Surface stress was first defined by
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
(1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. A similar term called "surface free energy", which represents the excess
free energy per unit area needed to create a new surface, is easily confused with "surface stress". Although surface stress and surface free energy of liquid–gas or liquid–liquid
interface
Interface or interfacing may refer to:
Academic journals
* ''Interface'' (journal), by the Electrochemical Society
* '' Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Linguistics''
* '' Int ...
are the same, they are very different in solid–gas or solid–solid interface, which will be discussed in details later. Since both terms represent a
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
per unit
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
, they have been referred to as "
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
", which contributes further to the confusion in the literature.
Thermodynamics of surface stress
Definition of surface free energy is seemly the amount of reversible work
performed to create new area
of surface, expressed as:
:
Gibbs was the first to define another surface quantity, different from the surface tension
, that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. Surface stress can be derived from surface free energy as follows:
One can define a surface
stress tensor that relates the work associated with the variation in
, the total excess free energy of the surface, owing to the strain
:
:
Now consider the two reversible paths showed in figure 0. The first path (clockwise), the solid object is cut into two same pieces. Then both pieces are elastically strained. The work associated with the first step (unstrained) is
, where
and
are the excess free energy and area of each of new surfaces. For the second step, work (
), equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces.
In the second path (counter-clockwise), the subject is first elastically strained and then is cut in two pieces. The work for the first step here,
is equal to that needed to deform the bulk volume and the two surfaces. The difference
is equal to the excess work needed to elastically deform two surfaces of area
to area
or:
:
the work associated with the second step of the second path can be expressed as
, so that:
:
These two paths are completely reversible, or W
2 – W
1 = W
2 – W
1. It means:
:
Since d(γA) = γdA + Adγ, and dA = Aδ
ijde
ij. Then surface stress can be expressed as:
:
Where δ
ij is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
and e
ij is
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
strain tensor.
Differently from the surface free energy γ, which is a scalar, surface stress f
ij is a second rank tensor. However, for a general surface, set of principal axes that are off-diagonal components are identically zero. Surface that possesses a threefold or higher rotation axis
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
,
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
components are equal. Therefore, surface stress can be rewritten as a scalar:
:
Now it can be easily explained why f and γ are equal in liquid-gas or liquid-liquid interfaces. Due to the chemical structure of liquid surface phase, the term ∂γ/∂e always equals to zero meaning that surface free energy won’t change even if the surface is being stretched. However, ∂γ/∂e is not zero in solid surface due do the fact that surface atomic structure of solid are modified in
elastic deformation
In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
.
Physical origins of surface stress
Origin of surface stress could be understood by nature of chemical bonding of atoms at the surface. In metallic materials, atomic chemical bonding structure at the surface is very different from in the bulk. Therefore,
equilibrium interatomic distance between surface atoms is different from bulk atoms. Since surface and bulk atoms are structurally
coherent
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deri ...
, the interior of the solid can be considered as applying a stress on the surface.
For illustration, figure 1 shows a simple picture of bond charges near the surface of a 2D
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
with charge (election) density around sphere atoms. Surface atoms only have two nearest neighbors compared with bulk atoms, which have four (for this example case). The loss of neighbors which results from the creation of a metal surface reduces the local
electron density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
around the atoms near the surface. Surface atoms then sit in a lower average electron density than bulk atoms. The response of these surface atoms would be to attempt to reduce their interatomic distance in order to increase surrounding charge density. Therefore, surface atoms would create a positive surface stress (
tensile
In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as ...
). In the other words, if the surface charge density is the same as in the bulk, surface stress would be zero.
Surface stress, which created by redistribution of electron density around surface atoms, can be both positive (
tensile
In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as ...
) or negative (
compressive
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elon ...
). If the surface is not clean meaning there are atoms sitting on a flat surface (adsorbates), charge density would then be modified leading to a different surface stress state compared with a perfect clean surface.
Measurement of surface stress
Theoretical calculations
Surface stresses normally calculated by calculating the surface free energy and its derivative with respect to elastic strain. Different methods have been used such as
first principles
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
, atomistic potential calculations and
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
simulations. Most of calculations are done at
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
of 0 K. Following are tables of surface stress and surface free energy values of
metals
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
and
semiconductors. Details of these calculations could be found in the attached references.
FCC Metal surfaces (111)
More metal surfaces
Semiconductor surfaces
III-V compounds
Experimental measurements
In the early time, several experimental techniques to measure surface stress of materials had been proposed. One was determining surface stress by measuring
curvature of a thin membrane of the material as it bends by gravitation through its own weight. This method turned out to be difficult since it requires a complete
homogeneous single crystal
In materials science, a single crystal (or single-crystal solid or monocrystalline solid) is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries.RIWD. "Re ...
surface. An alternative way to measure
absolute surface stress is to measure the elastic extension of the length of the thin wire under an applied force. However, this method had many limitations and wasn’t used popularly. While determination of the absolute surface stress is still a challenge, the experimental technique to measure changes in the surface stress due to external interaction is well established using “cantilever bending method”. The principle of the measurement is shown in figure 2. In this case, stress of one surface is changed upon deposition of material which results the bending of the cantilever. The surface wants to expand creating a compressive stress. The radius of curvature R is measured as the change of the gap of a capacitor by
. Figure 2b shows the two
electrodes
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials dep ...
of the
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
formed by the sample and a capacitor electrode c. The capacitor electrode is surrounded by a guard electrode in order to minimize the effects of stray capacitances. The sample b is clamped on one end in the sample holder a. The bending can also be measured with high sensitivity by
deflection
Deflection or deflexion may refer to:
Board games
* Deflection (chess), a tactic that forces an opposing chess piece to leave a square
* Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers
Mechanics
* Deflection ...
of the beam of a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
using a position sensitive detector. To use this method, it requires that the sample is thin enough. Some experiment measurement values are listed in table 5.
