- R X U Q D O 1 D P H · - R X U Q D O 1 D P H Supplementary Information: Elastic cavitation and...

$: - R X U Q D O 1 D P H · - R X U Q D O 1 D P H Supplementary Information: Elastic cavitation and fracture via injection† Shelby B. Hutchens,a Sami Fakhouri,b‡ and Alfred J. Crosbyb$
Journal Name

Supplementary Information: Elastic cavitation andfracture via injection†

Shelby B. Hutchens,∗a Sami Fakhouri,b‡ and Alfred J. Crosby∗b

1 Equilibrium Stretch Ratio

The initial equilibrium void size, Λeq = A/A0 may be calculatedfrom Pl p + Pelas = 0 corresponding to the equilibration of surfacetension and elastic restoring force. Since P = −∂F/∂V , non-dimensionalizing F (free energy) by µA3

0 and V (volume) by A30

allows us to write

∂Fl p/µA30

∂a/A0+

∂Felas/µA30

∂a/A0= 0 (S1)

The expression for the surface free energy is Fl p/(µA30) =

4π(a/A0)2(γ/µA0). The strain energy in the elastomer Felas is cal-

culated as in Zhu et al. 1

Felas

µA30= 4π

((a

A0

)3−1

)∫ a/A0

1

λ 2W (λ )

µ(λ 3−1)2 dλ , (S2)

where W is the free energy function for the material model cho-sen. For a neo-Hookean solid, substituting the derivatives of theexpressions for the free energy into Eq. S1 and noting that we aresolving for what we are calling the initial void size A, yields

5(

AA0

)4+4

γ

µA0

(AA0

)3−4(

AA0

)3−1 = 0. (S3)

The second term in this equation can be re-written as 4 γ

µA0

(AA0

)4

yielding an equation that can be solved for the equilibrium stretchratio Λeq = A/A0 as a function of the parameter γ/µA.(

5+4γ

µA

)Λ

4eq−4Λ

3eq−1 = 0 (S4)

a Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign, Urbana, IL USA. Tel: 1-217-300-0412; E-mail: [email protected] Department of Polymer Science and Engineering, University of Massachusetts Amherst,Amherst, MA USA. Tel: 1-413-577-1313; E-mail: [email protected]† Electronic Supplementary Information (ESI) available: Supplementary document.See DOI: 10.1039/b000000x/

Λeq is the real solution between zero and one. In the case of themodified Yeoh model (Eq. 7) the expression is non-linear (involv-ing logΛeq).

Accounting for the initial equilibrium stretch ratio enablesreplication of the distinctive pressure versus applied compressionobserved experimentally (Fig. 1). In other words, relative pres-sure at the onset of compression is zero as in Figure S1. Addi-tionally, bubble stretch values at cavitation are greater than onecorresponding to bubble growth prior to cavitation as seen exper-imentally.

CF0 0.005 0.01 0.015

P/7

0

1

2

3

4

5

Fig. S1 As the syringe plunger is depressed, the compression fractionCF defined as 1−Vc/Vc0 increases. The increasing compression in thechamber leads to an increase in pressure. This figure illustrates thepressure-versus compression response for a gas mediated cavitation.(Vc0 = 1 mL, µ = 250 Pa, A = 25 µm, γ = 27.7 mN/m, neo-Hookean)

Journal Name, [year], [vol.],1–10 | 1

Electronic Supplementary Material (ESI) for Soft Matter.This journal is © The Royal Society of Chemistry 2016

2 Yeoh Constitutive ResponseThe modified Yeoh model (Eq. 7) was used to approximate strainhardening while still resulting in neo-Hookean behavior in thelimit of zero strain hardening (C3/µ → 0). As is apparent in Fig-ure S2, increased strain hardening is observed for increased inC3/µ although these are accompanied by a slight increase in stiff-

$

2 4 6 8 10

P/7

0

2

4

6

8

10C

3/7 = 0

C3/7 = 10-3

C3/7 = 10-1

C3/7 = 10

Fig. S2 Pressure-stretch response for a spherical void within an infiniteelastic solid following the modified Yeoh constitutive model (Eq. 7).Decreasing saturation corresponds to decreasing stretch at hardeningas determined by an increase in C3/µ. (γ/µA = 3)

ness at small strain. This increase in small stretch stiffness haslittle effect on Pc from fracture and a relatively small contributionto G, making it a useful constitutive relation due to the closedform for Felas that can be achieved with this relation as comparedto, for example, Gent’s.

