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Shelby B.
Hutchens
*^{a},
Sami
Fakhouri
^{b} and
Alfred J.
Crosby
*^{b}
^{a}Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign, Urbana, IL, USA. E-mail: hutchs@illinois.edu; Tel: +1-217-300-0412
^{b}Department of Polymer Science and Engineering, University of Massachusetts Amherst, Amherst, MA, USA. E-mail: crosby@mail.pse.umass.edu; Tel: +1-413-577-1313

Received
18th August 2015
, Accepted 22nd January 2016

First published on 25th January 2016

The cavitation rheology technique extracts soft materials mechanical properties through pressure-monitored fluid injection. Properties are calculated from the system's response at a critical pressure that is governed by either elasticity or fracture (or both); however previous elementary analysis has not been capable of accurately determining which mechanism is dominant. We combine analyses of both mechanisms in order to determine how the full system thermodynamics, including far-field compliance, dictate whether a bubble in an elastomeric solid will grow through either reversible or irreversible deformations. Applying these analyses to experimental data, we demonstrate the sensitivity of cavitation rheology to microstructural variation via a co-dependence between modulus and fracture energy.

Rapid, 3-D mechanical characterization of soft solids provides information necessary for the elucidation of health-related mechanisms, e.g., mechanotransduction in cells and tissue property evolution during onset of disease. A recently explored technique referred to as cavitation rheology (CR)

To date, only elementary analyses have been applied to these observations, with a limited quantitative understanding as to when and how the observed deformation mechanisms relate to the measured data.^{2} Although the morphology of the expanded voids clearly indicate that fracture can be a dominant growth mechanism for certain materials, application of the simple ‘cavitation equation’^{2} based only on elastic deformations can typically yield modulus values having less than 15% error. In contrast, correlating P_{c} to the fracture toughness, G_{c}, results in much greater error. This result points to the need for further understanding into how materials properties, including nonlinear behavior, control unstable void growth in an elastomeric solid. This insight is critical for enhancing the implementation of CR in biological tissues, where nonlinear mechanics play a dominant role. More generally, resolving apparent conflict between the observed deformation path and the critical pressure is important for understanding how damage is incurred in soft materials, a topic that has been studied extensively, most notably by Alan Gent, though many questions remain.

In their 1959 paper, Gent and Lindley^{12} observed a discontinuity in the load-displacement response of rubber disks under uniaxial tension. Noting that the predominant stress state in this testing geometry is hydrostatic, they attributed the discontinuity to the rapid expansion of pre-existing spherical defects or voids within the sample. These observations ignited a discussion of cavitation (and fracture) within an elastomeric solid (e.g.ref. 13–15) that continues even now.^{16,17} Our extension of this discussion in the context of CR provides new understanding of how elasticity, fracture toughness, and the full system thermodynamics, control void growth mechanics in soft solids.

In the following work, we utilize a simplified, spherical geometry under quasi-static loading in order to quantitatively determine the interplay between elastic and fracture mechanisms and the material parameters and behaviors that affect that interplay, in particular strain hardening. Knowledge gained from understanding these mechanisms individually is then unified into a mechanism map of anticipated behavior from which predictions for P_{c} are made for three distinct experimental systems. Specifically, we follow previous derivations for a spherical void in an incompressible neo-Hookean solid.^{18–20} We incorporate energy-based fracture criteria for the void in order to determine the bubble deformation at which fracture occurs. From these models, we construct mechanism maps for two limiting-case loading scenarios. The resulting maps predict the critical pressure P_{c} as a function of the non-dimensionalized elastocapillary length γ/μA and relevant geometric and material parameters. The elastocapillary length γ/μ is the ratio of the surface tension, γ, to the elastic shear modulus, μ. A is the initial radius of the void, which we assume is equivalent to the needle radius used in CR. The relationship between these length scales is thus essential for interpreting experimental P_{c} data as an intrinsic material property.

