Low-temperature Chemistry in Uniform Supersonic Flows

M. FOURNIER, S. D. LE PICARD AND I. R. SIMS

1.1 Introduction

Enormous progress has been achieved over the past several decades in the determination of gas-phase rate constants for elementary reactions, especially at high temperatures relevant to combustion (T>ca. 500 K). At sub-ambient temperatures progress has been slower, despite strong interest in reaction kinetics data from the atmospheric and astrochemical communities, as well as for fundamental tests of rate constant theory. This is due in no small measure to the considerable experimental difficulties associated with low-temperature kinetics measurements, but perhaps also to the commonly-held belief that the temperature-dependent rate constants k(T) of chemical reactions are almost universally described by the well-known Arrhenius equation

k(T) = Aexp(-Eact/RT), (1.1)

where A is the Arrhenius pre-exponential factor, R is the universal gas constant, T the temperature, and Eact is the activation energy, or at least by the modified form of this equation in which additional temperature dependence is allowed for in the pre-exponential term

k(T) = A! Tm exp(-E'act/RT). (1.2)

The origin of and justification for these equations, which, though bearing Arrhenius' name, owe more to the work of Van't Hoff, have been discussed recently by Ian W. M. Smith. The temperature dependence of the rate constant for a reaction obeying the Arrhenius equation may be represented by the much-used Arrhenius plot of the log of k(T) as a function of the inverse of the absolute temperature, as in Figure 1.1. A linear dependence is obtained with a negative slope corresponding to -Eact/RT.

Indeed, most chemical reactions between stable molecules slow to a halt at very low temperatures because the reactants lack the energy needed to surmount activation barriers. However, many reactions involving radicals do not possess a barrier along the minimum energy path (MEP) joining reagents to products over the potential energy surface (PES) that governs the reaction. These so-called barrierless reactions, which are often already very fast at room temperature, may remain rapid or even become faster as the temperature drops down to the very low temperatures prevailing in extraterrestrial atmospheres such as that of Saturn's giant moon Titan (ca. 70–150 K) or interstellar clouds (ISCs; ca. 10–100 K). The chemistry of cold ISCs, including regions where stars and planetary systems form, was once thought to be completely dominated by ion-molecule reactions, which are barrierless owing to strong attractive forces. However, in no small measure due to the measurements described in this chapter, it is now recognized that neutral-neutral reactions play an important role in such environments, and it is on this class of reactions that we focus our attention.

A wide range of experimental techniques exists for studying reactivity at low and ultra-low temperatures, as can be seen in the other chapters of this book. However, many techniques do not enable the measurement of absolute rate constants at low temperatures. In chemical kinetics, the principal method used to do this is based upon the use of so-called pseudo-first-order conditions, where for a bimolecular reaction A + B -> products, the rate constant is determined by placing one of the two reagents (usually the most stable one) in large, known excess, and following the time-dependent concentration of the other reagent concentration. Because under these conditions the concentration of one reactant, in the present example reactant B, can be assumed to be constant, the standard second-order reaction rate eqn (1.3)

d[A]/dt = -k[A][B] (1.3)

then reduces to the pseudo-first-order eqn (1.4) when [B] >> [A], with k1st = k[B]:

d[A]/dt = -k1st [A] (1.4)

This technique effectively isolates the elementary reaction under study from other cross-reactions or self-reaction of the minority reagent A, and yields first-order exponential decays of [A] (or something directly proportional to it) which are analysed to yield values of k1st as a function of coreagent concentration [B]. An important advantage of this technique is that knowledge of the absolute concentration of A is not required. A plot of k1st as a function of [B] should yield a linear graph with slope k, the required bimolecular rate constant. Variants on this methodology exist, notably the use of flow-tube and relative rate techniques, but in essence all are based on this method. Knowledge of the absolute concentration of the B co-reagent enables the determination of an absolute rate constant, which is useful for comparison with theoretical calculations as well as for input to models of gas-phase chemical environments.

The determination of absolute rate constants for gas-phase reactions at temperatures much below 200 K is very difficult, principally owing to condensation on the cold walls of the reactor, such that maintaining a sufficiently high and known concentration of the more abundant reactant (B in the example above) becomes a problem. The CRESU technique,6K 8 which is the subject of this chapter, solves this by creating a wall-less flow of cold gas by expansion through Laval nozzles. (CRESU is a French acronym standing for cinétique de réaction en ecoulement supersonique uniforme, or reaction kinetics in uniform supersonic flow). Using this method, many neutral-neutral reactions have been found to have rate constants that either remain very fast or even increase as the temperature falls below 300 K, down to values as low as 6 K.

