## General Model

For a composite interface built of two fractions with the fractional areas off1 and f2 (so that f1 ? f2 = 1) the contact angle is given by the Cassie equation cos h = f1 cos h1 + f2 cos h2 (7.3)

where h1 and h2 are the contact angles of the fractions. If a composite material has a matrix and reinforcement with the volume fractions of fm and fr (so that fm ? fr = 1) forming a rough surface the contact angle is then given by cos h = Rfm(1 — fr) cos hm + Rfrfr cos hr (7.4)

where hm and hr are the contact angles for the matrix and reinforcement materials, and Rfm and Rfr are the corresponding roughness factors. Note that for spherical reinforcement particles the roughness factor is equal to the ratio of half of the sphere's area 2pR2 to the cross-sectional area pR2 or Rfr = 2. Solving for the reinforcement fraction yields the volume of the reinforcement fraction providing the desired contact angle h f = cos h — Rfm cos hm (7 5)

Further assuming Rfr = 2, Rfm = 1 (no roughness expected from the reinforcement particles), and h = 180° (the superhydrophobic limit) yields

2 cos dr — cos dm which has a solution (fr < 1) if dr > 120°. Thus it is difficult to produce a composite interface by only using the reinforcement roughness.

If water forms partial contact with the solid (composite or Cassie-Baxter) interface with the fractional solid-liquid contact areas /SLm and fSLr, the contact angle is given by cos d = Rfm (1 — fr)JsLm COS dm + Rfr/r/sLr COS Or — 1 + JrfsLr +(1 — fr )JsLm (7.7)

Solving for the reinforcement fraction yields the volume of the reinforcement fraction providing the desired contact angle d j = cos d — RfmfsLm cos dm + 1 — fsLm (7 g)

r Rfr/sLr cos Or — R/m/sLm cos dm + JsLr — fsLm '

Making the assumptions of Rfr = 2, Rfm = 1, /SLr = 1, and d = 180° yields j = _—fSLm — fSLm cos dm__(7 9)

Let us apply Eqs. 7.3-7.9 to the metallic (aluminum or copper) matrix with amorphous graphite reinforcement. For that end, we need to substitute the material properties of these materials. We measured the water contact angle with graphite, aluminum, and copper, which were used to produce MMC samples at the UWM Center for Composite materials, using the rame-hart Model 250 goniometer/tensiometer. We found the values of the contact angle equal to 140°, 47.2° and 47.7°, respectively. The results for metal matrices are close to those available in the literature [19]. To measure the water contact angle of amorphous graphite, we compressed it at first to obtain a smooth surface. The main reason for high water contact angle of graphite is that the surface still remains rough even after compressing. Since we need the water contact angle of smooth graphite, we used the result of Fowkes and Harkins [20]. They measured the contact angle of water on smooth graphite using the tilting plate method and found a value of 86°. A similar result (84°) is reported by Morcos [21]. Figure 7.2 shows the variation of the contact angle of water droplet on surface of metal matrix composite reinforced by graphite particles versus reinforcement volume fraction (dm = 47°, dr = 86°), as obtained from Eq. 7.8 for various values of fSLm. As seen in this figure, for fSLm > 0.4, the contact angle increases with increase in the reinforcement volume fraction, whilst for fSLm < 0.4, it decreases with increase in the reinforcement volume fraction. Figure 7.3 shows the variation of fSLm versus the reinforcement volume fraction, fr as obtained from Eq. 7.9 for cases where d is equal to 150o and 180o. For this case, it is assumed that the matrix is made of aluminum whereas the reinforcement is made of a material with the water contact angle of 140o. As observed from this figure,

Fig. 7.2 The water contact angle as a function of graphite reinforcement volume fraction for different M

Reinforcement volume fraction

Fig. 7.3 The variation of the reinforcement volume fraction, /r, versus /SLm

the reinforcement volume fraction, /r, is proportional to /SLm. The area between two lines of h = 150o and h = 180o is called the superhydrophobicity area.

The effect of wear on a composite material is twofold. First, the matrix roughness factor, Rfm, can be changed due to material removal and evolve to a certain ''equilibrium value'' [18]. This can affect the solid-liquid fractional area, /SLm. Second, the reinforcement particles can be removed as matrix surface layers are removed due to the deterioration. However, new particles come in contact so it is expected that the values of Rfr and/SLr do not change significantly. To decouple the effect of reinforcement and matrix roughness we investigated experimentally wetting of composite materials with initially smooth surface and with the matrix roughness by etching, as described in the next section.

