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Letter Cite This: ACS Macro Lett. 2019, 8, 492−498

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Effect of Polymer Volume Fraction on Fracture Initiation in Soft Gels at Small Length Scales Shruti Rattan and Alfred J. Crosby* Polymer Science and Engineering Department, University of Massachusetts−Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States

ACS Macro Lett. Downloaded from pubs.acs.org by UNIV AUTONOMA DE COAHUILA on 04/09/19. For personal use only.

S Supporting Information *

ABSTRACT: The influence of polymer volume fraction, ϕv on fracture initiation via puncture is studied in self-assembled triblock copolymer gels. Spherically tipped indenters of radii varying over a wide range were used to characterize puncture at length scales on the same order of magnitude as the elasto-capillary length (∼μm) and significantly below the elasto-fracture length (∼mm) for ϕv = 0.12−0.53. Critical energy release rate, Gc for ϕv = 0.12−0.30 was found to be in agreement with the predicted scaling from the classical Lake-Thomas model modified for gel fracture via the failure mechanism of chain pull-out and plastic yielding of micelles (Gc ∼ ϕ2.2 v ). Interestingly, we demonstrate that fracture initiation energy, Γ o, from puncture scales as Γ o ∼ ϕv, thus, indicating the role played by different fundamental mechanisms governing fracture initiation in soft gels. Additionally, gels with ϕv = 0.53 show deviation from experimental scalings for Gc and Γ o, likely due to a change in micellar morphology leading to anomalous fracture behavior.

T

fracture process zone formed beneath an indenter tip prior to puncture is governed by the indenter tip radius.14 The size of the fracture process zone is crucial in determining the failure regime, that is, whether failure is limited by a critical stress or a critical energy. In an energy-limited regime, the critical force for puncture, Pc, scales linearly with the radius of the indenter tip, R. This scaling relation is derived assuming that the resistance to deformation and failure during puncture is mainly contributed by the elasticity of the gel. However, there are several characteristic length scales at which forces due to surface tension and adhesion can possibly influence crack initiation and growth (schematically depicted in Figure 1).4,16−18 For example, the mesh size of the network, ξ ∼ (kbT/G′)1/319,20 (where kb is the Boltzmann constant, T is temperature, and G′ is the small strain shear modulus). In addition, other material structure relevant length scales that may play a role in the interaction between an indenter and a soft material are the elasto-capillary length and the elasto-

he fracture of soft polymer gels has been widely characterized with traditional testing methods;1−5 however, limited attention has been given to the critical force and energy involved in the initiation of a crack in soft solids. Understanding the mechanics behind crack nucleation is not only crucial to advance the current scientific knowledge on soft materials fracture but also is relevant to numerous health care applications such as percutaneous needle insertion (biopsies, blood sampling, and anesthesia),6 robot-assisted surgeries,7,8 and design of advanced surgical instruments.9,10 One approach for studying crack nucleation (or fracture initiation) in soft solids is through understanding the forces and the energy involved in the process of puncturing a soft polymer gel with sharp indenters.8,11−15 Previous studies on puncture of soft gels have primarily focused on understanding the mechanics of deep indentation to the point of puncture. Yet, little is known about the effect of intrinsic material properties governed by polymer concentration on the fracture initiation process in soft polymer gels. Furthermore, very few experimental studies have examined fracture initiation in soft gels at material-specific length scales. In our recent studies on puncture of soft gels with spherically tipped indenters, we discussed that the size of the © XXXX American Chemical Society

Received: January 30, 2019 Accepted: April 3, 2019

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DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498

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Figure 1. Schematic depicting various characteristic length scales, namely, mesh size of the network, ξ, the elasto-capillary length, ϒ/E, and the elasto-fracture length, Gc/E at which adhesion and surface energy can possibly affect the force required to nucleate a crack or initiate a fracture during a puncture event. (b) is adapted with permission from ref 22. Copyright 2013 Springer Nature. (b) is also adapted with permission from ref 18. Copyright 2016 Elsevier.

