University of Amsterdam
MSc in Physics
Theoretical physics
Master Thesis
credited 60 ECTS , conducted in the period nov 2013 to aug 2014
Nonlinear elastic behavior of branched networks
Rapid strain stiffening as a critical phenomenon
by Robbie Rens 6089879 Supervisors: A. Sharma F. C. MacKintosh Examiners: F. C. MacKintosh P. Schall
Vrije Universiteit LaserLab Amsterdam Theoretical Physics of Complex Systems
Abstract
Naturally occurring biopolymers such as collagen and actin form branched elastic networks. Such networks are found in the cytoskeleton of animal cells and are the main component of the connective tissue. It is known from experiments that these networks exhibit non trivial stiffening behavior that is not encountered in synthetic gels. For any small deformation the network acts as a soft material and complies with the applied deformation. However, there exists a finite amount of deformation beyond which the network becomes highly rigid. The stiffness rapidly increases by orders of magnitude. It is this intriguing feature that has been investigated in this master project. In this research a minimalistic model is used to model these biopolymer networks for various network architectures. The mechanical response of the individual filaments is assumed to be linear in deformation. Nevertheless, the stiffening behavior of the network remains highly non-linear. It is found that the transition from a soft to a rigid regime is an emergent effect of the microscopic rearrangements within the network. It is explored how the main features of this stiffening behavior can be understood in terms of the microscopic interactions and how it is influenced by the details of the network structure. Strong similarities are observed between the strain stiffening behavior of elastic networks and the ferromagnetic phase transition. Drawing parallels with the Ising model of ferromagnets, it is hypothesized that the strain stiffening transition is a critical phase transition that is driven by the applied strain. Tools from the theory of critical phenomenon along with numerical simulations were used to validate this hypothesis. Indeed it is found that independent of the microscopic and structural details of the network, rapid strain stiffening can be understood within the framework of critical phenomenon.
2
Populair wetenschappelijke samenvatting (dutch)
In de natuur en voornamelijke in levende organismes, vindt men zeer diverse materialen met unieke eigenschappen. Deze materialen zijn vaak dusdanig opgebouwd dat ze zich kunnen aanpassen aan de zeer dynamische omgeving waarin ze zich bevinden. Het zorgt ervoor dat onder zeer diverse omstandigheden de functionaliteit behouden blijft. Een voorbeeld van zulk adaptief gedrag is de mechanische respons van de menselijke huid. De menselijke huid voornamelijk opgebouwd uit collageen vezels is mechanisch gezien een zeer interessant materiaal. Relatief kleine deformaties van de huid zijn zonder grote weerstand uit te voeren. Het is materiaal is soepel en geeft mee met de vervorming. Wordt de huid te ver uitgerekt in een specifieke richting, dan zal grote weerstand gevoelt worden en de huid rekt zich amper meer uit. Deze overgang van een soepel materiaal naar een stevig materiaal is nodig om enerzijds beweging van lichaamsdelen toe te laten, maar anderzijds te zorgen dat het material geen drastische vervormingen zal ondergaan.
Dat dergelijk weefsel en andere biologische materialen dergelijke verharding vertonen is al langer bekend. Het precieze mechanisme dat dit gedrag veroorzaakt is minder bekend. Een volledig begrip van dit bijzondere gedrag is wenselijk zodat het mogelijk is zulke materialen kunstmatig te ontwerpen en de eigenschappen aan te passen aan specifieke wensen. Zo vragen we ons af hoe de transitie in de stijfheid wordt be¨ınvloed door de materiaal eigenschappen zoals de interacties tussen de microscopische elementen of de geometrische plaatsing van deze elementen. Waardoor wordt de maten van deformatie bepaald, die het materiaal stijf maakt? Wat zorgt ervoor dat de materiaal soepel doch stabiel is voor kleine deformaties ?
Een eenvoudig model voor dit type biologische weefsels wordt beschreven. Een model waarin de buiging en rek van de individuele filamenten een lineaire kracht uitoefenen. Verwacht wordt dat dit relatief eenvoudig model genoeg is om de belangrijkste eigenschappen beter te begrijpen. In de simulaties wordt de mechanische respons van dergelijke materialen bepaald door opzoek te gaan naar de configuratie van het systeem dat energetisch het meest voordelig is. In dit verslag worden geometrisch zeer verschillende netwerken gebruikt om de invloed hiervan te bestuderen.
Allereerst wordt het verkregen resultaat vergeleken met vergelijkbare experimenten. Een overeenstemming is gevonden tussen het model en de experimenten. Daarna vergelijken we zeer verschillende netwerkstructuren met elkaar en komen tot de conclusie dat de on-derliggende details van het systeem de mechanische respons kwalitatief niet be¨ınvloeden. Het impliceert het bestaan van een meer fundamentele verklaring voor het waargenomen gedrag. Een verklaring wordt gezocht door het verstijven in netwerken te vergelijken met het mag-netisch worden van ferromagmag-netische materialen. Ondanks dat de systemen totaal niet gerela-teerd zijn bestaat er een verbazingwekkende gelijkenis tussen overgang die in beiden systemen voorkomt. De transitie in magnetisme is veel beter bestudeerd en begrepen en helpt daardoor sterk ons begrip van de stijfheid in biologische netwerken te vergroten. We laten zien dat de data verkregen uit simulaties kan worden geanalyseerd met behulp van de bestaande theorie voor fase overgangen. Hiermee worden de vermoedens bevestigd dat de verstijving een fase overgang in het materiaal is.
Contents
1 Introduction 4
2 Network model with branched structure 8
2.1 Microscopic interaction . . . 8
2.2 Inter-fiber connections: towards a network structure . . . 9
2.3 Networks . . . 11
2.3.1 Mikado network . . . 12
2.3.2 Regular lattice . . . 12
2.3.3 Adding disorder to the lattice . . . 13
2.3.4 Boundary conditions . . . 15
2.4 Method of analyzing mechanical properties . . . 16
2.4.1 Simple shear deformation . . . 18
2.5 Relating simulation and experimental parameters . . . 19
3 Nonlinear stiffening: A geometrical phase transition 20 3.1 Criticality in athermal networks . . . 21
3.2 Differential non-affinity and geometric phases . . . 23
4 Results 25 4.1 Important features of nonlinear elasticity . . . 25
4.2 Comparison with experimental data . . . 27
4.3 Comparison of different architectures . . . 28
4.4 Critical behavior in athermal networks . . . 31
4.4.1 Scaling ansatz: Widom collapse of stiffening curves . . . 31
4.4.2 Differential non affinity . . . 32
4.5 Evolution of the critical exponents . . . 34
5 Discussion and conclusion 37 6 Outlook 39 6.1 Continuation of the mean- field argument . . . 39
6.2 Temperature dependent network structures of collagen fibers . . . 39
6.3 Behavior of Molecular motors as sources of network-stress . . . 40
7 Acknowledgements 41
1 Introduction 4
1
Introduction
Living matter is highly complex and dynamical in nature. Living systems can be studied on many different levels: as information processing systems, electrical systems, mechanical systems and so on. One of the defining characteristics of such systems is their capacity to adapt and react to their dynamical environments. In nature, remarkably often, network structures are a major component in this adaptive machinery.
