Frictional healing of quartz gouge under hydrothermal conditions: 2. Quantitative interpretation with a physical model



[1] The companion paper by Nakatani and Scholz [2004] shows that a hydrothermal frictional healing mechanism results from local solution transfer. Here we evaluate this mechanism with the model of Brechet and Estrin [1994], which assumes that the healing occurs by stress-driven asperity creep. The absence of a clear temperature dependence of the healing parameter b in the narrow tested range of 100–200°C is consistent with the model's prediction. The analysis also indicates that the mechanism involves a high stress assist parameter Ωσ = 200 kJ/mol, which is consistent with the contact stress being the indentation hardness, σ ∼ 10 GPa, and the activation volume Ω being the molar volume, both of which are reasonable. For this to be consistent with the observed temperature enhanced kinetics of healing also requires that the activation energy exceed 200 kJ/mol. This is much higher than the 20–70 kJ/mol known for low contact stress pressure solution. The analysis of several previously published studies of hydrothermal healing of hard silicates yielded the same results. Hence, if the underlying process is stress driven, it must have a different mechanism at high stress than at low stress. Alternatively, a solution transfer mechanism driven by something other than stress could be the underlying mechanism, but this is inconsistent with other aspects of our experimental results. On the other hand, the same analysis of phenomena that are independently inferred to proceed under relatively low contact stress yielded the parameter values consistent with low-stress pressure solution.

1. Introduction

[2] Time-dependent healing of frictional strength [e.g., Dieterich, 1972] is one of the key mechanisms that govern earthquake cycles. It is also a cornerstone of rate and state friction laws, which have been successfully applied to explain a wide spectrum of earthquake phenomena [e.g., Scholz, 1998]. In the laboratory, many classes of materials [e.g., Rabinowicz, 1965] including rocks [e.g., Dieterich, 1972; Beeler et al., 1994; Nakatani and Mochizuki, 1996] show that frictional strength μ increases linearly with the logarithm of quasi-stationary contact time t:

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Here, μ0 is the frictional strength at the beginning of stationary contact and in slide-hold-slide tests may be evaluated as the sliding friction in the slide period prior to the hold. In addition to the healing phenomena due to solid-state deformation of contact asperities usually observed in nominally dry tests at room temperature, equation (1) can also fit some of the healing phenomena that relies on solution transfer processes [e.g., Fredrich and Evans, 1992; Karner et al., 1997; Nakatani and Scholz, 2004].

[3] Although qualitative understanding of frictional healing as an increase of real contact area has been there all along [e.g., Dieterich, 1972; Scholz and Engelder, 1976], there had been no explanation for the particular functional form (logarithmic) of equation (1) until Brechet and Estrin [1994] proposed a general physical model that can produce it. Importantly, the Brechet and Estrin (BE) model provides expressions for the observational healing parameters (b and tc in equation (1)) in terms of the properties of the underlying mechanism: assumed to be asperity creep driven by the contact normal stress. The effects of temperature and contact stress are included by assuming that the underlying creep mechanism is a stress-aided thermally activated process.

[4] In this paper, the values of b and tc of the solution transfer healing phenomenon reported by Nakatani and Scholz [2004, hereinafter referred to as paper 1] will be examined in terms of the BE model. The suite of experiments in paper 1 established that the healing results from local solution transfer; its specific mechanism being yet to be determined. Although the BE model is applicable to any stress-driven healing process, we will primarily examine the quantitative agreements of the data with that model in which pressure solution is assumed as the particular underlying mechanism. We do this because the properties of pressure solution are relatively well known and because the mechanism is known to be active under the moderate hydrothermal conditions studied in paper 1 [e.g., Rutter, 1983; Tada and Siever, 1989; Dewers and Hajash, 1995]. The same analysis, for comparison, will be applied to other well-documented time-dependent growth phenomena that appear to also result from solution transfer processes.

[5] We note that the BE model, which assumes stress-driven creep as the underlying mechanism, is not a unique model for time-dependent healing phenomena. Mechanisms of local solution transfer driven by something other than stress may also cause healing phenomena [e.g., Smith and Evans, 1984; Hickman and Evans, 1992; Bos and Spiers, 2002], to which BE theory is not applicable. Such possibilities will be separately discussed.

2. Model

[6] Here, we briefly summarize the Brechet and Estrin [1994] model for logarithmic time-dependent healing. The model is based on the classic Bowden and Tabor [1964] adhesive theory of friction in which the macroscopic frictional strength is assumed to be in direct proportion to the real contact area. Healing is hence assumed to result from the time-dependent increase of real contact area, as is widely accepted [e.g., Scholz, 1990]. For healing due to solid-state deformation, this has been confirmed directly for many materials including hard silicates [e.g., Dieterich and Kilgore, 1994]. In applying the model to solution transfer healing mechanisms, we simply assume that the proportionality between the real contact area and frictional strength holds independently of the mechanism that brings about the change of real contact area.

