Effective visco-elastic models of tough, doubly cross-linked, single-network polyvinyl alcohol (PVA) hydrogels

Hydrogels are an interesting material, essentially consisting of cross-linked macromolecules containing hydrophilic functional groups while forming a three-dimensional polymer network, being abundantly swollen with water and intrinsically owing application potentials in such as hygienic products, food additives, bio sensors and in tissue engineering [1, 12, 14, 18, 53, 69, 77, 80, 82] and, more recently, as dampers and vibration isolators [72]. Tissue engineering includes fabrication of artificial cartilages, blood vessels, tendons, bladders, muscles and ligaments. Early fabricated hydrogels are displaying a poor mechanical strength while containing a single polymer network chemically cross-linked by covalent bonds. However, more recent hydrogels are containing a double network; the first network is tightly cross-linked, while the second is loosely cross-linked, as successfully developed by Gong et al. [18, 19] and other [2, 23, 76]. Those recent hydrogels are displaying a substantial increase in mechanical strength needed for the previously mentioned artificial body parts.

However, double-network hydrogels with only chemical bonds display permanent strain softening and loss of mechanical strength at repeated loading and unloading due to permanent rupture of chemical cross-links, mainly located at the tightly cross-linked network. The permanent mechanical strength is substantially increased for double-network hydrogels where chemical cross-links, located at the tightly cross-linked network, are substituted by physical cross-links [6‐8, 14, 25, 28, 38, 47, 52, 53, 63, 73, 75, 77, 82]. In principle, the physical cross-links broken dissipate energy and act as sacrificing cross-links at loading while being reassembled at unloading, thus leading to recovery, where the second, chemically cross-linked network provides a memory of the initial undeformed state and prevents propagation of cracks [6‐8, 14, 25, 28, 38, 47, 52, 53, 63, 73, 75, 77, 82]. Physical cross-links may include hydrogen bonds, hydrophobic, \(\pi \,\)‐\(\,\pi \), van der Waals and ionic; ion–ion, ion–dipole, dipole–dipole interactions, see e.g. Refs. [1, 45, 69, 80, 82].

Large deformation visco-elasticity models including Mullins effect have been developed for double-network hydrogels [44, 47, 79]. A fractional time power-dependent visco-elastic model for hydrogels has been developed by Keshavarz and co-workers [35], a more generic fluid–solid mechanical-based model by Birgersson and co-workers [9, 37] and a model which also includes poroelasticity has been developed by Wang and Wong [70].

An interesting alternative to the double-network hydrogels is doubly cross-linked, single-network hydrogels simultaneously containing both chemical and physical cross-links [11, 13‐15, 22, 24, 34, 39‐43, 48‐51, 53, 58, 69, 74, 78, 80‐83] and, similarly, both strong and weak physical cross-links [14, 27, 45, 60, 64, 65]. An attractive observation by e.g. Czarnecki et al. [15] is that it is possible to tune the strength of the physical cross-links by selecting a proper metal ion to modify the kinetics and thermodynamics of the adhesion–deadhesion activity of the physical cross-links resulting in a very broad potential relaxation time range of the temporary physical cross-links. This in turn results in a broad frequency range where within it is possible to tune the angular frequency \(\omega _{\mathrm {a}|\mathrm {d}}\) for maximum loss shear modulus \(\max (\mathfrak {I}\hat{\mu })\) of the hydrogel, where \(\hat{\mu }\) is the shear modulus, \(\mathfrak {I}\) is the imaginary part, and subscripts a and d stand for the adhesion and deadhesion of the physical cross-links, respectively. For angular excitation frequencies well below \(\omega _{\mathrm {a}|\mathrm {d}}\), the time is sufficiently long for the physical cross-links to debond (deadhesion) and for angular excitation frequencies well above \(\omega _{\mathrm {a}|\mathrm {d}}\), the time is too short for the physical cross-links to debond; that is, almost only chemical cross-links of the hydrogels are active in the former case, while both chemical and physical cross-links are active in the latter case, see Fig. 1 [15, 49, 58].

