Computational Materials Science 98 (2015) 24–33

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Co-continuous polymer systems: A numerical investigation

http://dx.doi.org/10.1016/j.commatsci.2014.10.0390927-0256/� 2014 The Authors. Published by Elsevier B.V.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

⇑ Corresponding author at: School of Mechanical and Materials Engineering,University College Dublin, Belfield, Dublin 4, Ireland.

E-mail address: d.carolan@imperial.ac.uk (D. Carolan).

D. Carolan a,b,⇑, H.M. Chong a, A. Ivankovic b, A.J. Kinloch a, A.C. Taylor a

a Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UKb School of Mechanical and Materials Engineering, University College Dublin, Belfield, Dublin 4, Ireland

a r t i c l e i n f o

Article history:Received 26 August 2014Received in revised form 13 October 2014Accepted 17 October 2014

Keywords:Cahn–HilliardPhase separationFinite volume methodMechanical properties

a b s t r a c t

A finite volume based implementation of the binary Cahn–Hilliard equation was implemented using anopen source library, OpenFOAM. This was used to investigate the development of droplet and co-contin-uous binary polymer microstructures. It was shown that the initial concentrations of each phase definethe final form of the resultant microstructure, either droplet, transition or co-continuous. Furthermore,the mechanical deformation response of the representative microstructures were investigated underboth uniaxial and triaxial loading conditions. The elastic response of these microstructures were thencompared to a classic representative microstructure based on a face centred cubic arrangement ofspheres with similar volume fractions of each phase. It was found that the numerically predicted compos-ite Young’s modulus closely followed the upper Hashin–Shtrikman bound for both co-continuous andclassical structures, while significant deviations from analytical composite theory were noted for the cal-culated values of Poisson’s ratio. The yield behaviour of the composite microstructures was also found tovary between the co-continuous microstructures and the representative microstructure, with a moregradual onset of plastic deformation noted for the co-continuous structures. The modelling approach pre-sented allows for the future investigation of binary composite systems with tuneable material properties.� 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://

creativecommons.org/licenses/by/3.0/).

1. Introduction

Structural adhesives are increasingly used for bonding compo-nents within critical load bearing engineering structures such asaerospace and automotives. Typically these adhesives are basedon epoxy polymers. Epoxies are inherently brittle due to theirhomogeneous microstructure and highly cross linked nature. Thus,there has been much research focused on improving the fracturetoughness of epoxy polymers by incorporating a second minorityphase at the nano-scale. These modifiers fall into one of two maincategories: inorganic additives, e.g. silica [1,2], glass [3], alumina[4], nano-clays [5] and carbon nanotubes [6,7] or organic, usuallyrubber particles. Rubbery additives can be either core–shell rubberparticles [8–10] or can form during curing via reaction inducedphase separation mechanisms [11,12]. The primary energy dissipa-tion mechanisms for rubber toughened epoxies are known to beboth plastic void growth and shear band development [13]. It hasalso been shown that a combination of the above additives to cre-ate a hybrid material can provide synergistic toughening effects,e.g. carbon nanotubes and silica nanoparticles [14] or rubber withsilica nanoparticles [15–17].

Kinloch et al. [12] and Brooker et al. [18] have studied themorphology and fracture toughness of thermoplastic toughenedepoxy polymers. They noted a change in morphology from a spher-ical-particulate morphology in an epoxy rich continuous phase atlow concentrations of thermoplastic phase. The morphology chan-ged to a co-continuous structure as the percentage of thermoplas-tic toughener was increased. Finally at very high concentrations ofthermoplastic phase, a phase-inverted structure was observedwhere the epoxy was dispersed as spherical particulates in a con-tinuous thermoplastic phase. The fracture toughness was found toincrease with increasing thermoplastic content, although this wasnot attributed to the change in morphology but rather as a functionof the percentage of the thermoplastic phase in the polymer blend.

