Fast recovery of non-Fickian moisture absorption parameters for polymers and polymer composites.
Polymers and polymer composites possess numerous attractive attributes, including a high strength-to-weight ratio, corrosion resistance, and ease of processing along with a reasonable lifetime cost [1, 2], These features have led to the widespread use of polymeric materials in large-scale structural components which are often exposed to humid and wet environments. The detrimental effects of absorbed moisture have been well-recognized and studied extensively during the last several decades [3, 4], For instance, Perez-Pacheco et al.  reported that a moisture content as low as 0.75 wt% results in reduction of the tensile strength, elastic modulus, and interfacial shear strength of a carbon/epoxy composite by 25%, 38%, and 11%, respectively. Other researchers also observed property loss in glass transition temperature , tensile properties [7, 8], and interlaminar shear strength and flexural properties [9-11] for different polymeric materials.
Environmental conditions experienced by these materials during their service life often consist of large variations of humidity and temperature over long periods of time. The deleterious effects of moisture on the mechanical behavior and durability of polymers and polymer composites underscore the need to accurately predict absorption behavior and moisture content, especially for structural load-bearing applications designed for long service life. The one-dimensional Fickian diffusion model is most often used for predicting absorption behavior and moisture content due to its simplicity and mathematical tractability . Although a primarily Fickian diffusion is often observed during the initial moisture uptake, most polymeric materials, in the long-term, exhibit a non-Fickian absorption behavior [13-20]. This non-Fickian behavior, also referred to as anomalous or hindered diffusion, is frequently observed after a pseudo equilibrium, where the initial high slope of moisture uptake is substantially reduced. Moreover, polymers may exhibit Fickian and non-Fickian behaviors at different temperatures, and/or when they are exposed to different hygrothermal history . It is worth noting here that the common practice for absorption characterization is to adhere to the ASTM standard, which is based on the Fickian diffusion model. However, a priori assumption of Fickian behavior may lead to: (a) premature termination of the thermogravimetric absorption experiments; (b) missing the non-Fickian behavior, which often becomes apparent in the long-term absorption experiments; and (c) inaccurate estimates of the maximum moisture absorption, critical in determining thermomechanical property reductions.
Recognizing the limitations of Fickian theory, several researchers have proposed different models for moisture absorption, to account for the anomalous, or hindered, diffusion behavior. These models include time-varying diffusion coefficient models ; two-phase polymer dual-diffusivity models, also known as Jacobs-Jones models [14, 15]; coupled diffusion-relaxation models [16, 17]; as well as dual-mode sorption models, referred to as the Langmuir-type or hindered diffusion models [18-20], The characterization of the absorption behavior of a specific polymeric material using any one of the models mentioned earlier would require experimental identification of a set of material properties, generally referred to as diffusion or absorption parameters. In a comparative study, Glaskova et al.  investigated the effectiveness of these models in representing the non-Fickian behavior of an epoxy system. The authors reported the Langmuir model to be particularly effective. Other researchers [19, 20, 22-32] also found the one-dimensional hindered diffusion model (HDM) to better describe the anomalous deviation from a Fickian behavior, and have demonstrated the ability to successfully predict both short- and long-term moisture absorption in polymeric materials. A 5-year validation study showed that the HDM parameters recovered from 2-year long absorption experiments allowed accurate prediction of 5-year absorption behavior of 6-, 12-, and 40-ply, quartz/BMI composites .
The hindered diffusion model builds on the Langmuir framework and is based on the assumption that moisture absorption in polymeric materials is partially related to the availability of molecular-sized interstice in the polymer structure and the polymer-water affinity. The availability of interstice, or free volume, within the polymer depends on its microstructural morphology and crosslink density . Polymer-water affinity, on the other hand, is related to the availability of certain hydrophilic functional groups such as hydroxyl or amine within polymer chains. Part of the absorbed water molecules would be inclined to strongly couple with polar groups or hydrogen bonding sites within the polymer. These molecules would cease to be a part of the ongoing diffusion and are often considered as bound, or immobile. In contrast, water molecules that are free to move through the interstice are often called unbound, or mobile. Other evidence supporting this approach includes measurements of bound and unbound water molecules in epoxy resin and epoxy composites [22, 24, 25, 28, 34-36] using nuclear magnetic resonance and Fourier transform infrared spectroscopy.
In thermogravimetric experiments of polymeric materials, the faster, Fickian-dominated initial moisture uptake is often followed by a slower absorption rate, where the diffusion is hindered by the increase in bound water molecules. The time needed to reach moisture equilibrium can be considerably delayed and depends on the exchange rates between the bound and unbound water. The difference in these rates can describe a vast range of absorption behavior in two-time scales and can be accurately captured by the hindered diffusion model.
