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Chapter 9
Self-Assembly and Mechanical Properties of a Triblock Copolymer Gel in a Mid-block Selective Solvent Santanu Kundu,* Seyed Meysam Hashemnejad, Mahla Zabet, and Satish Mishra Dave C. Swalm School of Chemical Engineering, 323 Presidents Circle, Mississippi State University, Mississippi State, Mississippi 39762, United States *E-mail:
[email protected].
Polymer gels are used in many applications including in bioimplants, tissue scaffolds, oil recovery, and drug delivery. In these applications, gels often undergo mechanical deformation when subjected to tensile, compressive, shear, and mix-mode loading. At sufficiently large-strain, the gel deformation can be non-linear and can often lead to failure of the material. The mechanical responses of gels depend on their structure at various length scales. These structures form through the chemical bonding and/or physical assembly of constituting gelator molecules. Subjected to mechanical loading, the underlying structure of gels undergoes association, dissociation, and bond-breaking process, although macroscopic mechanical responses are often similar. To elucidate the link between the gel structure and mechanical properties, here we consider a self-assembled gel consisting of a triblock copolymer [ABA] in a mid-block selective solvent. The triblock copolymer is poly (methyl methacrylate)- poly (n-butyl acrylate)- poly (methyl methacrylate) [PMMA-PnBA-PMMA] with midblock length much longer than the end-blocks. 2-ethyl-1-hexanol and n-butanol have been selected as the midblock selective solvents. Below gelation temperature, as investigated by small-angle scattering, the end-blocks formed spherical aggregates. These aggregates were connected by the midblock bridges leading
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to a three-dimensional network. Altering the self-assembly process by incorporating graphene nanoplatelets resulted in a decrease of gelation temperature. During shear-deformation, the midblock bridges were stretched without being pulled out of the aggregates, as the end-blocks were strongly associated. As a result, a distinct strain-stiffening behavior has been observed for these gels and such behavior was successfully captured using large amplitude oscillatory shear (LAOS) experiments. To understand the failure behavior of these gels originated from a defect within a gel, a custom developed cavitation rheology technique was used. The pressure vs time responses from the cavitation experiments were analyzed using neo-Hookean and Gent constitutive equations. Although a cavitation or snap-through expansion like deformation behavior was observed, the critical pressure was higher than that predicted by the Gent and neo-Hookean constitutive equations. It was likely that the chain pull-out from the aggregates took place during the cavitation process, which contributed to the additional pressure.
Introduction Gels are a class of soft solids consisting of a large amount of solvent (1, 2). In these materials, the solvent molecules are entrapped /immobilized in a volume-spanning three-dimensional network. The network is formed by polymer or oligomer chains, low-molecular-weight gelators, etc. (1–3). To form the network, the constituting molecules are often physically or chemically crosslinked, however, topological interactions of the self-assembled structures can also display a gel-like behavior (1–7). In hydrogels, the solvent is water, whereas, in organogels, the solvent is an organic liquid. Physical properties of gels, including mechanical and interfacial properties can be altered in many ways, for example, by choosing suitable chemical structure of the gelators, adopting different synthesis strategies for forming chemically crosslinking networks, dictating the self-assembly processes for physically crosslinked gels, and incorporation of nanoparticles in the gels. Further, gels can be rendered responsive to external stimuli such as temperature, pH, electrical field, and light (1, 2, 8–15). Because of these attractive features of gels, these materials are being used or have potential applications in many areas including in bioimplants, tissue engineering, drug delivery vehicles, superabsorbent, gel-based sensors, soft robots/machines (1, 2, 8–21). In fact, gels are ubiquitous in daily used products, particularly, contact lenses, jello, hair gels, etc. (22, 23) Gels formed by the self-assembly of a block-copolymer segment in a specific solvent are of significant interest, as the association and dissociation of the structure can often be controlled by the application of stimuli such as temperature and pH (4, 5, 24–30). Similarly, for the low-molecular-weight gelators, the small molecules can associate to form fibers or other self-assembled structures leading 158
