A. Indentation results

Indentations of all five compositions were analyzed by first applying the Garcia method for composition 3 (M7.5, R25), as it can be considered the most “standard” composition of the five based on the number of previous literatures surrounding it.38,48,52,56,5738. J. M. Urueña, A. A. Pitenis, R. M. Nixon, K. D. Schulze, T. E. Angelini, and W. Gregory Sawyer, Biotribology 1–2, 24 (2015). https://doi.org/10.1016/j.biotri.2015.03.00148. A. C. Dunn, J. M. Urueña, Y. Huo, S. S. Perry, T. E. Angelini, and W. G. Sawyer, Tribol. Lett. 49, 371 (2013). https://doi.org/10.1007/s11249-012-0076-852. E. P. Chan, Y. Hu, P. M. Johnson, Z. Suo, and C. M. Stafford, Soft Matter 8, 1492 (2012). https://doi.org/10.1039/C1SM06514A56. Y. Hu, X. Zhao, J. J. Vlassak, and Z. Suo, Appl. Phys. Lett. 96, 121904 (2010). https://doi.org/10.1063/1.337035457. A. A. Pitenis, J. M. Urueña, K. D. Schulze, R. M. Nixon, A. C. Dunn, B. A. Krick, W. G. Sawyer, and T. E. Angelini, Soft Matter 10, 8955 (2014). https://doi.org/10.1039/C4SM01728E Using the Garcia method, we found that the contact behavior for this composition does not exhibit behavior indicative of a single suitable contact model or a single power-law fit. Immediately after the point of first contact, we observed the greatest variability in the data and any potential exponential fit, with values ranging from 0.3 to 0.8. We predict that this region of variability is due to squeeze-out of the water held by the loose polymer segments, which causes a ramping-up force response due to pressurization of the released water while it escapes the contact. After this initial variability, we could fit a portion of the contact to the Fredrickson high-penetration brush model,58,5958. G. H. Fredrickson, A. Ajdari, L. Leibler, and J. P. Carton, Macromolecules 25, 2882 (1992). https://doi.org/10.1021/ma00037a01559. D. R. Williams, Macromolecules 26, 5096 (1993). https://doi.org/10.1021/ma00071a018 which had n=0.67 (Fig. 6). In this model, the force response is related to the shear modulus of the polymer chains G and their grafting density H [Eq. (7)]. After this interval, the next portion of each indentation curve fits well to a Winkler “bed-of-springs” contact model [see supplementary material, Eq. (2)], suggesting that the compression of the outermost brush segments did not induce significant strain in the material adjacent to it.6060. P. Põdra and S. Andersson, Wear 207, 1–2 (1997). https://doi.org/10.1016/S0043-1648(96)07468-6 Beyond the Winkler-fitting region, we were able to fit a Hertz contact model to the rest of the points, which implies that the compressed gradient-density surface layer had begun to emulate the response of the well-cross-linked bulk. Previous indentations on pAam hydrogels showed that hydrogels molded against glass surfaces exhibited Hertzian contact mechanics,39,52,56,6139. J. M. Urueña, E. O. McGhee, T. E. Angelini, D. Dowson, W. G. Sawyer, and A. A. Pitenis, Biotribology 13, 30 (2018). https://doi.org/10.1016/j.biotri.2018.01.00252. E. P. Chan, Y. Hu, P. M. Johnson, Z. Suo, and C. M. Stafford, Soft Matter 8, 1492 (2012). https://doi.org/10.1039/C1SM06514A56. Y. Hu, X. Zhao, J. J. Vlassak, and Z. Suo, Appl. Phys. Lett. 96, 121904 (2010). https://doi.org/10.1063/1.337035461. A. A. Pitenis and W. G. Sawyer, Tribol. Lett. 66, 113 (2018). https://doi.org/10.1007/s11249-018-1063-5 supporting our assumption that the latter portions of each indentation would fit well to a Hertzian contact model, F=192π2GR2H3(d−do)3.(7)The intersection between the Winkler-fitting region and the Hertz-fitting region occurs at a point we define as dmatch. We can establish the contact-perceived thickness of the gradient layer as the probe indentation depth from the video-derived point of first contact dcontact until the first point of the Hertzian-fitting region dmatch—we term this distance as Δd. The magnitude of Δd can be considered the penetration depth after first contact before the Hertz contact model applies. Because it spans the substrate thickness before Hertz contact fits, it provides a rough estimate of the thickness of the gradient layer, which is how we will refer to this measure hereafter. This shows that the thickness of the gradient layer for our standard composition hydrogel is roughly 17.1 μm thick (see Table II), which is within the proposed 10–20 μm range estimated via neutron reflectometry.34,3534. Y. A. Meier, K. Zhang, N. D. Spencer, and R. Simic, Langmuir 35, 15805 (2019). https://doi.org/10.1021/acs.langmuir.9b0163635. Y. Gombert, R. Simič, F. Roncoroni, M. Dübner, T. Geue, and N. D. Spencer, Adv. Mater. Interfaces 6, 1901320 (2019). https://doi.org/10.1002/admi.201901320 Of this thickness, approximately 80% of this distance comprises the brush-fitting and Winkler-fitting region, which indicates the degree of “brushiness” as a function of the penetration depth: a higher percentage corresponds to the presence of loosely cross-linked chains further into the depth. The Gemini contact had a gradient-layer thickness of 25.3 μm—47% larger than the soft-substrate case, and the portion of each indentation corresponding to a brush-fitting or Winkler-fitting region dropped to 67% for a Gemini contact configuration.

