Localització

Both the Fickian and hindered diffusion models contain a model specific set of material properties generally referred to as “absorption parameters” or “diffusion parameters” which are needed for the characterization of liquid absorption in a particular polymer or composite. The one-dimensional Fickian diffusion model uses only two distinctive absorption parameters: through-the-thickness diffusion coefficient, 𝐷, and the maximum amount of penetrant that the material can absorb, 𝑀_∞, often given in terms of weight percentage. In the event that edge effects and/or anisotropy are present, different directions might have different diffusion coefficients, usually referred to as: through-the- thickness diffusion coefficient, 𝐷_𝑧, and planar diffusion coefficients, 𝐷_𝑥 and 𝐷_𝑦 in the direction of 𝑥 and 𝑦 , respectively. When the diffusion is non-Fickian, the hindered diffusion model introduces two additional parameters to the ones used in the Fickian model: the rate of bound molecules becoming mobile per unit time, 𝛽, and the rate of mobile molecule becoming bound per unit time, 𝛾. Bound molecule means that the liquid penetrant is chemically interacts and binds the medium, whereas in the case of unbound molecule penetrant is mobile and free to diffuse into the medium.

The most accurate and versatile method available to recover the absorption parameters is the one proposed by Aktas et al. [61]. The absorption parameters of a material can be recovered simultaneously from experimental moisture absorption data by using a gradient optimization method to match the data with either the exact analytical or the approximate solution. The error to be minimized in the matching algorithm is:

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𝐸(𝑡) = ∑[𝑀𝑗(𝑡) − 𝑀𝑒𝑥𝑝,𝑗(𝑡)]2 𝑛

𝑗

(39)

where 𝑀(𝑡) and 𝑀_𝑒𝑥𝑝(𝑡) are theoretically predicted and experimentally measured mass gains, respectively. The error function, 𝐸(𝑡), is calculated via the summation of the errors from each one of the 𝑛 data points on the thermogravimetric curve.

A modified version of the steepest descent method [61] is used to iteratively find the set of absorption parameters that minimize the function 𝐸(𝑡). The gradient vector in the steepest descent optimization method is always towards the local maximum. Therefore, the model parameters are adjusted in the opposite direction of the gradient vector at each 𝑘𝑡ℎ _{iteration in order to find the minimum as given in Eqs. (40-41).}

[𝑈𝑖]𝑘+1= [𝑈𝑖]𝑘− [𝜌𝑖∇𝐸̅. 𝑈𝑖]𝑘 (40) ∇𝐸̅ = 𝜕𝐸 𝜕𝑈𝑖 √∑ (_𝜕𝑈𝜕𝐸 𝑖) 2 𝑛 𝑖=1 (41)

where 𝜌_𝑖 is a vector of parameters chosen to accelerate the minimization of 𝐸(𝑡), 𝑘 is the number of iterations, 𝑈𝑖 is the absorption parameter to be recovered (e.g., 𝐷, 𝛽, 𝛾, and 𝑀∞ for Langmuir diffusion model), and 𝑛 is the number of absorption parameters (e.g. 𝑛=2 for Fickian and 𝑛=4 for Langmuir diffusion model).

To start the algorithm, initial values should be assigned to unknown absorption parameters. Selection of the initial guess should be performed carefully, as the rate of convergence is highly sensitive to these values. Consequently, absorption parameters of a similar material systems could be useful in this selection. A recent approach developed by Guloglu et al. [1] is to use the parameters obtained from the approximate solutions as

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the initial guess. This novel approach could serve as excellent initial values and likely reduce the number of iterations needed, thus leading to the rapid recovery of parameters as demonstrated later in this section.

The minimization algorithm recalculates each of the absorption parameters based on the normalized gradient of the previous iteration. Using this algorithm, the set of absorption parameters that minimizes the function 𝐸(𝑡) can be found after the specified error convergence is achieved. Minimizing 𝐸(𝑡) results in the best possible match between the model prediction and each data point, and thus generates the best possible set of absorption parameters that represent the thermogravimetric experimental data.

