Applications of a continuum model for brush gels

Abstract

Brush gels, defined by their branched molecular architecture, exhibit unique mechanical adaptability and tunable properties. A continuum model is adopted to describe their mechanical behavior, governed by a parameter $\beta$ related to the mean-squared end-to-end distance of the underlying polymer chain, which controls the transition from tensile-dilating to tensile-contracting behavior under uniaxial tension. This and other topological parameters of brush gels can be engineered to produce mechanical behavior like that of linear polymer gels and fibrous gels. Adjustments in branch topology of the polymer, such as grafting density and side-chain length, allow for tuning the stiffness while holding the shape of the force-stretch curve fixed. The rate-dependence of the force-stretch curve is more pronounced under compression due to osmotic effects. Rotational shear tests reveal the impact of chemical potential gradients driven by normal stress variations and it is shown that pre-compression can enhance network stiffness. Furthermore, brush gels exhibit increased resistance to drying-induced volume changes, attributed to their network architecture, which enables rapid equilibration. These findings underscore the potential of brush gels in applications requiring mechanical adaptability and stability under drying conditions.

1. Introduction

Brush gels have attracted increasing attention in recent years due to their highly tunable mechanical properties and unique molecular architecture, which consists of grafted polymer side chains extending from a central backbone (see Fig.1). The brush strands in a brush gel can have different shapes depending on the grafting density of the polymer chains. With lower grafting density the strands form a comb-like structure, while at higher grafting densities they take on a bottlebrush shape with densely packed side chains[1]. The latter are also called bottlebrush gels. These structural features enable brush gels to exhibit a range of desirable mechanical behaviors, such as enhanced elasticity, toughness, and strain-stiffening[2, 3]. This bottlebrush architecture is also observed in mucin gels[4, 5, 6], naturally occurring biopolymer networks composed of glycoproteins with dense, brush-like side chains, which are essential for lubrication, hydration, and the protection of epithelial surfaces.

Fig. 1.

(a) An RVE of a brush gel when the grafting density is low. The topology of the brush polymer network is characterized by $n_{\text{x}},n_{\text{g}}$ and $n_{\text{sc}}$. The red circles represent cross-links. The branches have $n_{\text{sc}}$ monomers, the separation between branches is $n_{\text{g}}$ monomers and the length of a chain between cross-links is $n_{\text{x}}$ monomers. The space of the cube is fully filled with liquid at all times, which means that our RVE is saturated. Note that in this case, the radius of the fiber is approximately equal to the diameter of the backbone monomer. (b) For a brush molecule with high grafting density, the radius of the fiber is given by the sum of the backbone monomer diameter and the side-chain length.

The versatility of brush gels has to do with the ability to control key parameters like the degree of polymerization of both the backbone and side chains, as well as the grafting density. This tunability makes brush gels particularly useful for a variety of advanced applications. For instance, in the field of energy storage, bottlebrush-based polymer gels have demonstrated notable improvements in mechanical strength and electrochemical performance, as highlighted by Jia et al[7]. In their work on lithium metal batteries, bottlebrush gels offer a promising solution to long-standing challenges in battery technology, where both flexibility and mechanical integrity are paramount. Beyond industrial applications, the unique mechanical properties of brush gels make them highly relevant for biomimetic and biomedical uses. Their ability to replicate the mechanical characteristics of soft biological tissues, such as the brain, spinal cord, and adipose tissues, has made them appealing for applications in areas like tissue engineering and wearable electronics. Studies, such as the one conducted by Keith et al.[8], have shown how bottlebrush elastomers can bridge the gap between soft gels and firmer biological tissues, exhibiting a strain-adaptive stiffening that mimics the behavior of natural tissues (e.g. brain and spinal cord). This ability to combine softness with mechanical resilience is critical in applications where materials must remain compliant under low strains but become resistant to deformation under higher strains. Furthermore, the work of Nakagawa and Yoshie[3] adds to this understanding by demonstrating how bottlebrush elastomers exhibit significant backbone alignment and strain stiffening under uniaxial deformation, with the side chains providing structural support that resists entanglement and maintains elasticity. In biomedical contexts, the use of bottlebrush polymers in injectable hydrogels, as described by Vashahi et al.[9], provides materials that are capable of mimicking the deformation response of soft tissues like adipose and brain tissue, making them ideal for applications in reconstructive surgery and soft tissue repair. Similarly, research by Xu et al.[10] underscores the potential of bottlebrush elastomers in ultra-soft electronics, demonstrating their capability to achieve tissue-comparable mechanical properties while maintaining high conductivity and stability. Beyond these applications, certain bottlebrush gels incorporate zwitterionic side chains—bearing both cationic and anionic groups—which endow them with exceptional lubrication and antifouling properties, broadening their potential for biomedical coatings and ion-conductive systems[11, 12]. Molecular modeling, including coarse-grained approaches, has been developed to study brush-like gels such as human gut mucus, enabling a deeper understanding of nanoparticle diffusion for drug delivery applications[13].

Despite these advances in understanding the mechanical behavior of brush gels, the development of constitutive models that can accurately capture their unique mechanical responses remains under-explored. Hydrogels in general, with their three-dimensional network structures and ability to absorb large amounts of water, have been the subject of extensive modeling efforts[14, 15, 16]. These models typically use continuum mechanics frameworks that couple fluid diffusion with network deformation, accounting for the complex interplay between these two processes[17, 18, 19]. A variety of models, such as neo-Hookean and Mooney-Rivlin, together with the Flory-Rehner theory for two-phase interactions, have been applied to cross-linked linear polymer hydrogels [20, 21], yet they may not adequately describe the intricate structure of brush gels which have a more complex topology. Separate continuum models have been proposed for fibrous gels, such as fibrin networks in blood clots [22] and collagen networks in the extra-cellular matrix, but even these models are meant for simpler networks consisting of cross-linked long fibers. The highly branched nature of brush polymers, with their densely packed side chains and flexible backbones, leads to mechanical behaviors that are distinct from those of linear-chain hydrogels[9, 23] and fibrous gels.

The lack of well-developed constitutive models for brush gels is a significant limitation in both theoretical understanding and practical applications. While some continuum models have been proposed to illustrate the swelling mechanisms of brush gels, such as those described by Jacobs et al.[24], the full extent of three dimensional deformations have yet to be explored using these models. Developing a comprehensive constitutive model for brush gels would not only improve our theoretical understanding, but also enable more accurate predictions of their behavior in various applications, from biological to industrial materials.

The goal of this paper is to propose a 3D constitutive model for brush gels considering their complex network topology. Our model builds on the work of[24] and it explores a suite of deformation mechanisms that are commonly used in experiments. We treat the gels as a homogeneous material containing both solid and liquid phases. We compare our model with traditional models of linear polymer gels to examine how brush gels are different, and how the topology of brush gels can affect their mechanical behavior even when parameters such as solid volume fraction and/or the small strain shear modulus are held fixed. Section 2 presents the kinematics and balance laws of poro-elasticity, formulated with respect to a stress-free swollen reference state. In section 3, the constitutive model for the brush gel is developed based on the Flory-Rehner theory. The performance of the proposed model is tested under various conditions, including tension, compression, shear, and drying, as detailed in section 4 through 7. Throughout this work, both direct tensor notation (represented by bold symbols) and Cartesian components of tensors are used, with the Einstein summation convention employed.

