Crosslinked polymer networks under shear

Willem van Jaarsveld

July 26, 2007

Master’s thesis,Institute for Theoretical Physics, University of Utrecht,

Supervisor: Prof G.T. Barkema.

Abstract

A model system is developed that consists of crosslinked semiflexiblepolymers. We address in detail the generation of three-dimensional ho-mogeneous networks using the model. The developed methods are thenused to investigate the relation between the shear modulus of the net-works and the crosslink density and bending and stretching moduli of thenetwork. We find a clear stiffening as the prestress increases. We alsofind a crossover from affine, stretching dominated response to nonaffine,bending dominated response.

2

Contents

1 Introduction 4

2 Methods 72.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Finding a realistic topology . . . . . . . . . . . . . . . . . . . . . 112.3 Local Minimization Method . . . . . . . . . . . . . . . . . . . . . 192.4 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Extracting Mechanical properties . . . . . . . . . . . . . . . . . . 21

3 Results 223.1 The generated networks . . . . . . . . . . . . . . . . . . . . . . . 233.2 Numerical properties of minimization . . . . . . . . . . . . . . . . 243.3 Response to shear . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Discussion 334.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Mechanical properties of the networks . . . . . . . . . . . . . . . 35

5 Outlook 36

3

Figure 1: The Cytoskeleton Pellet from Peas isolated by CSB method. Imageobtained using laser confocal microscopy. Source: http://web-mcb.agr.ehime-u.ac.jp/english/bunnshi/default.htm, (may 19 2007).

1 Introduction

In certain materials, polymer strains are connected to each other to form poly-mer networks. These polymer networks have a number of unique properties thatsets them apart from other materials. The exact mechanisms that give thesenetworks their properties are not well understood.This lack of understanding has not withheld industry from the usage of thesenetworks. They are used for numerous applications, from soft lenses to the coat-ing of optical fibers. Controlling the properties of the networks is often done bytrial and error. A better understanding of the networks would enable an easierway of obtaining the desired result.Polymer networks are also often used in food industry. For instance, transglu-taminase is an enzyme that is used to form crosslinks between protein molecules.

4

Figure 2: The cytoskeleton of myoblasts. Image obtained using scan-ning electron microscopy. Source: http://www.fbs.leeds.ac.uk/staff/peckham/unconventional-myosins.htm, (june 3 2007).

This has a lot of applications. A recent overview of these applications is givenin Ref. [1].An important part of the cell is the cytoskeleton. In figures 1 and 2 images ofcytoskeletons are shown. The cytoskeleton is a polymer network consisting ofactine filaments, among a great number of other macromolecules. It providesstability against stress besides being important for a lot of other functions inthe cell. From the astonishing behavior of macrophages, changing their form tocrawl through pores in arterioles to reach the place where they are needed, itis clear that this network is able to adapt its properties to a great extent. Athorough understanding of the passive actine networks will clearly be requiredto be able to grasp the more complex interactions that enable the cells to showthis behavior. This approach fits in a new way to try to unravel the cell me-chanics, the bottom up approach. A recent overview of this approach is givenin Ref. [2].We also find polymer networks in the tunica media of arteries, another placewhere the unique properties of the networks are used to ensure our health[3].The arteries can only be elastic to some extent, as their stiffness enables theblood to flow through the body. On the other hand, arteries are flexible anddurable enough to be used for years without rupture. Actine networks are re-sponsible for this. A better understanding of them would allow us to use theproperties for industrial applications.Other places were polymer networks are encountered are the cornea [4], the lungparenchyma [5] and plasma clots [6].As already remarked, the properties of polymer networks are poorly understood.One of the most interesting properties of the polymer networks is the responseof the networks to mechanical stress. The careful preparation of a polymer net-work in vitro allows experimentalists to examine the mechanical properties ofthe networks using microrheology. Even though the preparation of pure actine

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networks is hard [7], this approach has led to important understanding of thenetworks[8, 9, 10]. However, it is hard to use these experiments to extract aprecise value of the mesh size[11], which would make the analytical modelling alot easier.Assuming that the networks deform in an affine way on a certain microscopiclength scale, one can use the classical theory of rubber elasticity [12] to derive atheory that links microscopic and macroscopic properties of the networks [13].In this approach the mechanical response is assumed to be a consequence ofthe affine stretching of single polymer strings of length Le. Polymer stringsof greater length are assumed to be left untouched by the shear. Extensionsto this approach are able to give a good description of the nonlinear responseof biopolymer gels and bundle networks[14, 15, 16]. However, all these mod-els make assumptions on how the macroscopic strain field corresponds to theanisotropic strain at the level of individual filaments. Even though these as-sumptions are quite plausible, for better understanding we need precise quanti-tative information about this strain propagation.To shed more light on this question, a model system for the networks is proposedin Refs. [17] and [18]. These networks consist of straight polymers depositedrandomly on a two-dimensional surface. The polymers can stretch and bend,both at the cost of an increasing energy. Links are made at the spots where thepolymers cross. This model is further investigated in Refs. [19] and [20].This model has led to important understanding of the way the strain is dis-tributed over the network. The most important conclusion is that within thismodel there is a regime where affinity dominates and a regime where nonaffinitydominates. The exact mechanisms are still under discussion, but it is clear thatas the number of crosslinks decreases, the networks response becomes less affine.Nonaffinity has profound effects on stress response: the properties of the net-work change as more stress is applied. For nonaffine networks, it is hard torely on analytical techniques to find the relation between the microscopic andmacroscopic properties of the system. Sparse systems with few crosslinks areencountered in practice, for instance the cytoskeleton mentioned above.These networks are thus expected to behave in a nonaffine way. To understandthem analytical techniques alone will not be sufficient; numerical studies willalso be important.From Ref. [17] it is clear that a number of important qualitative conclusionsfrom the two-dimensional networks carry over to three dimensions. However,the dimensionality of the system must be regarded as a limitation, as the strainpropagation through the system changes as we increase the dimension. Anotherlimitation of two-dimensional networks is that the fibre density and crosslinkdensity are correlated.In Ref. [18] further investigation towards three-dimensional networks is there-fore proposed as the next logical step. This will be the topic of this thesis. Itis not straightforward to give a generalization of the method to generate thenetworks used in [17] and [18] to three dimensions. This is because in three di-mensions, strains of polymers do not cross, and therefore the crosslinks cannotbe made in the same way.

