Diederik Korteweg, Pioneer of Criticality

Dutch mathematician Diederik Johannes Korteweg (1848–1941), pictured in figure 1, is best known for the 1895 paper he wrote with his student Gustav De Vries. That paper, on the propagation of soliton waves in channels, has, in recent decades, been widely recognized as a groundbreaking work. ¹ Korteweg also studied the stress—now called the Korteweg stress—resulting from density gradients at an interface between two fluids. ² Korteweg’s contributions to the thermodynamics of phase transitions and criticality in fluid mixtures are far less known. They include honing the tools of differential geometry to characterize phase separation and criticality of fluid mixtures, developing a method of deformation of surfaces, and performing the first detailed analysis of the phase behavior of a special case of the van der Waals equation for binary fluid mixtures. ³

Korteweg was born and raised in the southern Dutch province of Brabant. Fascinated by mathematics, he entered the Polytechnical School at Delft, but the engineering courses quickly discouraged him. He left, and took mathematics and mechanics courses that qualified him to teach. While a high-school teacher, he studied for the state exam in Latin and Greek, a requirement for access to a university education. He also published his first papers and corresponded with Johannes Diderik van der Waals (1837–1923) on the volume excluded by a collection of hard spheres.

After passing the university admission exam in 1876, Korteweg studied mathematics at the University of Utrecht for a year, and then entered the newly founded University of Amsterdam. His ascent there was meteoric. He defended his doctoral thesis on the propagation of waves in elastic tubes in 1878. His was the first doctorate granted by the young university, and physics professor van der Waals bestowed the degree. In September 1881, Korteweg assumed a professorship of mathematics with the inaugural address, “Mathematics as an Auxiliary Discipline.” He lived by his conviction that mathematics plays an important role in science, as proved by his work in thermodynamics, kinetic theory, and hydrodynamics. From 1900 to 1920, he edited volumes 11–15 of the collected papers of Christiaan Huygens. Korteweg was the thesis adviser of L. Egbertus Brouwer, who became famous for his work in topology and on the foundations of mathematics, ⁴ and to whom Korteweg voluntarily ceded his professorship in 1913.

In 1891, Korteweg published two seminal papers in the same volume of the Archives Néerlandaises des Sciences Exactes et Naturelles. The first concerned critical points, the second was about folds on surfaces. ⁵ The French version of van der Waals’s paper on the molecular theory of mixtures ⁶ appeared in the same volume. Van der Waals had worked on his theory for many years, and received Korteweg’s assistance with the mathematical aspects.

The van der Waals equation

Why was van der Waals so interested in the phase behavior of fluid mixtures? Fluid criticality had been recognized since 1822, and the critical points of many fluids had been measured by the time van der Waals formulated his equation of state in 1873. Reasonable approaches had been developed for describing the properties of gaseous mixtures and liquid solutions, but there remained a no man’s land at the intermediate densities where criticality typically occurs. (For a tutorial on criticality and its relation to phase transitions, see the box on page 49.)

Thomas Andrews published his experimental exploration of the criticality and condensation behavior of carbon dioxide in 1869. Four years later, van der Waals’s equation of state offered a satisfactory formulation of both vapor and liquid properties. ⁶ By 1880, Andrews, Louis Cailletet, and even van der Waals himself had experimented with the condensation of pressurized gas mixtures, but they were at a loss to interpret the results.

One could argue that the theoretical problem of phase behavior in fluid mixtures had already been solved. The seminal work of Josiah Willard Gibbs, published between 1873 and 1878, provided the theoretical tools. ⁷ Gibbs’s achievement was as formidable in its concepts and completeness as it was forbidding in its abstract approach. Among the few scientists grasping its significance were the omniscient James Clerk Maxwell and van der Waals. Nevertheless, even van der Waals did not know how to apply Gibbs’s theory to the mixture he had studied. Not until 1890 did van der Waals build a bridge between theory and the laboratory by generalizing his 1873 equation of state so that it could be applied to two-component systems.

Before considering those relatively complicated systems, we review some important properties of the simpler one-component systems. The van der Waals equation of state for a mole of fluid is $$\left(P+a/V^2\right)\left(V-b\right)=RT,$$ where P is the pressure, T the temperature, V the molar volume, and R the molar gas constant. The equation contains two substance-specific constants: b, which represents the volume excluded by the molecules, and a, which measures the attraction between molecules. Figure 2(a) shows P−V isotherms of the van der Waals equation for temperatures above, at, and below the critical value.