2. Kinematics and balance laws for gel model

2.1. Kinematics

As mentioned earlier, polymeric hydrogels contain hydrophilic chains and interstitial liquid. Microscopically, hydrogels are not homogeneous since the topology of the polymer network and the concentration of other solutes in the liquid is not the same everywhere. However, since our interest is in length scales that are much larger than the gel micro-structure’s representative length scale, we treat the gel as a homogeneous material. Let $\mathit{x}(\mathit{X},\mathit{t})$ and $\mathit{X}$ denote the current and reference positions of a specific material point respectively, then the deformation gradient and Jacobian are:

$$F_{ij}=\frac{\partial x_i}{\partial X_j},J=\text{det}\mathit{F}.$$ (1)

The material point we consider is a representative volume element (RVE) that contains both polymer network and liquid. In this study the brush structure of the polymer network is characterized by three topological parameters, namely: $n_{\text{sc}},n_{\text{g}}$ and $n_{\text{x}}$, as shown in Fig.1. $n_{sc}$ is the the length of a side-chain in monomer units, $n_g$ is the distance along the backbone between neighboring side-chains in monomer units and $n_x$ is the length of polymer backbone between cross-links in monomer units. All monomer units are assumed to be identical. Moreover, our initial (reference) configuration is the swollen state that the polymer network forms with liquid at equilibrium. We now invoke standard continuum mechanical principles to obtain the constitutive behavior of the gel. A list of variables appearing in the continuum derivation appears in Appendix C. Since the material behavior is thought to be isotropic, the stored energy density only depends on the following invariants of the left and right Cauchy–Green tensors:

$$I_1=\text{tr}\mathit{C}={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2,$$ (2)

$$I_2=\frac{1}{2}\left[(\text{tr}\mathit{C}{)}^2-\text{tr}{\mathit{C}}^2\right]={\lambda }_1^2{\lambda }_2^2+{\lambda }_1^2{\lambda }_3^2+{\lambda }_2^2{\lambda }_3^2,$$ (3)

$$I_3=\text{det}\mathit{C}={\lambda }_1^2{\lambda }_2^2{\lambda }_3^2,$$ (4)

where the principal values of $\mathit{C}$ are denoted by ${\lambda }_i^2$ and $\text{det}\mathit{F}=J=\sqrt{I_3}=\frac{dV}{d{V}_0}$ is the volume change relative to the reference state. Once equilibrium is reached, the equilibrium swelling stretch ratio ${\lambda }_0$ and the corresponding reference solid volume fraction ${\varphi }_{\text{s}}^{\text{ref}}$ are related by ${\varphi }_{\text{s}}^{\text{ref}}=\frac{1}{{\lambda }_0^3}$. The initial solid volume fraction in a swollen hydrogel can be 0.01 [25] or lower.

We assume that the network and the interstitial liquid are both incompressible and that the volume change of the material is caused by the diffusion of liquid only. Let $C(\mathit{X},t)$ denote the volume fraction of liquid per unit reference volume, then microscopic incompressibility of the solid material and the liquid together with an assumption of fully saturated gel imply:

$$C\left(\mathit{X},t\right)=J-{\varphi }_{\text{s}}^{\text{ref}},$$ (5)

where ${\varphi }_{\text{s}}^{\text{ref}}$ denotes the volume fraction of the polymer in the reference configuration.

2.2. Mass balance

Consider a volume $\mathit{V}$ with a normal $\mathit{N}\left(\mathit{X}\right)$ and surface $\mathit{S}\left(\mathit{X}\right)$ in the reference configuration, then mass balance in the absence of any liquid source can be written as:

$${\int }_{\mathit{V}}{\rho }_w\frac{\partial C}{\partial t}dV+{\int }_{\mathit{S}}{\rho }_w\mathit{Q}\cdot \mathit{N}dS=0,$$ (6)

where $\mathit{Q}$ denotes the liquid flux per unit reference area and ${\rho }_w$ denotes the density of the liquid. Utilizing the divergence theorem, we can rewrite this equation in localized form as:

$$\frac{\partial C}{\partial t}+\text{Div}\mathit{Q}=0.$$ (7)

Here $\text{Div}\mathit{Q}$ represents the divergence of liquid flux in the reference configuration.

2.3. Balance of linear and angular momentum

If we consider the loading to be quasi-static, which means that the load is applied very slowly, we can write our momentum balance equation expressed in the reference configuration as:

$$\text{Div}\mathit{P}=0,$$ (8)

where $\mathit{P}$ is the first Piola–Kirchhoff stress tensor. The balance of angular momentum is satisfied by the symmetry of Cauchy stress tensor.(i.e. $\mathit{\sigma }={\mathit{\sigma }}^{\text{T}}$).

3. Constitutive model for brush gels

3.1. Free energy density

Based on Flory-Rehner theory, the total free energy of a polymer and solvent mixture can be written as a sum of the free energy (Helmholtz free energy per unit reference volume) due to the deformation of the network ${\Psi }_{\text{net}}\left(\mathit{F}\right)$, and the free energy of the hydrophilic interaction of two phases ${\Psi }_{\text{int}}\left(C\right)$.

$$\Psi \left(\mathit{F},C\right)={\Psi }_{\text{net}}\left(\mathit{F}\right)+{\Psi }_{\text{int}}\left(C\right).$$ (9)

Through standard continuum mechanical derivations that use the dissipation inequality, it can be shown that $\Psi$ can be differentiated to give [26]:

$$\frac{\partial \Psi }{\partial \mathit{F}}=\mathit{P},\frac{\partial \Psi }{\partial C}=\mu ,$$ (10)

where $\mu$ is the chemical potential of the liquid per unit reference volume. Gradients in $\mu$ are the driving force for liquid mass transfer. Utilizing a Legendre transformation, we can define a new free energy as:

$$\hat{\Psi }=\Psi \left(\mathit{F},C\right)-\mu C,$$ (11)

which can be differentiated to get:

$$\frac{\partial \hat{\Psi }}{\partial \mathit{F}}=\mathit{P}-\frac{\partial \hat{\Psi }}{\partial \mu }=C.$$ (12)

This shows that the proper variables for $\hat{\Psi }$ are $\mathit{F}$ and $\mu$, therefore, the first Piola–Kirchhoff stress can be calculated as:

$$P_{ij}=\frac{\partial \hat{\Psi }}{\partial F_{ij}}=\frac{\partial {\Psi }_{\text{net}}}{\partial F_{ij}}+\left(\frac{\partial {\Psi }_{\text{int}}}{\partial J}-\mu \right)J{F}_{ji}^{-1}$$ (13)

Here $\pi =-\frac{\partial {\Psi }_{\text{int}}}{\partial J}$ is called the osmotic pressure which arises due to the free energy of mixing and plays a significant role in gel swelling. Here, $\pi$ for a semi-dilute polymer solution can be written as[27, 28]:

$$\pi =\alpha \frac{RT}{M_{\text{m}}}\left(\frac{C_{\text{m}}}{N}\right){\left(\frac{C_{\text{m}}}{C^{*}}\right)}^{\frac{5}{4}},C_{\text{m}}={\rho }_p{\varphi }_{\text{s}}={\rho }_p\frac{{\varphi }_{\text{s}}^{\text{ref}}}{J},$$ (14)

where $\alpha$ is an undetermined numerical factor of order one for many polymers. $\alpha$ could be used to account for the hydrophillic or hydrophobic nature of the polymer network. $R$ and $T$ are the gas constant and temperature, respectively, $M_{\text{m}}$ is the monomer molecular weight for the polymer and $N$ is the number of monomers in one polymer chain. $C_{\text{m}}$ denotes the mass concentration of the polymer and $C^{*}$ is a characteristic concentration which depends on $N$. Conventionally, $C^{*}$ is semi-quantitatively defined to account for the overlap concentration, which is order of magnitude $N^{-\frac{4}{5}}/\bar{V}$ [27]. Here $\bar{V}$ is the specific volume of the polymer, and it is in fact the reciprocal of the polymer density ${\rho }_p$. In this study we set $C^{*}=N^{-\frac{4}{5}}/\bar{V}$. Utilizing eqn.(14), the free energy of mixing ${\Psi }_{\text{int}}$ can be written as:

$${\hat{\Psi }}_{\text{int}}\left(J\right)=\frac{4}{5}\frac{\alpha RT{\rho }_p{\left({\varphi }_{\text{s}}^{\text{ref}}\right)}^{\frac{9}{4}}}{M_{\text{m}}}{J}^{-\frac{5}{4}}.$$ (15)