Carter and Kibler  developed an exact analytical solution for the one-dimensional hindered diffusion model (1D HDM). They also proposed an analytical approximate solution and two "useful" approximate solutions for two distinct regions of the moisture uptake curve: (a) for the initial absorption region and (b) for the long-term absorption region. Because of their relative simplicity compared to the exact solution, the majority of researchers have used these "useful" approximate solutions to recover the absorption parameters [23, 26, 27, 29-31, 37-40]. However, the approximate solutions of the 1D HDM are valid only under certain conditions. In addition, the procedure suggested by Carter and Kibler  to recover the absorption parameters, explicitly summarized by other authors , can be inaccurate and lead to incorrect characterization of the absorption. For example, if the final thermogravimetric data point (i.e., last weight gain data) is used as the maximum moisture intake as suggested , any additional moisture absorption cannot be accounted for. Consequently, premature and ad hoc termination of absorption experiments will be detrimental to the accurate characterization of all absorption parameters, including erroneously reporting a much lower value for the maximum moisture absorption. Instead, the maximum moisture absorption should be left as an unknown parameter and obtained from the theoretical model parameters that yield the best match to the experimental data .
It needs to be emphasized that very few researchers have used the exact solution of the HDM--or any other Fickian and non-Fickian model--to characterize the absorption behavior [24, 41]. This might be due to the mathematical difficulty and considerable computational effort involved in the recovery of absorption parameters. Using the exact solution would also require solving the classical inverse problem by developing a search algorithm to determine the set of absorption parameters which generate the absorption curve that matches the collected experimental thermogravimetric data with the minimum error. Searching for this set of absorption parameters may require a large number of iterations, depending on the initial guess and the convergence rate of the search algorithm. Within this framework, there is a need for a robust and faster computational method for the recovery of the absorption parameters using the exact analytical solutions of the absorption models.
In this article, we present a number of example cases to illustrate the proper use of both the approximate and exact solutions of the 1D HDM. A parameter recovery technique, used to simultaneously obtain absorption parameters, is also presented. This method is then used to recover the absorption parameters from experimental thermogravimetric data collected from two material systems: (a) nanoclay/epoxy composites immersed in distilled water and (b) published data for thin epoxy laminates immersed in salt water . The absorption behavior predicted by the recovered parameters is then validated by experimental data not used in the recovery process. In addition, a novel method is proposed to computationally accelerate the recovery of the absorption parameters. The new technique uses the approximate absorption parameters as the initial guess, which can be determined without any iteration. It is shown that using these approximate parameters substantially reduces the computational effort by decreasing the number of iterations without compromising from accuracy.
In both the Fickian and hindered diffusion models, a set of material parameters characterize the penetrant absorption. This set reduces to only two distinctive absorption parameters for the one-dimensional Fickian diffusion model: through-the-thickness diffusion coefficient, D, and the maximum amount of penetrant that material can absorb, [M.sub.[infinity]], often given in terms of weight percentage. In non-Fickian processes, the anomalous effects are captured by the hindered diffusion model via two additional parameters: the rate of bound molecules becoming mobile, [beta], and the rate of mobile molecules becoming bound, [gamma]. Bound molecules are the penetrant molecules that chemically interact and bind to the medium. Unbound penetrant molecules, on the other hand, are mobile and free to diffuse into the medium. The Langmuir model, proposed by Carter and Kibler , essentially modifies the Fickian diffusion kinetics by accounting for the hindrance introduced by chemical interactions between the penetrant and the medium. The governing set of differential equations of the one-dimensional hindered diffusion is written as:
D[[[partial derivative].sup.2]n/[partial derivative][x.sup.2]] = [[partial derivative]n/[partial derivative]t] + [[partial derivative]N/[partial derivative]t]. (1a)
[partial derivative]N/[partial derivative]t = [gamma]n - [beta]N, (1b)
where n represents the unbound molecules per unit volume (equivalent to the Fickian moisture concentration), N represents the bound molecules per unit volume, [gamma] is the rate at which unbound molecules become bound, and [beta] is the rate of bound molecules becoming unbound.
The hindered diffusion model expresses the total concentration as the sum of two terms: bound moisture concentration N (x, t) and unbound moisture concentration n(x, t), proportional to material parameters [gamma] and [beta] at equilibrium. In the hindered diffusion model, if the rate of unbound molecules becoming bound, [gamma], is equal to zero, the concentration of bound molecules, N(x, t), goes to zero, and thus, n(x, t), the concentration of unbound molecules, becomes equivalent to the Fickian moisture concentration. In other words, the hindered diffusion simplifies to Fickian behavior if there are no bound molecules. The ability to simplify to Fickian behavior is an additional advantage of the hindered diffusion model, where a priori knowledge of the absorption behavior is not needed. The moisture absorption continues until no more moisture molecules diffuse into the polymer, in other words until the unbound moisture, N(x, t), no longer changes with time. The equation describing this equilibrium condition can be written as:
[N.sub.[infinity]]/[n.sub.[infinity]] = [gamma]/[beta], (2)
where [N.sub.[infinity]] is the equilibrium bound moisture contents, and [n.sub.[infinity]] is the equilibrium unbound moisture content. Equation 2 indicates that the ratio of the bound to unbound equilibrium moisture becomes constant throughout the sample, regardless of the location. In other words, Eq. 2 suggests that when the moisture equilibrium is reached, there is a generalized microstructural ratio between the bound and unbound molecules at any location within the material.
