Computational design of anisotropic nanocomposite actuators Open Access

The development of soft, biocompatible actuators with properties comparable to those of mammalian muscles is a key step toward the next generation of soft robotic systems and bio-integrable prosthetic devices.¹ Several types of soft actuating materials have been investigated, which rely on different actuation mechanisms, such as electroactive polymers, piezoelectric actuators, shape-memory composites, humidity-responsive actuators, strain-programmable fibers, and molecular nanomachines.^2–4 However, neither of these materials offers a combination of softness, good temporal responsiveness, adaptability, scalability, and ease of manufacturing.¹ 

Stimuli-responsive hydrogels (SRHs) have gained attention as actuating materials, due to their ability to undergo large deformations in response to various triggers.⁵ These materials consist of a stimuli-responsive polymer network and a high fraction of water. By changing the solubility of the polymer via temperature, light-exposure, or changes in pH or ionic strength, the amount of water in the network can be controlled, causing large volumetric variations. Hydrogels are soft and shape-compliant, straightforward to process, and they can be prepared from biocompatible building blocks.⁵ However, they generally exhibit poor mechanical resistance and isotropic expansion.⁵ To overcome these limitations, SRHs have been combined with rigid, macroscopic elements that increase the strength and stiffness and confer directionality to the actuation.⁴ This approach has mainly been used to create devices in which the actuation consists of controlled folding rather than muscle-like expansion and contraction.⁴ 

The unique properties of animal muscles, such as high force-density, large and directional strokes, and fast response emerge from an interplay between hierarchical (anisotropic) mesoscale organization and molecular-scale actuation.⁶ Inspired by these design strategies, artificial muscles (AM) are proposed that combine the osmotic actuation principle of SRHs with the structural features of anisotropic colloidal liquid crystals. By orienting and subsequently cross-linking anisotropic colloids grafted with stimuli-responsive macromolecules, actuating nanocomposites having an anisotropic structure can be produced (Fig. 1). The cross-linked polymer forms a responsive hydrogel that enables actuation by swelling or contraction, while the aligned colloidal particles provide directionality and increase strength and stiffness. The design is modular and different types of colloids, polymers, and mesophases can be combined to tailor mechanical characteristics, processability, and actuation properties.

This study focuses on predicting the actuation properties of AMs based on this design. Cellulose nanocrystals^7,8 (CNCs) were selected as the model colloidal particles, because of their high aspect ratio, outstanding mechanical properties, biocompatibility,⁹ renewable nature, and the possibility to modify their surface with polymers.^10,11 Importantly, CNCs are also well-known to form chiral nematic liquid-crystalline phases in water.¹² A carboxylic-acid-containing, pH-responsive polymer network is chosen as the second component, as such materials can undergo large volumetric variations,⁵ which can, in principle, be remotely controlled using reversible photobases.^13,14 The combination of these elements holds the potential to yield a new generation of soft yet strong and remotely controllable biocompatible actuators.¹⁵ 

A computational framework based on the Scheutjens–Fleer self-consistent field (SF-SCF) method^16,17 has been developed to simulate the proposed AMs and to determine key performance parameters, such as stroke and force. The SF-SCF method is particularly suitable to accomplish this goal, because it allows a fast estimation of the equilibrium properties of polymer/colloid/solvent mixtures,^18–22 based on the mutual dependence between the spatial distribution of the components in a mixture and the total free energy of the mixture.

The SF-SCF theory uses an iterative procedure to calculate both parameters starting from an initial guess of the components’ distribution, the molecular architecture of each component, and pair-wise interaction parameters known as Flory–Huggins parameters (χ_ij). The latter can be either measured or estimated using established methods.^23–25 Electrostatic interactions can also be considered.²⁶ To simplify the computations, the components of the systems are coarse-grained in a lattice. By selecting a lattice with the appropriate symmetry, it is possible to reduce the dimensionality of the problem and to compute 3D properties using 1D calculations. This benefit results in very short computation times, making the SF-SCF theory an ideal tool for large systematic investigations.

