scattering methods [19] allow for time- and space-resolved measurements of the structure
and dynamics on macroscopic portions of the sample and for a large variety of systems (Fig.
2C). However, spatial mapping is coarsened over tens of microns and data interpretation is
less straightforward than for imaging. Ultrasonic velocimetry maps the flow field with
resolution 20-100 µm [20] (Fig. 2B). Mechanophore molecules incorporated in the system
can report optically the local stress [21] (Fig. 2E) or bond breaking events in networks [22]
(Fig. 2D), but they require system-specific, sophisticated synthesis. Future developments
will likely involve combining several of these methods.

3) Advances are also needed in sample design and preparation. Model systems allow for easier data interpretation and comparison with theory and simulations. For example, DNA-based systems afford great control on the interactions among constituents and the resulting architecture, e.g. in DNA hydrogels [21], but it is still difficult to synthesize them in amounts suitable for mechanical tests. A related issue is that of sample initialization: most soft solids are out-of-equilibrium and their LMD behaviour depends greatly on sample history, as exemplified by recent simulations of amorphous packings of soft particles, whose failure behaviour (ductile vs fragile) depend on the degree of annealing [23] (Fig. 2F).

4) In numerical works, a major challenge is to cover the wide range of time- and space- scales relevant to LMD. A promising approach consists in coupling different methods, from molecular dynamics or Monte Carlo at the molecular level, to mesoscopic finite-element models based on a visco-elasto-plastic formalism [24]. The ability of dealing with hydrodynamic interactions is also a key advance [25].

5) On the theoretical side, current approaches are mostly phenomenological [7], or restricted to a mean field approach, e.g. as in soft glassy rheology [6], making the connection with the microscopic level difficult. Strategies to go beyond these approaches include drawing analogies with well-known statistical physics models, e.g. likening yielding to a phase transition in the random field Ising model [23].

Accepted for publication but not yet final.
<figure><img src="image_23.png" /><figcaption>**Figure 2:** (A) Impact of a drop of viscoelastic fluid on a repellent surface, yielding a large and fast deformation of the material (bottom image). (courtesy of Carole-Ann Charles, L2C, CNRS & U. Montpellier). (B) Experimental spatiotemporal map of the flow velocity of a colloidal gel sheared at a fixed rate in a Couette cell, as measured by ultrasonic velocimetry [20]. (C) Experimental maps of the local dynamics over 1 s for a biopolymer under creep, measured by time- and space-resolved dynamic light scattering. Labels: time before failure. Adapted from [26]; (D) Bond breaking events reported by fluorescent mecanophores in (multiple) elastomer networks [22]; (E) Fluorescent signal reporting the local strain in DNA-based hydrogels [21]; (F) Color maps of the one-cycle nonaffine deformation after 10 and 30 shear cycles, for two different sample preparations, as obtained by molecular dynamics simulations of a dense packing of spheres. Adapted from [27].</figcaption></figure>

**Concluding Remarks** While many key questions are still open, the recent progress in experimental tools, numerical methods and theoretical approaches suggests that this field should develop vigorously in the forthcoming years. Drawing a common scenario by unveiling the common features across different systems and driving modes will be, in our view, the greatest challenge in the field.
**Acknowledgements** We thank the French CNES and ANR (grants No. ANR-14-CE32-0005, FAPRES, and ANR-20-CE06-0028, MultiNet) for financial support. LC acknowledges support from the Institut Universitaire de France.

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[7] Derec C, Ajdari A and Lequeux F 2001 Rheology and aging: A simple approach The European Physical Journal E **4** 355–61

[8] Fielding S M and Olmsted P D 2004 Spatiotemporal oscillations and rhechaos in a simple model of shear banding Phys. Rev. Lett. **92**

[9] Datta S S, Ardekani A M, Arratia P E, Beris A N, Bischofberger I, Eggers J G, López-Aguilar J E, Fielding S M, Frishman A, Graham M D, Guasto J S, Haward S J, Hormozi S, McKinley G H, Poole R J, Morozov A, Shankar V, Shaqfeh E S G, Shen A Q, Stark H, Steinberg V, Subramanian G and Stone H A 2021 Perspectives on viscoelastic flow instabilities and elastic turbulence arXiv:2108.09841 [cond-mat, physics:nlin, physics:physics]

[10] Fuchs M and Cates M E 2002 Theory of Nonlinear Rheology and Yielding of Dense Colloidal Suspensions Phys. Rev. Lett. **89** 248304

[11] Hadde E K, Cichero J A Y, Zhao S, Chen W and Chen J 2019 The Importance of Extensional Rheology in Bolus Control during Swallowing Sci Rep **9** 16106

[12] Cipelletti L, Martens K and Ramos L 2020 Microscopic precursors of failure in soft matter Soft Matter **16** 82–93

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[15] Pascua Y, Koç H and Foegeding E A 2013 Food structure: Roles of mechanical properties and oral processing in determining sensory texture of soft materials *Current Opinion in Colloid & Interface Science* **18** 324-33

[16] Stokes J R, Boehm M W and Baier S K 2013 Oral processing, texture and mouthfeel: From rheology to tribology and beyond *Current Opinion in Colloid & Interface Science* **18** 349-59

[17] Geri M, Keshavarz B, Divoux T, Clasen C, Curtis D J and McKinley G H 2018 Time-Resolved Mechanical Spectroscopy of Soft Materials via Optimally Windowed Chirps *Physical Review X* **8**

[18] Vermant J, Walker L, Moldenaers P and Mewis J 1998 Orthogonal versus parallel superposition measurements *Journal of Non-Newtonian Fluid Mechanics* **79** 173-89

[19] Cipelletti L, Trappe V and Pine D J 2016 Scattering Techniques *Fluids, Colloids and Soft Materials* ed A Fernandez-Nieves and A Puertas (John Wiley & Sons, Inc.) pp 131-48

[20] Liberto T, Le Merrer M, Manneville S and Barentin C 2020 Interparticle attraction controls flow heterogeneity in calcite gels *Soft Matter* **16** 9217-29

[21] Merindol R, Delechiave G, Heinen L, Catalani L H and Walther A 2019 Modular Design of Programmable Mechanofluorescent DNA Hydrogels *Nature Communications* **10** 528

[22] Ducrot E, Chen Y, Bulters M, Sijbesma R P and Creton C 2014 Toughening Elastomers with Sacrificial Bonds and Watching Them Break *Science* **344** 186-9

[23] Ozawa M, Berthier L, Biroli G, Rosso A and Tarjus G 2018 Random critical point separates brittle and ductile yielding transitions in amorphous materials *Proceedings of the National Academy of Sciences* **115** 6656-61

[24] Nicolas A, Ferrero E E, Martens K and Barrat J-L 2018 Deformation and flow of amorphous solids: Insights from elastoplastic models *Rev. Mod. Phys.* **90** 045006

[25] Swan J W and Brady J F 2011 The hydrodynamics of confined dispersions *J. Fluid Mech.* **687** 254-99

[26] Pommella A, Cipelletti L and Ramos L 2020 Role of Normal Stress in the Creep Dynamics and Failure of a Biopolymer Gel *Phys. Rev. Lett.* **125** 268006

[27] Yeh W-T, Ozawa M, Miyazaki K, Kawasaki T and Berthier L 2020 Glass Stability Changes the Nature of Yielding under Oscillatory Shear *Phys. Rev. Lett.* **124** 225502
# Section 3.3 – Suspension Rheology: A Smooth Past, Rough Present and Sticky Future?

Wilson C. K. Poon and James A. Richards

Edinburgh Complex Fluids Partnership, School of Physics and Astronomy, The University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Edinburgh EH9 3FD, UK

**Status** Thomas Kuhn famously describes fundamental shifts in perspective in science as changes in paradigm.¹ Paradigm changes share certain features. For example, one-time heresies become orthodoxy and previously unrelated areas become unexpectedly unified. After a new paradigm is seen to bring new insight in understanding to one or more exemplars, the members of the relevant community begin to rethink their entire field under the new perspective. There has been a recent paradigm shift in suspension rheology. Previous orthodoxy says that suspended particles do not touch, because lubrication forces between closely-approaching particles diverge and keep them apart.² The same forces render it difficult for nearby particles to separate, giving rise to ‘hydroclusters’ that cause shear thickening in suspensions.

The ‘no touching’ rule is true for perfectly smooth particles with ideal stick boundary conditions. However, the surfaces of two suspended spheres (radius R) in shear can approach³ to where the simplification of ‘smoothness’ must break down in practice, e.g., due to roughness at $\lesssim 10^{-4}$ R. In the new paradigm, the (previously unthinkable) frictional interactions between contacting particles dominate suspension rheology, especially in the non-Brownian size regime.⁴ This realisation partly emerged from the insight that aspects of the physics of dry granular materials could be ‘borrowed’ for wet suspensions. This led to a rather unexpected, and fruitful, convergence of these two hitherto unrelated areas.⁵ There followed an intense effort to understand the exemplar of shear thickening. The emerging picture is coherent enough that the new frictional paradigm has won widespread acceptance, and its wider implications are now being explored. Below, we review this paradigm shift and indicate key areas of suspension rheology that need to be re-thought in its light to move towards a predictive understanding of this industrially relevant class of materials, found in the processing of chocolate⁶ through to cement.⁷

For dry monodisperse spherical grains, it is long known that if interparticle contacts are frictionless, mechanical stability requires each particle to contact on average $Z = 6$ neighbours. At this ‘isostatic point’ with volume fraction $\phi = \phi_{rcp} \approx 0.64$, or random close packing, the system is jammed. With static friction coefficient $\mu \to \infty$, jamming occurs at $Z = 4$ and $\phi \approx 0.55$, sometimes identified as random loose packing ($\phi_{rlp}$). The transition from $Z = 6 \to 4$ as $\mu$ increases essentially occurs over the range $\mu \approx 0.1$ to $1.8$ In 2014, Wyart and Cates (WC)⁹ ‘borrowed’ this physics to explain shear thickening in wet hard-particle suspensions. They assume that suspension viscosity diverges as $\phi \to \phi_J$, a stress-dependent jamming point. As the applied stress increases, particles transition from having lubricated ($\mu = 0$) contacts to having frictional ($\mu \gtrsim 1$) contacts at some characteristic ‘onset stress’ $\sigma^*$. (This is in principle a local tensorial quantity, but in the scalar WC model it is typically taken to be the external applied shear stress, $\sigma$.) At $\sigma \ll \sigma^*$, $\phi_J = \phi_{RCP}$, while $\phi_J \approx \phi_{rlp}$ at $\sigma \gg \sigma^*$. At a fixed $\phi$, the suspension viscosity therefore increases as the system moves closer to jamming at higher $\sigma$.
Quantitatively, one uses a phenomenological expression for the relative viscosity of the suspension (i.e., normalised by that of the solvent) such as $\eta_r = [1 - \phi/\phi_j(\sigma)]^{-2}$ and a linear ansatz $\phi_j = f(\sigma)\phi_{rlp} + [1 - f(\sigma)]\phi_{RCP}$, with some interpolating expression $f(\sigma)$ for a changing fraction of frictional contacts to connect the two endpoints. Shear thickening data from sterically-stabilised polymethyl methacrylate (PMMA)⁴ and charge-stabilised silica suspensions¹⁰ can be fitted using $f(\sigma) = \exp[-(\sigma/\sigma^*)^{-\alpha}]$ with $\alpha \approx 1$. The WC model correctly predicts that above some $\phi_{DST}$ just below $\phi_{rlp}$, $\sigma(\dot{\gamma})$ becomes S-shaped, so that the viscosity jumps discontinuously, potentially by orders of magnitude, when a critical shear rate ($\dot{\gamma}$) is reached: the phenomenon of discontinuous shear thickening (DST). At $\phi_{rlp}$, there is no flow above some maximum applied stress; such shear jamming can be observed transiently in experiments.¹¹

<figure><img src="image_24.png" /><figcaption>**Figure 1** A schematic of the paradigm shift in suspension rheology from hydrodynamic lubrication to contact mechanics. The latter was first 'borrowed' for suspension rheology from dry granular mechanics in the WC model, which then inspired the consideration of multiple coarse-grained constraints on inter-particle motion that are agnostic to microscopic mechanisms at multiple length scales (hydrogen bonding, polymer interdigitation, surface asperities, particle softness, ...</figcaption></figure>

The onset stress, $\sigma^*$, for PMMA suspensions corresponds to the right order of magnitude of the force needed to collapse their stabilising 'hairs'. Importantly, $\sigma \sim R^{-2}$, so that for non-Brownian particles with $R \gtrsim 10$ µm, all practical stresses will drive the particles into frictional contact, and the suspension will respond as a shear-thickened system. It is therefore almost impossible to pack non-Brownian suspensions to $\phi_{RCP}$: they will almost always jam at the lower $\phi_{rlp}$. Simulations¹² confirm that the viscosity divergence in a suspension in which particles always interact frictionally occurs at a $\phi_j$ that smoothly varies from $\phi_{RCP}$ to $\phi_{rlp}$ as $\mu$ varies from 0 to $\infty$, giving a whole family of viscosity plots $\eta_\mu(\phi)$. Shear thickening is then a transition from the lowest branch of this family to one of the higher branches at fixed $\phi$. Simulations also reveal compressional 'force chains' in thickened suspensions comparable to jammed dry granular materials.¹²
The role of friction has been verified by a variety of more¹³ or less¹⁴ direct experiments, and many aspects of the rheology of shear-thickening suspensions beyond steady shear, e.g., shear reversal¹⁵ or sudden stress¹¹ (key for 'running on cornstarch') are being investigated under this new paradigm. Behaviour beyond the frictional jamming point ($\phi_{rlp} < \phi < \phi_{rcp}$) is also being elucidated, such as the response to vibration that allows flow at high $\phi^{16}$ or granulation (where boundary interactions due to a confining surface tension prevent flow).¹⁷

**Current and future challenges** The phenomenology of shear thickening is robust. Similar $\eta_r(\phi)$ plots are reported in suspensions of quasi-monodisperse spheres⁴,¹⁰, bidisperse spheres,¹⁸ irregularly-shaped mineral crystals,¹⁹ and mixtures of glass spheres and shards.¹⁷ Interestingly, the WC model is able to account for data and deliver physically-meaningful fit parameters whenever the particle size distribution is single-peaked. However, it fails badly for highly bidisperse (large-small) suspensions. This is because WC is a mean-field model that treats large-large, large-small and small-small particle contacts 'democratically', while in fact the large-large contacts dominate the rheology and force transmission.¹⁸ Similarly, in mixtures of bumpy and smooth particles, a small number of smooth particles have a disproportionate effect in breaking up the force network.²⁰ Indeed, mean field models should not work for suspensions, because $O(10)$ nearest neighbours do not justify averaging out local details. A theory including these details that treats local stress propagation tensorially is still to be developed. Progress is being made through considering local force balance²¹ and contact anisotropy.²² In parallel, the transition at $\phi_{DST}$ has been fruitfully compared to classical critical phenomena.²³

So far, we have not defined 'friction' beyond specifying that it should be *static* friction that prevents relative sliding and lowers the number of contacts needed to reach mechanical stability (= jamming) due to a 'constraint counting' argument. Precisely how sliding constraints generate extra dissipation is the subject of ongoing research. These constraints may act on the molecular level, e.g., hydrogen bonding;²⁴ mesoscopic surface 'bumpiness' much larger than tribological asperities also suffice,²⁰ Fig. 1. Even hydrodynamic interactions under special conditions²⁵ will give essentially the canonical shear-thickening phenomenology, with DST at $\phi \approx \phi_{rlp}$. As hydrodynamic interactions disappear in the strict static limit ($\dot{\gamma} \to 0$), this raises the question as to whether enhanced tangential interaction at $\ll \dot{\gamma}R$ suffices for 'static friction'. Indeed, measurement of 'static' friction remains outstanding, requiring either a reanalysis of existing dynamic techniques¹³,²⁰ or new methods. Such measurements may resolve the apparent discrepancy that dynamically measured friction coefficients are too low to explain observed values of $\phi_{rlp}$ that consistently suggest $\mu \gtrsim 1$.

The possibility of constraints other than sliding arising from a variety of interactions, Fig. 1, is now being explored.²⁶ Stress-activated constraints that prevent rolling in addition to sliding could enhance shear thickening and lead to shear jamming at $\phi \ll \phi_{rlp}$.²⁷ Alternatively, adding *adhesive* constraints to inter-particle rolling that are operative at zero applied stress, but are broken beyond some characteristic stress predicts a 'zoo' of experimentally-realistic flow curves.²⁶ In this scenario, a finite yield stress emerges above some 'adhesive loose packing', $\phi_{alp} < \phi_{rlp}$, and then increases to divergence at $\phi_{rlp}$. Significantly, this yield stress is not due to adhesive forces alone,
but such forces stabilising frictional contacts along force chains,¹⁹ so that studying yield-stress suspensions may illuminate the role of rolling resistance.

