Quantifying the bending of bilayer temperature-sensitive hydrogels

Abstract

Stimuli-responsive hydrogels can serve as manipulators, including grippers, sensors, etc., where structures can undergo significant bending. Here, a finite-deformation theory is developed to quantify the evolution of the curvature of bilayer temperature-sensitive hydrogels when subjected to a temperature change. Analysis of the theory indicates that there is an optimal thickness ratio to acquire the largest curvature in the bilayer and also suggests that the sign or the magnitude of the curvature can be significantly affected by pre-stretches or small pores in the bilayer. This study may provide important guidelines in fabricating temperature-responsive bilayers with desirable mechanical performance.

1. Introduction

Stimuli-responsive hydrogels have been the subject of much scientific attention recently [1–5]. This category of hydrogels can respond to very small changes in temperature, pH, humidity, etc. [6–12], with dramatic changes in their properties, which can therefore be potentially harvested for diverse applications. For example, stimuli-responsive hydrogels with homogeneous swelling/shrinkage are desirable for applications such as artificial muscles [13] and soft machines, while those with inhomogeneous swelling/shrinkage can be employed as chemical valves [14–17], manipulators[18–21], crawlers [9,22], etc.

Since temperature can be easily controlled, temperature-responsive hydrogels are quite popular in some studies. With a change in the temperature, these hydrogels can undergo a phase transition, upon which their volume can discontinuously change by up to 1000 times [23]. For the application of temperature-responsive hydrogels [24–29], both good responsive properties and good mechanical properties are necessary to generate desirable forces or deformations.

Unfortunately, poor responsive or poor mechanical properties of most fabricated temperature-responsive hydrogels have severely limited their practical use [30]. However, recently, by designing an asymmetrical distribution of nanoclays across the hydrogel thickness, a bilayer poly-(N-isopropylacrylamide) (PNIPAM)-clay hydrogel with both good responsive bending properties and good mechanical properties was successfully fabricated [30]. It was clearly demonstrated that the bilayer hydrogel can tightly capture and easily release target objects via thermo-triggered bending [30].

The bending of bilayer temperature-responsive hydrogels depends on many factors, such as shape, size, inhomogeneous architecture, etc. [31]. To facilitate a structural design with desirable bending properties or precise control of their mechanical performance when in use, here we develop a mechanics theory to investigate the evolution of the curvature of bilayer temperature-responsive hydrogels made of PNIPAM when subjected to temperature change. Our subsequent theory prediction indicates that there can exist an optimal thickness ratio of the bilayer to acquire the largest curvature. Our analysis also suggests that pre-stretches or small pores can significantly affect the evolution of the curvature of the bilayer with temperature.

2. Theory

As shown in figure 1a, a beam is formed by two layers, layer a and layer b, of hydrogels made of PNIPAM but with different cross-link densities. The bilayer beam is initially submerged in water at T₀ and is flat. The initial principal stretches on hydrogels relative to dry polymers, which can be, for example, due to free swelling, are denoted as ${\lambda }_1^{i0(\mathrm{a})}$, ${\lambda }_2^{i0(\mathrm{a})}$ and ${\lambda }_3^{i0(\mathrm{a})}$ for layer a, and as ${\lambda }_1^{i0(\mathrm{b})}$, ${\lambda }_2^{i0(\mathrm{b})}$ and ${\lambda }_3^{i0(\mathrm{b})}$ for layer b.

Figure 1.

(a) A bilayer beam composed of two layers of hydrogels made of PNIPAM with different cross-link densities is initially flat at temperature T₀. (b) When the temperature varies, the bilayer can bend. (c) The relative volume of free-swelling hydrogels changes with temperature. (Online version in colour.)

When the temperature changes, the bilayer beam may bend with a curvature of κ = 1/R along the interface between the two layers, as illustrated in figure 1b. In the local coordinates, the corresponding principal stretch along the x-direction at the interface in the current state relative to that in the initial state is ${\lambda }_{10}^{\mathrm{\prime }}$. The corresponding principal stretches along the y-direction at the interface in the current state relative to that in the initial state for the two layers are assumed to be the same, denoted as ${\lambda }_2^{ci}$. Those along the z-direction are ${\lambda }_3^{ci(\mathrm{a})}$ in layer a and ${\lambda }_3^{ci(\mathrm{b})}$ in layer b, which are assumed to be independent of z.

