Modeling Study of the Valid Apparent Interface Thickness in

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Modeling Study of the Valid Apparent Interface Thickness in Particulate Materials with Ellipsoidal Particles Wenxiang Xu,*,†,‡ Huisu Chen,‡ Wen Chen,† Xia Tian,† and Haitao Zhao§ †

Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China Jiangsu Key Laboratory of Construction Materials, School of Materials Science and Engineering, Southeast University, Nanjing 211189, China § College of Civil and Transporting Engineering, Hohai University, Nanjing 210098, China ‡

ABSTRACT: When interfacial layers are viewed as a separate phase, the interface thickness plays an essential role in assessing physicomechanical properties of particulate materials. However, the actual interface thickness is difficult to determine because of the opacity of particulate materials. The apparent interface thickness obtained from sectional analysis is often overestimated, due to the irregularity of surface texture of grain that gives rise to the normal of a cross-sectional plane nonperpendicular to the grain surface. Hence, the determination of the overestimation degree between the apparent value and preconfigured value is very critical to quantify precisely the valid apparent interface thickness. The majority of previous works focused on the overestimation degree for threedimensional (3D) spherical, two-dimensional (2D) elliptical and rectangular particles, whereas little is known about 3D ellipsoidal particles. In this work, a numerical scheme is proposed to obtain the overestimation degree of the interface thickness around ellipsoidal grains. On the basis of the quantitative stereology and geometrical probability, the valid statistical mean apparent interface thickness and overestimation degree are analyzed theoretically. The deduced results show that the two values primarily subject to an unknown coefficient. A sectional analysis numerical algorithm is then implemented to derive the coefficient for an ellipsoidal grain. To test the proposed numerical model, valid theoretical data for spherical and ellipsoidal particles is selected for comparisons. Finally, by the developed numerical model, the effects of geometrical characteristics of ellipsoidal grains and the preconfigured thickness on the overestimation degree and valid apparent interface thickness are investigated in a quantitative manner.

1. INTRODUCTION It is well-known that relative porous interfacial layers exist around grains in soft matter and particulate materials, such as colloids, cementitious, and ceramic materials, due to the wall effect of the grains.1−4 It has been experimentally and numerically investigated that microstructural characteristics of interfacial layers such as the interface thickness and volume fraction play an important role in macroscopic physical properties of materials.5−7 Nevertheless, the actual interface thickness is difficult to capture because of the opacity of particulate materials. Normally, sectional analysis is employed to assess the apparent interface thickness in different experimental and numerical schemes.2,8−10 For instance, Scrivener et al.2 investigated the backscattered electron (BSE) image of 2D sections of concrete by sectional analysis for measuring directly the apparent interface thickness. However, they concluded that this experimental scheme is very time-consuming and that the apparent interface thickness measured on 2D sections are on average greater than the actual value in the 3D microstructure. Furthermore, it is a very difficult task to evaluate quantitatively the overestimation degree from the 2D section investigation by experimental schemes. On the other hand, Stroeven and Hu8 applied a sectional analysis to obtain the curve of the volume fraction of the solid phase composed of the random packing of hard spherical particles, in which the apparent interface thickness is defined as the distance with respect to the range of the ascending region of such a curve.2,3,8 Subsequently, we introduced two sectional analysis numerical algorithms to derive the curves of the volume fraction of the solid phase composed of the random packings of ellipsoidal and dodecahedral particles for reflecting the © 2013 American Chemical Society

