For free-radical photopolymerization with a photobleaching initiator, we develop an unsteady one-dimensional model accounting for initiator consumption and optical attenuation and derive relationships for the spatial and temporal variation of the local initiator concentration and initiation rate. With increasing absorbance, the local initiation rate becomes increasingly nonuniform and assumes the form of a highly localized traveling wave propagating with speed φI 0/CA,0 (where φ, I0, and CA,0 are the quantum yield of photoinitiator consumption, incident intensity, and initial photoinitiator concentration, respectively), independent of photoinitiator absorption coefficient. At high attenuation, the maximum photoinitiation rate (as a function of position) is asymptotically one-fourth the initial rate at the front of the layer. Qualitative effects of nonuniform initiation in free-radical photopolymerizations with simple and more complicated kinetics are discussed in terms of the model and results.
We present experimental results supporting physics-based ejecta model development, where our main assumption is that ejecta form as a special limiting case of a Richtmyer–Meshkov (RM) instability at a metal–vacuum interface. From this assumption, we test established theory of unstable spike and bubble growth rates, rates that link to the wavelength and amplitudes of surface perturbations. We evaluate the rate theory through novel application of modern laser Doppler velocimetry (LDV) techniques, where we coincidentally measure bubble and spike velocities from explosively shocked solid and liquid metals with a single LDV probe. We also explore the relationship of ejecta formation from a solid material to the plastic flow stress it experiences at high-strain rates ($1{0}^{7} ~{\mathrm{s} }^{\ensuremath{-} 1} $) and high strains (700 %) as the fundamental link to the onset of ejecta formation. Our experimental observations allow us to approximate the strength of Cu at high strains and strain rates, revealing a unique diagnostic method for use at these extreme conditions.
The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.
For free-radical photopolymerizations with a photobleaching initiator in an initially uniform layer illuminated from one direction, we incorporate nonuniform photoinitiation into a simple kinetic model to show how the degree of monomer conversion varies spatially and temporally with the incident light intensity (I 0), the absorption coefficient (αA) and initial concentration (C A,0) of the photoinitiator, and the propagation (k p) and termination (k t) rate constants (taken to be independent of the degree of conversion). We show that the spatiotemporal variation of monomer conversion depends on two dimensionless parameters: the initial absorbance γ = αA C A,0 L, where L is the layer thickness, and β = k p[fC A,0/(φα A I 0 k t)]1/2, where φ is the quantum yield of photoinitiator consumption, and 0 ≤ f ≤ 2 is the number of primary radicals produced for each photoinitiator molecule consumed. For each value of γ, there is a minimum value of β beyond which a specified layer-averaged extent of monomer conversion is assured. As β decreases, so does the extent of monomer conversion, with the final degree of conversion being lowest near the optical “front” of the layer, where the light absorption and photoinitiation rates are initially highest. The extent of nonuniformity decreases with increasing β until β ≈ 2, beyond which monomer conversion is essentially complete regardless of the initial absorbance. The results are discussed in terms of the spatiotemporal distributions of the primary radical and radical chain concentrations.
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