to gel formation through topological interactions (3, 6, 7, 31, 32). For polymeric systems, the choice of solvent plays an important role, as the individual blocks of a block copolymer can interact with the solvent differently (33). Thermoreversible gels can be obtained by harnessing the change of solubility of one or more blocks of a block copolymer in comparison to the other blocks as a function of temperature (34–37). In triblock copolymer gels, triblock copolymers can form a gel in end-block and mid-block selective solvents (5, 25, 27, 38, 39). In pluronic gels, one of the most commonly known triblock copolymer gels, poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) or PEO-PPO-PEO forms a gel in water, an end-block selective solvent (38, 39). In these gels, transition from sol to gel phase occurs with increasing temperature. With increasing temperature, the PPO blocks collapse forming a micellar structure. The gelation behavior of pluronic gels can be altered by adding other components, such as chitosan (38, 39). These gels are commonly used in pharmaceutical formulations and in cosmetics (39–41). In another system, the end-blocks of triblock copolymers associate to form a gel-like material in mid-block selective solvents. Here, the gelation takes place with decreasing temperature. The decrease in the solubility of the end-block resulting in a reduction of interaction between the end-blocks and solvents, and formation of aggregates due to the association of the end-blocks. The aggregates are connected by the mid-blocks forming a three-dimensional network. Many gels of this type have been reported, particularly, polystyrene-rubber-polystyrene [PS-rubber-PS] in oil, which is a good solvent for the rubbery block (26–28). Rubbery blocks such as poly(ethylene/propylene)(PEP), poly(ethylene/butylene) (PEB), and polyisoprene (PI) have been investigated (26, 27, 29, 42, 43). In addition to PS-rubber-PS systems, other systems that have been considered are acrylic block copolymers, such as poly (methyl methacrylate)–poly (tert-butyl acrylate)–poly (methyl methacrylate) [PMMA-PtBA-PMMA] and poly(methyl methacrylate)–poly(n-butyl acrylate)–poly(methyl methacrylate) [PMMA-PnBA-PMMA] in butanol and 2-ethyl-1-hexanol (5, 24, 25, 36, 44). Most of these gels are thermoreversible in nature, i.e., these gels can be converted into liquid and vice versa by changing temperature. Structure and mechanical properties of gels consisting of ABA triblock copolymers in mid-block selective solvents depend on many factors such as the absolute and relative copolymer block lengths, the copolymer concentration in solution, pH, and temperature, which affect the polymer-solvent interactions significantly. Many different types of gels have been reported in the literature and we do not aim to present a complete review of all those gels. Rather, we present the self-assembly process and mechanical properties of a self-assembled triblock copolymer gel consisting of PMMA-PnBA-PMMA triblock copolymer in two mid-block selective solvents, n-butanol and 2-ethyl-1-hexanol. If not otherwise mentioned, the triblock copolymer considered here has a PMMA end-block molecular weight of 9,000 g.mol-1, whereas the molecular weight of the PnBA mid-block is 53,000 g.mol-1. In this chapter, this polymer is referred as A9B53A9. As the average entanglement molecular weight of PnBA is about 20,000-30,000 g.mol-1 (45), where a significant entanglement is not expected. 159
For a self-assembled system, achieving a control on the self-assembly process is an important consideration. For triblock copolymer gels, homopolymers resembling one of the blocks can be added to alter the self-assembly process. Alternatively, the self-assembly process can be altered by adding graphene nanoplatelets, as reported here. However, how these nanoplatelets affect the self-assembly process is an open question, particularly, if the size of the nanoplatelets is bigger than the aggregates and the pore size of the gels. Most of the literature reports on these gels have been limited to a small temperature range, near the gelation temperature, mostly at the room temperature. However, as the solvent quality continues to change with decreasing temperature, the self-assembly process is expected to continue. This, in turn, can affect the properties of these gels. Although important for practical applications of these materials, such understanding is limited and will be presented here. Another important question for these self-assembled gels is how these gels fail subjected to a mechanical load. Failure in these gels can primarily take place through chain pull-out from the aggregates, rather than chain scission. Strength of association in the aggregates, solvent quality, and mid-block length in tandem dictate the fracture process. Nonlinear rheology is increasingly being used to characterize soft materials and can capture the large-strain deformation behavior of these gels leading to failure. Failure behavior of these gels can be further investigated using cavitation rheology, a simple yet powerful technique. Both nonlinear rheology and cavitation rheology can provide fundamental understanding regarding the gel failure mechanism.
