# Reinforcement Efficiency of Cellulose Microfibers for the Tensile Stiffness and Strength of Rigid Low-Density Polyurethane Foams

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Foam Production

#### 2.3. Foam Characterization

## 3. Model

#### 3.1. Mechanical Reinforcement Efficiency of Foams

_{E}is defined as the ratio of composite foam modulus E

_{cf}and neat foam modulus E

_{f}, the latter corresponding to the density ρ

_{cf}and geometrical anisotropy R

_{cf}of composite foams [31]:

_{f}[28,31],

_{s}and ρ

_{s}designate the stiffness and density of the solid monolithic foam strut and wall material, f

_{E}(R) is a function reflecting the effect of cell shape anisotropy on foam stiffness, and c

_{E}, n

_{E}are constants to be determined via modeling [28] or by approximating the experimental data by Equation (2) [31]. Assuming that the dependence of stiffness of composite foams on their apparent density and geometrical anisotropy can be described by the same relation, Equation (2), upon substitution of neat solid material characteristics by the modulus E

_{cs}and density ρ

_{cs}of solid composite cell strut material:

_{E}is obtained:

_{s}and E

_{cs}from Equations (2) and (3), respectively, and substituting into Equation (4), the mechanical reinforcement efficiency for foam stiffness is derived in terms of foam characteristics as follows:

_{σ}can be defined as the ratio of composite, σ

_{cf}, and neat foam strength, σ

_{f}, with neat foams having the same density and geometric anisotropy as the composite foams:

_{s}of the solid strut material [28]. The foam strength can be expressed as

_{σ}, n

_{σ}are foam morphology-related constants, and function f

_{σ}(R) allows for geometrical anisotropy effect on foam strength [28]. Assuming as above that relation Equation (7), upon replacing the relevant neat polymer characteristics by those of the composite solid, holds also for composite foam strength, it follows that the strength reinforcement efficiency according to Equation (6) is given by

#### 3.2. Stiffness and Strength of Foam Struts

#### 3.2.1. Young’s Modulus

_{cs}of a fiber-reinforced composite strut can be related to the Young’s modulus of polymer matrix E

_{s}, axial modulus of the reinforcing fibers E

_{A}, and fiber volume fraction ν

_{f}by a rule of mixtures type of relationship:

_{oE}designates the fiber orientation factor and η

_{lE}is the fiber length efficiency factor. The latter can be expressed by a shear-lag model via the aspect ratio (i.e., length-to-diameter ratio) κ of the reinforcing fibers and stress transfer rate between the fiber and matrix β as follows:

_{A}and r are fiber shear modulus and radius, respectively, G

_{s}denotes the shear modulus of the matrix, D designates a stiffness parameter of the fiber/matrix interface, and $\chi =0.009$. is a numerically determined constant ensuring the accuracy of the expression for β at low fiber volume fractions [35].

#### 3.2.2. Strength

_{cs}of a composite foam strut, the Fukuda and Chou model [37] in its modified form [38] is applied, expressing σ

_{cs}as a weighted sum of fiber axial strength σ

_{A}and matrix strength σ

_{s}:

_{o}

_{σ}, and length, η

_{l}

_{σ}, efficiency factors for strength. The latter is presented in terms of reinforcing fiber length l and critical length l

_{c}as

_{o}

_{σ}is also given by Equation (13) [38].

#### 3.2.3. Fiber Orientation Distribution

## 4. Results and Discussion

_{E}, n

_{σ}and shape anisotropy functions ${f}_{E}$, ${f}_{\sigma}$ entering the respective expressions need to be specified. For stiffness and strength in the direction transverse to the foam rise, the latter can be expressed in the form ${f}_{E}\left(R\right)=\left(R+1/{R}^{2}\right)/{\left(R+2\right)}^{2}$ and ${f}_{\sigma}\left(R\right)=\left(R+1\right){\left(R/\left(R+2\right)\right)}^{3/2}/{R}^{2}$ respectively by the rectangular-cell model; see [28,34]. As concerns the exponents of the density dependence of foam characteristics, we evaluated them by fitting Equations (2) and (7) to the experimental data for neat foams of the same formulation reported in [34]. The respective data are shown in Figure 2 together with best-fit approximations corresponding to n

_{E}= 1.9 and n

_{σ}= 1.2. Notably, the value of the power-law exponent for the density dependence of foam stiffness is very close to the one derived for low-density open-cell foams and amounting to n

_{E}= 2, while the exponent for strength is lower than the predicted n

_{σ}= 1.5 [28].