In different places and on different scales in these living systems, networks are found to play an important role. Throughout the whole body specialized cells called neurons form a network structure that is responsible for fast transmission of signals from and to the brain [1]. Similar to how our road system is constructed with different layers, this kind of network is optimized for the transportation of information. This kind of network clearly serves an informational role in the body. At the scale of the individual cells one can identify another kind of networks, with a completely different function. In the cell near its periphery, a protein called actin forms actin filaments, which can attach to each other, linked by other proteins. The resulting network is the main component of the cytoskeleton [2]. In figure 1 a) an image of the cytoskeleton of the cell together with the cell nucleus is shown [3], with the actin fibers stained in red. The cytoskeleton governs the mechanical behavior of the cell [4]. It impacts structural integrity to the cells. At the same time, the cytoskeleton undergoes dynamic structural changes allowing the cell to move, undergo division and change shape. Another component of the cytoskeleton is the tubular polymers called microtubules, which are shown in green in figure 1 a). They form a network in the interior of the cell and form a connection between the periphery to the cell nucleus. In addition to providing mechanical rigidity to the cytoskeleton, the network of microtubules is used for transport of proteins to and from the cells, which is vital to the cells existence. The cells are themselves embedded in another important network which is called the extracellular matrix[5]. This extracellular matrix is primarily composed of the protein collagen which assembles into bundles. An image of a network, consisting of collagen proteins, is shown in figure 1 b). It can be seen in this figure that collagen assembles into a random network constraining the movement of the cells to hold them in place.
Figure 1: a) Microscopic picture of one isolated cell [3]. By fluorescent labeling different structural components can be distinguished. In red is in actin, microtubules are labeled green and the cell nucleus is shown in blue. b) Extracellurlar matrix imaged using backscattering of light (in grey). In red are two cells embedded in the extracellular matrix [6].
These naturally appearing biopolymer networks exhibit highly non-trivial mechanical properties [7, 8, 9]. The resistance of those materials to externally applied deformations shows very interesting characteristic behavior. In figure 2 the typical response of stiffness to applied deformation is schematically shown for a general biopolymer network. These char-acteristic features are qualitatively known to most people from everyday experience. For instance the human skin; if one applies a small deformation to the skin by pulling on it, the skin complies. However, the compliance is rather limited. Beyond a certain deformation, the resistance of the skin increases dramatically requiring very large forces to cause any incre-mental deformation. Such a behavior in which the resistance increases dramatically with the deformation is a hallmark of many tissues, for e.g., lungs, heart and bones etc. That we can move our limbs without putting in too much effort is in part due to the compliance of skin towards relatively small deformations. If the skin tissue were a very rigid structure, such as a metal sheet, only very limited movement would be possible. Due to the rapid increase in the resistance of the tissue with increasing deformation, relatively large forces are required to cause significant deformation. That such large deformations are naturally suppressed is an amazing adaptive property of bio-materials.
Figure 2: (left) The stiffness is a measure of the resistance to a deformation. This curve shows how in general the stiffness of a polymer network develops as function of the deformation. The dashed line marks the onset of nonlinear stiffening. (right) A schematic of a polymer network and the configuration of that network as it undergoes a shear deformation.
On a very qualitative level, stiffness of a material is understood as the amount of resistance to a certain deformation. However, it is desirable to have an unambiguous quantitative definition of a material’s stiffness. An illustrative system to compare our networks to is the Hookean spring, which is an ideal spring that follows Hooke’s law over the full range of deformation. Hooke’s law gives the relation between the restoring force i.e resistance and the extension of the Hookean spring i.e the amount of deformation and is generally stated as
~
F = −k~u, where ~F is the restoring force acting on the spring and ~u the extension. Here k, known as the spring constant, has a constant value and is a measure of the stiffness of that particular spring. In reality, for any physical spring the linear relation between F and u will break down beyond a certain extension. In other words the stiffness of the material starts to change and k is no longer a constant. In that case, stiffness is defined as a differential increase in the force for a small deformation. Analogous to the stiffness of a single spring, one can assign an effective spring constant to an entire material, which in scientific literature is referred to as the elastic modulus. Elastic modulus is a macroscopic mechanical property of any material. It can be considered as an emergent property of the network.
The effective spring constant i.e. the stiffness, is determined by the configuration of the filaments in the network as well as the interactions involved and can therefore develop non trivially. For example, most materials exhibit the so called linear regime in which, similar
1 Introduction 6
to a Hookean spring, the stiffness is constant. It is a priori not clear whether a material should exhibit such a regime. An illustrative example of a network with no linear regime is shown in figure 3. The rope in figure 3 a), which could be regarded as a network containing only one filament, clearly will fail to resist the displacement shown in figure 3 b) if it is ensured that the bending of the rope does not come with an energetic penalty. Therefore, the modulus of this network is exactly zero over a certain range of deformation. If however, the network is deformed sufficiently, the rope will straighten up as shown in figure 3 c). At this point the rope is precisely straightened out but has still no energy stored in it. For any incremental deformation beyond this point, the rope-network exhibits finite modulus. The jump in the modulus in indicated in figure 3 d) in which the stiffness is indicated as function of the deformation. It follows that this simple rope-network is mechanically unstable for small deformations. In order to make it stable, one needs to include additional interactions, for instance, bending interactions such that any shape-change of the rope costs energy. In figure 3 e). it is shown how the stiffness of the network becomes finite if bending interactions of the fibers are taken into account. Since the the network is stiff for any deformation, the interactions are stabilizing the network configuration.
Figure 3: A network consisting of a single rope spanning from point A to B. a) The rest configuration of the network: the rope has no energy stored. b) Point B is moved over a certain distance. Movement came without any energy cost as the rope can freely bend. c) Point B is now moved to the point that the network resists any further deformation. d)This network has zero stiffness for small deformations up to point c. A jump in the stiffness occurs. e) The stiffness for a similar deformation where the rope has been stabilized by a bending interaction.
For the class of materials in which thermal fluctuations can significantly alter the configu-rations of the fibers, a well established theory exists which is able to explain the origin of the rigidity in the linear regime[10, 11]. In materials consisting of athermal fibers, for example fibers consisting of collagen proteins, a similar linear regime is observed [12, 13]. However, the origin of the linear regime in such materials can not be attributed to the thermal fluctuations of the fibers as the fibers are composed of thick bundles which are relatively insensitive to
thermal fluctuations. It is likely that the origin of linear regime in athermal networks lies in the bending interactions. However, this needs to be thoroughly investigated both theoret-ically and experimentally. Moreover, little is known about the mechanism of stiffening for these athermal materials. Since the most abundant protein in mammals, collagen, is also an athermal polymer, it is desirable to have a better understanding than there is so far.
