## Abstract

Active networks are omnipresent in nature, from the molecular to the macro-scale. In this study, we explore the mechanical behaviour of fire ant aggregations, closely knit swarms that display impressive dynamics culminating with the aggregations’ capacity to self-heal and adapt to the environment. Although the combined elasticity and rheology of the ant aggregation can be characterized by phenomenological mechanical models (e.g. linear Maxwell or Kelvin–Voigt model), it is not clear how the behaviour of individual ants affects the aggregations’ emerging responses. Here, we explore an alternative way to think about these materials, describing them as a collection of individuals connected via elastic chains that associate and dissociate over time. Using our knowledge of these connections—e.g. their elasticity and attachment/dissociation rates—we construct a statistical description of connection stretch and derive an evolution equation for the corresponding stretch distribution. This time-evolving stretch distribution is then used to determine important macroscopic measures, e.g. stress, energy storage and energy dissipation, in the network. In this context, we show how the physical characteristics and activities of individual ants can explain the elasticity, flow and shear thinning of the aggregation. In particular, we find that experimental results are matched if the detachment rate between two individuals increases with tension in the connection.

## 1. Introduction

For the past few decades, biological assemblies of active particles (or swarms) have captured the attention of biologists, physicists and engineers for their capacity to display intelligent dynamical behaviour despite the lack of a centralized organization. Classical examples include fluid-like swarms such as schools of fish or flocks of birds whose collective behaviours have been well characterized by experimentalists and theoreticians alike [1–6]. However, surprisingly little attention has been given to solid-like aggregations such as those found in slugs of slime mould cells [7], cancer cell spheroids [8] or dynamic aggregations of fire ants [9]. In contrast to the fluid-like swarms, these assemblies are characterized by the presence of physical connections between individuals, which endow them with solid-like properties. The connections are usually highly dynamic, allowing the swarm to not only reorganize and self-heal over time but also display a fine balance of viscous and elastic response to external forces [10]. These intriguing properties have inspired material scientists and roboticists to produce active synthetic particles [11,12] and aggregations mimicking the response of their biological counterparts [13,14] for applications in active and intelligent materials.

The peculiar properties of solid aggregations have recently been exemplified by studies on fire ants that are capable of building dynamic structures such as rafts, bridges or towers [15,16], as illustrated in figure 1*a*,*b*. Such aggregations can be seen as dynamic networks wherein ants (the nodes) form transient connections by linking their tarsi together via mechanical claws or adhesion via sticky pads located at the end of each tarsus [9]. These networks possess attributes of a soft solid at the time scale of seconds and those of a fluid (with viscosity comparable to high-viscosity silicon oil (approx. 100 Pa · s) [9]) at the time scale of minutes. In order to characterize the viscous fluid-like behaviour of the aggregation, Tennenbaum *et al.* [10] performed controlled shear rate experiments or creep tests, where the stress on the aggregation is measured for different rates of applied shear strain rates. Interestingly, it is shown that the viscosity of live aggregations decreases with shear rate (or shear force), a behaviour that classifies them as a shear-thinning fluid. Furthermore, observations from linear oscillatory rheology suggest that the visco-elastic behaviour of aggregations is also sensitive to the activity and the density of the aggregation [17]. As we accumulate more data on these phenomena, a general understanding of the causal relationship between individual and swarm response can be constructed. In this context, a number of studies on collective behaviour [8] suggest that complex emerging behaviour often results from seemingly simple rules at the local level. In social insects, for instance, it was shown that the presence of feedback interactions between two individuals may lead to swarm intelligence by either weakening or strengthening a phenomenon at length scales much greater than that of individuals. Such dynamics are found, for instance, in the formation of trails by foraging ants [18], the collective migration of locusts [19] and the aggregation of roaches [20]. Understanding the local interactions between individuals and their role on the swarm behaviour is a daunting task that can hardly be achieved by experimentation alone. For this reason, the use of mathematical models may become extremely useful in exploring and subsequently identifying arrays of potential rules and their impact on the bulk behaviour. For instance, simple but powerful models such as the Vicsek model [21,22] and others based on the self-propelled particle method [23,24] were instrumental in identifying how different types of interactions and conditions may affect the flow of active particles [25,26]. These approaches could further authenticate the sensitivity of these nonlinear systems, i.e. slight variations in particle interactions can yield drastic changes in the swarm behaviour [27,28]. The vast majority of existing models are for active fluids and the study of solid aggregations is still in its infancy. There are indeed drastic differences between a fluid-like swarm and a solid aggregation, mainly due to the nature of the interactions (i.e. distant versus physical contact interactions) and the behaviour (i.e. fluid versus solid). Thus, in contrast with the flow behaviour of fluid swarms, the response of an aggregation is expressed in terms of stress, strain, strain rate, viscosity and elasticity. A research effort is therefore necessary to derive a mathematical theory that can explain and conceptualize the wealth of experimental data gathered on solid aggregations.

