Stick-slip kinetics in a bistable bar immersed in a heat bath

Abstract

Structural transitions in some rod-like biological macromolecules under tension are known to proceed by the propagation through the length of the molecule of an interface separating two phases. A continuum mechanical description of the motion of this interface, or phase boundary, takes the form of a kinetic law which relates the thermodynamic driving force across it with its velocity in the reference configuration. For biological macromolecules immersed in a heat bath, thermally activated kinetics described by the Arrhenius law is often a good choice. Here we show that ‘stick-slip’ kinetics, characteristic of friction, can also arise in an overdamped bistable bar immersed in a heat bath. To mimic a rod-like biomolecule we model the bar as a chain of masses and bistable springs moving in a viscous fluid. We conduct Langevin dynamics calculations on the chain and extract a temperature dependent kinetic relation by observing that the dissipation at a phase boundary can be estimated by performing an energy balance. Using this kinetic relation we solve boundary value problems for a bistable bar immersed in a constant temperature bath and show that the resultant force-extension relation matches very well with the Langevin dynamics results. We estimate the force fluctuations at the pulled end of the bar due to thermal kicks from the bath by using a partition function. We also show rate dependence of hysteresis in cyclic loading of the bar arising from the stick-slip kinetics. Our kinetic relation could be applied to rod-like biomolecules, such as, DNA and coiled-coil proteins which exhibit structural transitions that depend on both temperature and loading rate.

1. Introduction

The study of mechanical behavior of macromolecules is important to understand and manipulate DNA, proteins, enzymes, etc. Some of these molecules, such as DNA and coiled-coil proteins have a rod-like structure and are known to undergo structural transitions when subjected to large forces [1, 2, 3, 4]. These structural transitions are important for the function of these molecules: for example, intermediate filaments have a coiled-coil structure and they help maintain the integrity of a cell by unfolding (a structural transition) under large forces, but coming back to their original configuration when the force is released [5, 6]. It has been observed in numerical experiments that at high strain rates, the transition proceeds by the propagation of an interface, while at low strain rates random nucleation is the main mechanism of transition [7]. One or two interfaces between different phases of DNA have been observed in single molecule experiments [8, 9], and moving interfaces between α-helix and β-sheet phases of coiled-coils have been observed in molecular dynamic simulations [10, 11], although the dicreteness at the nano- scale makes it difficult to define a sharp interface [12, 13] as assumed in some continuum theories. We note here that both dsDNA and α-helical coiled-coils are double stranded molecules; when these undergo structural transitions the new phase may have both strands bonded to each other (through hydrogen bonds, for example) or not. Our analysis in this paper is applicable to the case when the two strands are bonded, as in the B-DNA to S-DNA transition in dsDNA [14, 8] and the α-helix to β-sheet transition in coiled-coils[10], both of which occur when the respective molecules are loaded in tension along the length. Our analysis may not be applicable if the strands are not bonded to each other (as in a B-DNA to single-stranded DNA transition [14, 8], or if the loading applied to cause the structural transition is of shear type[15].

One way to study phase transitions in long molecules is to use continuum theory. To do so, the macromolecule is modeled as a 1D continuum with an energy storage function that has multiple wells representing different phases which may be separated in the continuum by propagating interfaces[16, 17]. Such interfaces have been observed in phase change nanowires [18] and in some macromolecules as mentioned above. When the balance laws of mass, momentum and energy are applied to this continuum it is found that they are not sufficient to fully describe the propagation of phase boundaries; an additional kinetic relation is required to close the equations and usually must be supplied as constitutive information [19]. The choice of an appropriate kinetic relation has been the subject of much research. Truskinovsky and Vainchtein [20] have analytically derived a kinetic relation for a lattice model in which inertia forces drive the phase transition, and this kinetic relation has been verified numerically by Zhao and Purohit [21]. Zhao and Purohit [21] have also shown that the kinetic relation is temperature dependent, at least for low temperatures, in a solid in which inertia forces are dominant and damping is small. However, this kinetic relation cannot be applied to macromolecules since inertia forces are negiligible for macromolecules immersed in water. Instead, the viscous force due to surrounding fluid and thermal fluctuations are dominant in structural transitions of macromolecules. Hamiltonian dynamic simulations of a bi-stable chain have been carried out by Efendiev and Truskinovsky [22] with the objective of studying its thermalization and latent heat release during phase transitions. They study the equilibrium response of the chain and compute the partition function which is utilized to calculate free energies. They do not extract a kinetic relation from their simulations because they do not see phase boundaries propagating in an organized fashion unless next-to-nearest-neighbor interactions are added. Benichou and Givli [23, 24] consider overdamped dynamics of a chain with tri-linear springs and show that a single non-dimensional quantity that captures the competition between loading rate, temperature and energy barriers can predict the amount of hysteresis under various conditions. These authors also do not comment on the kinetic relation for propagating phase boundaries in the chain, and furthermore they confine their analysis to situations in which the damping force due to fluid viscosity in the heat bath is small. Finally, a recent study by Caruel and Truskinovsky [25] examines how a micromechanical device relying on ‘snap-through’ or bistability can retain its functionality in an environment dominated by fluctuations by using cooperativity of several degrees of freedom. They analyze the statistical mechanics of a parallel bundle of bistable elements and are hence not concerned with propagating phase boundaries.

An appropriate kinetic relation for phase transitions in an overdamped thermal environment is based on the Arrhenius law [19]. Indeed, analytical models of structural transitions in macromolecules based on Arrhenius kinetics have been shown to agree with molecular dynamic simulations over several decades of strain-rates [10, 26]. Arrhenius kinetics also predicts the strain-rate dependence of the plateau force in a structural transition in macromolecules [16]. More recently, Arrhenius kinetics was applied to a 1D analytical model of grain boundary motion in a solid and it was shown that it predicts the evolution of stresses in a molecular dynamic simulation without any fitting parameters [27]. Yet, Arrhenius kinetics may not always be appropriate to describe the evolution of phase fractions. In a recent experiment Dittmore and Neuman [28] found that the transition from straight DNA to plectonemic DNA in a single molecule tension-torsion experiment proceeds in a jerky fashion characteristic of frictional behavior [29]. Clearly, this cannot be described within an Arrhenius kinetic law, so ‘stick-slip’ type kinetic laws have been proposed that have been derived for phase boundary motion through a row of imperfections [30], for propagation of a front by kink motion [31], and for phase boundary motion through a heterogeneous solid [32]. By ‘stick-slip’ we mean that the phase fraction evolves only when the driving force exceeds a threshold value as in Bingham type visco-elastic fluid [33, 34]. In a different context, Atkinson and Cabrera [35] obtained a relation between driving force and defect velocity by considering the radiation (of energy) from a dislocation moving at constant velocity through a 1D chain of particles interacting through a periodic potential. They computed a threshold force to start the defect moving, effectively deriving a stick-slip type kinetic law. Similar 1D lattice models [36] with a periodic potential are also used to study nanoscale friction (as occurs when an atomic force microscope tip is dragged over a crystalline solid) and it is shown how a threshold force is required for motion as is well-known in the study of static friction. However, in these papers temperature and thermal fluctuations play no role. There are models of grain-boundary motion in which temperature dependent stick-slip kinetics have been explained [37], but it is not clear how these models are connected to phase boundary propagation in 1D continua, such as, the biological macromolecules mentioned earlier in this paper. Stick-slip type evolution of phase fraction in biological macromolecules was discussed in detail within a continuum theory in a recent paper by De Tommasi et al. [38]. The authors focused in this paper on macromolecules whose force-extension relation displays a saw-tooth pattern characterized by discrete events, and showed how a continuum limit leads to a force-extension relation with a plateau characterizing the folding/unfolding transition of the macromolecule. They also put their theory in the context of flow-rules used in plasticity [39] and explained how a theshold driving force must be reached for the phase fraction (treated as an internal variable in [34]) to evolve. If there is a threshold driving force for the evolution of phase fraction then there is hysteresis even in quasistatic loading/unloading of a phase-changing bar [19]. As a result, hysteresis in the model of De Tommasi et al. has both a rate-independent and a rate-dependent component. However, viscous drag due to the fluid bath surrounding a molecule and temperature dependence of phase boundary kinetices are not discussed in [38] either.

