Determination of the Failure Stresses for Fluid-filledMicrocapsules that Rupture Near the Elastic Regime

R. Mercadé-Prieto & R. Allen & D. York & J.A. Preece &

T.E. Goodwin & Z. Zhang

Received: 28 August 2011 /Accepted: 21 February 2012 /Published online: 8 March 2012# Society for Experimental Mechanics 2012

Abstract The encapsulation of liquids within an externalwall or shell is an important technology often utilized in theproduction of many commercial products. The mechanicalcharacterization of such microcapsules is paramount in or-der to fully understand their performance in their targetenvironment. Some microcapsules, with wall materials suchas inorganic based compounds, rupture at small deforma-tions, commonly near the elastic regime. The study hereinpresents a general methodology that enables calculation ofthe failure stresses leading to the elastic-like rupture ofmicrocapsules under parallel compression testing. Two sce-narios of failure, brittle and ductile, were considered. Anal-yses of the critical stresses present within the microcapsulewall during different degrees of fractional deformation wereobtained using finite element modelling, resulting in similarvalues for both the brittle and ductile scenarios. The

correlations presented were used to determine the failurestresses of tetraethoxyorthosilane-methyltrimethoxysilane(TEOS-MTMS) microcapsules with a model core oil, whichare 11–14±10 MPa. The data were further analyzed usingWeibull distributions.

Keywords Elastic behaviour . Finite element modelling(FEM) . Fracture stress . Micromanipulation .Microcapsule

Introduction

The microencapsulation of liquids is both a common anduseful technology which may be utilized in order to enhancethe stability of the encapsulated material, whilst also con-trolling its release. The mechanical testing of such micro-capsules to the point of rupture is often performed in orderto characterize the final product, and investigate its suitabil-ity for applications whereby triggered release of the liquidcore material by mechanical forces is desired [1]. Develop-ing an understanding of the mechanical properties of micro-capsules is therefore critical in this endeavour. Themechanical testing of microcapsules is commonly per-formed between two parallel plates using a unique apparatuswhich has been designed specifically for the compressionand analysis of microcapsules of 1–1000 μm, the use ofwhich is referred to as a micromanipulation technique [2].One of the most studied microcapsule shell materials for theencapsulation of liquids is melamine-formaldehyde (MF),particularly in the encapsulation of perfume oils. MF micro-capsules containing liquid oils commonly break (rupture,burst, fail or fracture) at large fractional deformations(~0.4–0.7) ε, defined as the displacement of the microcap-sule divided by its diameter, and it has recently been shownthat this occurs after extensive plastic deformation [3–5].

Electronic supplementary material The online version of this article(doi:10.1007/s11340-012-9605-5) contains supplementary material,which is available to authorized users.

R. Mercadé-Prieto : R. Allen : Z. Zhang (*)School of Chemical Engineering, University of Birmingham,Edgbaston,Birmingham B15 2TT, UKe-mail: Z.Zhang@bham.ac.uk

D. YorkProcter and Gamble Technical Centre,Newcastle upon Tyne NE27 0QW, UK

J.A. PreeceSchool of Chemistry, University of Birmingham,Edgbaston,Birmingham B15 2TT, UK

T.E. GoodwinENCAPSYS®, Appleton,P. O. Box 359, Appleton, WI 54912-0359, USA

Experimental Mechanics (2012) 52:1435–1445DOI 10.1007/s11340-012-9605-5

On the other hand, when inorganic materials are used forthe encapsulation of liquids, the fractional deformationswhere breaking is observed (εB) are smaller, i.e. <0.3. Forexample, fluid-filled CaCO3 microcapsules break at εB~0.08 [6]. Micromanipulation compression experiments read-ily provide two parameters: the previously mentioned εB,and the breaking force FB. These parameters, however, arespecific to the testing method, and so intrinsic mechanicalproperty parameters (such as the elastic modulus, yieldstress, and failure stress) are also required in order thatresults can be applied to end-use applications, and compar-isons may be made of microcapsules prepared using differ-ent formulations and various processing conditions.

Two examples are available in the literature where failurestresses in the wall of a core-shell structure are estimatedusing finite element modelling (FEM), one for single yeastcells with a core-shell structure similar to that of micro-capsules [7, 8] and one for single carbon microballoons[9]. Failure was considered to occur within the elastic re-gime, and therefore the failure stresses were determinedfrom the FEM results at εB and the population averageelastic modulus E. The analysis from previous studies how-ever, can not be applied to liquid-filled microcapsules thatrupture at small deformations (εB<0.35).

