Correcting mean-field approximations for spatially-dependent advection-diffusion-reaction processes

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair of an organism; on the macroscale, such effects are important in the sustainability of a population in its environment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may arise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative impact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological descriptions, such as the logistic model, are used to model population growth. In this work we outline some methods for analyzing exclusion processes which include agent proliferation, death and motility in two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description for these types of processes is of logistic form; we show that, under certain parameter conditions, such systems may display large deviations from the mean field, and suggest computationally tractable methods to correct the logistic-type description

Vacuum-Ultraviolet negative photoion spectroscopy of SF5Cl

Using vacuum-UV radiation from a synchrotron, gas-phase negative ions are detected by mass spectrometry following photoexcitation of SF$_5$Cl. F$^-$, Cl$^-$ and SF$_5^-$are observed, and their ion yields recorded in the range 8-30 eV. F$^-$ and Cl$^-$ show a linear dependence of signal with pressure, showing that they arise from unimolecular ion-pair dissociation, generically written AB + h$v$ $\rightarrow$ C$^-$ + D$^+$ (+ neutral(s)). F$^-$ is the strongest signal, and absolute cross sections are determined by calibrating the signal intensity with that of F$^-$ from SF$_6$ and CF$_4$. Resonances are observed, and assigned to transitions to Rydberg states of SF$_5$Cl. The Cl$^-$ signal is much weaker, despite the S-Cl bond being significantly weaker than the S-F bond. Appearance energies for F$^-$ and Cl$^-$ of 12.7 ± 0.2 and 10.6 ± 0.2 eV are determined. The spectra suggest that these ions form indirectly by crossing of Rydberg states of SF$_5$Cl onto an ion-pair continuum

Vacuum-UV negative photoion spectroscopy of gas-phase polyatomic molecules

This Review describes recent experiments to detect anions following vacuum-UV photoexcitation of gas-phase polyatomic molecules. Using synchrotron radiation in the range 10-35 eV at a resolution down to 0.02 eV, negative ions formed are detected by mass spectrometry. The molecules studied in detail include CF$_4$, SF$_6$ and CH$_4$; the CF$_3$X series where X = Cl,Br,I; the CH$_3$Y series where Y = F,Cl,Br; and SF$_5$Z where Z = CF$_3$,Cl. Spectra and raw data only are reported for other members of the CH$_x$F$_y$, CH$_x$Cl$_y$ including CCl$_4$, and CF$_x$Cl$_y$ series where ($x$+$y$) = 4; and saturated and unsaturated members of the C$_m$H$_n$ and C$_m$F$_n$ series up to m = 3. Anions detected range from atomic species such as H-, F- and Cl- through to heavier polyatomics such as SF$_5^-$, CF$_3^-$ and CH$_2$Cl$^-$. The majority of anions display a linear dependence of signal with pressure, showing that they arise from unimolecular ion-pair dissociation, generically written as ABC + h$v$ $\rightarrow$ D$^-$ + E$^+$ + neutral(s). In a few cases, the anion signal increases much more rapidly than a linear dependence with pressure, suggesting that anions now form via a multi-step process such as dissociative electron attachment. Cross sections for ion-pair formation can be put on to an absolute scale by calibrating the signal strength with those of F$^-$ from SF$_6$ and CF$_4$, although there are difficulties associated with the determination of H$^-$ cross sections from hydrogen-containing molecules unless this anion is dominant. Following normalisation to total vacuum-UV absorption cross sections (where data are available), quantum yields for anion production are obtained. Cross sections in the range ca. 10$^{-23}$ to 10$^{-19}$ cm$^2$ , and quantum yields in the range ca. 10$^{-6}$ to 10$^{-3}$ are reported. The Review describes the two ion-pair mechanisms of indirect and direct formation and their differing characteristics, and the properties needed for anion formation by dissociative electron attachment. From this huge quantity of data, attempts are made to rationalise the circumstances needed for favourable formation of anions, and which anions have the largest cross section for their formation. Since most anions form indirectly via predissociation of an initially-excited Rydberg state of the parent molecule by an ion-pair continuum, it appears that the dynamics of this curve crossing is the dominant process which determines which anions are formed preferentially. The thermochemistry of the different exit channels and the microscopic properties of the anion formed do not appear to be especially significant. Finally, for the reaction ABC + h$v$ $\rightarrow$ A$^-$ + BC$^+$ , the appearance energy of A$^-$ can be used to determine an upper limit to the bond dissociation energy of AB (to A + BC), or an upper limit to that of ABC$^+$ (to A + BC$^+$). Where known, the data are in excellent agreement with literature values

Models of collective cell motion for cell populations with different aspect ratio: diffusion, proliferation & travelling waves

Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this work we provide a new link between individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and population-level partial differential equation models. We work in an exclusion process framework, considering aligned, elongated cells that may occupy more than one lattice site, in order to represent populations of agents with different sizes. Three different idealisations of the individual-level mechanism are proposed, and these are connected to three different partial differential equations, each with a different diffusion coefficient; one linear, one nonlinear and degenerate and one nonlinear and nondegenerate. We test the ability of these three models to predict the population-level response of a cell spreading problem for both proliferative and nonproliferative cases. We also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two-dimensional spreading and invasion. Our results show that each model can accurately predict density data for nonproliferative systems, but that only one does so for proliferative systems. Hence great care must be taken to predict density data with varying cell shape

Models of collective cell spreading with variable cell aspect ration: a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multiscale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (PME). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the PME to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

The 1973-1984 Solar Modulation of Cosmic Ray Nuclei

As a continuation of the program of solar modulation studies, new measurements were carried out with the cosmic ray telescope on the Earth satellite IMP-8, of the intensity time variations and the energy spectra of galactic cosmic ray protons, helium, carbon and oxygen from 1980 through 1984 including the recent solar maximum. In order to test the applicability of a steady state model of solar modulation during a period which includes times of rapidly changing modulation, these fluxes were equated with the predictions of a conventional model of solar modulation which assumes equilibrium between modulation mechanisms. It is found that for a reasonable range of variations of the diffusion coefficient the model predictions can be made to agree with the measurements at essentially all times during the studied period. The model can account also for the observed hysteresis effects between cosmic rays of different rigidities

Vacuum-UV negative photoion spectroscopy of CH3F, CH3Cl and CH3Br

Using tunable vacuum-UV radiation from a synchrotron, negative ions are detected by quadrupolar mass spectrometry following photoexcitation of three gaseous halogenated methanes CH$_3$X (X = F,Cl,Br). The anions X$^-$, H$^-$, CX$^-$, CHX$^-$ and CH$_2$X$^-$ are observed, and their ion yields recorded in the range 8-35 eV. The anions show a linear dependence of signal with pressure, showing that they arise from unimolecular ion-pair dissociation, generically described as AB + h$v$ $\rightarrow$ A$^-$ + B$^+$ (+ neutrals). Absolute cross sections for ion-pair formation are obtained by calibrating the signal intensities with those of F$^-$ from both SF$_6$ and CF$_4$. The cross sections for formation of X$^-$ + CH$_3$$^+$ are much greater than for formation of CH$_2$X$^-$ + H$^+$. In common with many quadrupoles, the spectra of $m$/$z$ 1 (H$^-$) anions show contributions from all anions, and only for CH$_3$Br is it possible to perform the necessary subtraction to obtain the true H$^-$ spectrum. The anion cross sections are normalised to vacuum-UV absorption cross sections to obtain quantum yields for their production. The appearance energies of X$^-$ and CH$_2$X$^-$ are used to calculate upper limits to 298 K bond dissociation energies for D$^o$ (H$_3$C-X) and D$^o$ (XH$_2$C-H) which are consistent with literature values. The spectra suggest that most of the anions are formed indirectly by crossing of Rydberg states of the parent molecule onto an ion-pair continuum. The one exception is the lowest-energy peak of F$^-$ from CH$_3$F at 13.4 eV, where its width and lack of structure suggest it may correspond to a direct ion-pair transition

The ionization structure of the Orion nebula: Infrared line observations and models

Observations of the (O III) 52 and 88 micron lines and the (N III) 57 micron line have been made at 6 positions and the (Ne III) 36 micron line at 4 positions in the Orion Nebula to probe its ionization structure. The measurements, made with a -40" diameter beam, were spaced every 45" in a line south from and including the Trapezium. The wavelength of the (Ne III) line was measured to be 36.013 + or - 0.004 micron. Electron densities and abundance ratios of N(++)/O(++) have been calculated and compared to other radio and optical observations. Detailed one component and two component (bar plus halo) spherical models were calculated for exciting stars with effective temperatures of 37 to 40,000K and log g = 4.0 and 4.5. Both the new infrared observations and the visible line measurements of oxygen and nitrogen require T sub eff approx less than 37,000K. However, the double ionized neon requires a model with T sub eff more than or equal to 39,000K, which is more consistent with that inferred from the radio flux or spectral type. These differences in T sub eff are not due to effects of dust on the stellar radiation field, but are probably due to inaccuracies in the assumed stellar spectrum. The observed N(++)/O(++) ratio is almost twice the N(+)/O(+) ratio. The best fit models give N/H = 8.4 x 10 to the -5 power, O/H = 4.0 x 10 to the -4 power, and Ne/H = 1.3 x 10 to the -4 power. Thus neon and nitrogen are approximately solar, but oxygen is half solar in abundance. From the infrared O(++) lines it is concluded that the ionization bar results from an increase in column depth rather than from a local density enhancement