Nanocomposite Gels with Permanent and Transient Junctions under

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Nanocomposite Gels with Permanent and Transient Junctions under Cyclic Loading A. D. Drozdov* and J. deClaville Christiansen* Department of Materials and Production, Aalborg University, Fibigerstraede 16, Aalborg 9220, Denmark S Supporting Information *

ABSTRACT: The toughness of double-network gels is conventionally evaluated by comparison of their hysteresis energies under uniaxial cyclic deformation. A shortcoming of this approach is that it does not allow comparison of mechanical properties of gels prepared by various routes, as the energy dissipated per cycle depends strongly on maximum elongation ratio, strain rate, degree of swelling, etc. As an alternative approach, constitutive modeling is suggested of the mechanical behavior of hydrogels under cyclic deformation. We develop a model for the viscoelastic and viscoplastic responses of double-network gels and apply it to the analysis of observations on covalently and noncovalently cross-linked nanocomposite gels reinforced with graphene oxide nanosheets, hectorite clay nanoplatelets, zirconium oxide nanoparticles, cellulose nanocrystals and nanofibrils, and metal ions. Numerical simulation demonstrates that the model describes adequately stress−strain diagrams with various shapes, and its parameters evolve consistently with concentration of nanofillers and strength of matrix−filler interactions. (ii) hydrophobic association,13,14 (iii) supramolecular interactions of host−guest pairs,15,16 and (iv) reinforcement with nanoparticles.17,18 To assess experimentally the enhancement of toughness of hydrogels driven by the presence of temporary junctions, tensile cyclic tests with various maximum elongation ratios kmax are conventionally performed in which the energy dissipated per cycle (calculated as the area between the loading and unloading paths on a stress−strain diagram) is plotted versus kmax. Connections between the fracture energy and the energy dissipated under cyclic deformation are discussed in the Supporting Information with reference to recent studies.8,19,20 An advantage of this method is that it allows the energy dissipated due to breakage of sacrificial bonds under uniaxial tension−compression to be evaluated directly. Its shortcoming is that the results of calculation depend strongly on deformation mode, strain rate, maximum elongation ratio per cycle, degree of swelling, etc. This limits applicability of this method for comparison of the efficacy of various toughening strategies. An alternative approach that allows the influence of different types of sacrificial bonds on the mechanical behavior of doublenetwork gels to be compared consists in development of a mathematical model for the response of these gels under cyclic loading. Within such a model, the entire stress−strain diagram is described by a set of material constants available for

1. INTRODUCTION Hydrogels are three-dimensional networks of hydrophilic chains bridged by chemical and physical junctions. The mechanical response of gels subjected to swelling has recently attracted substantial attention as these materials demonstrate potential for a wide range of smart applications including biomedical devices, carriers for targeted drug delivery, scaffolds for tissue engineering, sensors, and soft actuators.1,2 A shortcoming of conventional (single-network) covalently cross-linked gels is that they are relatively weak (the elastic modulus of 100 kPa) and brittle (the fracture energy of 1−10 J/m2). The latter is traditionally explained by the inhomogeneity of polymer networks with randomly distributed chain lengths between cross-links, weak interactions between chains, and the lack of an efficient mechanism for energy dissipation.3 Mechanical properties are improved significantly when a double-network structure is formed in a gel that involves two interpenetrating networks of chains.4 The most promising strategy to prepare gels with superior mechanical properties (elastic modulus up to 10 MPa5 and fracture energy exceeding 103 J/m2 6) consists in introduction of sacrificial (dynamic) bonds between chains governed by reversible interactions.7 The design of double-network gels (with chains in a permanent network bridged by chemical cross-links and chains a transient network linked by noncovalent junctions) with high stiffness, strength, toughness, and fatigue resistance has become a focus of attention in the past few years.8,9 Several mechanisms have been proposed for development of sacrificial bonds in a polymer network:9 (i) electrostatic interactions (formation of polyion complexes6,10 and metal-coordination complexes11,12), © XXXX American Chemical Society

Received: December 21, 2017 Revised: February 5, 2018

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DOI: 10.1021/acs.macromol.7b02698 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules comparison after they are determined by fitting experimental data. Derivation of constitutive equations for double-network gels is complicated by the fact that these relations should account for a number of phenomena: (i) viscoelasticity (an increase in stress with strain rate in tensile tests and a decay in stress with time in relaxation tests21,22), (ii) viscoplasticity (an increase in residual strain with maximum elongation ratio under cyclic deformation6,10,23), (iii) recovery (a strong reduction in residual strain with time after loading−unloading6,10), and (iv) the Mullins effect (observed under multicycle deformation with monotonically growing maximum elongation ratios24,25). A number of models have recently been suggested for the analysis of time- and rate-dependent responses of doublenetwork hydrogels under cyclic loading (see refs 26−34), to mention a few. These studies concentrate on the viscoelastic behavior of hydrogels modeled within the concept of transient polymer networks35,36 and account for their inelastic response by means of the pseudoelasticity theory. This treatment demonstrates good quality of fitting observations in tests with moderate finite strains (several tens of percent)21,27 but ensures only qualitative agreement with experimental data under cyclic deformation with large (of order of 10) elongation ratios.28,29,34 The objective of this study is threefold: (i) to develop a model for the mechanical behavior of double-network gels under cyclic deformation that accounts for their viscoelastic and viscoplastic responses, (ii) to find adjustable parameters in the governing equations by fitting observations in cyclic (loading− unloading) tests on covalently and noncovalently cross-linked composite gels reinforced with graphene oxide nanosheets, cellulose nanocrystals and nanofibrils, nanoclay platelets, and zirconium hydroxide nanoparticles, and (iii) to establish correlations between shapes of the stress−strain diagrams, on the one hand, and concentration and type of nanofiller, on the other. For definiteness, the polymer network in a gel is treated as a combination of permanent and transient networks. Dissipation of energy under cyclic deformation is induced by two processes: (i) breakage and re-formation of temporary junctions driven by thermal fluctuations (viscoelasticity) and (ii) slippage of permanent junctions with respect to their reference positions in a gel (viscoplasticity). The former mechanism for dissipation of energy is standard for transient networks,35,36 where a dangling chain adopts the actual state of a gel at the instant of its reattachment as the reference (stressfree) state (relaxation of stresses in dangling chains). The latter mechanism, recently suggested in ref 37, presumes that a permanent junction becomes unbalanced when one of the chains connected by this junction is transformed from the active state into the dangling state. As a result, the junction begins to slide with respect to the network (plastic flow) until it reaches a new equilibrium state. The novelty of our approach consists in the description of both phenomena (viscoelasticity and viscoplasticity) within a unified constitutive model.

