**2. Generality on hydro/hygrothermal transfer**

#### **2.1 Terminology**

#### *2.1.1 Adsorption, absorption, and sorption*

Absorption is a phenomenon of filling the pores of a solid with liquid water; it occurs by capillary condensation. Whereas adsorption is a process of attachment of water molecules, water vapor, to the surface of materials by intermolecular forces. These forces are of the same nature as those responsible for the imperfection of real gases and the condensation of vapors [1]. We, therefore, talk about the absorption of liquid water (hydrothermal aging) and the adsorption of water vapor (hygrothermal aging). As for sorption, it encompasses both phenomena.

#### *2.1.2 Relationship between relative humidity and water activity*

Under isothermal conditions, the water activity (*aw*) in a material is defined by the ratio of the water vapor pressure at its surface (*pvs*) to the pure water vapor pressure (*pve*). When the material is in equilibrium with its environment or there is no mass or heat transfer, *pvs* equals the water vapor pressure of air (*pva*). Therefore, the water activity of a material in equilibrium with its environment is equal to the relative humidity of the air (*RHa*).

$$\mathfrak{a}\_{\textit{uv}} = HR\_{\textit{st}} = \frac{P\_{\textit{vv}}}{P\_{\textit{vv}}(T)} = \frac{P\_{\textit{vau}}}{P\_{\textit{vv}}(T)} \tag{1}$$

#### *2.1.3 Condensation capillaire*

This is the phenomenon where water vapor condenses into a liquid phase in a pore at an *aw* of less than 1, where *aw* = 1 represents the case of pure water.

#### *2.1.4 Sorption isotherms*

Under isothermal conditions, the curve representing the equilibrium moisture content (EMC) of a material as a function of *aw* is called the sorption isotherm if it was *Hydro/Hygrothermal Behavior of Plant Fibers and Its Influence on Bio-Composite Properties DOI: http://dx.doi.org/10.5772/intechopen.102580*

**Figure 1.** *IUPAC classification of physisorption isotherms (1) and hysteresis loops (2) [1].*

determined from the dry state of the material, and the desorption isotherm if the material was saturated at the initial state [2].

Isotherms are classified according to the new IUPAC classification [1] into six types (**Figure 1**(1)). The shape of the isotherm provides information about the pore size (macropores, mesopores, or micropores) and the types of adsorption (monolayer or multilayer). When the desorption process is different from the sorption process, a hysteresis loop is observed. These loops are also classified by IUPAC [1] into five categories (**Figure 1**(2)).

### *2.1.5 Pore size*

According to the IUPAC pore classification [1], macropores, mesopores, and micropores are, respectively, those pores whose width exceeds 50 nm, between 2 and 50 nm, and do not exceed 2 nm, respectively.

### *2.1.6 Aging*

Aging is defined according to Berges [3] as an evolution of one or more properties of the material through a modification of the structure, the composition, or the morphology of the constituents. These changes can be temporary and dependent on the presence of the aging source (reversible aging), or permanent (irreversible aging) [4]. Furthermore, the ability of a material to resist irreversible aging is defined by its durability.

#### **2.2 Sorption and diffusion of water molecules in polymers**

Generally, polymers are penetrable by water molecules. The latter propagate progressively through the polymer macromolecular network when it is in direct contact with water molecules (gaseous or liquid) [5]. Over time, this leads to a weight gain that continues until the material reaches a saturation plateau [5]. This adsorbed amount is related to the total amount of hydrophilic sites, polymeric areas sensitive to receive water molecules, available in the polymer chains [6]. This also means that the chemical potentials of water in the polymer and in the ambient environment are equal [7]. The kinetics of water sorption and the amount of water adsorbed/absorbed depends on the polymer nature (hydrophobic or hydrophilic), the characteristics of the water (pH, deionized or salted water), and other thermodynamic parameters [4].

The most common method for evaluating water sorption processes in polymers is the recording of mass gain versus time data. The gravimetric curve represents the plot of this data; the mass of water absorbed/adsorbed (*Mt*) versus time (LF curve in **Figure 2**). This curve contains important information: the linear part of this curve informs about the penetration rate of water (the diffusion), while the saturation level presents the mass adsorbed/absorbed at infinity (*M*∞) [7].

The diffusion behavior of polymers is often based on the Fick model. Nevertheless, some materials may exhibit non-Fick behavior in the presence of anomalies (curves A B, C, and D in **Figure 2**). The diffusion behavior can be identified mathematically using Eq. (2) [9]. If the parameter n is close to 0.5, this indicates that the diffusion of water, in this case, follows Fick's law.

$$\frac{M\_{\rm f}}{M\_{\rm os}} = kt^n \tag{2}$$

Where *k*, *n* represents the diffusion kinetic parameters.

In the case of bio-composites, we can see in the literature that the gravimetric curve of this kind of material shows a Fickian behavior at the macroscopic scale, see Section 4. However, at the microscopic scale, the problem is more complicated, water molecules can penetrate bio-composites by three different mechanisms: between polymer chains (matrix and fiber), by capillary action in micro-voids, and at the interface level (between fibers and the matrix) when chemical adhesion is absent.