The elastic regime—wherein the deformation of micro-capsules is considered to be fully elastic—has been deter-mined in two different ways. Experimentally, compressionand holding, loading-unloading, and repeated compressionexperiments have been performed to establish whether theelastic behaviour occurs until the point of fractional defor-mation tested [4, 10, 11]. These methodologies are not wellsuited to accurately determine the fractional deformationabove which non-elastic behaviour is observed (termedεLE) for a large number of microcapsules (for statisticalsignificance), and particularly for microcapsules with smallεB, as εLE~<εB. A different approach has been used todetermine εLE, via the comparison of experimental compres-sion force versus displacement data with predictions fromtheoretical elastic models [12] or FEM [3].

In the present study, the elastic regime and εLE of micro-capsules that break at small deformations will be determinedby the comparison of experimental force-displacement datawith FEM simulations. Due to the reverse engineering meth-odology to determine the elastic modulus E of the micro-capsule wall [13], and the experimental accuracy of the data,the predicted elastic regime primarily represents the linear-like profile of the stress–strain curve of the wall material,see Fig 1. In some instances therefore, depending upon thestress–strain curve, the predicted elastic regime may beoverestimated.

The ultimate aim of this study is to estimate the failurestress of the microcapsule wall, σB for the hypotheticalmaterial shown in Fig. 1, occurring at a strain HB. σB can

only be found for such a material if the complete stress–strain profile is known before failure, but such a complexreverse engineering problem has not yet been solved formicrocapsules with any kind of stress–strain curve. The onlytwo material models that have been studied in detail are thatof purely elastic and elastic-perfectly plastic [3, 13], whichwill be also applied in this study. If the material is com-pletely elastic until failure, then σB ¼ σB;elastic ¼ σLE. If thematerial is elastic–plastic before failure, the elastic-perfectlyplastic results will be used to determine σLE, thus allowingestimation of the breaking stress within the interval σLE<σB<σB,elastic, as shown in Fig. 1.

Materials and Methods

FEM Simulations of Elastic Microcapsules

Spherical capsules compressed between two parallel plates(a top probe and a bottom substrate) were simulated usingAbaqus® 6.5 as described in detail elsewhere [3, 13], aschematic diagram is shown in Fig. 2. In summary, thecapsule wall is simulated as an axisymmetric object in they plane, with 1440 CAX4RH elements (4-node bilinear,reduced integration with hourglass control, hybrid with con-stant pressure), with 4 element layers to model the shell forall thicknesses tested. The mesh chosen was validated usinga denser mesh, with 20 element layers to model the shellthickness. In agreement with previous studies [9, 14], amesh of 4-element layers provides both accurate and rapid

Fig. 1 Schematic representation of a hypothetical true stress—truestrain curve of a microcapsule wall material until rupture (shown asB). Straight line shows the likely estimated elastic modulus using areverse engineering method [13]. The possible overestimation of theelastic regime from the real yield point (σY and HY) to the estimatedelastic limit (σLE and HLE) is shown. For a rupture strain HB, thecorresponding rupture stress assuming a fully linear elastic behavior,σB,elastic, is larger than the real rupture stress σB

1436 Exp Mech (2012) 52:1435–1445

results. Different shell thickness h to radius r ratios weresimulated, between 0.6–9.1%. The liquid core of the micro-capsules was designed using incompressible FAX2 elements(2-node linear axisymmetric hydrostatic fluid) with a densi-ty of 1000 kg m−3, and the cavity volume was kept constant.The capsule shell was assumed to be incompressible (Pois-son ratio 0.5), and the contact interaction between the platesand the wall was considered to be frictionless. For moreinformation and validation of the simulations, refer to Mer-cadé-Prieto et al. [5].

Formation of TEOS-MTMS Microcapsules

Microcapsules with an inorganic wall material were made inorder to obtain microcapsules that failed near the elasticregime at small fractional deformations. The wall materialwas based on tetraethoxyorthosilane (TEOS, >98%, Fisher,UK) and methyltrimethoxysilane (MTMS, >98%, Fisher,UK), while for the liquid core a model oil, hexyl salicylate(HS, >98%, SAFC, USA), was used. Microcapsules weremade as follows: equimolar weights of MTMS, TEOS andwater were placed in a round bottomed flask equipped witha condenser and thermal control. The mixture was stirred at500 rpm and heated for 4.5 h at 100°C with alcohol beingsimultaneously distilled. The resulting mixture (silica pre-condensate) was cooled to ambient temperature whilst stir-ring was maintained at 500 rpm, before being transferred toan airtight container.

