Development of a Multiphysics Model to Characterize the Responsive

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Development of a Multiphysics Model to Characterize the Responsive Behavior of Magnetic-Sensitive Hydrogels with Finite Deformation Qimin Liu, Hua Li, and Khin Yong Lam J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 12 May 2017 Downloaded from http://pubs.acs.org on May 15, 2017

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The Journal of Physical Chemistry

Development of a Multiphysics Model to Characterize the Responsive Behavior of Magnetic-Sensitive Hydrogels with Finite Deformation Qimin Liu, Hua Li* and K. Y. Lam School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore ABSTRACT: A novel multiphysics model is developed in this paper for simulation of the responsive behavior of the magnetic-sensitive hydrogel, with the effects of magnetochemo-mechanical coupled fields, which is termed the multi-effect-coupling magneticstimulus (MECm) model. In this work, the magnetic susceptibility for magnetization of the general magnetic hydrogel is defined as a function of finite deformation, instead of a constant for an ideal magnetic hydrogel. The present constitutive equations, formulated by the second law of thermodynamics, accounts for the effects of the chemical potential, the externally applied magnetic field, and the finite deformation. In particular, a novel free energy density is proposed with consideration of the magnetic effect associated with finite deformation, instead of volume fraction. After examination with published experimental data, it is confirmed that the MECm model can capture well the responsive behavior of the magnetic hydrogel, including the deformation and its instability and hysteresis under a uniform or non-uniform magnetic field. The parameter studies are then carried out for influences of the magnetic and geometric properties, including the magnetic intensity, shear modulus, and volume fraction of the magnetic particles, on the behavior of the magnetic hydrogel, for a deeper insight into the fundamental mechanism of the magnetic hydrogels.

*

Corresponding author, Tel.: + 65 6790 4953; Fax: +65 6792 4062 Email address: [email protected] (Hua Li)

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1. Introduction As a novel smart material, magnetic hydrogel is composed of a soft polymeric matrix, in which water and magnetic particles are filled at micro- or nano- level.1,2 Subjected to an externally imposed magnetic field, the magnetic hydrogel undergoes a finite deformation. Recently the magnetic hydrogel attracts considerable attention due to its unique characteristics, including the fast response, low friction, biocompatibility, and remote actuation.3 Therefore, the magnetic hydrogel is used for wide-range applications, such as chemical adsorbent,4 drug delivery,5 and microfluidic valve.6 However, fundamental understanding of the magnetic hydrogel is still preliminary. In particular, most relevant investigations were experiment-based and thus time-consuming, such that they may not fully capture all the effects on the behavior of the magnetic hydrogel. As an alternative option, theoretical modeling is expected to predict well the responsive characteristics of the magnetic hydrogel, but most of the reported models focused on the magnetic elastomers that were placed in air environment, and assumed an incompressible elastomer with constant volume.7-9 However, the model developed in this paper focuses on the deformable hydrogel placed in a solvent environment, where the hydrogel imbibes large amounts of solvent, which is characterized by a chemical field, and significant volume change subsequently characterized by a mechanical equilibrium field.

A literature review reveals that modeling of the magnetic elastomers may be categorized by different length scales. At the macroscopic level, a complete set of continuum dynamic theory was formulated by the hydrodynamics concept for isotropic ferrogels10 and uniaxial magnetic gels.11 Recently several macroscopic magneto-


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mechanical models at macroscopic scale were also developed, where the Maxwell’s equations were coupled with the linear/nonlinear elasticity.12-16 However, at the level of mesoscopic, usually the polymeric matrix was regarded as an elastic continuum and the discrete magnetic particle regarded as a dipole, such that the effects of the particle concentration and spatial distribution on the magnetic elastomers were investigated by the magnetic dipole model.17-21 In order to capture more details at microscopic scale, including structure of the polymer as well as the coupling between the magnetic particles and individual polymer chain, the microscopic dipole-spring model was developed,8,22 and further refined to study the conformation of flexible magnetic filaments.23 Recently, the scale-bridging approach was proposed to bridge across multiple scales from the microscopic to macroscopic scales.24,25

In terms of the magnetic hydrogel, the literature search reveals that so far only one theoretical work for a magnetic hydrogel immersed in a solvent, subjected to a uniform magnetic field at equilibrium state, has been published to predict the swelling via its volume fraction ratio only,26 meaning that the present hydrogel swelling deformation is a scalar, instead of a vector. As such, it is unable to characterize the deformation behavior of the magnetic hydrogel appropriately. Recently relevant experimental works were conducted for the deformable magnetic hydrogel. However, the results seem to be contradictory each other qualitatively. For example, some of them showed that the hydrogel in a uniform magnetic field elongated along the field direction and contracted in the transverse direction,27,28 while another experiment demonstrated elongation or contraction in the field direction for the hydrogel with different magnetic particle concentrations.29 The contradiction remains unclear, probably due to the lack of an 3 ACS Paragon Plus Environment


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efficient model to characterize the fundamental mechanism of the performance of the magnetic hydrogel.

In order to address the above limitations, a multi-effect-coupling magnetic-stimulus (MECm) model is developed for simulation of the responsive behavior of the magnetic hydrogel, which is placed in solvent subject to the externally imposed magnetic field. In the MECm model, the magnetic susceptibility for the magnetization of the general magnetic hydrogel is defined as a function of finite deformation, instead of a constant for an ideal magnetic hydrogel.7 The present governing equations are based on the conservation of mass and momentum. The constitutive equations are formulated by the second law of thermodynamics to account for the effects of the chemical potential, the magnetic field, and the finite deformation. In addition, a novel free energy density is proposed with consideration of the magnetic effect. For validation of the present MECm model, several comparisons are made between the simulation results and the experimental data published in open literature for the magnetic hydrogels in response to the uniform and non-uniform magnetic fields respectively, and good agreements are achieved. Furthermore, several parameter studies are carried out, including the shear modulus, magnetic intensity, and volume fraction of the magnetic particles, on the behavior of the magnetic hydrogel, for a better insight into the fundamental mechanism of the magnetic hydrogels.


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2. Formulation of the MECm Model 2.1.