Figure 2. (a) Average small strain shear moduli, G′, and (b) critical energy release rate, Gc, plotted against polymer volume fraction, ϕv. The best 2.2 power law fit ∼G′ ∼ ϕ2.3 v and ∼Gc ∼ ϕv (fitting performed with weighted error bars), respectively, are shown as a guide for the eye in each plot. The measured values for Gc for ϕv = 0.53 deviate from the observed experimental scaling for lower gel concentrations.

fracture length. The elasto-capillary length, ϒ/E, is determined by the surface tension of the solvent, ϒ, and small strain Young’s modulus, E.16,21−23 The elasto-fracture length, Gc/E, is given by the ratio of the critical energy release rate for crack propagation (or the energy released at the crack tip to propagate a crack per unit area), Gc,24 and E.4,25 The objective of this paper is to probe fracture initiation properties at length scales relevant to material structure in a well-characterized model material system, acrylic triblock copolymer gels.3,26−31 We prepared a library of materials with varying ξ, ϒ/E, and Gc/E by changing the polymer volume fraction, ϕv in the range of 0.12−0.53, resulting in gels with tunable E (or G′) and Gc. Specifically, we investigate the effect of gel concentration on fracture initiation energy, Γ o, via puncture with indenter tip sizes on the same order of magnitude as ϒ/E and significantly below Gc/E (Figure S2). We utilize the knowledge gained from traditional mechanical property tests, namely, small strain oscillatory shear rheometry and pure shear, to provide valuable insight into the role played by materials parameters in the unique failure process of puncture. Acrylic triblock copolymer gels were chosen as model materials for this study due to their high elasticity up to large

strains and thermoreversible gel transition.3,26,32 The preparation of the gels is described in detail in the Materials section of the Supporting Information (Figure S1). Small-angle X-ray scattering (SAXS) experiments were performed on the gels to study the evolution of micellar morphology with increasing ϕv. Scattering patterns for all gel concentrations are shown in Figure S3. Although a detailed analysis of the obtained SAXS pattern is the subject of future research, we observe that a peak at q ∼ 0.3−0.4 nm−1 for lower gel concentrations, ϕv = 0.12− 0.30 is clearly absent in the SAXS pattern for ϕv = 0.53, and a new peak emerges at q ∼ 0.54 nm−1. While the SAXS data may not be rigorously conclusive, it suggests a change in micellar morphology at ϕv = 0.53. The mechanical response of gels with the highest ϕv = 0.53 are compared and contrasted with gels of lower ϕv = 0.12−0.30 (ϕv < 0.53) in the following sections. Small strain oscillatory shear rheometry was employed to measure G′ for each gel concentration investigated in this study (Figure S4). At T = 22°C, G′ is approximately independent of frequency (0.1−10 Hz), and G′ values (averaged over the frequency range for n = 3) are plotted as a function of ϕv in Figure 2a. Although, a linear dependence of G′ with ϕv is expected from the theory of rubber elasticity (G′ = νkbT, where ν is the number of elastically effective chains per 493

DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498

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Table 1. Summary of Characteristic Length Scales Defined by Material Parameters Estimated for Each Gel Concentration vol fraction, ϕv

elastic modulus, E = 3G′ (kPa)

mesh size, ξ ∼ kbT/G′1/3 (nm)

elasto-capillary length, ϒ/E (μm)

0.12 0.16 0.20 0.30 0.53

8.0 ± 0.1 9.9 ± 0.8 16.5 ± 2.9 41.7 ± 10.8 162.3 ± 13.0

11.7 ± 2.1 10.9 ± 3.2 9.2 ± 3.6 6.8 ± 3.0 4.3 ± 1.3

3.4 ± 0.1 2.8 ± 0.2 1.7 ± 0.3 0.7 ± 0.2 0.20 ± 0.01

elasto-fracture length, Gc/E (mm) 1.46 1.78 1.30 1.81 3.84

± ± ± ± ±

0.02 0.14 0.23 0.47 0.31

Figure 3. (a) Schematic of a typical experimental setup used to study puncture with spherically tipped indenters in soft gels. (b) Optical microscopy images of indenters, with R = 0.4−66.0 μm. (c) Empirical parameter k′E that represents effective moduli obtained from puncture is calculated for ϕv = 0.12−0.53 for puncture experiments, with R = 5.0, 15.0, 35.0, and 50.0 μm, and is plotted as a function of small strain elastic storage moduli, E, obtained from shear rheometry. A good correlation between k′E and E is observed with the coefficient empirically determined as k′ = 0.37 ± 0.01.

