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Out-of-equilibriummolecular systems hold great promise as dynamic, reconfigurable matter thatexecutes complex tasks autonomously. However, translating molecular scaledynamics into spatiotemporally controlled phenomena at mesoscopic length scalesremains a challenge. In living cells, reliable positioning processes such asthe centering of the centrosome involve forces that result from dissipativeself-assembly. We demonstrate how spatiotemporal positioning emerges in syntheticsystems where self-assembly is coupled to molecular fluxes originating fromconcentration gradients. At the core of our system are millimeter longself-assembled filaments and Marangoni flows induced by non-uniform amphiphiledistributions. We demonstrate how repulsive and attractive forces that aregenerated as filaments organize between source and drain droplets sustain autonomouspositioning of dynamic assemblies at the mesoscale. Our concepts provide a newparadigm for the development of non-equilibrium matter with spatiotemporal programmability.

A profoundly fundamental question at the interface between physics and biology remains open: what are the minimum requirements for emergence of complex behaviour from nonliving systems? Here, we address this question and report complex behaviour of tens to thousands of colloidal nanoparticles in a system designed to be as plain as possible: the system is driven far from equilibrium by ultrafast laser pulses that create spatiotemporal temperature gradients, inducing Marangoni flow that drags particles towards aggregation; strong Brownian motion, used as source of fluctuations, opposes aggregation. Nonlinear feedback mechanisms naturally arise between flow, aggregate and Brownian motion, allowing fast external control with minimal intervention. Consequently, complex behaviour, analogous to those seen in living organisms, emerges, whereby aggregates can self-sustain, self-regulate, self-replicate, self-heal and can be transferred from one location to another, all within seconds. Aggregates can comprise only one pattern or bifurcated patterns can coexist, compete, endure or perish.

Order, diversity and functionality spontaneously emerge in nature, resulting in hierarchical organization in far-from-equilibrium conditions through stochastic processes, typically regulated by nonlinear feedback mechanisms1,2. However, current understanding of the fundamental mechanisms and availability of experimental tools to test emerging theories on the subject are lacking. Most current understanding is from model systems3,4,5 that are either too simple to generate rich, complex dynamics collectively2 or so artificial that they have little relevance to actual physical systems. On the other hand, real-life systems, living organisms being the ultimate examples, are so complicated that it is difficult to isolate the essential factors for emergence of complex dynamics1,2. Specific instances of characteristically life-like properties, such as self-replication or self-healing, have been demonstrated in various microscopic systems6,7,8,9,10,11, but they were never observed collectively in a single system that is simple enough to allow identification of mechanisms of emergence.

Dissipative self-assembly is a practical experimental platform to study the fundamental mechanisms of emergent complex behaviour by providing settings akin to those found in nature: far-from-equilibrium conditions12,13,14,15,16, a time-varying external energy input12,13,14,15,16,17, nonlinear feedback mechanisms16,18,19,20,21,22, fast kinetics15,16,22,23, spatiotemporal control15,16,22,23 and a medium to efficiently dissipate the absorbed energy12,13,14,15,16,17. However, previous experimental demonstrations either relied on specific interactions between the building blocks and the external energy source24,25,26 or were limited to certain materials and/or sizes21,26,27,28. Furthermore, most of them were strongly limited by their slow kinetics14,29 and there was little room for fluctuations (Brownian motion was usually weak), where the nonlinear feedback mechanisms were often neglected, unemployed or unidentified.

Here, we report far-from-equilibrium self-assembly of tens to thousands of colloidal nanoparticles with fast kinetics that exhibits complex behaviour, analogous to those commonly associated with living organisms, namely, autocatalysis and self-regulation, competition and self-replication, adaptation and self-healing and motility. We do not use functionalized particles or commonly employed interaction mechanisms, such as optical trapping, tweezing, chemical or magnetic interactions. Instead, we designed a simple system that brings together the essential features: nonlinearity to give rise to multiple fixed points in phase space (hence, possibility of multiple steady states), each corresponding to a different pattern and their bifurcations2; positive and negative feedback to cause exponential growth of perturbations and their suppression, respectively18,19,22; fluctuations to spontaneously induce transitions through bifurcations1; and finally, spatiotemporal gradients to drive the system far from equilibrium, whereby the spatial part allows regions with different fixed points to coexist and the temporal part leads to dynamic growth or shrinkage of these regions.

