The Helmholtz free energy, or simply free energy, is another thermodynamic potential. The specific free energy f (in the wider literature the letter a is also used) is defined as

Specific free energy is an intensive variable; the free energy �=�−�� is an extensive variable. Its differential form again follows from the first law in differential form, analogous to the derivation of the differential of the enthalpy. We find

The free energy has as natural variables the temperature and the volume. Free energy plays a central role in statistical mechanics where it is natural to consider systems with a given temperature and volume, so that their free energy is fixed. A typical technique of statistical mechanics is to maximize the entropy S of a system (a measure of how probable a particular state is according to the Boltzmann definition of S) for a fixed total energy U. We then introduce a Lagrange multiplier β so that we maximize �−��. This can be interpreted as the (negative) free energy if β is interpreted as the inverse temperature. In this way, the microscopic world of statistical mechanics is linked to the macroscopic world of thermodynamics; see also Section 4.7. In atmospheric science free energy is perhaps used less often.

For the first derivatives of the free energy we find

The corresponding Maxwell relation is

Free energy or Gibbs free energy G, is the energy available in a system to do useful work and is different from the total energy change of a chemical reaction. Thus,

where:

G is Gibbs free energy (kJ mol⁻¹⁾

H is the heat of combustion, enthalpy (kJ mol⁻¹)

T is temperature (°Kelvin), and

S is entropy (J°K⁻¹)

To illustrate the concept of free energy, let us return to the familiar example of glucose oxidation. When 1 mol of glucose combines with 6 mol of carbon dioxide and 6 mol of water, the heat of combustion, ΔH, amounts to 673,000 calories:

Since there is no volume change, the total energy change, ΔE, is equal to the change in enthalpy, ΔH. The heat of formation, H, of glucose from the basic elements of CO₂ and H₂O can be calculated by subtracting the heat of combustion of glucose from the heats of formation of 6 mol of CO₂ and 6 mol of H₂O. The oxidation of 1 g-atomic-weight of solid carbon (graphite) to 1 mol of CO₂ gas yields a heat of combustion, ΔH, of −94,240 cal. The heat of formation of 1 mol of water by the combustion of 1 mol of hydrogen gas with ½ mole of oxygen gas amounts to −68,310 cal. The heat of formation, H, of glucose is then calculated as:

6 mol CO₂ (g) = 6 × −94,240 cal = −565,440 cal

6 mol H₂O (L) = 6 × −68,310 cal = −409,860 cal

Heats of formation of products of glucose combustion: 975,300 cal

Heat combustion of 1 mol glucose (g) = −673,000 cal

Heat of formation of glucose from CO₂ and H₂O = −302,300 cal

The free energy change, ΔG, in the formation of glucose can be calculated from the previous equation:

Entropy values for the various atoms can be found in the Handbook of Physics and Chemistry. Essentially, the values given are relative heat capacities at very low temperature equivalent to absolute zero (−273°C):

One gram of glucose represents −50.7 entropy units.

Therefore:

This is the free energy change in glucose formation. The free energy involved in the oxidation of glucose is the difference between the free energy of formation of the combustion products and that of glucose, because by definition the free energy of O₂ is zero.

G in 6 mols CO₂ = 6 × (−94,100) =  = −564,600 cal

G in 6 mols H₂O = 6 × (−56.560) =  = −339,360 cal

∑G in combustion products = −903,960 cal

G of glucose = −215,800 cal

ΔG in combustion of glucose = −688,166 cal

The precipitation of a solid phase comprises two distinct stages: nucleation, first, and crystal growth, afterwards. Nucleation is called homogeneous if the nuclei form in the bulk solution, whereas it is termed heterogeneous if they form on a solid surface. To describe the process, it is instructive to consider the variation of the free energy of precipitation, ΔG_n, as a function of the radius of a hypothetical single crystal (Fig. 6.3).

Sign in to download full-size image

Figure 6.3. Schematic diagram of the free energy of precipitation, ΔG_n, vs. the radius, r, of a hypothetical single crystal. Note that ΔG_n increases up to a maximum during nucleation and decreases afterwards during crystal growth. The effect of the degree of oversaturation, Ω, is also shown.

The free energy of precipitation depends on the balance of two terms:

(i)

The interfacial free energy, which represents the energy associated with the formation of the interface between the growing crystal and the aqueous solution or, in other terms, the work required to generate this surface; this term is positive and depends on the square of the radius for a spherical crystal.

(ii)

The bulk free energy, which is negative and depends on the cube of the radius for a spherical crystal; besides, it is a function of the degree of oversaturation Ω (see Section 6.4 for the definition of Ω).

