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The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific inquiry. However, a large body of natural phenomena incorporates memory effects that are best described via fractional integro-differential equations (FIDEs), in which the integral or differential operators accept non-integer orders. Addressing the challenges posed by nonlinear FIDEs is a recognized difficulty, necessitating the application of generic methods with immediate practical relevance. This work introduces the Universal Fractional Integro-Differential Equation Solvers (UniFIDES), a comprehensive machine learning platform designed to expeditiously solve a variety of FIDEs in both forward and inverse directions, without the need for ad hoc manipulation of the equations. The effectiveness of UniFIDES is demonstrated through a collection of integer-order and fractional problems in science and engineering. Our results highlight UniFIDES' ability to accurately solve a wide spectrum of integro-differential equations and offer the prospect of using machine learning platforms universally for discovering and describing dynamical and complex systems., Comment: 27 pages, 9 figures, regular article

Attractive colloidal gels exhibit solid-like behavior at vanishingly small fractions of solids, owing to ramified space-spanning networks that form due to particle-particle interactions. These networks give the gel its rigidity, and as the attraction between the particles grows, so does the elasticity of the colloidal network formed. The emergence of this rigidity can be described through a mean field approach; nonetheless, fundamental understanding of how rigidity varies in gels of different attraction strengths is lacking. Moreover, recovering an accurate gelation phase diagram based on the system's variables have been an extremely challenging task. Understanding the nature of these fractal clusters, and how rigidity emerges from their connections is key to controlling and designing gels with desirable properties. Here, we employ well-established concepts of network science to interrogate and characterize the network of colloidal gels. We construct a particle-level network, having all the spatial coordinates of colloids with different attraction levels, and also identify polydisperse rigid fractal clusters using a Gaussian Mixture Model, to form a coarse-grained cluster network that distinctly shows main physical features of the colloidal gels. A simple mass-spring model then is used to recover quantitatively the elasticity of colloidal gels from these cluster networks. Interrogating the resilience of these gel networks show that the elasticity of a gel (a dynamic property) is directly correlated to its cluster network's resilience (a static measure). Finally, we use the resilience investigations to devise [and experimentally validate] a fully resolved phase diagram for colloidal gelation, with a clear solid-liquid phase boundary using a single volume fraction of particles well beyond this phase boundary.

The concept of a Deborah number is widely used in study of viscoelastic materials and to represent the ratio of a material relaxation time to the timescale of observation, and to demarcate transitions between predominantly viscous or elastic material responses. However, this construct does not help quantify the importance of long transients and non-monotonic stress jumps that are often observed in more complex time-varying systems. Many of these non-intuitive effects are lumped collectively under the term thixotropy; however, no proper nouns are associated with the key phenomena observed in such materials. Thixotropy arises from the ability of a complex structured fluid to remember its prior deformation history, so it is natural to name the dimensionless group representing such behavior with respect to the ability to remember. In Greek mythology, Mnemosyne was mother of the nine Muses and the goddess of memory. We thus propose the definition of a Mnemosyne number as the dimensionless product of the thixotropic time scale and the imposed rate of deformation. The Mnemosyne number is thus a measure of the flow strength compared to the thixotropic timescale. Since long transients responses are endemic to thixotropic materials, one also needs to consider the duration of flow. The relevant dimensionless measure of this duration can be represented in terms of a mutation number which compares the timescale of experiment/observation to the thixotropic timescale. Collating the mutation number and the Mnemosyne number, we construct a general two-dimensional map of thixotropic behavior, and quantify these ideas using canonical thixotropic models.

We report experimental and computational observations of dynamic contact networks for colloidal suspensions undergoing shear thickening. The dense suspensions are comprised of sterically stabilized poly(methyl methacrylate) hard sphere colloids that are spherically symmetric and have varied surface roughness. Confocal rheometry and dissipative particle dynamics simulations show that the shear thickening strength scales exponentially with the scaled deficit contact number and the scaled jamming distance. Rough colloids, which experience additional tangential and rolling constraints, require an average of 1.5 - 2 fewer particle contacts as compared to smooth colloids, in order to generate the same shear thickening strength. This is because the surface roughness enhances geometric friction in a way that the rough colloids do not experience a large change in the free volume near the jamming point. In contrast, smooth colloids must undergo significant reduction in the free volume to support an equivalent shear stress. The available free volume for different colloid roughness is related to the deficiency from the maximum number of nearest neighbors at jamming under shear. Our results further suggest that the force per contact is different for particles with different morphologies.

The aggregation of attractive colloids has been extensively studied from both theoretical and experimental perspectives as the fraction of solid particles is changed, and the range, type and strength of attractive or repulsive forces between particles varies. The resulting gels consisting of disordered assemblies of attractive colloidal particles, have also been investigated with regards to percolation, phase separation, and the mechanical characteristics of the resulting fractal networks. Despite tremendous progress in our understanding of the gelation process, and the exploration of different routes for arresting the dynamics of attractive colloids, the complex interplay between convective transport processes and many-body effects in such systems has limited our ability to drive the system towards a specific configuration. Here we study a model attractive colloidal system over a wide range of particle characteristics and flow conditions undergoing aggregation far from equilibrium. The complex multiscale dynamics of the system can be understood using a Time-Rate-Transformation diagram adapted from understanding of materials processing in block copolymers, supercooled liquids and much stiffer glassy metals to direct targeted assembly of attractive colloidal particles.

We identify the sequence of microstructural changes that characterize the evolution of an attractive particulate gel under flow and discuss their implications on macroscopic rheology. Dissipative Particle Dynamics (DPD) is used to monitor shear-driven evolution of a fabric tensor constructed from the ensemble spatial configuration of individual attractive constituents within the gel. By decomposing this tensor into isotropic and non-isotropic components we show that the average coordination number correlates directly with the flow curve of the shear stress vs. shear rate, consistent with theoretical predictions for attractive systems. We show that the evolution in non-isotropic local particle rearrangements are primarily responsible for stress overshoots (strain-hardening) at the inception of steady shear flow and also lead, at larger times and longer scales, to microstructural localization phenomena such as shear banding flow-induced structure formation in the vorticity direction., Comment: 15 pages, 4 figures

The flow of dense suspensions, glasses, and granular materials is heavily influenced by frictional interactions between constituent particles. However, neither hydrodynamics nor friction has successfully explained the full range of flow phenomena in concentrated suspensions. Particles with asperities represent a case in point. Lubrication hydrodynamics fail to completely capture two key rheological properties - namely, that the viscosity increases drastically and the first normal stress difference can switch signs as volume fraction increases. Yet, simulations that account for interparticle friction are also unable to fully predict these properties. Furthermore, experiments show that rheological behavior can vary depending on particle roughness and deformability. We seek to resolve these apparent contradictions by systematically tuning the roughness of model colloids, investigating their viscosity and first normal stress differences under steady shear, and finally generating a rheological state diagram that demonstrates how surface roughness influences the transition between shear thickening and dilatancy. Our simulations, which are in good agreement with the experiments, suggest that friction between rough particles is significant. In addition, we find that roughness progressively lowers the critical conditions required for the onset of shear thickening and dilatancy. Our results thus provides a major contribution in the field of suspension rheology with broad relevance to granular and particulate materials. For instance, particle geometry can be tuned to increase the efficacy of materials that turn solid-like on the application of stimuli. On the other hand, engineers who work with concentrated slurries can now use images of the constituent particles to estimate optimal flow processing conditions.