Formulation of the MECe Governing Equations

MECe) Model for Electric-Sensitive Hydrogel

3.2 Development of the MECe Model

3.2.1 Formulation of the MECe Governing Equations

In order to overcome the drawbacks of the above models, a novel multiphysics model is developed mathematically in this chapter, called the multi-effect-coupling electric-stimulus (MECe) model, which is based on the work done by Hon et al.

(1999). Compared with the triphase model of Hon et al. (1999), the presently devel- oped MECe model defines a computational domain as covering both the hydrogel and the surrounding solution. The electric potential is also considered in the nonlin- ear coupled partial differential governing equations. By the multi-field formulation, the MECe model characterizes the mechanical deformation of the hydrogels more accurately and explicitly, making it more convenient for steady and transient sim- ulations. As such, the multiphysics MECe model is a more precise mathematical formulation with capability of providing more reliable simulation of the multiphase smart hydrogel responding to externally applied electric voltage.

It is assumed in the MECe model that the mixture of the hydrogel is com- posed of the three phases, the polymeric network matrix solid phase denoted by superscript s, the interstitial water/fluid phase by w and the mobile ion phase by k (k=1,2,. . .,Nion) where Nionis the number of total mobile ion species. If ø^α (α= s, w, k) is defined as the volume fraction of the phaseα

φ^α = dV^α

dV (α = s, w, k) (3.1)

where V^α is the true volume of the phase αand V is the volume of the hydrogel mixture; the saturation condition of the hydrogel mixture can be written as follows:

α=s,w,k

φ^α =1 (3.2)

If infinitesimal deformation of the mixture occurs, the apparent volume ratio of the polymeric network matrix solid phase is given by

J= dV0

dV = 1

1+tr(E) (3.3)

in which V0 is the volume of hydrogel mixture at reference configuration and E the elastic strain vector of the polymeric solid phase. The volume fraction of the polymeric network matrix solid phase is thus derived as

φ^s= dV^s dV = dV^s

dV0·dV0

dV =φ0^sJ= φ₀^s

(1+tr(E)) (3.4) whereφ₀^sis the volume fraction of the polymeric solid phase at reference configura- tion. Due to the extremely small volume of the ion phase, it is reasonable to neglect ø^kwhen compared with ø^sand ø^w. The volume fraction of the water/fluid phase is expressed as

φ^w≈1−φ^s=1− φ₀^s

(1+tr(E)) (3.5)

If chemical reaction is negligible, each phase should follow the law of mass conservation

∂ρ^α

∂t + ∇ ·(ρ^αv^α)=0(α = s, w, k) (3.6) where v^α(α =s, w, k) is the velocity of the phase αandρ^α(α =s, w, k) is the apparent mass density of the phaseα. It is known that the apparent mass densityρ^α can be expressed by the respective true mass densityρ_T^α, namelyρ^α=ρ_T^αφ^α(α=s, w, k). Meanwhile, on the basis of incompressibility restriction,ρ^α_T may be assumed reasonably to be constant. Equation (3.6) is then rewritten as

∂φ^α

∂t + ∇ ·(φ^αv^α)=0(α = s, w, k) (3.7) Considering Eqs. (3.2) and (3.7), the continuity condition of the mixture of the hydrogel is obtained as

∇ ·

⎛

⎝

α=s,w,k

φ^αv^α

⎞

⎠=0 (3.8)

By the tensor analysis, one can have

∇ ·(φ^αv^α)=φ^α∇ ·v^α+v^α· ∇φ^α =φ^αI:∇v^α+v^α· ∇φ^α (3.9) Equation (3.8) is thus rewritten as

α=s,w,k

(φ^αI:∇v^α+v^α· ∇φ^α)=0 (3.10)

In addition, the rate of kinetic energy of the hydrogel mixture is given by K˙ =

α=s,w,k

K˙^α =

V

⎛

⎝

α=s,w,k

v^α· ˙v^αρ^α

⎞

⎠dV (3.11)

If the internal energy of the hydrogel mixture U can be expressed by the Helmholtz energy function F as

U=F+TS (3.12)

the rate of internal energyUis written as follows:˙ U˙ = ˙F+(TS˙ +TS)˙ =

V

α=s,w,k

(˙^α+ ˙Tη^α+Tη˙^α)ρ^αdV (3.13)

where T is the absolute temperature, S the entropy of the system,^α andη^α are the density of Helmholtz energy and the entropy per unit mass for the phaseα, respectively.