More recently, significant advances [19] in polymer synthesishas allowed researchers and manufacturers the ability to createspecific copolymers. There are many possible combinations ofblock copolymers (BCP), and amphiphilic BCPs are of particularinterest for toughening epoxy polymers. The BCP modified epoxieshave been shown to have a lower viscosity than conventional rub-ber modified epoxies [20] which make them attractive for resininfusion processes. Several researchers have already shown thatthese BCPs can form complex nanostructures via self-assembly[21,22] or reaction-induced phase separation [23] to significantlyincrease the mechanical and fracture performance of epoxy poly-mers. Some of these hierarchical substructures include spherical

1.5 1 0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

F (

)

PurePhase 1

PurePhase 2

D. Carolan et al. / Computational Materials Science 98 (2015) 24–33 25

micelles, vesicles and worm-like micelles [24]. Of particular inter-est in this work is the formation of a co-continuous, or bicontinu-ous gyroid, structure [25]. Co-continuous structures are structureswhich contain two or more interpenetrating and self-supportingphases. These types of structures have been observed for thermo-plastic modified epoxies but the energy dissipation mechanismsfor such a complex structure were not fully understood[12,26,27]. Most of the early work on BCP modified epoxies hasbeen primarily focused on experimentally observing the morphol-ogy and resulting mechanical properties. The study of the toughen-ing mechanisms and mechanics of deformation of themicrostructure is required in order to fully understand the struc-ture/property relations.

Furthermore, such complex designed binary ordered structurescan provide outstanding combinations of material propertiesincluding stiffness, toughness and strength that would otherwisenot be possible in a homogeneous material and have been the sub-ject of intensive research efforts over the last number of years.Wang et al. [28] studied the energy dissipation in binary compositesystems with the phase interface closely approximating a triplyperiodic minimal surface. Similarly, work by Lee et al. [29] hasshown that the energy dissipated per unit volume in a voidedepoxy nanostructure can be significantly enhanced for a systemof lower relative density by close control of the nano-framegeometry. Torquato [30] has studied the developed of optimisedco-continuous structures with targeted properties such as minimalthermal expansion. It is therefore clear that understanding theunderlying physics of phase separation and the subsequentinfluence of the morphology on the mechanical properties of thecomposite blend, is key to guiding the future design of improvedmaterials for structural applications.

In this work, we use the well-known Cahn–Hilliard equation todescribe the temporal evolution of a binary phase field of twomixed polymers and subsequently investigate the mechanicalresponse of the evolved microstructures under different loadingconditions.

2. Cahn–Hilliard model

The Cahn–Hilliard equation was originally proposed in 1958[31] to model phase separation phenomena in binary alloys. Sincethen, it has been applied to a wide number of other diverse usesincluding the evolution of arbitrary microstructures [32,33],tumour growth [34] and image in-painting [35]. The Cahn–Hilliardequation can be written as:@c@t¼ r � ðMrlÞ ¼ MDl ð1Þ

where c is the concentration of one phase in the mixture, t is time,M is defined as the mobility, and l is the local chemical potential,more commonly defined as:

l ¼ F 0ðcÞ � cDc ð2Þ

In Eq. (2), FðcÞ represents the free energy density of a homogeneousmaterial of concentration c, and c is a constant related to the thick-ness of the interfacial regions between phases. The free energy, FðcÞ,is based on a Ginzburg–Landau functional of the form:

�ðcÞ ¼Z

XFðcÞ þ c2

2jrcj2

� �dx ð3Þ

The first term on the right hand side of Eq. (3) represents thehomogeneous free energy while the second term penalises largegradients. The free energy is generally deduced via the Flory–Huggins model [36]:

FðT; cÞ ¼ RT ð1� cÞ lnð1� cÞ þ c ln c½ � þ hcð1� cÞ ð4Þ

where h is a variable representing the Flory–Huggins interactionparameter, and R and T are the universal gas constant andtemperature respectively. From Landau theory, an alternativequartic approximation can be written as [36]:

FðcÞ ¼ aðc � bÞ2ðc � cÞ2 ð5Þ

where a; b and c can take suitable values. In order to derive aneasily implementable numerical solver for Eq. (1), we first definean order parameter / which represents the concentration differencebetween phase A and B at any point in the domain of interest.Therefore, we have / ¼ cA � cB with cA þ cB ¼ 1. It is easy to observethat / ¼ 2c � 1. Eq. (1) can then be rewritten as:

@/@t¼ MD /3 � /� cD/

� �ð6Þ

It can be seen in Fig. 1 that the new definition of Fð/Þ ¼ 14 ð/

2 � 1Þ2 isa double well potential with local minima at / ¼ f�1;1gcorresponding to the pure phases.

