Determination of Phonon Density of States from Constant-Pressure Heat Capacity Data of Soft Organic Materials in the Glassy and Crystalline States by Using the Real-Coded Genetic Algorithm

Abstract

A method was proposed to derive the phonon density [g(ω)] of states of materials from their heat capacity data by using Real-Coded Genetic Algorithm (RCGA) with Just Generation Gap + Real-Coded Ensemble Crossover. The performance of the method was confirmed by testing whether or not the RCGA reproduces a reasonable g(ω) by analyzing the set of heat capacity data evaluated from an initially assumed model g 0 (ω) composed of Debye and optical modes. As an example, constantpressure heat capacities (C P s) were measured for soft molecular materials, diphenyl phosphate (DPP) and diphenylphosphinic acid, in the condensed state, and their g(ω)s were determined from the C P data by applying the RCGA. The unusual behavior that the C P value of glass was smaller than the one of the crystal in the temperature range from 10 to 70 K was observed in DPP; the behavior is contrary to that expected ordinarily for the glass as compared with the crystal. The g(ω)s determined by the RCGA demonstrated that the unusual behavior was attributed to the blue shift in g(ω) of ω = 30−240 K in the glass compared with the crystal. The blue shift and other effects were discussed reasonably as originating from the competitive concurrence of strong and weak intermolecular hydrogen bonds in DPP, with the help of determination of their intramolecular vibrations for the isolated molecule by the density functional theory calculation. It was concluded that the method using the RCGA is of value for obtaining the microscopic information of g(ω) from the precise heat capacity data and for investigating any difference between the details of g(ω)s in different phases of materials.

Reliability of the Phonon Density of States Determined by Real-Coded Genetic Algorithm from Heat Capacities of Benzoic Acid Crystals

“…This roughly agrees with that of 3/4 = 0.75 evaluated from the results of spectroscopic measurements. In view that the value in the range 0.7–0.75 is a relatively small fraction of the whole degrees of freedom below ω = 400 K, the flat distribution of g (ω) potentially reflects the following situation: previously, we carried out model analyses which tried reproducing g (ω) using the RCGA from the sets of the heat capacity data computed from the assumed g 0 (ω) distributions involving two optical phonon modes and reported that the optical phonon in the high-frequency side in g (ω) became much broadened to have a wide distribution compared with that assumed in the initial g 0 (ω) . The reason remaining unclear, it is desired to develop the RCGA analysis in future without such a broadening effect.…”

“…g (ω) derived from each set of C P data for BA h and BA d crystals was optimized using the RCGA − with just generation gap (JGG) + real-coded ensemble cross-over (REX) method, , as reported previously . To calculate with a computer, the numerical integration in eq was replaced with summation, as given in eq , where Δω, being a digitization unit regarding ω, was taken to be 3 K. A set of candidate parameters for solutions, represented as a genotype in GA, is an array of real numbers with n length, and the REX­( n + k ) is a multiparental cross-over model, where n + k denotes the number of parents.…”

“…The C P – C V correction was approximately performed based on the Nernst-Lindeman relation with fitting parameter A as C P = C V + ATC P 2 . The presentations of the genotype for g (ω) and the C P – C V correction were the same as the method reported previously . The contribution attributed to a proton transfer in BA was represented by the Schottky heat capacity.…”

“…C P , cal was thus calculated as the sum of all the contributions in the genotype. The fitness, u , of derived g (ω) was evaluated by the same fitness function, given by eq , using C P , cal and C P , exp , as reported previously where μ is the experimental inaccuracy and n is the total number of data points: μ was determined as 0.003 and 0.002 below and above 35 K, respectively, by the previous heat capacity measurement of BA crystals …”

“…However, low-energy g (ω) has not been determined in most materials so far because such experiments encounter difficulties in determining the absolute value of g (ω) directly. Calorimetry is superior to the spectroscopic methods in that all the vibration modes contribute equally to C V according to their degrees of freedom, as given in eq , and the absolute value of g (ω) is determinable from a set of C V data by solving the inverse problem. − …”