surface energy

In surface science, surface free energy (also interfacial free energy or surface energy) quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energetically favorable than the bulk of the material (the atoms on the surface have more energy compared with the atoms in the bulk), otherwise there would be a driving force for surfaces to be created, removing the bulk of the material (see sublimation). The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding at the two surfaces.

Cutting a solid body into pieces disrupts its bonds and increases the surface area, and therefore increases surface energy. If the cutting is done reversibly, then conservation of energy means that the energy consumed by the cutting process will be equal to the energy inherent in the two new surfaces created. The unit surface energy of a material would therefore be half of its energy of cohesion, all other things being equal; in practice, this is true only for a surface freshly prepared in vacuum. Surfaces often change their form away from the simple " cleaved bond" model just implied above. They are found to be highly dynamic regions, which readily rearrange or react, so that energy is often reduced by such processes as passivation or adsorption.

Determination of surface energy

Measuring the surface energy with contact angle measurements

The most common way to measure surface energy is through contact angle experiments. 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.

There are several different models for calculating the surface energy based on the contact angle readings. 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.

Measuring the surface energy of a solid – other methods

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, , a quantity of work, , is needed (where 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 is the surface energy density of a cylindrical rod of radius and length at high temperature and a constant uniaxial tension , then at equilibrium, the variation of the total Helmholtz free energy vanishes and we have
:$\delta F = -P~\delta l + \gamma~\delta A = 0 \quad \implies \quad \gamma = P\frac$

where is the Helmholtz free energy and is the surface area of the rod:
:$A = 2\pi r^2 + 2\pi r l \quad \implies \quad \delta A = 4\pi r\delta r + 2\pi l\delta r + 2\pi r\delta l$

Also, since the volume () of the rod remains constant, the variation () of the volume is zero, that is,
:$V = \pi r^2 l \text \quad \implies \quad \delta V = 2\pi r l \delta r + \pi r^2 \delta l = 0 \quad\implies\quad \delta r = -\frac\delta l ~.$

Therefore, the surface energy density can be expressed as
:$\gamma = \frac ~.$

The surface energy density of the solid can be computed by measuring , , and 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.

Calculating the surface energy of a deformed solid

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:
:$\gamma = \frac \left(E_1 - E_0\right)$.

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.

Calculating the surface formation energy of a crystalline solid

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

:$\gamma = \frac$

where
: is the total energy of surface slab obtained using density functional theory.
: is the number of atoms in the surface slab.
: is the bulk energy per atom.
: 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. 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. 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.

Estimating surface energy from the heat of sublimation

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:

:$\gamma = \frac$

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

Surface area can be determined by squaring the cube root of the volume of the molecule:
:$a_0 = V_\text^\frac = \left(\frac\right)^\frac$

Here, corresponds to the molar mass of the molecule, corresponds to the density, and 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:

:$\Delta_\text H = -\fracW_\text N_\text z_b$

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:

:$\gamma \approx \frac$

Interfacial energy

The presence of an interface influences generally all thermodynamic parameters of a system. There are two models that are commonly used to demonstrate interfacial phenomena: the Gibbs ideal interface model and the Guggenheim model. In order to demonstrate the thermodynamics of an interfacial system using the Gibbs model, the system can be divided into three parts: two immiscible liquids with volumes and and an infinitesimally thin boundary layer known as the Gibbs dividing plane () separating these two volumes.