Surface energies

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"The comprehension of the laws which govern any material system is greatly facilitated by considering the energy and entropy of the system in the various states of which it is capable. As the difference of the values of the energy for any two states represents the combined amount of work and heat received or yielded by the system when it is brought from one state to another, and the difference of entropy is the limit of all the possible values of the integral <math>\int{\frac{dQ}{t}}</math>, (dQ denoting the element of the heat received from external sources, and t the temperature of the part of the system receiving it,) the vary values of the energy and entropy characterize in all that is essential the effects producible by the system in passing from one state to another."

J. Willard Gibbs. "On the equilibrium of heterogeneous substances." Trans. Conn. Acad., III, pp. 108 - 248, Oct 1875 - May 1876, and pp. 343 - 524, May 1877 - July, 1878.

The Gibbsian layer: " the transition zone between two phases there exits a field of electric and molecular forces that decreases with distance into each of the contiguous phases. We shall refer to these forces as surface forces (not to be confused with the term 'surface forces' used in mechanics to denote forces applied to a surface)." B.V. Derjaguin and N.V. Churaev, Surface forces, 1987, pp. 4-5.
The non-Gibbsian layer: "In contrast to Gibbs' interpretation, the excess of entropy and mass of the components related to the interlayer depend on h (with small values of h), and here it is impossible to distinguish the excesses on each interface." B.V. Derjaguin, Theory of stabiity of colloids and thin films, 1989, p.23-24.

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Surface energy and measurements

Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. For a liquid, the surface tension (force per unit length) and the surface energy density are identical. Water, a special case, has a surface energy density of 0.08 J/m2 and a surface tension of 0.08 N/m.

As first described by Thomas Young in 1805 in the Philosophical Transactions of the Royal Society of London, it is the interaction between the forces of cohesion and the forces of adhesion which determines whether or not wetting, the spreading of a liquid over a surface, occurs. If complete wetting does not occur, then a bead of liquid will form, with a contact angle which is a function of the surface energies of the system. Surface energy is most commonly quantified using a contact angle goniometer and a number of different methods.

Thomas Young described surface energy as the interaction between the forces of cohesion and the forces of adhesion which, in turn, dictate if wetting occurs. If wetting occurs, the drop will spread out flat. In most cases, however, the drop will bead to some extent and by measuring the contact angle formed where the drop makes contact with the solid the surface energies of the system can be measured.

Young established the well-regarded Young's Equation which defines the balances of forces caused by a wet drop on a dry surface. If the surface is hydrophobic then the contact angle of a drop of water will be larger. Hydrophilicity is indicated by smaller contact angles and higher surface energy. (Water has rather high surface energy by nature; it is polar and forms hydrogen bonds). The Young equation gives the following relation,




\theta_\mathrm{c}=\arccos\left(\frac{r_\mathrm{A}\cos{\theta_\mathrm{A}}+r_\mathrm{R}\cos{\theta_\mathrm{R}}}{r_\mathrm{A}+r_\mathrm{R}}\right) ~;~~ r_\mathrm{A}=\left(\frac{\sin^3{\theta_\mathrm{A}}}{2-3\cos{\theta_\mathrm{A}}+\cos^3{\theta_\mathrm{A}}}\right)^{1/3} ~;~~ r_\mathrm{R}=\left(\frac{\sin^3{\theta_\mathrm{R}}}{2-3\cos{\theta_\mathrm{R}}+\cos^3{\theta_\mathrm{R}}}\right)^{1/3} </math> where <math>\gamma_{\mathrm{SL}}</math>, <math>\gamma_{\mathrm{LV}}</math>, and <math>\gamma_{\mathrm{SV}}</math> are the interfacial tensions between the solid and the liquid, the liquid and the vapor, and the solid and the vapor, respectively. The equilibrium contact angle that the drop makes with the surface is denoted by <math>\theta_\mathrm{c}</math>. Even in a perfectly smooth surface a drop will assume a wide spectrum of contact angles ranging from the so called advancing contact angle, <math>\theta_\mathrm{A}</math>, to the so called receding contact angle, <math>\theta_\mathrm{R}</math>. The equilibrium contact angle (<math>\theta_\mathrm{c}</math>) can be calculated from <math>\theta_\mathrm{A}</math> and <math>\theta_\mathrm{R}</math>.

Electrostatic and induction energies

Charge-charge Angle and temperature independent <math>\frac{q_{a}q_{b}}{4\pi \varepsilon _{0}r}\,\!</math>
Charge-dipole Averaged over all orientations <math>-\frac{kT}{3}\frac{q_{a}^{2}\mu _{b}^{2}}{(4\pi \varepsilon _{0})^{2}r^{4}}\,\!</math>
At the maximum <math>-\frac{q_{a}\mu _{b}}{4\pi \varepsilon _{0}r^{2}}\,\!</math>
Charge-quadrupole Averaged over all orientations <math>-\frac{kT}{20}\frac{q_{a}^{2}Q_{b}^{2}}{(4\pi \varepsilon _{0})^{2}r^{6}}\,\!</math>
Dipole-dipole Averaged over all orientations <math>-\frac{2kT}{3}\frac{\mu _{a}^{2}\mu _{b}^{2}}{(4\pi \varepsilon _{0})^{2}r^{6}}\,\!</math>
At the maximum <math>-\frac{2\mu _{a}\mu _{b}}{4\pi \varepsilon _{0}r^{3}}\,\!</math>
Charge-induced dipole Angle and temperature independent <math>-\frac{q_{a}^{2}\alpha _{b}}{8\pi \varepsilon _{0}r^{4}}\,\!</math>
Dipole-induced dipole Averaged over all orientations <math>-\frac{\mu _{a}^{2}\alpha _{b}}{4\pi \varepsilon _{0}r^{6}}\,\!</math>
where charge on particle a (C) = <math>q_{a}\,\!</math>
interparticle distance (m) <math>r\,\!</math>
dipole moment of particle a (C-m) = <math>\mu _{a}\,\!</math>
polarizability of particle a (m3) = <math>\alpha _{a}\,\!</math>
quadrupole moment of particle a (C-m3) = <math>Q_{a}^{{}}\,\!</math>
permittivity of free space = <math>\varepsilon _{0}\,\!</math>
Molecular polarizabilities are often estimated from the sum of bond polarizabilities.

