Anomaly in the coefficient of performance of the volume phase transition process of poly(N-isopropylacrylamide) gels induced by mechanical stress

Introduction

It is well known that poly(N-isopropylacrylamide) (PNIPA) hydrogels show the volume phase transition if the crosslink density and polymer content are appropriately controlled.^{1, 2, 3} In our previous paper,⁴ we defined the coefficient of performance (the efficiency) (c) for a gel actuator driven by the volume-phase transition. During the swollen-to-collapsed (that is, shrinking) transition of PNIPA gels due to weights, heat is absorbed, and the shrinkage generates mechanical work.^{4, 5, 6, 7} Hereafter, we refer to this latent heat as the transition enthalpy (ΔH) because a constant pressure condition is assumed for the transition. Using ΔH, c in this isothermal process can be defined as⁴

where, f is the force acting on the gel, and Δl is the length change of the gel due to transition. The product fΔl corresponds to the work performed by the weights (ΔW): ΔW=fΔl. During the shrinking transition of PNIPA gels due to weights, Δl is negative while ΔH is always positive. Thus, c>0. We also demonstrated that Δl and ΔH are related to each other, as shown below.⁴

Here, T is the temperature, and the subscript ‘coex’ refers to the coexistence curve for the swollen and collapsed phases of the gels. The temperature T in equation (2) is the transition temperature. By combining equations (1) and (2), we obtain⁴

This equation relates c of the gel actuators driven by the shrinking transition to the f-dependent transition temperature of the gels.

We have experimentally estimated the values of c for a PNIPA gel actuator.⁴ Although the values of c were very small and appeared to be far from the maximum value, through experimentation we noticed the possibility that c becomes negative. The negative value may be regarded as an anomaly, which originates because the PNIPA gel (linear) actuator utilizes the volumetric (three-dimensional) change during the transition. In this paper, using model analysis and experimentation, we investigate the effects of mechanical stress on the volume phase transition of PNIPA gels.

Model

Here, we consider the volume phase transition of a cylindrical gel. Let l_s0 and S_s0 be the length and the cross-sectional area, respectively, of the cylindrical gel in the swollen state without a load. Then, we designate the length and the cross-sectional area in the collapsed state without a load as l_c0 and S_c0, respectively. To account for the change in dimensions of the gel due to the transition, we assume that

where α is the liner swelling ratio during the transition, and, typically, α>1. Volume change due to the transition without a load occurs isotropically. On the other hand, we designate the length and the cross-sectional area in the swollen state under a load as l_s and S_s, respectively. Similarly, the length and the cross-sectional area in the collapsed state under a load are written as l_c and S_c. The deformation due to the application of weights is described by the stretch ratios as

where λ_s is the stretch ratio in the swollen state and λ_c is the stretch ratio in the collapsed state. For the uniaxial deformation, we assume incompressibility for both states. Thus, we obtain l_s0S_s0=l_sS_s and l_c0S_c0=l_cS_c.

Regarding the stress that acts on the gel, the true stress in the swollen state (σ_s) and that in the collapsed state (σ_c) are written as

Now that we defined stretch ratios and stresses, we introduce the constitutive equation, which connects the stretch ratio to the stress, applicable to both states. A simple equation that is applicable to large deformations must have the form given by the classical theory of rubber elasticity:⁸

Equation (7a) refers to the constitutive relation of the swollen gel and equation (7b) refers to the gel in the collapsed state. Furthermore, G_s and G_c in the equations are the moduli of the gels in the swollen and collapsed states, respectively. A simple relationship exists between G_s and G_c, if the number of crosslinks is conserved before and after the transition.⁹ However, physical crosslinks other than original chemical crosslinks are introduced in the collapsed phase for the PNIPA gels.¹⁰ Thus, we regard them as independent and therefore define the ratio G_c/G_s as β (specifically, β=G_c/G_s). For β we can expect β>1.

