Introduction

Stimuli-responsive polymers (SRPs) have emerged as materials to be exploited in the development of a new generation of stimuli-responsive actuators that do not require sensors or processors (Zhang and Serpe 2017; Hu et al. 2020; Wang et al. 2022). In this context, the design of humidity-responsive self-actuators with hygroscopic materials (Okuzaki and Kunugi 1996; Burgert and Fratzl 2009; Shen et al. 2010; Okuzaki et al. 2013; Ma et al. 2013; Rubinger et al. 2013; Ochoa et al. 2013; Ionov 2014; Reichert et al. 2015; Taccola et al. 2015; Yao et al. 2015; Holstov et al. 2015a; Barozzi et al. 2016; Wang et al. 2016, 2017; Weng et al. 2016; Jian et al. 2017; Castaldo et al. 2019; Dingler et al. 2021; Si et al. 2022) to mimic the humidity-induced actuation of plants is not an exception (Burgert and Fratzl 2009; Reyssat and Mahadevan 2009; Reichert et al. 2015; Holstov et al. 2015a; Dingler et al. 2021). Some studies have already used a bilayer composite coupling a hygroscopic material and a non-hygroscopic one to exploit the different dimensional variations of the two layers induced by a change in the environmental humidity. However, for some of the hygroscopic materials reported in the literature, the moisture absorption and desorption processes differ significantly at the same temperature, and the desorption process is commonly quickened by heating (Greco et al. 2011; Zhou et al. 2014; Taccola et al. 2015; Dingler et al. 2021; Rivadeneyra et al. 2021).

A bilayer composite humidity-responsive actuator, schematically drawn in Fig. 1, exploits the significant length variation of the thin hygroscopic material layer (active layer), caused by a variation in the environmental humidity level, while the non-hygroscopic material substrate (passive layer) has a negligible length variation compared to the active one. If the active layer is constrained on one side by the passive one, the mismatch in deformation results in bending of the bi-layered composite.

Fig. 1
figure 1

Schematic representation of humidity-induced bending of a bilayer composite (Berry and Pritchet 1984). ΔLa and ΔLp indicate the linear expansion of active and passive layers due to an increase in the environmental humidity level, respectively. ha and hp indicate the thickness of active and passive layers, respectively (ha < hp). R represents the radius of curvature

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The bending effect is similar to that of bi-metal strips under heating described by Timoshenko (1925), which is based on Stoney’s work on induced stress in metallic films (Stoney 1909). Berry and Pritchet ( 1984) showed how Timoshenko's bi-metal strip bending theory can be adapted to describe moisture-induced deformation in a bilayer composite. The bending radius induced by a change in relative humidity, can be estimated by Eq. (1)(Berry and Pritchet 1984), in which the coefficient of hygroscopic expansion (α) and moisture concentration variation (ΔC) are used in place of the coefficient of thermal expansion (CTE) and temperature variation, respectively.

$$\frac{1}{R}=\kappa =\frac{\Delta \alpha \Delta C f(m,n)}{{h}_{tot}} ,\quad f\left(m,n\right)=\frac{6{(1+m)}^{2}}{3{(1+m)}^{2}+(1+mn)({m}^{2}+\frac{1}{m})}$$
(1)

where Δα is the difference between the coefficient of hygroscopic expansion of the active layer material (subscript a) and that of the passive one (subscript p) (Δα = αa–αp ≠ 0) and htot is the thickness of the bilayer composite (htot = ha + hp). f (m, n) is a function of the layers thicknesses (m = hp/ha), materials elastic modulus (E), and Poisson ratio (ν) (\(n=\frac{{\acute{E} }_{p}}{{\acute{E} }_{a}}\), where \({\acute{E} }_{a}=\frac{{E}_{a}}{(1-{\nu }_{a})}\) and \({\acute{E} }_{p}=\frac{{E}_{p}}{(1-{\nu }_{p})}\)). In brief, the induced bending curvature of a bilayer composite is a function of the environmental conditions (relative humidity), the geometry of each layer (thickness), as well as the mechanical and hygroscopic properties of the materials. The thickness of each layer can be designed according to the requirements of each bilayer product in relation to the properties of the hygroscopic material, namely Young’s modulus (Ea) and coefficient of hygroscopic expansion (αa).