_{s}= 2.3 GPa and ρ

_{s}= 1210 kg/m

^{3}[34]. The fiber volume fraction ${\nu}_{f}$ in the solid polymer is expressed via fiber weight fraction c (of neat polymer weight) as ${\nu}_{f}=c/\left(c+{\varrho}_{MCC}/{\varrho}_{s}\right)$ with ${\varrho}_{MCC}$ denoting the density of MCC fibers. Since the foam expansion ratios, calculated using the foam density values of Table 2, exhibited little variation ranging between 33.5 and 36.4, the average expansion ratio was applied to evaluate the fiber orientation factor by Equations (18) and (19), yielding η

_{o}= 0.69. The resulting dependence of Γ

_{E}on MCC fiber loading according to Equation (4) is plotted in Figure 3a by a solid line. A good agreement of the theoretical prediction with the efficiency factor values derived from test results is seen.

_{σ}derived from foam tests and from composite strength estimates agree well and are both rather low - the predicted reinforcement efficiency factor for 10 wt % loading amounts to 1.06, while the experimental Γ

_{σ}values are ca. 1.09 and 1.03 for 7 wt % and 10 wt % fiber content, respectively. For comparison, the predicted reinforcement efficiency for η

_{o}= 1 is also displayed in Figure 3, showing the level of Γ

_{E}, Γ

_{σ}attainable at a perfect fiber alignment with the strut axis.

_{σ}is calculated in the same way upon the substitution of ${\Gamma}_{E}$ by Γ

_{σ}in Equation (20). The results are presented in Table 3. It is seen that the relative RMS error is less than 5% for the MCC fiber-filled foams.

^{3}for neat foams to 82 kg/m

^{3}for foams with the highest whisker content.

_{E}= 2 and n

_{σ}= 1.5 derived using rectangular unit cell model [28]. Since the presence of whiskers apparently affected cell size while no effect on the shape anisotropy of cells was reported in [42], R

_{cf}= R was assumed. The mechanical reinforcement efficiency factors evaluated by Equations (5) and (9) are presented in Figure 4. It is seen that the maximum gain in nanocomposite foam stiffness, when corrected for density variation, becomes ca. 40%, and for strength—14%.

_{cel}= 1600 kg/m

^{3}. Nanocomposite foams had slightly larger density than the MCC-filled ones; hence, the orientation factor calculated using the average expansion ratio of foams was smaller, amounting to η

_{o}= 0.64. The predicted efficiency factors Γ

_{E}and Γ

_{σ}of nanocomposite foams according to Equations (4) and (8) for η

_{o}= 0.64 are plotted in Figure 4 by solid lines. The modest increase in the reinforcement efficiency for strength with nanocellulose loading predicted via the strut strength model agrees very well with the Γ

_{σ}values derived from foam tests, as seen in Figure 4b, as also indicated by the respective relative RMS error of prediction shown in Table 3. However, the mechanical reinforcement efficiency of foam stiffness is closer to that predicted for fully aligned whiskers (η

_{o}= 1).

_{cf}= R was assumed when estimating the reinforcement efficiency factors by Equations (5) and (9). Substantial gains in the foam stiffness and strength in tension, bending, and compression were achieved, as shown in Figure 5. The variability of stiffness reinforcement efficiency among different loading modes seen in Figure 5a is likely to reflect the inter-batch variability of foams, the rate of growth of the bending and compressive stiffness with CNF loading being similar. However, the specific tensile stiffness at the highest filler loading was reported to increase by a factor of 10 [45], which appears to be inconsistent with the rest of data; hence, the respective data point was excluded from further analysis as an outlier. The reinforcement efficiency for strength, Figure 5b, exhibits marked scatter, tensile strength of foams benefiting from filler the least and compressive strength - the most. Such an effect is apparently caused by differing sensitivity in tension, bending, and compression to the stress concentrations caused by filler agglomerates in cell struts, defects in foam morphology, and superficial flaws introduced during the cutting of specimens.