In the case of ion-molecule reactions, fairly simple considerations involving capture on long-range potentials dominated by electrostatic interactions enable, in the vast majority of cases, a fairly reliable theoretical estimate of the overall temperature-dependent rate constant k(T), even if branching into individual product channels remains an important subject for investigation. This is not, however, generally the case for neutral-neutral reactions which show more subtle effects in the temperature dependence of their overall rate constants.

Figure 1.2 shows three general cases for exothermic bimolecular reactions. The first case (a) illustrates a reaction with a substantial barrier caused by the necessity to break chemical bonds. The upper panel schematically represents the potential energy (the stored internal energy) of the system as the reaction takes place, as a function of the progress of reaction (the reaction coordinate), while the lower panel shows what an Arrhenius plot for such a reaction might look like. The reaction slows down rapidly as the temperature drops (towards the right of the plot). While roughly linear, a slight curvature is expected and is more evident at lower temperatures. The curvature is on account of the fact that the pre-exponential A factor is actually slightly temperature dependent and the occurrence of quantum mechanical tunnelling, which enables the system to cross through the barrier to reactants at energies below the summit of the barrier. If both reactants are radicals, for example CN and O2, then there may be no barrier on the potential energy curve, as shown schematically in Figure 1.2(b). The potential curve displays a deep well as the unpaired electrons on each radical combine to form a chemical bond. In this case the reaction generally becomes faster as the temperature drops, as depicted in the lower panel of Figure 1.2(b). In fact, not only reactions between two radicals, but also reactions between radicals and closed shell molecules have been shown to be rapid. One of the earliest reactions to be studied at very low temperatures by the CRESU technique, CN + C2H6 -> HCN + C2H5, showed a behaviour which is in some ways a combination of the two 'extremes' represented by Figure 1.2(a) and (b). At high temperatures it displays positive activation energy, but as the temperature decreases the rate constant passes through a minimum and then increases by almost an order of magnitude between 200 K and 23 K. This behaviour is characteristic of a system represented schematically in the upper panel of Figure 1.2(c), with an initial relatively weakly bound complex, which then goes on to cross a potential barrier to proceed to products. The interplay between the formation and redissociation of the initial complex versus subsequent crossing of the barrier has been invoked to explain the observed behaviour. Further experimental and theoretical studies have shown that many radical-molecule reactions fall into this same category, but the minimum in the rate constant is not always observed in the temperature range of existing measurements as it depends strongly on the interplay between the initial well depth and subsequent barrier height as represented by the red arrows in the upper panel of Figure 1.2(c).

The temperature dependence of neutral-neutral chemical reactions down to very low temperatures is controlled as we have seen by a subtle interplay of barrier heights (and associated entropic effects) as well as the possible effects of quantum mechanical tunnelling in the case of relatively small barriers. Measurements of low-temperature rate constants provide therefore an exacting challenge both for quantum chemical calculations of ab initio PESs as well as for quantum dynamical scattering calculations or other theoretical methods such as transition state theory designed to provide reaction rate constants.

Cryogenically cooled cells have been used to provide reaction rate constants in a few favourable cases down to below 100 K, especially where the co-reagent is not easily condensable, as is the case for O2. Studies of collisional energy transfer with bath gases such as He or H2 also fall into this category, and the collisional cooling technique invented by Willey and De Lucia, and later adapted and renamed buffer gas cooling by Doyle et al., has been used to obtain information on rotational energy transfer below 5 K in such cases: however, its application to date has been limited to inelastic collisions.

In order to investigate the kinetics of condensable species at very low temperatures it is necessary therefore to use expansion-based cooling methods. While free jet expansions and molecular beams have been used with great success in spectroscopy to prepare rotationally cold samples of isolated molecules, their use for the study of cold bimolecular reactions is limited by the very low densities and strong density gradients which result in a loss of the local thermodynamic equilibrium (non-Boltzmann distributions over rotational states, for example) and a lack of collisions within the expansion. The CRESU technique was originally invented by Rowe, Marquette, and co-workers to study ion-molecule reactions by using a special type of expansion technique, which maintains thermalization at all times. The next section describes this technique in detail, after which a number of case studies will be detailed to illustrate its application to problems in low-temperature chemical kinetics.