In this work, we employ statistical mechanics to derive a quantitative understanding of solid ant aggregation mechanics in terms of the elementary interactions between individuals. For this, we view an ant aggregation as an elastic network whose intra-connectivity may change over time due to the dynamic detachment and reattachment (turnover) of neighbouring ants. The viscoelasticity of these transient networks may be related to both the elasticity of connections (legs) and their turnover rate via a statistical description [29,30]. Using this approach, we aim to comprehend the origin of the complex viscoelastic response of live fire ant aggregations—specifically, their shear-thinning behaviour and the solid–fluid transitions they undergo at different rates of deformation. We further aim to better understand the local rules leading to these complex emerging behaviours. The study is organized as follows: we start by discussing the physics of fire ant aggregations and introduce a statistical mechanics framework that can be used to bridge the local and global mechanical response. Then—using this framework—we explore the rheological response of live aggregations concentrating on their visco-elasticity under creep and oscillatory shear conditions. Finally, this allows us to formulate a new hypothesis involving local feedback mechanisms that can explain the adaptation of the swarm to external forces and deformations.

## 2. Statistical mechanics of active ant aggregations

In live fire ant aggregations, individuals form voluntary connections using their legs to attach to the legs, antennae, bodies and even mandibles of other ants as illustrated in figure 1*d*. In doing so, they maintain, on an average, some effective separation distance from one another that gives rise to a corresponding preferential density of the overall aggregate. We define the preferred density of the agggregation as equivalent to an effective volume fraction of ants that is above the random packing fraction but below a jamming transition as defined in [10]. We further suppose that, at this density under no external forces, the aggregation is approximately stress free. Additionally, the connections or cross-links between ants are transient, reversible bonds that may dissociate and reform under lower stress configurations in the event of deformation, or residual internal stresses. This has been the prevailing explanation for fire ant aggregations’ self-healing, transient properties at longer time scales. This may also explain why ant aggregations behave like elastic solids at short time scales up to 10^{2} s order of magnitude: since it takes time for ants to break and reform active connections with their neighbours under stress, they act like a highly cross-linked material under fast shear rates or at short time scales. In any case, from a mechanical viewpoint, a live aggregation may be described as a network linked via elastic chains, i.e. legs, whose connectivity is dynamic (figure 1*e*). In this work, we therefore describe these insect aggregations as networks of nodes (the insects’ body) and cross-links (the insects’ legs) that are connected at their ends by physical attachments. When the insects are not under large compressive forces, these attachments are transient and lead to a reorganization of the network when subjected to mechanical forces. Our goal is to develop a general theory, based on statistical mechanics, to better understand how the emerging responses of such aggregations are related to the elementary behaviour of individuals.

