**5. Diffusion of water in thermosets and FRP**

The design of a marine structure is fully dependent on the mechanical properties of the candidate material in various modes. In addition, it must consider the environment in which the object has to perform. Therefore, there is a third consideration of timescale of the service. A very simple example is a static beam under a constant bending load in a building that should carry the load for a long period, for instance, 2–4 decades. Therefore, the design input must be the properties of the material after aging for that service period in the atmospheric environment, especially moisture, carbon dioxide, ultraviolet ray, oxygen, and ozone. While it is not possible to have a data for such a long period for designing an object, it is best to make a prediction of the extent of degradation/aging and degraded properties after a target period of service. This simulation is quite difficult because all environmental and load conditions cannot be simultaneously considered in the mathematical predictive equations. However, a preliminary knowledge or previous study might help in deciding the conditions of fastest degradation due to aging effect. For example, it is known that polypropylene degrades in sunlight due to UV much faster than any other environmental conditions. Therefore, the service life is better decided upon aging under UV of varying intensity.

For marine structures and vessels, the most important considerations to decide the service life are sea water aging, fatigue due to vibration, constant load, and also degradation due to microbial activities. Cyclic sorption-desorption along with a prestress was studied by Burla [42], which gave more information on the repeated sorption phenomenon of the cloisite 10A nanocomposites of epoxy, vinyl ester, and unsaturated polyester.

Atmospheric aging due to ozone, UV, etc., is also important for the objects or part of the structures above the water line. In all the factors, sea water aging is most severe because of dissolved salts and alkalinity. The pH of sea water is about 8.3 on an average, and it also contains chlorides, bromides, iodides, sulfates, and carbonates of sodium, magnesium, potassium, calcium, and also traces of heavy metals such as Iron, manganese, cadmium, lead etc. Therefore, diffusion of sea water, and the effect thereof, is the most relevant study for deciding the degradation in mechanical properties of FRP for marine application. It is well known that the extent of sea water uptake and its effect is quite different from potable water or industrial process water.

#### **5.1 Fickian model: constant diffusivity**

The diffusion phenomenon in pure thermosets and corresponding FRPs can be generally described by a fundamental theory of diffusion by Fick's Law:

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

where *C* is the instanteneous concentration of the diffusing molecule, *D* is the coefficient of diffusion, commonly called the diffusivity, *t* is the time, and *x*, *y*, *z* are the three Cartesian coordinates.

Considering only unidirectional diffusion of sea water in a thin panel, (thickness less than 2% of length and breadth), Eq. (1) can be solved to obtain a fractional mass gain (*G*) at any instant "*t*" [43]:

$$G = \frac{M\_l}{M\_\infty} = 1 - \sum\_{j=0}^{\infty} \frac{8}{\left(2j+1\right)^2 \pi^2} \exp\left[-\frac{D(2j+1)^2 \pi^2 t}{h^2}\right] \tag{2}$$

where *M* is the mass, suffixes indicate at any time "t" and the final value, *D* is diffusivity, *j* is a simulation factor, *h* is the thickness. The parameter *j* actually refines the output more for higher value. For example, *j* = 10<sup>6</sup> gives more accurate result than *j* = 10<sup>5</sup> .

The diffusivity can be directly calculated from a simple experiment of water uptake by a panel till saturation, using the following equation:

$$D = \frac{\pi}{16} \left( \frac{\mathcal{M}\_t / \mathcal{M}\_\infty}{\sqrt{t} / h} \right)^2 \tag{3}$$

If *Mt/M*<sup>∞</sup> is plotted against √*t*, *D* is obtained from the slope of the initial linear portion of the water absorption curve.

Eq. (1) can be approximated as [43]:

$$G = 1 - \exp\left[-7.3\left(\frac{D\_x t}{s^2}\right)^{0.75}\right] \tag{4}$$

If the sample is exposed to the sea water on both sides, then *s=h* (thickness of the panel), if one side is insulated by an impermeable coating, then *s=2h*.