This silica precondensate was dispersed in 9 g HS prior tothe addition of 70 mL 0.1 M HCl (aq). The solution wasimmediately homogenised (Model L4RT, Silverson

Machines Ltd., UK) at 1500 rpm for 30 min to produce anoil-in-water emulsion. The resulting emulsion was stirredwith a magnetic stirrer at 400 rpm for 1 h prior to theaddition of 3 g of MTMS, followed by further stirring for1 h. The average diameter of the microcapsules D[3, 4] was17.6 μm, measured using a Mastersizer 2000 (Malvern)based on a refractive index of 1.43 [15]. The wall thicknessof the microcapsules was measured using transmission elec-tron microscopy (TEM), JEOL 1200EX, operating at 80 eV,as described previously [16].

Microcapsules were compressed between two flat surfa-ces using a micromanipulation rig, described in detail else-where [2, 17]. The microcapsules were added to a Perspexchamber containing deionized water. A ϕ 69 μm glass probewas mounted on a force transducer with a maximum load of5 mN (Model 403A, Aurora Scientific Inc., Canada), andused to compress single microcapsules at a speed of 2.2 μm/s. Compression experiments were carried out at room tem-perature (22±1°C).

Failure Criteria

It is of interest to discuss the possible mechanism of failureof microcapsules that burst within or near the elastic regime.Such an elastic-like fracture usually occurs in organic orinorganic materials after small deformation strains, whichwould result in small fractional deformations at rupture εB.In biological systems such as yeast cells, an elastic failure isobserved at εB~0.6–0.7 [11]. The scope of the present study,however, is limited to εB<0.35, the range a priori mostrelevant for polymeric and inorganic microcapsules. Micro-capsules break because a failure criterion has been fulfilled;we discuss subsequently two classic criteria that are appli-cable here.

(a) Maximum principal stress (Rankine) theory. For brittlematerials that fail by brittle fracture, failure is common-ly observed when the largest maximum principal stress(σMax,P) in the system equals the elastic limit stress insimple tension [16].

(b) Distortion energy (von Mises) theory. This is usuallyapplicable to ductile failure, when failure is due to themaximum shear strain energy component in the systembeing equal to that at the yield point in tensile con-ditions [16].

Both failure criteria can occur during the compression ofa microcapsule. Further information would therefore berequired in order to determine if the failure is brittle orductile. However, evidence has not yet been obtained forindividually compressed microcapsules due to their smallsizes (with capsule wall thickness in the order of hundredsof nanometers). In the present study the general scenario is

Fig. 2 Schematic diagram showing the parallel plate compression of amicrocapsule with a liquid core, before and after a displacement d ofthe top plate

Exp Mech (2012) 52:1435–1445 1437

considered, and therefore both failure criteria will be devel-oped. The aim is to obtain critical stresses that define eitherthe point at which rupture actually occurs (if the process isinstantaneous), or the point at which rupture is initiated withbreakage occurring soon after, the most likely scenario.

Brittle Failure

Failure of brittle materials is commonly initiated by thepresence of microcracks, flaws or discontinuities that actas local stress raisers [16]. The effect of the material imper-fections on the strength of the microcapsule wall will beconsidered using linear elastic fracture mechanics (LEFM)[18]. Figure 3 inset shows a typical σMax,P distribution for acompressed microcapsule. Usually the area of maximumstress is at the point of maximum internal curvature of thewall, i.e. where it is bent, and this is where failure is mostlikely to occur (a close up is shown in Fig. S1 in theSupplementary Information). The largest stresses are ob-served in the circumferential direction (Out-of-Plane stressor S33 in Fig. S2(c)), which would result in meridionalfissures. In the case of thick-walled capsules at small ε,however, the stresses are concentrated in the area of the wallwhich is in contact with the flat plates, see Fig. 4 for h/r04.8and ε00.06, as discussed previously by Carlisle et al. [9].Externally, wrinkling appears in the part of the wall directlyin contact with the plates causing compressive stresses [13,19], whilst the interior surface is stretched. The largest σMax,P

under these conditions are found within these stretchedregions of the wall.