### 2.1. Statistical description

Insect aggregations are constituted of a large number of individuals, and it is often easier to represent their physical state in terms of macroscopic or average variables. For this reason, let us focus on a small representative volume of the swarm (hundreds of individuals) and explore its mechanical response when subjected to external forces. In this context, we first note that the elastic response of this volume depends on the deformation and elasticity of just the connected legs of each individual insect. In other words, a fairly accurate knowledge of the aggregation’s mechanical state can be gained by knowing the configuration, or end-to-end vector , of all active (or attached) legs in the network (figure 2*a*). Owing to the potentially large number of legs, it is preferable to express this knowledge in terms of a statistical distribution that defines the density of connected legs whose end-to-end vector exists between and in the elementary reference volume (figure 2*b*). For convenience, this distribution can be decomposed into (i) the density *c*(*t*) of attached connections, which is typically lower than the total concentration *C* of potential connections, and (ii) the probability density function that indicates the likelihood of finding an active connection whose end-to-end vector exists between and . Realizing that this function can be interpreted as the normalized distribution function, one can relate it to *ϕ* and the concentration *c* by
2.1where integrated over all chain configurations. From this knowledge, one can further determine the density *c*_{d}(*t*) = *C* − *c*(*t*) of potential connections that are detached, i.e. those that do not contribute to the overall mechanics of the network. Assuming here that the orientation of ant legs in the aggregation does not have any preferred directions before deformation, i.e. isotropic, the probability density of the end-to-end vector follows the well-known Gaussian or normal distribution with a zero mean. The end-to-end distance, however, follows a Maxwell–Boltzmann distribution, *p*(*r*), as schematically shown by figure 2*a*, around a mean value *R* corresponding to the distance between the coxae (figure 1*c*) of two connected ants. Assuming that the mean distance *R* is much larger than the end-to-end distance of a force-free leg in an unconnected ant, *r*_{0} (figure 2*a*), the normal distribution *f*_{0} for the vector can be written in terms of *R* as
2.2where the subscript 0 designates the distribution in its pre-deformed state. The corresponding distribution for the magnitude together with its two-dimensional representation are shown in figure 2*a*,*b*, respectively.

### 2.2. Elastic energy and forces

Based on this statistical description, it is now possible to assess the deformation energy stored in the network. For this, we make the assumption that leg connections occur in the tarsus region located at the legs’ extremities as shown in figure 1*d*. This means that all connections can be treated as equal in terms of their average length and elasticity. As a consequence, when a tensile (or compressive) force is applied along the direction of the connection, the participating legs are stretched (or compressed) and the relationship between the force and the end-to-end distance can be expressed in terms of an elastic potential . Assuming a harmonic potential similar to that of polymer chains in a network, the stored elastic energy in a leg can be expressed in terms of the stiffness *K* as
2.3In obtaining the above equation, it is assumed that the minimum energy or force-free state of the leg occurs at a negligible end-to-end distance. In other words, the natural leg configuration of an ant is bent such that the end-to-end distance between the coxa and the tip of the leg is small (figure 2*a*). Stretching a leg, therefore, comes at the cost of an axial force that is expressed in terms of the potential function as . If the connected leg distribution *ϕ* is known, it is possible to estimate the elastic energy per volume in the network by integrating over all configurations. Doing so, the energy density function *Ψ* per nominal volume can be estimated by the integral . Since the energy does not necessarily vanish in the aggregation’s relaxed state, the energy potential can be redefined as the difference between the elastic energy in the current and relaxed state as follows:
2.4where *ϕ*_{0} = *c f*_{0} may be thought of as the distribution of the end-to-end distance when the aggregation is in its relaxed (or nominal) state. This definition further enables the computation of the current (or Cauchy) stress tensor in the aggregation [29]
2.5where the symbol ⊗ denotes the dyadic product.

