# Assessing the Viscoelasticity of Photopolymer Nanowires Using a Three-Parameter Solid Model for Bending Recovery Motion

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Considerations

#### 2.1. The Three-Parameter Solid Model

_{1}, E

_{2}, and an intrinsic viscosity η

_{i}. The model defines the stress–strain constitutive relation of the photopolymer material. The corresponding creep compliance and relaxation modulus functions can be found in basic textbooks [30]. The standard linear solid model was used previously to characterize the viscoelasticity of macroscopic photopolymer materials [31].

#### 2.2. The Mechanical Model of the Microstructure

_{1}and k

_{2}, and the intrinsic damping coefficient δ can be expressed as follows (see the Supplementary Information):

^{3}b/12.

#### 2.3. The Bending Recovery Motion

_{1,2}and the two decay times τ

_{1,2}can be determined from the experiments. Furthermore, these parameters can be calculated when the four model parameters k

_{1,2}, δ, and γ are known [27].

#### 2.4. The Inverse Problem

_{1}and k

_{2}, and the intrinsic damping coefficient δ can be determined experimentally. Moreover, when combined with Equations (1) and (2), one can measure the material properties of the viscoelastic nanowires, as defined by the standard linear solid model. It is noted that the right sides of Equations (5)–(7) only depend on the ratio of the two amplitudes A

_{1}/A

_{2}. Consequently, the measured cantilever deflection does not require calibration in the experiment. The results are independent of the extent of the initial deflection; relative deflection measurements are sufficient to determine the three model parameters. On the other hand, the damping coefficient γ is not excluded from these formulas and must be calculated separately from Equation (3).

## 3. Materials and Methods

_{0}was applied to the endpoint of a straight beam. The obtained deflection was about 20% higher in this case (the blue curve in Figure 2e). The experimental deflection curve could be reproduced by lowering the applied force to 0.85 times F

_{0}(see the green curve in Figure 2e). After this force correction, the general shape of the deflection distribution was almost identical for the straight and curved beams. It was concluded that accounting for the proposed force correction, the straight beam theory, and the corresponding Equations (1) and (2), represented an accurate approximation for the studied curved nanowire system.

## 4. Results and Discussion

_{1,2}and the corresponding amplitudes A

_{1,2}were used to calculate the stiffnesses k

_{1,2}and the intrinsic damping coefficient δ, as defined by Equations (5)–(7). Representative results, obtained for pure water and 400 mg/mL glucose solution, are presented in Table 1. The equilibrium stiffness k

_{eq}of the cantilever, also included in Table 1, is obtained as a combination of the two spring constants: 1/k

_{eq}= 1/k

_{1}+ 1/k

_{2}. This value refers to the static response of the nanowire and determines its strength in the slow-motion limit. Interestingly, the obtained equilibrium stiffness is in the order of 0.002 pN/nm, which proves the extreme flexibility (softness) of the studied nanowires and points towards possible future applications in the field of sensitive force measurements. The intrinsic damping δ of the studied cantilever can be compared with the drag coefficient γ for the bead motion in the surrounding medium (Table 1). Notably, for the present experimental conditions, the intrinsic damping (as defined by the mechanical model), is more than an order of magnitude higher than the drag coefficient of the solution acting on the microsphere.

_{1}, E

_{2}, and the intrinsic viscosity η

_{i}are plotted in Figure 4 as a function of the solution viscosity. Selected parameter values are given in Table 1. The equilibrium elastic modulus E

_{eq}, defined as: 1/E

_{eq}= 1/E

_{1}+ 1/E

_{2}, was also calculated and is plotted in Figure 4a by the dashed line. The nanowire elastic moduli can be analyzed in terms of the bulk Young modulus, which is approximately E

_{bulk}= 1 GPa (producer data, [29]). For the studied conditions, the measured equilibrium elastic modulus of the nanowire is about 300 times smaller (E

_{eq}= 3.5 ± 0.6 MPa, see Table 1). Significantly reduced elastic moduli of nano- and micro-structures prepared by TPP-DLW were observed for other photopolymers immersed in different solutions [9,10,12]. These previous measurements were performed at static conditions. The present dynamic studies, analyzed by the three-parameter solid model, provide a more nuanced representation of the material. The equilibrium modulus E

_{eq}is determined predominantly by the E

_{1}values, which are substantially lower than E

_{2}.

