Development of the MECpHe Model

MECe) Model for Electric-Sensitive Hydrogel

4.2 Development of the MECpHe Model

The hydrogels responsive to the pH-electric coupled stimuli are able to convert chemical energy to mechanical one. They are often employed for the controlled drug delivery, where the solution pH and the externally applied electric field affect the release pattern (Qiu and Park, 2001). For simulation of the responsive behaviours of the hydrogels subject to the pH-electric coupled stimuli, a novel model is presented in this chapter, called the multi-effect-coupling pH-electric-stimuli (MECpHe) model. The contributions of the presently developed MECpHe model include the reformulations of the fixed charge density and the large deformation of the hydrogels.

For simplicity of describing the flux of the kth ionic species in solution, the con- vective transport of the ionic species is neglected here. If the Nernst–Planck theory is considered for ionic transportation, the flux of the kth ionic species in the system consisting of the hydrogel and the surrounding solution can be characterized by

Jk= −[Dk]

grad(ck)+^z_RT^kFckgrad(ψ)+ckgrad( lnγk)

(k=1,2,. . ., Nion) (4.1)

Fig. 4.1 Computational flowchart of the MECpHe model

where Jk, [Dk], ck and zk are the flux (mM/s), the diffusivity tensor, the concen- tration (mM) and the valence number of the kth diffusive ionic species.ψ is the electrostatic potential (V) andγk is the chemical activity coefficient. Nion is the number of total diffusive species in the system. F, R and T are the Faraday’s constant (9.6487×10⁴ C/mol), the universal gas constant (8.314 J/mol·K) and the absolute temperature (K), respectively.

The three terms on the right-hand side of Eq. (4.1) represent the diffusive flux due to the concentration gradient, the migration flux due to the gradient of electrical potential and the chemical flux associated with chemical activity coefficient.

According to the law of mass conservation, the change of the species k contained in volume with respect to time t can be characterized by the dif- ference between the fluxes entering and leaving the reference volume. If the chemical flux is neglected, the Nernst–Planck type of the mass conservation is

derived as

∂ck

∂t +div(Jk)= ∂ck

∂t +div

−[Dk]

grad(ck)+zkF

RTckgrad(ψ)

=0 (k=1,2,. . .,Nion)

(4.2)

which is coupled with the following Poisson equation to describe the spatial distribution of the electric potential in the domain:

∇²ψ= − F εε0

k

zkck+zfcf (4.3)

where ε is the relative dielectric constant of the surrounding medium and ε0 is the vacuum permittivity or dielectric constant (8.85418×10^–12 C²/Nm²). zf is the valence of the fixed charge groups attached onto the polymeric network chains of the hydrogel. For example, zf = −1 if the carboxylic acid groups are used as the fixed charges on the polymer chains. It is well known that the electroneutrality and constant field hypotheses are in fact the special cases of the Poisson equation.

Based on the Langmuir adsorption isotherm theory (Grimshaw et al., 1990), a relation between the fixed charge and the diffusive hydrogen ion concentration is presented to complete the Poisson–Nernst–Planck (PNP) system, whereby the concentration of the fixed charge group is determined by

cf =c⁰_f −c^b_f =c⁰_f − c⁰_f ·c_H+

K+c_H+ = c⁰_f ·K

K+c_H+ (4.4)

where cfand K are the concentration and the dissociation constant of the fixed charge groups attached onto the polymeric network chains within the hydrogel. c⁰_f is the total concentration of the ionizable groups in the hydrogel, c_H+ is the concentra- tion of diffusive hydrogen ions H⁺ within the hydrogel, and H is termed the local hydration of the hydrogel.

In the present MECpHe model, the total concentration of the fixed charge groups within the hydrogel at the relaxed state is defined as

c⁰_{f ,s}= n

V^s (4.5)

Then the total concentration of the ionizable groups in the hydrogel is obtained by c⁰_f = n

V = n

V^s+V^w = n V^s

V^s

V^s+V^w = n V^s

V^s

V^s+HV^s = n V^s

V^s

(H+1)V^s = c⁰_{f ,s} H+1

(4.6) Substituting Eq. (4.6) into Eq. (4.4), one can have

cf = c⁰_f ·K

K+cH = 1

H+1 · c⁰_{f ,s}·K K+cH

(4.7)

where the local hydration of the hydrogel is defined as H=V^w/V^s, namely 1+H=1+V^w

V^s = V^s+V^w V^s = V

V^s = 1

φ^s = 1

1−φ^w (4.8)

The volume fractions of the interstitial water and polymeric solid phases are thus written as

φ^w= H

1+H (4.9)

φ^s= 1

1+H (4.10)

Since the volume fraction of the ion species øⁱis negligibly small when compared with ø^wand ø^s, the saturation equation is simplified to

φ^w+φ^s≈1 (4.11)

The relation between the volume fractions of the interstitial water and polymeric network solid phases is then given by

φ^w≈1−φ^s =1−V^s

V =1−V^s V0

V0

V =1−φ^s₀·J (4.12) where J=dV0/dV is the volume ratio of apparent polymeric network matrix solid phase and may be formulated by the Green strain tensor E of the apparent polymeric solid phase as follows (Hon et al., 1999):

1 J =)

1+2F1(E)+4F2(E)+8F3(E) (4.13) where F1(E)=tr(E), F2(E) and F3(E) are the first, second and third invariants of Green strain tensor E, respectively.

