Formation and Growth of Image Crystals by Helium ... - ACS Publications

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Formation and Growth of Image Crystals by Helium Precipitation Hiroyuki Serizawa,*,† Junji Matsunaga,‡ Yoshinori Haga,§ Kunihisa Nakajima,# Mitsuo Akabori,† Tomohito Tsuru,† Yoshiyuki Kaji,† Shinji Kashibe,‡ Yuji Ohisi,∥ and Shinsuke Yamanaka∥ †

Nuclear Science and Engineering Directorate, Japan Atomic Energy Agency (JAEA), Tokai Research and Development Center, Nuclear Science Research Institute, 2-4 Shirane Shirakata, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan ‡ Research Department, Fuel Group, Nippon Nuclear Fuel Development Co., Ltd., 2163 Narita-cho, O-arai-machi, Higashi-ibaraki-gun, Ibaraki-ken 311-1313, Japan § Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai Research and Development Center, Nuclear Science Research Institute, 2-4 Shirane Shirakata, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan # Nuclear Science and Engineering Directorate, Research Group for High Temperature Science on Fuel Materials, Japan Atomic Energy Agency (JAEA), O-arai Research and Development Center, 4002 Narita-cho, O-arai-machi, Higashi-ibaraki-gun, Ibaraki-ken 311-1393, Japan ∥ Department of Nuclear Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan ABSTRACT: We found that porous UO2 grain was formed by the precipitation of helium injected by HIP. Scanning electron microscopy analysis showed that polyhedral negative crystals were formed in the sample. The shape of the negative crystal changes dramatically with the conditions of helium precipitation. A truncated octahedron-type, an octa-triacontahedrontype, and a pentacontahedron-type negative crystal were observed. Our study implies that the shape of the negative crystal should change depending on the helium inner pressure enclosed in the negative crystal. It is difficult to arbitrarily control the shapes of these polyhedral negative crystals embedded in a solid medium. However, the shape of the negative crystal can easily be controlled by the helium injection method. In this article, we call the shape controlled negative crystal an image crystal. Here, we report a relationship between the surface energy and the shape and discuss the transformation mechanism of the image crystal. Our detailed observation indicates that the growth process of the image crystal can be explained by a step free energy model rather than an attachment energy. We could not find a cuboctahedron-type negative crystal of which the surface area is larger than that of a pentacontahedron-type negative crystal with the same volume. nation, which is of prime importance.11−16 Morphological studies6,7 have demonstrated that the equilibrium shape of voids formed in a UO2 matrix is a truncated octahedron bounded by the {111} and {001} facets. However, as the figure shows, the cavity we found has the shape of a much more complex polyhedron. Precipitated fission product gas such as xenon or krypton is well-known to form spherical gas bubbles,17−21 and we discovered that helium has an unusual ability to form symmetrical-shaped cavities. Castell made an interesting suggestion on this point.6 Researchers interested in the Wulff shape of UO2 generally do not consider the cavity as being filled with gas at any significant pressure. However, gas at appropriate pressure enclosed within the cavity would affect the cavity’s shape. In the reference, he described that the issue remains a point of further investigation. Considering every factor, we thought that the figure precisely reflects the influence

1. INTRODUCTION This investigation was triggered by the unexpected discovery of a polyhedral cavity in a UO 2 matrix during another investigation aimed at reducing the high radiotoxicity of minor actinides accumulated through nuclear power plant operation. Figure 1 shows the field emission scanning electron microscopy (SEM) image that caught our interest. The figure shows the fracture surface of a single-crystal UO2 that was heat treated in helium at 90 MPa, followed by annealing at 1573 K for 1 h. The cavity in the center of the figure is bounded by well-grown facets. Although previous researchers have reported the presence of a polyhedral cavity, called a negative crystal,1−3 in UO2,4−10 the formation mechanism is still unclear. As is known, surface and interfacial phenomena are ubiquitous and constitute a topic of current interest in many research fields, ranging from phenomena such as nucleation, wetting and crystal-faceting transitions to lattice gauge theory. Therefore, such cavities always attract the interest of material researchers because the measurement of the concaved facet allows them to study pure surfaces while avoiding the problem of contami© XXXX American Chemical Society

Received: January 24, 2013 Revised: April 29, 2013

A

dx.doi.org/10.1021/cg4001335 | Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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Figure 2. External image of single-crystal sample employed in this study.

