Network Science Insights in Dense Suspension Rheology

Department of Macromolecular Science and Engineering, Case Western Reserve University, Cleveland, Ohio, USA.
E-mail: axs2601@case.edu

Abstract

Suspension rheology is a cornerstone of the Society of Rheology, as evidenced by its meetings and publications. Decades of work have focused on the non-Newtonian behavior of these materials, including yielding, shear thinning, and shear thickening. Yet dense suspensions continue to surprise us. A seemingly simple mixture of particles and liquid can abruptly become harder to flow, yield from a solid-like state, or jam outright when sheared. Increasingly, these behaviors are understood not only by where the particles are but also by how they are mechanically connected. Under deformation, particles form, break, and reorganize the contact and force networks that carry stress across the material. Network science offers a useful language for describing these mesoscale structures. In this article, I introduce this emerging perspective and discuss how concepts such as force chains, loops, rigid clusters, and percolating contact networks are providing new ways to think about yielding, discontinuous shear thickening, and shear jamming. I close with what remains open, especially the challenge of extending these ideas to the three-dimensional, polydisperse, and chemically complex suspensions that arise in practice.

Introduction

Particulate suspensions are everywhere—in industry and in nature. They appear as paints, cement, and ceramic slurries; as foods such as mayonnaise and molten chocolate; in pharmaceutical formulations and 3D-printing inks; and as mud, blood, and many other materials familiar to rheologists. Despite their apparent simplicity, particle–liquid mixtures exhibit remarkably rich flow behavior. Squeeze a bottle of ketchup, and it suddenly flows; strike a cornstarch suspension, and it solidifies¹. The same action can produce opposite responses. More generally, a suspension that flows easily under one condition may resist deformation, or even behave as a solid under another, spanning a wide range of macroscopic responses that include yielding, shear thinning, shear thickening, and shear jamming^2–4.

These diverse phenomena share a common thread: each can be framed as a transition in which rigidity is gained or lost within the material’s underlying structure during deformation (Fig. 1). Yielding is the transition from a soft viscoelastic solid to a flowing liquid when the applied stress exceeds the material’s yield stress σ_y. It reflects the breakdown of an internal structure supported by a microscopic particle network and has been the subject of several topical reviews^5–7.

The reverse can also occur. Increasing the applied stress or deformation rate can increase viscosity and trigger shear thickening, either continuously (CST) or abruptly (DST). At sufficiently high solid concentrations, this can culminate in a shear-jammed solid (SJ) above a characteristic stress σ_sj. As Fig. 1 illustrates, yielding and shear jamming are similar rigidity transitions viewed from opposite directions: yielding breaks down an elastic backbone of interparticle interactions, whereas shear jamming builds a stress-bearing network.

FIG 1. Typical rheological transitions related to loss or emergence of rigidity of the underlying force network. Loss (left) or emergence (right) of rigidity under deformation: yielding (left) and shear jamming (right). Both transitions are associated with the breakage (left) or formation (right) of an elastic backbone in the network formed by interparticle interactions. Shaded regions denote the solid-like behavior.

This article presents a perspective on both transitions through a shared lens: the loss of rigidity during yielding and the emergence of rigidity during shear jamming. It asks not only how the macroscopic response evolves under deformation but also which microscopic interactions and mesoscale structures drive it. How do interparticle interactions organize into networks that resist or fail under stress? Many rheologists already reach for this picture in their own systems: a polymer rheologist thinks in terms of transient entanglement networks, a gel rheologist in percolating clusters, and a granular physicist in force chains. Network science offers a way to make that shared instinct more quantitative across systems. I draw on how these ideas have been developed for polymer networks^8,9, colloidal gels^10–15, granular systems¹⁶, and, more recently, dense suspensions.

