24 research outputs found
Analytical solutions of the Arrhenius-Semenov problem for constant volume burn
Analytical solutions to the Semenov thermal ignition problem for constant
volume burn governed by Arrhenius reaction kinetics are derived. Specifically,
an approximate analytical solution technique for the Arrhenius-Semenov
differential equation is derived for reaction orders
and exact solutions are also constructed for reaction orders . The approximation technique relies on expansion of the respective
nondominant terms in the differential equation at the lower and upper bounds of
the reaction progress variable in order to create a pair of integrable series.
The two integrated series are then connected to create a single continuous
analytical solution. Excellent agreement is observed between the analytical
approximation and solutions obtained numerically. The presented approximation
constitutes a simple and robust strategy for solving the Arrhenius-Semenov
problem analytically
Persistence of transition state structure in chemical reactions driven by fields oscillating in time
Chemical reactions subjected to time-varying external forces cannot generally
be described through a fixed bottleneck near the transition state barrier or
dividing surface. A naive dividing surface attached to the instantaneous, but
moving, barrier top also fails to be recrossing-free. We construct a moving
dividing surface in phase space over a transition state trajectory. This
surface is recrossing-free for both Hamiltonian and dissipative dynamics. This
is confirmed even for strongly anharmonic barriers using simulation. The power
of transition state theory is thereby applicable to chemical reactions and
other activated processes even when the bottlenecks are time-dependent and move
across space
Chemical reactions induced by oscillating external fields in weak thermal environments
Chemical reaction rates must increasingly be determined in systems that
evolve under the control of external stimuli. In these systems, when a reactant
population is induced to cross an energy barrier through forcing from a
temporally varying external field, the transition state that the reaction must
pass through during the transformation from reactant to product is no longer a
fixed geometric structure, but is instead time-dependent. For a periodically
forced model reaction, we develop a recrossing-free dividing surface that is
attached to a transition state trajectory [T. Bartsch, R. Hernandez, and T.
Uzer, Phys. Rev. Lett. 95, 058301 (2005)]. We have previously shown that for
single-mode sinusoidal driving, the stability of the time-varying transition
state directly determines the reaction rate [G. T. Craven, T. Bartsch, and R.
Hernandez, J. Chem. Phys. 141, 041106 (2014)]. Here, we extend our previous
work to the case of multi-mode driving waveforms. Excellent agreement is
observed between the rates predicted by stability analysis and rates obtained
through numerical calculation of the reactive flux. We also show that the
optimal dividing surface and the resulting reaction rate for a reactive system
driven by weak thermal noise can be approximated well using the transition
state geometry of the underlying deterministic system. This agreement persists
as long as the thermal driving strength is less than the order of that of the
periodic driving. The power of this result is its simplicity. The surprising
accuracy of the time-dependent noise-free geometry for obtaining transition
state theory rates in chemical reactions driven by periodic fields reveals the
dynamics without requiring the cost of brute-force calculations
Energy transport between heat baths with oscillating temperatures
Energy transport is a fundamental physical process that plays a prominent
role in the function and performance of myriad systems and technologies. Recent
experimental measurements have shown that subjecting a macroscale system to a
time-periodic temperature gradient can increase thermal conductivity in
comparison to a static temperature gradient. Here, we theoretically examine
this mechanism in a nanoscale model by applying a stochastic Langevin framework
to describe the energy transport properties of a particle connecting two heat
baths with different temperatures, where the temperature difference between
baths is oscillating in time. Analytical expressions for the energy flux of
each heat bath and for the system itself are derived for the case of a free
particle and a particle in a harmonic potential. We find that dynamical effects
in the energy flux induced by temperature oscillations give rise to complex
energy transport hysteresis effects. The presented results suggest that
applying time-periodic temperature modulations is a potential route to control
energy storage and release in molecular devices and nanosystems