To date, information on rolling has come from optical-trap measurement of the critical bending moments of particle chains in a strong colloidal gel. Such experiments suggest that mechanical contacts between micron-sized suspended particles may strengthen with time.²⁸ This raises intriguing possibilities. For example, under prolonged stressing, nominally hard particles may be pushed beyond 'frictional contact' into softer, even adhesive, contact. A transiently jammed state may no longer be 'fragile', i.e. only load-bearing in the direction in which the suspension was originally shear jammed,¹⁵ with softness leading to the possibility of fully (or isotropically) jammed states.²⁹ This would profoundly influence how a jammed region behaves in unsteady flow, where currently a variety of system specific behaviours are seen in spatially-resolved experiments measuring density³⁰ or velocity.³¹ This suggests combining such tests with transient rheological protocols, e.g., shear reversal, to probe the nature of jammed regions. Recent large amplitude oscillatory shear work on local stresses already points in this direction.³²

**Concluding remarks** Once contacts are admitted, wide-ranging research topics emerge. Consider, e.g., issues raised by the move away from non-contacting *attraction* arising from potential interactions (Van der Waals, Coulomb) to contact *adhesion*; the origin for static sliding friction; the nature of various kinds of stress-dependent rolling resistance that enhances shear thickening; and how to control the response of the inhomogeneous DST state. Resolving these issues may finally give the ability to predict and tune the rheology of non-Brownian suspensions in industrial settings. Beyond such developments, the extension of frictional rheology to suspensions of highly anisotropic particles such as flexible rods³³ will considerably increase its range of applicability, e.g., to cellulose suspensions, a sustainable alternative additive to synthetic polymers in certain products.³⁴

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[12] Mari, R., Seto, R., Morris, J. F. & Denn, M. M. Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions. *J. Rheol.* **58**, 1693–1724 (2014).

[13] Comtet, J. et al. Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions. *Nat. Commun.* **8**, 15633 (2017).

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## Section 3.4 – Driven liquid to solid transitions

Itai Cohen

Department of Physics, Cornell University, Ithaca, NY, USA

**Status** Search the internet, and you will find hundreds of videos showing people running, jumping, and even riding bicycles on the surface of liquid mixtures of cornstarch and water. These fun demonstrations illustrate a ubiquitous property of nearly all dense suspensions, the tendency to undergo a liquid to solid transition when driven at high enough stresses [1] (Figure 1 Left). Such transitions continue to be industrially important for numerous applications including high-throughput processing of dense suspensions such as cement and concrete, 3D printing, and conching during the production of chocolate.

A seminal advance in this field has been the discovery that, in addition to hydrodynamic coupling between the suspension particles that causes a relatively minor increase in viscosity, the onset of thickening is stress dependent and arises from the repulsion, frictional interactions, and/or interdigitation between the particle surfaces [2,3] (Figure 1 Right). Indeed, studies have shown that thickening can be controlled by modifying the particle surfaces to be more or less frictional, smoother or rougher, more or less chemically attractive etc. The origin of these interactions continues to be a very active area of research and has obvious ties to many decades of activity in the field of surface science. Theoretically, the importance of stress dependent surface interactions was intuitively summarized by the Wyart and Cates model, which postulated that the fraction of frictional interactions between particles increases with stress, capturing the idea that for stable suspensions, there is barrier that must be overcome before surface interactions can arise [5].

More recently, attention has been turned towards abstracting away the specific origins of these interactions and thinking of them in terms of the physical constraints they impose on the particle motions. In this approach one becomes agnostic about the origin of say frictional interactions and simply acknowledges that the suspension particles are subject to sliding constraints [6]. This abstraction has focused the community on formulating questions about and addressing a different scale in the thickening problem: the formation, structure, and manipulation of the force network formed by the suspension particles exhibiting these surface interactions [7].
<figure><img src="image_25.png" /><figcaption>**Figure 1** Shear thickening suspension of cornstarch and water. Left, photograph illustrating the liquid to solid transition as the surface is scraped (Image courtesy of Neil Lin). Right, normalized viscosity versus shear rate in experiments and numerical simulations adapted from [3] and [4].</figcaption></figure>

**Current and Future Challenges** A major challenge for the field is to develop experimental techniques, models, and theories that can accurately address various properties of the force network including heterogeneity, anisotropy, and its evolution under imposed flows. In the Wyart and Cates model, for example, such effects are ignored and the suspension is assumed to be isotropic with a viscosity that is related to the suspension's proximity to a stress dependent jamming volume fraction: as surface interactions increase, isotropic particle networks can support themselves at lower volume fractions and the system is driven closer to a solid phase. To address this challenge in experiments, it will be necessary to move beyond standard rheometry, and develop approaches capable of measuring the heterogeneity of the stress response and its full tensorial nature. Techniques such as angle resolved rheometry where the suspension is strained along a primary flow direction, to set up the force network, and then probed at various angles are quite promising in this regard. Simulations capturing hydrodynamic, frictional, and interdigitation interactions have played a major role in elucidating the origins of thickening. Moving forward, however, it will be necessary to abstract some of these interactions to simulate larger systems, suspension states that are closer to shear jamming, and the evolution of the force networks in 3D. Finally, there is a desperate need for statistical theories that can elucidate the formation and evolution of the force network.

One avenue for developing such theories entails adopting ideas from equilibrium critical phenomena to the inherently out-of-equilibrium flows associated with thickening. Towards this end, a recent analysis of experiment data showed it is possible to collapse the viscous response of different suspensions (cornstarch in glycerol and silica in glycerol), over a range of volume fractions (0.16 to 0.58) onto a single universal curve with two separate scaling regimes [8]. The collapse
indicates that thickening is a transition from a regime where suspension properties are governed by isotropic frictionless jamming to one where they are governed by frictional shear jamming. The analogous process in thermodynamic systems is crossover scaling between two critical points belonging to different universality classes. Should this analogy hold, it may be possible to bring over 50 years of developments in the statistical mechanics of phase transitions to bear on the problem of thickening.

**Advances in Science and Technology to Meet Challenges** Developing a satisfying analogy between the statistical mechanics governing thermally equilibrated systems and those governing shear thickening non-thermal suspensions will require addressing several critical issues. For example, it will be necessary to determine how ergodicity, the ability of a system to explore a high dimensional phase space in a uniform fashion, must be modified to address driven non-thermal systems. It remains a deep and fascinating question whether trajectories between points in the phase space associated with different geometric contact networks in thickening systems are analogously ergodic, more akin to hopping between local attractors in a process governed by Levy statistics, or are characterized by some new, yet to be discovered, mode of phase space sampling. Moreover, given the steep potentials associated with the particle interactions, it will be necessary to confront the problem that for nearly identical particle configurations, the stress distribution could be completely different. As such, there may be important analogous questions relating to the way-imposed flows allow non-thermal systems to explore constraint landscapes associated with each geometric contact network. It is possible, therefore, that understanding thickening may require solving for system trajectories through a nested phase space.

The development of such theories will undoubtedly benefit from experiments that enable manipulation of these phase space trajectories and the resulting suspension properties. Several promising techniques for dramatically tuning the shear viscosity of thickened suspension have recently been developed. These include orthogonal shear perturbations [9] enabled by biaxial rheometers and the application of acoustic perturbations via attached piezos [10] (Figure 2). It is likely that additional techniques due to chemical, electrical, and magnetic activity will enable similarly powerful manipulations. Many of these techniques are applicable in more complex flow geometries and could be combined to enable important technical applications. As the number of parameters necessary to control the resulting flows increases, it will become useful to incorporate the latest advances in measurement science and even machine learning to determine how to efficiently scan an increasingly unwieldy parameter space to achieve the desired suspension properties.
<figure><img src="image_26.png" /><figcaption>**Figure 2** Using acoustic perturbations to rapidly tune thickening. a) A Piezo is used to apply amplitude modulated (AM) acoustic perturbations. b) The instantaneous viscosity response of a suspension with 53% volume fraction at four different strain rates. As the amplitude of the acoustic perturbations is increased, the viscosity rapidly decreases. Taken from [9].</figcaption></figure>

**Concluding Remarks** The rapid pace with which the understanding of this driven liquid to solid transition has advanced, points to a future where we could have full control over the resulting suspensions properties in arbitrarily complex flow geometries. Such control could, in addition to reducing the stresses associated with thickening suspensions, enable the development of completely new materials. For example, if we can manipulate the resulting suspension viscosity on a time scale smaller than that associated with the global flows, it may be possible to create viscosity metamaterials, where the average viscosity arises from a suspension that is only thickened for part of the time. More broadly, the breadth of technological applications, new experimental geometries, and deep analogies rooted in statistical theories of phase transitions speak to the richness of the thickening problem and its importance to the canon of soft matter phenomena.

**Acknowledgements** This work was supported by NSF CBET award numbers: 2010118, 1804963, 1509308, and an NSF DMR award number: 1507607
References

[1] Wagner, Norman J., and John F. Brady. "Shear thickening in colloidal dispersions." *Physics Today* 62, no. 10 (2009): 27-32.

[2] Brown, Eric, and Heinrich M. Jaeger. "Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming." *Reports on Progress in Physics* 77, no. 4 (2014): 046602.

[3] Morris, Jeffrey F. "Shear thickening of concentrated suspensions: Recent developments and relation to other phenomena." *Annual Review of Fluid Mechanics* 52 (2020): 121-144.

[4] Cwalina, Colin D., and Norman J. Wagner. "Material properties of the shear-thickened state in concentrated near hard-sphere colloidal dispersions." *Journal of Rheology* 58, no. 4 (2014): 949-967.

[5] Wyart, Matthieu, and Micheal E. Cates. "Discontinuous shear thickening without inertia in dense non-Brownian suspensions." *Physical review letters* 112, no. 9 (2014): 098302.

[6] Guy, B. M., J. A. Richards, D. J. M. Hodgson, E. Blanco, and W. C. K. Poon. "Constraint-based approach to granular dispersion rheology." *Physical review letters* 121, no. 12 (2018): 128001.

[7] Sedes, Omer, Bulbul Chakraborty, Hernan A. Makse, and Jeffrey F. Morris. "K-core analysis of shear-thickening suspensions." *arXiv preprint arXiv:2108.07261* (2021).

Gameiro, Marcio, Abhinendra Singh, Lou Kondic, Konstantin Mischaikow, and Jeffrey F. Morris. "Interaction network analysis in shear thickening suspensions." *Physical Review Fluids* 5, no. 3 (2020): 034307.

[8] Ramaswamy, Meera, Itay Griniasty, Danilo B. Liarte, Abhishek Shetty, Eleni Katifori, Emanuela Del Gado, James P. Sethna, Bulbul Chakraborty, and Itai Cohen. "Universal scaling of shear thickening transitions." *arXiv preprint arXiv:2107.13338* (2021).

[9] Lin, Neil YC, Christopher Ness, Michael E. Cates, Jin Sun, and Itai Cohen. "Tunable shear thickening in suspensions." *Proceedings of the National Academy of Sciences* 113, no. 39 (2016): 10774-10778.

[10] Sehgal, Prateek, Meera Ramaswamy, Itai Cohen, and Brian J. Kirby. "Using acoustic perturbations to dynamically tune shear thickening in colloidal suspensions." *Physical Review Letters* 123, no. 12 (2019): 128001.
# Section 3.5 – Electric and magnetic field control

Eric M. Furst

Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, DE, USA

**Status** External fields affect the flow behaviour, the transmission of electromagnetic energy, or the ultimate organization of a soft material (whether equilibrium or non-equilibrium). The scientific and engineering interests in external field control include magneto- and electrorheology, ferrofluid flow in magnetic fields, magnetic microfluidic mixers, and the rapid optical switching in liquid crystal and e-ink display technologies. Other applications of fields and field control relevant to soft materials include processing using external fields, such as electrospinning polymer nanofibers and the magnetic or electric directional ordering of nanoscale structure in block copolymers and polymer-based nanocomposites. It is clear from these examples that electric and magnetic fields drive a material's underlying nano- or micro-structure—externally and often rapidly relative to thermal or solvent annealing, drying, or changes in temperature or other physical chemical conditions. But phenomena that occur in field-driven soft materials are rich and sometimes unexpectedly universal, like the analogy between the Landau-Peierls fluctuation-driven interactions of polarizable particle chains and those of Abrikosov vortex lines in high-temperature superconductors. [1] Interesting analogies also exist with the behaviour and assembly of particles in nematic liquid crystals (see section 1.5) and on curved fluid interfaces. [2]

A few of the most exciting recent developments use electric and magnetic fields to direct self-assembly, realize new methods of dissipative self-assembly, and drive emergent dynamics and organization in active soft matter by dynamic self-assembly (Figure 1). [3] In directed self-assembly, fields nudge or guide the kinetics of self-assembly through nonequilibrium states to an equilibrium structure. Its aim is to provide direction and control without losing the scalability of thermodynamically driven self-organization. Using directed self-assembly, one can organize the direction of block co-polymer nanoscale structures by weak diamagnetic forces [4] or guide anisotropic nanoparticles to form close-packed crystals in electric fields. [5] Entirely new forms of non-equilibrium assembly, termed dissipative self-assembly, occur when fields are used to rapidly oscillate interaction energies, especially in dispersed materials like suspensions. [6, 7] Similarly, the energy input by oscillating or periodic fields can drive motion in a soft material that leads to self-organization—a form of the dynamic self-assembly of active matter (see section 3) which resembles the organization of insect swarms, bird flocks, and schools of fish.
<figure><img src="image_27.png" /></figure>

**Figure 1** Electric and magnetic fields are used to drive directed, dynamic, and dissipative self-assembly. (A) Dumbbell nanoparticles assembled into close-packed crystals through a directed self-assembly process. The inset shows the probability of crystallization with alignment for different aspect ratio particles. Adapted with permission from J. D. Forster et al., "Assembly of Optical-Scale Dumbbells into Dense Photonic Crystals," ACS Nano, vol. 8, pp. 6695-6700-6700, 2011. Copyright 2011 by the American Chemical Society. (B) Organized, rotating bands of silica particles in an ac electric field. This dynamic assembly is attributed to secondary electrokinetic flows around each particle. Adapted with permission from F. Katzmeier, et al., Phys. Rev. Lett., vol. 128, no. 5, p. 58002, 2022. Copyright 2022 by the American Physical Society. (C) Phase separation during the dissipative assembly of paramagnetic colloids in a toggled magnetic field. Adapted with permission from H. Kim, et al., Langmuir, vol. 36, pp. 9926-9934, 2020. Copyright 2020 by the American Chemical Society.

**Current and Future Challenges** Despite its promise—scalable assembly, potentially new morphologies, and dynamical responsiveness and control—there are several key challenges in the field. For one, there is a general lack of design rules for directed, dissipative, and dynamic self-assembly by electric and magnetic fields. Little is still understood about these processes, and much work remains in experiment, theory, and simulation. Perhaps the greatest challenge of electric and
magnetic field control of soft matter lies in the inverse design problem: given a desired functional structure, what are the fundamental building blocks and processes to achieve them?

In both *directed* and *dissipative* self-assembly, knowledge of thermodynamic states—including mechanical and thermal equations of state—are necessary. As a case in point, for the directed assembly of dispersions a significant design challenge is the need to know the equilibrium phases and nonequilibrium states, especially for complex particle shapes with field-mediated interactions. Relaxation timescales should be known, too, to capture the kinetics of transitions and select favorable assembly conditions. For some shapes, like highly anisotropic or asymmetric particles, the existence of multiple phases and dynamic modes (rotation, translation) complicate this theoretical understanding, and few modeling efforts have described the interactions between even more complex metallo-dielectric and patchy particles that are designed to couple to fields. [8, 9] Magnetic and electric interactions are long-ranged and subject to mutual polarization. Properly incorporating these interactions is challenging and in simulations requires sophisticated algorithms. [10] The computational expense of computing long-range interactions limits both the system size and simulation duration.

Where external fields have been used to guide particles into close-packed configurations, they orient anisotropic particles and move them from low concentrations to high concentrations through gradients [11]. With dumbbell nanoparticles (Fig. 1A), the role alignment is clear and explained by the low probability that particles form close-packed structures in the absence of a directing field because of combined arrest in their translational and rotational motion at high concentrations. [12] Without alignment, the suspension particles get trapped in a glassy, disordered state. For now, though, this process of field-guided assembly is entirely empirical. Nor is it clear whether the equilibrium close-packed structure is really achieved, or what controls the type and frequency of defects. Theoretical studies of dumbbell nanoparticles identified their glass lines, but what is the corresponding state diagram for particles *aligned* by a field? Furthermore, fields induce polarization and interactions, so there must be a trade-off between alignment and other field-driven microstructural transitions (even gel-like states) found in electro- and magneto-rheological fluids.