The initial concentration of each phase in the mixturedetermines the subsequent pattern evolution. Specifically, for nearequal concentrations of phase, a co-continuous pattern emerges.However, in the case where one phase is present in the mixturein a much greater concentration than the other, a droplet patternemerges. In this case, one phase can be considered as the matrixand the other as dispersed spherical inclusions.

Eq. (6) was implemented in foam-extend-3.0 [37,38]. FOAM is afully 3-dimensional, finite volume, object oriented C++ library. It isprimarily used to create numerical solvers for multi-physicsproblems. Originally developed for computational fluid mechanics,it has recently found application in other areas such as solidmechanics [39–42] including fracture mechanics [43–46], solidifi-cation problems [47] and fluid–structure interactions [48,49].

2.1. Two-dimensional cases

Using the model described, we can examine the evolution of theconcentrations of two phases with a random perturbation in atwo-dimensional system. First, we prescribe the concentrationdifference between each phase as an initial condition (the concen-tration difference as defined by Eq. (6) can take any value between�1 and 1). To initialise the random perturbation, we chose aGaussian normal distribution about the mean concentration witha standard deviation of 0.1, i.e. 5% of the total range.

A unit square domain with equal sides of 100 lm with a cellsize of side length 1 lm was used. A time step of 0.1 ms was usedin all cases while the mobility, M, was set to 0.01 m2/s, with c(which controls the width of the interface region) equal to

Fig. 1. The double well potential Fð/Þ ¼ 14 ð/

2 � 1Þ2.

(a) t = 1 (b) t = 10 (c) t = 40

Fig. 2. Evolution of phase variable at different values of t with initial concentration c ¼ 0:3.

26 D. Carolan et al. / Computational Materials Science 98 (2015) 24–33

1 � 10�4 m2/s. A number of different simulations were performedat various volume fractions to examine the role of volume fractionon the resultant morphology. The boundary conditions were set tobe periodic. Fig. 2 presents the temporal evolution of the phasemorphology with an initial mean concentration of 30%. It can beseen that the mixture decomposes into discrete phases, comprisingof a matrix (blue1 phase) and dispersed droplets (red phase). Thegreen phase demonstrates that in the Cahn–Hilliard implementationthe phase boundaries are not discrete, but rather are diffuse over agiven length. For very small particles the gradient term in Eq. (3)becomes dominant and the homogeneous mixture becomes thefavourable energy state. This results in the preferential growth of lar-ger particles over time as can be observed at t ¼ 10 and t ¼ 40.

Fig. 3 presents the temporal evolution of a binary mixture withan initial concentration of 50%. A co-continuous morphology isobserved, rather than the discrete decomposition observed inFig. 2. Again, coarsening of the phases over time is observed. Thecharacteristic length of each phase therefore changes over time.

The phase structures modelled can be observed experimentally.Fig. 4 shows the phase images of a block copolymer modifiedepoxy at 2 concentrations obtained using atomic force microscopy(AFM). The epoxy polymer is a polyetheramine (Jeffamine D-230from Huntsman, UK) cured diglycidyl ether of bisphenol A (DGEBA)resin (Araldite LY556 from Huntsman, UK), and the modifier is asymmetric triblock copolymer (Nanostrength M52N from Arkema,France). The block copolymer was received as a dry powder anddissolved into the epoxy precursor by mechanical stirring at120 �C. The curing agent was then added and the mixture degassedin a vacuum oven. The epoxy is cured at 75 �C for 3 h, followed by aramp of 1 �C/min to the post-cure temperature of 110 �C for 12 hand cooled to room temperature in the oven.