### 2.3. Reorganization and dissipation in the aggregation

It is clear from (2.4) and (2.5) that the evaluation of stored elastic energy and stress in the aggregation requires knowledge of the distribution of active connections. This means that one needs to determine how the deformation of chains evolves in time as a function of macroscopic forces (or deformation). Based on the fact that leg connections are dynamic entities periodically detaching and reattaching to their neighbours, we have shown in previous work [29] that the evolution of the distribution *ϕ* is given by a continuity equation that conserves the number of attached connections in the chain space. This may be understood as an interplay between three physical processes: (i) change in chain stretch, (ii) attachment of new chains, and (iii) detachment of active connections. The first physical process is the distortion of chains at a rate denoted by that results from macroscopic deformation of the network. The second physical process is the attachment of new chains to the network with association rate *k*_{a} occurring in nearly a force-free, i.e. unstressed, state. These new connections are nearly unstressed because they are inactive just prior to the association event. Accordingly, it can be assumed that chains attach in a random configuration that follows the stress-free probability density function *f*_{0} defined in (2.2). The third physical process is the detachment of active connections in their stretched configuration with dissociation rate *k*_{d}. In summary, the evolution equation for consists of the three physical processes described above in the form a flux term (chain deformations), a source term (new attachments) and a sink term (detachments), taking the form of a differential equation [29]
2.6where is the divergence operator in the chain space, *Ω*_{r}. Under the assumption of affine deformations [31], the chains’ stretch rate is related to the macroscopic velocity gradient on the network, , as (for more details on this relationship, the reader is referred to [29]). The macroscopic velocity gradient can in turn be obtained from the macroscopic velocity field as , where is the gradient operator in the material space of the network. Equation (2.6) can be solved for the leg connection distribution if subjected to initial condition and boundary conditions *ϕ* → 0 as *r* → ∞. We also note that (2.6) implicitly depends on the concentration of active connections , making it an integro-differential equation that is better solved numerically.

A few important observations can be made based on the analysis of equation (2.6). First, when leg turnover vanishes (i.e. *k*_{a} = *k*_{d} = 0) only the left-hand side remains, and the aggregation becomes purely elastic. The flux term induces stretch of the distribution (figure 3), an operation that may be interpreted as the elastic stretch of active connections with imposed deformation, . As there is no energy dissipation in this case due to leg detachments, the deformation of the aggregation is stored as elastic energy that is obtained from equation (2.4). Second, when the terms *k*_{d} and *k*_{a} are activated, active connections in their stretched state may detach at a frequency *k*_{d}, which induces stress relaxation in the aggregation. Furthermore, since reattachment occurs in a nearly stress-free state, we observe a net loss of mechanical energy that can be quantified by the rate of dissipation per unit volume in the form [29]
2.7It can be seen that dissipation increases at a rate associated with that of leg detachment *k*_{d} and is proportional to the elastic energy stored in the legs *ψ*. On a final note, if the detachment rate *k*_{d} is independent of leg stretch λ, equation (2.6) is decoupled from the mechanical response of a single leg. This means that the rate at which the network relaxes becomes independent of its level of deformation. By contrast, when *k*_{d} becomes a function of stretch, the system becomes coupled with deformation and we may see, for instance, an acceleration of leg detachment with stress. We will see later in this study that this mechanism may occur in ant aggregations and could be responsible for their complex rheological response.

## 3. Rheology and adaptation of fire ant aggregations

Fire ant aggregations are known to display an atypical elasto-rheological response that may originate from the transient and force-sensitive connections occurring between individuals. Using the above statistical mechanics framework, we aim to gain a deeper understanding of the individual-to-aggregation relationship in fire ant populations.

### 3.1. Ant aggregations are at the limit between solids and fluids

Using the above statistical description, we first aim to illustrate the characteristic elasto-rheological response of fire ant aggregations and more precisely their elasticity, stress relaxation and flow under different conditions. For the sake of simplicity, we first concentrate on the case where *k*_{d} and *k*_{a} are independent of force.