Solving for the pressure from equation (1) and integrating it with respect to volume yields, apart from an unspecified function of temperature, a free energy that van der Waals (following Gibbs) called the ψ function. Nowadays it is called the Helmholtz energy, denoted A: $$A\left(V,T\right)=-RT\ln \left(V-b\right)-a/V.$$ Van der Waals preferred this free (Helmholtz) energy because he could derive it explicitly from his equation of state.

Figure 2(b) displays the Helmholtz function along a subcritical isotherm. By splitting into two phases of variable proportions and following the tie line instead of the curve, fluid states lower their Helmholtz energy.

The T−V diagram in figure 2(c) shows the coexistence curve, or connodal, the locus of the molar volumes of coexisting vapor and liquid as a function of temperature. Those molar volumes become equal at the critical point C. Inside the connodal lies the spinodal, which bounds the region in which the system is thermodynamically unstable, the pressure increasing with volume.

Generalization to two components

Van der Waals generalized his equation of state from one to two components by postulating that a and b depend on the relative amounts of the two components. ⁶ The mole fraction X of the second component is one way to parameterize the relative quantities. The equation of state for a two-component mixture is thus $$\left(P+a_X/V^2\right)\left(V-b_X\right)=RT.$$ (1) For the applications we discuss, the excluded volume b _X is taken to be a constant independent of the mole fraction. The attraction parameter a _X is assumed to have the form $$a_X={\left(1-X\right)}^2/a_1+2X\left(1-X\right)a_{12}+X^2{a}_2,$$ (2) with the subscripts 1 and 2 indicating the two components. The positive constant a ₁₂ denotes the strength of the molecular attraction between the two components. The Helmholtz energy of the mixture is given by $$A\left(V,X,T\right)=-RT\ln \left(V-b_x\right)-a_X/V+RT\left[X\ln X+\left(1-X\right)\ln \left(1-X\right)\right]$$ (3) The last term in equation (5) represents Gibbs’s ideal-mixing term. ⁷

Thus, the Helmholtz energy of a binary mixture is a function of three variables: V, X, and T. According to Gibbs’s phase rule, the system will have critical curves instead of critical points, coexistence surfaces instead of coexistence curves, and up to four coexisting phases. The result is some very complicated phase behavior.

In the one-component case, as seen in figure 2(b), a line doubly tangent to the loop in the isothermal A(V) curve for the van der Waals equation yields two coexisting phases of different molar volumes. Similarly, in the two-component case, the isothermal A(V, X) surface for the van der Waals mixture equation may be dented to allow a plane to touch it in more than one point. The system lowers its free energy by splitting into two phases that differ both in molar volume and composition. A double-tangent plane guarantees that Gibbs’s thermodynamic equilibrium conditions are fulfilled—that is, the temperature, pressure, and chemical potentials of the two phases are equal. ^7,8 The connection between a phase split and the existence of a double-tangent plane to an appropriate thermodynamic free-energy surface originated with Gibbs and continued with Maxwell. The application to A(V, X), however, was due to van der Waals.

Figure 3 shows a typical region of a smooth surface z(x, y) near an ordinary critical point. At a point where the surface is bowl-shaped, the curvature is positive: All nearby points are on the same side of the tangent plane. At a point in the saddle-shaped region, the curvature is negative: There are nearby points on both sides of the tangent plane. Thermodynamic free energies are always unstable at points of negative curvature. The spinodal in figure 3 is the boundary between regions of positive and negative curvature. At spinodal points, the curvature is zero. A double-tangent plane rolling over the surface traces out a coexistence curve, or connodal. We use the word fold to describe the region bounded by the connodal. The fold shown in figure 3 ends in a critical point where the two tangent points merge. Korteweg and Maxwell credit Arthur Cayley for first defining such a point.

Van der Waals was able to show that, when his mixture equation was applied to components not greatly dissimilar, ordinary vapor–liquid phase separation occurs at sufficiently low temperature. Thus, the isothermal surface A(V, X) has a fold running from X = 0 to X = 1 that separates a large-molar-volume gas phase and a dense liquid phase. Such a fold is called a transverse fold. Separation of liquid phases occurs if the attraction between the two components is weaker than an appropriate average, that is, if $a_{12}<\left(a_1+a_2\right)/2$ , assuming b_X is constant. A longitudinal fold then separates two dense phases that differ in composition. As temperature decreases, folds usually increase in size, and may start interfering. This interference can lead to formation of a triple-tangent plane and a region in which three phases—two liquid, one vapor—are in equilibrium.