2 | 1–10Journal Name, [year], [vol.],

3 Parametric Variation of Mechanism Mapsin Both Gas and Incompressible FluidLimits

Similar to Figure 3, increasing applied displacement can lead toan elastic cavitation or fracture event (Figure S3). Repeating thiscalculation over a range of parameters results in a mechanismmap similar to Figure 5, only in this case we assume an incom-pressible fluid as the pressurizing liquid and a linear system com-pliance k to give Figure S4 for a neo-Hookean solid. Note thatthere is no empty space when an incompressible fluid is used;it is assumed that the applied volume Vapp can always be largeenough to cause the spring to reach Pc due to either elastic cavi-tation or fracture.

As discussed, the inseparability of two independent lengthscales, the needle size, A, and the elastocapillary length γ/µ ne-cessitate the generation of a mechanism map for a series of onelength scale by holding the other length scale constant. Figures S5and S6 demonstrate the variation of the relationship between in-stability mechanisms for the gas and incompressible fluid limits,respectively.

Briefly, system compliance in the case of the incompressiblefluid limit was measured by blocking the needle exit and load-ing the syringe/pressure sensor by displacing the syringe plungerusing a syringe pump. Results of one of these experiments alongwith the linear fit of the observed slope are shown in Figure S7.This fit value is shown via a gray line in Figs. S4 and S6.


$

1 2 3 4 5

" F

tot/7

A

-40

-35

-30

-25

-20

-15

-10

-5

$

1 2 3 4 5

P/7

0

1

2

3

single equil: pre-cavitationabsolute stability limitmetastability limitsingle equil: post-cavitationneo-Hookean void response

a) b) c) ΛF < Λc Λc < ΛF < Λcf Λcf < ΛF ΛF < Λc Λc < ΛF < Λcf Λcf < ΛF

Vapp

A3= 5200

5400

Vapp

1 2 3 4 5-14

-13.5

-13

-12.5

-12

Fig. S3 a) Pressure-stretch response for a neo-Hookean void within an infinite elastic solid (dashed line). (A = 50 µm, µ = 250 Pa, γ = 27.7 mN/m)Overlays of decreasing saturation correspond to increasing displaced volume Vapp (i.e., depressing the syringe) for compression levels reachingsingle, absolute, and meta- stability limits. (k = 1.1 kPa/µm, P0 =1 atm) b) Free energy-stretch response for the combined void/syringe system plottedin a) illustrating the onset of absolute and meta-stability. (∆Ftot = Ftot(Λ)−Ftot(1)) c) Schematic illustrating the applied or displaced volume Vapp.

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

Fig. S4 Map detailing the mechanism leading to observation of a Pcduring a CR experiment under the incompressible fluid assumption. ‘×’corresponds to fracture, ‘ ’ to cavitation, ‘�’ to cavitation followed byfracture, and ’F’ to a negligible pressure drop during cavitation such thatPc is governed by fracture. The magnitude of Pc is shown via colorgradient corresponding to the colorbar on the right. This map wasgenerated assuming a constant needle radius of 50 µm, Gc = 20γ,γ = 27.7 mN/m (air-2-ethyl hexanol interface), and P0 = 1 atm for aneo-Hookean solid. (Gray line corresponds to k = 1.1 kPa/µL.)


