Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations

In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).     

Specific heat of electron gas of arbitrary dimensionality

The exchange contribution to the specific heat of an electron gas in d dimensions is calculated and is shown to lead to a logarithmic temperature dependence at low temperatures in all dimensions (d > 1).     

Lateral intermolecular forces in the physisorbed state: surface field polarization of benzene and n-hexane at the water/ and mercury/vapor interfaces

The available experimental data on the dependence of the surface tensions of water and mercury on the adsorption of benzene and hexane from the vapor phase are critically analyzed and interpreted to obtain the two-dimensional second virial coefficients [B(2)(T)] for these adsorbed nonpolar molecules. Calculations based on the unperturbed Lennard-Jones (L-J) 12-6 formalism for benzene and the related 12-5 Salem formalism for long chains in two dimensions for hexane require that B(2)(T) should be negative for both adsorbates. On water, the experimental data indicate that B(2)(T) for both molecules is less negative than expected from the unperturbed L-J and Salem estimates, and on mercury the B(2)(T) values from experiment are positive. These findings are analyzed first in terms of a possible reduction in the attractive component of the potential of mean force between physisorbed molecules arising from their frequency-dependent interaction with their electrostatic images in the bulk phases, as described by McLachlan. It is concluded that the McLachlan corrections are small for these molecules and surfaces. A second analysis considers the effect of an extra repulsion between the adsorbed molecules arising from the induction of dipoles normal to the interface by the surface electric field. Surface field polarization (SFP) accounts reasonably well for the experimental results, leading to estimates of the surface fields at the mercury and water surfaces which are consistent with estimates from contact potentials for mercury and computation from modeling the water surface. SFP may have a wide impact in determining the form of physisorption isotherms.     

High temperature thermal properties of KH2PO4: Phase transitions and decompositions

Phase transitions and the thermal decomposition of KH₂PO₄ have been examined from room temperature to above 300°C by means of hot-stage microscopy, isothermal gravimetry and differential scanning calorimetry. Phase transitions at 198 and 242°C are confirmed, with corresponding enthalpy changes 4.2 and 2.3 kJ mole⁻¹, but no evidence has been found of a transition reported near 110°C. The thermodynamic and other evidence suggest a structural change at 198°C while the change at 242°C is less profound, perhaps involving only changes in the form of the hydrogen bonding. Thermal decomposition occurs in four stages, under conditions of free vapour escape, with the loss of one-quarter of a mole of water per formula unit of KH₂PO₄ in each stage. The products of each stage of decomposition are tentatively identified.     

Standard absolute entropy, S degrees 298 values from volume or density. 1. Inorganic materials

Standard absolute entropies of many inorganic materials are unknown; this precludes a full understanding of their thermodynamic stabilities. It is shown here that formula unit volume, V(m)(), can be employed for the general estimation of standard entropy, S degrees 298 values for inorganic materials of varying stoichiometry (including minerals), through a simple linear correlation between entropy and molar volume. V(m)() can be obtained from a number of possible sources, or alternatively density, rho, may be used as the source of data. The approach can also be extended to estimate entropies for hypothesized materials. The regression lines pass close to the origin, with the following formulas: For inorganic ionic salts, S degrees 298 /J K(-)(1) mol(-)(1) = 1360 (V(m)()/nm(3) formula unit(-)(1)) + 15 or = 2.258 [M/(rho/g cm(-)(3))] + 15. For ionic hydrates, S degrees 298 /J K(-)(1) mol(-)(1) = 1579 (V(m)()/nm(3) formula unit(-)(1)) + 6 or = 2.621 [M/(rho/g cm(-)(3))] + 6. For minerals, S degrees 298 /J K(-)(1) mol(-)(1) = 1262 (V(m)()/nm(3) formula unit(-)(1)) + 13 or = 2.095 [M/(rho/g cm(-)(3))] + 13. Coupled with our published procedures, which relate volume to other thermodynamic properties via lattice energy, the correlation reported here complements our development of a predictive approach to thermodynamics and ultimately permits the estimation of Gibbs energy data. Our procedures are simple, robust, and reliable and can be used by specialists and nonspecialists alike.     

Parameters influencing the templated growth of colloidal crystals on chemically patterned surfaces

The influence of various experimental parameters on the vertical deposition and structure formation of colloidal crystals on chemically patterned surfaces, with hydrophilic and hydrophobic areas, was investigated. The pattern dimensions range from about 4 to 400 microm, which is much larger than the individual particle size (255 nm), to control the microscopic crystal shape rather than influencing the crystal lattice geometry (as achieved in colloidal epitaxy). The deposition resolution and selectivity were tested by varying the particle concentration in the suspension, the substrate withdrawing speed, pattern size and orientation, and wetting contrast between the hydrophilic and hydrophobic regions. The evolution of colloidal crystal thickness with respect to the pattern dimensions and deposition parameters was further studied. Our results show that the pattern size has a rather strong influence on the deposited number of colloid layers and on the crystal quality. Better results are obtained when the lines of a stripe pattern are oriented parallel to the withdrawing direction rather than perpendicular. The deposition resolution (defined as the minimum feature size on which particles can be deposited) depends on the wetting contrast and increases with lower average hydrophobicity of the substrate.     

