Abstract
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
The threedimensional anisotropic moisture absorption behavior of quartzfiberreinforced bismaleimide (BMI) laminates is investigated by collecting 21 months of experimental gravimetric data. Laminates of six, twelve, and forty plies and various planar aspect ratios are used to determine the threedimensional anisotropic diffusion behavior when exposed to full immersion in distilled water at 25°C. The longterm moisture absorption behavior deviates from the widely used Fickian model, but can be accurately captured by the threedimensional, anisotropic hindered diffusion model (3D HDM). Excellent agreement is achieved between experimental gravimetric data and the 3D HDM for all laminate thicknesses. Recovered model parameters are shown to slightly vary with laminate thickness due to the small changes in the curedply thickness. However, model parameters identified for a given laminate thickness are observed to accurately predict the absorption behavior of samples with different planar dimensions. Equilibrium moisture content of 1.72, 1.69, and 1.84% and corresponding diffusion hindrance coefficients of 0.807, 0.844, and 0.671 are recovered for six, twelve, and fortyply laminates, respectively, thus confirming strong nonFickian behavior. Moisture absorption parameters may be determined successfully at 16.5 months of immersion, before reaching approximately 85% of the equilibrium moisture content at 21 months. Subsequent gravimetric measurements up to 21 months are consistent with the predicted longterm behavior. POLYM. ENG. SCI., 54:137–146, 2014. © 2013 Society of Plastics Engineers
INTRODUCTION
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
Bismaleimide (BMI) is often used as the matrix material in composites expected to operate in extreme environmental conditions, such as high temperature and high humidity. BMI resins are widely used in highperformance composites due to their excellent corrosion resistance, high glass transition temperature, and excellent mechanical property retention in hygrothermal conditions [1]. Further, BMI resins are often used in the electronic and aerospace industries due to their excellent dielectric properties, relatively low susceptibility to moisture absorption, and good flammability characteristics. These properties make BMI an ideal matrix material for quartzfiber reinforcement as part of the radar protecting structure, or radome, on commercial and military aircraft [2].
Moisture ingress has been widely shown to degrade mechanical properties of polymeric composites significantly, such as tensile, interlaminar shear [3], compressive, and flexural strength [4]. Detrimental effects of moisture ingress have been reported for a wide variety of fiber/matrix combinations and manufacturing techniques [513]. In addition, reduction in dielectric properties has been attributed to introduction of water molecules into a composite structure [14]. Therefore, the accurate characterization of the moisture absorption properties of BMI resins and their composites is critical to the determination of the mechanical and electrical performance of the material. This characterization is especially important for radome materials, which must have excellent mechanical, dielectric, and heatresistance properties [2].
During the last three decades, the moisture absorption behavior of epoxy resin and its composites has been investigated thoroughly [1530]. In comparison, research into the moisture absorption characteristics of BMI resin and its composites is not as extensive, though the topic has been addressed directly [3135] or indirectly [2, 36] by several articles. In addition, a few articles have addressed the moisture absorption behavior of epoxy resins modified with BMI [37, 38]. The broader popularity of epoxy resin compared with the more specialized BMI is the likely cause for the relative lack of moisture absorption studies. The available literature on BMI composites, to the best of our knowledge, addresses only onedimensional isotropic diffusion behavior. Further, the lack of extensive moisture absorption studies on fiberreinforced BMIs prevents a consensus from being drawn regarding the longterm moisture uptake behavior. The few available studies that directly address the moisture absorption behavior of BMI indicate nonFickian moisture absorption, which is compared with a twostage model [31], a dualdiffusivity model [32], and the onedimensional “Langmuirtype” model [35].
In a study of the moisture absorption behavior of a carbonfiberreinforced BMI, Bao and Yee [31] observed nonFickian diffusion moisture uptake as evidenced by deviation from the Fickian equilibrium plateau at long times. The authors suggest that the gradual increase in moisture content is due to the longrange cooperative motion of polymer chains, which occurs at an appreciable rate due to the relatively large amount of moisture in the matrix. Thus, the presence of moisture contributes to plasticization of the matrix, lower glass transition temperature, and dramatically enhances segmental mobility in the network. In summary, the authors suggest that the secondstage of moisture absorption, in which a gradual increase in moisture content is observed, is controlled by the rate of polymer network rearrangement. No equilibrium moisture content is assigned by the authors to this linear secondstage. The firststage was assumed to be Fickian in nature, and the onedimensional diffusivity of the composite under full immersion in liquid water was reported for various temperatures.