Most of the literature, dedicated to the characterization of bilayer humidity-responsive self-actuator composites, considers Eq. (1)—or an adapted analytical model—to evaluate the bending curvature at equilibrium conditions for changes in relative humidity. Some studies (Holstov et al. 2015b; Wang et al. 2017; Abdelmohsen 2019) compare only the initial shape and the final deformation of the bilayer at the equilibrium state, when the actuator is fully saturated and does not deform anymore. Other studies, however, took a step forward by experimentally characterizing the evolution of the bending curvature as a function of time using Eq. (1) (Reyssat and Mahadevan 2009; Yao et al. 2015; Holstov et al. 2015a). Nonetheless, neglecting the kinetics of moisture or water diffusion leads to uncertainty about the time for the diffusion process to reach the concentration at saturation (equilibrium state).

Considering the authors’ previous publications on the characterization and modeling of moisture diffusion (Khoshtinat et al. 2021) and hygroscopic strain induced by moisture absorption (Khoshtinat et al. 2022) of cellulose acetate membranes, this study is dedicated to the characterization and finite element modeling of a humidity-responsive self-actuator bilayer composite made of a cellulose acetate membrane and a non-hygroscopic ply. To the best of the authors' knowledge, no systematic study on the bending curvature evolution in time, accompanied by the relevant finite element modeling of a bilayer composite actuator, has been presented in the literature. In particular, there seems to be no study that includes highly hygroscopic materials such as cellulose acetate, which respond quickly to both increases and decreases in the environment's relative humidity. The finite element model presented in this study reproduces the peculiar hygroscopic behavior of cellulose acetate by using experimentally characterized material features, i.e., coupling the moisture diffusion process and the relevant induced deformation of a cellulose acetate membrane, while available analytical models (Yoon et al. 2007; Wong 2010) reformulate heat transfer theoretical models by just substituting thermal features with the diffusion counterparts.

A summary of previously characterized features of the cellulose acetate membrane employed in this study is presented first. Then, the experimental results of the bending curvature evolution induced by humidity variation in the cellulose acetate-based self-actuator composite are described. Finally, the two-physics finite element modelling is detailed, and the experimental data are compared to numerical predictions and to the commonly adopted analytical model (Eq. (1)).

Materials and experimental procedures

Cellulose acetate membrane

Mazzucchelli 1849 S.p.A. supplied cellulose acetate (CA) powder (53.3% acetylation). A 20% w/w solution of Cellulose Acetate was prepared by gradually adding the CA powder to ethyl lactate (≥ 98% purity, acquired from Sigma-Aldrich). An RCT basic IKAMAGTM safety control magnetic stirrer was used to mix the solution at 80 °C for 90 min (300 rpm), followed by another 90 min at 80 °C (200 rpm). The solution was poured onto a glass substrate after reaching room temperature, and a film of about 500 µm thickness was cast by a K Control Coater. First, the cast solution was placed in a closed container with no air flow for 3 h to limit solvent evaporation at the beginning of the drying process and to avoid the creation of bubbles. It was then dried for 4 h in a Vuototest Mazzali vacuum oven to speed up solvent evaporation. Finally, the CA membranes were peeled from the glass substrate and dried for 24 h at 125 °C in an oven (Mazzali Thermair) to guarantee full evaporation of any leftover solvent. More details on the process of membrane preparation and the obtained final thicknesses are presented in Khoshtinat et al. (2021, 2022).

Bilayer composite

The bilayer composite was prepared using the previously characterized cellulose acetate membranes as its active layers (Khoshtinat et al. 2021, 2022). The geometry and material characteristics of the passive substrate layer have been selected to fulfill the assumptions of Berry and Pritchet's analytical model (Berry and Pritchet 1984) and to guarantee a valid comparison between the experimental and finite element results with the analytical model. Therefore, the substrate: (1) must be thicker than the CA membrane (hp > ha), (2) must have a negligible hygroscopic expansion coefficient compared to cellulose acetate (αp <  < αa), and (3) must be resistant to humidity with good adhesion to the CA membrane.

