# Linking attractive interactions and confinement to the rheological response of suspended particles close to jamming

## Abstract

We study the response to simple shear start-up of an overdamped, athermal assembly of particles with tuneable attractive interactions. We focus on volume fractions close to the jamming point, where such systems can become disordered elastoplastic solids. By systematically varying the strength of the particle–particle attraction and the volume fraction, we demonstrate how cohesion and confinement individually contribute to the shear modulus and yield strain of the material. The results provide evidence for the influence of binding agents on the rheology of dense, athermal suspensions and describe a set of handles with which the macroscopic properties of such materials can be engineered.

### Keywords

Suspension Rheology Attraction Confinement## 1 Introduction

Yield-stress fluids have a vast range of uses in everyday life and in industry, from cement and ceramic pastes to shaving gel and mayonnaise [1, 2]. Their microscopic structure enables them to flow plastically if they are submitted to a stress above some critical value, otherwise they deform in a finite way, resembling an elastic, structurally disordered solid. This behaviour makes them ideal materials for a host of applications, and consequently they have long been the subject of extensive research [3, 4, 5, 6, 7]. There is an ongoing effort to further enhance the rheological properties of such materials, focussing on the specificity with which attractive interactions might be tailored to tune the yield stress behaviour. Recent experimental developments have made possible materials with particle–particle attractions that may be precisely tuned, and selectively ‘switched’ on and off using external stimuli [8, 9, 10]. This can provide an extra handle with which the macroscopic properties of dense suspensions can be controlled. Particles functionalised with single-stranded DNA, for example, can be designed to bind only with particles coated with the complementary strand, and if the surface functionalisation can be added in such a way as to make it mobile on the particle surface [11] then very complex and selective interactions can be realised. Predicting the stress response of such designer materials under shearing, with the goal of designing and creating the ideal material for given applications remains an exciting challenge to soft matter physicists and engineers.

At the same time, the emergence of yield-stress behaviour during the transportation and storage of nominally dry granular materials remains a related engineering challenge across manufacturing and processing industries [12, 13]. In such scenarios, attractive interactions may arise between particles as a result of electrostatic interactions or liquid bridging due to environmental moisture, for example. The material may then be described at a mesoscopic level as a very dense assembly of sticky particles, which are themselves plastically deformable [14]. Although various studies have explored the role of particle adhesion at this mesoscopic level with respect to specific industrial operations including powder processing and tablet formation, a thorough understanding of the influence of cohesion in dense grains under simple shear remains elusive.

Though seemingly distinct at an application level, the fundamental challenge of understanding yield stress behaviour in designer soft solids and cohesive grains shares many parallels, not least the need to reconcile the competing roles of attraction and confinement at high volume fractions. This challenge is reflected in the recent literature: suspensions of repulsive particles have been the subject of a considerable amount of research over recent years, but attention is increasingly turning to the role of attractive interactions in determining the rheology of such fluids [15, 16, 17, 18, 19]. Indeed, the phase diagram for attractive *dry* granular materials has been revealed recently by Refs. [6, 20], who identified adhesive close and loose packing volume fractions, analogous to their repulsive-particle counterparts. That jamming can be achieved at such low volume fractions naturally reflects the link between adhesive non-Brownian systems and colloidal gelation. While there is mounting evidence of the role that hydrodynamics can play in gelation [21], it remains unclear whether similar effects are relevant to saturated wet granular systems.

In the present article, discrete element simulations are used to model disordered assemblies of suspended particles very close to the point of marginal rigidity, which may be reached either through increased particle–particle connectivity or through increased confinement. These simulations account for differing particle–particle attraction strength and varying volume fractions, and we study the mechanical response to simple shear across the transition at which the system becomes solid. Distinguishing between solid and liquid phases in attractive systems is in general nontrivial, and a complete rheological characterisation requires a consideration of the complex moduli as functions of the frequency and amplitude of the applied strain. For simplicity, we define a flowing region in which the start-up stress is roughly linear in the strain rate, and a jammed region where the start-up stress is roughly linear in the strain magnitude. We do not, though, discuss in detail the non-Newtonian nature of each of these regimes.

Our systematic approach provides a guide for understanding the rheological consequences of emergent yield stresses during industrial processing, and tuning the yielding behaviour of soft materials more generally.

## 2 Methods

In each simulation we consider a three dimensional, periodic system of 5400 spheres, with the dimensions of the enclosing box being varied to achieve desired volume fractions \(\phi \). To avoid crystallisation, each system contains a 1:1 mixture of particles with diameters *d* and 1.4*d*. The particles are subjected to simple shear flow with strain \(\gamma \) at fixed rate \(\dot{\gamma }\), with the flow direction in *x* and the gradient in *y*. Operating at fixed shear rate allows us to prevent shear banding, which might otherwise arise in such attractive particle suspensions [22]. We compute the Newtonian dynamics for all particles at each timestep, with particles being subjected to hydrodynamic and contact forces.