The computational analysis performed in this work illustrates the effect of various relevant parameters on the actuation properties and the results provide a guiding framework for the design of actuating materials with predetermined properties so that a trial-and-error approach can be avoided.

Numerical SF-SCF lattice computations were performed with the SFbox program provided by Prof. Leermakers (Wageningen University, The Netherlands). In the SFBox, the computational setup is defined via an input file; see examples in the open data folder accompanying the paper.²⁷ The technical aspects of the SF-SCF method have been extensively discussed in the literature^16,17 and are briefly presented in the supplementary material, Sec. 1.

According to the design, the polymer-grafted CNC colloids composing the AM are aligned along the same “axial” direction while the polymer chains extend in the radial direction (the actuation direction) and are connected with each other via chemical cross-links (Fig. 1). Although this design provides a simplified description of the chain-end cross-linking, it represents the best trade-off between accuracy and complexity. Chemical cross-linking would most likely result in the formation of a partially connected network rather than the bridging of two opposite chain ends. However, such a structural difference is not expected to impact the performances of the actuators significantly and is, therefore, neglected.

To reproduce the actuator geometry, computations were performed in the cylindrical lattice geometry (implemented in the SFBox) while accounting for gradients in the radial direction. The computational setup adopted for all calculations is schematically depicted in Fig. 2(a). The SCF cylindrical lattice is composed of Λ concentric layers in the radial direction and one layer in the axial direction. In both directions, the lattice layer size is set to l = 0.3 nm. This value can be considered a reasonable estimate for both water molecules and small functional groups.

The position on the lattice in the radial direction is indicated with x, which equals the distance from the rod center. A solid is placed in the center of the cylindrical lattice and its radius corresponds to the radius of the rod-like particle (i.e., a CNC) with radius R = r ⋅ l. For all calculations, we assumed R = 2.5 nm, hence the solid was placed at lattice layers x = 1, 2 … 8, while the end segments of the polymer chains were grafted at the ninth lattice layer.

The lattice region x < r is forbidden to any component of the system and does not contribute to the energy calculations. Polymer molecules made of N_P monomers, each occupying h_P lattice sites, are grafted on the particle surface (the wall at x = r) with a grafting density σ_P (in number of polymers per squared lattice unit). The grafting density per nm² can be calculated from σ_P as $σP̄=σP/l2$⁠. The total number of polymer molecules in the lattice (M) is given by

Finally, a mirror is placed at the outer lattice layer (x = Λ), where the other ends of the polymer chains are also grafted. The presence of the mirror accounts for other polymer-grafted particles around the simulated one. In a previous investigation on colloidal interactions between spherical micelles by means of the SF-SCF theory, it has been shown that the use of a mirror reproduces quantitatively the hexagonal crystalline lattice.²⁸ Similarly, the use of a cylindrical mirror was used here to reproduce the hexagonal packing of (infinitely long) cylinders. It is noted that this configuration is difficult to realize experimentally over large, as partially random orientation of the cylinders should be expected.

As the polymer molecules are grafted to the particle surface and to the last lattice layer, the value of Λ must be restricted to a defined interval. The upper limit is given by Λ_max = h_PN_P + r. The lower limit, Λ_min, is the Λ value at which all the available lattice sites are occupied by polymer chains and their counterions (all solvent molecules are expelled from the system). The total number of lattice sites occupied by the polymer (including counterions) is

In Eq. (2), φ_I is the fraction of ionized monomers in the polymer chain and is incorporated to account for the presence of counterions. It is noted that Eq. (2) is valid only if all monomers have one ionizable group. From Eq. (2), Λ_min can be obtained as

The polymer molecules are covalently bound to the CNC and cannot exit the lattice; therefore, Λ_min is the minimum allowed number of lattice layers. In fact, condition Λ < Λ_min violates the incompressibility limit.

