U-Mo alloy fuel is a prime candidate for conversion of high-power research and test reactors, from HEU cores to LEU cores, thanks to its high density and advantageous irradiation performance. Two forms of U-Mo alloy fuel have been studied in the U.S. GTRI-CONVERT program (formerly known as RERTR): One is U-Mo particle dispersion in an Al matrix (U-Mo/Al dispersion fuel); the other is direct bonding of U-Mo thin foil to Al-alloy cladding (U-Mo monolithic fuel) (Fig. 1). In this paper, because modeling the interaction layer growth between the U-Mo particles and Al matrix was concerned, we focused only on the dispersion fuel form.
One of the intrinsic characteristics of the U-Mo/Al dispersion fuel is perhaps the interaction layer (IL) growth between the U-Mo fuel particles and matrix Al. Interaction layer growth is due to interdiffusion between the fuel particle (U and Mo) and matrix (Al). Hofman and co-workers  reported the first observation of IL growth in U-Mo/Al from an irradiation test, where they also proposed a correlation
for IL growth expressed as a function of the fission rate, temperature, and time. Hayes  later incorporated this model in a computer model PLATE. Ryu and coworkers  systematically measured the kinetics of IL growth and obtained the activation energy for IL growth for out-of-pile heating tests.
U-Mo/Al dispersion fuel performed well in irradiation tests at lower power and burnup, so its qualification appeared near . However, later tests adopting higher power, i.e., higher fission rate and higher burnup, showed enhanced fuel-meat swelling, which was caused by massive IL growth, and in some cases, by large pore formation in the fuel meat ,. The pores were distinguishable from the fission gas bubbles in the fuel particles, in that they mostly formed outside of fuel particles, i.e., in the ILs, although both were commonly filled with fission gases. The formation of IL results in a net volume increase and a net thermal conductivity decrease in the meat. The low thermal conductivity of IL provides adverse feedback for higher IL growth by further increasing the fuel meat temperature. While U-Mo particles and Al are crystalline, ILs formed between U-Mo and Al are amorphous during irradiation ,. The amorphous nature of the ILs is believed to be the reason for large pores formed outside of fuel particles . Fission gas diffusivity in the amorphous ILs is higher than in its crystalline state, and viscosity is lower, both of which promote fission-gas-pore formation in the ILs.
A remedy was needed to reduce the IL growth rate and strengthen the IL to resist against pore formation and growth. Noticing the low IL growth between U3Si2/Al dispersion that did not form pores in the IL until full LEU burnup (see ,), it was proposed at ANL that a small amount of silicon be added to the aluminum matrix ,. An ample number of studies, both out-of-pile (-) and in-pile (-), have been conducted to examine the effectiveness and better application of the remedy. The common findings were that Si addition to Al not only reduced IL growth, but also delayed pore formation and growth in the IL. Hence, overall performance of this fuel was improved.
Since IL growth is one of the deciding performance variables for U-Mo/Al dispersion fuel, an accurate model is also required. The studies for interaction layer (IL) growth modeling for in-pile applications includes two cases: one for the pure Al matrix case ,,,, and the other for the Si-added matrix case ,. The most up-to-date correlation for the pure Al matrix case was by Wachs , which was confirmed as accurate by Kim . The correlation for the Si-added matrix case, developed mostly on low-temperature miniplate test data, was applied for all Si contents larger than 2 wt%. There was no need to provide a separate factor to consider the effect as a function of the Si content, because in the low temperature tests the addition of 2 wt% Si virtually suppresses IL growth to the lowest possible level. However, it has been noticed that the modeling under-predicts the high temperature test results from the RERTR-9 and KOMO-4. The high temperature test KOMO-4 showed that a Si content higher than 2 wt% had different IL growth kinetics ,. Therefore, a model applicable to the high temperature regime and a Si content higher than 2 wt% was necessary. Another objective of the present work is to unify the existing separate models for the pure Al matrix and the Al matrix with Si addition.
An interaction layer can also grow during fabrication and the follow-on blister test when the fuel plates undergo heat processes. An IL growth model in this step was also required.