**Advances in Science and Technology to Meet Challenges** There are several immediate investments that can be made to meet the above challenges. First, better, and more accessible computational tools, simulations, and theories are needed. Experimentalists need theoretical and computational models that accurately capture the physics of directed, dissipative, and dynamic self-assembly.

In recent years, several groups produced encouraging results toward a theoretical basis of dissipative assembly (Figure 2). Tagliazucchi et al. developed a first principles understanding of suspensions with particles interacting by periodic, oscillatory potentials. [6] In the limit of fast oscillations, the resulting distribution of dissipative self-assembling states is exactly equivalent to that created by time-averaged effective potentials. These potentials have no equilibrium counterparts. But their work demonstrated that, in some cases, dissipative self-assembly can be predicted and analyzed in terms of equilibrium self-assembly. Sherman and co-workers studied
nanoparticle suspensions with periodic *toggled* interactions that led to structures that are not near the global minimum in free energy, and they found that toggling can stabilize nonequilibrium phases. [7, 13] Their brilliant use of the sticky hard sphere model connects equilibrium with toggled steady-state structures. [7] Both research groups provide clear design rules for experimental work—target oscillation frequencies, field strengths, and duties (for toggled fields) that optimize the assembly kinetics and control structures; both also lay the groundwork for more robust non-equilibrium thermodynamic theory.

Second, investigators need rational design and synthesis of field-responsive molecular and colloidal building blocks, especially those that use first-principles designs aided by computer modeling. Some part of these materials could be dedicated to a functional property while the other is optimized purely for the ability to align the functional domains, as described for field-responsive block copolymers. [4] For both polymeric and colloidal soft materials, there has been little optimization to enhance the efficacy of field direction and control, although there are promising studies that use metallodielectric patchy particles, noted earlier.

There have been few studies using combinations of fields, or combining fields and other driving forces, such as flow. The parameter space is vast, and there is plenty of opportunity to advance the experimental and theoretical work in the area. In related studies, investigators used orthogonal oscillatory fields to generate "isotropic" potentials (analogous to a Keesom dipole-dipole interaction) between paramagnetic colloids. [14] This effect relies on the short Néel relaxation time of the magnetic nanoparticles used to produce paramagnetic responses in otherwise large (and slow) polymer latex particles.

Finally, fabricating materials on a commercial scale by bottom-up self-assembly requires well-characterized and efficient manufacturing techniques. How these processes will be scaled up or incorporated into existing or new manufacturing is an open question. This is closely tied to the "inverse design" problem highlighted at the beginning of this survey. Possibly the most pressing need is to develop a strong feedback loop from simulation and theory to aid the design of new field-responsive materials that achieve targeted, functional structures.
<figure><img src="image_28.png" /></figure>

**Figure 2** Developments toward a theory of dissipative assembly include (A) nonequilibrium structures formed in an oscillatory potential modeled by changes in pH. The parameter C denotes a strength of interaction. Adapted with permission from ref [6]. (B) Nonequilibrium phases due to similar toggled attractive interactions are sensitive to frequency and duty. Reprinted with permission from Z. M. Sherman and J. W. Swan, "Dynamic, Directed Self-Assembly of Nanoparticles via Toggled Interactions," ACS Nano, vol. 10, pp. 5260–5271, 2016. Copyright 2016 American Chemical Society. (C) Spherical (left) and ellipsoidal (right) metallo-dielectric particles designed to couple and respond to electric fields. Adapted with permission from ref. [8] Copyright 2009, Royal Society of Chemistry and adapted by permission Springer Nature: Nature Materials, "Actuation of shape-memory colloidal fibres of Janus ellipsoids," A. Shah, B. Schultz, W. Zhang, S. C. Glotzer, and M. J. Solomon, 2014.

**Concluding Remarks** The use of electric and magnetic fields in soft matter much holds significant promise by enabling new modes of directed, dissipative, and dynamic assembly into functional materials. Fields can improve the speed of assembly, quality of the final product, and ease of control, while also leading to new and useful nonequilibrium structures. The focus of this survey is
on self-assembly, but new functional soft materials are also being developed as extensions of magnetorheology, including magnetic elastomers. [15] Intriguing and important applications of these composites include the development of controllable soft actuators—grippers or artificial cilia; open questions abound, such as the role of the interphase region between magnetic particles and the surrounding matrix. Further advances in magnetic and electric field control will require investments in experiment, simulation, and theory.

**Acknowledgements** This material is based upon work supported by the National Science Foundation under Grant No. CBET-1637991.

**References**

1. T. C. Halsey and W. Toor, "Fluctuation-induced couplings between defect lines or particle chains," *J. Stat. Phys.*, vol. 61, pp. 1257-1281, 1990.

2. M. J. Cavallaro, L. Botto, E. P. Lewandowski, M. Wang, and K. J. Stebe, "Curvature-driven capillary migration and assembly of rod-like particles," *Proc. Natl. Acad. Sci.*, vol. 108, no. 52, pp. 20923-20928, 2011.

3. F. Katzmeier, B. Altaner, J. List, U. Gerland, and F. C. Simmel, "Emergence of Colloidal Patterns in ac Electric Fields," *Phys. Rev. Lett.*, vol. 128, no. 5, p. 58002, 2022.

4. H. Hu, M. Gopinadhan, and C. O. Osuji, "Directed self-assembly of block copolymers: a tutorial review of strategies for enabling nanotechnology with soft matter," *Soft Matter*, vol. 10, pp. 3867-3889, 2014.

5. J. D. Forster et al., "Assembly of Optical-Scale Dumbbells into Dense Photonic Crystals," *ACS Nano*, vol. 8, pp. 6695-6700-6700, 2011.

6. M. Tagliazucchi, E. A. Weiss, and I. Szleifer, "Dissipative self-assembly of particles interacting through time-oscillatory potentials," *Proc. Natl. Acad. Sci.*, vol. 111, no. 27, pp. 9751-9756, 2014.

7. Z. M. Sherman and J. W. Swan, "Dynamic, Directed Self-Assembly of Nanoparticles via Toggled Interactions," *ACS Nano*, vol. 10, pp. 5260-5271, 2016.

8. S. Gangwal, A. Pawar, I. Kretzschmar, and O. D. Velev, "Programmed assembly of metallodielectric patchy particles in external AC electric fields," *Soft Matter*, vol. 6, no. 7, pp. 1413-1418, 2010.

9. A. Shah, B. Schultz, W. Zhang, S. C. Glotzer, and M. J. Solomon, "Actuation of shape-memory colloidal fibres of Janus ellipsoids," *Nat. Mater.*, vol. 13, Nov. 2014.

10. Z. M. Sherman, D. Ghosh, and J. W. Swan, "Field-directed self-assembly of mutually polarizable nanoparticles," *Langmuir*, vol. 34, pp. 7117-7134, 2018.

11. T. Ding, K. Song, K. Clays, and C.-H. Tung, "Fabrication of 3D Photonic Crystals of Ellipsoids: Convective Self-Assembly in Magnetic Field," *Adv. Mater.*, vol. 21, pp. 1-5, 2009.

12. R. Zhang and K. S. Schweizer, "Theory of coupled translational-rotational glassy dynamics in dense fluids of uniaxial particles," *Phys. Rev. E*, vol. 80, no. 1, p. 011502, 2009.

13. H. Kim, M. Sau, and E. M. Furst, "An Expanded State Diagram for the Directed Self-Assembly of Colloidal Suspensions in Toggled Fields," *Langmuir*, vol. 36, pp. 9926-9934, 2020.

14. D. Du, D. Li, M. Thakur, and S. L. Biswal, "Generating an in situ tunable interaction potential for probing 2-D colloidal phase behavior," *Soft Matter*, vol. 9, no. 29, p. 6867, 2013.

15. S. J. R. Morillas and J. D. Vicente, "Magnetorheology: A Review," *Soft Matter*, vol. 16, pp. 9614-9642, 2020.
## Section 3.6 – Stimuli-responsive materials

Alshakim Nelson<fnref n="1" /> and Stephen L. Craig<fnref n="2" />

**Status** The term “stimuli-responsive” (SR) refers to materials that undergo a meaningful change in properties (the response) when subjected to a change in external environment (the stimulus). The term is intrinsically redundant – a response by definition has a stimulus that triggered it, and the delivery of energy or matter is only a stimulus if it triggers a response. The broad, if linguistically uneconomical, use of “stimuli-responsive” likely reflects a desire to emphasize the breadth available on either side of the stimulus-response relationship. Stimuli include various forms of energy (e.g., thermal, photons, electromagnetic waves and fields, or mechanical) and the introduction or removal of matter (e.g., solvents, reagents, acids/bases, salt, or electrons), and each can in principle be coupled to an ever-increasing range of responses (e.g., change in shape, assembly, colour, luminescence, mechanical properties, thermal transport, material transport, conduction, or uptake/release of cargo). The interest in, and study of, such materials has captured commercial and academic interest for decades. Excellent reviews of stimuli-responsive materials are available [1-3].

The ability to change properties differentiates SR materials from those whose properties are largely static. As a result, for a couple of decades it has been fashionable to refer to such materials as “smart” [4]. A more appropriate term for the state-of-the-art, however, is probably “obedient” [5]. The scientist wants a material that exists in state A under condition *a*, and in state *B* under conditions *b*, and any “smarts” come into play by finding the material compositions in which *A/B* switching occurs at a desired *a/b* threshold. The material is instructed through chemical synthesis as to how it should respond. The complexity of the approach, and the range of applications for which it is relevant, can be extended by introducing additional states to form a “multi-stimuli-responsive” material. Each state has its own corresponding stimulus thresholds which can be simultaneously or independently addressed. This form of multi-stimuli-responsive material engineering is likely to remain an active area of research.

**Current and future challenges** Material systems with spatially distinct, individually responsive components hold the potential for a different level of “smarts,” because of the combinational possibilities of spatially distributed SR events. Further, when the right feedback loops are incorporated, materials might become capable of adapting their structure to arrive at a final system-wide state that meets the challenge of a given environmental condition, even when the optimum final state is not known in advance. That is, the material is “smarter” than the scientist who created it, creating a complexity in response that can be guided by machine learning models in the future.

<footnote n="1">¹Department of Chemistry, University of Washington, Seattle, WA</footnote>

<footnote n="2">²Department of Chemistry, Duke University, Durham, NC</footnote>
Here, we illustrate examples that employ SR elements on both the macro and molecular scale, with mechanical properties chosen for illustrative purposes.

On the **macro** scale, microstructured materials (metamaterials) include those with novel quasi-static and dynamically responsive mechanical properties [6], and there a few emerging examples of their intersection with SR materials, such as those made from magnetically response material components [7]. For the case of mechanical properties, the combination of metamaterials and SR materials holds the possibility of highly amplified system-level changes in mechanics due to a change in properties of only a small fraction of the overall object. An example can be seen in Maxwell lattices, where the removal of a single edge can have a dramatic impact on lattice mechanics (Fig. 1). An SR response in an edge can similarly trigger a similar transition from a stable to a floppy lattice across large length scales, even though the region of SR change is comparatively small. The local responses can, in principle, be used to switch between multiple stable lattice states, including extreme properties brought about by non-linearity. Opportunities include the potential for directional response(s) depending on which location(s) within the lattice is triggered – the exact same material response can give a dramatically different metamaterial response depending on the relative orientation of the triggered edge.

<figure><img src="image_29.png" /><figcaption>**Figure 1** The Maxwell lattice shown is rigid under shear when SR components (red) are intact and stiff, but the entire lattice becomes soft in shear when those components disappear or soften.</figcaption></figure>
On the **molecular** scale, consider polymer networks, in which the edges of a lattice are replaced by the spring-like behaviour of fluctional non-crystalline (disordered) polymer strands. The ultimate strength of polymer materials is <1% of the potential collective strength in their strands, and evenly distributing load across all strands in a network could increase ultimate strength of the material by over 100-fold (e.g., rubber bands or even hydrogels that are as strong as steel but still easily stretched). The challenge in achieving those properties arises from the fact that as a network is strained, a few strands within the broad distribution act as stress concentration sites that initiate failure. The “statistical distribution problem” cannot be overcome with traditional strand or network synthesis; purely monodisperse and fully reacted strands still have a Gaussian distribution of initial end-to-end lengths. One can envision, however, how a mechanically triggered response that irreversibly lengthens a single strand at a critical force of tension [8] might rectify the molecular force imbalances (Fig. 2) to create “self-perfecting” networks.

<figure><img src="image_30.png" /><figcaption>**Figure 2** Areas of high tension (red) in a polymer network are sites of likely damage initiation on the molecular level. Mechanoresponsiveness at the molecular level, however, might lead to the irreversible release of stored length that redistributes and balances the tension.</figcaption></figure>

Advances in science and technology to meet challenges

1. *Modeling of topological metamaterials as a function of the range of properties available in SR material components.* Such models should account for variations in properties that might arise from incomplete switching in real systems, any imprecision in the dimensions of the fabricated components, and the dynamics of switching.

2. *Molecularly precise synthesis of SR materials and improved fabrication of SR metamaterials.*
Opportunities abound from the macro and molecular scales, as advances in the breadth and precision of additive manufacturing [9] enable the pursuit of the former, while advances in precision synthesis and characterization enable an informed approach to the latter. Continued
expansion of these techniques to SR materials will provide an opportunity to test and refine the
development of multi-scale models.

3. *Robust strategies for complex feedback loops.* The range of available SR input and output combinations continues to increase. Mechanical property changes, for example, can be induced by swelling, light, redox, temperature, electromagnetic fields, etc... A particularly exciting opportunity is found, however, when the trigger mediates a change in the way its associated form of energy/matter is processed by the material/metamaterial, such as when the SR elements respond to mechanical signals by changing mechanical properties. It has been shown that the resulting feedback loops can lead to signal amplification, system-wide dynamic responses to local actions, and autonomous homeostasis [10].

4. *Data structures that capture chemical complexity, characterization, and metadata.* With increasing complexity, machine learning strategies may play a pivotal role in the development of these systems. A machine learning based design approach could accelerate the identification and optimization of SR input/output combinations with a fewer number of physical experiments.

**Concluding remarks** Mechanical properties are presented here only as an example of the broad possibilities. Related opportunities exist at the topology-SR material nexus for optical properties (responsive light traps, lenses, filters, or absorbers), electrical conductivity, thermal regulation, or the transport of diffusing species. Overarching questions include: when topology can be controlled precisely (e.g., in metamaterials), how can it be optimized to take greatest advantage of local switching in SR elements? Or when topology is intrinsically uncontrollable (e.g., in polymer networks made via random polymerization), how can responses at the molecular level be optimized to ameliorate performance limitations brought about by the topological disorder? These emerging opportunities will only be strengthened by continued development in traditional areas of SR material research, such as multi-response, faster and more complete switching, and continuing to increase the magnitude of property change available in an SR event.

**Acknowledgments** This work was supported by the NSF Center for the Chemistry of Molecularly Optimized Networks (MONET), CHE-2116298. We thank X. Mao and S. Wang for providing the graphics in Figures 1 and 2, respectively.

**References**

[1] P. Theato, B.S. Sumerlin, R.K. O'Reilly, and T.H. Epps III, "Stimuli responsive materials", *Chem. Soc. Rev.*, vol. 42, pp. 7055-7056, 2013, doi: 10.1039/C3CS90057F (Editorial).

[2] M.W. Urban, *Stimuli-Responsive Materials: From Molecules to Nature Mimicking Materials Design*. Cambridge, UK: Royal Society of Chemistry, 2019.
[3] F. Huang, X. Zhang, and B.Z. Tang, "Stimuli-responsive materials: a web themed collection", *Mater. Chem. Front.*, vol. 3, pp. 10-11, 2019, doi: 10.1039/C8QM90057D (Editorial).

[4] C.A. Rogers, "U.S. Army Research Office Workshop on Smart Materials, Structures and Mathematical Issues Held in Blacksburg, Virginia on 15-16 September 1988: Final Report", Virginia Polytechnic Institute, Blacksburg, VA, ADA218528, Jan. 1989.

[5] We have heard this characterization attributed to T. McCarthy.

