Published in Journal of Chemical Physics
02/14/2015

Abstract

Fluctuating rate coefficients are necessary when modeling disordered kinetic processes with mass-action rate equations. However, measuring the fluctuations of rate coefficients is a challenge, particularly for nonlinear rate equations. Here we present a measure of the total disorder in irreversible decay \(i\,A\to \textrm{products}\), \(i=1,2,3,\ldots n\) governed by (non)linear rate equations – the inequality between the time-integrated square of the rate coefficient (multiplied by the time interval of interest) and the square of the time-integrated rate coefficient. We apply the inequality to empirical models for statically and dynamically disordered kinetics with \(i\geq 2\). These models serve to demonstrate that the inequality quantifies the cumulative variations in a rate coefficient, and the equality is a bound only satisfied when the rate coefficients are constant in time.

Rates are a way to infer the mechanism of kinetic processes, such as chemical reactions. They typically obey the empirical mass-action rate laws when the reaction system is homogeneous, with uniform concentration (s) throughout. Deviations from traditional rate laws are possible when the system is heterogeneous and there are fluctuations in structure, energetics, or concentrations. When traditional kinetic descriptions break down [insert citation], the process is statically and/or dynamically disordered [insert Zwanzig citation], and it is necessary to replace the rate constant in the rate equation with a time-dependent rate coefficient. Measuring the variation of time-dependent rate coefficients is a means of quantifying the fidelity of a rate coefficient and rate law.

In our previous work a theory was developed for analyzing first-order irreversible decay kinetics through an inequality[insert citation]. The usefulness of this inequality is through its ability to quantify disorder, with the unique property of becoming an equality only when the system is disorder free, and therefore described by chemical kinetics in its classical formulation. The next problem that should be addressed is that of higher order kinetics, what if the physical systems one wishes to understand are more complex kinetic schemes, they would require a modified theoretical framework for analysis, but should and can be addressed. To motivate this type of development systems such as...... are all known to proceed through higher ordered kinetics, and all of these systems possess unique and interesting applications, therefore a more complete kinetics description of them should be pursued[insert citations].

Static and dynamic disorder lead to an observed rate coefficient that depends on time \(k(t)\). The main result here, and in Reference[cite], is an inequality \[\mathcal{L}(\Delta{t})^2 \leq \mathcal{J}(\Delta{t})\] between the statistical length (squared) \[\mathcal{L}(\Delta{t})^2 \equiv \left[\int_{t_i}^{t_f}k(t)dt\right]^2\] and the divergence \[\frac{\mathcal{J}(\Delta{t})}{\Delta{t}} \equiv \int_{t_i}^{t_f}k(t)^{2}dt\] over a time interval \(\Delta t = t_f - t_i\). Both \(\mathcal{L}\) and \(\mathcal{J}\) are functions of a possibly time-dependent rate coefficient, originally motivated by an adapted form of the Fisher information[cite]. Reference 1 showed that the difference \(\mathcal{J}(\Delta t)-\mathcal{L}(\Delta t)^2\) is a measure of the variation in the rate coefficient, due to static or dynamic disorder, for decay kinetics with a first-order rate law. The lower bound holds only when the rate coefficient is constant in first-order irreversible decay. Here we extend this result to irreversible decay processes with “order” higher than one. We show \(\mathcal{J}-\mathcal{L}^2=0\) is a condition for a constant rate coefficient for any \(i\). Accomplishing this end requires reformulating the definition of the time-dependent rate coefficient.

In this work we extend the application of this inequality to measure disorder in irreversible decay kinetics with nonlinear rate laws (i.e., kinetics with total “order” greater thane unity). We illustrate this framework with proof-of-principle analyses for second-order kinetics for irreversible decay phenomena. We also connect this theory to previous work on first-order kinetics showing how the model simplifies in a consistent manner when working with first order models.

