The formation of bubbles, or cavitation, has been studied classically due to its wide applicability to critical processes, ranging from foaming polymers to the onset of traumatic damage in biological tissues.1–3 The inflation of a cavity requires work to be done on the system equaling or exceeding the resistance provided by the material surrounding the cavity. For a liquid, this resistance is provided by the surface tension between the liquid and the fluid used to generate the cavity. Specifically, when a cavity is created at the tip of a syringe needle, this work relates to the development of a pressure defined by Pc = 2γ/r, where γ is the surface tension between the injecting fluid and surrounding liquid, and r is the inner radius of the syringe tip. In a related manner, the inflation of a bubble, or cavity, in an elastic solid requires work to be done equal to the sum of the elastic energy of deforming the solid and the surface tension.4 At the tip of a syringe, this balance of work has been shown to define a critical pressure for the spontaneous inflation of the cavity, Pc = 2γ/r + 5E/6, where E is the elastic modulus of the solid.5–8 Although most bubbles or cavities are formed by the insertion of immiscible fluids that have relatively high surface tensions with the surrounding medium, for example, air in water or air in a gel, the critical pressure required for an elastic solid suggests that cavities can be created with fluids that have zero surface tension with the surrounding medium. This observation has an important implication in the context of measuring mechanical properties within an elastic solid. Specifically, the critical pressure associated with the onset of the elastic instability of cavitation is directly proportional to the elastic modulus and, most importantly, independent of length scale. In other words, measuring the critical pressure for cavitation can provide a simple, quantitative measurement of mechanical properties at virtually any length scale, thus, providing an opportunity to gain critical knowledge of how mechanical properties develop in complex hierarchical materials, such as biological tissues.
We recently developed an experimental technique, cavitation rheology (CR), that takes advantage of these simple relationships between the critical pressure for cavitation and materials properties.9 This technique involves inducing a single cavity via a syringe needle within a soft material and relating the critical pressure of the elastic instability of cavitation to the mechanical properties of the material at the syringe tip. As the inner radius of the syringe needle, r, dictates the length scale that is probed, it is straightforward to probe the mechanical properties over a wide range of length scales. In the first demonstrations of this technique, air was the cavitating fluid. Simple and convenient air cavitation presents limitations in sensitivity to mechanical properties at small-length scales, where r is smaller than the materials-defined length scale of γ/E. This length scale of γ/E defines the transition of relative importance between surface properties and bulk mechanical properties, and thus plays an important role in virtually all mechanical property measurements at small-length scales.
In taking advantage of the unique aspect of cavitation in elastic materials, in this article, we demonstrate that fluids with negligible surface tensions with the surrounding solid can be used to induce the elastic instability of cavitation. Accordingly, the critical pressure for inducing this instability can be directly related to the local mechanical properties of the elastic solid. To demonstrate this, we perform CR experiments using water to induce cavitation in hydrogels of poly(vinyl alcohol) (PVA). We compare these measurements to CR measurements using air as the cavitating fluid in the same hydrogels. This comparison demonstrates the important concept of inducing elastic cavitation with fluids that are miscible with the surrounding material, thus eliminating the length scale dependence of mechanical property measurements and creating exciting opportunities for gaining new knowledge of structure–property relationships in complex materials over a range of length scales.
PVA hydrogels were prepared by heating the polymer (Mw = 13–23 kg/mol, 99.9% hydrolysis, Aldrich) in a mixed solvent of dimethyl sulfoxide and water in a 40:60 ratio with 2% boric acid as a crosslinking agent at 90 °C. Once the polymer was fully dissolved in solution, the sample was cooled to room temperature and stored in airtight containers for 6 days until testing. For all experiments, the weight fraction of polymer to solvent was fixed at 0.10.
The CR technique was used as previously described.9 In summary, it involves inducing a cavitation event at the tip of an arbitrarily placed syringe needle in the sample (microtip syringe needles, ri = 15, 5, and 2.5 μm were purchased from WPI; larger syringe needles, ri = 203, 127, and 51 μm were purchased from Fisher). Cavitation is induced by increasing the pressure of a closed system via a syringe pump (New Era Syringe Pump NE1000) and monitoring the pressure via a pressure transducer in conjunction with a custom-written program within the National Instruments LabView software. Although images of the deformed material at the tip of the syringe are not required for determining the material modulus, images are collected to provide further insight into the specific deformation mechanism.