Stress–strain models for doubly cross-linked single-network hydrogels have been developed using finite deformation and fractional time power dependence [49], finite deformation and survivability functions using several material parameters [21, 41‐43], generalized Stokes–Einstein equation [50] and are reviewed by Creton [14] and Zhou et al. [82]. Of particular interest is the observed associative Rouse mode behaviour [59] of the contributing physical cross-link network part to the total shear modulus of the hydrogels showing a corresponding relaxation function with a time dependence of fractional order \(-1/2\) and equivalently a shear modulus with a frequency dependence of fractional order 1/2 [42, 43, 45, 49]. In this paper and based on the latter observation, a visco-elastic model is derived by assuming nearly incompressible doubly cross-linked hydrogels with elastic spherical response and visco-elastic deviatoric response, using a Mittag–Leffler relaxation function [36] of order 1/2 where generalized relaxation time is tunable, thus resulting in a tunable frequency \(\omega _{\mathrm {a}|\mathrm {d}}\) for maximum loss shear modulus, in addition to a tunable equilibrium elastic modulus and relaxation intensity. It is possible to additively split the resulting fractional derivative-based shear modulus model into chemical and physical cross-link parts. Fractional calculus has found a broad application area including visco-elasticity, see e.g. Refs. [3‐5, 10, 17, 29, 30, 33, 36, 54‐56, 61, 62, 66‐68, 71], generally displaying a close agreement with those of measurements over a broad frequency range, in chemical and physical ageing modelling of elastomers [31, 32] and in other fields, as shown in Machado et al. [46]. The derived model results are compared to those of measurements of doubly cross-linked, single-network polyvinyl alcohol (PVA) hydrogels of two different borax concentrations from Refs. [49, 78]. Finally, the model is refined by additively including also an Abel relaxation function of order 1/2 into the deviatoric convolution integral to take into account the superimposed Rouse-type behaviour found in the measurements of the chemical cross-links in addition to their dominating elastic response. The simple, effective visco-elastic models developed are suitable in predicting the mechanical properties of tough, doubly cross-linked, single-network polyvinyl alcohol (PVA) hydrogels, including the separate chemical and physical cross-link contributions, with application potentials in tissue engineering and, more recently, as vibration isolators and dampers.

The hydrogels are assumed homogeneous, isotropic, non-ageing and nearly incompressible with deformation confined to infinitesimal strain. The total second-order stress tensor \(\varvec{\sigma }\) is additively decomposed into an elastic spherical partand a visco-elastic deviatoric partrespectively, using a convolution integral with a kernel of fractional standard linear solid[36], such aswhere \(\mathrm {\mathbf {u}}\) is first-order displacement tensor, t is time, \(a \gg 1\) is a nearly incompressible material constant (with \(a\rightarrow \infty \) being the incompressible limit), \(\mu _{\infty }\) is the equilibrium elastic modulus, and \(\triangle \) is a non-dimensional relaxation intensity,is the Mittag–Leffler function, \(\Gamma (z) = \int _{0}^{\infty } s^{z - 1} e ^{-s} {\text {d}}s\), \(z > 0,\) is the Gamma function, \(\tau _{\mathrm {rel}}\) is generalized relaxation time, \(0< \alpha < 1\) is a material constant, \(\mathrm {\mathbf {i}}\) is second-order unit tensor, \({\nabla }\) is the nabla operator, and the operators \({\text {tr}}\left[ \,\cdot \,\right] \), \({\text {div}}\left[ \,\cdot \,\right] \) and \({\text {dev}}\left[ \,\cdot \,\right] \) denote trace, divergence and deviator, respectively.