High wettability and inherently low surface energy of cellulose-based materials (Heinze et al. 2018), cellulose acetate in our case, lead to a poor interfacial bonding/adhesion. Although treatments such as chemical modification or plasma treatment can increase cellulose acetate's ability to adhere, they might affect the hygroscopic behavior by changing the concentration and/or arrangement of the hydroxyl groups on the membrane surface. Since the goal of this investigation is the characterization of the response of cellulose acetate layered composites to changes in humidity level, rather than the improvement of the CA adhesion, an adhesive tape (Tesa® 64621) that allows to satisfy all the mentioned requirements was chosen as the substrate. Tesa® 64621 is a double-sided adhesive tape with a PP-carrier and a synthetic rubber adhesive. It is not hygroscopic and ensures very good bonding on polar and non-polar surfaces. Since, the adhesive layer has a low stiffness, the release liner (silicon-coated paper) was preserved, and the two together are considered as a homogeneous material. The Young's modulus of Tesa® 64621, including its release liner, was 4.62 ± 0.03 GPa, measured by uniaxial tensile tests (strain rate 6 × 10–3 min−1, T = 25 ± 1 °C).

As observed in a previous study (Khoshtinat, et al. 2021), cellulose acetate responds quickly to variations in relative humidity. During preliminary attempts, the adhesive tape substrate was coupled to the CA membrane soon after it was removed from the oven (24 h, 125 °C drying procedure). However, due to thermal shrinkage induced by cooling from 125 °C to room temperature and the hygroscopic expansion in the free membrane's surface during the adhesion time (almost 30 s), a curvature was noticed in the composite. This could have been avoided if the composite assembly was carried out in a dry environment; however, such a facility was not available for this study. Therefore, the most practical approach was to couple the layers in a climatized room, at a known relative humidity, once the CA membrane had reached its moisture saturation state. Flat bilayer composites were thus obtained by coupling the passive layer (thickness including the release liner hp = 145 ± 1 µm) with a cellulose acetate membrane (ha = 86 ± 3 µm) equilibrated in the climatized room at a temperature (T) of 25 °C and a relative humidity (RH) of 35%. Specimens of 60 × 12.5 mm2 were then hollow punched, and the bending induced by changes in RH was measured in different ranges (see Table 1).

Table 1 Specimen dimensions and experimental conditions
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Experimental setup and image analysis

The bending deformation of the bilayer composite induced by a change in the relative humidity was measured using digital images recorded at a predefined frequency, which followed an extensively used approach (Reyssat and Mahadevan 2009; Correa et al. 2015; Holstov et al. 2015a; Vailati et al. 2018; Abdelmohsen et al. 2018, 2019; Abdelmohsen 2019). Specimens previously equilibrated at 17% and 35% of RH were placed within a polymethylmethacrylate closed chamber designed and built to have a constant humidity environment. Levels of relative humidity between 77 and 80% were obtained inside the chamber at room temperature (T = 25 °C), by using salt solutions of sodium chloride in distilled water (about 2/1 w/w). To stabilize the relative humidity, the salt solution was placed in the chamber 72 h in advance. The humidity level of 17% was obtained with a silica-gel desiccator. An RH level of 35% was, as already stated, the humidity in the laboratory thermostatic room. Three specimens, obtained from three different cellulose acetate membranes with the same thickness of about 86 ± 3 µm, were tested to assess the reproducibility, as detailed in Table 1.

Figure 2 depicts a schematic representation of the experimental setup for image acquisition at a constant relative humidity and temperature (T = 25 °C). The specimen (60 mm long) was clamped between two metal supports, leaving an overhanging length of 40 or 50 mm and placed in front of a vertical grid paper. The procedure of setting the specimen in the chamber takes about one minute. A digital camera (Nikon D70) was positioned perpendicular to the grid paper, at a distance of about 50 cm from the specimen. The time required for a 86 µm thick membrane of dry cellulose acetate (RH = 0%) to reach saturation at a relative humidity of 80%, calculated by the finite element simulation presented in authors’ previous work (Khoshtinat et al. 2021), is about 60 min when the absorption occurs from only one of its surfaces. Considering the climatized room’s relative humidity (RH = 35%) and the fact that the cellulose acetate membrane is not dry, a longer duration for the experimental procedure was hypothesized; therefore, the evolution of the bending of the self-actuator bilayer composite was tracked for 100 min. The camera used had a frequency acquisition rate of 0.016 Hz (one image every minute). The video showing the experimentally recorded bending deformation of specimen S1 with humidity variation from 35 to 80% is available as Online Resource 1.