Neglecting thermal fluctuations makes the study of elastic behavior of athermal networks relatively simple. Despite the relative simplicity, there are several interesting aspects that need careful theoretical analysis. For instance, it is not clear as to how an athermal network composed of linear elements exhibits highly nonlinear elastic behavior. Moreover, what is the impact of different geometries on the linear regime? Is the nonlinear regime also dependent on the geometrical details? Is the microscopic interactions within network crucial in determining the nonlinear elastic behavior of athermal networks?
In our study, we take a numerical and semi-analytical approach to address these ques-tions. By comparing the stiffness of different geometrically arranged networks, the influence of structure of the network is researched. As mentioned above, it is unclear how sensitive the stiffening behavior is to the way microscopic-interactions arise in the network. Therefore in this research, the obtained stiffening curves are compared to previous studies in which a different interaction was modeled for athermal filaments [14]. More specifically, quantifying the effect of varying the material properties on the characteristics of the stiffening behavior, such as the softness of the material in the linear regime, the amount of deformation for which rapid stiffening occurs and the approach of the very stiff regime. The ultimate goal would be to find an universal theory based on a very general mechanism, that would predict the stiffness of a material based on the polymer and network properties. Experimentally, a lot of progress has been made both in the detailed imaging of biological material [15], as in probing the mechanical response of athermal soft materials [16]. Studies have been performed in which rheology on and imaging of the material could be performed simultaneously [17]. With the increasing precision of experimental evidence, a solid theoretical model that is in agreement with this data is ever-so needed to settle these open questions.
In this thesis, we motivate and test a theoretical model that we believe is able to explain the intriguing properties of the nonlinear stiffening behavior of athermal polymer networks. In the next chapter, we introduce the model that has been used to resemble a generic polymer network, as well as the method that is used to test the mechanical properties of such networks. In the third chapter, all important methodology is introduced that is key to understanding the characteristic transition in the rigidity of these biological networks. By drawing analogy between rapid strain stiffening in networks and ferromagnetic phase transition in magnetic materials, we motivate the theory of stiffening as a critical phase transition. After the intro-duction to the model and theory, in chapter 4 the data obtained from simulation is presented and analyzed. A discussion of the data follows, in which it is compared to both the motivated theory as well as the experimental data on these biological networks. Finally, the conclusions of this discussion are summarized in the last chapter, together with suggested possibilities of continuing this research to address remaining research questions.
2 Network model with branched structure 8
2
Network model with branched structure
In this chapter, we introduce the model that is used throughout the thesis to understand the elastic behavior of athermal networks. We first describe the interactions on a single filament level. We refer to these interactions as the microscopic interactions within the network. Subsequently, we describe how a network is composed out of individual filaments and how one obtains the relaxed configuration of a network for a globally applied external deformation.
2.1 Microscopic interaction
There are many different types of filaments present in the living system. The different struc-tures arise due to the different interactions that exist between the constituting elements of a filament. Independent of details of the interactions on the single element level, for any kind of filament there will be an energy cost for stretching and compressing the filament and there will be energy associated with bending of filaments [18].
In figure 4, on the left, a single bond is drawn from point i to j at a distance of l0. The bond can be a segment of a filament that is part of a full network. Due to some deformation of the network the distance between point i and j might be forced to increase or decrease. Since the bond resembles a physical connection constituted of polymers, the elongation or compression of the bond invokes resistance by the interactions of the constituting elements. The schematic on the right of figure 4, shows an example of elongation δl. The bond is drawn in red, indicating that the bond will be in a stressed state. The question rises how the stress is related to the elongation. For the purpose of this study a linear relation between force and extension over the whole range will be a justifiable assumption because large stretching deformation are not very likely to happen. The measured force extension curves of athermal polymers mostly show nonlinear behavior for very large extensions only. Therefore, for the stretching of an individual bond i − j in the network the contribution to the hamiltonian is defined as Hstretchi,j = µ 2 1 l0,ij (| ~Xi− ~Xj| − l0,ij)2, (1) where i and j are two connected nodes. Here, µ is the effective modulus for the multiple cross-linked fibers that act together as one filament. The µ is assumed to be the same for each bundle i.e. a uniform bundle thickness throughout the network. l0 is the rest length of a segment of a filament connecting two nodes, defined as the end to end distance between two nodes in the network in its rest state when no external deformation is applied. As can be seen, the effect of both compression and extension of bonds with respect to this rest length, is symmetric and linear for any deformation.
Figure 4: Schematic of two nodes i and j connected by one bond. As the end to end distance is increased the bond becomes stressed and increases its energy according to equation 1, in which the δl is expressed in the coordinates of the two nodes.
In figure 5(a) it is shown that a pair of bonds i, j and j, k are connected at an angle θ0,ijk implying a rest configuration that is non straight. In figure 5(b) it is indicated how a change
in the position of one of the point can change the angle between the two bonds. The deviation from the rest angle will induce a restoring force. Resistance to bending deformations is feature which is consistently observed in all polymer fibers, relative strength and angle dependence of this interaction varies from one type to the other. Again the interaction is assumed to show linear response to a change in angle. Therefore the bending part of the hamiltonian is expressed as
Hbendi,j,k = κijk 2
θijk− θ0,ijk l0,ij + l0,jk
2
, (2)
where κ is the bending rigidity of the fiber. i,j and k are the nodes that are connected by a pair of connected bonds. Each adjacent pair of bonds on a filament has a mutual bending interaction. θ0,ijk is the rest angle between this pair of bonds, which is π for perfect straight bond pairs but can take any other value depending on the structural details.
Figure 5: Schematic of three nodes i, j and k connected by two bonds. As the angle between the two bonds deviates from the rest angle for the pair of bonds, stresses develop and the energy of the full system is increased according to equation 2.
The model for the microscopic interactions of fibers presented above is a very general one. In our model the parameters µ and κ along with l0 determine the scale of energy cost associated with an applied deformation. The model can be adjusted to fit with the properties of different types of fibers, by choosing the proper values for µ, κand l0. In our study we have assumed that bending is possible only on length scales larger or equal to l0. However, buckling of an individual segment can still occur depending on the amount of deformation applied as well as model parameters. It is useful to understand the above in terms of the persistence length of a fiber. Persistence length of a fiber is a measure of the distance over which the fiber orientation is highly correlated. Our assumption of bending occurring on length scales ≥ l0 is equivalent to the persistence length of fiber longer than the average distance between adjacent nodes in the network. This assumption is a plausible one for biologically relevant athermal networks. However, in networks where thermal fluctuations cannot be ignored, this assumption is invalid and hence our study does not apply to them. For many biological occurring fibers the bending energy scale of the fiber is much smaller than the stretching energy scale the fiber. In our model this corresponds to the dimensionless parameter κ/µl20 1. Different biopolymers exhibit a wide variation in this parameter.