From the FEM results, the σMax,P at each ε was deter-mined, as well as its position within the microcapsule wall,

shown in Fig. 5. Figures 3 and S1(a) show the plane inwhich the σMax,P is located when the bent region is that ofmaximum stress. Under these conditions, it is observed thatthe normalized σMax,P/E with ε is fairly independent of thewall thickness to radius ratio h/r (Fig. 5(a)). All of the

Fig. 3 von Mises stress and maximum principal stress (shown in theinset) distributions for the compression of a microcapsule with h/r04.8%, E0100 MPa and ε00.18. The thickness planes of the four failurestresses discussed in the text are also shown

Fig. 4 Diagram of the stresses in the horizontal direction (direction 1in the axis shown) for a microcapsule (h/r04.8%, E0100 MPa) with asmall deformation of ε00.06. Note that the area of large von Misesstress which is in contact with the probe is generated by compressivehorizontal stresses that arise due to wrinkling

Fig. 5 Values of the normalized largest maximum principal stressfound in microcapsules of different shell thickness to radius ratios h/r(shown in symbols) subjected to different compressive deformations.The location of the largest stress is shown in (b), where the position ofσMax,P in degrees relates to the non-deformed shape of the microcap-sule as shown in the inset. The best fits using equations (1) and (2) thatdescribe the stresses in the bent region (Fig. 3) are also shown, and thedashed lines are the 95% CI

1438 Exp Mech (2012) 52:1435–1445

conditions tested are well represented with the followingequation (ε<0.35, 95% Confidence Interval (CI) ±0.0034)

σMax;P

E¼ 0:146"2 þ 0:293" ð1Þ

However, for thick-walled capsules at small ε values,σMax,P/E increases linearly with the fractional deformationfor displacements <h. Thereafter it remains fairly constantwith ε until the largest σMax,P in the bent area surpasses thatin the contact area (Fig. 4), and the resulting sudden changein position of σMax,P is shown in Fig. 5(b). In a similarmanner, the position of the largest σMax,P at each ε doesnot differ significantly if only the bent area is considered.The data for all h/r values tested are well represented usingthe following polynomial, whereby the position of σMax, P indegrees (°) is for the initial non-deformed microcapsule,shown in Fig. 5(b) inset (95% CI ±4º).

Position of σMax;P�ð Þ ¼ �2185:7"3 þ 1471:3"2

� 408:4"þ 90 ð2ÞIn summary, for not too thick-walled microcapsules (e.g.

h/r<9%) with not too small εB (e.g. >0.05), the Rankinefailure criterion of a microcapsule can be estimated withequation (1).

Ductile Failure Under Perfect Plasticity

If the microcapsule wall material deforms plastically prior torupture, a ductile failure will eventually occur. There aremany plastic models but only the simplest one will beconsidered here as discussed in “Introduction”, i.e. that ofperfect plasticity. The complexity that arises with moreelaborate plastic models is left for future studies. In perfectplasticity the stress is constant above the yield stress, andunlike in Fig. 1 here σY ≈ σLE, hence the stress at rupture ofan elastic—perfectly plastic capsule is equal to the yieldstress, although the yield strain (HY ≈ HLE) will be differentfrom the rupture strain (HB) [5]. The critical stress obtainedfrom a perfectly plastic scenario (σY ≈ σLE in Fig. 1) will beequivalent to the lower threshold for other plastic materials;the upper threshold could be found using the elastic scenarioat the rupture deformation discussed in the previous section(ie. σB,elastic in Fig. 1).

In the case of plastic deformation, the stress to use is thevon Mises stress (σMIS). Figure 3 shows the σMIS distribu-tion of a compressed microcapsule, whereby two regions ofhigh stress are observed, i.e. under the compression plateand within the bent region. The former is a valid failureregion for a brittle failure case, but it is not for a ductilefailure. Figure 4, as well as Fig. S2, show that the largestσMIS in the wall under the compression plate are caused bycompression stresses. Compressive stresses are an unlikely

cause of failure of microcapsules due to the typically highercompressive strength of materials when compared to tensilestrengths, and are therefore unlikely to result in ductile fail-ures, even for plastics [16, 19]. It is for this reason that thevalues of σMIS located under the parallel plates, which are ofa compressive nature, will be shown but will not be consid-ered in subsequent analysis.

As in the previous section, the key area is the bent region,and an enlarged image of it is shown in Fig. S1(b). Firstlythe position of largest σMIS obtained from the FEM results isconsidered, termed σMax,MIS. This point is found in the innerside of the wall, in a plane that is closer to the compressingplates than that found for σMax,P. Analysis of σMax,MIS, seediagrams in Fig. S2, shows that there is a large compressivecomponent, which again is judged unlikely to be a goodpredictive parameter for ductile failure. For this reason, twoother stress parameters are proposed.