7.2.2 Underwater Oleophobicity

If an oil droplet is placed on a solid surface in water, the contact angle of an oil droplet in water, hOW, is given by Young's equation:

cos Qow

where QO and QW are contact angles of oil and water with the solid surface, yOA, yWA and yOW are interfacial energies for oil-air, water-air, and oil-water interfaces. As a consequence, a superhydrophobic (in air) surface can become oleophobic when immersed in water, under certain circumstances, which are summarized in Table 7.1. An oleophobic surface repels organic liquids and thus prevent organic contaminants from accumulation and decrease the adhesion of bacteria, thus facilitating antifouling properties.

Similar to the superhydrophobic surfaces, besides the homogeneous solid-oil interface (Wenzel state), a composite solid-oil-water interface (Cassie-Baxter or Cassie state) with water pockets trapped between the solid and the oil droplet can exist. The contact angle is then given by

Where 0 < fSO < 1 is the fractional solid-oil contact area and Rf is the roughness factor. The rules of the Cassie-Wenzel wetting regime transition are the same as in the case of superhydrophobic surfaces, as discussed in the preceding chapters. A more complex four-phase solid-oil-water-air system can form if both water and air bubbles are present at the composite interface. It is expected that with time air will dissolve so that fSA ? 0.

A wetting regime transition, similar to the Cassie-Wenzel transition can apparently occur at the solid-oil-water interface. The evidence of this is presented in Fig. 7.4 showing solid-oil-water system with the same aluminum alloy of different surface roughness. In our experiments samples with lower roughness having Root-mean-square (RMS) Ra = 0.2 im, showed low contact angles of h = 43.10° and h = 24.96°, whereas samples with higher roughness (etched by an acid) Ra = 0.8 im showed contact angles of h = 140.78o and h = 141.30o. We attribute such an abrupt change of roughness to the fact that the Wenzel state (homogeneous solid-oil interface) was realized for the low roughness samples, whereas the Cassie-Baxter state (composite solid-oil-water interface) was realized for high roughness samples.

### 7.2.3 Reinforcement-Induced Roughness

In this section we investigate wetting properties of a composite material with particulate or fiber reinforcement. The reinforcement, such as graphite, modifies wetting properties of the matrix material, such as a metal, by providing heterogeneity and surface roughness.

cos QOW

Table 7.1 Oleophobic and oleophilic interfaces

Interface

Solid-air- Hydrophobic ySA > ysw Hydrophobic ySA < ysw water

Solid-air- Oleophilic if Oleophobic if ySA < yso Oleophilic if ySA > yso Oleophobic if ySA < yso oil ySA > 7so

Solid- Oleophilic Oleophilic if Oleophobic if Oleophilic if Oleophobic if water- yOAcos(90 < ywAcos^w [email protected] > [email protected] [email protected] < [email protected] [email protected] < [email protected]

 o t < 0 seconds etching: fit= 0.2 jam 6=43 10' rrr 0 seconds etching: R„= 0.2 20 seconds etching: R0 = 0.8 |im 9=24,96" Jim, 6=141,30° c o ra F*nc

Fig. 7.4 Solid-oil-water system with low roughness (left) and high roughness, A dramatic change of the contact angle is attributed to the Cassie-Wenzel wetting regime change

Fig. 7.4 Solid-oil-water system with low roughness (left) and high roughness, A dramatic change of the contact angle is attributed to the Cassie-Wenzel wetting regime change

7.2.3.1 Homogeneous (Wenzel) Solid-Liquid Interface

The Wenzel model defines the homogeneous (solid-liquid) wetting regime. According to that model, water penetrates inside all crevices and there are no air gaps left between the liquid and solid (Fig. 7.1a).

### Particle Reinforcement

Now we model our cases starting with particle reinforcements. For this case, we assumed that the spherical particles make roughness on the surface. We considered a layer of matrix which contains randomly distributed particles with radii r (Fig. 7.5).

The reinforcements volume fraction, fv, is obtained through dividing the total volume of all particles inside the layer, Vr, by volume of layer, Vtot v»-

Vtot

4npr3

Fig. 7.5 The schematic of a matrix layer passes through spherical particles

Fig. 7.6 The schematic of changing h due to wear

where Atot is the top surface of the layer and n is the total number of particles inside the layer.

Substituting (7.13) and (7.14) into (7.12), one can show that

3/vVtot 4pr3

The height of each particle out of matrix layer is defined by h. Since the metal matrices are usually softer than reinforcements, this height can be changed due to wear during the time. We assumed that 0 < h < r (Fig. 7.6). therefore, fr = fv (7 .16)

where the total area of reinforcements in contact area, Ac, can be obtained by

We can find the fm from the following equation:

We define averaged Wenzel roughness factor as follows:

From Wenzel and Cassie-Baxter equations:

Fig. 7.7 Contact angle of 1801-

water droplet versus graphite particles volume fraction in metal matrix (Wenzel model)

Reinforcement volume fraction

Substituting (7.16), (7.18), and (7.19) into (7.20) we have:

Let us apply Eq. 7.21 to the metallic (aluminum or copper) matrix with graphite reinforcement. For that end, we need to substitute the material properties of these materials. We measured the water contact angle with graphite, aluminum, and copper, which were used to produce MMC samples at the UWM Center for Composite materials, using the rame-hart Model 250 goniometer/tensiometer. Figure 7.7 shows the variation of the contact angle of water droplet on surface of metal matrix composite reinforced by graphite particles versus reinforcement volume fraction. As seen in this figure, the contact angle increases with increase in the reinforcement volume fraction.

