A general solution for one dimensional chemo-mechanical coupled hydrogel rod

Xing-Quan Wang·Qing-Sheng Yang

Introduction

Hydrogel is a kind of materials constructed by crosslinked polymeric chains,it can absorb a massive volume of water and significantly swelling.Smart hydrogels are also capable of perceiving the small changes of external stimuli,and its susceptibility response to the stimulations is often demonstrated by volume shrinking or swelling.Usually the volume transformation is reversible when the external environmental stimuli disappear.In addition to reversible swelling/deswelling behavior,the smart hydrogels also exhibit many other magnificent features,like,biocompatibility,sorption capacity,permeability,and novel mechanical property.Due to these prominent properties,the smart hydrogels are proverbially used.The character of large and reversible deformation makes it an excellent material for micro-valves[1],actuators[2],and sensors[3],whereas the biocompatibility and softness assure it well applied in the area of drug delivery system[4]and tissue engineering[5].

As mentioned above,the smart hydrogel has fascinated an increased attention due to its notable application prospect in engineering.Using a parallel-plate stress-relaxation experiment,the viscoelasticity and poroelasticity properties of a polyacrylamide-alginate interpenetrating hydrogel were probed separately by Wang et al.[6].The experiment of swelling and deswelling behavior of chitosan hydrogels with pH sensitivity was studied by Kang et al.[7],and the deswelling process was explained by a stress-diffusion mechanism.Kim et al.[8]observed that the bending speed and bending angel of the PVA/chitosan IPN hydrogel under electric filed enlarged with the increase of NaCI concentration.However,Li and Sonja[9]found that the time required to achieve the equilibrium bending angle depends on the mobility of the cations,the electric field and the gel thickness but not on the salt concentration.

The experiments are always time-consuming and inefficient in comparison,thus the theoretical study involved multi filed coupling effects are increasingly arousing people’s emphasis.Thomas et al.[10]presented a transport model,which coupled the Nernst–Planck equations with the Poisson equation and the mechanical equilibrium equation to simulate the multi-field problem involving chemo-electro-mechanics.A model for transient deformation of temperature sensitive hydrogel was employed by Birgersson et al.[11]which considered the conservation of momentum,energy and mass for the solid polymer and fluid phase.Li et al.[12–14]developed a series of multi-effect-coupling(MEC)models for simulation of responsive behavior of different stimuli-responsive hydrogels.Based on a nonequilibrium thermodynamic theory,Hong et al.[15,16]developed a theoretical formalism for coupled mass transport and large deformation of the hydrogel.Further,the electrical effect was also taken into account in polyelectrolyte gel where the effects of stretching the network,mixing the polymers with the solvent and ions were considered as well as polarizing the gel.Drozdov[17]treated a polyelectrolyte gel as a three phase medium that consists of polymer network,solvent and solute,and the corresponding constitutive model is developed and mechanical responses of PH-sensitive cationic gels under swelling were studied.

Numerical simulations of hydrogels have also been done by many researchers using various approaches.The meshless Hermit–Cloud method has been employed by Li et al.[12,13]to study the multi-effect-coupling electric-stimulus and pH-electric–stimuli hydrogel models.Hong et al.[18]investigated inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load by invoking a userdefined subroutine in ABAQUS.Yang etal.[19]developed a coupled finite element procedure to simulate the multi-field behavior of the hydrogels.The phenomenons like swelling,shrinking and redistribution of ions were studied.Toh et al.[20]proposed an analogical model which used a heat transfer element to represent the diffusion of solvent molecules within the gel to investigate the transient process of a pH-sensitive hydrogel.Using these simulation methods,some interesting work have been done.Duan et al.[21]studied the effects of large deformation and material nonlinearity on gel indentation,and found that the poroelasticity is a good approximation for the gel.Liu etal.[22]simulated the deformation patterns of the membrane gel in buckling then the deformation processes of nature plant in growth or senescence were also explained.The mechanism for tough and tunable adhesion of hydrogels on solid material was explored by Zhang et al.[23],the simulation and experiment showed that the interfacial toughness is contributed by energy dissipation of the bulk hydrogel and intrinsic work of adhesion.

However,the analytical solutions for hydrogels are relatively few,Liu et al.[24]presented analytical solutions of swelling-induced instability of various slender beam and thin film gel structures,the buckling and wrinkle conditions as well as critical stress values are derived.Yang et al.[25]derived a series of closed-form solutions for a one dimensional chemo-mechanical coupling problem of intelligent hydrogel under a constant chemical stimulus and time-dependent mechanical load.Mazaheri et al.[26]established a model of the pH/temperature sensitive hydrogel,based on this model the inhomogeneous swelling behavior of the hydrogel spherical shell with a rigid core is analytically solved.Abdolahi et al.[27]solved the swelling induced bending of temperature responsive hydrogel bilayer under plain strain condition by an analytical method.Asymptotic method were used by Chen and Dai[28]to study the homogeneous swelling of a gel annulus attached to a rigid core,the approximate analytical solutions were obtained.Recently,the buckling in biodegradable photo-cross-linked poly(ethylene glycol)(PEG)hydrogel which was presented in the shape recovery and swelling of the pre-stretched dry hydrogel upon wetting in water was studied by Salvekar et al.[29],in which a simplified analytical solution was in good agreement with experimental results,some beneficial results were concluded.