Chains in the former network are bridged by permanent junctions, while chains in the latter network are connected by temporary junctions that rearrange (break and re-form) being driven by thermal fluctuations. According to the affinity hypothesis, deformations of the permanent and transient networks coincide with macro-deformation of the gel. The reference (stress-free) state of the permanent network before application of external loads coincides with the asprepared state of the gel. According to the multiplicative decomposition formula, the deformation gradient F for transition from the initial (undeformed dry) state into the actual (deformed swollen) state is given by F = Fe ·Fp

(1)

where Fe and Fp are the deformation gradients for elastic and plastic deformations, and the dot stands for inner product. Under the conventional hypothesis that the plastic spin vanishes, the rate-of-strain tensors for plastic deformation (dp in the unloaded state and Dp is the actual state) read d p = Fṗ ·Fp−1,

Dp =

1 (Fe ·d p·F−e 1 + F−e T ·d p·FeT) 2

(2)

where the superscript dot stands for the derivative with respect to time and T denotes transpose. Two mechanisms of plastic deformation (sliding of junctions between chains with respect to their initial positions) are introduced:38 (i) flow induced by macro-deformation (with the rate-of-strain tensor dm) and (ii) flow driven by interchain interaction (with the rate-of-strain tensor di). The rate-of-strain tensor for plastic deformation is determined by

d p = d m + di

(3)

The rate-of-strain tensor Dm (an analogue of dm in the actual state) is proportional to the rate-of-strain tensor for macrodeformation

Dm = ϕ D

(4)

where the non-negative function ϕ obeys the following conditions: (i) ϕ vanishes in the initial state (sliding of junctions does not occur at infinitesimal strains); (ii) ϕ increases with macro-deformation (which reflects acceleration of the sliding process under loading) and tends to its ultimate value ϕ = 1 at very large strains. With reference to39 we set ϕ ̇ = ±a(1 − ϕ)2 |D|α ,

ϕ(0) = 0

(5)

where the signs “+” and “−” correspond to loading and unloading, respectively, |D| =

1/2

( 23 D: D)

is the equivalent

strain rate (the colon stands for convolution), and a and α are material parameters. To describe the response of a transient network, we denote by τ an instant when an active chain (both ends are connected to the network) is bridged with the network by a temporary junction and distinguish chains that joined the network under polymerization of a pregel solution (τ = 0) and those attaching the network under deformation (τ > 0). The reference (stressfree) state of a chain with τ = 0 coincides with the as-prepared state of the gel. The reference state of a chain with τ > 0 coincides with the actual state of the gel at instant τ (stresses in chains totally relax under rearrangement). Under the conventional assumption that the number of active chains is not affected by mechanical factors,36 rearrangement of a temporary

2. MODEL A gel is treated as a two-phase medium composed of an equivalent polymer network and water molecules. The solid and fluid phases are modeled as immiscible interpenetrating continua. Deformation of the network and concentration of water molecules are connected by the molecular incompressibility condition. For definiteness, a polymer network is thought of as a superposition of two networks: permanent and transient. B


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Macromolecules kė k̇ k̇ = (1 − ϕ) − i , ke k ki

network (detachment of active chains and attachment of dangling chains) is described by the only parameter: the rate of breakage (detachment from temporary junctions) γ. Constitutive equations for the mechanical response of a double-network gel under an arbitrary deformation with finite strains are developed in the Supporting Information. These relations are derived by means of the free energy imbalance inequality for an arbitrary specific mechanical energy of the permanent network

⎛ k3−1 k3− ki̇ = P ⎜V e −R i ki ke ki ⎝

⎛1 ⎞ S1̇ = γ ⎜ 2 − S1⎟, ⎝k ⎠

w(Iτ1 , Iτ 2 , Iτ 3)

k(0) = 1

(14)

ϕ ̇ = ±a (1 − ϕ)2 k−α , ϕ(0) = 0 (15) * α with a* = aϵ̇ describes sliding of junctions in the permanent network driven by macro-deformation. Equations 9−13 contain six material constants: G stands for the elastic modulus of a gel, K−1 is the Gent constant accounting its strain hardening,41 κ is the ratio of the modulus of the transient network to that of the permanent network, R is the ratio of parameters Gp and Ge characterizing the energies stored in and dissipated by the permanent network, a* is the rate of sliding of junctions induced by macro-deformation, and α describes slowing down of the sliding process under large deformations. The governing equations involve two adjustable functions P and γ. According to eq 12, the coefficient P characterizes rate of plastic flow in the permanent network driven by interchain interaction. This coefficient is presumed to vanish under loading and to grow monotonically under retraction

(6)

(7)

(8)

