Transient swelling behaviour of dual stimuli sensitive nanocomposite hydrogels

Journal Pre-proof Transient swelling behaviour of dual stimuli sensitive nanocomposite hydrogels Saeed Mousazadeh, Mehrdad Kokabi PII:

S0032-3861(20)30119-1

DOI:

https://doi.org/10.1016/j.polymer.2020.122280

Reference:

JPOL 122280

To appear in:

Polymer

Received Date: 14 October 2019 Revised Date:

30 January 2020

Accepted Date: 11 February 2020

Please cite this article as: Mousazadeh S, Kokabi M, Transient swelling behaviour of dual stimuli sensitive nanocomposite hydrogels, Polymer (2020), doi: https://doi.org/10.1016/j.polymer.2020.122280. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

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Transient Swelling Behaviour of Dual Stimuli Sensitive Nanocomposite Hydrogels Saeed Mousazadeha, Mehrdad Kokabi a∗ a

Department of Polymer Engineering, Faculty of Chemical Engineering, Tarbiat Modares University, P.O. Box 14115-114, Tehran, Islamic Republic of Iran.

∗

Corresponding author’s e-mail: [email protected] (M. Kokabi)

Tel/Fax: +98-21-8288-3340

1

ABSTRACT Smart nanocomposite hydrogels can undergo large deformation under external stimuli. Several coupled physical processes simultaneously affect the transient swelling behaviour of pH and temperature-sensitive hydrogels. Besides, the hydrogel properties change in response to the external stimuli in the presence of nanoparticles, which affect its transient deformation and swelling behaviour. This work focused on transient swelling modeling of pH and temperaturesensitive polymer/clay nanocomposite hydrogels. The time-dependent processes were considered simultaneously with material properties updating in each time-step during the finite element procedure. A set of experiments was designed to evaluate the numerical solving results in the free and constrained swelling conditions. The results illustrate that the proposed developed model successfully predicts the transient swelling behaviour of pH and temperature-sensitive either for hydrogels or nanocomposite hydrogel counterparts, which can offer a guideline to the abundant usage of these soft materials.

Keywords: Dual stimuli sensitive hydrogel; Finite element method; Modeling; Nanocomposite; Transient swelling.

2

1. Introduction The ability to respond to the external stimulus of hydrogels made a new kind of material which is known as stimuli-responsive or smart hydrogels. The smart hydrogels have appealed much attention in recent years [1]. The accurate knowledge of the transient swelling behaviour during the process is valuable in some specific applications of hydrogels, while in some others, only equilibrium state requires to consider [2]. An overall view of hydrogel behaviour due to an external stimulus could be reached without any empirical tests by employing the mathematically predicting the deformation and swelling behaviour of hydrogels. Stimuli-responsive hydrogels, especially temperature and pH-sensitive hydrogels, are of great importance in a wide variety of applications such as lab-on-chip, microfluidic controller valves, contact lenses, biosensors, and drug delivery [3-6]. The transient swelling behaviour prediction of these hydrogels may be used for those who wish to design hydrogels for specific applications. The swelling process of a dry temperature and pH-sensitive hydrogel in a dilute electrolyte solution consists of one or more of the subsequent occurrences. •

Wetting the gel surfaces with the surrounding fluid.

•

Solvent penetration into the gel.

•

Relaxation of the polymeric network.

•

Diffusion of ionic spices (mobile ions) into the gel.

•

Deformation of the polymeric network due to the presence of the counterions for neutralization of the dissociated pending groups of the polymeric network.

•

Change in polymer-solvent interactions due to changes in the temperature.

•

Change in the fixed charge density of the polymeric network via a change in the presence of counterions and ionic strength of the surrounding fluid. 3

Due to the widespread application of hydrogels, the models, which could predict the swelling and deformation behaviour of hydrogels, are valuable. Over the last decades, numerous mathematical models with various approaches and different degrees of complexity have been proposed for the prediction of the volume change of hydrogels [7-11]. These models can divide into two categories consist of the equilibrium and transient models. In the equilibrium state, there are no solvent or ion migrations and energy transfer between the hydrogel and outside. Therefore, mechanical, thermal, and chemical equilibria are considered using the equilibrium models, simultaneously. Most of the thermodynamic models for prediction of the equilibrium swelling have been developed based on Flory-Huggins mean-field lattice theory [12-14]. According to this theory, the change of the total free energy is assumed to be the sum of the hydrogel network chains– solvent mixing, elastic deformation of the network chains, and ionicfree energies change [15]. The transient models consider at least one of mechanical energy, thermal energy, and chemical potential changes, while the hydrogel has not yet reached a steady state. The swelling behaviour and deformation of the hydrogel at both steady and transient state could be predicted using the transient models. Different approaches were elaborated for transient modeling of swelling kinetics of the hydrogels. In some models, the hydrogel considered as two phases[9, 16], while in some others, the hydrogel was treated as a continuum phase [17, 18]. During recent years, several studies were performed for modeling and simulation of the transient swelling behaviour prediction of hydrogels based on the non-equilibrium thermodynamic theory proposed by Hong et al. [19], which uses the coupled diffusion and large deformation for prediction of gel swelling [20-22]. Li et al. presented multi-effect coupling models for the prediction of swelling and deformation of smart hydrogels in response to different external

4

stimuli [23]. These models are based on the Poisson–Nernst–Planck equation for the ionic concentrations profile determination and the force balance formulation for evaluation of the deformation in hydrogels. Multi-effect coupling models predictions had good agreement with the published experimental data available in the literature [24]. Recently, transient swelling prediction of stimuli-sensitive hydrogels attracted much attention because of their versatility applications [9, 25, 26]. Liu et al. characterized the large deformation of the magnetic-sensitive hydrogel via the fully coupled arbitrary Lagrangian-Eulerian (ALE) method [3]. Drozdov developed a constitutive model for swelling prediction of a temperaturesensitive gel subjected to transient swelling [27]. Due to the numerous advantages of using hydrogels, their low mechanical strengths limit their applications. Hence, polymer/clay nanocomposite hydrogels were developed for overcoming this drawback of hydrogels. Their supreme characteristics have appealed much attention to widespread applications of polymer/clay nanocomposite hydrogels. Despite the prevalent experimental studies on the polymer/clay nanocomposite hydrogels swelling [28-31], there is no model development or modification for prediction of swelling behaviour of this kind of soft matter at the transient state. The only model modification for prediction of the equilibrium swelling behaviour of nanocomposite hydrogels is the study of Aalaie and Vasheghani-Farahani. They used their experimentally obtained equation for the elastic modulus of the sulfonated polyacrylamide/clay nanocomposite hydrogel for the cross-link density evaluation in the swelling behaviour prediction [32]. In the present work, transient modeling for swelling and deformation prediction of dual stimuli (pH and temperature) sensitive polymer/clay nanocomposite hydrogels (with exfoliated and randomly oriented clay nanolayers) was presented. In the proposed developed model, we