apparent interface thicknesses around ellipsoidal and dodecahedral particles, respectively.9,10 However, in those experimental and numerical schemes, a normal sectional plane/line is seldom perpendicular to the grain surface, owing to the complexity of surface textures of irregular-shaped grains, so that the derived results of the apparent thickness trend to overestimation. Therefore, to access the correct information on the apparent interface thickness, the overestimation degree of the apparent interface thickness from sectional analysis needs to be determined. In the past decade, some efforts have been devoted to evaluating the overestimation degree of the interface thickness by theoretical and modeling approaches. By virtue of the quantitative stereology, Stroeven11 investigated the overestimation degree for a 2D circular particle equivalent to π/2. Afterward, the overestimation degrees for 2D elliptical and rectangular particles were also assessed by computer simulation.12 Although this important research may provide guidance for understanding the overestimation degree of the interface thickness, these 2D investigations cannot reflect truly the overestimation degree of the interface thickness around 3D grains in particulate materials. It is worth mentioning that, by applying the geometrical probability, Chen et al.13,14 put forward a general expression for the overestimation degree for a convexshaped grain. Nevertheless, the solution is subjected to an Received: Revised: Accepted: Published: 17171

September 12, 2013 November 11, 2013 November 11, 2013 November 11, 2013 dx.doi.org/10.1021/ie403009c | Ind. Eng. Chem. Res. 2013, 52, 17171−17178

Industrial & Engineering Chemistry Research

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unknown so-called correction factor α (which will be introduced below), and the correction factors for 2D circular and 3D spherical grains were only given in that work. As described by Chen et al.,14 “it is a difficult task to derive the exact expression of the correction factor for 2D or 3D convex-shaped grains.” To the authors’ knowledge, at present, there is still no related information on the correction factor and overestimation degree of interface thickness around 3D nonspherical particles. It is of particular interest, as a more suitable approximation for real grains, that ellipsoidal particles have been extensively used to simulate grains in the modeling study of materials.7,9,15−19 It is our intention in the present work to address this gap. This article attempts to develop a numerical model for the overestimation degree and the valid apparent interface thickness around an ellipsoidal grain with an arbitrary aspect ratio and to present a simulation scheme to derive the correction factor. The remainder of this work is outlined as follows: In section 2, a geometrical analysis on the valid statistical mean apparent interface thickness around an ellipsoidal grain is presented. In section 3, a numerical simulation scheme for the correction factor for an ellipsoidal particle is elaborated. The present numerical model is tested in section 4. In section 5, with the developed numerical model, the effects of various factors on the overestimation degree and valid apparent interface thickness are discussed in a quantitative manner. The final section displays our conclusions.

Figure 1. Schematic view of an interfacial layer around an ellipsoidal grain E1. L1i and L2i are one of the intercept lengths from sectional lines intersecting E1 and E2.

A cross-sectional plane intercepting the grain is equivalent to a set of parallel lines in that plane; in other words, the mean apparent interface thickness from the cross-sectional plane analysis is consistent with the statistical mean intercept length from those of sets of parallel lines intersecting the interfacial layer, as shown in Figure 2. Thus, the statistical mean apparent interface thickness ta around E1 can be obtained by respectively calculating the mean intercept lengths L1 and L2 from sectional lines intersecting E1 and E2. Let us now consider the mean intercept length L1. It is supposed that a set of cross-sectional planes and sectional lines in each cross-sectional plane parallel to the XY-plane and Y-axis of a global Cartesian coordinate system and traverse E1, respectively, as shown in Figure 2. According to Cauchy’s

2. GEOMETRICAL ANALYSIS At the microscopic scale, particulate materials are normally regarded as a three-phase composite structure, composed of the matrix, ellipsoidal grains and interfacial layers with a preconfigured constant thickness t coated around the grains.2−4,11−14,20−22 Also, the surface spacing between neighboring grains is considered to be large enough to neglect the interference between adjacent interfacial layers. As an introductory example, a spheroidal grain E1 is considered, and its size is represented by an equivalent diameter Deq1 that is defined as the diameter of a sphere having the same volume as the spheroidal grain,7,9,17,18 as given the following equation. Deq1

1/3 ⎧ κ1 < 1 ⎪ 2a1κ1 =⎨ ⎪ −2/3 κ1 ≥ 1 ⎩ 2a1κ1

Figure 2. Schematic view of a set of cross-sectional planes and sectional lines in each cross-sectional plane intersecting the ellipsoidal grain E1. S1xz is the projected area of E1 mapping onto the XZ-plane.