P = 0 (stretching),

P = P exp[χ (σ − σ )] (retraction) * *

(16)

where σ* and P* stand for the tensile stress and the value of P at the instant when unloading starts. It follows from eq 13 that γ determines the rate of rearrangement of junctions in the temporary network. We suppose that the rearrangement process slows down with plastic deformation induced by interchain interaction, and the corresponding dependence is described by γ = γ exp( −Λϵi), ϵi = ki − 1 (17) * An advantage of the constitutive model is its ability to describe correctly experimental stress−strain diagrams under cyclic deformation with various shapes (see the next section). To reach this goal, mutual dependencies are introduced between macro-deformation and plastic flows (eqs 15 and 16) and between the viscoplastic and viscoelatic responses (eq 17). As a result, the total number of parameters grows up to 10 (G, K, κ, R, a*, α, P*, χ, γ*, and Λ). Although this number is not small, it remains lower than the number of adjustable constants in other models for cyclic deformation of polymers.43,44

(9)

(ii) the kinematic equation for elongation ratio k under macrodeformation

k ̇ = ±ϵ̇,

S1(0) = S2(0) = 1

accounts for strain hardening under large stretching, and the function ϕ, whose evolution is governed by eq 5

where Ge, Gp, g, and K are constants. With reference to ref 40 the function We is adopted in the Gent form (6) to describe strain-hardening of gels under large deformations.41 Following ref 38, the energy of interchain interaction Wp is accepted in the neo-Hookean form (7), which serves as the first term in the formal expansion of the function Wp into the Taylor series with respect to its arguments. Equation 8 was derived in ref 42 within the concept of entropic elasticity. Constitutive equations for a nanocomposite gel equilibrated before loading and subjected to uniaxial tensile cyclic deformation with a constant strain rate ϵ̇ involve (i) the formula for engineering tensile stress ⎡ ⎛ k3−1 S ⎞⎤ σ = G⎢(1 − ϕ)V e + κ ⎜S1k − 22 ⎟⎥ ⎝ kke k ⎠⎦ ⎣

S2̇ = γ(k − S2),

−1 ⎡ ⎞⎤ 1⎛ 2 − 3⎟⎥ V = ⎢1 − ⎜ke 2 + ⎢⎣ K⎝ ke ⎠⎥⎦

and the energy of active chains in the transient network 1 g[(Iτ1 − 3) − ln Iτ 3] 2

(12)

In the above relations, the function

the energy of interchain interactions

w=

ki(0) = 1

(13)

The specific energy We stored in chains of the permanent network depends on the principal invariants Iem (m = 1, 2, 3) of the Cauchy−Green tensor for elastic deformation Be = Fe·FTe . The specific energy of interchain interaction Wp is treated as a function of the principal invariants Iim (m = 1, 2) of the Cauchy−Green tensor for plastic deformation Bi = Fi·FTi . The energy w stored in an active chain of the transient network is presumed to depend on the principal invariants Iτm (m = 1, 2, 3) of the Cauchy−Green tensor bτ = fτ·fTτ , where fτ(t) = F(t)· F−1(τ) is the deformation gradient for transition from the actual state at time τ to the actual state at time t. In the analysis of observations, we focus on uniaxial tensile cyclic deformation of a gel with the strain energy density of the permanent network

1 Gp(Ii1 − 3) 2

1⎞ ⎟, ⎠

and (v) the kinetic equations for the functions S1 and S2 that describe rearrangement of chains in the transient network

and an arbitrary strain energy of chain in the transient network

Wp =

(11)

(iv) the kinetic equation for elongation ratio ki under plastic deformation

We(Ie1 , Ie2 , Ie3) + Wp(Ii1 , Ii2)

⎤ ⎛ I − 3⎞ 1 ⎡ ⎟ + ln I ⎥ We = − Ge⎢K ln⎜1 − e1 e3 ⎝ ⎠ ⎦ 2 ⎣ K

ke(0) = 1

(10)

(iii) the kinematic equation for elongation ratio ke for elastic deformation C


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Macromolecules

3. FITTING OF OBSERVATIONS Our aim is to demonstrate the ability of the model to describe stress−strain diagrams under cyclic deformation and to show that adjustable parameters can be found with the accuracy sufficient to compare the influence of the nature and content of filler on the mechanical response of nanocomposite gels. We begin with the analysis of experimental data on gels prepared by cross-linking polymerization (12 h at 60 °C) of acrylamide monomers (AAm, mAAm = 3.4 M) by using N,N′methylenebis(acrylamide) (BIS, mBIS = 2.4 mM) as a crosslinker and ammonium persulfate (APS) (0.1 mol % of AAm) as an initiator.45 Synthesis was conducted in aqueous dispersions of graphene oxide (GO) with concentrations C ranging from 0.5 to 4.0 mg/mL. Tensile cyclic tests were performed with a constant strain rate ϵ̇ = 0.042 s−1. In each test, a specimen was stretched with the strain rate ϵ̇ up to the maximum elongation ratio kmax = 4 and retracted with the same strain rate down to the zero stress. Observations in mechanical tests and results of simulation with the material parameters listed in Table S-1 are depicted in Figure 1, where the engineering stress σ is plotted versus

sheets), and the ratio R decreases strongly with concentration of GO (which reflects weakening of interchain interactions due to the presence of GO sheets in a pregel solution). We proceed with matching observations on nanocomposite gels prepared by cross-linking polymerization (4 h at 60 °C) of AAm monomers (mAAm = 1.4 M) by using BIS (mBIS = 2.4 mM) as a cross-linker and APS (1 wt % of AAm) as an initiator.46 Free radical polymerization was performed in aqueous dispersions of graphene oxide (GO) with concentrations C ranging from 1.0 to 3.0 mg/mL. Tensile cyclic tests were conducted with a constant strain rate ϵ̇ = 0.033 s−1 and kmax = 3. Observations in mechanical tests are depicted in Figure 2 together with results of simulation

Figure 2. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data46 in cyclic tests with ϵ̇ = 0.033 s−1 and kmax = 3.0 on acrylamide gels (mAAm = 1.4 M) cross-linked with BIS (mBIS = 6.5 mM) and reinforced with various concentrations C mg/mL of graphene oxide. Solid lines: results of simulation.

