Crushing strength of porous ice-mineral bodies-relevance for comets

Adv. Space Res. Vol. 14, No. 12, pp. (12)207-(12)216, 1994 Copyright © 1994 COSPAR Printed in Great Britain. All rights reserved. 0273-1177/94 $7.00 + 0.00

Pergamon

CRUSHING STRENGTH OF POROUS ICE-MINERAL BODIES-RELEVANCE FOR COMETS H. Thomas, L. Ratke and H. Kochan DLR, Institute for Space Simulation, P.O. Box 90 60 58, 5000 K~ln 90, Germany

ABSTRACT In laboratory investigations with fluffy, highly porous ice and ice-dust bodies a n e w mechanism could be identified which strengthens the porous bodies. The process takes place under isothermal conditions and leads to the formation of ice bridges between the ice {dust} particles. It is driven solely by the dependence of the partial pressure of water vapour on the curvature of the particles. This mechanism is generally called "sintering". A theory for the crushing strength of a porous ice and ice-dust agglomerate is developed which describes the experimental results on isothermal changes in strength due to sintering quantitatively well. The relevance for the evolution of comets is discussed. INTRODUCTION Since Whipple proposed his model of comets being ice-mineral conglomerates/1, 2 / t h e r e is ample evidence that the core of comets indeed consists of ice-mineral mixtures. They are not compact, dense bodies but are fluffy agglomerates of high porosity. Simulations of cometary analogous samples in the space simulation chamber within the "Kometensimulation"-project {KOSI} at DLR during the last years have definitively shown, that fluffy icedust agglomerates exhibit slow structural modifications during insolation with an artificial sun. The strength of the initially loosley packed ice-mineral bodies increased due to insolation and always a strong crust was formed at the surface of the samples. It was proven that the hardness of the crust and of the material below depends on the illumination profile {irradiation-intensity and -time} and on the composition of the sample {types of ice used water, methanol, carbondioxide - and mineral content}/3, 4/. The mechanisms leading to the observed strength increase and the structural modfications are partly understood, at least in qualitative t e r m s / 4 , 5/. The objectives of the investigations, which are reported here, were the identification of mechanisms leading to strength affecting structural modifications of loosely packed agglomerates of water-ice and water-ice dust mixtures and the development of a quantitative theory of the strength of these bodies. We report here only on strength changes occuring under isothermal conditions, Le. we present the first experimental evidence for sintering processes in such materials. In the 02)2o7

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H. Thomas et al.

subsequent sections we first outline the theory of sintering and present the experimental verification of this process for ice particles. Then we describe experiments on the strength evolution of ice-dust agglomerates under isothermal conditions and present a brief sketch of a strength theory developed for these materials and a comparison with e x perimental results. In the last section we present some consideration concerning the relevance of sintering for the evolution of comets. THE PROCESS OF SINTERING Generally the t e r m "Sintering" is used to describe processes which transform a powder particle agglomerate with a large amount of specific surface area into a solid body which has a minimum in specific surface area (namely only the external boundary) without reaching the melting point of the material used. The driving force for all sinter processes is the reduction of the total surface free energy. Compare a loosely packed agglomerate of particles of radius r. Then its total surface free e n e r g y is: AG agglomerate = 4nr: N ~ , w h e r e N is the number of particles and y the specific surface free energy. Compaction of such an agglomerate into a solid body without any internal surface yields AG agglomerate 0. To understand the process of sintering in more detail and to derive the kine~'~cs of this process we look for local gradients in the free e n r e g y of a particle agglomerate. Consider the Gibb "s free e n e r g y of an ice-particle agglomerate, which can be written as: G = G (T,p} + E ~'i Ai {1} G{T,p) is the Gibbs free e n e r g y of massive ice, T the absolute temperature and p the partial pressure, and Yi is the surface tension of the i-th particle and A i its surface area. The chemical potential of a single particle with radius r which is in thermodynamic equilibrium with its vapor depends linearly according to equation (1} on the particle curvature: ~-

Uo

-

dG dV

o

= "c o

K

(2)