The stimuli-responsive monomers (P) were modeled as composing two segments (h_P = 2), one of which is always neutral while the other can be either neutral (ψ = 0) or negatively charged (ψ = −1) to simulate the protonated or deprotonated states of a carboxylic acid group. A certain amount of monovalent background salt is present in the solution and mediates electrostatic interactions between the polymer chains. The bulk volume fraction of anions, ϕ_A, is specified in the system. This automatically sets the bulk volume fraction of cations since the system is defined as electroneutral. The local volume fraction profiles near the surfaces are results of the SCF computations, with the given bulk concentrations as the boundary condition. The value of ϕ_A can be calculated from the salt molar concentration C_salt as

where N_Av is the Avogadro number and the factor 10³ converts the units of C_salt to mol ⋅ m⁻³.

The lattice sites, which are not occupied by polymer chains or salt ions, are occupied by water. The average volume fraction of water in the system (including the CNCs) is

where θ_A is the number of lattice sites occupied by anions. It is noted that the polymer counterions are included in ϑ_P.

The SCF method uses Flory–Huggins²⁴ interaction parameters χ_ij to describe the short-range non-electrostatic interactions. The system comprises five components: CNCs, polymer chains (P), water (W), anions (A), and cations (C). χ_CNC–P = 1, corresponding to a repulsive interaction, was set to simulate polymer chains that do not absorb on the CNC surface. The interaction parameter χ_P–S, which describes the interaction between the polymer in the neutral state (ψ = 0) and the solvent, is one of the investigated parameters and is varied between χ_P–S = 0 (soluble polymer chains, only entropic interactions) and χ_P–S = 1 (insoluble polymer chains). When other parameters are systematically varied, χ_P–S = 0.4 is used, corresponding to soluble polymers with some excluded volume interactions. All other interaction parameters are set to 0 because they are not expected to significantly influence the behavior of the system when compared to electrostatic interactions. The relative permittivity of W, A, and C was set to ɛ_R = 80, while the permittivity of all other components was set to ɛ_R = 5. For most of the computations, the grafting density was set to $σP̄=1$ nm⁻². Achieving such grafting densities is possible²⁹ but synthetically challenging for long polymers (N_P > 200); therefore, this value should be considered as a sort of upper limit.

The procedure for determining the actuation properties of the proposed materials, sketched in Figs. 2(b)–2(d), consists of three steps. In the first step, partial open free energy $E$⁠) curves as a function of Λ are computed in the range Λ_max < Λ < Λ_min using the SCF procedure for a given set of r, χ_ij, ϕ_A, σ_P, N_P, and ψ. Partial open free energies are defined as the SCF Helmholtz free energy minus the chemical potential times the number of all non-grafted molecules (see the supplementary material, Sec. 1, for details). A normalized strain parameter, 0 < δ < 1, is calculated from Λ as follows:

This parameter is useful to compare results obtained with polymers of different N_P because δ = 0 when Λ = Λ_min and δ = 1when  Λ = Λ_max.

The energies obtained at different Λ can be plotted against δ as shown in Figs. 2(b) and 2(c), and the equilibrium strain value δ_eq can be estimated at the free energy minimum solving ∂E/∂δ = 0. In the second step, the internal pressure (⁠$Π$ of the system is calculated from the SCF free energy curves as Π = −∂E/∂V, with V = πΛ²l³ being the total volume of the lattice in m³ [Fig. 2(d), see the supplementary material for details]. The internal pressure is an extremely useful parameter, because it is an intensive property related to the actuation performance of the material. Multiplication of Π by the surface area on which such pressure is applied yields the total force exerted by the AM. A positive Π value corresponds to a tendency to swell (pushing), while a negative Π value indicates tendency to contract (pulling). Naturally, at equilibrium, $Πδeq=0$⁠. In the third and final step, for a given set of r, χ_ij, ϕ_A, σ_P, N_P, the $Πδ$ curves obtained at ψ = 0 (neutral state) and ψ = −1 are compared to determine the actuation properties of the material in response to a change in the charge state of the polymer chains.