By combining equations (4a and 4b), (5a and 5b), (6a and 6b) and (7a and 7b), the constitutive relations for the swollen and collapsed states are obtained. The stress ratio (σ_c/σ_s) yields

By solving the above equation with respect to λ_c, we obtain the exact solution for λ_c:

Because Δl is given as

we have

where,V_s0=S_s0l_s0, and the expression of f, as a function of λ_s, obtained from equations (6a) and (7a) is written as

Because equations (10) and (11) are commonly expressed by the parameter λ_s, c can be regarded as a function of f if we assume that ΔH is independent of f. This simplified assumption for ΔH was used because, currently, there are no data on the f-dependent ΔH (more preferably, ΔS (transition entropy)). In addition, the f-dependence curve of c provides information on how T varies with f because the f-dependence of T is written as

where T(f) stands for the transition temperature at f and T₀ is the transition temperature at f=0. This is an approximated equation derived from equation (3). It is valid as long as (T−T₀)/T₀<<1. On the basis of equation (12), the formal expression of T becomes

It should be noted that (dT/df)=0 holds at f giving c(f)=0.

Materials and methods

PNIPA gel was prepared by copolymerization of N-isopropylacrylamide (NIPA) and N,N'-methylenebisacrylamide (BIS) in water at 25 °C. Ammonium peroxodisulfate and N,N,N',N'-tetramethylethylenediamine were used as an initiator and as an accelerator, respectively. The ratio of molar concentrations (NIPA)/(BIS) was set at 120, and the total monomer concentration of these reagents was kept at 10 wt%. The experimental procedure was identical to the previously published procedure.^{4, 6, 7} The temperature dependence of length and diameter of the cylindrical PNIPA gel was measured using a laboratory-made water bath. The accuracy of temperature control was ~±0.02 °C. A combination of two different steel spheres with different dimensions (12.7 mg/sphere and 29.4 mg/sphere) and with an applied buoyancy correction was used for loading. A cyanoacrylate adhesive was used to glue spheres to the ends of gels (or between the spheres). The length and diameter of the gel specimen with weights were recorded using a combination of a CCD camera and a DVD recorder. The gel dimensions with a small sphere were 9.93 mm (l, length) and 0.49 mm (d, diameter) at 33 °C (starting temperature). The linear swelling ratio at the transition temperature α for this gel was 1.39.

Results and Discussion

In the preceding section, we presented the exact solution of c and the formal expression of T. These are powerful for numerical calculations but may have difficulties. On the basis of these expressions, it is difficult to see how the f-dependence curves behave. Here, we build the approximated expressions, and then determine the key parameter controlling the global shape of the c and T curves.

In the small f region, where λ_s≅1 and λ_c≅1, Δl becomes Δl≅l_s0((1/α)−1) (see equation (9)), which becomes a negative constant. This indicates that c is a linearly increasing function of f, which begins from zero because c ∝−fΔl. In the large f region, where f∝λ_s, c behaves as c ∝ f² because we expect λ_c≅(α²/β)λ_s (equation (8a)) and Δl≅l_s0((α/β)−1)λ_s (equation (9)). By combining the behavior of c when f is large with that when f is small, c can be written as

where a and b are the numerical constants and b>0. Once c is obtained, the f-dependence curve for T is written from equation (12) as

Equations (14) and (15) retain the essence of exact solutions (that is, equations (10) and (13)). However, equations (14) and (15) are simple and can be easily applied to the experimental data analysis. In addition, once the parameters for the f-dependence curve of T are determined, the curve for c is automatically determined.