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Representative molecular properties

Lanndolt-Bornstein, Vol. 1, Part 3, Springer, 1951

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Short and long range forces

Dispersion attraction:Long range; primarily dependent on particle properties. Dispersion attractions arise when charge fluctuations in a molecule give rise to instantaneous dipole moments even when the time averaged dipole moment is zero. Such induced dipole-induced dipole interactions are known as London dispersion forces. Such interactions are even present between hydrocarbons which may not interact via the other interactions described in this list. It is interesting to note that these are the only interactions that occur between noble gas atoms. Without dispersive attraction between noble gas atoms, noble gases would never be found in liquid form. Dispersive attractions become larger as the atom or molecule gets larger in size. Larger molecules tend to be easier to polarize.
Electrostatic repulsion: Intermediate range; heavily dependent on solution properties. Electrostatic repulsion arises when particles with like charges repel each other.
Steric (entropic) repulsion: Short range, primarily dependent on solution properties. This repulsion arises from the volume that a particle occupies. Debye shielding is an example of such a force. This interaction is particularly important is pharacology and biology where molecules may be very large. Such shielding may influence what reactions may occur or limit the circumstances under which the reaction may take place.
Two particles; two kinds of forces; over two distance scales. In many systems, there are multiple interactions that have differing effects depending on the length scale at which particles are observed. While long range forces may be observed at short scales, in this region short range forces tend to be larger and dominate the interactions between the particles.

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Derjaguin approximation for spheres

RIsraelachvili, Fig. 10.3
Israelachvili's text did a fantastic job illustrating this, the interaction forces versus energies, starting at 10.5 in "Intermolecular and Surface Forces". The final product of the Derjaguin approximation is to show the "force between two spheres in terms of the energy per unit area of two flat surfaces at the same separation D" (p.163).

First, Israelachvili relates force F(D) between two curved surfaces with the interaction free energy W(D) of two flat plates. Using a previously derived equation for the free energy of a sphere-surface interaction, he shows that the force value for a sphere near a flat surface is:

<math>F(D)= -\frac{\partial W(D)}{\partial D} = -\frac{4\pi^2 C\rho^2 R}{(n-2)(n-3)(n-4)D^{(n-4)}}</math>

Which is simplified to just the interaction free energy of two planar surfaces (flat disks!) per unit area:

<math>F(D)_{sphere}=2\pi RW(D)_{planes}</math>

He goes on to the derive the Derjaguin approximation to show that this relationship is valid for any type of force law (incredible) if the separation distance is D is much less than the radii of the spheres.

Calculate the force between two spheres, by integrating over force between thin disks: <math>F(D)=\int\limits_{Z=D}^{\infty }{2\pi xdxf\left( Z \right)}</math>

Now, looking at the diagram and remembering the Chord Theorem,

<math>x^2 \approx 2R_1z_1 = 2R_2z_2</math>

So, from page 162:

<math>Z = D + z_1 + z_2 = D + \frac{x^2}{2} (\frac{1}{R_1}+\frac{1}{R_2})</math>

<math>dZ= (\frac{1}{R_1}+\frac{1}{R_2})xdx</math>

Plugging back in to the integral over force and ignoring edge effects, Derjaguin found:

<math>F\left( D \right)\approx 2\pi \left( \frac{R_{1}R_{2}}{R_{1}+R_{2}} \right)W\left( D \right)</math>

Where W(D) is the energy of interaction of two flat plates.

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Destabilization of colloidal mixtures

Colloidal mixtures can become unstable, resulting in particle movement towards one another and aggregating in flocs. Flocculation is a process in which a solute comes out of the solution in the form of flakes or floc. This process can result in creation of photonic glasses using one of the following methods:

- Removing the electrostatic force that keeps the forces separate. This can be accomplished by changing the pH of the solution to effectively neutralize the charge in the solution. In this manner, Van Der Waals attractive force will dominate the electrostatic repulsion and the particles will flocculate.

- Physically deforming particles in the mixture. Doing so may allow Van Der Waals attractive force to "win" over other repulsive forces that we mentioned earlier (electrostatic, steric ...)

- Adding polymer of opposite charge compared to the colloids in the mixture. Added polymers can "bridge" two colloids, attracting two or more towards each other and thus aggregating.

- Adding nonadsorbed polymers (depletants) to the mixture can cause aggregation due to steric effects.

Below is an SEM image of a sample of "photonic glass" on the left and regular disordered media on right.

Photonic glasses.jpeg

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Cheese Making

Cheese.jpg Cow.jpg

Colloidal chemistry is connected with the cheese making process. Colloidal suspensions of casein micelles aggregate during a process called acidifcation where calcium ion concentrations increase thereby decreasing the net negative electric charge on the casein micelles causing them to collapse. In their new form they aggregate instead of being dispersed and form a gel related to the curd part of cheese!!!! So the calcium ions cause changes in surface charge and surface forces along with decreased repulsion between micelles allowing us to enjoy the wonders of dairy in the soft matter form. Tons of information including plots sort of like phase diagrams defining cheese making processes are available online in the two links below.

[1] [2]

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