Fig. 2
figure 2

Schematic representation of the experimental setup

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Figure 3 shows, as an example, the image of specimen S1 equilibrated at a relative humidity of 80% and the reference system for the measurement of the curvature. The recorded digital images were imported into Rhinoceros® as a picture frame for post-processing. Beside the main bending, a minor twist of the specimen was noted (Fig. 3), which could have been motivated by 3 µm variation in the thickness of the CA membrane. To neglect this small twist, the origin of the X–Y reference frame was set at the clamp between the metal supports (P0), and the bent shape of the specimen was fitted with an arc of a circumference from point P0 to point P1 positioned on the specimen side closest to the camera (Fig. 3), setting tan(P0) = 0. The bending curvature (κ) was obtained by Eq. (2), where x and y are the coordinates of P1. This process was performed on all the collected frames for each specimen.

Fig. 3
figure 3

Bilayer composite prepared at RH = 35% and kept for 100 min in the closed chamber at a relative humidity of 80%. The reference system used to evaluate the curvature of the P0–P1 arc is also shown

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$$\kappa =\frac{1}{R}=\frac{2x}{{y}^{2}+{x}^{2}}$$
(2)

Finite element modelling

Finite element simulations were performed by COMSOL Multiphysics® 5.6. Figure 4 depicts some features of the adopted finite element model. In the numerical analyses, three geometric parameters have been kept constant, namely the width (W = 12.5 mm) and the active and passive layers’ thicknesses (ha = 0.086 mm, hp = 0.145 mm). The length (L) has been varied based on the overhanging length of each specimen (Table 1). The concentration at saturation (Csat (g mm−3)) has been defined as a function of relative humidity by Eq. (3), according to the previous research (Khoshtinat et al. 2022).

Fig. 4
figure 4

Some features of the finite element model

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$${C}_{sat}=1.19\times {10}^{-10}\times {RH}^{3}- 7.97\times {10}^{-9}\times {RH}^{2}+8.09\times {10}^{-7}\times RH$$
(3)

Two 3D prismatic layers have been defined, one for the passive substrate (L × W × hp) and the other for the cellulose acetate membrane (L × W × ha). The two layers have been considered perfectly bonded by a tie constraint to guarantee continuity at the interface. Two user-defined materials have been set using the material properties in Table 2, where ρ, E, ν, β and D are: density, Young’s modulus, Poisson's ratio, relaxation factor, and diffusion coefficient, respectively. As previously mentioned, the mechanical properties are supposed to be insensitive to relative humidity. This hypothesis might not be realistic for some hygroscopic materials, and it will be topic for future developments of the present study.

Table 2 Materials properties used for the finite element simulations
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The coefficient of hygroscopic expansion (α) for cellulose acetate has been defined as a function of relative humidity (RH) according to Eq. (4) (Khoshtinat et al. 2021).

$$\alpha =\frac{803\times {RH}^{2}-51400\times RH+1.16\times {10}^{6} }{1.19\times {RH}^{2} - 79.7\times RH + 8090}$$
(4)

The bending deformation of the self-actuator bilayer composite is caused by a mismatch between the mechanical properties and the linear hygroscopic expansion of the active layer (cellulose acetate) and the passive layer (εhygro(passive) = 0). Since the passive layer is a non-hygroscopic material, the input parameters for the diffusion process were set only for the active layer (cellulose acetate membrane) considering its non-Fickian behavior (Khoshtinat et al. 2021). The initial dry condition was assumed to be a null initial moisture concentration, C0 = 0 g·mm−3. The concentration boundary condition was enforced on the faces of cellulose acetate membrane exposed to the humid environment, as a function of time in Eq. (5) (Khoshtinat et al. 2021).

$${C}_{t}={C}_{sat}(1-{e}^{-\beta t})$$
(5)

An external face of the model was fixed to reproduce the cantilever, see Fig. 4. From the top face of the passive layer (adhesive tape) to the bottom face of the cellulose acetate membrane, a user-controlled swept mesh was created, forming a discretized volume of 25 × 50 × 10 8-nodes hexahedral elements (total of 12,500 finite elements).