When the system is neutral and in equilibrium, the actuator is characterized by a certain strain value $δeqψ=0$ and a vanishing internal pressure $Πψ=0(δeqψ=0)=0$⁠. Upon fast deprotonation (for instance, by means of a photobase^13,14 or a sudden pH jump), assuming the deprotonation timescale to be much smaller than the strain relaxation timescale, the pressure of the system will suddenly increase to $Πψ=−1δeqψ=0>0$⁠. This change will trigger a strain relaxation resulting in swelling of the material, until a new equilibrium strain $δeqψ=−1$ is reached, where $Πψ=−1δeqψ=−1=0$⁠. Neutralization of the material will result in a negative internal pressure value $Πψ=0δeqψ=−1<0$⁠, followed by a relaxation to the initial state $Πψ=0(δeqψ=0)=0$⁠. Hence, the pressure difference that can be generated is $ΔΠdep=Πψ=−1δeqψ=0$ upon deprotonation and $ΔΠpro=Πψ=0δeqψ=−1$ upon protonation of the polymer chains. The procedure for determining ΔΠ_dep and ΔΠ_pro is graphically summarized in Fig. 2(d).

The stroke, representing the maximum radial displacement of the AM in the deprotonated state relative to the protonated state, is instead given by

while the water volume fractions at equilibrium in the two protonation states, $ϕWψ=0$ and $ϕWψ=−1$⁠, can be calculated from Eq. (5).

The procedure for determining ΔΠ_dep, ΔΠ_pro, $ϕWψ=0$⁠, $ϕWψ=−1$⁠, and ɛ is repeated for systematic variations of χ_P–S [Figs. 3(a) and 3(e), S1 and S5(a)], N_P [Figs. 3(b) and 3(f), S2 and S5(b)], C_salt [Figs. 3(c) and 3(g), S3 and S5(c)], and σ_P [Figs. 3(d) and 3(h), S4 and S5(d)] to study how these parameters affect the actuation properties of the AMs.

A set of calculations has also been performed to evaluate the performance of an actuator based on CNCs bearing neutral temperature-responsive polymer chains. The internal pressure differences upon swelling ΔΠ_swell and collapse ΔΠ_collapse of the polymer chains are calculated similarly to ΔΠ_dep and ΔΠ_pro, but instead of varying the charge state of the ionizable monomers from ψ = 0 to ψ = −1, the value of χ_P–S is varied from χ_P–S = 0.4 (swollen state) to χ_P–S = 0.8 (collapsed state) with ψ fixed to ψ = 0.

The properties of the AMs proposed in this work can be characterized via five parameters, which are

• the change in internal pressure upon deprotonation, ΔΠ_dep;

• the change in internal pressure upon protonation, ΔΠ_pro;

• the water volume fraction in the deprotonated state at equilibrium, $ϕWψ=0$⁠;

• the water volume fraction in the protonated state at equilibrium, $ϕWψ=−1$⁠;

• the stroke ɛ, which describes to which degree the material can linearly expand in the direction perpendicular to the alignment direction of the colloidal building blocks upon deprotonation.

Details on the definition and the calculation of these parameters via the SF-SCF procedure can be found in the Methods section. These five performance parameters can be modulated by acting on four system parameters:

• the solubility of the polymer chains in the deprotonated state, quantified via the polymer–solvent interaction parameter χ_P–S;

• the (average) degree of polymerization of the stimuli-responsive polymers, N_P;

• the background (monovalent) salt concentration C_salt;

• the grafting density of the polymer chains at the colloid surface σ_P.

Figure 3 summarizes the SF-SCF predictions for internal pressure differences [panels (a)–(d)] and stroke [panels (e)–(h)] obtained for systematic variations of χ_P–S [panels (a) and (e)], N_P [panels (b) and (f)], C_salt [panels (c) and (g)], and σ_P [panels (d) and (h)]. For almost all simulated scenarios, the internal pressure differences were on the order of 10–100 MPa upon swelling (ΔΠ_dep) and on the order of 0.1–1 MPa upon contraction (ΔΠ_pro). This comparison shows strikingly that the system can exert much larger forces in pushing than in the pulling mode. The differences between ΔΠ_dep and ΔΠ_pro are given by the different steepness of the free energy curves (Fig. S1–S4), which arise from different actuation mechanisms. Expansion is, in fact, mostly driven by osmotic forces induced by counter-ion condensation, while contraction is caused by the entropy-driven recoiling of the polymer chains (entropic elasticity). Interestingly, the ΔΠ_pro values are of similar magnitude to the bulk contraction force densities of biological muscles.³⁰ 