When the sum of the bound moisture concentration N(x, t) and the unbound moisture concentration n(x, t) is integrated over the material thickness, the resulting exact analytical solution gives the mass fraction of the absorbed moisture , and is expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)
M(t) and [M.sub.[infinity]] are the mass percent of absorbed moisture at time t and at equilibrium, respectively. The one-dimensional hindered diffusion model has an analytical approximation for cases where 2[beta] and 2[gamma] are very small when compared to K :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)
Equation 5 may be further simplified into "useful" approximations both for very short or very long times .
M(t) = [4/[[pi].sup.3/2]]([beta]/[beta] + [gamma])[M.sub.[infinity]][square root of (Kt)], 2[gamma], 2[beta] [much less than], t < [0.7/K]. (6)
M(t) = [M.sub.[infinity]][1 - ([gamma]/[beta] + [gamma])exp[-[beta]t]], 2[gamma], 2[beta] [much less than] K, t [much greater than] 1/K. (7)
When the required conditions for Eqs. 5-7 are satisfied, these approximations can been used to recover the absorption parameters faster and without any iterative process. However, the accuracy of parameters determined from these approximations may change depending on the extent of experimental data available. In addition, these approximations are only valid when 2[beta] and 2[gamma] are much smaller than K=([[pi].sup.2] D)/[h.sup.2]. The validity of these assumptions is not known a priori, and thus, indiscriminate use of these approximations without any justification may lead to erroneous results. Therefore, using the exact analytical solution given in Eq. 3 might be essential for the identification of absorption parameters as the accuracy of approximate solutions is not known.
PROPER USE OF EXACT AND APPROXIMATE MODEL SOLUTIONS TO RECOVER ABSORPTION PARAMETERS
Despite its prevalence in literature, there has been little discussion on how exactly one should implement the model developed by Carter and Kibler  to accurately recover absorption parameters (i.e., [M.sub.[infinity]], D, [gamma], and [beta]). A number of researchers have used the approximate solutions of the 1D HDM to recover the absorption parameters without a priori error estimate [23, 26, 27, 29-31, 39, 40]. This approach, recently summarized by Joliff et al. , gained popularity due to: (a) its relative simplicity; (b) potential to characterize the anomalous absorption behavior; and (c) convenience compared to using the exact analytical solution. The first step of this method involves determining the equilibrium moisture, [M.sub.[infinity]], from the experimental data, which is usually taken as the last data point of the weight gain curve. Next, pseudo-equilibrium moisture level, [M.sub.1], is identified by visual inspection of the thermogravimetric curve. The ratio of [M.sub.1] to [M.sub.[infinity]] can be used in Eq. 5 to calculate the approximate value of [beta]/(y + [beta]), which enables the calculation of [beta] from long-time solution given in Eq. 7. The last two unknown parameters, [gamma] and D, are then obtained from [beta]/(y + [beta]) and short-time solution given in Eq. 6, respectively. Although the absorption parameters are recovered quickly and without the use of an iterative method, the values of the recovered parameters are highly sensitive to the last thermogravimetric data point recorded. The ad hoc selection of the last data as [M.sub.[infinity]] coupled with the approximations inherent in the recovery method may lead to substantial errors in the absorption parameters.
Considering the general practice of using the last weight gain data as [M.sub.[infinity]], the criterion used in terminating the absorption experiment is critical. For example, the ASTM D 5229 standard for equilibrium conditioning of polymeric materials stipulates that the effective moisture equilibrium is reached when the average moisture content changes by less than 0.01% within the span of the reference period. This approach might be satisfactory for engineering applications where the focus is not on the accurate characterization of absorption dynamics. However, it may not be suitable for the identification of absorption parameters, which are essentia] for any predictive computational simulation of absorption behavior. Particularly, in the case of anomalous, non-Fickian behavior, adopting ASTM D 5229 may lead to premature termination of the thermogravimetric experiments [13, 32]. In such cases, moisture absorption often decelerates after the initial intake, reaching a pseudo-equilibrium where the condition set by ASTM D 5229 standard might be satisfied. However, absorption continues at a lower rate, and in most non-Fickian cases, the equilibrium moisture content, is reached at a much higher level. In fact, numerous moisture absorption tests have been terminated on an ad hoc basis and, beyond shortening the duration, one may not find any sound motive for stopping these thermogravimetric experiments early, before the equilibrium is reached.
Realizing the importance of obtaining accurately, a few studies [26, 28, 30] considered Mx and other absorption parameters as unknown to be determined from the approximate solutions for longer and shorter times, given in Eqs. 6 and 7, respectively. This procedure uses rudimentary curve fitting methods and determines unknown parameters iteratively in Eqs. 6 and 7. Laplante et al. , on the other hand, used the approximate solution given in Eq. 5 to simultaneously recover the absorption parameters of different epoxy systems. In Ref. 25, the Fickian contribution in Eq. 5 is further approximated to avoid the summation required in calculating the absorption level. More recently, Grace and Altan [19, 20] proposed an approximate solution for the three-dimensional, anisotropic HDM that would simplify to an expression mathematically equivalent to Eq. 5 for one-dimensional cases. Their method simultaneously recovers all absorption parameters using the approximate solution.