5

evaluate the kinematics of large-deformation based on Li’s approach and Flory’s mean-field theory with the local equilibrium assumption at each time steps of swelling. Subsequently, the finite element method was employed for numerical solving of the set of model equations. Finally, the experimental results of swelling of poly (vinyl alcohol) /poly (acrylic acid)/ montmorillonite nanocomposite hydrogel systems were compared with model predictions. 2. Mathematical modelling Here, we briefly summarize our proposed model for transient swelling behaviour of an arbitrarily-shaped dual stimuli (temperature and pH) nanocomposite hydrogel immersed in aqueous solution at different temperatures and pH. 2.1. Mass conservation law

The model comprised ionic conservation of mass (see Equation 1) for determination of the mobile ions concentration distributions inside the considered volume of nanocomposite hydrogel at each time step during swelling. The modified Nernst-Planck equation based on the chemical potential is used to describe the ionic species flux [33]. + ∇. −

.

∇

Where ci, Di, and

= 0 ,

= 1, 2, …

(1)

are the concentration, diffusion coefficient, and chemical potential of species

i, respectively. T, t, and R are the absolute temperature, time, and universal gas constant, respectively. N is the number of total mobile ions in the considered volume. The ion transfer within hydrogels occurs by the water-filled areas specified by the polymeric mesh. Therefore, the swelling process will affect the ion diffusion coefficient due to the increase in the size of the polymeric mesh. In general, according to Tsai and Strieder, the diffusion coefficient of an ion through hydrogels is proportional to the volume fraction of water and polymer within the hydrogel [34]. Based on the obstruction theory, the polymer chains cause an increase in the path 6

length for ion transfer, considering the polymer chains in hydrogels as impenetrable mesh. Therefore, the Tsai and Strieder model used for the determination of the diffusion coefficient of ion within the water-filled areas specified by the polymeric mesh (

), which is expressed as

Equation 2 [35]. !"

= 1+



(2)

Where α is defined according to Equation 3. = # $ wherein

(

%& ' (

+

)

)

*

(3)

is the diffusion coefficient of the ion into the solvent. α is a constant determined by

the volume fraction of polymer in the hydrogel (# ), the ion radius (rs), and polymer chains diameter (dp). In addition to the effect of the polymer chains on ion diffusion, the dispersed nanoparticles in the polymer/clay nanocomposite hydrogels increase the pathway for ion transfer by making an impenetrable mesh. So, the Tsai and Strieder model (Equation 2) could be used for determination of the effect of increasing the pathway for ion transfer on the diffusion coefficient of ion within the water-filled areas specified by the clay nanoparticles mesh with replacing polymer chains diameter (dp) with the clay nanoparticles thickness (dn) and replacing the volume fraction of polymer in the hydrogel (# ) with the volume fraction of the clay nanoparticles in the

hydrogel (#- ). Also, the dispersed Na-MMT-nanoparticles in various ionic aqueous solutions may contain different ion species. Therefore, the presence of these ion species could make an electrostatic friction force for ion transfer, which affects the diffusion coefficients of ions. The models based on the hydrodynamic theory could be used for the determination of the diffusion coefficient of ions with consideration of the ions transfer-resistant due to the electrostatic frictional drag in the water-filled areas specified by the clay nanoparticles mesh [34]. The 7

diffusion coefficient of ion within the water-filled areas specified by the clay nanoparticles mesh -

( .

) is finally determined according to Equation 4.

= /1 + #- $

( %& ' . ) (. )

* +

12.23) %&) 0. ( ) +.4 . )

12.23) %&) 0. ( ) +. " . )

+5

+.7

6 8

!"

(4)

The diffusion coefficient of ion within the polymer/clay nanocomposite hydrogels (

9

) is

defined as Equation 5. "

:

=

"

;

+

"

<

(5)

The chemical potential of the ion species i for an electrostatic field is written as Equation 6 [36]. =

+

+ =. >. ln1A . ) + B . C. D

In which

+

(6)

, γi, and zi are the chemical potential of a reference state, the chemical activity

coefficient, and valence number of the ion species i, respectively. Whereas, ψ and F are the electrostatic potential and Faraday’s constant. With the substitution of the chemical potential (Equation 6) into Equation 1, the ionic conservation of mass obtained as Equation 7 [36], which is coupled with the Poisson equation (Equation 8) for determination of the electric potential in the hydrogel [23]. + ∇. E−

9

F1 + G

G

∇ D = − MM N∑Q PR" BP K

H∇ − P

I

.J .K

+ BS S T

. ∇DL = 0 , = 1, 2, …

(7) (8)

Where ε0, ε, and zf are the dielectric constant of vacuum, the relative dielectric constant of the solvent, and the valence of the fixed charge groups within the hydrogel, respectively [23]. The concentration of counterions (cf), which is proportional to fixed charge groups in polymer chains is stated as Equation 9, for anionic and cationic hydrogels.

8

c

f

 0 Ka c f  H K a + c H+  =  0 c H+ c f  H K a + c H+

Where H,

+ S,

Anionic

(9)

Cationic

and Ka are the hydrogel hydration, concentration of total ionizable fixed charge

groups within the dry gel, and ionic dissociation constant, respectively. The hydrogel hydration (H) is defined as the ratio of the interstitial solvent volume to the solid volume of the hydrogel. Whereas, the concentration of total ionizable fixed charge groups within the dry gel can express as the ratio of the moles of the ionizable group to the volume of dry gel [23]. According to the Debye - Hückel theory, the chemical activity coefficient of an ion is given by Equation 10 [37].   − AZi2 I  log(γ i ) =   − AZi2 I   1+ I

I < 0.02

(10)

I ≥ 0.02

In Equation 10, A is a factor that depends on the absolute temperature and the relative dielectric constant of the medium, which is defined as Equation 11 [37]. U=

".V W×"+Y 1M× )2.Z

(11)

Also, the ionic strength of the solution (I) as a measure of the concentration presented in the solution is expressed as Equation 12 [37]. [ = ∑Q PR" BP "

P

(12)

2.2. Mechanical force balance

Due to the coupling effects of mass transfer and external stimuli, hydrogels usually undergo large deformation. The nonlinear deformation theory provides adequate computational accuracy

9

for the determination of hydrogels deformation. Therefore, the momentum equilibrium equation in the absence of external forces for large deformation is written as Equation 13.

