Surface energy

The most common way to measure surface energy is through contact angle experiments.[1] In this method, the contact angle of the surface is measured with several liquids, usually water and diiodomethane. Based on the contact angle results and knowing the surface tension of the liquids, the surface energy can be calculated. In practice, this analysis is done automatically by a contact angle meter.[2]

There are several different models for calculating the surface energy based on the contact angle readings.[3] The most commonly used method is OWRK which requires the use of two probe liquids and gives out as a result the total surface energy as well as divides it into polar and dispersive components.

Contact angle method is the standard surface energy measurement method due to its simplicity, applicability to a wide range of surfaces and quickness. The measurement can be fully automated and is standardized.[4]

In general, as surface energy increases, the contact angle decreases because more of the liquid is being "grabbed" by the surface. Conversely, as surface energy decreases, the contact angle increases, because the surface doesn't want to interact with the liquid.

The surface energy of a liquid may be measured by stretching a liquid membrane (which increases the surface area and hence the surface energy). In that case, in order to increase the surface area of a mass of liquid by an amount, δA, a quantity of work, γ δA, is needed (where γ is the surface energy density of the liquid). However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy.

The surface energy of a solid is usually measured at high temperatures. At such temperatures the solid creeps and even though the surface area changes, the volume remains approximately constant. If γ is the surface energy density of a cylindrical rod of radius r and length l at high temperature and a constant uniaxial tension P, then at equilibrium, the variation of the total Helmholtz free energy vanishes and we have

${\displaystyle \delta F=-P~\delta l+\gamma ~\delta A=0\quad \implies \quad \gamma =P{\frac {\delta l}{\delta A}}}$

where F is the Helmholtz free energy and A is the surface area of the rod:

${\displaystyle A=2\pi r^{2}+2\pi rl\quad \implies \quad \delta A=4\pi r\delta r+2\pi l\delta r+2\pi r\delta l}$

Also, since the volume (V) of the rod remains constant, the variation (δV) of the volume is zero, that is,

${\displaystyle V=\pi r^{2}l{\text{ is constant}}\quad \implies \quad \delta V=2\pi rl\delta r+\pi r^{2}\delta l=0\quad \implies \quad \delta r=-{\frac {r}{2l}}\delta l~.}$

Therefore, the surface energy density can be expressed as

${\displaystyle \gamma ={\frac {Pl}{\pi r(l-2r)}}~.}$

The surface energy density of the solid can be computed by measuring P, r, and l at equilibrium.

This method is valid only if the solid is isotropic, meaning the surface energy is the same for all crystallographic orientations. While this is only strictly true for amorphous solids (glass) and liquids, isotropy is a good approximation for many other materials. In particular, if the sample is polygranular (most metals) or made by powder sintering (most ceramics) this is a good approximation.

In the case of single-crystal materials, such as natural gemstones, anisotropy in the surface energy leads to faceting. The shape of the crystal (assuming equilibrium growth conditions) is related to the surface energy by the Wulff construction. The surface energy of the facets can thus be found to within a scaling constant by measuring the relative sizes of the facets.

In the deformation of solids, surface energy can be treated as the "energy required to create one unit of surface area", and is a function of the difference between the total energies of the system before and after the deformation:

${\displaystyle \gamma ={\frac {1}{A}}\left(E_{1}-E_{0}\right)}$.

Calculation of surface energy from first principles (for example, density functional theory) is an alternative approach to measurement. Surface energy is estimated from the following variables: width of the d-band, the number of valence d-electrons, and the coordination number of atoms at the surface and in the bulk of the solid.[5]

In density functional theory, surface energy can be calculated from the following expression:

${\displaystyle \gamma ={\frac {E_{\text{slab}}-NE_{\text{bulk}}}{2A}}}$

where

E_slab is the total energy of surface slab obtained using density functional theory.

N is the number of atoms in the surface slab.

E_bulk is the bulk energy per atom.

A is the surface area.

For a slab, we have two surfaces and they are of the same type, which is reflected by the number 2 in the denominator. To guarantee this, we need to create the slab carefully to make sure that the upper and lower surfaces are of the same type.