Compared to approximate methods, one would expect the exact analytical solution of the one-dimensional hindered diffusion model to yield more accurate characterization of the moisture absorption behavior. Nevertheless, the exact solution has been rarely used to describe the absorption dynamics [24, 41]. For instance, Aronhime et al.  seem to have used the exact solution of the 1D HDM to characterize the absorption behavior of an epoxy resin at multiple relative humidity levels. However, the authors estimated the equilibrium moisture content directly from the last experimental data point.
Comparing the efficiency and accuracy of methods that use approximate and exact solutions would be highly beneficial to the research community. Of particular interest is the quantification of the error induced by the approximate solutions. Yet, to the best of the authors' knowledge, a comparative assessment of these methods has not been available.
RECOVERY OF ABSORPTION PARAMETERS BY MATCHING THEORETICAL MODEL TO EXPERIMENTAL DATA
The most accurate and versatile method available to recover the absorption parameters is the one proposed by Aktas et al. . This technique, outlined in detail below, is used later in this article to compare the accuracy and efficiency of both the exact and approximate solutions.
The absorption parameters of a material can be recovered simultaneously from experimental moisture absorption data 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 given as:
E = [nn.summation over (j)][[[M.sub.j](t) - [M.sub.exp,j](t)].sup.2], (8)
where M(t) and [M.sub.exp](t) are theoretically predicted and experimentally measured mass gains, respectively. The error function. E, 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  is used to iteratively find the set of absorption parameters (i.e., D, [beta], [gamma], and [M.sub.[infinity]]) that minimize the function E. 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 to find the minimum as given in Eqs. 9 and 10.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)
where [[rho].sub.i] is a vector of parameters chosen to accelerate the minimization of E, and k is the number of iterations.
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 system could be useful in this selection. Another approach, which has not been explored before, is to use the parameters obtained from the approximate solutions as the initial guess. This novel approach could serve as an excellent initial values and likely reduce the number of iterations needed, thus leading to the rapid recovery of parameters as demonstrated later in this article.
The minimization algorithm recalculates each of the absorption parameters based on the gradient of the previous iteration, normalized by 5. Using this algorithm, the set of absorption parameters that minimizes the function E can be found after the specified error convergence is achieved. Minimizing E 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.
In this algorithm, [[rho].sub.i] are chosen to accelerate the minimization process by acting as a multiplier to the gradient for a faster convergence. [M.sub.[infinity]] and D generally converge faster than the [gamma] and [beta], which often requires setting [[rho].sub.2] and [[rho].sub.3] to higher values than [[rho].sub.1] and [[rho].sub.4].
The main advantage of this optimization method is that it minimizes E using all parameters simultaneously by changing their values at each iteration according to their individual effect on the error function E. 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.
EXACT SOLUTION OF THE ONE-DIMENSIONAL HINDERED DIFFUSION MODEL: EXPERIMENTAL VALIDATION AND COMPARISON TO THE APPROXIMATE SOLUTION
To assess the accuracy and efficiency of both exact and approximate HDM solutions in recovering the absorption parameters, thermogravimetric data from the following two materials systems are used:
a. Nanoclay/epoxy composites with 0.5 and 3 wt% nanoclay immersed in water at 25[degrees]C.
b. Desiccated and undesiccated epoxy laminates immersed in water at 20[degrees]C and 60[degrees]C .
Experimental Validation for Nanoclay I Epoxy Composites
Commercially available DGEBA epoxy resin, EPON 862, and a non-MDA aromatic amine curing agent, Epikure W, from Momentive Performance Materials, Inc. are used in this study. Nanomer; I.30E nanoclay, a surface modified montmorillonite mineral that contains 25-30 wt% of trimethyl stearyl ammonium, is used as the reinforcing material. The nanoclay is shear mixed with the epoxy resin at the desired nanoclay content of 0.5 and 3 wt% at 5,000 RPM for 30 min. Then, the epoxy-nanoclay mixture and curing agent is mixed at a ratio of 100:26.4 by weight at 106 RPM for 15 min. The mixture is then degassed for 4 h at room temperature to remove any trapped air. Subsequently, thin epoxy laminates are gravity cast in a mold, preheated to 100[degrees]C. The mold is kept at 100[degrees]C for 1.5 h and then heated to 179[degrees]C with a ramp rate of 1.1[degrees]C/min and held constant for 2 h. After 2 h, the part is cooled down to room temperature at approximately 2.8[degrees]C/min to complete the cure cycle. The resulting two laminates have an average thickness of 0.97 mm. Eight 31.8 x 31.8 mm-specimens are cut from each laminate, and are dried in a vacuum oven. The specimens are then immersed in a distilled water bath at 25[degrees]C. The thermogravimetric data obtained from the moisture conditioning experiments, shown in Figs. 1 and 2, are used to recover the absorption parameters of both the 0.5 and 3 wt% nanoclay/ epoxy composites.