Rearranging Eq. (4) we can get the time required to attain a certain water content due to unidirectional steady-state diffusion in a thermoset and FRP as:

$$t = \frac{s^2}{D\_c} \left[ -\frac{1}{7.3} \ln \left( 1 - G \right) \right]^{\frac{1}{10.5}} \tag{5}$$

Eq. (5) is used for predicting the time required to attain any level of water uptake for different thicknesses (*s*) in cases of a structure with varying contour where the thickness varies sectionwise, and it might not be possible to carry out experiments with all such thicknesses. Typical examples are marine boats, eliptical underwater objects, composite valves in pipelines carrying water, etc. Shen and Springer [44] showed a similar calculation with 12.7 mm thick panel subjected to moisture exposure.

The one-dimensional diffusion equation is valid for thin panels, where the diffusion from edges is not significant. In case of pure thermoset plaques, the edge effect is not very important, but FRP composites are anisotropic materials and hence the edges are to be protected. This is ensured in FRPs by applying the thermoset resin coating on all edges of the panel. However, there can be an edge correction too, to be more precise on unidirectional mass transfer, provided the sample is homogeneous in diffusivity in all directions [43]:

$$D\_c = D\_x \left( \mathbf{1} + \frac{h}{l} + \frac{h}{w} \right)^{-2} \tag{6}$$

where *Dz* is the measured diffusivity and *Dc* is the edge corrected value and *l, w* and *h* are length, breadth, and thickness of the panel.

#### *5.1.1 Example*

**Figure 1** shows an example of water diffusion data for an epoxy GFRP composite of dimensions *l* = 100 mm, *w* = 100 mm, and *h* = 4 mm experimentally determined and a theoretical value calculated by Eq. (4). The Diffusivity calculated from the *Mt* vs *t* 0.5 was corrected for edge error using Eq. (6). The Fickian model clearly does not validate the experimental data, hence the diffusion process might have either more than one diffusive process as the water ingress progresses or there can be an effect of molecular rearrangement in the epoxy thermoset due to absorbed water.

In order to determine the time required for water absorption to the extent of 90% of the saturation for a 12 mm thick FRP panel, assuming identical conditions, Eq.(5) is used and the time calculated as:

$$t = \text{30}, \text{944} \, h = \text{4.58 years}$$

The life prediction can be done on the basis of the minimum strength required by a designer of the FRP item. Suppose a minimum Flexural strength of 175 MPa is required for the designer to design an underwater vessel hull. The service life of an epoxy-GFRP hull of 12 mm thickness is to be predicted.

Taking the same FRP composition, the laboratory flexural strength data at various times of sea water aging was observed at 35°C for 10,000 h, and **Figure 2** shows the combined data of fractional water absorption and flexural strength with time. The flexural strength measured in 3-point bending test of the original, cured GFRP at 20°C was about 238–245 MPa.

From the source data of **Figure 2**, it is known that the fraction of saturation is 95.4% corresponding to the flexural strength of 175 MPa. Therefore, time required for the 12 mm thick panel at 0.954% saturation as calculated using Eq. (5) is:

*t* ¼ 2397 days ¼ 80 months at a sea water temperature of 35°C

*t* = 2397 days = 80 months at a sea water temperature of 35°C

The above solution of prediction is obviously approximate, as the theoretical curve is not in exact agreement with the experimental data till 7500 h (300 days). However, the theoretical prediction of the diffusion curve in **Figure 1** shows better agreement at longer period of exposure. In addition, considering the good fit in **Figure 2** for the fractional saturation (*Mt*/*M*∞) vs. log(time), with R2 = 0.9786, over the data range of 10–416 days of lab experiment, the data pair of Flexural strength of 175 MPa and 0.954 fractional saturation is fairly accurate compared with experimental value of 0.933 fractional saturation for same strength.

