Julian H. Gibbs

Molecular theory of surface tension

A molecular theory of surface tension is developed for a liquid–gas interface of a one component system. The Helmholtz free energy, the quantity minimized in the van der Waals approach, is obtained here from a rigorous expansion in powers of derivatives of density ρ and is minimized by the calculus of variations. The coefficient A (ρ) of the term in the square of the density gradient is (kT/6) Fdr r2C (r,ρ), C being the direct correlation function. In the case in which ρ varies in one direction x only, the solution of the Euler–Lagrange differential equation is analyzed in detail. This describes the cases of a single phase and of two coexisting phases and leads to the equal area Maxwell construction. The effect of an external field on the solution is discussed. The Euler–Lagrange differential equation provides a differential statement of Bernoulli’s theorem. In a three dimensional treatment the stress tensor formula is obtained from the corresponding Euler–Lagrange partial differential equation. A (differ...

Chain Stiffness and the Lattice Theory of Polymer Phases

General formulas for the thermodynamic properties of amorphous polymer phases are obtained from statistical mechanics, with the aid of the lattice model, in a manner which avoids the use of restrictive assumptions concerning the nature of the individual polymer chains.Certain results, such as prediction of a second‐order transition for systems of semiflexible chains and the Flory‐Huggins formula for the entropy of mixing with monomeric solvent, are thus shown to be independent of the precise nature of the model assumed for the polymer chains.More complete information may be obtained by application of the general formulas to models descriptive of the molecular chains in question. As an example, the results of Flory for semiflexible linear chains whose stiffness arises exclusively from intramolecular nearest neighbors are obtained as a special case. (The conventional thermodynamic properties of polydisperse systems of chains of this type are shown to depend on the number average molecular weight.)

The Composition Dependence of Glass Transition Properties.

The configurational entropy theory of glass formation is used to derive the composition dependence of the glass transition temperature (Tg) and of supercooled liquid transport properties for binary mixtures which obey the laws of regular solutions. The relations between expressions derived subject to specified approximations and known empirical equations are shown, and the parameters of the latter are thereby identified with defined measurable quantities. The predicted compostion dependence of Tg is compared with experimental data from binary mixtures with satisfactory agreement.

Electric Polarization of Solutions of Rodlike Polyelectrolytes

A statistical‐mechanical model for electrical polarizability of solutions of rodlike polyelectrolytes has been developed. The polyion is treated as a linear array of sites for adsorption of counterions. It is essentially the Rice—Harris model with an applied electric field. Calculation of the partition function of this system is performed by the matrix method and predictions are made for the effect on the low‐frequency dielectric constant of added salt, degree of polymerization, and degree of neutralization. The results are in approximate agreement with measurements of solutions of polymethacrylic acid and deoxyribonucleic acid.

A molecular theory of interfacial phenomena in multicomponent systems

The van der Waals theory of surface tensions is generalized to multicomponent systems. The local free energy density consists of a ’’local equilibrium’’ free energy (i.e., equilibrium free energy of a uniform mixture having species densities equal to the local species densities) plus a quadratic form in the gradients of the species densities. The coefficients in this quadratic form depend on the local species densities through the density dependence of the second moment of the local multicomponent direct correlation function. The requirement that the free energy be a minimum yields a system of partial differential equations (one for each component). A particular linear combination of the differential equations is the condition for mechanical equilibrium. It can be interpreted as a microscopic statement of the multicomponent Young–Laplace equation for the pressure variation across a curved interface. For two component systems the theory is a generalization of the treatment of Cahn and Hilliard in that it allows for pressure variations. If the local pressure fluctuations are suppressed, the differential equation for the concentration is very similar to theirs, except that the total density may vary across the interface. Similarly, when the theory is applied to liquid–vapor equilibrium in a binary system, the differential equation for the total number density reduces to that of a single component system when the local chemical potential difference (μ=μ1−μ2) is held constant.

The hard sphere ’’glass transition’’

We investigate the existence of a glass transition for the systems of hard spheres studied in computer simulations of molecular dynamics. An empirical best‐fit equation of state is established for the metastable (supercooled or supercompressed) hard sphere liquid. Configurational thermodynamic properties are calculated from this, and their significance is discussed. Evidence is presented to support the proposition that a system of hard spheres exhibits a phenomenon similar to a glass transition in which ’’glass formation’’ is purely a kinetic phenomenon. For hard spheres there is no underlying second order transition temperature T2. The effective ’’cooling rate’’ associated with these computer simulations of molecular dynamics is investigated and found to be enormous (∼1011 deg/sec). The implications of these results with respect to the configurational entropy theory of the glass transition are discussed.