With the plane section assumption of the beam theory [32], the principal stretch along the x-direction in the current state relative to a dry polymer in layer a is given by

$${\lambda }_1^{c0(\mathrm{a})}(z)={\lambda }_{10}^{\mathrm{\prime }}{\lambda }_1^{i0(\mathrm{a})}(1-{\lambda }_3^{ci(\mathrm{a})}\kappa z)z\le 0.$$ 2.1

The other two principal stretches in the current state relative to a dry polymer in layer a are given by

$$\begin{array}{rl}{\lambda }_2^{c0(\mathrm{a})}& ={\lambda }_2^{ci}{\lambda }_2^{i0(\mathrm{a})}\\ \text{and}{\lambda }_3^{c0(\mathrm{a})}& ={\lambda }_3^{ci(\mathrm{a})}{\lambda }_3^{i0(\mathrm{a})}.\end{array}\}$$ 2.2

Similarly, the principal stretch along the x-direction in the current state relative to a dry polymer in layer b is given by

$${\lambda }_1^{c0(\mathrm{b})}(z)={\lambda }_{10}^{\mathrm{\prime }}{\lambda }_1^{i0(\mathrm{b})}(1-{\lambda }_3^{ci(\mathrm{b})}\kappa z)z>0.$$ 2.3

The other two principal stretches in the current state relative to a dry polymer in layer b are given by

$$\begin{array}{rl}{\lambda }_2^{c0(\mathrm{b})}& ={\lambda }_2^{ci}{\lambda }_2^{i0(\mathrm{b})}\\ \text{and}{\lambda }_3^{c0(\mathrm{b})}& ={\lambda }_3^{ci(\mathrm{b})}{\lambda }_3^{i0(\mathrm{b})}.\end{array}\}$$ 2.4

The Helmholtz free energy density of a temperature-responsive hydrogel is given by [33,34]

$$\begin{array}{rl}W& =\frac{1}{2}NkT\left[{{\lambda }_1^{c0}}^2+{{\lambda }_2^{c0}}^2+{{\lambda }_3^{c0}}^2-3-2\log ({\lambda }_1^{c0}{\lambda }_2^{c0}{\lambda }_3^{c0})\right]\\ & +\frac{kT}{\mathrm{\Omega }}\left[({\lambda }_1^{c0}{\lambda }_2^{c0}{\lambda }_3^{c0}-1)\log \left(1-\frac{1}{{\lambda }_1^{c0}{\lambda }_2^{c0}{\lambda }_3^{c0}}\right)+\chi \left(1-\frac{1}{{\lambda }_1^{c0}{\lambda }_2^{c0}{\lambda }_3^{c0}}\right)\right],\end{array}$$ 2.5

where ${\lambda }_1^{c0}$, ${\lambda }_2^{c0}$ and ${\lambda }_3^{c0}$ are the three principal stretches at the current state relative to a dry polymer, N is the nominal density of the polymer chain, k is the Boltzmann constant, T is the absolute temperature, Ω is the volume per water molecule, and χ is a dimensionless measure of the strength of pair-wise interactions between species, given by [35]

$$\chi (T,\varphi )={\chi }_0+\frac{{\chi }_1}{{\lambda }_1^{c0}{\lambda }_2^{c0}{\lambda }_3^{c0}},$$ 2.6

where ${\chi }_0=A_0+B_{0}T$ and ${\chi }_1=A_1+B_{1}T$ with A₀ = −12.947, B₀ = 0.04496/K, A₁ = 17.92 and B₁ = −0.0569/K, as fitted from experiments for PNIPAM hydrogels [36].