[7] The essence of the BE model is to describe a concrete way in which the increase of real contact area occurs in a specific form such as equation (1) and thereby to give physical meaning to the observational parameters. The increase of real contact area is modeled as asperity creep at the contacts driven by the local normal contact stress (σ). Under a constant applied load, σ is inversely proportional to the real contact area. Hence healing slows down as it proceeds. Using direct proportionality between the real contact area and the frictional strength (i.e., adhesive friction model), normal contact stress can be expressed in terms of the amount of healing Δμ ≡ μ − μ0

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where σ0 ≡ σ(t = 0). Small healing (Δμ ≪ μ0) was assumed in deriving equation (2). The indentation hardness provides an order of magnitude estimate for σ0, independent of the externally applied nominal normal stress σn [Bowden and Tabor, 1964; Greenwood and Williamson, 1966; Dieterich and Kilgore, 1994]. Although applied shear load also may have some effects on asperity creep [e.g., Nakatani and Mochizuki, 1996; Berthoud et al., 1999], we do not discuss it because the experiments we analyze in this paper did not examine the systematic effects of shear load.

[8] The particular logarithmic form of slowing down of healing requires an exponential stress dependence of the rate of the underlying creep process.

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Here, equation image is the strain rate of the creep normal to the contact, equation image is a preexponential factor, and S is a constant denoting the magnitude of characteristic stress change that results in an e-folding change of strain rate. Different deformation mechanisms observed at high driving stress, such as dislocation glide, stress corrosion, and pressure solution, show such an exponential stress dependence. Further, equation (3) follows a theoretical rate equation of thermally activated rheology. Details will be discussed later in this section.

[9] By integrating equation (3), coupled with equation (2) and the relation between increments of normal strain and contact area constrained by volume conservation, one obtains

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which is equation (1) when

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Equation (5), when rearranged as

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implies that b is the magnitude of healing that brings about an e-folding decrease of healing rate through a reduction of contact stress by S. (Compare equation (7) with the second term of equation (2) with b inserted to Δμ.) So, b is proportional to S. Notice also that b is inversely proportional to σ0. This is because, as seen from equation (2), the higher the contact stress level σ0 is, the smaller the fractional increase of contact area (b0) is necessary for the reduction of contact stress by S. On the other hand, equation (6) can be rearranged as

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The right-hand side of equation (8) is the normal creep strain rate at t = 0, so (b0)/tc may be regarded as a sort of reaction rate constant after correction for the transience of the healing phenomenon due to the changing driving stress. The physical meaning of this relation will become clearer when the parameters in the constitutive relation (3) are expanded in terms of thermal activation. Berthoud et al. [1999] confirmed that relations (5) and (6) roughly hold for logarithmic healing of polymer glass surfaces at various temperatures, where the constitutive parameters in equation (3) and the yield strength that approximates σ0 were known from independent uniaxial testing of the bulk material.

[10] The constitutive relation given in equation (3) has its origin in an Arrhenius-type relation, and may be further expanded in the form of the Eyring equation:

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The preexponential factor A, the activation energy Q, and the activation volume Ω, are constants specific to the process. In general, Ω should be regarded as a phenomenological parameter describing the stress sensitivity of the process rate rather than some physical volume. T is the absolute temperature and R is the gas constant. This type of exponential dependence on driving stress is generally seen for thermally activated mechanisms at high stress such as dislocation glide [e.g., Frost and Ashby, 1982] and stress corrosion cracking [Scholz, 1972]. Pressure solution, a natural candidate for the solution transfer healing reported in paper 1, also has an exponential stress dependence when the driving stress is high [e.g., Rutter, 1976; Dewers and Hajash, 1995]. Although a linear stress dependence is often used for the rate law of pressure solution, this is actually the small stress approximation of the exponential dependence that comes directly from the stress dependence of chemical potential, the driving force of pressure solution [e.g., Patterson, 1973; Shimizu, 1992, 1995]. In this case, Ω will be the molar volume of the solid. On the other hand, there is some uncertainty in activation energy Q for pressure solution, because different subprocesses can be the rate-limiting stage depending on the condition. Theoretical considerations [Rutter, 1976; Raj, 1982; Shimizu, 1995] show that the activation energy for pressure solution should be in the range of the activation energies associated with the dissolution-precipitation kinetics. Depending on the rate-limiting process the activation energy for dissolution, precipitation, or the equilibrium constants is relevant. These values for quartz are 70, 50, and 20 kJ/mol, respectively [Rimstidt and Barnes, 1980]. In the case of diffusion control, the activation energy of the diffusion coefficient (15 kJ/mol [Nakashima, 1995]) must be added to the activation energy (20 kJ/mol) of the equilibrium constant. In any case, the activation energy possible for the usual pressure solution is thought to be less than ∼70 kJ/mol. Consistently, Dewers and Hajash [1995] obtained an activation energy Q = 73 kJ/mol for their compaction experiments with quartz sand at 100–200°C, the range same as the experiments of paper 1. Unfortunately, most other experiments obtained only activation enthalpy (HQ − Ωσ0).