∇. \ = ]^_

(13)

Wherein P, ^_, and ρ are the first Piola-Kirchhoff stress tensor, acceleration, and hydrogel density. As well known, because of the absence of symmetry in the first Piola-Kirchhoff stress tensor (P), it is not used in constitutive equations. Therefore, the first Piola-Kirchhoff stress tensor substituted with the second Piola-Kirchhoff stress tensor (S) as the symmetric stress for large deformation (P=SFT) in Equation 13. Finally, Equation 13 was rewritten as Equation 14 by using the modified second Piola-Kirchhoff stress tensor for hydrogels (S=CE-πTI) [23].

∇. [1ab − c d)e ] = ]^_

(14)

In which C, E, I, and F are the material properties tensor, Green Lagrangian strain tensor, unit

tensor, and deformation gradient tensor, respectively. Also, c , ^_, and ρ are the total osmotic

pressure, acceleration, and hydrogel density, respectively. The total free energy change of ionic hydrogel (∆GT) is the sum of the mixing-free energy change (∆Gmix), the elastic-free energy change (∆Gelas) and the ionic-free energy change (∆Gion) [38]. For polymer/clay nanocomposite hydrogels, either neutral or ionic, as a ternary system (solvent, polymer, and clay), the presence of clay nanoparticles could affect the mixing-free energy due to the mixing and interactions between components. In this study, the effect of clay nanoparticles in the mixing-free energy change for polymer/clay nanocomposite hydrogels are evaluated by generalizing the cubic lattice model for a ternary system. In summary, for a ternary system containing n1 solvent molecules, n2 polymer molecules contain y2 segments (each segment equal to a solvent molecule), and n3 clay layers consist of y3 segments, we used a cubic lattice having n0 cells (n0= n1+ y2n2+y3n3). According to the procedure

10

described in the lattice theory by Flory [15], the contribution of each component in the determination of the total entropy change of mixing (∆Smix) is equal to −gh < ln # . Where kB is

the Boltzmann constant, and # is the volume fraction of each component in the system (solvent

(#i ), polymer (# ), clay(#- )).

The enthalpy change of mixing (∆Hmix) for a polymer/clay nanocomposite hydrogels following the procedure described in the lattice theory could be determined by introducing a new parameter as Ψ (see Equation 15). This parameter specifies the total available contacts for segments of a clay layer with adjacent cells in a cubic lattice, according to appendix A in the supplementary data file. j = k ∗ F8 2

i )1p

−2

i

n2 'n) 'no n2 n) no

+

W &o i )T − n n n H 2 ) o

&

n2 n) no

N1p" − 2

i )1p

−2

i)

+ 1p" − 2

i )1p

−2

i) +

1p −

(15)

Ds is the solvent molecule diameter, and L1, L2, and L3 are the length of a clay layer in three directions of the cartesian coordinates. Therefore, ∆Hmix is finally written as Equation 16. ∆:r s = gh > F<" # t" +

u 1<" #- t" vwo

+ < #- t )H

(16)

Where t" , t" , and t are the solvent-polymer, solvent-clay, and polymer-clay interaction

parameters, respectively. An appropriate expression for the mixing free energy of polymer/clay nanocomposite hydrogels could be obtained by substitution of the number of polymer molecules and clay layer equal zero owing to the absence of individual polymer molecules and clay layers in the network structure [15]. Finally, the mixing-free energy change of a polymer/clay nanocomposite hydrogels could be written as Equation 17, according to appendix B in the supplementary data file. ∆xr s = gh > F<" ln #i + <" # t" +

u

vwo

1<" #- t" )H

11

(17)

The chemical potential change (∆µ) of a polymer/clay nanocomposite hydrogel could be obtained using the derivative of the total free energy change with respect to the moles of solvent (ns) at constant T, P, and moles of polymer and clay [38]. Furthermore, the total osmotic pressure of the system could be evaluated using the chemical potential change of the system as Equation 18 [39, 15]. c =−

∆y z&

= cr s + c{| + c }-

(18)

The molar volume of the solvent is expressed as Vs in Equation 18. The elastic contribution of the osmotic pressure for the Gaussian chain is evaluated using the phantom network model [40, 41]. The ionic contribution of free energy in a neutral polymer/clay nanocomposite hydrogel may be originated from two different parts. The first contribution is arisen from the ionic nature of clay layers due to the presence of ionic groups on their surfaces. The second contribution may be occurred due to the mobile ionic concentration difference in the nanocomposite hydrogel and the surrounding solution, which indirectly depends on the chemical potential of ion species (see Equation 1). Donnan theory was used to derive an expression for describing the ionic contribution of clay layers in total ionic-free energy and chemical potential of nanocomposite hydrogel [15]. According to this theory, the ionic contributions of the osmotic pressure of polymer/clay nanocomposite hydrogel could be interpreted by Equation 19. |{ c }- = c }- + c r}• = −=> €E }|~w

•.

‚ƒ,.

L #- + ∑QR"1 ̅ − )…

(19)

Where, ρn and Mw,n are the density and molecular weight of clay layers, respectively. Also, ̅ is the mobile ions concentration in the surrounding solution. In pH-sensitive nanocomposite hydrogel, the ionic nature of polymer chains (anionic or cationic) could affect the osmotic pressure [42]. For anionic (if pH>pKa) or cationic (if pH
12

that neutralize the ionized groups on the polymer chains of the hydrogel (see Equation 9), will be added to Equation 19, according to the Donnan theory. The counterion contribution of the osmotic pressure could be written according to Equation 20.    c0    f RT   For anionic nanocompos ite hydrogels (if pH > pKa)  ( pKa − pH ) )   H (1 + 10    π counter =  ion    c0    f RT   For cationic nanocompos ite hydrogels (if pH < pKa)    H (1 + 10 ( pH − pKa ) )    

(20)

The pH and pKa are the negative logarithms of the local hydrogen ion concentration (

' 9)

in the

hydrogel and dissociation constant (Ka) of ionizable groups on the polymer chain, respectively. Thus, the total osmotic pressure for isotropic swelling of anionic (if pH>pKa) or cationic (if pH € "

!"

ln #i + 11 + t" 11 − #i )# + 11 +

z&

# L − E . L #- + ∑QR"1 − ̅ ) + ‚ •

ƒ,.

ŠŒ‹•.Žˆ• ‹.