Effect of Solubility Parameter on Self-Assembly Gelation of triblock copolymer takes place as a result of a change in the interaction between the polymer blocks and the solvent, i.e., the change of solvent quality with the changing temperature. The solvent quality can be interpreted by the Flory–Huggins interaction parameter (χ) (1, 46). To understand the gelation of PMMA-PnBA-PMMA in mid-block selective solvents, such as n-butanol and 2-ethyl-1-hexanol, the temperature dependency of the χ parameter for the individual homopolymers has been studied. Figure 1a represents the images of PMMA and PnBA, in n-butanol and 2-ethyl-1-hexanol at different temperatures. At 80 °C, PMMA was soluble in the above-mentioned solvents resulting in transparent solution, whereas, the turbidity of the samples at 4 °C illustrates the PMMA phase separation. With a further reduction of temperature, precipitation of polymer occurred demonstrating a strong temperature dependency of χ parameter. The solubility of PnBA did not change over the temperature range of -66 °C to 80 °C, as the solution remained transparent. χ parameter as a function of the temperature can be obtained experimentally. The χ parameter and θ temperature for the PMMA samples with various molecular weights in different alcohols have been reported by Llopis et al. (47) The θ temperature was found to be 85 °C and 83.2 °C in n-butanol and n-propanol, respectively. The χ parameter as a function of temperature, T (in K) has been given as (47) χMS = −2.82 + 1189.058/T. The subscripts “M” and “S” represent PMMA 160
and the solvents, respectively (Figure 1b). χ 0) or strain-softening (e3 < 0), and shear-thickening (υ3 > 0) or shear-thinning (υ3 < 0) behavior of a material. Figure 10 displays the experimental data and the results analyzed using LAOS framework for a 5 vol% triblock gel (A9B53A9) at 22 °C and at a frequency of 1 rad/ s (5). The first harmonic storage modulus, G′1, and loss modulus, G″1, are shown in Figure 10a. G′1 values were much higher than G″1, confirming the soft-solid like the behavior of the gel. At low strain, G′1 values were almost independent of strain but at large strain, G′1 values increased with increasing strain amplitude, indicating the strain-stiffening behavior of this gel. The corresponding stress-strain curves (Lissajous-Bowditch curves) are shown in Figure 10c. The evolution from an elliptical stress-strain curve to a distorted one with increasing strain-amplitude is clearly visible, indicating a transition from linear to the non-linear region. The third-order Chebyshev coefficients, e3, were estimated using MITlaos software and are shown in Figure 10b. The e3 values became positive in the non-linear region, confirming the strain-stiffening response of the gel. Both G′1 or e1, and e3 increased rapidly at large-strain. These results indicate that the strength of the physical association (PMMA aggregates) was significant. During the deformation process, it was possible to stretch the chains to near to their full length before being pulled out of the aggregates. This behavior resulted in distinct strain-stiffening responses observed here. During strain-sweep experiments, a fracture in the gel sample was observed at a strain value of ≈350%, as a result, a drop in modulus value was observed (point 19 on Figure 10a). Fracture of triblock gel subjected to simple-shear has been related to the strain-localization or non-homogeneous strain field (89). Note that the strain value for fracture was very close to the maximum strain value corresponding to the maximum chain extensibility.