A fold is as well characterized by its spinodal as it is by its connodal. The spinodal is the easier of the two to calculate, because it follows from a local condition on the surface: a change of the sign of the curvature. The connodal follows from a global condition—tangency in more than one point—and was rarely calculated by Korteweg.

Folds on surfaces

In his 1891 papers on criticality, ⁵ Korteweg analyzed the properties of critical points and the evolution of folds. In the earlier of the two papers, he studied the properties of critical points on a surface. He performed a Taylor expansion of a surface z(x, y) at the critical point. The coordinates are defined so that, at the critical point, the tangent plane is the xy plane and the y-axis is tangent to the connodal. Because the tangent plane at the critical point is the limit of double-tangent planes, the surface is very flat in the y direction near the critical point. The Taylor expansion near that point is therefore quite different from an expansion at any arbitrary point on the surface.

From the expansion, Korteweg derived the form of the connodal and spinodal at the critical point, and found that, in first approximation, they were parabolas in xy coordinates. By imposing additional conditions on the expansion coefficients, Korteweg defined and examined double critical points. He then derived a set of three simultaneous equations, which, when solved, yield all critical points on an arbitrary surface.

Next he assumed that z(x, y) depends smoothly on an additional parameter—in the application to fluid mixtures, the parameter may be the temperature or an adjustable interaction parameter. His objective was to study local deformations, that is, the effect of small changes of the parameter on the surface or on the coefficients of its Taylor expansion near the origin. Several things can happen. For instance, an ordinary or double critical point may appear or vanish, and a double critical point may split into two ordinary critical points. In modern geometry one speaks of unfoldings of z(x, y).

The Taylor expansion of the free energy at a critical point is generally credited to Lev Landau. ⁹ Korteweg, however, thoroughly explored such an expansion almost half a century before Landau’s classic paper.

In Korteweg’s second paper, the first objective is the global study (not restricted to the local neighborhood of a point) of surfaces, their folds, critical points, spinodals, and connodals. The next, and more difficult subject is global deformations. If the surface depends on a parameter, how do folds, critical points, spinodals and connodals evolve with changes of the parameter? When should one expect double-, triple- and quadruple-tangent planes, representing the coexistence of two, three, or even four phases? Korteweg obtained his results by exploiting the local theory of his first paper.

A special but important result described by Korteweg is the accessory fold, which is formed from a double critical point on the spinodal of an existing fold. The accessory fold may be fully formed before it protrudes beyond the connodal of the existing fold. Thus, in applications to a thermodynamic free-energy surface, folds forming in the unstable region deserve close attention.

Symmetric binary mixtures

General surfaces may have very complicated folds. In the last part of his general-theory paper, Korteweg introduced and analyzed in detail the van der Waals model for a symmetric binary mixture. That model yields interesting folds that can be completely understood. For the symmetric mixture shown in figures 4 and 5, the attraction coefficients satisfy a ₁ = a ₂ ≡ a. The Helmholtz energy surface of the two-component system is characterized by the single parameter κ ≡ a ₁₂/a. If κ > 1, the only fold characterizing the system is a transverse vapor–liquid fold appearing at temperatures below the common critical temperature of the two components. The interesting case is κ < 1, in which additional phase separation occurs in the dense phase. Korteweg labeled the different ranges of κ he studied by the letters A (κ near zero) to E (κ just below unity).

In case E, a transverse fold forms, as for κ > 1, but as the temperature is lowered, separation begins in the liquid phase. As the temperature further decreases, the transverse and longitudinal folds begin to interfere, and a triple tangent plane is formed. Figure 4(a) shows a region in which two liquid phases and one vapor phase are all in equilibrium.

If, as in Korteweg’s case A, the interaction between the two components is very weak, a longitudinal fold forms even in the absence of a transverse fold (see figure 4(b)). When the temperature drops to a particular value, a double critical point forms on top of the existing critical point. Such a coincidence of three critical points, clearly recognized by Korteweg, ⁵ is presently called a tricritical point. ¹⁰ The tricritical point marks the end of the simple longitudinal fold and the beginning of the two new transverse folds. As the temperature decreases further, the double critical point splits into two ordinary critical points, creating two accessory folds (see figure 4(c)). Those two folds move to the sides so that, finally, a full transverse fold is formed from the longitudinal fold.