In a second study on the moisture absorption behavior of carbonfiberreinforced BMI, Bao and Yee [32] investigated the effect of fiber architecture, specifically woven and woven/uniweave reinforcement. Unlike in the initial study, short term diffusion behavior was found to be nonFickian. The authors developed a dualdiffusivity model that was able to successfully describe the experimental weight gain curves. The model describes two independent diffusion processes with very different diffusivities. The first process is associated with diffusion in cureinduced cracks and voids, and proceeds relatively quickly. The second, slower process is attributed to the diffusion of mobile molecules in the resin due to random molecular motions, as in the Fickian model. Equilibrium moisture content of the fiberreinforced BMI was found to be approximately 1.3%.
Li et al. [35] studied the moisture diffusion behavior of a neat BMI resin subject to hygrothermal aging conditions. The nature of the diffusion process was investigated using Fourier transform infrared spectroscopy and swelling experiments. The moisture absorption behavior was modeled successfully using “Langmuirtype” model of diffusion [39], which is mathematically equivalent to the onedimensional version of the hindered diffusion model (HDM). According to the authors, the widely used Fick's law is not appealing in this case due to the fact that it does not account for the interaction between water and polar groups in thermoset polymers. The BMI resin used in this study, like most BMI resins, contains polar groups such as hydroxyl groups, which can form hydrogen bonds with water during the diffusion process. In an effort to quantify the interaction between the polymer and the diffusing molecules, two different cure schedules were used to prepare BMI resins with different network structures. The specimens were referred to as LBMI (lower temperature postcure, 218°C), and HBMI (higher temperature postcure, 230°C). Specimens were exposed to 100% relative humidity at 70°C and 100°C. The 12°C difference in postcure temperature had a measureable effect on all the diffusion behavior, including diffusivity, probability of molecular binding and unbinding, and equilibrium moisture content.
The objective of this article is to address the threedimensional anisotropic moisture absorption behavior of quartzfiberreinforced BMI laminates. Threedimensional effects are particularly important for this material. Quartzreinforced BMI laminates up to fortyply are known to be used as primary structures on various aircraft, providing significant edge surface area for moisture ingress. As a result, onedimensional throughthickness diffusion models may not be applicable or accurate. To the best of our knowledge, the threedimensional and anisotropic nature of moisture absorption has not been described for a BMI resin or its composites.
EXPERIMENTAL
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
The material used in this study is a BMI resin, trade name HexPly® F650, reinforced with an eightharness satin weave quartz fabric. Test panels were constructed of six, twelve, and forty plies with dimensions of 36, 36, and 18 inches square, respectively. Test panels were cured in an autoclave at 190°C (375F) and 85 psi for four hours, followed by an 8h postcure at 232°C (450F) according to the manufacturer's recommendations [40]. The reported laminate properties for this cure procedure and reinforcement are 55.4% fiber volume, 31.4% resin by weight, 0.7% void content and specific gravity of 1.78. The equilibrium moisture content reported by the manufacturer for the neat resin is 4.3% by weight, which corresponds to 1.38% by weight for the laminate properties described. Individual test samples of multiple dimensions were cut from the original test panels using a wet diamond saw, which is a typical machining method in a composite manufacturing setting. To more accurately quantify the anisotropic diffusion characteristics of this material, samples of different planar dimensions were machined from the original test panels. Laminate samples were prepared with planar aspect ratios of 4, 3, 2, or 1. Five samples of each geometry were prepared to decrease experimental uncertainty, thus providing a total of 20 sixply samples, 15 twelveply samples, and 10 fortyply samples. The dimensions of each sample used in this study are shown in Table 1.