Figure 1. SEM image of image crystal formed in single-crystal UO2.

of the inner pressure on the shape of the cavity. In this article, we call such shape-controllable cavity bounded by facets an image crystal and reveal that the helium enclosed in the image crystal affects the formation of the complex symmetrical shape. We also discuss the growth process of the image crystal and show that the attachment energy method is not valid to describe the faceted void shape, but the step free energy model is more applicable.

Figure 3. Metallography of polycrystalline sample.

3. METHOD OF SURFACE AREA MEASUREMENT BY IMAGE ANALYSIS 3.1. Truncated Octahedron-type Image Crystal. The surface areas of the {111} and {001} facets on the truncated octahedron-type image crystal were determined as follows. A boundary was drawn around the {111} facet which was almost normal to the electron beam used for SEM, and the area contained within it was measured. This was then taken as a representative size for the eight {111} facets; therefore, to obtain the total {111} area of the image crystal, the measured area was multiplied by eight. On the same image crystal, the horizontal dimensions of the three {001} facets were measured and squared to determine the areas of the individual {001} facets. The three {001} areas were averaged, and the average was multiplied by 6 to obtain the total because there are six {001} faces. This was repeated for 30 images of image crystals of different sizes. The measurements were performed only on selected image crystals having a relatively symmetrical shape. 3.2. Pentacontahedron-type Image Crystal. The surface areas of the {111} and {001} facets were obtained by the same method as described in section 3.1. The surface areas of facets with higher indexes, {110} and {311}, were evaluated as follows. Boundaries were drawn around these facets, and the areas contained within the enclosure were measured. As described in the section below, these two facets are arranged around the {111} faces; three faces for each facet can be seen in the SEM image. Three individual faces were measured, and the averaged values were adopted as the representative value for the faces. These two facets appear considerably foreshortened because of the angle with respect to the electron beam; the representative value is the area of the facet projected on the {111} face. The values were then corrected according to the angle these facets form with the {111} face (35.26° and 29.5°, respectively). This procedure was repeated for 30 images of the pentacontahedron-type image crystal. To obtain the total area

2. EXPERIMENTAL PROCEDURES The single-crystal UO2 employed in this study was prepared by a chemical vapor transport method using UO2 powder as the source material and tellurium tetrachloride (TeCl4) as the transport reagent22 as follows:

UO2 (s) + 2Cl 2(g) ↔ UCl4(g) + O2 (g) The transport was performed in a sealed quartz ampule (internal diameter: 17 mm, length: 200 mm). The sample was heat treated for one week at a temperature between 1273 and 1348 K. The sample was examined by X-ray diffraction analysis (Mo Kα: 0.71 Å) with a curved imaging plate area detector (Rigaku, R-AXIS Rapid). The Bragg point covered less than 0.5°, indicating that the sample was well-grown single-crystal UO2. The polycrystalline sample was prepared by sieving a high-density pellet (approximately 97% TD) to produce 300−500 μm fragments. This sample was also examined using an X-ray diffractometer (Panalytical, X’Pert PW3040/3050). The lattice parameter evaluated from the diffraction chart was 0.547 nm, which exactly matches that of stoichiometric UO2.23 External views of single crystal sample and metallography of polycrystalline sample are given in Figures 2 and 3. The grain diameter of the polycrystalline sample evaluated by the intercept method is 12.04 μm, which is markedly small, compared to the grain size of the single-crystal sample shown in Figure 2. Helium was infused by a hot isostatic pressing method. The infusion conditions were as follows: 1473 K for 100 h under a helium pressure of 91 MPa. Both the heating and the cooling rates were 20 K/ min. Following the helium injection, the sample was annealed at a temperature of more than 1573 K in helium at atmospheric pressure for 1 h to grow facets on the surface of the cavity. The cross section of the single-crystal sample before and after the helium treatment was examined by focused ion beam microscopy (Hitachi High-Tech FB2100). The fracture surface of the annealed sample was examined using SEM (Hitachi High-Tech HD-2300A) to observe the structure of the image crystal. B



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Figure 4. Cross section of the single-crystal sample examined by secondary ion mass spectrometry. (a) Before helium injection. (b) After helium injection.

Figure 5. SEM images of fracture surfaces of annealed samples after helium treatment. (a) Single-crystal sample annealed at 1573 K for 1 h. (b) Single-crystal sample annealed at 1773 K for 1 h. (c) Single-crystal sample annealed at 1973 K for 1 h. (d) Polycrystalline sample annealed at 1573 K for 1 h. (e) Polycrystalline sample annealed at 1773 K for 1 h. (f) Polycrystalline sample annealed at 1973 K for 1 h.