with the material parameters collected in Table S-1. Calculations are conducted by presuming a* = 0 (the response under stretching is purely elastic) and χ = 0 (the rate of plastic flow under retraction P is constant). Two material parameters evolve with concentration of GO in a pregel solution: the elastic modulus G and the rate P* of plastic deformation. Changes in these quantities with concentration of GO are illustrated in Figure S-2, where the data are approximated by eq 18. The following conclusions are drawn: (I) No hysteresis is observed in the response of the covalently cross-linked AAm gel. The energy dissipated per cycle (defined as the area of the hysteresis loop) grows with concentration of GO. This increase is reflected in the model as the growth of P* with C. (II) The elastic modulus G increases linearly with concentration of GO. This increase may be attributed to formation of permanent bonds between polymer chains and GO sheets. The coefficient G1 that describes the growth of G in eq 18 is strongly affected by molar fraction of monomers in a pregel solution. (III) All parameters (except for K and χ) in Table S-1 adopt similar values. The parameter K−1 in the Gent model is a measure of extensibility of chains: it adopts small values for long chains (large molar fraction of monomers, mAAm = 3.4 M) and large values for short chains (small molar

Figure 1. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data45 in cyclic tests with ϵ̇ = 0.042 s−1 and kmax = 4.0 on acrylamide gels (mAAm = 3.4 M) cross-linked with BIS (mBIS = 2.4 mM) and reinforced with various concentrations C mg/mL of graphene oxide. Solid lines: results of simulation.

elongation ratio k. Calculations are conducted by presuming a* = 0 (the response under loading is purely elastic) and χ = 0, Λ = 0 (the rate of plastic flow P and the rate of rearrangement of chains γ remain constant under retraction). Three material parameters are affected by concentration of GO nanosheets in a pregel solution: the elastic modulus G, the rate P* of plastic flow under retraction, and the ratio R characterizing the energy of interchain interactions. The effect of concentration of GO on these quantities is illustrated in Figure S-1. The data are approximated by the equations G = G0 + G1C ,

P = P 0 + P 1C , * * *

log R = R 0 − R1C (18)

with coefficients calculated by the least-squares method. Equations 18 mean that the modulus G increases with concentration of nanofiller following the rule of mixture, the rate P* of plastic flow under retraction grows with C (this increase is induced by sliding of chains with respect to GO D


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Macromolecules fraction of monomers, mAAm = 1.4 M). The coefficient χ describes stress-induced acceleration of plastic flow and characterizes the shape of the stress−strain diagram under retraction. For relatively small χ, this curve is convex at small elongation ratios and concave at large k (Figure 1). When χ becomes relatively high, the unloading diagram is concave at all elongation ratios (Figure 2). We now fit observations on nanocomposite gels prepared by cross-linking polymerization (24 h at 35 °C) of AAm monomers (mAAm = 2.25 M) by using BIS (mBIS = 2.9 mM) as a cross-linker and potassium persulfate (KPS) (0.3 wt % of AAm) as an initiator.47 Synthesis was conducted in aqueous suspensions of cellulose nanocrystals (CNC) with concentrations C ranging from 0.1 to 0.6 vol %. Tensile cyclic tests were conducted with a constant strain rate ϵ̇ = 0.0167 s−1 and kmax = 5. Observations in these tests are depicted in Figure 3 together with results of simulation with the

Figure 4. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data47 in cyclic tests with ϵ̇ = 0.0167 s−1 and various kmax on acrylamide gel (mAAm = 2.25 M) cross-linked with BIS (mBIS = 2.9 mM) and cellulose nanocrystals (C = 0.6 vol %). Solid lines: results of simulation.

To assess the difference between the responses of gels with covalently and noncovalently cross-linked networks, we match observations on nanocomposite gels prepared by free radical copolymerization (2 h at 50 °C) of acrylamide (AAm) and poly(ethylene glycol) diacrylate (PEGDA) monomers (mAAm = 1 M, molar ratio of AAm and PEGDA equals 100:3) by using KPS (2.8 wt % of AAm) as an initiator.48 Synthesis was conducted in aqueous suspensions of cellulose nanocrystals (CNC) with concentrations C ranging from 0.2 to 2.0 vol %. Observations in tensile cyclic tests with a constant strain rate ϵ̇ = 0.05 s−1 and kmax = 6 are presented in Figure 5 together

Figure 3. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data47 in cyclic tests with ϵ̇ = 0.0167 s−1 and kmax = 5.0 on acrylamide gels (mAAm = 2.25 M) cross-linked with BIS (mBIS = 2.9 mM) and various concentrations C vol % of cellulose nanocrystals. Solid lines: results of simulation.

material parameters listed in Table S-2. Unlike AAm−GO nanocomposite gels, both mechanisms of plastic deformation (flow under loading with the rate a* and under retraction with the rate P) are taken into account in calculations. Despite rather complicated shapes of the stress−strain diagrams in Figure 3, each curve is determined by the only parameter G. The effect of volume fraction of CNC on the elastic modulus is illustrated in Figure S-3, where the data are approximated by eq 18. To examine the influence of loading conditions on parameters of the model, we match observations on nanocomposite gel with C = 0.6 vol % in cyclic tests with kmax ranging from 2.8 to 5.0. Experimental data and results of simulation are reported in Figure 4. Each retraction curve in this figure is determined by two parameters, P* and Λ. Evolution of these quantities with kmax is shown in Figure S-4, where the data are approximated by the equations

Figure 5. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data48 in cyclic tests with ϵ̇ = 0.05 s−1 and kmax = 6.0 on acrylamide−poly(ethylene glycol) diacrylate gels (mAAm = 1 M, molar ratio 100:3) with various concentrations C vol % of cellulose nanocrystals. Solid lines: results of simulation.

