**Increased Wettability and Surface Free Energy of Polyurethane by Ultraviolet Ozone Treatment**

Ping Kuang and Kristen Constant

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60798

#### **Abstract**

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The wettability of polyurethane (PU) was altered using ultraviolet ozone (UVO) treatment. The effect of UVO treatment on PU surface chemistry was investigated with various experiments. The direct measurement of sessile drops was employed to quantify the static contact angle of different wetting liquids on homogeneous PU films with various UV ozone treatment times. The contact angle of DI water droplets was decreased to 17.2º from 70.04º after 5min UVO treatment. The surface free energy of PU films was 51.46mN/m prior to treatment and was increased to 71.5mN/m after being fully treated. X-ray Photoelectron Spectroscopy (XP) analysis shows a signifi‐ cant amount of polar functional species (C-O and C=O bonding) were formed on the PU surface by UVO treatment. Atomic Force Microscopy (AFM) characterization shows the PU surface morphology was different before and after UVO treatment. The effect of water washing on UVO treated surface was also investigated. An aging effect study indicates the UV ozone modification can sustain the improved wettability with limited hydrophobic recovery, where the DI water contact angle remains constant at around 22º after the UVO treatment.

**Keywords:** UV ozone treatment, Surface energy, Contact angles, Surface wettability

#### **1. Introduction**

Polyurethane (PU) is a versatile polymeric material which, due to its wide range of molecular weights, can exist in different solid forms. In industry, PU is commonly used as rigid and

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flexible foams, adhesives, coatings, and molds [1-2]. PU coatings and molds are formed by the reaction of organic isocyanates, high molecular weight polyols, and low molecular weight chain extenders, and they are usually elastomers [3]. Because elastomeric PU is frequently used in contact with various materials, optimizing the PU surface properties is critical for enhancing their performance. Furthermore, complete understanding and characterization of PU surface properties, such as its wettability, is helpful for various practical applications. Moreover, a wide range of novel applications can be realized if the PU surface properties can be modified and tailored [4].

One novel application of PU is for fabricating patterned microstructures using microtransfer molding, which is one of various soft lithography techniques [2]. It was shown that, when using PU microstructures as molds in soft lithography, modification of the surface is important to achieve improved wetting for the infiltration of slurries or sol-gel materials. The purpose of this work was to utilize a simple and economical way to improve the surface wettability of PU molds for the infiltration of aqueous solutions including slurries of oxide nanoparticles.

Surface treatments are frequently used to modify the surface chemistry and improve wetting characteristics of polymers prior to use in a range of technological applications. Such treat‐ ments are necessary because the intrinsic activities of the polymer surfaces are frequently too low to allow satisfactory adhesion of surface coatings or laminates. Oxidation processes in the gaseous phase introduce a range of functionalities at the polymer surface therefore increasing the polar component of free energy. Gas phase treatments require the impingement of reactive species, such as oxygen radicals, on the polymer surface. In order to alter PU membranes or coatings to obtain a PU surface with better wettability, different surface modification techni‐ ques such as multicomponent poly-addition reactions, oxygen plasma, and ultraviolet (UV) irradiation have been used [3-5].

Another simple yet effective method for polyurethane surface modification is UV ozone (UVO) treatment. UVO treatment has been used to modify the surface wettability of various polymers [6-13]. A summary of selected previous work appears in Table 1. However, no experimental work had been done on polyurethane before our work.

Here, we show that UVO can have a much more significant impact on polyurethane surface. In UVO treatment for polymer surface modification, there are two different wavelengths of UV radiation present, 184.9 nm (λ1) and 253.7 nm (λ2) [14]. When the molecules on the surface of, as an example, a thin PU film are exposed to UV light, the shorter-wavelength UV radiation (λ1) will induce excitation and dissociation of the polymeric molecules. This is known as a photo-sensitized oxidation process [11]. Wettability of the film surface may be changed if the excited and dissociated PU molecules acquire different surface energies during this process. In addition, atomic oxygen is simultaneously generated when the oxygen molecules in air are dissociated by the 184.9 nm UV light and the ozone molecules by the 253.7 nm UV light. Upon dissociation of the oxygen molecules in air by the 184.9 nm UV radiation, the atomic oxygen will readily react with molecular oxygen to form ozone. Most hydrocarbons in PU and the ozone molecules can absorb the 253.7 nm UV radiation. Photolysis of ozone occurs and more highly reactive oxygen atoms are produced. Therefore, in UVO treatment when both wave‐


**Table 1.** List of previous work of UVO treatment on various polymers

flexible foams, adhesives, coatings, and molds [1-2]. PU coatings and molds are formed by the reaction of organic isocyanates, high molecular weight polyols, and low molecular weight chain extenders, and they are usually elastomers [3]. Because elastomeric PU is frequently used in contact with various materials, optimizing the PU surface properties is critical for enhancing their performance. Furthermore, complete understanding and characterization of PU surface properties, such as its wettability, is helpful for various practical applications. Moreover, a wide range of novel applications can be realized if the PU surface properties can be modified

One novel application of PU is for fabricating patterned microstructures using microtransfer molding, which is one of various soft lithography techniques [2]. It was shown that, when using PU microstructures as molds in soft lithography, modification of the surface is important to achieve improved wetting for the infiltration of slurries or sol-gel materials. The purpose of this work was to utilize a simple and economical way to improve the surface wettability of PU molds for the infiltration of aqueous solutions including slurries of oxide nanoparticles.

Surface treatments are frequently used to modify the surface chemistry and improve wetting characteristics of polymers prior to use in a range of technological applications. Such treat‐ ments are necessary because the intrinsic activities of the polymer surfaces are frequently too low to allow satisfactory adhesion of surface coatings or laminates. Oxidation processes in the gaseous phase introduce a range of functionalities at the polymer surface therefore increasing the polar component of free energy. Gas phase treatments require the impingement of reactive species, such as oxygen radicals, on the polymer surface. In order to alter PU membranes or coatings to obtain a PU surface with better wettability, different surface modification techni‐ ques such as multicomponent poly-addition reactions, oxygen plasma, and ultraviolet (UV)

Another simple yet effective method for polyurethane surface modification is UV ozone (UVO) treatment. UVO treatment has been used to modify the surface wettability of various polymers [6-13]. A summary of selected previous work appears in Table 1. However, no experimental

Here, we show that UVO can have a much more significant impact on polyurethane surface. In UVO treatment for polymer surface modification, there are two different wavelengths of UV radiation present, 184.9 nm (λ1) and 253.7 nm (λ2) [14]. When the molecules on the surface of, as an example, a thin PU film are exposed to UV light, the shorter-wavelength UV radiation (λ1) will induce excitation and dissociation of the polymeric molecules. This is known as a photo-sensitized oxidation process [11]. Wettability of the film surface may be changed if the excited and dissociated PU molecules acquire different surface energies during this process. In addition, atomic oxygen is simultaneously generated when the oxygen molecules in air are dissociated by the 184.9 nm UV light and the ozone molecules by the 253.7 nm UV light. Upon dissociation of the oxygen molecules in air by the 184.9 nm UV radiation, the atomic oxygen will readily react with molecular oxygen to form ozone. Most hydrocarbons in PU and the ozone molecules can absorb the 253.7 nm UV radiation. Photolysis of ozone occurs and more highly reactive oxygen atoms are produced. Therefore, in UVO treatment when both wave‐

and tailored [4].

86 Wetting and Wettability

irradiation have been used [3-5].

work had been done on polyurethane before our work.

lengths are present, oxygen atoms are continuously created and ozone is continuously created and destroyed. Most importantly, the highly reactive gaseous species, the atomic oxygen and ozone molecules, are oxidizing agents that may react with polymer surfaces to form peroxy and hydroxyl radicals, hydro-peroxide, carbonyl, and carboxyl functionalities, which are responsible for the increased wettability of treated polymer surfaces [5, 12]. The number of these radical functionality groups produced during the treatment is dependent on the ozone concentration and exposure time, the presence of water vapor, and the distance between the PU surface and the UV radiation source, among other factors [12, 14]. While the surface may be greatly modified, the bulk of the PU films remain unchanged. The purpose of this study is to characterize the change of wettability of PU films under different UVO treatment times. It has been well established that contact angle goniometry can be used to examine the wettability and measure the change in surface energy of a polymeric surface [15]. Static contact angle measurement was used to investigate the changes of the wettability in this study. The surface chemistry and morphology were studied using X-ray photoelectron spectroscopy (XPS) and atomic force microscopy (AFM). The results of these investigations show that the UVO treatment significantly changes the surface properties of the PU films.

#### **1.1. Wettability study of PU films by contact angle measurements**

For static sessile droplet contact angle measurement, PU films were placed on the sample stage of the contact angle measuring system, shown in Figure 1. Distilled water and diiodomethane (CH2I2) (99%, Sigma-Aldrich) liquid were used for the contact angle measurements.

**Figure 1.** Contact angle measuring system

Table 2

**Liquid**

Contact angles of the two liquids on PU films with no UVO treatment and with maximum UVO treatment at 600 seconds are shown in Figure 2. It is clear that after UVO treatment, the water contact angle is dramatically reduced from ~70° to ~18°. For the non-polar CH2I2, with UVO treatment, the contact angle is moderately increased from ~16° to ~39°. treatment, the water contact angle is dramatically reduced from ~70° to ~18°. For the nonpolar CH2I2, with UVO treatment, the contact angle is moderately increased from ~16° to ~39°.

 Figure 2 Contact angle images of (a) DI water droplet on non-treated PU film, (b) DI water droplet on 600sec UV-ozone treated PU film, (c) CH2I2 droplet on non-treated PU film, and (d) CH2I2 droplet on 600sec UVozone treated PU film **Figure 2.** Contact angle images of (a) DI water droplet on non-treated PU film, (b) DI water droplet on 600sec UVozone treated PU film, (c) CH2I2 droplet on non-treated PU film, and (d) CH2I2 droplet on 600sec UV-ozone treated PU film

Average values of contact angles (degrees) on PU films with different UVO treatment times

5

**UVO time** 0sec 10sec 20sec 40sec 60sec 80sec 100sec 300sec 600sec

Distilled water *71.04 49.21 48.61 43.40 33.76 25.23 16.84 17.17 17.64*  Diiodomethane *17.20 46.80 48.95 48.34 46.64 43.30 41.61 40.06 39.95* 


**Table 2.** Average values of contact angles (degrees) on PU films with different UVO treatment times

**1.1. Wettability study of PU films by contact angle measurements**

Light source

**Figure 1.** Contact angle measuring system

ozone treated PU film

**Liquid**

Table 2

film

~39°.

88 Wetting and Wettability

For static sessile droplet contact angle measurement, PU films were placed on the sample stage of the contact angle measuring system, shown in Figure 1. Distilled water and diiodomethane

Contact angles of the two liquids on PU films with no UVO treatment and with maximum UVO treatment at 600 seconds are shown in Figure 2. It is clear that after UVO treatment, the water contact angle is dramatically reduced from ~70° to ~18°. For the non-polar CH2I2, with

 Figure 2 Contact angle images of (a) DI water droplet on non-treated PU film, (b) DI water droplet on 600sec UV-ozone treated PU film, (c) CH2I2 droplet on non-treated PU film, and (d) CH2I2 droplet on 600sec UV-

**Figure 2.** Contact angle images of (a) DI water droplet on non-treated PU film, (b) DI water droplet on 600sec UVozone treated PU film, (c) CH2I2 droplet on non-treated PU film, and (d) CH2I2 droplet on 600sec UV-ozone treated PU

Average values of contact angles (degrees) on PU films with different UVO treatment times

treatment, the water contact angle is dramatically reduced from ~70° to ~18°. For the nonpolar CH2I2, with UVO treatment, the contact angle is moderately increased from ~16° to

**(a) (b)**

**(c) (d)**

5

**UVO time** 0sec 10sec 20sec 40sec 60sec 80sec 100sec 300sec 600sec

Distilled water *71.04 49.21 48.61 43.40 33.76 25.23 16.84 17.17 17.64*  Diiodomethane *17.20 46.80 48.95 48.34 46.64 43.30 41.61 40.06 39.95* 

UVO treatment, the contact angle is moderately increased from ~16° to ~39°.

**1 mm**

Microscope Video camera

Monitor

(CH2I2) (99%, Sigma-Aldrich) liquid were used for the contact angle measurements.

**Figure 3.** Contact angles of distilled water and diiodomethane on PU films (Insert shows the middle regime when UVO time is between 20 and 100 seconds)

For quantitative study, values of the average contact angles for two liquids, distilled water, and diiodomethane, on PU films with various UVO treatment times, were measured and tabulated in Table 2. The UVO treatment times were chosen as intervals from 0 to 600 seconds for different samples to determine the saturation time. Each sample was only treated once for the specific UVO treatment time. At least three data sets (three droplets) were taken for each sample, which consisted of both left and right angles. Therefore, at least six contact angle values for each UVO treatment time were recorded and the average value was used for data analysis. For distilled water droplets on the PU surface, the contact angle is 71.04º before any UVO treatment. Therefore, without any surface modification, the PU surface is close to hydrophobic and the water droplets only partially wet the surface. Values measured here are within the range of contact angles reported in the literature, which vary between 65º and 75º for this system [16, 17]. When the PU surface is UVO treated, the water contact angle decreases. It was found that as the UVO treatment time was increased, the contact angle of distilled water decreases continuously, up to a point. Therefore, the PU surface becomes less hydrophobic and more hydrophilic. In contrast, before UVO treatment, the contact angle is 17.2º for the diiodomethane liquid. Hence, diiodomethane wets the PU surface well without any surface modification. However, after UVO treatment, the contact angle for diiodomethane increases. The standard deviation of each data set was less than 2º for distilled water and 1º for diiodo‐ methane.

The contact angles for both DI water and diiodomethane on PU are plotted in Figure 3 for different UVO treatment time. Even at a very short UVO treatment time (10 seconds), the contact angles were changed significantly for both liquids (an increase of ~30º for diiodome‐ thane and an increase of ~22º for DI water). This implies that in the very beginning of the UVO exposure, when the PU film was placed in the UVO chamber, the ozone quickly reacts with the PU surface and changes its surface chemical composition immediately. For DI water liquid, when 20-second UVO treatment was carried out on a different PU film, the contact angle was further reduced but at a slower rate. This is because the ozone reaction on the PU surface is very quick and reaches its maximum effect within a very short period of time (< 20 sec), and the decrease in contact angle is not proportional to the ozone exposure time. However, for longer UVO exposure times, for example, between 20 and 100 seconds, it can be seen that there is a linear relationship between the DI water contact angle and the UVO treatment time (Fig. 3 insert). This is mainly due to the effect of UV light radiation on the PU surface. Because the UV light dose is linearly proportional to exposure time, the PU surface chemistry is altered in the same fashion. In contrast, when the UVO exposure is less than 20 seconds, the ozone reaction is dominating so the linear correlation between the time of UV light radiation and the DI water contact angle cannot be clearly seen. However, for the treatment times between 20 and 100 seconds, UV light radiation becomes the dominant source for UV surface chemistry modification, and the decrease in contact angle is linearly proportional to the treatment time, with the water contact angle reaching the minimum of ~ 17°. For UVO treatment times greater than 100 seconds, the PU surface is completely changed by both ozone molecules and UV light, and the contact angle remains at ~ 17°. On the other hand, for the diiodomethane liquid, the contact angle at first increases from 17.2° to a maximum value of 48.9° with 20-second treatment time. This indicates that the ozone gas also has the maximum effect on the PU surface and the effect is saturated within 20 seconds. Also, the PU film surface becomes less wettable for diiodomethane. However, for UVO treatment times between 20 and 100 seconds, the contact angle of diiodomethane droplets decreased as the treatment time increased, and the decrease is also linearly proportional to the treatment time. This indicates that the PU surface also becomes more wetting for diiodomethane after longer UV light exposure. However, the rate of decrease is lower than that of DI water, and the contact angle approaches the constant value of ~ 40° for UVO treatment greater than 100 seconds, when the UVO treatment effect is saturated.

Therefore, we can conclude that there are three regimes in UVO treatment on PU surfaces. The first stage (< 20 seconds) is when the ozone effect is dominant. For the second stage, the ozone effect diminishes and a linear UV effect takes over (20–100 seconds). Finally, in the third stage, the PU surface is fully treated (> 100 seconds).

#### **1.2. PU surface free energy calculation**

and more hydrophilic. In contrast, before UVO treatment, the contact angle is 17.2º for the diiodomethane liquid. Hence, diiodomethane wets the PU surface well without any surface modification. However, after UVO treatment, the contact angle for diiodomethane increases. The standard deviation of each data set was less than 2º for distilled water and 1º for diiodo‐

The contact angles for both DI water and diiodomethane on PU are plotted in Figure 3 for different UVO treatment time. Even at a very short UVO treatment time (10 seconds), the contact angles were changed significantly for both liquids (an increase of ~30º for diiodome‐ thane and an increase of ~22º for DI water). This implies that in the very beginning of the UVO exposure, when the PU film was placed in the UVO chamber, the ozone quickly reacts with the PU surface and changes its surface chemical composition immediately. For DI water liquid, when 20-second UVO treatment was carried out on a different PU film, the contact angle was further reduced but at a slower rate. This is because the ozone reaction on the PU surface is very quick and reaches its maximum effect within a very short period of time (< 20 sec), and the decrease in contact angle is not proportional to the ozone exposure time. However, for longer UVO exposure times, for example, between 20 and 100 seconds, it can be seen that there is a linear relationship between the DI water contact angle and the UVO treatment time (Fig. 3 insert). This is mainly due to the effect of UV light radiation on the PU surface. Because the UV light dose is linearly proportional to exposure time, the PU surface chemistry is altered in the same fashion. In contrast, when the UVO exposure is less than 20 seconds, the ozone reaction is dominating so the linear correlation between the time of UV light radiation and the DI water contact angle cannot be clearly seen. However, for the treatment times between 20 and 100 seconds, UV light radiation becomes the dominant source for UV surface chemistry modification, and the decrease in contact angle is linearly proportional to the treatment time, with the water contact angle reaching the minimum of ~ 17°. For UVO treatment times greater than 100 seconds, the PU surface is completely changed by both ozone molecules and UV light, and the contact angle remains at ~ 17°. On the other hand, for the diiodomethane liquid, the contact angle at first increases from 17.2° to a maximum value of 48.9° with 20-second treatment time. This indicates that the ozone gas also has the maximum effect on the PU surface and the effect is saturated within 20 seconds. Also, the PU film surface becomes less wettable for diiodomethane. However, for UVO treatment times between 20 and 100 seconds, the contact angle of diiodomethane droplets decreased as the treatment time increased, and the decrease is also linearly proportional to the treatment time. This indicates that the PU surface also becomes more wetting for diiodomethane after longer UV light exposure. However, the rate of decrease is lower than that of DI water, and the contact angle approaches the constant value of ~ 40° for UVO treatment greater than 100 seconds, when the UVO treatment effect is

Therefore, we can conclude that there are three regimes in UVO treatment on PU surfaces. The first stage (< 20 seconds) is when the ozone effect is dominant. For the second stage, the ozone effect diminishes and a linear UV effect takes over (20–100 seconds). Finally, in the third stage,

methane.

