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Mesoscopic Reaction-Diffusion Fronts Control Biomorph Growth Pamela Knoll, Elias Nakouzi, and Oliver Steinbock J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09559 • Publication Date (Web): 31 Oct 2017 Downloaded from http://pubs.acs.org on November 6, 2017

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The Journal of Physical Chemistry

Mesoscopic Reaction-Diffusion Fronts Control Biomorph Growth

Pamela Knoll, Elias Nakouzi+, and Oliver Steinbock* Florida State University, Department of Chemistry and Biochemistry, Tallahassee, FL 32306-4390, USA

ABSTRACT

Biomorphs are inorganic assemblies of crystalline nanorods that form non-crystalline

microshapes such as helices, funnels, and leaf-shaped sheets. This nano-to-microscale self-organization occurs from simple reactants in aqueous solution and suggests new engineering methods as well as insights into biomineralization; to date, however, the underlying mechanisms are not understood. Here we describe a reaction-diffusion model for sheet growth that reproduces the experimentally observed biomorph shapes. The sheet edges are logarithmic spirals caused by the propagation failure of the crystallization front and not by a curling-induced arrest as proposed earlier. The resulting defect motion is dynamically related to nonlinear wave dynamics in subexcitable media such as the photo-inhibited Belousov-Zhabotinsky reaction. An expanded model includes a nematic-like director field that sets the average nanorod orientation irreversibly during crystallization. The growth front of worm-like biomorph helices also obeys logarithmic spirals suggesting future applications of reaction-diffusion models for the simulation of three-dimensional biomorphs.

+ New address: Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, WA 99352, USA

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1. INTRODUCTION Chemical self-assembly and self-organization hold substantial promise for innovations in materials science and engineering as both uniquely bridge the length-scale gap between the microscopic world of molecules and the macroscopic realm of hierarchically structured matter and devices. A key challenge for research in this area is to create life-like spatial complexity by utilizing and controlling far-fromequilibrium conditions, concentration gradients, and transport phenomena.1-7 To date, only little of the underlying physical chemistry is understood and it is hence important to carefully study and analyze the few known chemical examples.8-13 These major opportunities for modern chemistry also have a profound impact on the identification of earliest (or non-terrestrial) microfossils and their reliable distinction from inorganic structures that merely mimic biological shapes.14-16 A thorough understanding of what non-euhedral structures can form from abiotic reactions is clearly of importance in both contexts.17 Ideally these efforts will also result in mathematical models that fuel the interplay between experiment and theory while guiding synthetic efforts targeting specific performance features and microshapes. A promising candidate for these research thrusts is a type of polycrystalline assembly called “biomorphs”. These inorganic microstructures are smoothly curved sheets, stalked cones, coral-like shapes, various single and double helices, as well as urns and funnels spanning up to about 1 mm in length.3,18-22 They consist of thousands of high-aspect ratio, co-aligned nanorods and form in alkaline solutions of barium chloride and sodium silicate under the influx of gaseous carbon dioxide. The individual nanorods have a diameter of 30-40 nm, are crystalline witherite (BaCO3) with silicate inclusions, and their alignment causes the—three orders of magnitude larger—sheets to polarize light.14,21 In addition, the biomorph microshapes are not limited to barium, but similar noncrystallographic morphologies have been observed during the silicate-assisted crystallization of SrCO3 and CaCO3.15,20 Recent studies also demonstrate that the growth process can be carried out in closed, single-phase systems if CO2 is replaced by dissolved Na2CO3.23,24 This change greatly reduces acidification, eliminates vertical gradients, and increases the duration of the pattern-forming growth phase from hours to weeks. Beyond the basic aspects of the initial crystallization process, which involve successive branching of carbonate crystallites according to the rod−fractal dumbbell pathway17,25, no mechanistic explanation for the occurrence of the observed microshapes exists. In 2017, Kaplan et al. developed a kinematic model that is based on the motion of a space curve in the presence of a parameter field that was 2 ACS Paragon Plus Environment

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interpreted as spatial variations in the solution pH.26 While this approach reproduces some of the observed morphologies and emphasizes the importance of the local biomorph curvature, it does not provide physical insights into the underlying growth mechanisms and does not explain the observed shapes of flat biomorphs. Additional mechanistic hints were recently reported by Nakouzi et al. who described band-like and patchy height variations of some biomorph sheets.27 These topographic oscillations correlate with out-of-plane, directional variations of the nanorods and have an amplitude of about 0.5 µm (about half the average sheet thickness) at a characteristic wavelength of 6.6 µm. 2. METHODS AND RESULTS All of our experiments employ solutions of 8.4 mM sodium silicate and 5.0 mM barium chloride at a pH of 10.6. At ambient conditions, these solutions are exposed to air which triggers biomorph formation within three to six hours (see Supporting Information for additional details). Telltale geometric aspects of the biomorph sheets are illustrated in Figure 1. The optical micrograph in (a) shows several leafshaped sheets that nucleated from initial globular precipitates (dark regions). The larger sheets are in direct contact with the glass substrate, but one of the smaller leaves demonstrates that sheets can also grow on top of pre-existing biomorphs. Garcia-Ruiz et al. reported that the specific leaf shape is the result of defects affecting the otherwise isotropic and steady advance (speed vN) of the crystallization front.18 These defects were described as sheet edges curling up and, thus, arresting the forward motion locally while spreading along the growth line at a tangential speed vT. The tip of each leaf is caused by the collision of a defect pair. Our analyses of the biomorphs sheets show that their edges are well described by logarithmic spirals r = a eb φ

