A viscoelastic bonded particle model to predict rheology and mechanical properties of hydrogel spheres

Abstract

The use of hydrogels has exponentially increased in recent years in many fields, such as biology, medicine, pharmaceuticals, agriculture, and more. These materials are so widely used because their mechanical properties change drastically with the different chemical compositions of the constituent polymer chains, making them highly versatile for different applications. We introduce a numerical simulation tool that relies on the Discrete Element Method to reproduce and predict the behavior of hydrogel spheres. We first use a benchmark test, namely an oscillatory compression test on a single hydrogel, to calibrate the model parameters, obtaining a good agreement on the material’s rheological properties. Specifically, we show that the normal modified storage and loss moduli, E’ and E”, obtained in the simulation match the experimental data with a small relative error, around 3%, for E’ and 11% for E”. This result aligns with recent work on numerical modeling of hydrogels, introducing a novel approach with bonded particles and a viscoelastic constitutive relation that can capture a wide range of applications thanks to the higher number of elements. Moreover, we validate the model on a particle-particle compression test by comparing the simulation output with the contact force in the compression direction, again obtaining promising results.

Graphical abstract

Appendix: Validation of the bonded generalized Maxwell model

The simulation is defined to resemble the single particle compression-extension experiment under dry conditions as accurately as possible, where the effect of the discretization needs to be investigated. When discretizing a larger object with DEM spheres, both the spatial configuration and the particles size play a role. In general, the higher the number of particles, the higher the computational cost of the simulation, but the better accuracy is achieved and vice-versa. Some models try to apply a so-called ”coarse-graining”, which aims to reduce the number of particles in the simulation without losing the information a larger system gives. Similarly, in this case, we want to check the effect of particle size and their configuration on the variables computed in a test simulation. We start by checking the effect of particles size by means of a simple test case. We simulate an oscillatory compression-extension motion, similar to the one used in the cyclic compression experiment, but applied to a cube this time. The cube is formed by a Face Centered Cube lattice, as shown in Fig. 15.

Fig. 15

Geometrical representation of the FCC lattice cube used for the convergence study and validation of the model

Three different particle sizes are used to study the effect on the output. The size is computed such that the total volume of the cube is kept constant according to the number of particles on one side of the cube. The particles aligned vertically have a distance of $\delta l = 1.001\cdot \sqrt{2}\cdot d$ between each other, accounting for a total length of the cube side of $L = \delta l\cdot (N-1)+d=0.005$ m, with N being the number of particles on the side of the cube and d being the diameter of the particles. It is straightforward to conclude that, given the fixed length of the cube and, hence, the volume, the relative diameter of the DEM particles can be calculated by changing the number of particles.

If one blindly computed the system’s stress response and compared it to the analytical solution, one would obtain a great difference in magnitude. This is because the lattice configuration used here, together with the direction of the deformation, would produce both normal and shear force components that are not aligned with the deformation, which means that the shear component of the force must be reduced using the ratio introduced in Section 2.2 between the normal and shear parameters.

To determine the optimal parameter $\alpha$, the classical approach used in determining DEM simulation parameters is used, which consists of simulating with a first guess parameter, evaluating the macro-properties to compare with analytical or experimental solutions, computing a residual and repeating the loop until the residual is minimized. The latter is a well-known and established method to calibrate parameters in a model. However, it lacks generality as it is usually optimized for specific applications, and the procedure must be repeated each time something in the original setup is changed. The cost function used in this case is similar to the one used in (9), with the difference that now we do not have a frequency sweep but only one frequency, and instead of comparing the modified storage and loss modulus, we are comparing the amplitude and the phase shift of the stress response, since those are the quantities that we want to minimize the difference of, giving

$$\begin{aligned} f_{cost} = \left( \dfrac{\sigma _{0,an}-\sigma _{0,sim}}{\sigma _{0,an}}\right) ^2 + \left( \dfrac{\varphi _{an}-\varphi _{sim}}{\varphi _{an}}\right) ^2, \end{aligned}$$

(A.1)

Fig. 16

Zoomed particular of the stress response of the FCC lattice cube under cyclic compression, when using the parameters of Table 1, with relative errors for phase and amplitude showing on the inset graph. All errors are measured against the analytical solution given by (A.2), resulting in values well below 3%, making the model valid for predicting the stress response in oscillatory deformation, regardless of particle size. To note the relative errors for both phase shift and amplitude in the inset graph. Stress is normalized w.r.t. the amplitude of the analytical stress $\sigma _{0,an}$

Fig. 17

Stress response of the BCC lattice for the same parameters of the FCC case. Here, the value of $\alpha$ found for the FCC lattice can not be used, showing how this parameter needs to be optimized for each specific system when changing either the lattice configuration and/or the particle size

where the analytical solution, in this case, is known as [23]

$$\begin{aligned} \sigma (t) = \sum _j^4\dfrac{k_j\gamma _0\omega \tau _j}{1+(\omega \tau )^2}(\omega \tau _j\sin {\omega t}+\cos {\omega t}). \end{aligned}$$

(A.2)

The stress in the DEM simulation is computed as the average force on the top layer of particles of the cube, divided by the face area

$$\begin{aligned} \sigma _{sim}(t) = \dfrac{1}{L^2}\sum _i^{N_{top}}f_i(t). \end{aligned}$$

(A.3)

with $N_{top}$ being the number of particles on the top layer of the cube. Once the stresses are known, the analytical and the simulated phase lags are computed by fitting the relative stresses with a function of the form $y=A\sin (\omega t+\varphi )$.

To perform the calibration, the smallest particle size is used, returning a value of $\alpha = 9.1$. This parameter is then used with the three different particles sizes to produce the plot of Fig. 16, which is a zoomed-in version of the total stress response, to observe better the effect of the particles’ size on the amplitude and the phase of the stress signal. In detail, no effect is observed on the phase shift error $\chi _{\varphi }$, as shown in the inset graph of Fig. 16, with a relative error that is almost constant and below $2\%$. A more significant effect is observed for the amplitude error $\chi _{\sigma }$. In conclusion, smaller particles give more accurate results, without the need to change the model parameters.

To check the effect of the particles spatial configuration, a body-centered cube lattice is now used to compute the stress response of the same volume. The same model parameters are used as in the FCC configuration, and the same test is performed together with the same post processing to compute the stress, resulting in the plot of Fig. 17. In this case, the particle size used is computed according to $d=\dfrac{L}{(N-1)*1.001+1}$, giving $d=5\cdot 10^{-4}$ m for N=10. As it can be easily observed, a large difference in the amplitude is observed between analytical and simulated stress. This suggests that the scaling of the shear parameters w.r.t. the normal ones needs to be adjusted, resulting in another calibration process to obtain the optimal $\alpha$ parameter. In conclusion, when changing the spatial configuration, one needs to adjust the $\alpha$ parameter accordingly but, once that is found, a change in particle size would improve the accuracy of the model without the need to re-calibrate its parameters.