…

u‡2o vwo

11 − #i )#- − 1 −

Sˆ

‰{ EN# T

2 o

N#+ T

) o

−

(21)

In Equation 21, ve and fe are the effective cross-link density and the average functionality parameter of the network, respectively. The effective cross-link density (ve) could be evaluated using calculated average molecular weight between cross-links via the suggested equation by Flory and Rehner [43]. For the evaluation of the clay nanoparticles effect as a cross-linking agent on the functionality parameter of the network, the average functionality parameter in the presence of clay nanoparticles can be estimated from the molecular weight of the clay nanoparticles (•‘,- ), the volume fraction of cross-linked polymer (# + ), and the clay nanoparticles concentration (Cn) as Equation 22 [41, 44]. ’{ = 2“{ # +

‚ƒ,. ”.

(22)

13

For the determination of the material properties tensor (see Equation 14), Young’s modulus of nanocomposite hydrogel is needed. It could be expressed using the rule of the mixture as Equation 23. •9 = •- #- + •i #i

(23)

Where EH, Enc, and Es are Young’s modulus of nanocomposite hydrogel, polymer/clay nanocomposite, and solvent, respectively. Also, #- is the volume fraction of polymer/clay nanocomposite hydrogel. Due to the assumption of the zero modulus for the solvent, the term •i #i eliminates from Equation 23.

In polymer/clay nanocomposites, the interphase region has an important role in mechanical properties determination. Experimental assessment of modulus of the interphase region in nanocomposites is almost impossible [45, 46]. Thus, in this study to have a realistic estimation of Young’s modulus, the Ji model (Equation 24) was used for Young’s modulus estimation of polymer/clay nanocomposite [47]. According to the literature, the Ji model can suggest accurate values for Young’s modulus of polymer/clay nanocomposites [45, 46, 48-51]. •- = •–— ˜11 − ) + = ¤ 2 n + 1 #S n

1™!š)

›1œ•2) žŸ œ

1"!™)'

+ 1›•

š

)1œ•2) ¡ 'š . ) ¡ ¢

£

!"

(25)

¥ = ¦#S g =

(24)

(26)

§ §¨¢

(27)

Where, Ei, Li, and φf are the interphase modulus, interphase thickness, and the volume fraction of clay in the polymer/clay system, respectively. Also, Epg, En, and Enc are Young’s modulus of polymeric gel without water, clay nanolayers, and polymer/clay nanocomposite, respectively. The interphase modulus depends on the distance from the clay nanolayers. So, the mean 14

interphase modulus for polymer /clay nanocomposite was calculated according to Equation 28 [45]. • =

n " © •ª n !nS nS

n •ª 1p) = •–— + n

1p)«p n !n

n !nS

(28) . )

E•- − •

n —n

¬

L

(29)

Where n, L, and Lf are the interfacial enhancement index, variable distance, and mean clay nanolayers width, respectively. Initial Young’s modulus of polymeric gel (Epg), with the phantom network assumption, is related to the gel cross-link density (ve), the average junction functionality parameter of the network (fe), the volume fraction of cross-linked polymer (# + ), Poisson’s ratio (vpr), and the absolute temperature of the gel according to Equation 30 [41, 52]. •

—

2.3.

= 2N1 + “ % T 1 −

Sˆ

=>“{ # +

(30)

Energy conservation law

The energy conservation law for the nanocomposite hydrogel (Equation 31) must be written for accounting the effect of the temperature change on the electric potential, ion concentrations, and mechanical force coupled effects for the transient swelling prediction of the temperaturesensitive nanocomposite hydrogel. ]-

= ∇. 1® {SS ∇>)

(31)

In Equation 31, the convection term is neglected due to the assumption of low velocity of fluid diffusion. The density (ρ) and heat capacity (Cp) of nanocomposite hydrogel could be calculated using the rule of mixture. Keff is the effective thermal conductivity coefficient of nanocomposite hydrogel, which could be determined using Equation 32. ® {SS = 11 − #- )® i + #- ® -

(32) 15

The thermal conductivity coefficient of polymer/clay nanocomposite was determined according to the MG-EMA model as Equation 33 [53]. This model can satisfactorily predict the thermal conductivity coefficient of a low range of layered silicates into the polymer matrix in which the layers randomly dispersed into the matrix [53]. ¯ .Œ ¯

=

g² =

'0¬ N P° 'P " T ! 0¬ P °

¯. !¯ ¯. '¯





(33) (34)

g " = ® - − ®

(35)

In Equations 32-35, Ks, Kp, Kn, and Knc are the thermal conductivity coefficient of solvent, polymer, clay nanolayers, and polymer/clay nanocomposite, respectively. 3. Numerical solution We perform a MATLAB code for the two-dimensional finite element solution of model equations with moving boundary conditions to consider the correctness of model prediction of transient swelling of the polymer/clay nanocomposite hydrogel. The finite element solving method at Lagrangian configuration with re-meshing geometry due to solvent permeation and hydrogel deformation coupled problem has been chosen for transient swelling simulation. Triangular elements were used to approximate the geometry. The decoupled formulation was used to solve time-dependent model equations in which the time and spatial coordinates are assumed to be separable. Galerkin method was employed for the generation of the weak form of the model equations. Additionally, Galerkin’s α-family approximation scheme was used for firstorder time derivative approximation in mass and energy conservation equations (see Equations 7 and 31). Also, the Newmark family approximation scheme with Galerkin parameters was used for second-order time derivative approximation in the mechanical force balance equation (see Equation 14) [54]. The discretized coupled model equations solved in each selected time step. 16

Then, the results were used in the next time step until the equilibrium condition for attaining acceptable simulation results for transient swelling. As a result of solving the ionic conservation of mass and energy conservation with defined initial and boundary conditions for the first time step, the ions concentration and temperature profiles in the nanocomposite hydrogel were determined. The results were used for the determination of the osmotic pressure profile at the first time step. The resulted osmotic pressure in each nodal points was used for the determination of nodal displacement by solving the mechanical force balance equation. Due to applying the displacement into the nodal points at the first time step, new geometry with new volume was generated. Then, all variables consist of the component volume fractions, effective diffusion coefficient, density, effective heat capacity, effective thermal conductivity coefficient, hydration, and Young’s modulus of nanocomposite hydrogel were updated and used for next time step. After re-meshing the new geometry, the new ions concentration and temperature at each nodal point were used as initial conditions for solving model equations at the next time step. For comparison of the experimental and model results, the free swelling of immersed cubic and spherical gels in a solution bath with specified pH and temperature were evaluated. Also, a junction was set for consideration of the cylindrical gels swelling. The 2D nanocomposite hydrogel geometry was discretized with higher mesh density nearby the interfaces among the external solution and the gel boundaries. Initial and boundary conditions for ionic conservation of mass (Equation 1), mechanical force balance (Equation 14), and energy conservation law (Equation 31) are presented in Table 1. Table 1 Where T0 is the initial temperature of hydrogel (ambient temperature=296 K), and pH0 is the pH of PVA/PAA/MMT solution before physical cross-linking (pH0 = 4). TE and n are the external