A direct comparison between experimental and numericalresults is not possible as the numerical procedure presented is amodel structure for simplicity. The model assumes that the viscos-ity and hence mobility of each phase is both constant and equal toeach other. Experimentally, this is not the case, allowing for theemergence of co-continuous structures at a much lower concentra-tion than in the numerical cases presented here. ComparingFigs. 4(a) and 2(a), there are some clear similarities for the low ini-tial concentration formulations. Spherical particles with a meanradius of approximately 117 ± 6 nm were measured from theAFM micrographs. At 10 wt% (see Fig. 4(b)), the co-continuousstructure is comparable to that in Fig. 3(a). The high conversionrate of the epoxy results in a relatively quick curing time, thusthe phases do not have the time to grow in size (see Figs. 2(c)

1 For interpretation of colour in Fig. 2, the reader is referred to the web version ofthis article.

and 3(c)). Moreover, the block copolymer in Fig. 4 is present inmuch lower concentrations than that investigated numerically.

It is important to be able to measure the characteristic length ofeach phase as the polymer morphology can significantly influencethe resultant bulk mechanical behaviour. To this end, two pointcorrelation functions were used to characterise the phasemorphology as a function of time. These functions have been usedpreviously by a number of researchers to quantify microstructures[50–52]. First, the diffuse phase boundaries are converted to abinary representation via an image thresholding operation. A setof functions, Pij, is then defined which denote the probability thata randomly placed vector of a given length, r, begins in phase iand ends in phase j (i; j = 0 or 1). The lighter phase in Fig. 5 isdefined as phase 0, while the darker phase is designated as phase1. We define f as the volume fraction of phase 1 and note that atthe limits r ! 0 and r !1, the correlation functions arecompletely defined by f [53,54], i.e.:

limr!0

P00ðrÞ ¼ 1� f ð7Þ

limr!0

P11ðrÞ ¼ f ð8Þ

limr!0

P01ðrÞ ¼ P10ðrÞ ¼ 0 ð9Þ

limr!1

P00ðrÞ ¼ ð1� f Þ2 ð10Þ

limr!1

P11ðrÞ ¼ f 2 ð11Þ

limr!1

P01ðrÞ ¼ P10ðrÞ ¼ f ð1� f Þ ð12Þ

Additionally, it can be shown that the minimum value of Pii, or max-imum value of Pij gives a reasonable estimate of the characteristiclengths of the microstructure. In the case of a dispersed structurethis corresponds to the diameter of the dispersed phase or thedistance between dispersed particles, while for a co-continuousstructure, it represents the average thickness of each phase. Fig. 5plots the two point correlations for a co-continuous structure fora range of values of r (r=rmax is the ratio of the random vector lengthto the side length of the overall domain).

The coarsening process was first formalised by Lifshitz andSlyosov [55] and Wagner [56] which predicts that the average par-ticle size is linearly proportional to the cube root of time. Fig. 6plots t1=3 versus normalised structural length scale. It is clearlyseen that the coarsening process in our numerical simulationsobeys the Lifshitz–Slyosov–Wagner model. It can also be notedthat the growth rate of the characteristic length is greater forc ¼ 0:5 than for c ¼ 0:3. The deviation from the straight linegrowth relationship can be attributed to the fact that the correla-tion function does not take into account the periodic nature ofthe system and that only 1000 random vectors were sampled ateach value of r.

(a) t = 1 (b) t = 10 (c) t = 40

Fig. 3. Evolution of phase variable at different values of t with initial concentration c ¼ 0:5.

Fig. 4. AFM phase images of cured poly(methyl methacrylate)–poly(butyl acrylate)–poly(methyl methacrylate) block copolymer modified epoxy.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/rmax

Pij

P00

P11

P01

Characteristic length of Pii

Fig. 5. Two point correlation function for a co-continuous morphology. In this casethe characteristic length is 0.18rmax .

1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r/r m

ax

t1/3

c = 0.3c = 0.5

Fig. 6. Characteristic length evolution during phase decompositions in twodimensions for c ¼ 0:3 and c ¼ 0:5.

D. Carolan et al. / Computational Materials Science 98 (2015) 24–33 27

In order to check that the numerical model has been imple-mented correctly and is valid, it is necessary to ensure that it ismass conservative, i.e. there must be no spontaneous creation ordestruction of one phase over the other, and that the free energyin the system should be strictly non-increasing. Fig. 7 plots thenormalised discrete energy calculated via Eq. (3) and the mass frac-tion of the minority phase versus time. This demonstrates that thedeveloped numerical solver is both mass conservative and that thetotal energy in the system is always decreasing.