Based on the observed associative Rouse mode behaviour [59] of the contributing physical cross-link network displaying a relaxation function with a time dependence of fractional order \(-1/2\) and equivalently a shear modulus with a frequency dependence of fractional order 1/2 [42, 45, 49], the material constant for the hydrogel is set to \(\alpha = 1/2\) being a generalization of the associative Rouse mode behaviour displaying a relaxation function with an approximative time dependence of fractional order \(-1/2\) in the long time range and shown below. The visco-elastic deviatoric stress Eq. (2) becomeswhich is possible to reformulate [26] aswhere the scaled complementary error functionbeing a numerically stable function reducing possible under- and overflow during its numerical estimation, and the error functionUsing an asymptotic expansion of the relaxation function [57], that is, of the scaled complementary error function, results inas \(t^{*}\rightarrow \infty \) where normalized time \(t^{*}=t/ \tau _{\mathrm {rel}}\). As a result, the relaxation function for the hydrogels displays an approximative normalized time dependence of \(1/\sqrt{t^{*}}\) as \(t^{*} \rightarrow \infty \), that is, of a fractional order \(-1/2\) in the long time range while reducing to that of the observed associative Rouse mode behaviour [59]. In passing, it is noted that e.g. Bagley and Torvik [4] and Wharmby and Bagley [71] apply half-order derivatives in their stress–strain relations, while identifying the Rouse mode behaviour, being the long time/low-frequency asymptote of the present model. Furthermore, the long normalized time approximation of the present relaxation function is identical to the Abel operator kernel [36]as \(t^{*}\rightarrow \infty \), where the specific value of the Gamma function \(\Gamma (0.5)=\sqrt{\mathrm {\pi }}\). On the other hand, the relaxation function displays [57]as \(t^{*}\rightarrow 0^{+}\), where \(\Gamma (1.5)=\sqrt{\mathrm {\pi }}/2\); that is, the relaxation function at a short normalized time shows a stretched exponential function behaviour with a very fast decay at increasing normalized time.

Temporal Fourier transformations \([\widetilde{\,\cdot \,}]=\int _{-\infty }^{\infty }[\,\cdot \,]\exp (-\mathrm {i}\omega t) {\text {d}}t\) of the constitutive Eqs. (1) and (5), where \(\mathrm {i}\) is the imaginary unit, result inandwhere the normalized angular frequency \(\omega ^{*}= \omega / \omega _{\mathrm {a}|\mathrm {d}}\) and the material parameter \(\omega _{\mathrm {a}|\mathrm {d}}\) is related to the generalized relaxation time as \(\omega _{\mathrm {a}|\mathrm {d}} = 1/\tau _{\mathrm {rel}}\). It should be noted, as stated in the Introduction section, that fractional derivative models have been applied previously also for visco-elasticity. It is the additive split in chemical and physical cross-link contributions that is the study focus. This is considered next.