It is noted that both the internal energy U and the Helmholtz energy F are the state functions depending on their state variables, such as the absolute temperature T, the elastic strain tensor E, the apparent densitiesρ^α and the fixed charge den- sity c^f. Based on such constitutive consideration, the Helmholtz energy density is expressed by

^α =^α(T,E,ρ^s,ρ^w,ρ^k,c^f)(α = s, w, k) (3.14) The rate of Helmholtz energy density is thus derived as

˙^α =∂^α

∂T T˙ +∂^α

∂E E˙+

β=s,w,k

∂^α

∂ρ^βρ˙^β+∂^α

∂c^f c˙^f (α = s, w, k) (3.15) From Eq. (3.9) one can know

˙

ρ^β ≡ Dρ^β

Dt = −ρ^βI:∇v^β (3.16)

and with the factJ˙ = −J∇ ·v^s, we have

˙

c^f = −c^f₀JI:∇v^s (3.17)

Considering the relationE˙ =(F^s)^T· ∇v^s·F^s, one can obtain

∂^α

∂E :E˙ =∂^α

∂E :((F^s)^T· ∇v^s·F^s)=

F^s·∂^α

∂E ·(F^s)^T

:∇v^s (3.18)

where F^s and (F^s)^T are the deformation gradient tensor and its transpose, respectively.

By substituting Eqs. (3.16), (3.17) and (3.18) into Eq. (3.15), we have ˙^α = ∂^α

∂T T˙+

F^s· ∂^α

∂E ·(F^s)^T−c^f₀J∂^α

∂c^f I

:∇v^s−

β=s,w,k

ρ^β∂^α

∂ρ^βI:∇v^β (3.19) Substituting Eq. (3.19) into (3.13), we have

U˙ =

V

α=s,w,k

∂^α

∂T T˙+

F^s· ∂^α

∂E ·(F^s)^T−c^f₀J∂^α

∂c^f I

:∇v^s−

−

β=s,w,k

ρ^β∂^α

∂ρ^βI:∇v^β + ˙Tη^α+Tη˙^α

⎞

⎠ρ^αdV

(3.20)

Rate of total workW consists of two portions, the rate of work done by external˙ forcesW˙eand the rate of work done by pressureW˙p

W˙ = ˙We+ ˙Wp (3.21)

The rate of work done by external forcesW˙eis defined as W˙e=

α=s,w,k

V

ρ^αf^α·v^αdV+

S

t^α·v^αdS

(3.22)

where f^αis the body force per unit mass of the phaseα, t^α =σ^α·v is the drag force applied on the surface,σ^αthe stress tensor of the phaseαand v the external normal on the surface. Due to the symmetry of stress tensor and the Gaussian gradient formula, we have the following transformation:

S

t^α·v^αdS=

S

(σ^α·v^α)·vdS=

V

∇ ·(σ^α·v^α)dV

=

V

((∇ ·σ^α)·v^α+σ^α:∇v^α)dV

(3.23)

Equation (3.22) is thus rewritten as W˙e=

α=s,w,k

V

((ρ^αf^α+ ∇ ·σ^α)·v^α+σ^α:∇v^α)dV (3.24)

The rate of work done by pressureW˙pis defined as

W˙p= −pV˙ (3.25)

Considering that the continuity equation (3.10) is adopted at the incompressible conditionV˙ =0, one can get

W˙p=

V

−p

α=s,w,k

(φ^αI:∇v^α+v^α· ∇φ^α)dV (3.26)

Substituting Eqs. (3.24) and (3.26) into Eq. (3.21), we obtain W˙ =

α=s,w,k

V

((ρ^αf^α+ ∇ ·σ^α−p∇φ^α)·v^α+(σ^α−pφ^αI):∇v^α)dV (3.27)

Rate of the heat transferred into the mixture is defined as Q˙ =

α=s,w,k V

ρ^αγ^αdV−

S

q^α·vdS

(3.28) whereγ^α is the rate of heat generation per unit mass of the phaseαand q^α the heat flux vector. Similarly, by the Gaussian gradient formula we have

