Thermodynamics of Surfaces

Thermodynamics of Surfaces

• Surface atoms differ from atoms in the bulk due to fewer neighbors, leading to a different chemical environment.
• The thermodynamics of the surface can be distinct from the thermodynamic properties of the bulk.
• Multicomponent systems see selective accumulation of atoms on the surface, affecting surface properties.
• The curvature of small drops and capillary action are due to the balance of energies at the surface.

Surface Thermodynamic Functions

• For a crystalline solid with N atoms and surface planes, thermodynamic properties are found using a term for excesses from surface properties.
• Let E° and S° represent the solid's energy and entropy per atom, respectively
• Es is specific surface energy in energy per unit area.
• E = NE° + aEs where:
  • N = number of atoms in the solid
  • E = total energy
  • a = surface area
• The total entropy is S = NS° + aSs.
• Total free energy: G = NG° + aGs
• Surface work content: As = ES - TSS
• Surface free energy: Gs = Hs – TSs

Work Needed to Create a Surface of One Component: Surface Tension

• Creating a surface involves moving atoms from the bulk and displacing surface atoms.
• At constant temperature (T) and pressure (P), the reversible surface work (δW) needed to increase the surface area (a) by an amount (da) for a one-component system is δW = γda.
• γ is surface tension, equivalent to a 2D pressure; in other words, like pressure but with the same units as force/area.

Units

• Pressure (P) = force per unit area = N/m².
• Surface tension (y) = force per unit length = N/m = J/m².
• A rough estimate of surface tension can be obtained, by assuming the work is of the same magnitude as heat of sublimation.
• The AHsub is about ~105 cal/mol (4.18x105 J/mole) for many metals.
• Using unit conversion via Avogadro constant value of 1 mole (6x1023 atoms) and substituting gives [4.18/6]x105/1023=0.694x10-18~7x10-19 J/atoms,
• This is ~7x10-19 J/atom
• Typical concentration of atoms on a surface: 1015 atoms/cm² = 1019 atoms/m²
• Estimated surface tension: Yest ≈ 7x10-19 J/atom x1019 atoms/m² ≈ 7 J/m²

Relationships

• Correlations exist between heat of sublimation and surface tension for metals.
• Experimentally the equation is best described as Yest ≈ 0.16·ΔHsub
• Relaxation allows surface atoms to move, maximizing bonding with neighbors.
• Surface atoms in a less symmetric state change their equilibrium positions.
• Surface atoms tend to contract toward the bulk, decreasing surface tension.
• To estimate surface pressure or tension (2D Pressure P), consider surface tension distributed over a 1 m² surface a few atomic layers thick, where P = y/d.
• Assuming y = 1 N/m, and an anisotropic surface environment influences bonding in the top atomic layers with thickness d=1 nm (10 A°).
• P= γ /d=[1J/m]/[1 nm]=
• P= 10⁹ (J/m²) or 104 atmospheric.
• This explains the atoms in a metal surface subjected to large compressive forces.
• Lower surface tension requires less energy to produce a new unit area and also produces more compressible.

Surface Tension and Work

• The table provides experimental values of surface tensions for different metals and substances:

  • W (solid): 2.900 J/m² at 1727 °C
  • Au (solid): 1.410 J/m² at 1027 °C
  • NaCl: 0.227 J/m² at 25 °C
  • Water: 0.07275 J/m² at 20 °C
  • Ethanol (liquid): 0.02275 J/m² at 20 °C

• Estimating 3D pressure (P) for a solid surface dividing by the thickness of interaction assuming few atomic layers (d ≈ 1 nm)

  • For W: γ = 2.900 J/m²
  • W = 2.9x104 atm
  • Conclusion metal surface experiences very large compressive forces.