Here ~0 is the chemical potential above a plane surface, -¢ the surface free e n e r g y and the curvature K = 1/r 1 + 1/r 2, w h e r e r 1 and r 2 are the two principle radii of curvature (which is not necessarily a sphere). ~ is the molar volume. Hence it follows from this equation, that the vapor pressure p above or the concentration of vacancies, c, below a curved surface dependends on the curvature K. For the vapor pressure one easily derives the Kelvin equation: p(K,T} = poexp

~

po(l + ~ )

(3)

and for the concentration of vacancies the Gibbs-Thomson-Freundlich equation: d K , T) = c o e x p ( - ~ T K ) a

Co(1

-

"(¢IK RT ) '

(4)

Here Po and c o are the vapor pressure above respectively the concentration of vacancies a b o v e / b e l o w a plane surface and R is the universal gas constant. These equations state that the partial pressure above a curved surface is larger than that above a plane one and the concentration of vacancies below a curved surface is higher than that below a plane one. If there are surfaces of opposite curvature near each other, a gradient in vapour pressure exists leading to a mass transport from the convex to the concave surfaces {for vacancies it is vice versa}.

Porouslce-MinerdBodies

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For a theoretical description of the morphological changes by sintering often the idealized model of two spheres is used, depicted in Figure I. If two identical particles touch each other, they will develop a small neck area by spontaneous adhesion /6/. The binding energy depends on the loss of surface due to the contact,

tF Fig. I.The two-spheres model with differenttransport mechanisms {I. sublimation and recondensation, 2. surface diffusion, 3. grainboundary diffusion and 4. latticediffusion}.

the formation of a grain boundary between the particles and the stored elastic energy in the particles. These necks form zones of concave curvature in contrast to the convex curvature of the particles. Therefore there is a gradient of the chemical potentialbetween the different areas of the two-particle arrangement leading to mass transport from the spherical surface to the neck area. Figure I also shows schematically differentpossible modes of mass transport. Using equations (3)and (4}and calculatingthe transport current from the convex to the concave areas, assuming also that the neck area can always be described by a torus whose cross section is a circle, one can calculate the growth of the neck radius x as a function of the particle radius, the transport mechanism (transport through the vapor phase, surface diffusion, diffusion through the volume of the spheres or along their c o m m o n grain boundary} and thus on temperature, surface energy, etc.. For the mechanisms shown in Figure 1 the calculation was done , e.g., Exner/7/. The different transport mechanisms generate different time-dependecies for the neck growth which are all of a power-law type, x ~ t I/n { n=3 for vapour transport, 4
JASR 14:12-0

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H. Thom~etal.

Following the t w o - s p h e r e model used above to calculate the sintering rate as a function of time, we performed experiments with two ice particles under a microscope in a cold lab at -20°C. Figures 2a, b, and c show the formation of a sinter neck between the ice particles directly. Figure 3 shows the measured dependence of the relative neck radius x / r on sintering time. We find a neck growth proportional to t 1/3, i.e., the sublimation and recondensation process predominate at a temperature of -20°C.

I

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Fig. Z Sintering of two ice particles at-20°C (Sintering times Fig. 2a: 3min; 21): lday; 2c: lweek}

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Fig. & Dependence of the relative neck radius x/r on sinlvring time. The spheres are measured values of sintering of two ice-particles at -20°C taken with an imaging analymng syslvn~ Tne fit of the data gives a dependence of x/r to t1/3.

Porous Ice-lVl/neralB o d i e s

(12)211

STRENGTH OF POROUS ICE-MINERAL BODIES - EXPERIMENTAL RESULTS In order to validate the model of sintering of ice particles and to extrapolate the experimental results from two spheres to an agglomerate, we performed experiments with porous ice bodies and ice-mineral bodies. Spherical particles were made by spraying water and water-mineral suspensions into liquid nitrogen. The mineral used was Olivine of a mean particle diameter of 2 gn~ The liquid nitrogen was evaporated at -20°C while the ice-mineral powder was stirred mechanically to avoid the formation of sinter necks prior to the experiments. The ice-mineral powder was filled into copper cans which were thermalized for a period of days to -20°C in the cold lab. The strength increase due to sintering of this isothermal powder was examined through measurements of the crushing strength at different times and compared to the initially loosely packed powder bed. The strength was measured with a spherical indentor which moved into the sample with a constant speed. The force needed to crack the sample locally was recorded as a function of penetration depth of the indentor into the sample. Figure 4 shows the strength profiles of a pure water ice sample after different times. After an initially linear increase in stress a maximum is reached after which the stress oscillates around a mean value in a random manner.

