Viscous flow behavior of selenium

Journal of Non-Crystalline Solids 12 (1973) 199-206. © North-Holland Publishing Company

VISCOUS FLOW BEHAVIOR OF SELENIUM M. CUKIERMAN* and D.R. UHLMANN Department of Metallurgy and Materials Science, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. Received 6 February 1973

The viscosity of 99.999% pure selenium has been determined over the available range of temperature where crystallization could be avoided. A bending beam viscosimeter was used to obtain data in the high viscosity region, and a rotating cylinder instrument was employed for the low viscosity range. Also determined was the stress dependence of the viscosity at temperatures of 31.5 and 29.6°~C. The critical stress for non-Newtonian flow was found to lie in the range of hundreds of psi, which is appreciably smaller than that for oxide liquids in a similar range of viscosity.

1. Introduction The viscous flow behavior of selenium has been the subject of several previous investigations. In some cases [ 1 - 3 ] , these cover the high temperature range above the melting point, while in another [4], the flow behavior in the vicinity of the glass transition was explored. These studies dealt, however, with specimen materials of various purities, the temperature range covered by each was rather small, and significant variations are noted in the different sets of high temperature data where comparisons could be effected. It seemed advisable, therefore, to determine the viscosity over both ranges of temperature, extending these ranges, using the same sample material of high purity for both. In addition, observations will be reported of the variation of the viscosity with stress in the high-viscosity region.

2. Experimental procedure Two viscosimeters were employed in the present investigation. The first, used in the high viscosity (low temperature) range, was a beam-bending viscosimeter similar in principle to that described by Hagy [5]. As shown in fig. 1, the sample bar is * Based in part on a thesis submitted by M.C. in partial fulfillment of the requirements for the Ph.D. degree in Ceramics, MIT, 1973.

200

M. Cukierman, D.R. Uhlmann, Viscous flow behavior of selenium

1- Supports

Sample ±

~- Heat incj oil

Fig. 1. Beam-bendingviscosimeter used in the present investigation (shcematic). supported by two knife edges 3.5 cm apart in the specimen chamber (5 cm diam. X 5 cm high) of a large volume oil bath. The temperature control of the bath, filled with Dow Coming 560 silicone off, was provided by a proportional controler with a probe located in the bath. The temperature in the specimen chamber close to the sample was monitored with a chromel-alumel thermocouple and a recorder. After stabilization at a given temperature, the temperature was found to remain constant within -+ 0.05°C. A stainless steel rod was used as the force transmitting device. This was also connected to a transducer core, and the output of a DCDT (Hewlett Packard 24 DCDT 005) was recorded to give the deflection rate and hence the viscosity. The high-temperature viscosity was measured using a Haake-Brabender Rotovisco instrument. In this case, a stainless steel crucible containing the sample material was immersed in a constant temperature bath (crucible temperature maintained within -+ 0.1 °C after stabilization at a particular temperature). The shear stress on a cylindrical plunger located in the sample was recorded, and knowing the rate of rotation the viscosity was determined. The viscosimeter was calibrated using NBS standard oils. The specimen material used in the present investigation was selenium of 99~999% purity, obtained from Cerac Pure Inc,, Butler, Wisconsin. For the low temperature

M. Cukierrnan, D.R. Uhlmann, Viscous flow behavior of selenium

280 I

2.0

260 I

Temperature 240 I

(°C) 220 I

•

200 I

201

IBO 1

1.81.6-

• •

-

o

d

g-l.4--

-

(2_

o

1.2-

1.0

•

o•

-

÷

•+

-

-

0.80.6

t 1.7

I

T 1.8

i

I

L

I 2.0

1.9 1/T

t

I 2.1

I

I 2.2

(°K -t xlO 3)

