J. Phys. Chem. 1983, 8 7 , 4261-4264
4201
Posslbllity of Anomalous Relaxatlon due to the Charged Dlslocation Processt K. Itagakl U.S. Army COM Regions Research and Englneerlng Laboratory, Hanover, New Hampshke 03755 (Received: August 23, 1982; In Final Form: April 19, 1983)
The possible contribution of electrically charged dislocations to dielectric relaxation and the consequent effects were examined and compared with experimental results. A catastrophe caused by the positive feedback was found to be possible under normally attainable conditions.
1. Introduction Audio-frequency dielectric relaxation has been attributed to Bjerrum defects’ and dielectric relaxation lower than the audio frequency to electrode polarizaiton. However, careful studies on unstrained ice of low dislocation density and the effect of straining have indicated more complex features. For example, low-dislocation-density ice frozen onto a solid electrode frequently displayed negative conductarice, which disappeared after several weeks of annealing. Such observations led me to a series of studies on the properties of electrically charged dislocations in ice and their effects on dielectric and anelastic relaxation. 2. Electrical Charge on Dislocation in Ice As discussed in ref 2, dislocations in ice can be displaced between pinning points by an externally applied electric field. One can deduce the charge density from the segment length between pinning points, the amplitude of the displacement, and the applied electric field. A brief description of the study is given here. More complete discussion is given in ref 2. The equation of motion of a charged dislocation under a local electric field E’ can be described by
where A, = apb2, is the linear mass of the dislocation, q the displacement of any point on the dislocation, B the damping coefficient, C the line tension, p the linear charge density along the dislocation, and w the angular velocity of the applied field. The solution under the boundary condition of q(0,t) = q(1,t) = 0 where q = ~ ( x , t is ) 4pE’1 (2n l)TX X q=--C--sin A ,42n 1 1 exp[i(wt - a,)]
+
+
+ ( W B / A ) ~ ](2) ’/~ = a(2n + 1)(C/A)lI2,and
[(a2- w 2 ) 2
where 1 is the segment length, w, 6 , = tan-’ wB/A(wn2- w2). Since the high-frequency end of the present measurement ( = l o 2 ) is far lower than wo ( - l o 8 ) , q may be approximated by
pE’12/8C (3) The amplitude of vibration qmarand the distance between pinning points 1 can be obtained from Lang X-ray topography of ice under a known applied electric field E. The -‘ can be calculated from possible maximum local field E qmar N
A
‘Paper presented at International Symposium on Physics, Rolla,
MO, Aug 1982. This article not subject to
TABLE I: Calculated Charge Density from Vibrating Dislocations charge per a spacing charge density +e/2.4 a pmS, 1.5 x lo-’’ C/m +e/7.1 a ppmb, 5 x lo-’’ +e1108 a pmin, 3.2 x lo-’*
~~
~~
E and the dielectric constant, K , by using eq A5 in the Appendix. The minimum local field E’minis the applied external field. C is calculated from C = 2Ksb2/r,where K,is the energy factor for screw dislocation and b is the length of Burgers vector b. The size of charge concentration p is thus (4)
Seventeen boat-shaped images were found suitable for measuring 11/12 from the topographs. An example is shown in Figure 1. The upper and lower limits of charge density were calculated for ELi,, and ElOpBI shown in the first column of Table I. Petrenko and Whitworth3 indicated that dislocation in ice has at least one positive net charge in every 200 dangling bonds, indicating our pmh may better agree. On the other hand, the number of positive net charges can be much higher. Fukuda and Higashi4v5 found that screw dislocations having a / 3 ( 1120) Burgers vectors outnumber the others in ice. Yoshida and Wakahama6showed in their Figure 10 and Photos 14,15, and 16 that dangling bonds exist every a lattice spacing along a / 3 ( 1120) screw dislocations. If only simple statistics apply in this case, half the dangling bonds have free protons, a positive unit charge, making charge density slightly higher than the observed.,p Of course strong elastic and electric fields around the dislocations should modify the charge density.