with results of simulation with the material parameters listed in Table S-2. Each stress−strain diagram in this figure is determined by two parameters, G and Λ. The influence of volume fraction of CNC on these quantities is illustrated in Figure S-5. The data are approximated by eq 18 for G and the relation

log P = P 0 − P 1k max , log Λ = Λ 0 − Λ1k max (19) * * * with coefficients calculated by the least-squares technique. E


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Macromolecules log Λ = Λ 0 − Λ1C

(20)

with the coefficients found by the least-squares method. The following conclusions are drawn: (I) The stress−strain diagrams depicted in Figures 3 and 5 reveal similar shapes. This similarity is surprising because Figure 3 demonstrates the response of gels with covalent cross-links, while Figure 5 presents the behavior of gels with noncovalent junctions. It may be explained by the fact that PEDGA chains form strong hydrogen bonds with cellulose nanocrystals. Part of these bonds withstands cyclic deformation and serves as permanent junctions between chains. (II) An important difference between the stress−strain curves in Figures 3 and 5 is that the residual strain (measured under retraction down to the zero stress) is independent of volume fraction of CNC for the chemically crosslinked gel (Figure 3) and decreases monotonically with C for the physical gel (Figure 5). This difference is reflected by the model as a decrease in Λ with C (Figure S-5). Bearing in mind that Λ characterizes the effect of plastic deformation on slowing down of rearrangement of temporary junctions, the reduction in Λ is tantamount to acceleration of the rearrangement process with concentration of CNC. (III) For nanocomposite gels with covalent and noncovalent bonds, the elastic modulus G increases with concentration of CNC following the rule of mixture (eqs 18). This growth is rather strong for the nanocomposite gels with chemical cross-links (Figure S-3) and becomes modest for those with physical cross-links (Figure S-5). The latter may be explained by a low molar fraction (3 mol %) of PEDGA monomers that form hydrogen bonds with cellulose nanocrystals. To demonstrate that shape of the stress−strain diagrams under cyclic loading can be strongly affected by strength of hydrogen bonds between nanofiller and polymer chains, we analyze the response of acrylamide-co-2-acrylamido-2methylpropanesulfonic acid (AMPS) gel reinforced with zirconium hydroxide Zr(OH)4 nanoparticles. The gel was prepared by free radical polymerization (72 h at room temperature) of AAm and AMPS monomers (mmon = 2.0 M, molar ratio 7:3) in colloidal solution (6 wt %) of Zr(OH)4 particles (diameter 10 nm) by using KPS as an initiator (0.45 wt % of monomers) and N,N,N′,N′-tetramethylethylenediamine (TEMED) (3.6 μL per gram of monomers) as a catalyst.49 Tensile cyclic tests were conducted with a constant strain rate ϵ̇ = 0.083 s−1 and maximum elongation ratio kmax = 4. Observations in these tests are depicted in Figure 6 together with results of simulation with the material parameters collected in Table S-3. Each retraction curve is determined by two parameters, P* and Λ. Evolution of these quantities with kmax is demonstrated in Figure S-4. The data are approximated by eq 19 for P* and the linear equation for Λ

Λ = Λ 0 − Λ1k max

Figure 6. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data49 in cyclic tests with ϵ̇ = 0.083 s−1 and various kmax on acrylamide-co-2-acrylamido-2-methylpropanesulfonic acid (mmon = 2.0 M, weight ratio 70:30) with 6 wt % of zirconium hydroxide nanoparticles. Solid lines: results of simulation.

pronouncedly. Comparison of Tables S-2 and S-3 reveals that the observed distinction in shapes is reflected by the differences in rates of plastic flow: under loading, the value of a* in Figure 5 exceeds that in Figure 6 by a factor of 60, while under retraction, the value of P* in Figure 6 exceeds that in Figure 5 by 2 orders of magnitude. The high rate of plastic flow a* under stretching of AAm− PEGDA−CNC gels may be explained as follows. Because of large concentration of OH groups on the surfaces of cellulose nanocrystals, these crystals form a secondary network (skeleton) in an undeformed gel. Two types of irreversible deformation occur under loading: (i) sliding of chains along the surfaces of CNC and (ii) slippage of nanocrystallites with respect to each other. This explains why a* of AAm−PEGDA− CNC gels exceeds strongly that of AAm−AMPS−NP gel, where zyrconium hydroxide nanoparticles are separated and plastic deformation is induced by sliding of chains along their surfaces only. The same reasoning can be applied to explain the difference in rates of plastic flow under retraction. As concentration of OH groups on the surfaces of Zr(OH)4 nanoparticles is small (compared with that on the surfaces of CNCs), chains slide easily along their surfaces. The latter is reflected in the model as a substantially higher value of P* for AAm−AMPS−NP gels compared with AAm−PEGDA−CNC gels, where the sliding process is severely restricted by a large concentration of hydrogen bonds on the surfaces of nanocrystallites. We proceed with matching experimental data on acrylic acid (AAc) gel prepared by free radical polymerization (48 h at 30 °C) of AAc monomers (mmon = 2.9 M) in colloidal solution (2.1 mg/mL, 1 wt % with respect to monomers) of cellulose nanofibrils (CNF) by using APS (0.2 wt % with respect to monomers) as an initiator and TEMED (3.3 μL per gram of monomers) as a catalyst.50 The pregel solution contained 1 mol % (with respect to monomers) of ferric chloride hexahydrate FeCl3·6H2O. Polymer chains are physically cross-linked by (i) hydrogen bonds between carboxyl groups COOH of AAc chains and hydroxy groups OH at the surfaces of CNFs and (ii) electrostatic interactions between Fe3+ cations and ionized functional groups COO− belonging to polymer chains and