In the above we used ${\hat{\Psi }}_{\text{int}}$ because we are writing the interaction energy in terms of $J$, rather than $C$. The free energy density of the pure bottle-brush network (with no liquid) can be written as [24]:

$${\psi }_{\text{net}}\left(I_1\right)=G_{\text{bot}}\left[\frac{I_1}{6}+{\beta }^{-1}{\left(1-\frac{\beta I_1}{3}\right)}^{-1}\right].$$ (16)

In the above $I_1$ is the first invariant of the left or right Cauchy-Green tensor. $\beta =⟨{R_{\text{in}}}^2⟩/{R_{\text{max}}}^2$ is the ratio of the mean-squared average end-to-end distance between cross-links in the undeformed network to the square of the end-to-end distance of the fully stretched strand. It is worth noting that a constant must be added to the free energy density above to make it zero when $I_1=3$. However, this will not matter because we work with derivatives of the free energy density (stress and chemical potential). $G_{\text{bot}}$ is the shear modulus of the brush network, which can be expressed as [24]:

$$G_{\text{bot}}=C_{\text{tp}}{k}_{\text{B}}T\frac{{\rho }_0\beta }{n_{\text{x}}{b}_{\text{K}}\left(1+{\chi }_1\right)R_{\text{max}}},$$ (17)

where $C_{\text{tp}}$ is a factor that accounts for the network topology and cross-link functionality. $k_{\text{B}}$ and $T$ are the Boltzmann constant and the absolute temperature, respectively. ${\rho }_0$ is the monomer density of the brush polymer and $b_{\text{K}}$ is the Kuhn length. ${\chi }_1=\frac{n_{\text{sc}}}{n_{\text{g}}}$ reflects the topology of the brush structure. We add a term proportional to ln $J$ to account for free energy changes due to volumetric deformation of the polymer network, following [29, 30]. This term, originally introduced by Flory [31], has been shown to capture the mechanical response of polymer networks phenomenologically in a wide range of experimental settings [32]. Then, we write ${\Psi }_{\text{net}}$ as:

$${\Psi }_{\text{net}}\left(\mathit{F}\right)={\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}\left[\frac{I_1}{6}+{\beta }^{-1}{\left(1-\frac{\beta I_1}{3}\right)}^{-1}+{\alpha }_{\text{bot}}\text{ln}J\right],$$ (18)

${\alpha }_{\text{bot}}$ is chosen such that the stress is zero in the swollen reference configuration. Combining eqn.(15) and eqn.(18), the free energy density per unit reference volume of the gel is:

$$\hat{\Psi }(\mathit{F},\mu )={\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}\left[\frac{I_1}{6}+{\beta }^{-1}{\left(1-\frac{\beta I_1}{3}\right)}^{-1}+{\alpha }_{\text{bot}}\text{ln}J\right]+\frac{4}{5}\frac{\alpha RT\rho }{M_{\text{m}}}{\left({\varphi }_{\text{s}}^{\text{ref}}\right)}^{\frac{9}{4}}{J}^{-\frac{5}{4}}-\mu \left(J-{\varphi }_{\text{s}}^{\text{ref}}\right).$$ (19)

Utilizing eqn.(13), and remembering that $\mathit{\sigma }=J^{-1}\mathit{P}{\mathit{F}}^{\text{T}}$, we can now compute the Cauchy stress as:

$$\mathit{\sigma }=2J^{-1}\left[\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{6}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{3{\left(1-\frac{\beta I_1}{3}\right)}^2}\right]\mathit{B}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}}{J}\mathit{I}-(\pi +\mu )\mathit{I},$$ (20)

where $\mathit{B}=\mathit{F}{\mathit{F}}^{\mathbf{T}}$ is the left Cauchy-Green tensor, and $\mathit{I}$ is the identity tensor. This can also be expressed in index notation as:

$${\sigma }_{ij}=2J^{-1}\left[\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{6}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{3{\left(1-\frac{\beta I_1}{3}\right)}^2}\right]B_{ij}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}}{J}{\delta }_{ij}-(\pi +\mu ){\delta }_{ij},$$ (21)

Initially, the osmotic pressure is balanced by the mechanical response of the dilated network, and the chemical potential is zero throughout the gel. So, we can find ${\alpha }_{\text{bot}}$ by setting ${\left.\mathit{\sigma }(\mathit{I},0)\right||}_{t=0}=0$. Consequently, we get

$${\alpha }_{\text{bot}}=\frac{{\pi }_0}{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}-\frac{1}{3}-\frac{2}{3(1-\beta {)}^2}.$$ (22)

where ${\pi }_0$ is the initial osmotic pressure in the reference configuration. In this study, we assume that the solid volume fraction in the reference configuration, ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, which implies that the total number of monomers per reference volume remains fixed. $n_{\text{g}}$ and $n_{\text{sc}}$ will be varied to reveal the effects of topology changes, but $n_{\text{x}}$ will be held constant. Unless otherwise specified we choose $C_{\text{tp}}=0.5,n_{\text{sc}}=40,n_{\text{g}}=10,n_{\text{x}}=200,N=1000,\alpha =0.5,R=8.314\text{J}/\text{mol}\cdot \text{K},M_{\text{m}}=200\text{g}/\text{mol},k_{\text{B}}=1.381\times {10}^{-23}\text{J}/\text{K},T=300\text{K},{\rho }_p=2\text{g}/\text{ml}=\frac{{\rho }_0{M}_{\text{m}}}{6.02\times {10}^{23}},\beta =0.1,b_{\text{K}}=2.5l,l={10}^{-10}\text{m}$, where $l$ is the link length of the chain. These parameters do not represent any particular brush gel, but similar numbers can be found in [33]. Notice that $N,n_{\text{sc}},n_{\text{x}},n_{\text{g}}$ satisfy the following relation:

$$N=n_{\text{x}}\left(1+\frac{n_{\text{sc}}}{n_{\text{g}}}\right).$$ (23)

In real brush gels the parameters $n_{\text{x}},n_{\text{sc}}$, etc., will not be constants, rather they will be polydisperse and described by probability density functions. The values chosen above could be considered as mean values of these quantities. Yet, since $G_{\text{bot}}$ and $\beta$ depend on $n_{\text{x}},n_{\text{sc}}$, etc., we expect that polydispersity will affect the mechanical behavior of brush gels (see [34] for effects of polydispersity in linear polymers). Here, we stick to a deterministic continuum theory without accounting for the effects of polydispersity for the sake of simplicity and brevity.