Therefore, supersaturated helium would typically diffuse along a concentration gradient toward the grain surface, at which the helium concentration should be low.24,26 However, when the sample cools very rapidly, a substantial amount of helium that cannot escape the grain clumps together to form gas bubbles. Thus, the cavities seen in Figure 4b are theorized to be formed by precipitated helium. The mechanism which drives the cavity creation had been discussed by Kashibe et al.18 They examined the formation of a gas bubble in UO2 by fission product gas and found that enhanced coarsening of bubbles was observed near the grain boundary. They described that a sufficient vacancy is supplied from external vacancy sources such as free surfaces, grain boundaries, or irradiation-induced subgrain boundaries for huge bubbles. However, as can be seen in Figure 4b, such kind of coarsening is not observed in this study. Though the gas bubble formation process is also worthy of note, the mechanism has not yet been established. The typical fracture surfaces of the

for each facet, the averaged values for the {110} and {311} faces were multiplied by factors of 12 and 24, respectively.

4. RESULTS AND DISCUSSION 4.1. Structural Analysis of Image Crystal. The cross sections of the single-crystal sample before and after helium treatment differ significantly (Figure 4, panels a and b, respectively). The sample before helium injection is fine; no visible defect was observed, whereas many round cavities are observed in Figure 4b. The figure indicates that the cavities were formed during high-pressure heat treatment in helium. Our recent study using Knudsen effusion mass spectrometry showed that the amount of helium dissolved in the singlecrystal UO2 was 4 × 10−2 cm3 (STP)/gUO2.24 The solubility of helium decreases as the temperature decreases,25 and the diffusion coefficient of helium in UO2 is sufficiently high for it to move in the grain at temperatures greater than 1273 K. C



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Table 1. Information on the Cavity on the Fracture Surface Examined by Image Analysis sample polycrystalline sample

single-crystal sample

a

heat treatment (K)

detected number of the cavity (μm)

mean size of the cavity

standard deviation (μm)

area of field (μm2)

density × 1012 (n/m2)

Acavityb/area of field

B. H.a

139

0.369

0.229

441.8

0.31

0.0336 ± 0.130

1573 1773 1973 B. H.a

76 90 90 341

0.474 0.394 0.492 0.225

0.435 0.27 0.484 0.138

441.8 441.8 441.8 141.6

0.17 0.2 0.2 2.41

0.0303 0.0248 0.0387 0.0957

1573 1773 1973

592 518 352

0.19 0.218 0.193

0.136 0.134 0.17

141.6 141.6 141.6

4.18 3.66 2.49

0.119 ± 0.0607 0.137 ± 0.0516 0.0727 ± 0.0564

± ± ± ±

0.0256 0.0117 0.0375 0.0360

Before heat treatment. bArea occupied by cavity.

Figure 6. SEM images of typical shape of image crystals.

Figure 7. Estimated shapes of image crystals. (a−c) Models of the image crystals formed by annealing at 1573, 1773, and 1973 K, respectively.

samples annealed at temperatures greater than 1473 K in helium at atmospheric pressure are shown in Figure 5. The results of image analysis of the cavity formed on the surface are summarized in Table 1. The cavity density in the single-crystal sample is about 10 times as high as that in the polycrystalline sample. The fourth column of the table shows that the diameter of the cavity in polycrystalline sample is slightly larger than that in the single-crystal sample. However, the distinction between

them is not clear since the scattering of the data is so large. As seen in the eighth column, the ratio of the area occupied by the cavity of the single-crystal is evidently larger than that of the polycrystalline sample. The cross sections of the single-crystal samples in the figure are {111} cleavage surfaces. The lines running across the images are a result of the cleavage process. The symmetrical features of the cavity, called an image crystal, initially attracted our interest. Although the bright symmetrical D



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Figure 8. Bunching and growth of facets with higher indices. (a) SEM image of image crystal observed in single-crystal UO2 annealed at 1773 K. (b) High-magnification image of field of view A shown in panel a.