With equations (2.1)–(2.6), the Helmholtz free energy density in layer a, W^(a), and that in layer b, W^(b), can be obtained. The total Helmholtz free energy in layer a, H^(a), is given by

$$H^{(\mathrm{a})}=\frac{L_1^{i0}{L}_2^{i0}}{{\lambda }_1^{i0(\mathrm{a})}{\lambda }_2^{i0(\mathrm{a})}{\lambda }_3^{i0(\mathrm{a})}}{\int }_{-L_3^{i0(\mathrm{a})}}^0{W}^{(\mathrm{a})}\hspace{0.17em}\text{d}z,$$ 2.7

and that in layer b, H^(b), is given by

$$H^{(\mathrm{b})}=\frac{L_1^{i0}{L}_2^{i0}}{{\lambda }_1^{i0(\mathrm{b})}{\lambda }_2^{i0(\mathrm{b})}{\lambda }_3^{i0(\mathrm{b})}}{\int }_0^{L_3^{i0(\mathrm{b})}}{W}^{(\mathrm{b})}\hspace{0.17em}\text{d}z.$$ 2.8

The total free energy of the system is then

$$H=H^{(\mathrm{a})}+H^{(\mathrm{b})},$$ 2.9

which is a function of five variables to be determined, including ${\lambda }_{10}^{\mathrm{\prime }}$, ${\lambda }_2^{ci}$, ${\lambda }_3^{ci(\mathrm{a})}$, ${\lambda }_3^{ci(\mathrm{b})}$ and κ.

It can be shown that the partial derivatives of H to each of these five variables are given by

$$\begin{array}{rl}\frac{\mathrm{\partial }H}{\mathrm{\partial }{{\lambda }^{\mathrm{\prime }}}_{10}}& =L_1^{i0}{F}_1-L_1^{i0}\kappa M,\\ \frac{\mathrm{\partial }H}{\mathrm{\partial }\kappa }& =-{{\lambda }^{\mathrm{\prime }}}_{10}{L}_1^{i0}M,\\ \frac{\mathrm{\partial }H}{\mathrm{\partial }{\lambda }_2^{ci}}& =L_2^{i0}{F}_2,\\ \frac{\mathrm{\partial }H}{\mathrm{\partial }{\lambda }_3^{ci(\mathrm{a})}}& =L_3^{i0(\mathrm{a})}{F}_3^{(\mathrm{a})}\\ \text{and}\frac{\mathrm{\partial }H}{\mathrm{\partial }{\lambda }_3^{ci(\mathrm{b})}}& =L_3^{i0(\mathrm{b})}{F}_3^{(\mathrm{b})},\end{array}\}$$ 2.10

where

$$\begin{array}{rl}{F}_1& =L_2^{i0}{\lambda }_2^{ci}{\lambda }_3^{ci(\mathrm{a})}{\int }_{-L_3^{i0(\mathrm{a})}}^0{\sigma }_1^{(\mathrm{a})}\text{d}z+L_2^{i0}{\lambda }_2^{ci}{\lambda }_3^{ci(\mathrm{b})}{\int }_0^{L_3^{i0(\mathrm{b})}}{\sigma }_2^{(\mathrm{b})}\text{d}z,\\ M& =L_2^{i0}{\lambda }_2^{ci(\mathrm{a})}{\lambda }_3^{ci(\mathrm{a})}{\int }_{-L_3^{i0(\mathrm{a})}}^0{\sigma }_1^{(\mathrm{a})}{\lambda }_3^{ci(\mathrm{a})}z\text{d}z+L_2^{i0}{\lambda }_2^{ci(\mathrm{b})}{\lambda }_3^{ci(\mathrm{b})}{\int }_0^{L_3^{i0(\mathrm{b})}}{\sigma }_2^{(\mathrm{b})}{\lambda }_3^{ci(\mathrm{b})}z\hspace{0.17em}\text{d}z,\\ F_2& =\frac{L_2^{i0}{\lambda }_3^{ci(\mathrm{a})}}{{\lambda }_1^{i0(\mathrm{a})}}{\int }_{-L_3^{i0(\mathrm{a})}}^0{\lambda }_1^{c0(\mathrm{a})}{\sigma }_2^{(\mathrm{a})}\text{d}z+\frac{L_2^{i0}{\lambda }_3^{ci(\mathrm{b})}}{{\lambda }_1^{i0(\mathrm{b})}}{\int }_0^{L_3^{i0(\mathrm{b})}}{\lambda }_1^{c0(\mathrm{b})}{\sigma }_2^{(\mathrm{b})}\hspace{0.17em}\text{d}z,\\ {\overline{F}}_3^{(\mathrm{a})}& =\frac{1}{{\lambda }_3^{ci(\mathrm{a})}{L}_3^{i0(\mathrm{a})}}{\int }_{-L_3^{i0(\mathrm{a})}}^0{\lambda }_3^{ci(\mathrm{a})}{L}_1^{i0}{L}_2^{i0}{\lambda }_1^{ci(\mathrm{a})}{\lambda }_2^{ci(\mathrm{a})}{\sigma }_3^{(\mathrm{a})}\hspace{0.17em}\text{d}z\\ \text{and}{\overline{F}}_3^{(\mathrm{b})}& =\frac{1}{{\lambda }_3^{ci(\mathrm{b})}{L}_3^{i0(\mathrm{b})}}{\int }_0^{L_3^{i0(\mathrm{a})}}{\lambda }_3^{ci(\mathrm{b})}{L}_1^{i0}{L}_2^{i0}{\lambda }_1^{ci(\mathrm{b})}{\lambda }_2^{ci(\mathrm{b})}{\sigma }_3^{(\mathrm{b})}\hspace{0.17em}\text{d}z.\end{array}$$