(21)

with the coefficients determined by the least-squares method. Polymer networks in the gels, whose behavior is reported in Figures 5 and 6, are formed by the same physical mechanism: hydrogen bonds between segments of chains and nanoparticles containing large amount of OH groups on their surfaces. However, shapes of their stress−strain diagrams differ F


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Macromolecules CNFs. The effect of interactions between mobile metal cations and ionized functional groups attached to polymer chains on the mechanical response of metal-containing gels has recently been reviewed in refs 51 and 52. Tensile cyclic tests were performed with a constant strain rate ϵ̇ = 0.05 s−1 and kmax ranging from 4 to 13. Observations in these tests are depicted in Figure 7 together with results of

Figure 8. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data53 in cyclic tests with ϵ̇ = 0.111 s−1 and various kmax on acrylic acid (AAc) gel (mAAc = 3.7 M) cross-linked with BIS (mBIS = 0.81 mM) and 0.5 mol % of Fe3+ ions. Solid lines: results of simulation.

rate of plastic flow P* (see Figures S-7 and S-8). Another interesting conclusion is that the elastic modulus of the AAc gel reinforced with CNFs is substantially higher than that of the covalently cross-linked gel. This means that concentration of physical junctions between polymer chains and cellulose nanofibrils that remain unbroken during cyclic deformation (Figure 7) exceeds strongly concentration of permanent junctions in the gel cross-linked with BIS (Figure 8). We now approximate observations on acrylamide-co-acrylic acid (AAm−AAc) copolymer gels impregnated with Fe3+ ions and physically cross-linked with hectorite nanoclay54 (Laponite XLG) and GO nanosheets.55 The gels were synthesized by free radical polymerization (16 h at room temperature for AAm− AAc−NC and 48 h at 40 °C for AAm−AAc−GO) of AAm and AAc monomers (mAAm = 3.0 M, molar ratio of AAm and AAc 100:15) in aqueous suspensions of NC (1.0 wt % with respect to monomers) and GO (0.5 mg/mL) by using APS (0.1 wt % with respect to aqueous suspensions) as an initiator and TEMED (20 μL for AAm−AAc−NC only) as a catalyst. After polymerization, the gels were immersed into ferric chloride hexahydrate solution (0.06 M) to develop physical bonds between mobile Fe3+ ions and carboxyl groups of AAc chains. Tensile cyclic tests were conducted with a constant strain rate ϵ̇ = 0.083 s−1 and maximum elongation ratios kmax ranging from 2 to 9 (AAm−AAc−NC gel) and 3 to 11 (AAm−AAc− GO gel). Observations in mechanical tests are presented in Figures 9 and 10 together with results of numerical simulation with the material parameters collected in Table S-6. Each curve in Figures 9 and 10 is determined by the only parameter P*. Evolution of this quantity with kmax is illustrated in Figure S-9, where the data are approximated by eq 19. The following conclusions are drawn: (I) The similarity of the stress−strain diagrams in Figures 9 and 10 results in close values of the material parameters collected in Table S-6, which confirms that the fitting procedure is stable. (II) The curves log P*(kmax) in Figure S-9 coincide practically for the two gels, which means that the plastic flow under retraction is weakly affected by the nature of interactions

Figure 7. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data50 in cyclic tests with ϵ̇ = 0.05 s−1 and various kmax on acrylic acid gel (mAAc = 2.9 M) cross-linked with 1 mol % of Fe3+ ions and 1 wt % of cellulose nanofibrils. Solid lines: results of simulation.

simulation with the material parameters collected in Table S-4. Each curve in Figure 7 is determined by the only parameter, the rate of plastic flow under retraction P*. The influence of kmax on this quantity is shown in Figure S-7, where the data are approximated by eq 19. To examine the effect of CNFs on the mechanical response of nanocomposite gels, observations in Figure 7 are compared with experimental data on a covalently cross-linked AAc gel impregnated with Fe3+ ions. The gel is synthesized by free radical polymerization (24 h at 30 °C) of AAc monomers (mAAc = 3.7 M) in an aqueous solution of ferric nitrate nonahydrate Fe(NO3)3·9H2O (0.5 mol % with respect to monomers) by using BIS as a cross-linker (mBIS = 0.81 mM) and APS (0.05 wt % of AAc) as an initiator.53 Tensile cyclic tests were conducted with the strain rate ϵ̇ = 0.11 s−1 and maximum elongation ratios kmax ranging from 3 to 11. Experimental stress−strain diagrams are reported in Figure 8 together with results of simulation with the material parameters listed in Table S-5. Each retraction curve in Figure 8 is determined by the only parameter P*. The effect of kmax on the rate of plastic flow under unloading is illustrated in Figure S-8, where the data are approximated by eq 19. Comparison of Figures 7 and 8 implies that the presence of a small concentration of covalent cross-links leads to a noticeable increase in stress σ at large elongation ratios k (strain hardening under stretching induced by extension of chains bridged by permanent junctions) and a pronounced decrease in dissipated energy (defined as the area between stress−strain diagrams under loading and retraction). Strain hardening of the AAc gel cross-linked with BIS is observed as a substantial increase in parameter K−1, whereas the decay in its toughness is reflected as a strong (by more than an order of magnitude) reduction in the G


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These values may serve as indicators of the effect of nanofiller on the rate of plastic flow under retraction in metal-containing hydrogels. The data reported in Table S-6 and Figure S-9 show that the nature of filler (NC versus GO) affects weakly the mechanical response of anionic polyelectrolyte gels impregnated with metal cations. To demonstrate that this influence can be substantial when the matrix is neutral, two sets of experimental data are analyzed. First, we match observations on N-isopropylacrylamide (NIPA) gels physically cross-linked with hectorite nanoclay (Laponite XLG). The gels were prepared56 by photopolymerization (2 h of irradiation with a UV lamp) of NIPA monomers (mNIPA = 0.98 M) in aqueous suspensions of NC with concentrations C ranging from 0.02 to 0.15 wt % by using α-ketoglutaric acid (2 wt % of monomers) as photoinitiator. Tensile cyclic tests were performed with a constant strain rate ϵ̇ = 0.01 s−1 and maximum elongation ratio kmax = 5.9. Observations in these tests are depicted in Figure 11 together

Figure 9. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data54 in cyclic tests with strain rate ϵ̇ = 0.083 s−1 and various kmax on acrylamide-co-acrylic acid gel (mAAm = 3.0 M, molar ratio 100:15) cross-linked with Fe3+ ions (aqueous solution with 60 mM of FeCl3) and nanoclay (1 wt %). Solid lines: results of simulation.