*i*and

*j*with diameters \(d_i\) and \(d_j\)) with centre-to-centre vector \({\varvec{r}}_\alpha \), relative normal velocity (along \({\varvec{r}}_\alpha \)) \({\Delta }\mathbf {v}_\alpha \), separated by surface-to-surface gap \(h = |{\varvec{r}}_\alpha | -\frac{1}{2} (d_i + d_j)\), the hydrodynamic force \(\mathbf {F}^h_\alpha \) is computed according to

*d*is the diameter of the smaller particle. We choose 0.001

*d*as an asperity length scale at which the lubrication forces cease to diverge. Hydrodynamic forces are set to zero when \(h>0.05d\).

*i*are further subjected to a Stokes drag force \(\mathbf {F}^h_i = 3\pi \eta _{f} d_i (\mathbf {v}_i - y_i\dot{\gamma })\) due to their relative velocity with the background streaming flow. The contact force \(\mathbf {F}^c_\beta \) represents a nonhydrodynamic interaction experienced by two particles \(\beta \) when they come into contact with one another, which occurs when the distance between the particle centres is less than \(d_{\beta }\) (which is given by \(0.5(d_i+d_j)\)). For a particle pair with centre-to-centre vector \({\varvec{r}}_\beta \), we define the overlap as \(\delta = d_\beta -|{\varvec{r}}_\beta |\). A contact occurs wherever \(\delta >0\), and we quantify the average number of such contacts per particle as \(\langle Z \rangle \). The interaction force is then simply given by

*k*is the constant stiffness, which we set to 10,000 [units of force per unit distance]. The second term represents subtraction of a constant normal force with magnitude \(F_{A}\), which has the effect of introducing a simple, short ranged, isotropic attraction between the particles. Not wishing to tie ourselves to a specific form of attractive potential [23], we demonstrate that this simplistic form is sufficient to obtain the desired crossover from liquid-like to solid-like properties. This approach paves the way for more complex interactions to be modelled between the particles in ongoing works, for example frictional forces, elastoplastic potentials or anisotropic attractive potentials. Particle trajectories are evolved with time according to the above forces using a Velocity–Verlet scheme implemented in LAMMPS [24].

For the purposes of these simulations, the dimensional parameters and their units are particle diameter *d* [length], fluid viscosity \(\eta _{f}\) \([\hbox {mass}/(\hbox {length}\times \hbox {time})]\), particle stiffness \(\textit{k}\) [mass/time\(^{2}\)], particle and fluid density \(\rho \) [mass/length\(^3\)] and shear rate \(\dot{\gamma }\) [1/time]. We express \(F_{A}\) in units of *kd* throughout. Particle inertia is mitigated throughout by setting \(\rho \dot{\gamma }d^2/\eta _{f}\sim 10^{-2}\), and we set \(\dot{\gamma }d/\sqrt{k/\rho d}\sim 10^{-4}\).

*V*is the volume of the system, \(\mathbf {r}_\alpha \) and \(\mathbf {r}_\beta \) represent vectors pointing from centre-to-centre of hydrodynamically interacting and contacting particles respectively, and \(N^h\) and \(N^c\) represent the total number of hydrodynamically interacting and contacting pairs, respectively. From the stress tensor \(\varvec{\sigma }\) the component corresponding to shear stress, that in the \(\textit{xy}\) plane, is identified and referred to as \(\sigma \) for simplicity. The stress values quoted herein have dimensions of \(\textit{k/d}\). For the cases where a suspension viscosity is given, this was calculated by dividing the stress response \(\sigma \) at large values of shear strain (after the system has yielded and the stress has reached its large strain value) by the shear rate \(\dot{\gamma }\). The viscosities are quoted as relative viscosities, \((\sigma /\dot{\gamma })/\eta _{f} = \eta /\eta _{f}\).

## 3 Results

From these plots we can also identify the regions in which the behaviour of the system resembles that of a jammed solid, or is flowing. As discussed in the Introduction, we identify jammed states as those systems with an elastic response at small strain, i.e. that are linear in \(\sigma \) versus \(\gamma \), and flowing states as those that are more linear in the strain rate \(\dot{\gamma }\). The boundary between the two is most clearly seen in (c) and (f), where the stress response has been plotted on a logarithmic axis. In this way we can approximate the minimum volume fraction (in the non-attractive case) or attractive strength (at fixed \(\phi \)) required to obtain a solid-like response at low shear strain. In the absence of attraction, the volume fraction required, \(\phi _c\), is between 0.64 and 0.65, close to random close packing as expected, and for a volume fraction \(\phi =0.62\) an \({F}_{\text {A}}\) of between 0.0001 and 0.001 *kd* is needed. For a glass sphere with Young’s Modulus \(\sim 10^{10}\) Pa this represents the force required to deform the surface by \(\sim 10^{-6}\) of the diameter. Comparing (c) and (f) we note that the incorporation of an attractive force \({F}_{\text {A}} = 0.01\) between particles for a system of volume fraction \(\phi =0.62\), which would otherwise flow like a liquid, can result in a solid-like material which deforms with a shear modulus and yield stress comparable to that of a material with \(\phi =0.66\) for \({F}_{\text {A}}=0\), which would be well within the jammed regime.