Surface Free Energy

• The total free energy change dG of a one-component system includes surface work (ydα): dG = -SdT + VdP + γda
• At constant T and P: dGT,P = γda
• Ways to increase surface area:
  • Adding new atoms from the bulk
  • Stretching the existing surface changes strain (changes γ), akin to a rubber band
• The change free energy becomes: dGT,P = d(aGs) or Surface work content in other words
• When changing the surface area by addition of atoms to the surface from the bulk, the specific surface free energy, Gs is independent of the surface area:
• dGr.p = Gºda
• The total free energy change of a one-component system is equal to surface tension, only when the surface area changes by movement of atoms to the surface from the bulk.
• Since surface tension is always positive, the surface free energy is always positive.

Creation of Surfaces

• Gs Positive free energy formation resulted.
• A reluctance defines many interfacial properties of condensed phases.
• Minimizing surface free energy causes solids form the lowest specific surface free energy/tension (y,) forming crystal faces with higher/closed packing of atoms.
• Surfaces with a high value of y tend to be covered to lower surface tensions.
• Metals are covered by oxide layers if the metal-gas interfacial energy Ym-g is more than the oxide-gas (Yox-g) and oxide-metal (Yox-m) interfacial energies.
• Water adsorbs and covers the oxide if Yox-g > YH2O-g + YH2O-ox
• Adsorbed water can be displaced by organic molecules with lower surface tension.
• Liquids minimize surface areas by assuming a spherical shape.
• Water adsorption occurs when Yox-g> УH20-д + H2O-ox.
• Surface coating happens when Ynew surface + Vinterface < Yold surface

Surface Tension

• Surface tension in a liquid arises from the forces of attraction between molecules.
• This mutual attraction causes liquids to coalesce into spherical droplets without other forces.
• Surface tension and interfacial tension are measured in newtons per meter (mN/m), often represented by σ or y , where 1 mN/m = 1 dyne/cm = 0.001N/m.
• Surface tension comes from an uneven distribution of molecular forces in a liquid.
• Liquid molecules at the surface attract each other, causing a net force that pulls them together.
• High surface tension means molecules interact strongly.
• Lower values mean the molecules do not interact as strongly
• Think of surface tension as the force holding a liquid together.
• Each molecule surrounded on all sides in equilibrium so the forces between balance out.
• Forces on the surface become imbalanced resulting in liquid behaving as if held together due to a stretching force

Liquids (Polar/Non-Polar)

• The more polar groups (e.g., O-H groups), the stronger the attractive forces.
• Stronger forces lead to high surface tension, forming discrete droplets on surfaces.
• Water's high surface tension is due to O-H groups.
• Alcohols have lower surface tensions due to alcohols smaller proportion of O-H groups.
• Non-polar liquid, like olive oil, have very low value.
• Lower surface tension creates an easier ability to create a satisfactory film.

Table of liquids

• Experimental Surface tension @ 20 °C in mN/m
• Mercury, 425.41
• n-Decane, 23.83
• n-Hexane, 18.43
• n-Hexane, 18.43
• Methanol, 22.7
• Perfluoroheptane, 12.85
• Perfluorohexane, 11.91
• Perfluorooctane, 14
• Water, 72.8

Surface Tension of Mercury

• Mercury's high surface tension is due to the metallic bonds.
• Intermolecular attraction between like molecules is cohesion, unlike molecules is adhesion.

Capillary Action

• Narrow tubes with strong liquid adhesion to walls causes surface tension to draw liquid up via capillary action.
• The height the column lifted by is:
  • h = 2Yla cos 0 / pgr units
    • h is the height the liquid is lifted,
    • Y la is the liquid-air surface tension,
    • p is the density of the liquid,
    • r is the radius of the capillary,
    • g is the acceleration due to gravity,
    • 0 is the angle of contact
• If 0 is over 90° (less wettability), the liquid will be depressed rather than lifted, as with mercury in glass.