,

(1)

where (r,φ) are the polar coordinates with respect to the center of the spiral, a is a geometric constant, and b = vN/vT. Logarithmic spirals are self-similar and stretching by a factor e2πb leaves them unchanged (28). Notice that the extreme cases of b = 0 and b → ∞ correspond to a circle and a half line, respectively. For comparison, we extract the coordinates of the individual sheet edges and perform least-square fitting of eq 1. The resulting fits are superposed in Figure 1a as colored symbols and are in very good agreement with the sheet edges. Figure 1b shows the corresponding half-logarithmic plots and yields b values between 0.55 and 1.14. This surprisingly large range—further substantiated by the histogram in Figure 1c—hints towards an unknown external variation or a “hidden” aspect of the 3 ACS Paragon Plus Environment


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crystallization front. In this context, we evaluated the dependence of b on the height of the biomorph sheet, which can vary from about 700 to 1300 nm, but found no correlation (see Figure S1 in the Supporting Information). Our experiments also reveal that the arrest of the growth front is not necessarily caused by the curling phenomenon discovered by Garcia-Ruiz et al.18 We frequently observe sheets with continuous edges that are only locally curled and many biomorph edges show no curling at all (Figure 1d-e). Furthermore, we find small differences between the b value distributions of curled (1.4±0.4) and flat biomorph edges (0.8±0.3) as color-coded in the histogram (Figure 1c). It is hence likely that the edge curling is a secondary phenomenon and not the cause of the arrest of the crystallization front. In the following, we propose a two-variable reaction-diffusion model for the growth of quasi-twodimensional biomorph sheets. This model links the front arrest in biomorph growth to chemical waves in two-dimensional excitable systems such as the photo-inhibited Belousov-Zhabotinsky reaction29-32, the CO oxidation on Pt(110) surfaces33, and the formation of rust on corroding steel34. In these systems, a spatial disruption of a wave pulse often nucleates a pair of rotating spiral waves; however, for certain parameter values, the wave ends do not curl into a spiral but rather erase the pulse by slow tangential movement.29,30 Accordingly, expanding circular wave pulses are stable, but any local disruption leads to lateral shrinkage and ultimately the self-annihilation of the pulse. We note that fragmented waves in such subexcitable media can be sustained (or their collapse slowed down) by external or internal noise.31,32 During biomorph growth such a source of stochasticity might be the statistical nucleation and termination of the nanorods; however, these factors will not be considered here. The equations of our dimensionless model are ∂u/∂t = ∇2u + ε -1 u (1−u) [u−(v+β )/α] ,

(2a)

∂v/∂t = u

(2b)

,

where the variables u and v represent the crystallization activity and the solid biomorph material, respectively. Notice that these specific rate laws do not result from a reaction mechanism and hence u has no unique interpretation. A possible candidate for the system’s autocatalytic process, however, was suggested by Montalti et al.25 in the form of the interplay of pH lowering silica and pH raising witherite formation. An alternative interpretation is the mere presence of a solid-liquid interface—the biomorph edge—that promotes growth from the over-saturated solution. The parameters are chosen as ε = 0.02,

α = 0.8, and β = 0.14735 but can be varied within a narrow band of the α,β -plane without loss of the 4 ACS Paragon Plus Environment