[11] With (9), the two constants in equation (3) are now expanded to

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Accordingly, the interpretations of the observational healing parameters (equations (5) and (8)) are now

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As Brechet and Estrin [1994] pointed out, equation (12) suggests that b is in direct proportion to absolute temperature if the level of contact stress dominated by the hardness remains roughly constant. This latter condition may hold up to considerable temperatures for hard silicates [e.g., Evans, 1984], whereas in the case of the polymer glass experiments of Berthoud et al. [1999] mentioned earlier, the temperature dependence of b was dominated by a large reduction of yield strength at temperatures near the glass transition and hence bT was not observed. On the other hand, equation (13) suggests that an Arrhenius-type temperature dependence, diagnostic of thermally activated processes, is expected of (b0)/tc. If the temperature dependence of b is only proportional to T as expected from equation (12), equation (13) suggests that the Arrhenius-type temperature effect on kinetics would be primarily manifested in 1/tc with an apparent activation enthalpy HQ − Ωσ0, as observed in paper 1.

3. Comparison of Experiments With BE Theory: Is it Pressure Solution?

[12] In this section, we apply the BE theory to the results of several experimental studies, listed in Table 1, where log time growth was observed. Because our primary motivation is to study the solution transfer healing mechanism reported in paper 1, we focus on experiments where the time-dependent phenomenon is thought to result from local solution transfer mechanisms. As mentioned earlier, we will mainly discuss the data in terms of consistency with the values expected from the BE model with pressure solution as the particular underlying mechanism, i.e., assuming that Ω is given by the molar volume of the material.

Table 1. Comparison of Various Time-Dependent Growth Phenomena Due to Local Solution Transfer
PhenomenonT, °Cb0tc, sΩσ0 = RT/(b0), kJ/molExpected σ0,a GPaH, kJ/molQ, kJ/mol
  • a

    Hardness of the material at the temperature for phenomena 1–5. From Pe dependence of the compaction rate [Dewers and Hajash, 1995] for phenomenon 6.

1, Frictional healing of quartz gouge [paper 1]100–2000.015–0.020103–1052001058 
2, Frictional healing of halite gouge [Bos and Spiers, 2000, 2001]250.079–0.14110233–181  
3, Frictional healing of quartz gouge [Fredrich and Evans, 1992; Karner et al., 1997]6360.013–0.036 566–1267.5119 
4, Long-dc frictional evolution effect after velocity change of quartz gouge [paper 1]2000.020 20010  
5, Long-dc frictional evolution effect after velocity change of granite gouge [Blanpied et al., 1998]250–3500.013–0.041 300–1008–10  
6, Compaction of quartz sand aggregate [Dewers and Hajash, 1995]150–2000.460–16.3106–107<10<0.5 73

3.1. Frictional Healing of Quartz Gouge at 100–200°C and Associated Long-dc Evolution Effect

[13] In paper 1, we identified a healing mechanism resulting from a local solution transfer process. The values of b obtained from the experiments at three temperatures (100, 150, 200°C) are plotted against absolute temperature in Figure 1. Error bars show one-sigma uncertainty in fitting the results with equation (1). According to equation (12), b is expected to be directly proportional to absolute temperature because μ0 did not significantly depend on temperature in those experiments and because σ0, which is dictated by the indentation hardness of quartz, is unlikely to vary significantly in this temperature range [Evans, 1984]. Within the limits of the large error bars, we can say that the data are consistent with bT.

Figure 1.

Observational parameter b from the healing tests of paper 1, plotted against absolute temperature. The straight line indicates a model prediction (equation (12)) with Ωσ0 = 200 kJ/mol.

[14] Further insights can be obtained from the absolute level of b. Using the observed value of b0 = 0.013/0.68 = 0.02 at 200°C, we find that Ωσ0 = 200 kJ/mol from equation (12). This corresponds to the line drawn in Figure 1. If pressure solution is the underlying mechanism, Ω will be the molar volume of solid quartz [e.g., Patterson, 1973; Shimizu, 1992]. Hence, taking Ω = 22.7 mL/mol, we obtain σ0 = 8.8 GPa, a value close to the indentation hardness of quartz. This agreement, however, does not demonstrate that the mechanism is indeed pressure solution, because a value of Ω in this range is likely for a variety of different mechanisms. Conversely, as long as the underlying mechanism is driven by contact stress, we may say that the large value (200 kJ/mol) of the product Ωσ0 shows that the driving stress for this healing mechanism is very high ∼10 GPa, regardless of the specific mechanism. Such a high stress level at frictional contacts of hard silicates is consistent with contact area measurements [Teufel and Logan, 1978; Dieterich and Kilgore, 1994] and a similar analysis of frictional shear creep [Nakatani, 2001].

[15] Next, we analyze the temperature dependence of tc, which decreased by orders of magnitude with temperature. Figure 2 shows an Arrhenius plot of (b0)/tc, the slope of which should yield the apparent activation enthalpy H = Q − Ωσ0 (equation (13)). Error bars in Figure 2 reflect one sigma uncertainty in determining b and tc at each temperature. Figure 2 indicates H = 58 kJ/mol. Because b and μ0 did not change much in this temperature range while tc changed by a factor of 45, this apparent activation enthalpy came almost entirely from the temperature dependence of tc(Figure 7 of paper 1). In other words, the apparent activation enthalpy arises primarily from the simple trading off of temperature with time following the Arrhenius relationship, as seen from equation (1), in which tc is the scaling factor for time.