analogue of the surface free energy.29 The equilibrium shape is determined by the angular variation in the surface free energy. On the other hand, the angular variation in the SFE determines the equilibrium shape of monolayer islands on a crystal surface. In addition, this SFE is proportional to the chemical potential of the monolayer islands and curved steps. Therefore, the SFE is a direct indicator of mass transport rates close to equilibrium.30,31 Assuming that the change in the shape is due mainly to mass transport from one facet to another, the driving force is considered to be the SFE. Although the theoretical value of the SFE in UO2 has not yet been reported, the growth rate is known to increase as the SFE decreases because the critical nucleus becomes smaller. Experimental results indicate that the MI of the faces in the growth process follows the sequence MI{111} > MI{001} > MI{311} > MI{110}; therefore, the SFE magnitude should decrease as follows: SFE{111} < SFE{001} < SFE{311} < SFE{110}. 4.2. Transformation of the Image Crystal. Note that the shape of the image crystal that formed in the polycrystalline UO2 differs from that in the single-crystal UO2 even though the annealing temperature was the same. This is attributed to a difference in the physical properties of the matrix that affects the behavior of helium in the matrix, namely, the monograin size (see Figures 2 and 3). A recent study on helium diffusivity in UO2 showed that the release behavior of helium in both single-crystal and polycrystalline samples can be successfully analyzed using an equivalent sphere model;32,33 in both cases, the diffusivity coefficient of helium lies within the scatter of the data in the literature.26 This result implies that the diffusivity coefficient of helium at the grain boundary is so high that helium in the polycrystalline sample, which consists of small grains, is easily released through the grain boundary. Hence, the concentration of helium in the polycrystalline sample should tend to be lower than that in the single-crystal sample when the heating parameters are the same. The results of the image analysis given in Table 1 also support the argument. As is mentioned above, the area ratio occupied by the cavity of the single-crystal is evidently larger than that of the polycrystalline sample, which means that a larger amount of helium was precipitated in the single crystal. It should be noted that the area ratio of the single-crystal sample annealed at 1973 K is smaller compared to those annealed at 1573 and 1773 K. The diffusion coefficient of helium in UO2 grain should increase with increasing temperature. Therefore, we consider that the decrease in the area ratio at high temperature reflects the decrease in the amount of helium in the grain since dissolved helium can move faster and escape from the grain surface easily. Considering that the activity of helium in the image crystal is approximately equal to that in the matrix, the inner pressure of

features varied depending on the heat treated condition, the image crystals that formed in the single crystal had the same shape and appeared to align with their edges parallel to a certain direction that might be related to the crystallographic orientation. An image crystal was also seen on the fracture surface of the polycrystalline sample. However, fewer image crystals appeared in the polycrystalline sample than in the single-crystal sample. Figure 6 shows typical images of the image crystals at high magnification. Their faceted nature is clearly visible. The SEM images of the single-crystal samples show the {111} cleavage surface, and the bottom face of the image crystals, which is almost parallel to the cleaved surface, is indexed as {111}. In the figure, the formation of other facets with higher Miller indices, {α}, {β}, and {χ}, is observed on the surface of the image crystal. According to the characteristics of crystals with Fm3̅m symmetry, the morphological importance (MI) of the facets follows the sequence MI{111} > MI{001} > MI{110} > MI{311}.7 This MI sequence was recently supported by a theoretical analysis of the attachment energy.27 On the basis of the attachment energy and SEM images, we modeled the shapes of the image crystals (Figure 7). The pentacontahedron (a) consists of the {111}, {100}, {110}, and {311} facets. The truncated octahedron (c) consists of the {111} and {001} facets; they are similar to the Wulff shape reported by Castell.6 The polyhedron (b) is an octa-triacontahedron bounded by facets indexed as {111}, {001}, and {311}. These models indicate that facets {α}, {β}, and {χ} in Figure 6 are indexed as {001}, {110}, and {311}, respectively. Figure 8a,b again shows SEM images of the image crystal formed at 1773 K; however, the emphasis is on the step bunching and growth of the {110} facets. In Figure 8a, a fullgrown {311} facet appears, whereas the {110} facet is almost invisible. Figure 8b is a high-magnification SEM image of area A of Figure 8a, where four {311} facets meet in the center of the image. Extensive steps and terraces appear around the small face that corresponds to the {110} facet. The complex nature of the steps is particularly evident in the region where the {110} facet meets two {311} facets. The rising sides of the steps are geometrically constrained to be close to the {111} and {001} facet orientations. These results imply that MI{311} > MI{110}, which disagrees with the morphology predicted by the attachment energy. However, considering that the image crystal in Figure 8 is seen during the growth process and far from equilibrium, the contradiction is not particularly surprising. As Sweegers et al.28 noted, the growth rate of a facet is not proportional to the attachment energy but depends on the step free energy (SFE). The SFE is defined as the free energy required to create a crystal step; it is regarded as a 2D or 3D E