17

solution temperature and the normal vector of the surface, respectively. The nodal displacement in the x-direction (u) and y-direction (v) were set to zero at the center point of geometry to prevent rigid body motion, while other nodes are allowed to swell. 4. Experimental A set of experiments were conducted for free and constrained swelling using the cubic, spherical, and cylindrical shape samples to prove the accuracy of the developed model for the deformation prediction of the nanocomposite hydrogel. Dual stimuli (temperature and pH) sensitive interpenetrating polymer network (IPN) nanocomposite hydrogels composed of poly (vinyl alcohol) (PVA: Mn~74800, 98%< hydrolyzed; Nippon Gohsei, Japan), poly (acrylic acid) (PAA: Mv~1250000, pKa=4.6; Sigma-Aldrich, USA) ), and 0, 4, 8, and 12 wt.% of Na-MMT (Cloisite Na+, CEC= 92.6 meq/100g; Southern Clay Products Inc, USA ) based on the mass of polymer and Na-MMT in water were prepared using the cyclic freezing-thawing method. In summary, an adequate amount of MMT was added to distilled water then placed under gentle stirring for 12 hr. PAA and PVA were gradually added to the MMT suspension under mechanical stirring for 4 hr at 90°C to attain complete dissolution. The resultant poured into silicone molds (sphere, cube, and cylinder shapes) and placed at -19°C for 24 hr to induce crystallization and gel formation. Subsequently, the thawing process was performed at room temperature for 3 hr. This procedure was repeated three times for each sample [55]. Then, the samples in the frozen state were removed from the molds and dried at refrigerator temperature (4±1°C) for one week to keep the shape of the mold. Free and constrained swelling tests were conducted in the aqueous solution with adjusted pH (2 and 7 using the diluted citrate-NaOH-HCl and phosphate buffer solutions, respectively) at different temperatures (296, 310 and 328 K). The experimental results (the swelling ratio as a quantitative measure of the ability of samples to absorb water (Equation 36)

18

and the geometry deformation) were compared to the model results to validate the numerical solution of the proposed model. ³ =

rŽ

r

(36)

In Equation 36, qt, m0, and mt are the swelling ratio, the initial mass of dried hydrogel, and mass of hydrogel at time t after immersing in solution with a specified pH and temperature. In this study, it is assumed that the nanocomposite hydrogel is isotropic and homogenous material. It could be assumed if the MMT-nanoparticles disperse homogenously and fully exfoliated with random orientation in the hydrogel. Therefore, for evaluation of MMTnanoparticles dispersion in the hydrogel, X-ray diffraction patterns were obtained using a Philips diffractometer (PW1710, Netherlands). Also, transmission electron microscopy micrographs were recorded with a Philips TEM apparatus (EM 208 S, Netherlands) operating at 100 kV tension. PVA/PAA/MMT nanocomposite hydrogel (VAMx) components properties that were used as input data are listed in Table 2. It should be noted that the “x” letter of VAMx represents the MMT wt.% in the PVA/PAA/MMT nanocomposite hydrogels. Citrate-NaOH-HCl and phosphate buffer solutions are used for adjusting the pH of the external solution at 2 and 7, respectively. So, the diffusion coefficient of H+, OH-, Na+, Cl, C6H5O7-3, and HPO4-2 into the water are set as 9.3e-9, 5.27e-9, 1.33e-9, 2.03e-9, 5.9e-10, and 8.78e-10 m2/s, respectively [8, 56, 57]. Also, the interaction parameter of solvent -polymer (χ12) and solvent- MMT (χ13) as a function of temperature were determined using Equation 37 [58]. t ´ 1>) =

z&

1µ − µ´ ) + 0.34

(37)

In which δi is the solubility parameter of component i. The solubility parameter of MMT was calculated using the Beerbower equation (Equation 38) [59] by knowing the surface energy of MMT. According to Helmy et al., the surface energy of MMT is 205.066 (mJ/m2) [60]. 19

Ai = 0.07147µ

2

1 ) ¹- o

(38)

In Equation 38, γs and Vn are the surface energy and molar volume of MMT nanoparticles. Table 2 5. Results and discussion 5.1. Morphology

The XRD patterns of as-received Na-MMT-nanoparticles and PVA/PAA/MMT nanocomposite hydrogels (VAM4, VAM8, and VAM12) are presented in Fig.1a. The XRD pattern of Na-MMT indicates a strong diffraction peak at 2θ = 7.6° that corresponds to the d-spacing equal to 11.61 Å. Fig. 1 The disappearing of the characteristic peak of MMT in the nanocomposite hydrogels XRD patterns implies the full exfoliation of MMT-nanoparticles in all nanocomposite hydrogels. The TEM graph of the VAM0 hydrogel (Fig.1b) demonstrates that the network with tetrafunctional junctions as the dominate junctions is formed due to the freezing-thawing cycles of the PVA/PAA solution. The average functionality parameter of the network is a measure of the number of sub-chains connected at a junction point of the hydrogel network [44]. Assuming a phantom network with tetra-functional junctions, fe for the VAM0 hydrogel is equal to 4. TEM graphs of the VAM8 nanocomposite hydrogel are illustrated in Fig.1c and 1d. The dark shadows in Fig. 1c corresponds to good dispersion and random orientation of MMT-nanoparticles within the nanocomposite hydrogel (VAM8). Fig.1d represents both exfoliated and intercalated MMT structures in the nanocomposite hydrogel. Therefore, the assumption of isotropic and homogeneity of nanocomposite hydrogels is reasonable. 5.2. Prediction of Transient Swelling Behaviour

