Swelling under Constraints: Exploiting 3D‐Printing to Optimize the Performance of Gel‐Based Devices

Stimuli‐responsive hydrogels that swell under constraints such as spatial geometric confinement are employed in many applications, including biomedical devices and actuators, to perform mechanical work. Due to the heterogeneous deformations that arise from these constraints, the computation of the swelling‐induced stress poses numerical difficulties. This work proposes a simple experimental method based on 3D‐printing technologies to characterize this stress. The swelling is investigated under two types of geometric confinements—transversely constraining and elastically constraining boxes. In the former, the box enforces uniaxial swelling of the gel. The results show that the longitudinal deformations decrease as the transverse stretches increase. The elastically constraining box comprises soft walls with various stiffnesses and sizes that deform in response to the swelling‐induced stress exerted by the gel. By employing elastic plate theory, a method to determine this stress is developed. The results reveal that: 1) the maximum volumetric deformation is achieved by free swelling and 2) the stress gels exert depends on the wall stiffness and non‐linearly decreases as the gel nears its freely swollen state. The insights from this work can be used to optimize the performance of swelling‐based systems and characterize the stress generated due to other stimuli such as pH and temperature.

Based on the above works, it is clear that the configuration of a gel that swells freely (i.e., in the absence of constraints) differs from that of a gel that swells under constraints. Accordingly, the swollen equilibrium configuration is governed by the interactions between a gel and its surroundings. These interactions mainly depend on the characteristics of the polymer network (such as the properties of the gel and the stage of swelling [42,43] ), the chemical environment (such as the liquid-gel interactions, the external pressure, and the temperature [27,44,45] ), and the size, the geometry, and the properties of any imposed mechanical constraints. [14,46,47] Broadly, swelling in the presence of constraints translates into a swelling-induced stress that a gel exerts on its environment. As an intuitive example, one can consider the swelling of a gel in a small elastic box. If the box inhibits the gel from reaching its freely swollen volume, the gel-box interactions give rise to stress which depends on the properties of these two constituents. The understanding of the relations between the properties of the gel and the box and the stress that develops are essential as they provide the tools to control, enhance, and optimize the performance of gel-based applications. Due to the heterogeneous deformations that the gel may experience, the determination of the swelling-induced stress is difficult and computationally expensive. To the best of the authors knowledge, simple methods that estimate the stress gels apply on their surrounding during the swelling process are lacking.
In this work, we propose an experimental set-up to measure the swelling-induced stress by gels that imbibe solvent under mechanical constraints. We investigate the influence of mechanical constraints in the form of spatial geometric confinement on the swelling of gels. To this end, we consider two types of constraints: 1) a transversely constraining box and 2) an elastically constraining box. In the former, open boxes with different sizes were 3Dprinted. The latter comprises of a 3D-printed box with four rigid walls and two soft walls.
To estimate the swelling-induced stress and determine the effects of constraints on swelling, polymeric water beads were placed in the boxes, which were then submerged in an aqueous bath. The gels were allowed to swell until equilibrium was reached. Next, the swollen configuration of the beads was characterized and investigated. In the transversely constraining box, the influence of the box size was studied. In the elastically constraining boxes the soft walls experienced deflections and bending. By employing tools from elastic plate theory, [48] a systematic method to quantify the stress that the gel exerted on the soft walls due to swelling was developed. The robustness of the proposed method enabled us to investigate the swelling-induced stress exerted by the polymeric beads in boxes with different sizes and compliant walls with various flexural rigidities.