Kinematics

In general, for a magnetic hydrogel initially undeformed in the absence of external mechanical force or magnetic field, the volume of the magnetic hydrogel may be defined as V0 enclosed by a boundary S 0 in a reference configuration, with material point labeled by the position vector X V0  S 0 . By the nonlinear elasticity theory,30 if the material point X moves to the spatial point x(X, t ) V  S , the deformation gradient is defined as

F  Xx 

x X

(1)

where  X () denotes the material gradient operator with respect to X . The Jacobian of the transformation is given by the determinant J  det(F) with the standard convention J  0.

2.2.

Magnetostatics

In the absence of free current and time-dependence effect, the magnetostatic field is governed by the classical Maxwell’s equations as 15 Ampere’s law:   h  0

(2)

Gauss’s law:   b  0

(3)



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where   () and   () denote the spatial curl and divergence operators with respect to x , respectively. Vectors b and h are the magnetic induction and intensity respectively

with respect to x , formulated by the following relation

b  0 (h  m)

(4)

where m is the magnetization vector and  0 the magnetic permeability in a vacuum.

Across the interface between the magnetic hydrogel and surrounding solvent, the magnetic induction b and intensity h are required to satisfy the following jump conditions 15 n  [[b]]  0 ,

n  [[h]]  0

(5)

where n is the unit outward normal to the surface S , the double square bracket denotes a quantity jump across the surface from the inside to outside of the material, for example,

[[]]  []  [] , where [] and [] are one-sided limit of the function () from the outward and inward normal directions, respectively. Equation (5) shows that the normal component of magnetic induction b and tangential component of the magnetic intensity

h are continuous across the material boundary, while the tangential component of magnetic induction b and normal component of the magnetic intensity h are discontinuous across the material boundary.15 After a pull-back to the reference configuration, the Eulerian Maxwell’s equations (2) and (3) as well as the relation (4) are presented respectively in the Lagrangian forms of 6 ACS Paragon Plus Environment

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X  H  0

(6)

X  B  0

(7)

B  0 JC1 (H  M)

(8)

where C  FT F is the right Cauchy-Green tensor, and the Lagrangian counterparts of the magnetic variables are given by31

B  JF 1b , H  FT h , M  FT m

(9)

Similarly, the Lagrangian counterparts of the magnetic boundary conditions (5) are rewritten as N  [[B]]  0 ,

N  [[H]]  0

(10)

where N is the unit outward normal to the surface boundary S 0 .

2.3.

Conservation of Mass

By the law of mass conservation for transport of mass species, it is required that the time rate of the mass change within the control volume equals to the rate at which mass enters into the volume plus the rate at which mass is generated or lost within the control volume due to the source or sink, namely

d { cs dV }    js  ndS   rs dV dt V(t ) S (t ) V (t )


(11)


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where c s is the true concentration of solvent, j s the true diffusion flux of solvent, rs the number of solvents supplied across the material element per unit time, and d () / dt the material time derivative. By the Reynolds transport and the divergence theorems,30 if the bulk flow is neglected, the local form of the mass conservation for solvent is written as

cs    js  rs

(12)

where cs refers to the material time derivative of c s .

For conversion of Equation (12) into an expression with nominal quantities, the nominal concentration C s is associated with the true concentration c s by Cs  cs det(F) , and the nominal flux J s with the true flux js by Nanson’s relation J s  JF 1 js .30 As a result, the conservation of solvent is rewritten in the Lagrangian form of 32

C s  X  J s  Rs

(13)

where Rs is the rate of mass supply per nominal material element and Rs  rs det(F) .

2.4.

Conservation of Momentum

By the conservation of momentum, the mechanical deformation of the hydrogel is given by

d {  h Vh dV }   σ  ndS    hf b dV dt V(t ) S (t ) V (t ) 8 ACS Paragon Plus Environment

(14)

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where  h is the true mass density of the hydrogel, Vh the deformation velocity, σ the Cauchy stress, and f b the body force per mass including the magnetic and mechanical body forces. In a magnetoelastic system, the magnetic body force is defined as

f m  0m  h .33 By the Reynolds transport and the divergence theorems,30 the mechanical equilibrium equation (14) is pulled back to Lagrangian configuration in a local form of

   P   f 0 V h X 0 b

(15)

where  0 is the nominal mass density of the polymer network and 0  J h , P the first Piola-Kirchhoff stress and P  JσFT .

2.5.

Energies of the Magnetic Hydrogel

The rate of total energy of the magnetic hydrogel  is defined as the rate of the integral of total energy density e over the volume V , and is contributed by four energy components, the thermal power Q , the mechanical power W , the chemical power C , and the magnetic power M . As a result, we have

 

d { edV }  Q  W  M  C dt V

(16)

It is known that heating is a main contribution to the change of internal energy. The heat supply is resulted from the heat flux q entering into the system and the heat generation rq within the system, such that the thermal power Q is given by 9 ACS Paragon Plus Environment


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Q   q  ndS   rq dV S

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(17)

V

The mechanical power W is contributed by the traction force t and the body force f b acting on the hydrogel, namely

W   t  Vh dS    hfb  Vh dV S

(18)

V

The chemical power C may vary due to the diffusion of solvent molecules into and out of the volume V , or mass generation rs by chemical reaction. Thereby, the power of chemical energy C is given by 34

C    s js  ndS    s rs dV S

(19)

V

where  s is the chemical potential. The magnetic power M is associated with the magnetization of the magnetic hydrogel by 14,35

M    m  b dV

(20)

V

By summation of Equations (17) to (20), the divergence theorem, the mass conservation equation (12), and replacing the traction force t with σ  n , the rate of total energy  is determined by Equation S4 (see Supporting Information)


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d { edV }   {rq    q  (  σ   hfb )  Vh  σ : Vh   s cs  js   s  m  b }dV (21) dt V V If E denotes the rate of the total energy per unit reference volume, the left-hand side of Equation (21) is rewritten as 35

d { edV }   E dV0 dt V V0

(22)

Similarly, if the right-hand side of Equation (21) is rewritten in Lagrangian forms, its material counterpart may be presented by Equation S10 (see Supporting Information), namely

 EdV   [( R 0

V0

q

  X  Q)  ( X  P   0f b )  Vh

(23)

V0

 (P  M  B  F

T

F

T

 ]dV  M  B) : F   s C s  J s   X  s  M  B 0

where Rq is the material heat generation and Rq  Jrq , and Q the nominal heat flux vector and Q  JF 1q . In general, the rate of total energy  is equal to the sum of the rate of kinetic energy

K and the internal energy U , namely

 EdV

0

V0

 dV   K  U    h Vh  V h 0 V0

 udV

0

V0

where u is the internal energy per unit reference volume.