20

The main objective of this section is to predict the deformation and swelling ratio of the PVA/PAA/MMT nanocomposite hydrogel samples during the water absorption and swelling process using a numerical solution of developed model with finite element method. The numerical results of PAA/PVA/MMT nanocomposite hydrogel swelling at different MMT content (0, 4, 8, and 12 wt.% based on the mass of the solids) and different swelling conditions (pH and temperature) are provided in this section. Consequently, the influence of variation of the swelling ratio, component volume fractions, effective diffusion coefficient, density, effective heat capacity, effective thermal conductivity coefficient and Young’s modulus on the nanocomposite hydrogel deformation was considered. During the swelling process of dual stimuli (temperature and pH) sensitive nanocomposite hydrogels in an external aqueous solution with adjusted pH using buffer solution and temperature control of the external solution, the swelling intensity may differ in different parts of the gel due to different temperature and ions concentration in the hydrogel before equilibrium. Consider an anionic pH-sensitive hydrogel immersed in a water bath with a specified pH using a buffer solution. At the initial time, the hydrogen ion concentration in the hydrogel is 10-4 (the pH of PVA/PAA/MMT solution before freezing-thawing was equal 4). When pH at a node becomes more than pKa due to the transfer of the hydrogen ion into the hydrogel, the ionizable pending groups in that node will be ionized. Then, the counterions diffuse into the hydrogel for neutralization of the ionized pending groups. }†- {% }†- {% ) will affect the swelling, while c }for So, the counterions part of osmotic pressure (c }-

nodes that their pH is less than pKa is zero. Also, the temperature difference among the different parts of hydrogel before equilibrium cause difference in the nodal osmotic pressures. Therefore, the swelling intensity is different in various parts of the hydrogel before reaching the equilibrium conditions. Fig. 2 represents the effect of the presence of nanoparticles in the nanocomposite

21

hydrogels on the predicted swelling ratio with the proposed developed model. The overall trend, in the absence or presence of MMT- nanoparticles, was a rapid increase in the swelling ratio at the beginning time of swelling followed by a slow increase in the swelling ratio after the intermediate time of swelling. As shown in Fig. 2, the swelling ratio of the hydrogels containing different amounts of MMT-nanoparticles was less than the net hydrogel (VAM0). As the nanoparticles content of the hydrogel increases, the gel becomes less swollen. Fig. 2 By incorporation of the nanoparticles in the hydrogels, the effective diffusion coefficient, density, effective heat capacity, effective thermal conductivity coefficient, and hydration of hydrogel were affected. Consequently, the mean osmotic pressure and Young’s modulus of hydrogel that are determinative parameters for the deformation of the system will be affected. Osmotic pressure is the main driving force for swelling of hydrogels, while the mechanical strength of hydrogel restricts the deformation of a hydrogel. Fig. 3 plots the mean local hydration (volume of interstitial fluid /volume of solid polymer) and the mean Young’s modulus change due to the presence of nanoparticles in the hydrogels during the swelling process. From the plots, which are shown in Fig. 3, we see the mean local hydration increased, while the mean Young’s modulus decreased during the swelling process. The reduction of the mean Young’s modulus during swelling is the result of hydrogel hydration and reduction of polymer volume fraction in hydrogel due to the swelling. Fig. 3 As showed in Fig. 3, the mechanical strength of hydrogel was increased with the incorporation of MMT-nanoparticles that could be related to the high modulus of MMT-nanoparticles [66] and their cross-linking effect on the hydrogel preparation [44, 67]. The cross-linking effect of MMT-

22

nanoparticles on the initial Young’s modulus of the nanocomposite hydrogels (see Equation 30) was applied with the increase of the average junction functionality parameter (see Equation 22) due to the creation of the additional effective junction points in compare with the net hydrogel. The model prediction of the cubic shape hydrogel (VAM0) swelling ratio and nanocomposite hydrogel (VAM12) at the free swelling conditions are compared with the experimental results in Fig. 4. We can see that there is good agreement among experimental results and model prediction. At the beginning of the swelling process, due to the lower polymer volume fraction and the reduction in the mixing part of osmotic pressure (πmix) with considering the presence of MMT nanoparticles, the mean osmotic pressure of the net hydrogel is higher than the mean osmotic pressure of the nanocomposite hydrogel (Fig. 4). At the intermediate times of the swelling process, the mean osmotic pressure of nanocomposite hydrogel is higher than the net hydrogel. It could be attributed to the slower ions diffusion in the presence of the nanoparticles [68] and subsequently increase of the ionic part of osmotic pressure (πion) in the nanocomposite hydrogel. Also, the higher mechanical strength of hydrogel affects the hydration and volume fraction of components, which directly affect the mixing and elastic parts of osmotic pressure. Fig. 4 The swelling ratio of PVA/PAA/MMT nanocomposite hydrogel (VAM12) at two different pH levels (above and below pKa of poly(acrylic acid)) and constant temperature (TE=296 K) were compared for evaluation of the pH sensitivity and accuracy of the model prediction in Fig. 5. The carboxyl groups of poly(acrylic acid) as an anionic polymer, become ionized at pH values above its pKa. Therefore, the counterions (from the aqueous surrounding solution) neutralize the ionized carboxyl groups of poly(acrylic acid) chains in the hydrogel and affect the osmotic

23

pressure [42]. As seen, the predicted swelling ratio of the nanocomposite hydrogel is affected by the pH level of the external solution. As shown in Fig. 5, the experimentally measured swelling ratio of the PVA/PAA/MMT nanocomposite hydrogel (VAM12) at pH=7 is much higher than its swelling ration at pH=2. The results showed that the model has a good prediction of the swelling ration and great sensitivity to the pH as the external stimulus. The pH and temperature sensitivity of the PVA/PAA interpenetrating network (IPN) was reported in the previous studies [69, 70]. The pH value directly affects the osmotic pressure and indirectly affects the component volume fractions, effective diffusion coefficient, density, effective heat capacity, effective thermal conductivity coefficient, hydration, and Young’s modulus of the hydrogel at each time step. Also, Fig. 5 demonstrates the simulation results that successfully revealed the effect of external solution temperature change on the swelling ratio of PVA/PAA nanocomposite hydrogel (VAM8) at constant pH=7. The results showed that the swelling ratio of the PVA/PAA nanocomposite hydrogel increased by increasing the temperature of the external solution. This behaviour was reported for PVA/PAA hydrogels in the prior researches [69-71]. The temperature directly affects the osmotic pressure and indirectly affects the other variables that influence the hydrogel swelling. Due to the consideration of the effect of temperature change on the osmotic pressure and the hydrogel properties, the model senses the temperature change as an external stimulus. Fig. 5 To evaluate the correctness of deformation prediction of the proposed developed model, the experimental and model prediction results of deformation at free swelling conditions were compared. Fig. 6 showed the experimental and model prediction results of transient swelling for PVA/PAA nanocomposite hydrogel (VAM8) at pH=7 and TE=310 k, and different times.

