Archimedean Spirals Form at Low Flow Rates in Confined Chemical Gardens.

We describe and study the formation of confined chemical garden patterns. At low flow rates of injection of cobalt chloride solution into a Hele-Shaw cell filled with sodium silicate, the precipitate forms with a thin filament wrapping around an expanding "candy floss" structure. The result is the formation of an Archimedean spiral structure. We model the growth of the structure mathematically. We estimate the effective density of the precipitate and calculate the membrane permeability. We set the results within the context of recent experimental and modeling work on confined chemical garden filaments.

Downward fingering accompanies upward tube growth in a chemical garden grown in a vertical confined geometry.

Chemical gardens are self-assembled structures of mineral precipitates enabled by semi-permeable membranes. To explore the effects of gravity on the formation of chemical gardens, we have studied chemical gardens grown from cobalt chloride pellets and aqueous sodium silicate solution in a vertical Hele-Shaw cell. Through photography, we have observed and quantitatively analysed upward growing tubes and downward growing fingers. The latter were not seen in previous experimental studies involving similar physicochemical systems in 3-dimensional or horizontal confined geometry. To better understand the results, further studies of flow patterns, buoyancy forces, and growth dynamics under schlieren optics have been carried out, together with characterisation of the precipitates with scanning electron microscopy and X-ray diffractometry. In addition to an ascending flow and the resulting precipitation of tubular filaments, a previously not reported descending flow has been observed which, under some conditions, is accompanied by precipitation of solid fingering structures. We conclude that the physics of both the ascending and descending flows are shaped by buoyancy, together with osmosis and chemical reaction. The existence of the descending flow might highlight a limitation in current experimental methods for growing chemical gardens under gravity, where seeds are typically not suspended in the middle of the solution and are confined by the bottom of the vessel.

Filament dynamics in vertical confined chemical gardens.

When confined to a Hele-Shaw cell, chemical gardens can grow as filaments, narrow structures with an erratic and tortuous trajectory. In this work, the methodology applied to studies with horizontal Hele-Shaw cells is adapted to a vertical configuration, thus introducing the effect of buoyancy into the system. The motion of a single filament tip is modeled by taking into account its internal pressure and the variation of the concentration of precipitate that constitutes the chemical garden membrane. While the model shows good agreement with the results, it also suggests that the concentration of the host solution of sodium silicate also plays a role in the growth of the structures despite being in stoichiometric excess.

Filament dynamics in planar chemical gardens.

Filaments in a planar chemical garden grow following tortuous, erratic paths. We show from statistical mechanics that this scaling results from a self-organized dispersion mechanism. Effective diffusivities as high as 10-5 m2 s-1 are measured in 2D laboratory experiments. This efficient transport is four orders of magnitude larger than molecular diffusion in a liquid, and ensures widespread contact and exchange between fluids in the chemical-garden structure and its surrounding environment.

Stokes, Tyndall, Ruskin and the nineteenth-century beginnings of climate science.

Although we humans have known since the first smokey campfires of prehistory that our activities might alter our local surroundings, the nineteenth century saw the first indications that humankind might alter the global environment; what we currently know as anthropogenic climate change. We are now celebrating the bicentenaries of three figures with a hand in the birth of climate science. George Stokes, John Tyndall and John Ruskin were born in August 1819, August 1820 and February 1819, respectively. We look back from the perspective of two centuries following their births. We outline their contributions to climate science: understanding the equations of fluid motion and the recognition of the need to collect global weather data together with comprehending the role in regulating terrestrial temperature played by gases in the atmosphere. This knowledge was accompanied by fears of the Earth's regression to another ice age, together with others that industrialization was ruining humankind's health, morals and creativity. The former fears of global cooling were justified but seem strange now that the balance has tipped so far the other way towards global warming; the latter, on the other hand, today seem very prescient. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Stokes at 200 (part 2).

We present the second half of the papers from the Stokes200 symposium celebrating the bicentenary of George Gabriel Stokes. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Stokes at 200: a celebration of the remarkable achievements of Sir George Gabriel Stokes two hundred years after his birth.

Sir George Gabriel Stokes PRS was for 30 years an inimitable Secretary of the Royal Society and its President from 1885 to 1890. Two hundred years after his birth, Stokes is a towering figure in physics and applied mathematics; fluids, asymptotics, optics, acoustics among many other fields. At the Stokes200 meeting, held at Pembroke College, Cambridge from 15-18th September 2019, an invited audience of about 100 discussed the state of the art in all the modern research fields that have sprung from his work in physics and mathematics, along with the history of how we have got from Stokes' contributions to where we are now. This theme issue is based on work presented at the Stokes200 meeting. In bringing together people whose work today is based upon Stokes' own, we aim to emphasize his influence and legacy at 200 to the community as a whole. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Chemobrionics: From Self-Assembled Material Architectures to the Origin of Life.