The shear modulus \(\hat{\mu }\) defined via \({\text {dev}}[\widetilde{\varvec{\sigma }}]=2 \hat{\mu } {\text {dev}}[{\nabla }\widetilde{\mathrm {\mathbf {u}}}]\) in Eq. (13) is possible to additively split intowhere \(\hat{\mu }_{\mathrm {Ch}}\) and \(\hat{\mu }_{\mathrm {Ph}}\) are the shear modulus derived from chemical and physical cross-links, respectively, see e.g. Refs. [15, 49]. In particular, the normalized shear modulus \(\hat{\mu }^{*}=\hat{\mu }/\mu _{\infty }\) with \(\hat{\mu }\) from Eq. (13) displaysfor \(\omega ^{*} \ll 1\), while using \(\sqrt{\mathrm {i}} = (1+\mathrm {i}\,) / \sqrt{2}\). That is, the normalized shear modulus derived from the chemical cross-links is approximately normalized frequency independent and elastic (only showing a non-vanishing normalized storage modulus), while the normalized shear modulus derived from corresponding physical cross-links displays an identical normalized storage and loss modulusfor \(\omega ^{*} \ll 1\), where \(\mathfrak {R}\) is the real part, being similar to associative Rouse mode behaviour [59] and other results [42, 45, 49], however, and most importantly, here with the coefficient in front of \(\sqrt{\omega ^{*}}\) (in Eq. (16)) given in terms of physically quantifying material parameter \(\triangle \). Furthermore, the maximum normalized loss modulusat an angular frequency for maximum dissipation \(\omega ^{*} = 1\) due to the maximum adhesion–deadhesion activity of the physical cross-links. In passing, it may be noted that the maximum normalized loss modulus \(\max \{ \mathfrak {I}{\hat{\mu }_{\mathrm {Ph}}^{*}} \}\) derived from the physical cross-links only and its normalized angular frequency locus are identical to those of the corresponding hydrogel \(\hat{\mu }^{*}\) derived from both chemical and physical cross-links due to the model applied, displaying a purely elastic, frequency-independent normalized shear modulus \(\hat{\mu }_{\mathrm {Ch}}^{*}\) derived from the chemical cross-links only and being identical to its normalized storage modulus \(\mathfrak {R}\hat{\mu }_{\mathrm {Ch}}^{*}\) and to the normalized equilibrium elastic modulus \(\mu _{\infty }^{*} \equiv 1\). Finally, the normalized shear modulus from Eq. (13) displaysfor \(\omega ^{*} \gg 1\). That is, the normalized shear modulus is showing an elastic, normalized frequency-independent behaviour being identical to the normalized storage modulus with vanishing loss modulus where relaxation intensity \(\triangle \) may physically be identified as the relative contribution of the physical cross-links at full activity to the normalized total shear modulus in relation to the corresponding contribution of the chemical cross-links. A simple estimation example of the relaxation intensity is \(\triangle = \nu _{\mathrm {Ph}}/\nu _{\mathrm {Ch}}\) for an ideally, isotropic, doubly and tetra-functionally cross-linked, single network with equal-functional and homogeneous distribution of chemical and physical cross-links, lacking any trapped entanglements, where \(\nu _{\mathrm {Ch}}\) and \(\nu _{\mathrm {Ph}}\) are the cross-link densities in mol per volume for chemical and maximally active physical cross-links, respectively [15]. Furthermore, simple and possible physical models to estimate the equilibrium elastic modulus include, but are not limited to, affine and phantom network models [16], reading \(\mu _{\infty }^{\mathrm {affine}} = \chi \nu _{\mathrm {Ch}} R T/2\) and \(\mu _{\infty }^{\mathrm {phantom}} = [1-2/\chi ] \chi \nu _{\mathrm {Ch}} R T/2\), respectively, where \(R = 8.314\,\)J/mol K is the universal molar gas constant, T is temperature, and \(\chi \) is cross-link functionality, that is, the number of chain segments emanating from each chemical cross-link, being \(\chi =4\) for tetra-functional cross-link networks. Physically, the increasing density of active physical cross-links at increasing normalized frequency and, thus, decreasing time for debonding of physical cross-links result in an increased total cross-link density, including both chemical and physical cross-links. This in turn results in a reduced network flexibility and reduced conformal freedom which in turn result in an entropy governed increase of the normalized storage modulus of the doubly cross-linked, single-network hydrogel [15].

Finally, the normalized shear modulus from Eq. (13), valid throughout the whole normalized frequency range, splits additively intowhere the chemical and physical parts from Eq. (14) are explicitly given and the normalized frequency \(f^{*}=f/f_{\mathrm {a}|\mathrm {d}}\). Note that the relation between the frequency f and the angular frequency \(\omega \) is \(2 \pi f = \omega \), while the normalized relation reads \(f^{*}=\omega ^{*}\). Note also that the normalized loss modulus \(\mathfrak {I}\hat{\mu }_{\mathrm {Ch}}^{*}\) derived from the chemical cross-links only is assumed to vanish. This is a plausible assumption as in general \(\mathfrak {I}\hat{\mu }_{\mathrm {Ch}}^{*} \ll \mathfrak {I}\hat{\mu }_{\mathrm {Ph}}^{*}\) for real doubly cross-linked hydrogels, see e.g. Refs. [15, 49, 58, 78].