#### **Figure 2.**

*Typical gravimetric absorption curves of a fluid: (LF) Fick behavior, (A) pseudo-Fick behavior, (B) 2-step diffusion, (C) diffusion with mechanical damage, and (D) diffusion with chemical damage [8].*

*Hydro/Hygrothermal Behavior of Plant Fibers and Its Influence on Bio-Composite Properties DOI: http://dx.doi.org/10.5772/intechopen.102580*

#### **2.3 Fick's mechanism**

Fick's laws were established by the analogy between conductive heat transfer and mass transfer [10]. In an isotropic case, Fick's second law is given by the equation below:

$$\frac{\partial \mathbf{C}}{\partial t} = D \left( \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{C}}{\partial \mathbf{y}^2} + \frac{\partial^2 \mathbf{C}}{\partial \mathbf{z}^2} \right) \tag{3}$$

Where: *D* is the diffusivity coefficient mm<sup>2</sup> /s, it is a scalar that defines the diffusion kinetics. *C* is the equilibrium water concentration.

The above equation could also be written in a case of radial diffusion in a solid cylinder of radius *r* as follows:

$$\frac{\partial \mathbf{C}}{\partial t} = D \left( \frac{\mathbf{1}}{r} \frac{\partial \mathbf{C}}{\partial r} + \frac{\partial^2 \mathbf{C}}{\partial r^2} \right) \tag{4}$$

In the case where a uniform concentration is imposed on the borders, the simplified solution for long times (0 < *Mt=M*<sup>∞</sup> <0*:*5) and short times (0*:*5< *Mt=M*<sup>∞</sup> <1) of the above relationship becomes, respectively, as follows [11]:

$$\frac{M\_t}{M\_\infty} = \frac{4}{\sqrt{\pi}} \boldsymbol{\tau}^{1/2} \left[ 1 - \frac{\sqrt{\pi}}{4} \boldsymbol{\tau}^{\frac{1}{2}} - \frac{\pi}{12} \dots \right] \tag{5}$$

$$\frac{M\_l}{M\_\infty} = 1 - \sum\_{n=1}^{\infty} \frac{4}{\infty\_n^2} \exp\left(-a\_n^2 \tau\right) \tag{6}$$

With *<sup>τ</sup>* <sup>¼</sup> *Dt=r*<sup>2</sup> and *<sup>α</sup><sup>n</sup>* the solutions of the Bessel equation of order *<sup>n</sup>*.

Equations (5) and (6) can be decomposed even more precisely according to the *Mt*/*M*<sup>∞</sup> intervals as follows [12]:

0< *Mt=M*<sup>∞</sup> <0*:*2 :

$$\frac{M\_l}{M\_\infty} = \frac{4}{\sqrt{\pi}} \pi^{1/2} \tag{7}$$

0*:*2< *Mt=M*<sup>∞</sup> < 0*:*5 :

$$\frac{M\_t}{M\_{\infty}} = \frac{4}{\sqrt{\pi}} \boldsymbol{\tau}^{1/2} \left[ \mathbf{1} - \frac{\sqrt{\pi}}{4} \boldsymbol{\tau}^{\frac{1}{2}} - \frac{\pi}{12} \dots \right] \tag{8}$$

0*:*5< *Mt=M*<sup>∞</sup> <0*:*7 :

$$\frac{M\_t}{M\_\infty} = 1 - 4\left(\frac{\exp\exp\left(-\infty\_1^2 \tau\right)}{\infty\_1^2} + \frac{\exp\exp\left(-\infty\_2^2 \tau\right)}{\infty\_2^2}\right) \tag{9}$$

0*:*7 < *Mt=M*<sup>∞</sup> < 1 :

$$\frac{M\_t}{M\_\infty} = 1 - 4\left(\frac{\exp\exp\left(-\alpha\_1^2 \tau\right)}{\alpha\_1^2}\right) \tag{10}$$

With *α*<sup>1</sup> ¼ 2*:*40483*etα<sup>n</sup>* ¼ 5*:*52008.

For the case of a plane plate of thickness *h* with a uniform initial concentration on both surfaces, the approximate analytical solution of Fick's law (Eq. (3)) for unidirectional diffusion along the thickness is expressed as follows [12]:

For the first half-absorption (*Mt=M*<sup>∞</sup> <0*:*5):

$$\frac{M\_t}{M\_\infty} = \frac{4}{h} \sqrt{\frac{Dt}{\pi}}\tag{11}$$

For the second half-absorption (*Mt=M*<sup>∞</sup> ≥0*:*5):

$$\ln\left(1-\frac{M\_t}{M\_\infty}\right) = \ln\frac{8}{\pi^2} - \frac{\pi^2 Dt}{h^2} \tag{12}$$

This is often used to illustrate the transfer of water vapor in continuous media and has been adopted by several authors to simulate the transfer of liquid water in materials assumed to be continuous at the macroscopic scale such as bio-composites and plant fibers, see Sections 3 and 4.
