice of $\dot{\mathbf{E}}$ and $\dot{\theta }$ values, the mechanical and thermal constitutive equations are obtained as [12] (11) ${\mathbf{K}}_{\mathrm{I}\mathrm{I}}={\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathbf{E}}:\dot{\mathbf{E}};s=-{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta };\mathcal{A}=-{\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathit{\alpha }}}}}$(11) where $\mathcal{A}$ denotes the affinity force tensor conjugate to the internal variable $\mathit{\alpha }$. The first equation of (11) implies that the free energy $\psi$ has the same operation as the strain–energy density function in hyperelasticity. The stress K_II is conserved throughout the body. The second equation yields a well-known thermal relationship.

The third equation indicates that the local or mesoscopic structural rearrangements relevant to plastic deformation are caused by the thermodynamic force $\mathcal{A}$, not an external force [11]. Accordingly, any plastic flow behaviour within a continuum system can be represented by introducing the corresponding internal variable $\mathit{\alpha }$, the components of which are uncontrollable [12].

Herein, we chose the dissipative strain tensor ${\mathbf{E}}_D$ as the internal variable $\mathit{\alpha }$. Accordingly, the entropy production rate is expressed as: (12) ${\rho }_0\theta {\dot{s}}_i=-{\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\mathbf{E}}_D}:{\dot{\mathbf{E}}}_D=\mathcal{A}:{\dot{\mathbf{E}}}_D}$(12) Subsequently, the specific free energy $\psi$ can be reconsidered as a function of $\mathbf{E}$ and ${\mathbf{E}}_D\left(=\mathit{\alpha }\right)$ through the recoverable component $\mathbf{E}-{\mathbf{E}}_D$ [34, 35]. The derivative chain rule leads to the coincidence of affinity $\mathcal{A}$ and external stress K_II [35]. Thus, (13) ${\mathbf{K}}_{\mathrm{I}\mathrm{I}}={\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\mathbf{E}}=-{\rho }_0{\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\mathbf{E}}_D}}}$(13) This coincidence indicates that the present system is a Maxwell-type linear viscoelastic system, which is a series combination of a dashpot and a spring, [13, 14] if $\mathit{\alpha }$ is chosen as a viscous strain. Alternatively, the Kelvin–Voigt model comprises a parallel combination of a dashpot and a spring. In the Kelvin–Voigt model, the quasi-conservative stress T is separated into the sum of the recoverable stress T_e and dissipative stress T_D [28–30, 35], and no internal variable is present, i.e. $\mathit{\alpha }=\mathbf{O}$.

Herein, we consider a polymeric body under uniaxial tension at $\theta \left(={\text{β}}^{-1}\right)$. Additionally, the undeformed body is assumed to be initially in equilibrium, and no temperature gradient is assumed to be present within the body. Accordingly, the second Piola-Kirchhoff tensor acting on a single flow unit is given by: (14) ${\mathbf{K}}_{\mathrm{I}\mathrm{I}}=\left(\begin{array}{ccc}{\sigma }^{\text{‡}}/\lambda & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$(14) where λ denotes the extension ratio resulting from the movement of the flow unit and ${\sigma }^{\text{‡}}$ indicates the nominal stress. In the simple uniaxial deformation mode, the Jacobian is given by $J={\lambda }^{1-2\nu }$, where $\nu$ denotes the Poisson’s ratio. The dissipative strain rate is expressed as (15) ${\dot{\mathbf{E}}}_D=\left(\begin{array}{ccc}\dot{\lambda }\lambda & 0& 0\\ 0& -\nu \dot{\lambda }{\lambda }^{-\left(1+2\nu \right)}& 0\\ 0& 0& -\nu \dot{\lambda }{\lambda }^{-\left(1+2\nu \right)}\end{array}\right)$(15) Therefore, the entropy production rate has the form (16) ${\rho }_0\theta {\dot{s}}_i=\text{β}{\mathbf{K}}_{\mathrm{I}\mathrm{I}}:{\dot{\mathbf{E}}}_D=\text{β}{\sigma }^{\text{‡}}\dot{\lambda }$(16) Consequently, the total entropy production Ω_p can be obtained by integrating both sides of the aforementioned equation in terms of the time and space occupied by the flow unit, as follows: (17) ${\mathrm{\Omega }}_p=\text{β}{\sigma }^{\text{‡}}{\upsilon }^{\text{‡}}=\text{β}f\delta l$(17) where f indicates the traction force acting on the flowing unit, $\delta l$ denotes the displacement of the flowing unit, and ${\upsilon }^{\text{‡}}\left(=a_0\delta l\right)$ represents the volume occupied by the movement of the flowing unit with a cross-sectional area $a_0$ (i.e. the activation volume). A positive value of f indicates extension, whereas a negative value indicates compression. Moreover, the total entropy of the uniaxially deformed body changes by $\text{β}\sigma {}^{\text{‡}}{\upsilon }^{\text{‡}}$ in response to the introduction of an internal variable. The $f\delta l\left(={\sigma }^{\text{‡}}{\upsilon }^{\text{‡}}\right)$ corresponds to dissipative work.

Macroscopic plastic deformation results from various mesoscopic processes that permit instantaneously reversible structural rearrangements. Such structural changes can be phenomenologically expressed using a kinetic transient process that causes the flow units to fluctuate or migrate by jumping into near-neighbour vacancies. Consequently, the principle of microscopic reversibility [18, 36] in kinetic/dynamic processes is applicable to describe the aforementioned plastic deformation behaviour.

