Thermomechanical Properties of Polyjet Voxel-Printed Parts and the Effect of Percolation

Abstract

The use of deformable materials in 3D printing has allowed for the fabrication of intricate soft robotics prototypes. Polyjet technology, with its ability to print multiple materials in a single print, has been popular in creating such designs. Vero and Agilus, the commercial materials provided by Polyjet, possess shape memory properties, making Polyjet ideal for high-precision and transformable applications. Voxel printing, where users assign materials to voxels, has allowed for the further expansion of design possibilities by tuning the properties of the jetted material. This study aims to investigate how different compositions of uniformly distributed Vero and Agilus voxels affect the thermomechanical properties of the voxel-printed part. In addition, high stiffness Vero droplets surrounded by a soft matrix of Agilus resemble polymer composites, thus calling for the examination of percolation, which is an important phenomenon in polymer composites. The study explores the presence of percolation in voxel-printed mixtures of Vero and Agilus and its impact on mechanical properties. Using dynamic thermomechanical analysis and thermomechanical analysis, the study characterizes the glass transition temperature ( $T_{g}$ ), maximum allowable strain, and modulus of the voxel-printed material at different compositions. The study found a highly linear relationship between $T_{g}$ and maximum yield strain with composition, and maximum yield strain occurs at 7°C above $T_{g}$ . On the other hand, there is a nonlinear relationship between the modulus and composition, which suggested that the percolation phenomenon might have altered the load distribution, therefore causing this inconsistency. So, in this study we used light microscopy, Monte Carlo simulations, and provided mathematical proofs to reveal the percolation threshold in voxel-printed parts, where Vero droplets suddenly form a single network that spreads across the material, altering the load distribution. This study is the first to highlight the percolation phenomenon in Polyjet voxel-printed parts and provides a useful guide for researchers in selecting suitable materials for their specific applications.

Introduction

3D printing is an additive manufacturing technology that enables the fabrication of complex geometry prototypes layer by layer with high resolution (1–50 μm).¹ When combined with the deformable characteristic of soft materials (generally with a modulus <10 MPa),² numerous impactful concepts in soft robotics are realized with 3D printing, such as biomimetic robots capable of locomotion,³ camouflage,⁴ jumping,⁵ and swimming⁶ as well as soft actuators^7–9 and soft sensors.^10–12

To further enhance the passive deformability of soft materials, there has been growing interest among researchers in printing shape memory polymers (SMPs) due to their ability to transform. Below the glass transition temperature ( $T_{g}$ ), it is rather stiff and “glassy,” while above this temperature it becomes deformable and “rubbery.” This has opened new avenues of research for integrating SMPs into soft robots, enabling proof of concept for morphing surfaces,¹³ active hinges,¹⁴ and variable stiffness actuators.¹⁵ These 3D printed prototypes have preprogrammed functionalities built into them, allowing the materials to transform independently over time, which is referred to as 4D printing,¹⁶ as time is an important additional dimension in the design process.

Most 3D printing technologies, including fused deposition modeling, direct ink writing, digital light processing, and inkjet printing, are capable of printing with SMPs. However, only Polyjet, an inkjet printing technology, could achieve multimaterial printing in a single print. The combination of Polyjet's high-resolution, multimaterial printing, and the smart properties of SMPs has led to the development of complex morphing systems.^17–20

However, the limited availability of SMPs for Polyjet printing is a challenge. Vero is a commercial thermosetting plastic that is widely used^{14,17,21–34} to integrate functional materials into morphing systems due to its drastic change in viscoelastic properties above room temperature, but the inability to use other types of SMPs limits the design space. An approach to address this issue is known as “voxel printing,” which uses the inkjet mechanism to assign the material of each jetted droplet. This technique allows for the creation of a new, homogeneous material with the properties of both materials by varying the distribution of Vero and other materials on a droplet scale. Voxel printing has the potential to solve the issue of limited SMP availability for Polyjet printing without requiring expertise in material science or expensive equipment.

Multiple studies have investigated the use of voxel printing and digital materials provided by Polyjet to mix Vero with other materials to achieve desired properties. Researchers have also explored how voxel printing could be used to tune the properties of SMPs. For instance, Akbari et al.¹⁴ characterized a series of digital materials that are mixtures of Vero and Agilus, an extensible and flexible elastomer provided by Polyjet. Akbari et al. evaluated the $T_{g}$ , Young's modulus, and breaking strain of these mixtures, which are important properties for robotics applications. Weeger et al.³³ designed a rod mesh with varying stiffness by voxel printing Vero and Tango, an elastomer that is less extensible than Agilus. To facilitate the design process, Weeger et al. characterized the Young's modulus as a function of the Vero portion in the material. Lumpe et al.²⁸ also proposed a similar approach, characterizing the Young's modulus as a function of the proportion of Vero in Agilus instead of Tango. Yuan et al.²³ investigated the effect of different layout strategies of Vero composition on a voxel scale.

However, abovementioned studies were more application focused and did not provide an explicit mapping of the effect of varying Vero volume fraction. Therefore, this article aims to unveil the thermomechanical properties of a voxel-printed mixture of Vero and Agilus and provide an explicit mapping between the thermomechanical properties and the Vero volume fraction.