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Figure 10. Linear and non-linear viscoelastic responses of a 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol. (a) The symbols are G′1 and G″1 estimated by the rheometer software. The solid line is e1, predicted from Gent model (Eq. 9) considering maximum stretch ratio (λm) of 3.8. (b) Closed symbols are the third Chebyshev coefficients (e3) estimated from experimental data and the solid line is the prediction of Gent model (Eq. 10). (c) Lissajous-Bowditch curves as a function of strain. Adapted with permission from Ref. (5).. Copyright 2015 Royal Society of Chemistry. (see color insert) Although a strain-stiffening behavior was observed for this gel, no negative normal stress was observed in dynamic shear-oscillation (Figure 11) (109). For parallel-plate or cone-plate geometries, the normal stress is estimated as σN = 2Fz/ πR2, where R is the radius of the plate, and Fz is the normal force measured by the rheometer (83). For parallel -plate, σN = N1 – N2 , where, N1 is the primary normal (N1 = σθθ – σzz) and N2 is the secondary normal stress difference (N2 = σzz − σrr). The magnitude of N2 is often smaller than that of N1.83 In comparison, an alginate gel, which also displays strain-stiffening behavior, exhibits negative normal stress. For semi-flexible polymer gels, strain-stiffening behavior and negative normal stress appear in tandem (79, 110). Such behavior has not been observed for the triblock gels likely because of flexible nature of the mid-block chain. 176
Figure 11. Shear stress and corresponding normal stress as a function of time during an oscillatory shear experiment for 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol (a); (b) for an alginate gel with alginate concentration is 10 mg/mL and [Ca2+] ≈ 12.5 mM. Applied frequency was 0.5 rad/s both sample, For triblock gel, the applied strain amplitudes was 150%, whereas, for alginate gel the strain amplitude was 30%. A parallel plate geometry was used. Adapted with permission from Ref. (109). Copyright 2016 Wiley.
G′1 and G″1 as a function frequency are shown in Figure 12. G′1 values were found to be much higher than that of G″1 over the frequency-range investigated here. G′1 also had a very week frequency-dependency, and such dependency decreased with temperature (22 vs 6 °C). Both these observations are typical of gel-like materials (31). Changes in the stress-strain responses with increasing frequency and strain amplitude provide us some interesting insights. Figure 13 displays the stressstrain plots as the frequency (ω) was increased from 1 rad/s to 30 rad/s and strain amplitude (γ0) increased from 10 % to 200 %. For each curve, the corresponding 177
e3 and G′1 values are shown. As expected, in the linear viscoelastic region, the e3 values were zero at small strain and became slightly positive at a higher frequency. However, with increasing frequency and strain amplitude, a significant increase in e3 (as high as 3 times) was observed. This indicates that the strain-stiffening response became further important with increasing frequency. It is anticipated that at high frequencies there would not be enough time for the exchange of PMMA end-blocks in and out of the aggregates and the mid-block stretching at large-strain had been manifested by the enhancement of strain-stiffening behavior.
Figure 12. G′1 and G″1 as a function of frequency for a 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol at (a) 6°C and (b) 22°C. Adapted with permission from Ref. (5). Copyright 2015 RSC.
Effect of GNPs on the strain-stiffening behavior has been investigated and the results are shown in Figure 14. The strain-stiffening behavior was observed for the pristine and graphene containing gels at 22 °C and 6 °C. At 6 °C, the difference between G′ and G″ was relatively independent of the concentration of graphene, whereas, at 22 °C the G′ and G″ were slightly different for different graphene concentration. These results were similar to that presented in temperature sweep data (Figure 9), in which it has been shown that below 10 °C, moduli for both pristine and graphene containing gels were similar, likely due to the completion of self-assembly process at that temperature. Also, GNPs had no effect on the onset of the nonlinear elastic behavior at these two temperatures, as the strain-stiffening behavior became important beyond the strain value of ≈100%. 178
Figure 13. Lissajous-Bowditch curves as a function of frequency and strain amplitude for a 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol. Red dashed lines represent the pure elastic stress responses of the gel. The e3 and G′1 values are indicated for each oscillatory test. Reproduced with permission from Ref. (5). Copyright 2015 RSC.
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Figure 14. G′ and G″ as a function of strain amplitude for a 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol with and without GNPs at (a) 22 °C and (b) 6 °C. The applied frequency was 1 rad/s. Adapted with permission from Ref. (4). Copyright 2015 RSC.