Self-organizing precipitation processes, such as chemical gardens forming biomimetic micro- and nanotubular forms, have the potential to show us new fundamental science to explore, quantify, and understand nonequilibrium physicochemical systems, and shed light on the conditions for life's emergence. The physics and chemistry of these phenomena, due to the assembly of material architectures under a flux of ions, and their exploitation in applications, have recently been termed chemobrionics. Advances in understanding in this area require a combination of expertise in physics, chemistry, mathematical modeling, biology, and nanoengineering, as well as in complex systems and nonlinear and materials sciences, giving rise to this new synergistic discipline of chemobrionics.

Filiform corrosion as a pressure-driven delamination process.

Filiform corrosion produces long and narrow trails on various coated metals through the detachment of the coating layer from the substrate. In this work, we present a combined experimental and theoretical analysis of this process with the aim to describe quantitatively the shape of the cross-section, perpendicular to the direction of propagation, of the filaments produced. For this purpose, we introduce a delamination model of filiform corrosion dynamics and show its compatibility with experimental data where the coating thickness has been varied systematically.

Wavy membranes and the growth rate of a planar chemical garden: Enhanced diffusion and bioenergetics.

To model ion transport across protocell membranes in Hadean hydrothermal vents, we consider both theoretically and experimentally the planar growth of a precipitate membrane formed at the interface between two parallel fluid streams in a 2D microfluidic reactor. The growth rate of the precipitate is found to be proportional to the square root of time, which is characteristic of diffusive transport. However, the dependence of the growth rate on the concentrations of hydroxide and metal ions is approximately linear and quadratic, respectively. We show that such a difference in ionic transport dynamics arises from the enhanced transport of metal ions across a thin gel layer present at the surface of the precipitate. The fluctuations in transverse velocity in this wavy porous gel layer allow an enhanced transport of the cation, so that the effective diffusivity is about one order of magnitude higher than that expected from molecular diffusion alone. Our theoretical predictions are in excellent agreement with our laboratory measurements of the growth of a manganese hydroxide membrane in a microfluidic channel, and this enhanced transport is thought to have been needed to account for the bioenergetics of the first single-celled organisms.

Exploding Chemical Gardens: A Phase-Change Clock Reaction.

Chemical gardens and clock reactions are two of the best-known demonstration reactions in chemistry. Until now these have been separate categories. We have discovered that a chemical garden confined to two dimensions is a clock reaction involving a phase change, so that after a reproducible and controllable induction period it explodes.

Reactive convective-dissolution in a porous medium: stability and nonlinear dynamics.

We investigate the effects of a dissolution reaction, A(aq) + B(s) → C(aq), on the gravitational instability and nonlinear dynamic behaviour of a diffusive boundary layer in a porous medium. Our linear stability and numerical results reveal that, unexpectedly, even when the density contribution of the soluble product C is smaller than that of the dissolved solute A, the chemical reaction can destabilize the layer and accelerate the onset of convection. However, for a very light product, the reaction stabilizes the layer. We show that these widely disparate characteristics of the reactive-diffusive layer are outcomes of the nonlinear competition between two reaction effects, the destabilizing sharpening of the solute concentration gradient and associated increase in the solute diffusive flux, and the stabilizing replacement of the solute by a less dense product near the interface.

Filament dynamics in confined chemical gardens and in filiform corrosion.

Two reaction systems that are at first sight very different produce similar macroscopic filamentary product trails. The systems are chemical gardens confined to a Hele-Shaw cell and corroding metal plates that undergo filiform corrosion. We show that the two systems are in fact very much alike. Our experiments and analysis show that filament dynamics obey similar scaling laws in both instances: filament motion is nearly ballistic and fully self-avoiding, which creates self-trapping events.

Acceleration of convective dissolution by chemical reaction in a Hele-Shaw cell.

New laboratory experiments quantify the destabilising effect of a second-order chemical reaction on the fingering instability of a diffusive boundary layer in a Hele-Shaw cell. We show that, for a given chemical system, the dynamics of such a reactive boundary layer is fully determined by two dimensionless groups, Da/Ra(2), which measures the timescale for convection compared to those for reaction and diffusion, and CBo', which reflects the excess of the environmental reactant species relative to the diffusing solute. Results of a systematic study varying CBo' in the range 0-0.1 are presented. It is shown that the chemical reaction increases the growth rate of a perturbation and favours small wavelengths compared to the inert system. A higher concentration of CBo' not only accelerates the onset of convection, but crucially also increases the transport of the solute by up to 150% compared to the inert system. This increase in solute transfer has important practical implications, such as in the storage of carbon dioxide in saline aquifers.