[6] K. Bertoldi, V. Vitelli, J. Christensen, and M. van Hecke, "Flexible mechanical metamaterials", *Nat. Rev. Mater.*, vol. 2, pp. 17066, 2017, doi: 10.1038/natrevmats.2017.66

[7] S. M. Montgomery, S. Wu, X. Kuang, C. D. Armstrong, C. Zemelka, Q. Ze, R. Zhang, R. Zhao, and H. J. Qi, "Magneto-mechanical metamaterials with widely tunable mechanical properties and acoustic bandgaps," *Adv. Funct. Mater.*, **2021**, *31*, 2005319. doi.org/10.1002/adfm.202005319

[8] Z. Wang, X. Zheng, T. Ouchi, T. B. Kouznetsova, H. K. Beech, B. H. Bowser, S. Wang, J. A. Johnson, J. A. Kalow, B. D. Olsen, J. P. Gong, M. Rubinstein, and S. L. Craig, "Toughening hydrogels through force-triggered chemical reactions that lengthen polymer strands," *Science*, **2021**, *374*, 193-196. doi: 10.1126/science.abg2689

[9] R.T. Shafranek, S.C. Millik, P.T. Smith, C-U. Lee, A.J. Boydston, and A. Nelson, "Stimuli-responsive materials in additive manufacturing," *Prog. Polym.*, vol. 93, pp. 36-67, 2019, doi: 10.1016/j.progpolymsci.2019.03.002

[10] X. He, M. Aizenberg, O. Kukensok, L.D. Zarzar, A. Shastri, A.C. Balazs, and J. Aizenberg, "Synthetic homeostatic materials with chemo-mechano-chemical self-regulation", *Nature*, vol. 487, pp. 214-218, 2012, doi: 10.1038/nature11223
# Section 4.1 – Linking Hard and Soft Matter Physics

Rajesh Ganapathy<fnref n="1" /> and Ajay Kumar Sood<fnref n="2" />

**Status** Any reasonable attempt at a roadmap linking the physics of hard and soft matter cannot possibly begin without bringing to the fore the landmark 1972 paper by de Gennes that brought out the striking analogies between the normal-to-superconducting transition in metals and the nematic-smectic A transition of mesogens [1]. In the absence of spatial inhomogeneities, the Ginzburg-Landau Hamiltonians describing both these transitions are similar, and the expulsion of the external magnetic field in the Meissner phase is analogous to the expulsion of twist in the smectic A phase. Building on this observation, Renn and Lubensky would later predict the twist-grain-boundary (TGB) phase - the liquid-crystalline analogue of the Abrikosov flux lattice of type-II superconductors [2]. Experiments on ferroelectric liquid crystals (LCs) observed the TGB phase just a year later [3]. Such striking analogies between hard and soft matter are not limited to superconductors and LCs; see, for instance, the mapping of polymer chain confirmations due to excluded volume interactions to a magnetic system [4], again due to de Gennes, or the Kovacs memory effect that was first observed in polyvinyl acetate and then in numerous other systems, including molecular glasses [5].

The link between soft and hard materials has further strengthened in the past two decades from the clever use of colloidal suspensions to investigate numerous physical phenomena encountered in atomic/molecular matter. Despite being ruled by entropy, the phase behavior of colloidal suspensions shares striking similarities with hard matter, where energetics dominates [6]. Furthermore, colloidal particles, being micrometer-sized, are amenable to particle resolved imaging of dynamics in real-space; particle shape and interactions can be tuned at will, and individual colloidal particles can also be manipulated. Probing physical phenomena in real-space with 'colloids as big atoms' presents a clear advantage over scattering-based approaches, which are ensemble-averaged, and has indeed been well-leveraged to test theory and numerical predictions [7]. See, for instance, the first observation of dynamical heterogeneities in supercooled liquids [8], the direct observation of thermal capillary waves [9], premelting at grain boundaries [10], observation of solid-solid phase transitions [11], kinetic roughening of driven grain boundaries [12], and recently, devitrification [13], to name a few.

Given the many bridges between soft and hard matter, each of which merits a detailed discussion, here, we restrict ourselves to three areas that we anticipate will continue to draw researchers.

**Exploiting parallels between atomic, nanoparticle, and colloidal epitaxy for self-assembly** Advances in understanding atomic epitaxial growth processes have hugely enhanced our ability to fabricate functional surface nano and microstructures [14]. More than two decades ago, van Blaaderen and co-workers extended these concepts to colloidal systems [15]. Subsequent studies found that while the island growth laws governing atomic and colloidal homoepitaxy were similar, there were also subtle differences [16]. For instance, the Ehrlich-Schwoebel step-edge barrier (ESB), which decides the ease of interlayer mass transport and the morphology of the

<footnote n="1">¹Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India</footnote>

<footnote n="2">²Indian Institute of Science, Bangalore 560012, India</footnote>
growing film, has an energetic origin in atomic systems. In contrast, in colloidal systems with short-range interactions, it is a purely kinetic barrier. Recent studies have found that for large molecules like C60, whose range of interactions is intermediate between atoms and colloids, both the energetic and kinetic ESB have a role during film growth [17].

Even while atomic homoepitaxy concepts carry over to colloidal systems [18], whether this is also the case with heteroepitaxy, i.e., when there is also a lattice misfit between the growing film and the substrate, remains largely unexplored. Exploiting strain-relief mechanisms in heteroepitaxy to organize matter on surfaces is extensively used in atomic and molecular systems; with nanoparticle and colloidal ones, the first steps have just been taken (see Fig. 1) [19-23]. The range of interparticle interactions is a crucial parameter in deciding how strain-relief occurs; for instance, with DNA-grafted nanoparticles, where the interactions are long-ranged, film growth remains pseudomorphic with the substrate even for very large misfits and is quite unlike in atomic systems [21]. On the other hand, for colloids with sticky interactions, Mondal et al. find striking parallels between the strain-relief mechanisms in colloidal and atomic systems for compressive misfits [22]. However, the level of self-organized growth seen in atomic systems remains unrivalled. See, for instance, the spectacular self-organized growth of metal nanostructure arrays on a strain-relief dislocation network achieved in the late 1990s [24]. If this level of self-organization with nanoparticle and colloidal systems is attained, one can then only envisage the richness that particle valence and shape can further bring [25].

<figure><img src="image_31.png" /><figcaption>**Figure 1** Hierarchically organised colloidal crystal arrays obtained in colloidal heteroepitaxy. Picture used with permission from Manodeep Mondal, Soft Matter Lab, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India. See Ref. [22] for details.</figcaption></figure>
**Glass Transition** Understanding how liquids turn into glassy solids on quenching is one of the grand challenges in condensed matter physics [26]. Theories of the glass transition are many, experiments that directly test theoretical predictions are few, and experiments that pit theories against each other, even fewer. The contribution of colloid experiments in gaining insights into glassy dynamics cannot be overstated (see the many Review Articles [27-28]). While helping visualize dynamics, the large size of the colloidal particles also means that the system falls out of equilibrium before reaching the deeply supercooled regime, which is also the dynamical regime of interest. The first steps towards overcoming the sample equilibration bottleneck have just been made. Using the Stimulated-Emission-Depletion (STED) microscopy technique, Royall and co-workers imaged nanoparticle liquids, where relaxation is substantially sped-up due to the smaller particle size [29]. This study provided real-space evidence of structural relaxation beyond the mode-coupling singularity and a direct link between the growth of locally favoured structures and dynamics. Competing glass theories remain untested in this newly accessible dynamical regime. Experiments that capitalize on the latest advances in imaging techniques to shed light on glass physics are bound to pay huge dividends.

As has been the case in the past, identifying the role of static structure in glassy dynamics will continue to remain a key research avenue. The numerical isoconfigurational ensemble (ICE) approach [30], which first identified an underlying structural basis of dynamical heterogeneities, remains untested in experiments, and for a good reason [31]. However, the difficulties plaguing colloidal systems may not apply to two-dimensional dense granular liquids, which is an avenue worth exploring. The causal link between structure and dynamics in supercooled liquids and amorphous solids has been made more robust with the advent of machine-learning-based approaches [32-33]. Moving forward, obtaining physical intuition into the machine-learned (ML) structure and determining how these fit within the various glass physics theories will be essential.

**Linking quiescent and driven soft and quantum matter** While much of our discussion hitherto has largely focussed on systems with a rather steep repulsion between the constituent particles, the link between hard and soft matter is equally rich when the particles are extremely squishy. There is now experimental evidence that soft matter systems made of highly deformable particles, such as DNA-based dendritic triblocks, form cluster crystals [34, 35]. These, as the name suggests, are spatially ordered arrangements of particle aggregates. What is even more remarkable is that these soft matter systems appear to share similarities with many quantum phases of matter [36,37].

Driving soft matter out of equilibrium is actively pursued due to fundamental science issues arising from the violation of equilibrium concepts like time-reversal symmetry and detailed balance and its relevance in industrial applications. Shearing soft matter as one of the many ways to drive the system has a rich history of fascinating phenomena like shear-thinning and -thickening, shear-jamming, unsteady chaotic flows [38], random organization separating the steady and fluctuating states [39], memory formation [40], to name a few. These findings have parallels in electrical current-driven vortex lattices [41], charge density wave, and skyrmion systems.

One future direction in linking soft and hard matter is the nonequilibrium physics of quantum soft matter, a state of correlated electron fluids that can exhibit electron nematic and smectic phases [42]. The former refers to a metallic phase that is translationally invariant with spontaneously
generated spatial anisotropy, whereas the latter has spontaneous breaking of both the symmetries. Recent experiments [43] have shown that the high mobility two-dimensional electron gas in a perpendicular magnetic field exhibits zero resistance state when driven by microwave radiation (ZRS). Interestingly, it has been shown [44] that the ZRS in the limit of zero magnetic field and short-range electron-electron interactions is identical to flocking transitions seen in the context of active soft matter. The connection between active matter and open quantum systems -- in particular between non-reciprocal interactions and non-Hermitian systems -- is a promising frontier [45,46]. We anticipate that microwave or terahertz radiation-driven quantum soft matter will be an exciting future direction with parallels to driven soft matter.

**Conclusions** The link between the physics of soft and hard matter is stronger than ever before, but there is room for more. This roadmap identifies potential problems in three subareas where a synergistic exchange of concepts between hard and soft matter systems is already present. At the same time, we acknowledge that these subareas are by no means representative. Yet another fascinating avenue of research bridging soft and hard materials, which we expect to thrive in the years to come, is unravelling the role of intermediates during crystallization, solid-solid phase transitions, and even devitrification.

**Acknowledgements** RG thanks Jawaharlal Nehru Centre for Advanced Scientific Research and the Department of Science and Technology (DST), Govt. of India (Swarna Jayanti Fellowship Grant and DST-Nanomission Grant) for financial support. RG also thanks members of the Experimental Soft Matter Research group at JNCASR for many discussions over the years. AKS thanks the DST, Govt of India for support under the Year of Science Professorship. AKS acknowledges gratefully the contributions of his group members over the years and Prof Sriram Ramaswamy for most enjoyable collaboration over the last 30 years.

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## Section 4.2 – Glasses and Gels

Francesco Sciortino

Department of Physics, Sapienza Universita' di Roma, Piazzale Aldo Moro, 5, 00185 Rome, Italy

**Status** Dynamically arrested disordered colloidal systems, either glasses or gels, are fairly common soft matter materials. With a small change in control parameters (either temperature, composition, concentration, shear stress) these systems transform from a fluid of diffusive particles to a material in which excluded volume (glass) or bonding (gels) freezes particle mobility. Arrested states provide additional structural and functional possibilities beyond those offered by crystalline states. At the same time, they also act as unwanted kinetic bottleneck in the self-assembly of a desired ordered lattice. This schizophrenic role mirrors recent and ongoing research, dedicated both to understand the conditions under which arrested disordered states are stable and to understand how to avoid kinetic arrest to facilitate the formation of ordered structures.

Colloidal solutions (Fig. 1) have provided model systems for theoretical investigations of the glass transition [1], starting from the paradigmatic hard-sphere case and progressively evolving with the modulation of the repulsive interaction, as in soft and in charged colloids. Goetze's mode coupling theory for the glass transition [2] (the progenitor of a long list of approaches [3–5] that seek to extend the theory to more realistic conditions or to clarify the underlying uncontrolled approximations) has the merit to highlight the important role of the self-reinforcing caging process on the increase in packing.

<figure><img src="image_32.png" /><figcaption>**Figure 1** Experimental and numerical images of colloidal gels: From left to right: a confocal microscope image of a depletion gel [9] and of a competing interaction gel (with the characteristic Bernal spiral in the inset) [10]. A simulation image of a low valence particle gel [13] and of a DNA-nanostar gel [27].</figcaption></figure>

Depletion interactions have added a new twist to colloidal systems, making it possible to complement repulsive interaction with an attractive tail of tuneable strength and, more important, interaction range. Guided by mode-coupling predictions [6], the experiments demonstrated the important role played by the combination of short-range attraction and hard-core repulsion [7]. At low packing, colloidal depletion gels originate from spinodal decomposition interrupted by glass formation [8, 9]. Similarly, the combination of a short-range attraction with a
long-range repulsion has provided a framework for understanding cluster phases and one-dimensional cluster growth [10, 11].

The synthesis of colloidal particles with specific binding sites on their surface (patchy colloids [12]) has opened the possibility of framing gel-formation in colloidal systems from a thermodynamic perspective. Gels of patchy particles — as opposed to glasses which are mainly driven by excluded volume interactions — are characterised by a network of bonds that is progressively formed with cooling. It has been shown that the ability to form such an equilibrium gel phase is made possible by the emergence of an intermediate density region between the density of the coexisting liquid and the that of the glass as the valence of the patchy particles decreases [13] (Fig. 2).

<figure><img src="image_33.png" /><figcaption>**Figure 2** Cartoon of the (disordered states) phase diagram in the temperature-density plane of isotropically attractive colloidal particles (as in depletion interactions [9]) and of the phase diagram of limited valence colloids (as in the DNA-nanostar case [27]). Note that in the last case, the phase-separation region is disjunct from the glass region, offering the opportunity to explore a window of densities in which the liquid progressively turns into an equilibrium gel (the network region).</figcaption></figure>

**Current and Future Challenges** Future theoretical research on colloidal glasses will focus on how to include hopping effects [14, 15] and non-gaussian dynamics [16, 17] in a quantitative way, improving our predictions on the long-time decay of the correlation functions, both in equilibrium and during aging. A comparable theoretical challenge in colloidal gelation is instead the formulation of a theory (with a predictive power similar to what mode-coupling can do for glasses, i.e., a prediction of the behaviour of the collective density correlation for different wave-vectors) for the gel dynamics, both approaching the gel state as well as in the fully developed gel. The difficulty arises from the much more complex dynamics that originates from bond-cages as compared to excluded-volume cages. In addition, clustering and network formation takes place in dilute solutions and requires a combined description of cluster diffusion as well as network breathing modes. While some attempts have been formulated for the self-dynamics, quantitative predictions for collective observables are largely unexplored.
Routes to gel formation are multiple. While we have now a clear understanding of depletion and equilibrium colloidal gels [18] (and for gels based on competing interactions [19, 20]), other gelation routes have been significantly less investigated. Particularly relevant, for their biological applications, are gels made by largely anisotropic particles and gels made by growth and branching of fibres. The thermodynamic stability and the spatial and temporal homogeneities of equilibrium gels [21] have been investigated in detail. It has been shown that limited valence gels can be the thermodynamic stable state even at quite low temperatures [22]. Now work is required to understand how these ideas can find applications to gels formed with other routes.

How to design particle interactions to self-assemble desired structures, avoiding kinetic traps will also be a prominent research theme. Up to now the bottom-up self-assembly, a powerful method, has been successful in important but limited cases. A general strategy for self-assembly into a well-defined crystal (avoiding polymorphs) or a precise finite-size structure [23] is clearly needed [24–26].

Finally, the connection between different forms of dynamic arrest (gel, glasses, jammed states) and the mechanical properties of the corresponding material and its response to external perturbations (yielding transition, shear-banding, viscoelasticity) are topics which are receiving (and will continue to receive) a significant attention. Diversity in dynamic arrest in gels and glasses surely leads to a rich variety of elastic and rheological behaviours relevant for industrial processes and products.

**Advances in Science and Technology to Meet Challenges** Understanding slow dynamics in colloidal solutions has proceeded step by step with both instrumental development as well as novel chemical synthesis. In the last years, particle with complex shapes and surface patterning have become available, significantly enlarging the arsenal of elementary building block. Among others, DNA functionalized particles [27], DNA constructs and DNA origami are emerging as examples of colloids with exquisite control of both particle shape and binding selectivity. Unfortunately, sometime the yield of these interesting novel colloidal particles is quite limited, optimized for academic application. The availability of bulk quantities of mono-disperse colloidal particles is often fundamental to precisely control theoretical predictions. An effort needs to be made to scale up synthesis and availability of the particle themselves. From the experimental side, several novel techniques [28, 29], mixing scattering and imaging, have been developed providing accessibility to longer time scales and larger distances. Super-resolution microscopy, a method based on photo-activatable fluorescent probes, allowing for a detailed picture at the nanoscale is also expected to significantly contribute to detailed characterization of soft-matter systems. From the computational side, efforts are ongoing to include hydrodynamic interactions in the computer models, an interaction which has a non-relevant role in the dynamics and in the coarsening of out-of-equilibrium gels.
**Concluding Remarks** Colloidal gels and glasses are central to material science (pastes, paints, foams), food preservation, household applications. The ability to encode in the particle (or in the chemical physic properties of the solvent) information about the desired material properties offers countless application possibilities. Understanding dynamic arrest will provide an important handle to control material properties from the bottom up. Despite significant efforts in the last decades (dynamic arrest in colloidal science has been central to soft-matter research), our current understanding of particle slowing down (resulting from a combination of experiments, numerical and theoretical studies) requires further efforts to unambiguously link particle properties to dynamic behaviour.