Emergence of complex behaviour from this plain system can be understood intuitively under the guidance of our toy model, numerical simulations and experimental observations. The laser-sustained thermal gradient not only keeps the system away from thermal equilibrium but, together with the boundary conditions imposed by the bubbles, also creates different local conditions corresponding to different fixed points: a given location can support, say, a square lattice of self-assembled particles, while a hexagonal lattice exists nearby. Each fixed point has a finite basin of attraction both in the phase space and real space, delineated by the spatially varying conditions. In response to perturbations, such as a shift of a bubble boundary or the omnipresent Brownian motion, the original pattern is recovered (self-healing) if the disturbed state remains within the basin of attraction. If the perturbation is large enough that the disturbed state falls outside of the basin of attraction, it switches to a different pattern (self-adaptation) or can be disassembled. A spatiotemporal gradient can also enlarge or shrink the region where a given pattern is the fixed point. In the former case, the pattern can grow (self-replication) or sustain itself (self-regulation). When two nearby regions supporting different patterns come into contact, competition ensues at their boundary: Brownian motion acting on each particle can displace it just enough that the particle leaves a pattern and joins the adjacent one if this stochastic perturbation is large enough and in the right direction. Consequently, the pattern boundaries are dynamic and if the conditions are favourable, one pattern can grow at the expense of another, demonstrating an analogue of interspecies competition. Similarly, motility can be understood as arising in response to temporal gradients that are small enough that the self-healing property can hold the aggregate together as it moves.

How to cite this article: Ilday, S. et al. Rich complex behaviour of self-assembled nanoparticles far from equilibrium. Nat. Commun. 8, 14942 doi: 10.1038/ncomms14942 (2017).

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Using different combinations of two or more beacons, we have also demonstrated distinct strategies to direct and enhance colloidal migration over distances that are hundreds of times further than typical equilibrium suspension interactions. Each of our different strategies comes with unique benefits, and one might be preferred depending on the specific application. The source-sink dipole system requires minimal raw material input (only one solute is needed) and is a clean system as solute is absorbed by the sink. Moreover, the system is conceptually straightforward and simple mass transport models capture the experimental nuances, allowing theoretical prediction of behavior.

In Sect. 2, we introduce a network conception of scientific theory. In Sect. 3, we use simple examples to outline Bayesian updating in a theoretical structure in the course of an evidence barrage. In Sect. 4, we demonstrate punctuated equilibrium in theoretical change across a variety of networks, with a further analysis of change dynamics in Sect. 5. Section 6 outlines philosophical implications. Section 7 concludes with three conjectures regarding dynamics within Bayesian models of scientific theory.

Our focus in what follows is the clear qualitative features of these typical patterns, an analysis of their dynamics, and what we take to be the philosophical significance of these with regard to theoretical change. Of particular note is the fact that these graphs show (a) distinctive periods of intense activity in terms of divergence between iterations under random evidence impact, together with (b) clear periods of flat calm, with very little divergence between iterations under evidence impact. Although evident in terms of credence change graphed node by node, patterns are particularly clear when viewed in terms of Brier divergence across the network as a whole. In Figs. 10 and 11 we extend iterations from 3000 to 30,000 in each case, demonstrating that with different intervals the qualitative pattern of punctuated equilibrium continues.

In the water sprinkler example, as in the simple linear network, patterns of change under an evidence barrage show distinct periods of (a) intense activity in terms of divergence from iteration to iteration together with (b) periods of calm: a pattern of punctuated equilibrium (Eldredge & Gould, 1972, 2007; Gould & Eldredge, 1977). 2b1af7f3a8