PHASE TRANSITION OF ICE Ic WITH IONIC DEFECTS

The oxygen arrangement of ice Ic exhibits a cubic structure of the diamond type. The arrangement of hydrogens is disordered within the Berna-Fowler rules. Ice Ic is expected to undergo order-disorder transition at low 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 symmetrical 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 C24 = 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 H3O+ 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 the two oxygens on this assumption. However, 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 αi (i = 1,…, 28), and the number of bonds for respective positions of a proton in bonds are denoted by βi (i=1, …, 8), there are some linear relations between them. Thus it is found that we can take as independent variables twenty two from among αi 's and βi '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 EC and the energy EGI of ionic defects (H3O+ or OH-) which is measured from the energy of water molecules. In this expression, the probability of neighbouring molecules having orientations specified by αi and βi is assumed to be proportional to αiβj. The number of configurations W of molecules and ions in the crystal is obtained after Takagi (2). Thus the free energy F=U-kBT ln 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 lawers ; for EGI= 10EC, the transition temperature is 3.05 EC/KB. The order of the transition remains of the first order. The polarization changes suddenly from O to the saturated value at transition temperature regardless of EGI. When EGI < 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 O 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 EGI ⩾ 30 EC, the influence of the ionic defects on the transition is negligible