Kaweesa and Meisel³⁵ and several other works^36,37 have reported that the modulus of voxel-printed parts does not increase linearly as the Vero volume fraction increases. This nonlinearity suggests that there may be microscopic mechanisms at work. Due to the major difference in stiffness between Vero and Agilus at room temperature, the voxel-printed part composed of these two base materials can be analyzed as a composite with a soft matrix and rigid fillers. In composites, percolation is a critical phenomenon that occurs when the filler fraction exceeds a critical threshold, resulting in a fundamental change in the magnetic polarity,³⁸ conductivity,³⁹ and load-bearing ability⁴⁰ of the fillers. Although studies have investigated the conductivity of jetted parts with added conductive fillers, there have been no reports on the percolation phenomenon in Polyjet voxel-printed parts, which is the focus of this study. The goal is to investigate the presence of the percolation phenomenon and its impact on the thermomechanical properties of the voxel-printed mixture of Vero and Agilus.

The article is structured as follows: in Introduction section, we describe the methods used for voxel printing and characterizing the thermomechanical properties of the voxel-printed material. We also investigate the state of Vero droplets using experimental and numerical methods. Materials and Methods section presents the effect of Vero composition on important properties relevant to robotics applications, and we compare the tested modulus data with a microscope image of Vero droplets and a Monte Carlo simulation to address the nonlinear map between the modulus and the composition of Vero. In Results and Discussion section, we analyze the different tendencies of the material properties as the Vero composition varies and compare them to different prediction models. The results are further employed to design a variable stiffness foldable structure. In Conclusions section, we provide the conclusions.

Materials and Methods

To investigate the thermomechanical characteristics of voxel-printed parts that are critical in robotics applications, it is necessary to assess specimens composed of various combinations of Vero and Agilus. This section will begin by outlining the fabrication process of a voxel-printed part before delving into the methodology employed to evaluate the parts' properties. To further understand the percolation phenomenon, we also conducted microscopic, probabilistic, and numerical analyses of the voxel-printed parts' microstructure.

Preparation of bitmaps for voxel printing

Polyjet technology is akin to traditional inkjet printing, wherein the printing equipment comprises multiple printing nozzles. During the printing process, a single material is sprayed from a nozzle onto the lift platform, and then a roller is used to mix each droplet with its neighboring unit to some extent. Subsequently, ultraviolet light is immediately irradiated to solidify the material that has been sprayed out (Fig. 1a). As the lift platform descends, the part is printed layer by layer based on the previous layer. In this study, Stratasys J750 was utilized, which provides a resolution of 14 μm in the z direction and resolutions of 423 and 846 μm in the x and y directions, respectively.

FIG. 1.

Polyjet voxel printing mechanism and part. (a) Schematic illustration of the Polyjet printing mechanism. The printhead is composed of multiple nozzles that dispense different materials. The roller flattens the jetted droplets for improved mixing, and then UV light is used to cure the material. (b) A graded material printed using Polyjet, consisting of Vero (black) and Agilus (transparent). The top surface, parallel to the xy-plane of the printing equipment, is examined using a light microscope, and the result is shown in (c). (c) Microscopic image of the voxel-printed part, displaying short, black whiskers of Vero. The minimum size of Vero droplets is observed in the region of ρ = 0.05, where a black dot of similar width and length is present, although it is slightly longer in the x direction. The microstructure of the voxel-printed part is similar to composites, with Vero acting as the filler and Agilus as the matrix. As the volume fraction of Vero increases, the droplets gradually connect from dispersed particles or fibers into an inseparable network.

Unlike traditional printing, where users provide an STL file of a part to the printer, voxel printing involves designating the material of each voxel by providing a series of bitmaps that are slices of the model, with materials differentiated by color. To achieve this process, a rasterization and dithering algorithm was developed based on MATLAB 2022a. The rasterization process involves acquiring the contour of the model at every 14 μm in the z direction, which corresponds to the J750's resolution. The planar contour is then discretized according to the resolution of the printer in the x and y directions, respectively. Macroscopically, to obtain arbitrary proportions of Vero with only discretized voxels of Vero and Agilus, we implemented the random dithering algorithm, as opposed to the commonly used Floyd–Steinberg dithering,^7,41 because the visual outcome is less important than ensuring a homogeneous part without artifacts that could lead to unexpected mechanical properties.

Characterization of thermomechanical properties

In this study, we conducted both dynamic thermomechanical analysis (DMA) and thermomechanical analysis (TMA) to characterize the important properties of voxel-printed parts with varying Vero volume fraction for robotics applications. We utilized a Waters Q800 to perform DMA on 30 × 2 × 1 mm specimens in tensile deformation mode at a frequency of 1 Hz and an amplitude of 0.1% of the remaining length after being clamped, which is 15 mm in our test (Fig. 2a). The tensile direction aligns with the x direction of the J750 printer. The specimens were equilibrated for 10 min at a temperature of at least 30°C above their estimated $T_{g}$ , and the temperature was then decreased by 2°C every 2 min. The results of the DMA test are presented in Figure 1.

FIG. 2.

DMA was conducted on a voxel-printed cuboid specimen to investigate the glass transition temperature ( $T_{g}$ ) at different compositions. (a) A sinusoidal strain was applied along the x direction, and the amplitude and phase offset of the repulsive stress were measured using DMA. Six volume fractions of Vero, ranging from 0 to 1, were tested. The figure shows the orientation of the specimen in accordance with the Polyjet equipment. (b) The tangent of each phase offset was measured under different temperatures. A single peak was observed for each composition, and the corresponding temperature was defined as the first type of $T_{g}$ in this study. (c) The storage modulus E′ was calculated as the ratio of the amplitude of the stress and strain times the cosine of the phase offset. The plot shows the E′ values at a logarithmic scale and can be divided into three distinct sections. The intersection of the asymptotic line (solid line) of the first and second sections is marked with a hollow circle and is referred to as the second type of $T_{g}$ in this study. The first type of $T_{g}$ is also marked with a hollow circle. (d) The two types of $T_{g}$ were scattered and fitted linearly, and the expression of the fitted result of the first type of $T_{g}$ is given. The two types of $T_{g}$ had a consistent difference of 21°C as the composition varied. DMA, dynamic thermomechanical analysis.