Free Swelling of Water Beads
Commercially available spherical polymeric water beads (made by iTrunk) were purchased from Amazon. In the dry state, the diameter and the mass of a dry bead were denoted by d p and m p , respectively. Typically, d p ≈ 5 − 6 mm and m p ≈ 0.13 − 0.15 gr. In these experiments, beads with d p = 5.61 ± 0.07 mm and m p = 0.14 ± 0.01 gr were chosen.
In the absence of constraints, the beads swell freely while maintaining a spherical shape. The swollen state was characterized by a diameter d f and a mass m f . In this simple case the deformations were homogeneous and characterized by the stretch f = d f /d p . This allowed to compute the volumetric deformation J f = 3 f , which is defined as the ratio between the volume of the swollen and the dry bead. The experiments revealed that the swollen diameter and mass were d f = 26.60 ± 0.50 mm and m f = 10.28 ± 1.16 gr, corresponding to a f ≈ 4.7 and J f ≈ 106. The dry and the freely swollen water beads are shown in Figure 1a.
Following common practice, in this work it was assumed that the increase in volume corresponds to the added mass due to water uptake. To verify this assumption, the mass of the added water m w = m f − m p was computed. Next, the added volume V w = m w / w , where w ≈ 1 gcm −3 is the density of water, was used to approximate the volumetric deformation through the added mass of water via where V p = ∕6 d 3 p is the volume of the dry bead. These findings revealed that J ≈ J f . This result implied that: 1) the volumetric deformations could be estimated by the measurements of the added mass and 2) the freely swollen hydrogel contained ≈ 99% water (estimated via V w /(V p + V w )).

Swelling under Mechanical Constraints
To determine the influence of geometric confinement on swelling, 3D-printing technologies which enable freeform fabrication were exploited. Specifically, a PolyJet based printer (Ob-jet260 CONNEX 3 by Stratasys), which is capable of printing structures from multiple rubber-like and ABS-like materials, was used. Two types of boxes were designed and 3D-printed : 1) a transversely constraining box with rigid walls and a rectangular cross-section L × L (Figure 1b,c) and 2) an elastically constraining box with four rigid walls, two compliant walls, and an edge length L. The latter was printed as an open box consisting of three rigid walls with small holes to allow water influx and two rubbery soft walls and a rigid closing lid, as shown in Figure 1d. The open box and the lid were printed as two separate structures to enable the insertion of the bead into the spatially confining box, as shown in Figure 1e,f.

Materials
The rigid walls were printed from commercially available stiff Digital ABS (combination of RGD515 and RGD531 by Stratasys). To investigate the influence of the stiffness of the compliant walls on the swelling-induced stress, five types of 3D-printed materials were used, namely rubber-like Agilus (Stratasys) and four Agilus and digital ABS-based digital materials (DMs). The DMs are referred to as FLX98XX by Stratasys, where XX = 40, 50, 60, and 70 denote the Shore A hardness. For convenience, the soft walls were henceforth referred to as SW-X, where X = 1 − 5, with SW-1 as the softest wall and SW-5 as the stiffest wall. The Young's moduli E and the Poisson's ratios of the soft materials were characterized with Instron 5943 and an AVE 2 video-extensometer. The properties are shown in Table 1. It is emphasized that the ratio between the Young's moduli of the digital ABS, which is assumed to be rigid, and the soft wall materials is >10 3 .

Geometric Confinement
To determine the overall swelling response and the swellinginduced stress at different stages of swelling, three representative edge lengths L = 15, 17.5, and 20 mm and soft wall thickness t = 1.5 mm were set. Note that the length to thickness ratio of the walls was chosen as L/t ⩾ 10. Before proceeding, it is pointed out that: 1) in boxes with L > 30 mm the gel reaches its freely swollen volume and no stress developes and 2) in boxes with L < 12.5 mm and wall thickness t = 1.5 mm the swelling-induced stress is sufficient to rupture the compliant walls.