Q˙ = −

α=s,w,k

V

(∇ ·q^α−ρ^αγ^α)dV (3.29)

Rate of energy dissipationD is defined as˙ D˙ =

α=s,w,k

V

^α·v^αdV (3.30)

where _α

is the diffusive momentum exchange between the phases, and it is a physical parameter indicating the diffusive resistance to the relative flow between the two phases._α

can thus be expressed by their relative velocities as follows:

^α =

β=s,w,k

f_αβ(v^β −v^α) (3.31)

where f_αβ is the diffusive drag coefficient between the phases α and β (or constituents) and f_αβ=f_βα. Obviously,_α

satisfies the following condition:

α=s,w,k

^α =0 (3.32)

Based on the first law of thermodynamics, we have the following relation of energy conservation for the irreversible thermodynamic system:

K˙ + ˙U− ˙D= ˙W+ ˙Q (3.33)

With substitution of Eqs. (3.11), (3.20), (3.27), (3.29) and (3.30) into Eq. (3.33), we obtain

−

V

α=s,w,k

(∇ ·σ^α+ρ^αf^α−ρ^αv˙^α+^α+p∇φ^α)·v^αdV

+

V

⎧⎨

⎩

⎡

⎣F^s·

⎛

⎝

α=s,w,k

∂^α

∂E

⎞

⎠·(F^s)^T −c^f₀J

⎛

⎝

α=s,w,k

∂^α

∂c^f

⎞

⎠I

⎤

⎦:∇v^s

+

α=s,w,k

⎡

⎣−σ^α−pφ^αI−

⎛

⎝

β=s,w,k

ρ^α∂^β

∂ρ^α

⎞

⎠I

⎤

⎦:∇v^α

⎫⎬

⎭dV

+

V

α=s,w,k

(∇ ·q^α+Tρ^αη˙^α−ρ^αγ^α)dV+

V

α=s,w,k

∂^α

∂T +ρ^αη^α

TdV˙ =0

(3.34) If the chemical potential is defined as

μ^α = ∂

∂ρ^α (3.35)

the chemical term in Eq. (3.34) becomes

β=s,w,k

∂^β

∂ρ^αρ^α =ρ^α∂('

β^β)

∂ρ^α =ρ^α ∂

∂ρ^α =ρ^αμ^α (3.36)

The mechanical term in Eq. (3.34) can be expressed by the second Piola–

Kirchhoff stress tensorτ_E^s and the Cauchy stress tensor σ_E^s, which are defined as follows:

τE^s =∂

∂E =

α=s,w,k

∂^α

∂E (3.37)

σE^s =F^s·τE^s ·(F^s)^T (3.38) In order to simplify Eq. (3.34), a chemical expansion stress TCis introduced to replace the complicated expression

TC=c^f₀J

⎛

⎝

α=s,w,k

ρ^α∂^α

∂ρ^α

⎞

⎠ (3.39)

By substituting Eqs. (3.36), (3.37), (3.38) and (3.39) into Eq. (3.34), a more complete formulation of Eq. (3.33) is obtained as

−

V

α=s,w,k

(∇ ·σ^α+ρ^αf^α−ρ^αv˙^α+^α+p∇φ^α)·v^αdV

+

V

⎧⎨

⎩(σE^s−TCI):∇v^s+

α=s,w,k

−σ^α−pφ^αI −

β=s,w,k

ρ^α∂^β

∂ρ^α

I

:∇v^α

⎫⎬

⎭dV

+

V

α=s,w,k

(∇ ·q^α+Tρ^αη˙^α−ρ^αγ^α)dV+

V

α=s,w,k

∂^α

∂T +ρ^αη^α

TdV˙ =0

(3.40) Due to the independence of the variables v^α,∇v^αandT satisfying Eq. (3.40), the˙ following equations are achieved:

Momentum equations:

∇ ·σ^α+ρ^αf^α−ρ^αv˙^α+^α+p∇φ^α =0(α = s, w, k) (3.41) Constitutive equations:

σ^s= −φ^spI+σ_E^s−ρ^sμ^sI−TCI (3.42) σ^α= −φ^αpI−ρ^αμ^αI(α = w, k) (3.43) Heat transfer equation:

∇ ·q^α+Tρ^αη˙^α−ρ^αγ^α =0(α = s, w, k) (3.44) By summation of Eq. (3.41), the momentum equation for the multiphase mixture of the hydrogel is written as

∇ ·σ+ρf −ρv˙=0 (3.45)

in which

σ =

α=s,w,k

σ^α, f= '

α=s,w,k

ρ^αf^α (

ρand v=

⎛

⎝

α=s,w,k

ρ^αv^α

⎞

⎠/ρ

If the body force f and the inertial force ρv are neglected, Eq. (3.45) is˙ simplified to

∇ ·σ =0 (3.46)

As well known, for an osmotic process at constant temperature, the relation between the chemical potential and the osmotic pressure posm can be derived by

the Gibbs–Duhem equation

dposm=

α=s,w,k

ρ^αdμ^α (3.47)

Integrating Eq. (3.47) we have posm=

α=s,w,k

ρ^α(μ^α−μ^α₀) (3.48)

By summation of Eqs. (3.42) and (3.43), the constitutive equation for the stress tensor of the hydrogel mixture is given as

σ =σE^s −(p+TC)I (3.49)

where the total pressure p includes the osmotic pressure posm. If the chemical expansion stress TCis neglected and the isotropic elastic material is assumed

σE^s =λstr(E)I+2μsE (3.50) in which λs andμs are Lamé coefficients of the polymer network solid matrix.

Equation (3.49) is rewritten as

σ = −pI+λstr(E)I+2μsE (3.51) By neglecting the body and inertial forces and considering Eqs. (3.41) and (3.43), the momentum equations of the water and ion phases in terms of their chemical potential are obtained as

ρ^α∇μ^α−^α = (α = w, k) (3.52) Substituting Eq. (3.31) into (3.52), one can derive the momentum equations in terms of the chemical potential and the velocities as follows:

−ρ^w∇μ^w+fws(v^s−v^w)+

N_ion

k=1

fwk(v^k−v^w)=0 (3.53)

−ρ^k∇μ^k+fks(v^s−v^k)+fkw(v^w−v^k)+

N_ion

j=1(j=k)

fkj(v^j−v^k)=0 (3.54)

Based on the work done by Lai et al. (1991), the following constitutive equations are obtained for the chemical potential of the water and ion phases:

μ^w=μ^w0 + 1 ρ_T^w

p−RT

Nion

k=1

^kc^k+Bwtr(E) (3.55) μ^k=μ^k₀+RT

M^kln (γkc^k)+z^kFcψ

M^k (3.56)

where μ^α₀ (α= w, k) is the chemical potential of the phase α at reference con- figuration. R is the universal gas constant, Fc Faraday constant, Bw the coupling coefficient,ψ the electric potential.^k, c^k,γk, M^kand z^k are the osmotic coeffi- cient, the concentration, the activity coefficient, the molar weight and the valance number for the kth ion species.

The works previously published by Hon et al. (1999) and Lai et al. (1991) have so far been summarized. On the basis of their work, the MECe model is developed as follows.

It is reasonably assumed that fskand fkjare negligible in comparison with fwsand fwkfor Eqs. (3.53) and (3.54). As such, the formulation for the momentum equation of the water/fluid phase is simplified to

fws(v^s−v^w)=

α=w,k

ρ^α∇μ^α (3.57)

Substituting the constitutive relations of the hydrogel mixture and each phase expressed by Eqs. (3.51), (3.55) and (3.56) into the momentum equations (3.46) and (3.57), we have

∇ ·(−pI+λstr(E)I+2μsE)=0 (3.58) fws(v^s−v^w)=φ^w

∇p+RT∇

k

(1−^k)c^k+Fc∇ψ

k

z^kc^k+Bw∇tr(E) (3.59) With the assumption that ø^kis negligibly small, and by Eqs. (3.2) and (3.8), one can write

∇ ·(φ^w(v^s−v^w))= ∇ ·v^s (3.60) Due to Eq. (3.60), Eq. (3.59) is rewritten as

∇∂u

∂t = ∇ (φ^w)²

fws

∇p+RT∇

k

(1−^k)c^k+Fc

k

z^kc^k∇ψ+Bw∇tr(E) (3.61) where u is the displacement of polymeric network matrix solid phase of the hydrogel.