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As we will show these problems are surmountable. In section 2 we will proposea method to generate highly homogeneous crosslinked polymer networks. Thepolymers in the model are strongly related to the extensible discrete persistentchain introduced in [21]. The polymers will be generated at finite temperature,after which the crosslinks are made. The resulting topology is frozen in. Theeffective Hamiltonian will be given, as well as a good method to find the prop-erties of the system at zero temperature. We will then examine some of themechanical properties of the resulting networks.

2 Methods

2.1 The model

We investigate a network consisting of long polymers which make crosslinks.There are a great number of models for polymers. Here, we will explain themodel used and the reasons for using it. The polymers that we would like tomodel have a few important characteristics. First of all, they are very thincompared to their length. Therefore, we can safely take their width to bevanishing in the model. Another important aspect of the polymers is that theyhave rigidity, i.e. they resist bending. Furthermore, polymers generally live inthree dimensions, so all models introduced below are in three dimensions. Thisalso means that all vectors in this report are assumed to be three-dimensional.Lastly, the polymers often have a form of discreteness, in the sense that they arebuilt up as a string of monomers. A model especially suited for our purposesis the extensible discrete persistent chain model(EDPC), which was proposedin Ref. [21]. In the following we explain in detail what reasoning led to theadoption of this model for our purposes. The discrete persistent chain usesfeatures from both the freely jointed chain and the worm-like chain, we willexplain these two models first.The freely jointed chain(FJC), illustrated in figure 3, is discrete in nature. It

consists of a string of straight segments, which are attached at bending points.The polymer can only bend around these points, the rest of the polymer isfully rigid. At the bending points, there is no energy involved in choosinganother angle. This means that the FJC does not have elastic properties asa consequence of energetic interactions. The only elasticity that is found, isa consequence of the fact that more states are accessible when beginning andendpoint of the FJC are close together.The FJC has no rigidity, while rigidity is an important aspect of the physicalpolymers that we are interested in. Therefore we turn to a model that does haverigidity: The worm-like chain(WLC) model, which is illustrated in figure 4. Themodel is continuous: the configuration of the polymer can be parameterized bya C2 function ~r(s), s ∈ [0, Ltot]. We denote

~t(s) =d~r(s)ds

.

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Figure 3: Illustration of a freely jointed chain. Monomers(black dots) are con-nected by bonds of fixed length. The angle between the two bonds is arbitrary.

Figure 4: Illustration of a wormlike chain. Configurations are smooth curves inspace. Configurations which have less bending are energetically more favorable.

8

Figure 5: Illustration of a discrete persistent chain. Configurations are identicalto those of a freely jointed chain(Fig. 3), but small angles are energeticallyfavorable.

The Hamiltonian for the WLC is given by:

HWLC =∫ Ltot

0

(A

2

∣∣∣∣dt̂(s)ds

∣∣∣∣2)

ds. (1)

This favors configurations which have little bending along the polymer. Often,we assume that the polymer has constant length Ltot, in which case one canchoose the parametrization, such that |~t(s)| = 1. If we choose kbT = 1, theparameter A is a measure of the stiffness of the polymer. It denotes the per-sistence length of the polymer. This length denotes the characteristic distanceover which the tangent to the polymer is correlated.This model is better equipped for our purposes, but it still has some disadvan-tages. The most important disadvantage is that it is continuous, in contrast toreal polymers which are discrete in nature. Another disadvantage of the modelbeing continuous is that numerical manipulation is hard on these models. So,what we really need is a discrete model with the same rigidity as the WLC.To do this, we will adapt the FJC with the rigidity of the WLC. This way, weget the best of both models. So, instead of assigning the same energy to anyconfiguration in the FJC, we will now punish nonaligned bending points with ahigher energy. Let us denote the location of the bending points ~Xi, i ∈ {1, .., N}.We denote ~Xi+1− ~Xi ≡ ~di, i ∈ {1, .., N−1} the vectors pointing from one bend-ing point to the next, and we denote the angle between ~di and ~di+1 by Θi,i+1,as in figure 5. We assume that there are no external forces. Then, Ref. [21]

9

suggests a bending energy equal to:

Hbend =A

2R

N−1∑

i=1

arccos2(

~di · ~di+1

|~di||~di+1|

)(2)

=A

2R

N−1∑

i=1

Θ2i,i+1.

The introduced parameter R denotes the distance between the bending pointson the polymer. This Hamiltonian makes sure that the FJC gets the stiffnessof a WLC. Physical polymers have this rigidity as an important property, andthe behavior of networks is influenced clearly by this rigidity. Therefore, thismodel is much better suited for our purposes. This model is called the discretepersistent chain.For our purpose, it has one minor disadvantage: using partial derivatives tofind the forces for the Hamiltonian (2) in the directions ~X gives complex ex-pressions which are computationally expensive. Therefore, we propose a relatedHamiltonian:

Hbend =A

R

N−1∑

i=1

(1−

~di · ~di+1

|~di||~di+1|

)(3)

=A

2R

N−1∑

i=1

(Θ2

i,i+1 +O(Θ4i,i+1)

).

We see that for small angles both Hamiltonians give the same result while eq.(3) is computationally faster. We can still show that at kbT = 1 the parameterA is the persistence length as in eq. (1). This means that this discrete polymerhas the same rigidity as the WLC model.In the DPC model, the distance between bending points is fixed at R. Thismeans the polymer has fixed length. For a number of reasons, we want torelax this condition. First of all, the condition is physically not realistic. Realpolymers are able to stretch. Furthermore, fixed distances lead to solutions in{ ~Xi}, i ∈ (1, ..N) space that are not feasible, i.e. no energy can be assignedto them. We’d rather have a model where we can assign an energy to anyconfiguration of the bending points, as this makes life much easier later on.Thus we relax the condition by introducing central forces that will make surethat the polymer will keep its initial length:

Hstretch =B

2R

N∑

i=1

(|~di| −R

)2

. (4)

For simplicity, we have chosen for the simplest stretching energy, quadraticaround the minimum |~di| = R. The parameter B denotes the stretching con-stant per unit of length. Note that the limit B → ∞ corresponds to the unre-laxed case addressed above. The model is easily adaptable to other, asymmetricenergies, possibly with more minima, which is sometimes seen in biological sys-tems.

