Evidence of materials degradation is all around us. With time, brick walls tend to crack, crumble and eventually fall. Paint on a house tends to crack and peel. Metals tend to oxidized/corrode. Our teeth tend to decay, require fillings, and may eventually require extraction. Semiconductor devices tend to degrade and eventually fail. All materials tend to degrade with time and thus, all devices built from such materials tend to eventually fail [1] .

In physics and engineering, materials degradation mechanisms occur under many names: corrosion, fatigue, cracking, delamination, stress relaxation, hysteresis, charge trapping, time-dependent dielectric breakdown, surface inversion, electro-migration, stress-migration, etc. The list of degradation mechanisms is seemingly endless and the consequences of degradation are very costly. For example, billions of dollars are spent each year in efforts to slow down just one of these degradation mechanisms: corrosion [2] . While materials degradation is usually described in macroscopic terms (such as the term corrosion), it will be shown that the root-cause of degradation generally starts at the microscopic/molecular level.

A material degradation rate (at a fixed temperature) is generally controlled by the activation energy required for the degradation process to proceed [1] [3] - [5] . The purpose of this work is to understand the underlying fundamental physics that controls this inevitable degradation process and learn how we might, at the very least, control the degradation rate. It will be shown that any external force/stress on material that increases the Gibbs Potential will serve to lower activation energy needed for the degradation process to proceed. Even a slight reduction in the activation energy can sometimes have a major impact on a materials degradation rate.

The internal energy U of a system of particles (solids, liquids, gases, plasmas) is the sum of the kinetic and potential energies of all the particles in the system. This internal energy can be increased (as described by the First Law of Thermodynamics) by doing work on the system and/or by the addition of heat to the system, as illustrated in Figure 1 . The First Law of Thermodynamics (energy conservation) can be stated simply: [6]

$\text{d}U=\delta \left(Work\right)+\delta \left(Heat\right)$. (1)

The use of $\delta$ on the right side of Equation (1) is a reminder that these terms are not exact differentials. They depend on the details of how the work is done and how the heat is added.

If the work done on the system is by a conservative force $\xi$, then the potential energy of the system will increase. Several generalized conservative forces and their differential work contributions to the system are shown in Table 1 .

The differential work contributions to a system by conservative forces can be written as:

$\delta \left(Work\right)={\displaystyle {\sum }_i{\xi }_i\text{d}{\epsilon }_i}=-p\text{d}V+F\text{d}L+\sigma \text{d}V+E\text{d}P+H\text{d}M+\cdots$. (2)

As for adding heat $\delta Q$ to the system, it will be assumed that the heat is added in a quasi-static reversible fashion such that it can be described by the Second Law of Thermodynamics:

$\delta \left(Heat\right)=T\text{d}S$, (3)

where T system temperature and $\text{d}S$ is the change in system entropy [7] . Note that as we add heat to the system, then the system entropy will increase. If heat is removed from the system, then the entropy of the system decreases. The total change in internal energy of the system can now expressed by the First Law:

$\Delta U=\delta \left(Work\right)+\delta \left(Heat\right)={\displaystyle {\sum }_i{\xi }_i\text{d}{\epsilon }_i}+T\Delta S$. (4)

The Gibbs potential G for a system of particles (plasmas, gases, liquids, or solids) is defined as:

$G=U-TS$. (5)

The Gibbs potential is defined in such a way that any natural/spontaneous changes in the Gibbs potential $\text{Δ}G$ must lead to a lower Gibbs potential. U can be lowered by the natural/spontaneous bonding of electrons and protons to form atoms, bonding of atoms to form molecules, and finally the bonding of molecules to form solids. Thus, at lower temperatures, bonded molecules in a solid will have a lower Gibbs potential. However, at elevated temperatures, an increase in entropy (chaos) can lead to a lower Gibbs potential. This is why, with ever-increasing temperatures, solids turn into liquids, liquids turn into gases, and gases turn into plasmas (these are generally referred to as the four states of matter).