Relaxation Behavior The relaxation behavior of both pristine and GNPs containing gels was studied over a temperature range of 6 °C to 25 °C to achieve further understanding of the self-assembly process (4). Here, a step strain of 5%, which falls within the linear viscoelastic region was applied and the evolution of time-dependent modulus G(t) as a function of time was captured (Figure 15). The decrease in G(t) with time represented the relaxation of samples with time. For samples far away from the gelation temperature, the relaxation process was slower than that near the gelation temperature, where the sample was more viscoelastic. For the physical gels, the stress relaxation process involves end-blocks pull-out of the aggregates, and exchange between the neighboring aggregates. Hence, both temperature and graphene platelets can alter the relaxation process. 180
A stretched exponential function has been used to fit the stress-relaxation data. This model is given as (4, 31, 111):
Here, is the shear modulus at time zero, τ is the relaxation time, and β is the stretching exponent. β =1 represents the Maxwell model, whereas, fractional value of β represents a distribution of relaxation time.
Figure 15. Stress relaxation of pristine gels (a) and gels with 0.12 mg/mL GNPs (b) over a temperature range of 6 to 25 °C. 5 vol% polymer (A9B53A9) was used in these gels. The symbols are experimental data, whereas the lines are model fitting (Eq.5). Aadapted with permission from Ref. (4). Copyright 2015 RSC. (see color insert) The stretched-exponential function captures the relaxation data reasonably well. The G0, τ, and β values for the samples without and with graphene at various temperatures are shown in Table 3. At lower temperature, chain pull-out or exchange from the less swollen aggregates was slower than that from the more swollen aggregates at higher temperatures. As a result, the relaxation time increased significantly with decreasing temperature for both these gels. The β has been found to be in the range of 0.2 – 0.3. These results are slightly different than that obtained for β = 0.33 by Erk and Douglas for a pristine gel (111). The results obtained here are similar to β = 0.2 by Hotta et al. for a triblock gel consists of polystyrene-polyisoprene-polystyrene (112). Interestingly, Drzal and Shull obtained β = 0.53 for PMMa-PtBA-PMMA gel (36). The differences are likely related to the different instruments and experimental protocols used in different studies. Both these gels display a similar trend in the stress-relaxation behavior. However, at a lower experimental temperature such as at 15 °C and 20 °C, the τ values for the graphene-containing gels were slightly lower than the pristine gels. The difference in response at 25 °C, was likely due to the liquid-like behavior of the graphene-containing samples at that temperature. 181
Table 3. Fitted Parameters for Stress-Relaxation Data Shown in Figure 15 for the Pristine Gel (a) and for the Gel with Graphene Concentration of 0.12 mg/mL (b) a)
b)
T
G0 (Pa)
τ (s)
β
T
G0 (Pa)
τ (s)
β
6 °C
202± 11
223±33
0.23
6 °C
218± 7
198±3
0.27
10 °C
196±8
20±34
0.22
10 °C
186±20
60±14
0.29
15 °C
175±12
21±6
0.23
15 °C
161±10
15±6
0.27
20 °C
135±12
7±2
0.25
20 °C
119±15
3±2
0.27
25 °C
78±15
4±2
0.3
Figure 16. Creep compliance for a 5 vol% gel 5 vol% gel consisting of A9B53A9 polymer in 2-ethyl-1-hexanol . Adapted with permission from Ref. (4). Copyright 2015 RSC.
Creep experiments have also been conducted on these samples with a constant applied stress of 100 Pa. The creep compliance data for a pristine gel is shown in Figure 16. A distinct creep-ringing was observed initially and that faded out gradually. A stretched exponential function can be reasonably used to the longterm creep compliance data (t > 5 s), i.e., beyond the creep-ringing. The fitted values for τ, and β are similar to the ones from the fitting of stress-relaxation data (Table 3). 182
Modeling Stress-Strain Behavior Dynamic rheology data captures the mechanical properties of triblock gels. The results are then fitted different constitutive models such as Gent model, Fung model, etc. (5, 37) Here, results are shown for Gent model, which considers the finite chain extensibility (5).