We show in this paper that stick-slip type kinetics can arise in an overdamped 1D continuum capable of phase transitions. A brief outline of our approach that led to this result follows. In section 2 we begin with a review of the continuum mechanical principles for a phase boundary in a one-dimensional bar, then we review Langevin dynamics calculations on a bilinear bi-stable mass-spring chain immersed in a heat bath. It has been shown in several papers in the literature that features in the evolution of phase transitions in 1D continua can be captured quite well using mass-spring chain models [21, 40, 19]. We verify that by a suitable choice of parameters in our mass-spring chain simulations we can capture the experimentally measured force-strain curves of phase changing nanowires and filamentous molecules. Next, we focus on the microscopic motion of a phase boundary in our mass-spring chains and show that plots of phase boundary position vs. time for a variety of loading rates and bath temperatures have a stair-case pattern characteristic of discrete systems. We show that the driving force computed using analytical formulae available for bilinear materials cannot be used to find a meaningful kinetic law that relates microscopic phase boundary velocity to the driving force. Nevertheless, it is possible to find a homogenized (or average) phase boundary velocity and a corresponding driving force using a balance of energy. The central result of our paper appears at the end of section 2 and it shows that the relation between the homogenized phase boundary velocity and the driving force is of the stick-slip type and is dependent on the bath temperature. In section 3 we verify through Langevin dynamic simulations of our mass-spring chain that this kinetic relation can predict the final position of the phase boundary in a problem with very different initial and boundary conditions. In section 4 we use the kinetic relation to solve initial-boundary value problems on a 1D bar capable of phase transitions and confirm that the force-extension curve of the bar agrees quite well with those obtained from Langevin dynamics calculations on a mass-spring chain (in which no kinetic relation was prescribed in the first place). We also show how thermal fluctuations in the force in our Langevin dynamic calculations can be quantified using statistical mechanical principles. In section 5 we explore the consequences of our kinetic relation on hysteresis in cyclic loading/unloading simulations on a chain, and connect to experiments that examine the temperature and loading rate dependence of hysteresis; then, we end with a brief conclusion in section 6.

2. Study of the kinetic relation

We begin with a brief review of the evolution of phase transitions in 1-D continua and show that it can be mimicked in simulations of mass-spring chains with bistable springs. Most treatments of 1-D continua capable of phase transitions ignore viscous drag forces which are relevant to macromolecules, but we account for these drag forces and connect them to the drag imposed by the fluid bath in our Langevin dynamic simulations of a mass-spring chain. Our central result in this section is the deduction of a relation between average phase boundary velocity in a bi-stable chain with the driving force acting on it.

2.1. Modeling of a rod-like molecule

We follow the development in Abeyaratne and Knowles [19]. Consider a 1-D bar occupying an interval [0, L] in the reference configuration. The displacement in the deformed configuration at x at time t is denoted by u(x, t). The corresponding velocity is $v(x,t)=\stackrel{.}{u}=\frac{\partial u}{\partial t}$ and the strain is given by $\gamma (x,t)=u_x=\frac{\partial u}{\partial x}$. Let σ(x, t) denote the force in the bar. Momentum balance requires a portion of the bar in [x₁, x₂] to satisfy:

$$\sigma (x_2,t)-\sigma (x_1,t)+{\int }_{x_1}^{x_2}\rho bdx=\frac{d}{dt}{\int }_{x_1}^{x_2}\rho \upsilon (x,t)dx,$$ (1)

where b is the body force and ρ is a mass per unit length in the reference configuration, which is assumed a constant in our model. If v, γ and σ are smooth enough (differentiable), then the momentum balance can be re-written as:

$$\frac{\partial \sigma }{\partial x}+\rho b=\rho \frac{\partial \upsilon }{\partial t}.$$ (2)

Also the differentiability of v and γ requires that:

$$\frac{\partial \upsilon }{\partial x}=\frac{\partial \gamma }{\partial t}.$$ (3)

These two equations together with the constitutive relation,

$$\sigma (\gamma )=W^{\prime }(\gamma ),$$ (4)

will close the system in the absence of any discontinuity. Here, W(γ) denotes strain energy per unit reference length as a function of strain γ.

Since our main interest is in the dynamics of a phase boundary (the moving interface separating two phases), we need to relax the continuity of some quantities. Here the displacement u is still assumed to be continuous, but $\stackrel{.}{u}$, u_x and σ may have isolated discontinuities. Assume there exists only one discontinuity at s(t) at time t in the reference configuration 0 ≤ s(t) ≤ L. The corresponding jump conditions at the discontinuity are

$$\begin{gathered} [|\sigma |]=-\rho \dot{s}[|\upsilon |], \\ [|\upsilon |]=-\dot{s}[|\gamma |]. \end{gathered}$$ (5)

Here [|ϕ|] = ϕ₊ − ϕ₋ = ϕ(s(t)⁺, t) − ϕ(s(t)⁻, t) and $\stackrel{.}{s}$ is the speed of the discontinuity in the reference configuration.

For a portion of the bar lying in the reference configuration between x₁ and x₂ the dissipation rate, which is defined as the difference between the rate of work done by the external forces acting on the bar and the rate of change of total mechanical energy within it, can be written as:

$$D(t)=\sigma \upsilon {|}_{x_1}^{x_2}-\frac{d}{dt}{\int }_{x_1}^{x_2}(\frac{1}{2}\rho v^2+W(\gamma ))dx.$$ (6)

Under the assumption that there is only one discontinuity in the bar and making use of the field equations and jump conditions, one can show that

$$D(t)=f(t)\dot{s}(t),$$ (7)

where f(t) is the driving force acting on the discontinuity. Note that the second law of thermodynamics requires D(t) ≥ 0. The driving force is determined by the strains immediately ahead and behind the phase boundary:

$$f={\int }_{{\gamma }^{-}}^{{\gamma }^{+}}\sigma (\gamma )d\gamma -\frac{1}{2}(\sigma ({\gamma }^{+})+\sigma ({\gamma }^{-}))({\gamma }^{+}-{\gamma }^{-}).$$ (8)

In order for the discontinuity to be a phase boundary the force σ cannot be a monotonically increasing function of strain γ. For simplicity, assume the constitutive relation of the bar is bi-linear:

$$\sigma (\gamma )=\{\begin{array}{ll}E\gamma ,& \gamma <{\gamma }_c,\\ E(\gamma -{\gamma }_t),& \gamma \ge {\gamma }_c.\end{array}$$ (9)

Also, for simplicity, assume ${\gamma }_c=\frac{{\gamma }_t}{2}$, then the stored energy landscape is a double-well:

$$W=\{\begin{array}{ll}\frac{1}{2}E{\gamma }^2,& \gamma <{\gamma }_c,\\ \frac{1}{2}E{(\gamma -{\gamma }_t)}^2,& \gamma \ge {\gamma }_c.\end{array}$$ (10)

The system is stable in each well, and thus we call γ < γ_c low strain phase, and γ ≥ γ_c high strain phase. By substituting the constitutive relation into the definition of driving force, one can find the following explicit form of driving force for a simple double-well material:

$$f=\frac{1}{2}E{\gamma }_t({\gamma }^{+}+{\gamma }^{-}-{\gamma }_t)$$ (11)

Our goal is to simulate a discrete version of a bar with the constitutive relation in eqn.(9) immersed in a constant temperature bath. We will perform impact experiments on the bar and follow the phase boundaries with the objective of getting an isothermal kinetic relation.

2.2. Constant temperature bath: Langevin dynamics

In order to realize a bar immersed in a constant temperature bath we simulate a mass-spring chain using Langevin dynamics [21]. Suppose we have N + 1 masses and interaction exists between only the nearest neighbors; the masses are then connected by N springs. The potential energy of the springs is double-welled,

$$W=\{\begin{array}{ll}\frac{1}{2}k{(x_i-a_0)}^2,& x<x_c,\\ \frac{1}{2}k{(x_i-a_t)}^2,& x\ge x_c,\end{array}$$ (12)

where x_i = u_i+1 − u_i is the length of the i-th spring, $x_c=\frac{1}{2}(a_0+a_t)$ for simplicity. We assume the chain is immersed in fluid with fixed temperature T. The equations of motion for the masses are:

$$m{\ddot{u}}_i=-\nu {\dot{u}}_i-\frac{\partial \varphi }{\partial u_i}+F_i(t).$$ (13)

The total potential energy of the chain is given by $\varphi ={\Sigma }_{i=1}^{N}W(x_i)$. ν denotes the drag coefficient due to the fluid. For a rod of length l and radius r in a Newtonian fluid with viscosity μ, the corresponding drag coefficient is given by $d_w=\frac{2\pi \mu }{\text{log}(l∕r)-0.8}$. This formula is valid when l/r > 10 and should work well for long rod-like DNA and protein molecules [16]. For mass-spring representation of a bar, ν is such that the drag force on a mass is equivalent to the drag force on the interval of [x_i, x_i + a₀] of the bar, ν = d_wa₀ where a₀ is the distance between two masses in the reference configuration. The force F_i(t) denotes a time-dependent random force exerted by the bath on each mass due to thermal fluctuation, which averages to 0 and is assumed to be delta-function correlated in time [41]:

$$\langle F_i(t)\rangle =0,\langle F_i(t)F(t^{\prime })\rangle =2\nu k_{B}T\delta (t-t^{\prime }),$$ (14)

where k_B is the Boltzmann constant, T is the absolute temperature and 〈·〉 denotes ensemble average.