As shown in Figs. 3(a) and 3(e), with increasing χ_P–S from χ_P–S = 0 (only entropic interactions) to χ_P–S = 1 (insoluble polymer chains), two distinct regimes can be observed. The transition between these two regions occurs at χ_P–S ≅ 0.5, a value that marks the transition from soluble to poorly soluble polymer chains.²⁴ For χ_P–S < 0.5, the actuation properties of the AMs are little affected by variations of χ_P–S. In this regime, the SCF method predicts ΔΠ_dep values on the order of 12 MPa and ΔΠ_pro of ≈ −0.7 MPa, water volume fractions at equilibrium of $ϕWψ=0≈0.83$ and $ϕWψ=−1≈0.97$ and strokes ɛ ≈ 2.5 (R = 2.5 nm, N_P = 100, $σP̄=1$ nm^–2, and C_salt = 0.1 mM).

For χ_P–S > 0.5, a collapse of the polymer chains in the neutral state is observed, as reflected by a steep decrease in $ϕWψ=0$ [Fig. S5(a)]. This is also clearly visible in the example SCF equilibrium concentration profiles plotted in Figs. S7(a) and S7(c). As a result, the AM is more contracted and expels more water in the neutral state, which yields larger strokes [Fig. 3(e)] and ΔΠ_dep values. Figure S7(c) further shows that water accumulates in the proximity of the CNC, while the polymer is depleted near the CNC surface. This is caused by an interplay between the good solvent-CNC affinity (χ_CNC–S = 0) and the poor polymer-CNC compatibility (χ_P–CNC = 1). The parameter $ϕWψ=−1$ [Fig. S5(a)] is less affected by the increase in χ_P–S, because for ψ = −1, the solubility of the polymer chains is mostly determined by long-range electrostatic interactions [Figs. S5(a) and S7(b) and (d)].

Figure 3(b) shows that both ΔΠ_dep and ΔΠ_pro decrease in absolute value while ɛ increases with increasing the polymer chain length $NP(R=2.5nm,χP−S=0.4,σP̄=1nm−2,Csalt=0.1mM)$⁠. This can be explained by the increasing dilution of the polymer brushes in the charged state (⁠$ϕWψ=−1$⁠) with increasing N_P, which can be observed in Fig. S5(b). As a result of this dilution, the number of counter ions that accumulate in the proximity of the polymer chains to balance their charge [Figs. S7(b) and S7(d) in the supplementary material] becomes lower and this leads to a smaller osmotic pressure difference between the neutral and the charged states. The lower average polymer coil density in the neutral state (increasing $ϕWψ=0$⁠) suggests that a reduction in the entropic elasticity of the polymer chains is likely responsible for a decreasing absolute value of ΔΠ_pro. In the investigated N_P range, ɛ follows the scaling relation $ε∼NP1/4$ [Fig. 3(f)]. This scaling relation follows from the ratio between the scaling regimes^31,32 of the equilibrium thickness of charged $∼NP$ and uncharged $∼NP3/4$ polymer brushes grafted on a cylindrical surface (see the supplementary material, Fig. S6). This finding suggests that chain end cross-linking does not affect the scaling behavior of cylindrical brushes significantly for sufficiently long chains (N_P > 70).

The energy of the charged state can be modified selectively by changing the salt concentration (Fig. S3), which determines the range of the charge interactions. Differently from the case of N_P, increasing C_salt leads to a concomitant (absolute value) reduction in ΔΠ_dep, $ΔΠpro,ϕWψ=0$⁠, $ϕWψ=−1$⁠, visible in Figs. 3(c) and 3(g) and S5(c). A decrease in ɛ is also observed at high salt concentrations, which is caused by a decreasing $δeqψ=−1$ resulting from charge screening effects.