3.2. Liquid flux

Following Darcy’s law [35], the liquid flux $\mathit{Q}$ is assumed to be linear in the chemical potential gradient:

$$Q_i=-\frac{K_{ij}}{\eta }\frac{\partial \mu }{\partial X_j},$$ (24)

where $\eta$ is the viscosity of the liquid and $\mathit{K}$ is a second order tensor. Since our material is isotropic, we will write the $\mathit{K}$ tensor as a scalar times the identity tensor in the reference configuration, so

$$Q_i=-\frac{k}{\eta }\frac{\partial \mu }{\partial X_i}.$$ (25)

Numerous experiments have been done to relate $k$ to the current solid volume fraction ${\varphi }_{\text{s}}$ and some other geometric parameters of the polymers by an empirical form. In this paper we adopt the following relation[36]:

$$k\left({\varphi }_{\text{s}},a\right)=\frac{a^2}{16{\varphi }_{\text{s}}^{1.5}\left(1+56{\varphi }_{\text{s}}^3\right)},$$ (26)

where $a$ represents the radius of fibers forming the network (see Fig.1). For example, for linear polymers $a$ will be on the order of a monomer size. For a brush gel with a very high density of side-chains $a$ will include both the backbone monomer size and the side-chain length because steric hindrances will cause the fiber to look relatively straight like a rod. For low side chain density $a$ will be intermediate between these extremes. Alternatively, $a$ may be regarded as a parameter that is fitted to experiments measuring the Darcy constant of brush gels with known solid volume fraction ${\varphi }_{\text{s}}$. The permeability decreases as the solid phase volume fraction ${\varphi }_{\text{s}}$ (in the current configuration) increases because this increases the network’s resistance to liquid flow. The permeability is first assumed to be homogeneous in the absence of loads due to a uniform solid volume fraction. After deformation, variation of the local ${\varphi }_{\text{s}}$ will cause the change of the local permeability $k$ according to eqn.(26).

3.3. Poisson ratio

The Poisson ratio of a gel can be easily measured in experiments with a parallel plate rheometer equipped with a camera. We assume that the Poisson ratio is given by the Hencky definition [37], i.e., $\nu =-\frac{\text{ln}{\lambda }_2}{\text{ln}{\lambda }_1}$. Notice that for small strain, $\nu =-\frac{\text{ln}\left(1+{ϵ}_2\right)}{\text{ln}\left(1+{ϵ}_1\right)}\approx -\frac{{ϵ}_2}{{ϵ}_2}$, which recovers the classical definition of Poisson ratio in the infinitesimal strain limit. We will compare results based on eqn.(19) with other widely known constitutive models for polymers. The first model is the classical neo-Hookean model [38]:

$${\Psi }_{\text{net}}\left(\mathit{F}\right)=\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{neo}}}{2}\left[I_1-3+{\alpha }_{\text{neo}}\text{ln}J\right],$$ (27)

and a second model is the Gent model[39]:

$${\Psi }_{\text{net}}\left(\mathit{F}\right)=-\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{gent}}{J}_{\text{m}}}{2}\left[\text{ln}\left(1-\frac{I_1-3}{J_{\text{m}}}\right)+{\alpha }_{\text{gent}}\text{ln}J\right],$$ (28)

where $G_{\text{neo}},G_{\text{gent}}$ are the shear moduli of the network and $J_{\text{m}}$ is the maximum value of $I_1-3$. ${\alpha }_{\text{neo}}$ and ${\alpha }_{\text{gent}}$ are chosen such that the reference state (swollen state) is stress free. The goal of this comparison is to highlight the differences between gels that consist of linear chains (with no side chains) and brush gels (with side chains). We assume that the load is applied slowly so that the gel is stretched in a quasistatic manner. Then, we can compute $\nu$ analytically for specified ${\lambda }_z$ as follows. Since the loading is quasistatic the chemical potential $\mu$ is set to zero throughout the gel. The only non-zero stress component is ${\sigma }_{\text{zz}}$ so we can compute the stretch $\lambda$ in the transverse direction by setting ${\sigma }_{\text{xx}}={\sigma }_{\text{yy}}=0$. Then, we find the Poisson ratio from Hencky’s formula, using different constitutive models. The results are shown in Fig.2.

Fig. 2.

Poisson ratio of a gel as a function of the stretch in the longitudinal direction for three different constitutive laws with the same shear modulus.

All three models exhibit a positive Poisson ratio for a long range of stretch. In tension, the Poisson ratio for the brush gel is larger than that of gels with linear chains. The difference between the three models becomes more pronounced as the stretch ${\lambda }_z$ increases. Under compression, the difference in Poisson ratio between the three constitutive laws is not as significant. Note that the material of the polymer network and the fluid are both incompressible but the Poisson ratio of the gel does not approach 0.5 due to the hydrophilic interactions between the liquid and the polymer and the volume changes caused by liquid flow. It is worth noting that the initial Poisson ratio, evaluated for ${\lambda }_z$ near unity, can be obtained analytically by linearization. The detailed derivation is provided in Appendix D.

4. Uniaxial tension of cylindrical specimens

Next, we focus on uniaxial tension of a cylindrical specimen of a brush gel immersed in a liquid bath. The simulations are performed using the finite element software ABAQUS with the Soil transient consolidation step and C3D8P elements. These elements have eight nodes, and the displacement and chemical potential (or the pore pressure) are both trilinear. The constitutive model is coded as a UHYPER user subroutine (see Appendix A,B).

Cylindrical specimens of height 25 mm and radius 5 mm are subjected to displacement-controlled uniaxial tension up to an axial stretch of ${\lambda }_z=2$. The initial chemical potential ${\mu }_0$ everywhere in the gel is set to 0. The chemical potential on the boundary is $\mu =0$ on the external side surfaces during deformation since the gel is immersed in a bath of pure liquid. The top and bottom surfaces of the cylindrical specimen are treated as impermeable, preventing any exchange of material across these surfaces. In some experiments [40] gels are polymerized between rheometer plates and they are stuck to both top and bottom plates, therefore, no slip boundary conditions (in-plane displacements set to zero) should be applied to model such experiments at the top and bottom surfaces. Here we will consider both no-slip and slip boundary conditions for comparison because a pure uniaxial stress state can be achieved only under slip boundary conditions. For polymeric gels containing linear chains, the radius of the chain is typically less than 10 nm [41], whereas for fibrous gels, the radius of the fiber is approximately 50 nm [36] or more. We will select $a=25\text{nm}$ for brush polymers because they lie somewhere in between slender polymer chains (e.g. alginate gels and polyacrylamide gels) and thick fibers (e.g. fibrin gels).

We will change the topology of the brush structures by varying the value of $n_{\text{sc}},n_{\text{g}}$ to investigate its effect on the overall mechanical properties of the gel. All other parameters except $\beta$ and $C_{\text{tp}}$ will be held fixed and remain the same as in the previous section. Note that the deformation of the cylinder is not homogeneous due to the variation of liquid concentration along radial direction, which results in a variation of $J$. The value of $J$ becomes uniform if the gel is held at a state of applied stretch and allowed to reach equilibrium.

Fig.3 illustrates the force measured as a function of time for slip (zero shear stress) boundary conditions and different constitutive models. Displacement is applied at the top surface of the cylinder with a loading rate of 10⁻³m/s for 25 seconds. All models share the same (dry) shear modulus, as derived from eqn.(17), and the value of $\beta$ is chosen as 0.1. Notably, the brush gel shows a stiffer response compared to the linear chain models. FE computations using no-slip boundary conditions were also performed; however, given the substantial height of our specimen, the variation in boundary conditions has little impact on the force-time curves.

Fig. 3.

Force versus time for gels with different network constitutive models. The loading rate is 10⁻³m/s and the shear modulus of the three models is kept the same, $G=9.98\times {10}^4\text{Pa}$ based on eqn.(17) when $n_{\text{sc}}=40,{\chi }_2=20$. Here $\beta$ is chosen as 0.1. The dimensions of the specimen are $r=5\text{mm}$ and $h=25\text{mm}$.

Fig.4 illustrates rate-dependent effects on the force-stretch curve. As the loading rate increases, the force measured at the top surface becomes larger. This is because a higher loading rate reduces the time available for liquid diffusion within the gel, making the gel deform in near isochoric manner. Notice also that the curves corresponding to high rates are close together. Further increasing the loading rate will barely change the force-stretch curve since fluid will not have time to diffuse and the network will deform purely elastically. However, we cannot go to arbitrarily high loading rates because we have neglected inertia. In situations where the loading rate is high, the chemical potential $\mu$ at the top center point of the gel decreases since the pore size expands rapidly. However, when the loading rate is reduced, liquid diffusion within the gel becomes more significant and the chemical potential at the top center point decreases first and then increases. This happens because the chemical potential on the side surfaces remains higher compared to the central region due to the boundary conditions, which results in a liquid flow inside the gel which ultimately increases the chemical potential in the bulk.