The first parameter is the largest average σMIS found withinthe thickness of the wall, termed σAv,MIS. Hence, this param-eter does not reflect localized stress conditions, but the aver-age stress to which the whole wall thickness is exposed. Thelocation of σAv,MIS, shown in Fig. 3 and S1, is similar to thatwhere shear stresses are the largest (Fig. S2(d)). The secondparameter is the largest minimum σMIS found within thethickness of the wall, termed σMin,MIS. It is determined fromFEM results as follows: the minimum value of σMIS is deter-mined for the cross section at each angle, and then the crosssection at which the minimum σMIS is greatest is selected. Allthe values of σMIS at that plane are therefore greater than, orequal to, σMin,MIS. The thickness plane of σMin,MIS is fairlyclose to where the shear stress is maximum, and is also in theregion where the largest out-of-plane stress is found (e.g. S33in Fig. S2(c)). The position of σMin,MIS is slightly closer to theequator than that of σAv,MIS. Both positions are only slightlydependent on h/r (in the bent region), and in general they arevery similar to that of σMax,P. The same equation (equation (2))for σMax,P, also represents the positions of σMin,MIS and σAv,MIS

well, shown in Fig. S3, which from an experimental point ofview would be indistinguishable.

The normalized values of σMin,MIS and σAv,MIS at differ-ent ε are shown in Fig. 6 (in the linear elastic regime, thenormalized values are constant). As previously, there is littleeffect of h/r on the stress values when only the bent region isconsidered. The best fitted polynomials are

σAv;MIS

E¼ 0:294" ð3Þ

σMin;MIS

E¼ 0:215"2 þ 0:225" ð4Þ

with a 95% CI of ±0.0027 and ±0.0019 for σAv,MIS/E andσMin,MIS/E, respectively. Whereas σAv,MIS is intuitively a

Exp Mech (2012) 52:1435–1445 1439

better failure parameter as σAv,MIS>σMin,MIS, σMin,MIS hasseveral interesting advantages due to the way in which it iscalculated. For instance, σMin,MIS easily provides values ofthe bent region, but not of the region in contact with thecompressing plates (see the positions of thick-walled capsu-les in Fig. S2). σMin,MIS is therefore less sensitive to h/r thanσAv,MIS under wider conditions of h/r and ε, which in turnincreases the robustness and accuracy of the parameter.Finally, this failure criterion agrees with the implication ofthe fractional deformation at the elastic limit (εLE) proposedby Mercadé-Prieto et al. [3], whereby the yield stress σY wasdetermined using the fractional deformation beyond whichthe compression force of an elastic–plastic capsule divergescontinuously from the elastic scenario. Equation (5) is pre-sented to fit all h/r data reasonably well [3].

σYE ¼ "LE�0:0354

3:13 0:01 < σYE < 0:07 ð5Þ

The good agreement of equation (5) with the results ofσMin,MIS (Fig. 6(b)) strengthens σMin,MIS as a failure param-eter because εLE represents the fractional deformation abovewhich the whole thickness of the most bent area has yielded

[3], which would lead at higher deformations to a plasticcollapse [20], when the material easily deforms until therupture strain is achieved.

In summary, three stress parameters have been presentedin order to determine the failure stresses of microcapsulesbased on brittle or ductile failures. The locations of thesestresses usually occur in almost exactly the same part of thebent region. In addition, despite σMax,P ≥ σAv,MIS ≥ σMin,MIS,the differences of all three stress parameters, whilst they arestatistically significant at some ε, are not major (see Fig. S4for a comparison of equations (1, 3 and 4)).

Determination of Failure Stresses from ExperimentalData of Compression

Previous work on the determination of rupture stresses,showed that the key parameter is the fractional deformationat rupture εB, from which either the rupture stress is calcu-lated (in the case of rupture within the elastic regime [8, 9]),or the rupture strain (in the case of elastic-perfectly plasticmicrocapsules that rupture at high εB [3, 5]). The formercase is of relevance here, however, in microcapsules it iscommon to observe that εB does not exactly fall within theelastic regime.

Figure 7 shows five examples of compression curves ofTEOS-MTMS microcapsules that will be analyzed in detaillater on; here it is intended to present a general methodolo-gy. The compression force (F) has been normalized by theelastic modulus of the wall (E, which for present purposeshas already been estimated, as explained subsequently), themicrocapsule radius (r) and the wall thickness (h). Thecontinuous line represents the scenario of a fully elasticcapsule as obtained by FEM [13], which represents theelastic regime mentioned in “Introduction”. The microcap-sules break with εB between 0.1–0.25, but there is a largedisparity of behaviours: some microcapsules break near theelastic regime, following the small deviation from the elasticmaster curve, such as capsules D and E in Fig. 7, whileothers (capsules B and C) break far from this regime. Inorder to quantify the degree of closeness to the elasticregime, the fractional deformation at which the force con-tinuously deviates from the ideal elastic scenario is firstdetermined. This, as mentioned earlier, corresponds to thedetermination of the elastic limit (εLE) as explained else-where [3]. FLE is the compression force at εLE. The ratios"B � "LEð Þ="LE and FB � FLEð Þ=FLE for the 5 examplecapsules are shown in the inset table in Fig. 7. Values of<10% would indicate that rupture occurs very near theelastic regime, and thus the theoretical models developedabove would be directly applicable. In some cases, forexample capsules B and C, the values of these ratios aregreater than 50%, and thus raise the question if an elastic-