2.2. Three-dimensional cases

In order to examine the energy dissipation mechanisms in thesepolymer structures, three dimensional (3-D) structures wereobtained via the solution of Eq. (1). A unit cube domain with equalsides of 100 lm was chosen and periodic boundary conditionswere specified between the appropriate faces. The cell size of1 lm was chosen to allow solution in a reasonable time. Threedifferent cases were examined, with c ¼ 0:26; c ¼ 0:4 and

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Simulation Time

Nor

mal

ised

ene

rgy/

Mas

s fr

actio

n

Normalised energyMass fraction

Fig. 7. Mass fraction and non-increasing normalised energy as a function of time.

28 D. Carolan et al. / Computational Materials Science 98 (2015) 24–33

c ¼ 0:5. For each system, both an early stage decomposition struc-ture and a late stage decomposition structure were chosen foranalysis as shown in Table 1. In each case the minority phase isshown in red with the majority phase in blue. It can be seen fromFigs. 6 and 7 that in the early stage structures, the characteristiclengths of the microstructures are much smaller and this couldinfluence the mechanical response of the composite. The choiceof early or late stage structures is also influenced by the fact thatin experimental studies the non-equilibrium co-continuousmicrostructures are trapped by gelation of the matrix phase. Theresultant microstructures were created by taking the concentra-tion field and thresholding c such that c ¼ v f , where v f is the vol-ume fraction of the minority phase. Furthermore, representativevolume elements comprising 1=8 of a face centred cubic structure,therefore consisting of spheres, corresponding to each value of v f

were also analysed to compare against the mechanical responseof the co-continuous structures as shown in Table 1. This unit cell

Table 1The three-dimensional structures investigated in the current work. The red phasedenotes the minority phase in all cases.

c;v f Early stage Late stage Representative structure

0.2

0.26

0.4

0.5

arrangement is popular amongst researchers as it is the simpleststructure which allows for investigation of phase interactionsand each particle is equidistant from each other [57]. In the caseswhere c ¼ 0:4 and c ¼ 0:5 a co-continuous structure was observedto have formed, whereas for c ¼ 0:26 an intermediate structurethat is between a co-continuous and discrete spherical particles,was observed. Fig. 8 details the difference between a co-continuousstructure and a so-called transition structure with the secondphase removed for clarity.

3. Stress analysis

Finite volume based stress analysis was carried out on each ofthe microstructures shown in Table 1 using foam-extend-3.0[38]. Phase 1 was treated as a linear elastic–plastic solid, whilephase 2 was treated as linear elastic, with the properties as givenin Table 2. It should be noted that the elastic and cohesiveproperties chosen are not indicative of any particular materialbut can be thought of as representing a generic epoxy–rubbersystem. In general the minority phase (red phase) represents therubber phase except when phase inverted structures are discussed.

The volume averaged stress, rc , and strain, ec , were found byaveraging the local stress and strain in each cell using Eqs. (13)and (14) [58]. This is known as homogenisation.

rc ¼1

VX

ZVX

rdV ð13Þ

ec ¼1

VX

ZVX

edV ð14Þ

where VX is the total volume of integration and r and e representthe local stress and strain at any point within the cell.

3.1. Elastic behaviour

Each microstructure was loaded using a fixed axial strain rate of_eu ¼ _eh ¼ 1� 10�3 s�1 in both uniaxial extension and triaxialextension configurations. _eu and _eh represent the uniaxial strainrate and hydrostatic strain rate respectively. This yields the com-posite bulk and constrained moduli respectively via appropriateapplication of Eqs. (13) and (14) and Hooke’s Law, allowing com-plete characterisation of each composite in the linear elasticregime. Hashin and Shtrikman [59] developed theoretical upperand lower bounds for the bulk modulus, k, and the shear modulus,l, of quasi-isotropic composites with an arbitrary phase geometry:

kl ¼ k2 þm1

1k1�k2

þ m2k2þl2

; ku ¼ k1 þm2

1k2�k1

þ m1k1þl1

ð15Þ

ll ¼ l2 þm1

1l1�l2

þ m2ðk2þ2l2Þ2l2ðk2þl2Þ

; lu ¼ l1 þm2

1l2�l1

þ m1ðk1þ2l1Þ2l1ðk1þl1Þ

ð16Þ

where l and u represent the lower and upper bounds respectively, mis the volume fraction and the subscripts 1 and 2 represent the twophases in the composite. From this it is possible to calculate upperand lower bounds for the Young’s modulus, E, and Poisson’s ratio, m,using the usual relations (for i equal to u and l):

