PHASE TRANSITION OF ICE Ic WITH IONIC DEFECTS

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PHASE TRANSITION OF ICE Ic WITH IONIC DEFECTS

I. Minagawa

To cite this version:

I. Minagawa. PHASE TRANSITION OF ICE Ic WITH IONIC DEFECTS. Journal de Physique

Colloques, 1987, 48 (C1), pp.C1-669-C1-670. �10.1051/jphyscol:19871102�. �jpa-00226243�

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JOURNAL D E PHYSIQUE

Colloque C1, suppl6ment au no 3, Tome 48, mars 1987

PHASE TRANSITION OF ICE I c WITH IONIC DEFECTS I. MINAGAWA

Kokugakuin University, Tokyo 150, Japan

Abstract : The oxygen arrangement of ice Ic exhibits a cubic structure of the diamond type. The arrangement of hydrogens is disordered within the Berndl-Fowler rules. Ice Ic is expected to undergo order-disorder transition at law temperatures. The interaction energy of neighbouring molecules is assumed to depend on their relative orientations. The difference of the two energies, that for less symmetrical relative orientations minus that for symnetrical orientations

,

^is

denoted by

EC.

In ice Ic without defects, it is studied that the order-disorder transition of the first order occurs at 3.28 EC/kB for EC

>

0 (1). There are two kinds of defects ; Bjerrum defects and ionic defects. This study deals with the influence of ionic defects on the phase transition of ice Ic taking nearest neighbour interaction of molecules into consideration in the mean field theory of Takagi (2).

There are C 4 = 6 ways of arranging two hydrogens of a water molecule in -the crystal provided that the correlation of protons with those of neighbouring molecules is ignored. There are four ways of arranging three hydrogens of a H30+ defect and four ways of arranging a hydrogen of a OH- defect on the same assumption. Since a unit cell of the diamond structure has two oxygens, there are 28 ways of arranging hydrogens around tlie two oxygens on this assunption. Hwever, the rule for the number of protons on a bond must be obeyed. Therefore, these 28 ways cannot be chosen without restriction. If the numbers of water molecules or ions in the crystal.

for respective orientations are denoted by

di

(i = 1,.

. . ,

²⁸⁾

,

and the number of bonds for respective positions of a proton in bonds are denoted by (3i (i=l,.

. ^.

^,8)

^,

there are some linear relations between them. Thus it is found that we can take as independent variables twenty two f rom among di s and p i s.

Since energy U of the crystal depends on the orientation of molecules and ions, and on the number of ions, it is expressed in term of the 22 independent variables, the energy

E

and the energy %I of ionic defects (H30+ or Off) which is measured from the energy of water molecules. In this expression, the probability of neighbouring molecules having orientations specified by di and

Pi

is assumed to be proportional to d i (j j

.

The n h r of con£ igurations W of molecules and ions in the crystal is obtained after Takagi (2). Thus the free energy F=U-kBT In W is given as a function of the 22 independent variables. It is minimized with respect to these variables by the simplex method.

When the energy EGI decreases, the transition temperature lavers ; for %I= lo%, the transition temperature is 3.05 EC/kB. The order of the transition remains of the first order. The polarization changes suddenly from 0 to the saturated value at transition temperature regardless of %I. When <30 E, the entropy in the paraelectric phase is larger than NkBln(3/2) with N being the number of oxygens in the crystal. The entropy in the ordered phase is zero in the temperature range from 0 K to the transition temperature. It makes a contrast with the entropy of ice Ic having Bjerrum defects which is not zero near the transition in the ordered phase

( 3 ) - When %I

>/

³⁰^Ec,the influence of the ionic defects on the transition is negligible.

References

(1) Minagawa, I., J. Phys. Soc. Jpn. 52 (1983), 1641.

(2) Takagi, Y., J. Phys. soc. Jpn. 3 (1984), 273.

(3) Minagawa, I., J. Phys. Soc. Jpn. 54 (1985), 4221.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19871102

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JOURNAL DE PHYSIQUE

COMMENTS

I wish to draw attention to the fact that the configuration-counting method used by Minagawa can be extended to calculate the contribution of the protons to the entropy of partially ordered Ice Ih or Ic. On the assumptions that the ice is homogeneous and that the probability f of a proton being at the "correctTT end of each bond is the same, the entropy is the same as NagleTs approximation (Ottawa conference, 1972) for f near to 1/2, and at f = 1, but is lower for intermediate values of f. The asymptotic slope of the entropy per molecule as f tends to I is -kb In Lr, which can be justified in this limit by an elementary argument. The calculation will be published in the near future.

Am I correct in assuming that the results given for free energy, entropy and polarization in this paper are for ice with ionic defects &,and that those in Minagawa(l985) are for Bjerrum defects only. If so what does one expect for ice with both present as given by the mass-action law you presented at the end of your presentation ?

Answer :

You are correct.

Ice with impurities can be treated theoretically.

One can expect that entropy in the ordered phase near the transition temperature is different from zero for the realistic concentrations of defects.