Standard rate theory methods have recently been combined with experimental microstructures to successfully reproduce measured swelling behavior in ternary steels around 400 C. Fit parameters have reasonable values except possibly for the recombination radius, R{sub c}, which can be larger than expected. Numerical simulations of void nucleation and growth reveal the importance additional recombination processes at unstable clusters. Such extra recombination may reduce the range of possible values for R{sub c}. A modified rate theory is presented here that includes the effect of these undetectably small defect clusters. The fit values for R{sub c} are not appreciably altered, as the modification has little effect on the model behavior in the late steady state. It slightly improves the predictions for early transient times, when the sink strength of stable voids and dislocations is relatively small. Standard rate theory successfully explains steady swelling behavior in high purity stainless steel

Okita, T.

Surh, M. P.

Wolfer, W. G.

UNT Digital Library

UCRL-CONF-200741Modified Rate-Theory Predictionsin Comparison to MicrostructuralDataMichael P. Surh, Taira Okita, Wilhelm G. WolferNovember 4, 200311th International Conference on Fusion Reactor MaterialsKyoto, JapanDecember 7, 2003 through December 12, 2003Disclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  1#04A P.O.43Modified Rate-Theory Predictions in Comparison to Microstructural DataMichael P. Surh, T. Okita and W. G. WolferLawrence Livermore National Laboratory7000 East Ave. Livermore, CA, 94550, USACorresponding author: Michael P. SurhTEL: +1-925-422-2087FAX: +1-925-423-7040E-mail: surh1@llnl.gov2AbstractStandard rate theory methods have recently been combined withexperimental microstructures to successfully reproduce measured swellingbehavior in ternary steels around 400 C. Fit parameters have reasonable valuesexcept possibly for the recombination radius, Rc, which can be larger thanexpected. Numerical simulations of void nucleation and growth reveal theimportance additional recombination processes at unstable clusters. Such extrarecombination may reduce the range of possible values for Rc. A modified ratetheory is presented here that includes the effect of these undetectably smalldefect clusters.  The fit values for Rc are not appreciably altered, as themodification has little effect on the model behavior in the late steady state. Itslightly improves the predictions for early transient times, when the sink strengthof stable voids and dislocations is relatively small. Standard rate theorysuccessfully explains steady swelling behavior in high purity stainless steel.Keyword codes: N0100, R0200, S0500, S1400, T01003IntroductionStructural materials for blankets in fusion reactors will likely be exposed tohigh-energy neutron fluxes over extended periods.  Predictive models for theresulting void swelling, irradiation creep, and changes in mechanical propertiesmay be needed to help design these components in the absence of long-termexperimental data.  In the past, simple rate-theory approaches have beenemployed for modeling irradiation-induced microstructural evolution to tens ofdpa. The simplest such model includes only three point defect reactions, namelyvacancy-interstitial recombination, monomer absorption at sinks, and thermalemission of vacancies from sinks.  Despite its approximations, rate theory cansuccessfully reproduce observed void swelling for the pure austenitic ternaryalloys using realistic model parameters that are consistent with expectations fromtheory and experiment [1].Swelling in Fe-15Cr-16Ni irradiated in FFTF at 408 – 444˚C has recentlybeen measured at two successive, cumulative doses for a wide range of doserates [3,4]. The swelling across the second cycle is ~ 1%//dpa, indicating that theexperiments have exceeded the incubation dose [5]. These swellingmeasurements are augmented by detailed microstructural observations for eachdata point, including size distributions and densities of cavities, dislocation loops,and network dislocations (not shown). These complementary sets of4observations have been used to fit or constrain rate theory models for irradiationswelling, as well. Based on the range of parameters obtained, rate theory suggests that thevacancy-interstitial recombination process may be more efficient than would beexpected from microscopic arguments [2].  