Fig. 2. Variation of viscosity with temperature for selenium in the high-temperature range. • = data of ref. [ 1 ] ; o = data of ref. [2] ; + = data of ref. [ 3] ; • = data of present investigation. viscous flOW study, charges of the starting material were placed in Vycor tubes, 6 mm or 9 m m in diameter; the tubes were evacuated and rotated during a 2 h heat treatment at 500°C; and the specimens were removed from the tubes after quenching into an ice bath. For investigating the stress dependence of viscosity in the low-temperature range, specimens 2 mm in diameter were prepared by plunging Pyrex tubes into molten selenium at 350°C and the tubes filled by aspiration. The tubes were then leached away with HF. In both cases, the specimens were annealed for several weeks at 20°C or 2 h at 40°C. After cutting to the desired lengths, the specimen bars were held at each test temperature for two hours prior to applying the load to ensure thermal equilibrium. After application of the load, a period of more than 50 times the characteristic relaxation time (viscosity/modulus) was allowed to elapse before measuring the steady-state flow rate to ensure that equilibrium viscosities were determined [6, 7]. For the high-temperature viscous flow study, pieces of the starting material were melted and stirred in the crucible for 3 h at 260°C to ensure homogeneity and remove residual gas. For runs at temperatures below the melting point (217°C), the sample was cooled directly to the test temperature from 260°C and reheated to 260°C before the succeeding run.

202

M. Cukierman, D.R. Uhlmann, Viscousflow behavior of selenium Temperature ( °C ) 14

55 I

50 I

45 I

40 I

I

I

35

30

I

I

I

I

13~o 12 -

I( -

93.00

I

3.05

I

3.10

3.15 3.20 3.25 3.30 3.35 ~/T(OK-I ~103)

Fig. 3. Vadation of viscosity with temperature for selenium in the low-temperature range. • -data of ref. [4] ; o = data of present investigation. 3. Results 3.1. Viscosity- temperature relation The high temperature viscosities determined in the present investigation are plotted in fig. 2. Also shown in the figure are the results of previous studies. As shown there, the present data are in close agreement with those of Shirai et al. [2] over the common temperature range, and are bracketed by results of the other two studies over their common ranges. The low temperature viscosities are plotted in fig. 3. Also shown are the low stress data points reported by Jenckel [4]. The latter investigator studied the stress dependence of the viscosity using a fiber elongation technique. The results of the two studies are seen to be in close agreement over the common temperature range, and indicate pronounced curvature in the log rl versus 1/T relation over the entire low temperature region. The combined data of the present investigation are shown in fig. 4, together with a reasonable interpolation over the intermediate temperature range (between 51 and 197°C), where the occurrence of crystallization precluded meaningful viscosity measurements. Significant curvature is noted in the overall log r~ versus 1/T relation. The limiting slope at high temperatures corresponds to an apparent activation energy of about 17 kcal • (g at) -1 , while that at low temperatures corresponds to about 170 kcal • (g at) -1 . These are, respectively, significantly smaller and significantly larger than the Se-Se bond strength of about 44 kcal • (g at) -1 ; but no direct physical significance should be attached to the magnitudes, for reasons which have been discussed in detail elsewhere (e.g. ref. [8] ).

M. Cukierman, D.R. Uhlmann, Viscous flow behavior o f selenium

260 2zo 16

I

I

180

Temperoture (°C) t40 I00

I

6o

I

I

203

zO

l

I

14-12--

/

I0-o

ss

8-

6

-

ss S

-

2 --

0

I

s

-

s

j 1.8

_

s

s 4--

•

--

~'SSSS I

2.0

I

2.2

t

I

I

2.4 2,6 2.8 l I T (°K-I xlO 3 )

I

I

1

3.0

5.2

:3.4

Fig. 4. Overall viscosity-temperature relation for selenium determined in the present investigation; 3.2. Stress d e p e n d e n c e o f the viscosity

