# An Extended Thickness-Dependent Moisture Absorption Model for Unidirectional Carbon/Epoxy Composites

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## Abstract

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_{o}) and thickness was characterized using a thickness-dependent model. A comparison with other materials revealed that all three non-Fickian parameters are able to be fitted using a power law. Nevertheless, the upper boundary for the applicability of this model was not determined in this study. The Weibull distribution plots indicate that the probability of non-Fickian moisture absorption is influenced by ϕ and α at approximately 62% within a normalized thickness range of 2–3. In regards to t

_{o}, it is 82% at a normalized thickness of 6. Therefore, the Weibull distribution is proposed for the assessment of non-Fickian moisture absorption based on the material’s thickness.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Specimens

^{3}, respectively. The nominal fiber volume fraction is 57.42%, and the nominal ply thickness is 0.15 mm. Three laminates were fabricated, with 4, 8, and 16 plies, with a nominal thickness of 0.6, 1.2, and 2.4 mm, respectively. All laminates were fabricated at Composites Technology Research Malaysia (CTRM) using an autoclave with the recommended manufacturing cycle.

#### 2.2. Moisture Absorption Test

^{2}size. The edges of all specimens were coated with water-resistant paint to avoid water penetration from the sides. Subsequently, the initial weight of all specimens was measured using a Shimadzu ATY224 four-decimal digital balance. They were subjected to continuous distilled water immersion in an HH-6 thermostatic water bath at a fixed temperature of 60 °C. Weight gains of all specimens were measured periodically. For each reading, three measurements were made to obtain the average moisture content, M. Figure 1 illustrates the specimens’ configuration and the aging condition for the moisture absorption test.

## 3. Thickness-Dependent Moisture Absorption Model

_{m}means the maximum moisture content of each specimen, D

_{z}refers to the Fickian diffusivity of the material, h indicates the thickness of the specimens, and t is any instant of the immersion time. Additionally, ϕ represents the ratio of the maximum moisture content at Stage I to the total maximum moisture content of the specimen (ϕ = M

_{m,F}/M

_{m}), α is the non-Fickian diffusivity per square thickness, and t

_{o}indicates the initiation time of Stage II. The Macaulay bracket, < > for the time delay term, <t − t

_{o}>, depicts that non-Fickian behavior only occurs after t ≥ t

_{o}. For this model to be applicable, Stage I of the experimentally reduced moisture absorption curves (M vs. √t/h) at all thicknesses must overlap. This behavior is observed because Stage I is assumed to exhibit Fickian diffusion behavior.

## 4. Experimental Results and Discussion

#### 4.1. Moisture Absorption Curves

#### 4.2. Fictitious Fickian Diffusion Curve

- The deviation of the reduced experimental curves in this study was estimated to be 60% of M
_{m,F}. From there, estimate the average M_{m,F}for the material (Table 1). A coefficient of variation (C.V) of 8.11% signifies good repeatability of the results; - Determine the apparent diffusivity, D
_{z}, of the fictitious Fickian curve using Equation (2). For the composite used in this study, D_{z}is estimated to be 4.25 × 10^{−2}mm^{2}/day;$${D}_{z}=\pi {\left(\frac{1}{4{M}_{m,F}}\right)}^{2}{\left(\frac{{M}_{2}-{M}_{1}}{\sqrt{{t}_{2}}/h-\sqrt{{t}_{1}}/h}\right)}^{2}$$ - Plot the fictitious Fickian curve (which is the first term in Equation (1)) using Equation (3). For the AS4/8552 carbon/epoxy composite, the unit ply thickness h
_{F}= 0.15 mm.$${M}_{I}(t)={M}_{m,F}\left\{1-\mathrm{exp}\left[-7.3{\left(\frac{{D}_{z}t}{{h}_{F}^{2}}\right)}^{0.75}\right]\right\}$$

#### 4.3. Determination of Non-Fickian Parameters

_{o}) are determined as follows:

- Calculate ϕ = M
_{m,F}/M_{m}for each thickness; - Plot the fictitious Fickian curve, together with the experimental curves (Figure 5). Estimate the deviation point (√t
_{o}/h) from the figure. From there, determine t_{o}for each thickness; - Subtract the experimental moisture content M(t) from the analytical M
_{I}(t) (which is determined in the previous section). This step gives the experimental M_{II}(t) term; - Plot the experimental W(t) (denoted as W
_{exp}(t)) using Equation (4) below:$${W}_{\mathrm{exp}}(t)=\frac{M(t)-{M}_{I}(t)}{{M}_{m}-{M}_{m,F}}$$ - Apply logarithms of both sides twice for the analytical W(t) term:$$\mathrm{ln}\{-\mathrm{ln}[1-W(t)]\}=0.75\times \mathrm{ln}\langle t-{t}_{o}\rangle +0.75\times \mathrm{ln}\alpha $$
- Plot the curve of ln[−ln[1 − W
_{exp}(t)]] versus 0.75 × ln<t − t_{o}>, and fit the data with a straight line. From the ordinate intersection (0.75 × ln α), determine the value of α. As an example, Figure 6 labels the slope and y-intercept of the 16-ply composite.

#### 4.4. Generalization of Non-Fickian Parameters

_{F}. For ϕ, it is already in the normalized form (ϕ = M

_{m,F}/M

_{m}). Both α and t

_{o}were normalized with respect to the normalized thickness values, where h’ = 2. The value of h’ = 1 corresponds to the Fickian behavior, so α and t

_{o}(which are non-Fickian parameters) do not exist.