Strength of adhesive contacts is determined by the work of adhesion which is also called relative surface energy of two contacting bodies.[6] The relative surface energy can be determined by detaching of bodies of well defined shape made of one material from the substrate made from the second material.[7] For example, the relative surface energy of the interface "acrylic glass – gelatin" is equal to 0.03 N/m. Experimental setup for measuring relative surface energy and its function can be seen in the video.[8]

To estimate the surface energy of a pure, uniform material, an individual region of the material can be modeled as a cube. In order to move a cube from the bulk of a material to the surface, energy is required. This energy cost is incorporated into the surface energy of the material, which is quantified by:

Cube model. The cube model can be used to model pure, uniform materials or an individual molecular component to estimate their surface energy.

${\displaystyle \gamma ={\frac {\left(z_{\sigma }-z_{\beta }\right){\frac {1}{2}}W_{\text{AA}}}{a_{0}}}}$

where z_σ and z_β are coordination numbers corresponding to the surface and the bulk regions of the material, and are equal to 5 and 6, respectively; a₀ is the surface area of an individual molecule, and W_AA is the pairwise intermolecular energy.

Surface area can be determined by squaring the cube root of the volume of the molecule:

${\displaystyle a_{0}=V_{\text{molecule}}^{\frac {2}{3}}=\left({\frac {\bar {M}}{\rho N_{\text{A}}}}\right)^{\frac {2}{3}}}$

Here, M̄ corresponds to the molar mass of the molecule, ρ corresponds to the density, and N_A is the Avogadro constant.

In order to determine the pairwise intermolecular energy, all intermolecular forces in the material must be broken. This allows thorough investigation of the interactions that occur for single molecules. During sublimation of a substance, intermolecular forces between molecules are broken, resulting in a change in the material from solid to gas. For this reason, considering the enthalpy of sublimation can be useful in determining the pairwise intermolecular energy. Enthalpy of sublimation can be calculated by the following equation:

${\displaystyle \Delta _{\text{sub}}H=-{\frac {1}{2}}W_{\text{AA}}N_{\text{A}}z_{b}}$

Using empirically tabulated values for enthalpy of sublimation, it is possible to determine the pairwise intermolecular energy. Incorporating this value into the surface energy equation allows for the surface energy to be estimated.

The following equation can be used as a reasonable estimate for surface energy:

${\displaystyle \gamma \approx {\frac {-\Delta _{\text{sub}}H\left(z_{\sigma }-z_{\beta }\right)}{a_{0}N_{\text{A}}z_{\beta }}}}$

Surface energy comes into play in wetting phenomena. To examine this, consider a drop of liquid on a solid substrate. If the surface energy of the substrate changes upon the addition of the drop, the substrate is said to be wetting. The spreading parameter can be used to mathematically determine this:

${\displaystyle S=\gamma _{\text{s}}-\gamma _{\text{l}}-\gamma _{\text{s-l}}}$

where S is the spreading parameter, γ_s the surface energy of the substrate, γ_l the surface energy of the liquid, and γ_s-l the interfacial energy between the substrate and the liquid.

If S < 0, the liquid partially wets the substrate. If S > 0, the liquid completely wets the substrate.[9]

Contact Angles: non-wetting, wetting, and perfect wetting. The contact angle is the angle that connects the solid–liquid interface and the liquid-gas interface.

A way to experimentally determine wetting is to look at the contact angle (θ), which is the angle connecting the solid–liquid interface and the liquid–gas interface (as in the figure).

If θ = 0°, the liquid completely wets the substrate.

If 0° < θ < 90°, high wetting occurs.

If 90° < θ < 180°, low wetting occurs.

If θ = 180°, the liquid does not wet the substrate at all.[10]

The Young equation relates the contact angle to interfacial energy:

${\displaystyle \gamma _{\text{s-g}}=\gamma _{\text{s-l}}+\gamma _{\text{l-g}}\cos \theta }$

where γ_s-g is the interfacial energy between the solid and gas phases, γ_s-l the interfacial energy between the substrate and the liquid, γ_l-g is the interfacial energy between the liquid and gas phases, and θ is the contact angle between the solid–liquid and the liquid–gas interface.[11]

The energy of the bulk component of a solid substrate is determined by the types of interactions that hold the substrate together. High-energy substrates are held together by bonds, while low-energy substrates are held together by forces. Covalent, ionic, and metallic bonds are much stronger than forces such as van der Waals and hydrogen bonding. High-energy substrates are more easily wetted than low-energy substrates.[12] In addition, more complete wetting will occur if the substrate has a much higher surface energy than the liquid.[13]