Gc ∼ νLNU

unit volume) and experimentally confirmed for triblock copolymer gels in the unentangled regime,33 acrylic triblock copolymer gels used in this study follow a continuous scaling of G′ ∼ ϕ2.3 v for concentrations in the range, ϕv ∼ 0.2−0.53. This scaling is similar to the scaling predicted for swollen polymer networks in a good solvent in the entangled regime, G′ ∼ ϕv2.25,34 and has been experimentally verified for entangled triblock copolymer gels.35,36 In our gel system, midblock entanglements are not expected to form for the range of concentrations, ϕv = 0.12−0.20. At ϕv ≥ 0.3, classical theories predict entanglements to develop (ne ∼ Neϕ5/4 v , where Ne is the number of midblock entanglements in the melt, and ne is the number of midblock entanglements at various volume fractions). Entanglement molecular weight of neat PnBA is 22 kg/mol. Hence, the ϕv at which midblocks form one entanglement for PMMA12-b-PnBA94-b-PMMA12 gels is ∼0.3. It is important to note here that the scaling theories are derived for swelling behavior of gel networks in a good solvent and are not well suited to describe the G′−ϕv relationship for networks prepared at different gel concentrations. The experimentally measured scaling between G′ and ϕv indicates ν ∼ ϕ2.3 v . We will use this relationship to derive a theoretical prediction for the dependence of critical energy release rate, Gc on ϕv in the following section. The Classical Lake and Thomas model is the most popular model for predicting fracture energies of elastomers and gels.37−42 According to this model, the minimum energy required to break the bonds per unit area of the fracture plane equivalent to Gc is given by

(1)

where ν is the number of elastically effective chains per unit volume, N is the degree of polymerization of the network strand, U is the energy required to fracture a monomer unit, and L is the length over which energy is reversibly dissipated during fracture. The length scale, L, is generally assumed to be of the order of the end-to-end distance of the undeformed network strand and is given by L ≈ Ro ≈ aN0.5ϕ−1/8 for swollen v polymer networks in a good solvent (where a is the Kuhn length).34,43 The above equation is a generalized form of the fracture energy predicted by Lake and Thomas and was provided by Akagi et al.44 Previous studies for fracture in a similar gel system (acrylic triblock gels of similar chemical composition but different molecular weights) have shown that failure does not occur by chain scission but rather via yielding of the polymer chains crossing the crack-plane until they are completely pulled out of the gel network. In particular, it has been argued that chains pullout from the PMMA nanodomains as they yield at a stress that is lower than stresses required for chain scission.3,15,45 Using the dependency of ν on ϕv and assuming N, a, and U are independent of ϕv, eq 1 can be rewritten as Gc ∼ ϕv2.2

(2)

This scaling compares favorably to our experimental results for Gc for ϕv < 0.53 from pure shear tests (described in the Mechanical Testing section in SI), as shown in Figure 2b. The fact that the simple Lake Thomas model predicts the scaling 494

DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498

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Figure 4. (a) Critical force for puncture, Pc, are plotted against indenter tip radius, R, for puncture with spherically tipped indenters for gel concentrations, ϕv = 0.12−0.53. The data is best described with linear fits (least-squares regression), as illustrated with lines, and the slope (Pc/R) gives Γ o, a measure of the fracture initiation energy. (b) Pc are plotted against R, ranging from 0.4 to 66 μm for ϕv = 0.53, showing a possible transition in Pc vs R scaling at R ∼ 30.