Fig. 4.

Force (a) and the chemical potential (b) as a function of longitudinal stretch when $\beta =0.1$. The chemical potential is measured at the top center point of the gel. The dimensions of the specimen and the topological parameters are the same as in Fig.3.

Both Fig.3 and Fig.4 show a concave-down shape for the force-stretch curve, which has been observed in certain experiments on brush gels [7]. However, other experiments have reported an opposite concave-up shape, attributed to the strain-stiffening behavior of the brush polymer[9]. To account for this observation, various values of $\beta$ are used to test the mechanical behaviors of the brush gel in tension, as shown in Fig.5. Clearly, as $\beta$ increases, the force curve changes from a concave shape to a convex shape, which shows that $\beta$ is a key parameter that controls the strain-stiffening behavior of brush gels. A brush gel with small $\beta$ shows a monotonic tensile-dilating behavior which is similar to gels made with linear chains. As $\beta$ increases, the volume change curve shows a non-monotonic trend – the volume starts to increase first, reaches a peak, and then decreases, much like the results shown in [21] for fibrous gels. In Garyfallogiannis et.al[21], ${\Psi }_{\text{net}}=G_1\left(I_1-3\right)+G_2{\left(I_1-3\right)}^2+\alpha \text{ln}J$ and they showed that tensile-dilating vs. tensile-contracting behavior of the gel is determined by the ratio $\frac{G_2}{G_1}\cdot \frac{G_2}{G_1}<1$ gave tensile-dilating behavior while $\frac{G_2}{G_1}\gg 1$ gave tensile-contracting behavior. The $G_2$ parameter in Garyfallogiannis et al., is somewhat similar to the $\beta$ parameter here since both of them characterize the strain-stiffening behavior of gels. The tensile mechanical behavior of brush gels occupies an intermediate position between that of gels composed of thick fibers, such as collagen, and those containing linear polymer chains, such as alginate and polyacrylamide [42]. For various types of brush gels, by controlling the topology of the brush network, one can achieve tensile behavior similar to fibrous gels or linear polymeric gels by adjusting $\beta$.

Fig. 5.

Force (a) and relative volume (b) as a function of time when the loading rate is 10⁻³m/s. Different values of $\beta$ affect the curvature of both the force and volume change curve. In these simulations the value of $C_{\text{tp}}$ is chosen such that the shear modulus for each $\beta$ is the same (i.e. $C_{\text{tp}}\beta$ is held fixed). The dimensions of the specimen and the topological parameters are the same as in Fig.3.

Fig.6 shows the effect of different topology on the force measured and relative volume change, both as functions of time when $\beta$ is fixed. Here ${\chi }_2=\frac{n_{\text{x}}}{n_{\text{g}}}$ is a topology parameter that characterizes the grafting density of the brush structure. By changing the value of $n_{\text{sc}}$ and $n_{\text{g}}$, the shear modulus can be changed according to eqn.(17). As the grafting density ${\chi }_2$ and the number of side chains $n_{\text{sc}}$ increase, the stiffness of the gel decreases. This reduction in stiffness is attributed to a decrease in the shear modulus of the network, which arises from stronger steric repulsion between the grafted chains. What’s more, gels with the lowest stiffness $(n_{\text{sc}}=60,{\chi }_2=30)$ exhibit the smallest relative volume change, while gels with the highest stiffness $\left(n_{\text{sc}}=40,{\chi }_2=10\right)$ exhibit the highest relative volume change. This indicates that differences in monomer distribution within the gels can result in variations in stiffness and relative volume change. However, it is important to note that the change in shear modulus resulting from variations in topological parameters does not alter the force-displacement curves from a concave to a convex shape when $\beta$ is fixed to a relatively small value. This observation indicates that traditional hyperelastic models, such as the neo-Hookean and Gent models, are insufficient to capture the strain-stiffening behavior observed in brush gels by merely adjusting the shear modulus.

Fig. 6.

Force (a) and relative volume (b) as a function of longitudinal stretch when the loading rate is 10⁻³m/s and $\beta =0.1$. Notice that both the value of $n_{\text{sc}}$ and $n_{\text{g}}$ affect the overall mechanical behavior of brush gels. The dimensions of the specimen are the same as in Fig.3.

Fig.7 provides a detailed representation of the stress and chemical potential distribution within the cross-sectional area of the gel when subjected to a tensile loading rate of 0.1 mm/s when ${\lambda }_z=2$. The figure illustrates the spatial variation in both the normal stress component, ${\sigma }_{zz}$, and the chemical potential, $\mu$, across the material. There is a marked difference in color between the bulk of the material and the region near the boundary, which visually indicates the presence of pronounced gradients. These gradients are shown quantitatively in Fig.8 as a function of radial position for different loading rates and topological configurations when ${\lambda }_z=2$. The figure reveals that the magnitudes of ${\sigma }_{zz}$ and $\mu$ are non-uniform and they decrease along the radial direction. As the loading rate decreases, the gradients of both ${\sigma }_{zz}$ and $\mu$ along the $r$ direction become less steep. At lower loading rates, the gel has more time to equilibrate, allowing the liquid to diffuse more evenly throughout the material, which results in smaller stress and chemical potential gradients. Moreover, the negative values of the chemical potential $\mu$ in the bulk of the material suggest that liquid is being absorbed from the external bath into the gel since the chosen value of $\beta$ corresponds to a tensile-dilating material.

Fig. 7.

Chemical potential (a) and stress ${\sigma }_{zz}$ (b) distribution when loading rate is 1mm/s at the time instant when ${\lambda }_z=2,\beta =0.1$. The dimensions of the specimen and the topological parameters are the same as in Fig.3.

Fig. 8.

Stress and chemical potential gradients under different loading rate for tension at the time instant when ${\lambda }_z=2,\beta =0.1$. The dimensions of the specimen and the topological parameters are the same as in Fig.3.

5. Uniaxial compression of cylindrical specimens

Unlike the tensile test, the gel specimens usually used for compression in a parallel plate rheometer are relatively thin. Here our specimen is chosen as a cylindrical disk with height of 1mm and a radius of 5mm, and all other parameters except the topological constants remain the same as in section 3. Following the same methodologies as in the last section, we fix the displacement of the bottom surface in z-direction and compress the gel using the same C3D8P elements. All other boundary conditions are the same as in the tensile test.

Fig.9 plots the result of the measured force as a function of time for a compression-relaxation process for slip vs. no-slip boundary conditions. The gel is compressed for 500 seconds with a loading rate of 10⁻⁶m/s, then the top surface is held fixed and the gel relaxes. Due to the bulging on the sides caused by the no-slip boundary condition, the force is larger and relaxation time is shorter in Fig.9a compared to those in Fig.9b. Like in the tensile test, the brush gel is stiffer than linear polymer gels. Besides, since the value of $I_1-3$ is much smaller than that in the tensile test the locking effect does not come into play, which makes the results predicted by neo-Hookean and Gent models quite similar. Unlike the tensile test, liquid in the bulk region is pressurized, which results in a positive $\mu$, as shown in Fig.10b. This chemical potential gradient causes liquid to leave the gel, which increases the contribution of the osmotic pressure to the force so that rate-dependence is more prominent (Fig.10a). Furthermore, a higher loading rate leads to a larger compressive force, just as in tensile experiments. Also, the curves corresponding to higher rates are closer together, again as in the tensile experiments.