Computation of Fourier transform quantities in Hartree–Fock calculations for simple crystals

An efficient method is presented to compute the Fourier transforms of lattice sums over Slater-type orbital products that arise in crystal Hartree–Fock calculations. Introduction of one-dimensional integral representations for the Fourier transforms of the STO 'S enables a separation of the three infinite sums under the (two-dimensional) integral sign. This, in turn, makes it possible to calculate and store quantities associated with the three Fourier transform vector components separately. The orbital transform integrations are then performed numerically. The method is very advantageous when lattice sum transforms are needed for a large number of transform vectors. Formulas and results are presented for simple-cubic crystals when 1s, 2s, and 2p Slater orbitals are involved.     

Difference rule-a new thermodynamic principle: prediction of standard thermodynamic data for inorganic solvates

We present a quite general thermodynamic "difference" rule, derived from thermochemical first principles, quantifying the difference between the standard thermodynamic properties, P, of a solid n-solvate (or n-hydrate), n-S, containing n molecules of solvate, S (water or other) and the corresponding solid parent (unsolvated) salt: [P[n-solvate] - P[parent]]/n = constant = theta(P)[S,s-s], or n-S and other solvate, n'-S: [P[-solvate] - P[n'-solvate]]/(n - n') = [P[n-S ] - P[n'-S]]/(n - n') = constant = theta(P)[S,s-s] where P may be any one of: U(POT) (the lattice potential energy), V(m) (the molecular or formula unit volume), Delta(f)H degrees , Delta(f)S degrees , Delta(f)G degrees or (the standard thermodynamic functions of formation and the absolute entropy), and n can be noninteger. The constants, theta(P)[S,s-s], for each property, P, of solvate of type S, are established by correlation of the available set of experimental data. We also show that, when solid-state data for a particular solvate is sparse, theta(P)[S,s-s] can be reliably predicted from liquid-state values, P[S,l], or even gas-state values, P[S,g]. This rule offers a powerful means for predicting unknown thermodynamic data, extending the compass of currently known thermodynamic information. Systems considered involve the following solvates: H(2)O (hydrates), D(2)O, NH(3), ND(3), (CH(3))(2)O, NaOH, CH(3)OH, C(2)H(5)OH, (CH(2)OH)(2), H(2)S, SO(2), HF, KOH, and (CH(CH(3))(2))(2)O. Detailed examples of usage are given for hydrates and for SO(2).     

Studies in hydrogen bonding: Association in ammonia using an empirical potential

The empirical potential EPEN /2 has been used to establish the structures of isolated hydrogen-bonding ammonia clusters. The most stable forms of the dimer have a linear or near-linear structure. The trimer has a closed structure with zero dipole moment. Two stable tetramer forms were found: one with a closed structure and zero dipole moment in agreement with experimental findings, and one with a pyramidal structure with nonzero dipole moment which may be an artifact of the EPEN /2 potential. The relation of the dimer structures to the limited available experimental information is discussed.     

Chitin Derivatives III Formation of Amidized Homologs of Chitosan

The kinetics of the formation of amidized homologs of chitosan from the ionic complexes of chitosan with four alkanoic acids, formic, acetic, propionic and butyric acid, have been studied. Their degrees of substitution (DS) and their thermal transitions have been investigated. The results suggest that heating of N-acylates of chitosan produces amidized chitosan homologs with DS ranging between 0.1 and 0.6 as determined by solid state NMR. Thermal analysis by DMTA revealed that the ionic complexes of chitosan with formic, acetic, propionic, and butyric acids, as well as the respective amidized chitosan homologs, display two transitions designated as - and -relaxation. There was no indication of melting. The -relaxations of the ionic complexes were more pronounced than those of their respective amidized chitosan derivatives. The ultimate T _g of the amidized chitosan homologs declined stepwise with acyl substituent size. The T _g-behavior was attributed to both substituent size and DS. Kinetic analysis using T _g-changes with heating time can be used to describe the kinetics of amidization. The formation of amidized chitosan homologs followed first order rate kinetics in a two-stage process, as was reported previously. The activation energy for amidization was constant irrespective of the acid used in the chitosan complex as well as the DS at ultimate T _g. It was 14±1kcal/mol initially and 21±2kcal/mol in the second stage. Since the transition from first to second stage occurs after the change from rubbery to glassy state (vitrification), vitrification is suspected to be responsible for the two-stage kinetics.