Inorganic Materials, Vol. 39, No. 12, 2003, pp. 1324–1325. Translated from Neorganicheskie Materialy, Vol. 39, No. 12, 2003, pp. 1527–1528. Original Russian Text Copyright © 2003 by Aleksandrov, Postnikov.
Effect of DC Electric Field on the Thermodynamic Parameters of Solidification of Molten Dielectrics V. D. Aleksandrov and V. A. Postnikov Donbass State Academy of Civil Engineering and Architecture, ul. Derzhavina 2, Makeevka, Donetsk oblast, 86023 Ukraine e-mail:
[email protected] Received February 7, 2003; in final form, June 13, 2003
Abstract—The thermodynamic parameters of solidification of molten dielectrics in an electric field are analyzed. It is shown that the Gibbs energy, enthalpy, and temperature of the phase transformation, supercooling, critical nucleus size, and work of nucleation are only sensitive to electric fields no lower than 107–108 V/m.
Consider a dielectric crystal–melt system in a uniform dc electric field E at constant pressure (p = const). The Gibbs energy of the ith bulk phase in this field is [1]
sponding phases, and ∆H0 is the enthalpy increment upon the phase transformation), we obtain dT = ( 2αT L V /∆H )EdE. 0
Gi ( T, E ) =
0 Gi ( T )
+ U i ( E ),
(1)
0
(5)
Integrating Eq. (5) between T L at E = 0 and T L at E ≠ 0, we find 0
0 G i (T)
where is the temperature-dependent Gibbs energy of the ith phase in zero field, and Ui(E) is the additional energy due to the electrification of the dielectric: U i ( E ) = ( ε 0 ε i E /2 )V . 2
(2)
Here, ε0 is the dielectric constant, εi is the dielectric permittivity of the ith phase, and V is volume. The energy difference between the solid (U1) and liquid (U2) phases is then given by U 2 – U 1 = αE V , 2
(3)
where α = (ε0/2)(ε2 – ε1): α > 0 for ε2 > ε1 and α < 0 for ε2 < ε1 . From Eqs. (1) and (3), the condition for phase equilibrium during solidification in an electric field, E E G 2 (T, E) = G 1 (T, E), has the form G 2 ( T ) = G 1 ( T ) + αE . 0
2
0
(4)
Differentiating Eq. (4) with respect to E and taking 0 0 0 0 into account that (∂ G 2 /∂T)p, E = S 2 , (∂ G 1 /∂T)p, T = S 1 , and S 2 – S 1 = ∆H0/ T L (where T L is the melting point in zero field, S1 and S2 are the entropies of the corre0
0
0
0
E
T L = T L ( ∆H + αE V )/∆H . E
0
0
2
0
(6)
Designating the enthalpy increment by ∆H = ∆H + αE V , E
0
2
(7)
we obtain the field-dependent temperature of the phase E transformation, T L , and heat of transformation, LE, T L = T L ( 1 + α'E /L ) ,
(8)
L = L + α'E ,
(9)
E
0
E
2
0
0
2
where α' = αρ and [L] = J/kg. Let us analyze the Gibbs energy change per unit volE ume (∆qE = D G V /V) upon the formation of a crystal from the melt in an electric field. It follows from Eqs. (1) and (3) that ∆q = ∆q + αE , E
0
2
(10)
where ∆q0 is the energy change per unit volume upon melt solidification in zero field (E = 0). The ∆q0 value is
0020-1685/03/3912-1324$25.00 © 2003 MAIK “Nauka /Interperiodica”
EFFECT OF DC ELECTRIC FIELD ON THE THERMODYNAMIC PARAMETERS
related to melt supercooling ∆T0 relative to the melting 0 point T L by [2] ∆q = ρL ∆T 0
0
0
0 /T L ,
(11)
where ρ is the density of the substance. In an electric field, ∆q = ρL ∆T E
E
E
E /T L .
E
1 2
3
1 4
(12) 0
10
20
30
40
50
3 60 lc , nm
Gibbs energy as a function of the nucleus size for water crystallization in (1) zero field and in an electric field of (2) 108, (3) 5 × 108, and (4) 8 × 108 V/m.
∆q = ∆q ∆T /∆T , 0
∆G × 1016, J 4
2
Substituting (8) and (9) in (12) and taking into account (11), we obtain E
1325
0
or ρL E E - ∆T . ∆q = -------0 TL 0
(13)
The term ∆ G F in Eq. (15) can then be written in the form
Using Eqs. (10), (11), and (13), one can relate the supercooling in an electric field, ∆TE, to that in zero field, ∆T0:
p 2 E E 0 (17) ∆G F = σ SL F = 6 σ SL – ----2-0 E l . a Finally, the Gibbs energy change (15) upon the formation of a cubic nucleus is given by p 0 E E 3 2 (18) ∆G = – ∆q l + 6 σ SL – ----2-0 E l . a
∆T = ∆T + αE T L / ( ρL ). E
0
2
0
0
(14)
Thus, the supercooling of a molten dielectric depends on the difference in dielectric permittivity between the solid and liquid phases (∆ε = ε2 – ε1) and the electric field. The quantities ∆qν, TL, L, and ∆T appear in expressions for the critical nucleus size and the work of nucleation. Nucleation in an electric field is accompanied by E E a Gibbs energy change ∆GE(∆GE = G S – G L ), which can be represented in the form [3] ∆G = – E
E ∆G V
+
E ∆G F
±
E ∆G D .
(15)
The term ∆ G D , representing the contribution of structural defects and mechanical stresses, will be left out of consideration here. The volume contribution is E given by ∆ G V = ∆qEV, where V is the nucleus volume, and the surface contribution is ∆GF = σEF. Consider a cubic nucleus (F = 6l2, V = l3) which is isotropic in zero field, so that the surface tension is the same on all of its faces. The field effect on surface tension can be represented by E
σ SL = σ SL + σ , E
0
E
(16)
where σ SL is the surface tension at the solid–liquid interface in zero field, and σE = –p0E/a2– is the electrification energy per unit surface area (p0 is the molecular dipole moment, parallel to the electric field, and a is the lattice parameter). 0
INORGANIC MATERIALS
Vol. 39
No. 12
2003
E
From the minimum condition ∂(∆GE)/∂l|E = const = 0, we have p 0 4 σ SL – ----2-0 E a . (19) l c = -------------------------------E ∆q The figure shows the plot of the Gibbs energy versus nucleus size for water crystallization in an electric field. It can be seen that ∆G is only sensitive to fields above 107 V/m. Increasing the electric field reduces the work E E of critical nucleus formation, A c = ∆ G max , and the critical nucleus size lc . Using Eqs. (8), (9), and (14), we estimated the melting points, heats of fusion, and critical supercoolings of several dielectrics. The results indicate that the electric-field effect is only significant starting at E ⯝ 107–108 V/m. REFERENCES 1. Poplavko, Yu.M., Fizika dielektrikov (Physics of Dielectrics), Kiev: Vyshcha Shkola, 1980. 2. Chalmers, B., Principles of Solidification, New York: Wiley, 1964. Translated under the title Teoriya zatverdevaniya, Moscow: Metallurgiya, 1968. 3. Kuraksina, O.V. and Aleksandrov, V.D., Analysis of the Gibbs Energy Change due to the Formation of Imperfect Nuclei in Supercooled Molten Metals, Metallofiz. Noveishie Tekhnol., 1996, vol. 18, no. 3, pp. 3–7.