TABLE II. Elastic moduli, gradient-layer thickness Δd, and portion of the gradient-layer thickness interval corresponding to a Winkler model for each contact setup.

Contact setupElastic modulus E (kPa)Gradient-layer thickness Δd (μm)Winkler % of ΔdSoft-substrate33.817.179.5Gemini33.125.366.9When the indentations are plotted in a log-log format, it becomes apparent that measurement error alone cannot account for the non-Hertzian region. In Fig. 6, error bars appear nonsymmetric due to the log-log scaling, which means that it is more difficult for a low dF/du response, such as what is demonstrated in the figure, to be an erroneous measurement of a dF/du response roughly half a magnitude higher. Thus, usage of the Garcia method strongly reinforces the non-Hertzian determination of the initial response from the gradient-surface layer.It is important to note that the contact area observed throughout each indentation also deviated from a single contact model estimation. For example, using the probe displacement to predict a contact area using Hertzian contact mechanics would underpredict the actual contact area of these gradient-layered hydrogels by as much as 44% (see Fig. 7). If we instead use a piece-wise analytic model, where we use a Winkler model for the probe displacement d spanning the gradient-layer thickness Δd and a Hertzian contact model for values after that, we obtain a prediction of the contact radius that is much closer to that observed experimentally (Fig. 8). We note that this fit is not perfect since we have assumed a Winkler contact model through the initial indentation regime where brush contact and fluid squeeze-out, not Winkler contact, are the prevalent behavior; deriving contact areas for these phenomena would improve the predicted contact area fit.Next, the indentations of the other four compositions with variations in monomer and cross-linker were analyzed using the Garcia method to infer the influence of hydrogel composition. Elastic moduli for each composition were calculated using the Garcia analysis method’s K coefficient, which was found using a single-parameter least-squares fitting of a Hertzian contact model to the high-displacement data. The resulting elastic moduli are listed in Table III. The total polymer concentration M had a larger effect on the elastic modulus compared to the cross-linker ratio R. In fact, the effect of increasing cross-linker concentration (decreasing R) appears to have diminishing returns: a 67% increase in the available cross-linker increased the modulus by only 19%, while a 90% reduction in the cross-linker reduced the modulus by 92%. This suggests that cross-linking is potentially inhibited by self-linking of the bisacrylamide, which has been predicted to be the predominant polymer structure for hydrogels with high concentrations of the cross-linker.6262. Y. Gombert, F. Roncoroni, A. Sánchez-Ferrer, and N. D. Spencer, Soft Matter 16, 9789 (2020). https://doi.org/10.1039/D0SM01536A

TABLE III. Composition number and their corresponding elastic moduli determined from indentation experiments. The indentation-derived gradient-layer thickness, Δd, is also listed for each contact setup.