We consider the irreversible reaction types \[i\,A \to \mathrm{products}\quad\quad\textrm{for}\quad i=1,2,3,\ldots,n\] with the nonlinear differential rate laws \[\frac{dC_i(t)}{dt} = k_i(t)\left[C_i(t)\right]^i.\] Experimental data is typically a concentration profile corresponding to the integrated rate law. If the concentration profile is normalized, by dividing the concentration at a time \(t\) to the initial concentration, it is called the survival function \[S_i(t) = \frac{C_i(t)}{C_i(0)},\] the input to our theory. Namely, we define the effective rate coefficient, \(k_i(t)\), through an appropriate time derivative of the survival function that depends on the order \(i\) of reaction \[k_i(t) \equiv \begin{cases} \displaystyle -\frac{d}{dt}\ln S_1(t) & \text{if } i = 1 \\[10pt] \displaystyle +\frac{d}{dt}\frac{1}{S_i(t)^{i-1}} & \text{if } i \geq 2. \end{cases}\]

These forms of \(k(t)\) satisfy the bound \(\mathcal{J}-\mathcal{L}^2 = 0\) in the absence of disorder, when \(k_i(t)\to\omega_i\). This is straightforward to show for the case of an \(i^{th}\)-order reaction (\(i\geq 2\)), with the traditional integrated rate law \[\frac{1}{C_i(t)^{i-1}} = \frac{1}{C_i(0)^{i-1}}+(i-1)\omega_i t.\] and associated survival function \[S_i(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega_i tC_i(0)^{i-1}}}.\] In traditional kinetics, the rate coefficient of irreversible decay is assumed to be constant, in which case \(k(t)\to\omega_i\), but this will not be the case when the kinetics are statically or dynamically disordered. In these cases, we will use the above definitions of \(k(t)\).

The statistical length and divergence can also be derived for these irreversible decay reactions. The time-dependent rate coefficient is \[k_i(t) \equiv \frac{d}{dt}\frac{1}{S_i(t)^{i-1}} = (i-1)\omega_i C_i(0)^{i-1}\] The statistical length \(\mathcal{L}_i\) is the integral of the cumulative time-dependent rate coefficient over a period of time \(\Delta{t}\), and the divergence is the cumulative square of the rate coefficient, multiplied by the time interval. For the equations governing traditional kinetics, both the statistical length squared and the divergence are \((i-1)^2\omega_i^2\left(C_i^{i-1}(0)\right)^2\Delta t^2\): the bound holds when there is no static or dynamic disorder, and a single rate coefficient \(\omega_i\) is sufficient to characterize irreversible decay.

The nonlinearity of the rate law leads to solutions that depend on concentration. This concentration dependence is also present in both \(\mathcal{J}\) and \(\mathcal{L}\).

Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to second order irreversible decay. The integrated second-order rate law gives the survival function \[S_2(t) = \frac{C_2(t)}{C_2(0)} = \frac{1}{1+\omega tC_2(0)}\] and the time-dependent rate coefficient \(k_2(t) = \omega_2 C_2(0)\).

\(S(t)\) has been changed to fit a second order model of irreversible decay. From this definition of \(k(t)\), we define a statistical distance. The statistical distance represents the distance between two different probability distributions, which can be applied to survival functions and rate coefficients.[cite] Integrating the arc length of the survival curve , \(\frac{1}{S(t)}\), gives the statistical length \(\mathcal{L}_2(\Delta{t}) = \frac{1}{S(t)}\big|_{S_(t_f)}^{S_(t_i)}\), which measures the cumulative rate coefficient, same as in first order irreversible decay. As seen in first order, the statistical length is also dependent on the time interval, with the statistical length being infinite in an infinite time interval. In first order irreversible decay, the inequality between the statistical length squared and Fisher divergence determines when a rate coefficient is constant, which is only when the inequality turns into an equality.[cite] A similar inequality is found in second order kinetics. Putting in the time dependent rate coefficient for a second order irreversible decay, the inequality becomes \[\omega^2 C_A(0)^2\Delta{t}^2-\left(\omega C_A(0)\Delta{t}\right)^2 \geq 0\] This result is very similar to that of first order irreversible decay(\(\omega^2\Delta{t}^2)-\left(\omega\Delta{t}\right)^2\geq0\), the only difference being a dependence on the initial concentration of the reactant. This initial concentration dependence serves to cancel the concentration units in the second order rate coefficient, making the statistical length and Fisher divergence dimensionless. When there is a time independent rate coefficient and there is no static disorder, the equality holds \(\omega^2 C_A(0)^2\Delta{t}^2 = \left(\omega C_A(0)\Delta{t}\right)^2\).