_{o}= 0.65). It is seen that the predicted dependence of reinforcement efficiency on filler loading reasonably closely reflects the experimental trend, although the relative RMS error values are considerably larger, see Table 3, which is mainly due to the scatter among results for different loading modes. The predicted reinforcement efficiency for η

_{o}= 1, shown by broken lines in Figure 5, demonstrates the Γ

_{E}, Γ

_{σ}values theoretically attainable at a perfect CNF alignment.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**SEM images of foam cross-section in a plane aligned with the foam rise direction (the vertical direction in the pictures) at MCC fiber filler content of: (

**a**) 0 wt % (neat foams); (

**b**) 1 wt %; (

**c**) 3 wt %; (

**d**) 5 wt %; (

**e**) 7 wt %; (

**f**) 10 wt %.

**Figure 2.**Variation of neat foam (

**a**) modulus and (

**b**) strength under tension in the transverse direction with foam density [34] and approximations of the data by Equations (2) and (7), as shown by dashed lines.

**Figure 3.**Mechanical reinforcement efficiency factors of composite foam (

**a**) stiffness and (

**b**) strength versus MCC fiber loading.

**Figure 4.**Mechanical reinforcement efficiency factors of composite foam (

**a**) stiffness and (

**b**) strength versus cellulose whisker [42] loading.

**Figure 5.**Mechanical reinforcement efficiency factors of composite foam (

**a**) stiffness and (

**b**) strength versus cellulose nanofibril [45] loading.

**Table 1.**Geometrical characteristics of foam cells as a function of microcrystalline cellulose fiber loading.

Fiber Loading, wt % | Cell Length, μm | Cell Width, μm | Shape Anisotropy R |
---|---|---|---|

0 | 745 (130) ^{1} | 480 (73) | 1.56 (0.18) |

1 | 567 (104) | 364 (55) | 1.56 (0.17) |

3 | 514 (166) | 345 (90) | 1.48 (0.18) |

5 | 565 (130) | 362 (65) | 1.56 (0.18) |

7 | 502 (105) | 333 (56) | 1.51 (0.17) |

10 | 498 (149) | 337 (82) | 1.47 (0.16) |

^{1}Standard deviation is given in parentheses.

Fiber Loading, wt % | Foam Density, kg/m^{3} | Young’s Modulus, MPa | Tensile Strength, kPa | Strain at Failure, % |
---|---|---|---|---|

0 | 33.1 (1.3) ^{1} | 3.36 (0.76) | 127 (20) | 6.6 (0.3) |

1 | 33.0 (0.5) | 3.10 (0.27) | 137 (12) | 7.6 (0.6) |

3 | 34.3 (0.5) | 3.90 (0.29) | 142 (7) | 6.4 (1.0)) |

5 | 33.4 (0.5) | 3.66 (0.25) | 134 (12) | 5.9 (0.4) |

7 | 35.6 (0.7) | 4.74 (0.67) | 152 (13) | 5.6 (0.7) |

10 | 35.8 (0.5) | 4.86 (0.47) | 148 (16) | 5.0 (0.6) |

^{1}Standard deviation is given in parentheses.

**Table 3.**Relative root mean square error of prediction of the mechanical reinforcement efficiency by cellulose micro- and nanofibers.

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**MDPI and ACS Style**

Andersons, J.; Kirpluks, M.; Cabulis, U.
Reinforcement Efficiency of Cellulose Microfibers for the Tensile Stiffness and Strength of Rigid Low-Density Polyurethane Foams. *Materials* **2020**, *13*, 2725.
https://doi.org/10.3390/ma13122725

**AMA Style**

Andersons J, Kirpluks M, Cabulis U.
Reinforcement Efficiency of Cellulose Microfibers for the Tensile Stiffness and Strength of Rigid Low-Density Polyurethane Foams. *Materials*. 2020; 13(12):2725.
https://doi.org/10.3390/ma13122725

**Chicago/Turabian Style**

Andersons, Jānis, Mikelis Kirpluks, and Ugis Cabulis.
2020. "Reinforcement Efficiency of Cellulose Microfibers for the Tensile Stiffness and Strength of Rigid Low-Density Polyurethane Foams" *Materials* 13, no. 12: 2725.
https://doi.org/10.3390/ma13122725