2.2 Inter-fiber connections: towards a network structure
A minimalistic model for the interactions can be used to model different semi flexible polymers. To obtain a material from these fibers we need to define how the fibers are connected to form a network structure. In nature there are several ways that fibers connect to each other. The type of connection is mainly determined by the interactions of the constitutive elements. Some fibers experience a weak attractive self-interactions, others can be bounded chemically.
2.2 Inter-fiber connections: towards a network structure 10
Some fibers are connected by external connective agents that provide linking between one one filament and another. Different type of connections can give rise to different constraints on the correlated fiber motion.
Many of the filaments in biological systems are connected by linkers, these cross-linkers are specific proteins that can attach to two filaments[19]. As long as the cross-linker is attached to the filaments, the filaments are geometrically constrained to stay together at this point. The direction of both fiber, i.e, the relative orientation, can be either free or come with a energetic penalty with respect to a desired orientation. In discrete fiber models the connection of two fibers is modeled by identifying a node on one filament with a node on the other filament. As the nodes are taken to be identical the motion of the fibers is constrained. In this model one can define an additional term in the hamiltonian that accounts for interactions due to the fiber orientation. However, in many studies the fibers are modeled as freely hinged at the cross-linked points.[20, 21, 22, 23]
A different set of polymers is able to construct a network from the constitutive elements, without forming separated fibers first. For this type of materials, the elements assemble in bundled layers along in a specific direction. In figure 6 a) an example of such a structure is given. Since the inner structure of such a bundle is layered, the bundle can split into two separated bundles, both containing some of the layers. The bundles continue in two separate directions and are now connected to one main branch. Here, it is no longer possible to distinguish individual fibers, only notion of branches in these networks exist. When the process of branching is continued it is possible to constitute a full network out of branched bundles of polymers.
Collagen, a bio-polymer which is of high biological relevance, is an example of a branched structure that is found in nature. A long collagen bundle constitutes of smaller collagen fiber that partly overlap and are cross-linked to form a coherent bundle. At a branching point the individual fibers are going in different directions. In figure 6 a) a high resolution microscopic image of collagen in lung-tissue [24] is shown. With the achieved resolution one can distinguish the individual fiber inside bundles. In figure 6 b) a schematic of the arrangement of the material at a branching point is drawn. As some of the fibers run from one fiber to the other, the orientation of the branches will effect the bending of the these fibers. In other words, for branching structures the orientation of all the branches effect the energy of the system. The blue curved lines in this figure indicates the possible stress that is involved in bending the branches. Since there are no fibers running from branch two to three, there is no such marking for this angle.
In the model the branching interacting is taken into account by assigning a bending interaction for both pair of bonds at a triple connected point. The bending rigidity κ at a branching point could be given a different value than for the non branching points. However, in this study they were assumed to be the same. As illustrated in figure 6 b) not necessarily every pair of bonds at a branching point are exhibiting a mutual bending interaction. Therefore, one out of the three bending rigidities at each branched point is set to zero as indicated in 6 c). In this study the rigidity of the smallest initial angle is set to zero as it naturally comes that this is the direction of branching.
Now we defined all the interactions and connection in the network properly and one should combine those results to obtain the full formal description of the energetics of the network. The Hamiltonian of the network will be a summation over all the stretching interactions on a single bond level described by equation 1, all the bending interaction between adjacent pair of bonds described by equation 2. By summation over all the adjacent bonds we already include the branched structure. The hamiltonian is written as
Hnetwork = µ 2 X i,j 1 l0,ij (| ~Xi− ~Xj| − l0,ij)2+ κijk 2 X <ijk> θijk− θ0,ijk l0,ij+ l0,jk 2 , (3)
In which i,j and k are indices of the nodes in the network. The summation over i an j includes all the pair of nodes connected by a bond. The summation over < ijk > runs over every adjacent pair over bonds.
In this research we aim to understand the elastic behavior of athermal networks. As mentioned above, collagen, the most abundant protein in connective tissue assembles into a branched athermal network. Motivated by this, in this study we focus on the branched networks. Previous studies on athermal networks have focused on network with straight filaments arranged either on a lattice or randomly in space. However, it is not a-priori clear whether results from non-branched networks are applicable to branched networks. By focusing on branched networks and comparing it with previous studies, we can gain a better understanding of the role of network architecture in determining the elastic behavior.
Figure 6: a) Fluorescent labeled collagen found in lung tissue. b) Schematic of a branching point in the network. c) There are three bond-pairs at a branching point. To each bond pair a bending rigidity κijk is assigned, where i, j and k are node-indices. The bond-pair connecting nodes 2 − 0 and 0 − 3 is assigned κ203= 0.
2.3 Networks
One way to look at a biopolymeric material is as following. It is composed of a the network which is collection of filaments together with their spatial arrangement. The network is modeled with nodes connected by compressible bond, the bonds themselves have infinitely bending rigidity. Multiple bonds together model a filament with discretized freedom to adapt its curvature. In other words, only at the nodes it is possible to bend the filament to which a finite bending rigidity is assign. In the picture of the network as discretized system it is easy to count the degrees of freedom in the system. Namely, the position of each node in the network is by d coordinates, where d is the dimensionality of the space in which the network lays. Therefore for a system containing Nd nodes the degrees of freedom are simply d · Nd. However, every bond is constraining the possibilities of the positions of the nodes in this space. In fact, every bond lowers the degree of freedom in the system by exactly one. Therefore, d · Nd connections are needed to remove all the internal freedom in the system. The coordination number, z, is a measure for the average number of connections per node. Since every bond is connecting two nodes it increases the local co´’ordination number of two nodes by one. In terms of coordination number of the system the degrees of freedom can be expressed as d.o.f. = Nd· (d − z/2). In other words if z is equal to two time the dimensionality the system has lost all its freedom. This general result was first obtained by J.C. Maxwell
2.3 Networks 12
back in 1864 [25]. At this point a change in the mechanical behavior of the material occurs, therefore this special point is called the isostatic point. All structure that are connected above this isostatic point are rigid for any deformation, because any movement of a node will violate at least one constraints of fixed node distances. If the node to node distance of two connected nodes change, stretching of the bonds occurs.The other mechanical phase is referred to as sub-isostatic regime. In this phase the material is not mechanically stable since there is freedom in the system to move nodes without elongated bonds. However, as shown in the introduction, it is possible to make any sub-isostatic network stable by applying sufficient deformation. To emphasize, if the network is sub-isostatic, it is only possible to get a linear soft stiffness by some stabilizing interaction. As discussed the networks in biological systems are constructed by either connecting filaments or by the branching of fibers. Both the branching and the establishment of connections are stochastic process that add disorder in biopolymer networks. To match the model with the reality, disorder can be incorporated into the model by using one of the many algorithms that generate network randomly. However, modeling the networks as regular lattice structures has many other benefits. Computationally, it is relatively easy to define the connection in such a lattice. More importantly, in a regular lattice the properties can be adjusted in a more controlled fashion, such as the number of connection per node. Finally, in regular lattices it is relatively easy to semi-analytically predict the mechanics of the material. In this section, the properties of several networks structures, both random and regular structures, are compared. Then a motivation for the used network follows.