24

This figure clearly shows that the deformation of the PVA/PAA nanocomposite hydrogels depends on the initial geometry of samples. After a while (about 60 minutes), the initially square geometry of the sample (2D view) is no longer square, which is due to the faster swelling near the corners at the result of the lower mechanical strength of hydrogel in these areas. The lower mechanical strength of hydrogel near the corners is due to the higher hydration and lower polymer volume fraction in these regions because of the deformation from two sides in the corners. As showed in Fig. 6, the initial circle geometry of the sample (2D view) is not changed during the transient swelling of the nanocomposite hydrogel (VAM8), which could be related to the absence of the edges in the sample with circle geometry. The same trends for deformation of initially square and circular geometries were observed during the transient swelling in the previous researches [72, 73]. Fig. 6 Fig. 7 illustrates snapshots of the numerical solution results of the deformation, hydrogen ions concentration, and temperature of the nanocomposite hydrogel (VAM4) as it is being contacted with the aqueous solution with different pH, and temperature at time 60 and 120 min. The effect of external solution temperature change from 296 to 328 K on the deformation of VAM4 at constant pH (snapshot at time= 60 min) could be observed form the comparison of color bars of Fig. 7a1 and 7b1. The comparison of Fig. 7a1 and 7b1 revealed that the deformation trend of the samples was the same at 296 and 328 K. The deformation was higher at 328K in comparison with 296K, which is related to the effect of the increase in the external solution temperature in the increase of the osmotic pressure. The comparison of snapshots in Fig. 7b1 and 7c1 shows the effect of pH change from 2 to 7 on the deformation of VAM4 at a constant temperature at time 60 min. It is demonstrated that the deformation of the anionic nanocomposite hydrogel (VAM4)

25

at pH=7 is higher than pH=2. It is due to the effect of the counterions presence for the neutralization of the ionized carboxyl groups of poly (acrylic acid) in the increase of the osmotic pressure. Also, the comparison of snapshots in Fig. 7c1 and 7d1 indicates the effect of time on the deformation of VAM4. It is showed that the deformation at time 120 min is lower than time 60 min, which is due to the lower osmotic pressure at time 120 minutes compared to time 60 minutes. The lower osmotic pressure is the result of the lower ions concentration difference and polymer volume fraction at time 120 minutes compared to time 60 minutes. The ions concentration difference and polymer volume fraction affect the ionic and mixing contribution of osmotic pressure, respectively. The difference between VAM4 deformation due to change of temperature and pH are the results of the coupling between the deformation, ions diffusion, and temperature. The effect of the temperature and pH changes on the hydrogen ions diffusion indicated in Fig. 7a2, 7b2, and 7c2. The effect of the external solution pH change on the hydrogen ions diffusion could be observed due to the comparison snapshots of Fig. 7b2 and 7c2, while the effect of the external solution temperature change on the hydrogen ions diffusion is not quite evident in snapshots of Fig. 7a2 and b2. The comparison of Fig. 7b2 and 7c2 revealed that the hydrogen ions concentration profiles are quite different at various pH. It is due to the higher concentration of hydrogen ions in the external solution with pH=2 than the initial concentration of hydrogen ions in the hydrogel (pH0=4) and the lower concentration of hydrogen ions in the external solution with pH=7 from the initial concentration of hydrogen ions in the hydrogel. The temperature profiles of VAM4 at different external solution temperature and pH are showed on snapshots of Fig. 7a3, 7b3, and 7c3. Fig. 7a3 shows the uniform temperature of the hydrogel at time 60 min, which is due to the same initial temperature of the VAM4 and external solution temperature. The difference among the temperature profiles of VAM4 at pH=7 and pH=2 could

26

be attributed to the higher volume fraction of water due to the higher swelling ratio of the VAM4 at pH=7 and the higher thermal conductivity coefficient of water in comparison to the polymers. Fig. 7 The ability of the developed model to capture the deformation, temperature, and ions concentration changes of the nanocomposite hydrogel across time is evident with the comparison of the Fig.7c (1-3) with 7d (1-3). The change of the nanocomposite hydrogel properties during transient swelling causes the difference between evaluated deformation, ions concentrations, and temperature profiles at each time step. It should be noted that calculation of the deformation, ions concentrations, and temperature profiles using new hydrogel properties and initial conditions determined in the previous time step, results in the successful prediction of the developed model. The swelling of cylindrical PVA/PAA hydrogel, which placed at a junction in contact with a stationary aqueous solution at defined pH and temperature, was considered for confirmation of the accuracy of deformation prediction of the proposed model at the constrained transient swelling condition. Fig. 8a illustrates a schematic representation of the constrained swelling of a cylindrical hydrogel sample. Also, Fig. 8b shows the constrained transient swelling of cylindrical PVA/PAA hydrogel (VAM0) during the water uptake process. Fig. 8 Fig. 8b demonstrates the displacements and the occurrence of hydrogel (VAM0) contacts with the junction walls during the transient swelling process. The displacements in the normal direction of the walls become zero when the hydrogel contacts with the walls, while the hydrogel slips in parallel with the wall surface. The comparison of the experimental and model-predicted results of the hydrogel deformation in Fig. 8b showed that the developed model had an appropriate prediction of the geometry change at the constrained swelling condition. The

27

accuracy of the model prediction of the deformation was evaluated using the comparison of the surface area of the experimental and model-predicted geometry. The boundary nodes coordinate of the experimentally measured geometries were determined via an image analysis program, and the surface areas of the experimental and model-predicted geometries were calculated using the developed MATLAB code using the boundary nodes coordinate. The surface area difference percentage was considered as a criterion for the accuracy of the deformation prediction (Fig. 9). As showed in Fig. 9, the simulation results of free swelling in Fig. 6 showed good agreement in the size and shape of nanocomposite hydrogel with experimental results (deviation < 8.46%). The deformation prediction of constrained swelling (Fig. 8b) had lower accuracy in comparison to free swelling (deviation < 10.45%). The lower accuracy of constrained deformation prediction is due to neglection of the friction force between the hydrogel and the junction during swelling. Fig. 9 6. Conclusions This study introduced a new parameter (Ψ) for consideration of the effect of clay nanolayers on mixing part of osmotic pressure due to the swelling of polymer/clay nanocomposite hydrogels. It is showed that the transient swelling behaviours of pH and temperature-sensitive nanocomposite hydrogels could be predicted using the proposed model. It is necessary to update the nanocomposite hydrogel properties (including the ions diffusion coefficients, thermal conductivity coefficient, heat capacity, osmotic pressure, and Young’s modulus) and using new initial conditions at each time step from previous time step in the numerical solution to obtain an accurate prediction. We demonstrated that we could have a good prediction of transient swelling behaviour and deformation of the nanocomposite hydrogels due to applying the effect of nanoparticles on the hydrogel properties, updating the nanocomposite hydrogel properties during

28

the swelling, consideration of the effect of hydrogen ions concentration change in pending groups dissociation and the temperature change on the osmotic pressure determination. We also show by applying the effect of counterions diffusion for neutralization of the dissociated pending groups on osmotic pressure at each nodal point, the pH dependence of transient swelling could be considered. Due to well agreement of the simulation and experimental results, the proposed model could be employed to design and analyze nanocomposite hydrogels with enhanced properties for many applications such as lab-on-chip, microfluidic controller valves, contact lenses, biosensors, and drug delivery.