(24)


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By Equation (24) and the mechanical equilibrium equation (15), Equation (23) is rewritten in the local form of

 u  Rq   X  Q  (P  M  B  F T  F T  M  B) : F  sC s  J s   X s  M  B

2.6.

(25)

Constitutive Equations

By the second law of thermodynamics, it is stated that the integral of entropy per unit volume  increases as rapidly as the sum of the volume and surface heating divided by the absolute temperature T , as presented by the Clausius-Duhem inequality below14

Rq

dV   T

V0

dV0   (

V0

S0

QN )dS 0 T

(26)

By the divergence law, and then substituting the energy equation (25) to the inequality (26) by eliminating the rate of heat generation Rq , the inequality is rewritten in the local form of

   C  J     Q  T (27) T  u  (P  M  B  F T  F T  M  B) : F  M  B s s s X s X T According to the thermodynamics, the density of Helmholtz free energy F is associated with the density of internal energy u via F  u  T . If the free energy density F is a function of state depending on the four independent variables, the absolute temperature T , the magnetic induction B , the solvent concentration C s , and the deformation gradient F , we have


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F F  F    )T  (P  M  B  F T  F T  M  B  ):F (  M)  B T F B F  Q ( s  )C s  J s   X  s    XT  0 C s T (

(28)

The inequality (28) is derived directly based on the second law of thermodynamics (26), which characterizes the present system, consisting of the magnetic hydrogel, magnetic field, and solvent environment, at arbitrary state including nonequilibrium and equilibrium. Therefore, the coefficient in any bracket vanishes, due to the arbitrariness of the variables, i.e. the absolute temperature T , the magnetic induction B , the solvent concentration C s , and the deformation gradient F . Theoretically it is supposed that there are the four dissipative terms contributed to the entropy production, given by Equations (27) and (28), due to the present four independent variables. In the present model however, both the deformation gradient F and magnetic induction B are neglected for the contributions to the entropy production, based on the two reasons: (1) the significant time-scale difference between the short-time mechanical equilibrium and the long-range solvent migration, 34,36,37 and between the short-time magnetic domain orientation and the long-time particle rotation,7 as well as (2) the reversible processes of elastic deformation and paramagnetic magnetization.38 As a result, the dissipation is mainly contributed by the two dissipative terms associated with the two independent variables, the absolute temperature T and concentration C s . According to the linear irreversible thermodynamics, the phenomenological equations are written as39,40

1 1 J s  Lss (  X  s )  Lsq ( X ) T T


(29)


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1 1 Q  Lqs (  X  s )  Lqq ( X ) T T

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(30)

where Lss , Lsq , Lqs , and Lqq are the Onsager coefficients, and Lsq  Lqs due to the Onsager relation. To make the inequality (28) held, the conditions for the coefficients

Lss  0 and Lqq  0 as well as for the off diagonal elements satisfy Lss Lqq  ( Lsq  Lqs )2 / 4 are required. Consequently, Equation (28) is always larger than or equal to zero. Therefore, we have the constitutive equation that consists of two groups, one is to characterize the solvent flux, given by Equation (29), and the other is for the thermodynamic force as given below.



F F F F , P  M  B  F T  F T  M  B, M   , s  T F B Cs

(31)

If the magnetization M is employed as the independent variable, a complementary version of F is required, represented by  , through the Legendre transformation

  F  M  B . As a result, the refined thermodynamic forces are summarized as

 

2.7.

    , P  M  B  F T  F T  M  B, B  , s  T F M Cs

(32)

Free Energy Density

In order to characterize the responsive behavior of the magnetic hydrogel, an explicit free energy density is required, and thus proposed here to account for the contributions from elastic deformation of the polymer networks  ela , mixing of the polymers and solvent

 mix , and the magnetization  mag , namely 14 ACS Paragon Plus Environment

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   ela   mix   mag

(33)

2.7.1. Elastic Energy Density Based on the Gauss statistical model for rubber elasticity,41 the elastic free energy density  ela is given as

 ela 

1 NK BT {tr(FT F)  3  2 ln[det(F)]} 2

(34)

where N is the effective number of polymer chains per unit reference volume, and K B the Boltzmann constant.

2.7.2. Mixing Energy Density By the Flory-Huggins polymer solution theory,41 the energy of mixing is expressed as

Fmix  K BT [ns ln(1  n )   H nsn ]

(35)

where ns is the number of solvent at reference configuration,  H the Flory-Huggins parameter to characterize the interaction between the solvent and polymer networks, and

 n the volume fraction of the polymer. If the volume of the hydrogel changes only due to solvent absorption/desorption, the condition of molecular incompressibility is given by 37

1  vs Cs  det(F)


(36)


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where vs is the volume per solvent species. In general, Equation (36) works as a constraint for the system at equilibrium state, where the total energy of the system reaches a minimum. For minimizing the system energy subject to a constraint, the constraint can be enforced by either a Lagrange multiplier  , or eliminating a variable, in which they produce the same results. The present technique, i.e. using a Lagrange multiplier, maintains the deformation gradient F and concentration C s as independent variables, and it employs the Lagrange multiplier  to prescribe the constraint.37,42,43 Therefore, the term [1  vs Cs  det(F)] is added into the total energy density  of the system (55). If  p denotes the volume fraction of magnetic particles for the magnetic hydrogel at reference state, the volume fraction of the polymer  n becomes n  (1   p ) / det(F) at equilibrium state after swelling. By Equations (35) and (36), the free energy density for mixing is formulated as

 mix 

vs C s   p 1  p K BT [vs C s ln( )   H vs C s ] vs det(F) det(F)

(37)

It is found that the mixing energy density  mix constructs a multiple coupling domain with the magneto-chemo-mechanical coupled fields, since the mixing energy density  mix is a function of the magnetic particle concentration  p , the solvent concentration C s , and the mechanical deformation gradient F .