In this paper,an analytical approach was applied to investigate the swelling and mechanical behavior of a hydrogel rod under two different boundary conditions.A poroelastical model for hydrogel of chemo-mechanical coupling which which was proposed by Yang et al.[30]was used.The model was simplified into a likely one dimensional heat conduction equation of ion concentration then analytically solved using the method of separation of variables.The analytical expressions of ion concentrations related to time were obtained and the displacement,stress,chemical potential were also deduced.It showed that the present chemomechanical theory can characterize the chemo-mechanical coupling behavior of intelligent materials.

1 Theory of chemo-mechanical coupling

Following Yang et al.[30],the constitutive equations of chemo-mechanical coupling for hydrogels in the isothermal process based on continuum mechanics can be expressed as

Assuming that the hydrogel was immersed in a solution which is dominated by a certain kind of ion concentration,other concentrations could be ignored.The Eq.(1)can be simplified as

The pressure caused by the chemical concentration difference is introduced asPion=R T c.Compared it with the effective stress of Terzaghi,it can be found that

whereRrepresents gas constant,Tis absolute temperature,andδijis Kronecker function.

According to the definition of chemical potential in classical chemistry,the coefficientsin Eq.(2)can be obtained

The governing equations are composed of the mechanical equilibrium equation and the chemical divergence equation which can be obtained by combining Eq.(4)with Fick’s second law,

The geometric equation and the chemical gradient equation are as follows

whereDijis ion diffusion coefficient.

Substituting Eqs.(2)and(6)into Eq.(5),the governing equations expressed by independent variables of displacementu iand ion concentrationccan be obtained

Assuming that the hydrogels considered here are isotropic and homogenous,the body force and inertial force are ignored.Substituting the constitutive equations for the stresses into Eq.(7),we find

Fig.1 Hydrogel rod under uniform pressure

whereGis the shear modulus,υis Poisson ratio,andεrepresents the volume strain.

2 The solutions for one dimensional problem

A one dimensional hydrogel rod immersed in a solution was considered.Assuming that there is only one solute in the solution and the concentration remains constant atc0.Forthe hydrogel rod,lateral expansions were ignored and the diffusion was supposed only happened at the top surface,while at the lateral or the bottom the flux of concentration remains zero.Two different boundary conditions were considered under the condition that the temperature remains unchanged.

Case 1 The hydrogel rod was under the pressure ofP0and the other end was fixed,showed in Fig.1.

Case 2 The hydrogel rod was fixed at both ends,showed in Fig.2.

The only component of displacement in one dimensional situation wasw.Both the displacement and chemical potential were functions of the coordinatezand the timet.So Eq.(8)could be written in the form of one dimensional.

2.1 Solution of Case 1

The hydrogel rod was under uniform pressure,the stress of the rod remained unchanged.Combined the stress boundary condition and Eq.(3)we could have the following relations

Fig.2 Hydrogel rod fixed at both ends

Derivate both sides of Eq.(10)with respect to time,

Substituting Eq.(11)into the second equation of Eq.(9)yielded

The boundary conditions about the ion concentration could be written as follows

The first condition expressed that at the top of the rod the ion concentration wasc0and remained unchanged.The second condition indicated that no ion concentration diffused through the bottom.At the initial time,the ion concentrations along the rod were zero,so the initial condition was

Under the boundary and initial conditions,the partial differential equation of Eq.(13)could be solved by the method of separation of variables.The solution of ion concentration written in a series form as follows

Then the total displacement inzdirection was

As for chemical potential,form Eq.(2)we knew that

Substituting the expression of concentration into Eq.(19),the chemical potential could be deduced.

2.2 Solution of Case 2

The hydrogelrod was fixed at both ends,so the displacement remains zero,Eq.(7)could be simplified as

wheref(t)was the constraining force of thezdirection,it changed with the time.

The boundary and initial conditions were the same as those in Case 1.In the same way,the second equation of Eq.(20)could be solved,

wherecn,βnare identical with the above.

The expressions of the stress and the chemical potential could also be deduced by substituting Eq.(21)into the constitutive equations.

3 Result discussion

Based on the formulas,some numerical examples were obtained.The values of the parameters we used are presented in Table 1,and the length of the rod was 100 mm.The numerical results of both cases were obtained.Figures 3–7 shows the change of ion concentration,displacement and chemical potential of Case 1 respectively.While Figs.8–11 reflect the change of the Case 2.Different pressure status were also considered in Case 1.