In general, crystal growth comprises several mechanisms and its rate is limited by the slowest one. In particular, the rate of crystal growth may be governed by either

(i)

the transport of solute particles from the bulk aqueous solution to the crystal surface and, in this case, it is termed transport-controlled, or

(ii)

one of the numerous reactions taking place at the surface of the growing crystal and, if so, it is called surface-reaction controlled or

(iii)

by a combination of these two different mechanisms.

The change in the concentration of the relevant solute(s) moving away from the crystal surface towards the bulk aqueous solution is different, depending on the mechanism controlling crystal growth, as shown schematically in Fig. 6.4.

Sign in to download full-size image

Figure 6.4. Plots (a) and (b) show schematically the change in the concentration of a relevant solute as a function of the distance from the crystal surface for (a) transport-controlled dissolution and precipitation and (b) surface-reaction-controlled processes. Plots (c) and (d) represent the change in concentration as a function of time in a generic batch experiment for (a) transport-controlled and (b) surface-reaction-controlled dissolution.

Following Nielsen (1964), the diffusion-controlled rate is computed by means of the following equation:

where r_c is the mean radius of the crystals; ν the molar volume of the precipitating substance; D_S the diffusion coefficient of solute particles in the aqueous solution; C_B and C_S are the concentrations of the solute in the bulk aqueous solution and close to the crystal surface, respectively; and t time. Assuming C_B to be constant, integration of equation (6-51) gives

where r_{c, t=0} is the average radius of the crystals at time “zero, ” i.e. at the beginning of crystal growth, i.e. at the end of the nucleation step. equations (6-51) and (6-52) apply to equidimensional crystals of regular shape (e.g. spheres, cubes, etc.) separated by at least five diameters.

What if one selects the same value for voids containing n vacancies? The relative relaxation volume, that is, the ratio ��VR/(�Ω)=3ɛ(�(�)), can now be computed with eqn [123] and the results are shown in Figure 23 by the solid curve. As it must, for n = 1 it reproduces the vacancy relaxation volume of −0.25Ω. In addition, it also agrees with the overall trend of the atomistic results of Shimomura.⁴⁸ Of course, the atomistic results for small vacancy cluster vary in a discontinuous manner with the cluster size. The surface stress model gives not only a reasonable approximation to these atomistic results, but also a valid extrapolation to relaxation volumes of large voids.

The chemical potential of vacancies for voids can now be computed with eqn [127] as F_C(R(n + 1)) − F_C(R(n)). Figure 24 shows the results for Ni as the solid curve. The vacancy chemical potentials for voids are significantly lower than the capillary approximation predicts with a fixed surface energy (dashed curve). The chemical potentials from atomistic simulations of voids in Ni have been obtained by Adams and Wolfer⁴⁹ using the Ni-EAM potential of Foiles et al.⁵⁰ These results converge to those predicted with the surface stress mode

One may of course ask if the values of the adhesion force and the detachment force are significantly different, and if a distinction between these two quantifies is meaningful. The answer is yes. For instance, in the case of two muscovite mica surfaces the adhesion force in dry air is about 1100 mN/m [8], and the adhesion force in water is about 30–50 mN/m [9]. In contrast, the detachment force in water is about 300 mN/m, i.e. more than a factor of three lower than the adhesion force in air, and about a factor of 10 larger than the adhesion force in water! The detachment force is lower than the adhesion force in air due to water adsorption that lowers the interfacial energy. The reason that it is larger than the adhesion force in water is that when the surfaces are brought into contact in water some water will remain between the surfaces also when they are “in contact”, i.e. the contact situation is different in the two cases [10]. This is illustrated by the schematic force curve in Figure 8.2. The detachment force is the magnitude of the force at point D, whereas during normal force measuring procedures the force is measured from large distances,over the force barrier (A), into an adhesive minimum at B (the magnitude of which is the adhesion force), and further in another very large force barrier is encountered (C). In most experiments, this force barrier is not overcome and the “contact” achieved is not absolutely dry (except when hydrophobic surfaces are used).

Some selected data where the detachment force is expressed in fractions of the adhesion force in air are shown in Figure 8.3. We note that the anionic Surfactants linear alkylbenzene sulfonate (LAS) and sodium dodecyl sulfate (SDS) show very similar effect on the detachment force. The anionic Surfactants alone are slightly more efficient than the non-ionic Surfactants alone in their lowering of the detachment force with one exception, the branched nonionic glucoside C₂C₆Glu that gives rise to a lower detachment force. The data in Figure 8.3 also show how addition of polymers can be beneficiai in lowering the detachment force. One polymer studied, called M4, is weakly cationic and carries graftedpoly(oxyethylene) chains. Alone, it is not able to reduce the detachment force significantly. However, when combined with an anionic Surfactant the mixture that contains only 20 ppm of polymer, reduces the detachment force significantly more than the polymer alone and the Surfactant alone.