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2.7.3. Magnetic Energy Density

As well known, when a magnetic hydrogel is placed in an external magnetic field H 0 , the hydrogel is magnetized and an inner demagnetization field H d is thus induced due to the magnetized poles on the material boundary, in analogy to the polarization field in the presence of electric fields. In the present MECm model, a homogeneous magnetic field within the hydrogel is assumed. Such a field may be generated by a relatively long solenoid.44 Therefore, the internal magnetic field H is the sum of demagnetization field

H d and the externally imposed field H 0 by H  H0  H d

(38)

where the demagnetization field H d is associated with magnetization M by H d   NM

(39)

where N is the demagnetizing factor of the hydrogel that depends on its geometry. If the internal magnetic intensity H is much smaller than that at saturation state H s , the magnetization M is linear in the magnetic field H , namely M  H . By Equations (38) and (39), the magnetization M is rewritten as

M

H 0 1  N

(40)

where  is the magnetic susceptibility to characterize the magnetizability of the magnetic hydrogel.



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The magnetic energy density  mag is given by 45

 mag  M  B

(41)

Considering the magnetic boundary condition given by Equation (10), the normal component of magnetic induction B is continuous across the material boundary,15 and thus B  0 H 0 .44 By Equation (40), the magnetization  mag is further written as

 mag 

 0 H 0 2 1  N

(42)

For specification of the magnetization, two variables, the demagnetizing factor N and magnetic susceptibility  , are required, which are associated with deformation.

Usually the demagnetizing factor N has three components and they obey the law below46

Nx  N y  Nz 1

(43)

For simple geometrical shapes, the demagnetizing factors of the hydrogels are constant. For example, N x  N y  N z  1 / 3 for a sphere or cube, N x  N y  0 and

N z  1 for a slab with infinite lateral (along x-, y- axis) dimensions. The exact demagnetizing factor is also achieved even for an ellipsoidal shape, since the demagnetizing field is uniform within the hydrogel.47 For other shapes, however, no generalized formula exists, due to the non-uniformity of the magnetic field within the


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hydrogel. An alternative way for the demagnetizing factor is based on the volumeaverage method or relevant tabulated data from experiments.46,48

The magnetic susceptibility in the magnetic hydrogel may be an analogy to the permittivity in the dielectric elastomer or gel. When the degree of crosslink is low, the elastomer or gel may behave like a liquid and then the permittivity is deformation independent. Such kind of material is also called ideal dielectric elastomer or gel.49-51 However, when the crosslink degree is not low, the permittivity of a general dielectric elastomer or gel may be associated with the deformation, which may change the responsive characteristic from compression to tension states.49,52 Similarly, the magnetic susceptibility is constant in an ideal magnetic hydrogel,7 whereas it may be deformationdependent in a general magnetic hydrogel, as demonstrated by the experimental work.44,53 In the MECm model, the magnetic susceptibility associated with the finite deformation is presented below.

If the magnetic particles are distributed homogeneously in polymeric matrix and linearly magnetized, the magnetic susceptibility of the hydrogel  is given by 21

   p p

hp H

(44)

where  p is the magnetic susceptibility of the magnetic particle, h p the magnetic field in the magnetic particle, h p

the average magnetic field over the positions of every

particle, H the magnitude of the internal magnetic field H .



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If no interaction occurs among the spherical magnetic particles, the magnetic field inside a particle h 0p is calculated by Equations (S11-S13) (see Supproting Information) via summarization of both the external magnetic and demagnetization fields, when subjected to a external magnetic field H 20

h 0p 

3 H 3 p

(45)

where “0” represents the initial state without interaction. With this approximation of h p  h p0 , and by Equation (44), the initial magnetic susceptibility of the hydrogel  0 is

given as

0 

3 p  p 3 p

(46)

which is the approximate result of the classical Maxwell-Garnett formula.19 It is known that the particles are magnetized and the particle-particle interaction may change the average field h p , if a magnetic field is imposed. Following Zubarev’s work,21 the average field h p is achieved via a regular method of reflection.

As shown in Figure 1, in the absence of the external magnetic field, the magnetic moments of the two particles oriented randomly, as illustrated by the dashed arrows. When the magnetic field is applied, the particles are magnetized and their moments align quickly along the field direction, due to the Néel or the Brownian mechanism, as illustrated by the solid arrows.8,22 Simultaneously, the particle-particle interaction occurs. The magnetic moment of the first particle generates an additional magnetic field in the 20 ACS Paragon Plus Environment

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second particle, which increases the magnetic moment in the second particle and vice versa. If the procedures recycle for k times, the magnetic field in a particle may be determined by Equation S20 (see Supporting Information), namely

3 cos 2   1 h p  h  h  h  h  (V p ) r3 k 1 

0 p

' p

0 p

k

0 p

(47)

where h 'p is the auxiliary magnetic field induced by another particle, V p the volume of a particle, r the distance between the two particles,  the angle between the field H 0 and the radius-vector r , and  

3 p . The power series in Equations (47) converges.21 4 3   p

For simplicity, only the first term of the series is employed in the present model.20

As the hydrogel deforms, the relative position is changed between the particles. The average magnetic field h p

is obtained by integration of the magnetic field over the

possible positions of the other particles, namely 21

h p  h 0p  h 'p  h p0 

p Vp

h

' p

(r ) g (r)dr

(48)

where g (r) is the spatial pair distribution function for the probability of finding the second particle at the distance r from the first particle. The function g (r) was also used to analyze the polarization field of a dielectric elastomer with a good agreement achieved between the experimental and numerical results.54 The function g (r) is given by

g  g 0  g 21 ACS Paragon Plus Environment

(49)


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where g 0 is pair distribution function for the present undeformed hydrogel, as given by55

rd

 0  3r r3  g 0  1  8 p (1   ) 3 4 d 16 d    1

d  r  2d

(50)

r  2d

where d is the diameter of the particle.

g is the change of the function g due to the hydrogel deformation, and is written as 54

g    ( g 0u)  ( g 0  u  u  g 0 )

(51)

where u is the relative displacement vector of the particles, and may be approximately taken as the hydrogel deformation, if an affine deformation is assumed. In other words, the macroscopic deformation of the hydrogel is affinely mapped to the displacement of each particle.17 Furthermore, no relaxation process is also assumed,51 in order to make Equation (51) held.