Figures 1 and 2 also depict the moisture absorption behavior predicted by the recovered absorption parameters using both the exact and approximate solutions. The last thermogravimetric data point at [square root of (t)] [congruent to] 90 [square root of (h)] is shown only for validating the longer term predictive ability of either solution, and was not part of the data used in the recovery process. For both nanocomposites, the exact solution shows excellent agreement with experimental data. Data collected over 4 months enabled the accurate prediction of the moisture content after almost 11 months. The moisture contents predicted by the approximate solution, however, deviate from the experimental values after the initial linear region and are unable to accurately capture the longer term absorption level, measured around 11 months.
The two sets of recovered absorption parameters are presented in Table 1 for both nanocomposites. The parameters recovered using the approximate and exact solutions are considerably different. For instance, the maximum mass intake of the 3 wt% nanoclay composite shows a 8.63% difference between the 2.11% value predicted using the approximate solution and 2.17% obtained using the exact solution. Diffusivity shows less than 2% variation when obtained using the exact or approximate solutions. The remaining parameters, [gamma] and [beta], exhibit the highest variability. Both show a reduction in the range of 80%-90% when recovered using the exact solution compared to the approximate solution. These high disparities in approximate and exact absorption parameters indicate that although curves predicted by both solutions seem to partially overlap, the diffusion kinetics described are fairly different. Furthermore, the root mean square (RMS) errors induced per data point by the approximate or exact solutions are also presented in Table 1. Recovering the absorption parameters using the approximate solution induces almost three- and five-fold increases in the RMS error per data point compared to the exact solution for the 0.5 and 3 wt% nanocomposites, respectively.
Increasing nanoclay content is observed to reduce all absorption parameters. The diffusivity and maximum moisture content drop by 1.4% and 2.6%, respectively. The hindrance coefficients [beta] and [gamma] decrease by 42.5% and 38.8%, respectively. These differences indicate that a six-fold increase in nanoclay content produces important changes in the absorption kinetics. To better understand these changes, the temporal evolutions of the bound and unbound moisture contents for both nanocomposites are presented in Fig. 3. Increasing the nanoclay content from 0.5 to 3 wt% is observed to decrease the maximum content of unbound moisture, causing a similar decrease in the maximum total moisture. The unbound water molecules are known to exist in the molecular-sized interstice, or free volume of the epoxy [6, 15, 21. 34]. Therefore, the higher presence of nanoclay seems to distort the microstructure of the epoxy, reducing the free volume available for unbound, diffusing water. On the other hand, the maximum content of bound moisture overlaps for both nanoclay contents, which indicates the same polymer-water affinity. However, the formation of bound water molecules is slightly hindered at the higher nanoclay content of 3 wt%. This delayed behavior has been reported in earlier studies for composites containing nanoclay, carbon nanotubes, and other nanofillers [27, 43].
Experimental Validation for Thin Epoxy Laminates
To further quantify the accuracy of the approximate solution in recovering the absorption parameters of polymers, the experimental thermogravimetric data recently published by Scott and Lees for thin laminates of epoxy polymer are used . The effect of premature discontinuation of moisture absorption experiments on the accuracy of the recovered parameters is also assessed.
Scott and Lees  investigated the absorption process of salt water into thin laminates of epoxy. The authors reported that the studied epoxy was an EPR4434/EPH943. The EPR4434 resin consists of 25%-50% 1,6-hexanediol diglycidyl ether, 25%-50% methylenedianiline, and 25%-50%, bisphenol F diglycidyl ether (DGEBF). The EPH943 hardener was 100% isophoronediamine. The authors conducted moisture experiments for different conditions, referred to as: U20 for undesiccated samples with no preconditioning before the experiment, D20 and D60 for desiccated samples, dried before immersion in water at 20[degrees]C and 60[degrees]C, respectively. The sample dimensions and the experimental procedure used in Ref. 30 are presented in Table 2. Desiccated samples were initially immersed for 100 days and the undesiccated samples immersed for 500 days. However, the samples were not saturated at that time, as indicated by measurements of desiccated and undesiccated samples taken at 800 and 1,800 days, respectively.
To recover the absorption parameters from thermogravimetric data given in Ref. 30, initial estimates are chosen according to previous experiments for similar material systems [28. 44]. The absorption parameters for U20, D20, and D60 are recovered using the exact and approximate solutions of the 1D HDM. To demonstrate the effect of ad hoc termination of the thermogravimetric experiments, the absorption parameters are recovered with and without using the single data points collected at later times.
First, the approximate and exact absorption parameters are recovered without using the last data point (Table 3). Figure 4ac depicts the moisture absorption behavior predicted by the corresponding absorption parameters for U20, D20, and D60, respectively. As depicted in Fig. 4a-c, a large mismatch is observed between the predicted moisture content at 800 and 1,800 days and the actual measured moisture content. Hence, discontinuing the absorption experiment prematurely is found to dramatically undermine the ability of both the exact and approximate solutions to accurately characterize the long-term absorption behavior for all three epoxy laminates. However, the moisture absorption predicted by the exact solution shows a lower deviation from the last experimental data point than its approximate counterpart.
