Kinetics of Thermally Activated Physical Processes in Disordered Media
Abstract
:1. Problem: Dynamic Evolution of Physical Quantities in Disordered Media
1.1. Under an External Action
1.2. Thermal Relaxation
2. Modeling
2.1. Exponential versus Non Exponential Kinetics (Introduction to the Theory)
2.2. Kinetics of Writing
2.2.1. First Group of Assumptions
- (1)
- We can easily imagine that the reaction leading to refractive index change contains an absorption step on a species (for instance Oxygen Deficient Center), followed sometimes by a relaxation to another excited level (Figure 20). Then, several pathways are possible (one to luminescence, another one to bond breaking, migration, and structural rearrangement). All of them finally contribute to refractive index change. The first assumption we make here is that only one elementary reaction is involved: the limiting reaction of the process. We can write the reaction A→B.
- (2)
- The second assumption is that it is thermally activated (with an activation energy E). The Arrhenius law can be used, the d[B]/dt rate is proportional to a reaction constant k0 like k(E, T) = k0exp(−E/kBT), where kB is the Boltzman constant and [B] is the concentration of the B species.
- (3)
- The third assumption is that activation energy is distributed. As a matter of fact, in glasses, E is not unique. The creation energy of B can vary with its atomic environment. The transition state energy (the bottleneck on the reaction pathway) is also sensitive to the disordered environment in such a way that the activation energy, which is the energy difference between B and the transition state, is distributed. This means that chemical pathways are variable depending on the configuration distribution of the transition state and/or the initial stable states throughout the glass. This is at the origin of ergodicity lost because space-time points are no more equivalent. The reaction is faster in some places than in others. This is also heterogeneity, but not at a macroscopic level, just at a microscopic one.
2.2.2. The Assumption on the Reaction (Connection with Time and Temperature)
- (1)
- It is worth noting that the reaction properties appear now only in Ed(t, T), on one hand (the bound of the integral), and, on the other hand, the structural disorder appears in the integrand.
- (2)
- Note that the demarcation energy approximation is valid only if g(E) varies slower than x(E, t, T). The width of is 2kBT. Otherwise, a correction has to be added to the integral [32]:
- (3)
- Most interesting, the differentiation of B against Ed() yields the shape of the distribution function.
- (4)
- As B is expressed with only one variable B(Ed). It exhibits the same shape as B(T) for given t or B(ln(t)) for given T. Thus, we deduce that if g(E) does not depend on T, isochrons and isotherms are equivalents:
2.2.3. Examples for Time Dependence Computation
- (Step 1)
- Integration using a Gaussian distribution as the following:
- (Step 2)
- The integration of a sigmoid differentiation distribution:
- (Step 3)
- with a Poisson distribution:
2.2.4. How to Obtain k0 and g(E) (Figure 23)
- (1)
- Measurement of several isochrons or isotherms writing kinetics and replotting against a new abscissa;
- (2)
- k0 fitting for curve collapsing and;
- (3)
- Differentiation .
2.3. Kinetics of Erasing
2.3.1. First Group of Assumptions
- (1)
- One elementary reaction (limiting process), A→B is the writing process;
- (2)
- Thermal activation, E−, ;
- (3)
- E− is distributed, i.e.,
- -
- variable chemical pathways;
- -
- the saddle point or the B state are sensitive to various structural configurations as decrived in Figure 24;
- -
- g(E) is the distribution function for backward reaction, it can be Gaussian for a thermal disorder or Poisson for a diffusional disorder or differentiation of a sigmoid, or also top hat function.
2.3.2. Second Group of Assumptions (towards the Computation of Time Dependence)
- (1)
- Most of the elementary reactions in solids are first order; some are second order, when two species associate, but these are much less likely. Thus, this is not so severe and we can write: , , is the advancement degree of the reaction.
- (2)
- Concept of demarcation energy, Ed
- Some Consequences of the Theory:
- (1)
- [B](t, T) can be expressed as a function of the unique variable called demarcation energy (). The curve is called the master curve (MC), as it is unique, regardless of what the (t, T) couple may be for a given value of Ed. We can note that T is equivalent to lnt in and, thus, isochronal ageing data are equivalent to isothermal ageing data for establishing MC. Note that from a practical point of view, most people consider that NICC() is the master curve. Now, the MC plot allows the user to predict the grating lifetime, providing that the anticipated conditions of BG (i.e., = f(tuse, Tuse) correspond to a point on the MC that has been actually sampled during the annealing experiment.
