# Surface free energy

Measurements of surface tension yield data, which directly reflect thermodynamic characteristics of the liquid tested. Measurement of contact angles yield data, which reflect the thermodynamics of a liquid/solid interaction. If you wish to characterize the wetting behavior of a particular liquid/solid pair you only need to report the contact angle. It is possible to characterize the wettability of your solid in a more general way. To characterize the thermodynamics of the solid surface itself more elaborate analysis is required. Various methods are used but the same basic principle applies for each. The solid is tested against a series of liquids and contact angles are measured.

Calculations based on these measurements produce a parameter (critical surface tension or surface free energy), which quantifies the characteristics of the solid and mediates the properties of the solid substrate. The critical surface tension or the surface free energy obtained in this way can be regarded as the "surface tension" of the solid substrate, which is a characteristic property of the solid in the same way as the surface tension is for a liquid.

Four different approaches are mainly used for determining the energy of solid substrates:

**1. Critical Surface Tension (Zisman)**

Using a series of homologous nonpolar liquids of differing surface tensions a graph of cos θ vs γ is produced. It will be found that the data form a line which approaches cos θ = 1 at a given value of γ. This value, called the critical surface tension, can be used to characterize your solid surface. It is often presented as the highest value of surface tension of a liquid which will completely wet your solid surface. This approach is most appropriate for low energy surfaces which are being wetted by nonpolar liquids. See references for details on procedure and limitations.

The other ways of characterizing a solid surface is by calculating the free surface energy from theories using slightly different approaches for the calculations. These approaches involve testing the solid against a series of well characterized liquids. The liquids used must be characterized such that the polar and dispersive components of their surface tensions are known.

**2. Geometric Mean (Fowkes)**

This approach divides the surface energy into two components, dispersive and polar, and uses a geometric mean approach to combine their contributions. The resulting equation when combined with Young's equation yields:

γ_{l} (1 + cos θ) = 2 [(γ_{l}^{p}γ_{s}^{p})^{1/2} + (γ_{l}^{d} γ_{s}^{d})^{1/2}]

This equation can be rearranged as by Owens and Wendt to yield:

γ_{l} (1 + cos θ) / (γ_{l}^{d})^{1/2} = (γ_{s}^{p})^{1/2} [(γ_{l}^{p})^{1/2} / (γ_{l}^{d})^{1/2}] + (γ_{s}^{d})^{1/2}

where θ is the contact angle, γ_{l} is liquid surface tension and γ_{s} is the solid surface tension, or free energy. The addition of d and p in the superscripts refer to the dispersive and polar components of each. The form of the equation is of the type y = mx + b. You can graph (γ_{l}^{p})^{1/2} /(γ_{l}^{d})^{1/2} vs γ_{l} (1 + cos θ) / (1+γ_{l}^{d})^{1/2} .The slope will be (γ_{s}^{p})^{1/2} and the y-intercept will be (γ_{s}^{d})^{1/2}. The total free surface energy is merely the sum of its two component forces.

**3. Harmonic Mean(Wu)**

This method utilizes a similar approach but uses a harmonic mean equation to sum the dispersive and polar contributions. Contact angles against two liquids with known values of γd and γp are measured. The values for each experiment are put into the following equation:

(1 + cos θ) γ_{l} = 4 (γ_{l}^{d} γ_{s}^{d} / γ_{l}^{d} + γ_{s}^{d} + γ_{l}^{p} γ_{s}^{p} / γ_{l}^{p} + γ_{s}^{p})

where γ refers to surface tension (surface free energy), the subscripts l and s refer to liquid and solid, and the superscripts d and p refer to dispersive and polar components. You then have two equations with two unknowns and can solve for γ_{s}^{d} and γ_{s}^{p}.

**4. Acid-Base (van Oss)**

Contact angles against at least three liquids with known values of γd, γ+ and γ- are measured. The values for each experiment are put into the following equation:

0.5 (1 + cos θ) γ_{l} = (γ_{s}^{d} γ_{l}^{d} )^{1/2} + (γ_{s}^{-} γ_{l}^{+})^{1/2} + (γ_{s}^{+} γ_{l}^{-})^{1/2}

where γ refers to surface tension (surface free energy), the subscripts l and s refer to liquid and solid, and the superscripts d, + and - refers to dispersive, acid and base components. You then have three equations with three unknowns and can solve for γ_{s}^{d}, γ_{s}^{+} and γ_{s}^{-}. The total surface free energy of the solid is then given by:

γ_{s} = γ_{s}^{d} + γ_{s}^{AB} , where γ_{s}^{AB} = 2 (γ_{s}^{+} γ_{s}^{-})^{1/2}

**Surface Free Energy using Theta Optical Tensiometer Video:**

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