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We denote the total surface areas of the {111} and {001} facets as s{111} and s{001}, respectively. Considering the geometric symmetry, the distances from the center of the truncated octahedron to each surface, d{111} and d{001}, are given as a function of the total surface areas of the facets respectively as

the cavity formed in the single-crystal sample should be higher than that in the polycrystalline sample. The release of the dissolved helium by heat treatment changes the activity of helium in the matrix and the inner pressure of the cavity decreases simultaneously. When the rate of mass transport of UO2 is sufficiently high, the shape of the image crystal will convert to the equilibrium one, the surface energy of which is proportional to the inner pressure. Unfortunately, we cannot confirm whether equilibrium was attained in our series of experiments. However, as Figure 6 shows, the shape of the image crystal in the polycrystalline sample is the same for annealing temperatures ranging from 1573 to 1973 K, whereas the image crystal in the single crystal was exceedingly deformed, which strongly suggests that the structure of the image crystal depends on the inner pressure rather than the annealing temperature. To grasp the relationship between the inner pressure and morphology of the image crystal, we examined its surface free energy. Despite zealous efforts by researchers,6,27,34 little is known about the morphology of UO2. Abramowski et al. examined the morphology of UO2 by first-principle calculation.27,34 They described that the calculated surface energies are 2.72 J m−2 for the {001} surface and 1.27 J m−2 for the {111} surface. The surface energy ratio calculated with these values is 2.14, which is much larger than the experimentally determined one, 1.42 ± 0.05. In addition, as is pointed out by Castell,6 an STM study on the UO2 (001) surface revealed that the surface has a maze-like (1 × 1) domain structure on different terraces. However, the complex structure did not take into account the theoretical model treated by Abramowski et al.35 Accordingly, the only credible knowledge regarding the surface energy of UO2, which is currently available, is the surface free energy ratio γ{001}/γ{111}, which makes it difficult to quantitatively investigate the surface free energy of the image crystal. In the following analysis, we assumed that the image crystal is deformed only by mass transport, which means that once it forms, its volume does not change throughout the growth process. First, we examined the truncated octahedron-type image crystal observed in the single-crystal UO2 annealed at 1973 K.

d{111} =

sin θβ S{001} 2 2

⎧ ⎪ + ⎨( 4 3 ) ⎪ ⎩

⎛ 3 3 S{001} S{111} ⎞ 1 ⎜⎜ + ⎟⎟ − 32 8 ⎠ 4 ⎝

⎫ 3S{001} ⎪ ⎬ sin θα 2 ⎪ ⎭ (1)

d{001}

⎧ ⎪ = 2⎨( 4 3 ) ⎪ ⎩

⎛ 3 3 s{001} s{111} ⎞ 1 ⎜⎜ ⎟⎟ − + 32 8 4 ⎝ ⎠

sin θβ

⎫ 3s{001} ⎪ ⎬ 2 ⎪ ⎭ (2)

where θα and θβ indicate interfacial angles of the adjacent surfaces (111) and (11̅ 1), and (111) and (001), respectively. Using these parameters, the volume, V, is expressed as V = 4 3 d{111}3 − 4( 3 d{111} − d{001})3

(3)

Because the numerical values of d{111} and d{001} vary with the size of the image crystal, we selected the ratio d{001}/d{111}, denoted as rd, as the primary parameter in the following analysis. When the volume and rd are fixed, the shape of the truncated octahedron is uniquely identified. The averaged ratio ⎛ d{001} ⎞ ⎟⎟ = 1.27 ± 0.067 rdexp = ⎜⎜ ⎝ d{111} ⎠exp

(4)

will be the value that is used from now on as the measured ratio for the truncated octahedron-type image crystal obtained in this study. As mentioned above, Castell examined the surface free energy ratio between the {111} and {001} facets of UO2 and estimated the ratio γ{001}/γ{111} as 1.42 ± 0.05 by measuring the surface area ratio of the Wulff-shaped voids.6 Moreover, it is known that for two faces with surface free energies γ1 and γ2, respectively, the distances d1 and d2 between the parallel planes h1k1l1 and h2k2l2 will be7,36 d1/d 2 = γ1/γ2

