Thesis for intensive discussion  from  Yurii  Baikov

Ioffe  Institute, St-Petersbutg, Russia

Historical View. The question on the charge and mass transfer in hydrogen-containing materials has started to discuss more than 200 years ago  [1]. The seeming clear understanding relatively liquid phases has been formed to the middle of XIX century. However, the “barrier” to understand the ionic mobility in strongly organized crystalline lattice has been overcome. Crucial conception on the role of defects for physicochemical properties was advanced by Ya.I.Frenkel  90 years ago in Ioffe Institute  [2]. Additional efforts were performed by Bernal & Fowler [3]and  Bjerrum [4] to form the model of hydrogen ion migration. Next steps of  the development of the interest to proton mobility  were both applied aspects (J.Bokris conception of hydrogen energy) and basic research taking into account too small size and mass of proton. For example, it may be indicate the attempt to find the proton tunneling in solids. In 1979 years was predicted the protonic Hall effect by theoretician from Ioffe Institute [5]. As the result  of such brain attack the three families of inorganic proton conductors  has been discovered factually simultaneously in 1980  -1982. (Fig.1)

Fig. 1

Omitting  the detail description of applied history of these families let us take into account some subtle features of hydrogen state (i.e. protons). To model the hydrogen behavior it is necessary to take into account that hydrogen is the “quest” in perovskites and on the contrary “host” in hydroxides and acidic salts. Of course the definition “host” for proton in CsHSO4 or in KOH means “the part of anion of “host”.

Proton behavior and stable content of hydrogen corresponds to experimental facts. Although there is different behavior relating to decomposition at heating to melt. However at first step this fact will be omitted.

The main aim of our model to evaluate the possibility to describe acidic salts and hydroxides as the composition of THREE particles,  namely 


TETRAHEDRONIC anions without protons,  


The basic relations and equations of such model keeping in mind CsHSO4

Anions + protons will be indicate for briefity as  HT, where Т = SO4.  For more detailed description let us introduce three states: normal and pair of defects:

normal state             (H1A 2T)-1(x)1);

double proton state  (H2A1T)0(•)2) ;

protonless state     (A3T)+1( ∕ ) 0).

Here:  Ai – “empty” or protonless positions on thtree vertexes of tetrahedron Charge states are indicated  by two versions: usual and  Kreger-Vink (late in margins). On the main line in marginas – atomic part of corresponding elements.

Naturally Σαi =1, and according to electro-neutrality rules  α2 = α0.

The part of free energy due to protonic subsystem  F for one mole of material include chemical potentials of proton-containing components  (μi ) and their pair interactions with energies εik :
F=Σαi μi εik αi αk ……………..(1).

Standard chemical potential of komponents will be parameters (μi0=Const ) (

In formulas (1) indexes i и k  are 0; 1; 2.

At considering the most important is the conditions of thermodynamic equilibrium ∂F/∂αi=0. To get it as the controlling parameter will be chose α2, i.d. the part of tetrahedrons with two protons. Such defects could be determined the protonic transport, allowing the idea of Frenkel [2] and Bjerrum[4]. For simplicity let us use  α  without index, i.d. α1=(1-2α), α2 = α0 = α.

F = {μ10 + αƒ1(μi ,εi)  + ½α2ƒ2(εi)} + kBT {(1-2α) ln(1-2α)+2α lnα}…….(2)

In (2) there are two parts: energetic one (first brace) and entropy second.

Let us also for convenience introduce the dimensionless energetic parameters   F, μ10,ƒ1(μi εi) и ƒ2(εi) dividing both part of (2) by kBTkB –Boltzman, but by default  keeping “previous” symbols. It is note that ƒ1(μi εi) contain standard chemical potentials and energy of interaction, but ƒ2(εi) only later. The question on sings                 ƒ1(μi εi) and ƒ2(εi) required special consideration. For example ƒ2 = ε2,0 – 2(ε2,1 +ε1,0). If ε2,0<0 (attraction) and  ε2,1>0 (repulson), then signƒ2 , and  ƒ1  are under?


Fig. 2

How to get the link between phenomenological approach  and the most  important characteristic of protonic conductors? 

Now we are only on first step and try to find the most remarkable “event” PHASE TRANSITION. Our model based on the many review like [6]. After mathematic analysis in agreement with physical sense has been found:                                   ƒ1(μi εi) +8 ±1 and ƒ2(εi) -12±1. It means for 300 – 400 K  +0.25  and  –0.35 эV. It was  revealed that maximum of disorder correspond α=1/3 (ΔS=ln3), stable state at α=0 (α1=1, ΔS=0 ) is the normal anions  HSO4-1  (H1A 2T)-1(x), but there is the third state with α=1/2 (ΔS=ln2), which could be considered as predecessor of dehydration  (H2SO4 0 + SO4-2).

These features are reflected on Fig.2. Free energy of protonic system (black line) first derivative(red) to reflect phase transformation. And second derivative to reflect stability of different phase (blue)  limiting  0.9<α1≤1 и      0.4≤α 2 ≤0.5. On the figures this boundaries could be seen as crossing blue line (second derivative) and grey line as Zeroth line for all curves.


Proton contribution in different part of free energy of model CsHSO4.



1. de Grotthuss C.J.T. // Ann. Chim. (Paris). 1806. T. LVIII. P. 54–74.

2.  Frenkel Ya.I/.// Z.Phys. 1926. V. 35.  S. 652.

3. Bernal J. D. and Fowler R. H. //J. Chem. Phys. 1933  V.1. P.515.

4. Bjerrum N.K. //Dan.Videwisk S.M.F.Medd. 1951,  V.276, N.1. P. 3-56

5. Азизян А.О., Клингер М.И.// Теор. и Мат.Физика.1980  Т.43, N 1. С.78-90

6. Baranov A.I.// Rus Crystallographie. 2003. Т.41. N6 .  С.1081-1107.

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