In experiments, the pressurizing fluid has been either air or water^{1–8,11} and in both cases unstable expansion at a critical pressure P_{c} is observed. Because the fluid reservoir has an inherent stiffness, k, associated with it (due to rubber o-rings and pressure sensors), we approximate the system as shown in Fig. 2c and d. This approximation leads to two limiting cases for loading conditions based on simplifications when the fluid is either a gas (ideal gas) or a liquid (incompressible fluid). In the case of the former, Zhu et al.^{20} previously showed that sudden elastic cavity growth (elastic cavitation) requires both a sufficiently large bubble surface energy, γ, and an initial fluid reservoir volume, V_{c0}, capable of achieving a pressure that can overcoming the surface energy contribution the bubble. Their system was comprised of a reservoir of ideal gas molecules and assumed no system compliance. Here we use this same framework, setting system stiffness equal to infinity (compliance equal to zero) in the case of a gas pressurizing fluid. This approximation is valid even for a system with finite stiffness due to the fact that the ratio of the system stiffness, k = dP/dV|_{spring}, (k ≈ 1 kPa μL^{−1}, Fig. S7, ESI†) to the ‘stiffness’ of an ideal gas, dP/dV|_{gas}, at the onset of cavitation is much greater than one in all cases except that of a high modulus material (∼1 MPa) combined with a small reservoir (<1 mL). Materials with such high moduli are readily characterized using traditional techniques and therefore outside the main scope of CR. This stiffness ratio is determined for a chamber volume V_{c} (initially having volume V_{c0} at atmospheric pressure P_{0}) evaluated at a critical pressure P_{c} that scales roughly with the material shear modulus so that kV_{c0}P_{0}/P_{c}^{2} ≈ kV_{c0}P_{0}/μ^{2} ≫ 1. Thus, for air as the pressurizing fluid, we will simply follow Zhu et al.^{20} derivation. However, we include a small redefinition of the initial state of the system that allows for experimental comparison as will be discussed in the next section.

As shown by Zhu et al.,^{20} the gas provides the only negative contribution to the free energy of the system just described (i.e., a system having an infinite stiffness spring) and thus enables its ability to cavitate. Application of this same model to an incompressible fluid^{4} therefore, cannot predict elastic cavitation. Because addressing both gas and system compliance contributions simultaneously becomes unwieldy, our second limiting case is simply that of a spring of finite stiffness only (no gas reservoir).^{19} We demonstrate that this system compliance provides the necessary negative free energy contribution to allow for elastic cavitation with an incompressible pressurizing fluid. In summary, our two limiting cases for loading conditions become: a reservoir of ideal gas molecules (no spring) and a linear spring of stiffness k = dP/dV. Note that in neither case are these load or displacement-controlled conditions. (Displacement-controlled, or more accurately volume-controlled, testing applies only to the case of a spring of infinite stiffness combined with an incompressible fluid.) Similar to a cantilever-type, ‘displacement-controlled’ compression test, the compliance of either the gas or the system (like the centilever) mediates any deformation of the test sample. In CR, that effect is an essential part of the observed, catastrophic response that leads to P_{c}.

(1) |

The pressure–stretch (P vs. Λ_{0}) and free energy–stretch (F_{tot}vs. Λ_{0}) responses^{20} previously derived without this correction are then simply modified through the substitution of Λ_{0} (eqn (1)) such that they become functions of Λ given that Λ_{eq} = Λ_{eq}(γ/μA, constitutive model parameter). For an incompressible neo-Hookean solid with strain energy function W = μ(I − 3)/2 (where I = 2Λ_{0}^{2} + Λ_{0}^{−4}: the first strain invariant of the right Cauchy–Green tensor) and assuming an ideal gas pressurizing fluid, these expressions are:

(2) |

(3) |

For the incompressible fluid approximation accounting for system stiffness, k, the expression for F_{gas} is replaced with one for the spring, F_{spr}.

