Transport/transpiration methods

Using Glushko et al. (1982) tables, partial pressures of monomer and dimmer have been calculated via the 3^rd law of thermodynamics and using equation (18), apparent pressures are deduced and compared to recent studies such as Venugopal et al. (1985) [41] and Cordfunke (1986) [48] as well as earlier Topor (1972) [49] determinations with the same experimental method (transport). Results are presented in figure III-3.

-6 -5 -4 -3 -2 -1 0

6 7 8 9 10 11 12 13

10⁴/T /K^-1 Log10 Papp / bar

Topor 1972 (Fit) Venugopal 1985 Cordfunke solid 1986

Glushko et al. 1982 table via 3rd law Cordfunke liquid 1986

T_m=905 K

Figure III-3: Apparent pressure determinations over solid and liquid CsI given by different authors using the transport method and comparison with recalculated values from Glushko et al. selected data.

Figure III-3 presents the decimal logarithmic of the apparent pressure of CsI over solid and over liquid as a function of the inverse of temperature given by different authors. Over CsI solid, Glushko et al. data remains slightly lower than the one determined by Cordfunke and Venugopal et al. Above the melting point (T_m = 905 K), Glushko et al. pressures are definitely lower than the one given by Cordfunke over liquid and, finally, at elevated temperatures join the equation proposed by Topor (1972). Indeed, this last agreement corresponds to the Glushko et al. selection at high temperature based on Topor’s measurements.

On the other hand, the analysis of Cordfunke’s pressure data around the melting point, show an abnormal trend in these data since a step-like is observed. Cordfunke related this fact to a sudden dimmer apparition, an interpretation which can not agree with thermodynamic

behavior –regular increase-. By comparison with Venugopal et al. pressures over the CsI solid, Cordfunke pressures are slightly lower especially close to the melting point and before the step-like. Three main features can explain this sudden increase:

• The difference of pressure can be due to a problem of evaporation rate which might be low because of “retarded” or “hindered” vaporization processes [50, 51]. Consequently, there may be no equilibrium reached over the solid phase and lower pressures are measured.

Rothberg et al. [52] determined free vaporization rates and by comparison with equilibrium rates proposed an evaporation coefficient for CsI total pressure α = 0.36 in the temperature range 757-772 K - α = p(measured) / p(equilibrium) by definition -. Cordfunke pressures data over CsI solid has been recalculated taking into account of this evaporation coefficient (P_app(recalculated) = P_app(cordfunke) / 0.36). Results are presented in figure III-4. This correction increases the pressure in the correct direction but seems too important, a feature which is explained because the steady state of vaporization in a transport reactor is not clearly free vaporization: indeed – as already studied for the effusion method [53] – the balance of evaporation and condensation flows at the sample surface increases the apparent net evaporation flow toward the equilibrium value as a function of the steady state flow in the reactor. Further in the experiment, the retarded vaporization observed over solid phase is normally non operating over the liquid phase because the evaporation coefficient is usually 1 for liquids [54] (and pressures are at equilibrium) due to no activation energy barrier for so disordered surfaces. The pressure should thus increase at once at the melting temperature.

• Note that the complex molecules – the dimmer in the present case – are more sensitive to vaporization kinetics due to difficulties in the adsorption stage reactions at the solid surface i.e. matching the surface structure to build the adsorbed dimmer before desorption.

Considering the retarded vaporization process for the only dimmer, this last might be the responsible molecule of the observed step as proposed by Cordfunke. Thus, considering that the proportion of the dimmer over the solid would be very low (evaporation coefficient ≈0 for instance), the pressure step at the melting would be mainly due to the sudden dimmer contribution in the total pressure. the experimental relative difference in apparent pressure between liquid and solid at 905 K is calculated according to the following relation,

32 . ) 0 ( )

( ) ( )

( ∆ =

− =

l p

p l

p

l p s p

app app app

app

app (20)

Relation in which P_app(s) and P_app(l) are respectively the apparent pressure over solid and over liquid at 905 K (obtained by least square fits).

The difference in apparent pressure ∆p_app can be written as following:

sol sol

liq

app p p p p p p

p =( ₁⁰+2 ₂⁰) −( ₁⁰ +2 ₂) =2 ₂⁰−2 ₂

∆ (21)

Relation in which p⁰1 and p⁰2 are the standard (pure compound) equilibrium pressures of the monomer and the dimmer at the melting temperature, liq for liquid phase and sol for solid phase. Since the equilibrium is not reached over the solid phase for the dimmer pressure, this pressure can be related to the equilibrium pressure through the evaporation coefficient α by the following relation:

0 2

2 p

p^sol =α (22)

and combining relations (21) and (22),

) 1 ( 2 2

2 ₂⁰− α ₂⁰ = ₂⁰ −α

=

∆p_app p p p (23)

and finally relation (20) becomes,

32 . ) 0 2 (

) 1 ( 2 )

( ₁⁰ ₂⁰

0

2 =

+

= −

∆

liq app

app

p p

p l

p

p α

(24)

The monomer/dimmer pressure ratio is obtained as, )

32 . 0 1

( 2

32 . 0

0 1 0 2

−

= −

α p

p (25)

For low evaporation coefficients, α → 0 the ratio dimmer to monomer is at its minimum value, i.e. 0.23, and increases up to 0.28 for α = 0.1. As a conclusion, the assumption of no dimmer over the solid phase in the Cordfunke experiments would lead to an estimate of at least 22% of dimmer molar fraction at equilibrium over the liquid at the melting temperature.

• The pressure step may come from temperature determinations. In fact, the necessary evaporation heat associated to the CsI molecules flow swept away by the carrier gas can cool down the vaporizing surface since in a such low temperature range mainly conduction heat tranfer in the solid phase as well as some heat contact resistance between the container and the solid CsI may occur. As the measured temperature is necessarily located between the furnace and the container, the determined pressure is reported to thermocouple temperature and not to the real vaporizing surface. By this way, the author attributes pressure data to the temperature of the liquid phase while these data belong to the temperature of the solid phase.

This is the meaning of the 11 K difference observed between the maximum temperature at the step-like (916 K) and the retained melting point (905 K) in the compilations. Cordfunke data in the temperature range 867-916 K were thus corrected using the following relation based on heat transfer relation,









−

⋅ −

=

0 max

11 0

T T

T

T_corr T (26)

relation in which Tcorr is the corrected temperature, T the Cordfunke measured temperature, T0 and Tmax are respectively the minimum and maximum measured temperature over solid.

Corrected data are presented in figure III-4.

-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2

9 10 11 12 13 14

10^4/T /K^-1 Log10 Papp/ bar

Cordfunke Solid phase Cordfunke Liquid phase Venugopal et al.

Cordfunke /0.36 (Rothberg) Cordfunke corrected T

Melting point

Evap.Coef.

Figure III-4: Corrected data from Cordfunke measured pressures below and close to the melting temperature according to two assumptions: evaporation coefficient and

According to the general evolution of the data in figure III-4, Cordfunke original pressures data over solid will be thus selected in the temperature range 779-855 K and these last corrected data will be retained in the temperature range 864-904 K. Over liquid, Cordfunke pressures will be used in the temperature range 916-1070 K.

Among the two preceeding above analysis, the second one (temperature gradient) seems likely reliable and we propose to retain the last corrected values.