Comp no.Elastic modulus E
(soft-substrate) (kPa)Elastic modulus E
(Gemini) (kPa)Gradient-layer thickness Δd
(soft-substrate) (μm)Gradient-layer thickness Δd
(Gemini) (μm)1 (R250)2.422.8443.79.62 (M5)7.266.3621.44.93 (M7.5,R25)33.833.117.125.34 (M15)169.7153.113.417.15 (R15)40.138.712.718.8After fitting the Hertzian regime of each indentation, we then determined regions of each indentation that corresponded to Winkler contact [see supplementary material, Eq. (2)] or Frederickson’s high-penetration brush model [Eq. (7)].5858. G. H. Fredrickson, A. Ajdari, L. Leibler, and J. P. Carton, Macromolecules 25, 2882 (1992). https://doi.org/10.1021/ma00037a015 In every experiment, we found portions of the indentation data that fit these models (see the supplementary material6868. See the supplementary material at https://www.scitation.org/doi/suppl/10.1116/6.0002047 for extended discussion of the tribometer setup, particle exclusion methodology, Garcia method application, and additional indentation results.), which means that changes to the composition cannot eliminate the “brushy” layer of pAam hydrogels molded against polystyrene. This additionally shows that the progressive contact modeling of the gradient layer for composition 3 holds true for all compositions. Early portions of the indentation data of the stiffer compositions 3–5 (R25, M15, R15) had a higher slope as large as n=0.9, which may be attributed to pressurization of water within the gradient layer that is unable to vacate the contact in time due to poroelastic pressure. The thickness of the gradient layer for each composition, Δd, was different for each composition (Table III). For all compositions and setups, the thickness was between 10 and 40 μm, which is similar to the order of magnitude predicted by Simič et al.6363. R. Simič, M. Yetkin, K. Zhang, and N. D. Spencer, Tribol. Lett. 68, 64 (2020). https://doi.org/10.1007/s11249-020-01304-x For changing monomer percent M and constant ratio R (compositions 2–4), the gradient-layer thickness was correlated with monomer percent M, with composition 2 having the largest gradient layer and composition 4 having the smallest one. The portion of the gradient layer corresponding to a brush or Winkler contact model was the largest for composition 2 at 89% and decreased to 67% for composition 4. Similar trends were obtained for a reduction in the monomer-to-cross-linker ratio R, where a gradient-layer thickness of 43.7 μm was found for composition 1 (R250), while composition 5 (R15) only had a layer thickness of 12.7 μm. The portion of the gradient layer that followed brush or Winkler contact modeling was 89% for composition 1 and 63% for composition 5. Thus, for all soft-substrate contact experiments, the proportion of the Δd interval corresponding to a brush-fitting or Winkler-fitting region increased for larger gradient-layer thicknesses. This also hints that both composition metrics M and R are influential on the resulting gradient-layer structure.

However, the results from the Gemini contact were not as predictable. While we saw larger gradient-layer thicknesses perceived for compositions 3–5 compared to their soft-substrate contact values, compositions 1 and 2 showed smaller gradient-layer thicknesses. The Gemini contact perceived gradient layers of 9.6 and 4.9 μm for compositions 1 and 2, respectively. The portion of each Δd thickness interval corresponding to a brush or Winkler model was 100% for both compositions; the initial brush response in composition 1 (R250) is stiffer than the bulk response. In soft-substrate contact, these two compositions showed thick gradient layers with a high percentage consisting of brushy behavior.

B. Creep results

Creep experiments confirmed an increase in the contact area with time, which corresponded to a pressure decrease with time (Fig. 9). The majority of contact area gain occurred within the first 200 s regardless of the contact setup. In general, smaller loads experienced greater area growth by percentage compared to larger loads. Area expansion was larger in the Gemini contact compared to the hard-probe setup. The softer two compositions (1,2) relaxed to a greater degree than the other compositions when under low loads but saw the greatest discrepancy between low-load and high-load relaxation (Table IV). Particularly in self-mated contact, we postulate that the initial probe deformation reduced the capacity for the contact to relax over time.