IL growth during fuel fabrication and any follow-on heating test is modeled in this section. The interaction layer growth between U-Mo and Al is a typical interdiffusion process between U and Mo atoms versus Al atoms. Hence, in this paper, interaction and interdiffusion are used interchangeably. In addition, the rate-controlling process in interaction layer growth is interdiffusion, not the reaction product formation . The out-of-pile IL growth takes the following modified Arrhenius correlation because interdiffusion is due to thermal activation:
where YAl is the average IL thickness (μm) in a pure Al matrix, i.e., Al matrix without any intentional Si addition, T is the fuel meat temperature in K, and t is the time in s. The constants have been fitted using the data taken from Ryu  (Table 1). Only data from dispersion fuel tests were used. The IL thickness was measured using optical micrographs of the cross section of a sample. From an optical micrograph, several IL thicknesses were measured, and the average value was used as the nominal value. The method to obtain the true IL thickness is explained elsewhere . Generally, in an optical micrograph, the common thinnest IL on the largest particle was taken to represent the true IL thickness.
The RERTR test-plates were fabricated using the hot rolling technology , in which the heat processes involved were much shorter, ~1 h, but at a much higher temperature, 500 ℃, than in-pile tests (Table 2, Table 3). For example, the test duration for the RERTR-4 test was 257 d, and the life-average fuel temperature was ~100 ℃.
IL growth during fabrication is hard to assess, so the correlation given in Eq. (1) is for an average value. The largest uncertainty during fabrication is due to the irregular contact pressure between particles with the matrix. For a typical rolling process used for RERTR test plates , the bonding between the U-Mo particles and matrix is poor during the pre-heating and first pass, so IL growth during these steps is virtually negligible. Meaningful bonding is achieved after the second pass and subsequent IL growth occurs during the last heating ~15 min (see  for detailed time and temperature for each step). The most IL growth takes place during the subsequent blister-annealing test at 485 ℃ for an hour. Using Eq. (1), we estimated an average of ~1.5 μm of IL growth during fabrication. The test
IL Growth between U-Mo a and Al from Out-of-pile Heating Tests, RERTR Tests and KOMO-4 Test during Fabrication and Follow-on Blister Test
samples of the KOMO-4 test from KAERI were produced using an extrusion process taking 0.5 hour at 400 ℃, for which Eq. (1) predicts negligible IL thickness. Table 2 compares the heat processes and IL growth involved during fabrication of some RERTR tests and the KOMO-4 test.
The Mo content of the U-Mo particles is in the range of 6 – 10 wt%. However, the most common Mo contents are 7 wt% and 10 wt% in the RERTR irradiation tests. In the Mo-content 10 wt%, the thermodynamic equilibrium phase of the U-Mo is the α-U + δ-U2Mo dual phase at temperatures below ~550 ℃. For other Mo content in the range of 6 – 10 wt%, the phase boundary is slightly higher and another phase (α+γ) exists. Therefore, during fabrication, some parts of the meta-stable γ-U particle transform to α-U. The γ-U has a lower swelling rate during irradiation than the α-U (see -for details) and has a lower IL growth rate . When the γ-U is partially transformed to the α-U during fabrication and the subsequent blister test, a thicker IL grows on the α-U while a thinner IL grows on the γ-U, which results in nodular type IL growth on the same U-Mo particle. In this situation, expressing IL in volume is more appropriate than in thickness.
Measured data for U-Mo/Al-Si dispersion, i.e., U-Mo dispersion in a Si-added Al matrix, are scarce from outof- pile tests and scatter with considerable uncertainties. Iltis  measured the IL thicknesses of Si-added plates fabricated at a lower fabrication temperature than typical RERTR plates (Table 2) and then heated for two or four hours. As given in Table 2, 4 wt% Si and 6i plates show no clear correlation between the Si content and IL thickness, implying that the effect of Si content on IL growth in the out-of-pile heating test is negligible. Keiser and co-workers  also reported an out-of-pile heating test result of plates with Si addition. Using images in Ref. , we measured IL thicknesses as given in Table 1. The Keiser data also confirm that there is no clear correlation between the Si content and IL thickness. This uncertainty is chiefly because of the irregular IL thickness, which is caused by the decomposition of the meta-stable γ-phase U-Mo particles.