90 Wetting and Wettability

saturated.

the PU surface is fully treated (> 100 seconds).

For a liquid droplet on a solid surface in equilibrium, the contact angle can be calculated by the well-known Young's equation:

$$\cos \theta = \frac{\mathcal{Y}\_{sv} - \mathcal{Y}\_{sl}}{\mathcal{Y}\_{lv}} \tag{1}$$

where *θ* is the contact angle, *γsv*, *γlv*, and *γsl* are the surface energy of solid against vapor, the surface energy of liquid against vapor, and the surface tension at the solid–liquid interface, respectively. Furthermore, Owens and Wendt proposed a general equation for calculating the surface free energy of solids based on Young's equation, which can also be applied for low surface energy materials such as polymers [18]. It can be seen that the surface tension at the solid–liquid interface, *γsl*, can be eliminated for the calculation from the equation:

$$1 + \cos \theta = 2\sqrt{\nu\_{sv}^d} \left(\frac{\sqrt{\nu\_{lv}^d}}{\mathcal{I}\_{lv}}\right) + \sqrt{\nu\_{sv}^h} \left(\frac{\sqrt{\nu\_{lv}^h}}{\mathcal{I}\_{lv}}\right) \tag{2}$$

where *γlv <sup>d</sup>* , and *γlv <sup>h</sup>* are the dispersive component (non-polar) and hydrogen bonding (polar) component of the surface energy of a given liquid against vapor, respectively (*γlv* =*γlv <sup>d</sup>* <sup>+</sup> *<sup>γ</sup>lv h* ), and *γsv <sup>d</sup>* , *γsv <sup>h</sup>* are the dispersive component and hydrogen component of the surface free energy of the solid against vapor. The dispersive component is contributed to by the dispersive van der Waals forces between the liquid and solid and the hydrogen component includes nondis‐ persive forces such as polar forces and hydrogen bonding forces. Additionally, (*γsv* =*γsv <sup>d</sup>* <sup>+</sup> *<sup>γ</sup>sv <sup>h</sup>* ). The surface free energy of the PU film with different UVO treatment times can be calculated, since the component and the total surface free energy values are known for two liquids (Table 3), and the contact angle for each liquid droplet on PU surface at different UVO treatment time was measured. The total surface energies for PU films at different UVO treatment times, as well as their dispersive and hydrogen component, are plotted in Figure 4. For PU film without any UVO treatment, the total surface free energy is 51.5 mN/m, and the dispersive and hydrogen component of the surface energy is 46.1 mN/m and 5.4 mN/m, respectively. Without any treatment, the PU surface consists of long-chain molecules and minimum amount of high energy hydrogen bonds and other radical groups. Therefore, the hydrogen component of the surface free energy is very small, and polar liquids, such as water, will not wet the surface well when they are placed in contact with PU. On the contrary, the contact angle of diiodomethane on a non-treated PU surface is very small (~17.2°). This is because diiodomethane is a non-polar liquid, so it easily wets the untreated, non-polar PU surface, which has a high dispersive (non-polar) component value. As soon as the PU film is placed into the UVO chamber (UVO time < 20 seconds), the ozone molecules immediately reacts with the PU molecules on the film surface, leading to a significant increase of the radical groups with hydrogen bonds. The direct result is an increased hydrogen component and a decreased non-polar component of the surface free energy. Chain scission at the PU surface happens quickly and the long chains of polymer molecules are broken by the highly reactive ozone gas. Therefore, a large number of high-energy polar bonds of the broken chains are generated and exposed on the surface. The direct result is a much improved wettability for DI water, which is evident by the much smaller contact angle. In contrast, when non-polar liquids such as diiodomethane are placed on the treated PU surface, the polymer surface is less wetting. In addition, the number of long chain molecules decreases as more polar ends and hydrogen radical groups are generated on the PU surface. Therefore, the dispersive component of surface free energy decreases. For UVO treatment time 20–100 seconds, when the PU surface is modified mainly by the UV radiation, the hydrogen component (23.2–39.1 mN/m) also increases linearly, corresponding to the linear decrease of the DI water contact angles. Similarly, the small increase in the dispersive component (30.1–32.5 mN/m) corresponds to the decrease of diiodomethane contact angles (Fig. 4). Increases in both components indicates that the UV ozone improves both van der Waals interactions and hydrogen bond interactions between the solid and liquid. At UVO treatment time longer than 100 seconds, the PU surface is completely modified and the surface free energy reaches the maximum value of 71.5 mN/m, which is an 38.8% increase compared to an untreated PU surface.


**Table 3.** Surface tensions (in mN/m) of the two testing liquids [19]

**Figure 4.** Surface energies of PU solid films at different UVO times

#### **1.3. Characterization of surface chemistry change**

groups with hydrogen bonds. The direct result is an increased hydrogen component and a decreased non-polar component of the surface free energy. Chain scission at the PU surface happens quickly and the long chains of polymer molecules are broken by the highly reactive ozone gas. Therefore, a large number of high-energy polar bonds of the broken chains are generated and exposed on the surface. The direct result is a much improved wettability for DI water, which is evident by the much smaller contact angle. In contrast, when non-polar liquids such as diiodomethane are placed on the treated PU surface, the polymer surface is less wetting. In addition, the number of long chain molecules decreases as more polar ends and hydrogen radical groups are generated on the PU surface. Therefore, the dispersive component of surface free energy decreases. For UVO treatment time 20–100 seconds, when the PU surface is modified mainly by the UV radiation, the hydrogen component (23.2–39.1 mN/m) also increases linearly, corresponding to the linear decrease of the DI water contact angles. Similarly, the small increase in the dispersive component (30.1–32.5 mN/m) corresponds to the decrease of diiodomethane contact angles (Fig. 4). Increases in both components indicates that the UV ozone improves both van der Waals interactions and hydrogen bond interactions between the solid and liquid. At UVO treatment time longer than 100 seconds, the PU surface is completely modified and the surface free energy reaches the maximum value of 71.5

mN/m, which is an 38.8% increase compared to an untreated PU surface.

*Water* 72.8 21.8 51.0 *Diiodomethane* 50.8 50.4 0.4

0 100 200 300 400 500 600

UVO time (sec)

*<sup>d</sup> γlv*

 gsv d

 gsv h

gsv

*h*

**Liquid** *γlv γlv*

**Table 3.** Surface tensions (in mN/m) of the two testing liquids [19]

0

**Figure 4.** Surface energies of PU solid films at different UVO times

10

20

30

Surface free energy (mN/m)

40

50

60

70

92 Wetting and Wettability

In order to better understand the change in the PU surface by UVO treatment, an XPS study was done for untreated and 5 min UVO treated PU films. Low resolution scans of both films have the characteristic peaks corresponding to carbon (C 1s), oxygen (O 1s), and nitrogen (N1s) (scans not shown) [20]. Furthermore, the scans show additional peaks corresponding to sulfur (S 1s), which is a common element in typical polyurethane. It also shows that, after 5 minutes of UVO treatment, the oxygen level on the PU surface was considerably increased.

Figure 5 High resolution XPS spectra showing the deconvoluted C1s envelopes for (a) untreated polyurethane film and (b) 5 min UVO treated PU film, and (c) 5 min UVO treated PU film after washing **Figure 5.** High resolution XPS spectra showing the deconvoluted C1s envelopes for (a) untreated polyurethane film and (b) 5 min UVO treated PU film, and (c) 5 min UVO treated PU film after washing

For detailed chemical analysis, Figure 5 (a) and (b) show the high-resolution spectra of the O 1s peaks for the untreated PU film and the 5min UVO treated PU film. The C 1s spectrum of

of the O 1s peaks for the untreated PU film and the 5min UVO treated PU film. The C 1s spectrum of the untreated PU film can be deconvoluted into four sub-peaks (Fig. 5(a)) [20].

For detailed chemical analysis, Figure 5 (a) and (b) show the high-resolution spectra

13

the untreated PU film can be deconvoluted into four sub-peaks (Fig. 5(a)) [20]. The highresolution spectra of the C 1s peaks for the samples were plotted in Figure 5 for further surface chemical characterizations. The C 1s spectrum of the untreated PU film can be deconvoluted into four sub-peaks (Fig. 5(a)) [20]. The peak at the lowest binding energy (285.0 eV) corre‐ sponds to (-C-C-) and (-C-H-) bonding (denoted **C1** in Fig. 1). The (-N-CO-O-) group corre‐ sponds to the second peak (**C2**) and is located at around 288.5 eV [4]. The third peak at 286.3 eV (**C3**) corresponds to the (-C-O-C-) group, where carbon atoms are single-bonded to oxygen atoms. A very small peak (**C4**) can be located at 287.6 eV, which corresponds to urea groups (-N-CO-N-). The C 1s spectrum of PU film shows significant chemistry change after 5 min UVO treatment (Fig. 5(b)). For the 5 min UVO treated PU surface, the oxygen-carbon double bonding (**C2**, **C4**) and single bonding (**C3**) peaks are much higher than those of the untreated PU surface. The significant increase of the oxygen content in PU surface is the direct result of UV ozone treatment, which breaks the long chain molecules, and atomic oxygen or ozone gas readily react and form oxygen-carbon single and double bonding groups. For the 5 min UVO treated PU, the number of oxygen-carbon bonding groups is saturated. Therefore, the XPS spectrum shows the maximum intensity of the peaks for various oxygen-carbon bonding species. The quantitative studies of the different species are shown in Table 4.


**Table 4.** Surface composition (area %) of untreated and 5 min UVO treated polyurethane thin films

Table 4 shows the surface chemical compositions for both untreated and treated PU films, represented by the integrated XPS intensity areas under each peak. The area of **C1** is reduced from 68% to 51%, while the areas of **C2**, **C3**, and **C4** increase for the 5 min treated sample. For comparison, the ratio of the areas of (C-O) and (C=O) bonds to the area of the (-C-C-) bonding (**C1/C2** and **C1/C3**) was taken for both samples. The ratio of the integrated areas **C1/C2** and **C1/C3** were 3.66 and 5.39, respectively, for the untreated sample. This is because carbon–carbon bonding is the dominant species on the pristine polyurethane surface. For the 5 min UVO treated PU surface, the ratios of **C1/C2** and **C1/C3** were reduced to 1.71 and 2.98, respectively. Therefore, the C-C bonds were reduced by about 50% after 5 min UVO treatment. The decrease in the area ratio after UVO treatment also implies the amount of C-O and C=O bonding species is increased. The increase of hydrophilic carbon-oxygen bonds is the main reason for the increased wettability of the PU films.

#### **1.4. Surface morphology change by UVO**

The 5µm x 5µm 3D AFM images in Figure 6(a) and (b) show the surface morphology of asprepared, untreated and 5 min UVO treated PU films, respectively. The AFM image shows a very smooth surface morphology for the pristine PU surface. The area roughness study showed the root mean square roughness, R*rms*, was 0.392 nm. The 5 min UVO treated sample showed a different surface morphology. The z-axis of the 3D image is 20 nm, compared to the 6 nm for the untreated sample. It shows many straight pillars with a height of around 3–8 nm. The R*rms* was also increased significantly to 2.073 nm, more than 5 times rougher than the untreated sample. The pillars are low molecular weight oxidized species and possibly the tips of broken chains of PU molecules created after UVO exposure. The AFM images (Figure 6) were taken within 1 hour of the 5 min UV ozone treatment in a dry environment. Therefore, the low molecular weight oxidized species could still be observed. It is intuitive to state that, from previous studies of different short UVO treatment times (Table 2), as the UVO treatment time increases (10–100 sec), more and more low molecular weight oxidized species were created on the surface, thus the contact angle of water on treated PU surface gradually decreases (from 49° to 17°). At treatment times longer than 100 sec, the low molecular weight oxidized species are saturated on the PU film surface, and the water contact angle stabilizes around 17°. Furthermore, the low molecular weight oxidized species can be identified in the XPS plot, which shows the fraction of oxygen-containing species is significantly increased for the 5 min UVO treated samples. Therefore, the oxygen groups (**C2**, **C3**, and **C4**) are created by the UV ozone where chain scission takes place and oxygen-containing free radicals are formed, and they are observed as straight pillars in the AFM image.

The advantage of UVO treatment is that it does not physically remove material from the surface. This is preferred to oxygen plasma etching, which can remove a significant amount of material [20]. UVO treatment only affects the surface of the PU film by breaking the long chain (-C-C-) bonds and inserting atomic oxygen and ozone molecules at the chain ends to create C-O and C=O bonding, resulting in a much more hydrophilic surface. This nondestructive surface modification method provides another simple and economical approach for polymer surface wettability functionalization.

#### **1.5. Effect of ultrasonic washing for the UVO treated sample**

the untreated PU film can be deconvoluted into four sub-peaks (Fig. 5(a)) [20]. The highresolution spectra of the C 1s peaks for the samples were plotted in Figure 5 for further surface chemical characterizations. The C 1s spectrum of the untreated PU film can be deconvoluted into four sub-peaks (Fig. 5(a)) [20]. The peak at the lowest binding energy (285.0 eV) corre‐ sponds to (-C-C-) and (-C-H-) bonding (denoted **C1** in Fig. 1). The (-N-CO-O-) group corre‐ sponds to the second peak (**C2**) and is located at around 288.5 eV [4]. The third peak at 286.3 eV (**C3**) corresponds to the (-C-O-C-) group, where carbon atoms are single-bonded to oxygen atoms. A very small peak (**C4**) can be located at 287.6 eV, which corresponds to urea groups (-N-CO-N-). The C 1s spectrum of PU film shows significant chemistry change after 5 min UVO treatment (Fig. 5(b)). For the 5 min UVO treated PU surface, the oxygen-carbon double bonding (**C2**, **C4**) and single bonding (**C3**) peaks are much higher than those of the untreated PU surface. The significant increase of the oxygen content in PU surface is the direct result of UV ozone treatment, which breaks the long chain molecules, and atomic oxygen or ozone gas readily react and form oxygen-carbon single and double bonding groups. For the 5 min UVO treated PU, the number of oxygen-carbon bonding groups is saturated. Therefore, the XPS spectrum shows the maximum intensity of the peaks for various oxygen-carbon bonding species. The

quantitative studies of the different species are shown in Table 4.

UVO treated DI water washed

94 Wetting and Wettability

increased wettability of the PU films.