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Though we made a few important adaptations, the model introduced here re-sembles very much the EDPC model introduced in Ref. [21].

Networks of rigid polymers have been studied before, but in two dimensions[18]. The generation of the initial network with crosslinks is then straightfor-ward, since in a sufficient density of polymers these polymers will always crosseach other. We can then simply make a rigid connection at these crossing points.In three dimensions this is no longer the case. Therefore, crosslinks will have tobe made in some other way. Instead of making rigid connections, we choose toattach bending points that we choose to crosslink with Hookean springs. Thisway, we circumvent the problem that we cannot simply make a rigid connec-tion, but with sufficiently large spring constants we obtain the same behavior.So, we create NC crosslinks (i, j) ∈ C between various bending points. Thesecrosslinks add an energy

Hcross =Bc

2

∑

(i,j)∈C

(~Xi − ~Xj

)2

. (5)

Note that again, other choices are possible here, and we can look at the effect ofbreaking crosslinks and other rich phenomena found in nature using the sameframework.In this section, we have introduced the Hamiltonian that will be used for thenetwork. The Hamiltonian is the sum of three components, eqs. (3), (4) and(5) and is thus given by:

Htotal =A

R

N−1∑

i=1

(1−

~di · ~di+1

|~di||~di+1|

)+

B

2R

N∑

i=1

(|~di| −R

)2

+Bc

2

∑

(i,j)∈C

(~Xi − ~Xj

)2

.

(6)

Together, they capture the spirit of polymer networks in nature, the rigidityand elasticity of the polymers, and the crosslinks.In conclusion, the model can be seen as a string of bending points. Thesebending points want to have a certain distance R to their neighbors, and theywant to be on a straight line as much as possible. Furthermore, between someof these bending points there are crosslinks. Bending points that are crosslinkedwant to be close to each other.We will now have to decide how to generate the starting configuration of thenetwork, and where to put the crosslinks. We will do this in the next section.There we will also put the network in a box with periodic boundary conditions,and connect beginning and endpoint of the polymer. This results in a numberof subtleties in the interpretation of the notation and Hamiltonian introducedhere. These subtleties will be clarified as well in the next section.

2.2 Finding a realistic topology

In this section we will describe how we generated the starting configuration ofthe network.

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For simplicity, we chose to let the network consist of one long polymer, whichmakes crosslinks with itself. Extension of our methods to the use of more poly-mers is straightforward.We generate the vectors {~di} representing the distances between bending pointsat fixed temperature T . We do this because the resulting network without thecrosslinks will then be locally at some temperature at the time the crosslinksare made. If we make the network big enough, choosing this temperature highenough will result in a homogeneous network. After generating the distances{~di} we can easily compute the coordinates { ~Xi} by fixing one of them.Since we are interested in big systems and not in edge effects, we identify pointsas to create periodic boundaries. We will redefine the notation to cope with theambiguities arising when identifying these points. To make the polymer as ho-mogeneous as possible, we then attach the endpoints of the polymer. Thereafter,we choose which bending points to crosslink based on their mutual distance inthe box, or over the boundaries of the box. The topology obtained in this wayis fixed; the links are not allowed to change anymore.To generate1 the distances {~di} we note that if we generate them one at a time,we can calculate the extra energy involved with each specific choice. Then, wecan make sure that the probability that a specific ~di is chosen is proportionalto the probability it has of being present at temperature T . We assume herethat kbT = 1; the procedure can be easily generalized to other values for T . Forsimplicity we split the length and the direction of the distance vectors {~di}:

~di ≡ Li~̂di.

We note that if ~di is given, we can calculate the additional energy for choosinga specific ~di+1 using (6):

∆E =A

R

(1− ~̂di · ~̂di+1

)+

B

2R(Li+1 −R)2. (7)

From this, we infer that at constant temperature the lengths Li are independentidentically distributed with probability density:

fLi(l) =

√B

2Rπexp

(− B

2R(l −R)2

). (8)

Stated otherwise, the Li are normally distributed with mean R and standarddeviation R/B. They can thus be easily generated. Note, however, that weoften choose B À R2 , so that in effect, the bending points start out with fixeddistance R.Furthermore, knowing ~̂di, we see that ~̂di+1 has probability density:

Cnorm exp(

A

R( ~̂di · ~̂di+1 − 1)

), (9)

1There exist methods to generate pseudo random numbers with virtually any given prob-ability distribution using a computer, so all we need is to find this probability distribution.

12

Figure 6: A small polymer with 50 bending points generated at fixed temper-ature with A = 5, B = 10, R = 1 and kbT = 1. The dots are the bendingpoints, the lines show the path of the polymer. For clarity, we have chosen touse for these figures two-dimensional small polymers while we used big three-dimensional polymers in our simulations.

where Cnorm is the normalization constant. Knowing this, we can initiate ~̂d1 atrandom on the surface of the sphere. Then we can use an accept/reject approach

to generate the rest of the ~̂di.Now, we have all ~di, it is easy to find all ~Xi by fixing for instance ~X1 = ~0. Wethen get:

~Xi =i−1∑

j=1

~dj . (10)

An example of what our network could look like now is given in figure 6. Lateron, we plan to attach beginning and endpoint of this polymer. Therefore a vector~dN is needed which is to become the distance ~XN − ~X1 after the attachment.However, simply generating ~̂dN from ~̂dN−1 won’t give good alignment between~̂dN and ~̂d1, because ~̂d1 was already fixed to obtain ~̂dN recursively. This will givea sharp kink in the polymer. We have come up with the following heuristic to

solve this. We want ~̂dN and ~̂d1 to have probabilities to be generated proportionalto

P (N, 1) = Cnorm exp(

A

R( ~̂dN · ~̂d1 − 1)

).