(1)

where a1 and κ1 are the semimajor axis and aspect ratio of the spheroid, respectively. κ1 = a1/c1 as the spheroid shape is prolate, or κ1 = c1/a1 as the spheroid shape is oblate, where c1 is the semiminor axis of the spheroid. As shown in Figure 1, an interfacial layer around the grain E1 can be denoted as the remaining region that the external spheroid E2 with the equivalent diameter Deq2 and the aspect ratio κ2 minus the grain. Actually, the external ellipsoid E2 originates from the grain E1 with a preconfigured thickness t, it means that the spatial position and orientation of E2 are the same as that of E1. Accordingly, the relationship between Deq2 and Deq1 can be given by eq 2.

Deq2

theorem,23,24 the average intercept length L1y along the Y-axis from sectional lines intersecting E1 can be denoted as

πDeq13 V1 L1y = = S1xz 6S1xz

(3)

where V1 is the volume of E1 equal to πDeq1 /6 and S1xz is the projected area of E1 mapping onto the XZ-plane, as shown in Figure 2. It is worth pointing out that L1y cannot replace L1, because the mean intercept length L1 mentioned above should be the average value over all possible directions, not just one parallel set. It indicates that the average projected area S1 of E1 mapping over all directions must be determined, that is 3

⎧ ⎡ D κ 2/3 + 2t ⎤−2/3 ⎪ 2/3 ⎢ eq1 1 ⎥ κ1 < 1 ⎪(Deq1κ1 + 2t )⎢ −1/3 ⎥⎦ + κ 2 D t ⎣ eq1 1 ⎪ =⎨ ⎪ 2/3 ⎡ ⎤1/3 ⎪(D κ −1/3 + 2t )⎢ Deq1κ1 + 2t ⎥ κ1 ≥ 1 ⎪ eq1 1 ⎢⎣ Deq1κ1−1/3 + 2t ⎥⎦ ⎩

L1 =

V1 S1

(4) 23

According to Cauchy’s theorem, the average projected area S of a convex particle is equal to the quarter of the surface area of the particle. Fortunately, for an ellipsoidal particle with its

(2) 17172

dx.doi.org/10.1021/ie403009c | Ind. Eng. Chem. Res. 2013, 52, 17171−17178


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equivalent diameter Deq and the aspect ratio κ, its surface area has been presented in our previous works.7,10,18 Thus, the average projected area S of the ellipsoidal particle can be given by S=

πDeq 2 A = 4 4n(κ )

(5)

where A and n(κ) are the surface area and sphericity of the ellipsoidal particle, respectively. n(κ) is defined as the ratio of the surface area between a sphere and an ellipsoid with the same volume,4,17,25 that is n(κ ) =

2κ 2/3 sin φ sin φ + κ 2 arctanh(sin φ)

(κ < 1)

Figure 3. Schematic view of valid sectional lines (thin black dashed lines) and invalid sectional lines (bold red dashed lines).

or

excluded region (i.e., the region untouched by the valid sectional lines) to the total volume of the region (E2 − E1) over all directions. The value of the coefficient is dependent on the grain shape and the preconfigured interface thickness.14 Thus, in eq 10, the numerator V2 should be replaced by Vv2. Consequently, the valid mean intercept length Lv2 from valid sectional lines intersecting E2 is given by

2/3

1

(κ = 1)

or

2κ tan φ tan φ + κ 2φ

(κ > 1)

(6)

where φ is defined as φ = arcos (η), η = 1/κ (κ > 1) or η = κ (κ < 1). On the basis of eq 5, the average projected area S1 of E1 mapping over all directions can be written as

S1 =

2n(κ1)[(1 − δ)Deq2 3 + δDeq13] Vv2 Lv2 = = S1 3Deq12

πDeq12 4n(κ1)

(7)

Therefore, according to eqs 8 and 12, the valid statistical mean apparent interface thickness tsa around E1 can be presented by

where n(κ1) is the sphericity of E1. Thus, when eq 7 is substituted into eq 4, the mean intercept length L1 from sectional lines intersecting E1 is given by L1 =