**1.4. Surface morphology change by UVO**

**Peak area (%) C1 C2 C3 C4 C1/C2 C1/C3** Untreated *68.35 18.69 12.68 0.28* **3.66 5.39** UVO treated *51.46 30.15 17.29 1.10* **1.71 2.98**

Table 4 shows the surface chemical compositions for both untreated and treated PU films, represented by the integrated XPS intensity areas under each peak. The area of **C1** is reduced from 68% to 51%, while the areas of **C2**, **C3**, and **C4** increase for the 5 min treated sample. For comparison, the ratio of the areas of (C-O) and (C=O) bonds to the area of the (-C-C-) bonding (**C1/C2** and **C1/C3**) was taken for both samples. The ratio of the integrated areas **C1/C2** and **C1/C3** were 3.66 and 5.39, respectively, for the untreated sample. This is because carbon–carbon bonding is the dominant species on the pristine polyurethane surface. For the 5 min UVO treated PU surface, the ratios of **C1/C2** and **C1/C3** were reduced to 1.71 and 2.98, respectively. Therefore, the C-C bonds were reduced by about 50% after 5 min UVO treatment. The decrease in the area ratio after UVO treatment also implies the amount of C-O and C=O bonding species is increased. The increase of hydrophilic carbon-oxygen bonds is the main reason for the

The 5µm x 5µm 3D AFM images in Figure 6(a) and (b) show the surface morphology of asprepared, untreated and 5 min UVO treated PU films, respectively. The AFM image shows a

**Table 4.** Surface composition (area %) of untreated and 5 min UVO treated polyurethane thin films

*58.92 23.92 16.11 1.05* **2.46 3.66**

The effect of water washing on the UVO treated PU film was also characterized. Similar studies have been done for other polymer materials with UVO treatment on the surface [10]. In this study, for a polyurethane film surface with 5 min UVO treatment, the sample was immediately placed in an ultrasonic DI water bath for 5 minutes right after UVO treatment. The sample was then dried by blowing air, and characterization was carried out within an hour. As shown in Figure 5 (c), the XPS spectrum of the washed sample has the same carbon (C 1s) peaks as the untreated sample and the 5 min UVO treated sample without washing. Compared to the treated sample without washing (Fig. 5(b)), the intensity of the peaks for oxygen-containing species (**C2, C3**) is considerably reduced for the washed sample. When compared to the pristine PU surface (Fig. 5(a)), the intensity of those peaks is still noticeably higher for the washed sample. Table 4 shows the quantitative surface composition study and indicates the **C1** area is increased to ~59% for washed sample, compared to 51% for the treated sample without washing. **C1/C2** and **C1/C3** ratios of the integrated areas were 2.46 and 3.66, respectively. This indicates that some of the low molecular weight oxygen-containing species created on the PU

Figure 6 The 5μm×5μm 3D AFM images of (a) untreated polyurethane thin film with rms roughness *w*= 0.392 nm, (b) polyurethane thin film treated by 5 min UV ozone with *w*=2.073 nm, and (c) 5 min UV ozone treated PU thin film after washing with *w*=0.321 nm. **Figure 6.** The 5µm×5µm 3D AFM images of (a) untreated polyurethane thin film with rms roughness *w*= 0.392 nm, (b) polyurethane thin film treated by 5 min UV ozone with *w*=2.073 nm, and (c) 5 min UV ozone treated PU thin film after washing with *w*=0.321 nm.

surface by UV ozone treatment were removed by water washing. Nevertheless, the washed sample still shows more oxygen content in the XPS spectrum than the untreated sample, and the oxygen content level is sustained days after washing. Furthermore, the AFM image of the UVO treated sample after washing is shown in Figure 6(c). Indeed, the surface roughness of the washed sample is quite similar to that of the pristine PU surface, and R*rms* is 0.321 nm, which is slightly lower than that of the untreated sample. This is because the low molecular weight oxidized species have been dissolved and removed by washing in the ultrasonic DI water bath, which resulted in a flat, smooth PU surface, as observed by the AFM studies. The XPS results confirmed that the oxygen content levels on the PU surface for the washed sample were lower than for the UVO treated sample. The DI water contact angle for the washed sample was ~48°, which is higher than the UVO fully treated sample (17°, Table 2), but it is still lower than the PU films without UVO treatment (70°). 16 The 5µm x 5µm 3D AFM images in Figure 6(a) and (b) show the surface morphology of as-prepared, untreated and 5 min UVO treated PU films, respectively. The AFM image shows a very smooth surface morphology for the pristine PU surface. The area roughness study showed the root mean square roughness, R*rms*, was 0.392 nm. The 5 min UVO treated

#### **1.6. Hydrophobic recovery**

Aging of the UVO treatment on PU surfaces was also investigated to study the hydrophobic recovery. The DI water contact angles on 5 min UVO treated PU films were measured at different aging times under ambient conditions (Fig. 7). The DI water contact angle on a 24 hour old PU film was 17.2º, which was nearly the same as the contact angle measured immediately after the UVO treatment (dotted line, Fig. 7). The contact angle increased to 19.8º after 2 days of aging and reached a constant value of about 21–22º after 3 days. Nevertheless, the increase in the water contact angle is negligible, since the contact angle for the untreated PU films is significantly higher at ~70° (Fig. 7 dashed line). This implies that the effect of UVO treatment is sustained with much improved wettability, and the hydrophobic recovery is insignificant. This is because in polar and hydrophilic species, the single and double carbonoxygen bonds stay intact on the PU surface permanently after UVO treatment. It was also noted that the transparent PU film yellowed after the 5 min UVO treatment, but the yellow color disappeared after a few days. The yellowing effect is possibly caused by the high UV light exposure on the PU surface during the UVO treatment.

**Figure 7.** Change of the water static contact angle values versus aging time for polyurethane thin films after 5 minutes of UV ozone exposure

#### **1.7. Wetting on PU grating structure**

surface by UV ozone treatment were removed by water washing. Nevertheless, the washed sample still shows more oxygen content in the XPS spectrum than the untreated sample, and the oxygen content level is sustained days after washing. Furthermore, the AFM image of the UVO treated sample after washing is shown in Figure 6(c). Indeed, the surface roughness of the washed sample is quite similar to that of the pristine PU surface, and R*rms* is 0.321 nm, which is slightly lower than that of the untreated sample. This is because the low molecular weight oxidized species have been dissolved and removed by washing in the ultrasonic DI water bath, which resulted in a flat, smooth PU surface, as observed by the AFM studies. The XPS results confirmed that the oxygen content levels on the PU surface for the washed sample were lower than for the UVO treated sample. The DI water contact angle for the washed sample was ~48°, which is higher than the UVO fully treated sample (17°, Table 2), but it is still lower than the

The 5µm x 5µm 3D AFM images in Figure 6(a) and (b) show the surface morphology of as-prepared, untreated and 5 min UVO treated PU films, respectively. The AFM image shows a very smooth surface morphology for the pristine PU surface. The area roughness study showed the root mean square roughness, R*rms*, was 0.392 nm. The 5 min UVO treated

Figure 6 The 5μm×5μm 3D AFM images of (a) untreated polyurethane thin film with rms roughness *w*= 0.392 nm, (b) polyurethane thin film treated by 5 min UV ozone with *w*=2.073 nm, and (c) 5 min UV ozone treated

**Figure 6.** The 5µm×5µm 3D AFM images of (a) untreated polyurethane thin film with rms roughness *w*= 0.392 nm, (b) polyurethane thin film treated by 5 min UV ozone with *w*=2.073 nm, and (c) 5 min UV ozone treated PU thin film after

16

PU films without UVO treatment (70°).

PU thin film after washing with *w*=0.321 nm.

washing with *w*=0.321 nm.

*4. Surface morphology change by UVO* 

96 Wetting and Wettability

**(a) (b)**

**(c)**

Microstructures with increased surface area and modulations possess more complicated surface wetting properties than flat surfaces. In general, due to the topography, such structures tend to be more hydrophobic with much larger apparent contact angles for different liquids. It is of particular interest to understand and manipulate the surface wettability of complex structures for specific surface wetting needs. Surface treatment on polymeric structures can increase the surface free energy and, thus, improve the wettability. For this investigation, a one-layer (1L) PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio were fabricated by microtransfer molding (Fig. 8). The water contact angle on the 1L PU grating is increased significantly to ~138° (Fig. 9 (a)) compared to ~70° of the PU flat surface. This indicates the PU grating has become much more hydrophobic due to the surface corrugation, approaching a superhydrophobic state. Such a hydrophobic surface poses as a serious problem for complete conformal wetting of the surface and for removal of air gaps and liquid infiltration of gratings. After UVO treatment (720 sec) was carried out on the 1L PU grating structure, the water contact angle was reduced to ~67°, and the surface has changed from hydrophobic to hydrophilic state. This is mainly caused by increased surface free radicals and polar bonds on the polymer chains on the PU grating surface. Hence, the water droplet spread out and easily penetrated into the trenches of PU gratings. It has been shown that this surface treatment can benefit the infiltration of liquids into 3D microscale polymeric molding structure [21]. Therefore, UVO treatment can be a simple and economical surface modification method for various surfaces and structures.

**Figure 8.** SEM image of a 1L PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio Figure 8 SEM image of a 1L PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio

 Figure 9 Water contact angles of 1L PU grating (a) without UVO treatment and (b) with 720 sec UVO **Figure 9.** Water contact angles of 1L PU grating (a) without UVO treatment and (b) with 720 sec UVO treatment

*8. Polyurethane fibrillar tip array as bio-inspired adhesives* 

treatment

 Recently, a microstructured PU surface has been investigated for improved wettability and adhesion for novel applications using UVO treatment [22, 23]. Animal perching systems are being mimicked to realize similar locomotion capabilities for landing on rough or smooth surfaces with various inclinations. Bio-inspired adhesives comprising fibrillar arrays are similar to those used by geckos, spiders, and flies. Fibrillar bio-inspired adhesives provide high adhesion against a great variety of surfaces with different moisture conditions. Such microscale fibrillar structures can be easily fabricated using standard soft

21

#### **1.8. Polyurethane fibrillar tip array as bio-inspired adhesives**

structures for specific surface wetting needs. Surface treatment on polymeric structures can increase the surface free energy and, thus, improve the wettability. For this investigation, a one-layer (1L) PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio were fabricated by microtransfer molding (Fig. 8). The water contact angle on the 1L PU grating is increased significantly to ~138° (Fig. 9 (a)) compared to ~70° of the PU flat surface. This indicates the PU grating has become much more hydrophobic due to the surface corrugation, approaching a superhydrophobic state. Such a hydrophobic surface poses as a serious problem for complete conformal wetting of the surface and for removal of air gaps and liquid infiltration of gratings. After UVO treatment (720 sec) was carried out on the 1L PU grating structure, the water contact angle was reduced to ~67°, and the surface has changed from hydrophobic to hydrophilic state. This is mainly caused by increased surface free radicals and polar bonds on the polymer chains on the PU grating surface. Hence, the water droplet spread out and easily penetrated into the trenches of PU gratings. It has been shown that this surface treatment can benefit the infiltration of liquids into 3D microscale polymeric molding structure [21]. Therefore, UVO treatment can be a simple and economical surface modification method for various surfaces and structures.

98 Wetting and Wettability

**Figure 8.** SEM image of a 1L PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio

**1 mm**

**(a) (b)**

*8. Polyurethane fibrillar tip array as bio-inspired adhesives* 

Figure 8 SEM image of a 1L PU grating structure with 2.5 µm periodicity and 1:1 aspect ratio

 Figure 9 Water contact angles of 1L PU grating (a) without UVO treatment and (b) with 720 sec UVO treatment

**Figure 9.** Water contact angles of 1L PU grating (a) without UVO treatment and (b) with 720 sec UVO treatment

 Recently, a microstructured PU surface has been investigated for improved wettability and adhesion for novel applications using UVO treatment [22, 23]. Animal perching systems are being mimicked to realize similar locomotion capabilities for landing on rough or smooth surfaces with various inclinations. Bio-inspired adhesives comprising fibrillar arrays are similar to those used by geckos, spiders, and flies. Fibrillar bio-inspired adhesives provide high adhesion against a great variety of surfaces with different moisture conditions. Such microscale fibrillar structures can be easily fabricated using standard soft

21

Recently, a microstructured PU surface has been investigated for improved wettability and adhesion for novel applications using UVO treatment [22, 23]. Animal perching systems are being mimicked to realize similar locomotion capabilities for landing on rough or smooth surfaces with various inclinations. Bio-inspired adhesives comprising fibrillar arrays are similar to those used by geckos, spiders, and flies. Fibrillar bio-inspired adhesives provide high adhesion against a great variety of surfaces with different moisture conditions. Such microscale fibrillar structures can be easily fabricated using standard soft lithography techni‐ ques with polymeric elastomers such as polyurethane (Fig. 10). Subsequently, the fibrillar tips are enlarged by an "inking process" [23].

**Figure 10.** (a) SEM image of the polyurethane fibrillar array integrated into the perching/bio-inspired adhesion mecha‐ nism, (b) higher magnification of fibril enlarged tip [23] (Permission from reference 23)

The fibrillar arrays are hydrophobic in nature due to the low surface energy of PU surface and its surface corrugation. Surface treatment is necessary to improve its wettability and increase its surface free energy for better adhesion. In this reference, PU flat films exhibit similar wettability for water as illustrated in section 1 of this work, where the water angle is ~70° before UVO treatment and <15-17° after UVO treatment. For the fibrillar array patterns, the water contact angle is ~120° before UVO treatment. With UVO treatment, the water contact angle is significantly reduced to ~15-17° as well (Fig. 11 (a)). Unlike the flat PU film, the shear and tensile adhesion capacities of PU fibrillar arrays after UVO treatment were also enhanced significantly acting as bio-inspired adhesives. For demonstration, a perching system was made with the UVO treated fibrillar arrays placed on top of four flat elastomer pads, which were attached to the toes of four shape memory alloy (SMA) wires. The perching system was placed at a flat and smooth polyethylene sheet surface and then inclined at 75° with respect to the horizontal direction. The system can be sustained on the surface without falling off even with a 20 gram mass.

 Figure 11 (a) Water contact angles of PU plain film and fibrillar array prior to and after UVO treatment. (b) Perched system with PU fibrillar array on four elastomer pads adhered to a smooth polyethylene sheet inclined at 75° with respect to the horizontal direction [23]. (Permission from reference 23) **Figure 11.** (a) Water contact angles of PU plain film and fibrillar array prior to and after UVO treatment. (b) Perched system with PU fibrillar array on four elastomer pads adhered to a smooth polyethylene sheet inclined at 75° with re‐ spect to the horizontal direction [23]. (Permission from reference 23)

#### **2. Conclusion and summary**

**Conclusion and Summary** Studies have shown that, by using UVO treatment as a surface modification method, the wettability of polyurethane film surface can be significantly improved. The two liquids tested in this study showed different wetting phenomena. It was observed that, in the very beginning of the UVO treatment (< 20 seconds), the ozone gas molecules had an immediate effect on the PU surface and the contact angle for DI water was noticeably reduced. In the second stage of the UVO treatment (20–100 seconds), the UVO effect is linearly related to the treatment time, and the contact angles for both testing liquids decrease in a linear fashion until the PU surface is fully modified. Both polar and non-polar interactions between the liquids are improved by the UVO treatment in the second stage, and the total surface free energy of the PU surface is increased linearly until it is fully treated. The total surface free energy of a fully treated PU surface is 71.5 mN/m, which is a ~38.8% increase over the Studies have shown that, by using UVO treatment as a surface modification method, the wettability of polyurethane film surface can be significantly improved. The two liquids tested in this study showed different wetting phenomena. It was observed that, in the very beginning of the UVO treatment (< 20 seconds), the ozone gas molecules had an immediate effect on the PU surface and the contact angle for DI water was noticeably reduced. In the second stage of the UVO treatment (20–100 seconds), the UVO effect is linearly related to the treatment time, and the contact angles for both testing liquids decrease in a linear fashion until the PU surface is fully modified. Both polar and non-polar interactions between the liquids are improved by the UVO treatment in the second stage, and the total surface free energy of the PU surface is increased linearly until it is fully treated. The total surface free energy of a fully treated PU surface is 71.5 mN/m, which is a ~38.8% increase over the untreated PU surface. XPS analysis shows significant hydrophilic C-O and C=O bonding species were created on the PU surface by UVO treatment, which is the cause for higher surface free energy and improved surface wettability. Surface morphology study by AFM shows the PU surface roughness is increased by UVO treatment with increased amount of low molecular weight oxidized species and broken ends of long chain molecules, which can be removed by strong agitation such as water washing in ultrasonic bath. But the oxygen content on the washed sample is still higher than a pristine PU surface. Additional experiments with distilled water were also conducted for one-layer PU grating molds (2.5µm pitch) made by the micro-transfer molding (µTM) technique. It was shown that on the structured non-treated one-layer PU mold, hydrophobic wetting occurred and the contact angle of a water droplet was about 138 degrees. When sufficient UVO treatment was done on the mold, the water droplet had a much smaller contact angle and was pulled into the channels because of the improvement on the wetting charac‐ teristics of the PU mold. This opened up a door for future studies on the infiltration of aqueous

23

untreated PU surface. XPS analysis shows significant hydrophilic C-O and C=O bonding

species were created on the PU surface by UVO treatment, which is the cause for higher

surface free energy and improved surface wettability. Surface morphology study by AFM

shows the PU surface roughness is increased by UVO treatment with increased amount of

low molecular weight oxidized species and broken ends of long chain molecules, which can

slurry into 3-D polymeric structures at micro-size, and for improved adhesion for novel applications [21, 23].

### **Acknowledgements**

The authors would like to thank J. Anderegg for the assistance on XPS analysis and K.S. Nalwa on AFM analysis, and for meaningful discussions with Professor Zhiqun Lin. This work at the Ames Laboratory was supported by the U.S. Department of Energy—Basic Energy Sciences under contract No. DE-AC02-07CH11358.