**Acknowledgements** I acknowledge the support from MIUR PRIN 2017 (Project 2017Z55KCW).

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## Section 4.3 – Memory in soft matter

Muhittin Mungan<fnref n="1" />,<fnref n="2" /> and Srikanth Sastry<fnref n="3" />

**Status** Memory effects in the properties of materials are abundant in nature, particularly in soft materials. Obvious examples are the properties of glasses, microstructure in metals, melt memory in polymers, the shape memory in some alloys, as well as the magnetization state of magnetic materials, each of which exhibit history dependence. In this road map we will use the term memory broadly to refer to the dependence of material properties on their past history (for a more extended discussion, see [1], and articles in a special collection on memory formation [2]). Such memory phenomena typically emerge in materials whose energy or free-energy landscape consists of multiple locally stable structures or states. Their dynamics is often associated with slow or glassy relaxation, characterized theoretically by concepts such as hysteresis, kinetic arrest, metastability and plasticity. Research in the past has focused primarily on understanding better the underlying physics (e.g. memory effects in glasses) and on improving upon performance (e.g. shape memory materials). There has been a renewed interest in memory effects in soft matter in recent years, with new perspectives which we attempt to highlight here.

These new directions of research have focused on understanding how specific memories are encoded in the properties of materials, how they can be accessed, and how the ability to do so can be harnessed towards the design of devices with targeted, often multiple, functionalities. Viewed from the perspective of designing such novel materials, these developments highlight the increasing role played by information and information processing, establishing relations with similar themes in the areas of self-assembly (see also Sections 1.1, 1.2, 3.6 and 4.4 of this roadmap) and learning networks [3], as well as machine learning inspired approaches in soft matter design [Section 25 of this roadmap]. The emphasis has been on systems whose properties can either be designed in detail through theoretical analysis [4,5,6] or tuned through accessible training protocols [7 – 15] (Fig. 1a). Some of these investigations, motivated by machine learning approaches, aim to develop physical alternatives to *in silico* learning platforms [3,15]. Many studies have focused on model systems investigated only computationally. However, there have also been several experimental investigations into fabricating, synthesizing or training soft materials that realize some of these ideas [4,5,6,7,9,10,14] (Fig. 1b). Such experimental explorations have established valuable proofs of principle and point the way to exciting future directions in soft matter physics.

<footnote n="1">¹Institute for Biological Physics, University of Cologne, Zülpicher Straße 77, 50937 Köln, Germany</footnote>

<footnote n="2">²Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany</footnote>

<footnote n="3">³Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkar Campus, 560064 Bengaluru, India</footnote>
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<figure><img src="image_34.png" /></figure>

**Figure 1(a) (Physics-driven learning network)** A network of resistors is allowed to evolve by changing values of the resistors based on local rules so that it can generate specified voltages at a set of target nodes (blue boxes), given input voltages at input nodes (red circles). Figure from ref. [15]. **(b) (Programmable metamaterial)** A holey metamaterial has three mechanically bi-stable elements with hysteresis, called hysterons, embedded in its centre, indicated by the red box in the top right part of the panel. The material is subjected to compression $\epsilon$. Denoting the states of each bi-stable element by 0 and 1, the transition graphs show the configuration changes as the compression is increased/decreased just enough to trigger a transition (blue and red arrows, respectively). By varying the relative angle $\alpha$ between the top and bottom plates, the hysteresis response of individual hysterons can be altered, giving rise to different history-dependent response to compression, as described by the different transition graphs at the bottom of the panel. Figure adapted from ref. [5] **(c) and (d) (Transition graphs and hysterons)** The microscopic states traversed by a sheared amorphous solid can also be represented through transition graphs which exhibit nested structure (c). Here the transitions are due to mechanical instability that trigger spatially localized particle rearrangements termed soft spots. Such soft spots act like hysterons and the configurations of the amorphous solid can be represented in terms of the individual hysteron states, as indicated by the binary sequences in (c). The state changes of individual soft spots are identified by numerals in black (forward transitions) or red (backward
transitions), and the corresponding particle rearrangements are shown (d). Soft spots do interact, however, as in the coordinated '23' reverse transition in panel (d), which leads to the presence of transients and coordinated avalanches. Such features are not present in the idealized Preisach model which assumes that hysterons are non-interacting. Figure adapted from [8]

**Current and Future Challenges** An example of a system in which memory encoding and retrieval have been analyzed in detail is an amorphous solid subjected to cyclic shear deformation [8]. Under such deformation, the solid evolves until eventually reaching limit-cycles consisting of microstates through which the system makes transitions in a periodic fashion and by repeatedly triggering a set of spatially localized mechanical instabilities called soft-spots, as shown in Figs. 1c and d. The limit-cycles encode information about the deformation they are subjected to. An idealized picture is given by the Preisach model of hysteresis which regards the system of soft-spots as non-interacting, bistable, hysteretic elements, called hysterons [16]. The Preisach model can help rationalize several aspects of memory behavior, as also observed, for example, in a metamaterial [5] shown in Fig. 1b, or in stacks of origami bellows [17]. It omits, however, the role played by interactions between the soft spots, which is important for understanding key aspects, such as the evolution of systems under repeated external perturbation to achieve memory encoding states. This type of evolution can be observed experimentally, e.g., in the mechanical response of crumpled elastic sheets to cyclic shear [9]. The interactions between hysterons also leads to more complex mechanisms, where the periodic response spans multiple driving cycles [18]. A satisfactory understanding of general mechanisms governing the evolution towards memory encoding cyclic response and the role of hysteron interactions is still lacking.

Another direction in exploring memory effects has aimed at the design of mechanical networks is to achieve desired properties or response, such as large negative Poisson ratios [10], or allosteric response [11]. Such properties have been achieved through computational design exploiting specific material properties [5,6], or through the evolution of the mechanical properties of network components under loading, for example through a process that has been described as *directed ageing* [10]. Such mechanical networks are capable of retaining multiple memories, such as multiple allosteric responses. Analogous approaches carry over to flow and electrical networks [13,15], with corresponding physical cost functions that the system may optimise, as well as (design) cost functions which represent the design goals. The ability to encode desired memories, however, requires specific properties of the network components – for example, encoding memories sequentially without erasing already existing ones has been shown to require nonlinear springs [12]. The properties of network components, and training protocols that will lead to the desired macroscopic behavior, are the subject of ongoing research.
Another important and recurring theme is that of memory capacity, which is the number of distinct memories that may be stored and retrieved. For systems described reasonably by noninteracting hysterons, the relationship with the nesting depth of the limit cycles, number of hysterons, and memory capacity has been investigated [16,19]. The standard point of reference for discussing memory capacity, the Hopfield model of associative memory, has also been the subject of recent investigations. Past, as well as recent work [20] has addressed the role of the learning rule in determining memory capacity in the Hopfield model. Interestingly, the nature of the dynamics associated with memory retrieval also appears to play a significant role, with active dynamics leading to enhanced memory capacity [21].

**Advances in Science and Technology to Meet Challenges** We outline some open questions and desirable directions to build upon recent progress along two broad themes:

(i) By taking advantage of the bistability underlying the mechanical response of the hysterons, features of the mechanical loading history can be digitally encoded. Such systems act therefore as analog-to-digital converters which sense aspects of their changing loading environment, which are digitally represented as hysteron configurations and thereby form memories. This type of memory is a consequence of the (loop) return-point memory property which gives rise to a hierarchical organization of configurations into cycles and sub-cycles of transitions [16]. It has been demonstrated in systems like the sheared amorphous solids [8] and crumpled elastic sheets under cyclic shear [9]. Common to both is their programmability under cyclic loading which can be understood as a process of self-organization leading to the establishment of a persistent system of interacting hysterons. Understanding better how the emergent system of interacting hysterons and the memory created thereby depends on the loading protocol, and how such information can be accessed and harnessed to create programmable materials, is a future challenge. Although powerful in the contexts mentioned, the mapping to discrete hysteron states is not obviously available for systems such as colloidal suspensions [7], gels [14] (and even mechanical networks), which nevertheless exhibit similar memory formation phenomenology. Useful alternate descriptions of memory encoding in these systems are thus highly desirable.

(ii) Several approaches have been explored to design [4,5,6] or train [10,11,12,13,15] networks (or metamaterials) with desired properties, starting from network elements with given properties, and a given network topology. The simplest formulation of the problem is through a (design) cost function whose minimization would correspond to achieving the design goal. Such an approach, easy to implement in silico, does not easily lend itself to experimental realization. An appealing alternative explored has been that of directed ageing, wherein the properties of network elements (e.g., stiffness) co-evolve with the state of the system (e.g., extensions) in response to a training protocol, optimizing a physical cost function (e.g., elastic energy). Such an approach has been shown
to be effective in several examples [10], but its generality remains to be further understood. An
alternative approach [13,15], combines optimization of a cost function with local updates of
network properties through feed-back. Exploration of the relative merits of these approaches and
of possible other approaches forms an exciting direction of future research.

**Concluding Remarks** We have described recent work exploring memory in soft matter along two broad themes: (i) Realizing hysteretic systems composed of identifiable bistable elements, or hysterons, and investigating interactions among them, and transitions between different states. (ii) Training of various networks to have desired properties, typically responses to specific stimuli. Work along theme (i) has been concerned with a system of hysterons whose presence is established a priori through design or training, and there has been relatively less attention to the transient behavior that leads to the memory encoding states. In theme (ii) the emphasis is on the transients and the microscopic properties of final states have been analyzed to a lesser degree. In several such cases, a discrete representation of the state of the system and memory formation in terms of hysterons is not applicable. How these two themes, related yet developing along parallel tracks, may be combined is a challenge for future work. As alluded to above, the research themes discussed here have obvious and exciting connections with active areas of self-assembly design, machine learning and neural networks, to name a few, which we hope will be explored more deeply in the future.

**Acknowledgements** S. S. acknowledges support through the JC Bose Fellowship (JBR/2020/000015) SERB, DST (India). MM was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Projektnummer 398962893, Projektnummer 21150405, and Projektnummer 390685813.

**References**

[1] NC Keim, JD Paulsen, Z Zeravcic, S Sastry, SR Nagel. Memory formation in matter, Reviews of Modern Physics **91** (3), 035002 (2019).

[2] Sidney R. Nagel, Srikanth Sastry, Zorana Zeravcic, Murugappan Muthukumar; Memory formation. *J. Chem. Phys.* **158** (21), 210401 (2023).

[3] M Stern, and A Murugan. "Learning without neurons in physical systems." *Annual Review of Condensed Matter Physics* 14 (2023): 417-441.

[4] J Mueller, JA Lewis, K Bertoldi. Architected Multimaterial Lattices with Thermally Programmable Mechanical Response, Adv. Funct. Mater. 32, 2105128 (2022).

[5] J Ding, M van Hecke. Sequential snapping and pathways in a mechanical metamaterial. J. Chem. Phys. 156, 204902 (2022).

[6] Carl Merrigan, Dor Shohat, Chaviva Sirote, Yoav Lahini, Cristiano Nisoli, Yair Shokef, Emergent Disorder and Mechanical Memory in Periodic Metamaterials, https://arxiv.org/abs/2204.04000.
[7] JD Paulsen, NC Keim, SR Nagel. Multiple transient memories in experiments on sheared non-Brownian suspensions. *Physical review letters* 113, no. 6 (2014): 068301.

[8] M Mungan, S Sastry, K Dahmen, I Regev, Networks and hierarchies: How amorphous materials learn to remember, Physical review letters 123 (17), 178002 (2019).

[9] D Shohat, D Hexner, Y Lahini. Memory from coupled instabilities in unfolded crumpled sheets, Proc. Natl. Acad. Sci. 119 (28), e2200028119 (2022).

[10] N Pashine, D Hexner, AJ Liu, SR Nagel. Directed aging, memory, and nature's greed, Sci. Adv. (5) eaax4215 (2019).

[11] Daniel Hexner, Andrea J. Liu and Sidney R. Nagel, Periodic training of creeping solids, Proc. Natl. Acad. Sci. 117 (50), 31690 (2020).

[12] M Stern, MB Pinson, A Murugan. Continual Learning of Multiple Memories in Mechanical Networks, Phys. Rev. X 10, 031044 (2020).

[13] K Bhattacharyya, D Zwicker, K Alim. Memory Formation in Adaptive Networks, Phys. Rev. Lett. 129, 028101 (2022).

[14] S Chattopadhyay, S Majumdar. Inter-particle adhesion induced strong mechanical memory in a dense granular suspension. *The Journal of Chemical Physics* 156, no. 24 (2022): 241102.

[15] JF Wycoff, S Dillavou, M Stern, AJ Liu, DJ Durian. Desynchronous learning in a physics-driven learning network, J. Chem. Phys. 156, 144903 (2022).

[16] MM Terzi, M Mungan. State transition graph of the Preisach model and the role of return-point memory. *Physical Review E* 102, no. 1 (2020): 012122.

[17] T Jules, A Reid, KE Daniels, M Mungan, F Lechenault, Delicate memory structure of origami switches. Phys. Rev. Res. 4, 013128 (2022).

[18] A Szulc, M Mungan, I Regev. Cooperative effects driving the multi-periodic dynamics of cyclically sheared amorphous solids. *The Journal of Chemical Physics* 156, no. 16 (2022): 164506.

[19] NC Keim, D Medina. "Mechanical annealing and memories in a disordered solid." *Science Advances* 8, no. 40 (2022): eabo1614.

[20] M Benedetti, E Ventura, E Marinari, Go Ruocco, Francesco Zamponi. "Supervised perceptron learning vs unsupervised Hebbian unlearning: Approaching optimal memory retrieval in Hopfield-like networks." The Journal of *Chemical Physics* 156 (10): 104107 (2022).

[21] AK Behera, M Rao, S Sastry, S Vaikuntanathan. *Enhancing associative memory recall in non-equilibrium materials through activity*, https://arxiv.org/abs/2203.03024v1.
**Section 4.4 – Soft Metamaterials: Adaptation and Intelligence**

Colin Scheibner<fnref n="1" />,<fnref n="2" />, Michel Fruchart<fnref n="1" />,<fnref n="2" />, Vincenzo Vitelli<fnref n="1" />,<fnref n="2" />,<fnref n="3" />,

**Status** Mechanical metamaterials are composite structures whose collective mechanical properties go beyond those of their individual constituents [1]. For example, auxetic materials, which contract in all directions when squeezed, can be engineered via appropriate arrangements of flexible bonds. *Topological* meta-materials are those that use notions from topology (such as winding numbers or other topological invariants) to ensure the existence of a particular feature, typically a localized deformation mechanism or vibrational mode [1]. For instance, the lattice in Fig. 1a is rigid everywhere, except near the colored defect where there is a soft mechanical mode protected by topology. Beyond materials with fixed properties, *programmable* metamaterials use elastic multistability to encode specialized responses to external actuation. In Fig. 1b, a pattern (e.g., a smiley face) is programmed to appear when the metamaterial is compressed. *Trainable* metamaterials turn programmability into a process that emerges from a sequence of experiences [2]. For instance, the elasticity of the network in Fig. 1c is tuned through a process called directed aging, during which bonds are strengthened and weakened in response to external stress, a phenomenon that may also occur in colloidal glasses and gels. The individual building blocks can also have an internal source of energy [3]. This is the case for the *active* and *robotic* metamaterial shown in Fig. 1d, in which a wall of motorized hinges steers the outgoing direction of a projectile. In such active media, the familiar symmetries and conservation laws of passive matter need to be revisited to write down effective elastic models [4, 5]. The concepts behind mechanical metamaterials, such as topology or continuum mechanics, often apply across length scales. For example, the world's smallest origami bird in Fig. 1e is a *deployable* metamaterial that uses minimal actuation along with geometric constraints to guide its global shape change. One can even envision metamaterials that build themselves - a possibility particularly interesting at small scales - like the *self-assembled* colloidal structure in Fig. 1f that mimics the atomic arrangement of diamond

<footnote n="1">¹James Franck Institute, The University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="2">²Department of Physics, The University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="3">³Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, IL 60637, USA</footnote>
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<figure><img src="image_35.png" /><figcaption>Figure 1: **Mechanical metamaterials.** **a.** A topological metamaterial exhibiting a robust soft mode bound to a crystalline defect, adapted from Paulose, et al. Nat. Phys. (2015). **b.** A programmable metamaterial using multistability to create a desired post-buckling form, adapted from Coulais, et al. Nature (2016). **c.** A disordered network of foam bonds can be trained via repeated compression and directed aging, adapted from Pashine, et al. Sci. Adv. (2019). **d.** An active metamaterial consisting of robotic building blocks with distributed energy sources, adapted from Brandenbourger, et al. arXiv:2108.08837 (2022). **e.** A deployable metamaterial undergoing dramatic, but controlled, shape changes using folding constraints and minimal actuation, adapted from Liu, et al. Sci. Robot (2021). **f.** A colloidal diamond exemplifies self-assembly at the micron scale, adapted from He, et al. Nature (2020).</figcaption></figure>

<figure><img src="image_36.png" /></figure>
**Current and Future Challenges** The design of trainable, deployable, or self-assembled metamaterials exploits dynamic pathways. Yet, the functionalities targeted by such dynamic pathways are often encoded in static structures (e.g., buckled or self-assembled states) with desired mechanical properties (like rigidity or a certain vibrational spectrum). Programmable active or robotic elements embedded in the metamaterial allow us to envision inherently dynamical functionalities such as self-sustained motion and learning.