The most interesting cases are at intermediate values of κ for which the transverse and longitudinal folds compete. Figure 5 shows phase diagrams at successively decreasing temperatures that Korteweg analyzed as part of case D. In figure 5(a), competing folds move in from three sides. As the temperature is lowered, opposing critical points on transverse and longitudinal folds meet in three pairs, not necessarily at the same temperature, then disappear by passage through double critical points. For the “central” value κ = 0.565, the three pairs of critical points merge at the same temperature.

In figure 5(b), the folds have met and regions of one, two, and three phases surround a small central one-phase region. As the temperature is further lowered, the central one-phase region shrinks. At one particular temperature, a tangent plane touches the free energy surface at four points, as shown in figure 5(c). At still lower temperatures, the central bowl-shaped region on the surface pulls up above the quadruple tangent plane. A triple tangent plane remains, though, together with three two-phase regions, as in figure 4(a). Korteweg calculated the range of κ values that admit four-phase regions and found 0.534 < κ < 0.67.

Modern work

By 1906, Dutch scientists had reached the limits of what they could derive, given their computational tools, from the van der Waals binary-mixture model. Progress had to wait for the advent of the computer. During that wait, much of Korteweg’s work was forgotten, although vestiges can be found in the phase diagrams that appear throughout fluid and materials science. The breakthroughs of modern physics around 1900 forced thermodynamics into the background until the rebirth of interest in critical phenomena in the mid-20th century. In the 1960s, Robert Scott and his student Peter Van Konynenburg initiated a thorough, computer-based analysis of the phase behavior of the general van der Waals binary mixture. ¹¹ Robert Griffiths and coworkers at Carnegie Mellon University rediscovered the region of four-phase coexistence in the context of a study of the three-state Potts model, which will now be introduced.

As described in the box on page 49, the two-state Ising model allows one to study criticality in, for example, a two-component incompressible liquid or a one-component compressible fluid. Likewise, a generalized lattice model in which each site can be in one of three states can represent a three-component incompressible liquid mixture or a two-component compressible fluid mixture. One such model is the symmetric three-state Potts model, which represents a mixture of three identical components. That model, widely studied in the 1970s, describes a system in terms of the temperature and one mutual-interaction parameter. The 1973 mean-field results for the model obtained by Joseph Straley and Michael Fisher at Cornell University revealed the existence of a four-phase region. The correspondence between their results and the phase diagrams that Korteweg presented some 80 years earlier is evident in figure 5. The analog of Korteweg’s diagram in figure 5(b) for phase separation of three-component mixtures has been rediscovered in various guises throughout the 20th century. ³ It was first described by Gibbs in 1876. ⁷

Griffiths and coworkers did extensive numerical studies of both the symmetric and the asymmetric three-state Potts model (including the four-phase region) in the mean-field approximation. In 1978, they looked for, and found, a four-phase region in the van der Waals binary mixture. Their lower bound and center of the four-phase region for the symmetric mixture agree with Korteweg’s 1891 results, but there is a discrepancy with Korteweg’s value for the upper bound. A symbolic-numerical solution of Korteweg’s conditions for the upper bound, by one of us (Levelt), unearthed a wrong sign in one term of Korteweg’s equations for that bound. After correction, Korteweg’s equations yield precisely the modern numerical result.

Just as physicists have overlooked Korteweg’s contributions to the thermodynamics of phase transitions, mathematicians seem to have completely forgotten Korteweg’s work on folds on surfaces. That is an amazing oversight, given the originality and generality of his results, and the notions and techniques he used—singularities, deformations, and unfoldings—that are quite characteristic of modern geometry. Korteweg exploited new mathematical ideas to describe, analyze, and understand phase equilibria in binary mixtures. The reappearance in modern singularity and catastrophe theory ¹² of problems Korteweg studied, particularly in the geometry of surfaces, confirms the importance of his new ideas.

Korteweg certainly would have benefited from the computational tools available to modern researchers. Symbolic computer software ¹³ would have allowed him to replace his many low-order approximations to physical quantities by expressions of arbitrarily high order and precision. Symbolic computation would have saved him much tedious labor in his analysis of the van der Waals model and enabled him to produce other visualizations in addition to the beautiful diagrams he published.