Chaos in oscillatory sorption of hydrogen in palladium

Abstract

Aperiodic oscillations in the sorption of hydrogen and deuterium in palladium have been observed. An expression relating the square of a function, with the derivative and integral with variable upper limit of the same function has been proved and proposed to be used as a base for a chaos-vs.-random test. The result of one "branch" of the test is a real number D ∈ [0, 2]; close to zero for the deterministic and smooth datasets, and approaching two for the random or discrete datasets. Another "branch" of the test, based on the same mathematical relation, produces two functions that appear to be convergent for deterministic and smooth datasets, but run totally divergent for random or discrete ones. The D-values yielded by deterministic time series, recorded in the periodic and quasiperiodic sorptions of H 2 or D 2 in Pd, are around 0.001. On the other hand, the databases that were presumably random or non-smooth yielded the test results from D = 0.2 to D = 1.9. Against these benchmarks, the experimental, aperiodic oscillations scored around 0.003 in D, which is much closer to the deterministic than to a random manner.

Oscillatory Behavior and Anomalous Heat Evolution in Recombination of H2 and O2 on Pd-based Catalysts

“…This is to say, we need to determine that the aperiodic, calorimetric time series in Figures and are a manifestation of a reaction dynamics and, therefore, that we observe a kinetic phenomenon of oscillating reaction rather than a stochastic noise of microcalorimetric recording. The underpinning mathematics as well as the testing procedure are described in detail in the ref , so they are only outlined here. Consider a function f ( x ), which is smooth, continuous and square integrable on the interval .…”

“…Consider a function f ( x ), which is smooth, continuous and square integrable on the interval . If the values of the function f ( x ) are zero at the points a and b , then the following relation holds (see ref for proof of this relation): Equation compares two definite integrals. The integrand in the left-hand side (LHS) represents a pointwise product, p ( x ), of two functions, namely: the first derivative of f ( x ) and the integral of f ( x ) with the variable upper limit.…”

“…For comparison, the D -values reported in ref for random databases such as a sequence of 3000 digits of π, and a random Fibonacci sequence were, respectively, 0.24783 and 1.0138, whereas on the other hand, a classic example of Lorenz chaos scored 0.0060459 in D . In the same article, the D -scores for the thermokinetic oscillations in sorption of hydrogen or deuterium in metallic palladium have been reported to be 0.00060501, 0.0010340 and 0.003041 for the periodic, quasiperiodic and chaotic oscillations, respectively . Hence, the score for the 8.6% round in Figure remains well within a range expected for determinism meaning that oscillations in this round are chaotic.…”

“…In fact, oscillatory oxidations of CO on metallic Pd or Pt are classic systems widely studied for their nonlinear dynamics, − also leading to the Nobel Prize in chemistry in 2007. As for the H/Pd system, a handful of results on oscillatory hydrogenations have been published. , Thermokinetic oscillations in the Pd/H system, i.e., the oscillatory rate of heat evolution accompanying the sorption of hydrogen and/or deuterium in metallic palladium have been reported recently. − Moreover, an incidence of anomalously high thermal effects has been observed in the H 2 /O 2 /Pd system, as well as in the H 2 /O 2 /Ni, and also with noble metals catalysts such as Au/TiO 2 , considerably exceeding the thermodynamically expected heat of water formation. − …”

Extracting the Oscillatory Component and Defining a Mean Amplitude of Thermokinetic Oscillations in the H/Pd System

“…In practical terms, the equilibrium with H 2 over the other metals settles almost instantaneously, whereas in Pd, it may take 20–30 min for its lattice to be saturated with hydrogen (under ambient conditions), during which period the system remains away from equilibrium. This is long enough for a nonlinear phenomenon such as self-sustained oscillations to develop, and indeed, apart from the hysteretic phenomena, the thermokinetic oscillations can be systematically observed in the H/Pd system. − Naturally, the sorption of H 2 in Pd seems to be the most elementary reaction convenient to investigate the hydrogen/metal adsorption far from equilibrium. Studying the thermokinetic oscillations observed in this reaction may help to understand nonlinearity in the other, more complex processes.…”