To perform TMA, we employed a ZwickRoell BT2 and followed the ASTM D638-22 standard by using type IV specimens (Fig. 3b). The temperature range for the TMA test was chosen based on the results of the DMA test, which will be explained in Relationship Between Composition and Glass Transition Temperature section. The results of the tensile test are shown in Supplementary File.

FIG. 3.

TMA was conducted on voxel-printed type IV specimens to investigate the yield strain according to ASTM D638-22 at different compositions. (a) Compositions ranging from 0.5 to 0.9 were tested at five different temperatures. Starting from the second type of $T_{g}$ every 7°C is tested. A total of 30 specimens were tested, and the maximum yield point is plotted. The two types of $T_{g}$ are indicated with hollow circles, and the black dashed line represents the fit explained in (b). (b) The geometry of the type IV specimen is shown in the top left. The maximum yield strain from (a) is plotted with its corresponding composition, and a linear fit is performed, with the expression given in the bottom right. By substituting the ρ in the above relation with the temperature 7°C above the second type of $T_{g}$ , we obtain the black dashed line in (a), as there is a bijection between ρ and $T_{g}$ . The Supplementary File provides the detailed results of TMA. DMA was performed on voxel-printed parts to investigate the storage modulus at different compositions. (c) The plot shows the storage modulus data from Figure 2c but in a linear scale. The vertical dashed line represents the second type of $T_{g}$ of the corresponding compositions. (d) The storage modulus is centered at their corresponding second type of $T_{g}$ , and all data above the second type of $T_{g}$ is fitted exponentially. The expression of the fitted result is provided at the lower right. (e) The storage modulus is plotted against the corresponding composition at room temperature. De Gennes' scalar model of elastic percolation is fitted with the data, and the expression of the fitted result and the goodness of fit are provided in the lower left. TMA, thermomechanical analysis.

Microstructure investigation

Metallograph

We used a LEICA DM6M metallographic microscope to investigate the surface of a voxel-printed cuboid parallel to the xy-plane of J750. The cuboid is 16.2 mm high and consists of six layers, including Vero volume fraction ρ = {0, 0.05, 0.1, 0.15, 0.2, 0.25} (Fig. 1b). In this study, we used black Vero with transparent Agilus for easy observation. After printing, the surface was ground with sandpaper of 500, 1000, and 2000 grids on a Phoenix4000 polishing machine and then polished with a suspension of alumina particles. The microscopic image is shown in Figure 1c.

Probabilistic and numerical analysis

To explore the voxel-printed part's internal characteristics and validate the presence of percolation, we conducted both a probabilistic analysis and a Monte Carlo simulation. To establish a probability space for modeling the voxelated and jetted Polyjet part, we envisioned an infinite cubic lattice, where each site corresponds to a Vero voxel with a probability equivalent to the portion of Vero material within the part.

Within this unbounded space, we identified two distinct scenarios: [VO] and [ $V \infty$ ]. In scenario [VO], all Vero clusters were entirely surrounded by Agilus voxels, while scenario [ $V \infty$ ] involved the presence of infinitely large Vero clusters. Analogously in finite space, [VO] represented the dispersion of Vero voxels throughout the part, while [ $V \infty$ ] indicated the connectedness of the majority of Vero voxels within the part.

To assess the probabilities of these events occurring at various Vero volume fractions, our study focused on determining a nontrivial value of $ρ$ , serving as the threshold for each event. Calculating the exact $ρ$ value necessitated a numerical approach. Hence, utilizing MATLAB, we simulated a 3D array (50 × 50 × 50) to represent the voxel-printed part. Each unit within the array represented a voxel and had a probability of $ρ$ to be designated as Vero.

Throughout the simulation, voxel units were considered connected only when their faces touched (utilizing a six-connected neighborhood). By adjusting $ρ$ , which directly correlated with the Vero volume fraction, we analyzed the emergence of the largest connected Vero region within the simulated part.

Results and Discussion

Relationship between composition and glass transition temperature

Shape memory phenomena do not occur in SMPs if the temperature during deformation is too low, and the fracture/yield strain decreases sharply if the temperature is too high. Therefore, determining the $T_{g}$ of a voxel-printed part is crucial for its application. DMA offers two major ways of determining $T_{g}$ . The first method is to identify the temperature at which the phase offset ( $tan δ$ ) of the stress and strain is the largest. The second method involves finding the intersection of two asymptotic lines of the storage modulus, as shown in Figure 2c. In this article, we will refer to these two methods as the first type of $T_{g}$ and the second type of $T_{g}$ , respectively. As shown in Figure 2b, the $tan δ$ exhibited a clear peak with temperature changes, indicating that the temperature corresponding to each peak is the $T_{g}$ of the respective composition. Moreover, the single peak confirms that the voxel-printed part is homogeneous, and it could be considered a single material. Otherwise, two peaks would have been observed.