Fig. 6 Normalized values of the largest average and minimum vonMises stresses, as described in the text. The best fits using equations(3–5) that describe the stresses in the bent region (Fig. 3) are alsoshown, and the dashed lines are the 95% CI. The legends represent theh/r of the microcapsules

1440 Exp Mech (2012) 52:1435–1445

like failure is applicable. Capsule A, with values of 10–20%,would correspond to an intermediate case.

Explanations for the deviation of the force from theelastic scenario before rupture are investigated subsequently.There are two main explanations for the two failure scenar-ios considered previously [21]. At this point, it should beemphasized that the failure mechanism of the capsule, brittleor ductile, is completely unknown. If a brittle failure isconsidered, the force would deviate from the elastic regimedue to the propagation of microcracks or imperfectionswithin the wall. At some point, the extensive microcrackscause rupture of the wall. Following linear elastic fracturemechanics [18, 22], there is a critical stress (σcr) abovewhich a given microcrack of length hcrack will propagate

σcr ¼ 2ΓEphcrack

� �0:5

ð6Þ

where Γ is the surface energy of the microcrack per area.Hence, from a brittle failure point of view, the key stress isσcr, as at higher values of stress the microcrack will quicklygrow until such propagation results in instability in the wallmaterial. In the present case, σcr corresponds to the largestprincipal stress at εLE, σMax,P.

The second approach utilized in the analysis of the devi-ation of the force from the elastic regime, is to consider thatthe material has yielded prior to rupture. The compressiondata of capsules B and C in Fig. 7 were compared assumingan elastic-perfectly plastic behaviour [3] as considered in“Ductile Failure Under Perfect Plasticity”. A good agree-ment was found between the capsules until failure, consid-ering a yield stress to elastic modulus ratio σY/E00.02 (thisvalue was not optimized, but was fitted qualitatively),

shown in Fig. 7. Under a perfectly plastic scenario, the stressvalues in the most stressed areas remain fixed at the yieldstress. Despite the fact that a fraction of the wall has yielded,the compressive force increases with ε because the contactarea between the microcapsule and the compressing platesincreases. Under perfect plasticity, the yielded section easilydistorts at higher fractional deformations until eventually therupture strain of the shell material is reached, causing therupture of the whole microcapsule [3, 5]. The key stress isthe yield stress, which facilitates deformation of the wall,and occurs for the first time throughout the whole thicknessof the wall at εLE. The difference between εLE and εB, as inHLE (≈HY) vs HB in Fig. 1, will be related to the plasticproperties of the wall material.

In summary, it has been shown that following two differ-ent failure mechanisms, the critical stresses of a brittlefailure and that found in an elastic-perfectly plastic wallbefore failure are the same, and are found at the elastic limit.

Example of Determining the Failure Stresses of TEOS-MTMS Microcapsules

In this section, the attainment of failure stress data fromcompression experiments of individual microcapsules usingparallel plates will be demonstrated. In the present study,compression experiments were performed using a microma-nipulation rig [2, 4], whereby single microcapsules werecompressed while immersed in water within a perspexchamber. Figure 8(a) shows three experimental results inwhich the compression force is normalized by the capsulesize (i.e. diameter). Capsule G is another example of failurenear the elastic regime, whilst capsule H is an example offailure far from the elastic regime. From such data, theproduct Eh is calculated as explained elsewhere [13], con-sidering a wall thickness of ~0.4 μm. With Eh, the elasticlimit εLE was determined as shown in Fig. 8(b) [3], while εBis easily determined from the compression profile.

