EVALUATION OF VOLUME-TEMPERATURE RELATIONSHIPS IN THE NEMATIC PHASE OF MBBA NEAR ITS NEMATIC-ISOTROPIC TRANSITION BY MEANS OF COMPUTER MINIMIZATION METHODS

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EVALUATION OF VOLUME-TEMPERATURE RELATIONSHIPS IN THE NEMATIC PHASE OF

MBBA NEAR ITS NEMATIC-ISOTROPIC TRANSITION BY MEANS OF COMPUTER

MINIMIZATION METHODS

R. Chang, J. Gysbers

To cite this version:

R. Chang, J. Gysbers. EVALUATION OF VOLUME-TEMPERATURE RELATIONSHIPS IN THE NEMATIC PHASE OF MBBA NEAR ITS NEMATIC-ISOTROPIC TRANSITION BY MEANS OF COMPUTER MINIMIZATION METHODS. Journal de Physique Colloques, 1975, 36 (C1), pp.C1- 147-C1-149. �10.1051/jphyscol:1975128�. �jpa-00215905�

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Classification Physics Abstracts

7.130

EVALUATION OF VOLUME-TEMPERATURE RELATIONSHIPS

IN THE NEMATIC PHASE OF MBBA NEAR ITS NEMATIC-ISOTROPIC TRANSITION BY MEANS OF COMPUTER MINIMIZATION METHODS

R. CHANG and J. C. GYSBERS

Rockwell International Science Center, Thousand Oaks, California, U. S. A.

Abstract. — The volume (F)-temperature (T) data for MBBA in the pretransition region slightly below its nematic-isotropic transition temperature are fitted to the five-parameter nonlinear equation

V= a + bT+c(Tc—T)»

be means of two independent methods : the Control Data Corporation PROSE constrained optimization method and an APL nonlinear minimization method.

Both methods yield global solutions. The computer minimization methods described here are expected to have wide application in data analyses pertaining to pretransition and second order transition in liquid crystalline systems.

In our investigation [1] of the pretransition and electromagnetic waves, etc. The background correc- critical phenomena in the nematic phase of MBBA tion, which for narrow temperature ranges can be the volume-temperature relationship was described approximated by a linearly temperature dependent by the following empirical equation : term, is included as a variational parameter in the

T, , , , , , ,„, rrr.n / i \ minimization treatment. The techniques we will

V = a + bT + c(Tc - T)n (1) , ., , .„ ^ ^ . . , „ .M ,

describe and illustrate in the following paragraphs where a, b, c are constants independent of temperature, can therefore be applied universally to related T^c a virtual critical temperature, and n an exponent investigations of the nature of pretransition and related to some critical index or indices. In our previous second order phase transition in liquid crystalline attempt to reduce the number of parameters which systems.

have to be fitted, the coefficient of volume thermal The volume-temperature data used in our compu- expansion of MBBA near room temperature was tation consist of thirty sets of data points from a single determined and assumed to be identical to the constant experiment carried out in our laboratory for distilled b of eq. (1). This assumption of b is quite arbitrary. In MBBA having a nematic-isotropic transition tempe- order to obtain unique solutions, we have decided to rature of about 316 K. The data points are compiled include the linear term bT as a variational parameter, in Table I, where the first column gives temperature The experimentally observed data are fitted to a five- readings in milli-volts from a copper-constantan parameter nonlinear equation according to computer thermocouple calibrated to give a linear temperature minimization procedures described below. We noted dependence (in the temperature region of interest) of that a considerable number of papers presented at the 24.26°/millivolt. The second column lists the corres- Fifth International Liquid Crystal Conference dealt ponding volume readings in arbitrary units of a with similar five parameter nonlinear equation, the 0.1 cm diameter capillary. The last data point corres- temperature-dependent physical property measured ponds to the temperature TB of the beginning of the being one of the following : density, volume, specific first order transition. We discarded all the data points heat, sound velocity, ultrasonic attenuation, bire- beyond TB, since the onset of the first order nematic- fringence, elastic properties, scattered intensity of isotropic phase transition leads to the co-existence of

Résumé. — Les mesures du volume Ket de la température Tdu MBBA au voisinage de la transi- tion nématique-isotrope servent à déterminer les paramètres de l'équation d'état :

V=a + bT+c(Te— T)n . Pour cela, on utilise deux méthodes différentes de calcul :

— Une méthode d'approximation PROSE et une méthode de minimisation non linéaire APL.

Ces méthodes pourraient être appliquées à de nombreux problèmes d'effets prétransitionnels et de transitions de phases du deuxième ordre dans les cristaux liquides.

JOURNAL DE PHYSIQUE Colloque C l , supplément au n° 3, Tome 36, Mars 1975, page Cl-147

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975128

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Cl-148 R, CHANG AND J. C . GYSBERS both the nematic and isotropic phases and more

abrupt volume changes in the CO-existence temperature region.

List of data points from direct experimental observations

eq. (1) suggests that a unique solution is very likely, since the sum of terms represented by eq. (1) is a mildly nonlinear function with no oscillating terms involved. To test the idea that a given solution is unique, we made runs with totally different starting points and observed that they converge to the same point. The constraints employed are the following :

(a) Tc in eq. (1 1) must be greater than the maximum temperature reading observed.

(b) n must be greater than zero and less than 1.0.