From a phase evolution perspective, the glass transition can be depicted by most “glassy” areas transitioning into a “rubbery” area. A classic model that depicts this transition is the normal distribution model,⁴² where the mean corresponds to the $T_{g}$ of an SMP. Since Vero and Agilus are both SMPs, their homogeneous mixture should be a proportional addition of the two distributions. Thus, a linear variation of the mean occurs as the fraction of Vero grows. This linearity is shown in Figure 2d, where the corresponding temperature of the peaks is plotted with a fitted line with an r² value of 0.989. Therefore, the first type of $T_{g}$ could be expressed with the following equation: $T_{g} = 59.3 ρ + 8.7$ (4.1)

This section continues by examining the relationship between composition variation and the second type of $T_{g}$ . Figure 2c depicts the logarithm of the storage modulus at various compositions. As the temperature rises, three distinct sections of the storage modulus can be observed, which can be approximated by asymptotic lines. The intersection of the first two asymptotic lines, represented by the hollow circle, corresponds to the second type of $T_{g}$ , while the other hollow circle represents the first type of $T_{g}$ . We observe that the second type of $T_{g}$ exhibits a linear variation with composition, as demonstrated by the lower line in Figure 2d. Additionally, we note that the values of $T_{g}$ obtained from the two types of analysis differ by ∼21°C for all compositions. Therefore, a range of glass transition temperatures is obtained, which can be utilized as a guide to test the yield/fracture strain.

Relationship between composition and yield strain

In practical applications, it is crucial to determine the allowable strain of a material before it yields. However, conducting TMA can be expensive, so it is essential to choose the appropriate range of composition and temperature for conducting tensile tests. To exhibit shape memory effects, deformation must occur when the SMP is in its “rubbery” state. Hence, Vero volume fractions lower than 0.5 should be excluded as their $T_{g}$ is below room temperature, making them unsuitable for practical applications. Based on the $T_{g}$ range obtained from the previous section, we conducted a series of tensile tests around it. The results are presented in Supplementary File. Starting from the second type of $T_{g}$ , each composition was tested at every 7°C for a total of five different temperatures.

As the yield point for polymers is not as well defined as for metals, we considered the maximum stress point as the yield point unless there was necking. If necking occurred, the maximum stress before necking was considered the yield point. The results are shown in Figure 3a. For every composition, the maximum yield strain occurred when the temperature was 7°C above the second type of $T_{g}$ . The yield strain then decreased sharply as the temperature continued to rise, except when the Vero volume fraction was 0.8.

At around $T_{g}$ temperature, the SMP is mainly composed of the “rubbery” phase, and most bonds between polymer chains have disappeared.⁴³ At this point, the net points hold the material's permanent shape. Therefore, the allowable strain of the material must be determined by the weakest net point. In a one-dimensional model where corresponding fractions of Vero and Agilus are connected in series, the maximum allowable strain would react linearly as the Agilus fraction varies linearly. While Agilus is more extensible than Vero at room temperature, the yield strain of Agilus decreases sharply at temperatures around the $T_{g}$ of Vero, whereas Vero has just reached its maximum allowable strain. Therefore, a linear variation of the composition should result in a linearly increasing maximum yield strain, which occurs at 7°C above each composition's second type of $T_{g}$ , as illustrated in Figure 3b. The data can be linearly fitted with the following equation with a R² of 0.9392: $ε_{Y} = 0.7375 ρ + 0.4386$ (4.2)

Moreover, if we substitute the $ρ$ in the equation above with the corresponding temperature (7°C above the second type of $T_{g}$ ), we obtain a relation between the maximum yield strain and temperature: $max ε_{Y} = 0.0124 T + 0.5038$ (4.3)

This equation is plotted in Figure 3a as the dashed black line, and it could predict the possible maximum yield strain of each composition especially at high compositions.

Relationship between composition and modulus

In practical applications, understanding the Young's modulus of a material at both its “glassy” and “rubbery” states is crucial as it reflects how the material interacts with its environment. For viscoelastic materials like SMPs, characterizing the Young's modulus without having to tear countless specimens with varying compositions and temperatures can be achieved through approximating it with the storage modulus acquired from DMA testing, provided that the deformation rate is not too high.

Figure 3c displays the variation in storage modulus with temperature for various compositions. The vertical dotted line on the graph represents the location of the second type of $T_{g}$ for each composition. It can be observed that the moduli of each composition show a good linear correlation with temperature below the second type of $T_{g}$ . Above the second type of $T_{g}$ , the moduli display a good exponential correlation with temperature. In fact, by setting the second type of $T_{g}$ as the origin, as illustrated in Figure 3d, each composition undergoes a similar change in storage modulus with temperature, and this process can be captured to a certain extent by the same exponential model, the red dashed line shown in Figure 3c. Therefore, by shifting the origin of the fitted exponent to the second type of $T_{g}$ corresponding to each composition, the modulus of that composition can be obtained. $E' = 24.3 e^{(- 1.74 \frac{T - T_{g} (ρ) - 25}{15})}$ (4.4)

The relationship between the modulus and composition was also analyzed due to the good linear relationship between the previous properties and composition. Figure 3e illustrates the relationship between composition and modulus at room temperature. It can be observed that unlike the previous properties, only the composition does not have a linear relationship with the modulus, which is consistent with previous studies.^35–37

Furthermore, a critical composition can be observed at approximately ρ = 0.2, where the change in modulus exhibits a different trend. It is highly likely that during the increase of Vero fractions, a large, connected network is formed from discrete Vero voxel droplets, shifting the load distribution. The addition of various forms of reinforcement materials (such as particles and short fibers) to the matrix material is a common method of changing overall properties in the field of composite materials.⁴⁴ If a new form of reinforcement material appears during the increase of filler proportion, such as replacing droplets with a network-like reinforcement material in the matrix, there will be a significant change in the modulus,²³ even if the volume fraction of the filler is the same.