- (2)
- Notice also that the distribution function is included in the MC differentiation as: .
- (3)
- Thermal stability increases along with the ageing time (hardening). This is due to the fact that the less stable sites are removed along the ageing.
- -
- (1) When the pathways are the same in both senses, the distribution function is the same for forward and backward reactions, as shown in the scheme below. If the writing kinetics are not saturated, the forward distribution function is not saturated and, thus, neither is the backward distribution. The stability is dependent on the writing (time and power density that is included in , see B0(Ed) in definition)
- -
- (2) When there is no correlation between the distribution functions because the pathways are not the same, there are two independent distribution functions, as shown in the scheme below. The B sites are randomly occupied whatever its stability. The complete distribution function of the backward reaction has to be used. The stability is independent of writing (i.e., on the initial grating strength), see Section 2.3.3.
2.3.3. A Few Examples of Time Dependence Computations Assuming B0(E) = B0
2.3.4. Incomplete forward Reaction in the Case of Dependent Backward and Forward Pathways (B0(E)) [36]
2.3.4.1. Gaussian Distribution
2.3.4.2. Sigmoid Differentiation Distribution
2.3.4.3. Poisson Distribution
3. Basic Experimental Analysis of the Erasure
3.1. Distribution Function Determination for the Reverse Reaction
- (1)
- note:
- (2)
- with the assumptions 2.3.1 and 2.3.2 (and even less, i.e., assumption on first order reaction is not necessary).
- (1)
- Measurement of several isochrons or isotherms:
- (2)
- k0 fitting for curve collapsing. We adjust the constant k0 for obtaining the collapse of the curves like in the Figure 29.
- (3)
- Differentiation of the previous curve. .
3.2. Practice (or application of the theory)
3.3. Stabilization
- -
- the non-passivated master curve after annealing is truncated and needs to be renormalized.
- -
- the origin of the time is usually taken from the beginning of the ageing after annealing and, thus, a time shift has to be applied on the previous demarcation energy. This is not negligible in the first 3% of ageing (see page 52 of [38]).
4. Peculiar Properties in the Frame of Assumption Validity (Action of the Reversed Reaction)
4.1. Writing-Erasure Connection (A⇒B, B⇒A)
- (1)
- Case 1 (same pathways): we will show that the distribution function is cut at large Ed at a more elevated temperature. Stability increases because more stable sites are filled [36] (see Section A below).
- (2)
- Case 2 (different pathways, two distribution functions): stability increases because less stable sites are removed in the reverse distribution function during writing (Section B below).
4.1.1. Writing in the Case of Reversible Reaction
4.1.2. Erasure in the Case of Reversible Reaction
4.1.3. Stability When Forward and Backward Reactions Are Independent
4.1.4. Writing in the Case of Independent Forward and Backward Reactions
4.2. Isotherm-Isochron Equivalence
- -
- the distribution function is not T dependent,
- -
- only one limiting reaction.
- Example of Equivalence Breaking
4.3. k0 Comparison
5. Beyond Simple Theory
5.1. Higher Order Reaction
5.2. Two Parallel Reactions
5.3. Serial Reactions
- Theory of Serial Reactions
5.4. Narrow and Steep Distribution Function
5.5. Distributed Non-Activated Kinetics (Photochromism, Luminescence)
5.5.1. “Writing like”
5.5.2. Erasure
6. Real Analysis of Bragg Grating Writing and Stability
7. Practical Formalism to Describe Generalized Thermally Activated Processes
8. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Poumellec, B.; Lancry, M. Kinetics of Thermally Activated Physical Processes in Disordered Media. Fibers 2015, 3, 206-252. https://doi.org/10.3390/fib3030206
Poumellec B, Lancry M. Kinetics of Thermally Activated Physical Processes in Disordered Media. Fibers. 2015; 3(3):206-252. https://doi.org/10.3390/fib3030206
Chicago/Turabian StylePoumellec, Bertrand, and Matthieu Lancry. 2015. "Kinetics of Thermally Activated Physical Processes in Disordered Media" Fibers 3, no. 3: 206-252. https://doi.org/10.3390/fib3030206