In the high temperature region, the viscous flow behavior is Newtonian - at least over the range of rotation rates from those used in obtaining the data of fig. 2 up to rates larger by a factor of three. In the low temperature region, significant departures from Newtonian behavior could be observed by increasing the stress from the low level used to obtain the results of fig. 3. The stress distribution in the bending beam technique is, however, not a simple one. For purposes of the present disqassion, we shall employ the simple engineering theory [9] to estimate the stresses rather than exact but more complicated treatments [10]. For reasonable values of Poisson's ratio, the two approaches agree within about 10%. According to the engineering theory, the maximum tensile stress in cylindrical bars loaded as in the present study may be expressed: Oqnax ~" P L /lrR 3,

(1)

where P is the load applied to the center of the bar of radius R and L is the length between the supports. This maximum tensile stress occurs at the mid-point of the bottom surface of the bar. The average tensile stress over the volume of the bar is: (~av ~" P L / 1 5 R 3 "

(2)

M. Cukierman, D.R. Uhlmann, Viscous flow behavior of selenium

204

13.2

I

I

I

I

t3.i ~

T:

29.6°C

-

12.9 12.e

~ 12.7

I

f

I

I

. ~ 12,7

I

I

I

I

12.6

"°~e~

- - o ~ . _

T = 3 1 . 5 °C

B

12.4 12.5 12.2 0

i

t

I

I

I 300

,

,

,

J

I J t 600

Sheor

I , 900 stress ( p s i ) ,

i

,

~

,

I , 1200

,

,

l

1500

Fig. 5. Variation of viscosity with stress for selenium a t 2 9 . 6 ° C a n d 3 1 . 5 ° C . A = present investigation, average shear stress; B = present investigation, m a x i m u m shear stress; C = d a t a o f ref. [ 4 ] , average shear stress; D = d a t a o f ref. [ 4 ] , m a x i m u m shear stress.

The maximum shear stress occurs at the centers of the top and bottom surfaces of the bar, oriented at an angle o f 45 ° to the direction of maximum tensile stress, and is approximately: 7.max ~ !2 0 m a x .

(3)

The average shear stress may for purposes of the present paper be taken as approximately: 7.max

,~,1

~ Oar.

(4)

The variation o f viscosity with stress, both ray and rmax, at two temperatures in the vicinity of the glass transition, 31.5 and 29.6°C, is shown in fig. 5. Also shown in the figure are the results of Jenckel [4], obtained in fiber elongation tests. Considering the differences in the test methods and sample materials used in the two investigations, the agreement between them in both the low stress level viscosities and the range of critical stress levels for non-Newtonian flow is rather good. The lowstress viscosities for these and other temperatures have been plotted in fig. 2. The stress levels determined in the present investigation for a 10% deviation from the low-stress viscosity are: rmax ~ 850 psi,

ray ~ 170 psi.

(5)

M. Cukierman, D.R. Uhlmann, Viscous flow behavior of selenium

205

4. Discussion The stress levels at which the flow of selenium becomes non-Newtonian are significantly lower than those reported previously for oxide glasses (e.g. ref. [ 11 ] ) in the vicinity of their glass transitions. The selenium values are somewhat higher than those often found for organic polymers in the high temperature, tow viscosity range (e.g. ref. [12]), and are comparable to those noted for such polymers in the region of their glass transitions [13]. The values for the critical stress for non-Newtonian behavior may be combined with the formalism of absolute rate theory to estimate the flow volume for selenium. This model fails to represent satisfactorily the observed variations of viscosity with temperature, but of all the standard models for flow, it is the only one which explicitly describes the stress dependence of viscosity. According to the model, the viscosity may be expressed [14] : A r exp ( ~ E / k T) r/= sin h ( r V o / 2 k T ) '

(6)

where A is a constant, r is the shear stress, zXEis the activation energy for flow, and V0 is the flow volume. From the stresses for a 10% deviation from the low-stress viscosity, the flow volumes are estimated as: for rmax, V0 ~ 1100 A 3,

for ray, V0 ~ 5400 A 3.