The main advantage of this optimization method is that it minimizes 𝐸(𝑡) using all parameters simultaneously by changing their values at each iteration according to their individual effect on the error function 𝐸(𝑡) . An advantage of the steepest descent optimization method is that when the function is differentiable and convex, this method is convergent. However, the number of iterations required and the convergence rate highly depend on the initial values supplied by the user.

i. Absorption parameters to be recovered

 Classical Fickian Model

[𝑈𝑖] = [_𝑀𝐷 ∞]

 Langmuir-type Diffusion Model (1D Hindered Diffusion Model)

[𝑈𝑖] = [ 𝐷 𝛾 𝛽 𝑀_∞ ]

54  3D Fickian Diffusion Model

[𝑈𝑖] = [ 𝐷_𝑥 𝐷𝑦 𝐷_𝑧 𝑀∞ ]

 3D Hindered Diffusion Model

[𝑈𝑖] = [ 𝐷_𝑥 𝐷_𝑦 𝐷𝑧 𝛾 𝛽 𝑀_∞]

To illustrate the recovery mechanism, a representative thermo-gravimetric data is generated. No approximate solution is used to get the initial guesses. In order to demonstrate the predictive capabilities of modified steepest descent method, initial guesses are deliberately chosen not to represent the generated moisture mass gain curve. Table 2 presents the one-dimensional hindered diffusion model absorption parameters used to generate the synthetic data, initialing parameters and final recovered parameters.

Table 2. Diffusion parameters of the synthetic data, initial guess, and model prediction

Synthetic Data Initial Guess Recovered Through-the-thickness

Diffusion Coefficient, 𝐷_𝑧 7.00 × 10

-4 _{1.00 × 10}-4 _{6.86 × 10}-4 BoundUnbound, 𝛽 1.00 × 10-3 _{4.00 × 10}-3 _{9.72 × 10}-4 UnboundBound, 𝛾 4.00 × 10-4 _{1.00 × 10}-4 _{3.82 × 10}-4

Maximum Moisture Content, 𝑀∞ 2.00 3.00 2.00

The recovery starts with the initial guesses given in Table 2 and continues until the least square error between the model prediction and generated data is minimized. To show how the recovery method minimizes the error, synthetic mass gain data, initial guess, 100th, 250th, 500th, and 750th iteration predictions, and final prediction is illustrated in

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Figure 14. The initial guess has completely different absorption kinetics than the generated data. Figure 14 shows that the model prediction quickly approaches to the synthetic data and slows down to fine-tune the absorption parameters.

Figure 14. Absorption parameter recovery from thermo-gravimetric synthetic data.

Figure 15 illustrates how the least square error per data point changes during the recovery mechanism. The error between the model prediction and the data quickly decreases. Once the model prediction and the mass gain data start to overlap, the error starts oscillating. Even though the direction of the error gradient is towards to a local maximum, high local error gradient of an absorption parameter may lead to overshooting the absorption parameter in the next iteration. Thus, once the error is low, it starts oscillating towards to local minima. Moreover, two separate plateaus can be observed in Figure 15 around iteration number ~300 and ~800. These two plateaus are due to the error

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gradient domination. Right after the first plateau, error gradient of 𝑀∞ and 𝐷𝑧 becomes the prevailing force to minimize the error. In the beginning of the second plateau, the recovery method already found the maximum moisture content of the synthetic data. The error gradient of 𝑀_∞, _𝜕𝑀𝜕𝐸

∞, is no longer contributing to the recovery method as much as

before. After 800 iteration, error gradient of 𝛾, 𝛽, and 𝐷𝑧 drives the recovery. It should be noted here that dominance of the error gradient of a constituent depends on the initial guesses and chosen vector parameters.

Figure 15. Variation in the least square error per data point during absorption parameter recovery

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