The change in Gibbs potential for a system (at a fixed temperature) can be written as

$\Delta G=\Delta U-T\Delta S$. (6)

Inserting Equation (4) into Equation (6), one obtains:

$\Delta G=\delta \left(Work\right)={\displaystyle {\sum }_i{\xi }_i\text{d}{\epsilon }_i}$. (7)

One can see that work done on a system (by conservative forces) will increase the Gibbs potential. This increase in Gibbs potential is the amount of energy that can be stored in a system and that can be used (at some later time) to do work on the environment. Examples of energy storage devices/materials are alluded to in Table 1 : batteries, fuel cells, pressure vessels, dielectrics, magnetics, elastic materials, etc. Because the stored energy is available to do work, it has historically been described as “free energy”. However, while in this stored-energy state (higher Gibbs Potential) the system is metastable and will look for ways of spontaneously/naturally finding lower Gibbs potential states. This is the fundamental physics behind materials degradation.

The rest of our degradation focus will be primarily be on degradation mechanisms in solids. From here on, we will refer to any solid in a state of higher Gibbs Potential (due to work done on the solid by conservative forces) as a stressed state. The stressed state will have a higher Gibbs potential and thus a lower activation energy is needed for the degradation to proceed (as illustrated in Figure 2 ). Any reduction in the activation energy will serve to accelerate the degradation rate given by:

$\text{Degradation}\text{Rate}\propto \exp \left(-\frac{\Delta G^{*}}{K_{B}T}\right)$, (8)

where $\Delta G^{*}$ is the activation energy needed for the reaction to proceed and $K_B$ is Boltzmann’s constant [8] .

For a conservative stress $\xi$ acting on a system (at a fixed temperature), the change in the Gibbs Potential can be expressed as a series expansion (see Appendix A):

$\Delta G\left(\xi ,T\right)=T{\displaystyle {\sum }_{m=1}^{\infty }{a}_m{\left(\frac{\xi }{T}\right)}^m}$. (9)

From here out, we will refer to Equation (9) as the generalized Gibbs Potential for a stressed system. Keeping terms in the expansion only through second order (justification will be given shortly), one obtains:

$\Delta G\left(\xi ,T\right)=\Delta U-T\Delta S=\delta \left(Work\right)=a_1\xi +\frac{a_2}{T}{\xi }^2$. (10)

One can see (from the above equation) that the linear and quadratic terms can have an impact on both the internal energy and the entropy. The impact of the stress $\xi$ on system entropy is given by:

$T\Delta S=-T{\left(\frac{\partial \Delta G}{\partial T}\right)}_{\xi }=\frac{a_2}{T}{\xi }^2$, (11)

and the impact of the stress $\xi$ on the system internal energy is given by:

$\Delta U=\Delta G+T\Delta S=a_1\xi +2\frac{a_2}{T}{\xi }^2$. (12)

One should note that the quadratic term has a Curie Law (1/T) dependence. The Curie Law will be present if the stress $\xi$ is having a significant impact on the change in system entropy. We will now investigate how well the generalized Gibbs Potential describes several actual systems.

In the above examples (for elastic materials, magnetic materials, and dielectric materials), the generalized Gibbs Potential [Equation (10)] was found to be successful in predicting a quadratic term in stress $\xi$ that would describe the amount of useful energy that could be stored in the system and is free/available to do work at some later time. The generalized Gibbs Potential was also successful in producing the Curie Law. These are significant accomplishments for the Generalized Gibbs Potential. However, $\Delta G$ was also expected to have both linear and quadratic terms in the stress $\xi$ variable. Does this mean that the linear term is not important for degradation? As we will now show, the answer to this question can be a strong no.

Let us reconsider the case of solid dielectric materials in more detail. Figure 3 shows a solid dielectric material with polar/ionic bonds. When we apply a field and do conservative work on the dielectric, a polarization (net dipole moment per unit volume) is produced, but at the expense of lattice distortion. Due to the polar/ionic bonding of the molecules in the solid dielectric, we see that the polarization (due to the electric field E) has produced alternating tensile and compressive molecular bonds within the dielectric material.

To further illustrate the impact of field on polar/ionic bonding (at the microscopic level), in Figure 4 , we show an example of a lattice-constrained molecule with tetrahedral polar/ionic bonding that occurs in a dielectric such as SiO₂.