(7)

These values for the flow volumes are much larger than the atomic volume of selenium estimated from density measurements [15, 16] of about 30 A 3 atom-1. If the present values are taken at face value, the structural units involved in flow would be estimated as consisting of about 37 atoms from rmax and 180 atoms from ray. For comparison, data on the stress dependence of viscosity for a R b 2 0 - S i O 2 glass [ 11 ] indicate an effective flow volume - estimated by applying eq. (6) to the critical stress for non-Newtonian flow - which is larger by about a factor of 5 than the molecular volume. The overall form of the viscosity-temperature relation for selenium is similar to that observed for a variety of other liquids having a wide range of structural features and types of bonding. In detail, the data are not well represented by a single set of WLF parameters, a result in keeping with those obtained on a number of other liquids (see discussion and references in ref. [ 17] ). Because of the occurrence of crystallization in the intermediate range of temperature, any elaborate discussion of the data in terms of various models for viscous flow seems unwarranted. When the glass transition temperature is used as a corresponding states parameter to effect a comparison of flow characteristics, the flow behavior of selenium is found to lie between that of oxide liquids and that of simple organic and metal alloy liquids.

206

M. Cukierman, D.R. Uhlmann, Viscous flow behavior of seleniurn

In a number of previous investigations, data on viscous flow in the molten range have been used to infer information about the structure of amorphous selenium. In the light of the similarity in form between the flow behavior of selenium and that of liquids having quite different structures, as well as the lack of reliable models for relating structural features to flow characteristics, caution must be employed in attaching weight to these inferences. Estimates of ring and chain populations obtained from studies of dissolution behavior and Raman spectroscopy should be much more reliable; and their suggestion of a ring component of 3 0 - 4 0 % seems to provide the best available value for this quantity.

Acknowledgements Financial support for the present work was provided by the National Science Foundation. This support is gratefully acknowledged, as are stimulating discussions with Professor David Turnbull and Dr. K. Kim of Harvard University.

References [ 1] [2] [3] [4] [5] [61 [71 [8] [9] [101 [11] [ 12 ] [13] [ 14 ] [15] [16] [17]

H. Krebs and W. Morsch, Z. Anorg. Allg. Chem. 263 (1950) 305. T. Shirai, S. Hamada and K. Kobayaski, J. Chem. Soc. Japan 84 (1963) 968. R.C. Keezer and M.W. Barley, Mat. Res. Bull 2 (1967) 185. E. Jenckel, Kolloid Zeit. 84 (1938) 266. H.E. Hagy, J. Am. Ceram. Soc. 46 (1963) 93. 1.L. Hopkins, Phys. Chem. Glasses 4 (1963) 173. J.H. Li and D.R. Uhlmann, J. Non-Crystalline Solids I (1969) 339. M. Goldstein, in: Modern Aspects of the Vitreous State, Vol. 3, ed. J.D. Mackenzie (Pergamon Press, New York, 1967). S.H. Crandall and N.C. Dahl, An Introduction to the Mechanics of Solids (McGraw Hill, New York, 1959) ch. 7. S. Timoshenko and J.N. Goodier, Theory of Elasticity, Second Edition (McGraw Hill, New York, 1951) ch. 12. J.H. Li and D.R. Uhlmann, J. Non-Crystalline Solids 3 (1970) 127. F. Rodriguez, Principles of Polymer Systems (McGraw Hill, New York, 1970) p. 166, etc. M. Cukierman and D.R. Uhlmann, to be published. S. Glasstone, K.J. Laidler and H. Eyring, Theory of Rate Processes (MCGraw Hill, New York 1941). J.D. Taynai and M.A. Nicolet, J. Phys. Chem. Solids 31 (1970) 1651. G.C. Das, M.B. Bever, D.R. Uhlmann and S.C. Moss, J. Non-Crystalline Solids 7 (1970) 251. W.T. Laughlin and D.R. Uhlmann, J. Phys. Chem. 76 (1972) 2317.