Finally, increasing the polymer grafting density $σP̄$ allows one to increase the absolute values of ΔΠ_dep and ΔΠ_pro, while selectively decreasing $ϕWψ=0$⁠, $ϕWψ=−1$⁠, and ɛ, as shown in Figs. 3(d) and 3(h) [see also Fig. S5(g)]. The value of ΔΠ_dep at low and medium grafting densities (⁠$σP̄<0.5$ nm⁻²) is consistent with the typical swelling pressure values of hydrogels, which is in the order of 1–10 MPa.³³ At larger grafting densities, an increasing steric and electrostatic repulsion between adjacent polymer chains leads to increasing $δeqψ=0$ values and ΔΠ_dep exceeding typical hydrogel swelling pressures.

These results show that tuning the polymer solvency in the neutral state is the most effective way to modulate the swelling force density and the stroke of the material, while the polymer grafting density is the most efficient parameter to tune the contraction force density. Moreover, balancing the polymer chain lengths and the salt concentration allows tuning the gap between swelling and contraction force densities and the stroke of the material independently. The SF-SCF guidelines can also help optimizing the temporal responsiveness of the actuators. Since the actuation rate of osmotic hydrogel actuators is limited by the flow rate of water into and out of the device,³⁴ faster actuation might be achieved by minimizing the difference between $ϕWψ=−1$ and $ϕWψ=0$ and maximizing ΔΠ_dep and ΔΠ_pro.

The actuators presented so far are based on monodisperse and fully crosslinked polymer chains whose state is either fully charged or fully uncharged. In an attempt to improve the realism of our model, additional calculations have been performed, which investigate the effect of polydispersity, degree of cross-linking, and degree of ionization. The results are presented in detail in Secs. S2, S3, and S4 in the supplementary material. Incomplete crosslinking appears to have a negligible effect on the stroke and on the actuation performances of the material upon expansion. However, the change in internal pressure upon protonation ΔΠ_pro, which affects the contraction force, registers approximately a 45% reduction when only half of the polymer chains are crosslinked. While expansion is caused by a change in osmotic pressure in the polymer brushes due to counter-ion condensation, which is almost independent of the cross-linking degree, contraction is driven by the entropic recoiling of the cross-linked chains and, therefore, strongly depends on the degree of cross-linking. Polydispersity has a similar effect because only the longer chains, which extend further in the bulk, can form inter-particle cross-links. On the contrary, a partial ionization of the polymer chains mostly affects ΔΠ_dep, which decreases with decreasing the number of charged groups, as expected. The other properties are less affected by partial ionization. Interestingly, ΔΠ_pro registers only an 85% reduction in absolute value when only 50% of the ionizable groups are charged. This shows that even a small degree of ionization is sufficient to ensure good traction performances.

Recently, it has been reported that confinement of hydrogels within an osmotic membrane allows improvement of actuation performance as compared to the free hydrogel case.³⁵ A set of computations has been performed to verify the effect of ion trapping in the proposed composite actuator system (see the supplementary material, Sec. 6 and Fig. S12). Ion trapping appears to enhance actuation performance, especially stroke and ΔΠ_pro. This effect is caused by the mismatch in osmotic pressure that is generated between the actuator and the surrounding water reservoir upon deprotonation of the polymer chains. Osmotic confinement seems to be a promising approach to tune the actuator performance, but its practical realizability and its effect on actuation dynamics remain to be evaluated.