By Equations (48) and (49), the average magnetic field h p

h p  h p0 

p Vp

'  hp (r) g 0 (r)dr 

p Vp

h

' p

(r)g (r)dr

By Equations (47), and (50) to (52), the average field h p

h p  [1 

is rewritten as

is further rewritten as

8  p (1  5 p )(x   y  2z )]h p0 15


(52)

(53)

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where  x ,  y , and  z are the stretches in the coordinate directions, respectively. Usually  p  1 and  p /(3   p )  1 ,9 by Equations (44) and (53), the susceptibility of the magnetic hydrogel  is thus rewritten as

2 5

  3 p [1   p (1  5 p )(x   y  2z )]

(54)

where the first term represents the magnetic susceptibility of the undeformed hydrogel,19 and the second term the change of susceptibility of hydrogel due to the deformation. Therefore, it is concluded that the magnetic energy density  mag  M  B , given by Equation (41), presents the magnetic field coupled with the mechanical field, since the magnetic energy density  mag is a function of the demagnetization factor N and magnetic susceptibility  , which are associated with mechanical deformation, due to the magnetization M  H 0 /(1  N ) , given by Equation (40).

2.7.4. Total Free Energy Density As expressed by Equation (33), the total free energy density  of the system is contributed by the elasticity, mixing, and magnetization. By summarization of Equations (34), (37), and (42), and considering molecular incompressibility (36),  is written below in details

 H 0 1 NK B T {tr(F T F)  3  2 ln[det(F)]}  0 2 1  N vs Cs   p 1 p K T  B [v s C s ln( )   H vs Cs ]  [1  v s C s  det(F)] vs det(F) det(F) 2




(55)


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where  is the Lagrange multiplier and also interpreted as the osmotic pressure37,42.

So far the formulation of the present MECm model has been completed theoretically. It consists of the Maxwell’s equations (6) and (7), the conservation of mass (13), and the conservation of momentum (15). By the second law of thermodynamics, the constitutive equation (32) is formulated. Furthermore, the present MECm model includes the free energy density presented by (55). To simulate the responsive behaviors of the magnetic hydrogel at the equilibrium state, the corresponding model reduction and boundary conditions are required and will be discussed in the following sections.

2.8.

Model Reduction and Boundary Conditions

2.8.1. A Uniform Magnetic Field

In a uniform magnetic field, no field-particle interaction exists and the particle-particle interaction dominates, due to the lack of a field gradient.56 If no mechanical load is considered for the equilibrium state, the mechanical equilibrium equation (15) is reduced to X  P  0

(56)

If no external constraint exists, the stress over the hydrogel-solvent interface vanishes, namely P  0 . In addition, the chemical potential  s inside the hydrogel, given by Equation (32), is equal to the external chemical potential  s*  0 .57 In the following validation and parameter studies, the performance of the magnetic hydrogel is investigated subject to a uniform magnetic field along the z-axis direction.


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2.8.2. A Non-uniform Magnetic Field

If a magnetic hydrogel is subjected to a non-uniform magnetic field generated by an electromagnet, four stages of responding process may exist, as illustrated in Figure 2: (1) the dry state of the hydrogel before immersed in the solvent, as shown in Figure 2(a); (2) the free swelling state of the hydrogel before imposing the non-uniform magnetic field, as shown in Figure 2(b); (3) the short-time responsive state of the hydrogel, as shown in Figure 2(c); and (4) the long-time responsive state of the hydrogel, as shown in Figure 2(d). It is noted that the solvent inside the hydrogel has no time to migrate at the shorttime scale, such that the concentration of solvent within the hydrogel remains unchanged and the hydrogel behaves like an incompressible elastomer.51,58 However, the solvent may migrate into the hydrogel gradually over a long-time scale, which allows the hydrogel to change both the shape and volume. All these states may induce large deformation, whereas different responsive behaviors may be included.

2.8.2.1.

Free swelling

The hydrogel swells isotropically if no constraint exists. When it reaches an equilibrium state, stress vanishes, namely Px  Py  Pz  0 , and the chemical potential  s is also equal to zero, due to the absence of chemical driving force.

By the constitutive relation (32) and the free energy density (55), the free swelling stretch is obtained by Px, y , z  NK BT (0  01 )   0  0 2


(57)


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where 0 is the free swelling stretch and x   y  z  0 .

2.8.2.2.

Short-time scale

If a non-uniform field is imposed, two distinct interactions exist, particle-particle and the field-particle interactions. The former results from the particle magnetization, and the latter from the magnetic field gradient. Generally, the deformation induced by the fieldparticle interaction is much larger than that by particle-particle interaction.56 As such, the magnetostriction due to the particle-particle interaction is negligible in a non-uniform magnetic field. If the inertial term is ignorable, the mechanical equilibrium equation (15) is rewritten as

 X  P  J0 F T  m   Xh  0

(58)

If the magnetic field along the z-axis direction is not perturbed by adding the magnetic hydrogel, the magnetic force is parallel to the field direction. Integrating Equation (58) gives

Pz 

J

z



ht

hb

0 mz  dhz  0

(59)

where m z and hz are the magnitudes of the magnetization and magnetic intensity, respectively, ht and hb the magnitudes of the magnetic intensity on the top and bottom surfaces of the hydrogel, respectively.

At a short-time scale, the volume of the hydrogel is kept unchanged. By Equations (32) and (55), the stress due to elasticity is given by 26 ACS Paragon Plus Environment

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Pz  NK BT ( z 

1

z 2

)

(60)

where  z is the hydrogel stretch along the magnetic field direction.

Inserting Equation (60) into Equation (59), we have

NK BT (z 

1

z

2

)  z

1 ht



hb

0 mz  dhz  0

(61)

For solution of Equation (61), the well-known Langevin function is used to associate the magnetization m with magnetic intensity h by 59

m   pm sat (coth   1 /  )

(62)

where m sat is the saturation magnetization of the magnetic particle,  the Langevin parameter and    0 m p hz /( K BT ) , where m p is the magnetic moment of the particle. Substituting Equation (62) into (61), the stretch  z at short-time scale is thus obtained by

NK BT ( z 

2.8.2.3.

1

z

2

)   p msat

sinh  t sinh  b K BT (ln  ln )0 m p t b

(63)

Long-time scale

Over a long time, the hydrogel is able to absorb additional solvent due to the imbalance of the chemical potential  s between the interior hydrogel and exterior solvent, and its



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volume is thus changed. Based on the constitutive equation (32), the stresses in transverse and field directions are obtained respectively as

Px  NK BT (x  x1 )   y z

(64)

Py  NK BT ( y  y1 )   x z

(65)

Pz  NK BT (z  z 1 )   x  y

(66)

If no constraint exists, the equilibrium stress in the transverse direction is zero. In the magnetic field direction, the stress Pz is required to equilibrate with stress by the magnetic effect given by

Pz   p msat

sinh  t sinh  b J K BT (ln  ln ) 3 m p t b

(67)

Therefore, the hydrogel deformation is obtained by Equations (64) to (67).