It can be inferred that F₁ corresponds to the resultant force along the local x-direction at the cross-section of the bilayer beam, that F₂ corresponds to the resultant force along the y-direction of the bilayer, M is the resultant bending moment along the y-direction at the cross-section of the bilayer beam, $F_3^{(\mathrm{a})}$ is the volume integration of the local ${\sigma }_3^{(\mathrm{a})}$ in layer a divided by its current thickness, and $F_3^{(\mathrm{b})}$ is the volume integration of the local ${\sigma }_3^{(\mathrm{b})}$ in layer b divided by its current thickness. Considering that $F_1\text{ = }{F}_2\text{ = }{F}_3^{(\mathrm{a})}\text{ = }{F}_3^{(\mathrm{b})}\text{ = }0$ and M = 0 for the bilayer beam, all five partial derivatives in equation (2.10) are then 0. With five equations for five unknowns, ${\lambda }_{10}^{\mathrm{\prime }}$, ${\lambda }_2^{ci}$, ${\lambda }_3^{ci(\mathrm{a})}$, ${\lambda }_3^{ci(\mathrm{b})}$ and κ can then be determined. Alternatively, these five unknowns can be determined by searching for values upon which H reaches the minimum, which is actually adopted in the current analysis. Default parameters used in the analysis are listed in table 1.

Table 1.

Default values for some parameters in the analysis.

parameter	value	parameter	value
N(a)Ω	0.01	Ω	3 × 10−29 m−3 [37]
N(b)Ω	0.02	$L_1^{i0}$	1 mm
${\lambda }_1^{i0(\mathrm{a})}$, ${\lambda }_2^{i0(\mathrm{a})}$, ${\lambda }_3^{i0(\mathrm{a})}$	2.771	$L_2^{i0}$	1 mm
${\lambda }_1^{i0(\mathrm{b})}$, ${\lambda }_2^{i0(\mathrm{b})}$, ${\lambda }_3^{i0(\mathrm{b})}$	2.418	T0	20°C
$L_3^{i(\mathrm{a})}+L_3^{i(\mathrm{b})}$	2 mm

3. Results

With the quantitative theory developed above, we first calculate the change in the relative volume, V/V₀, where V is the volume in the current state and V₀ is that of a dry polymer, versus the temperature of free-swelling hydrogels with equation (2.5). As shown in figure 1c, the relative volume dramatically decreases around a certain temperature, which is due to the phase transition. According to figure 1c, the phase transition occurs around 31.5°C when $N^{(\mathrm{a})}\mathrm{\Omega }=0.01$ and around 31°C when $N^{(\mathrm{b})}\mathrm{\Omega }=0.02$, as reported in a previous study [37].