#### 3.1.1. Elastic response and stress relaxation

As shown in figure 4*a*, ant aggregations can be temporarily assembled into a mound that can subsequently be used to measure the elastic response of the network under compression. In this context, experiments have shown that, when subjected to a fast, unconfined loading–unloading cycle, the aggregation quickly recovers most of its original shape after unloading, a response that is characteristic of elasticity. To simulate these conditions using the statistical approach, we used finite elements to reproduce the mound-like shape of the aggregation and its deformation under force. We modelled the compression of the mound between two plates as a change in interstitial space *δ* at a rate of . During deformation, the velocity of each point within the material space, , of the aggregation was determined and used to compute the local velocity gradient . Subsequently, the change in leg distribution (using (2.6)) and stress (using (2.5)) was computed at each material point in the aggregation. The velocities at the next time increment were then determined by ensuring the equilibrium of material points via the standard equation . For additional details on the numerical approach, readers are referred to [32,33]. Figure 4 shows the deformation of the aggregation and stretch of active connections as a function of time for the quick load and release performed experimentally. The emergent response of the aggregation is here characterized by its vertical reaction force *F* while the local ant mechanics are represented by the value and direction of the maximal stretch λ experienced by active connections. These quantities can be extracted from the distribution *ϕ* as shown in figure 4*c*,*d*. We clearly see here that, for vertical compression, active connections are mostly stretched in the centre of the aggregations in a horizontal direction. Generally, our simulations suggest that, when the loading rate is faster than the detachment rate *k*_{d}, ants do not have time to reorganize their connections and the aggregations’ deformation is predominantly elastic. A small amount of permanent deformation may still be observed during experiments and simulations as the loading rate is finite compared with the relaxation rate of the network (figure 4).

To test whether this observation is correct, we explored a different scenario in which we applied a quick compression step to a constant displacement *δ* to look for a stress relaxation response (figure 5). In this case, we see that under a constant displacement, the stretch in active connections decreases over time until they completely relax and return to a stretch-free state. This is the consequence of the detachment of stressed legs and their subsequent reattachment into a relaxed configuration. These local reorganization events eventually lead to a global stress relaxation as observed in figure 5*c*. We note here that the characteristic relaxation time *t*_{r} = 1/*k*_{d} is directly related to the detachment rate. Overall these results show that, depending on the normalized loading rate *W* = *L*/*k*_{d} (where *L* is the norm of the velocity gradient), the aggregation response ranges from a solid to a fluid behaviour. The coexistence of these modes occurs when the strain rate exists at values near or slightly below the natural attachment/detachment frequency *k*_{d} = *k*_{a} of the ants.

#### 3.1.2. The role of leg detachment on aggregation rheology

Here we use a similar finite-element model to investigate the pure rheology of the aggregation. For this, we simulate the motion of a heavy spherical ball placed on top of an aggregation held in a cylindrical container. Experimentally, the ball is observed to sink through the aggregation in a manner that is reminiscent of what would be observed if the ball was placed in a viscous fluid (figure 6*a*). Theoretically, the ball applies a force (its weight) to the swarm, which results in locally stretching the active connections as shown in figure 6*b* (where the maximum stretch and its directions are represented by a colour map and stream lines). As ants detach and reattach to nearby neighbours in more favourable states, we observe a slow creep of the aggregation, which is reflected by a downward sinking motion of the ball. We see that for constant values of attachment/detachment events the ball eventually reaches a steady-state velocity *v* that is proportional to *k*_{d} and the size of the ball. We note here that, when *k*_{d} = 0, the aggregation becomes purely elastic and the ball remains at equilibrium on top of the aggregation. One must bear in mind that this simulation represents only the most basic features of a dynamic aggregation; other physical mechanisms such as the affinity between ants and the surface of the ball or the change of attachment/detachment rates as a function of force (or stretch) have been neglected. We will see in the next section that the latter needs to be revisited in order to describe the nonlinear rheology of fire ant aggregations.

