Introduction
1.2 Historical Development of Modelling Hydrogel
1.2.2 Transient Modelling for Kinetics of Smart Hydrogels
1.2.2.1 Mathematical Models and Simulations
The kinetics process of swelling of the smart hydrogels is a complicated phe- nomenon, which generally involves three successive steps: (1) the diffusion of water molecules into the polymer system; (2) the relaxation of the hydrated poly- mer chains and (3) the expansion of the polymer network in surrounding aqueous solution. The swelling kinetics of the hydrogel may be further divided into more detailed steps. For example, the kinetics process of the swelling or shrinking of poly(N-vinylimidazole-co-sodium styrene sulfonate) hydrogels includes the follow- ing simultaneous steps (Valencia and Pierola, 2002): (1) the diffusion of water towards the hydrogel; (2) the chain disentanglement; (3) the sodium–proton inter- change through the external bath boundary; (4) approaching of chains to allow the interaction of sulfonate groups with neighbouring protonated imidazole moieties and (5) the diffusion of water outside the hydrogel. On the other hand, the wet- ting process of polymer dry gel with cylindrical shape is divided into three steps (Ji and Ding, 2002): (1) swelling of the gel in the radial direction with cusp-like patterns evolving on the surface; (2) shrinking of the gel in the radial direction and swelling in the axial direction and (3) the re-swelling of the gel to the final stage and eventually disappearing of patterns.
Generally, two processes are employed to describe the diffusion of water into the polymer matrix, namely the diffusion of the solvent into the swollen matrix and the advancement of the swollen–unswollen boundary as a result of polymer relaxation. When the rate-determining step is the diffusion of the solvent into the swollen matrix, there is a linear relation between the water uptake and time1/2, the system is regarded to exhibit the Fickian behaviour. In contrast, if the advancement of the swollen–unswollen boundary is slower than the diffusion of the solvent in the swollen polymer, the zero order of kinetics is achieved with the water uptake time, and the system is regarded to exhibit the non-Fickian behaviour. In pure water condition, the diffusion of water generally approaches to the Fickian behaviour.
However, the diffusion of water into the hydrogels under solution conditions fol- lows the non-Fickian behaviour due to the dominance of polymer relaxation. The swelling of the poly(NIPAAm) hydrogel in the gelated corn starch aqueous solution is determined mainly by the relation of hydrated polymer chains, instead of the water diffusion rate (Zhang and Zhuo, 2000c). The starch may act as the long graft-like chains of the poly(NIPAAm) hydrogels, which provides the channels for water to be released during the shrinking process. It is claimed that the molecular relaxation controls the velocity of water uptake of hydroxypropylmethyl cellulose (HPMC), carboxymethylcellulose-sodium (NaCMC) and mixed HPMC/NaCMC hydrogels irrespective of pH of the aqueous phase (Michailova et al., 2000), in which there is an inversely proportional dependence between the viscosity and the water uptake velocity.
Three typical transient models developed with different complexities are reviewed here for simulation of the kinetics of the volume variation of the smart hydrogels. They are the phenomenal model, the power law model and the multi-component diffusion model.
Phenomenal Model
The phenomenal model is based on experimental observation for describing the extensive polymer swelling (Schott, 1992) and is written as follows:
t
W =A+Bt (1.34)
where W is the water uptake at time t, B=1/W∞is the inverse of the maximum swelling W∞and A=1/(dW/dt)0is the reciprocal of the initial swelling rate.
Equation (1.34) implies the second order of kinetics of swelling for volume vari- ation of the hydrogels, which is expressed as dW/dt=kr(W∞−W)2. The specific rate kr is a constant related to the parameter A as kr = 1/(AW∞2). By substituting the swelling data into Eq. (1.34), it is found that the plots of t/W against t can give straight lines with good correlation coefficients for experimental observations.
Obviously, the phenomenal model cannot provide accurate information about the mechanism of the volume variation of hydrogel. In most cases, the initial phase of the volume variation of hydrogel follows the zero order of kinetics of swelling, instead of the second order of kinetics. This means that the model is suit- able for simulation of kinetics of the hydrogel volume variation after the initially fast swelling process. However, at least it provides a simple approach to correlate experimental data.
Power Law Model
As well known, the diffusion of water into the hydrogel system results in the volume variation of the hydrogels. The water transport in the hydrogels can be characterized either by Fick’s second law or by a more advanced equation of anomalous diffusion, which incorporates a diffusive term coupled with a pseudo-convective velocity term.
A simple method of describing water transport in polymers is given via the power law (Astarita, 1989)
Mt/M∞=ktn (1.35)
where Mt is the mass of water gained or lost at time t, M∞is the initial mass of water contained in the polymer, k is a constant and n is the diffusion exponent.