This seeming discrepancy might bereconciled in light of numerical simulations of void nucleation and growth.  Thesestudies reveal the influence of vacancy clusters that are undetectable to TEM, inaddition to the usual, visible cavities and dislocations. The modified theoryrequires the density and aggregate sink strength for these small clusters thatmust currently be obtained from detailed numerical simulations, since theclusters are not visible to TEM.  The unstable clusters slightly reduce thepredicted swelling rates. However, they have little effect on the simple rate theorymodel in the steady state regime. Their effects are most noticeable at low doses.Thus, standard rate theory should give reliable estimates of material parametersin the steady state, whether or not unstable defect clusters are included.For low temperatures, thermal emission may be neglected in the ratetheory, which is assumed here for 400-440 C in austenitic stainless steel.  Therate theory equations become:† dCidt= Pi -kDvCvDiCi - NuAuZiuuÂÊ Ë Á ˆ ¯ ˜ DiCi - NsAsZissÂÊ Ë Á ˆ ¯ ˜ DiCi - Nd Ad ZiddÂÊ Ë Á ˆ ¯ ˜ DiCidCvdt= Pv -kDvCvDiCi - NuAuZiuuÂÊ Ë Á ˆ ¯ ˜ DiCi-Dv NsAsZvs(Cv -sÂ Cveq(s))Ê Ë Á ˆ ¯ ˜ - Dv Nd Ad Zvd (Cv - Cveq(d ))dÂÊ Ë Á ˆ ¯ ˜ 5where subscripts v and i denote mobile vacancies and interstitials; D arediffusivities; and the vacancy interstitial annihilation rate constant is† k = 4pRc1Dv+1DiÊ Ë Á ˆ ¯ ˜ .Defect clusters are separated into stable (large) and unstable (small) voids anddislocations with size-dependent bias factors denoted Z.  The cross-section formonomer-cluster aggregation in the absence of long-range interactions is A.Dislocation parameters in the rate equations are denoted by superscripts d. Thecavity contribution is separated into stable and unstable populations, u and s,respectively. The experimentally measured cluster population gives Nh , thecavity density versus size, h, or radius, ah  for visible clusters only.Unstable clusters are included in the vacancy equation in anunconventional manner. It is important to recognize that the distribution ofunstable clusters is quasi-stationary.  That is, cluster size fluctuates very rapidlyfor the small clusters.  For all but the earliest part of the irradiation transient, thesub-critical population reaches a limit in which its aggregate volume neithergrows nor shrinks appreciably (compared to the rate of growth of the stablevoids).  These unstable clusters continue to absorb defect monomers, but inalmost exactly equal numbers of vacancies and interstitials, on average.  Thus,the net vacancy flux can be replaced with the expression for the interstitial flux.In effect, the unstable clusters catalyze an indirect recombination of vacanciesand interstitials, in an average sense.6A representative calculation for interstitial-biased sink strengths of stable,visible voids and unstable, undetectable clusters is shown in Fig. 1.  Thesimulation corresponds to high purity stainless steel at 420 C, with a dislocationcontent of 6¥1014 m-2. The elapsed time is not important; rather, what matters isthat the unstable sink strength is controlled by the instantaneous strength of thedislocations and stable voids. These unstable clusters are sustained by the non-equilibrium population of defect monomers, and are present in both theincubation and steady-state swelling regimes, depending on the sink strength ofother elements of the microstructure. For the experimental microstructures, theunstable cluster sink strength will lie between 6¥1014 m-2 and 9¥1014 m-2. Theunstable cluster value is set to 8¥1014 m-2, for all of the modified rate theorycalculations. This is reasonable for late times, but it may underestimate theireffect early in the irradiation, when the stable clusters have a relatively small sinkstrength.When the microstructure data obtained by the FFTF experiments are usedin the rate equations, some parameters remain undetermined.  