The conservative work done by the electric field acting on the dipoles shown in Figure 4 is given by $p_0\cdot E$. The work done by E on the top half of the lattice-constrained tetrahedral-molecule is given by

$w_{top}=\frac{1}{2}\left(p_{0}E+\text{Δ}{p}_{0}E\right)$. (25)

The work done by E on the bottom half of the molecule is given by:

$w_{bottom}=\frac{1}{2}\left(-p_{0}E+\text{Δ}{p}_{0}E\right)$. (26)

The total conservative work done by the electric field on the constrained molecule becomes:

$\delta w=w_{top}+w_{bottom}=\text{Δ}{p}_{0}E$. (27)

Since the induced dipole moment is given by $\text{Δ}{p}_0=\alpha E$, then

$\delta w=\alpha E^2$, (28)

where $\alpha$ is the molecular polarizability. The change in internal energy for the molecule becomes:

$\Delta u=\alpha E^2+T\Delta s$. (29)

The change in the Gibbs Potential becomes:

$\Delta g=\Delta u-T\Delta s=\alpha E^2$. (30)

As expected, from Equation (19) for dielectrics, we find that the macroscopic useful/free-energy stored in the lattice-constrained molecule depends quadratically on the field. However, as illustrated in Figure 5 , the degradation rate will be impacted strongly by the microscopic linear field term $p_{0}E$.

One can see in Figure 5 that we have two distinctly different potential energy states for the dipoles. Lattice constrained dipoles oriented antiparallel to the electric field will have a significantly higher Gibbs Potential and will therefore be much more unstable and more prone to bond breakage. The change in the Gibbs potential for the antiparallel portion of the molecule is given by

$\Delta g_{antiparallel}=\Delta u-T\Delta s=\frac{1}{2}\left(p_{0}E+\alpha E^2\right)$. (31)

This is consistent with the Generalized Gibbs Potential Model [Equation (10)] in that it predicts both a linear and quadratic term will be present for polar/ionic dielectrics. Previously, it has been shown that the linear term ( $p_{0}E$) is at least two orders of greater larger than the quadratic term ( $\alpha E^2$). [10] This means that the linear term in field E will dominate the degradation rate for polar/ionic dielectrics even though the stored free/useful energy in the dielectric is proportional to $E^2$. Note that the antiparallel dipoles have a higher Gibbs potential by $+2p_{0}E$ than the parallel dipoles. This makes the antiparallel dipoles much more unstable and much more prone to degradation/bond-breakage. All of this is of fundamental importance for describing time-dependent dielectric breakdown (TDDB) and is the microscopic physical basis for the thermochemical E-model. [10]

We must also discuss the role of the local electric field. The local electric field is the field that actually distorts the polar/ionic molecules. The local electric field is the sum of the external electric field E plus the dipolar field from neighboring molecules. The local electric field is described by the Lorentz relation:

$E_{loc}=E+L{\chi }_{e}E=\left[1+L\left({\epsilon }_r-1\right)\right]E$, (32)

where L is the Lorentz factor, ${\chi }_e$ is the electric susceptibility, and ${\epsilon }_r$ is the relative dielectric constant. [11] L =1/3 is normally used for very thin dielectrics but it has been shown more recently that L actually increases from 1/3 to asymptotically approaching 1 for thick dielectrics. [12] As shown in Figure 6 , the increase in $E_{loc}$ (with dielectric thickness) tends to explain the reason that thicker dielectrics have lower breakdown strength. [13] [14]

Also, as described by Equation (32), $E_{loc}$ increases with the relative dielectric constant ( ${\epsilon }_r$) and this tends to explain why high dielectric constant materials have lower breakdown strength as shown in Figure 7 . [15]

In summary, for polar/ionic dielectric degradation, we have shown that both the linear and the quadratic terms in the Generalized Gibbs Potential Model can be important for degradation. The linear term tends to appear at the microscopic level whereas the quadratic term tends to appear at the macroscopic level. As for degradation in polar dielectrics, the linear term can be orders of magnitude greater than the quadratic term. [10]

Before leaving this section, it is instructive to look at non-polar/non-ionic dielectrics. Such a non-polar solid dielectric is shown in Figure 8 . As is illustrated, the electric-field E induces a polarization simply by the shifting of the electronic cloud(s) about each atom, thus little/no lattice distortion occurs.