As well known, the pressure results from the difference of the diffusive ionic concentrations between the hydrogel and the surrounding solution. In the MECe model, the ionic concentration is determined by the Nernst–Planck flux as follows:

Ji= −Dkc^k_,i− Fc

RTz^kDkc^kψ,i+c^kvi(k = 1, 2,. . ., Nion) (3.62) where i denotes the spatial direction xi, the subscript i after a comma indicates partial differentiation with respect to the variable xi and Dk is the diffusive coefficient of

the ion species k. The diffusion equation of the ionic species k is given as

∂c^k

∂t = −Ji,i+rk (k = 1, 2,. . ., Nion) (3.63) in which rk is the source term resulting from the chemical conversion of the molecules.

By Eqs. (3.62) and (3.63), the following diffusion–convection equations are obtained:

(Dkc^k_,i),i+Fcz^k

RT (Dkc^kψ,i),i= ∂ck

∂t +(c^kvi),i(k = 1, 2,. . ., Nion) (3.64) where rkis neglected due to the assumption of ideal solution.

For the MECe model, it should be noted that the externally applied electric field has important effect on the deformation of the hydrogels, and it is determined by the Poisson equation as

∇²ψ= −Fc

εε0

_N ion

k=1

z^kc^k+z^fc^f

(3.65)

in which the fixed charge density c^fis given by c^f = φ₀^wc^f₀

φ^w(1+tr(E)) = c^f₀

(1+tr(E)/φ₀^w) (3.66) whereεis the dielectric constant,ε0the permittivity of free space, z^fthe valence of the fixed charge groups, c^f₀the fixed charge density at reference configuration and φ₀^wthe volume fraction of the water phase at reference configuration.

The formulation of the MECe model has thus far been completed. It consists of the Nernst–Planck type of diffusion–convection equations (3.64) for the diffu- sive ionic concentrations, the Poisson equation (3.65) for the electric potential, the continuity equation of the hydrogel mixture (3.61) for the fluid pressure and the momentum equation of the hydrogel mixture (3.58) for the displacement of the hydrogel. They are coupled together to form a set of nonlinear partial differen- tial equations and solved numerically by a hierarchical iteration technique. In the inner iteration process, the diffusive ionic concentrations c^kand the electric poten- tialψ are computed first by solving Eqs. (3.64) and (3.65) simultaneously. Then substituting the converged concentrations c^k and the electric potential ψ into the outer iteration, the fluid pressure p and the hydrogel displacement u are obtained by Eqs. (3.58) and (3.61), respectively. In addition, ø^wand c^f required for solving these equations are calculated by Eqs. (3.5) and (3.66). Following the computa- tional procedure, the presently developed MECe model can be used for both the steady-state and transient simulations for analysis of equilibrium and kinetics of the electric-sensitive hydrogels.

For convenience of coding and computing, non-dimensional variables are defined as follows:

ζ¯= ζ Lref

;u¯ =_L^u_ref;c¯k =_c^c_ref^k ;c¯f =_c^c_ref^f ;ψ¯ = _ψ^ψ_ref =_α^F_RT^ψ;p¯= _p^p_ref = _βcref^p_RT (3.67) where ζ denotes the spatial coordinate variable, α and β are non-dimensional adjustable parameters.

Therefore, the non-dimensional form of the partial differential governing equa- tions of the MECe model can be derived as follows:

(Dk¯c^k_,i),i+αz^k(Dk¯c^kψ¯,i),i=L²_ref∂c¯k

∂t +Lref(¯c^kvi),i (3.68)

∇²ψ¯ = −F²_cL²_refcref

εε0RTα _N

ion

k=1

z^kc¯^k+z^f¯c^f

(3.69) βRTcref∇ ·(pI)= ∇(λstr(E)I+2μsE) (3.70) L²_ref

crefRT∇∂u¯

∂t = ∇ (φ^w)²

fws

β∇ ¯p+RT∇

k

(1−^k)¯c^k+α

k

z^kc¯^k∇ ¯ψ + 1

crefRTBw∇tr(E)

(3.71)