Tanaka-Fillmore theory

The Tanaka-Fillmore theory (Tanaka et al. 1973 Tanaka and Fillmore 1979) is a purely mechanical theory based on the theory of elasticity (Lifshitz et al. 1986). The gel network is described by a displacement vector field uir). This field describes how every point in a certain state is shifted compared to the equilibrium reference state. A point that is at r in the reference state is shifted to r (r) = r -F u r) in the deformed state. In the reference state, u(r) is zero for all r. Expansion and contraction of the network corresponds to solvent flowing into or out of the gel. 

In the swelling theory by Doi (2009), like in the Tanaka-Fillmore theory, the hydrogel is treated as a eontinuous meehanieal objeet deseribed by a deformation field F and aceording inner tensions described by die stress tensor field ay. However, instead of ealeulating cy from values of the bulk modulus and the shear modulus that are assumed to be given, it is derived from the free energy density... 

One of the problems faced in applying PNlPAAm hydrogels is that the response rate to temperature changes is very slow, which restricts their wider applications, such as on-olf valves and artificial muscles. According to the Tanaka-Fillmore theory [168], where t, R, and D... 

Fig. 10. Plot of normalized approach to equilibrium thickness against the square root of time for a temperature-sensitive 10x4 PNIPAAm gel sheet swelling and shrinking between 10 and 25 °C. Shown are the curve fits to the kinetic data of theory developed from equations of motion (Tanaka and Fillmore theory) [60]. The equilibrium degree of swelling is 17.0 at 10°Cand 11.1 at 25 °C the diffusion coefficients obtained from the curve fits are 2.3 x 10"7 ctn2/s for swelling and 3.6 x 10 7 cm2/s for shrinking [121]...

Therefore advanced models have been presented in the twenty-first eentury that reconeile flie Flory-Rehner thermodynamie theory with the meehanieal approach of the Tanaka-Fillmore frteory. 

The swelling kinetics of stimuli-responsive hydrogels is described by the theory of cooperative diffusion (Sect. 3.2.2 and 3.2.3, (Tanaka and Fillmore 1979)). Besides the cooperative diffusion constant, which is specific for each solvent-polymer network combination (see Eq. (3.2.9)), the time behaviour of active hydrogel components can be as well influenced by further design-dependent aspects (Richter 2006, 2008b). 

The theory of Tanaka and Fillmore (Tanaka and Fillmore 1979) describes only the unloaded case. For actuator applications this case is almost impossible. 

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