### **Author details**

Ping Kuang1\* and Kristen Constant2

\*Address all correspondence to: kuangp2@rpi.edu

1 Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York, USA

2 Department of Materials Science & Engineering, Iowa State University of Science and Technology, Ames, Iowa, USA

#### **References**

23

Studies have shown that, by using UVO treatment as a surface modification method,

the wettability of polyurethane film surface can be significantly improved. The two liquids

tested in this study showed different wetting phenomena. It was observed that, in the very

beginning of the UVO treatment (< 20 seconds), the ozone gas molecules had an immediate

effect on the PU surface and the contact angle for DI water was noticeably reduced. In the

second stage of the UVO treatment (20–100 seconds), the UVO effect is linearly related to

the treatment time, and the contact angles for both testing liquids decrease in a linear fashion

until the PU surface is fully modified. Both polar and non-polar interactions between the

liquids are improved by the UVO treatment in the second stage, and the total surface free

energy of the PU surface is increased linearly until it is fully treated. The total surface free

energy of a fully treated PU surface is 71.5 mN/m, which is a ~38.8% increase over the

untreated PU surface. XPS analysis shows significant hydrophilic C-O and C=O bonding

species were created on the PU surface by UVO treatment, which is the cause for higher

surface free energy and improved surface wettability. Surface morphology study by AFM

shows the PU surface roughness is increased by UVO treatment with increased amount of

low molecular weight oxidized species and broken ends of long chain molecules, which can

 Figure 11 (a) Water contact angles of PU plain film and fibrillar array prior to and after UVO treatment. (b) Perched system with PU fibrillar array on four elastomer pads adhered to a smooth polyethylene sheet inclined at 75° with respect to the horizontal

**Figure 11.** (a) Water contact angles of PU plain film and fibrillar array prior to and after UVO treatment. (b) Perched system with PU fibrillar array on four elastomer pads adhered to a smooth polyethylene sheet inclined at 75° with re‐

Studies have shown that, by using UVO treatment as a surface modification method, the wettability of polyurethane film surface can be significantly improved. The two liquids tested in this study showed different wetting phenomena. It was observed that, in the very beginning of the UVO treatment (< 20 seconds), the ozone gas molecules had an immediate effect on the PU surface and the contact angle for DI water was noticeably reduced. In the second stage of the UVO treatment (20–100 seconds), the UVO effect is linearly related to the treatment time, and the contact angles for both testing liquids decrease in a linear fashion until the PU surface is fully modified. Both polar and non-polar interactions between the liquids are improved by the UVO treatment in the second stage, and the total surface free energy of the PU surface is increased linearly until it is fully treated. The total surface free energy of a fully treated PU surface is 71.5 mN/m, which is a ~38.8% increase over the untreated PU surface. XPS analysis shows significant hydrophilic C-O and C=O bonding species were created on the PU surface by UVO treatment, which is the cause for higher surface free energy and improved surface wettability. Surface morphology study by AFM shows the PU surface roughness is increased by UVO treatment with increased amount of low molecular weight oxidized species and broken ends of long chain molecules, which can be removed by strong agitation such as water washing in ultrasonic bath. But the oxygen content on the washed sample is still higher than a pristine PU surface. Additional experiments with distilled water were also conducted for one-layer PU grating molds (2.5µm pitch) made by the micro-transfer molding (µTM) technique. It was shown that on the structured non-treated one-layer PU mold, hydrophobic wetting occurred and the contact angle of a water droplet was about 138 degrees. When sufficient UVO treatment was done on the mold, the water droplet had a much smaller contact angle and was pulled into the channels because of the improvement on the wetting charac‐ teristics of the PU mold. This opened up a door for future studies on the infiltration of aqueous

direction [23]. (Permission from reference 23)

**2. Conclusion and summary**

spect to the horizontal direction [23]. (Permission from reference 23)

**Conclusion and Summary**

100 Wetting and Wettability


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102 Wetting and Wettability


#### **Chapter 5**

## **Wetting and Navier-Stokes Equation — The Manufacture of Composite Materials**

Mario Caccia, Antonio Camarano, Danilo Sergi, Alberto Ortona and Javier Narciso

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61167

#### **Abstract**

It is well known that there are several processes to manufacture composite materials, a large part of which consist in the infiltration of a liquid (matrix) through a porous medium (reinforcement). To perform these processes, both thermodynamics (wet‐ ting) and kinetics (Navier-Stokes) must be considered if a good quality composite material is sought. Although wetting and the laws that govern it have been well known for over 200 years, dating back to the original works of Young and Laplace, this is not the case with the Navier-Stokes equation, which remains so far unsolved. Although the Navier-Stokes equation, which describes the motion of a fluid, has been solved for many particular cases, such as the motion of a fluid through a pipe, which has resulted in the well-known Poiseuille equation, or the motion of a fluid through a porous media, described by the Darcy's law (empirical law obtained by Darcy), its general solution remains one of the greatest challenges of mathematicians today. Therefore, the objective of this chapter is to present the resolution of the Navier-Stokes equation with the laws of wetting for different cases of interest in the manufacture of composite materials.

**Keywords:** Wettability, Infiltration, Navier-Stokes

#### **1. Introduction**

Composite materials are an important and oftentimes used type of engineering materials with tuneable properties which depend on the materials combined. They are usually made of two

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

or more materials. The first phase, commonly referred to as "reinforcement" is constituted by a solid material, which can be employed in several morphologies such as fibers (short or continuous), particles, or as a co-continuous domain of a solid phase which occupies a fraction of the available volume (foams) [1-7]. The second phase, normally called "matrix," can be gaseous, liquid, or solid, and is in contact with the reinforcement through a proper interface. Composite materials are very common in nature, where a combination of fibers and matrix is often found. A good example is wood, which is composed of flexible fibers of cellulose embedded in a rigid matrix of lignin. Artificially produced composites are usually classified, according to the nature of their matrix, into metal, ceramic, or polymer matrix composites. Each type of composite presents its advantages and disadvantages, and together they cover a great scope of applications [1, 8-12].

Composite materials manufacturing can be accomplished through several techniques. Each technique is mainly the result of the trade-off between the target properties of the final product, the starting materials properties, and the final cost. In most of the cases, the matrix is in a liquid state during processing. A common action in composite materials manufacturing is "to infiltrate" the reinforcement with the matrix. When a liquid is used, the impregnation of the remaining free volume is highly dependent on how the first phase is placed in the space and on the physical properties of the liquid itself. In the case of polymeric matrix composites (PMC) techniques involving a liquid matrix are the only way to process them, as polymers are in a liquid state before their cross-linking in thermosets, or can be melted if they are thermoplastics. Also, ceramic matrix composites are processed through liquid routes (e.g., sol-gel techniques for oxide composites, or polymer infiltration and pyrolysis (PIP) [13-14] for ceramic matrix composites (CMC) produced by pre-ceramic precursors. In the latter case, where several impregnations are performed, the permeability of the porous materials decreases with the increasing of the final density of the composite under production). A widely utilized technique which stands among ceramic and metallic matrix composites (MMC) manufacturing is the socalled reactive silicon infiltration [15]. Its peculiarity is that, during processing, silicon is molten and by reacting with extra carbon present in the primary phase it yields silicon carbide. This method was not originally conceived to produce composite materials, but for the synthesis of bulk SiC, still, it proved to be a suitable methodology for ceramic matrix composites manu‐ facture as well.

Some industrial products obtained with this infiltration technology are pantographs, connect‐ ing rods, loudspeakers, piston rings, heat sinks, electro-brushes, engine cylinders, brake discs, etc. In most of these examples, primary phase (reinforcement) content is above 50% in volume. For reinforcement contents of such magnitude, the most commonly used technique for manufacturing composites is infiltration [1, 5-7, 16-18].

Actually, infiltration processes are not exclusive of materials manufacture, but can be observed in many everyday life activities, such as making coffee, where hot water is infiltrated into a porous bed of coffee particles. Perhaps one of the most important cases where infiltration phenomenon is observed is in the penetration of rainwater into the ground, which ends up filling aquifers that guarantee water supply to population in dry regions where the surface runoff is scarce. Because of the great importance of infiltration in geological sciences, it was in this field where the deepest and most detailed studies of this phenomenon were performed. Many advances in this area of knowledge have been expanded and applied to other scientific fields such as materials science. The clearest example is Darcy's law which is widely used in the design of composite materials. This empirical law was obtained by Henry Darcy whose original work published in 1856 is entitled: *"Les Fontaines Publiques de la Ville de Dijon"* (The Public Fountains of the City of Dijon). However, Darcy's law is not the only example that can be found in the literature. The relationship between water saturation and organic compounds in soils of different origins has resulted in the semiempirical equations of Brooks and Corey [19] and Van Genuchten [20], both widely used in composite materials.

The aim of the present chapter is to introduce the reader to the basic concepts of wetting and infiltration, and to combine these concepts to solve the equations governing the infiltration process in several particular cases that often arise in the composite materials manufacturing industry.

#### **2. Experimental**

or more materials. The first phase, commonly referred to as "reinforcement" is constituted by a solid material, which can be employed in several morphologies such as fibers (short or continuous), particles, or as a co-continuous domain of a solid phase which occupies a fraction of the available volume (foams) [1-7]. The second phase, normally called "matrix," can be gaseous, liquid, or solid, and is in contact with the reinforcement through a proper interface. Composite materials are very common in nature, where a combination of fibers and matrix is often found. A good example is wood, which is composed of flexible fibers of cellulose embedded in a rigid matrix of lignin. Artificially produced composites are usually classified, according to the nature of their matrix, into metal, ceramic, or polymer matrix composites. Each type of composite presents its advantages and disadvantages, and together they cover a

Composite materials manufacturing can be accomplished through several techniques. Each technique is mainly the result of the trade-off between the target properties of the final product, the starting materials properties, and the final cost. In most of the cases, the matrix is in a liquid state during processing. A common action in composite materials manufacturing is "to infiltrate" the reinforcement with the matrix. When a liquid is used, the impregnation of the remaining free volume is highly dependent on how the first phase is placed in the space and on the physical properties of the liquid itself. In the case of polymeric matrix composites (PMC) techniques involving a liquid matrix are the only way to process them, as polymers are in a liquid state before their cross-linking in thermosets, or can be melted if they are thermoplastics. Also, ceramic matrix composites are processed through liquid routes (e.g., sol-gel techniques for oxide composites, or polymer infiltration and pyrolysis (PIP) [13-14] for ceramic matrix composites (CMC) produced by pre-ceramic precursors. In the latter case, where several impregnations are performed, the permeability of the porous materials decreases with the increasing of the final density of the composite under production). A widely utilized technique which stands among ceramic and metallic matrix composites (MMC) manufacturing is the socalled reactive silicon infiltration [15]. Its peculiarity is that, during processing, silicon is molten and by reacting with extra carbon present in the primary phase it yields silicon carbide. This method was not originally conceived to produce composite materials, but for the synthesis of bulk SiC, still, it proved to be a suitable methodology for ceramic matrix composites manu‐

Some industrial products obtained with this infiltration technology are pantographs, connect‐ ing rods, loudspeakers, piston rings, heat sinks, electro-brushes, engine cylinders, brake discs, etc. In most of these examples, primary phase (reinforcement) content is above 50% in volume. For reinforcement contents of such magnitude, the most commonly used technique for

Actually, infiltration processes are not exclusive of materials manufacture, but can be observed in many everyday life activities, such as making coffee, where hot water is infiltrated into a porous bed of coffee particles. Perhaps one of the most important cases where infiltration phenomenon is observed is in the penetration of rainwater into the ground, which ends up filling aquifers that guarantee water supply to population in dry regions where the surface runoff is scarce. Because of the great importance of infiltration in geological sciences, it was in

manufacturing composites is infiltration [1, 5-7, 16-18].

great scope of applications [1, 8-12].

106 Wetting and Wettability

facture as well.

Although it will be discussed in detail in the upcoming sections of the chapter, it is important to clarify that infiltration, in most composite materials manufacturing processes, does not occur spontaneously (in the case of PMCs it does proceed spontaneously, but it is so slow that it is necessary to force it), so the liquid must be forced to infiltrate by applying an external pressure. From an engineering point of view, mainly two technologies have been developed for this purpose: the so-called gas-assisted pressure infiltration (GPI), where a gas is used to push the liquid; and another called squeeze-casting (SC) in which pressure is applied by means of a mechanical device. Each technique has its upsides and downsides, which will be discussed throughout this chapter. The main advantage of GPI over SC is that it allows a better control of physical parameters (pressure and temperature), and admits higher complexity in the manufactured parts. On the other hand, the main advantage of SC over GPI is that higher pressures can be achieved and the automatization of the process becomes easier.

First the GPI technique will be analyzed, and for this purpose, the example of a laboratory equipment developed at the University of Alicante will be used [10, 11, 21-29]. A scheme of the equipment is shown in Figure 1. The equipment consists of a pressure chamber (maximum admissible pressure at 300 °C of 5 MPa) equipped with a resistance furnace, which is thermally isolated from the chamber by the use of conventional refractory material (temperature of chamber wall is always below 100°C). The furnace has a power of 1500 W and can reach temperatures of 1000°C. For measurement and control of furnace temperature, K type thermocouples, connected to a temperature controller, are used. The pressure control is entirely performed by a system of pneumatic and electronic valves connected to a pressure controller. The pressure vessel is made entirely of stainless steel and consists of a cylindrical body with an internal diameter of 30 cm, an outer diameter of 34.6 cm, a height of 45 cm, and a wall thickness of 3 cm. The closing of the camera is made with 12 M16 mounting studs. Both, the electrical feed (to thermocouple input and power cables) and the gas inlet and outlet for pressurizing the chamber are positioned in the body. Additionally, the camera is connected to a primary vacuum pump, which allows the process to be performed in the absence of oxygen (low partial pressure).

The manufacturing process of a composite material using GPI can be described as follows: a preform, which consists of a porous bed of compacted particles, is placed into a quartz (or BNcoated graphite) crucible and a metal ingot is placed on top of it (see Figure 1b). The crucible is then introduced inside the pressure chamber and vacuum is applied during at least 30 min., until a pressure of about 100 Pa is reached. The chamber is heated under vacuum at a very low rate (3°C/min) up to 250°C, to remove humidity and gas adsorbed on particles. Subsequently, further heating is applied up to a temperature 50°C higher than the melting point of the metal at a rate of approximately 5°C/min. The maximum temperature is then maintained for 30 min followed by argon injection to reach the desired infiltration pressure. Once the preform has been infiltrated (approximately 2 min. under pressure) the crucible is pushed towards the bottom of the chamber, which is cold, and directional solidification occurs, ensuring a highquality composite (null porosity). Figure 2 shows a composite material obtained at the University of Alicante with this process, where it is possible to observe the absence of porosity and an intimate contact between reinforcement and matrix.

**Figure 1.** Schematic representation of a gas-assisted pressure infiltration (GPI) system developed at the University of Alicante and arrangement of the materials (reinforcement and matrix) inside the crucible.

At the University of Alicante, another equipment, in this case for squeeze-casting technique [30], was also developed. Figure 3 shows a schematic representation of the main body of this equipment and how it operates. The main feature of this apparatus is that it can work up to 450°C and at a maximum pressure of 150 MPa. To resist these operating conditions, special (W720 Bölher) steel was used to build the body of the equipment. Embedded resistances in the central body heat the system, and temperature is measured and controlled by K type thermo‐ couples connected to a temperature controller.

pressurizing the chamber are positioned in the body. Additionally, the camera is connected to a primary vacuum pump, which allows the process to be performed in the absence of oxygen

The manufacturing process of a composite material using GPI can be described as follows: a preform, which consists of a porous bed of compacted particles, is placed into a quartz (or BNcoated graphite) crucible and a metal ingot is placed on top of it (see Figure 1b). The crucible is then introduced inside the pressure chamber and vacuum is applied during at least 30 min., until a pressure of about 100 Pa is reached. The chamber is heated under vacuum at a very low rate (3°C/min) up to 250°C, to remove humidity and gas adsorbed on particles. Subsequently, further heating is applied up to a temperature 50°C higher than the melting point of the metal at a rate of approximately 5°C/min. The maximum temperature is then maintained for 30 min followed by argon injection to reach the desired infiltration pressure. Once the preform has been infiltrated (approximately 2 min. under pressure) the crucible is pushed towards the bottom of the chamber, which is cold, and directional solidification occurs, ensuring a highquality composite (null porosity). Figure 2 shows a composite material obtained at the University of Alicante with this process, where it is possible to observe the absence of porosity

**Figure 1.** Schematic representation of a gas-assisted pressure infiltration (GPI) system developed at the University of

At the University of Alicante, another equipment, in this case for squeeze-casting technique [30], was also developed. Figure 3 shows a schematic representation of the main body of this equipment and how it operates. The main feature of this apparatus is that it can work up to 450°C and at a maximum pressure of 150 MPa. To resist these operating conditions, special

Alicante and arrangement of the materials (reinforcement and matrix) inside the crucible.

and an intimate contact between reinforcement and matrix.

(low partial pressure).

108 Wetting and Wettability

**Figure 2.** Polarized light optical microscopy image of a composite material obtained by GPI. The composite material consists of needle coke (reinforcement) and copper (matrix).

**Figure 3.** Schematic representation of a squeeze-casting (SC) system developed at the University of Alicante. The oper‐ ating mode is indicated in the figure.

The manufacturing process of a composite material using SC can be described as follows: The system is first preheated to the operating temperature (between 350 and 450 °C depending on the alloy) to avoid early solidification of the metal matrix during infiltration. The preform is preheated (600 to 700°C) and the alloy is melted (150°C above the melting point) in two auxiliary furnaces. Thereafter, the preform is transferred to the main chamber, and the liquid metal is added. The desired pressure is applied as indicated in Figure 3. After infiltration, the system is cooled down and the manufactured material is removed as indicated in Figure 3.

The main problem with this technique is that the processing is complex because the times and temperatures must be very accurate to keep the metal from solidifying before penetrating into the preform. This disadvantage is also an advantage, since the high velocity of the process generates a finer microstructure and thus improves the mechanical properties of the final material.

Most importantly, it is possible to find in the market, both SC and GPI fully automated devices, but the basic operating principles are the same for the ones developed at University of Alicante with a scale factor between 2 and 10.

#### **3. Wetting basic laws**

One of the most relevant characteristics for the production of MMC through liquid processing routes is the wetting of the reinforcement by the liquid metals, which is quantified by meas‐ uring the contact angle between the reinforcement and matrix [31]. The contact angle between a flat substrate and a drop of liquid metal can be routinely measured by the method of the sessile drop. However, this measurement is not possible when the solid is in particulate form, in which case the wetting can be studied by means of liquid metal infiltration into compacted particulate samples [10, 11, 21-29].

Considering a solid, flat, non-deformable and chemically homogeneous surface in contact with a nonreactive liquid in the presence of a gaseous phase, if the liquid does not completely cover the solid, the liquid will form with the solid a contact angle θ, as depicted in Figure 4.

**Figure 4.** Configuration of a sessile drop on a substrate within which it forms a contact angle θ.

The equilibrium value of θ, which is used to define the wetting behavior of a fluid, can be written in terms of the balance of three energies present in the interfacial vapor-liquid-solid system, according to the well-known Young equation (1805):

The manufacturing process of a composite material using SC can be described as follows: The system is first preheated to the operating temperature (between 350 and 450 °C depending on the alloy) to avoid early solidification of the metal matrix during infiltration. The preform is preheated (600 to 700°C) and the alloy is melted (150°C above the melting point) in two auxiliary furnaces. Thereafter, the preform is transferred to the main chamber, and the liquid metal is added. The desired pressure is applied as indicated in Figure 3. After infiltration, the system is cooled down and the manufactured material is removed as indicated in Figure 3.