In a different approach, the diffusion-related life estimation can be realized if a time-temperature superposition is done from the data of sea water absorption and a functional property such as strength vs. immersion time at different fixed temperatures of the sea water. In an isothermal analysis with one temperature, the slow relaxation of larger segments of a thermoset polymer (*Tg* > ambient) is not realized in their contribution to the diffusion and other related properties in long-time aging behavior. There is a slow and continuous physical aging of the matrix at any

**Figure 1.** *Typical experimental data and corresponding Fickian model prediction.*

#### **Figure 2.**

*Flexural strength and fraction of saturation with immersion time in artificial sea water for an epoxy-GFRP at 35°C.*

temperature, even below the glass transition. The water absorbed reduces the *Tg* and if the test temperature is near the modified *Tg*, the rate of physical aging is enhanced, which may become significant after a long time for a composite of long service life, say more than a decade. It is possible to accommodate such physical process and its effect on properties by resorting to time-temperature superposition principle, since a property at a particular isothermal temperature after a period of aging is same for a different period of aging at another isothermal temperature. This is achieved by shifting the data of an isotherm to another reference isotherm. In classical theory, this relation can be Arrhenius expression as:

$$
\pi = \pi\_0 e^{E\_s/RT} \tag{7}
$$

where *t* is the relaxation time of the polymer segments. Eq. (7) is applicable for high temperatures above glass transitions since the exponential rise in relaxation time is possible when the movement of a segment or a molecule is not hindered by neighboring species. As the polymer is cooled below glass transition, the mobility is hindered by a close approximation of neighboring segments or molecules, and thus the above equation is not valid. In the glass transition region, the relaxation time would change by several decades on even one degree rise in temperature. A rather different expression is Vogel-Fulcher (VF) equation, which is somewhat valid even near glass transition [45]:

$$
\pi = A \mathfrak{e}^{\frac{B}{T - T\_0}} \tag{8}
$$

However, the value of the constants *A*, *B*, and *T*<sup>0</sup> would change as the temperature is lowered near *Tg* when more densification of the molecules would take place.

Williams, Landel and Ferry [46] relate the temperature-dependent events such as viscosity, relaxation time, or relaxation frequency with change in fractional free volume of the molecule or segments. The fractional free volume changes linearly with temperature. Accordingly, the relaxation time-temperature relationship is given as the famous WLF equation:

$$\log a\_T = \log \left(\frac{\tau}{\tau\_\text{g}}\right) = \frac{C\_1(T - T\_\text{g})}{C\_2 + (T - T\_\text{g})} \tag{9}$$

#### *FRP for Marine Application DOI: http://dx.doi.org/10.5772/intechopen.101332*

where log(*aT*) is the shift factor, *C*<sup>1</sup> and *C*<sup>2</sup> are constants,*Tg* is the glass transition temperature, and if it is the reference temperature, then *C*<sup>1</sup> = �17.44 and *C*<sup>2</sup> = 51.6, valid till a test temperature of *Tg* + 50°C, and these values are �8.86 and 101.6 respectively at any other reference temperature up to *Tg* + 50°C, and this is valid till a test temperature up to *Tg* + 100°C.

However, the best process of superposition is to shift the isotherms graphically in a data plot of the property (say strength) vs. time.

A typical *t–T* superposition is shown for a limited time of 8 months as an example to demonstrate the predicted value of a property after aging in sea water. The raw data was plotted in **Figure 3** as a flexural strength vs. time as isotherms at only 20°C, 30°C, 40°C, and 50°C and graphically shifted to the reference temperature 20°C, to predict the value of the strength at longer time than experimental time as an example. For a comprehensive study, longer period and more importantly, higher range of temperature should be used for a prediction at much longer period.

Subsequently, the shifted data are plotted as a master curve with a reference temperature of 20°C as shown in **Figure 4**. The best fit of the shifted data is approximately a second-order polynomial expression here. However, for a long period, the property will vary with the logarithm of time. The shift factors corresponding to the temperatures 30°C, 40°C, and 50°C were used to calculate new time (*tnew*) at the reference temperature of 20°C by following simple equation:

$$\log\left(t\_{new}\right) = \log\left(t\_{test}\right) + \log\left(a\_T\right) \tag{10}$$

The process of determination of shift factor from a graphical shifting is described in Ref. [47]. For example, shift factor log(*aT*) of 50°C is 0.301, and if we take test time at 8 months, then

log(*tnew*) = 0.9031 + 0.301 = 1.2041, therefore, *tnew* = 16 months.

This means that the strength at 50°C after 8 months of sea water aging corresponds to 16 months aging in sea water at 20°C. The shifted values can be approximately described by a polynomial fit:

$$
\sigma\_f = -0.0238t\_{new}^2 - 4.0t\_{new} + 222.72 \tag{11}
$$

where *σ<sup>f</sup>* is the flexural strength in MPa and *tnew* is in months.