As presented in Table 3, the difference between the maximum moisture content predicted by the exact and approximate solution is in the range of 5%-6% for the desiccated samples and 0.5% for the undesiccated ones. The remaining absorption parameters show a larger variation between the exact and approximate solutions. For instance, the diffusivity and hindrance coefficients show a two-fold increase for the D20 samples. The increase in the same parameters for the undesiccated samples is observed to be between 6% and 12%, as the moisture absorption curves predicted by both models are practically overlapping (Fig. 4a). Furthermore, using the approximate solution significantly increases the RMS error per data point compared to the exact solution. For instance, the RMS error per data point are observed to be 134% and 38% higher using the approximate solution for D20 and D60 samples, respectively.
Second, the absorption parameters for the three sample sets are recovered with the last data point included in the dataset. Figure 5a-c depicts the moisture absorption behavior predicted by the corresponding exact and approximate absorption parameters. As shown in Fig. 5a, the exact solution is considerably more accurate in characterizing the absorption parameters for the undesiccated samples. The moisture predicted by approximate solution at 1,800 days deviates by almost 3%, while the exact solution recovers that data point with virtually no error. Conversely, both the exact and approximate solutions seem, at first, to characterize the D20 and D60 laminates with the same accuracy as shown in Fig. 5b and c. Nonetheless, the errors induced using each approach show increased RMS error in approximate solutions. As shown in Table 4, using the approximate solution induces 101%, 81%, and 19% higher RMS error per data point than the exact solution for U20, D20, and D60 samples, respectively.
Figure 6 illustrates the effect of experimental variables on the absorption behavior of all the epoxy samples. Substantial differences in the absorption behavior are observed between the U20, D20, and D60 samples. Analyzing the bound and unbound moisture contents can provide more insight into understanding how the absorption kinetics are altered by desiccation and temperature increase from 20[degrees]C to 60[degrees]C.
Figure 7 shows the separate contributions of bound and unbound moisture to the total moisture uptake for undesiccated (U20) and desiccated (D20) samples. Prior presence of moisture in the epoxy laminates is observed to cause a considerable delay in moisture absorption. The slope of unbound moisture absorption is initially similar for both sample sets, but the desiccated samples continue to absorb unbound molecules well after undesiccated specimens reach saturation for unbound molecules. This is reflected in the maximum unbound moisture content, which is 25% higher for desiccated (D20) samples. Mobile molecules can only lodge in the molecular-size interstice, or free volume, of the polymer [21, 28, 35, 36], and thus saturation of mobile molecules would occur when the free volume is literally full. In contrast, the bound moisture shows distinctly different initial slopes. For D20 samples, the bound moisture is observed to occur earlier and at a higher rate compared to the U20 set. Preexisting moisture is reported to slow subsequent moisture absorption in polymers . This delay is attributed mainly to the increased interaction between free water molecules and the polymer chains.
Similarly, Fig. 8 illustrates the separate contributions of the bound and unbound moisture to the total moisture uptake for the sample sets immersed in 20[degrees]C (D20) and 60[degrees]C (D60) salt water baths. As expected, increasing the conditioning temperature is observed to considerably alter the absorption behavior. First, the unbound moisture absorption slope is notably higher at 60[degrees]C, as predicted by the Arrhenius law [6, 7, 38, 45]. Increasing the bath temperature is observed also to slightly increase the maximum unbound moisture content. This 5% increase might be induced by additional free volume created during either thermal expansion or plasticization. Bound moisture absorption curves are observed to have practically identical initial slopes for both temperatures. However, increasing the bath temperature from 20[degrees]C to 60[degrees]C causes a two-fold increase in bound moisture maximum content, yielding a 35% increase in the maximum total moisture content. This phenomenon is often reported for polymeric materials [21, 35, 45], and is attributed to the higher energy available at higher temperatures for the water molecules to bond with polymer chains.
FAST COMPUTATIONAL RECOVERY OF ABSORPTION PARAMETERS USING THE APPROXIMATE PARAMETERS AS INITIAL GUESS
Although the approximate solution is found to be less accurate in characterizing the absorption behavior, it still has an important role to play in the recovery of absorption parameters. A fast recovery method that uses approximate solutions to reduce the computational effort needed for the exact solution may be feasible. This new technique uses the approximate solution first to obtain the approximate absorption parameters. These values, obtained quickly and without an iteration, are then used as the initial guess to start the iterative search algorithm using the exact solution.
Tables 5-7 show the number of iterations before convergence for all the cases studied in this article. In the normal recovery scheme, the literature-inferred values are used as initial guesses, while in the fast recovery method, the values from approximate solutions are used. It is worth mentioning here that the initial values used for normal recovery are carefully defined from published values in the literature for similar epoxy systems and nanoclay/epoxy composites. Hence, the iteration numbers presented for normal recovery are distinctively lower compared to using random values as initial guesses. Nonetheless, even with this judicial choice of initial values, the number of iterations before convergence is still fairly excessive.