3. Dielectric Polarization Caused by the Displaced Charged Dislocation Since dislocations can be displaced by the external electric field, the contribution of their displacement to dielectric polarization would be expected. Detailed discussions in this chapter and chapter 4 are in ref 7. (1) P. V. Hobbs, “Ice Physics”, Oxford University Press, London, 1974. (2) K. Itagaki, CRREL Report 79-25, 1979. (3) V. F. Petrenko and R. W. Whitworth, Abstracts, Sixth International Symposium on the Physics and Chemistry of Ice, University of Missouri-Rolla, Aug 2-6, 1982, pp 77-8. (4) A. Fukuda and A. Higashi, “Physics of Ice”, Plenum Press, New York, 1969, 239-250. (5) A. Fukuda and A. Higashi, Jpn. J. AppE. Phys., 8, 993-9 (1969). (6)Z.Yosida and G . Wakahama, Low Temp. Sci. Ser. A , 20, 29-56 (1962).
U.S.Copyright. Published 1983 by the American Chemical Society
4262
Itagakl
The Awnal of pnvskcal Chetn&try, Vd. 87, No. 21, 1983
TABLE 11: Calculated Dielectric Constantr,
.Y
maximum probable minimum P , c/m 1.5 X lo-’’ 5 X lo-’’ 3.2 X lo-’’ assumed Eo’/Eo 1 3 45.5 KO
case I case I1
456.4 20721
152.8 2303
10.4 10.4
of the vibrating string. The contribution of charged dislocation polarization to the static dielectric permittivity do is found from eq 7a by setting w = 0; thus
Charge density p can be found in Table I. Dislocation density A was measured from the X-ray topographs and found to be of the order of 5 X 108/m2which agrees with the Fukuda and Higa~hi.~ Mean segment length was obtained from the 17 segments suitable for use in calculation of the q/L2relationship in charge density measurements. Flgure 1. Dislocation line vibrating (between two arrows) under a horitofltal fieM. A cumulative probability plot of these segment lengths is closer to a log-normal than to a normal distribution. AdPolarization P due to a displaced charged dislocation of mittedly such measurements tend to disregard the shorter segment length 1 is segments; however, L2 dependency in eq 9 and the closeness of a log-normal distribution reduce the error caused P = pql by the omission of shorter segments. where mean displacement can be obtained from eq 2 as Two assumptions were made to calculate the dielectric constants shown in Table 11. In case I, the same Eo’/Eo used to calculate charge density was assumed to be applicable to the dielectric constant. In case 11, E(/Eo was exp[i(ot - a,)] assumed to be 45.5;this local field is in an infinite volume f s i n (2n l) ?mx dx (5) [(a; - w2) + ( ~ d ) ~ ] ’0/ ~ 1 having a dielectric constant of 90. Clearly the calculated dielectric constant is highly dependent on the assumption. where d = B / A . By definition, the dielectric permittivity The major cause of this problem is the positive feedback K due to the polarization caused by the charged dislocations nature of the local field E{ which will be discussed in the for the uniform segment length 1 distribution in volume section 5. v is Relatively low anisotropy of the dielectric constant can be understood if the charged dislocation mechanism is assumed to be as follows. Observations by Fukuda and Higashi8 indicated that where c, is the dielectric constant of a vacuum. screw dislocation prevails in nonbasally grown ice with Then for the applied field E = Eo exp(iot) which most of the studies have been made. Fukuda and Higashi4observed that mechanical straining produces pure 8p2E’ exp(-ib,) dv screw dislocations having a / 3 ( 2110) Burgers vectors. K = l + O TAQO n (2n + [(on2- o ~+ )( ~~d ) ~ ] ” ~Samples used in all published data I have found were either prepared mechanically or electrodes were frozen into (6) ice, both processes which should generate extensive screw Since wo = (a/L)(C/A)II2 is on the order of 108, while the dislocations. Since the motion of screw dislocations both dielectric measurements were made in the range of w < within and out of basal plane is conservative (i.e., no 6 X l@,higher order terms can be dropped and we obtain volume change is involved), no restrictions is imposed in the motion in either direction. Such behavior should not be confused with the long-distance migration and multiplications involved in the creep of ice.
+
for uniform segment lengths where
BL2
.-.