**Figure 3.** *Isotherms of flexural strength vs. time of immersion of a GFRP based on epoxy resin.*

**Figure 4.** *Master curve for 20°C reference temperature: epoxy-GFRP aging in sea water.*

The polynomial fit can be used for determination of the property at extended period too. Therefore, the strength is calculated with Eq. (11) for longer period than the shifted data. **Figure 5** shows the data up to 50 months. The result is obviously an approximation, but gives one the idea of range of the degraded property (strength) for a long exposure time. The validation of the data is not possible unless an experiment is done for the similar period.

Similarly, 10 months data on water uptake by an epoxy-GFRP were studied at limited temperature range of 20°C, 30°C, 40°C, and 50°C. The data were plotted as isotherms and graphically shifted to the reference temperature 20°C. The shift factors were determined, and subsequently new time was obtained using the method already described, and a master curve of water uptake predicted at longer time was obtained. The plot is shown as **Figure 6** here only from 20 months of aging onward till 90 months.

A correlation of these two master curves for prediction of long-term properties as water uptake and flexural strength can be made with some approximation, in this case, because of the limitation of data.

Let us take strength and diffusion data at 48 months from **Figures 5** and **6** respectively. At 48 months, the Flexural strength is 85.6 MPa (calc.) and Water uptake is 6.61% (calc), at a sea water temperature of 20°C.

The example of evaluation of long-term property and water diffusion shown above does not simulate an actual FRP item. In practice, the thicknesses for underwater structures are much higher due to load requirements. Moreover, multiple

**Figure 5.** *Extrapolation of the master curve for longer period.*

*FRP for Marine Application DOI: http://dx.doi.org/10.5772/intechopen.101332*

**Figure 6.** *Long period prediction of water uptake constructed by graphical t-T shift.*

types of mats, chopped fibers, fabric with various weaving styles are used in thick composites where FEM analysis is resorted to design the layers.

An approach can be made for life estimation by calculating the diffusion time using Eq. (5) for an FRP of actual size and thickness from the laboratory experiment at different temperature of sea water aging with respect to time. Once a data table is made of Mt% vs. time for the actual size at various isothermal aging temperatures, the data can be used to obtain a graphically constructed master cure following a time-temperature superposition principle for a reference temperature, which is the actual sea water temperature of that geographical region. Since the composites often show dual Fickian behavior or non-Fickian behavior, the data for only long-term study can be taken from the master curve for a good fitting equation. The probability of error is minimized in this process, as graphical shift does not need any assumption such as glass transition temperature, values of activation energy, or the WLF constants, etc. However, a careful experimental determination of the value of diffusivity is required, which is a most critical parameter.

In experiments on diffusion, the panel thickness plays an important part. Although there is an edge correction method available, but it is best to use thin panels of maximum 4.0 mm thickness and edge sealing by a marine-grade vinyl ester resin tissue coat and gel coat of 1.0 mm thickness each. Number of layers of the fabric should be restricted by using fairly thick quality fabric and mats, but not very thick to make the resin infusion difficult. Nevertheless, similar materials such as the resin, curatives, catalysts, and type of mats and fabrics as actual FRP item would be best for a realistic prediction of service life.

#### **5.2 Dual-stage diffusion**

After observation of many experimental results on water diffusion process in thermosets and composites, it is certain that the diffusivity is not unique for a case and may vary according to the behavior of the polymer as the process of water ingress progresses. The water diffused in a polymer acts as a plasticizer to change the relaxation process, resulting in swelling, and also initiates some chemical reactions. Karter and Kibler [48] offered a theory that the water absorption is described by a simple diffusion with sources and sinks of diffusing water molecule and that the absorbed water is divided into mobile and strongly bound phases in the polymer. There is a continuous migration from mobile to bound phase and the reverse. There is an equilibrium of this interchange of bound and mobile water. The theory is somewhat similar to Langmuir theory of adsorption-desorption. Considering the

probabilities of the interchange of bound and mobile water molecules, the relative mass gain is given by the authors as:

$$\frac{M\_t}{M\_{\infty}} = 1 - \frac{\gamma}{\gamma + \beta} e^{-\beta t} - \frac{8\beta}{\pi^2 (\gamma + \beta)} \sum\_{0}^{\infty} \left( 2n + 1 \right)^{-2} e^{-k(2n+1)^2 t} \tag{12}$$

where *γ* is the probability per unit time that a mobile H20 molecule becomes bound, and *β* is the probability per unit time that a bound H20 molecule becomes mobile. The units of both are time�<sup>1</sup> .