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qualitative results. Figure 2a,b illustrates that the model has a propagating pulse solution that connects the stable steady state (u,v) = (0,0) to the product state (0,0.5) via an intermittent increase of u. The speed of the crystallization front is constant in one-dimensional simulations and also for planar pulses in two dimensions. Our model is inspired by the Barkley model of excitable media35 and differs from it only in eq 2b which does not allow for a recovery of v. Accordingly, the formation of the biomorph is irreversible and each location can undergo crystallization only once. In close analogy to the supersaturated solution in biomorph experiments22-24, the simulated system can remain free of biomorphs indefinitely. Biomorph growth requires a local perturbation, which in our simulations is accomplished by setting u = 0.5 within a small disk-shaped region. This seed starts a circular front that creates a steadily expanding biomorph disk. We further model the sheet-nucleating globules as regions with no-flux boundaries. In the simplest case of a box-shaped boundary and an initial perturbation at the center of one of the box edges, we first obtain an expanding half-disk which then detaches to generate two defects that mark a left and right termination point of the crystallization front. The points move forward with the main front speed but also tangentially along the front, thus, shortening it. These dynamics and their final termination via defect collision are shown in Figure 2c and generate the main leaf shape. We also note that the important detachment of the front from the boundary occurs because of its inability to perform a sufficiently sharp turn at the corner (see Figure S2 in the Supporting Information); effects of the external system boundaries do not exist in our simulations but could be relevant if the front collides with those walls. Many of the experimentally observed variations in the sheet shapes can be explained in terms of defect pair creation via front breakage or detachment. Four characteristic examples are shown in Figure 3, in which the defect pair is generated near the globule (a), far away from the globule (b), asynchronously (c), and synchronously (d). Our model reproduces these cases as shown in Figure 3e. The top row are simulations in which an expanding circular front is disrupted by small, inert heterogeneity with (u,v) = (0,0) = const (yellow disks, black arrow). The three simulations in this row differ only in the distance between the heterogeneity and the biomorph nucleation site creating distinctly different shapes that match Figure 3a,b. Axial asymmetry in the sheet stems from an asynchronous defect generation that can be simulated by different placements of the initiation site along a no-flux boundary (middle row). Moreover, elongated sheets give way to stubby sheets if the time of defect-free expansion is shortened (bottom row).



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As an example, Figure 3e includes least-square fits of logarithmic spirals to the edges of the lower, left biomorph. The fits yield b values of 1.15 which falls within the range of experimental values (Figure 1c). Most of the other simulated sheet edges are also well described by logarithmic spirals. However, this specific shape arises only if the initial crystallization front is a (full or partial) circle; otherwise, more complicated forms are generated. In the experimental system, sheets nucleate from small point-like regions and therefore trigger circular fronts, thus explaining the abundance of logarithmic spirals. Our modeling approach also allows for the description of the average nanorod orientation. For this, we describe the mean-field orientation of the nanocrystals by a director field similar to those used for the description of nematic liquid crystals.36 From earlier studies employing both scanning electron and optical polarization microscopy14, we know that the nanorods tend to grow normal to the crystallization front (a fact that also holds for helices and other non-planar structures). Although no direct experimental evidence exists, it seems self-evident that the average nanorod orientation is set during the crystallization event. Accordingly, we propose the model equation ∂ψ/∂t = − k u (∂v/∂x sinψ − ∂v/∂y cosψ),

(3)

where the local director angle ψ is defined with respect to an arbitrary reference direction (here the x axis). Notice that changes in ψ occur only within the crystallization front (u > 0). In addition, the expression within the parentheses is the z-component of the cross product (∇v × d) / |∇v| where d is the unit vector (cosψ , sinψ) and hence the sine of the angle between the two vectors. Notice that alternative expressions replacing ∇v with ∇u would induce unwanted directional reversals due to the pulse nature of u (compare Figure 2a,b). The rate constant k must be sufficiently fast to safely permit ψ to change by up to π during the pulse duration (here k = 100; see Supporting Information for details). The initial values of the ψ field carry no physical meaning. Figure 4a shows a representative electron micrograph of the surface-exposed nanorods of a biomorph sheet illustrating the coherence of the nanorod alignment. The nanorods consist of crystalline witherite with about 20% embedded silica that causes crystal strain as reflected in the shape of the Xray diffraction signal.37 Figure 4b shows the simulated director field for a symmetric biomorph sheet which matches the experimental observations. Our eq 3 could be used for further modeling efforts that include the feedback of the nanorod orientation on the dynamics of the crystallization pulse (eq 2a,b).


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Such a feedback is one possible mechanism behind the growth of biomorph funnels, urns, and other thin-sheet morphologies in three dimensions. 3. CONCLUSIONS Surprisingly, our finding that the edges of biomorph sheets are logarithmic spirals also extends to single helices (often referred to as “worms”). Figure 4c shows a scanning electron micrograph of a representative worm head. This top view reveals a distinct growth edge and its shape is in very good agreement with the superposed logarithmic spiral as further documented in the accompanying halflogarithmic plot of r(φ) (Figure 4d). The micrograph also shows a second, trailing spiral that might result from an earlier reported secondary growth in the backward direction.24 Overall the b values for the growth fronts of helices are smaller than for sheets, but the presence of logarithmic spirals suggests that a similar mechanism is at play. This interpretation is further supported by the fact that we frequently observe sheet growth on pre-existing biomorph sheets (e.g. Figure 1a and Supporting Information). However, if single helices are a continuous extension of the sheet motif into the third dimension, additional factors must select the diameter of the worm (typically around 50 µm). Encouraged by the results from our two-variable reaction-diffusion model, we speculate that this confinement is caused by an additional inhibitor variable. For example, Purwins et al. reported localized moving structures, socalled quasi-particles, in nonlinear reaction-diffusion systems with one activator and two inhibitor species.38,39 The dynamics of these reaction zones include simple translation, splitting cascades, and rotating bound states. The phenomena persist in spatially three-dimensional systems39 and could, with modifications similar to those in our eqs 2, lead to qualitative models of helical and other biomorphs that are not surface bound. Additional similarities exist between the coral-shaped biomorphs reported by Aizenberg et al.3 and the labyrinthine reaction-diffusion patterns in the Gray-Scott model40. Future work should also investigate the mechanistic importance of the underlying self-assembly processes at the nanoscale and the impact of the associated fluctuations on the mesoscale structures.