The main problem with this technique is that the processing is complex because the times and temperatures must be very accurate to keep the metal from solidifying before penetrating into the preform. This disadvantage is also an advantage, since the high velocity of the process generates a finer microstructure and thus improves the mechanical properties of the final

Most importantly, it is possible to find in the market, both SC and GPI fully automated devices, but the basic operating principles are the same for the ones developed at University of Alicante

One of the most relevant characteristics for the production of MMC through liquid processing routes is the wetting of the reinforcement by the liquid metals, which is quantified by meas‐ uring the contact angle between the reinforcement and matrix [31]. The contact angle between a flat substrate and a drop of liquid metal can be routinely measured by the method of the sessile drop. However, this measurement is not possible when the solid is in particulate form, in which case the wetting can be studied by means of liquid metal infiltration into compacted

Considering a solid, flat, non-deformable and chemically homogeneous surface in contact with a nonreactive liquid in the presence of a gaseous phase, if the liquid does not completely cover the solid, the liquid will form with the solid a contact angle θ, as depicted in Figure 4.

**Figure 4.** Configuration of a sessile drop on a substrate within which it forms a contact angle θ.

material.

110 Wetting and Wettability

with a scale factor between 2 and 10.

particulate samples [10, 11, 21-29].

**3. Wetting basic laws**

$$\cos \theta = \frac{\sigma\_{sv} - \sigma\_{sl}}{\sigma\_{lv}} \tag{1}$$

where *σij* is the interfacial energy between the phases *i* and *j* (s, l, and v represent the solid, liquid, and vapor phases, respectively). A contact angle of less than 90° identifies a liquid that wets the substrate, while larger angles identify liquids that do not wet it. An ideal wetting liquid will have an assigned value of θ equal to 0°.

The interfacial energy *σlv*, also called surface tension (*γlv*), can be measured with sufficient accuracy by a large number of techniques, e.g., maximum bubble pressure [32, 33], pendant drop [34], or great crucible [35, 36]. However, the surface energy of the solid-vapor phase (*σsv*) and the solid-liquid (*σsl*) phase can only be estimated. Consequently, the two variables that are unknown in the Young's equation can be grouped into only one, called work of adhesion (Wa), which expresses the strength of the solid-liquid bonds and was introduced by Dupré (1869):

$$\mathcal{W}\_a = \sigma\_{lv} + \sigma\_{sv} - \sigma\_{sl} \tag{2}$$

If now this equation is inserted in the Young's equation, the so-called Young-Dupré equation is obtained:

$$W\_a = \gamma\_{lv} \cdot (1 + \cos \theta) \tag{3}$$

All infiltration processes are driven by a differential pressure that allows the liquid matrix to penetrate into the porous reinforcement. This difference of pressure between the two fluids (liquid matrix and gas phase filling porosity) is defined as the capillary pressure (*Pc*) and described by the famous Young-Laplace equation for a straight circular capillary:

$$
\Delta P\_c = P\_A - P\_B = \frac{2 \cdot \gamma\_{lv} \cdot \cos \theta}{r} \tag{4}
$$

Where *γlv* is the liquid-vapour surface tension, θ the contact angle, and r the capillary radius. Figure 5 schematizes different infiltration situations based on the wetting or non-wetting behavior of the system.

**Figure 5.** Scheme of infiltration process for different wetting behaving systems.

The energy shift that occurs when immersing the capillary into a liquid can be described in terms of work as:

$$\mathcal{W}\_{l} = \sigma\_{sl} - \sigma\_{sv} \tag{5}$$

where *Wi* is the work of immersion. For wetting systems (*Wi* <0), infiltration proceeds spon‐ taneously without the necessity of applying an external pressure, as is the case of infiltration of silicon into carbon in the RBSC method. For non-wetting systems (*Wi* >0), infiltration does not occur unless an external pressure, superior to the threshold pressure, is applied. The threshold pressure can be written in terms of work of immersion as:

$$\mathbf{P}\_0 = \mathbf{S}\_i \cdot \mathbf{W}\_i \tag{6}$$

where *Si* is the particle surface area per unit of volume of liquid matrix. Since *σsv* =*σsl* −*γlv* · *cosθ*, The threshold pressure can be related to the contact angle as:

$$\mathbf{P}\_0 = \mathbf{S}\_i \cdot \mathbf{y}\_{lv} \cdot \cos \theta \tag{7}$$

Since the measurement of the surface area of the particles becomes rather inaccurate when the particle diameter surpasses a certain value, and the surface roughness is very difficult to estimate, it is possible to assume a particular geometry for the porous media, e.g., cylindrical pores with a radius r as an approximation, in which case the surface area can be written as *Si* <sup>=</sup> 2 · *<sup>λ</sup>* · (1 <sup>−</sup> *<sup>ϕ</sup>*) *<sup>ϕ</sup>* · *<sup>r</sup>* and the threshold pressure can be written as:

Wetting and Navier-Stokes Equation — The Manufacture of Composite Materials http://dx.doi.org/10.5772/61167 113

$$P\_0 = 2 \cdot \mathcal{A} \cdot \gamma\_{\rm lv} \cdot \cos \theta \cdot \frac{(1-\phi)}{\phi \cdot \mathbf{r}} \tag{8}$$

Where λ is a geometrical factor introduced to describe deviation from the assumed geometry of the pores and the surface roughness of the material (it usually oscillates between 2 and 4 [10, 21-23, 25, 29], γlv is the surface tension, *ϕ* the porosity of the sample, r the mean pore radius, and θ is the contact angle at the triple line. It is important to emphasize that the contact angle in eq. 5 is the equilibrium contact angle that can be measured with high accuracy in sessile drop experiments.

#### **4. The simplest case: Obtaining Darcy's law**

**Figure 5.** Scheme of infiltration process for different wetting behaving systems.

threshold pressure can be written in terms of work of immersion as:

*<sup>ϕ</sup>* · *<sup>r</sup>* and the threshold pressure can be written as:

*σsv* =*σsl* −*γlv* · *cosθ*, The threshold pressure can be related to the contact angle as:

<sup>0</sup> P ·· *i lv* = *S cos* g

terms of work as:

112 Wetting and Wettability

where *Wi*

where *Si*

*Si* <sup>=</sup> 2 · *<sup>λ</sup>* · (1 <sup>−</sup> *<sup>ϕ</sup>*)

The energy shift that occurs when immersing the capillary into a liquid can be described in

taneously without the necessity of applying an external pressure, as is the case of infiltration of silicon into carbon in the RBSC method. For non-wetting systems (*Wi* >0), infiltration does not occur unless an external pressure, superior to the threshold pressure, is applied. The

is the work of immersion. For wetting systems (*Wi* <0), infiltration proceeds spon‐

is the particle surface area per unit of volume of liquid matrix. Since

 q

Since the measurement of the surface area of the particles becomes rather inaccurate when the particle diameter surpasses a certain value, and the surface roughness is very difficult to estimate, it is possible to assume a particular geometry for the porous media, e.g., cylindrical pores with a radius r as an approximation, in which case the surface area can be written as

(5)

<sup>0</sup> P ·*i i* = *S W* (6)

(7)

*Wi sl sv* = s s

As previously mentioned, Darcy's law predicts how a fluid moves through a porous system. This empirical law is based on the study of water movement in sand beds, and represents an ideal model in the manufacture of a composite material. Darcy's law can be derived from the resolution of the Navier-Stokes equation, considering the ideality of the system. In this section, the basis for describing the process of manufacturing of composite materials will be provided, and the most important factors that affect the process will be identified.

As aforementioned, the flow of viscous liquids is governed by the Navier-Stokes equation. This differential equation does not have a general solution. However, in some cases, it is possible to solve, e.g., for an incompressible fluid, in laminar flow (low Reynolds number), assuming that the time derivative of the fluid velocity is much smaller than its spatial deriv‐ atives, and that the effect of gravity is negligible. In this case, the equation can be reduced to:

$$
\nabla \mathbf{P} = \eta \cdot \nabla^2 \mathbf{v} \tag{9}
$$

where v is the speed of the fluid, η is the dynamic viscosity, and P the pressure. Assuming unidirectional flow (which is not so realistic in a standard porous system), it is possible to reduce the expression to the Darcy's law. This equation can be rewritten in the most commonly used form to analyze the flow of incompressible fluids through porous media:

$$\mathbf{v}\_0 = -\frac{\mathbf{K}}{\eta} \frac{\mathbf{d}\mathbf{P}}{\mathbf{d}z} \tag{10}$$

where z is the flow direction, v0 the average superficial velocity of the fluid, dP/dz the pressure gradient along the infiltration front, and K the permeability of the porous media. The super‐ ficial velocity v0 can be converted into infiltration velocity (dz/dt) in the porous medium by means of the porosity ϕ:

$$\mathbf{v}\_0 = \phi \frac{\mathbf{dz}}{\mathbf{dt}} \tag{11}$$

Combining both equations and integrating, the Darcy's law, which shows the relationship between infiltration distance (h), time (t) and the pressure drop in the infiltrating liquid (ΔP=P-Po) is obtained. be rewritten in the most commonly used form to analyze the flow of incompressible fluids through porous media: v� � � <sup>K</sup> η � dP dz eq. 10

where z is the flow direction, v� the average superficial velocity of the fluid, dP/dz the pressure gradient along the

not so realistic in a standard porous system), it is possible to reduce the expression to the Darcy's law. This equation can

spatial derivatives, and that the effect of gravity is negligible. In this case, the equation can be reduced to:

$$\mathbf{h}^2 = \frac{\mathbf{2} \cdot \mathbf{K} \cdot \mathbf{t}}{\eta \cdot \phi} \Delta \mathbf{P} \tag{12}$$

It must be pointed out that the permeability of the porous media is proportional to the square of the average pore radius: dt eq. 11 Combining both equations and integrating, the Darcy's law, which shows the relationship between infiltration distance (h), time (t) and the pressure drop in the infiltrating liquid (ΔP=P‐Po ) is obtained.

v� ��� dz

$$\mathbf{K} = \mathbf{a} \cdot \mathbf{r}^2 \tag{13}$$

Darcy's law has been widely used in basic studies of manufacture of metal matrix composites. Despite its simplicity, it has proven to be very useful in metallurgical engineering. As an example, the results of infiltration of liquid aluminium into a porous bed of particles, obtained by Narciso et al., are shown in Figure 6 [10, 25]. Figure 6a provides the relationship between infiltration distance and applied pressure for liquid aluminium into a porous bed of alumina particles, while Figure 6b shows the results for a liquid infiltration of an aluminium alloy into preforms of graphite particles. As predicted by Darcy's law, the square of infiltration distance is proportional to the pressure drop (h2 vs ΔP) in these systems. It must be pointed out that the permeability of the porous media is proportional to the square of the average pore radius: K����� eq. 13 Darcy's law has been widely used in basic studies of manufacture of metal matrix composites. Despite its simplicity, it has proven to be very useful in metallurgical engineering. As an example, the results of infiltration of liquid aluminium into a porous bed of particles, obtained by Narciso et al., are shown in Figure 6 [10, 25]. Figure 6a provides the relationship between infiltration distance and applied pressure for liquid aluminium into a porous bed of alumina particles, while Figure 6b shows the results for a liquid infiltration of an aluminium alloy into preforms of graphite particles. As predicted by Darcy's law, the square of infiltration distance is proportional to the pressure drop (h2 vs ΔP) in these systems.

**Figure 6**. *Evolution of the infiltrated height as a function of pressure for liquid aluminium on a preform of alumina particles (Alumina particles AA10, Vp <sup>=</sup> <sup>59</sup> %, Temperature <sup>=</sup> <sup>700</sup> °C) (a). Evolution of the square of the infiltrated height for <sup>a</sup> liquid* **Figure 6.** Evolution of the infiltrated height as a function of pressure for liquid aluminium on a preform of alumina particles (Alumina particles AA10, Vp = 59 %, Temperature = 700 °C) (a). Evolution of the square of the infiltrated height for a liquid aluminium alloy as a function of pressure drop on a preform of carbon particles at temperature of 700 °C (see the appendix for the characteristics of the graphite particles) (b).

8 

As discussed in the preceding paragraphs, to solve the Navier-Stokes equation a full saturation and a laminar flow (Re < 2100) have been assumed. Besides, the effect of gravity has been considered to be negligible. Figure 6b shows a good correlation of the experimental data with Darcy's law, and thus, the assumed conditions should be valid. Reynolds number (Re) can be obtained from the following expression:

$$Re = \frac{\rho \cdot v \cdot D}{\eta} \tag{14}$$

Where *ρ* is the density of the liquid metal; D is the diameter of the channel, which in this case is the interparticle distance that can be assumed to be about 1/3 of the average size of the particles; v is the fluid velocity, which is comprised between 1 and 5 mm/s (in our equipments); and η the dynamic fluid viscosity. If the Reynolds number is calculated for the experiments shown in Figure 6, the values obtained are between 0.001 and 0.7, which means that the laminar flow assumption is indeed correct. The force exerted by gravity would be the metallostatic pressure of the melt, which is given by:

$$P = \rho \cdot g \cdot h \tag{15}$$

And considering that the liquid column is always less than 10 cm, in the case of aluminium for example, the pressure would be about 2 kPa, which is almost 3 orders of magnitude lower than the capillary forces, and therefore negligible.

In the following sections different real situations that can occur are considered and will be analyzed separately. The combination of all these particular cases simultaneously, although more realistic, is much more complex and goes beyond the scope of this chapter.

#### **5. The effect of saturation**

0

Po) is obtained.

114 Wetting and Wettability

in these systems.

of the average pore radius:

is proportional to the pressure drop (h2

v · dt <sup>=</sup> f

spatial derivatives, and that the effect of gravity is negligible. In this case, the equation can be reduced to:

<sup>2</sup> 2·K·t h ·P h f

> v� ��� dz

h� � 2�K�t

infiltration velocity (dz/dt) in the porous medium by means of the porosity :

(h), time (t) and the pressure drop in the infiltrating liquid (ΔP=P‐Po ) is obtained.

v� � � <sup>K</sup> η � dP dz

dz

Combining both equations and integrating, the Darcy's law, which shows the relationship between infiltration distance (h), time (t) and the pressure drop in the infiltrating liquid (ΔP=P-

where z is the flow direction, v� the average superficial velocity of the fluid, dP/dz the pressure gradient along the infiltration front, and K the permeability of the porous media. The superficial velocity v� can be converted into

where v is the speed of the fluid, η is the dynamic viscosity, and P the pressure. Assuming unidirectional flow (which is not so realistic in a standard porous system), it is possible to reduce the expression to the Darcy's law. This equation can be rewritten in the most commonly used form to analyze the flow of incompressible fluids through porous media:

It must be pointed out that the permeability of the porous media is proportional to the square

Combining both equations and integrating, the Darcy's law, which shows the relationship between infiltration distance

Darcy's law has been widely used in basic studies of manufacture of metal matrix composites. Despite its simplicity, it has proven to be very useful in metallurgical engineering. As an example, the results of infiltration of liquid aluminium into a porous bed of particles, obtained by Narciso et al., are shown in Figure 6 [10, 25]. Figure 6a provides the relationship between infiltration distance and applied pressure for liquid aluminium into a porous bed of alumina particles, while Figure 6b shows the results for a liquid infiltration of an aluminium alloy into preforms of graphite particles. As predicted by Darcy's law, the square of infiltration distance

Darcy's law has been widely used in basic studies of manufacture of metal matrix composites. Despite its simplicity, it has proven to be very useful in metallurgical engineering. As an example, the results of infiltration of liquid aluminium into a porous bed of particles, obtained by Narciso et al., are shown in Figure 6 [10, 25]. Figure 6a provides the relationship between infiltration distance and applied pressure for liquid aluminium into a porous bed of alumina particles, while Figure 6b shows the results for a liquid infiltration of an aluminium alloy into preforms of graphite particles. As predicted by Darcy's law, the square of infiltration distance is proportional to the pressure drop (h2 vs ΔP)

It must be pointed out that the permeability of the porous media is proportional to the square of the average pore radius:

vs ΔP) in these systems.

8 

700 °C (see the appendix for the characteristics of the graphite particles) (b).

 **(a) (b) Figure 6**. *Evolution of the infiltrated height as a function of pressure for liquid aluminium on a preform of alumina particles (Alumina particles AA10, Vp <sup>=</sup> <sup>59</sup> %, Temperature <sup>=</sup> <sup>700</sup> °C) (a). Evolution of the square of the infiltrated height for <sup>a</sup> liquid* **Figure 6.** Evolution of the infiltrated height as a function of pressure for liquid aluminium on a preform of alumina particles (Alumina particles AA10, Vp = 59 %, Temperature = 700 °C) (a). Evolution of the square of the infiltrated height for a liquid aluminium alloy as a function of pressure drop on a preform of carbon particles at temperature of

(11)

eq. 10

�P � η � ��v eq. 9

· = D (12)

dt eq. 11

<sup>2</sup> K a·r = (13)

<sup>η</sup> � � � �P eq. 12

K����� eq. 13

On systems where the liquid does not wet the solid, which is most common in MMCs, it must be taken into account that the permeability of the porous system is not constant [37-39] and that it depends on the saturation and this latter, and at the same time, on the applied pressure [21, 40-42].

The study of unsaturated flow is well known in geology and in 1931 it was described mathe‐ matically by Richards [34], whose partial differential equations are difficult to solve and usually solved numerically for each particular case. Numerous researchers have worked on this particular hydrodynamics and mathematical problem [44,45], and the books by Prof. Bear collect these study cases [46-48], which are essential reading for researchers working in this field. It is also noteworthy that Prof. Bear highlights in his books the goodness of Darcy's law, and that its application range is wider (more useful) than it was expected at first (i.e., unsatu‐ rated flow, high Reynolds number, etc.).

These studies were not limited to the case study of hydrology, but have crossed over to the field of metal matrix composites (MMC), performed especially by the group of Andreas Mortensen [40-42], but as mentioned above without analytical solution of the differential equations describing the phenomenon. The highlight to be found in the recent literature are two articles that show that close to the threshold pressure, infiltration is actually governed by percolation phenomena (instead of slug flow) [49,50].