Ei ¼9kili

3ki þ lið17Þ

mi ¼3ki � 2li

6ki þ 2lið18Þ

In the case of a voided epoxy structure, it is appropriate to use thescaling laws for cellular structures. Gibson and Ashby [60] statethat, for an open cell foam, the Young’s modulus, E�, of a compositeis approximately given by:

Fig. 8. A co-continuous, v f ¼ 0:50 (a) and dispersed, v f ¼ 0:26 (b) phase.

Table 2Material properties used in the current work.

Phase 1 2

Type Elastic–plastic ElasticYoung’s modulus E (GPa) 3 0.01Poisson’s ratio m 0.3 0.499Plastic modulus Ep (MPa) 1 n/aYield strength ry (MPa) 50 n/a

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

vf

Ec [G

Pa]

Eu

El

E*

EarlyLateSpheres

(a) Young’s modulus (Ec)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

vf

c

u

l

EarlyLateSpheres

(b) Poisson’s ratio (νc)

Fig. 9. Variation of Ec and mc with volume fraction, v f , of the rubbery phase for theepoxy–rubber composite microstructure. The upper and lower Hashin–Shtrikmanbounds and the empirical relationship (Eq. (20)) are also plotted.

D. Carolan et al. / Computational Materials Science 98 (2015) 24–33 29

E� ¼ CEsq�

qs

� �2

ð19Þ

where q� is the relative density of the foam, qs is the density of thefoam material and Es is the Young’s modulus of the foam material. Cis an empirical constant which is approximately equal to 1 for anequi-axed cellular structure. Furthermore, Gibson and Ashbysuggested that the Poisson’s ratio for an open cell foam can beadequately approximated as 0.33, although they concede that therecan be a large variability in this property between structures, whichthey attribute to differences in cell geometry. A simple adaptationof this formula to take into account the presence of the secondphase in the binary system, i.e. E� ¼ Er at q� ¼ 0 and E� ¼ Es atq� ¼ 1, can be written as:

E� ¼ Er þ ðEs � ErÞq�

qs

� �2

ð20Þ

where Er represents the Young’s modulus of the compliant phaseand the other variables have the same meaning as in Eq. (19).

Fig. 9 shows the effect of volume fraction on the Young’s mod-ulus and Poisson’s ratio of the different microstructures discussedin Table 1. The higher volume fractions of 0.6 and 0.74 wereobtained by simply inverting the material phases of themicrostructures with v f ¼ 0:4 and 0.26, except in the case of therepresentative face centred cubic structures where the soft rubberyphase was confined to the discontinuous phase at the corners ofeach microstructure. This is to maintain the stiff central columnsin the directions of loading [28]. It should be noted that it is notpossible to create a representative face centred cubic structurefor v f ¼ 0:8 as the maximum packing fraction for spheres of thesame size is approximately 0.74048 [61]. The Hashin–Shtrikmanbounds and the adapted empirical relationship (Eq. (20)) are alsoplotted in Fig. 9. It can be seen that the Young’s modulus, Ec , of eachof the structures investigated lies close to the upperHashin–Shtrikman bound. This is not surprising given that theelastic modulus response would be dominated by the much stiffer

component in the composite. There is no significant differencebetween the predicted Young’s modulus of the co-continuousstructures and the representative sphere based structures. How-ever there is a significant difference in the predicted Poisson’s ratio,mc , and this is due to the mutual constraints of each phase withinthe complex co-continuous geometry.