The Green strain tensor can be expressed in terms of displacement gradients (Belytschko et al., 2001),

Eij =1 2

∂ui

∂Xj + ∂uj

∂Xi +∂uk

∂Xi

∂uk

∂Xj

(4.14) where Xiand Xjare the components of the position vector in the initial configuration, ui, ujand ukare the displacements. In one-dimensional case,

E11= 1 2

∂u1

∂X1+ ∂u1

∂X1 +∂u1

∂X1

∂u1

∂X1

= 1 2

2du

dX + du

dX 2

(4.15)

Three invariants of the deformation gradient tensor are defined as Lai et al.

(1974),

F1=E11+E22+E33 (4.16)

F2=**

**E11 E12

E21 E22

****+**

**E11E13

E31E33

****+**

**E22 E23

E32 E33

**** (4.17)

F3=**Eij**=

****

**

E11E12 E13

E21E22 E23

E31E32 E33

****

** (4.18)

For one-dimensional case, F1=E11 =1

2

2du dX +

du dX

2

= du dX +1

2 du

dX 2

(4.19)

Using Eqs. (4.9) and (4.12), one can have φ^w= H

1+H =1−φ0^sJ (4.20)

The local hydration of the hydrogel H is rewritten as H= 1−φ₀^SJ

φ₀^SJ (4.21)

Substituting Eqs. (4.13) and (4.21) into Eq. (4.7), the density of fixed charge groups is finally derived as follows:

cf = c^s_m0·K·φ₀^s (K+cH)√

1+2F1(E)+4F2(E)+8F3(E) (4.22) By substituting Eq. (4.19) into the Eq. (4.22), the one-dimensional form of the fixed charge density is obtained as

cf = c^s_m0·K·φ₀^s (K+cH)

+

1+2_dX^du +

du dX

2 (4.23)

As well known, the first Piola–Kirchhoff stress tensor P is a kind of expatri- ate, living partially in the deformed (current) configuration x and partially in the reference (initial) configuration X where x = X+u, and it is unable to measure.

Because of the absence of symmetry in the first Piola–Kirchhoff stress tensor P, it is seldom used in constitutive equations. However, the second Piola–Kirchhoff stress tensor S is symmetric and is often used as the stress measure for large deforma- tion. The relation between the first Piola–Kirchhoff stress tensor P and the second

Piola–Kirchhoff stress tensor S is written as

P=SF^T (4.24)

where F is the deformation gradient tensor and defined as

F=I+ ∇u (4.25)

The second Piola–Kirchhoff stress tensor S is given by

S=CE−posmoticI (4.26)

where C is the material tensor and E is the Green–Lagrangian strain tensor used as the strain measure

E= 1

2(F^TF−I) (4.27)

The nonlinear mechanical governing equation for large deformation of the smart hydrogel is finally written as follows:

∇ ·[(CE−posmoticI)F^T]=0 (4.28)

For one-dimensional analysis,

(λs+2μs)

d²u dX² +3du

dX d²u dX² +3

2 du

dX 2

d²u dX²

−dposmotic

dX =0 (4.29)

So far the development of MECpHe model has been completed. It is composed of the Nernst–Planck diffusion equation (4.2) for the diffusive ion concentrations, the Poisson equation (4.3) with the fixed charge density (4.22) or (4.23) for the electric potential and nonlinear mechanical equation (4.28) or (4.29) for the large displacement of the smart hydrogel.

The MECpHe governing equations are associated with the boundary conditions of the diffusive ion concentrations and the electric potential, which are imposed at the edges of the surrounding solution

c|Anode= c|Cathode=c^∗ (4.30)

ψ|Anode=0.5Veandψ|Cathode= −0.5Ve (4.31) where c^∗is the initial ionic concentration of the bath solution and Vethe externally applied electric voltage.

Boundary condition of the mechanical deformation is imposed at the hydrogel–

solution interfaces

(λs+2μs)

du dX+1

2 du

dX 2

=RT

N_ion

k=1

(ck−c⁰_k) at X=(L±h)/2 (4.32) In order to prevent the hydrogel from undergoing rigid-body motion, a point constraint is requested in the middle of the hydrogel

u=0 at X=L/2 (4.33)