Another aspect to consider is that pressure also depends on the position along the flow path, and is usually assumed that the pressure drop between the front (P0) and head (applied pressure) is linear, which is reasonably sane. Nevertheless, in order for the differential equations to have an analytic solution, it will be assumed in this chapter that practically all the preform is at the same pressure, which in principle is something reasonable as working pressures oscillate between 1 and 5% of the threshold pressure (i.e., head=1.05P0; front=P0).

In the literature, several models showing the relationship between pressure and saturation have been described, but the most famous are the Brooks-Corey (BC) [19] and van Genuchten (VG) [20] models. In this section, the Navier-Stokes equation has been solved considering that the permeability is not constant, relating it to the saturation through the BC and VG models. Different authors have used these models to describe the infiltration process in MMCs [51]. However, previous works have shown that for low pressures (close to the threshold pressure) these models, especially the BC, do not describe the infiltration process correctly [52, 53].

Starting from reduced Navier-Stokes equation:

$$dz^2 = \frac{K \cdot t}{\eta \cdot \mathcal{Q}} dP\tag{16}$$

Where z is the infiltration distance, *K* the permeability, *t* the time of infiltration, *η* the dynamic viscosity of the fluid, ∅ the porosity of the porous body, and P the applied pressure. Perme‐ ability can be decomposed into two different terms, intrinsic permeability (Ki ), which depends exclusively on the nature of the solid phase; and relative permeability (*Kr*), which depends on the degree of saturation (S) of the system:

$$\mathbf{K} = \mathbf{K}\_{\mathrm{l}} \cdot \mathbf{K}\_{\mathrm{r}} \tag{17}$$

Relative permeability, depends on the degree of saturation according to:

$$K\_r = \mathcal{S}^n \tag{18}$$

Where n is a parameter that ranges from 1 to 3. In the case of composites fabricated by infiltration, it has been found that 1 is the most accurate value for n [51]. There are several models that relate saturation to the applied pressure (P) and the threshold pressure (P0) of the system. One of the most used models of this kind is the Brooks-Corey equation:

Wetting and Navier-Stokes Equation — The Manufacture of Composite Materials http://dx.doi.org/10.5772/61167 117

$$S = 1 - \left(\frac{P\_0}{P}\right)^k \tag{19}$$

Where *λ* is a parameter that depends on the solid material. Replacing K in eq. 16 with eqs. 17, 18, and 19 yields a modified reduced Navier-Stokes equation:

These studies were not limited to the case study of hydrology, but have crossed over to the field of metal matrix composites (MMC), performed especially by the group of Andreas Mortensen [40-42], but as mentioned above without analytical solution of the differential equations describing the phenomenon. The highlight to be found in the recent literature are two articles that show that close to the threshold pressure, infiltration is actually governed by

Another aspect to consider is that pressure also depends on the position along the flow path, and is usually assumed that the pressure drop between the front (P0) and head (applied pressure) is linear, which is reasonably sane. Nevertheless, in order for the differential equations to have an analytic solution, it will be assumed in this chapter that practically all the preform is at the same pressure, which in principle is something reasonable as working pressures oscillate between 1 and 5% of the threshold pressure (i.e., head=1.05P0; front=P0). In the literature, several models showing the relationship between pressure and saturation have been described, but the most famous are the Brooks-Corey (BC) [19] and van Genuchten (VG) [20] models. In this section, the Navier-Stokes equation has been solved considering that the permeability is not constant, relating it to the saturation through the BC and VG models. Different authors have used these models to describe the infiltration process in MMCs [51]. However, previous works have shown that for low pressures (close to the threshold pressure) these models, especially the BC, do not describe the infiltration process correctly [52, 53].

> <sup>2</sup> · · · *K t dz dP* h Æ

ability can be decomposed into two different terms, intrinsic permeability (Ki

Relative permeability, depends on the degree of saturation according to:

system. One of the most used models of this kind is the Brooks-Corey equation:

Where z is the infiltration distance, *K* the permeability, *t* the time of infiltration, *η* the dynamic viscosity of the fluid, ∅ the porosity of the porous body, and P the applied pressure. Perme‐

exclusively on the nature of the solid phase; and relative permeability (*Kr*), which depends on

Where n is a parameter that ranges from 1 to 3. In the case of composites fabricated by infiltration, it has been found that 1 is the most accurate value for n [51]. There are several models that relate saturation to the applied pressure (P) and the threshold pressure (P0) of the

<sup>=</sup> (16)

· *K KKi r* = (17)

*<sup>n</sup> K S <sup>r</sup>* = (18)

), which depends

percolation phenomena (instead of slug flow) [49,50].

116 Wetting and Wettability

Starting from reduced Navier-Stokes equation:

the degree of saturation (S) of the system:

$$dz^2 = \frac{K\_i \cdot \left[1 - \left(\frac{P\_0}{P}\right)^\lambda\right]^n \cdot t}{\eta \cdot \mathcal{Q}} dP$$

Solving this equation provides a model to estimate depth of infiltration as a function of applied pressure. In this form, there is no general analytical solution to the equation, so, to obtain an analytical model, a particular case must be considered. As discussed above, experimental data fit better when *n* =1, so to solve the equation, this particular case will be considered:

$$\frac{1}{2} \cdot h^2 = \frac{K\_i \cdot t}{\eta \cdot \mathcal{Q}} \int \left( 1 - \left( \frac{P\_0}{P} \right)^i \right) \cdot dP \tag{21}$$

$$\eta h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{O}} \left( P + \frac{P\left(\frac{P\_0}{P}\right)^\lambda}{\lambda - 1} + A \right) \tag{22}$$

Where A is the integration constant, and its value can be determined using initial conditions, i.e., for P = P0 and h = 0

$$0 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{Q}'} \left( P\_0 + \frac{P\_0 \left( \frac{P\_0}{P\_0} \right)^{\lambda}}{\lambda - 1} + A \right) \tag{23}$$

$$\begin{aligned} 0 &= P\_0 + \frac{P\_0}{\lambda - 1} + A\\ A &= -P\_0 - \frac{P\_0}{\lambda - 1} \\ P\_0 &\left( -\frac{1}{\lambda - 1} - 1 \right) \end{aligned} \tag{24}$$

Then the final expression for *h(P)* is:

$$h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{Q}} \left| P + \frac{P\left(\frac{P\_0}{P}\right)^\lambda}{\lambda - 1} - P\_0 \left(\frac{1}{\lambda - 1} + 1\right) \right| \tag{25}$$

It must be noted that if *λ* acquires a very high value, the model tends to Darcy's law:

$$
\lambda \to \infty
$$

$$
h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{QP}} \Big( P + 0 - P\_0 \cdot \left( 0 + 1 \right) \Big)
$$

$$
h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{QP}} \Big( P - P\_0 \Big)
$$

If, instead of using the Brooks-Corey equation, the Van Genuchten model is applied:

$$S = 1 - \left[1 + \left(\alpha \cdot (P - P\_0)\right)^{\ast 1}\right]^{-m} \tag{26}$$

$$m = 1 - \frac{1}{n1} \tag{27}$$

where *α* is a scale parameter inversely proportional to the mean pore diameter, and *n1* (called n1 to differentiate from *n* in eq. 18) and *m* are shape parameters. The reduced Navier-Stokes equation results in:

$$dz^2 = \frac{K\_i \cdot \left[1 - \left[1 + \alpha \cdot \left(P - P\_0\right)^{\circ 1}\right]^{-1 + \frac{1}{n\cdot 1}}\right]^n \cdot t}{\eta \cdot \mathcal{O}} \cdot dP} \cdot dP \tag{28}$$

As in the case of the Brooks-Corey model, a general analytical solution is not available, so the case of n = 1 will be considered again:

$$dz^2 = \frac{K\_i \cdot \left[1 - \left[1 + \alpha \cdot \left(P - P\_0\right)^{\text{tr}}\right]^{-1 + \frac{1}{n\cdot 1}}\right] t}{\eta \cdot \mathcal{Q}} \cdot dP} \cdot dP \tag{29}$$

Wetting and Navier-Stokes Equation — The Manufacture of Composite Materials http://dx.doi.org/10.5772/61167 119

$$\frac{1}{2} \cdot \hbar^2 = \frac{K\_i \cdot t}{\eta \cdot \mathcal{Q}} \cdot \left[ \left[ 1 - \left[ 1 + \left( \alpha \cdot (P - P\_0) \right)^{\circ 1} \right]^{-1 + \frac{1}{m\_1}} \right] \cdot dP \right. \tag{30}$$

$$\frac{1}{2}\frac{1}{2}h^2 = \frac{K\_\circ \cdot t}{\eta \cdot \mathcal{Q}} \cdot \left[1 \cdot dP - \int \left[1 + \left(\alpha \cdot (P - P\_0)\right)^{\circ 1}\right]^{-1} \overset{1}{}{\cdot} dP\tag{31}$$

$$\frac{1}{2} \cdot \hbar^2 = \frac{K\_i \cdot t}{\eta \cdot \mathcal{Q}'} P - \left[ \left[ 1 + \left( \alpha \cdot (P - P\_0) \right)^{\text{n1}} \right]^{-1 \ast} \right]^{\frac{1}{\text{n1}}} dP \tag{32}$$

The second integral does not have an analytical solution either, so in order to obtain a solution as general as possible, the value of n1=2 is introduced [42, 51]:

$$h^2 = \frac{2 \cdot K\_{\cdot} \cdot t}{\eta \cdot \mathcal{Q}} \left[ P - \iint \left[ 1 + \left( \alpha \cdot \left( P - P\_0 \right) \right)^2 \right]^{-1/2} dP \right] \tag{33}$$

$$\frac{1}{2} \cdot h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{Q}'} \left[ P - \int \frac{1}{\sqrt{1 + (\alpha \cdot (P - P\_0))^2}} dP \right] \tag{34}$$

Applying the substitution *u* =*α* · (*P* −*P*0), *du* =*α* · *dP* results in:

$$h^2 = \frac{2 \cdot K\_{\cdot} \cdot t}{\eta \cdot \mathcal{Q}} \left[ P - \int \frac{1}{\sqrt{1 + u^2}} \cdot \frac{1}{\alpha} \cdot du \right] \tag{35}$$

$$h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{D}} \left[ P - \frac{1}{\alpha} \cdot \left[ \frac{1}{\sqrt{1 + u^2}} \cdot du \right] \right] \tag{36}$$

$$h^2 = \frac{2 \cdot K\_{\parallel} \cdot t}{\eta \cdot \mathcal{O}} \left[ P - \frac{\text{arcsinh}\left(u\right)}{a} \cdot + \text{C} \right] \tag{37}$$

Replacing back *u* =*α* · (*P* −*P*0) :

.

Then the final expression for *h(P)* is:

118 Wetting and Wettability

equation results in:

2

*i*

hÆ

0

*<sup>P</sup> <sup>P</sup>*

 l

It must be noted that if *λ* acquires a very high value, the model tends to Darcy's law:

*K t P hP P*

2

2

h Æ

*i*

*K t h PP*

l ¥

®

= +- +

· 2· · <sup>1</sup> · ·1 · 11

æ ö æ ö ç ÷ ç ÷ æ ö è ø = +-+ ç ÷ - - è ø

l

0

è ø

 l

( ( ))

2· · · 0 ·0 1 ·

2· · · ·

h Æ

If, instead of using the Brooks-Corey equation, the Van Genuchten model is applied:

<sup>0</sup> 1 1 ( ·( )) *<sup>m</sup> <sup>n</sup> S PP* a

> <sup>1</sup> <sup>1</sup> 1

where *α* is a scale parameter inversely proportional to the mean pore diameter, and *n1* (called n1 to differentiate from *n* in eq. 18) and *m* are shape parameters. The reduced Navier-Stokes

0

As in the case of the Brooks-Corey model, a general analytical solution is not available, so the

0

· 1 1 ·( ) ·

é ù - +

*<sup>n</sup> <sup>n</sup> K PP t <sup>i</sup> dz dP* a

h Æ

· 1 1 ·( ) ·

*<sup>n</sup> <sup>n</sup> K PP t <sup>i</sup> dz dP* a

é ù - + -+ - é ù ê ú ë û ë û <sup>=</sup>

h Æ

*m*

2

2

case of n = 1 will be considered again:

=- + - é ù

*i*

*K t h PP*

= -

0

( )

1

<sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>

*n*

· ·

<sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>


· ·

ë û (26)

*<sup>n</sup>* = - (27)


0

(25)

(28)

$$h^2 = \frac{2 \cdot K\_{\cdot} \cdot t}{\eta \cdot \mathcal{D}} \left[ P - \frac{\text{arcsinh}\left(a \cdot (P - P\_0)\right)}{a} + C \right] \tag{38}$$

To estimate the value of C, initial conditions must be applied *P* =*P*0 and *h* =0.

$$0 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{D}} \left[ P\_0 - \frac{\text{arcsinh}\left(0\right)}{a} \cdot + \text{C} \right] \tag{39}$$

$$\mathbf{C} = -P\_0 \tag{40}$$

Then, the general solution for n1 = 2 is:

law is more universal than it was originally supposed.

radius affects the permeability of the porous media.

**6. The effect of variable radius**

Then, the general solution for n1 = 2 is:

To estimate the value of C, initial conditions must be applied ���� and � � 0�

0 �

2��� � �

polyethylene glycol is used) and the curve modified with the Van Genuchten equation.

$$h^2 = \frac{2 \cdot K\_i \cdot t}{\eta \cdot \mathcal{Q}} \left[ (P - P\_0) - \frac{\text{arcsinh}\left(a \cdot (P - P\_0)\right)}{a} \right] \tag{41}$$

In order to analyze the accuracy of the results, the evolution of the square of the infiltrated height as a function of pressure drop is plotted in Figures 7a and 7b. As shown, experimental results are perfectly framed between the line defined by Darcy's law (assuming that the liquid permeability is invariant and therefore permeability measured with polyethylene glycol is used) and the curve modified with the Van Genuchten equation. �� � 2��� � � ��� � ������� � arcsinh�� � �� � ��� � � � eq. 41 In order to analyze the accuracy of the results, the evolution of the square of the infiltrated height as a function of pressure drop is plotted in Figures 7a and 7b. As shown, experimental results are perfectly framed between the line defined by Darcy's law (assuming that the liquid permeability is invariant and therefore permeability measured with

**Figure 7**. *Evolution of the square of infiltrated height for liquid aluminium‐silicon alloy as a function of pressure drop on a preform of graphite particles P1 at 700°C) (a). Evolution of the square of the infiltrated height for liquid aluminium‐silicon alloy as a function of pressure drop on a preform of carbon particles P2 at 700°C) (b) Characteristics of the graphite particles are listed in the appendix. The coefficients for the BC model have been extracted from [53].* **Figure 7.** Evolution of the square of infiltrated height for liquid aluminium-silicon alloy as a function of pressure drop on a preform of graphite particles P1 at 700°C) (a). Evolution of the square of the infiltrated height for liquid alumini‐ um-silicon alloy as a function of pressure drop on a preform of carbon particles P2 at 700°C) (b) Characteristics of the graphite particles are listed in the appendix. The coefficients for the BC model have been extracted from [53].

The results shown here clearly demonstrate that it is not necessary to solve the Richards equation (modified Darcy's law to take into account saturation), which has only numerical solutions to predict infiltration in composite materials unlike what was suggested by Mortensen et al. [54]. It is noteworthy that many authors in soil mechanics indicate that Darcy's

Although the variation of the radius is not a very common phenomenon in the manufacture of composite materials, it is a process that can occur. The variation of the radius is a fundamental issue for example in the manufacture of SiC by the RBSC process or reactive infiltration and SiC derived CMCs. This is a rather complex situation since the variation of the

To solve the Navier‐Stokes equation it has been assumed that the reduction of the radius, caused by a chemical reaction, is controlled by diffusion. The equations presented so far are valid for infiltration in nonreactive systems. However, they are not valid for reactive infiltration because the pore radius, in this kind of systems, is not constant but decreases with

12 

The results shown here clearly demonstrate that it is not necessary to solve the Richards equation (modified Darcy's law to take into account saturation), which has only numerical solutions to predict infiltration in composite materials unlike what was suggested by Morten‐ sen et al. [54]. It is noteworthy that many authors in soil mechanics indicate that Darcy's law is more universal than it was originally supposed.

#### **6. The effect of variable radius**

<sup>2</sup> ( ) <sup>0</sup> 2· · arcsinh ·( ) · · *<sup>i</sup> K t P P h P C*

To estimate the value of C, initial conditions must be applied *P* =*P*0 and *h* =0.

0 2· · arcsinh 0 0· · · *K ti P C*

<sup>2</sup> ( ) <sup>0</sup> 0

2· · arcsinh ·( ) ·( ) · *<sup>i</sup> K t P P*

ë û

In order to analyze the accuracy of the results, the evolution of the square of the infiltrated height as a function of pressure drop is plotted in Figures 7a and 7b. As shown, experimental results are perfectly framed between the line defined by Darcy's law (assuming that the liquid permeability is invariant and therefore permeability measured with polyethylene glycol is

In order to analyze the accuracy of the results, the evolution of the square of the infiltrated height as a function of pressure drop is plotted in Figures 7a and 7b. As shown, experimental results are perfectly framed between the line defined by Darcy's law (assuming that the liquid permeability is invariant and therefore permeability measured with

**Figure 7**. *Evolution of the square of infiltrated height for liquid aluminium‐silicon alloy as a function of pressure drop on a preform of graphite particles P1 at 700°C) (a). Evolution of the square of the infiltrated height for liquid aluminium‐silicon alloy as a function of pressure drop on a preform of carbon particles P2 at 700°C) (b) Characteristics of the graphite particles are listed in the appendix.*

**Figure 7.** Evolution of the square of infiltrated height for liquid aluminium-silicon alloy as a function of pressure drop on a preform of graphite particles P1 at 700°C) (a). Evolution of the square of the infiltrated height for liquid alumini‐ um-silicon alloy as a function of pressure drop on a preform of carbon particles P2 at 700°C) (b) Characteristics of the graphite particles are listed in the appendix. The coefficients for the BC model have been extracted from [53].