3. Results and Discussion 3.1.

Validation of the MECm Model

In order to examine the MECm model, a comparison is carried out with the experimental measurement by Safronov et al. for the swelling of a square-shaped magnetic hydrogel,29 in which polyacrylamide (PAAm) hydrogels with different volume fractions of magnetite particles were immersed in water for the deformation of the hydrogels with and without a uniform magnetic field, as shown in Figure 3.29 Initially, the magnetic hydrogel was in free swelling state without a magnetic field, and its dimensions were labeled Lx 0 , L y 0 , 28 ACS Paragon Plus Environment

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Lz 0 , and Lx 0  Ly 0 . After imposing a magnetic field in the z-axis direction, the hydrogel deforms with dimensions of Lx , L y , Lz , and Lx  Ly . In the experiment,29 the deformation of the hydrogel is characterized by dimensional variation  i  ( Li  Li 0 ) / L0i , where i  x, y ,or z , the volume change thus by V  Lx Ly Lz / Lx 0 Ly 0 Lz 0 . Under the assumption of homogeneous deformation, the dimensional variations is rewritten as:  i  i  1 . The demagnetizing factor for the hydrogel along the z-axis direction is

approximated by N z  1 /( 2n  1) ,48 where n is the dimensional ratio between axial and lateral directions, namely n  Lz / Lx . If f is the initial geometric factor, defined as f  Lz 0 / Lx 0 , the factor N z is rewritten as N z  x /( 2 fz  x ) . The input parameters

required for the MECm model are tabulated in Table 1.

For a uniform magnetic field, the MECm model is examined in Figure 4 via comparison of the deformation of the magnetic hydrogel  i along the field and transverse directions, with different filled particle ratios Vparticle / Vpolymer , between the presented simulation results and the published experimental counterpart.29 For plotting the figure, the deformation of the magnetic hydrogel  i at the equilibrium state in a uniform magnetic field is determined by the constitutive equation (32), the free energy density (55), the governing equation (56), and the corresponding boundary condition P  0 and  s  0 . The equations are solved numerically by the commercial finite

element-solver, COMSOL/Multiphysics 4.4. It is observed that the hydrogel elongates along the magnetic field direction and contracts in the transverse direction, when the ratio

Vparticle / Vpolymer is lower than 0.17. Such responsive behaviors were also observed by other 29 ACS Paragon Plus Environment


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experimental work.27,28 It is also seen from Figure 4 that, both the magnitudes of the elongation and contraction reach their maximum when the ratio Vparticle / Vpolymer is around 0.1, then followed by a decreasing trend with the further increasing ratio Vparticle / Vpolymer . It is interesting that the hydrogel contracts along the field direction but elongates in the transverse direction, if the ratio Vparticle / Vpolymer exceeds about 0.17. Probably the inversion results from the competition between the change of two factors, demagnetizing factor N and magnetic susceptibility  .60 The former induces the elongation of the hydrogel along the field direction, while the latter stimulates either elongation or contraction.20 When the ratio Vparticle / Vpolymer is low, the demagnetizing effect dominates and the hydrogel elongates along the field direction. However, when the ratio

Vparticle / Vpolymer is high, the change of the magnetic susceptibility may play a more important role in the contraction along the field direction.61

For a non-uniform magnetic field, the MECm model is validated and demonstrated in Figure 5, through a numerical comparison between the presently computational deformation of the magnetic hydrogel  z , numerically solved by Equation (63) via the COMSOL/Multiphysics 4.4, and published experimental data.62 As shown in Equation (62), the effect of nonlinear magnetization is included in the present model, such that it is applicable for either low or high magnetic field. In other words, the present model can be reduced for the low magnetic field with linear magnetization, similar to the work by zriny et al.

62

for the magnetoelastic equation formulated by linear magnetization. In addition,

the effect of elasticity coupled with the magnetization induced by particle-particle interaction on the stress is also considered in the present model, given by Equation (15). 30 ACS Paragon Plus Environment

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In other words, the present model can be reduced for the case with elasticity only, similar to the work by zriny et al.62 In summary, the present model can be reduced to the theory62 if the particle-particle interaction and the nonlinear magnetization are ignored. The input data given in Table 2 includes the characteristic constant  for the field distribution, the current constant k I obtained by fitting the experimental curves.62 It is also found that the magnetic field maximizes near the magnet and decays exponentially away from the surface of the magnet, namely h  hmax e   z , where the maximum magnetic field hmax is associated with the electric current I by hmax  k I I .

Figure 5 illustrates the influence of initial magnet position z 0 on the deformation of the magnetic hydrogel  z with the electrical current intensity I . The magnetic hydrogel is stretched significantly up to 165%, and exhibits the instability and hysteresis phenomena. Initially, the elongation  z increases slightly with increasing current I , and it exhibits maximum when z 0  26.81mm , due to the decreasing magnetic field away from the magnet. When the current I increases up to a certain point, the increase of the elongation follows a step function with a significant change within a small range of the current I . As the current I increases further, the elongation increases linearly again and tends to the saturation gradually. Similarly, if the current I decreases, the hydrogel contracts and also exhibits the instability. However, the contraction does not follow the same path as the elongation, which is referred to as hysteresis. The phenomena of instability and hysteresis in the magnetic hydrogel are also found in other soft materials, such as photo-sensitive hydrogel,63 temperature-sensitive hydrogel,64 magnetic drop,65 and ferrogel.7 The present instability and hysteresis are analogy to the phase transition in 31 ACS Paragon Plus Environment