The normalized relaxation spectrum \(\gamma ^{*}\) to the normalized relaxation functions in Eq. (6) is defined viafor \(t^{*} \ge 0\), respectively, where \(\tau ^{*}=\tau /\tau _{\mathrm {rel}}\) is the normalized relaxation time. The normalized relaxation spectrum is possible to obtain from the normalized shear modulus [20] aswhere \(\breve{\mu }^{*} (\mathrm {i}\omega ^{*}) \equiv \widehat{\mu }^{*} (\omega ^{*})\), resulting inObviously, the normalized spectrum is continuous while displaying a maximum oflocated at the normalized relaxation time \(\tau ^{*} = 1\). Clearly, the generalized relaxation time \(\tau _{\mathrm {rel}}\) applied in the constitutive Eq. (6) is the relaxation time for maximum relaxation spectrum. Spectrum is useful while investigating visco-elasticity and molecular motion, being intelligible as the specific molecular time reaction range, involving broad time scales, thus making a natural logarithmic scale (20) most appropriate. Clearly, the generalized relaxation time for hydrogel is physically interpreted as the optimal relaxation time for adhesion–deadhesion of physical cross-links [58].

The normalized shear storage and loss modulus with \(\triangle = 10\) are shown in Fig. 2 versus the normalized frequency together with the loss factor, determined asClearly, the studied hydrogel is essentially displaying a fast normalized storage modulus increase with the normalized frequency, starting at a low-frequency normalized storage modulus plateau of about 1 and is clearly approaching a high normalized frequency plateau of \(1+\triangle =11\) at the high normalized frequency end; the latter observations are in accordance with the high normalized frequency approximation (18). The normalized loss modulus is peaking at \(f^{*}=1\), not surprising, as the hydrogel is tuned to peak at that normalized frequency while displaying a maximum normalized loss modulus of \(\triangle /(2+2\sqrt{2})=2.07\), in accordance with Eq. (17). The maximum loss factor of the studied hydrogels isthat is \(47\,\%\), while peaking at a lower normalized frequency than the corresponding normalized frequency for maximum normalized loss modulus, namely at \(f^{*}=1/[1+\triangle ]=0.09\).