*Metamaterials as dynamical systems.* A first challenge consists in engineering functionalities rooted in the nonlinear dynamics of the metamaterial. Figure 2a-c illustrates this objective: when stimulated, an initially homogeneous and undifferentiated piece of metamaterial transitions to a deployed state (dolphin- or bird-like states in panels a and c) that depends on the stimulus (purple and blue light in panel b). Crucially, this deployed state is not a static fixed point but a dynamic state that allows the metamaterial to act like a robot: it repeatedly performs actions that do work on their surroundings (swimming and walking). Formally, these dynamic states are limit cycles or more complex attractors in configuration space. This is shown in Fig. 2d-f, in which each axis represents a dynamical degree of freedom. Reaching this goal requires designing dynamical states that perform the required tasks, ensuring that they can be dynamically reached, and implementing them at the appropriate scales.

*Metamaterials as computers.* A second challenge consists in conceptualizing and implementing some amount of intelligence in metamaterials. The non-linear dynamics of the metamaterial can be harnessed to perform simple calculations in situ, learn and perform a variety of tasks from the same undifferentiated form, and adapt to external fluctuations. Materials that change over time, such as shape-memory alloys and spin glasses, naturally store representations of their previous experiences [2]. A key objective consists in creating materials that can manipulate and act upon this information in desired ways. Figure 2g-i illustrates the end goal: as the metamaterial is externally deformed under a colored light, it learns to perform a desired motion. This level of complexity could be achieved by metamaterials that effectively perform simple machine learning algorithms using only distributed physical processes. For instance, Hebbian-like learning strategies, similar to that used in Hopfield networks, could be applied to interparticle interactions in order to train an ensemble of particles to perform tasks by showing them examples of the desired dynamics [6]. Such a form of distributed intelligence would lead to materials that are truly adaptive, like army ants forming and maintaining a bridge with their bodies.
Advances in Science and Technology to Meet Challenges

*Fabrication and programming.* Three main goals include miniaturizing and mass-producing small elements that can be combined into a material-like structures; decentralizing control to remove the need of an external computer; and integrating computation and actuation so that part of the computation is delegated to the underlying physics. To achieve these goals, advances in manufacturing will need to be combined with the design of decentralized algorithms tailored to the limited abilities of the building blocks. Ideally, these building blocks should be able to repair and replicate themselves. In that respect, biology is a power- ful source of inspiration: neural tissues (Fig. 2j) are able to form and process complex representations, while in-vitro reconstitution of cytoskeletal proteins forming an active gel produce large-scale deformations (Fig. 2k). Integrating both the computation and actuation into a single metamaterial could be done using bio-inspired systems, such as engineered DNA molecules capable of implementing algorithms [7] and acting as robots that perform complex tasks [8]; or using micro-robots: small, electronically integrated machines, which can now be manufactured by the millions at the micron scale [9].

*Effective theories of adaptive dynamics.* Adaptive metamaterials are complex: they can adapt over long-time scales while sustaining continued microscopic motions at very short time scales. This presents a challenge in describing the intermediate time scales relevant to material functionalities. Hence, describing adaptive metamaterials requires going beyond approaches relying on the simple separation of time scales characteristic of hydrodynamic theories. Model reduction techniques developed in the context of statistics and dynamical systems, such as dimensionality reduction and invariant manifold reduction, can serve as a basis for developing such effective theories of adaptive dynamics. These effective theories must rise to the challenge of describing phenomena such as physical learning [6] and back propagation [10], which occur in non-linear glassy model systems. When these learning processes occur, the material effectively produces a representation of its past experiences which can be leveraged to perform some of the computations necessary to achieve desired functionalities.

*Machine learning for material intelligence and design.* Finally, machine learning can be used to accelerate and automate the inverse problem of design: how to go from target functionalities to the basic building blocks. For instance, machine learning can help establish relationships between tunable physical parameters and the dynamical attractors that define the dynamic functionalities of adaptive materials. Moreover, machine learning can facilitate dimensionality reduction techniques for forming effective theories, thereby offering the possibility of starting directly from experimental data.
<figure><img src="image_37.png" /><figcaption>Figure 2: **Adaptive and intelligent metamaterials.** **a-c.** Schematic picture of a dynamic and intelligent metamaterial. When exposed to a stimulus (represented by light), the metamaterial performs a computation, actuates, and reaches a dynamical steady-state that serves a function such as swimming or flying. **d-f.** From the point of view of dynamical systems, these dynamical steady-states are attractors (such as limit cycles). Functionalities emerge by engineering suitable attractors and bifurcations. **g-i.** An adaptive metamaterial performs computation and learns. For example, physical stimulation (panel g) is processed by the material, which acts as a distributed machine-learning algorithm (panel h). The outputs of this process are, for instance, adapted dynamical states (panel i). **j.** Neurons are biology's solution for intelligent distributed materials, reproduced from [https://www.flickr.com/photos/nichd/21086425615]. CC BY 2.0. **k.** Physicochemical processes such as ATP consumption by kinesin motors power large-scale motion like active nematic flows, adapted from Duclos, et al. Science (2020). **l.** Micromachines: the size of a single-cell organism provide a platform for intelligent and adaptive metamaterials, adapted from Miskin et al. Nature (2020).</figcaption></figure>

<figure><img src="image_38.png" /></figure>
**Concluding Remarks.** Adaptive and intelligent metamaterials raise the prospect of merging matter with computers, and hold promise for applications ranging from medical science to space exploration. Reaching these promises will require tackling conceptual and practical challenges in fabrication at the nano and microscales [7, 9] as well as answering theoretical questions about the nature of information storage and learning in systems out of equilibrium [5, 6, 8, 7].

References

[1] Katia Bertoldi, Vincenzo Vitelli, Johan Christensen, and Martin van Hecke. Flexible mechanical metamaterials. *Nature Reviews Materials*, 2:17066, oct 2017.

[2] Nathan C. Keim, Joseph D. Paulsen, Zorana Zeravcic, Srikanth Sastry, and Sidney R. Nagel. Memory formation in matter. *Rev. Mod. Phys.*, 91:035002, Jul 2019.

[3] Suraj Shankar, Anton Souslov, Mark J. Bowick, M. Cristina Marchetti, and Vincenzo Vitelli. Topological active matter. *Nature Reviews Physics*, 2020.

[4] Colin Scheibner, Anton Souslov, Debarghya Banerjee, Piotr Surówka, William T. M. Irvine, and Vincenzo Vitelli. Odd elasticity. *Nature Physics*, 16(4):475–480, mar 2020.

[5] Michel Fruchart, Ryo Hanai, Peter B. Littlewood, and Vincenzo Vitelli. Non-reciprocal phase transitions. *Nature*, 592(7854):363–369, apr 2021.

[6] Rituparno Mandal et al. (in preparation).

[7] Damien Woods, David Doty, Cameron Myhrvold, Joy Hui, Felix Zhou, Peng Yin, and Erik Winfree. Diverse and robust molecular algorithms using reprogrammable dna self-assembly. *Nature*, 567(7748):366–372, 2019.

[8] Anupama J. Thubagere, Wei Li, Robert F. Johnson, Zibo Chen, Shayan Doroudi, Yae Lim Lee, Gregory Izatt, Sarah Wittman, Niranjan Srinivas, Damien Woods, Erik Winfree, and Lulu Qian. A cargo-sorting dna robot. *Science*, 357(6356):6558, 2017.

[9] Marc Z. Miskin, Alejandro J. Cortese, Kyle Dorsey, Edward P. Esposito, Michael F. Reynolds, Qingkun Liu, Michael Cao, David A. Muller, Paul L. McEuen, and Itai Cohen. Electronically integrated, mass-manufactured, microscopic robots. *Nature*, 584(7822):557–561, 2020.

[10] Logan G. Wright, Tatsuhiro Onodera, Martin M. Stein, Tianyu Wang, Darren T. Schachter, Zoey Hu, and Peter L. McMahon. Deep physical neural networks trained with backpropagation. *Nature*, 601(7894):549–555, 2022.
## Section 4.5 – Machine learning for soft matter learning

S. A. Ridout<fnref n="1" />,<fnref n="4" />, M. Stern<fnref n="1" />, I. Tah<fnref n="1" />,<fnref n="3" />, G. Zhang<fnref n="1" />,<fnref n="2" />, Andrea J. Liu<fnref n="1" />,<fnref n="5" />,

**Status** Machine learning (ML) and soft matter have many natural connections [1-2]. ML can be used to analyze or augment soft matter data or evaluate models for describing data [3-9]. Soft matter ideas have also proven invaluable for understanding ML [10-11]. Here we concentrate on three other areas of connection that have received less attention in reviews. We focus on how ML can be useful for physical insight into soft matter problems.

<figure><img src="image_39.png" /><figcaption>**Figure 1** (a) Input data, such as the mobility of particles (top left) or the probability of observing certain microstates (center left [14] and bottom left [13]), may be fed into an ML algorithm to construct a model of the data. (b) In some applications, the structure of the model is directly analyzed to gain understanding, e.g. identifying a structural variable strongly correlated with dynamics in a glassy liquid, whose behaviour may then be studied (top center) [12], or identifying an RG transformation in lattice models (bottom center) [13]. (c) In other cases, the actual ML output (prediction) is used as an order parameter, as in the study of crystallization (right) [14].</figcaption></figure>

**Identifying key variables or equations** A primary aim in soft matter is to understand the microscopic origins of collective behaviour. Statistical physics is the workhorse for distilling immense amounts of microscopic detail into distributions of a few key variables or governing equations of motion to capture and explain emergent properties. However, many soft matter problems have resisted solution, often because they are far from equilibrium, exhibit highly nonlinear response, possess complex structure/behaviour over multiple length or time scales, and/or lack characterized forms of order. It is natural to turn to ML for help since it is designed for dimensional reduction. However, it is not easy to extract insight from it, either from the output or from the trained algorithm itself (Fig. 1) [12-19].

<footnote n="1">¹Department of Physics, University of Pennsylvania, Philadelphia PA 19104 USA</footnote>

<footnote n="2">²Department of Physics, City University of Hong Kong, 83 Tat Chee Ave., Kowloon Tong, Hong Kong</footnote>

<footnote n="3">³Speciality Glass Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata 700032, India</footnote>

<footnote n="4">⁴Department of Physics, Emory University, Atlanta GA 30322 USA</footnote>

<footnote n="5">⁵Flatiron Institute, Simons Foundation, New York NY 10010 USA</footnote>
**Solving inverse design** ML is a logical choice of tools for designing systems with desired properties because training a neural network to perform tasks is itself an inverse problem. Conventional approaches to inverse design require expensive repeated solution of the forward problem in the high-dimensional space of “learning degrees of freedom,” namely adjustable microscopic interactions/properties. ML can optimize the choice of forward problems for efficient solution of the inverse problem [20-21]. Alternatively, ML-inspired approaches adjust learning degrees of freedom to minimize a cost function whose global minimum corresponds to attainment of the desired property [22-23].

**Soft-matter learning machines** In addition to using machine learning to understand soft matter, one can use soft matter to understand machine learning by developing soft matter systems that solve machine learning tasks by evolving each learning degree of freedom independently according to direct physical influences [24-30] so that the system minimizes the desired cost function on its own (Fig. 2). Such systems, while currently less powerful than computational or biological neural networks, combine the adaptability and parallelization of the latter with the mathematical simplicity of the former, providing a novel window into how systems can learn [30,31].

**Current and Future Challenges**

ML is a new tool for physicists to use in order to gain physical understanding. Learning how to use it most effectively towards this goal is an important challenge.

**Symmetries** In applying ML, some architectures and models, due to their inductive biases, as in [32], may be more effective than others for a given problem. For example, algorithms that encode certain symmetries are more effective for problems that obey those symmetries [33]. In soft matter, building in such symmetries can be more challenging since the symmetries can be complex and different at different length and/or time scales. For example, block copolymer systems can be disordered at short and long length scales but have different symmetries at intermediate length scales. As another example, phase transition kinetics can lead to nuclei with structures of different symmetries at different timescales.
<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

**Figure 2** Machine learning inspires inverse design of metamaterials and learning machines. (a) High-yield self-assembling lattices can be obtained via numerical optimization of a cost function [23]. (b) Directed aging allows materials to adopt desired properties by lowering the energy cost of such properties [25]. (c) Local “learning rules” allow materials to be modified to minimize a general cost function and adopt desired properties [28]. (d) Such learning rules can be embedded in the physical dynamics of active matter systems, giving rise to learning machines [29].

**Specificity vs. universality (transferability)** In soft matter we often search for insight that transcends a given system and desire common ways of understanding phenomena across different systems. ML solutions are typically designed for specific tasks over narrow domains, so that different training is required for different conditions or systems. Understanding phenomena across conditions or systems requires transferring what is learned from training for one task to another task. Ref. [34] shows a successful example in which a ML model predicts nonlinear composition-structure relationships for glass compositions outside its training set.

**Prediction vs. model building** ML algorithms are typically used for prediction, not physical insight. While prediction is the aim for some soft matter problems, and some insight can be gained from prediction, more typically the aim is to construct a model as a caricature that captures only the essential features for understanding a system or phenomenon. It can therefore be more useful to use ML to identify key variables, and then construct models based on those variables. One example is a model that yields insight into microscopic mechanisms controlling ductility of disordered solids, built upon a machine-learned structural variable that predicts the propensity of constituent particles to change neighbours [35]. Another approach is to use an algorithm that directly learns a model in the form of a set of differential equations [36,37].

**Explainability** Most tools used in soft matter allow for straightforward physical interpretation, but many ML tools do not. Using the most interpretable (usually the simplest and least powerful) ML approach possible is helpful, but identifying what a machine learning algorithm learns to perform a task remains a major stumbling block. Insight into explainability in soft matter problems has been gained using dimensional reduction techniques like autoencoders [19] or physical representation learning [38].
**Physical learning** While physical learning machines show much promise [29] for smart metamaterials and for machine learning in hardware (neuromorphic computing), they are hard to realize experimentally in specific physical media. Physical learning machines that exploit contrastive learning need to be able to compare physical states with different boundary conditions (requiring memory and changing sign for weight updates) [24, 27, 29]. It is likely that each different system, be it an elastic network, flow network, etc., will require very different engineering solutions to implement the appropriate version of learning dynamics for that system.

**Advances in Science and Technology to Meet Challenges** Several advances are needed to harness machine learning more effectively towards our own ends in soft matter. ML methods [39-41] can be useful for performing computer simulations. Autodifferentiation is also powerful for inverse design [23]. Technological ML advances such as methods to accelerate automatic differentiation or optimization, or to train algorithms with less data can certainly benefit soft matter, especially for experiments [3]. In addition, technical advances that allow algorithms to build in varying symmetries across scales are needed.

For the challenge of transferability, we note that the ability to apply parameters learned from one task to another task varies among ML algorithms. The subfield of "transfer learning" in learning theory is devoted to developing methods that allow a computational neural network to retain memory of how to perform one task while adding memory related to a new task. Progress on developing highly transferable algorithms could potentially allow physical insight gained from ML applied to one system to be applied to another, enabling quantitative study of connections across phenomena and across systems.

The greatest challenge in wringing physical understanding from ML remains explainability. Fortunately, the difficulty of understanding what a trained ML algorithm has learned bedevils ML practitioners across all fields so there are widespread and well-funded efforts to address it. New mathematical tools are needed to understand precisely what algorithms—particularly those involving neural networks--pick up on when they learn a task. Approaches that have been applied to simple tasks [31] may be useful for neural networks, but this has yet to be determined.

While improving explainability is necessary, it is not sufficient to reach the level of understanding we typically aim for. Physical understanding requires an "all of the above" approach, guided by physical insight to formulate a useful question, determine whether ML will be helpful, choose the best ML algorithm, and use it and interpret it most effectively.

**Concluding Remarks** Machine learning is not a panacea for solving all of our soft matter problems, but neither should it be shunned as a temporary fad incapable of providing understanding to serious researchers. It is here to stay. Like any other good set of tools, machine learning is most effective when wielded with physical insight to answer well-posed questions.