Equations 7a and 7b indicate that the relaxation is of the Debye type if the segment length is uniform throughout the crystal. Note that the slope of the d vs. d ’ / w relationship, which is the dielectric relaxation time of this string model, coincides with r, the relaxation time (7) K. Itagaki, CRREL Report 82-7,1982.
4. Spectra Controlled by the Charged Dislocations Displacement of electrically charged dislocations can contribute to electrical polarization. In order to find the extent of this polarization involved and the spectra, dielectric relaxation of dislocation-freeice grown form vapor was measured. As shown in the inset of Figure 2, the Cole-Cole plot of dislocation-free ice is very small. The polarization strength and relaxation time is close to spectrum 2 of von Hippel et al. Scratching drastically modified the relaxation as shown by filled circles. Un(8) A. Higashi, M. Oguro, and A. Fukuda, J. Crystal Growth, 3, 4, 728-32 (1968).
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983 4263
Charged Dislocations in Ice
1201
I 8
0
A
I
I
I
ST-5 ST-6 ST-7 ST-8
i
A
AAA
;,.
i
1
'! 10:
$j/.
O A 30
35
.n,orp*m.nt
E
,
A 4.0
45
1
\
\
Kb
60
E'
Flgure 2. Very small Cole-Cole semicircle shown in inset, which was enormously enlarged by scratching. A hlgh dislocation density specimen showed a normal semicircle.
fortunately further annealing behavior did not follow due to quick sublimitation of the sample. However, it is not unreasonable to assume that the final Cole-Cole plot resembles the one made on the high dislocation density specimen since they were both grown in the same batch. Detailed discussion of this is given in ref 10. Although point defects produced by the dislocations may contribute to some of the spectrum, the following evidence supports the viewpoint that the audio- and slightly subaudio-frequency relaxation spectra are directly caused by the charged dislocation motion: 1. The X-ray irradiation effect described in the companion paper seems difficult to explain by such point defect processes. 2. The annealing behavior found by von Hippell' may best be understood by the dislocation process. 3. The effect of scratching on dislocation-free ice results in a drastic increase in the dielectric constant and loss (ref 10). 4. The slight depression usually observed in the Cole-Cole plot of spectrum 3 defined by von Hippelg can be explained by assuming that the dislocation segment length has a distribution of a normal or log-normal type.7 By using the standard deviation obtained from the X-ray topography, deviation from a pure Cole-Cole semicircle was studied. The results agree reasonably well with the measured semicircles of strain-free prepared samples. 5. Replacing L2 in eq 9, with L2 = a2Cr/B from eq 8, we obtain 8p2ACEo' Kd=1+ (10) a2CBs,.EoT indicating that T and KOI have a linear relationship. Such a relationships was found in the series of dielectric studies on strained ice. A set of examples supporting the existence of this relationship are shown in Figure 3. Strain-free prepared samples having orientations parallel and 45" to basal plane were compressed between electrodes parallel to the electric field under pressures of up to 300 P a . Points marked St. 5 and St. 8 were oriented to have axes parallel to the electric field and St. 6 and St. 7 were 45" cut samples. Both orientation samples showed the linear relationship between KOI and T . A detailed description can be found in ref 12. 5. Local Field Problem
Addition of one charged dislocation segment increases polarization, and then increases the dielectric constant. (9)A. von Hippel, D. B. Knoll, and W. B. Westphal, J. Chem. Phys., 54, 134-44 (1971). (10)K.Itapaki, J. Glaciol., 21, 207-17 (1978). (11)M. A. Maidique, A. von Hippel, and W. B. Westphal, J. Chem. Phys., 54, 150-60(1971). (12)S. F. Ackley and K. Itagaki, CRREL Technical Note, 1969.