As expected, using the approximate values of the absorption parameters as an initial guess helped substantially reduce the computational effort involved in the recovery algorithm. For instance (Table 5), the number of iterations drops by almost 50% for the 0.5 wt% nanocomposites, from 672 to 344. An even higher drop of 87% is attained for the 3 wt% nanocomposite samples, where the number of iteration went from 747 to 100. Significant reductions in the number of iterations are also achieved when characterizing moisture absorption behavior of the thin epoxy laminates with and without the last data point. As shown in Tables 6 and 7, more than 80% reduction is achieved in the number of iterations required to recover the absorption parameters for the U20, D20, and D60 sample sets. For instance, the number of iterations dropped from 4,765 to 893 for the D60 samples when recovering the absorption parameters using the last data point. Without this later reading, the number of iterations for the same D60 set goes from 1,973 to only 1 iteration, which indicates that, for this case, the approximate solution was just as good as the exact solution. Furthermore, this rapid recovery is achieved without impairing the accuracy of the characterization. As presented in Tables 5-7, the RMS error per data point for both the normal and fast recovery is in the same range. For instance, the difference in RMS per data point between the normal and fast recovery methods is 3.25% for the D60 absorption parameters. For the D20 thin laminates, on the other hand, the error recorded using the normal recovery is even 2.83% higher than using the fast recovery technique.
The 1D HDM, that is, Langmuir-type absorption, has been used to successfully capture both Fickian and anomalous diffusion processes in polymeric materials. In this article, proper use of both the approximate and exact solutions of the 1D HDM to characterize the moisture absorption behavior of nanoclay/epoxy laminates and thin epoxy laminates was demonstrated. A modified version of the steepest descent search algorithm was used to determine the absorption parameters that minimize the error between the experimental data and the exact solution. Hence, unlike approximate solutions, one can accurately recover all absorption parameters simultaneously from the experimental data. The absorption behavior predicted by the recovered parameters was then validated by the long-term experimental data not used in the recovery process. The errors induced by the approximate solution were found to be dependent on the materials system and could be significantly higher (up to five-fold) compared to the exact solution. In addition, terminating moisture absorption experiment prematurely was shown to significantly affect the accuracy of the recovery, particularly by underestimating the saturation moisture level. Furthermore, a novel method was introduced to computationally accelerate the recovery of the absorption parameters. This new fast recovery technique uses the absorption parameters obtained from approximate solution as the initial guess to start the search algorithm. Using approximate solution as the initial guess substantially reduced the computational effort required to match the exact solution to experimental data. Depending on the quality of the initial guess, the number of iterations was shown to be reduced by 50%100% without compromising from the accuracy.
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Gorkem E. Guloglu, (1) Youssef K. Hamidi, (1,2) M. Cengiz Altan (1)
(1) School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, Oklahoma
(2) Ecole Nationale Superieure Des Mines De Rabat, Rabat, Morocco
Correspondence to: Y. K. Hamidi; e-mail: firstname.lastname@example.org
Caption: FIG. 1. Comparison of the exact and approximate solutions of the hindered diffusion model for the 0.5 wt% nanoclay/epoxy composite.
Caption: FIG. 2. Comparison of the exact and approximate solutions of the hindered diffusion model for the 3 wt% nanoclay/epoxy composite.
Caption: FIG. 3. Effect of nanoclay content on moisture absorption kinetics.
Caption: FIG. 4. (a) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of U20 dataset without the data point, (b) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of D20 dataset without the last data point, (c) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of D60 dataset without the last data point.
Caption: FIG. 5. (a) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of the U20 whole dataset. (b) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of the D20 whole dataset. (c) Comparison of the exact and approximate solutions of the hindered diffusion model using the absorption parameters of the D60 whole dataset.
Caption: FIG. 6. Moisture absorption curves of the U20, D20. and D60 samples predicted using absorption parameters recovered with the exact solution of the one-dimensional hindered diffusion model. the increased interaction between free water molecules and the polymer chains.
Caption: FIG. 7. Effect of preconditioning on moisture absorption kinetics in thin polymer laminates.
Caption: FIG. 8. Effect of temperature on moisture absorption kinetics in thin polymer laminates.