100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

100

102

104

./7A10-1 100 101 102 103

Vc0

/(4/3:

A3 )

100

105

1010

b)

a) fixed µ fixed A

c) f)

d)

e)

Pc Pc

Pc Pc

Pc Pc

A = 5 µm µ = 25 Pa

A = 50 µm

A = 500 µm

µ = 250 Pa

µ = 2.5 kPa

Fig. S5 Series of mechanism maps for the pressurizing gas case demonstrating length scale dependence for either fixed needle size (a-c) or fixedelastocapillary length, fixed µ, (d-f). The gray line in each figure corresponds to Vc0 = 1mL. (Gc = 20γ, γ = 27.7 mN/m (air-2-ethyl hexanol interface),and P0 = 1 for a modified Yeoh solid with C3/µ = 0.01.)


100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

b)

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

a) fixed µ fixed A

100

102

104

./7A10-1 100 101 102 103

kA3 /7

10-6

10-4

10-2

100

c) f)

d)

e)

Pc Pc

Pc Pc

Pc Pc

A = 5 µm µ = 25 Pa

A = 50 µm

A = 500 µm

µ = 250 Pa

µ = 2.5 kPa

Fig. S6 Series of mechanism maps for the incompressible fluid/ finite system compliance case demonstrating length scale dependence for eitherfixed needle size (a-c) or fixed elastocapillary length, fixed µ, (d-f). The gray line corresponds to k = 1.1 kPa/µL. (Gc = 20γ, γ = 27.7 mN/m (air-2-ethylhexanol interface), and P0 = 1 for a modified Yeoh solid with C3/µ = 0.01.)


V [7L]0 5 10 15 20 25

Pga

uge [

kPa]

0

10

20

30

P =1.1 kPa7L V

datafit: P = mV

Fig. S7 Pressure increase with applied volume Vapp for the closedsyringe/pressure sensor system over a range of pressures exceedingthose attained during triblock CR experiments.

4 Triblock Co-Polymer GelsTriblock gels were prepared as a 7 wt% solution of PMMA-PnBA-PMMA with block lengths of 25, 116, and 25 kg/mol (Kuraray)dissolved in 2-ethyl-1-hexanol. The solution was covered andheated to 70◦C for at least 1 hour (longer for the initial prepa-ration) prior to allowing the solution to cool at room temperature(∼15 mL within a 20 mL glass vial). Samples were then broughtup to temperature in a water bath monitored by a thermometerand allowed to equilibrate for at least 30 minutes. Each cavitationexperiment was performed with the needle embedded approxi-mately 2 cm below the sample surface as estimated by a ruler. Aportion of the syringe was filled with water, leaving the needleand any remaining bubbles as the only air in the system. A volu-metric flow rate of 50 µL/min was used to pressurize the sampleas controlled with a syringe pump. The data gathered in this wayare shown in Figure S8.

Temperature [°C]10 20 30 40

Pc [

kPa]

100

105

110

115

Fig. S8 Absolute Pc values for 7 wt% PMMA-PnBA-PMMA triblock gelsusing a 30 g stainless steel needle, ID: 106 µm, OD: 266 µm.

Rheological data (TA Instruments) was gathered using

standard-size double cylinders in which the liquid solutionwas poured into the pre-heated, temperature controlled sampleholder. Data, shown in Figure S9, were taken at varying frequen-cies and an oscillation strain of 5×10−5 .

Temperature [°C]10 20 30 40 50 60

G' ,

G''

[Pa]

10-3

10-1

101

103

G' - 10 rad/sG'' - 10 rad/sG' - 1.0 rad/sG'' - 1.0 rad/sG' - 0.1 rad/sG'' - 0.1 rad/s

Fig. S9 Shear rheology data temperature sweeps for 7wt%PMMA-PnBA-PMMA triblock gels at a range of .