Figure 2.

Arrhenius plot of (b/Ωσ0)/tc from the healing tests of paper 1.

[16] Using the values Ωσ0 = 200 kJ/mol and H = 58 kJ/mol, the activation energy Q becomes 258 kJ/mol. This is much higher than the activation energy known of pressure solution (Q ≤ ∼70 kJ/mol, see section 2). Actually the “stress assist” Ωσ0 (=200 kJ/mol) alone far exceeds the expected activation energy for pressure solution of quartz. If pressure solution with such a small Q were the underlying mechanism, the healing kinetics would have a negative dependence on temperature (equations (9) and (13)), which is not the case. Therefore we conclude that it is difficult to attribute the solution transfer healing of paper 1 to pressure solution with its normally small Q. We will discuss alternative underlying mechanisms in section 4.

[17] We emphasize that the implication of the large Ωσ0 mentioned above should not be ignored because it was directly obtained from the observational fact that the decrease of driving stress by only 2% (=b0 = 0.013/0.68) led to the slowing down of the process by a factor of e. The choice of the particular form (exponential) of stress dependence is constrained by the fact that the healing was logarithmic with time. That is, the large Ωσ0 value is firm as long as the assumption that the healing was driven by the contact stress is correct, irrespective of the specific mechanism.

[18] Before proceeding further to the same analysis of other experiments, we note a few possible problems in the data from paper 1. First, the silica concentration was confirmed to stay at the equilibrium level only for experiments at 200°C. Hence healing data at lower temperatures may be affected by net dissolution or precipitation as well as by local solution transfer. However, as discussed in paper 1, we think that the effects are minor. Second, as pointed out by Nakatani [2001], cutoff time observed in slide-hold-slide tests can sometimes be the effective contact time at the beginning of the hold period, which is inversely proportional to the slip velocity [Dieterich, 1978; Dieterich and Kilgore, 1994; Marone, 1998] in the preceding slide period, rather than the temperature-dependent “intrinsic” cutoff time predicted by the BE model. We, however, expect that the tc observed in paper 1 is the intrinsic cutoff time because the slip velocity of 13 μm/s employed in the slide-hold-slide tests was higher than the upper cutoff velocity (1.3 μm/s) of the evolution effect in velocity step tests at 200°C (paper 1). In such a condition, the effective contact time set in the preceding slide period is less than the intrinsic cutoff time, and hence the hold period should start with the minimal initial contact area independent of the slip velocity in the preceding slide, allowing us to observe intrinsic cutoff time in slide-hold-slide tests. At lower temperatures, the cutoff velocity should be further smaller because of slower kinetics, giving more safety margin. This point could be more directly checked by further slide-hold-slide tests employing different slip velocities. (Intrinsic cutoff time should be independent of the slip velocity of the preceding slide, whereas the apparent cutoff time should be inversely proportional to the slip velocity of the preceding slide.)

3.2. Frictional Healing of Halite Gouge at Room Temperature

[19] Bos and Spiers [2000, 2002] studied healing of halite gouge in slide-hold-slide tests in saturated brine at room temperature. This healing showed a logarithmic dependence on hold time, with a large value of b of 0.118 or 0.064, depending on the interpretation of the nature of dilation upon reloading [Marone et al., 1990; Beeler and Tullis, 1997; Bos and Spiers, 2000, 2002]. According to equation (5) or (12), their large b can be understood as resulting from the much lower level of contact stress (σ0) expected for the softer material (the indentation hardness of halite is about 1/10 of that of quartz.) The stress assist Ωσ0 is calculated to be 18–33 kJ/mol, using the above mentioned value of b, their representative value of μ0 ∼ 0.84, and their experimental temperature of 25°C. This agrees well with the product of molar volume of halite (27 mL/mol) and its indentation hardness of ∼0.9 GPa inferred from its Mohs number [Bowden and Tabor, 1964]. Furthermore, the stress assist is smaller than the activation energy of Q ∼ 50 kJ/mol for NaCl dissolution at zero effective stress [Olander et al., 1982]. Hence there is no problem in attributing the Bos and Spiers' water-assisted healing to the usual pressure solution process of halite. Even though the experiment was at room temperature, Bos and Spiers [2000, 2002] could observe the healing with fairly short hold times, consistent with the much faster dissolution kinetics of halite compared with quartz. From Bos and Spiers [2000, Figure 5], tc seems to be ∼100 s. The strengthening was erased by subsequent sliding with a dc of a few hundred microns, similar to our results.