Fig. 9.

Force as a function of time for compression and relaxation process when loading rate is 10⁻⁶m/s. (a) Slip boundary condition, and (b) no-slip boundary condition with $\beta =0.1$. The dimensions of the specimen are $r=5\text{mm}$ and $h=1\text{mm}$ with $n_{\text{sc}}=40,{\chi }_2=20$.

Fig. 10.

Force (a) and chemical potential of the top central point (b) as a function of compression depth for different loading rate when $\beta =0.1$. The dimensions of the specimen and the topological parameters are the same as in Fig.9.

The effect of brush topology on the measured force and the relative volume change is shown in Fig.11. Similar to the behavior in tension, the stiffness of the network decreases due to an increase in grafting density. Brush gels with the lowest stiffness exhibit the smallest relative volume change, while those with the highest stiffness exhibit the largest relative volume change. Fig.12 shows the profiles of ${\sigma }_{zz}$ and $\mu$ for different loading rates when ${\lambda }_z=0.5$. Compared to the tensile test, the positive value of $\mu$ in the bulk indicates that liquid is leaving the gel. As the loading rate decreases, the gradients along the $r$-direction for both ${\sigma }_{zz}$ and $\mu$ decrease, as expected.

Fig. 11.

Force (a) and relative volume change (b) as a function of time for different branch topology when $\beta =0.1$. The loading rate is 10⁻⁵m/s and the dimensions of the specimen are the same as in Fig.9.

Fig. 12.

Stress and chemical potential gradients under different loading rate for compression at the time instant when ${\lambda }_z=0.5,n_{\text{sc}}=40,{\chi }_2=20,\beta =0.1$. The dimensions of the specimen and the topological parameters are the same as in Fig.9.

6. Shear of cylindrical specimen

In experiments gels are typically subjected to a two-step process where they are initially compressed before undergoing shear deformation [43]. This sequence is often employed to replicate physiological or application-specific conditions, as the initial compression ensures uniform contact with testing equipment; hence, in this work, we will examine the role that compressive strain plays in the gel’s stiffness during rotational shear. In our calculations the displacement in three directions of the bottom surface is held fixed, and a compression together with a rotational shear is applied on the top surface in sequence. All other boundary conditions remain the same as in the previous sections and the dimensions of the gel are the same as in section 5. Various compressive displacements are applied at a strain-rate of 1%/s on the top surface and then the gel is sheared rotationally by a displacement $u_{\theta }$ proportional to the radial distance. Results are shown in Fig.13.

Fig. 13.

(a)Torque versus rotation angle $\theta$ for various compression depths; the increase in compression depth results in an increase in stiffness. (b) Chemical potential gradients for various value of angular velocity of the top surface when no compression is applied at the top. The top plate is rotated to $\theta =0.5$. The topological parameters used here are $n_{\text{sc}}=40,{\chi }_2=20$ with $\beta =0.1$.

Fig.13a shows a dependence of the measured torque on the compression depth. As the compression depth increases, the stiffness of the gel increases, as evidenced by the steeper torque curves for higher compression levels. This highlights the significant role of pre-compression in enhancing the mechanical response of the gel under shear, likely due to the densification of the network structure and reduced slack in the polymer chains. In Fig.13b, the chemical potential versus radial distance exhibits a strong dependence on the angular velocity $\omega$. Higher angular velocities result in increased chemical potential gradients. Typically, one might assume that liquid diffusion plays a negligible role in shear tests of gels, as these tests generally do not involve volume changes that drive significant liquid flow. However, simulation shows that the chemical potential is not uniform along the radial direction. This indicates that liquid diffusion, although often considered unimportant in shear tests, can still occur due to the normal stress gradients throughout the gel. To confirm that chemical potential gradients are caused by normal stress gradients, we conducted a simple shear test as a comparison, as shown in Fig.14. The specimen used for the simple shear test has dimensions of 5 mm × 5 mm × 1 mm. The bottom surface is fixed in place, the side surfaces are maintained at zero chemical potential, and a lateral displacement is applied to the top surface. In this test, the normal stress due to the Poynting effect is uniform and tensile, and the resulting chemical potential is negligible. This outcome demonstrates that chemical potential gradients observed in other loading conditions can be directly attributed to variations in normal stress, rather than shear deformation effects.

Fig. 14.

(a) Rotational shear: Stress and chemical potential profiles immediately after the applied shear deformation is completed (solid lines) and at equilibrium (dashed lines). All stress components and the chemical potential exhibit significant gradients along the radial direction. (b) Simple shear: Stress and chemical potential profiles as a function of $y$ (see right inset). In this case, all stress components and the chemical potential show minimal variation along the lateral direction. The left inset shows a view of the specimen along the $z$ direction. The topological parameters used here are $n_{\text{sc}}=40,{\chi }_2=20$ with $\beta =0.1$.

Fig.15 demonstrates the significant strain-stiffening behavior of brush gels in shear tests, particularly as $\beta$ increases. In Fig.15a, with a small $\beta$, the brush gel shows a moderate nonlinear torque response compared to the nearly linear behavior of neo-Hookean and Gent models, which is similar to the results obtained by [44]. However, with a larger $\beta$, as shown in Fig.15b, the torque curve steepens significantly, highlighting stronger strain-stiffening. These results show that the strain-stiffening behavior of brush gels is seen both in tension and shear becoming more pronounced as $\beta$ increases.

Fig. 15.

(a)Torque versus rotation angle $\theta$ for three models. For the brush gel we choose $\beta =0.1\left(\text{a}\right)$ and $\beta =0.2\left(\text{b}\right)$. The strain stiffening behavior becomes stronger as $\beta$ increases. The initial shear modulus of all three models is kept fixed in all cases. The topological parameters used here are $n_{\text{sc}}=40,{\chi }_2=20$.

7. Drying problem

Gel drying occurs due to chemical potential differences between the interior and exterior of the gel, with solvent migration driven by a lower chemical potential outside. Solvent loss causes shrinkage and mechanical changes in the gel. When the gel is surrounded by air, the chemical potential of the water outside is governed by the relative humidity as [45]:

$${\mu }_{\text{vapor}}={\mu }_0+RT\text{ln}\text{RH},$$ (29)

where the relative humidity (RH) is defined as the ratio of the partial pressure of the water vapor $P$ to the saturation vapor pressure of water $P_0$ at a given temperature. We will take $\mu =0$ when relative humidity is 1, so ${\mu }_0=0$. When the gel is put in a bath containing solute (e.g. polyethylene glycol (PEG), dextran), the chemical potential of the water is equal to the negative value of the osmotic pressure[46]:

$${\mu }_{\text{solvent}}=-{\pi }_{\text{solvent}}=-\frac{RT}{M}c,$$ (30)

where $M$ is the molecular weight of the solute and $c$ is the concentration (in mass per unit volume). In the above a virial expansion up to the first order (also known as the vant Hoff formula) is used to estimate the osmotic pressure(Fig.16). We assume that the gel can shrink freely in all directions. When the gel reaches equilibrium with the bath, one can find the relative volume change by setting ${\sigma }_{\text{xx}}={\sigma }_{\text{yy}}={\sigma }_{\text{zz}}=0$ and solving for the stretch which is isotropic. The results of this exercise are shown in Fig.17.

Fig. 16.

A gel immersed in a polymer solution with a negative outside chemical potential will experience chemical potential gradients that cause the gel to dehydrate until equilibrium is reached. Dehydration can be achieved by drying or by osmotic shock.

Fig. 17.

Relative volume change when the gel is put in the air(a) and a solution bath(b). The parameters used in (b) are $\beta =0.1,n_{\text{sc}}=40,{\chi }_2=20$.