These include thecascade efficiency x and defect production rate, P (equal to x times the dose ratein dpa);  the vacancy migration energy Ev  (controlling the diffusivity); theinterstitial bias factors for faulted loops Zfl and network dislocation Zd; and therecombination radius for point defects Rc. These parameters must fall within arange of acceptable values, based on microscopic arguments (1.2 eV £ Ev£  1.4eV [6,7], 1.1 £ Z £ 2.0 [8-10], 2.0 a0 £Rc £ 7.0a0 [2], and 0.1 £ x £ 0.2 [11]).  The7bias factors for voids are approximated as equal to 1.0, as are the dislocation-vacancy bias factors.At least in pure ternary austenitic alloys, the standard rate theory canexplain the swelling rate quite well. Table 1 summarizes the results of the modelbased on a single, typical fit. Generally, variation in one parameter requireschanges in others. The fit procedure thus obtains multiple sets of theoreticallyplausible material parameters.  In some cases, the vacancy-interstitialrecombination radius is larger than expected. Values as large as 7.0 a0 arepossible, much larger than the estimation of 2.0 a0 obtained from theoreticalarguments.  This suggests that the vacancy-interstitial recombination process ismore efficient than calculations for vacancy-interstitial interactions would allow.This tendency is consistent with recent numerical simulations of cavitynucleation and growth, which include a population of sub-critical vacancyclusters. These invisible recombination sites can enhance the overall rate ofvacancy-interstitial annihilation.  A rate theory has been constructed for thismodel, which also agrees well with experiment. In practice, the rate theory is veryinsensitive to these changes, except possibly at the very earliest times during theirradiation transient.Results of the modified rate theory modelIn the quasi-stationary limit, the time-derivatives of the monomerconcentrations are essentially zero. The solution to their coupled, quadratic rateequations is then easily obtained [1]. Defining † Zd = Z ddÂ Ad N d , † Zs = Z ssÂ AsN s,8† Z = Z xx= s,dÂ AxN x , and † Zv Cveq = Z xx= s,dÂ AxN xCveq(x ) gives a quadratic equation withsolution:† DiCi =12M 2 + L - M( )M = Dv Cveq ZvZ i+(Pv - Pi)Z i+ZvK(Z iu + Z i )Z iL = 4PiKZvZ iThe vacancy concentration is obtained from:† DvCv =Pv - PiZv+ DiCiZ iZv+ Dv CveqThere is no production bias considered here, so that Pv=Pi.  The solutioncan be reduced to the standard rate theory by simply setting the unstable clustersink strength to zero.  They merely make an additive contribution to the total sinkstrength in the quantity M.Finally, the swelling rate is obtained from the net flux of interstitials (minusvacancies) to the dislocations.  All vacancy emission is neglected for lowtemperatures, so that:† ddtDVV= DiCi Z id - DvCv Zvdin terms of average quantities for the dislocation population.Measured swelling rates are shown in Table 1, along with the results of asingle fit parameter set. The measured swelling rate “Dswelling” is based on thevolume difference between the two successive observations.  Model predictions9are obtained at both the lower and upper dose points (both “standard” and“modified” rate theories).  The standard rate theory can agree quite well with theexperimental results. The modified rate theory similarly agrees with the availableswelling data. It gives a slightly lower swelling rate, which could becompensated by adjusting the model parameters. The unstable clusters doinfluence transient behavior, when the unstable cluster sinks can dominate thestable voids and dislocations. (It can be seen that the relative reduction inswelling rate is larger at the top of the table, which corresponds to samples withlower doses.) Here, the modified rate theory displays more realistic swellingbehavior.  