Acknowledgment The authors would like to thank Tarbiat Modares University and Iran Nanotechnology Initiative Council (INIC) for their supports.

Declaration of interest The authors declare that they have no known competing for financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary data (Appendices A and B)

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39

Table 1 Initial and boundary conditions of model equations Table 2 The nanocomposite hydrogel components properties used as input data for the proposed model

40

Fig. 1. (a) XRD patterns of MMT nanolayers and PVA/PAA/MMT nanocomposite hydrogels containing 4, 8, and 12 wt.% MMT; (b) TEM micrograph of the net hydrogel (VAM0) showing the functionality parameter of the network, (c and d) TEM micrographs showing exfoliated (e) /intercalated (i) MMT structure in the nanocomposite hydrogel (VAM8). Fig. 2. Effect of MMT content in the model predicted swelling ratio of the PVA/PAA/MMT nanocomposite hydrogels (VAMx) at TE=296 and pH=2. Fig. 3. Effect of MMT content in the hydration and Young’s modulus during swelling of the PVA/PAA/MMT nanocomposite hydrogels (VAM0 & VAM12) at pH=2 and TE=296 K. Fig. 4. Comparison of the experimental results and model prediction of the swelling ratio for VAM0 and VAM12 at TE=296 & pH=2. Fig. 5. Effect of pH level on the swelling ratio of VAM12 at TE=296 K and effect of temperature on the swelling ratio of VAM8 at pH=7. Fig. 6. Typical transient swelling evolution for a cubic and spherical nanocomposite hydrogels (VAM8) immersed in an aqueous solution (pH= 7, TE=310 K). Fig. 7. The snapshots of the hydrogel nanocomposite (VAM4) deformation (a1, b1, c1, and d1), hydrogen ions concentration (a2, b2, c2, and d2) and temperature (a3, b3, c3, and d3) immersed in aqueous solutions at different pH and temperature and times. Fig. 8. Constrained transient swelling for a cylindrical hydrogel: a) Schematic representation; b) Experimental and model-predicted results for the hydrogel (VAM0) in pH=7 and TE=310 K at different times. Fig. 9. The accuracy of the experimental and model-predicted deformation of cubic and spherical nanocomposite hydrogels for free swelling and cylindrical hydrogel for constrained swelling (pH= 7, TE=310 K).

41

Table 1 Initial and boundary conditions of model equations Mass conservation equation (Equation 1) Initial conditions Boundary conditions Ion Con. (mol/lit) Ion Con. (mol/lit) pH=2 + ! 9 H 10-2 9 º = 10 !1"W! 9 ) OH 10-12 »9 • = 10 Cl3.9×10-4 ”| • = 0 Na+ 4×10-4 Q~º = 0 -2 HPO4 ---¼½¾¿•) =0 -3 C H O =0 3.337×10-3 6 5 7 ÀY ¼Z ¾•o 3 Mechanical force balance equation (Equation 14) Initial conditions Boundary conditions ÂÁ Á=0 =0 Âà Ä. 1-• − c [) = 0 Å “=0 =0 Energy conservation equation (Equation 31) Initial conditions Boundary conditions > = >+ > = >§

42

pH=7 10-7 10-7 6×10-4 1.2×10-3 3.0005×10-4 ----

Table 2 The nanocomposite hydrogel components properties used as input data for the proposed model Property

Unit

PVA

PAA

Water

Na-MMT

Specific heat capacity

[J/(m3.k)]

1620[61]

1490[61]

4180[62]

799[63]

Density

[kg/m3]

1.25e3

1.22e3

1e3

2.86e3

Number average molecular weight

[kg/mol]

78

81

18.016e-3

540.46e-3

Thermal conductivity coefficient

[W/(m.K)]

0.31[61]

0.37[61]

0.614[64]

0.1

Young’s modulus

Pa

-

-

0

178e9[65]

43

Figure 1. (a) XRD patterns of MMT nanolayers and PVA/PAA/MMT nanocomposite hydrogels containing 4, 8, and 12 wt.% MMT; (b) TEM micrograph of the net hydrogel (VAM0) showing the functionality parameter of the network, (c and d) TEM micrographs showing exfoliated (e) /intercalated (i) MMT structure in the nanocomposite hydrogel (VAM8).

1

Figure 2. Effect of MMT content in the model predicted swelling ratio of the PVA/PAA/MMT nanocomposite hydrogels (VAMx) at TE=296 and pH=2.

1

Figure 3. Effect of MMT content in the hydration and Young’s modulus during swelling of the PVA/PAA/MMT nanocomposite hydrogels (VAM0 & VAM12) at pH=2 and TE=296 K.

1

Figure 4. Comparison of the experimental results and model prediction of the swelling ratio for VAM0 and VAM12 at TE=296 & pH=2.

1

Figure 5. Effect of pH level on the swelling ratio of VAM12 at TE=296 K and effect of temperature on the swelling ratio of VAM8 at pH=7.

1

Figure 6. Typical transient swelling evolution for a cubic and spherical nanocomposite hydrogels (VAM8) immersed in an aqueous solution (pH= 7, TE=310 K).

1

Figure 7. The snapshots of the hydrogel nanocomposite (VAM4) deformation (a1, b1, c1, and d1), hydrogen ions concentration (a2, b2, c2, and d2) and temperature (a3, b3, c3, and d3) immersed in aqueous solutions at different pH and temperature and times.

1

Figure 8. Constrained transient swelling for a cylindrical hydrogel: a) Schematic representation; b) Experimental and model-predicted results for the hydrogel (VAM0) in pH=7 and TE=310 K at different times.

1

2

Figure 9. The accuracy of the experimental and model-predicted deformation of cubic and spherical nanocomposite hydrogels for free swelling and cylindrical hydrogel for constrained swelling (pH= 7, TE=310 K).

1

Highlights: •

The proposed model satisfactorily predicted the transient swelling of pH and temperature-sensitive polymer/clay nanocomposite hydrogels.

•

A new parameter was introduced to consider the effect of clay nanolayer in osmotic pressure.

•

Two scenarios of swelling (free and constrained swelling) are revealed by numerical simulation.

•

Material properties of sensitive polymer/clay nanocomposite hydrogels changed due to the swelling process.

The authors declare that they have no competing and conflict of interest.