For the attractive systems we further define a ‘repulsive’ contact number which gives the number of contacts that have \(\delta '>0\) for \(\delta ' =(d_\beta -F_A/k)-|{\varvec{r}}_\beta |\), Figure 2b (Inset). This quantity identifies those particles that are in the repulsive part of the overall interaction potential. For \(F_A<0.1\), we find analogous behaviour to the standard definition of \(\langle Z \rangle \), though with the numbers reduced. Interestingly, the systems identified as ‘jammed’ previously have, according to this new definition, contact numbers lower than the critical value of 6. This suggests that the system is stabilised not by repulsive interactions alone, but by a combination of repulsion and attraction. For \(F_A=0.1\), we observe anomalous behaviour whereby the repulsive contact number is lower than for more weakly attractive systems. This is due to steric effects. In this case, particles must overlap by 10% of their diameter to reach the repulsive region. While this is permissible for isolated contacts, once a sphere has, say, 4 contacts at 10% overlap, it is more difficult to place a fifth or sixth sphere around the same central particle with 10% overlap because the radius of the central particle is effectively reduced by 10%.

We take a snapshot of the initial contact topology at \(\gamma =0\) and count all those initial contacts that remain intact as shearing proceeds. To do this, we first construct a list of pairwise particle–particle contacts at time \(t=0\), by performing a search of particles with centre-to-centre separation less than \(d_\beta \). We then repeat this computation at all subsequent timesteps, each time counting all those contacts from \(t=0\) that are still present. The result is presented in Fig. 2c, d. We can see from (d) that a larger attractive force between the particles causes, as expected, a greater percentage of contacts to remain over the duration of the simulation. Similarly, from (c), a lower volume fraction of particles with the same \({F}_{\text {A}}\) results in a higher number of initial contacts persisting, as the particles are better able to stick together due to fewer collisions with other particles, and increases adherence to the background streaming flow. We define the critical strain \(\gamma _c\) at the point where the number of remaining contacts drops below 99.85%. This was found to be a convenient choice at which the predicted values of \(\gamma _c\) closely correspond to the positions where the stress-strain curves begin to deviate from linearity for the jammed systems.

*k*/

*d*in each case. They are, however, of different origin, with Fig. 3a representing solidification by percolation of

*repulsive*interactions and Fig. 3b representing solidification by percolation of

*attractive*interactions. The shear moduli clearly increase with volume fraction, which can be understood in terms of the increasing particle density causing the particles to be forced into one another to a greater extent - resulting in a higher degree of stress for a given applied strain value. Similarly with increasing attractive force (the data shown is for a volume fraction \({\phi }=0.620\)), the particles are pulled towards each other more tightly and hence produce a larger elastic response upon shearing. In (c) and (d) it is clear that viscosity increases with volume fraction, which makes sense since the increasing particle density means that particles experience a greater resistance on average to any movement through the background liquid—with both contact and hydrodynamic contributions to this resistance. A larger \({F}_{\text {A}}\) will also increase the viscosity as the greater particle–particle attraction acts to impede motion through the system. Figure 3e shows the phase diagram for the results investigated here, and it illustrates how we can either move vertically and increase \({F}_{\text {A}}\), or move horizontally—increasing \({\phi }\)—to achieve solid-like behaviour at small shear strain.

## 4 Conclusion

We present here the findings of a study into how isotropic attractions between suspended particles and precisely controlled volume fractions can give extra handles through which one may control the macroscopic flow properties of densely packed athermal suspensions. This is relevant both to the design of future yield stress materials with industrial and consumer product applications, but also to understanding the challenging processing and transportation of attractive, dense systems such as powers, pastes and wet grains. A liquid-like suspension with a volume fraction marginally below jamming can be made to deform in a solid-like way by means of a simple (and rather weak) attractive force acting between the particles. Similarly if the attractions between particles in a suspension can be suppressed, for example by specific tuning of interactions or by dehumidification of processing atmosphere, the material will behave as if its density has been decreased, allowing for a more uninhibited flow of material. These findings pave the way for more material-specific simulations involving, perhaps, tuneable, surface-mobile and site-specific interactions between particles which will further enhance our ability to predict and control the behaviour of soft matter systems.

## Notes

### Acknowledgements

MJ carried out the research and wrote the manuscript as part of the EPSRC CDT in Nanoscience and Nanotechnology, supported by ESPRC Grant EP/L015978/1; CN is supported by the Maudslay-Butler Research Fellowship at Pembroke College, Cambridge.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest and that the research described in this manuscript complies with the Ethical Standards of the Journal. We had useful discussions with Alessio Caciagli and Erika Eiser.

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