2.3.1 Mikado network
Mikado network [26] is a special case of random networks which can be generated in an intuitive way. Rods of a fixed length are added one by one at a random position and with a random orientation into a fixed box. This process is continued until a desired concentration of rods is reach. Then, at each intersection point between a pair of rods a connection is added. Using the mikado algorithm the lengths between any two connected points on the same filament can vary from exactly 0 up to the fixed length of all filaments. The distribution in these lengths will be highly disperse by the random nature of the process. It makes the mikado network interesting model to study, since the same dispersion is found in real systems. Therefore, the elastic behavior of the Mikado lattice has been subjected to study [27, 26]. While the Mikado network is very successful in 2D it has its limitations for the implementation in 3D, since is much harder define the intersection between rods in 3D. 2.3.2 Regular lattice
In the class of regular lattice structure one can distinguish three different geometries with equal distance between connected nodes. The lattice with the highest number of connections per node that belongs to this class is the triangular lattice. For the full lattice the local connectivity is six for every node which is more than the degrees of freedom in a 2D network. It implies that in order to study the linear behavior of isostatic networks, it is necessary to remove constraints by removing bonds. The benefit of the triangular lattice is that it is that it is possible to consider straight lines in the network as straight filaments. Therefore the triangular lattice forms a good basis for the modeling of a filamentous network that are cross-linked at the intersection of the filaments.
The second member of this group is the square lattice with a local coordination number of four. So from the beginning it is isostatic, since the number of constraints are exactly canceling
Figure 7: Samples networks are shown for a) triangular lattice b) square lattice c) honeycomb lattice.
the number of degrees of freedom. So compared to the triangular lattice less constraints need to be removed to be able to explore the sub-isostatic regime. Just as the triangular lattice, in a square lattice it is possible to identify long straight fibers. An important difference is that now every fiber is either perfect perpendicular or parallel to all others. It is not only less realistic, it can also add unexpected effects to the modulus. It is for example known that the full regular square lattice has zero stiffness for small deformation even though it is exactly isostatic.
Another network structure with all bonds of same length is constructed with a local coordination number of three. The corresponding geometric structure is the hexagonal lattice, also referred to as a honeycomb lattice. The honeycomb lattice is the only one out of the three that naturally is sub-isostatic with all its connections. The honeycomb is also the structure without bonds lined up in one direction. Therefore it is not obvious to define filaments in this lattice. It is only possible to identify random paths of bonds as non straight filaments. It does on the other hand, forms an ideal basis for a model of branched fibers. At each branching point a part of the protein bundle splits up moving in a different direction. Therefore, at a branching point naturally the local coordination number is three. Only if branching occurs twice over a distance significantly smaller than the bond length, can one consider that there the connectivity is higher than three. While the triangular lattice has been subject of many rigidity studies [28], the mechanical properties of honeycomb structure are less extensively studied. These properties combined makes the honeycomb lattice an interesting structure to study and the focus will be particular on this kind of lattice.
2.3.3 Adding disorder to the lattice
Independent of the chosen lattice structure, it is highly debatable if the regularity of a lattice makes it justifiable to use them to model realistic biopolymer networks. For a regular lattice with no bonds missing, the stiffness of the network is that of one unit cell of the lattice, since symmetry ensures that every unit cell will exhibit the same movement as we deform the material. Only if irregularities (some randomness) are added to the lattice structure it is no longer possible to say that the networks stiffness is known by considering only a small part of the network.
One way of adding irregularity to any of the mentioned lattice structures, is by the dilution of the material in the structure. Simply by removing the connection between nodes in a random order, the resulting structure no longer has a repetitive pattern. It is common to express the amount of dilution in the dimensionless quantity of the number of removed bond over the number of bonds in full lattice. In figure 8 the honeycomb lattice is shown for different
2.3 Networks 14
Figure 8: Shown are multiple diluted versions of the full honeycomb lattice. With a protocol for random dilution disorder is added to the network. From a) to e) the dilution is varied from 5% up to 40%. Notice that in e) for the first time the network is disconnected from top to bottom.
amounts of dilution varying from 5% dilution tup to of 40%. It is clear how the variation is added to the network as the dilution of the bonds is increased. The distance between three coordinated points is getting more disperse and so does the void sizes in the system. For any lattice structure there is an limitation to how much we can dilute the system. Far before there is no material left in the system, due to the random nature of the dilution, it possible that by chance parts of the network start to disconnect from the rest of the network. If it so happens that the boundary to which stress will be applied is fully disconnected from the rest of the network the rigidity of the network will not be felt. The dilution probability for which this will happen for the first time is different for different lattice structures. There exists a whole separate field of study, called percolation theory, which is dedicated to problems that are related to this behavior. The theory of percolation is able to predict these values for the different lattice types and so gives us an upper bound on the maximum dilution.
Besides random dilution of a lattice it is also interesting to perform a more guided dilution. It is for example possible to obtain a honeycomb lattice from a triangular lattice by a very specific dilution of bonds. Another possibilities is to randomly dilute three out of the six bonds that start from one node in a triangular lattice. By continuing this process for every node in the lattice an irregular lattice is achieved that has a forced local connectivity of three every where. The interesting structure that is found by this specific dilution is shown in figure 9 c). It is another suitable candidate for modeling a branched network. It is very instructive to compare the stiffening of both lattice types to get an idea of what features of stiffening are general and which of them are due to the lattice geometry.
The dilution is an effective way of adding disorder in the connectivity of nodes and there-fore the local constraints in the network. However, the orientation and length of all bonds in the system remain unaltered by this method. The mono disperse distribution of bond lengths
Figure 9: Network schematic for bond probability p = 0.85. (a) and (c) are examples of undistorted lattices. (b) and (d) are corresponding distorted lattices. The applied distortion is d = 0.8. After applying distortion the obtained configuration is set to be the relaxed state of the network.
and initial angles in the system does not correspond with disperse nature of grown biopolymer networks. There is evidence that rapid strain stiffening is not limited to a specific geometri-cally ordered networks but is qualitatively similar to random networks, such as the Mikado network. However, it remains an open question how the stiffening is effected quantitatively by the disorder of these networks. In this thesis, we try to investigate this open puzzle by considering another way to add disorder to a network based on a regular lattice. The disorder is achieved by changing the regular positioning of the nodes in a lattice. Different protocols can be used to add the distortion in specific ways. In this thesis, a very general and robust protocol has been used to obtain disordered versions of the honeycomb en branched triangular network as described above. The simple distortion that is going to be applied consists of a displacement of every node in the network over a distance of δ · l0/2 in a random direction θ, where δ is a randomly chosen number between 0 and dmax. The maximum displacement dmax can be varied to control the amount of distortion in the network and is constrained to a maximum of one. By distorting the lattice in the way a structure is created which has the same connective properties as the lattice structure it originated from. However the symmetric features like equal lengths and a specific initial angle between bonds are removed. Instead we are left with a distribution of angles and lengths around a certain mean value. As mentioned above in this research the honeycomb and triangular lattice are used to study mechanical properties. In figure 9 b) and d) examples of the distorted equivalents are given.