### 3.2. Aggregation rheology and force-dependent dynamics

To obtain a more quantitative understanding of the aggregation rheology, let us now consider a standard creep test on the aggregation (as illustrated in figure 7*a*), similar to that conducted in a previous study on fire ants [10]. During the experiment, the aggregation is subjected to a constant shear strain rate in a rheometer and the stress *τ* is measured until a steady state is reached. In this situation, the velocity gradient becomes a tensor whose components are shown in figure 7*a*. Using this gradient in (2.6) and (2.5), one can determine the steady-state leg connection distribution *ϕ* and corresponding shear stress *τ* for different strain rates. Figure 7*c* thus represents the distribution with different Weissenberg numbers (10^{−4}, 10^{−2}, 10^{−1} and 0.4) while figure 7*b* shows the shear viscosity with respect to *W*. Interestingly, the theory indicates that while the normal stress ramps up nonlinearly at high values of *W*, the shear viscosity remains constant and is equal to *η* = *cKR*^{2}/*k*_{d} for any shear rate as long as the springs are linearly elastic [34]. This phenomenon can be explained as follows: as the aggregation undergoes shear, the legs of individuals get aligned and stretched in the direction of shear. The average leg stretch at low shear rates (*W* ≪ 1) remains fairly low as the ant connections can relax faster than they are stretched. However, as the shear rate approaches the turnover rate, the relaxation mechanism is unable to keep up with deformation and connections become highly stretched within the aggregation (figure 7*c*). As the legs have a constant turnover rate, the resistance to shear flow, i.e. viscosity, remains constant regardless of the magnitude of shear rate. However, the high leg stretch at high shear rates manifests in the normal stress creating a non-Newtonian effect. While not accounted for in the current model, it is clear that, as *W* approaches higher values, excessive stretch would yield leg rupture and overall damage to the aggregation [35]. Yet, experimental results show a very different story from these predictions and, in particular, indicate that fire ant aggregations exhibit a decrease in viscosity with shear rate (figure 8*c*). This shear-thinning behaviour indicates that the detachment rate *k*_{d} is not constant, and is likely to be a function of force as discussed next.

Previous studies have shown that ants are able to sense forces through their active connections [8]. To minimize damage, it is therefore likely that the insects increase their rate of detachment when subjected to excessive forces. Here, we propose to pursue this hypothesis and explore whether it could be used to explain the aggregation’s shear-thinning response. To build this hypothesis into the statistical model, active connections may be viewed as slip bonds [36,37], a type of physical bond that decreases its lifetime with force. The theory behind bond stability can be built from Kramer’s transition state theory [38], which describes how the energy barrier between a bond in its attached and detached states decreases with applied force (figure 8*a*). This argument has been used to derive a quadratic relationship between the natural rate of leg detachment and the force across the connection as [39]
3.1where is the detachment rate at zero force and ζ_{0} is the force at which *k*_{d} increases by a factor of 2. A lower value of ζ_{0} therefore indicates a higher sensitivity to force and results in a more abrupt detachment rate distribution with increased force.

Recalling that ζ = *Kr* and using the fact that the body weight of an ant is *w* ≈ 1.5 dynes [9], we now explore the outcome of force sensitivity on the aggregation’s response to an imposed shear rate. Figure 8*c* shows the shear rate–viscosity relation, i.e. *η* versus *W*, for three aggregations characterized by (i) a detachment rate that is insensitive to force (, ζ_{0}/*w* → ∞), (ii) a detachment rate that is weakly dependent on force (, ζ_{0}/*w* = 20), and (iii) a detachment rate that is strongly dependent on force (, ζ_{0}/*w* = 8). The distributions corresponding to the strongly force-dependent case are shown for specific strain rates (figure 8*d*). These results indicate that the force dependence of the leg detachment rate has a significant effect on the aggregation’s viscosity (figure 8*c*): a decrease in ζ_{0} yields an increasingly pronounced shear-thinning response of the aggregation. This trend may be explained as follows: for slow loading the leg stretch remains small (see distributions in figure 8*d*1,2) and the connections’ detachment sensitivities (i.e. ζ_{0}/*w*) to force do not impact the aggregation’s viscosity. However, as the strain rate (i.e. *W*) increases, legs become stretched (figure 8*d*2–4) and the connections start detaching faster. This phenomenon induces a reduction in both the connection density and the maximum stretch experienced by active connections (figure 8*d*). The global effect is a viscosity reduction with respect to increasing shear rate, i.e. shear thinning. These model predictions agree very well with experimental measurements presented by Tennenbaum *et al.* [10]. In figure 8, we show that the model matches experimental results for leg stiffness *K*/*w* = 19 mm^{−1}; and kinetic properties and ζ_{0}/*w* = 8. Furthermore, we obtained from the dissipation energy per unit volume and strain rate , the steady value of stress as approximately 73 kPa, which is consistent with that measured experimentally. This means that, on average, an ant in a stress-free environment detaches/attaches its leg every 0.7 s. When subjected to mechanical load, however, these kinetics increase such that *k*_{d} rises by a factor of 2 when the force on a single leg reaches eight times an ant’s body weight. This response to load may be interpreted as a feedback mechanism in which ants can sense forces and naturally adapt their detachment frequency as a protective mechanism. We have seen in figure 7 that, without this mechanism, at large strain rates connections can be excessively stretched, which would—in reality—result in damage. To further test the validity of our predictions, let us now turn to the aggregation’s response to oscillatory shear deformation.