5 Power Law Scaling of Gc

These scaling relation discussed here are direct extensions ofthose made by Kundu and Crosby 2 and apply to cross-linked gelsystems. From classical gel mechanics, µ ∼ kT

b3N φ where k is Boltz-man’s constant, T is the temperature, b is the Kuhn length, Nis the number of chain segments, and φ is the volume fractionof polymer. Following their arguments based on the Lake andThomas3 model for strain energy, Gc for a good solvent scales asGc ∼UN1/2φ 7/8/b2. For the PAAm gels characterized, contact me-chanics found that µ ∼ φ 2.3. Substituting this into the expressionfor µ, one obtains N ∼ φ−1.3, which when substituted back intothe expression for µ yields µ ∼ φ 2.3. Similarly substituting N intoGc results in Gc ∼ φ 0.225. Combining these two dependences onvolume fraction φ results in a power dependence:

Gc ∼ φ0.098. (S5)

Figure 7 demonstrates the effect of Gc scaling and strain hard-ening on Pc predictions for the case of the gas with a fixed nee-dle radius. Similar behavior is observed when the elastocapillarylength/µ is held constant as demonstrated in Figure S10 for anidentical set of parameters.

6 Modulus and Fracture Energy Compar-isons to Model Predictions

To determine the percent error of the cavitation equation predic-tions for modulus µcav, values were compared to µ obtained fromshear rheology data at a frequency of 0.1 rad/s or less (triblocksand HSA organogels) or from contact mechanics (PAAm). Theexpression, whose results are plotted in Figure S11a is:

%Err =µcav−µ

µ(S6)


./7A10-2 100 102 104

P c/7

100

102

104

106C3/7 = 1, Gc = 2.C3/7 = 1, Gc = 20.C3/7 = 0.01, Gc = 20.

Eq. 5

./7A10-2 100 102 104

P c/7

100

102

104

106

, = 0.2, - = 0, = 0.2, - = 1, = 0.2, - = 2, = 2, - = 2Eq. (4)

a) b)

5

Fig. S10 Normalized Pc values predicted for varying constitutive behavior and Gc for a parameter series at constant modulus (varying needle radius).The cavitation equation (Eq. 5) is the black solid curve in both plots. a) Effect of increasing strain-hardening corresponding to an increase in C3/µ.Light and dark blue curve illustrates associated effect of increases in Gc (increased Pc) combined with strain hardening. (µ = 500 kPa, Vc0 = 1 mL, γ =

27.7 mN/m) b) Predictions after incorporation of the power law dependence for Gc (Eq. 8). In contrast to Figure 7b, increases in α tend to shift the Pcwithin the fracture regime only slightly. Much larger increases in Pc are observed with increases in β . When µ is held constant, as in these plots,increases in β do not affect the slope within the fracture regime. (µ = 500 kPa, Vc0 = 1 mL, γ = 27.7 mN/m, C3/µ = 0.01)


./7 A10-2 10-1 100 101

Gc [J

/m2 ]

10-1

100

101

triblockPAAm: 207 7mPAAm: 125 7m

PAAm: 75 7mPAAm: 30 7mHSA organogel

./7 A10-2 10-1 100 101

Gc/.

100

101

102

103

triblockPAAm: 207 7mPAAm: 125 7m


./7 A10-2 10-1 100 101

% E

rr in

7

-5

0

5

10

15

20triblockPAAm: 207 7mPAAm: 125 7m


a)

b) c) Open symbols: Gc Filled symbols: Gc,lin

Fig. S11 a) Percent error (Eq. S6) in µ as predicted by the cavitation equation (Eq. 5) and compared to µ obtained using more standard techniques.b)-c) Gc (open symbols, Eq. 8) and Gc,lin (closed symbols, Eq. 9) as a function of γ/µA expressed relative to surface energy (b) and with units (c). Theformer is included in order to show fracture energy values relative to the minimum energy associated with the creation of two new surfaces at the tip ofa crack, 2γ.


References1 J. Zhu, T. Li, S. Cai and Z. Suo, J. Adhes., 2011, 87, 466–481.2 S. Kundu and A. J. Crosby, Soft Matter, 2009, 5, 3963.

3 G. J. Lake and A. G. Thomas, Proc. R. Soc. Lond. A. Math. Phys.Sci., 1967, 300, 108–119.


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