Table 1. Dimensions, plycounts, and curedply thickness of laminate samples used in this studyPlanar aspect ratio  Planar dimensions (mm) 

6ply  12ply  40ply 


4  40 × 10a  40 × 10a  40 × 10a 
3  45 × 15  –  – 
2  40 × 20a  40 × 20a  – 
1  30 × 30  30 × 30  30 × 30a 
Total thickness  1.34  2.63  10.6 
Curedply thickness  0.223  0.219  0.264 
Once cut to their final size, samples were dried in an oven at 90°C until a constant, moisturefree weight was achieved. Samples were then placed in an environmental chamber and maintained at 25°C and full immersion in distilled water. To gather gravimetric weight gain data at periodic intervals, samples were removed from immersion, dried with a nonlinting cloth, exposed to room temperature air for 5 min, and weighed with a highprecision analytical balance. The time required to dry and weigh the samples was subtracted from the moisture exposure time. All gravimetric data reported in this study for a specific sample size is an average of all five samples with uncertainty levels calculated using 95% confidence interval.
MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
Based on existing BMI moisture absorption studies, the moisture absorption behavior was not expected to conform to typical Fickian diffusion behavior [31, 32, 35]. Therefore, the threedimensional anisotropic hindered diffusion model (3D HDM) developed by Grace and Altan [41] was chosen to represent the moisture absorption behavior. The 3D HDM is a comprehensive model that has the capability to predict threedimensional anisotropic moisture diffusion while incorporating varying degrees of Fickian or nonFickian behavior. In this model, the concentration at time t and position x, y, and z satisfies the coupled pair of differential equations
 (1)
where n represents the mobile molecules per unit volume, N represents the bound molecules per unit volume, γ is the probability per unit time that a mobile molecule will become bound, β is the probability per unit time that a bound molecule will become mobile, and D_{x}, D_{y}, D_{z} are diffusivities in the three principal directions.
Assuming an initially dry slab of dimensions h × w × l, the boundary and initial conditions are
 (2)
and
 (3)
An analytical solution was developed by Grace and Altan [42] which is valid under the following conditions:
where
 (4)
The analytical mass gain function of the 3D HDM is given by [42]:
where
 (5)
where M(t) is the weight of absorbed moisture at time t given an equilibrium moisture weight percent, M_{∞}, and μ is the nondimensional diffusion hindrance coefficient, governing the nonFickian effects. The condition given in Eq. (4) is easily satisfied for most polymeric composites, including the quartz/BMI materials system used in this study. The dimensions and diffusion directions used throughout this article are presented in Fig. 1.
The 3D HDM was applied to the experimental data using via the minimization of
 (6)
where M(t) is the theoretical moisture content predicted by the 3D HDM and M_{exp(t)} is the experimentally determined moisture content. The objective is to minimize the error between the experimental data and the HDM by arriving at a set of hindered diffusion parameters that accurately capture the moisture absorption behavior of the laminate under consideration. The six model parameters (i.e., D_{x}, D_{y}, D_{z}, γ, β, M_{∞}) that minimize the function given in Eq. (6) are determined by using a modified version of the method of steepest descent, or Cauchy's method as in Ref.[42]. These six model parameters fully characterize the moisture absorption behavior of a composite laminate and, thus, can be used in conjunction with the HDM to predict moisture absorption behavior. An initial guess, E_{o}(t), for each of these parameters is provided to a search algorithm. The search is terminated when E(t) changes less than a set value in one iteration.
To compare the agreement of the 3D HDM and the 3D Fickian model with the experimental data, both are shown in Fig. 2. The experimental moisture absorption data exhibits a definite “pseudoequilibrium” at which moisture weight percent does not deviate more than 0.01% for over 100 days. Such behavior has the potential to cause premature termination of gravimetric measurements, on the assumption that the moisture absorption is Fickian in nature and equilibrium has been reached. In fact, the experimental gravimetric data easily meets the requirements for equilibrium as defined in ASTM D5229 [43]. The ASTM specification requires two consecutive measurements within a maximum time period of 7 days which do not change by >0.01% moisture weight percent. In this case, seven consecutive measurements over >100 days conform to this requirement. Moisture equilibrium determination per ASTM D5229, therefore, yields a maximum moisture content of 1.38%.
RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
In an effort to predict the diffusion properties of the BMI/quartz laminates, composite laminates of three different thicknesses (six, twelve, and fortyply) were used to obtain gravimetric moisture absorption data, which was subsequently used to recover the 3D HDM parameters. For each thickness, samples with two unique planar dimensions are used in recovering the HDM parameters. Each unique combination of planar dimensions and thickness contains five identical samples for increased accuracy, yielding a total of 30 samples and six unique sample sizes. The gravimetric data for each unique sample size is the average of the five samples of that particular size. The diffusion behavior was assumed to be orthotropic due to the orthotropic nature of the fiber reinforcement. Therefore, five parameters were recovered: throughthickness diffusivity (D_{z}), edge diffusivity (D_{e} = D_{x} = D_{y}), probability of a mobile molecule becoming bound (γ), probability of a bound molecule becoming mobile (β), and equilibrium moisture content (M_{∞}). The quality of the fit was measured by calculating the rootmean squared (RMS) error of the predicted moisture content relative to the experimental gravimetric data. The search algorithm for the five absorption parameters terminates when the RMS error between the experimental data and the 3D HDM prediction changes less than 5 × 10^{−5} in a single iteration. In some cases, such as lack of sufficient data points, multiple sets of model parameters which satisfy this termination criterion may exist. As such, the recovered model parameters may exhibit dependence on the initial estimate. To account for this potential dependence, multiple initial guesses are used for each parameter. Convergence to the actual model parameter is assumed when the recovered value does not vary more than one percent regardless of the initial estimate.
The sixply data set included gravimetric data from the 40 × 10 mm and 40 × 20 mm samples. The recovered 3D HDM parameters are shown in Table 2. The RMS error between the prediction and the experimental data is 2.5 × 10^{−3} and 3 × 10^{−3} for the 40 × 10 mm and 40 × 20 mm samples, respectively. The experimental data and the corresponding 3D HDM predictions are shown in Fig. 3. The quality of the prediction is excellent for both sample dimensions.
Table 2. 3D HDM parameters recovered from experimental data sets for sixply, twelveply, and fortyply laminates. For each ply thickness two different size sample sets (n = 5) are used  6ply  12ply  40ply 

D_{e} (mm^{2}/h)  1.98 × 10^{−3}  2.89 × 10^{−3}  4.07 × 10^{−3} 
D_{z} (mm^{2}/h)  1.73 × 10^{−4}  1.47 × 10^{−4}  3.12 × 10^{−4} 
γ  2.98 × 10^{−6}  1.22 × 10^{−6}  2.3 × 10^{−5} 
β  1.25 × 10^{−5}  6.64 × 10^{−6}  4.7 × 10^{−5} 
μ  0.807  0.844  0.671 
M_{∞} (%)  1.72  1.69  1.84 
The recovered hindered diffusion parameters from the twelveply data set are also shown in Table 2. A slightly higher equilibrium moisture content and edge diffusivity are recovered relative to the sixply samples. The twelveply data set again included gravimetric data from the 40 × 10 mm and 40 × 20 mm samples, and yielded RMS errors of 2.4 × 10^{−3} and 3.2 × 10^{−3}, respectively. The experimental data and the corresponding 3D HDM predictions are shown in Fig. 4. Again, the quality of the prediction is excellent for both sample sizes.
The longer times required to reach equilibrium moisture content for very thick samples, such as the fortyply samples used in this study, resulted in experimental data that is further from reaching equilibrium. As a result, the parameters recovered from the experimental data from the two fortyply sample sizes exhibits a slight dependence on the initial estimates. Therefore, initial estimates were set according to the parameters recovered from the twelveply experimental data. The best fit was obtained for equilibrium moisture content of 1.84%, which is slightly higher than the values obtained for the six and twelveply samples. The recovered parameters (see Table 2) show excellent agreement with the experimental data, as evidenced by RMS error of 2.4 × 10^{−3} and 2 × 10^{−3} for planar dimensions of 40 × 10 mm and 30 × 30 mm, respectively. The quality of the prediction as applied to the experimental data is shown in Fig. 5.