Table 1 Material parameters used in the numerical example[25]

Fig.3 Variation of ion concentration versus coordinate of Case 1

In Fig.3,the concentration versus coordinate of different times in Case 1were showed.Whent=0,the concentration through out the hydrogel rod are zero,except the vertex.It was strictly complied with the boundary and initial conditions.As time went on,the concentration of each point in the hydrogel rod gradually increased and finally reached a balance state where the concentrations through the rod are 15 mol/m3or very near that value.The existence of balance state is complied with previous experimental results as well as the FEM simulation[29].The figure also shows that the concentration changes more rapidly at the beginning,this phenomenon can be showed more intuitive in Fig.4,where the concentration versus time of some selected points are showed.The moment of the concentration began to change is increased with coordinate.It is also consistent with FEM simulation where the sample wetting from the outer and gradually moves inward[29].And the closer the points away from the top,the faster it would reach the balance state.When close to the balance state,the change of concentration also slows down.

The chemical potential of the hydrogel rod at different times when the pressure is 40 kPa is showed in Fig.5,while in Fig.6 the chemical potential versus time of three selected points were compared.These two figures of chemical potential are similar to those of concentration respectively.The rate of increase of chemical potential also slows down with time and the change trend is the same with that of concentration.It illustrates that the chemical potential raised with the increasing of concentration.Due to the effect of uniform pressure,the chemical potentials of the hydrogel rod are not zero while the concentrations are zero at initial time.

Figure 7 shows the deformations versus time of the rod inzdirection at different pressure states.When there was a pressure,the hydrogel rod was in a compressed state in the initial and the total displacement was positive.With the diffusion of the concentration the total displacements decreases.This illustrated that the diffusion of concentration aroused the expansion of the hydrogel rod.The expanding process has the same trend with the varying of the concentration,and also reaches a steady state.The curves under different pressure status are found to be parallel to each other,and the difference between them is caused by the pressures.

Based on the constitutive equations in Eq.(2)the parallel relationship which has been showed in Fig.7 could be proved.Equation(2)could be simplified into a one dimensional form,that is

Fig.4 Variation of ion concentration versus time of different points in Case 1

Fig.5 Variation of chemical potential versus coordinate of Case 1(P=40 kPa)

Fig.6 Variation of chemical potential versus time of different points in Case 1(P=40 kPa)

Fig.7 Variation of displacement at end of rod versus time for different applied pressures

whereσzwas equal to the pressure.The displacement inzdirection could be written in following integral form.

The first item in the integral of the right side shows that the total displacement is positive correlated with concentration,the changing trends of displacements versus time are similar though pressure status were different.The second item illustrates that the difference between the curves originates in the difference of pressure.

The hydrogel rod in Case2 is fixed at both ends,the figures for concentration are the same as Case 1 which have been showed in Figs.3 and 4.The images of the stresses in the hydrogel rod caused by concentration diffusion are showed in Figs.8 and 9.

Figure 8 shows the stresses versus coordinates of different times,the value of the compressive stress is increased from zero to nearly 37,000 Pa.The changing process of the stress of three selected points are showed in Fig.9,the changing trends of them are similar to their corresponding concentrations which are showed in Fig.4.It indicated that the diffusion of the concentration caused the increasing of compressive stresses of the hydrogel rod when both ends were restricted.

The chemical potential versus coordinate and versus time of case two are showed in Figs.10 and 11 respectively.They are similar to those curves in Figs.5 and 6 of Case 1,but some difference also exist.The starting value of Case 2 is zero because the hydrogel rod was in natural state at first.The equilibrium value of the chemical potentials in Case 1(The pressure was 40,000 Pa)is2481 J,while the equilibrium value of situation two is2475 J.These two equilibrium values are very closely compared to the stresses at equilibrium of these two cases,so we could find that the chemical potential increased with the stress while the impact of concentration for chemical potential is more obvious.

Fig.8 Variation of stress versus coordinate in Case 2

Fig.9 Variation of stress versus time of different points in Case 2

Fig.10 Variation of chemical potential versus coordinate of Case 2

Fig.11 Variation of chemical potential versus time of different points in Case 2

4 Conclusions

In this paper an analytical solution for one dimensional chemo-mechanical problem of a rod under two different situations has been proposed.The analytical solutions of concentration have been obtained in a series form,the expressions of displacement,chemical potential and the stress have also been deduced.Some numerical examples were given.It was found that the diffusion of concentration could lead to the expansion or the change of the stress of the rod.There is an obvious balance state in the diffusion of concentration,the chemical potential and displacement,stress also come to a steady value at last.This paper also found that the process of diffusion is independent to the stress,while the value of chemical potential can be influenced by the stress.

The solutions presented here can be further used to investigate the effects of material parameters on the expansion caused by diffusion.By changing the boundary or initial conditions,some particular cases of one dimensional chemo-mechanical coupling can also be solved.Nevertheless,these results obtained here were based on poroelastical model so there is a certain error for the case of large deformation,which needs to be further studied.

Acknowledgements The financial supports from the National Natural Science Foundation of China(Grants 11472020,11502007,and 11632005)and Hong Kong Scholars Program(Grant XJ2016021)are gratefully acknowledged.

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