Surface stress effects in materials science
Surface structural reconstruction
Structural reconstruction at the surfaces has been studied extensively by both theoretical and experimental methods. However, a question about surface stress is high enough to be a main
driving force of the reconstruction is still not very clear.
Most of metallic surface reconstruction exhibit in two genetic forms. On the original (100) surface, it would form a
hexagonal
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
overlayer which results in a considerably higher density of surface atoms by 20–25%. On the original (111) surface, since it already in closed-pack structure, the higher
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
is due to a contraction while the local coordination of the surface atoms remains a hexagonal one. Another way to explain the surface reconstruction phenomenon is called “soft phonon type of reconstruction”. The driving force for a change in the surface concentration associated with a contraction of the surface is proportional to the difference between surface stress and surface free energy. It corresponds to the amount of energy gained by structure transformation to over the surface stress. For semiconductor surface, forming dimer is the way for it to response to the tensile stress. Figure 3 shows an example of Si(100) surface reconstruction that create tensile stress.
Adsorbate-induced changes in the surface stress
As mentioned above, surface stress is caused by charge density redistribution of surface atoms due to lacking of nearest neighbor atoms. In case of introduction of adsorbates (atoms that land on surface), charge density would be then modified around these adsorbates, resulting different surface stress state. There are many types of reaction between adsorbates and the surface that cause different stress behavior. Here, two most common behaviors are shown:
Coverage dependence of the adsorbate-induced surface stress
Coverage effect of surface stress without surface reconstruction usually result a compressive stress (assuming clean surface as reference or zero stress). Induced surface stress of number of different coverages on Ni(100) and Pt(111) surface is shown in figure 4. In all cases, it shows an initially linear increase of the induced stress with coverage, followed by an increase larger than linear at higher coverages. The non-linear increase is first thought to be due to the repulsive interaction between adsorbates. The repulsive interaction should be proportional to the overlap integrals summed of non-bonding orbitals with exponential relationship:
S
ij & exp(-cr
ij)
where r
ij is distance between two adsorbate i and j
One can easily relate the mean distance between two adsorbates with square root of the coverage:
S
ij & exp(-c/√θ)
Then the stress induced by absorbates can be derived as:
∆τ=a.θ+b.exp(-c/√θ) (8)
where a, b, and c are fitting parameters. Figure 4 shows very good fits for all systems with the equation 8.
However, later research shows that direct repulsive interaction between absorbate atoms (as well as dipolar interactions) contribute very little to the induced surface stress. The stress can become large only if the distance between the adsorbed atoms becomes small so that φ
ij (repulsive pairwise interaction potential) becomes large. It rarely happens without very high gas pressure since adsorbated state become unstable with respect to desorption.
Adsorbate-induced stress and restructuring of surfaces
It shows that the tensile stress on clean surfaces can be so strong that the surface reconstructs to form an overlayer of higher charge density. In the presence of adsorbates, stress induced by could also be high enough for such reconstruction. The mechanism of the reconstruction of the two processes would be similar. The reconstruction due to adsorbates is easily recognized by deviation from stress-induced vs. coverage relationship. One example is shown in figure 5 and 6. It shows clearly the difference between stress-induced behavior of silicon compared with oxygen or carbon absorbate on Ni(100) surface. S/Ni(100) system reaches very high stress at the coverage of ~0.3. This stress then causes a reconstruction (figure 5) to increase the charge density of surface atoms in order to reduce the developed stress.
The surface stress due to the adsorption of alkanethiol/alkylthiols on thin gold films have been studied using experimental and computational approaches. Zhao et al.,
[Yue Zhao, Agnivo Gosai, Kyungho Kang, and Pranav Shrotriya. "Multiscale Modeling Reveals the Cause of Surface Stress Change on Microcantilevers Due to Alkanethiol SAM Adsorption",Journal of Chemical Information and Modeling https://pubs.acs.org/doi/abs/10.1021/acs.jcim.0c00146] showed that majority of the surface stress originated due to the reconstruction of the surface, resulting from the electron loss from the neighboring gold atoms of the surface adsorbed sulphur adatom.
See also
*
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
*
Surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
*
First principles
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
calculation
*
Molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
simulations
*
Nanocrystalline
A nanocrystalline (NC) material is a polycrystalline material with a crystallite size of only a few nanometers. These materials fill the gap between amorphous materials without any long range order and conventional coarse-grained materials. De ...
materials
*
Nanowire
A nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 metres). More generally, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less ...
materials
*
Nanoporous
Nanoporous materials consist of a regular organic or inorganic bulk phase in which a porous structure is present. Nanoporous materials exhibit pore diameters that are most appropriately quantified using units of nanometers. The diameter of pores in ...
materials
References
{{DEFAULTSORT:Surface Stress
Physical quantities
Materials science
Surface science