We use a symplectic method to integrate eqn.(13) [21, 42]:

$$\begin{gathered} v_{1,i,k}=g(\frac{\tau }{2};v_{i,k}), \\ {u}_{1,i,k}=u_{i,k}+\frac{\tau }{2}{v}_{1,i,k}, \\ {v}_{2,k}=v_{1,i,k}+\tau \frac{F(x_{1,i+1,k},x_{1,i,k})}{m}+\frac{{\tau }^{1/2}}{m}\sigma \xi , \\ {v}_{i,k+1}=g(\frac{\tau }{2};v_{2,i,k}), \\ {u}_{i,k+1}=u_{1,i,k}+\frac{\tau }{2}{v}_{2,i,k,}, \end{gathered}$$ (15)

where τ is the time step, k denotes the k-th time step, and i denotes the i-th particle, g(t, v) = ve^−νt/m, $\sigma =\sqrt{2\nu k_{B}T}$ and ξ is a Gaussian random number with mean 0 and variance 1. The above numerical scheme remains valid for T = 0 when thermal fluctuation is not considered or ν = 0 which corresponds to zero fluid drag. In fact, Zhao and Purohit [21] had used this scheme for an isothermal chain in which inertia forces played a significant role, but damping was small. The temperature T in those simulations was small enough that although thermal fluctuations were apparent the chain dynamics were similar to one with T = 0. As a result, the inferred kinetic relation was of a form derived analytically for a purely mechanical chain [20], but it showed a weak dependence on temperature [21]. In this paper we have the opposite situation in which the inertia forces are negligible and damping is significant as is the case for a long biomolecule immersed in water.

In our simulations the chain is fixed at one end (i = 1) and stretched at a constant rate v_p at the other end (i = N + 1). Such a case corresponds to the impact problem in a continuum bar, where a constant velocity is applied to a semi-infinite bar. It has been shown in molecular simulations [11] that the springs at the moving end first transit into the high strain phase, and a phase boundary appears at that end, then propagates towards the other end. We wish to obtain the kinetic relation for such a phase boundary.

2.3. Non-dimensionalization

Here we choose the initial length of springs a₀, the stiffness of the springs k, and the mass m as the independent variables. These three variables are all internal properties of the mass-spring chain. Once the material is chosen, the characteristic variables are determined. Let variables for the bar be the characteristic parameters:

$$\begin{array}{ll}\overline{c}=a_0\sqrt{\frac{k}{m}}& \hfill (\mathrm{characteristic}\mathrm{velocity}),\\ \overline{t}=\sqrt{\frac{m}{k}}& \hfill (\mathrm{characteristic}\mathrm{time}),\\ \overline{\nu }=\sqrt{km}& \hfill (\mathrm{characteristic}\mathrm{viscosity}),\\ \overline{E}=k{a_0}^2& \hfill (\mathrm{characteristic}\mathrm{energy}),\\ \overline{T}=\frac{k{a_0}^2}{k_B}& \hfill (\mathrm{characteristic}\mathrm{temperature}).\end{array}$$ (16)

We notice that the characteristic velocity is in fact the sonic wave speed of the mass-spring chain, $c=a_0\sqrt{k∕m}$. The corresponding dimensionless variables (variables with *) are given by:

$$u_i^{\ast }=\frac{u_i}{a_0},{\tau }^{\ast }=\frac{\tau }{\overline{t}},v_p^{\ast }=\frac{v_p}{c},{\nu }^{\ast }=\frac{\nu }{\overline{\nu }},T^{\ast }=\frac{T}{\overline{T}},W^{\ast }=\frac{W}{\overline{E}}.$$ (17)

It can be easily shown that these dimensionless variables satisfy the same governing equations and boundary conditions as the original equations. For example, the non-dimensional Langevin equation is:

$$\ddot{u_i^{\ast }}=-{\nu }^{\ast }\dot{u_i^{\ast }}-\frac{\partial {\varphi }^{\ast }}{\partial u_i^{\ast }}+F_i^{\ast }(t).$$

Since our interest is in long molecules, consider a coiled-coil protein [43]. The corresponding characteristic variables are:

$${\overline{a}}_0=2\times {10}^{-10}m,\overline{k}=5N/m,\overline{m}=4.3\times {10}^{-25}Kg.$$

These parameters are obtained from the force-strain relation, the length and the linear density of a coiled-coil protein [43]. The non-dimensional double-well potential is assumed to be (this may not be characteristic of a coiled-coil protein):

$$W^{\ast }=\{\begin{array}{ll}\frac{1}{2}{(x_i^{\ast }-1)}^2,& x^{\ast }<2.5,\\ \frac{1}{2}{(x_i^{\ast }-4)}^2,& x^{\ast }\ge \mathrm{2.5.}\end{array}$$

The ratio of length l and radius r is approximated as 20 [43]. We ignore the effect of temperature on the viscosity of the fluid and use the viscosity of water at 300K and 1 atm, ν = 1.01 × 10⁻³ [44]. The non-dimensional viscosity is ν* = 0.400. We used these parameters to carry out a simulation in which a mass-spring chain was pulled at a constant velocity. The results are plotted in Fig. 1 for two pulling velocities. We see the same mechanical behavior as was observed in experiments and molecular dynamic simulations [26, 43] on coiled-coil proteins. In particular, there are three regions of the force-strain curve – a low strain region in which all springs are in the low strain well, a high strain region in which all springs are in the high strain well and a plateau region in which some springs are in the low strain well and others in the high strain well. In particular, the slope of the plateau region increases with increasing pulling velocity as observed in molecular dynamic simulations [26] as well as in experiments [6]. Note also that there is significant fluctuation in the force F_e measured at the end being pulled. We will quantify these fluctuations later in the paper.

Fig. 1:

The force-strain relation for a chain with N = 100 at pulling rate of (a) $v_p^{\ast }=0.002$ and (b) $v_p^{\ast }=0.02$. We find three regions demarcated by dashed vertical lines above: a linearly increasing region in low strain phase, a plateau denoting phase transition process, and a second linearly increasing region in high strain phase. Such behaviors have been observed in experiments and molecular dynamic simulations. There is significant fluctuation in the force measured at the Nth mass. (c) Stress-strain curve of vanadium oxide sample undergoing phase transition (blue dots) [18] matched by our Langevin dynamics simulation of a mass-spring chain (red curve) with appropriate parameters. (d) Force-strain curves at two different pulling speeds of vimentin intermediate filaments immersed in fluid [6] matched by our Langevin dynamic simulations with appropriate parameters. We fit the parameters of the energy landscape (green curve) to the experimental data (red curve) at the lower pulling speed. Then, we change to higher pulling speed and show that our Langevin dynamics simulations (blue curve) match with experimental data (red curve). The experimental curves in this figure are averages of multiple pulls (see [6]), so fluctuations are not apparent. Here the strain $\gamma =\frac{L-L_0}{L_0}$ is the ratio of the extension L − L₀ and the initial length L₀ of the whole chain.

Although we will mostly confine ourselves to the non-dimensional spring paramaters chosen above while changing the pulling velocity v_p and bath temperature T, we emphasize that our mass-spring model can capture force-strain plots with plateaus in various materials. To demonstrate this we modify our energy potential slightly as shown below to account for different well depths in the unloaded state:

$$W^{\ast }=\{\begin{array}{ll}\frac{1}{2}{k}^{\ast }{(x_i^{\ast }-x_0^{\ast })}^2,& x^{\ast }<x_c^{\ast },\\ \frac{1}{2}{k}^{\ast }{(x_i^{\ast }-x_t^{\ast })}^2+W_t^{\ast },& x^{\ast }\ge x_c^{\ast }.\end{array}$$

We set k* = 120, $x_0^{\ast }=1$, $x_c^{\ast }=1.005$, $x_t^{\ast }=1.0055$, and $W_t^{\ast }=2.97\times {10}^{-4}$ and show in Fig. 1(c) that we can match the experimental stress-strain curves of Guo et al. for vanadium oxide nanowires [18]. The experimental stress-strain curves were obtained by in situ uniaxial tension of these nanowires in a transmission electron microscope. Guo et al. show by using X-ray diffraction techniques that there is co-existence of two phases in the plateau region of the force-strain curve, but there is only one phase in the low-strain region, hence establishing that a phase transition is responsible for the characteristic form of their stress-strain curve. In our simulations we used room temperature, a low drag coefficient, and a very slow pulling speed to mimic the experimental conditions of Guo et al. [18]. The output of our mass-spring simulations is a force at the pulled end but it can be converted to a stress since the cross-sectional area of the nanowires used in [18] can be inferred from their experimental daata. Next, we set k* = 1, $x_0^{\ast }=1$, $x_c^{\ast }=1.2$, $x_t^{\ast }=1.9$, and $W_t^{\ast }=-0.225$ and show in Fig. 1(d) that we can match experimental force-strain curves of Block et al. for vimentin intermediate filaments [6]. In this figure we show that the same set of parameters can reproduce the experimental data for two different pulling rates. The experiments of Block et al. are performed using optical traps within a micro-fluidic device and it is shown that an α-helix to β-sheet transition occurs during pulling. In our simulations we used room temperature and drag coefficient appropriate for a filament immersed in water.

Having shown the versatitility of our mass-spring simulations we turn our attention back to phase boundary propagation in the next section.