For both c and T, the shape of f-dependence curves on a global scale differs and depends on the Δl sign. At small f, Δl is always negative. However, when f is large, the sign of Δl is controlled by the value of (α/β) because Δl≅l_s0((α/β)−1)λ_s, when f is large. This condition is applicable to exact solutions of c (and also T) without a significant correction. If (α/β)<1, Δl remains negative. Thus, c is always positive, and the f-dependence of T is an increasing function of f over the entire region of f. Therefore, a>0 in addition to b>0. However, if (α/β)>1, Δl becomes positive and c becomes negative, which means that a<0. In this case, c behaves as c≅af(f−(b/a)). Therefore, c changes its sign at the point of f=(b/a), which we designate as f_inv (the force at the inversion point). At f>f_inv, c becomes negative. Because c is the coefficient of performance, this negativity appears to be anomalous. Qualitatively, this inversion originates from the situation that the decrease in cross-sectional area becomes more effective than the increase of rigidity in the collapsed state. Specifically, this anomalous phenomenon is related to the fact that the volume phase transition is a three-dimensional phenomenon. However, the lifting-up motion, which linear actuators utilize, is only one-dimensional. Concerning the T curve, we have (T−T₀)/T₀≅af(f−(2b/a))/2. This clearly shows that the T curve possesses a peak at f_inv.

The f-dependence curves of c and ΔT/T₀ calculated using equations (10) and (11) with α=2 and β=3 are shown in their reduced (dimensionless) forms in Figure 1. Here, ΔT stands for ΔT=T−T₀. Because the highly extensible gels usually have smaller values of α, this combination must be the simplest example for the case of (α/β)<1. The f-dependence curves of c and T in this case are concave. The f-dependence curves of T, which are obtained from experiments, never show this type of curvature,^{5, 6, 7} and, thus, the condition (α/β)<1 is not satisfied by the real gels. In Figure 2, similar curves with α=3 and β=2 (an opposite combination to what is shown in Figure 1) are shown as an example of (α/β)>1. Both curves are convex in shape. At large f, both curves are located in the negative zone. As stated previously, the fact that the c curve becomes negative at large f is an anomaly. The f-dependence curve of T shows a maximum at f_inv because the integrand in equation (12) becomes negative at f>f_inv.

Figure 1

Reduced coefficient of performance c (solid) and the reduced increment of transition temperature ΔT/T₀ (dashed) plotted as a function of the reduced force f for α=2 and β=3. The quantity c is reduced by (G_sV_s0/ΔH), ΔT/T₀ is reduced by (G_sl_s0/ΔH), and f is reduced by G_sS_s0. For the definition of parameters, see the text.

Figure 2

Reduced coefficient of performance c (solid) and the reduced increment of transition temperature ΔT/T₀ (dashed) plotted as a function of the reduced force f for α=3 and β=2. The definition of reduced parameters is the same as in Figure 1.

Figure 3 shows the f-dependence curve of T, which is obtained for the PNIPA gel used in this study. The investigated f-region corresponds to the reduced force range from 0 to 0.7 in Figure 2. The transition temperature was determined as an average value obtained during heating and cooling. The solid curve represents the best fit for the experimental data determined using equation (15) (giving a=−614.4 and b=0.001432) and agrees well with the experimental data. The transition temperature increases monotonously with increasing f, and the curvature of the dependence curve appears convex. The convex shape is common for the real PNIPA gels because, as stated previously, other examined PNIPA gels showed similar behavior.^{5, 6, 7} Figure 4 shows the f-dependence curve of c, which is created using equation (14) with the same values of a and b that were determined in Figure 3. The curve shows a maximum at approximately f=7 × 10⁻⁴ N, which is significantly involved in the f region that was examined for the T curve. The emergence of the maximum suggests that the anomaly, when c becomes negative, occurs for this PNIPA gel.

Figure 3

Force f-dependence of the transition temperature T for the PNIPA gel.

Figure 4

Coefficient of performance c of the PNIPA gel as a function of f.

Conclusions

We have presented a model that describes force dependence of the transition temperature as well as the coefficient of performance for the shrinking transition for the PNIPA gel actuators. By analyzing the asymptotic behavior of the c curve, the global nature of c and T curves is controlled by the α/β ratio, and the case of α/β>1 corresponds to real gels. The numerical simulation results for α/β>1 predict an anomaly that the coefficient of performance becomes negative when f is large. In addition, at f_inv for the c curve, the transition temperature has a maximum. The experimental data for T for a real PNIPA gel was explained well using the approximated model with α/β>1. The model calculation predicts that the anomaly occurs for the PNIPA gel.