Now we obtained four structural different lattices that can be subjected to a rigidity test. The variety of networks allows us to test the generality of the stiffening. The two distorted lattices that are structurally more similar to realistic biopolymer networks.
2.3.4 Boundary conditions
A different mechanical response off the material on the boundaries of the material is expected, since the topology differs from that of the core of this same material. This is due to different connectivity for a closed boundary. On length scales much larger than the average filament length, the effect of the presence of a boundary becomes less important compared to the response of the full material. In any finite system boundaries will play some role and should carefully be taken into account. Here, our goal is to study the response of the bulk of the material and therefore a big model system is desirable. Computationally however, computa-tion time rapidly grows with system size making it impossible to study any size even close to that of a real system. By applying proper periodic boundary conditions we are able to eliminate the open boundaries. Additional connections are created from the far left nodes to the far right nodes, equally from to bottom. Different way to understand this is that infinite projections of the network are created, translated in both x and y direction, by an integer number times the system size. In figure 10 it is shown how four images of the same network are connected to form a bigger network. Indicated is how the images of one specific node is translated. The coordinates of the nth image in x direction and,m th image in y direction is
~ Xim,n = ~Xi+ n √ 3 2 m ! L. (4)
If shear is applied to our system the periodic boundaries conditions need a correction. The correct way of projecting images of the network becomes
~ Xim,n = ~Xi+ n + γm√ 3 2 m ! L. (5)
2.4 Method of analyzing mechanical properties 16
These conditions are known as Lees-Edward’s boundary conditions [29].
While this method is of great help in eliminating the effect of free moving boundaries of a finite system and enables us to perform simulations on finite systems, the system will never fully agree with a infinite system. The repetitive nature of network adds a characteristic length scale to the system, where in a real infinite system, this length scale is brought to infinite as well. The presents of this internal length scale will influence the mechanical response of such a network. As an example, consider the possible fiber lengths in a box of size L by L. All lengths between 0 and L are possible lengths of the fibers. If the fiber of length L comes back onto itself by the implied boundary conditions the length of this filament is brought to infinity. However, for the L × L box no lengths between L and infinity can be found in the network. If we compare this to a real infinite box in principle it should be possible to find filaments of any length. Similar to the distribution of lengths in the system, more properties are effected by the intrinsic length scale. The collection of these effects are called finite size effects.
Figure 10: (left) The periodic boundary conditions are visualized by drawing the projected images of the network. (right) The Lees-Edwards boundary conditions project the network structure taking into account an additional translation in the direction of the shear propor-tional to the strain.
2.4 Method of analyzing mechanical properties
Once a specific network configuration is chosen and the microscopic interaction are properly defined, one can explore the mechanical properties of the material. If we consider a cubical piece of material such as is illustrated in figure 11 a), one can deform the material by exerting stresses on the boundaries. While the possibilities of deforming are endless, it is possible to consider intuitive basic deformations that are common to test a materials stiffness[30]. The modulus, a measure of the stiffness, is defined as the amount of stress per unit deformation that develops at the system boundary in response to the applied deformation. Firstly, a uniform stress on the surface corresponding to normal forces will extend or compress the material illustrated in figure 11 b). The corresponding mechanical response to this deformation is known as the Young’s modulus and is defined as KY = σn, where σn is the normal stress and is the amount of deformation. Another deformation one can consider is that of applying a uniform pressure on the material that will result in either compression or expansion of the bulk of the material, depending on the sign of the pressure difference across the boundary. In figure 11 c) a bulk expansion is shown for our cubical martial. The corresponding modulus
is called the bulk modulus of the material and is defined as Kbulk = V · dPdV. Another highly relevant deformation is to pull along the boundaries of the material as shown in figure 11 d) known as shear stress. The corresponding modulus to a shear deformation is the shear modulus K = dσs
dγ , where σs is the applied shear stress and γ the shear strain.
Figure 11: a) For a cubical material in rest consisting of a network of filaments three different deformations are illustrated. b) Extension of the material in one specific direction by a uniaxial normal stress c) Bulk extension by uniform pressure d) Shear stress applied on the boundaries of the material, the opposite forces move the two planes apart.
The above explained deformations methods can characterize the the mechanical behavior of a material under deformation. Depending on the specific problem it is more instructive to study one of the deformations above the others. For the biological networks of our interest it is most convenient to study the elasticity of the network by studying the response to an applied shear strain. In living systems the network is always embedded in an environment, mostly a viscous liquid with other biological objects suspended in that liquid. If the volume occupied by the network would be changed, it implies that that the liquid either has to flow in to or out of the system. For a viscous fluid it takes relatively long to to flow in or out of the network, as this time scale might get way larger then the time scale of performing the deformation, somehow the liquid resists the deformation. This will therefore increase the modulus and hence will contribute to the stiffness of the material in addition to the network. To eliminate all of these problems, it is best to consider non volume changing deformations only. A simple shear deformation is a volume preserving deformation. Another consideration, is the compatibility with experiment. It is only possible to test and compare the proposed theoretic model if the same properties are measured experimentally. Rheology, is the most common experimental technique to measure materials stiffness and is based on shearing materials.
Shear strain, denoted as γ, is the amount of deformation normalized to the system size. By combining our previous definition of the shear modulus with the relation
E = Z ~ F · d ~S = A Z σs(γ)dγ, (6)
it is possible to express the modulus in terms of the energy in the network K = dσs dγ = 1 A d2E dγ2. (7)
Equation 3 implies that once the energy of the system as function of applied shear strain is obtained, the shear modulus can easily be derived.