(c) The coefficient b must be positive.

A typical run with JOVE, depending on initial guesses of the parameters, will involve a thousand or so iterations to reach convergent solutions. At the end of each solution, the sum of squares of (V[experi- mentall-V[calculated]) of all thirty data points (SUMSQ) is calculated. The fitting is indeed excellent.

The results of five runs near the global minimum are tabulated in Table II(a).

2. The APL nonlinear minimization method. - Two 'Omputer minimization methods In this approach we first locate an approximate global were used. First is the Control Data Corporation minimum by means of a random minimization PROSE constrained optimization method. Second is technique. We write ^Q= SUMSQ as defined before : an IBM APL nonlinear minimization method slightly

modified by us to insure global solutions. The two methods will be described separately.

1. The prose constrained optimization method

(G JOVE >>). - The constrained optimization method JOVE ^%is based on penalty function techniques transforming the constrained optimum problem into a sequence of unconstrained problems for which a variety of methods is effective. In this method a new objective function is constructed from the original one by adding penalty functions with constraints and then the constraints are ultimately discarded. The augmented objective is optimized in a sequence of steps in which the contribution of the penalty functions are progressively diminished. In the limit, their contri- butions become negligible and the optimum of the augmented objective is also that of the original one.

Although there is in general no guarantee that a nonlinear least squares fit will have a unique solution, the nature of our fitting function represented by

where the constraints on b, Tc and n have been removed by defining b = d2, Tc = T,,,

+

^{p2, and}

and Tma, is the largest observed temperature. Diffe- rentiation of Q with respect to a, d, c, p and m and equating the derivatives to zero yield five more equation defining these f i v ~ parameters.

A pair of positive numbers (p, n) is selected from a uniform random populations with p less than 0.2 and n less than 1.0 respectively and the constants are then computed using the newly created derivative equations, the SUMSQ = Q1 for this combination of parameters is computed. A second random pair (p, n) is selected and the process is repeated to yield a value

e2.

If Q2 is larger than Q', the new set of parameters is TABLE IIa

PROSE (JOVE) printout near convergence

Initial 1 2 3 4 5

- - - - -

a 38.281 5 - 14.064 8 - 9.244 8 - 6.119 3 ^-5.728 0 - 5.704 6

b 26.094 7 41.147 8 36.646 6 34.805 6 34.578 5 34.564 9

C - 45.776 2 - 10.353 5 - 11.648 6 ^-13.277 8 - 13.484 3 - 13.496 6

n 0.332 1 0.285 9 0.594 5 0.640 9 0.645 7 0.645 9

Tc 2.028 7 1.784 9 1.765 3 1.764 4 1.764 3 1.764 3

SUMSQ 0.707 25 0.145 52 0.109 18 0.106 62 0.106 58 0.106 58

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STATE EQUATION OF MBBA

TABLE IIb

APL printout near convergence

Initial 1 2 3 4 5

- -

C( - 8.8348 - 1.9978 - 1.9957 - 1.9939 - 1.9924 - 1.9921

d 6.032 2 5.696 2 5.696 1 5.696 0 5.696 0 5.696 0

C -11.8662 -15.4509 -15.4519 -15,4528 -15.4536 -15.4538

m 1.233 9 1.453 4 1.453 4 1.453 5 1.453 5 1.453 5

P 0.023 4 0.004 563 0.004 559 0.004 555 0.004 552 0.004 551 2

SUMSQ 0.119 90 ---p--- not calculated - - 0.115 50

Note the following equalities :

b = d 2

n = m2/(1

+

m 2 )

(T,,,,, is the last temperature reading given in Table I.).

discarded. If, however Q2 is less than Q' the old set is discarded and the new set retained. Thus a monotone decending sequence of sums of squares are obtained together with the parameter point associated with the lowest value. The process ceases after a preassigned number of trials (usually 100 to 200). I n this way we are quite certain that the final set of parameters is near the true global minimum in the p-n two phase space.

The final set of parameters constitutes the initial guess for the APL minimization program employing a derivative free analog of the Levenberg-Marquart algorithm. The final values for a, d, c, p and m computed by this algorithm together with the sum of suqares (SUMSQ) near the convergence point are reproduced in Table II(b) for comparison purposes.

A comparison of the results shown in Table I1 suggest that both methods lead to the global solution of the nonlinear eq. (1) but the parameters obtained by the two methods differ slightly from one another.

The five parameters from the last iteration of the two methods are shown in the following :

Method

Parameter PROSE (JOVE) APL

- -

a - 5.704 6 - .1.992 1

b 34.564 9 32.444 4

C - 13.496 6 - 15.453 8

n 0.645 7 0.678 7

T c 1.764 3 1.764 3

SUMSQ 0.106 58 0.115 50

The slight differences in the optimized parameters obtained by the two methods suggest possibly that the global minimum in the five dimensional space is rather broad and the two methods are approaching this minimum from different directions. We fed the final PROSE parameters as initial guesses into the APL program and observed that after about 100 iterations, the same final set of APL parameters were reproduced.

This suggests that the length of the computer word employed may be an important factor with respect to the seIection of the best computer minimization method for nonlinear least squares problems.

Reference

[l] Pretransition and Critical Phenomena in the Nematic Phase of MBBA, Roger Chang, Solid State Comrnui:.

14 (1974) 403.