24 O-

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C 0 if)

c

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depth (cm) Fig. 4. Crushing ~ e n g i h of a porous icy body versus hhe penetration depth of the indentor afmr 3rain respectively 25h. Figure 5 shows the increase of the mean strength of an ice-mineral body with a mineral content of 3,4 Vol% as a function of sintering time.

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Fig. 5. Mean strength of a porous ice-mineral body with a mineral content of &4Vo~ as a function of ~ntering time. Figure 6 shows the strength versus the mineral content for different times {10min, 100min, 1000min}; there is a distinct effect of the mineral content on the strength of porous bodies. Approximately l0 volume percent olivine increases the strength by a factor of l0 compared to pure ice. i

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mi n e r a l c o n t e n t (Vol Z) Fig. 6. Mean crushing strength versus the mineral content for different times {lOmin, lOOmin, lO00min}.

To summarize, the results show an increase of strength of the porous ice-mineral bodies with sintering time and with increase of the mineral content.

Porous Ice-Mineral Bodies

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STRENGTH OF POROUS ICE-MINERAL BODIES - THEORY OF STRENGTH The strength of a single particle oHp depends on the particle diameter d=2 r according to the w e l l - k n o w n Hall-Petch relationsship/10/: drip = d0 +

k/-/-d-

(5)

.

% and k are material constants measured for ice down to -40 ° C / 1 1 / . In a two- particles configuration the strength is reduced at the neck, which acts like a notch. This can be taken into account by a so-called notch factor N(x,r), which is known from elasticity theory. If there are minerals embedded into the ice particles the minerals act as obstacles for a crack front to advance through the ice matrix; so that the strength of a particle increases with mineral content This effect has to be added to the strength of a pure ice particle (equation (6)). A theory of strength increase due to particles was developed by E v a n s / 1 2 / . The single-particle strength oHp has to be diminished in an agglomerate taking into account that the bridges (sinter necks) can only sustain part of the stress of a single particle. The m a x i m u m strength of an agglomerate is therefore multiplied with the relative neck area ~ x 2 / m -2. The strength of a porous ice-mineral body consisting of many particles sintered together can be calculated by the strength of a two- particle configuration multiplied with a factor containing the porosity ~) of the whole body/13/. Thus we arrive at an expression for the strength of porous ice-mineral bodies: o0+ k/4d N(x,r)

o =

1~60E.~1/2( V .~1/4 + k~) k]~-)

2

(x(t) / r )

3/2

(1 -

¢)

(6)

Here V is the volume content of mineral within the ice particles and E is Young "s modulus of water ice and a is the diameter of the indentor. subli m a t i o n / r e c o n d e n s a t i on ' """1

.-.

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10-2 10-I 100 101 102 103 104 105 106 10;' 108 ti m e (s) Fig. 7. Comparision of experimental and calculated crushing strengths of an icy body with a mineral content of &4VolZ. With this equation it is possible to calculate the strength of an icy body with known parameters (e.g. grain size distribution, mineral content, etc.) for different sintering stages (that

H. TnomasetaL

(12)214

means for differentratios of x/r) using the material constants given in the literature/14/. Figure 7 compares the experimental and calculated crushing strengths of an icy body with a mineral content of 3.4Vo17o. Since the particles in the icy body are not identical in size, but follow a certain size distribution due to the preparation mechanism, the calculated strength curves shown are those of the minimum and maximum value of the measured grain size distribution (dotted lines). The dots with error bars are the experimental values. The experimental results agree remarkably well with the theoretical prediction, even quantitatively. RELATIONS TO C O M E T S In the experiments described above w e have investigated the sintering behaviour of porous ice-mineral bodies simulating a cometary composition at -20°C. What happens at much lower temperatures, which are of interest for the origin and development of comets (T
lyear 1

10-1

lO00yem

lmioyems

E

I

surfacedif f ~ "

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~_

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10 6 10 7 10 8 10 9 1010 1011 1012 1013 1014 10 '15 1016

ti m e (s) Fig.& Neck growth of ice particles with radii of 0.1 and 1 Nn, respectively,calculated by surface diffusion for a temperature of 30K.