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

35

40

ii [mm/mm]

eq [M

Pa]

vf = 0.2

vf = 0.26

vf = 0.4

vf = 0.5

Fig. 11. Yield behaviour of the sphere based structure (solid lines) and the latestage co-continuous structure (dashed lines) for v f ¼ ½0:2;0:26;0:4;0:5� in triaxialloading configurations. The 0.2% offset proof stress is also shown for each of theloading cases (dots and open circles). The early stage co-continuous structures arenot plotted for clarity but behave similarly to the late stage co-continuousstructures.

35FCCCo continuous

30 D. Carolan et al. / Computational Materials Science 98 (2015) 24–33

The effective elastic properties of a voided structure were alsocomputed. These structures were achieved by removing the rub-bery phase prior to simulation. The numerical predictions for Ec

and mc are given in Fig. 10. The predictions for Ec lie close to theupper Hashin–Shtrikman bounds, but in this case the empiricalEq. (19) suggested by Gibson and Ashby gives a better fit to thedata. As in the case of the epoxy rubber microstructures, there islittle difference in Ec between the co-continuous structures andthe representative face centred cubic structure.

However, it can be seen that the agreement of calculatedPoisson’s ratio between the co-continuous structures and repre-sentative structures diverge significantly as the volume fractionof the voids increase. In the case of the representative microstruc-tures, the calculated Poisson’s ratio agree with the suggestion byGibson and Ashby that for any open cell structure, m � 0:33. Byextrapolating the Poisson’s ratio data for the co-continuous struc-tures investigated in the current work, it can be expected that anauxetic structure exists at high void volume fractions. Indeed,there is significant research interest in auxetic materials(�1 < m < 0) with microstructures similar to those investigatedin the current work [62].

In the case of the epoxy–rubber system, the high bulk modulusof the rubber phase provides an additional constraint to the epoxypreventing local deformation into the regions occupied by therubber phase. In the case of the voided structure, where a networkof voids replaces the rubber phase, no such constraints are presentand the epoxy phase is free to locally deform into the voided

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

vf

Ec [G

Pa]

Eu

El

E*

EarlyLateSpheres

(a) Young’s modulus (Ec)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

vf

c

u

l

EarlyLateSpheres

(b) Poisson’s ratio (νc)

Fig. 10. Variation of Ec and mc with volume fraction, v f , of voids for the voidedepoxy microstructures. The upper and lower Hashin–Shtrikman bounds and theempirical relationship (Eq. (19)) are also plotted.

0.2 0.25 0.3 0.35 0.4 0.45 0.520

25

30

vf

0.2%

[MP

a]

0.2 0.25 0.3 0.35 0.4 0.45 0.51

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

vf

0.2%

[%]

FCCCo continuous

Fig. 12. Variation in 0.2% proof stress and strain for each of the structures.

Fig. 13. Maximum local principal strains in the epoxy phase for sphere based microstructures (a and b) and co-continuous structures (c and d) with v f = 0.5. The scale bar isshown in (e). The rubbery phase has been removed for clarity.

D. Carolan et al. / Computational Materials Science 98 (2015) 24–33 31

network under a macroscopic load. This is the primary reason forthe different mechanical responses between the epoxy–rubbermicrostructures and the voided microstructures.

3.2. Post-elastic behaviour

In order to simulate the yield behaviour of the composite micro-structures, it is important to be able to understand the state ofstress under a range of different types of loading. The constraintfactor, H ¼ rH=req ¼ 1=3, for a bulk material under uniaxial load-ing conditions where rH is the hydrostatic stress and req is thevon Mises stress. For other forms of loading, H is typically greaterthan 1/3. Where a crack exists in a material, the crack tip region isnormally expected to be under a highly triaxial state of stress. Foran adhesively bonded tapered double cantilever beam geometry,Hadavinia et al. reported H values of between 1.5 and 2.3 for arange of bond-gap thicknesses [63]. Cooper et al. attributed thebond-gap thickness effect on fracture toughness for a rubbertoughened epoxy adhesive to the variation in constraint ahead ofthe crack tip rather than the variation in ‘far-field’ plasticity [64].

Therefore, in the current work, the yield and post-yield behaviourof each model microstructure is studied under triaxial extension.

Fig. 11 plots the homogenised applied strain versus von Misesstress for triaxial loading for both the sphere based structuresand the late stage co-continuous structures. The 0.2% offset proofstress is also shown for each case. The difference in behaviourbetween the face centred cubic structures and the physically basedco-continuous is emphasised in Fig. 12.

