Lunar and Planetary Science XXXVIII (2007) 2186.pdf

DIFFUSION CREEP OF ICE: CONSTRAINTS FROM LABORATORY CREEP EXPERIMENTS. D.L. Goldsby, Department of Geological Sciences, Brown University, 324 Brook Street, Providence, RI 02912, [email protected]

Introduction: The importance of diffusion creep fusion dominates, and a 1/d 3 dependence when bound- of ice as a rate-controlling creep mechanism in con- ary diffusion dominates. A hallmark of diffusion creep vective processes within icy satellites has recently is the linear dependence of creep rate on stress. been emphasized [1]. Though likely an important de- The diffusion creep rate of ice: The diffusion formation mechanism for planets, diffusion creep of creep rate of ice can be estimated using the diffusion ice has never been identified in the laboratory [2]. creep equation with known or estimated equation pa- Here I review classical diffusion creep theory and use rameters. The principal unknown for ice is Db , which, it to estimate the diffusion creep rate of ice. These as for most materials, and, unlike Dv , has not been estimates are then compared with creep data from ex- measured. The preexponential factor for boundary periments on fine-grained ice samples to provide the diffusion Do,b is assumed equal to that measured for -4 2 best available constraints on the diffusion creep rate. volume diffusion, Do,v = 9.1H10 m /s [11]. The acti- Background: Flow of materials at low stresses and vation energy for grain boundary diffusion is equated relatively high temperatures is often controlled by with Q measured in creep experiments on fine-grained grain size-sensitive creep processes which involve ice deformed in the GBS-limited creep regime, Q=49 grain boundary sliding (GBS), whereby neighboring kJ/mol [2] (interpreted by [2] as the activation energy grains in the material are translated along their mutual for grain boundary diffusion, Qb). This value is also grain boundaries. To provide compatible deformation equal to the value of Q measured for grain growth of of polycrystalline materials, GBS must be accommo- ice [12], a process often limited by boundary diffusion. dated by dislocation motion or diffusional flow. Dis- Remaining parameter values are (after [2,11] and ref- -29 3 location-accommodated GBS (DAGBS) has been posi- erences cited therein): B=14, Ω=3.27H10 m, Qv tively identified and extensively studied in laboratory =59.4 kJ/mol, and δ=0.9H10-9 m. experiments on fine-grained ice samples (with grain Comparisons of the theoretical diffusion creep sizes of 3 to 200 μm) [2-5], and resulting flow laws rate with creep data for fine-grained ice: Diffusion have been successfully employed in modeling a host of creep and DAGBS are parallel kinetic processes, such icy planetary phenomena [6-9]. Diffusion- that the faster process dominates the creep rate. The accommodated GBS, i.e., diffusion creep, of ice, in DAGBS flow law has the form contrast, has never been observed in the laboratory [2]. . n 1 ⎛ Q ⎞ Diffusion creep is expected to dominate the rheology = A σε exp ⎜− ⎟ p RT of ice at stresses lower than for DAGBS creep [10], d ⎝ ⎠ and so is anticipated to compete with that mechanism where A is a constant, n the stress exponent, p the for dominance in low stress planetary environments. grain size exponent, and Q the activation energy for The strain rate due to diffusion creep is described creep. DAGBS occurs via mutually accommodating, classically by the diffusion creep equation [11] serial kinetic processes, dislocation slip (on the basal slip system) and GBS. When GBS limits the creep . B σΩ ⎡ δD ⎤ ε = D + b rate, the flow law parameters are: A=0.0039 MPa-1.8 2 ⎢ v d ⎥ kTd ⎣ ⎦ m1.4 s-1, n=1.8, p=1.4, and Q=49 kJ/mol below 255 K where B is a constant which depends on grain geome- and ~190 kJ/mol above 255 K [2]. When basal slip try, Ω molecular volume, σ stress, d grain size, k limits the rate, the flow law parameters are A=5.5e7 -2.4 -1 Boltzmann's constant, T temperature, Dv the volume MPa s , n=2.4, p=0, and Q=60 kJ/mol [2]. diffusion coefficient, δ grain boundary width, and Db Experiments on fine-grained water ice deformed in the grain boundary diffusion coefficient. Diffusion the GBS-limited (n=1.8) and basal slip-limited (n=2.4) coefficients are of the form D=Doexp(-Q/RT), where creep regimes can be used to constrain, if necessary, Do is a constant, Q the activation energy for diffusion, the theoretical diffusion creep rate. For example, if a and R the gas constant. Volume diffusion and bound- stress exponent characteristic of GBS-limited creep, ary diffusion contribute independently to the creep n=1.8, is observed in a creep test, the parallel kinetic rate, i.e., are parallel kinetic processes, so that the relationship between diffusion creep and GBS-limited overall rate is dominated by the faster process, yielding creep means that diffusion creep is slower than GBS- a 1/d 2 dependence of the creep rate when volume dif- limited creep at the conditions of the experiment. Sub- Lunar and Planetary Science XXXVIII (2007) 2186.pdf