Since there is little/no lattice energy going into increasing Gibbs Potential (for non-polar/non-ionic materials), then the usually dominant linear $a_1\xi$ term in $\Delta G$ for polar/ionic dielectrics is not relevant for non-polar/non-ionic dielectrics.

This means that only the much weaker second-order quadratic $\frac{a_2}{T}{\xi }^2$ term will be present in $\Delta G$. Thus, non-polar/non-ionic dielectrics are much more stable and much less prone to TDDB. This is shown in Table 2 .

Whether a dielectric is polar or non-polar can be easily identified by comparing the relative dielectric constant ${\epsilon }_r$ with the refractive index $n^2$. If they are equal, then the material is non-polar. As for the role of leakage current through the dielectric (at elevated electric fields) possibly impacting TDDB, [16] current flow might be expected to help catalyzing the bond-breakage process and thus impacting TDDB [17] , but it cannot be the fundamental mechanism for TDDB. Current flow will occur in both polar and non-polar materials, but only polar materials experience TDDB. The same can be said to be true for hydrogen release TDDB models. [18] Hydrogen release (due to current flow) might be expected for both polar and non-polar materials, but only polar materials experience TDDB.

A generalized Gibbs potential model for degradation in solid materials is shown in Figure 9 . When a system is stressed, it forces the system into a state of higher Gibbs potential, making the system metastable and more prone to degradation. Using reaction rate theory, the rate of degradation will depend on the forward reaction activation energy $\text{Δ}{G}_0^{*}$ and the reverse reaction activation energy $\text{Δ}{G}_1^{*}$. One can see that the forward reaction rate (degradation rate) will be strongly impacted by activation energy lowering by $\Delta G\left(\xi ,T\right)$.

From Figure 9 , one can construct a net degradation rate:

$\begin{array}{c}{R}_{degradation}=R_{forward}-R_{reverse}\\ ={\upsilon }_0\left[\exp \left(-\frac{\Delta G_0^{*}-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)-\exp \left(-\frac{\Delta G_1^{*}}{K_{B}T}\right)\right],\end{array}$ (33)

where ${\upsilon }_0$ is the interaction/collision frequency with the potential barriers. If $\text{Δ}{G}_0^{*}\ll \text{Δ}{G}_1^{*}$, then it is rather obvious that the forward reaction dominates and the degradation rate is given simply by:

$R_{degradation}\cong {\upsilon }_0\exp \left(-\frac{\text{Δ}{G}_0^{*}-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)$. (34)

However, in Appendix B we discuss perhaps a more interesting case of $\text{Δ}{G}_0^{*}\cong \text{Δ}{G}_1^{*}$. In this case, the reverse reaction can be significant. For small values of stress [small values of $\Delta G\left(\xi ,T\right)$], Equation (33) reduces to:

$R_{degradation}\cong {\upsilon }_0\frac{\Delta G\left(\xi ,T\right)}{K_{B}T}\exp \left(-\frac{Q}{K_{B}T}\right)$. (35)

where Q is the average activation energy for the forward and reverse reactions. For large values of stress [ $\Delta G\left(\xi ,T\right)$ is on the same order of magnitude as Q], then Equation (33) reduces to:

$R_{degradation}\cong {\upsilon }_0\exp \left(-\frac{Q-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)$. (36)

For intermediate values of stress, a simple power law is a reasonable approximation to bridge the gap between low stress [Equation (35)] and high stress [Equation (36)] conditions:

$R_{degradation}\cong {\upsilon }_0{\left(\frac{\Delta G\left(\xi ,T\right)}{K_{B}T}\right)}^m\exp \left(-\frac{Q}{K_{B}T}\right)$. (37)

Because the activation energy Q is usually on the order of an eV, the exponential temperature dependence tends to dominate the smaller temperature dependence in the pre-factor. For this reason, the temperature dependence in the pre-factor is often ignored or simply incorporated into Q as an effective activation energy $Q_{eff}$. In practice (reliability engineering), a simplified expression is typically used for degradation analysis:

$R_{degradation}\cong B_0{\xi }^n\exp \left(-\frac{Q_{eff}}{K_{B}T}\right)$. (38)

Time-To-Failure TF models are easily determined since,

$TF\propto \frac{1}{R_{degradation}}=A_0{\xi }^{-n}\exp \left(\frac{Q_{eff}}{K_{B}T}\right)$. (39)