Experimental Protocol
The experiments were conducted as follows: First, the diameter d p and the mass m p of a dry bead were measured using a caliper and an analytical scale, respectively. In addition, the empty box was weighed and its mass was denoted m b . Next, the water bead was placed in a box, which was then submerged in a water filled container for 24 h. It was found that this time period is sufficient to achieve equilibrium. Subsequently, the mass of the swollen bead in the box m tot was measured. The mass of the swollen gel is m g = m tot − m b and the added mass due to water uptake is m w = m g − m p . Since the added mass corresponds to the volume increase, the volumetric deformation J is approximated via Equation (1). To ensure reproducibility, each experiment was repeated three times. Overall, nine transversely constraining boxes and 45 elastically constraining boxes were printed. The swelling-induced deformations of the beads in the confining boxes are highly heterogeneous. As a result, the determination of the stress that the gels apply on the box using standard computational methods such as finite elements is complex and computationally expensive. To obviate this challenge, a simple method is proposed to estimate the swelling-induced stress that the gel exerts on the box.  First, a coordinate system {x,ŷ,ẑ} was defined, wherex points along the normal direction to the open and the soft wall in the transversely and elastically constraining boxes, respectively (see Figure 2a,b). Once equilibrium was reached, the length along thê y andẑ-directions was determined by the length of the box L and the overall length along thex-direction of the box l ⩾ L was measured (see Figures 2 and 3). The average stretch along the transverse and the longitudinal directions are denoted bỹ and respectively. It is emphasized that in the case of transversely constraining boxes there were no constraints along thex-direction and thus no stress was generated. However, as will be shown later, the transverse constraints greatly influencẽx and the volumetric deformation J.
To determine the swelling-induced stress in the elastically constraining box, the overall distance l + 2t between the maximum deflection points of the soft walls along thex-axis was measured. As illustrated in the dry and the swollen states in Figure 3a,b, respectively, the maximum deflection of the soft walls was achieved at the center of the wall and approximated via = (l − L)/2.
Next, the soft walls were modeled as thin simply supported plates with a thickness t, where t ≪ L, and it was assumed that the gel applied a radial stress along the points of contact with the soft walls. Consequently, the stress could be projected onto thê x-direction to result in a roughly sinusoidal stress distribution that works toward bending the wall (see Figure 3c). Following the Kirchhoff-Love plate theory, the relation between the maximum stress q 0 and the maximum displacement is given by [48] is the flexural rigidity of a soft wall with a Young's modulus E, a Poisson's ratio , and a thickness t. Substitution of Equations (2) and (3) into Equation (4) yielded which relates the maximum stress to the average transverse and the longitudinal stretches.

Results and Discussion
In the following we determine the longitudinal deformation in transversely constraining boxes and the swelling-induced stress in elastically constraining boxes. To this end, boxes with three representative dimensions and soft walls with five different flexural rigidities are considered.

Transversely Constraining Boxes
We begin by studying the swelling of a water bead in a transversely confining box. Table 2 summarizes the transverse stretch t , the axial stretch̃x, and the volumetric deformation J of a bead that swells freely or in transversely constraining boxes with L = 15, 17.5, and 20 mm. For convenience, the data is also displayed in Figure 4.
As expected, in free swelling (i.e., in the absence of mechanical constraints) an isotropic response is observed such that̃x = t = f . In the transversely constraining boxes the swelling of the beads is hindered along theŷ −ẑ plane and, interestingly, our measurements reveal that increasing the dimension of the geometric confinement length L, and consequently the transverse stretches (̃t), results in smaller longitudinal stretches (̃x). For example, the measured transverse stretch is̃t ≈ 2.66 mm and t ≈ 3.56 mm while the longitudinal stretch is̃x ≈ 6.08 mm and x ≈ 5.14 mm for boxes with the dimensions L = 15 mm and L = 20 mm, respectively.
In general, the transverse constraints imposẽt < f and as a result the axial deformatioñx > f and the volumetric deformation J < J f . Thus, one can deduce that the volumetric deformation obtained from a free swelling experiment J f ≈ 106 denotes the largest volumetric deformation that the water bead can achieve.
The observed trends stem from a combination of the elongation and rotation of the polymer chains in the network. To understand this behavior, consider a unit element with representative chains, as illustrated in Figure 5a. In a bead that swells freely, the chains extend without rotating, as illustrated in Figure 5b. However, transverse constraints directly impose restrictions on chains. Specifically, chains with an end-to-end vectors along thê y andẑ directions experience smaller deformations than chains along the longitudinal (x) direction. As a result, it can be shown that chains that are off-axes rotate toward thex-direction and experience larger extensions than in a freely swollen experiment. [19] This is schematically illustrated in Figure 5c.
It is also noted that the general trends in our experimental findings can be captured by well-known swelling models. To obviate the computational difficulties associated with the heterogeneous deformation and stress fields that result from the constraints, we consider a representative homogeneous deformation. Section S1, Supporting Information, summarizes the main equations of a well-established model for swelling [49] that is based on the eightchain model. [50] In Section S2, Supporting Information, the axial stretch and the volumetric deformations are computed as a function of the transverse stretch. This simplified investigation predicts trends that are identical to the ones that we observed experimentally.