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The following iterative procedure is used. First, {~d1, .., ~dN} are generated asdescribed above. With probability P (N, 1) the vectors are accepted. If the

vectors are not accepted, there is apparently a kink between ~̂dN and ~̂d1. Then a

new ~̂d1 is generated using eq. (9) with ~̂dN . The new ~̂d1 is thus aligned with ~̂dN ,

but it might not be aligned with ~̂d2. So accept with probability P (1, 2). If we do

not accept, ~̂d2 is generated anew, using the new ~̂d1. We accept with probabilityP (2, 3). This procedure is continued until a configuration is accepted. This willgive us a much more homogeneous polymer. So now we have a value for ~dN .However, we cannot impose ~X1 = ~XN + ~dN , because ~X1 is already fixed. Wewill deal with this problem later on, because then we can attach beginning andendpoint while introducing virtually no strain.First, we address the problem that we can only deal with relatively small samplesin numerical approaches. We compensate for this problem by creating periodicimages of our polymer at regular distances on an infinite lattice. This gives aninfinite sample in practice, and we do not have to worry about boundary effectsanymore. We do this by identifying ~X with points of the form

~X + I ·

ijk

i, j, k ∈ Z (11)

for some lattice distance DL and identification matrix I. I is initially given by

I =

DL 0 00 DL 00 0 DL

. (12)

This means we have a cubic lattice with vertices DL. Using this procedure,each bending point in space gets an infinite number of periodic images. Notethat we assume that the periodic images will always obey eq. (11), so that weonly have to keep track of the position of one the images which we will call therepresentative. The position of the rest can then be inferred from eq. (11).One way to choose the representatives could be to take as representative thecoordinates of the bending points of the original polymer. We denote this choiceby ~Xo

i , so ~Xoi points to the coordinate of the ith bending point of the original

polymer. For visualization, we refer to figure 7. However, this choice is notthe most clarifying one, because using eq. (11) changes the system to a greatextent. A more illuminating choice for representatives will be explained next.Instead of choosing the original coordinates as representatives, we choose to takeas representative for each class the unique representative that is confined in thebox [0, DL)3, which we will denote by { ~Xb

i }. See for a visualization figure 7.However, we don’t want the distances ~di to change when we put the systemin the box, because we carefully generated them the way we want them tobe. In effect, in our new representation di = Xb

i+1 − Xbi does not hold. The

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Figure 7: The polymer of figure 6 put into a box with DL = 10. Only a part ofthe infinite system is shown. The polymer gets periodic images. The originalpolymer is marked with the thick black line, points on this line are the { ~Xo

i }.The squares mark the identification lattice. Note that in each square the sameconfiguration is present. The bending points in the middle box are the onesthat represent our system after we have put it in the box, i.e. { ~Xb

i }.

15

Figure 8: The coordinates ~Xbi which represent the system after we have put it

in the box.

transformation that will put our system in the box is

~Xbi = ~Xo

i − I⌊I−1 ~Xo

i

⌋∀i ∈ (1, N)

~di = ~Xoi+1 − ~Xo

i

= ~Xbi+1 − ~Xb

i + I(⌊

I−1 ~Xoi

⌋−

⌊I−1 ~Xo

i+1

⌋)∀i ∈ (1, N − 1) (13)

where I is given in eq. 12. We let the floor operation work on each componentof the vector separately.We will use the coordinates ~di in the Hamiltonian eq. (6). An importantreason for the transformation of the coordinates ~Xi is that after doing it, wecan visualize better the effect of using the different periodic images for thepolymer. We plotted the resulting system in figure 8. We see that our systemwhich consisted of one polymer has changed to a great extent, it now looks likea part of a polymer solution. Now that we have all the periodic images for thepolymer, we are in a much better position to attach beginning and endpoint ofthe polymer. This is because the distance | ~Xb

1 − ~XbN | is generally much smaller

than the distance | ~Xo1 − ~Xo

N |. Sometimes however, it is possible to choose aneven smaller distance across the boundaries of the box, as is shown in figure 9.So, we want to impose ~Xr

1 = ~XrN + ~dr

N , for one choice of representatives for thecoordinates. To do this we denote ~lb = ~Xb

1 − ~XbN − ~dN , which is the distance

between the position where we want ~X1 to be in the box and its actual position

16

Figure 9: Illustration shows that with periodic boundary conditions, the short-est distance is not always the distance within the box, but can be across theboundaries.

17

in the box. We choose the ~l with minimal length to obtain minimal distortionof the system:

~l = minbox(~lb) (14)

= ~lb −DL

1DL

~lb −

0.50.50.5

. (15)

Now, we perform a transformation of the ~Xi,~di:

~X ′i = ~Xi +

i− 1N

~l

~d′i = ~di +1N

~l.

After this transformation, ~Xb1 = ~Xb

N + ~dbN holds modulo the equivalence relation

(11).Since we use values for which the total polymer length is very big compared toDL we have that |~lmin| ≤

√3/4DL ¿ NR ≈ N |~di|, so |~lmin|/N ¿ |~di|. We

conclude that the change to the system is negligible.

Lastly, we will make the crosslinks. Intuitively, we adopt the following pro-cedure:Initially we have polymers of thickness 0, this is why they don’t touch. Byincreasing the thickness of the polymers, we can make sure that they eventuallywill touch. Gradually increasing thickness, we make crosslinks between poly-mers that touch, until a certain predetermined number of crosslinks per polymerlength is made. Then, we set the thickness back to 0 and attach springs to thecrosslinked polymers.We will denote the distance between bending point i and j by D(i, j). We takefor this distance the minimal distance within or across the boundaries of thebox, given by

D(i, j) = minbox( ~Xi − ~Xj), (16)

where minbox is given in eq. (14). Now, some width w of the polymer couldbe used and all bending points (i, j) with distances closer than w could beattached, gradually increasing w until we have the desired number of crosslinks.However, applying this procedure gives undesired results. First of all, it wouldin principle make crosslinks between points (i, j) that are neighbors or almostneighbors. This is not how we expect crosslinks to be made in real systems, inany case, it won’t give the system much strength. Pairs of bending points (i, j)that are considered to become crosslinked should not already be neighbors oralmost neighbors, i.e.

|i− j|(R− 2B/R) > (1 + ε)w,

where ε is some small number2.Furthermore, once we attach polymers i and j we should not attach neighbors

2We chose ε = 0.5.

18

of i to neighbors of j anymore, so that for attached bending points i, j it holdsthat:

∀l ∈ {i− n, .., i + n}, k ∈ {j − n, .., j + n} : D(i, j) ≤ D(k, l).