It is typical that considerable parts of the U-Mo surfaces have extremely thin ILs while some parts have thicker ILs, including some nodular type ILs. As discussed in sect. 2.1, this nodular type of IL is known to form on the parts of the U-Mo surface transformed to α+γ' during the heating steps. Because of the much higher IL growth rate of the α phase , the fuel surface with γ-to-α+γ' phase transformation shows higher IL growth. Once formed, the thick IL on the α-U precludes Si diffusion to the fuel
Heat Processes during Fabrication and Follow-on Blister Test and Predicted IL Growth before Irradiation for RERTR Tests and KOMO-4 Test
surface and formation of a Si-barrier on the fuel surface to block Al diffusion. This further contributes to the formation of more irregular ILs in U-Mo/Al-Si dispersions than U-Mo/ Al dispersions after in-pile tests. Hence, considerable uncertainty is inevitably involved in the IL thickness of a plate with Si addition.
Park et al.  annealed a U-Mo/Al-5Si dispersion sample at 580 ℃ for ~1 h and observed an average IL thickness of ~8 μm (see also Ref. ). Because the temperature was much higher, but still below the γ-phase boundary, it is probable that the decomposition of the meta-stable γ-phase U-Mo particles was higher in Park’s test, so his datum may be exaggerated. However, the extent of IL growth reported by Park is lower than those of U3Si/Al and U3Si2/Al dispersion fuels , suggesting that at the annealing temperature the U-Mo fuel surface forms an excellent barrier that effectively blocks Al diffusion into U-Mo. This result also implies that the effect of the presence of the Mo in U-Mo/Al-Si is to synergistically reduce IL growth as in U-Si intermetallic dispersion in Al with a small addition of Mo, as reported by Ugajin .
Some diffusion-couple test data are available in the literature ,. In a typical diffusion-couple test to measure interdiffusion between U-Mo and Al, a U-Mo block is welded with an Al block, and the metal couple, U-Mo versus Al, is isothermally heated. In the post-test analysis, the interaction layer thickness is measured. The contact pressure between U-Mo and Al in a diffusioncouple test is higher than that of a dispersion sample, so the IL growth rate is generally higher in the diffusioncouple test than in the out-of-pile test of dispersion fuel. For this reason, the diffusion-couple test data cannot be used with the dispersion test data.
Using the data provided in Table 1, we performed a data fitting to the Arrhenius equation and the IL growth correlation is obtained as follows (R2=0.67):
where YAl-Si is the IL thickness (μm) in an Al with a Si addition, T in K and t in s.
Using Eqs. (1) and (2), the IL growth during fabrication and the follow-on blister test for the RERTR test plates, and KOMO-4 test were calculated and given in Table 2. Once IL growth before irradiation is assessed by using Eq. (1) or Eq. (2), the total IL growth accumulating during fabrication and irradiation can be obtained by
where Y is the total IL thickness, Yfab is the fabrication IL thickness calculated by Eq. (1) or Eq. (2), and Yi is the IL thickness during irradiation. The model for IL thickness prediction during irradiation is described in the following section.
Test Data and Comparison of IL Thicknesses between the Measured and Predicted from RERTR-4, -5, -6, -7, -9 and KOMO-4 Tests
The irradiation test data used in the model development are summarized in Table 3. While the RERTR test samples are miniplates having the meat thickness of ~0.64 mm, KOMO-4 test samples have rod-type geometry with the meat diameter of 6.4 mm. Because of the much greater distance of heat transport and the higher heat flux, the KOMO-4 test samples have much higher temperatures than the RERTR test samples for the same power.
For the RERTR test plates, the fuel temperatures were calculated at BOL and EOL, and arithmetic averages were taken to represent the test samples (Table 3). For the KOMO-4 test, the time-dependent fuel temperatures were calculated and life-average values were obtained. The test-sample power, and hence the fission rate, decreases due to depletion of fissile atoms in the sample. On the other hand, fuel temperatures tend to increase as the meat thermal conductivity degrades due to IL growth and the formation of fission gas pores. These two factors somewhat compensate for each other, and as a result, the change in fuel temperature over test life is insignificant. In the RERTR miniplate tests, although the temperature and fission rate (i.e., power) are each independent variables to each other, they are still coupled. In this regard, the KOMO-4 test data are valuable because they have higher temperatures than the RERTR test data while the fission rates are similar.