Contacts and constraints: beyond a purely fluid-mechanical picture

Suspension rheology has traditionally been viewed through a fluid-mechanics lens^2,17, in which pairwise lubrication and long-ranged hydrodynamic interactions are the dominant contributions. In this picture, particle motion perturbs the surrounding fluid streamlines as the suspension deforms. For perfectly smooth, rigid particles, continuum hydrodynamics predicts that the lubrication force diverges as 1/h, where h is the gap between two particle surfaces: as the gap closes, it becomes ever harder to squeeze the intervening fluid out, and the particles should never quite touch¹⁷.

A more recent perspective suggests that, in real suspensions, the lubrication film between two particles must eventually break down. This reminds us that perfectly smooth, rigid particles are an idealization^18,19. Once that idealization fails, direct particle contacts can form. From this vantage point, the surprising non-Newtonian response of dense suspensions arises not from hydrodynamics alone but from the interplay between interparticle interactions and the applied stress^1,4,18. These interactions are governed by the solid–fluid interfacial chemistry of both phases, as well as by frictional interactions between particles, which depend on particle roughness and surface chemistry^4,20,21.

This framework can be recast in terms of a more general and abstract concept: constraints. The non-Newtonian response then arises from a competition between constraints and applied stress. Stress-released constraints reduce viscosity, leading to shear thinning; stress-activated constraints increase viscosity, leading to shear thickening. Framing the problem this way has, over the past decade, helped unify dry granular physics and suspension rheology^4,22–24.

The intuition behind constraints is simple: how many contacts does a particle need for the system to be mechanically stable? Each constraint on relative motion removes degrees of freedom (DOF). As additional modes of relative motion are constrained, fewer contacts per particle are needed to achieve mechanical stability. This threshold is captured by the isostatic coordination number Z_iso and the corresponding volume fraction at which the system jams, φ_J drops accordingly. This establishes a hierarchy. A purely lubricated particle experiences no constraints in sliding or rolling modes; if only excluded volume counts, isostaticity requires Z_iso = 2D (= 6 in 3D) with φ_J⁰ ≈ 0.64. If sliding is also forbidden, the requirement drops to Z_iso = D + 1 (= 4 in 3D) with φ_J^μ ≈ 0.57. If rolling is also constrained, it drops further to Z_iso = D(D+1)/(2D-1) (= 2.4 in 3D) with φ_J^μ ≈ 0.35. In this language, yielding and shear jamming can be viewed as departures from and approaches to isostaticity.^{4,18,21,23,24}.

The strength of the constraint picture is that what matters is the number of constraints, or equivalently the available (or lost) degrees of freedom, rather than only the precise physical or chemical origin of these constraints. Mean-field models built on this idea successfully predict steady-state, strain-averaged flow. By construction, though, they are largely agnostic to mesoscale rearrangements: how the contacts are spatially organized and how that organization carries stress. This is the aspect I focus on in the rest of this article.

From contacts to force-chain networks

Constraints that are activated, as in friction, or released, as in adhesive or attractive systems, provide a unifying framework for understanding shear jamming and yielding. But invoking the isostatic condition immediately raises a fundamental question: which contacts are present, and how are they arranged? Counting constraints tells us when a system can become rigid; it does not tell us what the rigid structure looks like or how stress is transmitted through it. Framed in terms of rigidity, the question becomes: what geometrical and topological features of the contact or force network produce the observed flow behavior?

This question has brought network- and graph-theoretic methods into the study of suspension rheology. These tools offer statistical descriptions of microstructure—and, ultimately, a possible route to controlling flow—that constraint-counting mean-field models cannot provide by construction. The approach is increasingly used across rheology and the broader soft-matter communities.

Why the frictional network?