Gent Model For elastic materials, strain energy density functions (W) relates the strain energy density of a material to the deformation gradient (1, 107, 113). For Gent model, W can be presented as (1, 107):
Where, E is Young’s modulus, J1 = λ12 + λ22 + λ32 – 3 (λis are the extension ratios in the principal stretch directions), Jm corresponds to the maximum extensibility or maximum chain extension, λm, Glin is the linear elastic modulus and is equal to E/3 for Poisson’s ratio = 0.5. If λm →∞, Jm →∞, i.e., if the chain extensibility approaching infinity, the Gent model approaches the neo-Hookean model (1, 107). Since the elastic modulus of the triblock gel was much higher than the viscous modulus, the viscous dissipation can be considered to be negligible. Therefore, the elastic shear stress is then approximately equal to the total stress (τelastic ≈ τ). In that case, Glin is equal to the first-harmonic storage modulus in the linear viscoelastic region. In a general term, the shear stress as a function of shear strain can be written as (92):
Where γ is the shear strain and f(γ) represents the functional dependence of shear modulus as a function of γ. At small strain, f(γ) approaches Glin. For simpleshear experiments, shear strain (γ) and extension ratio (λ) is related as γ = λ - 1/λ. For the Gent model:
Here, shear stress was obtained by taking the derivative of the strain energy function with respect to strain (Eq. 6) (1, 107). It has been shown that (92)
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Using these relationships, the e1 and e3 for the Gent model can be presented as:
Note that the third-order Chebyshev coefficient for Gent model is always a positive number, predicting a strain stiffening behavior. The summation of e1 and e3 is equal to f(γ) in Eq. 8. The Eqs. 9, 10 were fitted to the experimental data and are shown in Figures 10a and 10b. The maximum extension ratio for the 5 vol% copolymer gel was estimated based on the scattering data (24, 25, 37). The fully stretched length of the PnBA mid-blocks with the molecular weight 53,000 g/ mol has been estimated as ≈ 105 nm and the unstretched length obtained from the scattering data is ≈ 28 nm. This gives maximum extensibility, λm ≈ 3.8, which was used to fit the data. As presented in Figure 10, the model captured the experimental data reasonably well. As shown in the Eqs. 9 and 10, the coefficient, e3 captures the non-linear response of the gel, whereas, e1 has both linear and nonlinear contributions.
Investigation of Mechanical Properties Using Cavitation Rheology Cavitation phenomena, caused by elastic instability, is observed in pressure sensitive adhesives during the peeling-off process, and in biological materials such as in human brain subjected to a shock wave (114, 115). This phenomenon has been harnessed in developing cavitation rheology technique towards investigating local mechanical properties of soft solids (5–7, 95–103). Schematic of a cavitation set-up is shown in Figure 17a, in which a needle is inserted at any arbitrary location within the gel. As the needle radius can be varied, using this technique the gel deformation behavior can be investigated over a length scale of ~ 10 µm to 1000 µm. This technique has been used to investigate the mechanical properties of triblock copolymer gels with different volume fractions and at two different temperatures. Effect of polymer volume fraction: The pressure vs time response obtained in cavitation experiments for three gels with polymer (A9B53A9) volume fractions (φ) of 0.05, 0.07, and 0.10, respectively. With pressurization, the system pressure increased to a maximum pressure, also defined as a critical pressure, Pc, before a rapid drop of pressure took place (Figure 17b). Pc values increased with increasing polymer volume fraction, i.e., with increasing modulus. Gel deformation at the tip 184
of the needle at the critical pressure and beyond the critical pressure are shown in Figures 17c-e. For φ≈0.05 and 0.07, a rapid spherical cavity growth at Pc was observed, whereas, a fracture like behavior was observed for φ≈0.1. With sudden increases in cavity volume or fracture like process, a decrease in system-pressure was observed. If the expansion ratio or stretch ratio is defined as λ = (Ac/Ac0)½, where, Ac is the surface area of the cavity at any instance and Ac0 is the inner cross-sectional area of the needle (the initial area), λ was found to be as high as 15. The maximum stretchability of the PnBA chains considered here has been shown to be ≈ 3.8. This indicates that during the cavity growth process the PnBA chains most likely have been pulled out of the PMMA aggregates.