The results shown here clearly demonstrate that it is not necessary to solve the Richards equation (modified Darcy's law to take into account saturation), which has only numerical solutions to predict infiltration in composite materials unlike what was suggested by Mortensen et al. [54]. It is noteworthy that many authors in soil mechanics indicate that Darcy's

Although the variation of the radius is not a very common phenomenon in the manufacture of composite materials, it is a process that can occur. The variation of the radius is a fundamental issue for example in the manufacture of SiC by the RBSC process or reactive infiltration and SiC derived CMCs. This is a rather complex situation since the variation of the

To solve the Navier‐Stokes equation it has been assumed that the reduction of the radius, caused by a chemical reaction, is controlled by diffusion. The equations presented so far are valid for infiltration in nonreactive systems. However, they are not valid for reactive infiltration because the pore radius, in this kind of systems, is not constant but decreases with

12 

é ù - = -- ê ú

��� � ��� � arcsinh�0 �

hÆ

*h PP*

�� � 2��� � �

0 �

2��� � �

��� � ������� �

To estimate the value of C, initial conditions must be applied ���� and � � 0�

used) and the curve modified with the Van Genuchten equation.

polyethylene glycol is used) and the curve modified with the Van Genuchten equation.

 (a) (b)

*The coefficients for the BC model have been extracted from [53].*

law is more universal than it was originally supposed.

radius affects the permeability of the porous media.

**6. The effect of variable radius**

h Æ

é ù - =- + ê ú

h Æ

Then, the general solution for n1 = 2 is:

Then, the general solution for n1 = 2 is:

120 Wetting and Wettability

a

a

(38)

(39)

(41)

ë û

 a

ë û

é ù =- + ê ú

( )

a

a

arcsinh�� � �� � ��� �

*C P*<sup>0</sup> = - (40)

� � ��� eq. 40

� � ��� eq. 39

� � eq. 41

Although the variation of the radius is not a very common phenomenon in the manufacture of composite materials, it is a process that can occur. The variation of the radius is a funda‐ mental issue for example in the manufacture of SiC by the RBSC process or reactive infiltration and SiC derived CMCs. This is a rather complex situation since the variation of the radius affects the permeability of the porous media.

To solve the Navier-Stokes equation it has been assumed that the reduction of the radius, caused by a chemical reaction, is controlled by diffusion. The equations presented so far are valid for infiltration in nonreactive systems. However, they are not valid for reactive infiltra‐ tion because the pore radius, in this kind of systems, is not constant but decreases with the generation of new solid. As discussed above, the threshold pressure and the permeability depend on the radius, so they are not invariant over time.

So for reactive systems, it is necessary to re-solve the reduced Navier-Stokes equation consid‐ ering the variation of the radius:

$$
\phi \cdot \frac{dz}{dt} = -\frac{K}{\eta} \cdot \frac{dP}{dz} \tag{42}
$$

After integration, the following expression is obtained:

$$\frac{1}{2}h^2 = -\quad \frac{K}{\mathcal{D} \cdot \eta}dP \cdot dt\tag{43}$$

Usually in reactive systems, infiltration takes place spontaneously (without applying an external pressure), so Δ*P* = *P*0. Differentiating the expression for *P*0 results in:

$$\mathbf{dP} = -2 \cdot \mathbb{A} \cdot \boldsymbol{\gamma}\_{\mathrm{lv}} \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\Phi \cdot r^2} dr \tag{44}$$

Combining this expression with eq. 6 yields:

$$\frac{1}{2}\dot{\mathsf{h}}^2 = -\quad \frac{\mathsf{a}\cdot\mathsf{r}^2}{\mathscr{D}\cdot\mathsf{v}} \Big| \begin{pmatrix} -2\cdot\mathsf{A}\cdot\mathsf{y}\_{\mathsf{lv}}\cdot\cos\theta\cdot\frac{(1-\phi)}{\mathscr{D}\cdot\mathsf{r}^2} \\ \end{pmatrix} \mathrm{d}r\cdot\mathrm{d}t \tag{45}$$

Simplifying and rearranging:

$$h^2 = 4 \cdot \lambda \cdot a \cdot \gamma\_{lv} \cdot \cos\theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \quad dr \cdot dt \tag{46}$$

In principle, porosity of the system is not constant, but for a first approximation it can be considered so. To proceed with the solution of the equation, a relationship between radius and time is necessary. Assuming that the controlling step of the reaction is the diffusion of C through the SiC layer, then the model of decreasing nucleus developed in the decade of the 1950s can be used [55, 56]:

$$t = \frac{M\_{\rm Si} \cdot \rho\_{\rm SiC} \cdot r\_o^2}{6 \cdot M\_{\rm SiC} \cdot \mathcal{D}\_{\rm C/SiC} \cdot \rho\_{\rm Si}} \left[ 1 - 3 \left( \frac{r}{r\_o} \right)^2 + 2 \cdot \left( \frac{r}{r\_o} \right)^3 \right] \tag{47}$$

$$dt = \frac{M\_{\rm Si} \cdot \rho\_{\rm SiC} \cdot r\_o^2}{\Theta \cdot M\_{\rm SiC} \cdot \mathcal{D}\_{\rm C/SiC} \cdot \rho\_{\rm Si}} \left[ -\Theta \left( \frac{r}{r\_o^2} \right) + \Theta \left( \frac{2r^2}{r\_o^3} \right) \right] dr \tag{48}$$

Where *ρSiC* is the density of SiC, *r*0 the initial pore radius, *DC/SiC* the diffusion coefficient of C atoms through SiC, and *ρSi* the density of Si atoms in kg/m3 , MSiC and MSi the molar weight of SiC and Si, respectively.

Introducing these relations into eq. 46 and defining:

$$\mathbf{G} = 4 \cdot \mathbb{A} \cdot \mathbf{a} \cdot \boldsymbol{\gamma}\_{\mathrm{lv}} \cdot \cos \theta \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \frac{M\_{\mathrm{Si}} \, \rho\_{\mathrm{SiC}} \cdot \boldsymbol{r}\_{\mathrm{o}}^2}{\boldsymbol{\Theta} \cdot \mathcal{D}\_{\mathrm{C,SiC}} \cdot M\_{\mathrm{SiC}} \cdot \rho\_{\mathrm{Si}}} \tag{49}$$

Results in the following expression:

$$h^2 = \text{ G} \cdot \left[ -\Theta \left( \frac{r}{r\_o^2} \right) + \Theta \left( \frac{r^2}{r\_o^3} \right) \right] dr \cdot dr \tag{50}$$

Solving the first integral yields:

#### Wetting and Navier-Stokes Equation — The Manufacture of Composite Materials http://dx.doi.org/10.5772/61167 123

$$h^2 = \left. \mathbf{G} \cdot \prod\_{0}^{'} A - \mathbf{3} \cdot \left( \frac{r}{r\_o} \right)^2 + 2 \left( \frac{r}{r\_o} \right)^3 \right] dr\tag{51}$$

Where A is an integration constant. Solving the second integral results in:

$$h^2 = G \cdot \left( A \cdot r - \left( \frac{r^3}{r\_o^2} \right) + \frac{1}{2} \left( \frac{r^4}{r\_o^3} \right) + B \right) \tag{52}$$

where B is an integration constant. In order to obtain the final solution, boundary conditions must be used to find the values of A and B:

**a.** *r* = *ro* →*h* =0 →*t* =0 **b.** *<sup>r</sup>* = 0 →*<sup>h</sup>* <sup>=</sup>*<sup>h</sup> MAX* <sup>→</sup>*<sup>t</sup>* <sup>=</sup>*<sup>τ</sup>* <sup>→</sup> *dh* Where *<sup>τ</sup>* <sup>=</sup>*MSi* · *<sup>ρ</sup>SiC* · *ro* 2 6 · *MSiC*· D*<sup>C</sup>*/*SiC* · *CSi*

By applying condition a), the next expression is obtained:

*dr* =0

$$\mathbf{0} = \mathbf{G} \cdot \left( A \cdot r\_0 - \left( \frac{r\_0^3}{r\_o^2} \right) + \frac{1}{2} \left( \frac{r\_o^4}{r\_o^3} \right) + B \right) \tag{53}$$

Applying condition b) leads to:

$$h\_{\text{MAX}}^2 = \text{G} \cdot \text{B} \tag{54}$$

and

( ) <sup>2</sup>

 q

2 1 4· · · · · · · · *lv h a cos dr dt*

2 3 <sup>2</sup>

2 2

2 3

In principle, porosity of the system is not constant, but for a first approximation it can be considered so. To proceed with the solution of the equation, a relationship between radius and time is necessary. Assuming that the controlling step of the reaction is the diffusion of C through the SiC layer, then the model of decreasing nucleus developed in the decade of the

f

è ø

· 2· · · · · · 2 · · *lv*

æ ö - =- -ç ÷

*a r <sup>h</sup> cos dr dt*

lg

1 · 1

( ) <sup>2</sup>

 q h Æ

· · 1 3· 2·

*M r r r <sup>t</sup> M rr*

r

r

*SiC C SiC Si o o*

· · 2 · 6· 6· ·

é ù æö æ ö <sup>=</sup> ê- + ú ç÷ ç ÷ ê ú

*SiC C SiC Si o o M r r r dt dr M r r*

Where *ρSiC* is the density of SiC, *r*0 the initial pore radius, *DC/SiC* the diffusion coefficient of C

( ) <sup>2</sup> ·

,

2

*Si SiC o*

r

*M*

*C SiC SiC Si*

r


2

f

<sup>1</sup> · 4· · · · · · 6· · ·

2 3 · 6· 6· · *o o*

*r r h G dr dr r r*

é ù æö æö = ê- + ú ç÷ ç÷ ê ú ë û èø èø

*M r G a cos*

 q h Æ

é ù æö æö <sup>=</sup> ê ú - + ç÷ ç÷

lg

· /

· /

*lv*

lg

2

r

6· · *Si SiC o*

atoms through SiC, and *ρSi* the density of Si atoms in kg/m3

Introducing these relations into eq. 46 and defining:

r

6· · *Si SiC o*

Æ h 2


èø èø ë û <sup>D</sup> (47)

ë û èø è ø <sup>D</sup> (48)

, MSiC and MSi the molar weight of

(50)

(45)

*r* f

Æ

2

Simplifying and rearranging:

122 Wetting and Wettability

1950s can be used [55, 56]:

SiC and Si, respectively.

Results in the following expression:

Solving the first integral yields:

$$\frac{dh}{dr} = 0 = \frac{G\left(A - 6\left(\frac{r^2}{r\_o^2}\right) + 2\left(\frac{r^3}{r\_o^3}\right)\right)}{2\sqrt{G\left(A \cdot r - \left(\frac{r^3}{r\_o^2}\right) + \frac{1}{2}\left(\frac{r^4}{r\_o^3}\right) + B\right)}}\tag{55}$$

Solving eq. 55 leads to:

*A* = 0

Replacing the value of A in eq. 53 results in:

$$B = \frac{1}{2}r\_0$$

Replacing the value of B in eq. 54 provides the expression for hMAX:

$$h\_{\rm MAX} = \sqrt{\mathcal{A} \cdot a \cdot \gamma\_{\rm \upsilon} \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \cdot \frac{M\_{\rm Si} \, \rho\_{\rm SiC} \cdot r\_o^3}{\mathbf{3} \cdot \mathcal{D}\_{\rm C, SiC} \cdot M\_{\rm Si} \cdot \rho\_{\rm Si}}}\tag{56}$$

And after replacing the values of A and B, and rearranging the final expression h(r) is obtained:

$$\hbar^2 = \lambda \cdot a \cdot \gamma\_w \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{Q}^2} \cdot \frac{\rho\_{\text{SiC}} \cdot r\_o^2 \cdot r}{\mathbf{3} \cdot \mathcal{D}\_{C, \text{SiC}} \cdot \rho\_{\text{Si}}} \cdot \left[\left(\frac{r}{r\_0}\right)^3 - 2\left(\frac{r}{r\_0}\right)^2 + \frac{r\_0}{r}\right] \tag{57}$$

$$h^2 = \frac{h\_{\rm MAX}^2}{r\_0} r \cdot \left[ \left(\frac{r}{r\_0}\right)^3 - 2 \left(\frac{r}{r\_0}\right)^2 + \frac{r\_0}{r} \right] \tag{58}$$

$$h^2 = h\_{\rm MAX}^2 \cdot \left[ \left( \frac{r}{r\_0} \right)^4 - 2 \left( \frac{r}{r\_0} \right)^3 + 1 \right] \tag{59}$$

It must be taken into consideration that this equation is valid only as long as the modification of the porosity is negligible. For evaluating h(t), eq. 59 and eq. 47 must be combined.

Figure 8 shows the calculated infiltration distance with time for a carbon preform with an initial pore radius of 30 micron and 50% open porosity, infiltrated with pure Si, using Darcy's law and the variable radius model for cylindrical and spherical pores. The variable radius models predict a slower rate of infiltration, which is closer to the experimental data obtained by several authors [12, 57-61].

Another approach is to consider that the pore closure is limited by the chemical reaction that originates it. In this case, the equations of the decreasing nucleus model to be used are:

$$t = \frac{M\_{\text{Si}} \cdot \rho\_{\text{SiC}} \cdot r\_0}{k\_s \cdot \rho\_{\text{Si}} \cdot M\_{\text{SiC}}} \left(1 - \frac{r}{r\_0}\right) \tag{60}$$

**Figure 8.** Comparison of the square distance of infiltration (h2 ) vs. time, calculated using Darcy's law and variable radi‐ us models for a carbon with an initial pore radius of 30 micron, and 50% porosity, infiltrated with pure Si. For the variable radius model, the pore closure is assumed to be controlled by diffusion, and two different pore geometries are considered. Infiltration is simulated until the pore radius decreases by 30%.

$$dt = -\frac{\rho\_{\text{SiC}}}{k\_s \cdot \rho\_{\text{Si}}} dr\tag{61}$$

Introducing this equation into the reduced Navier-Stokes equation and solving the integrals results in:

$$\hbar^2 = -4 \cdot \lambda \cdot a \cdot \gamma\_{\nu} \cdot \cos \theta \cdot \frac{(1-\phi)}{\eta \cdot \mathcal{D}} \cdot \frac{M\_{\text{Si}} \cdot \rho\_{\text{SiC}}}{k\_s \cdot \rho\_{\text{Si}} \cdot M\_{\text{SiC}}} \cdot \left[ (r+A) \cdot dr \right] \tag{62}$$

$$\hbar^2 = -4 \cdot \mathbb{A} \cdot \mathbf{a} \cdot \gamma\_{lv} \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \cdot \frac{M\_{\text{Si}} \cdot \rho\_{\text{SiC}}}{k\_s \cdot \rho\_{\text{Si}} \cdot M\_{\text{SiC}}} \cdot \left(\frac{r^2}{2} + A \cdot r + B\right) \tag{63}$$

The integration constants A and B are calculated using contour conditions:

**c.** *r* = *ro* →*h* =0 →*t* =0

Replacing the value of A in eq. 53 results in:

124 Wetting and Wettability

Replacing the value of B in eq. 54 provides the expression for hMAX:

lg

*MAX lv*

*lv*

 q h Æ

2 2

lg

authors [12, 57-61].

0 1 · 2 *B r* =

( ) <sup>3</sup> ·

,

, 00

*Si SiC o*

r

*M*

*C SiC SiC Si*

r


*r rr*

èø èø ë û <sup>D</sup> (57)

(58)

(59)

(60)

2

And after replacing the values of A and B, and rearranging the final expression h(r) is obtained:

( ) 3 2 <sup>2</sup>

é ù æö æö

èø èø ë û

4 3

é ù æö æö = -+ ê ú ç÷ ç÷

èø èø ë û

0 0 · 2· 1 *MAX*

It must be taken into consideration that this equation is valid only as long as the modification

Figure 8 shows the calculated infiltration distance with time for a carbon preform with an initial pore radius of 30 micron and 50% open porosity, infiltrated with pure Si, using Darcy's law and the variable radius model for cylindrical and spherical pores. The variable radius models predict a slower rate of infiltration, which is closer to the experimental data obtained by several

Another approach is to consider that the pore closure is limited by the chemical reaction that originates it. In this case, the equations of the decreasing nucleus model to be used are:

0

æ ö <sup>=</sup> ç ÷ - ç ÷

· · ·1 · · *Si SiC s Si SiC M r <sup>r</sup> <sup>t</sup> kM r* r

r

0

è ø

of the porosity is negligible. For evaluating h(t), eq. 59 and eq. 47 must be combined.