(4) |

We obtain stability limits, and thus P_{c} and deformation at cavitation, Λ_{c}, from the expression for the free energy of the system (eqn (3)). Free energetic equilibria (dF_{tot}/dΛ = 0) correspond to the intersection of the void pressure and the gas pressure as a function of void deformation (Fig. 3), essentially amounting to a force balance. (The gas is replaced by a spring in the case of incompressible fluid loading, Fig. S3, ESI.†) In order for elastic cavitation to occur, there must be at least two equilibrium deformations such that it is thermodynamically favorable for the system to undergo a transition from the smaller to the larger deformed bubble while passing through a thermodynamically unstable (d^{2}F_{tot}/dΛ^{2} < 0) range of deformation.^{18}

Fig. 3 illustrates how an increasingly compressed reservoir encounters two types of limits for cavitation that satisfy these criteria. First, the system encounters equilibrium (the absolute stability limit), at which the free energy of the smaller and larger bubble become equal (d^{2}F_{tot}/dΛ^{2} > 0, dF_{tot}/dΛ = 0, F_{tot}(Λ_{small}) = F_{tot}(Λ_{large})), but are separated by an energetic barrier (d^{2}F_{tot}/dΛ^{2} < 0, dF_{tot}/dΛ = 0). Further compression leads to the metastability limit, in which the energetic barrier between the two states vanishes (d^{2}F_{tot}/dΛ^{2} = dF_{tot}/dΛ = 0). We use the latter to predict P_{c} as thermal fluctuations provide little perturbation to overcome an energetic barrier in a macroscale system. Thus, we determine the ability of the system to elastically cavitate for any given criteria, γ/μA, P_{0}/μ, and V_{c0}/(4/3πA^{3}) (or kA^{3}/μ for the incompressible fluid).

Non-zero surface energy is required in order to produce an elastic cavitation response in both loading condition cases. The minimum amount of surface energy is determined by the dimensionless ratio, γ/μA, which must be greater than ≈1.4953, the value at which a peak in the pressure (P)–deformation (Λ) curve (dP/dΛ = 0) no longer exists for real values of Λ (for an incompressible, neo-Hookean solid). (Examples of P vs. Λ for which a peak exists (black dashed) and at the limit where the peak disappears (gray dashed) are shown in Fig. 3a.)

When elastic cavitation is possible, we calculate P_{c}, Λ_{c} (the deformation at the onset of elastic cavitation), and Λ_{cf} (the final deformed state after cavitation). (The latter are shown with gray arrows in Fig. 3 where Λ_{cf} ≈ 11.) This model also allows us to determine the equilibrium void radius as a function of increasing compression, CF, as shown by the progression of increasing compression fraction curves in Fig. 3a. With this series of equilibirium deformation values, we replicate the experimentally observed pressure vs. compression curves displaying a sudden, discontinuous drop in pressure at P_{c} (Fig. 1 and Fig. S1, ESI†). Also notable from this analysis is the fact that P_{c} for elastic cavitation is well approximated by the simple ‘cavitation equation’,

(5) |

The model we implement here, following Gent and Wang^{14} and Williams and Schapery,^{24} is similar to Griffith's analysis, with a strain energy release rate G for an incompressible, neo-Hookean solid:

(6) |

Λ
_{F} depends on the constitutive response of the material and G_{c}. We address the motivation behind varying these elements in Sections 7 and 8, however we give a brief overview of their effect on Λ_{F} here. The incorporation of strain hardening, decreases Λ_{F} overall as a function of γ/μA (solid lines). Fig. 4 also demonstrates that the monotonic dependence of Λ_{F} on γ/μA disappears when G_{c} is no longer assumed to be constant over order of magnitude changes in γ/μA (due to a dependence on μ). Assuming that G_{c} increases with increasing μ, it follows that Λ_{F} eventually increases with decreasing γ/μA as shown.

Most CR experiments have been performed with relatively large gas volumes, putting them toward the top of Fig. 5, above the gray line that corresponds to V_{c0} = 1 mL. As discussed above, the elastic cavitation limit that exists at finite γ/μA due to the loss of a peak in the P vs. Λ response (Fig. 3a) clearly divides the mechanism governing P_{c}. Needle radii and/or μ must be small for P_{c} values to have a chance to fall within the elastic cavitation regime.

An implication for failure not only in CR but in soft solids in general that follows from this work is that even when deformation is accommodated by elastic cavitation events, it is likely that these elastically initiated events lead almost instantaneously to localized fractures. Whether elastic cavitation can occur without fracture depends on the compliance of the loading environment, e.g., the chamber stiffness k or volume V_{0}, sample size (in the case of dead-loading),^{18} or machine compliance,^{19} with less compliance/smaller samples leading to an increased likelihood of microfracture. The extent of the fracture that occurs is dependent on dynamics and fracture geometries not studied here.