### Fiber Reinforcement

For this case, we assumed that the aligned fibers perpendicular to the surface make roughness on it. The length of each fiber out of matrix layer divided by fiber radius is defined by a (Fig. 7.8).

The Wenzel roughness factor is defined as follows:

Ac Ac r where lp is the length of each fiber out of matrix layer. r and n are the radius and total number of fibers inside certain volume, respectively. One can easily prove the following equations:

So after calculating the following roughness factor is obtained:

Fig. 7.8 The schematic of a certain volume passes perpendicular to the reinforcement fiber length

Fig. 7.9 Variation of contact angle versus graphite aligned fibers volume fraction in metal matrix (Wenzel model)

Using Eq. 7.20 and substituting the values of fr, fm and Rf into it, we have the following equation for calculating the contact angle on composite surface:

hc = cos-1 [2a cos hf2 + (cos hr — cos hm)fv + cos hm] (7.25)

Then we apply Eq. 7.25 to the metallic (aluminum or copper) matrix with graphite reinforcement. Figure 7.9 shows the contact angle of water droplet on surface of metal matrix composite reinforced by graphite fibers versus reinforcement volume fraction. As it is seen in this figure, the contact angle increases with increasing the reinforcement volume fraction and the value of "a" as well.

7.2.3.2 Composite (Cassie-Baxter) Solid-Liquid-Air Interface

The Cassie-Baxter model defines the heterogeneous wetting regime. Based on Cassie-baxter model, there are some air gaps left between liquid and solid surfaces and we deal with a three-phase system of liquid-solid-gas (Fig. 7.1b).

In order to calculate the contact angle, h, for a rough surface in a manner similar to the previous section but for Cassie-Baxter model, the differential area of the liquid-air interface under the droplet, fLAdAc, should be subtracted from the differential of the total liquid-air area dALA, which yields cos h = Rf cos h0 — fLA(Rf cos h0 + 1)

Fig. 7.10 Contact angle of water droplet versus graphite particles volume fraction in metal matrix for different fractions of liquid-air in contact (C-B model)

such that the fLA is the fraction of contact area between liquid and air to total contact area and ho is contact angle between the water droplet and the solid with relatively smooth surface.

### Particle Reinforcement

For this case, we modeled our MMCs based on Cassie-baxter in case the spherical particles make roughness on the surface, so that the Cassie-Baxter equation is turned into

Cos6c = Rffr cos hr + fm cos hm — /la(R/fr cos hr + fm cos hm + 1) (7.27)

The values of R/, fm, and fr are defined by Eqs. 7.16, 7.18, and 7.19 and after substituting into Eq. 7.26 we obtained the following contact angle:

(1 — /la)( 1616f3f cos hr +(1 — fv) cos 6m + 1 J — 1

Figure 7.10 shows the contact angle of water droplet on the surface of aluminum matrix composite versus graphite particles volume fraction. As seen in this figure, the contact angle increases with increase in the reinforcement volume fraction and the value offLA as well.

### Fiber Reinforcement

For this case, we modeled our MMCs based on Cassie-baxter in case the aligned fibers make roughness on the surface, so that the Cassie-Baxter equation is the same as Eq. 7.20.

The values of R/, fm, and fr are defined by Eqs. 7.23 and 7.24 and after substituting into Eq. 7.27, we obtained the following contact angle:

Fig. 7.11 Contact angle of water droplet versus graphite aligned fibers volume fraction in metal matrix for different aspect ratio and fractions of liquid-air in contact (C-B model)

Oc = cos 1 [2a cos Or (1 — fLA)f2 +[cos Or — cos Om — /la(cos Or + 1)] fv + cos Om]

Figure 7.11 shows the variation of the contact angle of water droplet on the surface of aluminum matrix composite versus graphite fibers volume fraction inside certain volume. As seen in this figure, the contact angle increases with increase in the reinforcement volume fraction. These figures show that for a = 0.1, the contact angle is proportional to fLA, whereas for a = 0.5 and a = 0.9, it is proportional to the inverse of fLA.

Fig. 7.12 The schematic of surface pattern after etching for particles and fibers reinforcement

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