This phenomenon is often exploited in soft sensors,³⁹ where designers add conductive fillers such as carbon black and carbon nano tubes beyond a critical ratio to make the entire material suddenly conductive, a phenomenon referred to as percolation of the conductive fillers. The next section verifies this hypothesis.

Existence of percolation in a Polyjet voxel-printed part

Figure 1c illustrates a microscopic image featuring black fibrous Vero dispersed within a transparent Agilus matrix, with the short fibers oriented in the x direction. The Polyjet printing mechanism, considering the arrangement of the print head, would typically result in the SMP appearing as short fibers oriented in the y direction, due to the x direction having twice the resolution. However, a closer examination revealed that the fibrous Vero consists of droplets with similar length and width, slightly longer in the x direction. The Vero droplets' length and width were ∼850 μm, suggesting that the voxel droplets, initially rectangular (423 × 846 μm), are extended by the roller effect in the x direction, giving them a squarer appearance.

As the proportion of Vero increases in the printed part, it undergoes a transition from dispersed droplets to a connected network. This shift in the material's arrangement affects the load-bearing mechanism, which explains the observed trend in modulus. However, the critical fraction at which the trend of the modulus changes still requires explanation. To address this issue, we turn to the probability space constructed in Microstructure Investigation section to investigate the part's inner structure.

In an infinite space, the occurrence of both event [Vo] and [ $V \infty$ ] remains unaffected by switching the material of a finite set of voxels, making them tail events. According to Kolmogorov's 0–1 law, the probability of both events must either be 0 or 1 at any given fraction. Considering that the possibility of [ $V \infty$ ] can only increase as the fraction grows, and taking into account that [ $V \infty$ ]'s probability is exactly 0 and 1 at fractions 0 and 1, there must exist a critical fraction that serves as a threshold for both events. The remaining problem lies in proving that this critical fraction is neither 0 nor 1. This is accomplished by considering the special mechanisms of Polyjet voxel printing, as detailed in the Supplementary File.

Once the theoretical existence of percolation is established, its exact value can be estimated numerically. Figure 4a displays the result of a Monte Carlo simulation of the volume of the largest connected region at different Vero compositions. Notably, the volume of the largest connected region significantly increases at approximately ρ = 0.3, consistent with 3D site percolation theory. However, this assumption is not applicable to our case, as depicted in Figure 1c, where Vero extends from the designated voxel to the neighboring sites along the x direction due to the roller effect, leading to an additional site being filled with Vero. Consequently, the simulation is adjusted, and the critical value drops to ρ = 0.15 (Fig. 4b), aligning with experimental observations. Visualizing the largest connected region in the voxel-printed part, where the Vero network is marked with blue dots (Fig. 4c–f), demonstrates that dispersed Vero droplets suddenly connect at the critical fraction, forming a network that spans the material. This simulation provides deeper insights into the underlying mechanisms driving the observed changes in modulus with varying Vero composition.

FIG. 4.

Monte Carlo simulations were carried out on a 50 × 50 × 50 matrix to investigate the percolation phenomenon in Polyjet voxel-printed parts. (a, b) The volume of the largest connected region is plotted against the corresponding composition. The connection is determined when faces touch (six-connected neighborhood). In (a), the effect of the roller is not considered, resulting in a percolation threshold consistent with the theoretical value of the 3D site percolation threshold of a simple cubic lattice. In contrast, in (b), the impact of the Polyjet roller is considered, whereby every site assigned with Vero is also assigned as Vero to its neighbor in the x direction, resulting in a percolation threshold that agrees with the result of Figure 3e. (c–f) The visual results of (b) depicting the largest connected region at different compositions. (f) The majority of Vero sites are suddenly connected into a network that spreads across the material.

Effect of percolation on the thermomechanical properties

In the previous section, we discovered the percolation threshold in jetted parts, which changes the load distribution in voxel-printed parts. In this study, we aim to investigate how it can affect thermomechanical properties.

Both the first and second types of $T_{g}$ are not affected by percolation. The $tan δ$ curve shows a single peak, indicating that the voxel-printed part is well mixed on a macroscopic scale, even though there are two distinct phases on a microscopic level. Therefore, percolation can be viewed as a homogeneous material that becomes better at load bearing, without affecting the material's viscosity. As a result, we observe a linear correlation between $T_{g}$ and the composition. However, if the voxels are not well mixed, and the load is mainly distributed on either Agilus or Vero, percolation can affect $T_{g}$ . Chao showed different Vero layouts,²³ where the network layout was not well mixed but still exhibited a single peak in $tan δ$ , indicating that the load is mainly distributed on the Vero network, and the impact of Agilus on $T_{g}$ was less significant.

In this article, the characterization of yield strain is limited to the range of ρ larger than 0.5. Although 0.5 is far away from the percolation threshold of 0.15, it is still difficult to establish a clear relationship between percolation and the maximum yield strain.