The rupture force of 86 capsules analysed is shown inFig. 9(a). Contrary to previous published data, such asmelamine-formaldehyde microcapsules [4, 13], a linear re-lationship between FB and the capsule diameter is not ob-served. Scattered, but less than FB, εB has a mean value of15.9±6% (SD) (Fig. 9(b)), whereas εLE is even less scat-tered with a mean value of 9.7±2%. Note that neither εB norεLE shows any significant dependence on the capsule size, inagreement with many previous compression results [13].Following the above, it is no surprise that the individualEh values are also scattered (Fig. 9(c)), with a mean valueof 180±110 Nm−1. Assuming h~0.4 μm (see “WallThickness” for justification), the mean population Eis 450±275 MPa. Finally, Fig. 9(d) shows the ratio"B � "LEð Þ="LE, where values close to zero imply a rupture

Fig. 7 Different examples of compression experiments: some micro-capsules burst near the elastic regime (shown as the continuous curve)while others burst farther away. The different rupture and elastic limitparameters are shown for microcapsule A. The two ratio values arecalculated to indicate how close the bursting of the microcapsule is tothe elastic regime, shown in the inset table. Note that microcapsules Band C are well represented considering an elastic-perfectly plastic shellmaterial with a yield stress to elastic modulus ratio σY/E00.02 (shownas the dashed curve), from Mercadé-Prieto et al. [3]

Exp Mech (2012) 52:1435–1445 1441

near the elastic regime. Considering a cut off value of ~20%,such as capsule A in Fig. 7, approximately ~35% of thecapsules broke near the linear regime. Again, no trend withthe capsule size was observed.

The normalized failure stresses are calculated withεLE and the equations presented in “Failure Criteria”,obtaining: σMax;P=E ¼ 0:031� 0:008 SDð Þ , σAv;MIS=E ¼0:029� 0:007, and σMin;MIS=E ¼ 0:025� 0:007. In thepresent case, only the value of σMin,MIS/E is statisticallydifferent from those of σMax,P/E and σAv,MIS/E. The finalstep is to obtain the individual failure stresses using theindividual E calculated for each microcapsule. Theindividual results are shown for the case of σAv,MIS

in Fig. 10. Note that, due to the scatter of Eh shownin Fig. 9(c) the failure stresses are also highly scattered,with the following average values: σMax;P ¼ 13:8� 9 SDð ÞMPa,σAv;MIS ¼ 13:2� 9MPa, andσMin;MIS ¼ 11:1� 7MPa.In comparison, if the rupture deformation εB was used insteadof εLE, it would result in a value of σMax,P at ruptureof 18±10 MPa. This stress can be used as an upperthreshold value, for example, in capsules known to failin a ductile manner after a plastic hardening regime;the lower threshold stress would be σMin,MIS. Themechanical properties found here for TEOS-MTMSmicrocapsules are difficult to compare with those inthe literature, which were obtained using typical tensiletests, due to different formulations and testing condi-tions, but they are in the same order of magnitude, seeTable S1.

Fig. 8 (a) Three examples of compressions of TEOS-MTMS micro-capsules and the best fit of the elastic regime based on FEM [8]; thecapsule sizes and the calculated Eh are shown in the inset table. (b)Difference between the experimental normalized compression force andthat predicted for a fully elastic microcapsule, which was used to deter-mine the elastic limit εLE. Solid arrows show εB ; dotted arrows show εLE

Fig. 9 Experimental compres-sion results for TEOS-MTMSmicrocapsules, showing (a) therupture force, (b) the fractionaldeformation at rupture and theelastic limit, (c) the calculatedEh (error bars show the SD), and(d) the deviation of the rupturepoint from the elastic regime(low values imply ruptures nearthe elastic regime)

1442 Exp Mech (2012) 52:1435–1445

Weibull Distribution Analysis of Failure Stresses

The large dispersion of the failure stresses shown in Fig. 10highlights that a simple mean value of the microcapsulestested is an insufficient parameter to characterize the wholepopulation. In such instances, Weibull distributions are

commonly used to describe the mechanical properties ofbrittle materials [18, 23, 24]. Its applicability can be limitedto instances when flaws are few and do not interact with oneanother; failure occurs once any flaw initiates fracture (theweakest link assumption); and when the flaw distribution isunimodal [25]. Such information, however, is at present notavailable for microcapsules. Here it is assumed that it issufficient to use at this early stage of studying microcapsulefailures, if a Weibull distribution can represent the failuredata well. A three parameter Weibull distribution is consid-ered here [26]

CDFðxÞ ¼ 1� exp � x� x0xc

� �b" #

ð7Þ

where CDF(x) is the cumulative distribution function and xis the failure stress, x0 is the location parameter (i.e. thelower bound of the distribution), xc is the scale parameter(x0+xc is about the median value), and β is the shapeparameter, which is related to the properties of the material.These parameters were determined by fitting the failurestress data using the median ranks method explained else-where [27]. Figure 11(a) shows the typical fit of equation (7)to the normalized σAv,MIS/E failure data shown in Fig. 10, and

Fig. 10 Normalized and absolute values of the average von Misesfailure stress. The E value used is found using the individual Eh shownin Fig. 9(c) and a wall thickness value of 0.4 μm