_{o}’ (Figure 8c), it was noticed that the previous fitted relationship as described in reference [31] does not fit the value at h’ = 16. Therefore, the data were refitted using the power law to give a better fit. The best-fit parameters for all three non-Fickian parameters are shown in Figure 8. The regression fit, R

^{2}, of at least 0.92 suggests a good fit for all three parameters’ plots. Furthermore, from Figure 8c, it is apparent that the upper limit of t

_{o}has not yet been determined. More works are needed to determine the upper boundary of this parameter.

#### 4.5. Characterization of the Distribution Model for Non-Fickian Parameters

_{o}) obtained from the experimental data by extracting the statistical features using the probability distribution function (PDF) and cumulative distribution function (CDF). Therefore, the Weibull distribution quantifies and control the uncertainties of the non-Fickian parameters in moisture absorption behavior based on the materials’ normalized thickness. From the non-Fickian two-parameter Weibull probability distribution shown in Figure 9, the mean normalized values of ϕ = 0.53, α’ = 0.41, and t

_{o}’ = 11 were obtained (marked in Figure 8).

_{o’}, estimated at 0.82. This result suggests that the probability of non-Fickian moisture absorption is mainly influenced by ϕ and α at approximately 62% within a range of normalized thickness h’ = 2–3. Additionally, it is affected by the non-Fickian initiation time, t

_{o}, at around 82%, with a normalized thickness h’ = 6 (refer to Figure 8 for the corresponding normalized thickness).

## 5. Conclusions

_{z}= 4.25 × 10

^{−2}mm

^{2}/day and the single-ply maximum moisture content M

_{m,F}= 0.322%. By implementing the thickness-dependent moisture absorption model, the relationships between the non-Fickian parameters and normalized thickness were further generalized and extended to a larger normalized thickness (h’ = 16). Specifically, ϕ = 0.91h’

^{−0.68}, α’ = 3.85h’

^{−2.01}, and t

_{o}’ = 0.22h’

^{2.17}. Generalization of the non-Fickian parameters is beneficial for minimizing the amount of experimental work. Furthermore, Weibull distribution plots suggest that the probability of non-Fickian moisture absorption is mainly affected by ϕ and α at approximately 62% for a normalized thickness h’ = 2–3. Moreover, the non-Fickian initiation time t

_{o}results in approximately 82% absorption when h’ = 6. Further works are required to determine the upper limit of the thickness-dependent model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Experimental and fitted water absorption behavior of AS4/8552 carbon/epoxy composite laminates.

**Figure 8.**Variation of the non-Fickian parameters with the normalized thickness: (

**a**) Fickian to non-Fickian maximum moisture content ratio ϕ; (

**b**) normalized non-Fickian diffusivity per square thickness α’; and (

**c**) normalized non-Fickian initiation time t

_{o}’.

**Figure 9.**The probability distribution function of non-Fickian parameters for the normalized thickness: (

**a**) Fickian to non-Fickian maximum moisture content ratio ϕ; (

**b**) normalized non-Fickian diffusivity per square thickness α’; and (

**c**) normalized non-Fickian initiation time t

_{o}’.

**Figure 10.**The cumulative distribution function of non-Fickian parameters for the normalized thickness: (

**a**) Fickian to non-Fickian maximum moisture content ratio ϕ; (

**b**) normalized non-Fickian diffusivity per square thickness α’; and (

**c**) normalized non-Fickian initiation time t

_{o}’.

No. of Ply | M_{m,F} | Slope, (M _{2} − M_{1})/(√t_{2}/h − √t_{1}/h) | R^{2} |
---|---|---|---|

4 | 0.319 | 0.1356 | 0.9843 |

8 | 0.297 | 0.1486 | 0.9957 |

16 | 0.349 | 0.1646 | 0.9801 |

Average | 0.322 | 0.1496 | |

S.D | 0.0261 | 0.0145 | - |

C.V (%) | 8.11 | 9.71 | - |

h (mm) | 0.6 | 1.2 | 2.4 | |
---|---|---|---|---|

Parameter | ||||

ϕ | 0.29 | 0.23 | 0.14 | |

α (10^{−3} day^{−1}) | 19.3 | 20.1 | 4.2 | |

t_{o} (hours) | 24 | 71 | 269 |

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**MDPI and ACS Style**

Azizan, A.; Johar, M.; Karam Singh, S.S.; Abdullah, S.; Koloor, S.S.R.; Petrů, M.; Wong, K.J.; Tamin, M.N. An Extended Thickness-Dependent Moisture Absorption Model for Unidirectional Carbon/Epoxy Composites. *Polymers* **2021**, *13*, 440.
https://doi.org/10.3390/polym13030440

**AMA Style**

Azizan A, Johar M, Karam Singh SS, Abdullah S, Koloor SSR, Petrů M, Wong KJ, Tamin MN. An Extended Thickness-Dependent Moisture Absorption Model for Unidirectional Carbon/Epoxy Composites. *Polymers*. 2021; 13(3):440.
https://doi.org/10.3390/polym13030440

**Chicago/Turabian Style**

Azizan, Azisyahirah, Mahzan Johar, Salvinder Singh Karam Singh, Shahrum Abdullah, Seyed Saeid Rahimian Koloor, Michal Petrů, King Jye Wong, and Mohd Nasir Tamin. 2021. "An Extended Thickness-Dependent Moisture Absorption Model for Unidirectional Carbon/Epoxy Composites" *Polymers* 13, no. 3: 440.
https://doi.org/10.3390/polym13030440