1.2 The CRESU Technique

1.2.1 Uniform Supersonic Flows

When performing reaction rate measurements, as explained in the previous section, it is essential to know the absolute concentration of at least one of the reacting species. A common means of cooling a gas-phase reactor is to cool down the walls of the system by circulating a coolant or cryogen in a double-walled jacket around the gas-phase (slow) flow tube. However, the vapour pressures of most molecular species of potential interest for low-temperature chemistry, including hydrocarbons and many others, shrink to a negligible level at temperatures below 200 K, and so such species will condense on the walls of the reaction vessel. Even if some reagents can be introduced via heated injectors far away from the walls, their concentration will neither be homogeneous in the reaction zone, nor able to be easily determined. This means that cooling via the walls of the reaction cell is not a viable method for (very) low-temperature kinetics, and we must find another way to cool down the reactant species.

The most widely used technique for generating cold gas-phase molecules in chemical physics experiments is by expansion from a very high-pressure reservoir through a small orifice to a very low-pressure chamber. By this free jet expansion technique a very high degree of cooling can be achieved, and subsequent passage via an appropriate skimmer can yield molecular beams where the internal rotational and translational temperatures can be of the order of 1 K or even well below. Such molecular beams have been used for example in the spectroscopic observation of the helium dimer at a translational temperature of 1 mK, as well as in studies of reactivity by the use of the crossed molecular beam technique. However, their use in low-temperature chemical kinetics is limited by the very strong density gradients and loss of local thermodynamic equilibrium, with non-Boltzmann rotational distributions and out-of-equilibrium translational temperatures, linked to the very low rate of collisions in such beams. This severely limits the use of these environments for kinetics studies especially of neutral-neutral reactions, where it is necessary to establish thermal equilibrium after, for example, photolytic generation of a radical reagent, on a timescale that is short compared to reaction. These two requirements--expansion cooling to enable the rapid generation of highly supersaturated gas samples, and reduction of the strong density gradients associated with uncontrolled free jet expansions to achieve a higher density homogeneous environment where local thermodynamic equilibrium can be maintained by frequent collisions--have led to the use of uniform supersonic expansions in kinetics applications.

Laval nozzles, described in 1888 by Gustaf de Laval, are axisymmetric nozzles consisting of a convergent section followed by a divergent section. They have found application in the generation of supersonic flows, notably for rocket motors. A Laval nozzle can be used to expand any gas and generate a supersonic expansion.

The action of a Laval nozzle on a gas flowing through it is a two-stage process. In the convergent section of the nozzle (see Figure 1.3), the gas is accelerated since the cross-section becomes smaller as the gas moves towards the exit (but the gas is not compressed). If the gas flow is adequate for the nozzle, the gas reaches a Mach number of 1 at the throat. The gas further expands in the divergent section of the nozzle and reaches higher Mach numbers. The adiabatic expansion taking place through the nozzle reduces the temperature and density of the gas, while raising the velocity. If proper conditions are met, the flow is collimated and propagates along a cylindrical trajectory. If the pressure of the gas inside the flow is too high or too low, the flow is over-expanded or chocked and shockwaves will appear. This is to be avoided for a reaction rate study, since a local compression will be induced, changing the flow properties, and consequently the physical conditions encountered by the reactants.

In a Laval nozzle, the flow is compressible. It is possible to apply a variation of Bernoulli's equation, taking into account enthalpy and potential energy. However, the nozzle size is small (gravity effects are negligible) and the flow is isentropic. Thus, we can apply the following equations directly in such a nozzle.

[MATHEMATICAL EXPRESSION OMITTED] (1.5)

PV? = constant, (1.6)

where Cp is the specific heat capacity at constant pressure, v the fluid velocity, T the fluid temperature, V the occupied volume, and ? the ratio of specific heat capacities, defined by ? = Cp/Cv, where Cv is the specific heat capacity at constant volume. The "0" indices refer to reservoir values (static gas), while "flow" indices can refer to any point in the flow.

In a CRESU experiment, the nozzle is mounted on a reservoir whose volume is large compared to the nozzle's internal volume. Because of this, the fluid velocity in the reservoir, v0, is considered to be negligible compared to the speed in the Laval nozzle flow, vflow. The Mach number, introduced before, is the ratio of flow speed to the speed of sound in the same fluid. It is important to remember that the speed of sound is influenced by the mass density of the medium (the higher the density, the higher the speed). The Mach number M is defined as

[MATHEMATICAL EXPRESSION OMITTED] (1.7)