IL thicknesses were measured at several locations in an optical micrograph and the average value was used as the nominal value. The method to obtain the representative IL thickness is illustrated in Fig. 2. Two cross sections are shown for illustration. The cut surface A is off of the equator B. Therefore, the IL thickness L1 measured in the cut surface A is thicker than L2 measured in the equator. Generally, on a metallographic picture, the common thinnest IL is more likely to be the true IL thickness, because it is more likely to be on a plane close to the equator of the fuel particle.
In Fig. 3 and Fig. 4, posttest micrographs of RERTR- 6 and KOMO-4 samples are shown, respectively. The RERTR-6 is a test having relatively low temperatures and low fission rates. Although temperatures vary by ~18 ℃, all test plates consistently show that IL thickness decreases as the Si content increases. This trend is more noticeable in the KOMO-4 test, which has less temperature variability.
The ILs of the RERTR samples were measured at the plate transverse (width) center to exclude possible influence
of pores forming in the matrix, and fuel creep occurring in the fuel meat loaded closer to the ATR core center, a phenomenon reported previously ,. For the KOMO- 4 test, measurements were made at the meat center and meat mid-radius regions, by which two IL thicknesses at different temperatures with the same fission rate were obtained (Table 3).
The measurement errors, given in Table 3, stem largely from the irregular IL thicknesses. The errors tend to be larger for the samples with a Si-added matrix than the
pure Al matrix samples. PIE images also showed locally thick, nodular type, ILs, the cause for which originated from the α+γ' transformation during fabrication and the following blister test. On the other extreme, PIE images showed that ILs on some U-Mo particles were still thinner than the average IL thickness during fabrication. This indicates either that some parts of the fuel surface had poor contact from fabrication to the end of irradiation, or that large Si particles contacting the fuel particle effectively blocked Al diffusion. The as-fabricated IL thickness was subtracted from the post-irradiation IL, which means the predicted results by the new correlations are for those during irradiation.
3.2.1 Basic Correlation
The IL growth correlation during irradiation is a modified Arrhenius equation from Eq. (1), and an additional term with the fission rate is multiplied to account for fissionenhanced- diffusion (FED):
There have been several other correlations that take a similar form to Eq. (4) but with different constants , ,,. In order to fit both the RERTR and KOMO data, the fitting constants here were revised from those of Refs.  and . The best fit was obtained with A= 2.6x10-8 and q=3850 K for plates with U-7Mo in a pure Al matrix. In Table 3, the predicted and measured IL thicknesses are compared. Therefore, the correlation for the samples with a pure Al matrix is:
where Y is in μm, ḟ in f/cm3-s, T in K, and t in s.
The effects of Si addition and Mo content are considered by multiplying additional factors to Eq. (5) as follows:
where fSi is the reduction factor by Si addition in the Al matrix and fMo is the factor to assess the Mo content other than 7 wt% in the U-Mo alloy particles. The modeling of these factors is described in the following subsections.
3.2.2 Factor for Si-addition Effect, fSi
The addition of Si in the Al matrix results in an exponential reduction of IL thickness that varies with the Si content. Comparing the results from the lower temperature test RERTR-6 and the higher temperature test KOMO-4, it is noticeable that the Si addition effect on IL growth also depends on temperature. For the RERTR-6 test, the Si effect is virtually saturated at 2 wt% Si (Fig. 5). However, the KOMO-4 test shows that the IL reduction effect continues up to 8 wt% Si without saturation. Fitting the IL reduction versus the Si content for two different temperatures given in Fig. 5, we obtained the IL reduction factor by Si addition in Al as
where T is the fuel meat temperature in K (≤ 473 K) and WSi is the Si content in the matrix in wt% (≤ 8 wt%). fSi is in the range 0.002 ≤ fSi ≤ 1. The minimum occurs at the Si content of 8 wt%, whereas the maximum occurs for the pure Al.