Cone and Plate Rheology
Cone and plate shear rheometry (TA Instruments rheometer) was used to measure the shear moduli of the hydrogels for comparison with the moduli values measured with CR. A 2°, 40-mm acrylic cone was used for all samples. Stress-controlled frequency sweeps (0.1–2.0 Pa depending on the sample) were performed in a range of 0.01–100 Hz. The modulus value was taken at the strain rate of 0.05 Hz to match the experimental strain rate of CR experiments. Although not rigorously true for all hydrogels, we assume that our materials are isotropic and incompressible. This assumption is used to transform shear moduli data to linear elastic moduli for comparison with the CR data.
Figure 1 illustrates typical pressure history curves for CRT experiments using air as the cavitation media over various syringe radii for the PVA hydrogel. At time t = 0, the syringe pump begins compressing the air in the system, and the pressure in the system increases. As observed in the images provided in Figure 1, limited deformation occurs at the syringe tip until the sudden formation of a cavity. Upon cavitation, the slope of the pressure history curve changes sign with the maximum signifying the critical pressure, Pc, for cavitation. From Figure 1, it is apparent that Pc increases with decreasing syringe radii.
This trend is described by the pressure–growth relationship for a spherical void, growing from an initial flat surface at the tip of syringe, derived by balancing the energy required to create new surface area, dUS, and the energy required to elastically deform the material, dUE, where dW = dUE + dUS. Applying this equation to the present system, we obtain PdV = γdA + σdV, where γ is the surface energy and σ is the material stress defined by the appropriate strain energy function. As demonstrated previously, the neoHookean strain energy function provides a reasonable description of PVA hydrogels; therefore, the pressure–growth relationship for the growing cavity is described by:10
P is the pressure; r is the inner radius of the syringe needle; E is the elastic modulus; and λ is the extension ratio equal to (A/Ao)1/2, where A is the interfacial area between the cavitation media and the hydrogel. Initially, A = Ao = πr2 and as the experiment proceeds, A increases as a spherical cap develops at the tip of the needle. Evaluating the maximum pressure for different values of γ/Er, it can be demonstrated that:
Thus, the materials properties E and γ can be determined by a linear fit of Pc as a function of 1/r (Fig. 2). For the 10% PVA hydrogels, these measurements determine that E = 2.9 ± 0.80 kPa and γ = 0.032 ± 0.0018 N/m. This modulus value was confirmed with shear rheometry, where E is ∼0.71 kPa. The linear dependence of Pc with 1/r contrasts the predicted scaling for Pc associated with the initiation of fracture at the tip of the syringe needle, as discussed in a recent publication.10 In the case of fracture, Pc, which the data in Figure 2 does not follow. Accordingly, the initiation of the instability-derived deformation is associated with an elastic event, but the propagation may involve subsequent material fracture. The propagation, or resulting geometry, of this deformation zone will be the subject of a future manuscript.
By using both the E and γ determined by CR, the material length scale, γ/E, is found to be ∼11 μm. This estimation is consistent with the observed significant increase in Pc as the syringe radius changed from 15 to 2.5 μm in Figure 1, demonstrating an increased importance of surface energy contributions relative to bulk contributions at this length scale.
As discussed earlier, cavitation in an elastic solid does not require the cavitating fluid to have a significant surface tension with the surrounding material. To demonstrate this concept and minimize the length-scale dependence of Pc, we exchanged the cavitation media, air, with water, which has negligible interfacial tension with the PVA hydrogels.
The water cavitation experiments provide similar pressure history curves to those resulting from air cavitation, as seen in Figure 3(a,b). These figures show that the pressure increases as the system volume is compressed with little activity at the tip of the syringe needle before the instability, and on cavitation, the pressure drops suddenly. The most striking difference between CRT experiments when using water versus air is that there is little to no increase in cavitation pressure with decreasing syringe radius, even using radii as small as 2.5 μm. The plot in Figure 3(b) emphasizes this point by comparing the pressure history curves for both air and water experiments using the 5-μm syringe. Here, the critical pressure in the air experiment is almost eight times greater than that for water cavitation.