TABLE IV. Contact area expansion compared to the initial contact for both contact configurations and for the minimum and maximum loads. Minimum loads experienced larger contact growth % compared to maximum loads.

Comp no.Area expansion (min load)
(soft-substrate) (%)Area expansion (max load)
(soft-substrate) (%)Area expansion (min load)
(Gemini) (%)Area expansion (max load)
(Gemini) (%)169.244.368.317.6280.222.2103.151.1345.338.371.051.3454.568.139.647.4551.844.368.250.9When fitting the area growth curves to exponential functions and extracting time constants, we were able to show that the time constants varied significantly with the initial contact radius, which is a function of the applied load/initial pressure (see Fig. 10). The inverse of the slope created by the time constant trend with the contact radius produced a steady-state diffusion coefficient for each contact setup.5555. Y.-Y. Lin and B.-W. Hu, J. Non-Cryst. Solids 352, 4034 (2006). https://doi.org/10.1016/j.jnoncrysol.2006.07.007 For composition 3 in a soft-substrate contact setup, this was 12.4×10−10m2/s, while the Gemini contact had a coefficient value of 7.51×10−10m2/s. Both of these values are close to reported coefficients in previous studies of pAam hydrogel poroelasticity, which had a diffusion constant of 5×10−10m2/s.5252. E. P. Chan, Y. Hu, P. M. Johnson, Z. Suo, and C. M. Stafford, Soft Matter 8, 1492 (2012). https://doi.org/10.1039/C1SM06514A This, as well as the variation in the time constant magnitude with the initial contact radius, proves that poroelastic squeeze-out was the primary driver of the contact area expansion. This aligns with previous studies that have shown a low degree of viscoelastic relaxation versus poroelastic relaxation.4545. M. Galli, K. S. Comley, T. A. Shean, and M. L. Oyen, J. Mater. Res. 24, 973 (2009). https://doi.org/10.1557/jmr.2009.0129Creep experiments on the other four compositions showed a time constant variation with the initial applied pressure, also suggesting poroelastic-dominated relaxation made possible by a less-dense, less-cross-linked surface layer. The only exception to this behavior was composition 1, which we had initially predicted to show viscoelastic-dominant behavior.4646. J. Zhang, C. R. Daubert, and E. A. Foegeding, Rheol. Acta 44, 622 (2005). https://doi.org/10.1007/s00397-005-0444-5 Composition 1, with a cross-linker ratio of R = 250, showed consistent time constants with varying normal load/contact radii, indicative of viscoelastic-dominant relaxation behavior. This means that, while poroelastic squeeze-out was likely to also be occurring, it was not as significant of a contributor to the area expansion as the viscoelastic stress relaxation throughout the gel. Diffusion coefficients were calculated using the slope produced by the time constants (Table V). These coefficients give a measure of the relative mobility of the water within the gel and particularly within the gradient layer. In general, the steady-state diffusion coefficient decreased for stiffer compositions, i.e., for larger M and smaller R. This aligns with the results from indentation, which showed thinner gradient layers for stiffer compositions, which reduces the amount of water that can be exuded under pressures below the osmotic pressure of the bulk cross-linked structure. In composition 3, we see values 50%–100% larger than those found in the previous work on glass-molded pAam gels of identical chemical composition (5∗10−10m2/s);5252. E. P. Chan, Y. Hu, P. M. Johnson, Z. Suo, and C. M. Stafford, Soft Matter 8, 1492 (2012). https://doi.org/10.1039/C1SM06514A this further supports the theory of increased water mobility within the gradient layer.

TABLE V. Diffusion coefficients obtained from exponential fitting of the area relaxation of each experiment for both contact setups. Composition 1 experienced a far greater degree of viscoelasticity compared to the other four compositions.

Comp no.DSS (soft-substrate) (10−10 m2/s)DSS (Gemini) (10−10 m2/s)1N/A82.3210.420.0312.47.5143.1610.156.706.62