Two separate time-dependent analyses for the diffusion process and hygroscopic swelling were created to minimize computational time compared to a single analysis that included all physical processes. For both analyses, a time range of 100 min was set. First, the diffusion process simulation was run. Then, the hygroscopic swelling was performed, referring to the output of the first analysis and including geometric nonlinearity. The evolution of the bending curvature of the self-actuator composite for the simulated time (100 min) was determined by a built-in function considering the displacement of one external node (RP in Fig. 4). The predictions were compared to the experimental results as well as to the analytical model in Eq. (1).

The video showing the prediction of the non-Fickian moisture diffusion and the consequent hygroscopic deformation in the bilayer with humidity variation from 35 to 80% is available in Online Resource 2.

Results and discussion

Figure 5 shows the experimental results of the bending curvature (κ) evolution in time induced by relative humidity variations for specimens first equilibrated at RH = 35% and then placed into the closed chamber at RH = 77% or 80% (see Table 1 for the experimental conditions). As expected, by increasing the relative humidity, the final induced bending curvature in the bilayer composite increases as well. The initial point of all curves is the curvature of the specimen given by the first image captured by the camera, i.e. after one minute of exposure to a relative humidity of 77% or 80%, as detailed in sub-Sect. 2.3. During the first 20 min, the bending curvature increases rapidly. After this period, the variation of κ as a function of time decreases steadily, up to 40-min mark. Then, between 40 and 60 min, the curvature approaches a plateau. After 60 min, the constant curvature indicates the conclusion of the moisture absorption process. It should be noted that the observed behavior is similar to the trend of time dependency of moisture diffusion (Hansen 2010; Mensitieri and Scherillo 2012), emphasizing the influence of moisture absorption on the bending curvature. Even though three specimens from three different membranes were used and the test was replicated on each one with different conditions, the consistency of the results highlights the good repeatability and reproducibility of the measurements.

Fig. 5
figure 5

Experimental results of bending curvature (κ) of self-actuators bilayer composite induced by relative humidity change, from RH = 35% in a climatized room to 77% and 80% in a closed chamber

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To assess the effect of the initial relative humidity on the bending curvature and the cyclic behavior of this self-actuator composite, specimens were partially dried by keeping them for 24 h in a closed chamber containing dried silica gel (RH = 17%). This led to a negative bending curvature of -0.0199, as shown in Fig. 6 (images in center and the first point in the Second cycles graphs). Then, they were immediately put in the closed chamber with higher relative humidity. Figure 6 depicts the results of two specimens, S1 and S3, at relative humidity levels of 80% and 77%, respectively, both starting from 35% RH (see Table 1). Figure 6a shows the curvature evolution of specimen S1, which was preliminarily equilibrated at RH = 35% and was: (1) exposed to RH = 80% for 100 min (first cycle), (2) dried for 24 h at RH = 17%, and (3) exposed again to RH = 80% for another 100 min (second cycle). Figure 6b reports similar results for specimen S3, exposed to a humidity of 77%.

Fig. 6
figure 6

Experimental results (curvature evolution over time) for an initial relative humidity of a 80% and b 77%

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At RH = 80% for S1 and RH = 77% for S3, the ultimate bending curvature is consistent regardless of the starting relative humidity (17% or 35%). The starting RH influences the bending curvature captured 1 min after exposure. This effect is due to the difference in the initial bending curvature. For the specimen that has been stabilized at RH = 35%, the starting bending curvature was κ = 0; whereas for the same specimen stabilized at RH = 17%, the starting bending curvature was κ = −0.0199.

Figure 7a shows the deformation over time of the self-actuator composite due to the moisture absorption simulated by the finite element model for the specimen with the same geometry as S1 at the relative humidity of 80%. The overlapping of the self-actuator shape after 60 and 70 min indicates that no further deformation occurs during this period, suggesting that the moisture diffusion process in the cellulose acetate membrane has already terminated, as discussed in sub-Sect. 2.3. Simulation also highlights the fast response of the self-actuator composite to changes in relative humidity. The bilayer composite deforms considerably during the first two minutes of exposure to the higher relative humidity level, which points out that this self-actuator can be considered an alternative for applications such as indoor humidity control, as presented in some projects (Reichert et al. 2015; Menges and Reichert 2015; Correa et al. 2015; Holstov et al. 2015a).