3.3. Frictional Healing of Quartz Gouge at 636°C

[20] For quartz powder saturated with H2O, strong healing that is logarithmic with hold time has been found at 636°C healing temperature [Fredrich and Evans, 1992; Karner et al., 1997], while their control experiment without pore water did not show any strengthening. The pore fluid in those experiments is expected to be silica saturated, judging from their temperature, time duration, and surface area/water ratio. In these experiments, the hold was done isotropically (i.e., zero shear stress) under a 175 MPa effective confining pressure. Frictional strength was measured by imposing sliding at a lower temperature (270°C). This strengthening was followed by slip weakening, as in our results described in paper 1. The slip weakening dc could not be obtained due to insufficient machine stiffness. Their value of b is 0.045 if the healing is measured from the absolute value of peak friction, but is 0.010 if the healing is measured from the difference of peak friction from the subsequent dynamic friction level (because the dynamic friction level also increased with hold time). Extrapolation of the b from our healing results at 200°C to 636°C using equation (12) yields b = 0.027, which is in the range of b for their results. If we assume that their healing was driven by stress, this suggests a large stress assist (Ωσ0) similar to ours and hence driven by a similarly very high contact stress. Their value of b, taken together with their typical μ0 of ∼0.75, yields Ωσ0 = 126–566 kJ/mol, which is considerably higher, even without the activation enthalpy term included, than the activation energy for pressure solution of quartz. Therefore their healing, as ours, is difficult to attribute to pressure solution with a normally small activation energy. From the temperature (450–800°C) dependence of the peak strength for the same hold time (1 hour), they found that a ∼130°C rise in temperature corresponds to a decade increase in hold time. This translates to an apparent activation enthalpy of H = 119 kJ/mol, indicating an even higher activation energy (Q = H + Ωσ0) for their healing. Although this is higher than our value (H = 58 kJ/mol), it does not necessarily mean a difference in the underlying mechanism. Because of temperature softening of quartz [Evans, 1984] the contact stress level (σ0) in Fredrich and Evans' and Karner et al's experiments at 636°C is inferred to be lower by ∼25% than in our paper 1 experiments at 200°C. Hence stress assist (Ωσ0), which was 200 kJ/mol in our paper 1 experiments at 200°C, is inferred to be lower by ∼0.25 × 200 kJ/mol = 50 kJ/mol in Fredrich and Evans' and Karner et al.'s experiments at 636°C. This in turn makes the activation enthalpy (H = Q − Ωσ0) in Fredrich and Evans' and Karner et al.'s experiments higher by 50 kJ/mol than in our paper 1 experiments at 200°C, largely accounting for the observed difference in H (119 kJ/mol versus 58 kJ/mol), without assuming a change in Q.

[21] Thus, in terms of BE analysis, there is a good deal of coherence between our healing at 100–200°C and their healing at 636°C. However, we hesitate to conclude that the mechanisms are the same because the extent of sample induration was drastically different between these two experiments. Microscopic observation [Fredrich and Evans, 1992] showed their postexperimental samples were strongly indurated, which presumably is the cause of their healing. On the other hand, we did not find any significant signs of induration in our postexperimental samples as reported in paper 1. It seems difficult to ascribe this strong contrast to the difference in the extent of healing (10% strengthening in our case, 35% strengthening in their case).

3.4. Long-dc Evolution Effects in Velocity Step Tests

[22] In these experiments [paper 1, Blanpied et al., 1998], both with a layer of simulated gouge of hard silicates, a prominent evolution effect over a long dc (typically ∼500 μm) was observed, only under hydrothermal conditions, which seems to be a manifestation of solution transfer healing mechanisms, as discussed in paper 1. The value of b, obtained as −Δμ/(ln(Vnew/Vold), where Δμ is the magnitude of slip-dependent change of frictional coefficient following the velocity change, was 0.013 at 200°C in the quartz gouge experiments [paper 1] and was 0.009–0.029 at 250–350°C in the granite gouge experiments [Blanpied et al., 1998]. In the latter experiments, an evolution effect over a short dc (∼10 μm) was also observed which is presumably related to conventional Dieterich-type healing. The b we cited above is “b2” in Blanpied et al.'s [1998] Table 2 (note their typographical error) obtained from the Δμ associated with the long-dc evolution. Although b is obtained in the latter experiments at multiple temperatures, data scatter was considerable. The range of b cited above seems to be more like data scatter than a systematic change with temperature (Figure 3). Hence we interpret this result in the same way as we did on the temperature dependence of b from our healing tests; b did not vary very much with temperature, consistent with the theory (equation (12)). Above 400°C, their b2 turned negative and became more negative with increasing temperature up to their highest temperature (600°C). This behavior is totally incomprehensible in the context of BE model. As discussed in paper 1, this may reflect a change of sliding mechanism rather than the healing mechanism itself.

Figure 3.

Observational parameter b obtained from long-dc evolution effects in velocity stepping tests on quartz gouge [paper 1] and granite gouge [Blanpied et al., 1998] plotted against the absolute temperature. For comparison, b from the healing tests of paper 1 is also shown. The straight line indicates a model prediction (equation (12)) with Ωσ0 = 220 kJ/mol.

[23] The absolute level of b = 0.013 at 200°C observed in the velocity step tests of paper 1 is the same as that observed in the healing tests on the same system, so the value of stress assist is also the same, being Ωσ0 = 200 kJ/mol. Representative values from the Blanpied et al. velocity step tests, b = 0.015, μ0 = 0.7, T = 300°C, yield a stress assist value Ωσ0 = 220 kJ/mol that is similar to our value. This is shown as the line in Figure 3.