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ASSOCIATED CONTENT Supporting Information Experimental and numerical methods, study of height effects, and schematics illustrating differences between crystallization fronts and excitation waves. The Supporting Information is available free of charge on the ACS Publications website at DOI: xxxx AUTHOR INFORMATION Corresponding Author E-mail: [email protected]. Phone: +1 850-644-4824. ORCID Oliver Steinbock: 0000-0002-7525-6399 Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grant No. 1609495 to OS and a Graduate Research Fellowship No. 1449440 to PK. EN acknowledges the Mineralogical Society of America (MSA) and the International Centre for Diffraction Data (ICDD). The Condensed Matter and Material Physics (CMMP) User Facility at Florida State University provided access to the SEM instruments. We also thank Zhihui Zhang and Arash Azhand for discussions.


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Figure 1. (a) Optical image of five biomorph sheets. The superposed markers are fits of logarithmic spirals to the sheet edges. (b) Half-logarithmic plot of the same edges in polar coordinates and corresponding linear fits. The marker colors allow cross-referencing between (a,b). (c) Histogram of the b values for curled (cyan) and non-curled (yellow) sheet edges. The average b values for curled and noncurled edges are 1.4±0.4 and 0.8±0.3, respectively. Inset: histogram of the combined data sets; the average b value is 1.0±0.5. (d,e) SEM images of biomorph sheets demonstrating that edge curling is not necessary for growth termination. The absence of a curled edge is particularly clear in the inset of (e) that magnifies a detached, growth-terminated sheet edge. Scale bars are (a,e) 100 µm and (d) 20 µm.


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Figure 2. (a,b) Traveling crystallization pulse in a spatially one-dimensional version of the model (eqs 2). The two subsequent snapshots illustrate that the shape and amplitude of the pulse are constant. The variables u (red) and v (blue) correspond to the crystallization activity and the solid biomorph product, respectively. (c) Two-dimensional simulations of a crystallization front initiated near a no-flux boundary (border of yellow area). The initially half-circular u pulse (red) detaches at the corners of the boundary and initiates a defect pair (free ends of the red bands). The motion of the defects shrinks the crystallization front and shapes the biomorph sheet (blue). The three columns show the same simulation in terms of u only, v only, and a combination of the two variables.



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Figure 3. (a-d) Optical micrographs of biomorph sheets illustrating different morphologies. The nodular, black regions are witherite globules that form prior to sheet nucleation and constitute no-flux boundaries for the subsequent crystallization front. (e) Numerical simulations initiated from small circular or stripe-like regions (light blue). The yellow disks or boxes represent inert regions with no-flux boundaries. Different biomorph shapes are found when the shapes and distances of the initiation site and the heterogeneities are varied. The blue and red curves show that the edges of the lower left sheet are logarithmic spirals. Good agreement exists between the shapes in the top row of (e) and the top two optical images (a,b), the asymmetrical leaf in the center of (e) and the biomorph in (c), and the last row in (e) and the biomorph in (d). Scale bars: (a,b) 100 µm and (c,d) 50 µm.


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Figure 4. (a) SEM image of the nanorods at a non-curled biomorph sheet edge with a preferential direction normal to the growth front. The same area (red arrow) is shown at lower magnification in the inset. (b) Simulation based on the director field model (eqs 2,3) with k = 100. The white arrows represent the average, local nanorod direction and are in good agreement with the experimental findings. (c) Electron micrograph of the growing surface of a three-dimensional, single-helix biomorph. The blue and red points are logarithmic spiral fits with b = 0.4 and b = 0.3, respectively. (d) Halflogarithmic plot of the polar edge coordinates in (d) and the single helix in the optical micrograph (inset). The latter is a different sample and fitting yields b = 0.3 (green). The continuous lines represent logarithmic spirals (eq 1) and are linear fits of the experimental data. Scale bars are (a) 500 nm, (c) 10 µm, and (d, inset) 20 µm.



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TOC Graphic


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Mesoscopic Reaction-Diffusion Fronts Control ... - ACS Publications

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