Figure 11. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data56 in cyclic tests with ϵ̇ = 0.01 s−1 and kmax = 5.9 on N-isopropylacrylamide gels (mNIPA = 0.98 M) with various concentrations C wt % of nanoclay. Solid lines: results of simulation.

Figure 10. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data55 in cyclic tests with strain rate ϵ̇ = 0.083 s−1 and various kmax on acrylamide-co-acrylic acid gel (mAAm = 3.0 M, molar ratio 100:15) cross-linked with Fe3+ ions (aqueous solution with 60 mM of FeCl3) and graphene oxide (0.5 mg/mL). Solid lines: results of simulation.

with results of simulation with the material parameters collected in Table S-7. Each loading curve in Figure 11 is determined by three parameters, G, a*, and κ, while each retraction curve is characterized by the only parameter P*. Changes in these quantities with concentration of NC are shown in Figure S-10. The data are approximated by eq 18 for P* and the equations

between polymer chains and nanoplatelets (electrostatic interaction for NC versus hydrogen bonds for GO). (III) The slopes of the curves d log P*/dkmax adopt the values (−0.17 for NC and −0.19 for GO) that exceed slightly those for the other gels with Fe3+ ions (−0.12 for chemically cross-linked AAc gel in Figure S-8 and −0.13 for AAc−CNF gel in Figure S-7). This difference may be explained by slowing down of rearrangement of metal coordination bonds due to the presence of nanoclay platelets and GO nanosheets. (IV) On the basis of the data in Figures S-7 to S-9, we calculate P* at kmax = 1 and obtain P* = 7.3 × 10−4 s−1 for covalently cross-linked AAc gel, P* = 1.2 × 10−2 s−1 for AAc−CNF gel, P* = 2.7 × 10−2 s−1 for AAm−AAc−NC gel, and P* = 4.4 × 10−2 s−1 for AAm−AAc−GO gel.

log G = G0 + G1C ,

log κ = κ0 + κ1C ,

a = a 0 + a 1C * * * (22)

where the coefficients are calculated by the least-squares technique. Elastic moduli of permanent (G) and transient (κ) networks are determined by the logarithmic analogue (22) of the rule of mixture. The exponential dependence of moduli on concentration of nanofiller is in accord with refs 57 and 58 where experimental data on silica-reinforced hydrogels were compared with the Guth−Gold equation. An increase in a* with C may be explained with reference to the house-of-cards model.18 According to this concept, weakly H


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rated. Displacements of these platelets with respect to their initial positions in AAm-NC gel are severely restricted by entanglements between chains. This is reflected by relatively low values of the rates of plastic flow in AAm-NC gel (for example, P* for NIPA-NC gel exceeds that for AAm-NC gel by 2 orders of magnitude). (II) Exfoliation of stacks of NC platelets in an aqueous solution results in release of cations from the surfaces of nanoclay platelets.60 Because of association of mobile cations with water molecules, the concentration of cagelike structures formed by water molecules around hydrophobic segments of NIPA chains is reduced.61 Breakage of cages induces agglomeration of hydrophobic segments and formation of aggregates from which water molecules are expelled. Dehydrated hydrophobic aggregates serve as permanent (G) and temporary (κ) physical cross-links between hydrophilic segments of NIPA chains. This mechanism explains the growth of G and κ with C in NIPA-NC gel and the independence of these parameters of concentration of nanofiller in AAm-NC gel whose chains do not involve hydrophobic segments.

charged clay platelets bridged by electrostatic interaction form a secondary network in an undeformed gel. Plastic flow under loading reflects two types of irreversible deformation: (i) sliding of chains along the surfaces of NC platelets and (ii) slippage of nanoplatelets with respect to each other. The linear growth of a* and P* with concentration C is attributed to the latter mechanism. For comparison, we approximate experimental data on acrylamide gels physically cross-linked with nanoclay (Laponite RDS) modified with pyrophosphate ions.59 The gels were synthesized by free radical polymerization (72 h at 30 °C) of AAm monomers (mAAm = 2.8 M) in aqueous suspensions of NC with concentrations C ranging from 0.04 to 0.10 wt % by using KPS as an initiator and TEMED as a catalyst (with the mole ratios of monomers, initiator, and catalyst 100:0.263:0.453). Tensile cyclic tests were performed with a constant strain rate ϵ̇ = 0.033 s−1 and maximum elongation ratio kmax = 11. Observations in mechanical tests on nanocomposite gels with C = 0.04 and 0.10 wt % are reported in Figure 12 together with results of simulation with the material parameters listed in Table S-8.