### 3.3. Effect of force-dependent detachment rate on oscillatory rheology

We now ask whether the force-dependence hypothesis can explain the way by which a fire ant aggregation stores and dissipates energy in different dynamic regimes. For this, we simulate how the aggregation reacts when it is placed in a rheometer and subjected to oscillatory shear strains of different frequencies. We assume that the shear strain is harmonic in the form *γ*(*t*) = *γ*_{0}sin(*ω**t*), where *γ*_{0} is the strain amplitude and *ω* is the time frequency. In the linear regime, the corresponding shear stress *τ* typically follows a similar harmonic form with a delay *δ* such that *τ*(*t*) = *τ*_{0}sin(*ω**t* + *δ*). This delay indicates a loss in energy, such that we can introduce a storage modulus *G*′ = *τ*_{0}/*γ*_{0}cos(*δ*) characterizing the aggregation’s elasticity and a loss modulus *G*″ = (*τ*_{0}/*γ*_{0})sin(*δ*) characterizing its viscous behaviour. In order to focus on the linear stress–strain regime, we choose a small strain amplitude *γ*_{0} = 0.01 and explore how the force dependence, expressed by ζ_{0} in equation (3.1), affects the moduli for different frequencies.

As shown in figure 9, a shear strain was imposed in equation (2.6) to determine the connection distribution *ϕ* (figure 7*c*) at steady state. The resulting shear stress *τ* = σ_{12} was then recovered from equation (2.5). For a given frequency, the typical, normalized stress and strain time profiles are given in figure 9*a*. Performing these simulations for a wide spectrum of frequencies, ranging from 0.1 to 100 Hz, the changes in storage and loss moduli with frequency were determined. Figure 9*b* shows how the functions *G*′(*ω*) and *G*″(*ω*) are affected by the force dependence of the detachment rate. As expected, we observe that, when the detachment rate is force independent (ζ_{0} → ∞), the aggregation behaves like a visco-elastic material with a single relaxation time, where the cross-over between the two moduli takes place at the natural frequency *k*_{d}. At lower frequencies, *ω* ≪ *k*_{d}, the aggregation reacts like a viscous fluid, while at higher frequencies, *ω* ≫ *k*_{d}, its behaviour is that of an elastic solid. Strikingly, as the force dependence (ζ_{0}/*w*) decreases, this cross-over is pushed over to higher frequencies. In the range of frequencies previously studied experimentally [10], we remain in a regime where *G*′ is roughly equal to *G*″, i.e. the aggregation is equally storing and dissipating energy for all frequencies. To explain this observation, the authors in [10] speculated that there exists a wide spectrum of relaxation times in the aggregation. Although this variance in relaxation time may be attributed to the existence of different types of mechanisms within the network (e.g. body motion of an ant in the aggregation, collective rearrangements in time), we find here that this may also be a reflection of the force dependence of the detachment rate between two neighbouring ants. This result may be understood as follows: when the aggregation is subjected to low frequencies (lower than *k*_{d}), connections’ detachments occur faster than the loading rate, which allows for force relaxation in the legs. However, at high frequencies (higher than *k*_{d}), the leg detachments occur at an insufficient rate to allow for relaxation; thus, significant tensile forces are induced in the connections, which in turn accelerate the detachment rate. This mechanism has the effect of increasing the overall detachment rate as frequencies increase, thereby shifting the cross-over regions in figure 9*b* to the right. This result is consistent with the linear oscillatory rheology predictions for associative polymers in previous work using transient network theory [39,40]. Interestingly, we obtain a best fit of experimental results when the force sensitivity corresponds to ζ_{0} = 8*w*, as determined for the creep test. This observation, together with the prediction of shear thinning, therefore suggests that a force-dependent detachment rate with ζ_{0} = 8*w* governs the aggregation’s rheology.