2.4. Homogenized phase boundary velocity

In our simulations we find a single propagating phase boundary when pulling velocity v_p is sufficiently high. Although we focus here on situations with a single propagating boundary, we acknowledge that in both experiments and simulations one may also find multiple phase boundaries[8, 11, 26]. In particular, if non-nearest neighbor interactions between masses are included [45, 46] then the number of phase boundaries is determined by the boundary conditions as shown for biomolecules in [47]. In our simulations multiple phase boundaries are seen at low pulling velocities as shown in section 3, but we focus on one propagating phase boundary here because we want to deduce the kinetic relation for it. We have found that the average (or homogenized) velocity $\stackrel{.}{s}$ of this phase boundary is nearly constant throughout the simulation if the pulling velocity v_p is constant. There are small fluctuations about this average. To see this we refer the reader to Fig. 2(a) in which we plot the phase boundary position for three combinations of pulling velocity v_p and bath temperature T. In each case we find a ‘stair-case pattern’ for the phase boundary position as a function of time. The phase boundary moves in discrete steps which are not all of the same length. This may be understood by considering the following three time scales – (a) the time scale ${\tau }_{well}=\sqrt{\frac{m}{k}}$ for in-well minimization, (b) the time scale ${\tau }_p=\frac{a_0}{v_p}$ associated with the loading, and (c) the time scale ${\tau }_{bar}=\frac{\pi m}{k}[\frac{\nu }{m}+\sqrt{{(\frac{\nu }{m})}^2+4\frac{k}{m}}]{\text{e}}^{\frac{Q}{k_{B}T}}$ associated with crossing the energy barrier Q [48]. In our case τ_well « 1, τ_p « 10 and τ_bar « 10¹⁰. Since, τ_bar is much longer than the other two time scales our hypothesis is that the phase boundary jumps from the current mass to the next when the force in the spring ahead of that mass becomes large enough to make the barrier comparable to k_BT. If τ_bar was not too large compared to τ_p then barrier crossing would occur more easily as the chain is pulled and Arrhenius kinetics would be appropriate. This is not the case for the parameters in our mass-spring chain in which the energy barrier separating the two wells is high, hence the mass oscillates inside a well until the energy landscape is tilted sufficiently by the loading to cause a jump. This is why the microscopic motion of the phase boundary appears like a stair-case pattern. Nevertheless, the dashed lines through each stair-case show that there is a ‘homogenized’ constant velocity of the phase boundary as it moves from one end of the chain to the other. Let t₁ denote the time the phase boundary appears and t₂ the time it reaches the other end and vanishes, then by definition, the homogenized phase boundary velocity is given by

$$\dot{s}=\frac{N{a}_0}{t_2-t_1}.$$ (18)

Fig. 2:

Homogenized phase boundary velocity and driving force. (a) Phase boundary position plotted as a function of time for three cases (1) T* = 0.027, $v_p^{\ast }=0.05$ (red curve), (2) T* = 0.018, $v_p^{\ast }=0.08$ (blue curve), and (3) T* = 0.032, $v_p^{\ast }=0.11$ (green curve). In each case we see a stair-case pattern characteristic of a discrete system. The dashed line through each stair-case has a constant slope which shows that there is a homegenized phase boundary velocity that is constant for constant $v_p^{\ast }$. (b,c,d) Plots of the driving force computed using eqn. (11) versus time for each of the three cases in (a). This figure shows that a meaningful relation between driving force and microscopic phase boundary velocity cannot be extracted.

There are multiple ways for finding the driving force. Eqn.(11) gives an explicit form of driving force for double-welled potentials, where γ⁺ and γ⁻ denote the strain right behind and right in front of the phase boundary, respectively. Here, the strain is defined as ${\gamma }_i=\frac{x_i}{a_0}$. However, in practice, we find it difficult to evaluate the exact value of the two strains due to the inevitable thermal fluctuation from the surrounding heat bath and possible oscillation of the springs. In spite of this difficulty, the driving force evaluated using eqn. (11) is plotted as a function of time for each of three combinations of pulling velocity and bath temperature used Fig. 2(a) in the three panels Fig. 2(b,c,d). These plots show that the driving force obtained using eqn. (11) cannot yiled a meaningful kinetic law if plotted together with instantaneous phase boundary velocity. However, the driving force in these plots seems to be fluctuating around a constant value suggesting that it may be possible to evaluate an average driving force that is conjugate to the homogenized phase boundary velocity using a different method. The method we use here is based on energy balance which gives the dashed lines in Fig. 2(b,c,d) and is explained below.

2.5. Energy balance method

Consider a bar occupying an interval of [0, L] in reference configuration. At time t_s, an external force F_e is applied to the bar such that the x = L end is moving at a constant rate v_p in the deformed configuration while the x = 0 end is fixed. At time t_e we stop pulling so that both ends are fixed. Then the following energy balance holds:

$$W_e=\Delta K_m+Q,$$ (19)

where W_e denotes the work input to the system due to F_e, ΔK_m is the change in mechanical energy of the bar and Q is the total dissipation over time interval (t_s, t_e). For the mass-spring chain, applying Newton’s law on the (N + 1)-th mass, we get

$$F_e=\nu v_p+{\sigma }_N,$$

where σ_N is the tension in the N-th spring. Therefore, the total work done on the system is given by the integration of the force F_e with respect to displacement:

$$W_e={\int }_{t_s}^{t_e}{F}_e{v}_{p}dt={\int }_{t_s}^{t_e}(\nu v_p^2+{\sigma }_N{v}_p)dt.$$ (20)

We use the trapezoidal rule to do the numerical integration. Let j denote the j-th time step, then the total external work is the summation of work done for each step:

$$W_e=\sum _j\tau v_p(\frac{{\sigma }_{N,j-1}+{\sigma }_{N,j}}{2}+\nu v_p).$$

The mechanical energy is the sum of the kinetic energy of all masses and potential energy of all springs:

$$\Delta K_m=\{\sum _{i=1}^{N+1}\frac{1}{2}m{v}_i^2+\sum _{i=1}^{N}W(x_i)\}|{}_{t_s}^{t_e}.$$ (21)

The total dissipation involves two parts: (1) dissipation due to propagation of the phase boundary, and (2) dissipation due to the drag of the fluid. The dissipation rate for part (1) is given by eqn.(7), under the assumption that there is only one phase boundary propagating. Thus, the total dissipation due to phase boundary is

$$D_p={\int }_{t_s}^{t_e}f(t)\dot{s}(t)dt.$$

If the phase boundary velocity and the driving force are constant, the above equation can be simplified to

$$D_p=fL,$$ (22)

where L is the length of the bar in the reference configuration. The dissipation due to viscous drag is given by the following:

$$D_v=\sum _i{\int }_{t_s}^{t_e}(\nu {\dot{u}}_i^2-F_i{\dot{u}}_i)dt.$$ (23)

Note that the random force must be included in the equation above because thermal fluctuation and viscous dissipation are intimately connected through the fluctuation-dissipation theorem [41]. To see how, consider the following decomposition of velocity ${\stackrel{.}{u}}_i=\langle {\stackrel{.}{u}}_i\rangle +{\widehat{\stackrel{.}{u}}}_i$, where 〈·〉 denotes the ensemble average and $\langle {\widehat{\stackrel{.}{u}}}_i\rangle =0$. Taking the ensemble average of eqn.(23), we have:

$$\begin{gathered} \langle D_v\rangle =\sum _i{\int }_{t_s}^{t_e}(\nu \langle {\dot{u}}_i^2\rangle -\langle F_i{\dot{u}}_i\rangle )dt \\ =\sum _i{\int }_{t_s}^{t_e}(\nu \langle {\dot{u}}_i{\rangle }^2+\nu \langle {\widehat{\dot{u}}}_i^2\rangle -\langle F_i{\widehat{\dot{u}}}_i\rangle )dt \\ =\sum _i{\int }_{t_s}^{t_e}\nu \langle {\dot{u}}_i{\rangle }^{2}dt. \end{gathered}$$ (24)

The last step assumes an isothermal system, for which $\nu \langle {\widehat{\stackrel{.}{u}}}_i^2\rangle -\langle F_i{\widehat{\stackrel{.}{u}}}_i\rangle =0$, from the fluctuation-dissipation theorem in Langevin dynamics for free particles and harmonic oscillators [41]. Finally, the driving force can be obtained from the energy balance through:

$$f=\frac{1}{L}(W_e-\langle D_v\rangle -\langle \Delta K_m\rangle ).$$ (25)

Here D_v denotes the work done by viscous forces and is given by eqn.(24).

Our method of computing the driving force f using Eqn. (25) is reminiscent of an energy method used by Abeyaratne, Chu and James (ACJ)[49]. In the problem studied by ACJ the energy of the entire system E(s) is considered and the driving force is given by $f=-\frac{\partial E}{\partial s}$. The energy of their system consists of elastic energy in the bar, the energy of a dead loading device, the interfacial energy of the phase boundary and the energy of the imperfections. Then, the driving force is obtained by determining how much this total energy changes when s changes by an infinitesimal amount. In eqn.(25), we consider the energy of all masses and springs, but unlike the stress σ in ACJ which is constant all over the bar, the force in our springs is not the same in all springs due to viscous drag. Also, viscous drag leads to a dissipation other than that caused by phase boundary motion (which was the only dissipating mechanism in ACJ). Therefore, we subtract out the energy dissipated due to viscous drag D_v before computing how much the energy of the masses and springs changed as the phase boundary moved in eqn. (25). We let the phase boundary move the entire length L of the chain because we show in Fig. 2 that the plot of phase boundary position vs. time for several combinations of bath temperature and pulling speed shows a stair-case pattern suggesting that the microscopic phase boundary velocity fluctuates around a constant (homogenized) value and that the driving force computed using eqn. (11) also fluctuates around a constant (homogenized) value as the phase boundary traverses the entire chain. Here we caution the reader that although the microscopic motion of the phase boundary looks like a ‘stick-slip’ motion due to the discreteness of the chain, we use the term ‘stick-slip’ in this paper to refer to the macroscopic kinetic law relating a homogenized driving force to a homogenized phase boundary velocity, which is like a Bingham type viscoelastic fluid in which a threshold driving force must be reached for time evolution to occur.