4. DISCUSSION Analysis of observations in cyclic tests leads to the following conclusions for covalently and noncovalently cross-linked nanocomposite gels: Covalently cross-linked gels 1. A neat gel demonstrates purely elastic response (the loading and unloading paths coincide). Reinforcement of the gel with nanofiller results in an increase in its modulus G. The growth of G with concentration of filler is described by the rule of mixture (eqs 18). 2. The energy dissipated per cycle grows with nanofiller content. The increase is modest when chains are bridged with GO nanosheets (Figures 1 and 2) and Fe3+ ions (Figure 8) but becomes noticeable when cellulose nanocrystals are used as reinforcement (Figure 3). 3. Rather small dissipation of energy revealed by these figures can be ascribed to two factors: (i) negligible rate of plastic flow a* in Figures 1 and 2 and (ii) relatively low rate of plastic flow P* in Figures 3 and 8. Noncovalently cross-linked gels 1. The energy dissipated per cycle by a gel with physical junctions is comparable with the energy dissipated by a covalently cross-linked gel when physical cross-links are strong and their rate of rearrangement is substantially lower than the strain rate under deformation (see Figures 3, 5, and 12). 2. Hydrogels demonstrate large dissipated energy per cycle when their rates of plastic flow under loading a* and unloading P* are sufficiently high (Figures 6, 7, and 9−11). To satisfy this condition, polymer chains are to be connected by a large number of relatively weak physical junctions. 3. Toughness of a nanocomposite gel is strongly influenced by filler−matrix interactions. The energy dissipated per cycle remains unaffected by the nature of filler when these interactions are weak (Figures 9 and 10). On the contrary, the dissipated energy grows pronouncedly when these interactions result in development of a secondary network formed by nanofiller (Figures 11 and 12).

Figure 12. Engineering tensile stress σ versus elongation ratio k. Symbols: experimental data59 in cyclic tests with ϵ̇ = 0.033 s−1 and kmax = 11 on AAm-NC gels (mAAm = 2.8 M) with various clay contents C wt %. Solid lines: results of simulation.

Unlike NIPA-NC hydrogels with high toughness (Figure 11), the energy dissipated per cycle by AAm-NC gels is rather modest, and the stress−strain curves in Figure 12 are similar to those depicted in Figures 3 and 8. The growth of the NC content results in (i) a weak (by 6%) increase in the elastic modulus G, (ii) a decrease in the rate of plastic flow under stretching a*, and (iii) it does not affect concentration of transient junctions (characterized by the parameter κ) and the rate of plastic flow under retraction P*. The difference between the mechanical response of NIPANC and AAm-NC gels may be explained by two reasons: (I) Platelets of untreated NC Laponite XLG form a secondary network. Slippage of platelets with respect to each other under stretching and retraction results in pronounced plastic deformation of NIPA-NC gel (characterized by high values of a* and P*). Platelets of chemically modified NC Laponite RDS remain sepaI


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Macromolecules 4. Unlike gels with covalent cross-links whose moduli obey the rule of mixture (Figures S-1 to S-3 and S-5), the elastic moduli of noncovalently cross-linked gels grow strongly (exponentially) with concentration of nanofiller (Figure S-10). Observations in cyclic tests with various maximum elongation ratios show that the rate of plastic flow under retraction P* decreases exponentially with kmax (Figures S-4 and S-6 to S-9). This decay reflects evolution of internal structure of nanocomposite gels induced by plastic deformation under stretching. Although this study does not dwell on multicycle deformation, it is worth noting that this evolution (damage accumulation) explains why the energy dissipated by a hydrogel in subsequent cycles is noticeably lower than that dissipated along the first cycle of loading−retraction.49,50,54,55

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.D.D.). *E-mail: [email protected] (J.d.C.). ORCID

J. deClaville Christiansen: 0000-0001-5501-7019 Notes

The authors declare no competing financial interest.

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5. CONCLUSIONS A model is developed for the mechanical (viscoelastic and viscoplastic) responses of nanocomposite gels. A gel is treated as a double network of polymer chains bridged by permanent and temporary junctions. The characteristic feature of the model is the presence of two mechanisms for plastic flow: (i) nondissipative, when junctions slide with respect to their initial positions with the rate proportional to the strain rate under macro-deformation, and (ii) dissipative, with the rate of sliding determined by the energy of interchain interaction. Plastic deformation is presumed to be governed by the nondissipative mechanism under loading and unloading and by the dissipative mechanism under retraction. Numerical simulation demonstrates that (i) the model describes adequately experimental stress−strain diagrams on nanocomposite gels under cyclic loading and (ii) adjustable parameters in the governing equations evolve consistently with concentration of fillers (graphene oxide nanosheets, hectorite clay nanoplatelets, zirconium oxide nanoparticles, cellulose nanocrystals and nanofibrils, metal ions) and experimental conditions (maximum elongation ratio under stretching). Analysis of experimental data shows that two conditions are required to be fulfilled in order to ensure an extraordinary toughness of nanocomposite gels (assessed by the area between stress−strain diagrams under stretching and retraction): (i) high elastic modulus G and (ii) large ratios of the rates of plastic flow a* and P* (that characterize sliding of junctions between chains induced by nondissipative and dissipative mechanisms) to the strain rate ϵ̇. Although the model can be applied to analyze the response of a gel under an arbitrary loading program, the present study focuses on the mechanical behavior of nanocomposite gels observed in uniaxial loading−unloading tests with constant strain rates. To reach quantitative agreement with observations in tensile tests with different strain rates, as well as in relaxation and creep tests, the only relaxation time γ*−1 should be replaced with a relaxation spectrum11 following the approach.62 Selfrecovery in double-network gels and their behavior under stress- and strain-controlled multicycle deformation programs will be investigated in a separate study.

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Derivation of the constitutive model, discussion of connections between fracture toughness and dissipated energy, list of physical properties of nanofillers in the gels under consideration, tables with material constants for nanocomposite gels, figures showing the effects of filler content and maximum elongation ratio on adjustable parameters (PDF)

ACKNOWLEDGMENTS Financial support by the Danish Innovation Fund (project 5152-00002B) is gratefully acknowledged. A.D.D. is thankful to Prof. Z. Suo for an inspiring discussion.

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REFERENCES

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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/acs.macromol.7b02698. J


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