While explainability remains a major hurdle, soft matter researchers can also contribute to overcoming this challenge through, for example, the development of learning machines [27,29] and new analysis approaches [31].

The use of ML for going beyond prediction to gain understanding of soft matter is still in its infancy. In coming years we will discover the extent to which ML can be used to tackle long standing challenges, particularly in inverse-design problems such as non-equilibrium processes that avoid kinetic traps or that lead to desired equilibrium or metastable states, as well as recalcitrant far-from-equilibrium problems such as the glass problem, friction, fracture and turbulence.
**Acknowledgements** The Flatiron Institute is a division of the Simons Foundation, which also supported this work through the collaboration "Cracking the glass problem" via Grant No. 454945 (SAR and AJL) and Simons Investigator Grant No. 327939 (AJL). This work was also supported by the U. S. Department of Energy, Office of Science, DE-SC0020963 (AJL), and the National Science Foundation via NSF-DMR-2005749 (MS and IT), and the UPenn MRSEC NSF-DMR-1720530 (GZ).

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# Section 5.1 – Soft Materials for Energy

Chinedum O. Osuji

Department of Chemical and Biomolecular Engineering, University of Pennsylvania, USA

**Status** The generation, transduction, storage and deployment of energy rely on physical and chemical phenomena operating in materials. Conventionally, our attention first falls to hard condensed matter, or hard materials. The roles of metals, semiconductors and ceramics in energy lie in plain sight – for example, copper as a universal conductor for the transmission of electricity, silicon as the quintessential photovoltaic material for solar cells, and ceramics, whether employed for their dielectric strength in capacitors and insulators, or for lossless transport in superconducting heavy metal oxides in magnets. Soft materials however also play critical roles. Polymeric insulators for example are key enablers of low cost, safe electricity transmission and usage. That said, the growing impetus for improving sustainability in energy has brought with it a growing interest in developing and utilizing new soft materials.

Today, a broad variety of soft materials are the subject of research for energy applications. Here, the term “soft materials” is used to encompass solid polymers, gels, complex fluids (e.g., multicomponent particulate slurries), organic-inorganic hybrid nanomaterials, and molecular materials such as liquid crystals. Both small molecule and polymeric organic photovoltaics (OPVs) remain highly visible examples of soft materials used in energy (Figure 1).[1] A burst of activity over roughly 10-15 years saw the power conversion efficiency (PCE) of polymer-based bulk heterojunction devices increase over an order of magnitude from ~ 0.6% to more than 7% in the 2010s. These devices feature an interpenetrating network, a nanocomposite, of a p-type polymer donor and a n-type acceptor, as the active layer. The canonical materials were poly(3-hexylthiophene) (P3HT) and derivatives thereof, and a functionalized carbon nanomaterial, [6,6]-phenyl-C61-butyric acid methyl ester (PC61BM). Improvements driven largely by empirical optimization of nanostructure through processing, including the use of additives, eventually saturated. Subsequent efforts focused more deliberately on improving PCE through molecular engineering to optimize donor band gaps and band-alignment. So called “push-pull” conjugated polymers which rely on alternating electron-rich and electron-poor segments are now the current state of the art in low bandgap conjugated polymers and offer PCEs above 18%. Roughly comparable PCEs (~15% and higher) have been demonstrated in state-of-the-art small molecule OPVs. Significant progress has been made in the development of all organic solar cells that do away with the need for inorganic acceptors such as PC61BM.[2] Finally, the field has focused in recent years on amorphous or near-amorphous conjugated polymers that display high charge carrier mobilities despite their disordered structure. This represents something of a paradigm shift given the traditional association of poor electronic transport with poor structural order.

Beyond OPVs, three areas of particular interest regarding soft materials in energy are polymer membranes, ionic gels, and energy harvesting materials, Figure 2. Polymer membranes have been developed and used for a few decades in a variety of energy technologies and energy-adjacent applications, including water purification, gas separations, and as separators in batteries (e.g., Celgard™, a microporous polypropylene) and fuel cells (Nafion™, a sulfonated fluoropolymer, developed in the 1960s). There is a strong need however for improved performance of such membranes, particularly regarding the ability to design membranes with tailored high selectivity. Cost is also an issue, as seen with Nafion in fuel cells and other electrochemical devices. The use of membranes for harvesting energy from salinity gradients is at a nascent stage. Its potential future viability is reliant on the development of high-performance polymer membranes. Energy harvesting using soft materials and ionic gels is at a relatively early stage of development by comparison with polymer membranes, and by comparison with OPVs. The ability to realize stretchable soft materials that are simultaneously electrically and ionically conducting is a topic of contemporary interest,
particularly in the context of bioelectronics. While we do not cover complex fluids here in detail, it is worth noting that they have a substantial footprint in energy. The flow behaviour and thermophysical properties of particulate slurries are important as they affect both the manufacturing and performance of batteries, including current (e.g., Li-ion) and emerging (e.g., redox flow batteries) technologies (Figure 2). As dense, disordered suspensions, these materials are typically non-Newtonian and viscoelastic systems. They also may exhibit jamming, and time-dependent behaviour that are subjects of current scientific inquiry.

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /><figcaption>**Figure 1** Current state of the art all-polymer solar cells. Top: Cell composition. Bottom: Chemical structures of polymer donors and acceptors. Adapted with permission from Ref. [2].</figcaption></figure>

<figure><img src="image_46.png" /></figure>
**Figure 2** A: A redox flow battery based on particulate-polymer slurry. Adapted with permission from [3]. B: Nanoporous membranes made by self-assembly of crosslinkable amphiphiles into bicontinuous double gyroid structures. Adapted with permission from Ref. [4]. C, D: Conductivity performance of polymer electrolytes with desired room temperature performance indicated, and segmental vs. hopping motion of ions in polymers in liquid vs solid-like regimes. Adapted with permission from Ref. [5]

**Current and Future Challenges** On the basis of PCE alone, it is apparent that OPVs have experienced significant maturation. Nevertheless, challenges remain, particularly regarding durability of materials during extended use in non-inert conditions. Concerning polymer membranes, there are also challenges associated with durability under challenging environments, as well as persistent challenges to sharpen selectivity. For example, one of the principal impediments to the development of practical alkaline fuel cells is the poor performance of hydroxide ion-selective membranes. Many polymer membranes exhibit good OH- conductivity (> 100 mS/cm at < 90 °C), but their durability at high pH and elevated temperatures is poor.

Concerns regarding stability are also present in the field of organic solvent nanofiltration, which has attracted considerable interest as a means of circumventing the large energy footprint of conventional thermal hydrocarbon separations. In other contexts, such as gas separations and water purification, the acute need is to break the permeability-selectivity trade-off that is ubiquitous in the operation of current materials.[6] This calls for the development of membranes with ideal nanostructure – with a high degree of uniformity in the transport limiting feature sizes and chemistry. Such materials are also of interest in electrochemical applications such as redox flow batteries, to facilitate ion transport while preventing harmful crossover of relevant species by size and/or electrostatic exclusion. Self-assembly of soft materials provides an attractive route to achieving the desired nanostructure uniformity. Membranes derived from block copolymers, lyotropic surfactant mesophases and small molecule liquid crystals are of interest in this respect, in addition to systems employing discrete nanomaterials such as carbon nanotubes or transmembrane proteins (e.g. aquaporin) as transport structures.[7] There are significant challenges in this space associated with scalable fabrication of the relevant soft materials. Recent advances have been made however in highly-ordered membranes derived from self-assembled direct hexagonal lyotropic mesophases.[8] There are also non-trivial challenges associated with our still incomplete understanding of nanoscale transport under strong confinement in the presence of charge – molecular transport under nanoscale confinement and in the presence of electrostatic interactions is poorly understood from a fundamental perspective. There are numerous open questions regarding the impact on transport, and ultimately selectivity, due to pore wall dynamics, roughness and local structure, suppression and anisotropy of local dielectric permittivity, and ion-pairing, to name a few. Our ability to rationally design nanoporous ion selective materials for energy applications depends on developing new principles that can accurately describe such situations.[9] Such principles will provide clear insight regarding phenomena in nanoscale transport for which we currently lack satisfactory explanations and which we generically refer to as anomalous.[10]

Some of these same concerns regarding nanoscale transport apply to nanoporous inorganic materials used as electrodes in electrochemical devices. While these electrodes are made of hard materials, soft materials provide an attractive route for creating well-controlled high specific surface area constructs that are useful as templates to produce electrodes. Methods include the pyrolysis or carbonization of self-assembled soft mesophases, and the use of co-continuous bijel structures as templates for hard material deposition.[11] Polymers can be incorporated into conventional electrode materials to improve mechanical properties. They provide resilience against repeated volume changes during charge/discharge cycles and thereby increase the durability of the electrode.[12, 13] Significant challenges remain in terms of molecular design to realize electrochemically stable polymers and self-
healing systems that can lead to not just more durable static electrodes, but potentially also stretchable batteries.

Ion conducting gels, or gel polymer electrolytes (GPEs) have been employed in batteries and supercapacitors for energy storage.[14] In comparison to liquid electrolytes, they bring improved mechanical stability without overly compromising transport properties. Many of the challenges here are associated with molecular engineering to provide the required transport and electrochemical stability, as is also the case for liquid electrolytes. For example, for Li-ion batteries, a conductivity > 1 mS/cm at 25 °C is typically sought (Figure 2). Additionally, the landscape is still evolving regarding the ability of computational approaches and theoretical polymer/soft matter physics to accurately describe the thermodynamics and dynamics of ion containing polymers and gels.[15-18] There is intellectual synergy here with efforts on coacervation in polyelectrolytes and other charged macromolecular systems.[19] Methods to improve mechanical properties, such as the use of double network architectures, or the incorporation of high strength nanomaterials are topics of current interest.

Polymers and (electrolyte-impregnated) polymer gels incorporating stable radicals provide a rich landscape for new research in soft materials for energy, with the stable radicals providing redox-based electronic conduction mechanism.[20] The development of organic radical polymers (ORPs) has been driven by their potential as electrode materials in organic radical batteries (ORBs)[21] based on their high stability and fast redox kinetics. Ease of synthesis and processability, as well as optical transparency are key benefits, and ORBs offer performance that is competitive with Li-ion batteries in many respects (e.g. capacity ~ 140 mAh/g), but absent the adverse environmental impacts, highlighted for example by a recent report on biodegradable ORPs.[22] While the principal focus is on electrochemical energy storage in batteries and supercapacitors, applications of ORPs in biomedicine, as organic magnets, and as electrochemical transistors are also under study. Current and future challenges include processing and molecular engineering to improve conductivity, understanding dopant interactions, and improved first-principles prediction of properties.

There is an emerging class of devices, ionic diodes, that lies at the intersection of ionic membranes and GPEs. Appropriately engineered ionic heterojunctions – of membranes with asymmetric pore geometry and charge, or GPEs of differing charge – lead to rectification of ion currents, in a manner that is strongly reminiscent of biological ion channels. This is a compelling area of research that has challenges regarding fundamental nanoscale electrokinetics, as well as material and process development. GPEs or other stretchable soft materials that exhibit simultaneous ionic and electronic conductivity are of interest in bioelectronics – biological signals are communicated by ionic currents, while device logic is handled electronically.

Energy harvesting devices have been engineered from ion diodes, using osmotic gradients. Other energy harvesting mechanisms have also been leveraged from soft materials, involving triboelectric, piezoelectric and thermoelectric generation.[23] The use of thermomechanical actuation for locomotion, or the realization of artificial muscles has been a topic of interest, particularly given the large coefficients that can be obtained in soft materials such as semi-crystalline polymers and liquid crystal elastomers. A coupling of thermomechanical effects in shape memory soft materials and bimorphs with piezo/pyroelectric devices provides a route to energy harvesting.

**Advances in Science and Technology to Meet Challenges**

Addressing the challenges described above requires progress in both fundamental and applied research. As highlighted, some challenges are associated with gaps in fundamental knowledge regarding physico-chemical phenomena in new situations. Molecular transport under tight confinement in the presence of charged species represents
one such scenario. Here there is a need for advances in theory and computational approaches, as well as a need for high quality statistically meaningful experimental data against which theory and computational results can be evaluated. Properly elucidating molecular structure-function relationships in charge- and stable-radical containing soft materials is also reliant on advances in computational and theoretical approaches. In other areas, efforts aimed at scalability are needed to enable large area/volume processing of materials at low cost. Methods to effectively guide self-assembly of soft materials used in nanostructured membranes are needed. Finally, advancements are required to accelerate the rate at which new soft materials can be discovered and developed. Scaling up computational and experimental efforts to generate large high-dimensional data sets that can accelerate materials discovery through machine learning of other methods is notoriously difficult. Advances that enable (smart/directed) higher throughout experimentation, such as autonomous experimentation, are expected to be greatly enabling.

**Concluding Remarks** The increasing emphasis on sustainability is opening up new directions for soft materials in energy and accelerating already ongoing efforts. OPVs will continue to command attention in this regard, but this is alongside concerted efforts in other areas. A few key topics were discussed here. Conceptually, ion-containing and radical-containing soft materials represent a very rich landscape in aggregate – including charged nanoporous membranes, ion gels, polymerized ionic liquids, and related systems. There are important contributions to be made regarding the fundamental materials science, transport phenomena and physical chemistry of these materials, as well as in the development of new chemistries and processing methods to control their properties. These materials have the potential for significant societal impact in numerous energy and energy-adjacent applications.

**Acknowledgements** C.O.O. acknowledges financial support from NSF through DMR-1945966 and DMR-2223705

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# Section 5.2 – Soft Materials and Food Structure

Yuan Xu, Heather M. Shewan and Jason R. Stokes

School of Chemical Engineering, The University of Queensland, Australia

**Status** Soft matter includes soft materials and complex fluids found in everyday life, soft materials whose dynamics are governed by timescales of seconds. These include structured fluids, semi-solids, colloids, polymers, foams, emulsions, gels, and granular materials, which are all relevant to a wide range of naturally occurring and manufactured foods and beverages that contain hierarchical structures and multiple phases. Many of the building blocks in foods are polymeric and colloidal in origin. Considering soft matter theory in relation to soft materials and food provides a coherent basis from which to interpret complex structural and rheological properties occurring across several length, time, and energy scales. The relevance of soft matter physics to all classes of macronutrients (protein, fat, and carbohydrates) supports the development of causal relationships with food sensory perception that go beyond changes in food composition.

The definition of food as soft matter comes fundamentally from the physical and mechanical perspective [1] that differentiates them from pure, low-viscosity fluids and rigid solid materials. This is because the structural entities (such as proteins, polysaccharides, starch, microgels, micelles, colloidal fat, plant cells, and plant fibres) and included phases in foods form complex microstructures via self-assembly, thermal treatment and mechanical processing. The dynamics of the resultant structure is arrested because the interaction between entities makes them unable to move freely within a fluid phase. This generally leads to solid-like rheology that includes a yield stress and elasticity [2] to stabilise multiple phases (e.g., emulsions, particles). Typical physical structures in foods and soft materials include gel and soft glasses (Figure 1), at which structural entities are arrested by a percolated network or jamming, respectively. This definition covers a vast array of foods, such as yoghurt and cheese that may be considered densely packed suspensions of proteins in a microgel-like state; molten chocolate that is a concentrated suspension of solid particles (sugar, cocoa and milk solids) in cocoa butter; and potato chips and apple that are a packed array of plant cells. In addition, even foods that may not appear to follow this structural definition are transformed during oral processing into a swallowable bolus, which is typically a soft material in the form of a densely packed particle suspension in an aqueous physiological fluid, i.e., saliva.

Donald [3] has commented that many foods are complex self-assembled composites that demonstrate behaviours following soft matter physics laws. She also emphasises that, despite our familiarity with food and the plethora of empirical knowledge, research from the perspective of soft matter physics and chemistry is worthy and necessary to meet the 21st century challenges of sustainably producing nutritious, safe, and convenient foods. In addition to societal challenges, scientific challenges remain in fundamental soft matter physics and chemistry and their application to food design, synthesis, and characterisation. For example, due to the complexity of foods, they are often found far from equilibrium; despite this, universal structural scaling laws may still apply [2]. As such, the investigation of food within the framework of soft matter physics remains a rich research area with exciting opportunities for applying fundamental science to complex food structures and behaviours for translation into food manufacture and design. Future research directions include:

1. Understanding of food structures as basic soft matter states such as gels, liquid crystals, biopolymer complexes, foams, emulsions, and colloidal dispersions [1, 4]. Such interpretations of food structure help create model food systems to investigate the underlying mechanism of food functionality and perception, ultimately enabling the design of novel food structures for the delivery of sustainable nutrition.
II. Advanced techniques in characterising food structures and their soft matter behaviour. These include, for example, scattering [5], rheology [2], tribology [6] and microscopy/fluorescence.