(5)

Therefore, d{001}/d{111} for the truncated octahedron-type image crystal with the Wulff shape is evaluated as 1.42 ± 0.05. Here we define the Wulff-shaped image crystal with a fixed size that is used as a mold for making things. Assuming that the equilibrium values, deq{001} and deq{111}, are 1.42 and 1 μm, respectively, at which d{001}/d{111} = 1.42, the volume Veq is 6.807 μm3. Next, we consider the variations in d{001} and d{111} with decreasing rd while the volume calculated by eq 3 is maintained at 6.807 μm3 throughout the transformation. As the {001} facet grows, the surface area increases with a dramatic decrease in d{001}, which results in a simultaneous increase in d{111} with the contraction of the {111} facet. When the ratio further decreases to 0.8660, the two sets of facets enclose a cuboctahedron with two triangles and two squares meeting at each facet. The difference between the Wulff shape and the shape of the observed image crystal is quite evident. Because of the

Figure 9. Dependence of d{100} and d{111} on rd when the volume is kept constant. F



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evaluated by image analysis. Although the data obtained in this study show some scatter, the averaged and calculated values agreed, which implies that the premise of the analysis, that the image crystal is deformed only by mass transport, is a valid explanation of the growth of facets on the truncated octahedron-type image crystal. Previous researchers have discussed the mechanical force balance on a gas-filled bubble embedded in a solid medium.18,37 For an equilibrium bubble in a stress-free solid, the force balance takes the form p = 2γ/R, where p, γ, and R represent the inner gas pressure, surface free energy, and bubble radius, respectively. We consider two spherical gas bubbles having different surface energies but the same volume. In this case, the ratio of the inner pressures, p1/p2, is equal to that of the surface free energies, γ1/γ2. We cannot adopt this exact relationship because the image crystal is not a sphere with a homogeneous surface free energy. However, the surface energies of the different facet types appearing on the Wulff-shaped crystal are sufficiently close to each other. Thus, we regard the surface free energy of the image crystal as homogeneous and use this relationship to analyze the inner pressure in truncated octahedron-type image crystals. Therefore, we can consider the vertical axis of Figure 10 as the ratio of the inner pressure p(rd)/peq. As the figure shows, the Γ(rd)/Γeq value of the image crystal observed in this work is about 1.03, which implies that its shape is deformed to increase the surface free energy so as to balance the higher inner pressure originating in the helium enclosed in the image crystal. Earlier we considered that the image crystal changes the shape continuously from the truncated octahedron to the cuboctahedron with increasing inner pressure. However, we could not identify a cuboctahedron-type image crystal in this investigation. Instead of a simple shaped cavity, a pentacontahedron-type image crystal formed in the matrix. To clarify the transformation mechanism, we examined the structure of this complicated polyhedron using image analysis. The s{hkl }/s{111} value obtained by this analysis and the structural parameters simulated using the computer code SHAPE38 are summarized in Table 2. The distance parameter d{hkl }*/d{111} in the table

outstanding development of the {001} facet, the image crystal is more rounded than the Wulff shape. Using these two parameters, d{111} and d{001}, the relationship between the shape and the surface free energy can be clarified. However, because the absolute values of the surface free energies for the {111} and {001} facets are not available, we examine the relative value normalized by the surface free energy of the Wulff-shaped image crystal, which has the same volume. The areas of the two facets, s{111} and s{001}, are expressed as functions of d{001} and d{111}, respectively, as follows: s{111} = 12{ 3 d{111}2 − ( 3 d{111} − d{001})2 }

(6)

s{001} = 12{ 3 d{111} − d{001}}2

(7)

The surface free energy of the image crystal with the Wulff shape, denoted as Γeq, is defined as s{111}eqγ{111} + s{001}eqγ{001} Γeq = s{111}eq + s{001}eq =

18.76γ{111} + 1.169γ{001} (8)

19.929

where γ{hkl} is the surface free energy of the {hkl} facet per unit area. Although γ{hkl} is generally a function of temperature, we treat it as a constant because the surface of the image crystal consists of well-grown facets of which the surface entropy should be small enough for the temperature dependence of the surface free energy to be considered negligible. The surface free energy of the truncated octahedron-type image crystal Γ(rd) normalized by Γeq is thus expressed as