Rheologists routinely ask why a given behavior appears: what microstructure underlies it? The usual structural descriptors, the pair correlation function g(r) and the structure factor S(q), are invaluable for quantifying the spatial organization of particles^2,25,26. But the spatial arrangement of particles is not the same as knowing which particles bear the load. This is where the network view adds to an old intuition. The aphorism widely attributed to polymer rheologist Prof. Richard S. Stein—“Rheology without morphology is theology, while morphology without rheology is just zoology”—reminds us of the coupling between flow and structure. For dense suspensions, the morphology that matters is not only where the particles are but also how they are mechanically connected: the morphology of the contact and force network, not of particle positions alone.

Simulations that resolve both lubrication and frictional contact forces have made this concrete^27,28. Comparing shear rates just below and just above the onset of DST—that is, unconstrained versus constrained states- the traditional measures (g(r) and S(q)) and showed only minimal differences, even though the viscosity changed by nearly an order of magnitude. In contrast, analysis of the frictional contact network showed that the two states were remarkably different^18,28. These observations helped motivate the use of graph- and network-theory approaches in dense suspensions.

Motivation also runs deep in granular physics, where jamming has long been linked to enduring contacts that form force networks. These networks transmit stress and correlate motion over distances far larger than a single particle but smaller than the sample—a natural mesoscale structure. Force chains are a canonical example: filament-like strings of particles that carry most of the load. Cates et al. pioneered the concept of force chains and defined them as linear strings of rigid particles in point contact²⁹. This definition is useful because it highlights both the stability and fragility of these structures. Under imposed deformation, such as simple shear, the stress field can be decomposed into principal compressive and tensile directions. As a particulate system is deformed, force chains naturally emerge in the direction of compression. However, such structures require orthogonal support because contacts can break under continued deformation. If the external deformation is perturbed sufficiently, the structure can fall apart; in this sense, it is fragile.

Stabilizing these force chains requires orthogonal support from other force-bearing particles along the tensile direction. This hypothetical configuration can then become mechanically stable, or jammed, under external deformation and behave as a fragile solid with a finite modulus. Experiments on photoelastic disks and simulations of granular materials have corroborated these ideas, establishing both the heterogeneity of particle forces and stresses and the experimental validation of shear jamming^30,31.

For dense suspensions, this lens is particularly useful. The particle arrangement may shift only subtly during a rheological transition, while the frictional contact network reorganizes dramatically. This suggests that the key question is not only where the particles are, but also which particles are mechanically connected, how those connections carry stress, and whether (and how) the resulting network can resist external deformation.

From particle space to network space

Once frictional contact or the force network is recognized as an important mesoscale descriptor, the next task is to quantify it. A useful starting point is that the particle configuration and the network provide dual descriptions of the same system: particle positions indicate where particles are, whereas the network indicates which particles are mechanically connected. In this sense, the particle configuration and the contact network offer complementary views of the same problem (see Fig. 2).

Network science studies systems of interacting entities, and its natural representation is a graph G, with entities as nodes V and interactions as edges E; thus, a graph is represented mathematically as G ≡ {V, E}^32,33. An edge can be unweighted, recording only whether an interaction is present, or weighted, also carrying its strength.

Using the graph G, an adjacency matrix A can be constructed to represent the network. For an undirected and unweighted graph, A is defined as:
\[A_{ij}=\Bigg\{\begin{matrix}1,&\mathrm{if\ there\ is\ an\ edge\ between\ nodes\ }i\mathrm{\ and\ }j,\\0,&\mathrm{otherwise.}\end{matrix}\]

For a more general weighted network, the adjacency matrix is

\[W_{ij}=\Bigg\{\begin{matrix}w_{ij},&\mathrm{if\ there\ is\ an\ edge\ between\ nodes\ }i\mathrm{\ and\ }j,\\0,&\mathrm{otherwise,}\end{matrix}\]

where w_ij is the strength of the interaction between nodes i and j.

These definitions map naturally onto familiar objects: a frictional contact network is unweighted—it records only whether a frictional contact exists—whereas a frictional force network is weighted by the magnitude of the contact force. Force chains can then be viewed as filament-like structures composed of strongly weighted edges embedded in the force network. Formally, the network is encoded in an adjacency matrix (binary for the contact network and force-weighted for the force network; see Eqs. 1 and 2). Information about nodes, edges, and weights is central to the network description used here.