Reactive-convective dissolution in a porous medium: the storage of carbon dioxide in saline aquifers.

We quantify the destabilising effect of a first-order chemical reaction on the fingering instability of a diffusive boundary layer in a porous medium. Using scaling, we show that the dynamics of such a reactive boundary layer is fully determined by two dimensionless groups: Da/Ra2, which measures the timescale for convection compared to those for reaction and diffusion; and ßC/ßA, which reflects the density change induced by the product relative to that of the diffusing solute. Linear stability and numerical results for ßC/ßA in the range 0-10 and Da/Ra2 in the range 0-0.01 are presented. It is shown that the chemical reaction increases the growth rate of a transverse perturbation and favours large wavenumbers compared to the inert system. Higher ßC/ßA and Da/Ra2 not only accelerate the onset of convection, but crucially also double the transport of the solute compared to the inert system. Application of our findings to the storage of carbon dioxide in carbonate saline aquifers reveals that chemical equilibrium curtails this increase of CO2 flux to 50%.

Effects of kinetic and transport phenomena on thermal explosion and oscillatory behaviour in a spherical reactor with mixed convection.

Thermal explosions are often influenced by the complex interaction between transport and reaction phenomena. In particular, reactant consumption can promote safer, non-explosive operation conditions of combustion systems. However, in liquids or gases, the presence of forced convection can affect the behaviour of a system, instigating oscillations in the temperature, reactant concentration and velocity fields. This work describes the effect of reactant consumption on a simple, one-step, exothermic reaction occurring in a spherical reactor with both forced and natural convection, by means of numerical simulations. Regime diagrams characterised by ratios of timescales for each transport and reaction phenomena are presented and the explosion boundary is represented for several forced convection and reaction consumption intensities. Special attention is given to the oscillatory behaviour observed for moderate forced convection and oscillatory regions are represented on the regime diagrams. Parametric conditions for this new oscillatory regime are identified by extending the criticality condition developed by Frank-Kamenetskii for the effect of reactant consumption in diffusive systems to include the effects of both natural and forced convection.

Dynamics of the osmotic lysis of mineral protocells and its avoidance at the origins of life.

The osmotic rupture of a cell, its osmotic lysis or cytolysis, is a phenomenon that active biological cell volume regulation mechanisms have evolved in the cell membrane to avoid. How then, at the origin of life, did the first protocells survive prior to such active processes? The pores of alkaline hydrothermal vents in the oceans form natural nanoreactors in which osmosis across a mineral membrane plays a fundamental role. Here, we discuss the dynamics of lysis and its avoidance in an abiotic system without any active mechanisms, reliant upon self-organized behaviour, similar to the first self-organized mineral membranes within which complex chemistry may have begun to evolve into metabolism. We show that such mineral nanoreactors could function as protocells without exploding because their self-organized dynamics have a large regime in parameter space where osmotic lysis does not occur and homeostasis is possible. The beginnings of Darwinian evolution in proto-biochemistry must have involved the survival of protocells that remained within such a safe regime.

Bridging the Gray and Yang, Sal'nikov and one-step, non-isothermal reaction schemes for oscillatory and explosive behaviour.

Explosions may occur as a result of thermal and branching effects in combustion processes. The aim of this work is to study the effects of chain branching and termination in a "Gray and Yang" three-step, chain-thermal reaction scheme, and the role of thermal diffusion and convection in determining the global behaviour. The response includes steady, exothermic combustion, oscillatory reaction and explosion. This was done by solving numerically the governing equations for the underlying kinetics and the heat transport, and analysing solutions for the limits at which each transport phenomenon dominates. Using scales to describe the different transport mechanisms, a 3D regime diagram is used to characterise the behaviour of the system. The results show how branching and heating by reaction enhance the oscillatory and explosive behaviour. The integration of a two-step Sal'nikov reaction scheme and a one-step reaction model in the Gray and Yang scheme is presented and the Frank-Kamenteskii and Semenov limits are identified on the regime diagram. Comparison is made with some experimental, numerical and analytical results.

Effects of localised and global convection on thermal explosion in a parallel-plate geometry.

During an exothermic reaction in a fluid, convection may ensue on a local scale and then develop to the scale of the entire vessel. In this work, we study the effects of both localised and global convection on thermal explosions occurring between parallel plates. Analytical relations are derived for the various transitions in regimes of convective and thermal behaviours. We show that these relations agree well with previous numerical work and with new simulations in the present investigation. We also determine analytically the time for onset of convection, as well as the temperature increase at that time, for stable and explosive systems. The effects of the Prandtl number of the fluid on the transitions between regimes are noted.