A detailed study over the normalized moduli versus the normalized frequency with \(\triangle =10\) is shown in Fig. 3 while using Eq. (19), displaying the contributions from the chemical and physical cross-links and their normalized frequency dependencies. Clearly, the normalized storage modulus due to the contributions from the chemical cross-links only is displaying a normalized frequency-independent behaviour while fully generating the normalized total storage modulus at vanishing normalized frequency. In contrast, the normalized storage modulus due to the contributions from the physical cross-links only disappears at vanishing normalized frequency while showing a normalized frequency dependence of fractional order 1/2 in the low normalized frequency range. Additionally and for enhanced visual clarity, a function with a normalized frequency dependence of fractional order 1/2 is depicted in Fig. 3. Furthermore, the normalized storage modulus due to the contributions from the physical cross-links only coincides with its normalized loss modulus in the low normalized frequency range, in accordance with the preceding research results [49], being similar to the associated Rouse mode behaviour[59]. A conceivably ingenuous physical explanation of the dynamic behaviour of a doubly cross-linked single-network hydrogel over the normalized frequency range is depicted within the circles in Fig. 3, displaying the hydrogel consisting of chemically and physically cross-linked macromolecules abundantly swollen with water, while being evaluated at various normalized frequencies. At a very low normalized frequency (\(f^{*} \ll 0.001\)), almost at a quasi-static state, the period time of the mechanical oscillation is very long and all temporary physical cross-links are debonded (after bonding) within a time frame much less than the period time of the mechanical oscillation. Consequently, the influence of the temporary physical cross-links is negligible within the period time of the mechanical oscillation and only the chemical cross-links are active, symbolically indicated by the two red \(\bullet \) chemical cross-links in the circle above \(\ll 0.001\) in Fig. 3. The hydrogel is displaying essentially an elastic behaviour within this very low normalized frequency range with the normalized storage modulus mainly proportional to the chemical cross-link density. At increasing normalized frequency, the influence of the temporary physical cross-links increases; they still have time to debond (after bonding), though the period time of the mechanical oscillations is not any longer very long compared to all the temporary physical adhesion–deadhesion processes. That is, the physical cross-links start to be active while simultaneously contributing to the normalized storage modulus by the increased total effective cross-link density and to the normalized loss modulus by the dissipation activity connected to the adhesion–deadhesion operations of the physical cross-links. This is a plausible physical explanation of the simultaneous increase of the normalized storage and loss modulus with the increased normalized frequency; the higher the normalized frequency is, the larger is the increase due to the shortened mechanical oscillation time while approaching that of the time frame for adhesion–deadhesion activity of the physical cross-links. This results in a relatively more active physical cross-links within the period time of the mechanical oscillation, symbolically indicated by the one, five and the twenty yellow \(\bullet \) physical cross-links undergoing adhesion and deadhesion operations in the circles above 0.01, 0.1 and 1, respectively, in Fig. 3. At \(f^{*} = 1\), the period time of the mechanical oscillation matches perfectly the time frame for adhesion–deadhesion activity of the physical cross-links, resulting in an optimum dissipation and, thus, a maximum normalized loss modulus with maximum adhesion and deadhesion active physical cross-links within the period time of mechanical oscillation, symbolically indicated by the twenty yellow \(\bullet \) physical cross-links undergoing adhesion and deadhesion operations in the circle above 1 in Fig. 3. At further increased normalized frequency, the period time of the mechanical oscillation is shorter than the time frame for adhesion–deadhesion activity of the physical cross-links, resulting in physical cross-links that do not have sufficient time to debond (after bonding), thus resulting in a decreased normalized loss modulus due to the decreased adhesion–deadhesion activity of the physical cross-links, in addition to the increased normalized storage modulus due to the increased total effective cross-link density. The higher the normalized frequency is, the shorter is the period time of the mechanical oscillation and, thus, the higher is the activity of the non-debonding physical cross-links, symbolically indicated by the twelve and twenty green \(\bullet \) non-debonding physical cross-links in the circles above 10 and \(\gg 100\), respectively, in Fig. 3. At a very high frequency, \(f^{*} \gg 100\), the period time for the mechanical oscillation is very short and all possible temporary physical cross-links are non-debonding, symbolically indicated by the twenty green \(\bullet \) non-debonding physical cross-links in the circle above \(\gg 100\). This results in a high normalized storage modulus, being proportional to the total chemical and physical cross-link density, symbolically indicated by the two red \(\bullet \) chemical cross-links plus the twenty green \(\bullet \) non-debonding physical cross-links in the circles above \(\gg 100\). However, the normalized loss modulus is vanishing due to the disappearing adhesion–deadhesion activity of the physical cross-links, symbolically indicated by the vanishing number of yellow \(\bullet \) physical cross-links undergoing adhesion and deadhesion operations in the circle above \(\gg 100\). The normalized storage modulus at a very high normalized frequency is about 11 (\(1+\triangle \)) times that of the corresponding normalized storage modulus at vanishing normalized frequency, achievable by selecting a maximum physical cross-link density ten times higher than the corresponding chemical cross-link density, symbolically indicated by the ratio of twenty green \(\bullet \) non-debonding physical cross-links to two red \(\bullet \) chemical cross-links, that is \(20/2=10\), in the circles above \(\gg 100\). Although the normalized storage modulus due to the contributions from the physical cross-links only disappears at vanishing normalized frequency and the normalized frequency for the maximum normalized loss modulus is as high as \(f^{*}=1\), the normalized frequency for equal contribution from the physical and chemical cross-links to the normalized storage modulus is nevertheless very low, only about \(f^{*}=0.0207\). The main reason is the relatively high physical cross-link density compared to the corresponding chemical cross-link density, resulting in a fast increase of the normalized storage modulus with increased normalized frequency due to a relatively fast growing contribution from the physical cross-links. In passing, it is noted that other dissipation mechanisms may exist although adhesion–deadhesion activity of the physical cross-links prevails over those alterative mechanisms, thus making a purely elastic normalized storage modulus due to the contributions from the chemical cross-links only and with vanishing normalized loss modulus, a suitable model [15, 58]. Finally, similar physical explanations as above are given elsewhere [15, 49, 58], although the physical explanations, nevertheless, in this study are comprehensively clarified.