The modulus near the percolation threshold can be predicted using de Gennes' scalar model of elastic percolation⁴⁵ as follows: $E = c o n s t . {(ρ - ρ_{c})}^{t}$ (4.5)

Here, $ρ_{c} = 0.15$ is the percolation threshold, and t is the elastic percolation exponent. Fitting the data in Figure 3e, we obtained a constant of 1985 and t of 1.754, with an R² value of 0.9951. The resulting curve is plotted as a black dashed line in Figure 3e. Importantly, the percolation exponent of 1.754 is close to the 3D theoretical value of 1.833 proposed by de Gennes. Hence, we can conclude that the modulus of voxel-printed parts follows de Gennes' scalar model of elastic percolation theory.

Design of a variable stiffness foldable structure

By explicitly expressing the properties of voxel-printed parts, both applications using digital mixtures,¹⁴ and voxelated^28,33 ones can select materials according to their desired application properties. As a practical example, we demonstrate the design of a simple variable stiffness foldable structure, as depicted in Figure 5.

FIG. 5.

Variable stiffness structure design using the prediction of modulus in the rubbery phase. (a) A cubic, accordion-like foldable structure has dimensions of 20 × 20 × 10 mm³ and is voxel-printed as a single part using Polyjet technology to achieve the desired structural properties. The 3D model (upper left) depicts the definition of each crease, with crease AC embedded with nickel wires to provide heat for varying the stiffness. Note that the designation “crease AC” pertains to a total of eight creases occupying analogous positions within the structure. A compressive test was conducted on the structure, with the test performed under room temperature while the temperature of crease AC was set to 65°C. Both loading and unloading rates were 0.1 mm/s. (b) Results of the compressive test are presented with the dashed line. The blue line represents the prediction using a classic linear spring model and the fitted result (4.4). However, upon closer inspection, it was observed that the model required correction with a proportional term to align better with the experimental data. The red line represents the corrected model, and it demonstrates good agreement with the experimental data.

The objective is to create an accordion-like foldable structure with creases made of different materials to achieve specific properties. To realize this, Polyjet is a suitable method where the creases can be printed using various material compositions. For demonstration purpose, the desired properties are as follows: a reactant force lower than 4N at a compressive strain of 0.45 in the rubbery phase, and the highest possible shape fixity in the glassy phase. This foldable structure finds potential applications as the body of a robotic worm,⁴⁶ a constituent of a morphing surface,⁴⁷ or a robotic gripper.⁴⁸

Due to length limitations and scope considerations in this article, we refrain from presenting the modeling and design optimization process in detail. Instead, we regard the structure as rigid facets with torsional springs, employing a simplified yet classic model. Figure 5a illustrates the actual strucutre. Due to its possession of three distinct planes of symmetry, the definition of a single octave suffices. Here we define crease AC as the connection between vertex A and vertex C. Subsequent references to crease AC encompass all analogous creases situated within equivalent positions across the other octaves. Note that crease AC has a Vero fraction of 0.8, while the remaining creases are composed of pure Agilus.

With the model and Equation (4.4), we can predict the reactant force when loading the structure while the temperature on crease AC is 65°C, depicted in blue in Figure 5b, where the force remains just below 4 N. However, experimental data, indicated by the black dashed line, reveals a discrepancy between the model and the experiment. To address this discrepancy, we applied a proportional term to the model, resulting in the red curve. The adjusted model now aligns well with the experimental data, demonstrating its ability to predict the tendencies of the real condition. It is essential to note that viscoelastic materials can be influenced by loading speed, humidity, and aging, thus explaining why this result is considered satisfactory despite using an oversimplified model.

Conclusion

Polyjet 3D printing technology is being utilized by designers to integrate smart materials into soft robotics, which allows for the creation of prototypes with high precision and the ability to transform. This is combined with Polyjet's ability to achieve voxel printing, enabling researchers to fine-tune the thermomechanical properties of the printed part. The purpose of this study is to explore how changes in Vero/Agilus composition impact the thermomechanical properties of voxel-printed parts. Additionally, since percolation is a crucial factor that impacts composite properties, this study investigates whether percolation occurs in jetted parts and its impact on thermomechanical properties. The results of the study can be summarized as follows:

A linear relationship exists between Vero volume fraction and $T_{g}$ (both types), as well as the maximum allowable yield strain. The maximum yield strain occurs at 7°C above the second type of $T_{g}$ .

Percolation occurs at a Vero volume fraction of ρ = 0.15, where the Vero droplets form a single network that spreads throughout the material, resulting in a sudden change in modulus due to altered load distribution. This is verified through microscopic images, Monte Carlo simulation, and de Gennes' scalar model of elastic percolation theory.

The roller in Polyjet equipment extends the jetted rectangle voxels approximately twice its width for better mixing between neighbor voxels, resulting in a percolation threshold of 0.15, which differs from the 3D site percolation threshold of 0.31.⁴⁹

These results can guide future works in selecting the most appropriate Vero and Agilus composition for their voxel-printed parts. However, it is important to note that for viscoelastic materials, the strain rate is a crucial factor in its constitutive relation. Therefore, the results of this study should be used as a guide for the thermomechanical property variation tendency as the composition changes, and it should be adjusted accordingly to the strain rate of specific applications. Moreover, for applications that operate below room temperatures, the yield strain when the Vero fraction is lower than 0.5, which was intentionally avoided in this study, should be examined. Furthermore, exploring how different dithering algorithms can impact thermomechanical properties could be an interesting topic for future research.