{

γ

19.929 s{111}r d + s{001}r d γ{001}

}

Γ(rd) {111} = γ{001} ⎞ ⎛ Γeq ⎜18.76 + 1.169 ⎟(s + s{001}r d) γ{111} ⎠ {111}r d ⎝

(9)

The dotted line in Figure 10 shows the Γ(rd)/Γeq dependence on rd given by eq 9. The surface free energy of the truncated octahedron-type image crystal increases consistently along the dotted line as the shape changes continuously from the Wulff shape to a cuboctahedron. The solid circle and the error bar represent experimental data and the standard deviation

Table 2. Relative Surface Areas of Facets Bounding the Pentacontahedron-Type Image Crystal Formed in SingleCrystal UO2 facets

(s{hkl}/s{111})exp

s{hkl} (μm2)

s{hkl }*/s{111}*

d{hkl}* /d{1111}

no. faces

{111} {001} {110} {311}

1 0.1633 ± 0.0227 0.2148 ± 0.0571 1.435 ± 0.4008

6.382 1.043 1.365 9.160

1 0.1634 0.2139 1.435

1 1.080 1.129 1.059

8 6 12 24

was determined so that s{hkl}*/s{111}* agreed with (s{hkl}/ s{111})exp. The shape of the pentacontahedron is specified uniquely using these parameters. The total surface area of this model, which has a volume of 6.807 μm3, is 17.95 μm2; this value is obtained by summing the surface areas of the facets in the third column of the table. The variation in the total surface area of the image crystal arising from the transformation in its shape is shown in Figure 11. The dotted line is the theoretical value calculated using eqs 6 and 7. The solid circle and the error bar represent the experimental data and the standard deviation by image analysis. The pentacontahedron-type image crystal labeled (d) was

Figure 10. Surface free energy dependence on rd when the volume of the octahedron is kept constant. G



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The shape of the image crystal varies so as to decrease the surface area by the growth of facets with higher indices. However, the cuboctahedron-type image crystal could not be found.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +81(29)-282-6380. Fax: +81(29)-282-6122. E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. H.S. and S.Y. developed the concept and organized the project. J.M., Y.H., S.K., and K.N. carried out the experiments. M.A., T.T., and Y.K. carried out the simulation to determine the model of the truncated octahedron-type negative crystal. Y.O. performed the refinement of the structure parameter of pentacontahedron-type negative crystal. Figure 11. Surface area dependence on rd when the volume of the octahedron is kept constant.

Notes

drawn using the structural parameters listed in Table 2. The surface area decreases with decreasing rd but increases in the region where rd ≤ 1, whereas the relative surface area of the pentacontahedron-type image crystal is much smaller than that of the truncated octahedron and cuboctahedron. The surface free energy of the truncated octahedron-type image crystal increases with decreasing rd. However, the decrease in rd is limited; our experimental results indicate that the lowest value is 1.08, which is the value for truncated octahedron named (c) in Figure 11. When the inner pressure becomes much higher, the shape of the polyhedron changes so as to decrease the surface area by the growth of facets with higher indices to form a complex polyhedron rather than the cuboctahedron which has a larger surface area. As the inner pressure increases, the facets with much higher indices should appear on the surface of the image crystal to make the shape round. These results imply that the round shape of the cavity formed just after the helium treatment (Figure 4b) is attributed to the high inner pressure. We conclude that the image crystal is formed since the inner pressure decreased by the helium release during the subsequent heat treatment. We could produce the image crystal using HIP because the solubility of helium in UO2 is high enough to form the gas bubble. The next issue is how to inject helium into the sample in which helium solubility is low. We are now investigating the formation of image crystals in other ceramics, which can possibly lead to the development of a new functional material. In this study, we discussed mainly the transformation of the image crystal. However, the relationship between the growth mechanism of the cavity and a defect cluster is still in question. We consider that the clarification of the void formation process is another important point for further investigation.

ACKNOWLEDGMENTS This study is the result of the project “Development of common and fundamental technologies on the evaluation of nuclear fuel behavior for realizing MA recycling”, which was entrusted to the Japan Atomic Energy Agency by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). This study was also supported by JSPS KAKENHI Grant Number 23246174. The authors are grateful to Special Adviser to the President Dr. Sohei Okada (JAEA) for his encouraging advice and suggestions.

The authors declare no competing financial interest.

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ABBREVIATIONS SFE, CC step free energy REFERENCES

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Formation and Growth of Image Crystals by Helium ... - ACS Publications

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