The constant *k* is given as:

$$k = \frac{\pi^2 D}{h^2} \tag{13}$$

When the exposure time is short, an approximate equation can be used as follows:

$$M\_t = \frac{4}{\pi^{3/2}} \left( \frac{\beta}{\gamma + \beta} M \lhd \right) \sqrt{kt} \; ; \; \; 2\gamma, 2\beta \ll k \; ; \; \; t \le 0.7/k \tag{14}$$

Hence, *β* /(*γ* + *β*) can be calculated from the slope of a plot of *Mt/M*<sup>∞</sup> vs. *t* 0.5. and for a long exposure period, so that *kt* is much larger than 1. The following

approximate equation can be used:

$$M\_{\rm f} = M\_{\rm os} \left[ 1 - \frac{\chi}{\chi + \beta} e^{-\beta \mathfrak{r}} \right] \; ; \quad 2\gamma, 2\beta \ll k \; ; \quad t \gg 1/k \tag{15}$$

Eq. (15) can be rearranged, and after taking logarithm, it becomes:

$$\ln\left(\frac{M\_{\text{so}} - M\_{\text{f}}}{M\_{\text{so}}}\right) = \ln\left(\frac{\chi}{\chi + \beta}\right) - \beta t \tag{16}$$

Eq. (16) represents a straight line with �*β* as the slope of the curve and the intercept would give the value of *γ*, once *β* is calculated.

#### *5.2.1 Example*

A GFRP based on USP and chopped glass fiber is exposed to artificial sea water at 45°C for 8400 h. The size of the laminate was 80 mm � 12 mm � 4 mm (*l* � *w* � *h*), and the maximum water uptake was about 4%. The approximate equation Eq. (15) for long-term prediction was used for evaluating the parameters *β, γ*, and *k* from the time-dependent absorption data. The values are:

*β =* 0.0004 h�<sup>1</sup> , *<sup>γ</sup> <sup>=</sup>* 0.00233 h�<sup>1</sup> and *k =* 1.82 � <sup>10</sup>�<sup>05</sup> <sup>h</sup>�<sup>1</sup> . The corrected diffusivity (*Dc*) calculated at the initial slope was 2.95 � <sup>10</sup>�<sup>11</sup> m2 /h.

**Figure 7** shows the experimental data and the predicted data of absorption for long-term approximation considering a period beyond 2000 h as the long term.

#### **5.3 Dual Fickian model**

Due to the time-varying process of moisture absorption, it is assumed that instead of one constant diffusivity, two diffusivities can be used to describe the long-term water uptake, provided that there is no loss of small molecules as a product of hydrolysis and subsequent leaching out of the experimental panel. The *FRP for Marine Application DOI: http://dx.doi.org/10.5772/intechopen.101332*

**Figure 7.** *Long-term water uptake data: experimental and Eq. (15).*

initial diffusivity *D*<sup>1</sup> is determined by the initial slope, and the second diffusivity *D*<sup>2</sup> is determined by the subsequent part with a distinctly different slope taking only the linear portion. The water uptake is the summation of mass increase for both the diffusivity and the total mass increase *Mt* at any given time *t* is given by:

$$\begin{split} M\_{\mathbf{f}} &= \left( 1 - \frac{8}{\pi^2} \sum\_{n=0}^{\infty} \frac{1}{\left( 2n + 1 \right)^2} e^{\frac{-D\_1 (2n+1)^2 x\_{\mathbf{f}}^2}{4^2}} \right) M\_{1\infty} \\ &+ \left( 1 - \frac{8}{\pi^2} \sum\_{n=0}^{\infty} \frac{1}{\left( 2n + 1 \right)^2} e^{\frac{-D\_2 (2n+1)^2 x\_{\mathbf{f}}^2}{4^2}} \right) M\_{2\infty} \end{split} \tag{17}$$

where *2 l* is the length of the diffusion path (=thickness of the panel), and *M1*<sup>∞</sup> and *M2*<sup>∞</sup> are fractions of final mass uptake *M*∞.