Footnotes

Acknowledgments

The authors would like to express their gratitude to Jing Lee for the valuable discussions, as well as to Renjun Dang and Changping Hu for providing helpful suggestions during the revision process.

Authors' Contributions

C.H.: Conceptualization (lead); Methodology; writing—original draft; formal analysis; and writing—review and editing. J.B.: Conceptualization (supporting). J.X.: Supervision.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

No funding was received for this article.

Supplementary Material

References

Truby

, Lewis

. Printing soft matter in three dimensions. Nature, 2016; 540(7633):371–378.

Wallin

, Pikul

, Shepherd

. 3D printing of soft robotic systems. Nat Rev Mater, 2018; 3(6):84–100; doi: 10.1038/s41578-018-0002-2

Shepherd

, Ilievski

, Choi

, et al. Multigait soft robot. Proc Natl Acad Sci U S A, 2011; 108(51):20400–20403.

Morin

, Shepherd

, Kwok

, et al. Camouflage and display for soft machines. Science, 2012; 337(6096):828–832.

Lin

H-T

, Leisk

, Trimmer

. GoQBot: A caterpillar-inspired soft-bodied rolling robot. Bioinspir Biomim, 2011; 6(2):026007; doi: 10.1088/1748-3182/6/2/026007

Phamduy

, Vazquez

, Kim

, et al. Design and characterization of a miniature free-swimming robotic fish based on multi-material 3D printing. Int J Intell Robot Appl, 2017; 1:209–223.

Scharff

, Doubrovski

, Poelman

, et al. Towards behavior design of a 3D-printed soft robotic hand. In: Soft Robotics: Trends, Applications and Challenges: Proceedings of the Soft Robotics Week, April 25–30, 2016, Livorno, Italy. Springer: Switzerland; 2017; pp. 23–29.

Yap

, Ng

, Yeow

C-H

. High-force soft printable pneumatics for soft robotic applications. Soft Robot, 2016; 3(3):144–158.

Sydney Gladman

, Matsumoto

, Nuzzo

, et al. Biomimetic 4D printing. Nat Mater, 2016; 15(4):413–418.

10.

Sarwar

, Dobashi

, Preston

, et al. Bend, stretch, and touch: Locating a finger on an actively deformed transparent sensor array. Sci Adv, 2017; 3(3):e1602200; doi: 10.1126/sciadv.1602200

11.

Muth

, Vogt

, Truby

, et al. Embedded 3D printing of strain sensors within highly stretchable elastomers. Adv Mater, 2014; 26(36):6307–6312.

12.

Mannsfeld

, Tee

, Stoltenberg

, et al. Highly sensitive flexible pressure sensors with microstructured rubber dielectric layers. Nat Mater, 2010; 9(10):859–864.

13.

Wang

, Salazar

, Vahabi

, et al. Metamorphic superomniphobic surfaces. Adv Mater, 2017; 29(27):1700295; doi: 10.1002/adma.201700295

14.

Akbari

, Sakhaei

, Kowsari

, et al. Enhanced multimaterial 4D printing with active hinges. Smart Mater Struct, 2018; 27(6):065027; doi: 10.1088/1361-665X/aabe63

15.

Yang

, Chen

, Li

, et al. Novel variable-stiffness robotic fingers with built-in position feedback. Soft Robot, 2017; 4(4):338–352; doi: 10.1089/soro.2016.0060

16.

Tibbits

4D Printing: Multi-material shape change. Archit Des, 2014; 84(1):116–121; doi: 10.1002/ad.1710

17.

Kimionis

, Isakov

, Koh

, et al. 3D-printed origami packaging with inkjet-printed antennas for RF harvesting sensors. IEEE Trans Microw Theory Tech, 2015; 63(12):4521–4532; doi: 10.1109/TMTT.2015.2494580

18.

Zarek

, Layani

, Cooperstein

, et al. 3D printing of shape memory polymers for flexible electronic devices. Adv Mater, 2016; 28(22):4449–4454; doi: 10.1002/adma.201503132

19.

Nauroze

, Novelino

, Tentzeris

, et al. Inkjet-printed “4D” tunable spatial filters using on-demand foldable surfaces. In: 2017 IEEE MTT-S International Microwave Symposium (IMS). IEEE: Honololu, HI;, 2017; pp. 1575–1578; doi: 10.1109/MWSYM.2017.8058932

20.

Hingorani

, Zhang

Y-F

, Zhang

, et al. Modified commercial UV curable elastomers for passive 4D printing. Int J Smart Nano Mater, 2019; 10(3):225–236; doi: 10.1080/19475411.2019.1591540

21.

Gul

, Sajid

, Rehman

, et al. 3D printing for soft robotics–A review. Sci Technol Adv Mater, 2018; 19(1):243–262; doi: 10.1080/14686996.2018.1431862

22.

, Chen

, Cheng

, et al. 3D printing of highly stretchable hydrogel with diverse UV curable polymers. Sci Adv, 2021; 7(2):eaba4261; doi: 10.1126/sciadv.aba4261

23.

Yuan

, Wang

, Qi

, et al. 3D printing of multi-material composites with tunable shape memory behavior. Mater Des, 2020; 193:108785; doi: 10.1016/j.matdes.2020.108785

24.

Tao

, Xi

, Wu

, et al. 4D printed multi-stable metamaterials with mechanically tunable performance. Compos Struct, 2020; 252:112663; doi: 10.1016/j.compstruct.2020.112663

25.

