Materials for Biomedical Engineering. Mohamed N. Rahaman

are the specific energies (energy per unit area) of the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively, and θ is the contact angle between the liquid and the solid (the angle between the tangent to the liquid–vapor interface at the contact point with the solid surface). At equilibrium, dG/dA = 0 and Eq. (5.2) gives

(5.3)

Figure 5.3 Contributions to the Gibbs free energy change due to change in area dA of a liquid drop on a solid.

Equation (5.3), sometimes referred to as the Young and Dupré equation, is what we would obtain by taking the horizontal components of the interfacial tensions in Figure 5.3. Roughly, good wetting of a solid by a liquid is said to be characterized by a low contact angle ( θ < 90°) and poor wetting by a high contact angle ( θ > 90°). Complete wetting and spreading of a liquid occurs when θ = 0° (Figure 5.4).

Figure 5.4 Wetting behavior between a liquid and a solid showing (a) good wetting, (b) poor wetting, and (c) complete wetting for a liquid of contact angle θ .

      As the physiological fluid is aqueous in nature, the extent to which water will wet a biomaterial and spread over it has significant consequences for its interaction with the aqueous medium in vivo. A material that shows good wetting and spreading by water (low θ ) is referred to as hydrophilic (literally, water‐loving). If the solid has a higher surface energy than water, there is a thermodynamic driving force for wetting and spreading of the liquid in order to reduce the energy of the system. In comparison, a material that shows poor wetting by water (high θ ) is referred to as hydrophobic (literally, water‐hating). In this case, the material has a lower surface energy than water and, thus, wetting is thermodynamically unfavorable because it will lead to an increase in the energy of the system.

      5.2.1 Determination of Surface Energy of Materials

      According to Eq. (5.3), the surface energy γ_sv is related to the surface energy of the liquid γ_lv, the solid–liquid interfacial energy γ_sl and the contact angle θ . Whereas γ_lv and θ can be easily measured, γ_sv and γ_sl cannot. Consequently, methods for estimating γ_sv often involve a combination of measuring γ_lv and θ, and estimating γ_sl using theoretical analyses, some of which can be fairly complex. While a variety of methods have been proposed, a useful method for polymers, based on its simplicity and accuracy, involves measuring θ for a single liquid of known surface energy (surface tension) γ_lv and using a theoretical expression for γ_sl (Girifalco and Good 1957):

(5.4)

where Φ is expressed in terms of V_s and V_l, the molar volume (molecular weight divided by the density) of the molecules or molecular segments of the solid and the liquid, respectively, given by

(5.5)

For a polymer, the repeating unit in the polymer chain can be taken to approximate the molecular segment. Using Eqs. (5.3) and (5.4), we get

(5.6)

      Example 5.1

      The measured contact angle of a water droplet on polymethyl methacrylate (PMMA) is 70°. Determine the surface energy of PMMA, given that the densities of PMMA and water are 1.2 and 1.0 g/cm³, respectively, and the surface tension of water is 73.0 mN/m.

      Solution:

The molecular weight of water is 18.0 g/mol and the molecular weight of the repeating unit in PMMA is 100.1 g/mol, giving V_l = 18.0 cm³, and V_s = 83.4 cm³. Substituting in Eq. (5.5), we get Φ = 0.94. Then, substituting the values for Φ, γ_lv, and θ in Eq. (5.6) gives γ_sv = 37 mJ/m².

      Another method applicable to materials of low surface energy, such as polymers, is referred to as the Zisman method. Fox and Zisman (1950, 1952) found that cos θ for a variety of polymers was approximately a monotonic function of γ_lv, that is

(5.7)

where a and b are constants. Eq. (5.7) can be expressed in the form

      (5.8)

where c is a constant and γ_cr is a characteristic property of the material called the critical surface tension. The significance of γ_cr is that a liquid of surface tension less than or equal to γ_cr will wet and spread over the material. This follows from the limiting condition for spreading is θ = 0° and the fact that the contact angle cannot be less than 0°. The Zisman method involves measuring the contact angle θ for several liquids of known surface tension on a given polymer and plotting cos θ versus γ_lv (Figure 5.5). For many polymers, it is found that γ_cr is approximately equal to γ_sv, as exemplified by the data for PMMA in Figure 5.5. However, because of the time‐consuming nature of the experiments and the less than rigorous theoretical justification, the Zisman method has found only limited application in recent years.