Before we start our computations to get a kinetic relation, we need to first show that the energy balance method of eqn.(25) gives reasonable results for our mass-spring chain. To verify the energy balance method, we first consider the impact problem of a mass-spring chain with linear springs. In this case there is no phase boundary and the corresponding energy balance is given by

$$W_e-\langle D_v\rangle -\langle \Delta K_m\rangle =0.$$

In practice, we equilibrate the chain with the heat bath and mechanically stretch the chain at a constant rate. Then we compute the total external work and viscous work and compare them with the change of internal energy. The calculation is conducted at two different temperatures T = 460K and T = 260K (T* = 0.032 and T* = 0.018, correspondingly), and two different pulling velocities v_p = 0.05c and v_p = 0.11c ($v_p^{\ast }=0.05$ and $v_p^{\ast }=0.11$, correspondingly). Each case is repeated 300 times to make an ensemble average. We observe in our simulations that the change in kinetic energy of the chain is much smaller than the change in potential energy. Therefore, for simplicity, we ignore the kinetic energy in the mechanical energy change. Table 1 shows that the energy balance works well for a chain of linear springs.

Table 1:

Enegy balance for a chain of linear springs.

Temperature	pulling velocity	ΔKm × 10−3	We − 〈Dv〉 × 10−3	$\frac{(W_e-\langle D_v\rangle -\Delta K_m)}{\Delta K_m}\times 100\%$
T = 460K	vp = 0.05c	2.4295	2.4281	−0.056%
	vp = 0.11c	4.6999	4.6893	−0.011%
T = 260K	vp = 0.05c	2.4281	2.4291	−0.041%
	vp = 0.11c	4.6881	4.6892	−0.023%

Next, we come back to a chain with double-well springs and compare the driving force that we find from energy balance with that known analytically. As we have illustrated above, it is difficult to find the strains right behind and right in front of the phase boundary when the thermal fluctuation is large. Therefore, we set the temperature low enough such that the energy balance can be done with small non-zero thermal fluctuations.

We set $v_p^{\ast }=0.08$ and T* = 0.001. In this case there is only one phase boundary propagating (see Fig. 3(a)). The thermal fluctuation is small enough for us to find the strains right in front of and right behind the phase boundary, γ⁻ and γ⁺, respectively. We obtain the strains for each time step and average over all time steps and over all replicas of the system. The result shows γ⁺ = 3.504 and γ⁻ = 0.949. By eqn.(11) the corresponding driving force is f_c = 2.179. The driving force from the energy balance method eqn.(25) gives f_e = 2.153. The error is within 1%, which shows that the energy balance method agrees with analytical expressions, at least for small thermal fluctuations.

Fig. 3:

Snapshots of a chain with a phase boundary moving from right to left. (a) Snapshot for T* = 0.001, $v_p^{\ast }=0.08$. (b,c,d) Snapshots at three times showing the propagation of phase boundary for T* = 0.024 and $v_p^{\ast }=0.08$. In (b) the phase boundary is at mass number 48, in (d) it is at mass number 45. The instantaneous phase boundary velocity cannot be determined in a straight forward manner from snapshots b,c,d.

2.6. Kinetic relation

Having verified that our energy balance method works we now want to use it to get the kinetic relation for a bi-stable chain at various bath temperatures. We carry out a series of computations for the impact problem under different pulling velocities and different temperatures. The temperature of the heat bath varies from 260K to 460K (T* = 0.018 to T* = 0.032), and the pulling velocity varies from 90m/s to 200m/s ($v_p^{\ast }=0.05$ to $v_p^{\ast }=0.11$). We showed above that the energy balance method is valid in this range.

Under these conditions there is only one phase boundary propagating through the bar. The phase boundary (see Fig. 3(b), (c), (d)) moves in such a way that the spring right in front of the phase boundary jumps into the high strain state first and then the phase boundary “follows up” to the next mass. We had explained the reasons behind the jerky motion of the phase boundary which resulted in the plots of figure 2 earlier. The plots for the corresponding driving forces computed using eqn.(11) also fluctuated so much as to make it impossible to find the microscale kinetic law relating the driving force to the phase boundary velocity. However, we can still find the average driving force from eqn.(25) and homogenized phase boundary velocity from eqn.(18). These will enter a macroscopic kinetic relation just as emerges from spatial homogenization or time averaging in earlier works [49, 33, 34].

Fig. 4(a) and 4(b) show driving force and phase boundary velocities for two different temperatures T* as blue dots. We generate such data for a large number of T* values. We notice the following four facts: (1) at fixed pulling velocity, the phase boundary velocity suffers a tiny increase due to increasing temperature; (2) under same heat bath temperature, the phase boundary velocity increases with increasing driving force; (3) at the same phase boundary velocity, the driving force increases with increasing temperature, indicating a larger dissipation rate due to the phase transition; (4) at the same driving force, the phase boundary velocity decreases with increasing temperature, indicating a smaller dissipation rate due to the propagation of phase boundary. We postulate a kinetic relation in the following form:

$${\dot{s}}^{\ast }=M^{\ast }(T)(\frac{f^{\ast }}{{\widehat{f}}^{\ast }(T)}-1{)}^{\alpha }.$$ (26)

Here ${\stackrel{.}{s}}^{\ast }$ denotes the non-dimensional phase boundary velocity, f* is the non-dimensional driving force. M*(T) and ${\widehat{f}}^{\ast }(T)$ both depend on temperature T. Fits of this equation to the data from our simulations is shown in Fig. 4(a) and 4(b) as red lines. A similar fitting exercise was carried out for all bath temperatures tested. We find $\alpha \approx \frac{1}{3}$, which is independent of temperature. Here M* is a constant (it is like a mobility), and ${\widehat{f}}^{\ast }$ is the minimum driving force to activate the propagation, or the minimum driving force to maintain the phase boundary. We find that both M* and ${\widehat{f}}^{\ast }$ are linear functions of non-dimensional temperature T* as shown in Fig. 4(c) and Fig. 4(d), respectively:

$$M^{\ast }=0.1453T^{\ast }+0.007424,$$ (27)

$${\widehat{f}}^{\ast }={27.44T}^{\ast }+\mathrm{0.3083.}$$ (28)

The dimensional kinetic relation can be easily obtained from the above:

$$\dot{s}=M(T)(\frac{f}{\widehat{f}(T)}-1{)}^{1/3},M=0.006838T+5.063,\widehat{f}=1.894T+308.3,$$ (29)

where f is units of picoNewton, M is in units of m/s, and T is in units of Kelvin.

Fig. 4:

The kinetic relation at various temperatures. (a) T* = 0.018 and (b) T* = 0.032. The red lines are fitting based on eqn. (26). (c) M* is a linear function of temperature. The correlation coefficient is r² = 0.9974. (d) ${\widehat{f}}^{\ast }$ is also a linear function of temperature with correlation coefficient r² = 0.9993.

A power law kinetic relation whose plots look similar to those in Fig. 4(b) has been studied by Abeyaratne, Chu and James [30]. They analyze the propagation of the phase boundary through a row of imperfections and find that in the limit of a large number of small imperfections the effective kinetic relation has a stick-slip type form. Thus, there is a critical value of driving force, a, such that the phase boundary moves only when f > a. a is a measure of energy associated with the imperfections. The slope of the graph of $\stackrel{.}{s}$ versus f is infinite at f = a in the kinetic relation of Abeyaratne, Chu and James [30], just as it is at $f^{\ast }={\widehat{f}}^{\ast }$ in our analytical form in eqn.(26). Temperature does not enter the analysis of Abeyaratne, Chu and James [30], rather they assume that at the microscopic level the kinetic law governing the increase in a phase fraction is linear in the driving force. In a subsequent paper Abeyaratne and Vedantam [31] obtained a stick-slip kinetic law by analyzing a Frenkel-Kontorova type discrete model for kink motion in a crystalline solid and considering its implications at the continuum scale. They followed the methods of Atkinson and Cabrera [35] in which the dispersion relation of a defect moving at constant velocity through a lattice is analytically computed in the Fourier domain assuming that the lattice is subject to a constant externally imposed stress. Abeyaratne and Vedantam computed their own dispersion relation and specialized it to long wave-length waves (long in comparison to the lattice spacing) and showed how a stick-slip type kinetic law for phase boundary motion emerges. Inertia forces were important in their analysis, but the effects of temperature were not considered. A stick-slip type kinetic relation has also been obtained by Bhattacharya for phase boundary propagation in heterogeneous bodies [50]. He shows that irrespective of the microscopic kinetic law one ends up with a jerky motion of the phase boundary at macroscopic length scales if the material properties are heterogeneous. Temperature does not enter Bhattacharya’s analysis either. Finally, De Tommasi et al. [38] inferred a stick-slip type kinetic law for the folding/unfolding transitions in macromolecules, but even they did not consider the effects of temperature and viscous drag. However, bath temperature effects both M and $\widehat{f}$ in our simulations and our chain is homogeneous.