The reason we don’t want this is because a double crosslink between two stringsof bending points would not only attach the two strings, but also make it en-ergetically favorable to have the two strings aligned. Even though this kindof behavior is very interesting, this research will be restricted to the simplercrosslinks that only favor crosslinked strings to be close.An implementation to find which bending points to crosslink that is both fastenough and easy to implement is to list all pairs that have the last two charac-teristics and that are closer than some threshold. Then, from those pairs, theM pairs that are closest are selected to become crosslinked.We now have obtained a homogeneous network, that has a random topology.This topology is fixed. However, the coordinates {Xi} used to generate thetopology will change3. We will find a local minimum in the energy of the net-work with the given topology. This will be done in the next section.

2.3 Local Minimization Method

We have generated a network, but to calculate the shear modulus we need away to evaluate the energy and the response of the network to shear. This isdone by finding a minimum in the energy for the topology obtained in section2.2. This minimum corresponds to a specific coordinate for each bending point.To find this minimum we need the Hamiltonian eq. (6). For simplicity of no-tation all the coordinates of bending points will be bundled in ~Y . Note thatan initial vector ~Y is known, which is probably close to the optimum. Thisvector corresponds to the coordinates used to generate the topology. The force~F on the system can easily be computed using the partial derivatives of theHamiltonian. A number of derivative based techniques exist to find the localminimum of such analytical functions. Because we couldn’t find in literaturethe specific algorithm we used, we will describe it here. The main idea behindthe algorithm is to let the system perform a Newtonian motion. However, eachtime the potential energy starts to rise, we halt the system, so that all kineticenergy is lost. We do this until the total force becomes smaller than some pre-determined value, which we will denote by Ft. In detail:

• Initiate value for step size ∆t, calculate initial energy E and force ~F .

• While |~F | < Ft

– Update:

~V := ~V + ∆t ~F

~Y := ~Y + ∆t~V

3Of course, there are boundary conditions on the ~di, corresponding to the way the networkis oriented in the box.

19

– Calculate Enew and ~F from new ~Y .

– If potential energy Enew > E undo last step and halt particle:

~Y := ~Y −∆t~V

~V := ~0Enew := E

– Otherwise

E := Enew

• accept E and ~Y as final solution.

When many steps are not accepted, the value of ∆t should be decreased. Thisalgorithm is similar to steepest descent, but much faster.

2.4 Shear

Now, after generating the network, it is interesting to measure its properties.We are mainly interested in the effect of shearing the network.We first note that points in space are identified according to eq. (11), and wenote then that we can perform a shear by transforming each coordinate using ashearing matrix S, which can be for instance of the form:

S =

1 0 ε0 1 00 0 1

. (17)

We then have the transformation:

~X ′i + I

ijk

= S ·

~Xi + I ·

ijk

,

or

~X ′i = S · ~Xi

I ′ = S · I. (18)

We see from eq. (18) that we can perform our transformation on any repre-sentative of the coordinate of a particle, as long as we adapt our identificationmatrix.Note that we can use this method to perform other linear transformations onthe system as well. For instance, we can stretch the networks using the trans-formation matrix:

S =

1 + ε 0 00 1 00 0 1

. (19)

20

2.5 Extracting Mechanical properties

We want to use the methods described above to find the mechanical properties ofthe networks. The procedure we use is to generate a network, and subsequentlyfind the energy of this network. Then, we perform a shear on the network, andwe find the energy again. Since we know the volume V of the network we canobtain the energy density as a function of the shear

E(x)V

, x ∈ {−Nε,−(N − 1)ε, .., (N − 1)ε,Nε}. (20)

Using discrete analysis we can then find the force in the direction of shear(−∂E∂s )

and shear modulus density of the network:

F (x) =E(x− ε)− E(x + ε)

2εVx ∈ {−(N − 1)ε, .., (N − 1)ε} (21)

G(x) =E(x + ε)− 2E(x) + E(x− ε)

ε2Vx ∈ {−(N − 1)ε, .., (N − 1)ε}. (22)

Note that we expect the shear modulus to depend on the amount of prestress,since this is a general property of polymer networks as is shown in the Ref.[16]. Because we don’t want the results to be dependent on an overall factorin the hamiltonian, we will always normalise the quantities so that they areindependent of B. For a visualization of the relation between the three quantitiesE(x), F (x) and G(x), we refer to figure 14.We will also be interested in the value of Gaffine(x). This corresponds to theshear modulus the network would have if it responded totally affinely. It is forour model not straightforward to derive an analytical value for Gaffine. A reasonfor this is that the distances between crosslinked bending points is not known4.We therefore will use our simulation model to estimate Gaffine(0). This is doneby first minimizing the energy for the network at zero shear. This energy willbe denoted Eaffine(0) = E(0). After this, we perform a shear nε, and calculatethe energy again. Note that we do not minimize, since we are interested inthe affine response. This way, we can obtain the energy corresponding to affineresponse to different shears:

Eaffine

V(x), x ∈ {−Nε,−(N − 1)ε, .., (N − 1)ε,Nε}. (23)

We can then obtain an estimator for Gaffine(0) in the same way as used in eq.(22).A last quantity that we are interested in is the degree of bending. We will defineit as:

Dbend =Ebend

Etotal, (24)

where Ebend and Etotal are given by eqs. (3) and (6), respectively.4If we had rigidly attached the crosslinks, things would be different, but finding an exact

expression for Gaffine would still be hard.