Elastically Constraining Boxes
In this sub-section we report the stretches and the swellinginduced stress that a gel exerts as a result of swelling in an elastically constraining box. To this end, we investigate the influence www.advancedsciencenews.com www.advmattechnol.de  of two factors: 1) the size of the elastically constraining box; and 2) the flexural rigidity of the soft walls. Figure 6a plots the normalized stretch̃x∕̃t as a function of the flexural rigidity D (Equation (5)) for the five examined soft walls and the three selected box sizes. Recall that the dimensions of the box (i.e., L) limit swelling. Thus, we find that the swelling trends non-linearly depend on the stage of swelling. Specifically, the ratiõx∕̃t decreases as the length L increases and the gel nears its freely swollen state. This effect can also be appreciated from Figure 1e, which shows larger deformation of the soft walls in the smaller box.
This observation is explained as follows: to minimize its energy, the water bead swells with the aim of reaching a volume that is as close as possible to that of a freely swollen bead J f . Boxes with L = 15 mm significantly constrict swelling along thê y −ẑ plane and thus, to maximize the volume of the gel, relatively large deflections along thex direction are obtained. Since the volume of the gel increases (and its proximity to the freely swollen volume decreases) with the size of the box L, the deflection and the normalized stretch̃x∕̃t along thex-direction decrease. We note that these trends are similar to the stretch increase along thex direction of the smaller transversely constraining boxes. It is important to emphasize that as expected, the true value of̃x increases with L. Specifically,̃x is ≈3.24 − 3.48, ≈3.69 − 3.84, and ≈4.10 − 4.25 for boxes with L = 15, 17.5, and 20 mm.
As expected, comparison between the different soft wall materials reveals that the normalized stretch̃x∕̃t and the maximum deflection decrease as the flexural rigidity increases. The largest deflection was measured for the SW-1 box, as illustrated in Figure 1f.
Following our unique set-up, we determine and plot the swelling-induced stress q 0 (Equation (4)) that the gel exerts on the soft walls as a function of the longitudinal stretch̃x for three box sizes and five different soft walls in Figure 6b. The range of stress exerted by the gel decreases from q 0 ≈ 6.7 kPa and q 0 ≈ 1.9 kPa to q 0 ≈ 4.91kPa and q 0 ≈ 1.1 kPa in the smallest (L = 15 mm) and the largest (L = 20 mm) boxes, respectively. Accordingly, we find that the gel exerts extremely large stress in the initial stages of swelling. This stress decreases exponentially as the gel nears its freely swollen volume. This behavior stems from the non-linearities that are associated with the swelling-induced large deformations. [19,24,51,52] These general trends are also captured by the well-known model which is summarized in Section S1, Supporting Information. [49] Specifically, in Section S3, Supporting Information, we show that well-known frameworks for the swelling of polymers predict that: 1) the stress decreases non-linearly as the axial stretch increases; and 2) the swelling-induced stress is lower in elastically constraining boxes with higher transverse stretches.
Comparison between the different soft wall materials shows a slight decrease in the stress as the walls become softer. To understand this result, recall that the transverse stretch̃t is fixed and the softer walls are characterized by a lower flexural rigidity D (Equation (5)). Consequently, larger deflections and longitudinal stretches̃x are achieved in the gel. Following Equation (6), we conclude that the reduction in the flexural rigidity is more dominant than increase in the longitudinal stretch.
It is also pointed out that the above trends and conclusions for the swelling-induced stress q 0 can also be represented as a function of the volumetric deformation J, as shown in Section S4, Supporting Information.