Because the swelling curves are concave upwards, the predictedinstantaneous swelling should be lower at the first point than the finite differenceresult.  Similarly, the instantaneous rate should be higher at the secondobservation point.  This behavior is better achieved with the modified ratetheory.SummaryThe standard, simplified rate theory for swelling in Fe-15Cr-16Ni maysuggest a larger recombination radius then is expected from reaction-diffusioncalculations.  When the rate theory is extended so that recombination can occurat invisible defect sites, the modification is found to have little effect on the fit.This suggests that the rate theory is useful in its standard form at moderatetemperatures and higher and especially in the steady state, where theconcentration of small, unstable clusters is low.  The modified rate theory may10be needed more during the incubation period, when undetectable clusters aremore prevalent.11AcknowledgmentsThe authors gratefully acknowledge support of this work by a NERI grantfrom the U.S. Department of Energy under contract W-7405-ENG-48 with theLawrence Livermore National Laboratory.[1] T. Okita, and W. G. Wolfer submitted to J. Nucl. Mater.[2] W. G. Wolfer and A. Si-Ahmed, J. Nucl. Mater. 99 (1981) 117.[3] T. Okita, T. Sato, N. Sekimura, F. A. Garner, and L. R. Greenwood, J. Nucl.Mater. 307-311 (2002) 322.[4] T. Okita, W. G. Wolfer, F. A. Garner, and N. Sekimura, submitted to Philo.Mag.[5] F. A. Garner, “Irradiation Performance of Cladding and Structural Steels inLiquid Metal Reactors”, Materials Science and Technology: A ComprehensiveTreatment, VCH, 1994, Vol. 10 A, Chapter 6, pp. 419[6] O. Dimitrov and C. Dimitrov, Journal of Nuclear Materials 105. (1982) 39.[7]  E. Kuramoto, N. Tsukada, Y.Aono, M. Takenaka, H. Yoshida and K. Shiraisi,Journal of Nuclear Materials 133 & 134 (1985) 561.[8] J. J. Sniegowski and W. G. Wolfer, “ On the Physical Basis for the SwellingResistance of Ferritic Steels”, Proc. of Topical Conference on Ferritic Alloysfor Use in Nuclear Energy Technologies, Snowbird, Utah, June 19-23, 1983,[9] W. G. Wolfer, M. Ashkin and A. Boltax, ASTM-STP 571 (1975) 233.12[10] J. Tenbrink, R. P. Wahi and H. Wollenberger, J. Nucl. Mater. 155-157 (1988)850.13Table 1. Experimental swelling versus the model results for one possibleparameter set. The rates labeled “Dswelling” are obtained by finite differences ofthe two successive swelling measurements in paired samples.  The “standard”and “modified” results refer to the respective rate theory predictions for a single,reasonable parameter set (set #1 in Ref. [1]).doserateCumulativedoseswelling Dswelling(%/dpa)standard(%/dpa)modified(%/dpa)8.9 ¥10-92.2 ¥10-80.230.610.01060.101 0.24 0.490.780.100.272.7 ¥10-86.6 ¥10-80.711.870.07610.9340.74 0.751.100.270.649.1 ¥10-82.1 ¥10-72.366.360.5153.770.81 1.001.110.550.803.1 ¥10-73.0 ¥10-78.0511.12.215.140.96 1.091.160.770.895.4 ¥10-78.4 ¥10-714.028.83.9719.71.06 0.940.820.700.667.8 ¥10-79.5 ¥10-720.032.45.3616.90.93 0.790.870.620.7114Figure CaptionsFigure 1  Simulated sink strengths for stable voids (solid line) and unstableclusters versus dose for a type-316 stainless steel at 420 C.  The jaggedness ofthe lines results from discrete changes in the critical cluster size as the monomerconcentration evolves with time. Unstable cluster sinks dominate the system untilthe dislocation and stable voids rise to appreciable values.15Figure 1  Simulated sink strengths for stable voids (solid line) and unstableclusters versus dose for a type-316 stainless steel at 420 C.  The jaggedness ofthe lines results from discrete changes in the critical cluster size as the monomerconcentration evolves with time. Unstable cluster sinks dominate the system untilthe dislocation and stable voids rise to appreciable values

Modified Rate-Theory Predictions in Comparison to Microstructural Data

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Modified Rate-Theory Predictions in Comparison to Microstructural Data

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