Historically, the Gibbs potential has been very important for science and engineering. As we have seen, if one does reversible work on a system against conservative forces, then Gibbs potential/free-energy will rise (potential energy is stored in system). In this case, $\Delta G$ represents the maximum amount of useful energy that is available/free to do work at some later time. Thus, the Gibbs potential is rather ideal for describing useable energy storage. This has been the focus for the Gibbs potential since Gibbs first introduced it for chemical/fluid systems. However, what has received much less attention is the fact that the materials, in a Gibbs potential/free-energy storage device, are metastable and the materials will degrade with time. The materials degradation not only impacts the amount of available/free-energy in the storage device but, perhaps more importantly for reliability, the degradation of the material causes eventual storage-device failure. This is seemingly a fundamental law of nature and degradation physics—all materials/systems put into a state of higher Gibbs potential are metastable and will spontaneously/naturally degrade with time.

A generalized Gibbs Potential Model has been introduced here and it seems to be well suited for describing the degradation physics for materials under stress. At the most fundamental level of understanding, the Gibbs potential/free-energy has two competing terms: a cohesive term (where bonding is preferred) and dispersive term (where bonding disruption is preferred). At lower temperatures, the cohesive term dominates. At higher temperatures, the dispersive term dominates. When we apply a generalized stress $\xi$ to a material, it tilts the Gibbs potential/free-energy in a direction that favors the dispersive/disruptive term. We have presented a general approach to degradation physics that describes how the Gibbs potential/free-energy changes when a material is put under stress. The stress tends to lower the activation energy needed for an enhanced degradation rate.

Materials are often fabricated into metastable states that will inevitably degrade. For example, pure metals (used for their conductive properties) must be purified and fabricated for use. However, pure metals (with perhaps gold being only the exception) are not found in nature. Metals are only found in their metal-oxide state. This is because the metal-oxide state has a lower free energy than pure metal. Thus, a device fabricated with pure metal is likely to degrade by oxidation/corrosion. Important semiconductor materials such a Ge, Si, GaA, GaN, and SiC do not exist in nature and thus must be purified and fabricated. Pure dielectrics such as SiO₂ are not found in nature in a purified state. Thus, materials purification and fabrication are required. All semiconductor devices (which are so important for everyday life) are fabricated from metastable materials (semiconductors, metals, dielectrics). Fabrication of these materials into metastable states has thus put them in a state of higher Gibbs potential/free-energy and they will degrade. Thus, this inevitable degradation process must be controlled through design and process.

Perhaps without realizing it, we have been using Gibbs potential all along but possibly under a different name. The historical story of Newton seeing an apple falling from a tree limb to earth is a good example. Newton described the falling-apple event as due to the force of gravity (a gradient existed in the gravitational potential). Another way of looking at this event—the apple existed in a metastable state (higher gravitational potential energy) while hanging on the tree limb. While in this state of higher Gibbs potential, microscopic/molecular degradation occurred within the apple’s stem. The degradation within the stem continued until the stem could no longer support the weight of the apple and it fell. This would be a Gibbs potential explanation for the apple spontaneously/naturally falling from the tree.

Several of the driving forces commonly used for degradation analysis are illustrated in Figure 10 .

The historical focus of the Gibbs potential has been primarily concerned with how much recoverable energy is possible whenever a system is stored in a state of higher Gibbs potential (electrical, mechanical, chemical, etc.). While the amount of free/available energy is certainly important (from a practical engineering point of view), but perhaps something even more profound (from a reliability physics point of view) was not emphasized enough—any system put into a state of higher Gibbs potential is fundamentally more unstable and will degrade with time. This is apparently an important fundamental property of nature and is the driving force for degradation physics.

For a conservative stress $\xi$ acting on a material (at a fixed temperature T), the change in the Gibbs Potential can be expressed as a series expansion of $\xi$ and T. Keeping terms in the expansion only through second order, it was shown that the quadratic term tends to describe the amount of useful energy (Gibbs free energy) that can be stored in a material. In addition, it was shown that the empirical Curie Law is theoretically produce. While the quadratic term predicts the amount of useful energy that can be stored in a material (a macroscopic effect), the linear term (a microscopic effect) can often dominate the degradation rate for the material while in this state of higher Gibbs potential. It was shown that, for thermally activated degradation mechanisms, the degradation rate can be described well by either an exponential or a power-law dependence on the stresss $\xi$. This theoretical work tends to give a fundamental physics basis for the exponential and power-law time-to-failure models that are commonly used today in reliability engineering.