The normalized relaxation spectrum \(\gamma ^{*}\) with \(\triangle =10\) is shown in Fig. 4 versus the normalized relaxation time \(\tau ^{*}\), ranging over a very broad normalized time scale. Clearly, the normalized relaxation time locus for the maximum normalized relaxation spectrum is \(\tau ^{*}=1\) while displaying a maximum normalized relaxation spectrum of 1.59. Physically, the generalized relaxation time for hydrogel is of the same order as the optimal time frame for adhesion–deadhesion process of physical cross-links [58]. Indeed, the optimal time frame is \(T_{\mathrm {opt}} \equiv 1/f_{\mathrm {a}|\mathrm {d}}=2\pi \tau _{\mathrm {rel}}\), derived from the constitutive material parameter relation \(\omega _{\mathrm {a}|\mathrm {d}} = 1/\tau _{\mathrm {rel}}\), reading in normalized form \(T_{\mathrm {opt}}^{*}\equiv T_{\mathrm {opt}}/\tau _{\mathrm {rel}} = 2 \pi \). Furthermore, the relaxation spectrum in Fig. 4 is continuous, thus enabling a wider time scale most likely found in the real molecular networks and in the adhesion–deadhesion physical cross-links processes to be successfully taken into account by the applied Mittag–Leffler and the scaled complementary error relaxation function, while additionally giving a richer structure to the model.