Sign in to download full-size image

Figure 8.3. Detachment force divided by the adhesion force in air. The detachment force was measured at a surfactant concentration above the cmc using one hydrophilic mica surface and one hydrophobized mica surface. The surfactants were: C₁₂E₅, penta(oxyethylene) dodecyl ether; C_l0Glu, n-decyl β–d glucopyranoside; C₂C₆Glu, 2–ethylhexyl α–glucoside; Sterol ethoxylate, a sterol backbone connected with a 25 unit long poly(oxyethylene) chain; LAS, linear alkyl sulphate; SDS, sodium dodecyl sulphate, M4 a weakly cationic polymer with grafted EO–chains. The concentration of the polymer was 20 ppm

One of the systems studied¹⁵³ was a polystyrene-block-poly(ethylene/propylene) (37 300:59 700 M_n) copolymer in decane. Electron microscopy studies showed that the micelles formed by the block copolymer were spherical in shape and had a narrow size distribution. Since decane is a selectively bad solvent for polystyrene, the latter component formed the cores of the micelles. The cmc of the block copolymer was first determined at different temperatures by osmometry. Figure 13 shows a plot of π/cRT against c (where c is the concentration of the solution) for T = 97.1 °C. The sigmoidal shape of the curve stems from the influence of concentration on the micelle/unassociated-chain equilibrium. When the concentration of the solution is very low most of the chains are unassociated; extrapolation of the curve to infinite dilution gives �n−1 of the unassociated chains. On increasing the concentration of the solution the osmotic pressure decreases rapidly over a narrow concentration range as expected for closed association. The arrow indicates the cmc. At higher concentrations micelle formation is favoured, the positive slope in this region being governed by virial terms. Similar shaped curves were obtained for other temperatures.

The slope of the linear plot of ln(cmc) against T⁻¹ (see Figure 14) gave a value for ΔH of −130 kJ mol⁻¹. The values of ΔG and TΔS for T = 86 °C were −30 and 100 kJ mol⁻¹ respectively; the standard states are ideally dilute with c = 1 mol dm⁻³.

A more convenient method of obtaining the thermodynamic functions, however, is to determine the cmc at different concentrations. A plot of light-scattering intensity against concentration is shown in Figure 15 for a solution of concentration c = 3.8 × 10⁻⁵ g cm⁻³ and a scattering angle of 60°. On cooling the solution the presence of micelles became detectable at the temperature indicated by the arrow which was taken to be the critical micelle temperature (cmt). On further cooling the weight fraction of micelles increases rapidly leading to a rapid increase in scattering intensity at lower temperatures till the micellar state predominates. The slope of the linear plot of ln c against (cmt)⁻¹ shown in Figure 16, which is equivalent to the more traditional plot of ln(cmc) against T⁻¹, gave a value of ΔH = −141 kJ mol⁻¹ which is in fair agreement with the result obtained by osmometry considering the difficulties in locating the cmc by the osmometric method. Direct calorimetric measurements gave a value of 138 kJ mol⁻¹ for ΔH.

Figure 15. A plot of light-scattering intensity against temperature for a polystyrene-block-poly(ethylene/propylene) (37 000:59 700 M_n) copolymer in decane. The concentration of the solution was 3.85 × 10⁻⁵ gcm⁻³ and the angle of scatter 60°. The arrow indicates the cmt (reproduced from ref. 153)

Results obtained for a range of polymers are given in Table 3.^{154, 155, 159} The first two sets of results were obtained using light-scattering to determine the cmt. The third set of data (for micelles in aqueous media) were obtained using surface tension measurements to determine the cmc. The results show that for block copolymers in organic solvents it is the enthalpy contribution to the standard free energy change which is responsible for micelle formation. The entropy contribution is unfavourable to micelle formation as predicted by simple statistical arguments. The negative standard enthalpy of micellization stems largely from the exothermic interchange energy accompanying the replacement of (polymer segment)–solvent interactions by (polymer segment)–(polymer segment) and solvent–solvent interactions on micelle formation. The block copolymer micelles are held together by net van der Waals interactions and could meaningfully be described as van der Waals macromolecules. The combined effect per copolymer chain is an attractive interaction similar in magnitude to that posed by a covalent chemical bond.