Fig. 6
P
_{c}
vs. γ/μA for the three datasets: (purple-diamonds) triblock copolymer gels (PAA–PnBA–PAA), (orange-inverted triangles) HSA organogels,^{11} and (blue-left triangles) PAAm hydrogels.^{2} A neo-Hookean model prediction taken from Fig. 5a is shown with a gray line and symbols representing the governing mechanism as outlined in the Fig. 5 caption. The cavitation equation (eqn (5)) is shown with a solid black line. |

While the neo-Hookean constitutive relation provides a starting place for exploring hyperelastic behavior, its entropic spring assumption breaks down near material failure, as the network is fully stretched. Fig. 6 demonstrates a clear deficiency in the neo-Hookean model. Several experimental values lie above both the cavitation equation and the isochor prediction for V_{c0} = 1 mL. (The gap between these two predictions at high γ/μA arises purely from geometry; the cavitation equation accounts for a spherical-cap like bubble whereas our model is entirely spherical.) The neo-Hookean prediction (Fig. 6, gray line) uses G_{c} = 20γ, however, even for increasingly tough materials P_{c} can never exceed the cavitation equation whose magnitude aligns with the peak in the bubble's pressure–stretch relation (Fig. 3a). Thus, the data clearly support the necessity of accounting for strain hardening, especially for the triblock gels,^{25} prior to failure. Further, as written, the equations in Sections 3 and 4 assume fixed G_{c} over the set of conditions spanned by a mechanism map. Such an assumption of constant G_{c} is accurate for a constant μ map, but in the case of order of magnitude variations in μ in a constant A map, G_{c} will not remain constant, as becomes clear on an Ashby plot for soft solid materials.^{26} The next two sections detail the incorporation of both strain hardening and G_{c} variation.

(7) |

Further, Fig. 7a demonstrates that strain hardening leads to P_{c} predictions that lie above the cavitation equation threshold. Fig. 7a is generated using the assumption of constant G_{c} but spans a range of γ/μA values while ‘needle radius’ (corresponding to initial void radius) is held constant at A = 100 μm. The result of this assumption is that a negative slope in P_{c}vs. γ/μA (for constant A) cannot be predicted. In order to do so, and therefore capture the behavior observed in the triblock data shown in Fig. 6, we must consider the possibility of a G_{c} dependence on μ. (Experimental conditions are critical for interpreting P_{c} data. If μ, rather than A is held constant, no negative slope in P_{c}vs. γ/μA is observed, Fig. S10, ESI.†)

Fig. 7 Normalized P_{c} values predicted for varying constitutive behavior and G_{c} for a parameter series at constant needle radius (varying modulus). The cavitation equation (eqn (5)) is the black solid curve in both plots. (a) Effect of increasing strain-hardening corresponding to an increase in C_{3}/μ. Light and dark blue curve illustrates associated effect of increases in G_{c} (increased P_{c}) combined with strain hardening. (A = 100 μm, V_{c0} = 1 mL, γ = 27.7 mN m^{−1}). (b) Predictions after incorporation of the power law dependence for G_{c} (eqn (9)). Increases in α tend to shift the P_{c} within the fracture regime up. Increases in β tend to make the slope within the fracture regime negative. (A = 100 μm, V_{c0} = 1 mL, γ = 27.7 mN m^{−1}, C_{3}/μ = 0.01.) |

Fig. 8 Maps detailing the mechanism leading to observation of a P_{c} during a CR experiment using the modified Yeoh Model. ‘×’ corresponds to fracture, ‘●’ to cavitation, ‘■’ to cavitation followed by fracture, and ‘★’ to a negligible pressure drop during cavitation such that P_{c} is governed by fracture. The magnitude of P_{c} is shown via color gradient corresponding to the colorbar on the right. This map was generated for a modified Yeoh material with C_{3} = 0.01 and the same conditions as in Fig. 5. |