An example of a problem in which a phase fraction evolves in a jerky manner in a thermal environment has to do with plectoneme formation in DNA under high ionic concentrations [29]. This paper puts forth an explanation based on nanoscale friction for the experimentally observed jerky growth of the plectoneme phase when a long DNA molecule is twisted while subject to tension. The torque in the DNA increases as it is twisted while the plectoneme does not grow; the system is stuck at the bottom of an energy well. As the torque increases the energy barrier preventing exit from the well due to a random thermal kick becomes smaller. Min and Purohit [29] find a critical torque for causing plectoneme growth by equating the energy barrier to the thermal energy scale k_BT and show that all experimental observations can be captured. Another example concerns jerky motion of a grain boundary subject to shear loads in a thermal environment as has been observed in molecular dynamic simulations [37] . This paper also gives an explanantion for the jerky motion of the grain boundary by considering how energy barriers change in response to shear stresses and how a critical shear stress is required for the barrier to be surmounted at both zero or non-zero temperature. Motivated by these analyses we followed the methods of Puglisi and Truskinovsky [51] to obtain the free energy landascape from which we can obtain a critical force to move the phase boundary. However, this estimate of the critical force ignores the effects of fluid drag and thermal fluctuations on the chain and does not agree with the results from our simulations. We anticipate that an analysis based on the Fokker-Planck equation [52] will be necessary to explain the mechanisms behind our kinetic relation and it is beyond the scope of the present work.

3. Simulations with initial phase boundary

In the previous section we found a kinetic relation which seems to be a power law after a threshold driving force to initiate motion is crossed; the coefficients M(T) and $\widehat{f}(T)$ are both increasing functions of temperature. The main goal of this section is to verify that this kinetic relation is meaningful even if the boundary and initial conditions are different from the pulling (or impact) experiments considered before. Recall that $\widehat{f}$ is the minimum driving force that maintains a phase boundary. However, in our simulations it is almost impossible to observe such a low driving force. To reach low driving forces we need to decrease the pulling velocity, but when the pulling velocity is smaller than $v_p^{\ast }=0.02$, a second phase boundary appears (see Fig. 5(a)), and thus the assumption of one phase boundary fails. This is somewhat similar to the results of Atkinson and Cabrera [35] who show that multiple modes get excited at low defect velocities in their 1D chain, so that the relation between driving force and defect velocity is jagged for low velocities. However, our chain is different from that of Atkinson and Cabrera because it is immersed in a constant temperature bath (which imposes thermal kicks) and viscous drag (rather than inertia forces) dominates its mechanics. If we further decrease the pulling velocity then the phase transition is dominated by nucleation events (see Fig. 5(b)) in which random portions of the bar change to the high strain phase. The same phenomenon can be observed when we fix the pulling velocity and increase the temperature. For high temperatures and low pulling velocities the elongation is closer to a quasi-static process, resulting in a smaller dissipation rate.

Fig. 5:

Failure of the energy balance method at low pulling velocities. (a) At $v_p^{\ast }=0.01$, a second phase boundary appears at the left end and propagates rightward. It is cumbersome to apply the energy balance method with two propagating phase boundaries. (b) At $v_p^{\ast }=0.002$, the phase transition proceeds through nucleation at random positions. The kinetic relation is irrelevant in this case.

In order to verify that the previous kinetic relation is valid for driving force smaller than ${\widehat{f}}^{\ast }$, we study the motion of a phase boundary that is inserted into the bar as part of the initial conditions. Consider a bar occupying an interval of [−L, L] in the reference configuration. The motion of the bar is governed by eqns.(2),(3),(4). Initially, the left half of the bar is in the low strain phase and the right half is in the high strain phase:

$$\gamma (x,0)=\{\begin{array}{ll}{\gamma }_0^{-},& -L\le x<0,\\ {\gamma }_0^{+},& 0<x\le L.\end{array}$$ (30)

The motion of the bar satisfies the following boundary conditions:

$$\gamma (-L,t)={\gamma }_0^{-},\gamma (L,t)={\gamma }_0^{+}.$$ (31)

There is a phase boundary at x = 0 at t = 0. We would like to observe the evolution of this phase boundary and compare it with the prediction from the kinetic relation eqn.(26). To do this, we use the model of a mass-spring chain to simulate the motion of the bar. Suppose we have 2N+1 masses and 2N springs with an energy landscape as described by eqn.(12). The motion of the masses is governed by the Langevin equation (eqn.(13)). Initially, the left N springs are in low strain phase, and the right N springs are in high strain phase:

$$\begin{gathered} x_i=a_0(1+{\gamma }_0^{-}),i=1,2,\dots ,N, \\ {x}_i=a_0(1+{\gamma }_0^{+}),i=N+1,N+2,\dots ,2N, \end{gathered}$$ (32)

with the lengths at the two ends fixed throughout the simulation.

In our simulation we first fix the 2nd, the N + 1th, and the 2Nth masses and equilibrate the chain with the heat bath. Then, we set free the whole chain and observe the evolution of the phase boundary. We first notice that there exist steady-state solutions under the above boundary conditions by solving the averaged Langevin equations. The whole chain moves at a constant velocity in steady state:

$${\dot{u}}_i=\dot{u}=\frac{a_0({\gamma }^{+}-{\gamma }^{-})+a_0-a_t}{(2N-1)\nu }k,i=1,2,\dots ,2N+1.$$ (33)

However, without the kinetic relation, we cannot determine the position of the phase boundary, and thus the length of each spring in steady state is unknown. Let the (q + 1)th mass be the location of the phase boundary, then lengths of the springs are:

$$\begin{array}{cc}{x}_i=a_0(1+{\gamma }^{-})+\frac{\nu \dot{u}}{k}(i-1),& \hfill i=1,2,\dots ,q,\\ x_i=a_0(1+{\gamma }^{+})-\frac{\nu \dot{u}}{k}(2N-i),& \hfill i=q+1,q+2,\dots ,2N.\end{array}$$ (34)

In steady state, the phase boundary velocity must go to zero. Thus, from the kinetic relation eqn.(26), the driving force at the phase boundary must be f* at steady state. Combining eqn.(26) with eqn.(34) and eqn.(11), we find an analytical form for q:

$$q=N+\frac{f^{\ast }}{\mu \dot{u}}\frac{a_0}{a_t}-\frac{1}{2\mu \dot{u}}k{a}_0({\gamma }^{+}+{\gamma }^{-}-{\gamma }_t)=N-\frac{f_i-f^{\ast }}{\mu \dot{u}}\frac{a_0}{a_t},$$ (35)

where $f_i=\frac{1}{2}k{a}_t({\gamma }^{+}+{\gamma }^{-}-{\gamma }_t)$ denotes the initial driving force. Notice that if the initial driving force is too large, then the phase boundary will keep moving leftwards and finally disappear at the left end if N is not large enough.

Hoping to verify the above analysis we carried out the simulations with N = 100 for different choices of γ⁺ and γ⁻. Generally, we find three possibilities – (1) a moving phase boundary under large driving force, (2) a stationary phase boundary when driving force is smaller than the critical, and (3) nucleation dominated phase transition when the chain is close to mechanical equilibrium. Fig. 6 shows an example in which two of these possibilities are seen. The calculation is carried out at T* = 0.028, with initial strains γ⁻ = 0.0 and γ⁺ = 7.0. In this case, the steady state phase boundary position is q = 38 by eqn.(35). In the beginning, the phase boundary stands in the middle at q = 100 (the yellow line) and moves leftwards at a relatively large velocity. After t = 2000, the phase boundary slows down and nucleation events occur (see the blue line). The numerical results at t = 2000 and t = 3000 are both close to the theoretical solution eqn. (35), which implies that the kinetic relation eqn.(26) gives a good prediction.

Fig. 6:

An example of the verification of eqn.(35) at T* = 0.028. The black line shows the theoretical steady state solution given by eqn.(35). The yellow line represents the spring length initially. There is very little difference between t = 2000 (blue line) and t = 3000 (red line), which implies that the chain reaches steady state (black dashed line) at t = 3000.

4. Continuum and statistical mechanical analysis

Having verfied our kinetic relation we now want to input it as constitutive information for a continuum 1-D bar capable of phase transitions in an isothermal bath. Note that there is no reference to thermal fluctuations in the continuum mechanical description of the 1-D bar, so we expect that with the help of this kinetic relation we should be able to quantify the average force-extension response of the bar. We will check if the average force-extension relation of the bar matches that of a mass-spring chain immersed in an isothermal bath (in which no kinetic relation is specified).