Network-theory tools

Once the network is defined, a standard toolkit allows us to quantify its structure. Some diagnostics recur across particulate systems.

Degree: The simplest measure is the degree, i.e., the number of edges incident to a node. In the granular and suspension literature, this is the contact number, or coordination number Z, which largely controls mechanical stability and, through it, rheology^22,34. Tracing connected sequences of edges yields loops, or cycles: closed paths that return to their starting node. The order of a loop (l) is its number of edges, so the smallest possible loop is a triangle—three particles in mutual contact, or a third-order loop.

Community structure and detection: Communities are groups of nodes that are more densely connected to one another than to the rest of the network. In other words, nodes within a community have many edges among themselves and relatively few edges connecting them to nodes in other communities³⁵. Community-detection algorithms typically identify such groups by optimizing a quantity called modularity, which assigns nodes to the same cluster when they are densely connected within the community and weakly connected to nodes outside it³⁶.

Persistent homology: Persistent homology tracks how features, such as connected components and loops, are born and die as a force threshold (or edge weight) is swept from high to low. Picture lowering a water level over a landscape: islands appear, bridges form, and components merge³⁷. Persistent homology tracks the birth and death of these structures, enabling one to distinguish short-lived features from robust structures that persist across a wide range of force scales³⁸.

Rigidity theory: Rigidity theory raises another fundamental question: can a network of contacts resist deformation? Connectivity alone is not enough—a connected chain can still bend or slide unless stabilized by sufficient contacts. This is Maxwell’s constraint count: rigidity requires enough independent constraints relative to the available degrees of freedom^39,40. The pebble-game algorithm operationalizes this idea by partitioning a contact network into rigid and floppy regions⁴¹. Originally devised for frictionless packings, it has recently been extended to frictional ones⁴². For suspension rheology, the key message is that connectivity alone is insufficient: the contact network must be organized to be mechanically rigid and capable of supporting stress.
What insights does network science provide into suspension rheology?

FIG 2: Insights into suspension rheology: from particle to network space. (Left) A packing of particles; (Middle) representation of particles as a graph, with particles as nodes and contacts or forces as unweighted or weighted edges, followed by network-theory-based decomposition; (Right) Relative viscosity plotted against (top) number of edges and (bottom) number of edges in third-order loops.

A central question is whether network-based descriptions provide insight into suspension rheology beyond traditional particle-based measures. In particular, can the organization of contacts and forces explain why suspensions yield, shear-thicken, or jam?
Network science has been applied to a wide range of disordered soft matter systems, including polymer networks^8,9, colloidal gels^10–15, granular systems¹⁶. When a material is represented as a network, the loss or emergence of rigidity can be framed as a phase transition: the appearance of a rigid spanning cluster can signal the transition, and the cluster’s size or fractal dimension can serve as an order parameter.

These ideas were initially explored in frictionless two-dimensional granular packings within the framework of isotropic jamming. Below the jamming point, there is no persistent contact network; at the transition, a contact graph emerges, and every edge belongs to a single rigid cluster that spans the system with a diverging correlation length⁴³. Friction changes this picture qualitatively. A frictional system can support one large rigid cluster (non-spanning) alongside a few fleeting, smaller ones, set in a sea of non-rigid bonds and particles without contacts⁴².

Colloidal gels offer a complementary view: rigidity emerges at strikingly low solid concentrations.^{10–15,44,45} This behavior is linked to the growth of fractal clusters and their percolation into a single connected network⁴⁵, with the modulus arising from hierarchical structure and cluster-level elasticity^12,13. Rigidity-based analyses show that attraction enhances correlations and lowers the rigidity-percolation threshold⁴⁶, and network tools can resolve sub-clusters within a single giant network, thereby tracing the gel’s elasticity to the resilience of that network⁴⁴.