*r r*

2 0

3 2 <sup>2</sup> 2 0 00 0 · · 2· *MAX h r r r h r r r rr*

= -+ ê ú ç÷ ç÷

é ù - æö æö <sup>=</sup> ê ú ç÷ ç÷ - +

*r r r r <sup>r</sup> h a cos*

*C SiC Si*

r

<sup>1</sup> · · · · · · · · 2· · 3· · *SiC o*

*r r h h*

r

2

f f

<sup>1</sup> · ·· · · · · 3· · ·

*M r h a cos*

 q h Æ


$$0 = -4 \cdot \lambda \cdot a \cdot \gamma\_{lv} \cdot \cos \theta \cdot \frac{(1-\phi)}{\eta \cdot \mathcal{D}} \cdot \frac{M\_{Si} \cdot \rho\_{SiC}}{k\_s \cdot \rho\_{Si} \cdot M\_{SiC}} \cdot \left(\frac{r\_0^2}{2} + A \cdot r\_0 + B\right) \tag{64}$$

For condition b):

$$\frac{dh}{dr} = 0 = \frac{-4 \cdot \lambda \cdot a \cdot \gamma\_{\rm los} \cdot \cos \theta \cdot \frac{(1-\phi)}{\eta \cdot \mathcal{QP}^2} \cdot \frac{M\_{\rm Si} \cdot \rho\_{\rm SiC}}{k\_{\rm s} \cdot \rho\_{\rm Si} \cdot M\_{\rm SiC}} (0+A)}{2 \sqrt{-4 \cdot \lambda \cdot a \cdot \gamma\_{\rm los} \cdot \cos \theta \cdot \frac{(1-\phi)}{\eta \cdot \mathcal{QP}^2} \cdot \frac{M\_{\rm Si} \cdot \rho\_{\rm SiC}}{k\_{\rm s} \cdot \rho\_{\rm Si} \cdot M\_{\rm SiC}} \left(\frac{0}{2} + A \cdot 0 + B\right)}\tag{65}$$

$$A = 0\tag{66}$$

and replacing in condition a):

$$0 = -4 \cdot \lambda \cdot a \cdot \gamma\_{\rm{lo}} \cdot \cos \theta \cdot \frac{(1 - \phi)}{\eta \cdot \mathcal{D}^2} \cdot \frac{M\_{\rm{Si}} \cdot \rho\_{\rm{SiC}}}{k\_s \cdot \rho\_{\rm{Si}} \cdot M\_{\rm{SiC}}} \cdot \left(\frac{r\_0^2}{2} + 0 \cdot r\_0 + B\right) \tag{67}$$

$$B = -\frac{r\_0^2}{2} \tag{68}$$

and

$$h\_{\rm MAX} = \sqrt{\mathcal{A} \cdot \boldsymbol{\sigma} \cdot \boldsymbol{\gamma}\_{\rm tr} \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \cdot \frac{M\_{\rm Si} \cdot \boldsymbol{\rho}\_{\rm SiC}}{k\_s \cdot \rho\_{\rm Si} \cdot M\_{\rm SiC}} \cdot \boldsymbol{r}\_0^2} \tag{69}$$

Replacing the values of A, B and hMAX:

$$\hbar^2 = 4 \cdot \lambda \cdot a \cdot \gamma\_{\nu} \cdot \cos \theta \cdot \frac{\left(1 - \phi\right)}{\eta \cdot \mathcal{D}^2} \cdot \frac{M\_{\text{Si}} \cdot \rho\_{\text{SiC}}}{k\_s \cdot \rho\_{\text{Si}} \cdot M\_{\text{SiC}}} \left(\frac{r\_0^2 - r^2}{2}\right) \tag{70}$$

Figure 9 shows the square of the infiltration distance as a function of time for a carbon preform with an initial pore radius of 30 micron and 50% open porosity infiltrated with pure Si, using Darcy's law and the variable radius model for a chemically controlled pore closure. As seen, infiltration rate decreases with time, just as in the previous case (diffusion controlled pore closure). However, the experimental data [12, 42, 59-66] show that the infiltration rate is lower than the one predicted by these models; furthermore, infiltrated height is proportional to t instead of to t(1/2). Experiments indicate that infiltration is controlled by the reaction at the triple line and not by the viscous phenomena described here. However, both interpretations are compatible. It is probable that in the first stage of the process, infiltration is controlled by the reaction at the triple line, and after the SiC interface has been formed, the viscous flow becomes the controlling process. A completely different approach to the process of infiltration is performed by Sergi et al. [67, 68], in which they use Lattice-Bolztman models to describe infiltration in a single capillary and predict that infiltration distance is proportional to time when considering pore narrowing of the capillary due to the reaction. However, their predic‐ tions have not been validated experimentally up to date. It is possible to conclude that a deeper understanding of the reactive infiltration phenomenon should be gained in order to discern what is happening in the overall process.

For condition b):

126 Wetting and Wettability

( ) ( )

*M a cos A*

r

*a cos A B k M*


r

*s Si SiC Si SiC*

0

è ø

2 0

( ) <sup>2</sup> 2 0

*Si SiC*

r


ç ÷ è ø

*s Si SiC*

*k M*

r

( ) 2 2

*s Si SiC*

*k M*

r

Figure 9 shows the square of the infiltration distance as a function of time for a carbon preform with an initial pore radius of 30 micron and 50% open porosity infiltrated with pure Si, using Darcy's law and the variable radius model for a chemically controlled pore closure. As seen, infiltration rate decreases with time, just as in the previous case (diffusion controlled pore closure). However, the experimental data [12, 42, 59-66] show that the infiltration rate is lower than the one predicted by these models; furthermore, infiltrated height is proportional to t instead of to t(1/2). Experiments indicate that infiltration is controlled by the reaction at the triple line and not by the viscous phenomena described here. However, both interpretations are compatible. It is probable that in the first stage of the process, infiltration is controlled by the

*Si SiC*

r è ø

*A* = 0 (66)

*<sup>r</sup> <sup>B</sup>* = - (68)

(65)

(67)

(70)

*s Si SiC*

r

*Si SiC*

r

( )

<sup>1</sup> · 4· · · · · · · 0 · · · <sup>0</sup>

 q h Æ

 q h Æ

<sup>1</sup> · 0 4· · · · · · · 0· · ·· 2

f


*lv*

*dh k M*

*lv*

*dr M*

lg

*lv*

*MAX lv*

*lv*

 q h Æ

lg

Replacing the values of A, B and hMAX:

lg

 q h Æ

lg

= =

and replacing in condition a):

and

lg

2

( ) <sup>2</sup>

r

2 0 2

<sup>1</sup> · ·· · · · · · · ·

f

*<sup>M</sup> h a cos <sup>r</sup>*

 q h Æ

2 0 2 <sup>1</sup> · 4· · · · · · · · ·· 2

*M rr h a cos*

f


*Si SiC*

r

*s Si SiC M r a cos r B k M*

<sup>1</sup> · <sup>0</sup> 2· 4· · · · · · · ·0 · ·· 2

f 2

f


**Figure 9.** Comparison of the square distance of infiltration (h2 ) as a function of time, calculated using Darcy's law and the variable radius models for a carbon with an initial pore radius of 30 micron, and 50% porosity, infiltrated with pure Si. For the variable radius model, pore closure is assumed to be controlled by the chemical reaction. Infiltration is si‐ mulated until the pore radius decreases by 30%.

#### **7. Effect of variable viscosity: Curing polymeric resins**

The properties of PMCs depend strongly on the type of matrix used. Among the possible binders, thermoset resins present a singularity: their viscosity is time dependent [69, 70]. These kinds of polymers present an initial low viscosity that grows exponentially with time. During this process, the material undergoes a series of chemical reactions that cause the cross-linking of polymeric chains yielding a 3D structure with excellent thermal stability and solvent resistance. The process of cross-linking that takes place in these resins is known as curing. In polymer matrix composites in which the matrix cures with time, thus hardening and losing fluidity, viscosity becomes a limiting factor in infiltration.

The evolution of viscosity is usually expressed with the following equation:

$$
\eta = \eta\_0 \cdot e^{k\_n \cdot t} \tag{71}
$$

Where *η*0 is the initial viscosity and *kn* is a strongly temperature-dependent rheological kinetic constant [70]. Starting from the reduced Navier-Stokes equation:

$$
\phi \cdot \frac{dz}{dt} = -\frac{K}{\eta} \cdot \frac{dP}{dz} \tag{72}
$$

After integration, the following expression is obtained:

$$\frac{1}{2\cdot 2}h^2 = -\quad \frac{K}{\mathcal{D}\cdot \eta}d\mathcal{P}\cdot dt\tag{73}$$

Introducing eq. 71 and solving the integral yields a model for predicting infiltration distance as a function of time for curing polymeric resins:

$$\frac{1}{2\cdot}h^2 = -\quad \frac{K}{\mathcal{D} \cdot \eta\_0 \cdot e^{k\_\circ \cdot t}}dP \cdot dt \tag{74}$$

$$\frac{1}{2\cdot}h^2 = -\frac{K}{\mathcal{D}\cdot\eta\_0} \quad \frac{dP}{e^{k\_\ast\cdot t}}dt\tag{75}$$

$$h^2 = -\frac{\mathsf{Z} \cdot K \cdot \Delta \mathsf{P}}{\mathcal{Q} \cdot \eta\_0} \cdot \left[ e^{-k\_{\text{s}} \cdot t} \cdot dt \right. \tag{76}$$

$$h^2 = \frac{\mathbf{2} \cdot \mathbf{K} \cdot \Delta \mathbf{P}}{\mathcal{D} \cdot \eta\_0} \frac{e^{-k\_\pi \cdot t}}{k\_\imath} + A \cdot t + B \tag{77}$$

To calculate the values of integration constants A and B, initial conditions are applied:


For condition a):

$$\begin{aligned} 0 &= \frac{\mathbf{2} \cdot \mathbf{K} \cdot \boldsymbol{\Delta P}}{\boldsymbol{\mathcal{Q}} \cdot \boldsymbol{\eta}\_{0}} \cdot \frac{1}{\boldsymbol{e}^{k\_{u} \cdot \boldsymbol{0}}} + A \cdot \mathbf{0} + B\\ \mathbf{B} &= -\frac{\mathbf{2} \cdot \mathbf{K} \cdot \boldsymbol{\Delta P}}{\boldsymbol{\mathcal{Q}} \cdot \boldsymbol{\eta}\_{0} \cdot k\_{u}} \end{aligned} \tag{78}$$

for condition b):

Where *η*0 is the initial viscosity and *kn* is a strongly temperature-dependent rheological kinetic

· · *dz K dP dt dz*

<sup>1</sup> 2 · 2· · *<sup>K</sup> <sup>h</sup> dP dt* Æ h

Introducing eq. 71 and solving the integral yields a model for predicting infiltration distance

· 0 1 · 2· · · *nk t <sup>K</sup> <sup>h</sup> dP dt* Æ h

·

0 1 ·· 2· · *nk t K dP <sup>h</sup> dt* Æ h

·

*n K Pe <sup>h</sup> At B*

·0

*K P A B e K P <sup>B</sup>*

2· · · ·

Æ h

<sup>D</sup> = -

*nk*

0

*n*

*k*

0

D = ++

Æ h

2· · 1 0 · ·0 ·

To calculate the values of integration constants A and B, initial conditions are applied:

2 · 0 2· · · · · *<sup>n</sup> K P k t <sup>h</sup> e dt* Æ h

> 0 2· · · · · *nk t*

Æ h *k* - D

h

= - (72)

= - (73)

*<sup>e</sup>* = - (74)

*<sup>e</sup>* = - (75)

<sup>D</sup> - = - ò (76)

= ++ (77)

(78)

f

2

2

2

**a.** *t* =0;*h* =0

128 Wetting and Wettability

*dh dt* =0

For condition a):

**b.** *<sup>t</sup>* <sup>=</sup>*∞*;

constant [70]. Starting from the reduced Navier-Stokes equation:

After integration, the following expression is obtained:

as a function of time for curing polymeric resins:

$$\frac{dh}{dt} = 0 = \frac{-\frac{2 \cdot K \cdot \Delta P}{\mathcal{Q} \mathcal{I}} e^{-k\_n \cdot t} + A}{2 \cdot \sqrt{\frac{e^{-k\_n \cdot t}}{-k\_n} + A \cdot t + B}} \tag{79}$$
 
$$A = 0$$

Then the final equation results in:

$$\hbar^2 = \frac{\mathbf{2} \cdot \mathbf{K} \cdot \boldsymbol{\Delta P}}{\boldsymbol{\mathcal{Q}} \cdot \boldsymbol{\eta}\_0 \cdot \mathbf{k}\_n} \cdot e^{-\mathbf{k}\_n \cdot \mathbf{t}} - \frac{\mathbf{2} \cdot \mathbf{K} \cdot \boldsymbol{\Delta P}}{\boldsymbol{\mathcal{Q}} \cdot \boldsymbol{\eta}\_0 \cdot \mathbf{k}\_n} \tag{80}$$

$$h^2 = -\frac{2 \cdot K \cdot \Delta P}{\mathcal{D} \cdot \eta\_0 \cdot k\_u} (1 - e^{-k\_u \cdot t}) \tag{81}$$

It must be highlighted that the negative sign is only indicative of the direction of flow. Figure 10 shows the evolution of the square of infiltrated height for a preform infiltrated with curing resins with an initial viscosity of 100 Pa s at different temperatures. The drastic effect of temperature on infiltration distance becomes evident, since increasing temperature to 50°C reduces infiltrated distance by one order of magnitude.

**Figure 10.** Evolution of the square of infiltration height with time for curing resins into porous preforms at different temperatures.

#### **8. Concluding remarks**

The present chapter has introduced the reader to the processes involved in the manufacture of different types of composite materials (PMCs, MMCs, CMCs) via infiltration. Over 90% of the world production of composite materials is carried out by infiltration, especially in the case of PMCs where it is almost the only available processing technique, while in case of metal matrix, production via infiltration represents only 70% [7]. In order to accurately control the properties of the final material and to be able to optimize the production, it is key to identify and understand the different processes involved in the manufacture of these materials. For this purpose, in this chapter, different models have been elucidated, starting from the differ‐ ential equation of Navier-Stokes and taking into account the basic laws of wetting, by applying different real conditions often found in industry. For PMCs, the main discussed issue was the change in viscosity that thermoset matrixes experiment during infiltration. By introducing this phenomenon in the infiltration equations, it becomes clear that it can be a limiting factor during production since it shows a great deviation from Darcy's law. In the case of MMCs, the main phenomenon analyzed was the effect of saturation on infiltration. It became clear that variable saturation not only affects in a huge manner infiltration kinetics, but also changes the domi‐ nating phenomena. The last publications in this area have identified percolation as probably the leading phenomenon on infiltration in unsaturated systems at very low overpressure. Finally, for CMCs, the role of pore narrowing, an issue that has been widely discussed in the literature over the past 10 years was revised. This phenomenon is especially relevant in processing of ceramics and ceramic matrix composites via reactive techniques like the RBSC method used for SiC production. Even though the research on this area is vast, there is still not a full understanding of the effect of pore reduction on infiltration. Most models proposed still yield higher infiltration rates than the ones observed experimentally, because they do not consider the effect of chemical reactivity.

#### **Appendix**


Table 1 shows the main characteristic of the carbon particles used in this work.

**Table 1.** Main characteristics of the carbon particles used. Span is defined as D(90) − D(10)/D(50), where D(x) represents the diameter below which x% of particles are encountered. Vp is the particle volume fraction.

Table 2 shows the value of the parameters required to assess h(t) for the several cases exposed in this work. All parameters are in SI units.


**Table 2.** Value of the main parameters used in this work in SI units.

#### **Acknowledgements**

**8. Concluding remarks**

130 Wetting and Wettability

consider the effect of chemical reactivity.

in this work. All parameters are in SI units.

Table 1 shows the main characteristic of the carbon particles used in this work.

**Particle D(90) D(50) D(10) D(4,3) D(3,2) Span Vp P1** 25.8 13.4 7.0 15.1 11.6 1.39 0.511 **P2** 41.0 25.8 15.5 27.2 23.0 0.99 0.515 **P3** 95.1 62.0 36.8 64.0 53.0 0.94 0.518 **P4** 173.6 117.6 56.6 124.1 110.4 0.99 0.482

**Table 1.** Main characteristics of the carbon particles used. Span is defined as D(90) − D(10)/D(50), where D(x) represents the diameter below which x% of particles are encountered. Vp is the particle volume fraction.

Table 2 shows the value of the parameters required to assess h(t) for the several cases exposed

**Appendix**

The present chapter has introduced the reader to the processes involved in the manufacture of different types of composite materials (PMCs, MMCs, CMCs) via infiltration. Over 90% of the world production of composite materials is carried out by infiltration, especially in the case of PMCs where it is almost the only available processing technique, while in case of metal matrix, production via infiltration represents only 70% [7]. In order to accurately control the properties of the final material and to be able to optimize the production, it is key to identify and understand the different processes involved in the manufacture of these materials. For this purpose, in this chapter, different models have been elucidated, starting from the differ‐ ential equation of Navier-Stokes and taking into account the basic laws of wetting, by applying different real conditions often found in industry. For PMCs, the main discussed issue was the change in viscosity that thermoset matrixes experiment during infiltration. By introducing this phenomenon in the infiltration equations, it becomes clear that it can be a limiting factor during production since it shows a great deviation from Darcy's law. In the case of MMCs, the main phenomenon analyzed was the effect of saturation on infiltration. It became clear that variable saturation not only affects in a huge manner infiltration kinetics, but also changes the domi‐ nating phenomena. The last publications in this area have identified percolation as probably the leading phenomenon on infiltration in unsaturated systems at very low overpressure. Finally, for CMCs, the role of pore narrowing, an issue that has been widely discussed in the literature over the past 10 years was revised. This phenomenon is especially relevant in processing of ceramics and ceramic matrix composites via reactive techniques like the RBSC method used for SiC production. Even though the research on this area is vast, there is still not a full understanding of the effect of pore reduction on infiltration. Most models proposed still yield higher infiltration rates than the ones observed experimentally, because they do not

The authors would like to thank the University of Alicante for Danilo Sergi's scholarship under the program "Support for short stays of foreign researchers at the University of Alicante".

Financial support from the Generalitat Valenciana (PROMETEO II/2014/004- FEDER, Master grant "Santiago Grisolia" and PhD grant Vali+d), and the European Union's Seventh Frame‐ work Programme (FP7/2007-2013) under the HELM project (grant agreement no. 280464) is gratefully acknowledged.

#### **Author details**

Mario Caccia1 , Antonio Camarano1 , Danilo Sergi2 , Alberto Ortona2 and Javier Narciso1

\*Address all correspondence to: narciso@ua.es


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**Chapter 6**