It can be seen that as the volume fraction of the rubbery phaseincreases the 0.2% offset stress decreases, with a lower value forthe co-continuous structure being recorded in all cases. A similardecrease with increasing volume fraction is noted for both thesphere based FCC structures and the co-continuous structures.No significant effect is noted in the transition from droplet typeco-continuous structures to fully co-continuous structures. It isnot surprising that the co-continuous structure has a lower proofstress than the corresponding FCC structure. During the evolutionof the microstructures, the phase interface rarely has a constantcurvature as in the case of the FCC structures (curvature,K ¼ 1=R2 where R is the radius of the sphere). In some areas along

32 D. Carolan et al. / Computational Materials Science 98 (2015) 24–33

the phase interface the local curvature is much greater, leading toan increased stress concentration factor at that point. This leads tothe material yielding locally at a lower global stress andconsequently a lower proof stress is measured. This implies thatthe yield stress is only dependent on the local stress concentrationfactors at the phase interfaces within the microstructure.

For the sphere based FCC structures, the strain at which the 0.2%offset is reached is steadily predicted to decrease marginally from1.27% for v f = 0.2 to 1.16% for v f = 0.5. Whereas, in the co-continu-ous structures, the trend for the 0.2%, with an increasing value ofv f , initially follows that for the offset stress before increasing againat high volume fractions. This is directly attributable to the changein microstructure from a droplet type microstructure to a fully-co-continuous structure at high volume fractions. Therefore, the effectof a co-continuous structure appears to be to increase the proofstrain. On the other hand, the proof stress is independent of mor-phology since it is only dependent on the local stress concentrationfactors at the phase interfaces. The variation in 0.2% yield stressagrees with the trends noted by Wang et al. [28] in their study ofco-continuous structures defined by minimal periodic surfaces.

In the co-continuous structures localised plasticity occurs early,and at multiple locations in the epoxy phase, before graduallyspreading throughout the entire structure. This is illustrated bythe more gradual transition between the linear elastic regimeand the fully plastic regime compared to the representative struc-tures. The location of this early stage plasticity coincides with junc-tions in the geometry where multiple ligaments of the epoxy phaseconnect. In the case of the face centred cubic microstructures, plas-ticity does not occur until later in the loading history and thenspreads rapidly between each rubber particle. This is clearly dem-onstrated by the fringe plots in Fig. 13 which detail the maximumprincipal strains in both the face centred cubic microstructure andco-continuous microstructure at 1% and 2% axial applied strain.

Finally, it is worth noting that once plasticity has diffusedthroughout the entire structure, i.e. when eii ¼ 5%, the equivalentstress, req, in the co-continuous structure is higher than that inthe corresponding representative structure.

4. Conclusions

A method of generating physically realistic co-continuousmicrostructures based on the Cahn–Hilliard equation has beendeveloped and implemented within an open-source finite volumepackage. The method was shown to accurately capture the physicsof phase separation with a minimum of computational effort. Anumber of representative microstructures generated using theCahn–Hilliard equation have been investigated under differentmechanical loading conditions and compared to conventional rep-resentative structures, namely 1/8 face centred cubic (spherical)structures. The results reveal that the mechanical properties ofthe co-continuous structures can differ significantly from the clas-sic structures, and must be taken into account when analysing suchcomplex nanostructures. The results presented in this paper pro-vide a numerical roadmap for developing co-continuous polymercomposites with tuneable mechanical behaviour and energyabsorption properties. For example, Suo and Lu [65] demonstratethe possibility to create self assembling periodic structures byintroducing discontinuities at a coarse level in the concentrationfield in a binary system such as the one studied here. Clearly, suchstructures have the potential to offer a unique combination ofmechanical, thermal and electrical properties.

Acknowledgements

The authors would like to acknowledge the financial support ofthe Irish Research Council and Marie Curie Actions under the ELEVATE

fellowship scheme for Dr. D. Carolan. The authors gratefullyacknowledge the award of a PhD scholarship for Mr. H.M. Chongfrom the Department of Mechanical Engineering at ImperialCollege London.

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