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material and may be characterized by the energy-stretch relation.66 For example, if the initial magnet is positioned at z0  40.75 mm with the small or large current ( I  1 A or 5A), the hydrogel energy WT reaches a global minimum of the curve for total energy against hydrogel stretch, which corresponds to a stable phase, as shown in Figure 6(a). However, for the intermediate current ( I  2.6 ~ 3.6A ), a local maximum and two local minima occur, in which the local maximum represents a unstable phase, the lower minimum corresponds to a stable phase and the higher one to a metastable phase,64 as shown by the ABCD loop in Figure 6(b). It is found that Points E and F correspond to the current ( I  3.2A ) where the two minimum energies are equal each other. Since the energies along the paths AE and CF are smaller than those along the paths FD and EB, the stable equilibrium states are achieved along the paths AE and CF, and the metastable equilibrium states achieved along the paths FD and EB. The dashed path BD thus represents the unstable state. As the current I increases from 0 to around 3.8A, the hydrogel stretch  z increases from 1 to about 1.08 (at Point B), where the energy barrier between the two local minima vanishes and the metastable state of the hydrogel becomes unstable. The hydrogel stretch  z tends to the stable state (at Point C), such that an instability phenomenon occurs. If the current I increases continuously, the stretch  z increases further. However, when the current I decreases from 5.5A to about 2.6A, the hydrogel stretch  z decreases from 1.45 to about 1.35 (at Point D), where the energy barrier between the two local minima vanishes and the metastable stable of the hydrogel becomes unstable again. As a result, the hydrogel stretch  z tends to the stable state (at Point A), and accordingly another instability phenomenon occurs. Therefore, the


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magnetic hydrogel undergoes the instability with a hysteresis loop when subjected to a non-uniform magnetic field.

Based on the above comparisons, it is found that the simulation results agree well with the experimental findings qualitatively and quantitatively. Thereby, the present MECm model can predict well the responsive behavior of the magnetic hydrogel.

3.2.

Parameter Studies

3.2.1. A Uniform Magnetic Field

For further understanding of the fundamental characteristics and responsive behavior of the magnetic hydrogel immersed in the solvent when subjected to a uniform magnetic field, several parameter studies with inputs given in Table 1 are carried out for the effects of the shear modulus G , the magnetic induction Bz , and the magnetic particle volume fraction  p .

The first parameter study is conducted for analysis of the effect of shear modulus G on the variations of dimensional change in the field and transverse directions as shown in Figure 7(a), and of volume change as shown in Figure 7(b), via Equations (32), (55) and (56), against the magnetic particle volume fraction  p , where f  0.75 . As observed from Figure 7(a), the shear modulus G influences the dimensional change  significantly when the volume fraction  p ranges from 0.02 to 0.15. However, it has insignificant effect on the dimensional change  when  p is smaller than 0.02 or larger than 0.15. It is also seen from Figure 7(a) that the magnitude of  decreases in both the



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field and transverse directions with the increasing G . Probably the deformation of the hydrogel is contributed by the elastic, magnetic, and chemical effects. The larger shear modulus G strengthens the hydrogels and makes the polymer networks more compact. Since the osmotic pressure and the magnetic force as the driving sources are equilibrated by the elastic restoring force, the increase of G prevents the hydrogel from further stretching. Regarding the volume change V , it deswells at first, then reaches the minimum when  p is around 0.11, as shown in Figure 7(b). After that, it recovers back to its free swelling state progressively and then swells continuously, when  p increases further from 0.11 to 0.17.

By the constitutive equation (32), the free energy density (55) and the governing equation (56), Figure 8 is plotted for the variations of dimensional change  and volume change V with magnetic induction Bz subject to different shear modulus G , where f  0.75 and

 p  0.09 . As illustrated in Figure 8(a), the maximum dimensional

change  is stimulated by the larger magnetic induction Bz and the smaller shear modulus G . It is also seen from Figure 8(b) that the swelling behavior of the magnetic hydrogel is much dependent on the external magnetic field, which was also demonstrated in the published experimental work,27,67 where the volume swelling ratio decreases when the magnetic induction Bz increases. We may understand this phenomenon by the wellknown magnetization mechanism. Under an external magnetic field, the particle-particle interactions are enhanced, since the magnetic moment of the magnetic particle tends to align along the field direction by the Néel and Brownian relaxations.7,59 As the particles


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are attached tightly onto the polymeric networks, the translations of magnetic particles stimulate the deswelling of the hydrogel.

3.2.2. A Non-uniform Magnetic Field

In order to investigate the responses of the magnetic hydrogel immersed in solvent subject to a non-uniform magnetic field at a long-time scale, several parameter studies are conducted for the effects of the initial magnet position z 0 , the current density I , and the volume fraction of magnetic particles  p . For numerical simulation, the inputs of the parameters are given as follows.  H  0.5 , G  1.4kPa , vs  1028 m 2 , T  298K ,

 p  0.024 ,   0.4 , msat  571kA/m , m p  5.95  1020 A  m 2 , k I  7.32  104 m1 , and h0  122.1mm .

By Equations (64) to (67), Figure 9 numerically illustrates the influence of the initial magnet position z 0 on the stretch  z along the field direction with the electric current I at short- and long-time scales, respectively. As seen from Figure 9, the instability and hysteresis phenomena are observed at both scales. However, the instability at the longtime scale starts at a lower current I and the width of hysteresis is much smaller than that at short-time scale. It is also found that the stretch  z is larger at long-time scale than that at short-time scale, resulted from the diffusion of the solvent.

The influence of the initial magnet position z 0 is illustrated numerically in Figure 10 on the swelling ratio of the magnetic hydrogel, through Equations (64) to (67), against the electric current I at long-time scale. In the absence of the current I , the hydrogels are at 35 ACS Paragon Plus Environment


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the free swelling state and thus have the same swelling ratio. When the current I is applied, the swelling ratio also exhibits the instability and hysteresis, whereas the hysteresis width becomes smaller when the magnet position z 0 decreases from 50 to 30mm, and even vanishes if z0  20mm . It is also noted that the swelling ratio tends to a saturation state when I is high. Probably the magnetization of the magnetic hydrogel, characterized by the Langevin equation, becomes saturated at the high magnetic field.

Based on Equations (64) to (67), Figure 11 demonstrates the influence of the volume fraction of magnetic particles  p on the swelling ratio with the electric current I . As observed from the figure, in the absence of current I , the swelling ratio increases with the decrease of the particles volume fraction  p , which is consistent with the experimental measurement.67,68 Theoretically, the incorporation of the magnetic particles decreases the volume fraction of hydrophilic polymer networks, and thus reduces the ability of the hydrogel to imbibe solvent.