[24] At lower temperatures (100 and 150°C), however, Blanpied et al. [1998] consistently observed a smaller value of b = 0.004. As seen in Figure 3, this is significantly below the theoretically expected trend calibrated to the data at 250–350°C for the same experiments. These values are less than the half of the values of b observed from the healing tests of paper 1 at the same temperatures. At face value, this may suggest a higher stress assist value of Ωσ0 ∼ 400 kJ/mol and hence a different mechanism. However, we suspect that this may rather have resulted from the temperature-enhanced kinetics, which should result in cutoff time decreasing with temperature. The velocity range (0.1–1 μm/s) employed in these velocity step tests may have been, at these low temperatures, higher than the upper cutoff velocity for the evolution effect reflecting the solution transfer healing mechanism, resulting in Δμ < −bln(Vnew/Vold), which will lead to underestimation of b. The velocity step tests of paper 1 showed the cutoff velocity at 200°C was of order μm/s, which would be lower still at lower temperatures due to the marked increase of cutoff time for the healing mechanism both experimentally observed (paper 1) and theoretically expected (equation (13)).

3.5. Compaction of Quartz Sand Aggregate at 150–200°C

[25] Among previous compaction experiments of quartz sand aggregates, those of Dewers and Hajash [1995] are the easiest to compare with the healing experiments of paper 1. The temperatures were 200 and 150°C, and the sample was pure quartz sand. They held their sand packs for ∼1000 hours under isotropic effective confining pressures of Pe = 6.9–48.3 MPa, and observed that compaction evolving logarithmically with time. They obtained parameters corresponding to b and tc in equation (1). They analyzed their results by explicitly considering the effect of the decrease of contact stress as the compaction occurs, that is, using essentially the same model as the BE theory for logarithmic healing. The agreement of their model prediction with the observed time-dependent evolution of strain is excellent. Furthermore, their model assumed an inverse proportionality between total strain and contact stress under a given external loading condition. Therefore, in terms of their roles in the time-dependent evolution model, their strain and our frictional strength are equivalent. However, they did not discuss the value of stress assist Ωσ0, which we do below.

[26] They have two lines of observations from which we can estimate Ωσ0. The first is the value corresponding to b0 in our analysis, which is a measure of how much fractional decrease of contact stress is necessary to cause an e–fold decrease in process rate. Contrasting to the small value (b0 = 0.02) from the healing experiments of paper 1, they found much larger values of the corresponding parameter, bi, where ɛi is strain at t = 0. Their values of this parameter ranged from 0.46 to 16.3 for experiments at 150 and 200°C. This yields an estimated stress assist Ωσ0 of <7.7 kJ/mol. Using Ω = 22.7 mL/mol, this implies that the driving contact stress of the compaction was <330 MPa. This low contact stress is consistent with the fact that they observed no grain crushing in their experiments.

[27] Another line of evidence for small Ωσ0 comes from the dependence of compaction rate on the effective confining pressure (Pe). While the normal contact stress at a frictional interface is, to the first order, independent of the externally applied effective normal stress [Bowden and Tabor, 1964; Greenwood and Williamson, 1966], things are very different for contacts in sand compaction experiments. As indicated by the large flattened grain contacts in the postexperimental samples, the contacts are much larger than expected for elastic deformation (Hertzian contacts) or quasi-instantaneous plastic yielding at high contact stress [e.g., Dieterich and Kilgore, 1994; Masuda et al., 2000]. This seems to hold throughout the duration of the compaction measurement at a given target applied pressure because a considerable amount of time under hydrothermal conditions had elapsed before they set the pressure at the target level. For such well-mated grain contacts, the mechanism that keep the contact stress constant independent of the external normal load does not work. The experiments at 200°C showed that the compaction rate exponentially increases with Pe in the range 6.9–48.3 MPa. The characteristic pressure increment which produces an e-folding rate increase was found to be 20 MPa. On the other hand, the characteristic contact stress S of pressure solution is RT/Ω = 170 MPa at 200°C (equation (10)). This implies that the contact stress was magnified only by ∼9(= 170/20) times the nominal effective pressure. Hence, even for their highest Pe of 48.3 MPa, the contact stress was only 435 MPa, much less than the 8.5 GPa deduced for our frictional healing. With Ω = 22.7 mL/mol and σ = 435 MPa, the stress assist is 9.7 kJ/mol.

[28] As shown above, two lines of evidence show that Ωσ0 < 10 kJ/mol for the sand compaction experiments of Dewers and Hajash [1995]. This is much lower than the activation energy expected for pressure solution of quartz. Hence there is no problem in attributing their compaction to the pressure solution mechanism with a normally small Q. Dewers and Hajash obtained an activation energy Q = 73 kJ/mol from the temperature dependence of the “limiting strain rate,” defined as (b/tc)exp(−ɛi/b), where exp(−ɛi/b) is a correction factor for the difference in initial contact stress between runs, which varied considerably from test to test because their tests at different conditions were done sequentially without resetting the sample. This value is thought to be Q rather than H, because the second exponential factor in equation (13) is removed by that correction, although, in either case, Q and H would not differ much because Ωσ0 is small in this case.