A similar expression is a modified Jacob-Jones model [43, 49, 50]:

$$M\_{l} = M\_{1} \left\{ 1 - \exp \left[ -7.3 \left( \frac{D\_{1}t}{b^{2}} \right)^{0.75} \right] \right\} + M\_{2} \left\{ 1 - \exp \left[ -7.3 \left( \frac{D\_{2}t}{b^{2}} \right)^{0.75} \right] \right\} \tag{18}$$

Here, *M*<sup>∞</sup> *= M*<sup>1</sup> *+ M*2*,* and *b =* thickness = *2 l* of Eq. (18) above.

Comparing Eqs. (17) and (18), it is more convenient to use the latter, although the equation is an approximate one.

## *5.3.1 Example*

The same water diffusion data of Example 5.1.1, which did not show good fit using the Fickian model with one diffusivity, is tested for the dual Fickian model, taking modified Jacob-Jones expression as in Eq. (18). The diffusivities were calculated as *<sup>D</sup>*<sup>1</sup> = 2.25 � <sup>10</sup>�<sup>9</sup> <sup>m</sup><sup>2</sup> /h and *<sup>D</sup>*<sup>2</sup> = 7 � <sup>10</sup>�<sup>10</sup> <sup>m</sup><sup>2</sup> /h. The theoretical and experimental data are plotted in **Figure 8** below. The resulting theoretical prediction is more accurate compared with simple Fickian model (**Figure 1**).

#### **5.4 Diffusion relaxation models**

In some models, apart from initial diffusion process of Fickian type, relaxation of the polymer chain segments is also considered, as the water ingress progresses. The water has a plasticizing effect, and hence the relaxational phenomenon, which involves segmental motion of macro-Brownian type, increases with the progress of

**Figure 8.** *Dual Fickian model fitted to experimental data of example 6.1.1.*

diffusion. The relaxation process in a polymer is related to the slow rearrangements of the chain segments and therefore, distribution of the free volume in the polymer, considering large number of different sizes of the segments in the network. The diffusion and relaxation were combined in a single model by adding the relaxation terms to a classical Fickian diffusion model. The two diffusion processes were assumed to be independent of each other. The mass uptake at any time interval, *t,* is given by:

$$M\_t = M\_d(t) + M\_{R\circ\circ}(1 - e^{-rt})\tag{19}$$

where *r* is a relaxation parameter (inverse of time), *Md* is mass uptake for initial phase, calculated by classical equations of diffusion, and *MR*<sup>∞</sup> is the saturation water content due to segmental relaxation process.

The above model is only applicable where the relaxation process is approximately commensurate with the experimental timescale, since a short-term experiment may not result in actual effect of segmental motion and relaxation of a thermoset, which has a very high relaxation time at the experimental temperature.

### **6. Nanocomposite**

Nanometric-sized materials are presently used as reinforcing fillers with polymers. The nanoparticles are defined as those that has at least one dimension below 100 nm. Due to the tiny size, the nanoparticles have very high surface area compared with volume, and hence, their force of attraction with a polymer is much higher compared with common fillers. In addition, the shaped nanoparticles such as rods, platelets, stacked layers, fibers, etc., impart good resistance to diffusion of gas and liquids in polymers.

Needless to say that the intermolecular forces between the nanoparticle and the polymer much depend on homogeneity and polarity. The force of attraction may be Van der Waals, hydrogen bond, polar attraction, dipole-dipole, etc. Some fillers also form covalent bonds too. The secondary valence bonds are physical bonds and are reversible, unlike the covalent bond, which is a chemical bond. Most common nanoparticles are carbon nanotubes, nanorods, nanofibers, graphene and graphene oxide, clays such as montmorillonite, layered silica, nano particles of minerals such as nano titanium dioxide, nano ceramics, etc.