4.1. Using the kinetic relation in a finite difference scheme

In this section we use a finite difference method to simulate the stretching of a 1-D bar and compare the force-extension relation with the result from our mass-spring chain in a heat bath. The constitutive relation for the bar satisfies eqn.(9). The motion of the phase boundary is described by the kinetic relation that we found in the previous section (eqn.(26)). Here f is the driving force on the phase boundary, given by eqn.(11) for bi-linear materials and s(t) is the position of the phase boundary in the reference configuration. The 1-D continuum is discretized into several elements as described in Zhao and Purohit [53].

For nodes away from the phase boundary, the motion is governed by:

$$\frac{\partial \sigma }{\partial x}=d_w\dot{u},$$ (36)

where u(x, t) is the position of the material point x at time t, <(x, t) is the force at that point and d_w is a drag coefficient. We neglect inertia effects and assume that the surrounding fluid is static. The drag coefficient is the same as the one we used in a previous section on Langevin dynamics, $d_w=\frac{2\pi \mu }{\text{log}l∕r-0.8}$. We then discretize eqn.(36) for numerical calculation as:

$$\frac{{\sigma }_{j+\frac{1}{2}}^i-{\sigma }_{j-\frac{1}{2}}^i}{\Delta x}=d_w\frac{u_j^{i+1}-u_j^i}{\Delta t}.$$ (37)

Here i denotes the i-th time step and j denotes the j-th node. Δx is the element length and Δt is the time step. For the stability of the numerical method, the two must satisfy $\Delta t<\frac{(\Delta x{)}^2}{2}\frac{d_w}{k}$.

To numerically integrate the continuum equations of the bar, we need to consider two situations: (1) nodes away from the phase boundary, and (2) nodes close to the phase boundary. We assume that the forces in each element ${\sigma }_{j-\frac{1}{2}}^i$ at time instant i are known for j = 1, 2, 3, …, N + 1, together with the postion of the nodes $u_j^i$. Hence, the position $u_j^{i+1}$ can be computed using eqn.(37). For nodes away from the phase boundary, the forces in each element can be updated by eqn.(9). For nodes close to a phase boundary the above scheme does not work. Let s denote the position of the phase boundary in the reference configuration and assume that it is between node l and node l + 1. To describe the propagation of the phase boundary, we need equations to update the positions of nodes l, l + 1 and s. These can be obtained from: (1) the balance of the linear momentum which requires [|σ|] = 0 across the phase boundary; (2) the total reference length between nodes l and l + 1 is Δx; (3) the continuity of the displacement u; and (4) the kinetic relation that governs the evolution of the position s(t) of the phase boundary. Using these the numerical scheme near the phase boundary is given by [53]:

$$\begin{gathered} \Delta x_{-}^i+\Delta x_{+}^i=\Delta x, \\ {u}_l^{i+1}=u_l^i+\frac{\Delta t}{d_w\Delta x}({\sigma }_d^i-{\sigma }_{l-\frac{1}{2}}^i), \\ {u}_{l+1}^{i+1}=u_{l+1}^i+\frac{\Delta t}{d_w\Delta x}({\sigma }_{l+\frac{1}{2}}^i-{\sigma }_d^i), \\ \Delta x_{-}^{i+1}=\Delta x_{-}^i-{\dot{s}}^i\Delta t, \\ {u}_{l+1}^{i+1}-u_l^{i+1}=(\frac{{\sigma }_d^{i+1}}{k}+1)\Delta x_{-}^{i+1}+(\frac{{\sigma }_d^{i+1}}{k}+1+{\gamma }_t)\Delta x_{+}^{i+1}, \end{gathered}$$ (38)

where ${\sigma }_d^i$ denotes the force at the phase boundary and $\Delta x_{-}^i$ and $\Delta x_{+}^i$ denotes the length in reference configuration of part of this element in the low strain phase and high strain phase, respectively. $\stackrel{.}{s}$ is the phase boundary velocity obtained from eqn.(11) and eqn.(29). In figure 7 we show the results of integrating the continuum equations (black curves) together with those from the mass-spring calculations (blue lines) for various combinations of pulling velocity and bath temperature. The excellent agreement between the two curves indicates that the kinetic relation extracted using the energy balance method from the Langevin dynamics simulations is reliable.

Fig. 7:

The continuum simulation matches the results from Langevin dynamics on a mass-spring chain in a heat bath. The equilibrium approximation for the fluctuation of the extensional force agrees with the the numerical calculation for different temperatures and velocities. (a) T* = 0.032, v* = 0.05; (b) T* = 0.018, v* = 0.05; (c) T* = 0.032, v* = 0.10; (d) T* = 0.018, v* = 0.10;

4.2. Estimating the size of fluctuations

In this section we give a statistical mechanical method to estimate the magnitude of thermal fluctuation in the force-extension relation, given the ergodicity and local equilibrium assumption at the microscale. First, our mass-spring chain immersed in a bath is very similar to a Rouse chain model for polymers immersed in liquid[54]. We compare the time scale of pulling τ_p with the fastest mode of the Rouse model (i.e., the one with the shortest wavelength which will be the length of the springs a₀ in the unloaded configuration for our chain) τ_r:

$$\begin{gathered} {\tau }_p=\frac{a_0}{v_p}\approx 10, \\ {\tau }_r=\frac{a_0^2\nu }{3{\pi }^2{k}_{B}T}\approx 0.37. \end{gathered}$$

Since τ_p ≫ τ_r, our masses are in local equilibrium. Second, our computation of three time scales in section 2.4 showed that the time scale for in-well minimization is the shortest. Further, we notice that in our Langevin dynamics, the extensional force takes the form

$${\sigma }_N=\{\begin{array}{ll}k(x_N-a_0),& x_N<x_c,\\ k(x_N-a_t),& x_N\ge x_c,\end{array}$$

where x_N is the extension of the N-th mass and σ_N is the force on the N-th mass. When there is no phase boundary, the force is a linear function of the extension x_N . The fluctuation in the extensional force could then be written as:

$$\langle {({\sigma }_N-{\overline{\sigma }}_N)}^2\rangle =k^2\langle {(x_N-{\overline{x}}_N)}^2\rangle .$$

Clearly, the fluctuation of the force σ_N depends only on the fluctuation in the length of the n-th spring.

Now, the evolution of the chain is governed by the Langevin equation eqn. (13). If we decompose the motion as the average part plus the fluctuating part, $u_i=\langle u_i\rangle +{\widehat{u}}_i$ and make an ensemble average, we find the equation for the fluctuating part:

$$m{\ddot{\widehat{u}}}_i=-\nu {\dot{\widehat{u}}}_i-\left(\frac{\partial \varphi }{\partial u_i}-\langle \frac{\partial \varphi }{\partial u_i}\rangle \right)+F_i(t).$$

If we temporarily ignore the propagation of the phase boundary, since the potential is bi-quadratic and the stiffness of the springs in both high strain phase and low strain phase is identical, we can find:

$$m{\ddot{\widehat{u}}}_i=-\nu {\dot{\widehat{u}}}_i-\frac{{\partial }^2\varphi }{\partial u_i^2}{\widehat{u}}_i+F_i(t),$$ (39)

with boundary conditions ${\widehat{u}}_0={\widehat{u}}_N=0$. Next, we note that the fluctuation of the motion of each mass i of a length-fixed chain with the phase boundary fixed at a specific spring satisfies the same governing equation and boundary condition. For this reason we use the equilibrium fluctuation to approximate the fluctuation of the extensional force during the stretching.

We define a canonical ensemble Z(N, L, T) of the mass-spring chain. Here N is the number of degrees of freedom, which is the number of masses not located at the boundary. L is the total length of the chain. T is the temperature of the system. The microstate of the system is given by (p₁, u₁, p₂, u₂, …, p_N, u_N;L, T), with the energy:

$$H=\sum _{i=1}^n\frac{1}{2m}{p}_i^2+\varphi (u_1,u_2,\dots ,u_N;L).$$

Here ϕ is the potential energy stored in the springs. For our bi-linear mass spring chain satisfying eqn.(12), the potential energy is given by:

$$\varphi =\sum _{i=1}^{N+1}\frac{k}{2}{(u_i-u_{i-1}-a_i)}^2,u_0=0,u_{N+1}=L.$$

The probability for the system to occupy such a microstate in the phase space satisfies the canonical distri bution:

$$f(p_i,u_i:i=1,2,\dots ,N;L,T)=\frac{1}{Z}\exp (-\frac{H}{k_{B}T}),$$

where k_B is the Boltzmann constant. Z, the partition function, is defined as the normalization factor of the probability distribution:

$$Z(L,T)=\int \exp (-\frac{H}{k_{B}T})\text{d}{p}_1\dots \text{d}{p}_N\text{d}{u}_1\dots \text{d}{u}_N,$$ (40)

with the integration carried out over all degrees of freedom. The free energy of the system is given by:

$$\mathcal{F}=-k_{B}T\log (Z).$$ (41)

The average force exerted at the boundary and the variance of the force are given, respectively, by [55]:

$$\langle \sigma \rangle =\frac{\partial \mathcal{F}}{\partial L}=-k_{B}T\frac{\partial }{\partial L}\log Z,$$ (42)

$$\langle {(\sigma -\langle \sigma \rangle )}^2\rangle =k_{B}T\left(\langle \frac{{\partial }^{2}H}{\partial L^2}\rangle -\frac{{\partial }^2\mathcal{F}}{\partial L^2}\right).$$ (43)