We then calculate the effects of temperature on the evolution of the curvature of the bilayer. As shown in figure 2a, there generally exist four phases in curvature evolution. In phase I, the curvature varies little with temperature; in phase II, the curvature rapidly increases with temperature and reaches a positive maximum; in phase III, the curvature dramatically decreases with temperature and reaches a negative maximum; in phase IV, the negative curvature gradually decreases and the bilayer becomes flat again. As also seen from figure 2a,c, the ratio of the thickness, $L_3^{i(\mathrm{a})}\text{/}{L}_3^{i(\mathrm{b})}$, has a strong effect on the curvature. The peak value of the bending curvature dramatically changes with $L_3^{i0(\mathrm{a})}\text{/}{L}_3^{i0(\mathrm{b})}$ and there appears to exist an optimal $L_3^{i(\mathrm{a})}\text{/}{L}_3^{i(\mathrm{b})}$ with the highest peak value. The responsive relative length is the ratio of the length of the bilayer in the current state to its initial length. As indicated in figure 2b,d, the relative length of the layer rapidly decreases with temperature upon phase transition.

Figure 2.

Effects of temperature on the curvature of the bilayer (a,c) and on the relative length of the bilayer (b,d). In $(c,d),N^{(b)}\mathrm{\Omega }=0.03$ and ${\lambda }_1^{i}0(b)={\lambda }_2^{i}0(b)={\lambda }_3^{i}0(b)=2.233$. (Online version in colour.)

We also investigate the effect of pre-stretches on the curvature of the bilayer when subjected to a temperature change. In our study, either layer a or layer b is subjected to an equi-biaxial pre-stretch initially and the bilayer is flat. This pre-stretch is immediately relaxed so that the bilayer can develop a non-zero initial curvature. In the analysis, we just set the initial stretches on layer a or layer b to be different from the respective free-swelling stretches. In this way, equations (2.1)–(2.10) can be directly used for cases with pre-stretches. The results for pre-stretches applied to layer a are given in figure 3a–c, while those for pre-stretches applied to layer b are given in figure 3d–f. As can be seen from figure 3, pre-stretches can have a strong effect on the evolution of the curvature of the bilayer with temperature, which not only changes the amplitude of the curvature but also its sign. Increasing the magnitude of the pre-stretch generally shifts the curve to the negative side in figure 3a–c but to the positive side in figure 3d–f.

Figure 3.

Effects of pre-stretches on the bending of the bilayer. (a–c) Layer a is subjected to equi-biaxial stretches with $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = }0.5$ for (a), $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = 1}$ for (b), and $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = 2}$ for (c). In the analysis, ${\lambda }_1^{i0(\mathrm{a})}={\lambda }_2^{i0(\mathrm{a})}=\alpha {\lambda }_0^{(\mathrm{a})}$ with ${\lambda }_0^{(\mathrm{a})}=2.771$. (d–f) Layer b is subjected to equi-biaxial stretches with $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = }0.5$ for (d), $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = 1}$ for (e), and $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}\text{ = 2}$ for (f). In the analysis, ${\lambda }_1^{i0(\mathrm{b})}={\lambda }_2^{i0(\mathrm{b})}=\alpha {\lambda }_0^{(\mathrm{b})}$ with ${\lambda }_0^{(\mathrm{b})}=2.418$. (Online version in colour.)

4. Discussion

It is essentially due to the mutual constraints of the two layers within the bilayer that it can bend when subjected to temperature change. When the thickness of the top layer is so small that its constraint on the bottom layer is negligible, the bilayer does not bend. At the other extreme, when the thickness of the top layer is so large that its constraint by the bottom layer is negligible, there is no bending either. Therefore, there exists an optimal thickness ratio for the highest peak value of the curvature, as exactly suggested from the results given in figure 2a,c. As also seen from figure 2, the exact value of the optimal thickness ratio is affected by the cross-link densities of the bilayer.

The phase transition of temperature-responsive hydrogels can be strongly affected by the mechanical constraint. For example, a stiff shell can suppress the phase transition of a hydrogel core in a core–shell structure [38]. The pressure can lower the phase transition temperature [39]. As can be seen from figure 4, pre-stretches can not only affect the relative volume, but also affect the phase transition temperature. However, the temperature at which the curvature of the bilayer dramatically changes appears to be not far from that of the phase transition of free-swelling hydrogels, as indicated in figure 3.

Figure 4.