Key Notes on Capillary Rise in Surfaces

• A liquid in a small-radius tube develops a significant pressure difference between internal and external pressures ((Pint - Pext) = 2y/r)
• The liquid level rises until the hydrostatic pressure of the rising liquid column balances the internal pressure (ΔP = Δpgh =2y/r).
• Where:
  • -Δρ is is the density difference between liquid and gas phases,
  • -g is the gravitational constant
  • -h is equal to the height h of the capillary column

Vapor Pressure of Curved Surfaces

• The vapor pressure of a droplet depends on the radius of curvature (r.)
• At equilibrium, the the pressure difference across between the droplet is is expressed as (Pint - Pext) = 2y/r
• Transferring the molecules to one phase is exactly the same by also transferring molecules from the other phase to the liquid phase

Energy and Surface Tension

• Surface energy y is proportional to Gibbs Free Energy.

Surface Characteristics

• Small droplets form larger droplets to minimize the combination's surface-to-volume ratios
• Small metal particles that have been sintered tend to become solid mass, the driving force this is reduction of free energy is the driving force
• A loop of film on frame can be pulled to the right by a force (F) that creates a new surface of two surfaces, in the top and bottom and bottom of the film
• Balance of forces expressed as: 21 y dx = F dX
• y = F/2 1 (Where I is length)
• The units are newton per meter.
• y can be thought of as surface tension or surface energy, in liquids, the two is same, for solids are not.

Surface Tension and Energy: Solid Vs Liquid

• Liquid molecules are isotropic (evenly disturbed within)
• Solid molecules are a function of crystallographic plane that is exposed
• Unit surface energy does not change as as liquid is stretched as the molecules can either come to the surface or go into it
• For solids no such balance can take place

Calculation of Surface Energy

• Energy of one bond (ε) is expressed as: ε = ΔHs / 0.5ZN A
• Assumption: Bonding energy of an atom to a solid is from closest neighbours
  • ΔHs is molar enthalpy of sublimation
  • Z coordination
  • NA is Avogadro number
• There are 0.5 Z NA number of bonds per mole.
• Cleaving a FCC crystal along a (111) plane breaks 3 bonds per atom, creating 2 surfaces.
• This leads to work equals (3/2) ε per surface atom.
• The given formula for the work per surface atom becomes: if Z=12 then W = 3H / 4NA
• The surface energy y is then : y = ΔH/4NA * N/A Where NIA is no of atom per unit area
• For FCC structure, N/A : 4/(√3a^2)
• a, is lattice parameter.
• Enthalpy of sublimation of approximately 170,000 (J/ (g.mole) and lattice parameter a. is 3.615 A.
• Calculated work for free surface of Cu is typically 1400 ergs/cm2, while value is 1600 ergs/cm2
• An erg is the unit of measurement, calculated in the second tier of measurement, second

Wulff Thermo

• Problems are simplified if the surface energy is isotropic
• In crystalline solids the surface energy creates the anisotropic energy which is then used to find the limiting planes of the lowest possible surface energy

1901 and 1953

• The 1901 proposed theorem that equilibrium crystal exists such that its proportional to surface energy y(i)
• The 1953 Conyers Herring then created two exercises to prove
  • Start by plotting as function of is made
  • Known as gamma plot
• It then can be graphed to create lines from the origin of the gamma plat
• Planes parallel to the point intersects with the gamma plot
• Inner envelopes produces to crystals equilibrium point

Contact Angle

• The measurements calculate both the shape of water combined by G/ gravity
• The equation is then expressed on the screen, if water the solution
• Then the surface tension extends against surface to intersections of surface of liquid (S) and solid

Structural Information and Energy

• The energy and tension depends on crystal structures
• Bulk surface is far less than step and adatom

Equilibrium Crystal Shape

• Crystal shapes created by min surface energy
• For sphere liquids the faceted shapes are defined by the equilibrium shape in relation

Wulff theorem surface energy

• In thermodynamics bulk, Boltzman energy is f(NI,N2),at Temperature volume
• In this equation, Entropy is volume with p and m
• Schematic view to deviding the surface view