The shape of the specimen is kept same as in the compression test. Topological parameters and $\beta$ are varied in this section, while other parameters remain the same as those used at the beginning of this work. Fig.17 shows that as the relative humidity decreases, the volume shrinkage become larger since more water is leaving the gel driven due to larger chemical potential differences. Furthermore, as ${\chi }_2$ increases, the volume change decreases, which shows that the network topology affects the equilibrium state of the gel by changing the shear modulus. Osmotic shock experiments with solutes in the bath surrounding a gel have been performed on mucin gels. In an effort to mimic these experiments, Fig.17b shows that a larger molecular weight of the osmotic agent causes a smaller volume change of the gel since it results in smaller chemical potential difference according to eqn. (30). This contradicts the result obtained in [47] which is caused by the fact that the polymer in the bath in our calculations is not allowed to penetrate the network of the gel, while it diffused into the gel in experiments. Nonetheless, drying or osmotic shock experiments of the type shown in Fig.17 can be used to qualitatively detect changes in topology of brush gels.

Now we consider the transient case, we assume that only the top surface of the gel is permeable. Two types of boundary condition are considered – Dirichlet-type

$${\mu }_{\text{surf}}={\mu }_{\infty },$$ (31)

and Robin-type

$$\mathit{N}\cdot \mathit{Q}=H\left({\mu }_{\text{surf}}-{\mu }_{\infty }\right).$$ (32)

In the Robin-type boundary condition the resistance to liquid penetration is measured by the mass transfer coefficient $H$. Impermeable boundary conditions are recovered in the case of infinite resistance, or equivalently $H=0$, whereas permeable boundary conditions are recovered in the case of no resistance, or equivalently $H\to \infty$. In thermal problems, the convection process is significantly influenced by the surrounding environmental conditions, as well as by the material properties and geometric characteristics of the system. Therefore, the selection of the convective mass transfer coefficient $H$, must be approached with care to accurately capture the dynamics of the drying process. In this study we choose $H=3\times {10}^{-12}{\text{m}}^3/\text{N}\cdot \text{m}$ to induce a relatively large volume change when Robin type boundary condition is applied. The dimensions of the gel remain the same as in section 5. In some experimental setups, the top surface of the gel is permeable whereas other surfaces remain impermeable, and the displacement in the $z$-direction of the bottom surface is set to zero [48]. However, in other experimental setups, the gel is between two plates whose separation is held fixed, after which it undergoes drying. This procedure induces a tensile force between the plates as the gel contracts. Under these conditions, the top and bottom surfaces of the gel are impermeable, and the lateral surface (side surface) of the gel remains permeable.

Fig.18a–d shows finite element results for transient drying with different types of boundary conditions and models for the first setup (i.e., top surface permeable). When the chemical potential $\mu$ at the top surface, or ${\mu }_{\infty }$, decreases, the relative volume change becomes larger due to the increased driving force caused by chemical potential differences. Furthermore, brush gels exhibit a relatively smaller volume change compared to linear polymer gels (neo-Hookean model) when the volume fraction and shear modulus are kept the same, and the difference between the two models become larger as $\beta$ increases. This difference arises from the unique side-chain-dominated structure of brush gels, which resists deformation more effectively under drying conditions. Fig.18e–f illustrates the tensile force developed for different values of $\beta$ under Dirichlet boundary conditions. The results show that a higher absolute value of the chemical potential at the side surface induces a larger tensile force due to larger volume change. The primary distinction between brush and linear polymer gels during the drying process lies in the equilibrium drying time, especially when Robin-type boundary conditions are applied. Specifically, brush gels reach their equilibrium state significantly faster than linear polymer gels as the parameter $\beta$ increases. Fig.19 shows the shape of the gel after equilibrium is reached when top and bottom surface are fixed. The volume decrease due to liquid diffusion causes the shrinkage of the gel in lateral direction.

Fig. 18.

(a)-(d) Volume change of the gel as a function of time. The top surface of the gel is permeable, and $u_{\text{z}}$ at the bottom surface is zero. Dirichlet boundary conditions are applied in (a) and (b), while Robin boundary conditions are applied in (c) and (d). (e)-(f) Tensile force development during drying, with both the top and bottom surfaces fixed and a Dirichlet boundary condition applied to the side surface. Here $\beta$ is chosen as 0.3 for (a),(c),(e) and 0.5 for (b),(d),(f). The topological parameters for all the cases are $n_{\text{sc}}=40,{\chi }_2=20$.

Fig. 19.

Finite element simulation for drying with the top and bottom surfaces fixed when $\beta =0.3$. The side surface is permeable, with a negative chemical potential of $\mu =-1000\text{Pa}$ in (a) and $\mu =-3000\text{Pa}$ in (b).

8. Conclusion

This work builds on a recently developed continuum model for brush gels, highlighting their unique mechanical properties and topological adaptability. Brush gels are shown to be significantly stiffer for large deformations than linear polymer gels for the same small strain shear modulus, a behavior rooted in their branched molecular architecture. Acting as intermediates between linear polymer gels and fibrous gels, their mechanical response is governed by the parameter $\beta$, which controls key characteristics such as strain-stiffening and the curvature of force-displacement curves. We have shown that as $\beta$ increases, the mechanical behavior of brush gels undergoes significant changes. Specifically, the force-stretch curve transitions from concave to convex, reflecting a shift in the material’s response to tensile loading. This transition is accompanied by a change in the gel’s volumetric behavior – at lower values of $\beta$, brush gels exhibit tensile-dilating behavior, which shifts to tensile-contracting behavior as $\beta$ increases. This makes brush gels versatile materials that can mimic the mechanical behaviors of either linear or fibrous gels, depending on design needs. Furthermore, branch topology, including parameters such as grafting density and side-chain length, can modify the stiffness of the gel without affecting the convexity of tension or compression curves, as long as $\beta$ remains fixed. Furthermore, we have shown that the rate-dependence of the force-stretch curve is more pronounced in compression tests than in tension tests, due to the osmotic effects caused by liquid diffusion.

Shear tests of brush gels show that pre-compression can enhance the network’s stiffness by densifying the structure, and a strain stiffening phenomenon can be observed as well in the torque-rotation curve as $\beta$ increases. Additionally, we have shown that chemical potential gradients observed during rotational shear tests are driven by normal stress gradients within the gel. This conclusion is supported by a comparison with simple shear tests, where uniform normal stresses result in negligible chemical potential variations.

Lastly, we have demonstrated that changes in the topology of the brush network can result in different volume changes in drying and osmotic shock experiments. This sensitivity to topology provides a qualitative means of detecting structural changes in brush gels. Furthermore, brush gels exhibit greater resistance to volume changes during drying compared to linear polymer gels when they share the same small strain shear modulus, and their network architecture enables them to equilibrate more quickly, highlighting their practical advantage in applications requiring rapid stabilization under drying conditions.

Highlights.