TABLE 1. Absorption parameters of 0.5 and 3 wt% nanoclay/epoxy com- posite recovered using the exact and approximate solutions of the one- dimensional hindered diffusion model. Nanoclay content 0.5 wt% 3 wt% Parameters recovered by exact solution Diffusion coefficient, D (x[10.sup.-4] 7.57 7.47 [mm.sup.2]/h) Bound [right arrow] Unbound, [beta] (x[10.sup.-3] 14.67 8.43 [h.sup.-1]) Unbound [right arrow] Bound, [gamma] 2.18 1.33 (x[10.sup.-3] [h.sup.-1]) Maximum moisture, [M.sup.[infinity]] (wt%) 2.23 2.17 RMS error/data point (x[10.sup.-3]) 2.54 2.29 Parameters recovered by approximate solution Diffusion coefficient, D (x[10.sup.-4] 7.58 7.61 [mm.sup.2]/h) Bound [right arrow] Unbound, [beta] (x[10.sup.-3] 7.14 7.17 [h.sup.-1]) Unbound [right arrow] Bound, [gamma] 1.08 1.05 (x[10.sup.-3] [h.sup.-1]) Maximum moisture, [M.sub.[infinity]] (wt%) 2.19 2.11 RMS error/data point (x[10.sup.-3]) 10.04 13.66 TABLE 2. Sample information and experimental procedure for thin polymer laminates. Dimensions Notation ([mm.sup.3]) Testing period Last data point U20 40 x 35 x 0.3398 500 days 1,800 days D20 40 x 35 x 0.3488 100 days 800 days D60 40 x 35 x 0.3359 100 days 800 days Testing Number Notation temperature Preconditioning of samples U20 20[degrees]C Undesiccated 5 D20 20[degrees]C Desiccated 3 D60 60[degrees]C Desiccated 3 TABLE 3. Absorption parameters for the U20, D20, and D60 samples recovered without using the last data point. Sample set U20 D20 D60 Parameters recovered by exact solution Diffusion coefficient, D ([10.sup.-3] 6.05 5.44 30.23 [mm.sup.2]/day) Bound [right arrow] Unbound, [beta] 7.88 24.40 16.73 ([10.sup.-3] [day.sup.-1]) Unbound [right arrow] Bound, [gamma] 3.52 8.74 6.79 ([10.sup.-3] [day.sup.-1]) Maximum moisture, [M.sub.[infinity]] (wt%) 3.57 4.39 4.74 RMS error/data point (x[10.sup.-2]) 1.59 2.33 2.65 Parameters recovered by approximate solution Diffusion coefficient, D ([10.sup.-3] 6.43 10.95 40.08 [mm.sup.2]/day) Bound [right arrow] Unbound, [beta] 8.87 47.26 31.39 ([10.sup.-3] [day.sup.-1]) Unbound [right arrow] Bound, [gamma] 3.91 24.20 11.63 ([10.sup.-3] [day.sup.-1]) Maximum moisture, [M.sub.[infinity]] (wt%) 3.55 4.19 4.47 RMS error/data point (x[10.sup.-2]) 1.63 5.44 3.65 TABLE 4. Absorption parameters for the U20, D20, and D60 samples recovered using the whole dataset. Sample set U20 D20 D60 Parameters recovered by exact solution Diffusion coefficient, D (]10.sup.-3] 5.17 5.35 28.49 [mm.sup.2]/day) Bound [right arrow] Unbound, [beta] 1.57 14.14 5.39 ([10.sup.-3] [day.sup.-1]) Unbound [right arrow] Bound, [gamma] 1.12 6.08 4.56 ([10.sup.-3] [day.sup.-1]) Maximum moisture, [M.sub.[infinity]] (wt%) 4.67 4.71 6.37 RMS error/data point (x[10.sup.-2]) 2.50 2.26 2.58 Parameters recovered by approximate solution Diffusion coefficient, D (]10.sup.-3] 5.82 5.88 29.95 [mm.sup.2]/day) Bound [right arrow] Unbound, [beta] 3.83 14.75 6.49 ([10.sup.-3] [day.sup.-1]) Unbound [right arrow] Bound, [gamma] 3.21 8.94 6.01 ([10.sup.-3] [day.sup.-1]) Maximum moisture, [M.sub.[infinity]] (wt%) 4.53 4.71 6.35 RMS error/data point (x[10.sup.-2]) 5.04 4.08 3.06 TABLE 5. Effect of the fast recovery method on the number of iterations required to recover the absorption parameters of nanoclay/epoxy composites. Nanoclay content 0.5 wt% 3 wt% Number of iterations Normal recovery 672 747 Fast recovery 344 100 RMS error per data point Normal recovery (x[10.sup.-3]) 2.54 2.29 Fast recovery (x[10.sup.-3]) 2.67 2.30 TABLE 6. Effect of the fast recovery method on the number of iterations required to recover the absorption parameters of thin epoxy laminates using the whole data. Sample set U20 D20 D60 Number of iterations Normal recovery 327 369 4,765 Fast recovery 66 11 893 RMS error per data point Normal recovery (x[10.sup.-2]) 2.50 2.26 2.58 Fast recovery (x[10.sup.-2]) 2.54 2.26 2.57 TABLE 7. Effect of the fast recovery method on the number of iterations required to recover the absorption parameters of thin epoxy laminates without using the last data point. Sample set U20 D20 D60 Number of iterations Normal recovery 612 699 1,973 Fast recovery 16 93 1 RMS error per data point Normal recovery (x[10.sup.-2]) 1.59 2.33 2.65 Fast recovery (x[10.sup.-2]) 1.61 2.26 2.74
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|Author:||Guloglu, Gorkem E.; Hamidi, Youssef K.; Altan, M. Cengiz|
|Publication:||Polymer Engineering and Science|
|Date:||Sep 1, 2017|
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