21

3 Results

In the previous chapter we have developed a model for polymer networks. Wehave introduced a way to generate networks having certain properties. Fromthese networks, we now want to extract the mechanical properties.First of all, let’s sum up the control variables when generating and evaluatingthe networks. When looking at the energy, we have the control variables A,B and BC . The network is generated at a certain temperature T , giving thepolymer an initial persistence length

A

KBT. (25)

The simulations are performed in the regime B À RKBT so that the exactvalue of B does not have effect on the generation of the network5.Of course, once the network has been generated, an overall factor in the energyscales out. We restrict ourselves to BC = B

R , so that only one independentparameter remains6. As is already remarked in [19] we need in our three di-mensional system two quantities to fully describe the geometry of the randomnetwork, the density of crosslinks and the density of polymers. This in contrastto the two dimensional model where one parameter is sufficient since the poly-mer density fully determines the crosslink density.Another complication in comparing the results for our model to the results ob-tained in Refs. [17, 18, 19, 20] will be that a density in two dimensions does notcorrespond in a unique way to a density in three dimensions.The geometry of the network is governed by N , R , DL and NC . In principle,we can always choose networks large enough so that finite-size effects will notplay a role. This will leave us with three independent variables. First of all, thebending point density of the network:

ρ =N

D3L

. (26)

Then, the average number of bending points between the crosslinks across thepolymer, given by

lc =N

2NC. (27)

This leaves us one independent variable, which we choose to be R. Note thatfor fixed R, ρ, and lc, we can always scale up N , NC and DL so that we are ina region where finite-size effects play no role.In the following section the effectiveness of the different heuristics used in thegeneration will be investigated. Then we will take a look at what is our primaryinterest, the response to shear of the networks.

5It has however effect on the properties of the network under shear.6Tests have indicated that the results do not depend significantly on the parameter BC ,

as long as BC & B/R.

22

Figure 10: An example of a network generated with the procedure described insection 2.2. This figure shows that the network is homogeneous.

3.1 The generated networks

Choosing ρ = 1250, R/LC = 0.05 and A/kbT = 50, a network shown in figure10 is obtained. For this network we can plot the correlation of the directionacross the polymer, as given by:

c(dt) =⟨

~d(t) · ~d(t + dt)⟩

=1N

N∑

i=1

~di · ~dj(dt), (28)

where

j(dt) = mod (i + bdt/Rc, N).

This correlation for the network from figure 10 is plotted in figure 11. Weconclude that the different heuristics we used do not have an effect on thecorrelation across the polymer at the time the crosslinks are made.

23

200 400 600 800 1000dt

0.2

0.4

0.6

0.8

1

cHdtL

Figure 11: The correlation of the direction across the polymer, together with thereference line c(dt) = exp(−dt/50). The correlation across the generated poly-mer follows the theoretically expected correlation, apart from some deviationsthat can be attributed to the finite size of the sample.

3.2 Numerical properties of minimization

Another matter of interest is the effectiveness of the minimization procedure infinding a local minimum. In figures 12 and 13 it is shown that a minimum isindeed found. This is however not the case for all parameters. We are able tofind a minimum for parameters for which the following approximate inequalitieshold7:

NC & N

20, (29)

B & A. (30)

We suspect that we cannot find the minimum for the other parameters becausethey correspond to systems which have a great contrast in the energy dependenceof the different degrees of freedom, or to systems which are under-constrained.The minimization is numerically very expensive, even for moderately sized net-works. For a network of 10000 bending points, typically 3 milliseconds arerequired for a single step8. For a single minimization of a network with about10000 bending points we need between 2000 and 50000 steps, depending on thenumber of crosslinks and the persistence of the polymers.

7These values have been obtained by trial and error.8The program was run on the Venus server, property of the Universiteit Utrecht. For

specifications see http://www1.phys.uu.nl/helpdesk/linux/venus/default.htm.

24

500 1000 1500 2000 2500 3000steps

1.664

1.666

1.668

E

Figure 12: The energy during minimization as a function of the number of steps.After ∼ 2000 steps, the energy does not decrease significantly anymore. This isa weak indication that a local minimum has been found.

2.5 7.5 10 12.5 15 17.5 20-lnHÈFÈL

1.66

1.68

1.72

1.74

E

Figure 13: Scatter plot of Energy E and force strength |~F | during minimization.Each data point corresponds to the − log |~F | and E measured at a particularstep during minimization. After a while, the force keeps dropping with ordersof magnitude, while the energy stays the same. This indicates that a localminimum has been found.

25

The generation of the network does not cost much time, typically 5−10 secondsfor a network with 10000 bending points.We conclude that the size of networks that can be investigated is limited, whichmeans that finite-size effects can play a role.

3.3 Response to shear

We are interested in the response to shear of the networks that we have devel-oped. Using the methods described in section 2 the information we need can beextracted.In this discussion we will concentrate on the mechanical properties as the num-ber of crosslinks and the persistence of the polymer are varied. Networks withρ = 1250 and R = 0.1 will be investigated9. We choose

A

RkbT= 50. (31)

For these networks we are interested in the shear modulus as a function ofshear. This quantity is closely related to the force and energy as a functionof the shear. To visualize the relation between these three quantities and toshow the qualitative mechanical behavior for our networks, these quantities areplotted as a function of shear in figure 14. Note that in all figures, when givingresults, the energy, force or shear per unit of volume will always be used. Wealso divide by B to obtain a quantity that is independent of an overall factor inthe Hamiltonian.A matter of interest is the effect on the shear modulus when using different

values for A/B in the networks. The results are plotted in figure 15. Anothermatter of interest is the dependence of the response of the networks on thecrosslink density. The results are plotted in figure 16.

Now, attention will be focused to the shear modulus of the networks underzero prestress. The crosslink density and the persistence A/B will be varied.The results are plotted in the figures 17, 18 and 19.The dependence of the networks on the persistence on generation is shown infigure 20.

9Throughout, we will concentrate on networks with 10000 bending points.

26

-1.5 -1.0 -0.5 0.5 1.0 1.5Shear

-0.5

0.5

1.0

1.5

E,DE �DS,G

Figure 14: The energy, −∂E∂s , and shear modulus as a function of the shear of a

typical network. For this network A/B = 0.1 and 1/lc = 0.4. The × mark theenergy, the ¤ mark the force and the + mark the shear modulus. For bettervisualization, we scaled up the shear modulus by a factor 5. A clear stiffening ofthe network is observed as we apply more and more shear. If we apply enoughshear, we observe that the shear modulus becomes constant(not shown). Thisqualitative behavior is observed for all networks that were investigated. Theinterpolation between the data points does not reflect measurements, it is justthere to guide the eye.

27

-1.0 -0.5 0.5 1.0Shear

0.18

0.20

0.22

0.24

0.26

0.28

G

Figure 15: The shear modulus as a function of shear distance for networks ofdifferent A/B with fixed 1/lc = 0.4. The × mark A/B = 0.25, the ¤ markA/B = 0.05 and the + mark the A/B = 0.01. For small shear, the shearmodulus of the network depends on the persistence of the individual polymers.For bigger shear this dependence reduces and the curves collapse(not shown).There is some asymmetry in the curves, a consequence of the finite size of thenetworks.