Fig. 11 (a) Example of the fitting of a three parameter Weibulldistribution using the median ranks method. (b) Cumulative failureprobability (i.e. the CDF) using the σAv,MIS failure criterion, consider-ing the population average E or that determined from each individualcompression

Fig. 12 (a) Example of TEM image of TEOS-MTMS microcapsulesused to determine the wall thickness. (b) Average wall thickness ofseveral microcapsules. Error bars show the minimum and maximumthickness measured for each capsule

Exp Mech (2012) 52:1435–1445 1443

the good agreement between them justifies here the use ofWeibull distributions tomicrocapsules. However, it is σAv,MIS

and not the normalized value that is of primary interest. IfσAv,MIS is calculated using the population average E givenpreviously, of ~450 MPa, the shape of the CDF is the same asthat shown in Fig. 11(a) (i.e. β remains unchanged at ~2.2),and only the parameters x0 and xc vary. If the individual Ecalculated for each capsule, however, is used (from Fig. 9(c)),then the CDF of the σAv,MIS calculated is quite different fromthat obtained from using the population average E, with β~1.3(Fig. 11(b) and Table 1). The small value of β quantifies thelarge scatter observed for the breaking force shown in Fig. 9(a).

Wall Thickness

In the previous section, a uniform wall thickness value of~0.4 μm has been considered without evidence. TEMimages of the cross section of microcapsules show that thewall thickness was not uniform, and an example is shown inFig. 12(a). The minimum wall thickness of the two capsulesshown is ~0.4 μm, while the maximum thickness is 0.95–1.25 μm. Variations are not only observed within a capsulebut also as expected between different capsules. The aver-age wall thickness of TEOS-MTMS microcapsules was notobserved to change with the capsule size, as shown inFig. 12(b), with a mean value of 0.47±0.16 μm (SD, n0175, using a correction factor of 0.86 in order to considerrandom slicing effects [13]), whereas the population meanof the minimum thickness for each capsule was 0.33±0.13 μm.

It is expected that a share of the large scatter observed forEh (Fig. 9(c)) comes from the observed intra- and inter-capsule variability in the wall thickness, as h/r is required toaccurately estimate Eh. It is not known, however, whetherthe h variability will cause more or less scattered E values. Itis not currently possible to obtain accurate wall thicknessesof microcapsules after being compressed; hence populationmean values have to be considered. The average thicknesswould be the most appropriate to determine E values, yet theminimum wall thickness can control fracture [9]. For thisreason, we have considered a compromised thickness valueof 0.4 μm for simplicity. The mean average thickness to

radius ratio for microcapsules of 10–20 μm in diameter, thesize range considered in compression experiments (Fig. 9),was 5.5±2% (SD, n0106) from TEM images, while themean minimum thickness to radius ratio was 3.7±1.4%.These relatively low h/r values and εLE~0.1 justify usingthe bent region in Fig. 3 and S1 as the most likely failureregion, and not that in contact with the parallel plates. Theuncertainty of the wall thickness affects the determination offailure stresses primarily from the determination of E, as thefailure stress to E ratios are fairly independent of h/r for thefailure conditions of TEOS-MTMS microcapsules, seeFigs. 5 and 6.

Conclusions

The first general analysis to estimate the failure stresses ofmicrocapsules that burst near the elastic regime has beenperformed in the present study. Three simple equations havebeen developed using FEM in order to estimate the failurestresses of single microcapsules under parallel compression.Two different failure mechanisms have been considered,brittle or ductile failure. Importantly, due to the difficultyin experimentally determining the mechanism of failure ofmicrocapsules, all three critical stress scenarios presentedhave similar values, and occur at a similar location withinthe microcapsule wall. It has been argued that criticalstresses occur when the compressive force begins to deviatefrom the elastic scenario. Under the brittle failure scenario itwould be due to the propagation of flaws that weaken thestructure of the microcapsule wall, while in the ductilefailure it would be caused by the yielding of the wallmaterial. The model presented has been used to characterizethe failure stresses of TEOS-MTMS microcapsules, obtain-ing similar values to that expected from the literature. AWeibull analysis has been demonstrated to be useful inanalyzing failure data, particularly if brittle failure is areasonable scenario.

Acknowledgements The authors gratefully acknowledge financialsupport from the Engineering and Physical Sciences Research Council(EPRSC), UK, through grant number EP/F068395/1.

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Table 1 Weibull parameters (equation (7)) calculated for the failurestresses of TEOS-MTMS microcapsules

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σMax,P Individual 1.35 1.2 13.9

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1444 Exp Mech (2012) 52:1435–1445

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