Dense suspensions differ from these systems in one important respect: under steady shear, they reach a dynamic steady state in which frictional contacts continuously form, break, and reform. The relevant question is therefore not the structure of a fixed network, but how the topology and geometry of these networks govern rheology^47–54, as summarized in Fig. 2.

Over the past decade, the following picture has emerged. At low stress, below the thickening onset, the physics is governed by conservative, non-contact forces—repulsion or attraction, along with hydrodynamics. Frictional contacts are largely absent. As stress increases beyond the characteristic onset stress, frictional contacts are activated, and the FCN becomes tenuous and aligned with the compressive axis. Such a structure, supported only by sliding constraints, is difficult to sustain under steady shear.

One way to stabilize it is through the same orthogonal support observed in granular matter (as proposed by Cates et al.²⁹). As DST or shear jamming is approached by increasing either stress or particle concentration, frictional contacts also form along the tensile direction. These contacts stabilize the compressive force chains and, in doing so, they form closed loops. The appearance of loops in the FCN is therefore useful: it provides a structural signature of a network that has become capable of supporting stress, and it connects DST and SJ in suspensions to force-chain ideas from the dry granular literature^30,31,55.

Persistent homology quantifies these loops by asking not only whether they exist, but also how robust they are as the force threshold varies. In this measure, the persistence of loops in the force network is strongly correlated with viscosity and appears largely independent of solid concentration, applied stress, and even dimensionality⁴⁹. That last point is particularly striking and has motivated much of the recent two-dimensional work on the topology–geometry–rheology link; one likely reason is that, under simple shear, the dominant structure lies in the flow and gradient directions, with comparatively little variation along the vorticity axis. In the same spirit, DST has been shown to coincide with the emergence of third-order loops, i.e., triangles⁵⁰. The relative viscosity can then be expressed as a function of the number of such loops, and the correlation is independent of concentration, stress, and sliding-friction coefficient. This suggests that triangular motifs act as local building blocks of mechanically stable frictional networks^48,54 and brings graph theory and rigidity theory together: by Laman’s theorem, the triangle is the smallest rigid unit in a two-dimensional bar-and-joint framework—it does not deform under load⁵⁶.

Rigidity-based methods point to a similar conclusion from a different angle. The pebble game and related tools track mechanically stable regions in the FCN, provide information on their size distribution under shear,^48,57 and show that DST and SJ are accompanied by large, nearly system-spanning rigid structures. Independent network analysis shows a similar picture: the force-bearing structures in the DST regime are more strongly interconnected and mechanically supported than those in the CST regime⁵³. The emerging picture is that strong shear thickening is not simply about more contacts, but about contacts organized into structures that collectively resist deformation.

Three dimensions remain underexplored, but early results are consistent with this picture: shear thickening tracks the growth and eventual system-spanning connectivity of the FCN^47,51,54,58. The onset of DST appears to coincide with the emergence of contact structures capable of transmitting stresses over distances larger than a typical particle size within the suspension. These findings are broadly consistent with the picture developed throughout this article: rheology depends not only on the number of frictional contacts but also on how those contacts organize into larger stress-bearing structures. A recent caution is that much of this work relies only on sliding constraints. When rolling resistance is added, force chains can percolate along the compressive axis and stabilize without the need for tensile support, so suspensions with nearly identical viscosities can support markedly different network architecture⁵². The question of which features of the network are essential to mechanical stability and which are interchangeable remains open.

Conclusions and a word of caution

Interaction-network descriptions have provided a useful new window into rheological transitions across disordered soft matter---from polymer networks and colloidal gels to granular materials and, more recently, dense suspensions. In dense suspensions, these ideas have been particularly useful because they provide a mesoscale level of description: larger than individual particles yet smaller than the sample size. This mesoscale organization, encoded in frictional contacts and force networks, has been instrumental in understanding discontinuous shear thickening and shear jamming.