28

-1.0 -0.5 0.5 1.0Shear

0.20

0.22

0.24

0.26

0.28

G

Figure 16: The shear modulus as a function of the shear distance for networkswith a different number of links densities 1/lc. The × mark 1/lc = 0.8, the ¤mark 1/lc = 0.6 and the + mark the 1/lc = 0.4. A/B = 0.05 is kept fixed.The shear modulus of the networks increases with the number of crosslinks,regardless of the prestress. There is some asymmetry in the curves.

29

0.2 0.4 0.6 0.8 1.0 1.2Crosslink Density

0.5

1.0

1.5

2.0

G,Affinity ,Dbend

Figure 17: The shear modulus at zero prestress(+), the degree of bending(¤),and the affinity G(0)/Gaffine(0)(×) for networks of different crosslink density.Here and in all subsequent figures, we used the average of a great number ofnetworks to compute the curves. The asymptotic errors are smaller than thesymbols. A/B = 0.05 is kept fixed. Note that for large crosslinks density, thedegree of bending and affinity are constant, while for small crosslink density,the degree of bending increases and affinity decreases. In this nonaffine regimethe shear modulus depends stronger on the number of crosslinks than in theaffine regime. Furthermore, in the regime where response is affine, the shearmodulus increases monotonically as the number of crosslinks increase. We seethat G(0) approaches 0 as the crosslink density is decreased, even before 1/lcreaches 0. This is probably related to the rigidity percolation point, but becauseminimization becomes harder and harder as we decrease the number of crosslinkswe cannot extract more quantitative results.

30

0.2 0.4 0.6 0.8 1.0 1.2Crosslink Density

0.5

1.0

1.5

2.0

G

Figure 18: The shear modulus at zero prestress for networks with differentcrosslink density 1/lc. To investigate the dependence of the crossover on thepersistence, we varied the persistence. The × mark A/B = 0.05, the ¤ markA/B = 0.01 and the + mark the A/B = 0.002. The crossover starts at lowercrosslink density as the persistence is increased. This is also reflected in theaffinity and degree of bending(not shown). Note that in the affine regime, thecurves are parallel.

31

0.1 0.2 0.3 0.4 0.5Persistence

0.05

0.10

0.15

G

0.1 0.2 0.3 0.4 0.5Persistence

0.2

0.4

0.6

0.8

Dbend

Figure 19: The shear modulus (upper figure) and the degree of bending (lowerfigure) at zero prestress for networks with different persistence A/B. The rela-tion is investigated for three crosslink densities: 1/lc = 0.4(×),1/lc = 0.2(¤) and1/lc = 0.14(+). Regardless of crosslink density, the shear modulus approaches0 as A/B approaches zero. There is a strong increase in shear modulus as thepersistence increases. Also, the degree of bending decreases. Note that eventhough the degree of bending increases as A/B decreases, it does not approach1.

32

20 40 60 80 100A�KbT

6

8

10

12

14

GH0L

Figure 20: The shear modulus for networks generated with different initial ther-mal persistence lengths A/(kbT ). We keep A/B = 0.01 and 1/lc = 0.4 fixed.Initial increases in thermal persistence length have a large impact, while as thepersistence length increases, dependence decreases.

4 Discussion

The main goal of the thesis is to develop a way to generate realistic polymernetworks. We will first discuss the developed procedure, and recommendationsfor use and extension will be given.We will then discuss the investigation of the mechanical properties of the net-works.

4.1 The model

Our approach enables us to successfully generate random homogeneous net-works in three dimensions. We did this by generating the network at a finitetemperature. This approach has the advantage of being simple, intuitive andeffective.There is good resemblance between the polymer networks observed in nature(figure 2) and the generated networks (figure 10).Furthermore the procedure is fast. We were able to generate a network with20000 bending points and 7000 crosslinks on a standard desktop pc within 10seconds.In our approach the networks is build up from one giant polymer. We will dis-cuss here the reasons for using this approach.First of all, in nature networks often consist of very long polymers. In our ap-

33

proach, we investigate such networks. Additionally, using this approach assuresthat the resulting network is fully connected. Non fully connected parts of apolymer network do not contribute to the mechanical properties of the network.Also, dangling ends do not contribute to the mechanical properties. So the con-nected part of the networks is the only part that is of interest to us.

However, if one would like to investigate networks consisting of a mixture ofpolymers with different properties, it is necessary to use multiple polymers inthe model. The dangling ends and parts that are not connected to the rest ofthe network can complicate the procedures used to extract mechanical proper-ties of the networks. So they need to be removed.Furthermore, we do not recommend to attach beginning and endpoint of poly-mers when short polymers are used, because this cannot be done without intro-ducing much strain. We do not foresee any further problems using our methods.

To get rid of boundary effects the network is put in a box with periodic boundaryconditions. The beginning and endpoint were attached. To avoid a kink at theconnection a simple and effective rule was used. The attachment of beginningand endpoint has the advantage that the network becomes more homogeneous,and we avoid problems with dangling ends.The rule used to make the crosslinks is both simple and effective. It can beseen as a direct generalization of the rule used in Ref. [17], where crosslinksare made at points were polymers cross. We used exactly this approach. Weensured that the polymers cross by giving them a finite thickness. A big ad-vantage of this approach is that we can vary the number of crosslinks withouthaving to increase the polymer density, while in the model used in Ref. [17] thepolymer density and crosslink density could not be controlled independently.This however complicates the comparison of the results obtained in Ref. [17]with the results obtained here. This problem can not be fully overcome, forthere is no way to relate a density in two dimensions to a density in three di-mensions without arbitrary assumptions.We attached the crosslinks using Hookean springs. In principle, any connectioncan be used, even a rigid connection. We did not investigate the effect of chang-ing this connection.A method was developed to shear the network. We do this without using metalplates, which has the advantage that no arbitrary distance between those plateshas to be chosen. This method to apply the shear over the boundaries is alsouseful to investigate the effect of other linear transformations on the system.After the shear has been applied, the minimum of the Hamiltonian moves. Thismovement corresponds to the reaction of the network to the shear. This reactionis composed of two parts, the affine part, and the nonaffine part. We developeda way to distinguish between these parts, which is very simple to implementand works for general Hamiltonians.However, to find the nonaffine part of the reaction an expensive numerical mini-mization is needed. This limits the size of the networks that can be investigated.This is a severe limitation, because indications of finite-size effects are found in

34

some simulations. See for instance a slight asymmetry in figure 15. It is veryhard to assess the effect of these size limitations on the results, but we expectthat in any case the general qualitative results will also hold for bigger systems.Using numerical derivatives, we were able to find the mechanical properties ofthe networks from the energy.