Materials degradation is a fundamental property of nature. The degradation rate for a material (at a fixed temperature) tends to increase with the applied stress level (electrical, mechanical, or chemical). Using a Generalize Gibbs Potential, and reaction rate theory, it was shown that the degradation rate is expected to be thermally activated and linearly dependent on the stress level (for low values of stress) but exponentially dependent on the stress (for high values of stress). A power-law stress dependence and an effective activation energy is generally used for intermediate levels of stress.

For a single conservative stress $\xi$ acting on a system that produces a system response $\epsilon$, the change in internal energy U can be written:

$\text{d}U=\delta \left(Work\right)+\delta \left(Heat\right)=\xi \text{d}\epsilon +T\text{d}S$ (A1)

where T is the temperature and $\text{d}S$ is the change in entropy. Since U is a function of the extensive variables $\left(\epsilon ,S\right)$, one can write:

$\text{d}U={\left(\frac{\partial U}{\partial \epsilon }\right)}_S\text{d}\epsilon +{\left(\frac{\partial U}{\partial S}\right)}_{\epsilon }\text{d}S$. (A2)

Comparing Equations (A.1) and (A.2), one obtains:

$\xi ={\left(\frac{\partial U}{\partial \epsilon }\right)}_S$ (A3)

and

$T={\left(\frac{\partial U}{\partial S}\right)}_{\epsilon }$ (A4)

Using the Euler theorem, one can write:

$U={\left(\frac{\partial U}{\partial \epsilon }\right)}_S\epsilon +{\left(\frac{\partial U}{\partial S}\right)}_{\epsilon }S$. (A5)

Finally, using Equations (A.3) through (A.5), we obtain:

$U=\xi \epsilon +TS$. (A6)

The Gibbs potential/free-energy for a system (where no pdV work is being done) is defined as:

$G=U-TS$. (A7)

For the unstressed system, one has:

$G_0=U_0-T{S}_0$. (A8)

For the stressed system, one has

$G_{stressed}=U_0+\xi \epsilon -T{S}_{stressed}$. (A9)

Therefore, $\text{Δ}G$ becomes:

$\Delta G=G_{stressed}-G_0=\xi \epsilon -T\left(S_{stressed}-S_0\right)$. (A10)

The Gibbs potential is a function of intensive parameters $\xi$ and T. We will attempt to use the method of separation of variables and write Equation (A.10) as:

$\Delta G=\xi \epsilon -T\left(S_{stressed}-S_0\right)=f\left(\xi \right)g\left(T\right)$. (A11)

From Equation (A.10), we have the relations:

$\epsilon ={\left(\frac{\partial \left(\Delta G\right)}{\partial \xi }\right)}_T$ (A12)

and

$\left(S_{stressed}-S_0\right)=-{\left(\frac{\partial \left(\Delta G\right)}{\partial T}\right)}_{\xi }$. (A13)

Substituting Equations (A.12) and (A.13) into Equation (A.11), we obtain:

$\xi g\frac{\text{d}f}{\text{d}\xi }+Tf\frac{\text{d}g}{\text{d}T}=fg$. (A14)

Dividing both sides of Equation (A.14) by $fg$, we obtain a separation of variables equation:

$\frac{\xi }{f}\frac{\text{d}f}{\text{d}\xi }+\left(\frac{T}{g}\frac{\text{d}g}{\text{d}T}-1\right)=0$. (A15)

Since the independent variables are now separated, then the only way the two terms can add to zero is by both equaling some constant m:

$\frac{\xi }{f}\frac{\text{d}f}{\text{d}\xi }=m$ (A16)

and

$\frac{T}{g}\frac{\text{d}g}{\text{d}T}-1=-m$. (A17)

The solution to Equation (A.16) is:

$f={\xi }^m$, (A18)

and the solution to Equation (A.17) is:

$g=T^{1-m}$. (A19)

The full separation of variables solution becomes:

$\Delta G=T{\displaystyle {\sum }_{m=0}^{\infty }{a}_m{\left(\frac{\xi }{T}\right)}^m}$. (A20)

Since $\Delta G=0$ at $\xi =0$, then the separation of variables solution reduces to:

$\Delta G=T{\displaystyle {\sum }_{m=1}^{\infty }{a}_m{\left(\frac{\xi }{T}\right)}^m}$. (A21)

Finally, for a stressed system we have the series expansion:

$G_{stressed}=G_0+T{\displaystyle {\sum }_{m=1}^{\infty }{a}_m{\left(\frac{\xi }{T}\right)}^m}$. (A22)

The net degradation rate in Figure 9 is governed by the forward and reverse reaction rates:

$\begin{array}{c}{R}_{degradation}=R_{forward}-R_{reverse}\\ ={\upsilon }_0\left[\exp \left(-\frac{\Delta G_0^{*}-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)-\exp \left(-\frac{\Delta G_1^{*}}{K_{B}T}\right)\right],\end{array}$ (B1)

where ${\upsilon }_0$ = the fundamental vibrational/interactional-frequency for the atoms in the solid system and where $\text{Δ}{G}_0^{*}$ and $\text{Δ}{G}_1^{*}$ are the activation energies illustrated in Figure 9 . We note that if $\text{Δ}{G}_0^{*}\ll \text{Δ}{G}_1^{*}$, then the forward reaction dominates and the degradation rate is given simply as:

$R_{degradation}\cong {\upsilon }_0\exp \left(-\frac{\text{Δ}{G}_0^{*}-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)$. (B2)

Now, let us consider the case where $\text{Δ}{G}_1^{*}\cong \text{Δ}{G}_0^{*}$. Under these conditions, the reverse reaction $R_{reverse}$ can be significant. Rearranging Equation (B1), we obtain:

$R_{degradation}={\upsilon }_0\exp \left(-\frac{\Delta G_0^{*}-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)\left[1-\exp \left(-\frac{\Delta G_1^{*}-\Delta G_0^{*}+\Delta G\left(\xi ,T\right)}{K_{B}T}\right)\right]$. (B3)

Using the identity:

$2\exp \left(-\frac{x}{2}\right)\sinh \left(\frac{x}{2}\right)=1-\exp \left(-x\right)$, (B4)

One can write:

$R_{degradation}=2{\upsilon }_0\exp \left(-\frac{\Delta G_1^{*}+\Delta G_0^{*}-\Delta G\left(\xi ,T\right)}{2K_{B}T}\right)\sinh \left(\frac{\Delta G_1^{*}-\Delta G_0^{*}+\Delta G\left(\xi ,T\right)}{2K_{B}T}\right)$.(B5)

Since we are assuming that strong reverse reactions can occur only when $\text{Δ}{G}_1^{*}\cong \text{Δ}{G}_0^{*}$, then we can define an average activation energy for degradation as Q, where:

$Q=\frac{\text{Δ}{G}_1^{*}+\text{Δ}{G}_0^{*}}{2}$. (B6)

Assuming that Q is $\gg \Delta G\left(\xi ,T\right)$, then for small $\Delta G\left(\xi ,T\right)$ Equation (B5) reduces to:

$R_{degradation}\cong {\upsilon }_0\frac{\Delta G\left(\xi ,T\right)}{K_{B}T}\exp \left(-\frac{Q}{K_{B}T}\right)$. (B7)

For large values of $G\left(\xi ,T\right)$ (on the same order of magnitude as Q), Equation (B5) reduces to:

$R_{degradation}\cong {\upsilon }_0\exp \left(-\frac{Q-\Delta G\left(\xi ,T\right)}{K_{B}T}\right)$. (B8)

For intermediate values of $\text{Δ}G\left(\xi ,T\right)$, a power law is a reasonable approximation to bridge the gap between low stress (Equation (B7)) and high stress (Equation (B8)).

$R_{degradation}\cong {\upsilon }_0{\left(\frac{\Delta G\left(\xi ,T\right)}{K_{B}T}\right)}^m\exp \left(-\frac{Q}{K_{B}T}\right)$. (B9)