2.4 Method of analyzing mechanical properties 18
2.4.1 Simple shear deformation
A shear deformation is applied by moving internal boundaries in the direction parallel to these boundaries. The system has to take a new arrangement to fulfill the added global constrained. A transformation of the network can be performed that ensures that the global constraints of sheared boundaries are satisfied is the simple shear transformation. Let ~Xi be a two dimensional vector, containing the x and y position of node i. The positions of the nodes ~Xi after a applied simple shear strain γ, is obtained from the rest position of the node
~ Xi0 by the transformation ~ Xiγ=1 γ 0 1 ~ Xi0. (8)
The simple shear transformation is, besides a volume preserving deformation, also an affine transformation. A transformation is called affine if sets of parallel lines remain parallel and ratios of distances between points are preserved if they are lying on a straight line. In figure 12 a) is shown a honeycomb network in rest. In figure 12 b) it is shown how the network looks like after a applied simple shear deformation. An affine transformation does not necessarily preserve angles between lines or distances between points and will therefore increase the energy of the system. While this affine transformation ensures that the global shear constraints are satisfied, it does not ensure that the system is brought in the lowest energy state. The energy of system is obtained from the well defined Hamiltonian in equation 3. If we allow for local rearrangements in the material a more favorable energy of the network can be explored. The minimum energy state of the system for a given shear strain is found using a gradient descent minimizing algorithm from numerical recipes [31]. In figure 12 c) it is shown how the network rearranges after the shear deformation is applied. Once the configuration of the network is determined and the energy is calculated, the shear strain is increased in small steps and the procedure is repeated obtaining the energy of the network over a full strain range. How much the obtained configuration deviates from the network configuration after the affine transformation is going to depend on the applied strain. It is worth exploring the rearrangements in the material since that can help understanding what happens with the stiffening of the material. In figure 12 the non affine displacement of each node in the network is drawn. To express the amount of deviation in one number we quantify the non affinity in the system by summing up all the displacements in the displacement field. Γ(γ) ≡ l21
0N γ2 N P
i
( ~Xiaf f − ~Xi)2, where Γ is the non affinity, i sums over all N nodes in the system. γ is the applied shear strain and l0 the length of each bond.
Figure 12: A disorder 2D honeycomb network shown a) in its rest configuration, no external stresses applied as well as no internal stresses, b) after an affine displacement of every node according to a simple shear transformation. The arrows indicate the height dependence of the displacement of nodes c) After non-affine rearrangements of the nodes minimize the energy of the system, maintaining the imposed shear strain conditions. The arrows do not apply anymore since the displacement of the nodes is no longer affine. Only is it valid to indicate that over the length scale of the system the displacement is of the order of the applied strain. d) The non affine displacement vector field for the transition from b to c.
2.5 Relating simulation and experimental parameters
The model introduces three different parameters that define the materials properties. Bending rigidity κ, stretching rigidity µ, and the distance between nodes l0, setting the rest lengths of the bonds in the system. Experimentally the control variable is the concentration of the protein that assembles into a elastic network. However, in our computational model, the concept of concentration is not well defined. However, it is still possible to compare our simulation results to the experiment by carefully mapping concentration in experiments to the relevant parameters in our computational model. To that effect one can use the dimensionless parameter ˜κ = κ/µl20 in the following way. Under the assumption that changing concentration of proteins in the experiment only changes the typical void size (also known as the mesh size), we can consider that change in concentration can be captured in the computational model by changing the length scale l0. The change in l0, through the dimensionless parameter, is equivalent to changing ˜κ in simulations. The exact functional form of the dependence of l0 on the concentration of proteins is dependent on the underlying spatial dimensions. To summarize, the parameter ˜κ in our simulation model can be mapped to the concentration in experiments allowing us to compare our findings with the experiments.
In this chapter a full description of the model was presented. The microscopic interac-tions that were included in this model were motivated from the expected mechanical response on individual fiber level, neglecting details on the level of the constitute elements of a fiber. Then the description of the connective properties of different types of fibers were discussed. Important differences between two of the existing models were given to motivate the study of networks as branched structures. Different spatial arrangements of material were compared varying from random generated networks to regular lattice structures. Given the compu-tational efficiency, lattice based models of branched networks are developed. However, the lattice is diluted and distorted to introduce disorder in the network with the goal that the generated networks resemble the biologically appearing networks. After a full description of the material properties is given, different methods for testing the materials mechanical properties were presented. A motivation is given to use shear deformations to explore the mechanics of the material. A detailed description of the computational steps that need to be taken followed. Finally, it is explained how the model parameters are related to real physical properties that can be measured in experiments. This mapping allows us to compare results obtained from simulation with results obtained from rheology measurements.
3 Nonlinear stiffening: A geometrical phase transition 20
3
Nonlinear stiffening: A geometrical phase transition
In this chapter we aim to set the background for theoretical understanding of nonlinear stiff-ening in elastic networks. To that effect, we present a very interesting and useful analogy between nonlinear stiffening in athermal networks and ferromagnetic phase transition in mag-netic materials. Drawing parallels between the two we suggest that the nonlinear stiffening in athermal networks can be considered as a strain driven continuous phase transition. We mo-tivate using tools from the thoroughly-researched field of critical phenomenon to understand the emergence of nonlinear elasticity in athermal networks.
Figure 13: (left) generic stiffening curves, in terms of the modulus K, as function of applied shear strain γ. Rapid stiffening transition between a bending and a stretching phase. The transition point is denoted with γc (right) Magnetization curves of a ferromagnetic material, modeled by the Ising model. A transition occurs from an ordered spin phases to a disordered spin phase, as the temperature goes transits from below Tcto above Tc.
In figure 13 on the left we show stiffening curves obtained from a branched network with honeycomb geometry for a certain dilution probability. We plot the differential elastic modulus K as a function of the applied shear strain γ for different bending rigidities κ. We focus on the stiffening curves with κ < 10−2. One can identify three distinct regimes in each stiffening curve. The first one is the linear regime in which the elastic modulus is constant (independent of γ). This regime persists up to the strain γ = γ0. For γ > γ0, the elastic modulus starts increasing rapidly. γ0 is referred as the stiffening strain and it marks the onset of the nonlinear regime. This regime persists till γ = γc, and for γ > γc, the modulus starts converging towards a constant value. This regime γ > γcis referred to as the stretching regime. As indicated in figure 13 the modulus scales with the bending rigidity in the linear regime. In other words, for small deformations, the network elasticity is governed by bending of the fibers. In the stretching regime, the network elements start aligning in the direction of the applied strain and undergo stretching. The most interesting regime is the nonlinear regime in which the modulus undergoes transition from bend-dominated to stretch-dominated elasticity. A striking feature observed in these stiffening curves is that γ0 is independent of the bending rigidity. The independence of γ0 on κ implies that γ0 is not determined by competition of bending and stretching energy in the network but by the nonlinear growth in bending energy. The mechanism of such strain stiffening has been recently explored by
Property Ferromagnetic phase transition Rapid strain stiffening
Order-parameter Magnetization M Shear modulus K
Control parameter Temperature T Strain γ
External field Magnetic field h Bending rigidity κ
Divergent quantity Spin correlation length ξ Differential non-affinity (δΓ(γ)) Table 1: The comparison between several important system parameters from the model of rapid strain stiffening and the model of ferromagnetism
Licup et. al[14] in their study on athermal networks of straight filaments. There the authors highlight the role of collective rearrangements in the network as the source of rapid increase in stiffness. Even though our networks are branched, we expect an analogous mechanism as reported by Licup et. al to account for the nonlinear stiffening in branched networks. That collective rearrangements within the network become increasingly important for γ > γ0 in understanding the nonlinear elasticity suggests an intriguing analogy which is as following.