3.2.3 Factor for Mo Content Effect, fMo
It is known that as the Mo content increases, the IL growth rate decreases ,,, so cases with a Mo content other than 7 wt% need to be considered. The current understanding is that the metastable γ-phase U-Mo more readily decomposes to the α+γ' if the Mo content in the alloy is reduced. Although the α-phase transforms back to the γ-phase soon after irradiation starts ,,, once formed, the α-phase produces a thicker IL during fabrication and the subsequent blister test. The thicker IL formed on the a phase weakens the Si addition effect, and the result is an increase in the overall IL growth rate during irradiation. In the thick IL, as the Si concentration decreases, so does its activity . Mirandou  also found a weakened Si effect in the thick IL from an elevated-temperature out-of-pile test and argued that it was owing to the formation
[Fig. 5.] Comparison of IL Reduction Factors Obtained for the Low-temperature RERTR-6 Test (at Life-avErage T ~100 ℃) and the High-temperature KOMO-4 Test (at Life-average T ~180 ℃), where the IL Reduction Factor is Defined as the Square of IL Thickness Divided by the Square of IL Thickness for a Pure Al Matrix at the Same Irradiation Condition.
of the α-phase, on which U4Mo6Al43+UAl3 phases form, excluding Si.
In the RERTR experiment modeling , we observed that the IL growth rate decreased linearly with the Mo content in the range of 6 – 10 wt%. This effect was similarly formulated as follows:
where WMo is the Mo content in fuel particles in wt%. fMo varies between 0.85 (at 10 wt% Mo) and 1.05 (6 wt% Mo). A Mo-content beyond 10 wt% is not considered in the US RERTR program.
One of the objectives of the development of the present model was to unify the two independent IL growth correlations applicable either to a pure Al matrix, or to an Al matrix with up to 8wt% Si and temperatures up to 200 ℃. In addition, IL formation before irradiation is included, so the model can be used for fuels produced by various fabrication methods.
The observations from the KOMO-4 test  showed that Si tends to accumulate mostly at the IL-matrix interface due to Si diffusion from the matrix. It is known that Si contained only in the fission-fragment recoil zone produces Si diffusion flux. As the IL grows, the overlapping volume fraction of the recoil zones increases and, hence, there is a decrease in the effective amount of Si per unit-fuel-surface area available for diffusion.
For the RERTR-9 test, the fuel plates were made with smaller fuel particles and higher U-loading, which yielded more recoil-zone overlapping during the test, and in turn reduced the effective Si amount available per unit-fuelsurface area. Contrarily, the KOMO-4 test samples had larger fuel particles sizes and relatively lower U-loading in the meat (see Table 3). As a result, smaller recoil-zone overlapping was made, so the effective Si amount per unit-fuel-surface area was higher. As U-loading increases, not only the matrix volume decreases and the depletion of the matrix occurs earlier, but homogeneity in fuel particle spatial distribution also tends to decrease. Regions with higher U-Mo particle densities occur, where the effective Si amount per unit-fuel-surface area decreases. Hence, even for the same Si content in the matrix, the IL growth rate can be different.
In the meat, the homogeneity of fuel-particle spatial distribution may also affect the effective Si availability. For example, a higher homogeneity in fuel-particle spatial distribution improves the effect of Si addition.
An additional consideration in the efficacy of Si is how Si is added to the matrix Al. The finer the Si precipitates, the more effective is the Si addition: Hence, a lower IL growth rate. At present, there are two viable methods of adding Si to the matrix. One is adding Si to the Al as an alloying element, and the other is directly adding the Si powder to the Al powder during fuel fabrication. The former was considered the better method to provide finer Si particles in the matrix.
In order to have more accurate modeling, consideration of the parameters discussed above (U-loading, homogeneity in fuel-particle spatial distribution and Si precipitate size in the matrix) may also be necessary. In the present modeling, however, these parameters are not explicitly considered.
The Si concentration in the IL tends to decrease during irradiation. As the Si concentration decreases, the Si’s strength to reduce the IL growth rate also decreases. However, the change in the Si effect during irradiation is not included in the present model.
The comparison between the measured data and predicted IL thicknesses are given in Table 3. Considering uncertainties in temperatures and measurement errors, a fair agreement was generally achieved, except for some cases. The uncertainty in the meat temperatures is believed to be caused by the uncertainty in the meat thermal conductivity at high burnup. Measurement errors were primarily due to the irregular IL thickness.
IL growth data given in thickness are often more convenient than those given in IL volume, because they do not need supplemental data such as U-loading and U-Mo particle size. However, for high IL growth cases, the thickness is not only difficult to measure, but it is also less meaningful. For these circumstances, data on IL volume fraction is a better metric. Consequently, a conversion model from IL thickness to IL volume was developed, assuming that fuel particles had a uniform size and were distributed in an FCC array.