III. The relationship between food structure, oral processing and sensory texture perception [6, 7]. This assists in elucidating the influence of whole and masticated food structure on sensory texture and flavour during oral processing to enable the application of fundamental soft matter models and theories in rational food product design. This includes consideration for the transformation of food structure during oral processing.

<figure><img src="image_47.png" /><figcaption>**Figure 1** Schematics of soft glass and/or gel microstructure required to physically stabilise macroscopic phases (e.g. oil droplets) as generally observed in foods, building on ref. [2]. The circles indicate nano- and micro-scale structural entities. The multi-structure represents a situation where multiple structural entities give both soft glass and gel characteristics.</figcaption></figure>

**Current and Future Challenges** Natural and processed foods are complex compositionally, structurally, and rheologically, which presents a significant challenge in rational structure design, evaluation and manufacturing. This complexity occurs across time and length scales at the molecular-to-nano scale (e.g. protein molecular structure), microscopic or meso-scale (e.g. protein aggregation) and macroscale (e.g. phase separation). This is the case for even seemingly simple soft materials and foods; for example, butter is a water-in-oil emulsion derived from cream, and the oil phase is structured via a complex multi-dimensional fractal network made from self-assembly of fat crystalline nanoplatelets over a range of length scales [8], which is also a non-equilibrium structure resulting from varying degrees of lipid crystallisation that is a function of time and temperature [5]. Food can be challenging to characterise into a single fundamental classification, such as a simple gel or soft glass. Many manufactured food products are formulated as a combination of several physical states. Unlike many model systems studied in soft matter literature, foods may have a multi-structured state (Figure 1) where it is often unclear whether they have a soft glass or gel microstructure or both. Many foods have the closely packed structure of a soft glass yet also contain strongly interactive particles or polymers (proteins, polysaccharides) that form a gel; one example is mayonnaise products that have a dense suspension of oil droplets structured within an egg protein network[2]. Thus, the compatibility and interaction between components and phases complicates the structure and behaviour of soft matter foods.

"Soft" is not a definitive term from a physical and mechanical perspective, i.e., a material is considered soft only relative to a reference state and over a specific time scale. A challenge thus comes as the measured rheological properties of foods and soft materials are time-dependent [2], such that measurement of rheological parameters over a time scale may insufficiently represent the behaviour of food during processing and consumption. The microstructures within foods and soft materials are typically shear-sensitive. Thus, the structural state alters with any imposed shear, and microstructures may be unrecoverable and rheologically thixotropic. These issues also impose difficulties correlating the properties obtained from static (e.g., scattering) and dynamic (e.g., rheology) measurements.

Early progress in food research has related food structures with fundamental colloidal, polymeric, and physical sciences. However, recent works increasingly realise that the inherent properties such as
viscosity, density and stability cannot completely describe the behaviour of soft materials and foods under conditions they may encounter. For example, oral processing exposes soft foods to a combination of shear and squeeze flow, which causes the confinement of particulates and fluids to thin films between oral surfaces and exposure to saliva, as shown in Figure 2. Under these ‘thin film’ conditions, the continuum assumption in rheology measurement is no longer valid and complementary characterisation such as micromechanics, soft-tribology and interfacial properties are required. ‘Micromechanics’ refers to the mechanical analysis of composite or heterogeneous materials on the level of the individual constituents.

Plant-based substitution of animal products is a growing food research area, which needs investigation using a soft matter framework to design microstructures emulating those from animal components. For meat analogues, the key challenge is creating a soft matter matrix that exhibits multi-scale mechanical properties and textures close to their natural meat rivals to be accepted by consumers [9], which may require greater control over structural and mechanical anisotropy and other factors. The key challenge in replacing animal-based ingredients with plant-based equivalents, particularly proteins, is matching their techno-functional properties, including their interfacial and colloidal properties that contribute to stability and texture, whilst avoiding the negative sensory properties arising from poor solubility. Incomplete understanding of the relationship between food structure, mechanical/physical properties, and sensory textures makes this challenge more profound. Addressing this challenge is required to improve sustainability in food processing.

**Advances in Science and Technology to Meet Challenges** The hierarchical structure of foods is now widely accepted in food research. However, it is still more common to alter food composition rather than investigate food's hierarchical structures in the context of fundamental soft matter science. Food researchers and manufacturers also recognise that formulation, manufacturing, storage, and oral processing of foods is dependent on the physical/mechanical properties over a range of length and time scales. Below, we highlight how recent advances in science and technology empower researchers to measure and characterise multi-dimensional properties and the response of foods as soft matter systems.

Small angle scattering (SAS) is a powerful tool in soft condensed matter physics and is highly applicable to food research. The setup of SAS experiments and data interpretation can be complicated, as reviewed by Gilbert [5]. However, by selecting the proper radiation source, such as X-ray or neutrons, and experimental setup, SAS provides spatial information regarding how entities, from nanometre to micron length scales, are located relative to others within the material. Successful interpretation of SAS results requires complementary techniques to assist in determining mathematical form factors and structure factors. This multidimensional structure studied by SAS corresponds to the multiple scales of self-assembly within soft matter foods, such as the study of lipid self-assembly and lipid digestion. Thus, SAS provides unique value in food structure research.

Investigation into the mechanical and rheological response of soft materials and foods in the non-linear regime and under confinement forms another area in which progress has been made on model soft matter systems. Soft materials are characterised by their non-linear and time-dependent rheology, e.g. the measured yield stress and viscoelastic parameters are not definitive to the material but apparent over a certain deformation and time spectra [2]. These extremes of rheology are relevant to the high-shear environment and confinement experienced by foods during oral processing between tongue and palate or tongue and teeth and offer opportunities to understand the mechanisms driving sensory perception. Large amplitude oscillatory shear (LAOS) [10], gap-dependent rheology [6], and squeeze flow rheology are three such techniques.
Soft contact tribology is emerging as a valuable tool in the design of foods and beverages due to its relevance to oral processing and sensory perception [6]. This involves rubbing two highly deformable bio-mimetic surfaces with a soft matter or complex fluid phase as a lubricant. The soft lubrication behaviour of foods is a system variable strongly dependent on lubricants' rheology but further impacted by the speed, substrate elasticity, surface roughness and interfacial properties. Therefore, it provides complementary information to rheological measurements in efforts to correlate the physical properties of foods with their sensory texture (Figure 2).

<figure><img src="image_48.png" /><figcaption>**Figure 2** Multi-scale mechanical/rheological analysis techniques of soft matter foods. The length scale of each method and its 'correlation' with sensory textures are indicated. Builds on ref. [6].</figcaption></figure>

**Concluding Remarks** The research into food structure and its relationship with physical/mechanical properties is closely related to fundamental soft matter physics and colloidal science. Challenges in soft matter physics are also common in food research, e.g., formulation, synthesis, and advanced characterisation of soft materials with multi-dimensional structures and tailored properties under complex physiological and mechanical environments. The predominant approach in food science is to build empirical correlations between physical properties and sensory attributes of whole foods, which does not provide mechanistic understanding. The use of soft matter theory as a fundamental basis for food research promotes a focus on the relevant physics occurring in-mouth to provide insights to interpret and mechanistically predict and design products for desired performance, and sustainable food processing.

**Acknowledgements** Contributions to this article have been made possible due to research funded by the Australian Government through the Australian Research Council (ARC) Grants DP180101919 and LP160100239.
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[3] A. Donald, Food for thought, Nat Mater, 3 (2004) 579-581.

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[5] E.P. Gilbert, Small-angle X-Ray and neutron scattering in food colloids, Current Opinion in Colloid & Interface Science, 42 (2019) 55-72.

[6] J.R. Stokes, M.W. Boehm, S.K. Baier, Oral processing, texture and mouthfeel: From rheology to tribology and beyond, Current Opinion in Colloid & Interface Science, 18 (2013) 349-359.

[7] E.A. Foegeding, M. Stieger, F. van de Velde, Moving from molecules, to structure, to texture perception, Food Hydrocolloids, 68 (2017) 31-42.

[8] A.G. Marangoni, J.P.M. van Duynhoven, N.C. Acevedo, R.A. Nicholson, A.R. Patel, Advances in our understanding of the structure and functionality of edible fats and fat mimetics, Soft Matter, 16 (2020) 289-306.

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[10] H.S. Joyner, Nonlinear (Large-Amplitude Oscillatory Shear) Rheological Properties and Their Impact on Food Processing and Quality, Annu Rev Food Sci Technol, 12 (2021) 591-609.
# Section 6.1 – Active Biological Materials

Matthias Merkel<fnref n="1" />, Pierre Ronceray<fnref n="2" /> and Jean-François Rupprecht<fnref n="1" />

*All authors contributed equally to this article.*

**Status** The field of active matter physics arose almost 30 years ago from the desire to better understand the collective and dynamical properties of living systems. Most of the time, researchers have set out to better understand how many, equally behaving, energy-consuming agents can organize across different scales, e.g., molecules into cytoskeletal gels, cells into tissues, and animals into swarms.

Here, we focus on active biological materials, which we define as living biological systems for which material properties such as a viscosity or elastic moduli can be reasonably defined. This is often not the case for “dry” systems where active agents propel themselves with respect to a common substrate. Instead, this usually applies to “wet” active systems such as the cytoskeleton, swarms of bacteria, and biological tissues.

Despite this vast diversity of active biological materials, many fundamental properties have been understood from a theoretical perspective, using simulations of agent-based models, explicitly coarse-graining such models, and analyzing the resulting hydrodynamic descriptions (Marchetti et al., 2013). Experimentally, these properties have been demonstrated in many *in-vitro* and *in-vivo* systems (Sanchez et al., 2012; Duclos et al., 2018). So far, connecting between theories and experiments has, in most cases, only been possible in simplified setups, such as 2D reconstituted gels or cultured epithelial monolayers.

Despite this progress, our understanding of active biological materials is still limited. In some cases, active matter theory even opened new fundamental questions. For instance, active anisotropic tissue deformation is a key process during animal morphogenesis. However, the orientationally ordered state in wet active matter is generally unstable, consistent with *in-vitro* experiments (Duclos et al., 2018). How oriented tissue deformation can be robust during morphogenesis despite this instability is still unclear (Ibrahimi and Merkel, 2023). More generally, much remains to be done to bridge between active material theories and the full complexity of living organisms.

**Current and Future Challenges** Current theories focus on the description of idealized systems, which fail to capture several key aspects of the complexity of *in vivo* active matter. We discuss here three of these aspects.

First, many of the active matter systems studied so far had a flat 2D geometry (reconstituted cortices, motility assays, cell monolayers, etc.). However, both intracellular structures and living tissues *in vivo* can be bulk 3D structures (e.g. connective tissue, cytoplasm; Duclos et al., 2020), curved 2D structures (e.g. epithelia, membranes; Maroudas-Sacks et al., 2021; Randriamanantsoa et al., 2022; Bell et al. 2022; Gehrels et al., 2023), or combinations of both (e.g. embryos, organs, organoids; Tao et al., 2019; Brandstätter et al., 2023), each embedded in a complex 3D environment. For instance, in many systems, taking into account the curvature and its temporal dynamics is experimentally and mathematically challenging. Furthermore, the environment of active materials *in vivo*

<footnote n="1">¹ Aix Marseille Univ, Université de Toulon, CNRS, CPT (UMR 7332), Turing Centre for Living Systems (CENTURI), Marseille, France.</footnote>

<footnote n="2">² Aix Marseille Univ, CNRS, CINaM (UMR 7325), Turing Centre for Living Systems (CENTURI), Marseille, France.</footnote>
vivo often deforms in response to the forces exerted by the active material. This adds the challenge of dissecting the interplay between active matter mechanics, geometry, and mechanics of the environment.

Second, much of the past work studied average behavior of active biological materials, even though fluctuations can play a physiological role, e.g., in tissue fracture (Sonam et al., 2023). In addition, fluctuations can also be harnessed to extract more information about the dynamics driving the system. For equilibrium systems the fluctuation-dissipation theorem (FDT) paved the way to micro-rheology, allowing to extract rheological parameters from observed fluctuations. For active systems, weaker generalizations of the FDT exist (Rupprecht and Prost, 2016), and micro-rheology has been successful for some systems (Lau et al., 2003). However, additional measurements are generally required to extract the material rheology (Turlier and Betz, 2018). While a limited number of key material parameters can often be directly extracted from fluctuations (Blanch-Mercader et al., 2018), a deeper understanding of fluctuations in active materials will allow us to better harness the information contained in fluctuations.

Third, both *in vitro* and artificial active matter systems tend to be designed to be as clean, i.e., homogeneous and monodisperse, as possible, and most often, theories for active materials have been tested on such systems. But *in vivo* materials are not clean. The cytoskeleton is a heterogeneous, highly composite structure, while biological tissues contain multiple cell types, arranged in complex structures. Taking the “dirtiness” of *in-vivo* systems into account will be a major challenge.

Finally, an overarching difficulty in accounting for more complexity is the proliferation of parameters required to describe the systems. Keeping the number of parameters in check while moving towards a more realistic yet intelligible description of living materials is a key future challenge (Brunton et al., 2016).

**Advances in Science and Technology to Meet Challenges** To address these challenges, one must bridge between the complexity of living organisms and theories of active materials. We distinguish here four steps.

*Step 1 - from living organisms to images.* Significant progress has been made recently in imaging 3D systems, where light-sheet and two-photon microscopy have become standard techniques over the last decades (Wan et al., 2019). However, challenges remain such as light scattering, phototoxicity, and a limitation in the number of fluorophores for live imaging of protein dynamics.

*Step 2 - from images to data.* Image data often needs to be segmented and objects tracked over time before it can be compared to theoretical models. For such data analysis, new tools, including deep learning tools (Weigert et al., 2020), are emerging to analyze the local material structure. However, so far system-specific solutions are often required, but universal tools are starting to emerge (Pachitariu, 2022; Kirillov, 2023). Image analysis is also important for techniques to measure forces and rheological properties of living systems (Roca-Cusachs et al., 2017). While these techniques are currently being adopted more broadly, we believe that they should be combined more often with techniques to experimentally manipulate the 3D environment, such as microfluidics and micro-confinement.

*Step 3 - from data to models.* To compare data to a theoretical model, problem-specific data analysis tools often need to be developed. For noisy or incomplete data, the calibration of theoretical models
requires the use of statistical inference techniques (Frishman and Ronceray, 2020). Another example are theoretical tools to connect dynamics across different length scales, which allow calibrating macroscopic models using microscopic data (Merkel and Manning, 2017).

*Step 4 - from models to theories.* Finally, quantitative models for specific systems are not the end goal; the goal is to understand what unifying principles underlie the complexity of life. For 3D systems, general theoretical descriptions (Salbreux et al., 2022) and simulation tools (Okuda et al., 2018; Bächer et al., 2021; Borja da Rocha et al., 2022) have recently been developed. However, new theoretical developments are needed to account for the “dirtiness” of real biological materials. This includes, for instance, the combination of active matter physics with existing high throughput “omics” techniques (La Manno et al., 2018), to better understand, e.g. how cell type heterogeneity interacts with active mechanics during morphogenesis.

<figure><img src="image_49.png" /><figcaption>**Figure 1** (1) Well-studied active biological materials include, at the sub-cellular scale, reconstituted cytoskeletal networks (which can be modeled as complex spring networks) and, at the tissue-scale, epithelial monolayers (which can be modeled for example using vertex models). (2) Current challenges include generalization to complex geometries (curved, 3D, soft, evolving surfaces), to fluctuations (that can be active, non-Gaussian and colored), and to encompass the heterogeneity of materials. (3) A forward challenge is to develop a framework to understand the properties of active biomaterials within live organisms. One example is the epithelium in tubular structures (such as the gut); it combines the challenge of curved geometries, of flow fluctuations at the global and local scales, and of heterogeneities in cell types. All illustrations by Alexander Brodovitch (curved tissue image in (2) adapted from Sussman, 2020).</figcaption></figure>
**Concluding Remarks** The field of active matter was created to better understand the collective and out-of-equilibrium properties of living systems. After almost 30 years, we have arrived at a theoretical and experimental characterization of many fundamental effects in the dynamics and structure of biological materials, from intracellular gels to multicellular tissues. These advances, however, concern mostly idealized systems: to move towards an active material description of *in vivo* systems such as live animals, progress must be made to include the complexity of these systems, for instance by looking at 3D systems, better understanding fluctuations, and try to include more and more of the “dirtiness” of real systems. This will require an even closer collaboration between biologists and physicists.

Finally, improving our quantitative understanding of the collective properties of active biological materials opens the way for the control of their dynamical and mechanical properties using global fields or local cues (Cohen et al., 2014).

**Acknowledgements** M.M. thanks the Centre Interdisciplinaire de Nanoscience de Marseille (CINaM) for providing office space. M.M., P.R. and J.-F.R. received funding from the «Investissements d'Avenir» French Government program managed by the French National Research Agency (ANR-16-CONV-0001) and from the Excellence Initiative of Aix-Marseille University - A*MIDEX.

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