stituting the grain size, temperature and stress from rate. A transition to diffusion creep is expected at such an experiment into the diffusion creep equation, a lower stresses, and is suggested by the data at 259 K. theoretical diffusion creep rate can be calculated and compared with the observed creep rate. If the theoreti- 10-3 cal rate exceeds the observed rate, then the theoretical n=2.4 266 K rate is faster than the true diffusion creep rate, placing 10-4 259 K an upper bound on the diffusion creep rate. 236 K As another example, an experiment on a sample -1 10-5 with d=3 μm deformed in the n=2.4 regime at T=208 9 K 25 -7 -1 p, K and σ=1.57 MPa yields a creep rate of 8.9H10 s ree -6 f. c 10 dif [2]. Using the diffusion creep equation and the above e 9 K lum 25 strain rate, s vo p, ree parameter values along with values of d, T and σ for . c -7 iff . d this experiment yields a theoretical diffusion creep rate 10 g.b -8 -1 basal slip- Transition to (dominated by volume diffusion) of 2.7H10 s . Thus, limited flow diffusion creep? the theoretical diffusion creep rate is less than the ob- 10-8 law 259 K served creep rate. Similar analyses of the entire creep 10-2 10-1 100 101 data set for fine-grained ice samples deformed in both stress, MPa

the n=1.8 and n=2.4 creep regimes [2-5] yield theoreti- Fig. 1 - Data from creep tests on ice alloyed with 12 vol.% Al2O3 cal diffusion creep rates that are less than observed (size 0.3 µm). Basal-slip limited flow (n=2.4 regime) law for pure ice plotted for T=259 K for comparison with data in blue. creep rates in those creep regimes, even when the grain If particulates have a modest or even negligible ef- boundary diffusion creep rate is generously enhanced fect on the diffusion creep rate of ice, then the data in by a factor of 1000 above 255 K to account for Fig. 1 further constrain the diffusion creep rate, forc- premelting and short circuit diffusion effects [2, 13]. ing minimum decreases in volume and grain boundary Thus, the above best estimate of the diffusion creep diffusion creep rates by factors of 140 and 25, respec- rate over a wide range of conditions is in full agree- tively (assuming an ice grain size of 2 μm). If diffu- ment with all available creep data for ice. sion creep is enhanced above 255 K due to premelting, Where is the diffusion creep regime?: Compari- then required decreases in the diffusion creep rate are son of diffusion creep rates with DAGBS creep rates even larger. Other experiments on samples with 3 for all possible combinations of grain size, temperature vol.% alumina deformed via GBS-limited creep yield and stress indicate that in order to access the diffusion creep rates nearly identical to those for otherwise iden- creep field in the laboratory, i.e., in order for diffusion tically fabricated particle-free samples, deformed at the creep to be faster than DAGBS, samples with d <3 µm same T and σ. This suggests a minor effect of parti- must be deformed at T >248 K. This is a practical im- cles on GBS in the GBS-limited creep regime and in possibility, due to rapid grain growth at these tempera- the diffusion creep regime. This conclusion depends tures, particularly for fine grain sizes. The diffusion critically, however, on the ice grain size of the particle- creep regime will therefore never be accessed in the bearing samples, measurements of which are in pro- laboratory for pure ice samples. gress. New creep experiments on two-phased ice: In an References: [1] McKinnon W.B. (2006) Icarus, effort to circumvent this problem and thereby discover 183, 435-450. [2] Goldsby D.L. and Kohlstedt D.L. the diffusion creep field, techniques were developed (2001) JGR, 106, 11017-11030. [3] Goldsby D.L. and for fabricating impure ice samples with d <3 μm which Kohlstedt D.L. (1997) Scr. Mater., 37, 1399-1406. [4] contain an intergranular dispersion of Al2O3. Alumina Goldsby D.L. (1997) Ph.D. thesis, Univ. Minn., 155 pins the ice grain boundaries, allowing the samples to pp. [5] Durham W.B. et al. (2001) JGR, 106, 11031- be deformed at near-melting point temperatures. The 11042. [6] Dombard A.J. and McKinnon W.B. (2001) tacit assumption in this approach is that intergranular Icarus, 154, 321-336. [7] Dombard A.J. and particles have minor effects on the diffusion creep rate. McKinnon W.B. (2000) GRL, 27, 3663-3667. [8] Nye Results of such tests are shown in Fig. 1. The data J.F. (2000) 46, 438-444. [9] Pappalardo R.T. and Barr toward the RHS of the figure have a slope n=2.4, con- A.C. (2004) GRL, 31, doi:10.1029/ 2003GL019202. sistent with basal-slip limited creep [2]. The data [10] Mukherjee A. K. (1971) Mat. Sci. Eng., 8, 83-89. merge into near-vertical trends with decreasing stress, [11] Frost H.J. and Ashby M.F. (1982) Deformation likely due to limitation of Frank-Read basal dislocation Mechanism Maps, 166 pp. [12] Duval P. et al. (1983) sources at low stresses, drastically reducing the creep J. Phys. Chem., 87, 4066-4074. [13] Dash J.G. et al. (1995) Rep. Prog. Phys., 58, 115-167.