Fig. 6 illustrates the configuration of two neighboring fuel particles and the IL formed around them. The distance,
d, between the surfaces of two neighboring particles is
where a is the FCC unit cell edge length, and r is the fuel particle radius. The fuel volume fraction in the meat is defined as
where Vm is the meat volume and Vf is the fuel volume in the unit cell, respectively. Because there are four fuel particles in the unit cell, Vf is given by
where r is the average fuel particle radius known from the fabrication data. From Eqs. (9) – (11), d can be found by
where the applicable range of Vf is less than 0.74, which is the FCC packing fraction. The fuel volume fraction Vf is usually calculated with known fabrication data (U-loading), because Vf =wU/ρU where wU is U-loading, and ρU is U-density in the fuel particle. For example, Vf for 8 gU/cm3 of U-7Mo is calculated by Vf= wU/ρU = 8/16.3=0.49 where ρU = 17.5 * (1 – 0.07). The physical density of U-7Mo alloy is 17.5 g/cm3.
In Fig. 6, the overlapping part of the IL volume of the right-hand-side fuel particle, by the neighbor particle, is a partial sphere with height, h, of which the volume is
where h = Y – d/2 and R= r + Y. Here Y is the IL thickness. When the IL overlaps with neighboring fuel particles, i.e., Y is greater than d, as marked by the darker gray region in Fig. 6, and this volume must be subtracted from the large cap volume given in Eq. (13). Because there are two symmetric caps, two of these small caps must be considered.
where g = (Y– d)/2.
The IL volume can be calculated considering only one fuel particle surrounded by 12 neighboring ones by
The IL volume fraction vIL in the fuel meat can be calculated by
[Fig. 7.] Prediction of IL Volume Fraction as a Function of IL Thickness for 6 gU/cm3 and 8 gU/cm3 U-density in the Meat (Average Fuel Particle size = 70 μm) and for 5 gU/cm3 U-Density in the Meat (Average Fuel Particle Size = 150 μm).
where VIL is given by Eq. (15) and Vf given in Eq. (11).
Fig. 7 shows the model predictions for IL volume fraction versus IL thickness for 6 gU/cm3 and 8 gU/cm3 U-loading cases with a fuel particle size of 70 μm. Also included is a case for the KOMO-4 test, for which fuel loading is 5 gU/cm3 with fuel particle size of 150 μm. Fuel swelling and fuel consumption by the interdiffusion layer growth are not considered. However, these two factors somewhat tend to compensate for each other to give reasonably acceptable model predictions. A more detailed model is necessary for an accurate result, but the model described here is suitable for approximate estimates.
In the US GTRI-convert program (formerly known as RERTR), U-Mo alloy particle dispersion in Al is a primary fuel form being studied for use in converting HEU-fueled research and test reactors to LEU-fueled ones. In order to achieve sufficient total uranium densities, high fuel-volume loading is pursued. One of the largest obstacles to good fuel performance is interaction layer (IL) growth between the U-Mo particles and matrix Al, which needs to be minimized.
For an accurate fuel performance analysis, a prediction model for IL growth is therefore necessary. Two empirical IL growth correlations were developed for out-of-pile fabrication and possible subsequent testing: One is for the pure Al matrix, and the other for the matrix with Si addition. The IL growth correlations are Arrhenius equations.
The in-pile growth correlation, developed using the RERTR-4, -5, -6, -7 and -9 data, as well as KOMO-4 data, takes a form similar to the out-of-pile correlations. For the in-pile correlation, the out-of-pile IL growth was augmented by fission-enhanced diffusion (FED). The FED effect was modeled by the square-root of the fuel-particle fission rate.
The in-pile correlation was further enhanced to include the effect of Si addition in the Al and the effect of the Mo content. The model is applicable for Si contents in the matrix up to 8 wt%, for fuel temperatures up to 200 ℃, and for Mo content in fuel particles in the range of 6 – 10 wt%.
An approximate analytical method to convert IL thickness to IL volume fraction in the fuel meat was proposed. For a more accurate conversion, the volume changes in fuel and matrix caused by their consumption by IL growth must be considered together with the volume expansion by fission-product-induced fuel swelling, which was not considered in the present work.