For bi-linear materials satisfying constitutive relation eqn.(12) with the same stiffness for two phases, the partition function from eqn.(40) is given by the following [22]:

$$Z=N^{-\frac{1}{2}}(\frac{2\pi k_{B}T}{h}{)}^N(\frac{m}{k}{)}^{\frac{N}{2}}\sum _{q=0}^{N+1}\frac{(N+1)!}{q!(N+1-q)!}\exp \{-\frac{k}{2k_{B}T(N+1)}[L-(q{a}_0+(N+1-q)a_t]\}.$$ (44)

Further, if the phase boundary is at the q-th mass, with q springs in low strain phase and N + 1 − q in high strain phase, the partition function is:

$$Z=N^{-\frac{1}{2}}(\frac{2\pi k_{B}T}{h}{)}^N(\frac{m}{k}{)}^{\frac{N}{2}}\exp \{-\frac{k}{2k_{B}T(N+1)}[L-(q{a}_0+(N+1-q)a_t]\}.$$ (45)

Combining eqns.(41),(43),(45), we find that the variance of the extensional force, given by:

$$\langle {(\sigma -\langle \sigma \rangle )}^2\rangle =V_{\sigma }^2=\frac{Nk}{N+1}{k}_{B}T\approx k{k}_{B}T.$$ (46)

In figure 7 the red dashed lines above and below the black lines show σ_N ± V_σ. The dashed lines provide a good estimate of the force fluctuations seen in the Langevin dynamics simulations except when the phase boundary first appears when there is a large jump in the extensional force.

5. Cyclic loading and hysteresis

In most experiments on phase transforming materials both loading and unloading are performed in a cyclic manner to estimate the hysteresis [19]. Now we will explore the consequences of our stick-slip kinetic relation on loading rate and temperature dependence of hysteresis. Cyclic loading and unloading of an overdamped chain was recently studied by Benichou and Givli [23, 24] and a relationship between hysteresis, loading rate and energy barriers was explained. In our simulations the energy dissipation at the phase boundary depends strongly on the loading rate as well, but to unravel the hysteresis in cyclic loading we must study the kinetic relation in the third quadrant of the $f-\stackrel{.}{s}$ plane. In our simulations, we use the end state in the extension process for the mass-spring chain as the initial condition for the unloading. We first equilibriate the system at maximum extension, then we unload the chain at the same rate as loading. However, in our simulation if we use the constitutive relation given by eqn.(12), we unrealistically get a strain less than −1 in the low strain phase in many springs. To address this we modify our potential in the discrete mass-spring chain to the following:

$$W=\{\begin{array}{ll}\frac{{10}^{-3}}{2}k{a}_0^2\left({(\frac{a_0}{x_i})}^2-2\frac{a_0}{x_i}+730\right),& x<0.1a_0\\ \frac{1}{2}k{(x_i-a_0)}^2,& 0.1a_0\le x<x_c,\\ \frac{1}{2}k{(x_i-a_t)}^2,& x\ge x_c.\end{array}$$ (47)

Thus, the force in the springs is

$$F=\{\begin{array}{ll}{10}^{-3}k{a}_0\left({(\frac{a_0}{x_i})}^3-{(\frac{a_0}{x_i})}^2\right),& x<0.1a_0\\ -k(x_i-a_0),& 0.1a_0\le x<x_c,\\ -k(x_i-a_t),& x\ge x_c.\end{array}$$ (48)

The coefficients are chosen such that the potential energy and the force are continuous at x = 0.1a₀. The potential consists of a bi-linear part when x > 0.1a₀, and a non-linear part that goes to infinity when x → 0. This ensures that spring length will not be negative in our simulations.

We study the cyclic force-strain relation for the cases T* = 0.018, v* = 0.08 (blue curve) and v* = 0.05 (red curve) in Fig. 8(a). We notice that the force-strain curves for loading and unloading are different. The force in the plateau region is smaller in the unloading process than in the loading process at the same strain. Also, during unloading the phase boundary appears later on the right end than when it disappeared at the left end during loading. When the extension reaches 0, the phase boundary has not yet reached the left end. The above phenomenon leads to hysteresis in the loading-unloading cycle (see Fig. 8(a)(b)). The hysteresis has two parts, (a) the dissipation due to viscous drag of the fluid, and (b) dissipation due the propagation of the phase boundary. It is rate-dependent as can be seen from the red and blue loops in Fig. 8(a) with more hysteresis at higher pulling velocity. In Fig. 8(b) we plot the force-strain curves at two different temperatures but the same pulling speed and find that the hysteresis hardly changes. The finding that the magnitude of hysteresis is almost independent of temperature is consistent with experimental results on several shape memory alloys [56]; however, we must not ignore the fact that the hysteresis in our simulations is dominated by fluid drag (whose coefficient was not changed when we changed the bath temperature), not the dissipation due to phase boundary propagation. Therefore, in the inset of Fig. 8(b) we perform a simulation with low drag coefficient and low pulling speed and parameters for vanadium oxide used in Fig. 1. The inset confirms that hysteresis hardly changes with temperature for this case too in which viscous dissipation is very small and all the dissipation is due to phase boundary propagation. This inset also shows that our mass-spring model can capture rate-independent hysteresis (which implies a stick-slip form of the kinetic law). Finally, in Fig. 8(c) we account for the reduction in drag coefficient with temperature (due to decreasing viscosity of water at higher temperatures) and show that hysteresis can decrease with increasing temperature. A decrease in hysteresis with increasing temperature has been observed in experiments on whelk egg capsules by Miserez et al. [57]. They performed stress-strain experiments (at constant strain rate) while their specimen was immersed in water held at a fixed temperature. They showed, using wide angle X-ray scattering, that a reversible α-helix to β-sheet transition is the reason for the typical stress-strain curves (with plateaus) seen in their experiments. Although their observation of a decrease in moduli with increasing temperature may be the reason for the decrease in hysteresis with increasing tempeature, we speculate on the basis of Fig. 8(c) that the decreasing viscosity of water with increasing temperature should not be ignored.

Fig. 8:

Dependence of hysteresis on loading rate and temperature. (a) The force-strain relation for $v_p^{\ast }=0.08$, T* = 0.018 (blue curve) and $v_p^{\ast }=0.05$, T* = 0.018 (red curve). There is lower hysteresis at lower strain rate. (b) The force-strain relation at $v_p^{\ast }=0.08$ and T* = 0.024 (red curve) and T* = 0.021 (blue curve). The hysteresis is hardly affected by the change in bath temperature. The inset shows force-strain curves at two different temperatures with parameters same as in Fig. 1(c) in which we used a lower drag coefficient and low pulling velocity. (c) The force-strain relation for $v_p^{\ast }=0.08$, T* = 0.021 (blue curve) and T* = 0.024 (red curve) with the decrease in viscosity of water with increasing temperature accounted for. (d) Kinetic relation inferred from the Langevin dynamics calculations in the third quadrant of the $f-\stackrel{.}{s}$ plane for the unloading process.

During unloading the strains lie in the non-linear region in the low strain phase behind the phase boundary. Therefore, the kinetic relation in this case must be different from the extension. Also, eqn.(11) is no longer applicable for this non-linear potential. However, we can still use the energy balance method eqn.(25) to find the driving force. This allows us to plot the kinetic relation in the third quadrant of the $f-\stackrel{.}{s}$ plane. The result is shown in Fig. 8(d). The form of the kinetic relation in this quadrant is similar to that in the first quadrant. For example, at the same driving force the phase boundary moves slower at higher temperature. The kinetic relation also follows the form of eqn.(26) with power $\alpha =\frac{1}{4}$.

6. Concluding remarks

In this paper we have used energy balance to obtain the kinetic relation for a phase boundary in a bistable bar immersed in a heat bath. We performed Langevin dynamic simulations on a mass-spring chain with bistable springs at various pulling velocities and bath temperatures and revealed a jerky motion of the phase boundary due to the discreteness of the system and thermal fluctuations caused by the bath. Yet, we could find a relation between the homogenized phase boundary velocity and an average driving force on it even though the microscopic kinetic law was hard to discern. We found a stick-slip type kinetic relation for the homogenized phase boundary velocity which is characteristic of friction. Our kinetic law bears some similarity to those obtained for phase boundary propagation through a row of imperfections [30] and heterogeneous materials [50] , but we reveal a temperature dependence of the kinetics that had not been seen before. While inertia played an important role in early derivations of stick-slip kinetic laws [35], our chains are dominated by viscous drag and thermal fluctuations that are important for biological macromolecules. The form of the slip part of our kinetic relation is a power law which is different from the Arrhenius form which may be expected for thermally activated phase boundary motion [19]. A result of the stick-slip kinetics is that the force-extension relations are different for the loading and unloading paths which leads to hysteresis. We have also shed light on the role and magnitude of thermal fluctuations in the force-extension response of our bistable bar. The kinetic law derived here may be relevant to the mechanical response of macromolecules, such as, coiled-coil proteins that rely on force-induced structural transitions for their function.