0.12 at R = 3.3 μm (Figure S6a). For gels with ϕv = 0.53, 10 indenter tip sizes ranging from R = 0.4−66 μm were used to probe manifestation of surface tension and adhesion forces resisting fracture initiation in addition to elasticity in soft polymer gels. Representative P−δg curves for R = 0.4, 3.3, 5.0, 15.0, and, 50.0 μm shown in Figure S7a−c. In particular, a puncture test was conducted at R = 400 nm, an order of magnitude below the smallest length scale tested for ϕv < 0.53 due to absence of distinct peaks at the point of puncture for R < 3.3. Interestingly, a linear P−δg curve was measured for R = 400 nm compared to the typical, nonlinear response measured for indenter tip sizes R > 3.3 μm (Figure S7). This observation can be possibly understood from the idea that shear stresses along the sides of the indenter are likely dominating the elastic response to the point of failure. It is also probable that other factors influence the elastic response of ϕv = 0.53 due to the close proximity of the indenter tip radius to the elasto-capillary length scale (∼200 nm). We have previously described the large strain deformation behavior of a soft gel indented by flat and spherically tipped indenters with P = k″ERδg + k′Eδ2g ,14 where k′ is an empirical constant and k″ is a constant, depending upon indenter tip geometry, obtained from Hertzian contact mechanics for flat and spherically tipped indenters.48 This deep indentation equation was used to fit P−δg curves for triblock copolymers gels with ϕv = 0.12−0.53 to obtain effective elastic moduli values, k′E for all compositions and for R = 5.0, 15.0, 35.0, and 50.0 μm. k′E is a measure of the large-strain response of materials and has been shown to correlate well with low strain elastic moduli measurements from standard macroscale mechanical testing techniques, such as shear rheometry, for various polymer gels and elastomer materials.14,49 Figure 3c shows a linear correlation between k′E and E with the coefficient k′, empirically determined as 0.37 ± 0.01. The magnitude of Pc increases with R for each gel concentration, and a linear fitting between Pc and R describes the data in the best manner, as shown in Figure 4a. The relationship between Pc and R is similar to our previously observed linear scaling in the energy-limited regime for failure for an identical gel system with different midblock and endblock molecular weights.14 Figure 4b shows a transition in Pc versus R scaling from linear to quadratic at R ∼ 30 μm for ϕv = 0.53. We associate this transition to the role played by indenter

for crack propagation energy suggests that the elastic energy in a chain available for fracture is released upon end-block dissociation and any remaining entanglements do not contribute to fracture processes. Additionally, for ϕv = 0.53, it is observed that Gc does not follow the observed experimental scaling for lower ϕv, possibly due to a change in micellar morphology, which can cause a significant change in mechanical behavior, as discussed previously by Seitz et al. for a similar gel system.29 The measured values for small strain elastic storage moduli, E (calculated from G′), assuming gel incompressibility for quasi-static loading conditions. Poisson’s ratio is assumed to be 0.5,46 such that E = 3G′. Gc, and surface tension of 2-ethyl-1hexanol (ϒ ∼ 0.0276 N/m47) were used to estimate various characteristic length scales. Table 1 summarizes ξ, ϒ/E, and Gc/E for each gel concentration. It is interesting to note how the intrinsic material parameters governed by polymer concentration lead to the observed trends in ϒ/E and Gc/E with ϕv. ϒ/E decreases with increasing ϕv, since we assume ϒ to be the surface tension of the solvent, which is independent of ϕv, and E scales directly with ϕv. The observed trend aligns with the well-known concept in soft material mechanics that the elasto-capillary length scale increases as polymer gels become softer.16,21−23 Moreover, our experimental results have shown that Gc ∼ ϕ2.2 v and E ∼ ϕ2.3 v , which leads to Gc/E to be independent of polymer concentration, which is not widely reported for soft materials. Gc/E is traditionally associated with the size scale where transitions in the failure mechanisms are observed. For example, the size scale can provide guidance for the critical flaw size in soft materials. Our finding that this length scale is essentially independent of ϕv for our specific gel system suggests size-independent failure processes; however, more research will be required to fully understand the implications of this finding. Deep indentation and puncture experiments shown schematically in Figure 3a were conducted for all gel concentrations (described in Mechanical Testing section in SI). Figure 3b shows spherically tipped glass indenters with radius, R = 0.4−66.0 μm fabricated from glass capillary tubes as described in the Materials section in SI. For ϕv = 0.12−0.30, representative force−displacement (P−δg) data for three R is shown in Figure S6, noting that Pc was indiscernible for ϕv = 495

DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498

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Figure 5. (a) Fracture initiation energy, Γ o is plotted against polymer volume fraction, ϕv. The linear fit to the data shows the power-law relationship, Γ o ∼ ϕv. The value for Γ o for ϕv = 0.53 deviates from observed scaling for lower gel concentrations from experiments. (b) A bar graph comparing Γ o and Gc for each ϕv.

tip geometry and shape based on our current understanding and leave it as an open question for future investigation to establish conclusively. The slope of Pc versus R is denoted as the fracture initiation energy, Γ o. Γ o is plotted against polymer volume fraction, ϕv, and a linear relationship is demonstrated in Figure 5a, such that resistance to fracture initiation in swollen polymer networks increases linearly with increasing gel concentration. Unsurprisingly, Γ o for ϕv = 0.53 does not follow Γ o ∼ ϕv observed for ϕv < 0.53, further supporting our previous reasoning of different mechanical response likely being associated with the change in micellar morphology. Moreover, Γ o does not follow the same scaling relation with ϕv as Gc, suggesting that different fundamental mechanisms come into play in fracture initiation other than, or in addition to, those governing crack propagation in soft polymer gels. Figure 5b shows a comparison of Γ o and Gc for each ϕv clearly indicating that fracture initiation energy is approximately an order of magnitude more than crack propagation energy. This observation is similar to the results we obtained previously, suggesting greater threshold energy requirement for the fracture initiation process.14 While no previous studies have quantified both fracture initiation and crack propagation energy for soft materials, a limited number of studies have looked at the nucleation of cracks in hard materials.50−52 Interestingly, we observe Gc values approaching Γ o with increasing ϕv, pointing toward an additional role played by gel compressibility on fracture initiation. A systematic future work to study the variation of gel compressibility due to poroelastic relaxations during deep indentation experiments in various gel concentrations can provide further insight into this observation.53 Figure 6 shows that the critical displacement of the indenter at puncture, δg,c scales linearly with R for each ϕv. The results indicate that puncture consistently occurs at a displacement of δg,c ∼ 27R, possibly depending only on indenter geometry and shape and independent of gel concentration. These results are in contrast to the dependence of Gc and Γ o on concentration discussed in this paper and opens a wide range of questions for future investigation. This study presents new insight into the effect of polymer concentration on low strain deformation behavior and crack propagation energy characterized by conventional measurements and the rarely explored process of fracture initiation in

Figure 6. δg,c follows a continuous linear scaling with R for various ϕv, suggesting that critical depth at puncture is independent of gel concentration.

triblock copolymer gels. Our results indicate that a single parameter, that is, fracture initiation energy, Γ o, can be used to describe the process of puncture, related to fracture initiation, across all length scales studied in the polymer gels. Moreover, Γ o varies significantly across several volume fractions in the same material system suggesting that network structure plays an important role in fracture initiation. The energy associated with fracture initiation was found to be greater than the critical energy for crack propagation, with both failure processes shown to scale differently with polymer concentration and micelle morphology, shedding light on the uniqueness of the failure mechanism associated with puncture. An understanding of the influence of polymer gel concentration on the resistance to crack initiation and propagation will enable new synthetic strategies for development of fracture resistant materials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.9b00086. Materials: Triblock copolymer gels preparation and fabrication of spherically tipped indenters; Small-angle X-ray scattering (SAXS) data; Mechanical Testing: oscillatory shear rheometry, pure shear test, and 496

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puncture experiments procedure, which includes representative force−displacement curves for different R for each gel concentration (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1-413-5771313. ORCID

Shruti Rattan: 0000-0001-7629-8809 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.R. would like to thank Benjamin Yavitt for help with SAXS data collection and C. W. Barney, R. K. Bay, and, C. Chen for their help with editing this document. The authors would like to thank National Science Foundation for funding (NSF-DMR 1609940) and Kuraray Co., Ltd. for kindly providing acrylic triblock copolymers.

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DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498

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DOI: 10.1021/acsmacrolett.9b00086 ACS Macro Lett. 2019, 8, 492−498