Finally, the model results are compared to those of the measurements from Mayumi et al. [49] and Zhao et al. [78]. The doubly cross-linked, single-network hydrogels consist of poly(vinyl alcohol) (PVA) chemically cross-linked in an aqueous solutions with glutaraldehyde (GA) at a PVA concentration of 12 wt% and a GA concentration of 5.5 mM corresponding to 0.2 mol% of GA to monomer unit of PVA. The physical cross-links are borate ions acting as transient cross-links between the PVA chains and prepared by immersing the chemically cross-linked PVA hydrogels into an aqueous solutions of borax and NaCl of concentrations of 1 mM [49], 50  mM [78] and 90 mM [49], 460  mM [78], respectively, corresponding to 1.5 mol % [49] and 73 mol% [78], respectively, of borate to monomer unit of PVA. The reader is referred to Mayumi et al. [49] and Zhao et al. [78] for the details regarding the sample preparations and measurements of the various shear moduli. The results are in Fig. 5 showing the normalized total storage modulus, physical storage modulus, physical loss modulus, chemical storage modulus and the normalized chemical loss modulus versus the normalized frequency \(f^{*}\), for the model where \(\triangle ^{(\mathrm {a})} = 12.3\) and \(\triangle ^{(\mathrm {b})} = 28.4\), respectively, and for the measurements [49]\({}^{(\mathrm {a})}\), [78]\({}^{(\mathrm {b})}\), however, displaying a courser measurement normalized frequency mesh than those in Mayumi et al. (Fig. 2) [49] and Zhao et al. (Figs. 5a and 10a) [78] for visual clarity. Other materials parameters applied read \(\mu ^{(\mathrm {a})}_{\infty } = 1500\,\)N/m\(^{2}\), \(\mu ^{(\mathrm {b})}_{\infty } = 1700\,\)N/m\(^{2}\) and \(f^{(\mathrm {a})}_{\mathrm {a}|\mathrm {d}} = 0.6/2 \pi \,\)Hz, \(f^{(\mathrm {b})}_{\mathrm {a}|\mathrm {d}} = 5.0/2 \pi \,\)Hz, respectively. Clearly, the model results and those of the measurements match remarkably well over the broad normalized frequency ranges while using only three physically quantifiable material parameters: the frequency for maximum loss modulus, the equilibrium elastic modulus and the relaxation intensity. In particular, the model results and those of the measurements of the normalized total and physical storage modulus almost overlap each other, the normalized loss modulus displays identical pattern although differing slightly more in Fig. 5a, while almost matching in Fig. 5b. Moreover, the elastic, normalized frequency-independent chemical storage modulus is clearly a suitable model displaying an almost perfect overlap between measurement results and those of the model over the normalized frequency range with a small deviation at the higher normalized frequency range. Furthermore, the normalized chemical loss modulus measurement results display \(\mathfrak {I}\hat{\mu }_{\mathrm {Ch}}^{*} \ll \mathfrak {I}\hat{\mu }_{\mathrm {Ph}}^{*}\) throughout the normalized frequency range, thus making a vanishing normalized chemical loss modulus \(\mathfrak {I}\hat{\mu }_{\mathrm {Ch}}^{*} \approx 0\) a suitable model. Finally, the shear modulus (19) was generalized to allow for other \(\alpha \) values than 1/2 via temporal Fourier transform of Eq. (2) while subsequently minimizing the relative least square difference between the measured \(\hat{\mu }_{n}^{*\mathrm {meas}}\) and modelled \(\hat{\mu }_{n}^{*\mathrm {mod}}(\alpha )\) normalized shear modulusby a Matlab\(^\circledR \) constrained nonlinear multi-variable programming solver fmincon, where \(n_{\mathrm {tot}}\) is the total number of measurement points, resulting in an optimum \(\alpha ^{\mathrm {optimum}} = 0.55\) for the measurement in Fig. 5a. Consequently, the shear modulus (19) with \(\alpha = 1/2\) is an appropriate model while also showing a Rouse mode behaviour of the physical storage and loss modulus with the fractional half-order frequency dependence at low normalized frequencies. It should be noted that other models with more parameters such as 7-parameter [42, 43] and 4-parameter [21] large strain models including survivability functions have been tested for doubly cross-linked, single-network PVA hydrogels also showing a good fit.

The derived visco-elastic 3-parameter model for doubly cross-linked, single-network polyvinyl alcohol hydrogels:Physically, the wide spectrum of the normalized time scales in the applied relaxation Mittag–Leffler/scaled complementary error function, as compared to the ordinary exponential relaxation function, allows for a wider range of normalized relaxation times most likely found in the real, doubly cross-linked, single-network hydrogels at adhesion–deadhesion activities of the physical cross-links, thus giving a richer structure to the model. Interestingly, the 3-parameter model (19) covers the full normalized frequency range and matches measurements results remarkably well while applying only three material parameters that are physically quantifiable, namely The 3-parameter model is extended into a 4-parameter model by including a fourth parameter; The resulting 4-parameter model (28) covers the full normalized frequency range and matches measurements results remarkably well while also displaying a Rouse-type behaviour of the normalized chemical loss modulus with a normalized frequency dependence of the fractional order 1/2 throughout the normalized frequency range and a slight increase of the normalized chemical storage modulus with normalized frequency in accordance with those of measurements.

The simple, effective visco-elastic 3- and 4-parameter models are suitable in predicting the mechanical properties of tough, doubly cross-linked, single-network PVA hydrogels with application potentials in tissue and noise abatement engineering. The 4-parameter model is preferable while more accurately modelling the normalized frequency dependence of the normalized chemical loss modulus. Nonetheless, the 3-parameter model results in satisfactory results since the normalized chemical loss modulus is substantially smaller than the other normalized modula (physical and chemical) over the considered normalized frequency range while, in addition, evidently not requiring any preparations nor measurements for the determination of the material parameter \(\mathcal {C}\).