Wang

, Yuan

, Wang

, et al. A phase evolution based constitutive model for shape memory polymer and its application in 4D printing. Smart Mater Struct, 2020; 29(5):055016; doi: 10.1088/1361-665X/ab7ab0

26.

, Dunn

, Qi

, et al. Active origami by 4D printing. Smart Mater Struct, 2014; 23(9):094007; doi: 10.1088/0964-1726/23/9/094007

27.

Lumpe

, Shea

. Computational design of 3D-printed active lattice structures for reversible shape morphing. J Mater Res, 2021; 36(18):3642–3655; doi: 10.1557/s43578-021-00225-2

28.

Lumpe

, Tao

, Shea

, et al. Computational design and fabrication of active 3D-printed multi-state structures for shape morphing. Smart Mater Struct, 2022; 32; doi: 10.1088/1361-665X/aca5d6

29.

Wang

, Xu

, Wang

, et al. Design of 3D printed programmable horseshoe lattice structures based on a phase-evolution model. ACS Appl Mater Interfaces, 2020; 12(19):22146–22156; doi: 10.1021/acsami.0c04097

30.

Ding

, Yuan

, Peng

, et al. Direct 4D printing via active composite materials. Sci Adv, 2017; 3(4):e1602890; doi: 10.1126/sciadv.1602890

31.

Zhang

, Zhang

, Hingorani

, et al. Fast-response, stiffness-tunable soft actuator by hybrid multimaterial 3D printing. Adv Funct Mater, 2019; 29(15); doi: 10.1002/adfm.201806698

32.

, Yuan

, Ding

, et al. Multi-shape active composites by 3D printing of digital shape memory polymers. Sci Rep, 2016; 6:24224; doi: 10.1038/srep24224

33.

Weeger

, Kang

YSB

, Yeung

, et al. Optimal design and manufacture of active rod structures with spatially variable materials. 3D Print Addit Manuf, 2016; 3(4):205–215; doi: 10.1089/3dp.2016.0039

34.

Ren

, Ji

, Tao

, et al. SMP-based multi-stable mechanical metamaterials: From bandgap tuning to wave logic gates. Extreme Mech Lett, 2021; 42:101077; doi: 10.1016/j.eml.2020.101077

35.

Kaweesa

, Meisel

. Quantifying fatigue property changes in material jetted parts due to functionally graded material interface design. Addit Manuf, 2018; 21:141–149; doi: 10.1016/j.addma.2018.03.011

36.

Cheng

, Wang

, Sun

, et al. Centrifugal multimaterial 3D printing of multifunctional heterogeneous objects. Nat Commun, 2022; 13(1):7931; doi: 10.1038/s41467-022-35622-6

37.

Zolfagharian

, Jarrah HR

S X

, avier

, et al. Multimaterial 4D printing with a tunable bending model. Smart Mater Struct, 2023; 32(6):065001; doi: 10.1088/1361-665X/accba8

38.

Kokkinis

, Schaffner

, Studart

. Multimaterial magnetically assisted 3D printing of composite materials. Nat Commun, 2015; 6(1):8643; doi: 10.1038/ncomms9643

39.

Jung

, Choi

, Mocanu

, et al. Modeling electrical percolation to optimize the electromechanical properties of CNT/polymer composites in highly stretchable fiber strain sensors. Sci Rep, 2019; 9(1):20376; doi: 10.1038/s41598-019-56940-8

40.

Tsurusawa

, Leocmach

, Russo

, et al. Direct link between mechanical stability in gels and percolation of isostatic particles. Sci Adv, 2019; 5(5):eaav6090; doi: 10.1126/sciadv.aav6090

41.

Guy

, Morris

, Mirjalili

. Toward emulating human movement: Adopting a data-driven bitmap-based “Voxel” multimaterial workflow to create a flexible 3D printed neonatal lower limb. 3D Print Addit Manuf, 2022; 9(5):349–364; doi: 10.1089/3dp.2021.0256

42.

Guo

, Liu

, et al. Constitutive model for a stress- and thermal-induced phase transition in a shape memory polymer. Smart Mater Struct, 2014; 23(10):105019; doi: 10.1088/0964-1726/23/10/105019

43.

Behl

, Lendlein

. Shape-memory polymers. Mater Today, 2007; 10(4):20–28; doi: 10.1016/S1369-7021(07)70047-0

44.

S-Y

, Xu

, Mai

Y-W

. On the elastic modulus of hybrid particle/short-fiber/polymer composites. Compos Part B Eng, 2002; 33(4):291–299; doi: 10.1016/S1359-8368(02)00013-6

45.

De Gennes

PG.

On a relation between percolation theory and the elasticity of gels. J Phys Lett, 1976; 37(1):1–2; doi: 10.1051/jphyslet:019760037010100

46.

Onal

, Wood

, Rus

. An origami-inspired approach to worm robots. IEEE/ASME Trans Mechatron, 2013; 18(2):430–438; doi: 10.1109/TMECH.2012.2210239

47.

Robertson

, Murakami

, Felt

, et al. A compact modular soft surface with reconfigurable shape and stiffness. IEEE/ASME Trans Mechatron, 2018; 24(1):16–24.

48.

Kim

S-R

, Lee

D-Y

, Ahn

S-J

, et al. Morphing origami block for lightweight reconfigurable system. IEEE Trans Robot, 2020; 37(2):494–505.

49.

Stauffer

, Aharony

Introduction to Percolation Theory. CRC Press: London; 2018.

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