The relationship between the recovered edge and throughthickness diffusivity further highlights the importance of including threedimensional anisotropic diffusion effects in any comprehensive model of moisture absorption in polymeric composites. Diffusion through the edges of the laminate (along the fiber) proceeds at 11.4, 19.7, and 13 times faster than through the thickness (transverse to the fiber) for the six, twelve, and fortyply samples, respectively. These values are basically consistent with those reported for other materials in the literature. Blikstad [30], for example, observed a 10fold increase in diffusion through the edges of a graphite/epoxy laminate relative to throughthickness diffusion. In addition, Aronhime et al. studied the anisotropic moisture absorption behavior of unidirectional Kevlar/epoxy and found that the ratio of diffusion along the fiber to diffusion transverse to the fiber varied from 3 to >100, depending on the fiber volume fraction [44]. The contribution of moisture absorption into the Kevlar fiber itself was assumed to play a role. In addition, Arao et al. report edge diffusivities between two and four times greater than throughthickness diffusivity for a woven carbonfiberreinforced epoxy laminate [45].
The differences in the values of recovered 3D HDM parameters for each laminate thickness highlight the effect of laminate construction on diffusion properties. Despite simultaneous layup, cure, and postcure under identical conditions, discernible differences in diffusion parameters exist depending on the plycount. Variations in the recovered diffusion properties may be the result of different fiber volume fractions and void content, or the existence of degree of cure gradients within thick laminates.
One of the most likely contributors to the variation in diffusion properties among these laminates is the change in fiber volume fraction as measured by the curedply thickness and as evidenced by scanning electron microscopy of sample cross sections. The six, twelve, and fortyply laminates have an average thickness of 1.34, 2.63, and 10.55 mm, respectively. Therefore, the average curedply thickness for the six, twelve, and fortyply laminates is 0.223, 0.219, and 0.264 mm, respectively as shown in Table 1. These variations in curedply thickness exist despite the simultaneous cure of all three laminates under the same conditions. The specific gravity reported by the manufacturer of 1.78 is consistent with the six and twelveply samples, which have an average specific gravity of 1.75 and 1.77, respectively. The fortyply samples, however, have an average specific gravity of 1.69, which is indicative of higher resin content. The slightly higher predicted equilibrium moisture content for the fortyply laminate, therefore, is to be expected. The quartz fibers are assumed to be impermeable, and all absorbed moisture is contained within voids, the fiber/matrix interface, or the resin itself. Not surprisingly, the predicted equilibrium moisture content for the six, twelve, and fortyply laminates correlates well to the curedply thickness. The highest curedply thickness and highest equilibrium moisture content is observed in the fortyply laminate. Whereas, the lowest perply thickness and lowest equilibrium moisture content is seen in the twelveply laminate. The sixply laminate has a slightly higher perply thickness relative to the twelveply laminate, in addition to slightly higher predicted equilibrium moisture content.
In addition to different fiber volume fractions, temperature gradients that develop during the cure process of thick composites may lead to variations in the degree of cure throughout the thickness of the laminate [46]. A number of studies have been performed on the variation in the degree of cure of thick polymer composites due to the existence of temperature gradients [4651]. In addition, thick laminates may experience internal temperatures exceeding the cure temperature due to the exothermic crosslinking reactions of the polymer. In some cases, this internal temperature may reach levels that are likely to induce degradation of the polymer matrix [51]. These variations in the structure of the polymer through the thickness of the laminate may contribute to changes in diffusivity as thickness increases. Specific studies on this phenomenon are very limited. However, the degree of cure and crosslink density was found to influence the moisture absorption properties of an epoxy in a study performed by Perrin et al. [52]. Specifically, a fully crosslinked epoxy network was found to absorb more water and at a faster rate than a less crosslinked network. No specific qualitative analysis was performed on the BMI/Quartz sample crosssections to identify and characterize polymer structural variations between samples of different thickness. However, it is likely that any such variations in the polymer structure will influence the diffusion parameters, specifically the probabilities of molecular binding and unbinding.