1. The mechanical behavior of brush gels is intermediate between elastomeric gels and fibrous gels

2. The topology of brush gels strongly affects their tensile, compressive, shear and drying behavior.

3. The strain stiffening behavior of brush gels is linked to their volumetric response under tensile and compressive loading

Appendix A. Permeability

In ABAQUS, Forchheimer’s law[49] is used to calculate the liquid flux as follows:

$$\mathit{f}(1+\beta \sqrt{\left|\mathit{Q}\right|})=\frac{-k_s\left(s\right)}{{\gamma }_w}\hat{k}\left(\frac{\partial \mu }{\partial \mathit{x}}-{\rho }_w\mathit{g}\right)$$ (A.1)

Where $\mathit{f}=sn\mathit{Q}$ is the effective velocity of the wetting liquid, $\beta$ is a velocity coefficient that depends on the void ratio of the material, $k_s\left(s\right)$ is the dependence of permeability on saturation of the wetting liquid such that $k_s=1$ at $s=1$; and $\mathit{g}$ is the acceleration due to gravity. $n$ is the porosity which is the volume fraction of the liquid. Since in our case the effect of gravity is negligible (i.e. ${\rho }_w\mathit{g}\ll \frac{\partial \mu }{\partial \mathit{x}}$), the velocity of the liquid is low and the gel is fully saturated, so we drop ${\rho }_w\mathit{g}$ and set $\beta =0,s=1$. Notice that $\hat{k}$ is what we need to enter into ABAQUS. In this way we can write eqn.(A.1) as:

$$\mathit{Q}=-\frac{\hat{k}}{{\gamma }_{w}n}\frac{\partial \mu }{\partial \mathit{x}}$$ (A.2)

Comparing eqn.(A.2) with eqn.(25), we get:

$$\hat{k}=\frac{k\left({\varphi }_s,a\right)}{\hat{\nu }}gn=\frac{\tilde{k}(e,a)}{\hat{\nu }}\frac{g}{e+1},$$ (A.3)

where $e$ is the void ratio and $\hat{\nu }$ is the kinematic viscosity of the liquid. In this study we choose $\hat{\nu }={10}^{-6}{\text{m}}^2/\text{s}$ and $g=9.8\text{m}/{\text{s}}^2$. This is how eqn.(25) is entered into ABAQUS.

Appendix B. UHYPER implementation

ABAQUS requires the user to program within the UHYPER subroutine the stress-deformation relation of the material and its derivatives with respect to the isochoric invariants. Specifically, one can write eqn.(19) with isochoric invariants as:

$$\hat{\Psi }(\mathit{F},\mu )={\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}\left[\frac{J^{\frac{2}{3}}{\bar{I}}_1}{6}+{\beta }^{-1}{\left(1-\frac{\beta J^{\frac{2}{3}}{\bar{I}}_1}{3}\right)}^{-1}+{\alpha }_{\text{bot}}\text{ln}J\right]+\frac{4}{5}\frac{\alpha RT{\rho }_p}{M_{\text{m}}}{\left({\varphi }_{\text{s}}^{\text{ref}}\right)}^{\frac{9}{4}}{J}^{-\frac{5}{4}}-\mu \left(J-{\varphi }_{\text{s}}^{\text{ref}}\right).$$ (B.1)

Since the last term results in chemical potential (which is also called pore pressure), one can drop it and calculate $\frac{\partial \hat{\Psi }}{\partial {\bar{I}}_1},\frac{\partial \hat{\Psi }}{\partial {\bar{I}}_2},\frac{\partial \hat{\Psi }}{\partial J}$ based on eqn.(9) and enter them into ABAQUS. Notice that any derivatives with respect to ${\bar{I}}_2$ are zero for all the three models used in this paper.

Appendix C. Notation and meaning of variables

Symbol	Definition
$G_{\text{bot}},G_{\text{neo}},G_{\text{gent}}$	Shear moduli for three type of gels
${\varphi }_{\text{s}}^{\text{ref}}$	Volume fraction of solid in reference configuration
$C_{\text{tp}}$	Topological constant
$n_{sc}$	Side chain length
$n_g$	Distance between adjacent side chains
$n_x$	Length of a chain between cross-links
${\chi }_1$	$\frac{n_{\text{sc}}}{n_{\text{g}}}$
${\chi }_2$	$\frac{n_{\text{x}}}{n_{\text{g}}}$
$N$	Total number of monomers in one molecule
$\alpha$	Numerical factor for osmotic pressure
$R$	Gas constant
$M_m$	Molecular weight of a monomer
$k_B$	Boltzmann constant
$T$	Absolute temperature
${\rho }_0$	Monomer density
${\rho }_p$	Polymer density
${\rho }_w$	Liquid density
$\beta$	$⟨{R_{\text{in}}}^2⟩/{R_{\text{max}}}^2$
$b_K$	Kuhn length of the brush polymer
$l$	Monomer length of the brush polymer

Appendix D. Derivation of Poisson ratio

We use the brush gel model as an illustrative example to obtain the small strain Poisson ratio. The same derivation can be applied also to the Gent and neo-Hookean gels. Assuming the gel is subjected to uniaxial tension along direction 1 and allowed to reach equilibrium, we have $\mu =0$ and ${ϵ}_2={ϵ}_3$. Under these conditions, ${\sigma }_{22}$ can be written as:

$${\sigma }_{22}=2J^{-1}\left[\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{6}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{3{\left(1-\frac{\beta I_1}{3}\right)}^2}\right]B_{22}+\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}}{J}-\pi ,$$ (D.1)

where

$$J=1+{ϵ}_1+2{ϵ}_2,$$ (D.2)

$$I_1=3+2{ϵ}_1+4{ϵ}_2,$$ (D.3)

$$B_{22}={\left(1+{ϵ}_2\right)}^2,$$ (D.4)

$$\pi ={\pi }_0{J}^{-9/4}.$$ (D.5)

For small ${ϵ}_1,{ϵ}_2$, we use the binomial expansion $(1+x{)}^p\approx 1+px$ to obtain:

$$J^{-1}\approx 1-{ϵ}_1-2{ϵ}_2,$$ (D.6)

$$J^{-9/4}\approx 1-\frac{9}{4}\left({ϵ}_1+2{ϵ}_2\right),$$ (D.7)

$$B_{22}\approx 1+2{ϵ}_2.$$ (D.8)

$${\left(1-\frac{\beta I_1}{3}\right)}^{-2}\approx \frac{1}{(1-\beta {)}^2}\left[1+\frac{4\beta }{3(1-\beta )}{ϵ}_1+\frac{8\beta }{3(1-\beta )}{ϵ}_2\right].$$ (D.9)

Substituting these expansions into the ${\sigma }_{22}$ formula and keeping only linear terms in ${ϵ}_1,{ϵ}_2$, we define:

$$C_0=\frac{{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}}{6}+\frac{{\varphi }_s^{\text{ref}}{G}_{\text{bot}}}{3(1-\beta {)}^2},$$ (D.10)

$$D_1=\frac{4\beta }{9(1-\beta {)}^3}{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}},$$ (D.11)

$$D_2=\frac{8\beta }{9(1-\beta {)}^3}{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}.$$ (D.12)

Then, we can write ${\sigma }_{22}$ as:

$${\sigma }_{22}\approx K_0+K_1{ϵ}_1+K_2{ϵ}_2,$$ (D.13)

$$K_0=2C_0+{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}-{\pi }_0,$$ (D.14)

$$K_1=2D_1-2C_0-{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}+\frac{9{\pi }_0}{4},$$ (D.15)

$$K_2=2D_2-2{\varphi }_{\text{s}}^{\text{ref}}{G}_{\text{bot}}{\alpha }_{\text{bot}}+\frac{9{\pi }_0}{2}.$$ (D.16)

Setting ${\sigma }_{22}=0$ yields:

$$0=K_0+K_1{ϵ}_1+K_2{ϵ}_2,$$

Therefore, the Poisson ratio for small strain is:

$$\frac{{ϵ}_2}{{ϵ}_1}=-\frac{K_1}{K_2}.$$ (D.17)

Since ${\alpha }_{\text{bot}}=\left(\frac{{\pi }_0}{\varphi G_{\text{bot}}}-\frac{1}{3}-\frac{2}{3(1-\beta {)}^2}\right)$, it follows that $K_0=0$. Consequently, the Poisson ratio at small strain for the brush gel is given by $-\frac{K_1}{K_2}$ in our model. Substituting the parameter values yields 0.1738, which matches precisely the value observed in Fig. 2 when ${\lambda }_z$ is close to 1.