4.2 Mechanical properties of the networks

There are a number of indications that our networks indeed capture importantaspects of polymer networks in nature. First of all, in figure 14 it is shown thatthe shear stiffening observed in nature is also observed in the model networks.The networks have strongly decreasing shear modulus as the crosslink densitygoes to zero. Real networks become fluid as the crosslinks are removed, whichalso means they have zero shear modulus.The developed networks show the strain stiffening that also is found in practice,and that is explained in the article [16]. This gives confidence that the modelresembles physical polymer networks.It was shown that for increasing prestress, the dependence on the persistenceof the polymers decreases. Furthermore, the number of crosslinks has influenceon the shear modulus regardless of the prestress. With precise control of thenumber of crosslinks this behavior should be observable in vitro.This makes clear why we find polymer networks everywhere in nature: adaptingthe number of crosslinks and the persistence of individual polymers, networkswith virtually any nonlinear response can be obtained.After making these observations on the response of the networks to prestress,we turn to the behavior of the networks for zero prestress. It was shown that forfixed lb, for sufficient crosslink density we are in an affine regime, where the re-sponse to shear is mainly due to stretching. Decreasing the crosslink density wefind a gradual decrease of the shear modulus, staying in a regime where stretch-ing is dominant. Then we come to a point where further decrease in crosslinkdensity has a drastic effect. Simultaneously the degree of bending increases andthe affinity G/Gaffine decreases. We call this regime the stretching dominatedregime. In this regime, the dependence of the shear modulus on the number ofcrosslinks is enhanced.Most of these observations are in parallel to the observation made on the two-dimensional model in e.g. Ref. [19], where also observe two distinct regimeswere reported depending on the crosslink density. In one regime, the affinestretching dominates, while in the other regime, nonaffine bending dominates.However, there is one distinction: in the two-dimensional model the dependenceon crosslink density is stronger in the affine regime than in the nonaffine regime.We obtained less dependence on crosslink density in the affine regime.This distinction could be due to the fact that crosslink density and polymerdensity are closely related in the two-dimensional model, while they are in-dependent in our model. We keep the polymer density fixed while increasingpolymer density, they increase both at the same time.There is no easy way out of this problem, because densities in two dimensionsdo not correspond to densities in three dimensions, so there is no way to mimic

35

the approach in two dimensions. However, an extension of the model usedin Ref. [19] where crossing polymers are crosslinked with some probability pwould enable a better comparison of the two methods, because then the num-ber of crosslinks could be varied without varying the density.In any case, the observations give confidence that the crossover between affineand nonaffine response is also relevant for real polymer networks, because it hasnow been observed in two very different model systems.The crosslink density at which the crossover takes place depends on the persis-tence of the polymers. With increasing persistence, the crossover is at a lowercrosslink density. However, it is hard to characterize the crossover in a objec-tive way. Moreover, it is not clear on what parameters the crossover dependsin our model. Therefore, we will not try to find a quantitative relation betweenpersistence and crossover.If we decrease the number of crosslinks too much, the minimization proceduredoes not converge anymore. This is probably due to a fluidization of the net-work, very much like the rigidity percolation point in two dimensions. We didnot find a precise characterization of the point where minimization was no longerpossible. This would be very interesting for future research.We also looked at the dependence of the network on the persistence for fixedcrosslink density. If we set the persistence too close to zero, we are not able tominimize the energy anymore. This is probably because setting the persistencetoo close to zero makes the system too floppy.For these small values of the persistence the shear modulus depends to greatextent on the persistence, approaching zero as the persistence approaches zero.By increasing the persistence, we come in a regime where the stretching domi-nates. In this regime, the shear modulus does not depend much on the persis-tence, and response is mainly affine.It can be concluded that the affine regime can be reached for every fixed crosslinkdensity by increasing the persistence length. For any fixed persistence length,the affine regime can be reached by increasing the number of crosslinks.We have shown that the shear modulus of the polymer becomes smaller andsmaller as the persistence on generation is decreased. This probably is due tothe ”folded” polymer strands between crosslinks for these networks.A deeper investigation of this effect would be very interesting.

5 Outlook

The model that was described in this thesis is one of a number of models thatwill be used to gain a better understanding of polymer networks, in particularthe strain propagation in polymer networks. Because it is sometimes unclearwhether observed behavior on a model system is generic or model specific, it isalways important to compare the outcome of different models.The models that will be compared will have a number of characteristics in com-mon. These are the polymers and the crosslinks.However, in each model the network is generated in a completely different man-ner. This way it is possible to identify which behavior is model specific and

36

which behavior is generic for crosslinked polymer networks.The main conclusions for this research were the strain stiffening, and the crossoverfrom a bending to a stretching dominated regime. We therefore recommend thatthe results regarding this will be compared to the results obtained with differ-ent models. In Ref. [17] the crossover from an affine to a nonaffine regime wasalready identified. Because the model used in Ref. [17] was in two dimensionsa quantitative comparison was not possible.There are also a number of interesting aspects of polymer networks that havebeen neglected in this research. These are the dependance of the networks ontheir density, and the response of the networks to uniaxial strain. It is clearthat these aspects are important to take into account in subsequent research.Finally, we would like to point out that an early version of the model exhibitedbehavior very similar to the bundling as observed in many physical polymernetworks. In this version, after the network was generated some crosslinks weremade between bending points with a distance smaller than some threshold, afterwhich the energy was minimized. Then more crosslinks were made, after whichagain minimization again took place. This gave networks which had bundles ofpolymers. It would be interesting to investigate the mechanical properties ofthese networks.

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