As expected, variation in the recovered probabilities of molecular binding and unbinding based on the laminate thickness is also observed. However, the diffusion hindrance coefficient is relatively consistent, ranging from 0.67 to 0.84 with an average value of 0.79. A hindrance coefficient of unity indicates Fickian diffusion, in which mobile molecules are free to diffuse by random molecular motion. A lower hindrance coefficient indicates a greater degree of disruption of the diffusion of mobile molecules. In this case, the recovered hindrance coefficient of 0.67 indicates that moisture absorption in the fortyply laminates is more hindered than in the other laminates. However, the recovered values are consistent with hindered diffusion reported in BMI resins. Li et al., for example, report γ and β values corresponding to hindrance coefficients of 0.78, 0.80, 0.83, and 0.88 depending on postcure temperature and immersion bath temperature [34]. The hindrance coefficient may also be influenced by the changes in the structure of the polymer as a result of a temperature gradient during the cure cycle. Kumosa [53] suggest a correlation between sample thickness and the values of γ and β for a unidirectional glassreinforced polymer composite exposed to 80% relative humidity at 50°C. The extent of laminate compaction during the cure process that led to different curedply thickness likely contributes to the change in hindrance coefficient. The lower resin flow and higher resin content observed in the fortyply laminate may lead to changes in the structure of the polymer, such as increased microvoids. The microvoids may contribute to the increased diffusion hindrance in the fortyply laminate by acting as moisture storage sites. In addition, changes in the polymer structure due to degree of cure gradient may contribute to increased diffusion hindrance, though additional qualitative analysis is required to verify any structural differences due to degree of cure.
APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
The ability of the threedimensional HDM to capture the moisture absorption behavior of two differentsized samples from the three different laminates was demonstrated in the previous section. To further validate the recovered diffusion parameters and the applicability of the 3D HDM, the model parameters determined in the previous section were applied to the gravimetric data from a new set of samples having different planar sizes. In this way, the ability to accurately predict the moisture absorption behavior of other laminate sizes, which have not been used in determining the model parameters, can be validated.
Additional gravimetric data from sixply samples of 45 × 15 mm and 30 × 30 mm planar dimensions and twelveply samples of 30 × 30 mm planar dimensions were not used in determining the absorption parameters for either plycount. The recovered sixply parameters given in Table 2 were applied to the two other sixply sample sizes, 45 × 15 mm and 30 × 30 mm. The associated RMS error of 3.9 × 10^{−3} and 3.5 × 10^{−3}, respectively, is only slightly higher than the RMS error associated with the 40 × 10 mm and 40 × 20 mm sample sizes. In addition, the twelveply parameters were applied to the gravimetric data from the twelveply 30 × 30 mm sample with excellent agreement. The RMS error in this case was 3.7 × 10^{−3}, again only slightly higher than the error associated with the gravimetric data from which the parameters were recovered. The gravimetric data from the two sixply samples and one twelveply sample are shown in Fig. 6, along with the corresponding 3D HDM predictions.
DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
The maximum moisture content of a composite in a specific environment plays a critical role in engineering design decisions due to the relationship between moisture and material property degradation. The pseudoequilibrium moisture content of 1.38% illustrated in Fig. 2 corresponds with the equilibrium moisture content reported by the manufacturer for this material when fabricated according to the manufacturer's recommendations. However, experimental gravimetric data indicates that moisture absorption continues at a slower rate after this point and clearly deviates from the Fickian equilibrium plateau. The moisture content reached experimentally after 21 months of exposure is 1.46%, which is higher than the value reported by the manufacturer, but still lower than the final equilibrium moisture of 1.72% predicted by the 3D HDM. The ability to accurately predict equilibrium moisture content before experimentally reaching the saturation point was demonstrated by Grace and Altan [42] using simulated experimental data. In this case specifically, early prediction of equilibrium moisture content is critical. The twelveply 30 mm × 30 mm sample, for example, will take approximately 17 years to reach 99% of equilibrium saturation. As suggested in Ref.[42], the 3D HDM was applied to experimental data at various times throughout the moisture absorption study. The accuracy with which the model parameters may be recovered is affected by the length of the experimental time frame and the proximity of the experimental moisture content to the actual moisture equilibrium. The initial, relatively fast moisture uptake is largely controlled by diffusion of mobile molecules by random molecular motion. As a result, the threedimensional diffusivities are relatively immune to changes in the initial estimates when experimental data is used that reaches or exceeds Fickian “pseudoequilibrium.” The recovered diffusion hindrance coefficient and equilibrium moisture content, however, may vary widely based on initial estimates if sufficient experimental data is not used.