Conclusions
In this work we propose an experimental method to measure the swelling-induced stress that gels exert on their environment during the swelling process. To this end, we exploit 3D-printing technologies to design boxes that transversely or elastically constrict the swelling of commercially available hydrophilic water beads. By considering boxes with different sizes, we investigate the equilibrium swollen configuration of the beads at various stages of swelling. In the elastically constraining boxes, we employ tools from the elastic plate theory to estimate the stress that the gels exert on the walls of the box. It is underscored that since the deformation state of a bead that swells in a box is highly heterogeneous, numerical investigations are complex and computationally expensive. Specifically, in such analyses one must consider the swelling of the gel, contact mechanics that account for the gelbox interactions, and the deformation of the soft walls in the box. The method presented here allows to easily explore the swellinginduced stress and quantifying the influence of mechanical constraints in the form of geometric confinement on the swelling of gels.
Our findings show that under transverse constraints the axial elongation of gels exceeds that which is obtained under free swelling (i.e., in the absence of constraints). However, the overall volumetric deformation in these cases is smaller than the freely swollen volume. In addition, the longitudinal elongation exhibited a non-linear dependence on the box size. As expected, these findings suggest that the gel aims to achieve the maximum volumetric deformation possible under constraints. This trends stem from the stretching and the rotation of polymer chains due to water uptake. In a free swelling experiment, all chains are expected to stretch equally with an average rotation angle of zero. However, swelling in transversely constraining boxes results in smaller stretches but larger rotations toward the direction that is free of constraints.
To determine the stress that gels exert on their environment, an elastically constraining box with four rigid walls and two soft walls was designed. Based on the measurements of the deformation of the soft walls and elastic plate theory, one can determine the swelling-induced stress. Our findings reveal that stiffer walls and smaller geometric confinements lead to larger swelling-induced stress. While the former has a relatively minor effect, the latter is significant. Specifically, the swelling-induced stress is very high if the volumetric deformation of the swollen gel is far from the freely swollen configuration. This stress nonlinearly decreases as the gel nears the freely swollen volume. It is also worth mentioning that the longitudinal stretch in transversely confining boxes, which essentially represents a box with soft walls of zero stiffness, sets the upper limit for the deformation of the gels in elastically confining boxes with the same size.
The method proposed in this work also serves to validate the commonly employed swelling models. Since the two types of constraints we considered result in heterogeneous deformation fields, which make it difficult to numerically analyze the swelling-induced stress, we employed a well-known swelling model to solve a representative problem characterized by homogeneous deformations that is expected to lead to similar behaviors. The trends obtained from the model predictions agree with our experimental findings.
To conclude, in this work we developed and introduced a method to estimate the swelling-induced stress that gels can apply to their environment in the presence of mechanical constraints. Our findings reveal that the swollen equilibrium configuration of a gel that swells under mechanical constraints depends on: 1) the type of geometric confinement; 2) the stage of swelling; and 3) the properties (or flexural rigidity) of the constraints. Our findings can be implemented to design, enhance, and even optimize the performance of swelling-based systems in various fields such as drug delivery, actuators, sensors, and soft robotics. While hydrophilic beads were used, the conclusions from this work can also be applied to other types of gels. Last, the proposed method can be used to study the influence of constraints on materials that experience volumetric deformations in response to other stimuli such as pH and temperature.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.