Fig. 7
figure 7

a Self-actuator deformation by finite element simulation, b comparison between experimental data, analytical (Eq. (1)), and finite element model (FEM)

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The discrepancy between the S1 specimen's predicted final shape in Fig. 7a and the experimental one (Fig. 3) is mainly due to the preparation of the bilayer composite at the relative humidity of 35%, which imparted a hygroscopic expansion of the cellulose acetate membrane. As mentioned in sub-Sect. 2.2, it was not possible to reach a relative humidity of 0% in the laboratory, and the adhesive tape-based passive layer was coupled to cellulose acetate membrane at 35% RH. Since the bending curvature at equilibrium is unaffected by diffusion kinetics, the curvature from dry (RH = 0%) to 80% relative humidity is equal to the sum of the curvatures from dry to 35% and from 35 to 80% (Holstov et al. 2015a). Therefore, the variation of bending curvature from the relative humidity of 35% to 80% (Δκ RH (35–80)) can be retrieved as:

$$\Delta {\kappa }_{RH(35-80)}= {\kappa }_{RH(0-80)}-{\kappa }_{RH(0-35)}$$
(6)

Figure 7b shows a comparison of the evolution over time of the bending curvature of the experiment and the numerical simulation for the specimen S1 at 80% RH. Moreover, the analytical model prediction for the final bending curvature (Eq. (1)) is compared, assuming the relative humidity dependent coefficient of hygroscopic expansion according to Eq. (4), setting n = Ép/Éa = 3.84, m = hp /ha = 1.69 and htot = 0.228 mm (see Tables 1 and 2). Both the numerical and analytical predictions have been adjusted according to Eq. (6). Figure 7b shows a lower rate of growth of experimental bending curvature compared to finite element modeling. This might be related to the experiment's initial condition (RH = 35%) and thus the slightly lower diffusion coefficient of the cellulose acetate, since it has been partially plasticized by the absorbed moisture compared to the dry one.

In terms of the ultimate bending curvature, the experimental results (κ = 0.0511 mm−1, R = 19.56 mm) and the finite element simulation (κ = 0.0519 mm−1, R = 19.26 mm) are in very good agreement with each other. Bending radius R difference is approximately just 0.3 mm. The analytical model, on the other hand, estimates a final curvature of only 0.0645 mm−1 (R = 15.5 mm), resulting in a 3.43 mm difference in bending radius from the experiment. The analytical model's curvature overestimation is the result of its simplifications and assumptions. It must be mentioned that the bending curvature in the experiment is also dependent on the plasticization of the cellulose acetate membrane caused by moisture absorption, which results in a decrease of the Young's modulus of the cellulose acetate, which has not been considered in the analytical and numerical models.

Conclusions

A humidity-responsive self-actuator composite, coupling a non-hygroscopic adhesive tape and a cellulose acetate membrane, has been proposed. The experimental setup and procedure for the characterization of the induced bending deformation in the self-actuator bilayer composite have been presented for tests performed at T = 25 ± 1 °C, and at relative humidities of RH = 17, 77, and 80%. A finite element model, implementing the peculiar hygroscopic behavior of cellulose acetate, was exploited for predicting the evolution of the non-Fickian moisture diffusion and the relevant induced hygroscopic expansion. Experimentally evaluated material properties, such as the absorbed moisture concentration at saturation (Csat), the hygroscopic strain (εhygro), the diffusion coefficient (D), and the relaxation factor (β) of the cellulose acetate, among other properties, were used as inputs of the finite element model. Attention was dedicated to the relative humidity level for the bilayer composite preparation and its effect on the self-actuated bending curvature evolution.

The proposed numerical model not only accurately predicts the equilibrium bending curvature, but it also provides the transient bending curvature of the cellulose acetate-based self-actuator composite. The comparison between the numerical model and the experimental measurements also highlights how the experimental initial conditions affect the performance of the self-actuator. The agreement between the experimental measurements and finite element numerical predictions shows the accuracy of the model, which can be adopted for the design and tuning of the proposed hygroscopic self-actuator composite for different applications. Future investigations should be aimed at evaluating the effects of relative humidity and temperature on the plasticization and mechanical properties of the cellulose acetate to improve the accuracy of the finite element model.