In accordance with the microscopic reversibility condition, the system obeys the canonical distribution at the temperature θ of the heat bath and the ratio of the transition probability $\omega \left(+\mathrm{\Gamma }\right)$ of the forward path +Γ movement of a unit to the neighbouring empty site and $\omega \left(-\mathrm{\Gamma }\right)$ of its reverse path −Γ back to the original site is quantitatively related to the entropy production Ω_p within the system [18, 36]. The representative equation has the following form: (18) ${\displaystyle \frac{\omega \left(+\mathrm{\Gamma }\right)}{\omega \left(-\mathrm{\Gamma }\right)}={\mathrm{e}}^{{\mathrm{\Omega }}_p}}$(18) The system initially exists in an equilibrium state and is later drawn out of equilibrium via uniaxial stretching on an external traction force f. In the initial state (${\mathrm{\Omega }}_p=0$), any forward process and its reverse occur at the same rate, where the path in the reverse direction must be the reverse of the path in the forward direction, yielding a canonical form $\omega \left(0\right)={\omega }_0{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}$in accordance with (1). Consequently, the forward and reverse transition probabilities under tension have the following respective forms. (19) $\{\begin{array}{c}\omega \left(+\mathrm{\Gamma }\right)={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\left(\text{β}\mathrm{\Delta }{F}^{†}-{\mathrm{\Omega }}_p/2\right)\right]\\ \\ \omega \left(-\mathrm{\Gamma }\right)={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\left(\text{β}\mathrm{\Delta }{F}^{†}+{\mathrm{\Omega }}_p/2\right)\right]\end{array}$(19) The Eyring equations (2) can be immediately derived by substituting (17) into (19), assuming that the activation work ${\sigma }^{\text{‡}}{\upsilon }^{\text{‡}}$ is equal to ${\sigma }_0{\upsilon }^{†}$. The system shifts towards a new equilibrium state under tension. The dominant characteristic time τ [24] required to attain the new equilibrium state is expressed as: (20) $\tau ={\displaystyle \frac{1}{2}\text{β}\hslash \mathrm{e}\mathrm{x}\mathrm{p}\left(\text{β}\mathrm{\Delta }{F}^{†}\right)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\text{β}{\sigma }^{\text{‡}}{\upsilon }^{\text{‡}}/2\right)}$(20) The value of τ reflects the distance from the corresponding equilibrium state [37]. Considering that this mechanical process occurs at any representative volume element (RVE), the τ parameter can be experimentally determined through macroscopic stress-relaxation measurements.

The second law of thermodynamics ensures that the ensemble average value$〈{\mathrm{\Omega }}_p〉$ over the duration τ is a non-negative value. Consequently, the second law requires the following inequality: (21) $f〈\delta l〉\ge 0$(21)

because $\text{β}(>0)$ and f are fixed in accordance with the protocol. The second law ensures that the ensemble average of the displacement $\delta l$ has the same polarity as the force f. Accordingly, the second law states that the macroscopic displacement of the flow units during plastic deformation under uniaxial tension always proceeds in the direction of the applied force. The ensemble average of ${\mathrm{e}}^{-{\mathrm{\Omega }}_p}$ yields $〈{\mathrm{e}}^{-{\mathrm{\Omega }}_p}〉=1$ through Jarzynski’s theorem or detailed fluctuation theorem [38, 39]. The second law of thermodynamics, $〈{\mathrm{\Omega }}_p〉\ge 0$, is readily derived using Jensen’s inequality: $〈{\mathrm{e}}^{-{\mathrm{\Omega }}_p}〉\ge {\mathrm{e}}^{-〈{\mathrm{\Omega }}_p〉}$. Various types of fluctuation theorems have been identified as yet, and several studies have focused on experimentally validating these fluctuation theorems [40–43].

The deformation flux J(Γ), which corresponds to the strain rate, is given by $\omega \left(+\mathrm{\Gamma }\right)-\omega \left(-\mathrm{\Gamma }\right)$. Under the first-order approximation of the exponential function ${\mathrm{e}}^{\mathrm{x}}\approx 1+x$, the following expression is obtained: (22) $J\left(\mathrm{\Gamma }\right)={\omega }_0{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}{\mathrm{\Omega }}_p={\displaystyle \frac{1}{\hslash }{\mathrm{e}}^{-\text{β}\mathrm{\Delta }{F}^{†}}{v}^{\text{‡}}{\sigma }^{\text{‡}}}$(22) This corresponds to Onsager’s linear non-equilibrium process and represents the case closest to equilibrium. In this case, the flux is proportional to the applied nominal stress (or affinity), which is a Newtonian flow. Recently, it has been shown that Onsager’s reciprocity process can be derived based also on multiple independent transport processes [44]. The linear relation (22) will be derived also by a simple transport process out of the microscopic reversibility principle.

4. Final remarks

Eyring rate theory for mechanical plastic deformation was investigated within the framework of non-equilibrium thermodynamics in a continuum body. This study demonstrated that the governing equation of the Eyring rate theory is derived from the principle of microscopic reversibility in a non-equilibrium process, and that the entropy production owing to external work leads to plastic displacement. The empirical fact that the plastic deformation of any polymer solid followed the Eyring rate theory is attributed to the second law of thermodynamics in the non-equilibrium state.

Disclosure statement

No potential conflict of interest was reported by the author(s).