SF-SCF calculations can be used to predict the performances of artificial muscles made with temperature-responsive (uncharged) polymers as well. In fact, it is possible to map the swelling–deswelling behavior of temperature-responsive polymers, such as poly(N-isopropylacrylamide), into temperature-dependent χ_P–S parameters.^36,37 The performances of neutral temperature-responsive actuators were, therefore, compared to the performances of actuators bearing well-soluble (χ_P–S = 0.4) and poorly-soluble (χ_P–S = 0.8) ionizable polymers. Under the same conditions (N_P = 100, R = 2.5 nm, $σP̄=1$ nm⁻², and C_salt = 0.1 mM), temperature-responsive actuators exhibit superior traction performance while actuators with poorly soluble ionizable chains exhibit the larger strokes and swelling pressures (Fig. S11). This suggests that temperature-responsive polymers are the best choice to maximize traction, while poorly-soluble pH-responsive polymers remain the best choice to maximize stroke and push. Uncharged stimuli-responsive polymers are also a good choice when the presence of unwanted mono- or multivalent ions is an issue, as the absence of long-range electrostatic interactions makes them much less sensitive to variations in ionic strength.

As a final consideration, the properties of a fully oriented AM are compared with the properties of an actuator bearing randomly oriented colloidal subunits. Using a simple geometrical model for a cubic actuator (see the supplementary material, Sec. S5), it can be shown that the stroke of the aligned actuator is ɛ^1/3 times larger than the isotropic one, and that expansion work performed along a single direction by a fully oriented muscle (⁠$wor$ with respect to the work made by an isotropic muscle (w_is) is

The experimental realization of a fully oriented actuator is challenging and a degree of misalignment is to be expected in real systems. This misalignment can be minimized by using orienting fields prior to cross-linking of the building blocks. In the ideal case, described by Eq. (8), a fully oriented actuator can perform 1.5 times the work of an isotropic actuator under the same conditions and, therefore, is 50% more energy efficient.

A computational protocol based on the Scheutjens–Fleer self-consistent field theory has been developed to predict the actuation properties of nanostructured composite AMs made of aligned anisotropic colloids grafted with stimuli-responsive polymer chains. The protocol is fast and versatile, allowing for systematic variations of several parameters. The procedure was used to investigate model actuators composed of thin cylindrical particles, for instance, cellulose nanocrystals, grafted with pH-responsive polymers, such as poly(acrylic acid), but its adoption to simulate colloidal responsive units of different shapes is straightforward.

The results of this systematic study show that balancing the solubility of the polymer chains in the deprotonated state, the degree of polymerization of the stimuli-responsive polymers, the background salt concentration, and the grafting density of the polymer chains makes it possible to finely tune each of the key characteristics of the AMs independently to meet different types of needs. The SF-SCF results can be used as guidelines to build AMs with predetermined properties. The model predicts the pulling force density of these osmotic AMs to be comparable to that of human muscles, with the additional capability of exerting strong active pushing motions. Biological muscles, in fact, are incapable of active pushing, which is the reason why in living systems, motion is always controlled via pairs of antagonist muscles. Finally, alignment of the actuating building blocks is important as it enables larger strokes and a remarkable improvement of the device energy efficiency compared to the unaligned case. Overall, the results of this theoretical investigation provide further support for the high potential of the proposed technology and pave the way for a new type of soft osmotic AMs.

See the supplementary material for details on the SF-SCF theory and for additional discussion on the effect of pH, polydispersity, degree of cross-linking, ion trapping, and colloid orientation.

This work was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie “SAM,” Grant Agreement No. 891084, and by the EIC Pathfinder Open “INTEGRATE,” Grant Agreement No. 101046333. The authors gratefully acknowledge funding from the Adolphe Merkle Foundation. The authors thank A. Ercoli for the advice and the useful discussions on the comparison between isotropic and anisotropic actuators.

The authors have no conflicts to disclose.

A.I. conceived this study and performed the simulations. All authors contributed to the data analysis and to the interpretation of the results. The manuscript was written by all authors.

Alessandro Ianiro: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (equal). José Augusto Berrocal: Formal analysis (equal); Writing – review & editing (equal). Remco Tuinier: Formal analysis (equal); Writing – review & editing (equal). Michael Mayer: Formal analysis (equal); Writing – review & editing (equal). Christoph Weder: Formal analysis (equal); Funding acquisition (supporting); Resources (supporting); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Zenodo at https://zenodo.org/record/6304515#.YhzgM5Yo-Uk.