20
-
-
Temp: 10°C I
I
KO'
I
-
I
2+F =2-F
Therefore K,,' goes to infinity when F = 2, and a catastrophe can occur. Using the values obtained in section 3, we obtain F values of 455, 50.5, and 0.2, with values of p = 1.5 X 10-lo, 0.5 X 10-lo,and 3.2 X C/m, respectively. If we assume that the only variable is mobile dislocation density, dislocation densities necessary to make F = 2 are 2.2 X lo6,2.0 X lo7,and 4.8 X 109/m2for charge densities of 1.5 X 10-lo,0.5 X 10-lo,and 3.2 X lo-'?- C/m, respectively. These dislocation densities are within the observed range. These arguments may be overly simplified. For instance, the contributions of dislocation segments to the overall local field may depend on their orientation and Burgers vector, but have been simply averaged in the calculation of local fields here. Dislocation densities in certain directions can be considerably higher or lower locally than other^,^ making the local field nonuniform. Also the line tension term in eq 1 is applicable to only small amplitude vibrations. Higher order term have to be taken into account in the high field strength case found in near catastrophes. However, a positive feedback situation persists, although the amount of feedback may depend on the assumption of a local field. Whether such a system can reach a cata-
J. Phys. Chem. 1083, 87, 4264-4267
4264
I
P cos2 0 sin I)ds
I
Concluding Remarks Studies on the charge of dislocations and their possible contribution to the dielectric constant lead to the possibility of local field catastrophes. Since the effective local field becomes complex in nonstatic conditions, a dielectric relaxation anomaly may appear to be lower than the audio-frequency relaxation usually attributed to electric polarization. A crude computer simulation resulted in two Cole-Cole semicircles somewhat similar to the semicircles appearing in dielectric relaxation studies on unstrained ice. Further experimental and theoretical studies are in progress. Appendix: Mosotti Type Field on the Core of a Cylindrical Cavity Let us assume a cylindrical cavity around the dislocation line as shown in Figure 4. The electrical charge l appearing on the surface element S of the cavity, small enough that only one dislocation line can be included in the center, is caused by polarization P and is given by
dE2
(A2)
=
The total field E'acting on the point A is then
E ' = Eo + E,
+ E2 =
K-
V O vO - ( K - 1)L L
- 1 vO +-2 L K
K +1 vO
645) 2 L Eo(KV/L)is the applied field, and El ( ( K - l)Vo/L) is the depolarization field where the dipole contribution inside the cavity is disregarded. Problems caused by omission of the dipole contribution may be somewhat nonrigorous in the present case since the major source of the field is discrete dislocation and the cavity is assumed to be filled with low dielectric constant materials.
Stability of the Vitreous Phase in Water-Ethylene Glycol Mlxtures Studied by Internal Friction N. Alberola, Laboratolre de Biqohyslque, Facult6 de MBdecne Alexis CARREL, 69372 Lyon Cedex 2, France
J. Perez,' J. Tatlbouet, and R. Vassollle Group8 dEtudes de M6bIhtrgb physique et de physique des Met6riaux (LA 34 1) INSA de L YONsBit. 502, 6962 1 Vllewbanne Cedex, France (Recelved: August 23, 1982; In Flnal Fwm: February 15, 1983)
A low-frequency (1Hz) internal friction technique is used to determine quantitatively the stability of the vitreous phase distributed in an ice matrix and formed in water-ethylene glycol mixtures (5/1000 v/v glycol) during rapid cooling. The determination of the crystallization kinetics allows us to establish a time-temperaturetransformation (TITdiagram. ) A simple inspection of this diagram gives the limit of stability of the vitreous fraction. By isochronal and isothermal annealing treatments, nucleation and growth phenomena can be distinguished. The nucleation rate is a maximum in the 160-165 K temperature range and the total devitrification rate is maximum about 180 K. In the 160-165 K temperature range, the growth process can be detected. Nucleation probably occurs at the ice (matrix)-vitreous islets boundaries. The low-frequencyinternal friction technique appears to be of use for determining the behavior of water-alcohol mixtures of interest in cryobiology.
Introduction Since the discovery of the cryoprotective effects of glycero1 by Polge et d.1it has been found that some organic substances act as protective agents against freezing and (1) Polge, C.; Smith, A. U.; Parkers, A. S. Nature (London)1949,164, 666.
0022-3654/83/2087-4264$01.50/0
thawing injury. Glycerol is a biological substance, and it is well-known that some frost-resistant insects are able to accumulate glycerol in their bodies.2 On the other hand, (2) Tanno, K. Reprinted f q m contributions from the Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan, Series B, No. 16, 1970, p 1.
0 1983 American Chemical Society
