Shock transition

In the irreversible limit R < 0.1), the adsorption front within the particle approaches a shock transition separating an inner core into which the adsorbate has not yet penetrated from an outer layer in which the adsorbed phase concentration is uniform at the saturation value. The dynamics of this process is described approximately by the shrinldng-core model [Yagi and Kunii, Chem. Eng. (Japan), 19, 500 (1955)]. For an infinite fluid volume, the solution is ... 

To reiterate, the development of these relations, (2.1)-(2.3), expresses conservation of mass, momentum, and energy across a planar shock discontinuity between an initial and a final uniform state. They are frequently called the jump conditions" because the initial values jump to the final values as the idealized shock wave passes by. It should be pointed out that the assumption of a discontinuity was not required to derive them. They are equally valid for any steady compression wave, connecting two uniform states, whose profile does not change with time. It is important to note that the initial and final states achieved through the shock transition must be states of mechanical equilibrium for these relations to be valid. The time required to reach such equilibrium is arbitrary, providing the transition wave is steady. For a more rigorous discussion of steady compression waves, see Courant and Friedrichs (1948). 

The diagnostics applied to shock experiments can be characterized as either prompt or delayed. Prompt instrumentation measures shock velocity, particle velocity, stress history, or temperature during the initial few shock transits of the specimen, and leads to the basic equation of state information on the specimen material. Delayed instrumentation includes optical photography and flash X-rays of shock-compression events, as well as post-mortem examinations of shock-produced craters and soft-recovered debris material. 

On a different note, after some 50 years of intensive research on high-pressure shock compression, there are still many outstanding problems that cannot be solved. For example, it is not possible to predict ab initio the time scales of the shock-transition process or the thermophysical and mechanical properties of condensed media under shock compression. For the most part, these properties must presently be evaluated experimentally for incorporation into semiempirical theories. To realize the potential of truly predictive capabilities, it will be necessary to develop first-principles theories that have robust predictive capability. This will require critical examination of the fundamental postulates and assumptions used to interpret shock-compression processes. For example, it is usually assumed that a steady state is achieved immediately after the shock-transition process. However, due to the fact that... 

Grady, D.E., Temperature and Deformation Micostructure in the Shock Transition, in Shock Waves in Condensed Matter—1983 (edited by Asay, J.R., Graham, R.A., and Straub, G.K.), North-Holland Physics, Amsterdam, 1984, pp. 363-367. 

Figure 2.13 shows a summary of transformation pressures in iron alloys. The present accepted value for the iron shock transition is 12.5 GPa, whereas under static pressure the accepted value is 12 GPa. 

A metal plate ("driver ) of mild steel or brass was propelled explosively against a similar plate ("target ) on which was resting a sample layer of explosive backed by a further layer of an inert solid. When the driver plated velocity was sufficiently high, this process generated a steady "overdriven detonation wave in the explosive unless (or until) it was overtaken by the rarefaction from the rear of the driver plate. The shock transit times thru each layer of the system were measured to determine tne transmitted shock or detonation velocities. 

Shock processes in the DDT are discussed under "Detonation (and Explosion), Development (Transition) from Burning (Combustion) or Deflagration and also under "Detonation from Shock Transition (See also "Shock Processes and Initiation , in Ref 4, p 194-96 and Ref 10, pp 17b 17c)... 

Evaluation of the density at the front, together with the Rankine-Hugoniot relations and the measured front velocity, determines the pressure and particle velocity there. In practice, this requires an additional assumption, which will be made throughout. Since the reaction zone is much smaller than the foil spacing, the reaction is treated as instantaneously complete within the shock transition, and the final state to which the Rankine-Hugoniot equations apply is taken to be the equilibrium state at the end of the reaction zone. No evidence of a reaction zone can be detected either in the analysis of the foil data or on the radiographs. 

For a nonlinear system the behavior depends on the shape of the isotherm. If the isotherm is unfavorable (Fig. 11),/(c) increases with concentration so that w decreases with concentration. This leads to a spreading profile, as illustrated in Figure 12b. However, if the isotherm is favorable (in the direction of the concentration change), an entirely different situation arises. ThenJ(c) decreases with concentration so that w increases with concentration. This would lead to the physically unreasonable overhanging profiles shown in Figure 12a. In fact, what happens is that the continuous solution is replaced by the equivalent shock transition so that response becomes a shock wave which propagates at a steady velocity vv given by... 

Still, todays computer power does not allow us to hilly resolve the shock transition region. We can, however, explain the non-linear kinetic plasma dynamics that generate the electromagnetic fields needed for transmission of momentum between the colliding plasma populations. Here, we report on 3D PIC simulations of the shock formation in the counter-streaming region of two colliding plasma shells. 

Shock-compressed carbon dioxide exhibits a strong slope change in the Hugoniot (recall Fig. 1), a clear indication of chemical reaction, at around 40 GPa and estimated temperature of 4500 K [1]. The previous theoretical calculation has confirmed that it is indeed due to chemical dissociation of carbon dioxide to elementary products such as diamond and oxygen. In recent diamond-anvil cell experiments [74], the similar dissociative products, lonstaleite diamond and oxygen have also been observed from the quenched products after laser-heating of CO2 samples at 67 GPa. The transition temperatures were estimated to be about 2500 K at 35 GPa, substantially lower than the estimated shock transition temperature 4500 K. 

The shock diagnostic laser pulses typically examine the aluminum surface opposite the substrate. If we call the substrate side the front of the target, then the shock emerges from the back at some time after the drive pulse is absorbed at the front of the aluminum layer. The aluminum layer thickness and the shock velocity determine the shock transit time. Each shock driving laser pulse (ca. 100 pm in focal diameter at the target) destroys the sample (after times much longer than our experiments) at that particular location. A series of experiments can be... 

Figure 8 Diagram of thin film structure and time dependent thicknesses as shock transits sample firom right to left. Shock velocity is Usand Al interface velocity is Up. Arrows indicate path of light partially reflected off interfaces, leading to thin film interference.

Fig. 5. Shock front rise time in a 700 nm thick sample of polycrystalline anthracene, (a)-(e) Coherent Raman (CARS) spectra of the V4 transition, which blueshifts and broadens when shocked at 4 GPa. The cartoons at right illustrate the progress of the shock front through an impedance-matched sandwich. When the shock front is midway through the anthracene layer, two separate peaks are seen in the CARS spectrum, representing ambient and shocked anthracene. The shock front risetime is considerably shorter than the shock transit time of 180 ps through the 700 nm layer. Detailed analysis shows that tr < 25 ps. Adapted from ref. [35].

The detonation state, B, lies on the Hugoniot curve as shown. Its location is determined by the reaction rate and by the transport coefficients that apply to the shock transition layer. The flow behind the shock is supersonic consequently, once a steady shock is established the boundary conditions have no influence on its behavior. The determination of the detonation state can be understood by considering the behavior of the integral curve in the specific volume-reaction coordinate plane, or phase plane. 

The reverse procedure, that is, elution of a column initially at state C with a solution corresponding to point A, leads to a shock wave profile. Since component 2 is the more strongly adsorbed species, the simple wave velocity increases along C D so this transition must be replaced by the chain-dotted curve representing the corresponding shock transition. Similarly the transition D->A is replaced by the shock transition D ->A and the profile consists of two shocks separated by a plateau corresponding to D. ... 

---