Effects of equi-biaxial stretches on the variation in the relative volume of a hydrogel with temperature: (a) NΩ = 0.01 and (b) NΩ = 0.02. (Online version in colour.)

In fabricating a bilayer beam of temperature-responsive hydrogels, the polymerization of the second layer by UV irradiation took place only after the first layer was already formed [30]. Such a fabrication procedure can induce pre-stretches within the bilayer so that the bilayer manifests with an initial pre-curvature [30]. As seen in figure 3, the effects of pre-stretches on the evolution of the curvature of the bilayer with temperature are significant. A high level of pre-stretch can completely change the sign of the bending and the bending/unbending becomes highly asymmetric. As also indicated in figure 3, the pre-stretch may also change the amplitude of the bending/unbending curvature. Thus, pre-stretches can be very important in the design of a bilayer temperature-responsive hydrogel for desirable mechanical performance. It should be pointed out that PNIPAM-clay hydrogels employed in [30] contained nanoclays and their Helmholtz free energy density upon mechanical stretches may deviate from equation (2.5), which was previously formulated for PNIPAM hydrogels only [33,34]. This explains why the prediction of the current theory is not exactly the same as the experimental results [30].

We also see how other mechanical designs can affect curvature evolution of the bilayer. For example, we homogeneously added small pores on one side of the bilayer, as schematically shown in the insets of figure 5. These pores would lower the total Helmholtz free energy in the corresponding layer. For example, with a pore density of $v_{\alpha }$ in layer a, the total free energy of the system will be given by

$$H=(1-v^{(\mathrm{a})})H^{(\mathrm{a})}+H^{(\mathrm{b})}.$$ 4.1

With equations (2.1)–(2.8) and (4.1), we analyse the effects of the pores and find that these pores can also induce a highly asymmetric bending and significantly affect the bending amplitude of the bilayer upon temperature changes, as shown in figure 5.

Figure 5.

The effects of small pores on curvature evolution of the bilayer with temperature: (a) $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}=1$ and (b) $L_3^{i(\mathrm{b})}\text{/}{L}_3^{i(\mathrm{a})}=2$. (Online version in colour.)

Several previous studies have provided important insights into the understanding of the bending of a bilayer system. Cai [40] considered a hybrid beam with a gel layer bonded onto an elastic substrate, where the gel was assumed to apply moment to the substrate. In Cai's approach, the gel did not contribute to the bending stiffness of the beam [40]. Morimoto & Ashida [41] investigated the bending of a bilayer hydrogel consisting of a non-swellable layer and a temperature-sensitive hydrogel layer, and found that the predicted bending of the bilayer based on a finite-deformation theory was stronger than that based on a linear theory. Different from these previous works, here we consider a bilayer with both layers being temperature responsive and swellable. Based on the plane section beam assumption, we develop the finite-deformation theory to quantify the bending of bilayer temperature-responsive hydrogels and our subsequent theoretical analysis successfully reveals the significant effects of the thickness ratio of the bilayer, pre-stretches and pores on its curvature evolution with temperature. Nevertheless, we admit that the inertia effect, which is not considered in our theory yet, might be important in cases when temperature changes rapidly.

5. Conclusion

The temperature-responsive bending of a bilayer hydrogel in water with switching temperature can be reversible and repeatable, which has the potential for use in stimuli-responsive manipulators, grippers, sensors, etc. Here, a theory framework is developed to quantify the evolution of the curvature of a bilayer hydrogel made of PNIPAM when subjected to a temperature change. The theory is then employed to investigate the effects of thickness ratio, pre-stretch and pores on the bending of the bilayer. Our analysis indicates that there can exist an optimal thickness ratio of the bilayer to acquire the highest variation of curvature. Our analysis also indicates that the sign or the magnitude of the curvature can be affected by pre-stretches and small pores on the hydrogels. This study might provide important guidelines in fabricating or precisely controlling stimuli-responsive smart structures with excellent mechanical performance.

Authors' contributions

B.C. designed the work; C.D. performed the simulations; C.D. and B.C. analysed the data and wrote the manuscript. Both authors reviewed the manuscript and gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This work was supported by the National Natural Science Foundation of China (grant nos. 11372279, 11572285 and 11621062).