In this case, the model parameters are consistently recovered within 1% after approximately 16.5 months of exposure as illustrated in Fig. 7, regardless of initial estimates. Subsequent measurements up to 21 months confirm the validity of the predictions. Figure 7 also depicts that before 16.5 months, however, a range of model parameters may be recovered which exhibit equivalent RMS error. An attempt to recover the model parameters based on 4.5 months of experimental data, just after Fickian “pseudoequilibrium” exhibited significant dependence on the initial parameter estimates. A range of hindrance coefficients from unity to 0.56 and moisture equilibrium from 1.4% to 2.3% were recovered, depending on the initial guess. Within these upper and lower bounds, an infinite set of model parameters exist which yield RMS error of <3 × 10^{−3} relative to experimental data. The recovered hindrance coefficient of unity indicates unimpeded Fickian diffusion of mobile molecules, while 0.56 indicates a highly hindered diffusion process. The range of recovered model parameters narrows as the experimental data increases. At 6 months, the window for hindrance coefficient and equilibrium moisture content narrows to 0.61–0.89 and 1.57–2.14%, respectively. The window narrows further at 11.5 months, where the hindrance coefficient may fall between 0.77 and 0.82 and equilibrium moisture content may fall between 1.69% and 1.74%. After 16.5 months of exposure time, the parameters vary little with changes in initial estimates. The parameters determined from 21 months of experimental data are tightly bounded by the parameters recovered at 16.5 months. The hindrance coefficient of 0.807, for example, is between the range of hindrance coefficient values recovered at 16.5 months, which are 0.804 and 0.808. Similarly, the equilibrium moisture content of 1.723% falls between the 16.5 months of 1.716 and 1.725%. The results of the parameter recovery based on experimental data from 4.5, 6, 11.5, and 16.5 months are presented in Fig. 7 along with the gravimetric data from the sixply, 40 × 10 mm samples.
CONCLUSIONS
 Top of page
 Abstract
 INTRODUCTION
 EXPERIMENTAL
 MODELING MOISTURE ABSORPTION BEHAVIOR OF BMI/QUARTZ LAMINATES
 RECOVERING 3D HDM PARAMETERS FROM EXPERIMENTAL DATA
 APPLICATION OF RECOVERED DIFFUSION PARAMETERS TO SAMPLES OF DIFFERENT PLANAR DIMENSIONS
 DETERMINATION OF ANISOTROPIC ABSORPTION PROPERTIES OF COMPOSITE LAMINATES BEFORE REACHING MOISTURE EQUILIBRIUM
 CONCLUSIONS
 ABBREVIATIONS
 REFERENCES
Gravimetric moisture absorption data from six, twelve, and fortyply BMI/quartz laminate specimens immersed in water at 25°C was collected over 21 months. The moisture absorption behavior was shown to deviate from the widely used Fickian diffusion model. The threedimensional HDM was successfully implemented in the recovery of orthotropic diffusion parameters. Excellent agreement is achieved between experimental gravimetric data and the 3D HDM for each laminate thickness, regardless of the planar size of the samples. Recovered model parameters show slight differences in diffusion properties between six, twelve, and fortyply laminates, possibly due to variations in curedply thickness. Equilibrium moisture content of 1.72, 1.69, and 1.84% and corresponding diffusion hindrance coefficients of 0.807, 0.844, and 0.671 are recovered for six, twelve, and fortyply laminates, respectively, thus indicating strong nonFickian behavior. Edge diffusivity of 1.98 × 10^{−3}, 2.89 × 10^{−3}, and 4.07 × 10^{−3} mm^{2}/h was recovered for six, twelve, and fortyply laminates, respectively. Hence, diffusion through the laminate edges (in the fiber direction) occurs 11.4, 19.7, and 13 times faster than diffusion through the thickness (transverse to the fibers) for six, twelve, and fortyply laminates, respectively. Moisture absorption parameters were shown to be successfully recovered after 16.5 months of exposure, before experimentally reaching equilibrium moisture content. Subsequent gravimetric data up to 21 months were consistent with the predicted behavior.