The above described features of the nonlinear stiffening reminds one of the features ob-served in the behavior of a ferromagnet as a function of temperature. In the right graph of figure 13 simulated data of the Ising model for a ferromagnet is shown [32]. In a magnetic material, one considers the net magnetization as the order parameter. The ferromagnetic phase transition corresponds to spontaneous emergence of magnetization at (and below) a certain temperature referred to as the Curie temperature Tc. In the absence of an externally applied magnetic field, the magnetization of the ferromagnet is zero if the temperature of the material is above Tc, while for T < Tc, the spins in the material are aligned and give rise to a net magnetization of the material. For T > Tc, the magnetization scales linearly with the externally applied magnetic field. We now consider the following analogies. The net mag-netization m is analogous to the elastic modulus K. The externally applied magnetic field h is analogous to the bending rigidity ˜κ and the inverse temperature T−1 is analogous to γ. With these analogies one can immediately see the following parallels. In absence of bending rigidity, the stiffness of the network is zero for strains γ < γc. If the bending rigidity is finite the network is stabilized in the linear regime such that the elastic modulus, K ∼ ˜κ analogous to m ∼ h for T > Tc. For γ γc, K becomes independent of the bending rigidity analogous to m becoming independent of h for T Tc. Just as Tc marks the onset of magnetization in a ferromagnetic material, γc marks the onset of rigidity in a network without bending in-teractions. With the correct mapping of the relevant parameters of the system it seems that the same terminology can be used to describe the main features of the two systems. In table 1 we present an overview of the mapping of relevant parameters between the Ising model of ferromagnetism and Nonlinear stiffening in athermal networks.
3.1 Criticality in athermal networks
In the Ising model of ferromagnets the obtained transition in magnetization is second-order transition between two distinct phases[33]. The two phases are the state of the system found far left and right of Tc, In the right panel of figure 13 shown is simulated data of the Ising model. In this figure these two phases are indicated as well as the transition point Tc. The phase on the left of the transition point is an ordered state of the material in which the spins are aligned giving rise to a magnetization. On the right, the phase of disorder is found, where the spins are directed in a random direction. Due to the increased temperature the entropy becomes more relevant, making the higher entropy state favorable. The transition from the
3.1 Criticality in athermal networks 22
Figure 14: The phases transition predicted in the Ising model visualized in simulations[34]. For temperatures below Tc the ordered phase of the system is found. At Tc it seen that varying cluster sizes can be found. As the temperature gets far above Tcthe net magnetization becomes zero and the relative spin orientations get uncorrelated.
ordered to disordered state is a critical second order phase-transition. Critical behavior is an important characteristic of second-order phase transitions. The transition is in general expressed in terms of the order parameter, in the example of the Ising model this is the magnetization. One of the properties of such a transition is the existence of a quantity in the system that diverges at the point of transition. This quantity is typically a measure for the correlation of the interactions in the system. Even if in the Hamiltonian, there are no long-range interactions, at the critical point collective behavior of the constituting elements, for example that of spins in a ferromagnet, leads to an infinitely ranged correlation. Such correlations lead to divergent fluctuations in the order parameter at the critical point. In figure 14 it visualized how correlations are increased by the time the critical temperature is approached. In an athermal network there cannot be fluctuations in the elastic modulus at the critical strain because the modulus is either zero or finite with no variation for a given configuration of the network. However it is still possible to create a quantity that is analogous to divergent fluctuations. That quantity is called the differential non-affinity and is defined below. Most importantly, near the critical point the behavior of the order parameter exhibits a scale-free transition. The behavior has a power-law dependence on the distance to the critical, which has no explicit length scale associated with it. The implication is that microscopic details of the system do not effect the way the critical point is approached.
The remarkable similarity between the seemingly unrelated systems raises interesting ques-tion about the understanding of the rapid stiffening transiques-tion. If the nonlinear stiffening is indeed a critical phenomenon than is the microstructure of the network irrelevant near the critical strain? If yes, which property of any elastic network is the most relevant property that determines the nonlinear stiffening behavior? Is the appearance of rigidity in a zero κ network with applied strain a continuous phase transition? Can one consider the bending rigidity as a field or is it just a coupling constant in the hamiltonian describing the energetics of a network? What is the analogue of divergent fluctuations in an elastic network? Can one identify and quantify collective rearrangements in an elastic network? And probably the most relevant question is whether one can use the tools from the theory of critical phenomenon to provide a theoretical framework for understanding nonlinear elastic behavior of branched
networks?
3.2 Differential non-affinity and geometric phases
Differential non-affinity is a measure of the internal rearrangements that occur in a network in response to an incremental change in the applied strain. It is given as
δΓ(γ) = 1 l2 0N dγ2 N X i (d ~Xiaf f − d ~Xi)2, (9)
where dγ is a differentially applied strain, d ~Xiaf f − d ~Xi is the change in the non affine dis-placement vector of node i, as consequence of the differential strain. Were i is summed over all the nodes in the system. Near the critical strain, for κ = 0, δΓ(γ) is expected to be a function of |γ − γc| only. The divergence of differential non-affinity in an infinite size system would represent the long range collective rearrangements that occur in a network near the critical strain.
If the analogy is not only a visual one, but also agrees quantitatively with the model it suggests that the rapid stiffening in sub-isostatic networks is a critical transition between two phases of the system. Similar to the Ising model, we can qualitatively identify two distinct phases. As mentioned before, it is clear from the curves that there exists a transition from a bending phase to a stretch phase for increasing shear strain. In figure 16 b) the ratio of bending to stretch energy is indicated. It can be seen that at the critical strain, there is a sharp transition in the fraction of bending and stretching energies. However, is it plausible to say that these two phases are again an ordered and disordered phase of the material? Here, it is claimed that there is a concept of order and disorder in the system in terms of the fiber orientation in the system. For the disordered lattices described in previous chapter, there is no correlation between the orientation of different bonds as the network is in rest. If relatively large strain is applied to the network more and more fibers are forced to be directed in the direction of the applied shear strain. The two states of the network are geometrically distinguishable by the alignment of the bonds. In figure 15 one can see how the fiber alignment occurs as the strain increases to way above γc. In this research investigated is the rapid strain stiffening, it is hypothesized that the result is not only visually similar but can be understood and mathematically described as a strain driven phase transition between a disordered bending phase and an ordered stretching phases.
3.2 Differential non-affinity and geometric phases 24
Figure 15: The configuration of network is shown for three different states. For small strains way below the critical strain the network is bend dominated. Right at the critical point for the first time stretching interactions are percolating. For strains far above the critical strain the network is clearly stretching dominated and visible is how the fibers aligning in the direction of the applied shear strain.
In this chapter we presented an analogy between ferromagnetic phase transition and non-linear stiffening in athermal networks. We made a mapping from the parameter set of Ising model to the elastic parameters in an athermal network. Based on the analogy we suggest that the nonlinear stiffening in athermal networks can be regarded as geometrical phase transition. Motivated from the theory of critical phenomenon we expect that the nonlinear elastic modu-lus can be expressed as function of the distance to the critical strain and the bending rigidity. We identified differential non-affinity as an effective measure of divergent rearrangements in a network in response to an incremental strain.