Pocous lee-Mineral Bodies

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Whether sintering takes place in astronomically relevant times does depend strongly on the surface diffusion constant Ds . Unfortunately there are no experimental values for Ds of water ice at very low temperatures. We have taken the following dependency given by Gjostein /15/ as a result of a scaling fit performed for a large variety of metals and ceramics: forT/Tm< 0 . 7 5 : Ds=l.410 - 2 exp( 54"3Tml RT i Here T m is the melting temperature. The apparent activation energy for surface diffusion, Qs=54.3-Tm=14.8 kJ/mole fits well to the known activation energy Qb for grain boundary diffusion in ice (Qb=38 kJ/mole/17/), considering that t h e n u m b e r of nearest neighbours for a diffusing molecule at a surface is approximately half that in a grain boundary. Figure 8 shows that particles of radii of 0.1gin form sinter neck radii of x=0.1r in a few thousend years. Consequentaly a strength increase of this material due to sintering takes also place at very low temperatures. The neck growth rate is negligible for particles r>lgm. The larger particles require a higher temperature (about 50K) to obtain a strength increase on timescales relevant for comets. From these considerations and the help of equation (6} one can calculate a mean strength of comets of about 10 kPa due to sintering alone. This value agrees with calculations made from observations of the sun grazing comets /16/. SUMMARY In laboratory experiments it could be shown that 1. ice and ice-dust particles form ice bridges at their points of contact under isothermal conidtions. At the temperature used (-20"C) the transport of water occurs by sublimation and recondensation. The process of particle neck formation was described within the framework of sintering theory, which is well known from powder metallurgy. 2. Experiments with ice and ice-dust agglomerates of high porosity were performed, which held the initially fluffy, porous powder beds at constant temperature. The strength of these materials was measured with an indentation technique. The experiments showed that the strength increased with increasing sintering time and increasing volume content of mineral. The experimental results could be described quantitatively with a newly developed theory for the strength of brittle, porous bodies. 3. The process of sintering of ice particles can also occur at very low temperatures via surface diffusion of water molecules. Within astronomically reasonable times comets could thus have reached strengthes of the order of a few 10 kPa. REFERENCES 1. F. Whipple, Astrophys. Journal, 111,375 (1950) 2. F. Whipple, Astrophys. Journal, 113, 464 (1951) 3. E. Grfin et al, Comets in the Post-Halley Era, Kluwer Academic Publishers, 1, 277-298 (1989) 4. H. Kochan, K. Roessler L. Ratke, M. Heyl, H. Hellmann, and G. Schwehm, Proc. Int. Workshop on Physics and Mechanics of Cometary Materials, MOnster FRG, ESA SP-302, 115-119 (1989} 5. L Ratke, H. Kochan and H. Thomas, ACM Proceedings (in press) (1992) 6. F.B. Swinkels, and M.F. Ashby, Acta Metall., 29, 259-281 (1981) 7. H.E. Exner, GebrOder Borntr/iger Berlin-Stuttgart (1978) 8. H. Thomas, diploma thesis, University of Cologne [1992}

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9. H. Thomas, L. Ratke and H. Kochan, Ann. Geoph. (to be published) 10. B.R. Lawn and T.R. Wilshaw Fracture of Brittle Solids, Cambridge University Press (1975) 11. E.M. Schulson, Journal de Physique, Colloque C1, 207-220 (1987) 12. A.G. Evans, Phil. Meg., 26, 1327 (1972) 13. L.J. Gibson and M.F. Ashby, Cellular Solids, Pergamon Press, Oxford (1988) 14. B. Michel, Ice Mechanics, Les Presses de l'Universite Laval, Quebec (1978) 15. N.A. Gjostein, Surfaces and Interfaces, Syracruse University Press, 271 (1966) 16. D. Mtihlmann, Kometen, Akademie Verlag Berlin, 47-51 (1990)