The past decade has shown that frictional contact networks are critical for strong shear thickening. A recurring lesson is that higher viscosity does not arise simply from more contacts. Rather, what matters is how those contacts are arranged into stress-bearing structures. Network-science tools have made this picture more quantitative: loops, rigidity, persistence, and community structure each, in its own way, show that the organization of contacts is closely tied to the suspension’s ability to resist deformation. Loop motifs, nearly system-spanning rigid clusters, and percolating subnetworks therefore provide complementary descriptions of the mechanical stability of contact networks. Together, these observations suggest that the macroscopic rheology of dense suspensions is shaped not only by microscopic particle interactions but also by the topology and rigidity of the mesoscale networks of these interactions.

Although most of the discussion here has focused on viscosity, the same mesoscale view may also help connect network structure to other rheological observables, especially normal stresses. Normal stresses reflect the anisotropy of the stress-bearing microstructure and are central to dense-suspension rheology. From a network perspective, this raises a useful question: which features of the contact or force network control shear resistance, and which control stress anisotropy, dilation, and normal-stress differences? Establishing such connections remains relatively open, but it could provide a more complete test of whether a proposed network measure captures the mechanics of the suspension rather than only correlating with viscosity.

At the same time, this emerging picture warrants caution. The link between frictional contact networks and suspension rheology is still developing, and it is too early to draw firm universal conclusions. Many current studies focus on idealized systems, often in two dimensions and with a single dominant interaction, such as frictional contact. Real-world suspensions are usually three-dimensional, polydisperse, and chemically complex. Their particles may interact through friction, adhesion, repulsion, attraction, hydrodynamics, or combinations of these mechanisms. In such systems, the relevant object may not be a single subnetwork but an embedded or multilayered one, in which different interaction types coexist and compete.

This matters because similar bulk rheology need not imply similar mesoscale organization. When additional constraints, such as rolling resistance, are included, suspensions with comparable viscosities can exhibit markedly different network architecture⁵². Thus, a network measure that tracks rheology in one system may not apply to another. To my mind, an important open question is which network features are universal, and which are system-specific.

Looking ahead, I see an important opportunity. Network science offers a framework for linking microscale, particle-level interactions to macroscopic flow via an intermediate mesoscale description. To make this perspective more broadly useful, new approaches will be needed to extend network-based analysis to three-dimensional, polydisperse, chemically complex, and industrially relevant suspensions. If successful, this could lead not only to a better description of suspension rheology but also to the ability to predict---and eventually design---the network structures that influence how these materials flow.

Acknowledgements

I am especially grateful to Bulbul Chakraborty, Karen Daniels, and Emanuela del Gado, who introduced me to network-science methods and generously shared their expertise. I thank Jeff Morris, Emanuela del Gado, Safa Jamali, Vivek Sharma, Sam Root, Benjamin Yavitt, Meera Ramaswamy, Shweta Sharma, and Ria Duggal for insightful comments and valuable discussions during the preparation of this article. I gratefully acknowledge financial support from Case Western Reserve University startup funds.

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Abhinendra (Abhi) Singh is an Assistant Professor of Macromolecular Science and Engineering at Case Western Reserve University, where he has been since 2022. He earned his PhD in Mechanical Engineering from the University of Twente (The Netherlands) and completed postdoctoral training at the Levich Institute at the City College of New York and the University of Chicago. His research group focuses on the physics and rheology of dense suspensions, granular materials, and complex fluids, using computational, theoretical, and data-driven tools. A major focus of his recent work is linking microscopic particle-scale interactions to macroscopic flow behavior through mesoscale frictional contact networks, using network science, topology, and machine-learning-based models. His group’s research is supported by early-career awards from the ACS Petroleum Research Fund Doctoral New Investigator program and the NSF CAREER program.