Equations of surface energy.
T.Tokoro

Using the contact angle q and balancing the resolved interfacial tensions in the plane of the surface, the final equilibrium condition is the well-known Young's equation:

g_SL + g_LV cos q = g_SV . (1)

Here g_SL, g_LV and g_SV are an interfacial free energy ( or surface tension ) of solid-liquid, liquid-vapor and solid-vapor interfaces, respectively. For our study q >10 then g_S and g_L, the surface energy of Nylon in vacuum and of water with its own saturated vapors, are equal to g_SV and g_LV. Therefore (1) is,

g_SL + g_Lcos q = g_S. (1a)

If q 0 then p_e the equilibrium spreading pressure of the vapor on the substrate is not negligible. p_e is therefore the decrease of surface energy due to vapor adsorption.

p_e = g_S - g_SV.

Using the free energy of the work adhesion W_SL, Dupre equation is,

g_SL = g_S + g_L - W_SL. (2)

Combination of (1a) and (2) yields the Young-Dupre equation,

g_L ( 1 + cos q ) = W_SL. (3)

For systems that are polar, the surface free energies are assumed to be composed of two parts: dispersion (g_SD and g_LD ) and non-dispersion (g_SH and g_LH ), i.e.,

g_S = g_SD + g_SH. (4)

g_L = g_LD + g_LH. (5)

Here, g_SD and g_LD are dispersion parts of free energy of Nylon and water and g_SH and g_LH are non-dispersion parts of free energy of Nylon and water, respectively.

For Harmonic-mean equation, equation (5) for water is,

72.8 = 22.1 + 50.7 (in milli J/m²).

The harmonic-mean equation for solid and liquid is,

g_SL = g_S + g_L- 4 g_SD g_LD / (g_SD + g_LD )

- 4 g_SH g_LH / (g_SH + g_LH ). (6)

If g_SD > g_SH then g_S can calculate without knowing the values of g_SD and g_SH, i.e.,

g_{S =} g_{SD =} g_Lg_LD(1+cosq )/{4g_LD-g_L(1+cosq )} (7)

Using equations (2) to (6) and the measured value of contact angle q =72, we can get the values of W_SL, g_SL, g_SD and g_SH.

W_SL= 95.3 x 10^-3 J/m².

g_SL = 23.5 x 10^-3 J/m².

g_S = g_SD + g_SH.

46.0 = 31.1 + 14.9 (in milli J/m²).