(8) |

To account for this co-dependency, we can assume a relationship between G_{c} and μ, based on experimental data (e.g.ref. 26 and 31):

G_{c} = αμ^{β} | (9) |

System |
C
_{3} (λ_{F}) |
α | β |
G
_{c} [J m^{−2}] range |
G
_{c} from the literature |
---|---|---|---|---|---|

Triblock (7 wt%) | 0.01 (2–4^{7,25}) |
3 × 10^{−7} |
3.0 | 2–20 | For 10 wt%, 10–20 J m^{−2} (ref. 27) |

HSA organogels (0.25 wt%) | 10 (1.06^{28}) |
0.8 | 0.08 | ∼1 | None |

PAAm (vol. frac. 0.027–0.088) | 0.01 (5^{29}) |
4 | 0.1 | 7–10 | For vol. frac. ∼0.08, 10–20 J m^{−2} (ref. 30) |

It is important to note that for the modified Yeoh relation used here, the fit value α and thus G_{c} is dependent on C_{3}; while the parameter β is relatively insensitive to C_{3}. This makes the accurate choice of constitutive relation an important part of the interpretation of a quantitative G_{c}. For this reason our reported G_{c} values should still be taken as estimates, though, as stated above, we believe these estimates to be more accurate than using the linear approximation. To clarify the role that β plays in replicating experimental results, one must understand its effect on the deformation at fracture, Λ_{F}. For increasing power, β, Λ_{F} becomes increasingly large at low γ/μA (Fig. 4). This increased Λ_{F} corresponds to an increased P_{c} in a strain hardening material model as shown in Fig. 7b. For very large values of β, Λ_{F}A^{3} becomes large with respect to the initial chamber size V_{c0} and our model predicts that no fracture and thus no P_{c} will be observed.

As an aside, the unexpected cubed relation between G_{c} and μ that is responsible for the large deviation of the triblock P_{c} data (Fig. 9) from the cavitation equation (the initial driving force for this work) is likely due to the thermally reversible nature of this material's crosslinks. The glass transition temperature, T_{g} for the PMMA endblocks is just above room temperature, likely resulting in the large change in P_{c} as a function of temperature right around 20–25 °C (Fig. S8, ESI†) that leads to β = 3.0. We interpret this as the result of a sharp decrease in chain pullout forces over a region of constant elastic response due to relatively unchanged chain configuration between crosslinks (Fig. S9, ESI†). The G_{c} dependence on temperature is significantly stronger than has been predicted via simulations of pull-out from a purely glassy state,^{33} suggesting a transition from pull-out in a condensed state near the gel point to pull-out from a glassy state at lower temperatures.

a | Deformed bubble radius |

A | Initial bubble radius (equivalent to needle radius) |

A
_{0}
| Zero strain energy bubble radius |

α | Power law coefficient |

β | Power law power |

C
_{3}
| Phenomenological parameter in Yeoh constitutive relation |

CF | Gas compression fraction, 1 − V_{c}/V_{c0} |

F
_{tot}
| Total free energy of the bubble + system + surroundings |

G | Strain energy release rate |

G
_{c}
| Critical strain energy release rate |

γ | Surface energy |

I | First strain invariant of the right Cauchy–Green tensor |

k | Stiffness of the fluid pressurizing setup |

λ
_{F}
| Experimentally measured strain at failure |

Λ | Deformation of the bubble: a/A |

Λ
_{0}
| Hoop stretch; zero strain energy bubble deformation: a/A_{0} |

Λ
_{c}
| Deformation of the bubble at P_{c} due to cavitation |

Λ
_{cf}
| Final deformation of the bubble after elastic cavitation |

Λ
_{eq}
| Zero strain energy bubble deformation due to surface energy: A/A_{0} |

Λ
_{F}
| Deformation of the bubble at P_{c} due to fracture |

μ | Linear elastic shear modulus |

P | Pressure |

P
_{0}
| Pressure of the environment |

P
_{c}
| Critical pressure at which sudden bubble expansion occurs |

V
_{c}
| Chamber volume |

V
_{c0}
| Initial chamber volume |

W | Strain energy function |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5sm02055g |

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