_{2}elastic modulus increases towards higher glucose concentrations. We observe the same tendency for E

_{1}and E

_{eq}. These data indicate that glucose molecules stiffen the nanowires made of Ormocomp.

_{i}changes from 4.1 to 10 MPas, when the surrounding solution viscosity is varied between 0.9 and 4.5 mPas. There must be a high affinity of glucose molecules to the porous polymer structure. The prolonged time, ca. 15–20 min, needed to restore the mechanical properties of the nanowire cantilever when moving back from glucose solutions to pure water (not shown) supports this assumption. A similar effect was observed recently for commercial polyurethane foam immersed in dextran solutions [39]. It would be desirable to study the Ormocomp–glucose interaction in more detail, focusing on the solvent permeability and the glucose penetration depth, which is beyond the scope of the present work.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The scheme showing the interconnection of the used models: (

**a**) the 3-parameter solid model of the photopolymer material; (

**b**) the mechanical model of the cantilever system immersed in a Newtonian liquid.

**Figure 2.**Top-view brightfield images of microstructures immersed in: (

**a**) OrmoDev developer; (

**b**) water; (

**c**) air. (

**d**) Side-view SEM image of the dried microstructures, the inset shows the spherical bead. (

**e**) Steady-state deflection of the nanowire plotted against the distance along the cantilever beam. The experimental beam deflection (solid points) is compared with the results of straight and curved beam simulations. Inset: the nanowire with the bead immersed in water. (

**f**) Deflection of straight and curved cantilevers simulated by finite element method.

**Figure 3.**The experimental bending recovery curves of the cantilever nanowire system were measured at different glucose concentrations. The data points were fitted by double-exponential time dependencies (solid lines).

**Figure 4.**The nanowire material properties: the elastic moduli E

_{1}, E

_{2}, and E

_{eq}(

**a**), and the intrinsic viscosity η

_{i}(

**b**), are plotted against the glucose solution viscosity.

**Table 1.**Representative values of the mechanical model parameters (stiffnesses k

_{1,2}and the intrinsic damping coefficient δ) and the corresponding material properties (elastic moduli E

_{1,2}, and the intrinsic viscosity η

_{i}) are evaluated for pure water and dense glucose solution conditions. The equilibrium stiffness k

_{eq}is calculated as: 1/k

_{eq}= 1/k

_{1}+ 1/k

_{2}. The equilibrium elastic modulus E

_{eq}is obtain analogously.

k_{1}pN/nm | k_{2}pN/nm | k_{eq}pN/nm | δ 10 ^{−6} kg s^{−1} | γ 10 ^{−6} kg s^{−1} | E_{1}MPa | E_{2}MPa | E_{eq}MPa | η_{i}MPa s | |
---|---|---|---|---|---|---|---|---|---|

water | 0.0017 | 0.011 | 0.0015 | 2.1 | 0.042 | 3.4 | 21 | 2.9 | 4.1 |

400 mg/mL glucose | 0.0024 | 0.017 | 0.0021 | 5.4 | 0.21 | 4.6 | 33 | 4.1 | 10 |

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**MDPI and ACS Style**

Kubacková, J.; Slabý, C.; Horvath, D.; Hovan, A.; Iványi, G.T.; Vizsnyiczai, G.; Kelemen, L.; Žoldák, G.; Tomori, Z.; Bánó, G.
Assessing the Viscoelasticity of Photopolymer Nanowires Using a Three-Parameter Solid Model for Bending Recovery Motion. *Nanomaterials* **2021**, *11*, 2961.
https://doi.org/10.3390/nano11112961

**AMA Style**

Kubacková J, Slabý C, Horvath D, Hovan A, Iványi GT, Vizsnyiczai G, Kelemen L, Žoldák G, Tomori Z, Bánó G.
Assessing the Viscoelasticity of Photopolymer Nanowires Using a Three-Parameter Solid Model for Bending Recovery Motion. *Nanomaterials*. 2021; 11(11):2961.
https://doi.org/10.3390/nano11112961

**Chicago/Turabian Style**

Kubacková, Jana, Cyril Slabý, Denis Horvath, Andrej Hovan, Gergely T. Iványi, Gaszton Vizsnyiczai, Lóránd Kelemen, Gabriel Žoldák, Zoltán Tomori, and Gregor Bánó.
2021. "Assessing the Viscoelasticity of Photopolymer Nanowires Using a Three-Parameter Solid Model for Bending Recovery Motion" *Nanomaterials* 11, no. 11: 2961.
https://doi.org/10.3390/nano11112961