4. Conclusion A multi-effect-coupling magnetic-stimulus (MECm) model has been developed for numerical characterization of the magnetic hydrogel that is immersed in solvent subject to the external magnetic field. The present MECm model allows for finite deformation, and includes the governing equations for mass and momentum conservations. Based on the second law of thermodynamics, the constitutive equations are formulated to consider the effects of the chemical potential, the magnetic field, and the finite deformation. In the MECm model, the magnetic susceptibility for magnetization of the general magnetic 36 ACS Paragon Plus Environment

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hydrogel is defined as a function of the finite deformation, instead of a constant for an ideal magnetic hydrogel.7,69 Moreover, a novel free energy density is proposed to account for the magnetic effects. The MECm model is validated by comparison of the equilibrium responsive behavior of the magnetic hydrogel between the simulation results and the published experimental data, in which very good agreements are achieved. Furthermore, several parameter studies are conducted for influences of the magnetic and geometric properties on the responsive behavior of the magnetic hydrogel.

Acknowledgement The authors gratefully acknowledge the financial support from Nanyang Technological University through the project (No: M4081151.050) and NTU Research Scholarships.

Supporting Information 1. Derivation of total energy in Lagrangian form. 2. Derivation of the magnetic field inside a particle without interaction 3. Derivation of the magnetic field inside the particle after k times reflections. References.



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List of Tables Table 1. Input parameters of the MECm model for the magnetic hydrogel subject to a uniform magnetic field. Table 2. Input parameters of the MECm model for the magnetic hydrogel subject to a non-uniform magnetic field.


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List of Figures Figure 1. Schematic illustration of the interaction between two magnetic particles. Figure 2. Four stages of response processes for the magnetic hydrogel at (a) the dry state, (b) the free swelling state, (c) the short-time responsive state, and (d) the longtime responsive state. Figure 3. Hydrogel swells at (a) the free swelling state, and (b) the current state with magnetic effect. Figure 4. Comparison of the present theoretical simulation and published experiment for the PAAm magnetic hydrogel subject to a uniform magnetic field.29 Figure 5. Comparison of the present theoretical simulation and published experiment for the PVA magnetic hydrogel subject to a non-uniform magnetic field,62 where the dots denote the experimental data, the solid and dashed lines the simulation results at stable and unstable states respectively. Figure 6. Total energy of the magnetic hydrogel WT as a function of stretch  z at several current intensities I when z 0  40.75mm , where the open and solid circles denote the local minima at small and large stretch respectively (a); Instability and hysteresis phenomena under a non-uniform magnetic field, where the arrows above the curve denote the direction of contraction and those below the curve denote the direction of extension (b). Figure 7. (a) Influence of shear modulus G on the variation of dimensional change  with particle volume fraction  p , (b) Influence of shear modulus G on the variation of volume change V against particle volume fraction  p . Figure 8. (a) Influence of shear modulus G on the variation of dimensional change  with magnetic induction B z , (b) Influence of shear modulus G on the variation of volume change V against magnetic induction B z . Figure 9. Influence of initial magnet position z 0 on the variation of stretch  z along the field direction with current I under a non-uniform magnetic field.



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Figure 10. Influence of initial magnet position z 0 on the variation of swelling ratio with current I under a non-uniform magnetic field. Figure 11. Influence of magnetic particle volume fraction  p on the variation of swelling ratio with current I under a non-uniform magnetic field.


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Table 1. Input parameters of the MECm model for the magnetic hydrogel subject to a uniform magnetic field. parameters

symbol

value

Boltzmann constant

KB

1.38 10-23 J/K

absolute temperature

T

298K

volume per solvent molecule

vs

10 28 m3

volume fraction of magnetic particles

p

0 ~ 0.2

density of PAAm

 gel

1350kg/m3 70

shear modulus of PAAm

G

0.43 ~ 55.3kPa

Flory-Huggins parameter

H

0.12 ~ 0.23

vacuum magnetic permeability

0

4 10 7 N / A2

initial geometrical factor

f

0.1 ~ 10


29

29

71


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Table 2. Input parameters of the MECm model for the magnetic hydrogel subject to a non-uniform magnetic field.

parameter

symbol

Value*

volume fraction of magnetic particles

p

0.024 62

magnetic distribution coefficient



0.4 62

saturation magnetization

msat

571kA/m 72

magnetic moment

mp

5.95 10 20 A  m 2

current constant

kI

7.32  10 4 m 1

shear modulus

G

1.2 ~ 2.5 kPa 62

initial height

h0

122.1mm 62

*

62

62

All the parameters listed above are cited from the published experimental

works in open literature for the present model validations


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Figure 1. Schematic illustration of the interaction between two magnetic particles.

Figure 2. Four stages of response processes for the magnetic hydrogel at (a) the dry state, (b) the free swelling state, (c) the short-time responsive state, and (d) the long-time responsive state.

Figure 3. Hydrogel swells at (a) the free swelling state, and (b) the current state with magnetic effect.



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Figure 4. Comparison of the present theoretical simulation and published experiment for the PAAm magnetic hydrogel subject to a uniform magnetic field.29

Figure 5. Comparison of the present theoretical simulation and published experiment for the PVA magnetic hydrogel subject to a non-uniform magnetic field,62 where the dots denote the experimental data, the solid and dashed lines the simulation results at stable and unstable states respectively.


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(a)

(b) Figure 6. Total energy of the magnetic hydrogel WT as a function of stretch  z at several current intensities I when z 0  40.75mm , where the open and solid circles denote the local minima at small and large stretch respectively (a); Instability and hysteresis phenomena under a non-uniform magnetic field, where the arrows above the curve denote the direction of contraction and those below the curve denote the direction of extension (b). 51 ACS Paragon Plus Environment


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(a)

(b) Figure 7. (a) Influence of shear modulus G on the variation of dimensional change  with particle volume fraction  p , (b) Influence of shear modulus G on the variation of volume change V against particle volume fraction  p .


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(a)

(b) Figure 8. (a) Influence of shear modulus G on the variation of dimensional change  with magnetic induction B z , (b) Influence of shear modulus G on the variation of volume change V against magnetic induction B z .



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Figure 9. Influence of initial magnet position z 0 on the variation of stretch  z along the field direction with current I under a non-uniform magnetic field.

Figure 10. Influence of initial magnet position z 0 on the variation of swelling ratio with current I under a non-uniform magnetic field.


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Figure 11. Influence of magnetic particle volume fraction  p on the variation of swelling ratio with current I under a non-uniform magnetic field.



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Development of a Multiphysics Model to Characterize the Responsive

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