On a final note, figure 9*c* shows that the accuracy of the model’s predictions tends to decrease at low frequencies with the model resembling a Maxwell fluid while experiments resemble the rheology of polymers around the gel point (power law with exponent ). In this regime, the model overestimates the loss modulus and underestimates the storage modulus. This disagreement may arise from several sources, such as the quadratic dependency of *k*_{d} on force, the assumption that all connections can be treated equally or that the detachment rate does not depend on the rate of stretch (which may induce a dependency on frequency for oscillatory deformations). While such features can be added to a model to better match experiments, one may think of them as a motivation for additional experiments that can assess the mechanical behaviour and feedback mechanisms at the scale of a single ant and their roles on network dynamics.

## 4. Summary and concluding remarks

To summarize, we have introduced a statistical mechanics approach to explore the relationship between the local rules governing interactions between fire ants and the macroscopic mechanical response of the resulting aggregation. The methodology rests on three pillars: (i) a statistical description of the aggregation in terms of the connections’ (or legs’) end-to-end vectors; (ii) a Boltzmann-type equation that describes the evolution of the end-to-end distribution due to macroscopic deformation and the turnover of connections over time, and (iii) a relationship between the end-to-end distribution and the macroscopic stress, stored elastic energy and energy dissipation in the aggregation. Our analysis points out that ant aggregations are an excellent example of an active and reconfigurable material that can readjust itself as a function of environmental conditions. In particular, the viscoelastic properties of an aggregation of fire ants may be traced back to the collective behaviour of individual ants following simple, local rules as follows: active connections are characterized by a rate of detachment that increases with force. We hypothesize that these feedback mechanisms (force sensing and reaction) are essential to protect ants from sustaining too much stress in their attachments, thereby mitigating the risk of irreversible damage. As a result, the aggregation displays a shear-thinning behaviour within a select range of shear rates.

We note that this study only represents a first attempt in understanding the behaviour of insect aggregations from a mechanics stand-point. It is limited to describing the behaviour of an aggregation of fire ants under shear loading and does not delve into the active (or voluntary) deformations observed in aggregations that enable them to build and modify structures such as towers, rafts and bridges. Furthermore, we note that the statistical approach taken here assumes a continuum description of individuals and so it is appropriate to describe the network dynamics that involve a large ant population. When applied to the behaviour of small populations, such as the escape droplets formed by some Argentine ants or localized phenomena such as transport of small objects in large populations, where the relevant length scale is comparable to that of a few ants, the current approach is likely to be inaccurate [41]. Discrete methods such as the Monte Carlo method may be required in such cases to decode the link between local and global responses. Ants are also known to display a time-dependent response to stimuli that may manifest via sudden and local events that can propagate through the aggregation [17]. Such phenomena are still poorly understood; however, the statistical framework presented here will hopefully provide a useful route to understanding them in a comprehensive, local-to-global context. This approach may also extend beyond the scope of insect aggregations since it is applicable to a variety of dynamic networks found in nature and engineered systems. Examples of such networks include acto-myosin networks in cells [42]; dynamic polymers (containing reversible cross-links); bioresorbable hydrogels [43,44]; and a large array of plant and animal tissues.

## Data accessibility

All data from experiment and model are presented in the paper.

## Authors' contributions

F.J.V. conceived and designed the study; and wrote the initial manuscript. F.J.V. and S.L.S. contributed to the derivation of the theoretical formulation. T.S. carried out all *in silico* studies. F.J.V., S.L.S., T.S. and R.J.W. contributed to the interpretation of simulation data. All authors contributed to, read, and corrected earlier versions of the manuscript, as well as approved the final version.

## Competing interests

We declare we have no competing interests.

## Funding

The authors acknowledge the support of the National Science Foundation under the NSF CAREER award 1350090.

- Received August 21, 2018.
- Accepted October 8, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.