Usually the mechanism of the water transport in the hydrogels may be charac- terized by the diffusion exponent n and also by two limited cases of this power law model, n=0.5 and n=1. When n=0.5, the water transport is controlled exclu- sively by the chemical potential gradient, little or no volume variation occurs during the transport. This is referred to as Case I diffusion or Fickian diffusion, and then the volume variation can be described by a diffusion coefficient. If n=1, the rate of the volume variation of the hydrogels is proportional to time t, in which the stress relaxation controls the kinetics of the volume variation of the hydrogels and the vol- ume variation counteracts the mechanical stresses produced by water transport. This
is termed Case II diffusion, where the velocity of the waterfront describes the kinet- ics process of the volume variation. The intermediate case called the anomalous diffusion occurs when the water transport is proportional to tn, where 0.5<n<1.
Actually the water transport in the hydrogels may result from a mixture of Case I and Case II processes, in which both the waterfront velocity and diffusion coeffi- cient are required to characterize the volume variation process (Chou et al., 2000).
A model proposed can account for Case I, Case II and anomalous water transport processes (Harmon et al., 1987), in which the total flux J consists of two compo- nents, One is due to the diffusion with a concentration gradient and the other is due to the stress relaxation of polymer chains with a propagation speed v. For example, for one-dimensional volume variation of the hydrogels
J= −D∂C
∂X +v(C−C0) for 0≤ |X| ≤1 (1.36) where C =C(X) and C0 =C(0) are the concentrations at points X and X = 0, respectively. The diffusion coefficient D and the velocity v come from Case I and Case II water transport processes. According to the law of mass conservation, one can have
∂C
∂t =D∂2C
∂X2 −v∂C
∂X for 0≤ |X| ≤1 (1.37)
After integrating the concentration over the domain with the boundary condition of constant surface concentration C0at X= ±1, the variation of water mass Mtat time t associated with the equilibrium water mass M∞is written as follows (Harmon et al. 1987):
Mt
M∞ =1−2 ∞ n=1
λ2n
1−2 cosλnexp
−2Dvl
βn4
1−2Dvl cos2λn
exp
−βn2Dt l2
(1.38)
where
λn= vl
2Dtanλn (1.39)
βn2= v2l2
4D2 +λ2n (1.40)
where the rootsλn (n=1, 2, 3, ..., ∞) of Eq. (1.39) are used in Eqs. (1.38) and (1.40), respectively.
For a special case where v is equal to zero, Eqs. (1.39) and (1.40) are simplified toλn=(n+1/2)πandβn=λn, respectively. Equation (1.38) is then reduced to
Mt
M∞=1−∞
n=1
8
(2n+1)2π2 exp
−(2n+1)2π2 Dt
l2
(1.41)
The short-time limited expression for Eq. (1.41) is thus obtained as Mt
M∞ = 4
π0.5 Dt
l2 0.5
(1.42) It is observed from the above equation that the relation between Mt/M∞ and t0.5 yields a straight line. Based on the slope of the graph, the value of diffusion coefficient D is calculated. As a good example of applications, the experiment of the water transport in crosslinked 2-hydroxyethyl methacrylate (HEMA) hydrogels is in excellent agreement with the prediction by the model (Chou et al., 2000).
In order to achieve better approximation, the exact solution of Eq. (1.37) is derived, termed the Berens–Hopfenberg model (Enscore et al., 1977; Berens and Hopfenberg, 1978), and it is expressed as follows:
Mt
M∞=φF
1−
n=∞
n=1
8
(2n+1) π2exp
−D(2n+1)2t 4 l2
+φR(1−exp(−kt))
(1.43)
where k is the first order of relaxation constant, øFand øRare the fractions of water uptake contributed by the Fickian diffusion and the chain relaxation, respectively.
This heuristic model can be used for analysis of the overall water uptake in terms of the Fickian and non-Fickian contributions, which leads to the determination of both the diffusion coefficient D and the characteristic relaxation timeτ that is defined as the reciprocal of the constant k.
It is noted that the model mentioned above does not take into account the interac- tions between the polymer and the solvent, which significantly influence the water diffusion and the stress relaxation of polymer chains. In addition, the diffusion coef- ficient D is assumed to be constant in the model. In fact, the hydrogel network structure always changes with the volume variation of the hydrogel, which results in a varying diffusion coefficient. However, at least when 0 < Mt
M∞ < 0.60, the model can predict well the kinetics process of volume variation of the smart hydrogels.
Multi-field Model
The kinetics of the volume variation of the hydrogels involves the water diffusion and the mechanical deformation simultaneously. For the ionic hydrogels, the volume variation of the hydrogels is strongly dependent upon the diffusions of all species and the variation of electrical potential. A good model for the volume variation of the hydrogels should be able to incorporate the multi-field effects.
Modelling of the ion transport could be conducted by many models within var- ious theoretical frameworks. It is sufficient to model the ion transport as follows:
∂ck
∂t = ∂
∂x
Dk∂ck
∂x
+ ∂
∂x
cZkFDk
RT
∂φ
∂x
(1.44)
where ck is the concentration of the ion species k, Dkis the binary diffusion coef- ficient of the ion species k, c is the total concentration, zk is the valence of the ion species k, ø is the electrostatic potential, F is the Faraday constant, R is the universal gas constant and T is the absolute temperature. The concentration of ionized pen- dant groups in the polymer network is always in equilibrium with the concentration of hydrogen ions in the polymer network.