Fluorescence as a conformational probe of beef liver phenylalanine transfer RNA

Abstract 1. 1. Purified beef liver phenylalanine tRNA (tRNA Phe ) exhibits a 4-fold increase in fluorescence intensity upon the addition of low concentrations (0.3–1.0 mM) of Mg 2+ . This enhancement of fluorescence by Mg 2+ is reversible and ribonuclease-sensitive. It occurs instantaneously and completely at both 1.4 and 27.0° and is accompanied by a 4% hypochromicity (260 nm) at the latter temperature. 2. 2. Simultaneous measurement of the optical rotatory dispersion (ORD) revealed that tRNA Phe exhibits a correspondingly sharp increase in optical rotation over the same concentration range of Mg 2+ . This transition was accompanied by a blue shift of the optical rotatory dispersion spectrum of approx. 3 nm. 3. 3. On the basis of the coincidence of these Mg 2+ -induced fluorescence and rotational changes, it is concluded that fluorescence spectroscopy is a valid and sensitive probe for the study of conformational changes in tRNA Phe . 4. 4. High concentrations (2.0 M) of NaCl mimic to a certain extent (61%) the effect of these lower concentrations of Mg 2+ on the fluorescence of tRNA Phe . NaCl competitively inhibited the enhancement of the fluorescence intensity of tRNA Phe by Mg 2+ . 5. 5. These results are discussed in relation to the effect of Mg 2+ and Na + on the biological activity of tRNAs in general.

Toward a model for liquid water

A new model is proposed for liquid water. It is obtained by consideration of the two transitions (melting and boiling) which define the liquid phase. These transitions are discussed with the aid of two analogies to well-known phenomena in polymer physical chemistry. In analogy to the helix-coil transition in polypeptides and polynucleotides, the melting of ice is viewed as a process consisting essentially of the destruction of the orderly interconnected small rings of hydrogen bonds characteristic of the crystal. The fact that the breakup of interconnected small rings is cooperative, even when unaccompanied by the breaking of bonds which are not parts of rings, is clearly seen by inspection of the theory for the putatively analogous helix-coil transition. The condensation of water vapor is viewed in analogy to gelation in reversibly polymerizing systems, an analogy which interprets its cooperativity. Taken together, these interpretations of the phase transitions indicate that the liquid can be viewed as an infinitely and randomly branched “gel” of (rapidly interchanging) hydrogen bonds in which closures of rings (primarily large rings) are present at random but in which there is no significant preference for an ordered array of small rings. These concepts also lead naturally to an interpretation of the triple point and sublimation. The random gel model is seen to be consistent with most of the known properties of liquid water. In particular, the radial distribution function, infrared and Raman spectra, dielectric properties, density maximum, and properties of the supercooled region are discussed briefly here.

Mean molecular size distributions and the sol–gel transition in finite, polycondensing systems

A new theory is presented for the molecular size distributions and the sol–gel transition in a model, nonlinear, polycondensing system. The theory differs from the earlier theories of Flory and of Stockmayer in that the molecular size distributions are calculated as averages over the ensemble of all states of the finite system rather than as the ’’most probable’’ distribution in the limit as the size of the system goes to infinity. This modification is necessary because the mean molecular size distribution for a finite system is not described adequately over the range of large molecular sizes, at any extent of reaction, by the correctly normalized most probable distribution. This is clearly indicated by the failure of the most probable distribution to describe the gel explicitly. Numerical evaluations of the mean and the most probable distributions are shown to agree over the range of smallest molecular sizes at all extents of reaction. Beyond the gel point a second peak, clearly descriptive of the gel, i...

Theory for the influence of gravity on liquid-vapor interfaces

The van der Waals theory of surface tensions in the presence of gravity is discussed in detail. The physical two‐phase solution in the gravitational field is shown to exist and be unique. The relation between the chemical potential and the position of the interface in the gravitational field is elucidated. The solution approaches the solution given previously and the chemical potential approaches its coexistence value as the strength of the gravitational field is diminished. The gravitational effects are seen to be especially enhanced in the neighborhood of the critical point.