[29] The values of tc observed in Dewers and Hajash's [1995] experiments were very long, ranging from several hundreds to several thousands of hours, which are two to three orders of magnitude larger than for our healing at the same temperature. This may reflect a difference in operative stress level between the two phenomena (e.g., equations (6) and (13)), as is also suggested by the large differences in b0 mentioned earlier.

4. Discussion

[30] In this section, we discuss what solution transfer mechanisms were underlying the time-dependent growth phenomena examined in the previous section (Table 1), using the results of BE analysis.

[31] The value of the observational parameter b showed reasonable agreement with the model prediction (equation (12)) in all six cases. The product Ωσ0 was directly obtained from the observable quantities as RT/(b0). Then, assuming a molar volume of the material for Ω, a driving stress levels σ0 were obtained. It has turned out that σ0 is consistent with the expected contact stress level for each case, which is the level of indentation hardness (∼10 GPa for hard silicates, ∼1 GPa for halite) for the five frictional phenomena (rows 1–5 in Table 1). For the quartz sand compaction experiments (row 6 in Table 1), the same analysis suggested σ0 < 0.5 GPa, consistent with an independent estimate of σ0 from the dependence of the compaction rate on the applied effective confining pressure in the same experiment. Such a low contact stress level, much lower than the material hardness usually assumed for frictional interface, was consistent with the large flattened grain contacts observed in these long-term compaction experiments. Thus, in terms of the stress sensitivity of the phenomena, manifested in b, all the examined cases, which differ in temperature, material, and the observed physical variable, are consistent with the underlying mechanisms being pressure solution, or some stress-driven mechanism with a Ω equal to the molar volume of the material. Such coherency is noteworthy; the large variation in b among the six cases is, in this theory, interpreted to reflect the orders of magnitude differences in the level of contact stress σ0. In addition, for the two cases where b was obtained at different temperatures (rows 1 and 5 in Table 1), the observed temperature dependence of b was small, as expected from equation (12).

[32] However, pressure solution is precluded as the underlying mechanism for the four cases (rows 1 and 3–5 in Table 1) including the healing of paper 1, all of which are associated with frictional healing on hard silicates. This is because the value of stress assist Ωσ0 obtained for these cases were much larger than the activation energy of pressure solution (Q < 70 kJ/mol). If that activation energy was assumed, it would predict a negative activation enthalpy (H = Q − Ωσ0), conflicting with the temperature-enhanced kinetics inferred directly or indirectly from the observations for each case. We discuss below what solution transfer mechanism is possible for these cases. For the other two cases (rows 2 and 6 in Table 1) where a relatively small Ωσ0 (<33 kJ/mol) was obtained, there is no problem in thinking that the underlying mechanism is pressure solution.

[33] Two possible interpretations may be proposed for the cases for which high Ωσ0 values were obtained in the above analysis. The first is that there may be a stress-enhanced dissolution mechanism that operates under high driving stresses with an activation energy much higher than the usual pressure solution process. Because of its high activation energy, such a mechanism, if it exists, would be hard to observe except at extreme conditions of very high stress (lowering H = Q − Ωσ0) or high temperature (strong temperature enhancement due to large H). Such conditions are usually avoided in studying pressure solution processes because other mechanisms of deformation can take place (e.g., stress corrosion cracking for high stress, crystal plasticity for high temperature). This high activation energy may imply that the dissolution occurs through a reaction path different from that of usual dissolution. The stress dependence of the process, however, would be similar to that of the usual pressure solution as the value of Ωσ0 obtained here coincides with the product of the molar volume of quartz and the expected contact stress at the frictional contacts of hard silicates. This does not necessarily mean that the stress dependence in this case comes from the same reason as that of the usual pressure solution because Ω for different mechanisms often has a similar value.

[34] The second possibility is a local solution transfer process driven by something other than stress. The only such mechanism known to date is one driven by the curvature dependence of the chemical potential. This process was observed for neck growth of halite contacts [Hickman and Evans, 1992] and the time-dependent disappearance of microcracks in quartz [Brantley et al., 1990; Smith and Evans, 1984]. Hickman and Evans [1992] specifically raised the possibility that neck growth causes the time-dependent healing of friction. Since the rate of that process does not depend on contact stress, the above argument against the usual pressure solution based on the large stress assist value is not applicable. Provided that there is no role of stress in this process, the activation energy is obtained directly by the trading off of temperature with time. The value obtained this way for the healing phenomena described in paper 1 was Q = 54 kJ/mol, which is in the range of the apparent activation energy for crack healing of quartz (35–80 kJ/mol [Brantley et al., 1990]). It is also in the range of activation energies related to the dissolution-precipitation kinetics of quartz. Therefore the healing phenomena of paper 1 are consistent with the curvature-driven solution transfer mechanism in terms of its activation energy. The value of Q = 119 kJ/mol, also obtained by a simple temperature-time trading for the healing of quartz gouge at higher temperatures [Karner et al., 1997], is somewhat too high, but it was a crude estimation.