Similar to the air cavitation measurements, we can determine E and γ from a linear fit of Pc as a function of 1/r (Fig. 2) for the water cavitation data. We determine E = 2.2 ± 0.24 kPa and γ = 0.0011 ± 0.0005 N/m. This measurement of E is within reasonable agreement with the best-fit measurement provided by the air cavitation measurements, and γ between water and the 10% PVA hydrogel is significantly less than the air/hydrogel surface tension. Furthermore, the materials length scale, γ/E, for the water cavitation measurements is ∼500 nm.
These measurements confirm that the elastic instability of cavitation can be induced in elastic materials using fluids that have negligible surface tensions with the surrounding material. As illustrated in Figure 2, the critical pressure for cavitation in these cases is nearly independent of length scale, at least over the two orders of magnitude explored for this material. This independence with length scale has important implications for measuring the mechanical properties of polymer networks on length scales that approach network and molecular characteristic dimensions, such as the mesh size.
Although zero or negligible surface tension between the cavitating fluid and surrounding material is possible for inducing cavitation in an elastic solid, a significant challenge exists in the development of pressure under these interfacial conditions. In other words, cavitation, either in liquids or solids, typically involves the use of an immiscible fluid to induce cavitation. Under these conditions, pressure increases at the tip of a syringe due to the compression of the cavitation fluid between the syringe plunger and the surrounding material, as the fluid and material cannot mix. This compression is measured as the development of pressure at the tip of the syringe. For fluids and materials that have zero or negligible surface tensions, mixing may occur as one is injected into the other. This mixing, or diffusion, of the cavitating fluid would limit the development of pressure, which is required to exceed the critical value, Pc, for cavitation to occur.
Therefore, for pressure to develop at the tip of the syringe, the pressure driven mobility of the injecting fluid into the surrounding material or the relaxation processes associated with poroelastic mechanisms in the gel must be significantly slower than the time scale of the CR test. Although a complete investigation will be presented in a future paper focused on the use of CR to probe poroelastic properties of swollen gels, here we present a straightforward experiment to confirm the maintenance of a subcritical pressure at the tip of the syringe over time scales that are significantly larger than the time scale for CR experiments. In this experiment, a 5-μm radius syringe is inserted into a 10% PVA hydrogel and water is used as the cavitation media. The pressure in the system was increased to approximately half of the critical pressure to cavitate, and then the syringe compression was fixed at a constant value. Under this condition of fixed compression, the pressure was monitored for ∼600 s. After the dwell time of fixed compression, the syringe compression was reinitiated, further compressing the system until the cavitation event was observed. As seen in Figure 4, cavitation occurred at the same value of Pc as with the experiments run without a dwell time. Furthermore, the pressure relaxed minimally during the dwell time over the time scale of 600 s, with the ratio of 1.61 kPa at 600 s to 1.67 kPa at the initiation of dwell equal to 0.964. This maintenance of pressure at the tip of syringe for water-based CR experiments demonstrates the robustness of these measurements and presents opportunities for future experiments on poroelasticity and diffusion into swollen gels at subcritical and postcritical cavitation strains.
In summary, we have shown that fluids with negligible interfacial tensions with a surrounding material can be used to induce the elastic instability of cavitation in that material. Accordingly, under these conditions, the critical pressure to induce cavitation is directly related to the elastic modulus of the surrounding network, and is nearly independent of length scale. This independence of size scale has important implications in the use of CR for the characterization of mechanical properties across a broad range of length scales, allowing future opportunities to span from molecular to macroscopic lengths with a single experimental approach. We anticipate that this broad range of length scales and the ability to probe specific locations within a material local to the tip of the syringe will allow CR to be used to gain new knowledge of how mechanical properties develop in hierarchical soft materials, such as biological tissues.
The authors thank the NSF-IGERT and the NSF-MRSEC for funding and Ken Shull for his insightful discussions.