The kinetics of swelling of the ionic hydrogels induced by pH change is modelled by considering the diffusion of hydrogen ions, which is governed by the chemical diffusion equation together with the mechanical equation (Chatterjee et al., 2003).
This model assumes that the mechanical deformation of the polymer network occurs instantaneously with the diffusion of hydrogen ions. The mechanical equation takes into account the deformation of the polymer network that occurs during the diffusion of hydrogen ions into the hydrogel, namely
k =φ
−Dk∂ck
∂x −μkzkck∂ψ
∂x
+ckU (1.45)
wherekis the flux of the kth ion, ø is the gel porosity and U is the area-averaged fluid velocity relative to the polymer network. The diffusion rate inside the hydro- gel is related to the diffusion in aqueous solution through the obstruction model as follows:
Dk
Dk
= K
2+H 2
(1.46) where H is the hydration of the hydrogel and it is defined as the ratio of fluid volume to polymer volume.
Continuity condition for the divergence of each ionic flux is given by
∂
∂t
HCk+Hcbk
= −∂ (αk)
∂X (1.47)
where cbk is the concentration of the ion k that can be reversibly bound to the polymeric fixed charge, X is the Lagrangian coordinate system associated with the hydrogel andαis the total hydrogel area normalized to its initial area.
The concentration of the ion k reversibly bound to the polymeric chains (cbk) in the presence of chemical reactions is calculated by (Grimshaw et al., 1990)
cbk =c0f H
ck
(K+ck) (1.48)
where c0f is the total concentration of ionizable groups within the hydrogel before the volume variation.
By rearrangement of the above equations, a nonlinear diffusion–reaction equa- tion for the concentration of H+ions in the hydrogel is obtained as
∂
∂t
cH
H+ c0f
K+cH =−∂
∂X
αφ
DH∂cH
∂x +μHzHcH∂ψ
∂x
−αcHU
(1.49)
whereD¯His the diffusion rate of hydrogen ions within the hydrogel, cHis the inter- nal concentration of hydrogen ions and cbH is the concentration of hydrogen ions reversibly bound to the fixed charges of the hydrogel.
The presence of buffer in the solution increases the apparent diffusion rate of hydrogen by providing an alternate path for diffusing hydrogen ions between the hydrogel and bath solution. Buffer augmented transport of hydrogen ions under cer- tain conditions may result in the apparent diffusion rates of hydrogen, which is several orders of magnitude higher than the diffusion coefficient of hydrogel alone (Ohs et al., 2001). It is found that the kinetics of the water uptake in the buffered solutions is markedly faster than that in the unbuffered solutions for the hydropho- bic ionizable copolymer gels composed of PEG, AAc and styrene, though the water uptake is contributed by both the mechanisms of diffusion and polymer chain relax- ation (Bajpai and Shrivastava, 2001). The influence of the buffer on the transport of hydrogen ions can be modelled by including additional terms in the continuity equation of hydrogen ions within the hydrogel
∂
∂t
HcH+HcbH+HcHB
= −∂(αH+αHB)
∂X (1.50)
where cHBis the concentration of hydrogen ions bound to the buffer,His the flux of hydrogen ions andHBis the flux of hydrogen ions bound to the buffer.
cHB= cTcH
(KB+cH) (1.51)
where KB is the dissociation constant of the buffer and cT is the total buffer concentration.
The flux of the buffer is proportional to the flux of the hydrogen ions:
HB= DHB
DH
cT
KB+cH
H (1.52)
where DHBis the diffusion rate of buffer in the hydrogel.
The Poisson equation is used to calculate the electrostatic potential
∂2ψ
∂x2 = − F εε0
N
k=1
zkck+zfcf (1.53)
The chemical diffusion equation then becomes
∂
∂t
HcH+ c0fcH
K+cH + HcTcH
KB+cH
= − ∂
∂X
α H
1+H 1+DHB
DH
cT
KB+cH
DH∂cH
∂x
(1.54)
The change in the hydration of the hydrogel is calculated by the following mechanical equation that describes the forces and fluid flow:
∂H
∂t = ∂
∂X
ak
−∂ (Mε)
∂x +zfcfF∂ψ
∂x
(1.55) where kis the hydraulic permeability of the hydrogel, M is the bulk modulus of the hydrogel andεis the compressive strain.
The advantage of this model is the capability of providing a good evaluation of the volume variation of the hydrogel by integration of the chemical, electrical and mechanical interactions. It captures the variations of the concentration distributions of all ionic species with time. However, this model is valid only for the ionic hydro- gels, and the deformation of the polymeric network matrix is characterized by the volume-based hydration H, instead of the displacement vectors. The mathematical complexity also limits its extensive applications.
1.2.2.2 Key Parameters in Transient Modelling for Kinetics of Hydrogels