Figure 17. Cavitation rheology experiments. (a) Schematic of the experimental set-up. (b) Pressure as a function of time for three polymer (A9B53A9) volume fractions, φ = 0.05, 0.07, and 0.10. (c-e) Photomicrographs of cavity growth at and after the critical pressure for (c) φ = 0.05, (d) φ = 0.07, (e) φ = 0.10. Needle radius, rs = 156 µm. Experimental temperature was 22 °C. Adapted with permission from Ref. (5). Copyright 2015 RSC. 185
Effect of Compression/Pumping Rate Mechanical response of a viscoelastic material depends on the applied strainrate. Cavitation experiments were conducted at different pumping/compression rates (μ) directly related to the different strain-rate gel deformation at the needletip. Pumping rate was varied from 0.01 to 25 mL/min. For a polymer volume fraction of φ = 0.05, Pc for μ = 0.01 mL/min was 850±50 Pa, which was lower than 1500±100 Pa observed for μ = 0.5 mL/min. Pc did not change significantly beyond that pumping rate. This difference in Pc resembled the weak frequency dependence observed in frequency sweep data (Figure 12).
Effect of Temperature To investigate the temperature dependence of cavitation process, experiments were conducted by cooling the sample in an ice bath corresponding to the sample temperature of ≈ 6 °C. Results for 6 and 22 °C for φ = 0.05 have been compared. As shown in Figure 12, G′ increased with decreasing temperature from 22 °C to 6 °C. Similarly, increase in Pc has also been observed. However, the increase in G′ was about two times, in comparison to 1.5 times increase in the Pc.
Model Prediction for Pressure Responses The pressure vs. time responses presented above can be captured analytically approach and finite-element model (FEM). In addition to Gent model, which captures the rheological data for triblock copolymer gels, the neo-Hookean model has also been considered. To obtain pressure response analytically, cavitation phenomena has been approximated as the growth of a spherical cap at the needle-tip. Here, the pressure at any instance can be given by, P =ΓdA/dV +σ (97), where Γ is the surface energy, σ is the mechanical stress the gel at the needle-tip is subjected to. A and V are the surface area and volume of the cap, respectively. The stress σ can be estimated based on various strain energy functions. For Gent strain energy function the critical pressure for cavitation (97, 107):
For neo-Hookean solid, i.e., for Jm → ∞, the critical pressure for cavitation is given by (97, 108, 109):
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For neo-Hookean solid analytically it has been shown that
or in dimensionless form
In Eq 13b, the is defined as the elastocapillary number or ECN. The first term of the equation 13a represents the elastic contribution of the gel resisting the cavity growth and the second term represents the surface energy contribution, necessary to overcome the increasing surface area during cavity growth. As shown by Gent, for a cavity growth in dry rubber, where the surface tension is negligible, . Such relationship is also obtained for very small values ECNs (Eq 13b), that can be obtained, for example, if rs is large. FEM has been used to investigate cavitation phenomena for a neo-Hoookean solid (108). FEM results and that obtained for the analytical method are compared in Figure 18. Simulations have been conducted over a range of ECN = 0 to 100, capturing the cases without any surface tension to a significant surface tension present, respectively. In these simulations, pressure was increased in a step-wise manner, in pressure-control mode. FEM could only capture the deformation up to the maximum pressure but not beyond that. The maximum pressure, which was same as Pc for this case, was about 0.4E higher than that obtained from analytical prediction. A similar increase has also been observed by Hutchens and Crosby (99). Therefore, the needle has an effect on the critical pressure likely due to the stress concentration at the needle corner. The cavity shapes at the maximum pressure for different ECNs are shown in Figure 18b (108). No significant change in cavity shape has been observed although the pressure magnitude increased significantly. For ECN =0, no pressure maximum was observed, rather the pressure asymptotically reached a critical pressure. In this case, the pinning of the cavity at the needle tip has been observed. Figure 18c summarizes Pc and λc as a function of ECN. Note that λc was small at the critical pressure, except for the case where ECN is very small. Although a slight increase in critical pressure was observed for the presence of needle, such increase was not significant with respect to the contribution of the surface tension effect. The above equation captures the maximum or critical pressure, but not the pressure evolution. However, using the Eqs. 11, 12 and considering the closed volume confined by the syringe plunger and the gel at the tip of the needle filled with an ideal gas, and no diffusion of air into the gel during pressurizing, the following equations can be obtained (5, 108). For neo-Hookean solid:
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For Gent gel:
where, P0 and the V0 represent the initial system pressure and system volume, respectively. µ is the pumping rate and t is the time. By solving the above equations for each time step, pressure vs time response can be obtained, as discussed in Ref (5). The results are summarized in Figure 19 displaying the pressure and extension/stretch ratio (λ) as a function of time for neo-Hookean and Gent gels (5). In the neo-Hookean gel, pressure increases to a critical (maximum) pressure followed by a rapid drop in pressure. λ increases slowly initially until a certain jump at the critical point (λc = 1.4) caused by elastic instability. For neo-Hookean gels, λm→∞ and the cavity volume can increase without any bound. This rapid increase in cavity volume or snap-through expansion results in a sudden drop of pressure. This phenomenon is defined as the cavitation phenomena. For Gent gels, because of finite chain extensibility (λm ≈3.8, for the present case), the increase of cavity volume is restricted at the instability point. As a result, the decrease in pressure is small at the instability point. The pressure continues to rise beyond that instability point with increasing compression. Both neo-Hookean and Gent models predict the similar response up to the critical pressure, but the polymer chain extensibility dictates whether a snap-through expansion is expected. In fact, cavitation in Gent gels is possible, if λm is very large. The increase of pressure in Gent gel cannot be unrestricted as the failure or fracture of the gel takes place beyond a certain pressure. Cavitation or snap-through expansion was observed for triblock gels, although the chain extensibility has been considered to be equal to of 3.8. Therefore, the cavity growth observed here likely involved the fracture process, where the chains were pulled out of the aggregates. The critical pressure corresponding to fracture is a function of elastic modulus, needle radius, and critical energy release rate (Gc) (5, 85, 97). For a first order approximation, if a linear elastic material is considered, the critical pressure for fracture scales as (5, 85, 97). For Gent gels, such functional relationship is not precisely known. This equation predicts Gc ~ 0.1 J/m2. This value is of the same order obtained by Seitz et al. for triblock gels (88). Further theoretical and computational studies are necessary to obtain a quantitative relationship between the critical pressure and the materials parameters of the gels.
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Figure 18. Comparison of simulation and analytical results for a neo-Hookean solid. (a) Normalized pressure vs stretch ratio for different ECNs. Solid lines display the analytical prediction (Eq. 13b) and the symbols represent the FEA results. (b) Cavity shape at the maximum pressure obtained from FEA. For ECN = 0, cavity shape is plotted for λ= 4. (c) The critical pressure, Pc, and critical stretch, λc, plotted as a function of ECN. Adapted with permission from Ref. (108). Copyright Elsevier 2018. (see color insert)
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Figure 19. Predicted pressure and extension ratio (λ) as a function of time for (a) a neo-Hookean gel with λm → ∞; (b) for a Gent gel with λm = 3.8; (c) the zoomed-in view of (b). Adapted with permission from Ref. (5). Copyright 2015 RSC.
Concluding Remarks and Future Directions Here, we have reported the structural evolution, and mechanical properties of a thermoreversible, triblock copolymer gel consisting of PMMA-PnBA-PMMA in mid-block selective solvents. The self-assembly process has been altered by adding GNPs, and such has been manifested by a decrease in gelation temperature. The gel displays strain-stiffening behavior and such behavior has been captured using a Gent constitutive equation with finite chain extensibility. Although a major understanding has been achieved in this gel system, many questions still need to be addressed. Based on the present literature, no other self-assembled triblock copolymer gels display strain-stiffening behavior, similar to the biological network. Interestingly, in most of these gels, the mid-block molecular weight is larger than the entanglement molecular weight. Therefore, a systematic study with the varying molecular weight of the blocks will provide further insights into the mechanical properties of these gels. Simulation studies indicate that that the size and the number of aggregates change with the applied frequency and strain (116, 117), however, the results from the experimental investigations are not conclusive (25). The fracture process of these gels needs to be further investigated, as this process is influenced by many factors such as the block-length, solvent quality, the strength of association, time and length scale associated with the gelation process. As graphene nanoplatelets have been successfully incorporated in these gels, these gels can be rendered responsive subjected to an electrical field. Such behavior can lead to new applications of these gels.
Acknowledgments The authors gratefully acknowledge the financial support from National Science Foundation through CAREER Award [DMR 1352572], and EPSCoRTrackII funding [IIA-1430364]. We also acknowledge supports from Mississippi State University. 190
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