[35] Although the rate of curvature-driven solution transfer does not depend on contact stress, it does slow down as the contact grows due to the blunting of the curvature at the edge of the contact. The radius of the halite contact at 50°C grew as t0.25t0.33 [Hickman and Evans, 1992]. The shortening of microcracks in quartz proceeded as t0.4 in the temperature range 400–600°C [Brantley et al., 1990]. In Figure 4, we show the fitting of the healing results from paper 1 with a power law. The results at 100°C is fit equally well with a logarithmic or power law. At 200°C, the data are better fit logarithmically. The power law fit overpredicts Δμ for shorter hold times (t < 4000 s) and predicts an upward curvature that is not seen in the data for longer hold times (t > 10,000 s). Although we need a wider range of hold times to confidently distinguish the logarithmic from the power law evolutions, the logarithmic function is preferred at present. This is not only because of the better fit, but also because the so obtained tc scaled with temperature following an Arrhenius relationship (Figure 7 in paper 1). We also point out that the exponent of the power law fits systematically decreased with temperature, whereas no such a tendency was observed in the crack healing of quartz [Brantley et al., 1990]. However, we note that the time dependence of curvature-driven solution transfer process can strongly depend on the contact geometry, which can differ significantly between the complex frictional interface of the present experiments and the simple geometry in the earlier studies [Brantley et al., 1990; Hickman and Evans, 1992].

Figure 4.

Fitting of the healing data from paper 1 with a power function of hold time (thick lines.) Gray scale denotes experimental temperature. Fitted curves with logarithmic function (from Figure 5 of paper 1) are also shown for comparison (thin lines).

[36] In appraising the curvature-driven solution transfer process as the cause of the healing phenomena reported in paper 1, we cannot say that our observations firmly rule out this possibility. One good way to discriminate between the curvature-driven and stress-driven solution transfer mechanisms would be to observe time-dependent compaction, as pointed out by Bos and Spiers, [2002]. The stress-driven mechanism (pressure solution) transfers the mass from the inside of the contact to the open face, leading to compaction, while the curvature-driven mechanism transfers the mass within the open face and is not accompanied by compaction [Hickman and Evans, 1992].

5. Summary and Conclusion

[37] Following Brechet and Estrin's [1994] physical model of frictional healing, which assumes that the underlying mechanism is driven by contact stress, we analyzed the healing of quartz gouge at 100–200°C described in paper 1. From the observed value of b, we obtained a high value of stress assist (Ωσ0 = 200 kJ/mol) for this process. Although this value agrees with the product of the contact stress (σ0) expected of frictional interface of hard silicates and the activation volume (Ω) for pressure solution, a natural candidate of the underlying solution transfer mechanism, it far exceeds the activation energy value expected of pressure solution (Q ≤ 70 kJ/mol). This implies that if pressure solution with this small activation energy were the underlying mechanism, the apparent activation enthalpy (H = Q − Ωσ0) would be negative, which conflicts with the observed temperature-enhanced kinetics manifested mainly by the temperature dependence of tc.

[38] Hence we infer two possibilities regarding the underlying mechanism. The first possibility is that some stress-driven dissolution mechanism with a high activation energy might operate under the very high stress conditions imposed. Although such a mechanism is not known at present, we think this to be a viable possibility because the value of b was, as mentioned above, consistent with an exponential stress dependence with Ω approximately molar volume, a commonly known range for such deformation mechanisms under high stress. Also, the direct proportionality of b to absolute temperature expected from BE model was consistent with the observation that b did not change significantly with the temperature in the narrow temperature range tested. The second possibility is that the underlying solution transfer mechanism was driven by something other than stress. In this case, the BE model becomes irrelevant. At present, one such mechanism, driven by the curvature dependence of the chemical potential, is known to exist. The activation energy known for such processes is consistent with our observed value of activation enthalpy (H ∼ 54 kJ/mol), which is the same as the activation energy (Q) because there is no stress assist in this case. Although the observed form of time-dependent healing seems to be different from those known for curvature-driven solution transfer processes, we cannot be certain of this with the currently available data.

[39] The same conclusion may also apply to the other three experimental studies of frictional healing of hard silicates (evolution effects in velocity stepping tests [paper 1, Blanpied et al., 1998], hold tests at >600°C [Fredrich and Evans, 1992; Karner et al., 1997]), all of which showed similarly small b values and hence yielded similarly high stress assist values, consistent with the high contact stress expected at the sites of these phenomena, whereas temperature-enhanced kinetics is inferred for all of them.

[40] On the other hand, BE analysis suggests a relatively small stress assist (Ωσ0) for the two other phenomena (frictional healing of halite gouge [Bos and Spiers, 2000, 2002] and long-term compaction of quartz sand [Dewers and Hajash, 1995]). This is again consistent with the contact stress levels for these phenomena inferred from independent evidence, which are relatively low (<1 GPa). Because the stress assist was smaller than the activation energy of pressure solution, there is no problem in interpreting them as resulting from usual pressure solution with a normally low activation energy, as those authors did.


[41] We are indebted to Jim Rice for his valuable suggestions. Reviews by Nick Beeler and an anonymous reviewer are appreciated. This study was supported by USGS grants 99-HQ GR0046 and 01HQGR0053. M. Nakatani was supported by JSPS fellowship for research abroad for a part of the period of this study. LDEO contribution 6567.