2 Deq1n(κ1) 3

tsa =

(8)

πDeq2 2 4n(κ2)

n(κ1)(1 − δ)(Deq3 2 − Deq3 1) tsa Od = == t 3tDeq2 1

(9)

where n(κ2) is the sphericity of E2. Consequently, by means of the above scheme, the mean intercept length L2 from sectional lines intersecting E2 can be displayed by L2 =

πDeq2 3 V2 2 = = Deq2n(κ2) S2 6S2 3

(13)

(14)

From eqs 2 and 14, when the size and shape of E1 are known, the overestimation degree mainly depends on the coefficient δ and the preconfigured interface thickness t. Unfortunately, the value of the coefficient for a 3D nonspherical grain remains unknown. In the following section, a numerical algorithm will be presented to determine the coefficient for an ellipsoidal particle with an arbitrary aspect ratio.

(10)

where V2 is the volume of E2 equal to πDeq2 /6. Generally speaking, the statistical mean apparent interface thickness ta around E1 may be derived by eqs 8 and 10, that is, ta = 0.5(L2 − L1). Nevertheless, this value is not the valid apparent interface thickness around E1. The reason is that, for the sets of sectional lines, only the ones that intersect with the grain E1 are valid, those of sectional lines that locate merely within the region of the interfacial layer are invalid, as shown in Figure 3. Hence, the valid average projected area of E2 from the valid sectional lines should be S1 rather than S2, so that the denominator S2 should be replaced by S1 in eq 10. Furthermore, the valid volume Vv2 of E2 from the valid sectional lines intersecting E2 should not be the whole volume V2 of E2 but should be the whole volume V2 of E2 minus the region occupied by invalid sectional lines, that is 3

Vv2 = V2 − δ(V2 − V1)

n(κ1) 1 (Lv2 − L1) = (1 − δ)(Deq2 3 − Deq13) 2 3Deq12

Further, the overestimation degree Od of the interface thickness around E1 is characterized as the ratio of the valid statistical mean apparent thickness to the preconfigured thickness of the interfacial layer:

Similarly, according to eq 5, the average projected area S2 of E2 mapping over all directions is expressed as

S2 =

(12)

3. NUMERICAL SIMULATION FOR δ In the section, the coefficient δ for the ellipsoidal grain E1 with the equivalent diameter Deq1 and aspect ratio κ1 is considered. According to eq 11, the valid volume Vv2 from valid sectional lines intercepting E2 is a key parameter for deriving the coefficient δ. Thereby, the following numerical algorithm is implemented to derive Vv2 and δ, and the flowchart of the numerical algorithm is presented in Figure 4. (1) Randomly generate the center (x1, y1, z1) and three Euler angles (φ1, ϕ1, γ1) of E1. According to the relationship of coefficients of a quadratic surface equation and nine degrees of freedom of an ellipsoid,9,18 the geometrical models of E1 and E2 can be represented by a quadratic surface equation:

(11)

XTQ 1X = 0

where δ is a coefficient smaller than 1 that is actually the correction factor described by Chen et al.,14 and its geometrical interpretation is exactly the mean ratio of the volume of the

T

X Q 2X = 0 17173

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(4) In the valid cross-sectional plane pi, let a set of sectional lines at equal spacing parallel to Y-axis, as shown in Figure 2. Likewise, in the set of sectional lines, it exists at a minimum value (Y = lmin) and maximum value (Y = lmax) in the direction of Y-axis to traverse the section ellipse SEi1. It indicates that valid sectional lines as {Y = lij, 1 ≤ j ≤ M} in the direction of Y-axis must be located in the range of [lmin, lmax], where M is the number of valid sectional lines. According to the theory of analytical geometry26 and eq 18, lmin and lmax are denoted as lmin,max =

−b′ ±

b′2 − a′c′ a′

(20)

where a′ = D12 − A1B1 b′ = D1(G1 + Fp 1 i ) − A1(H1 + E1pi ) 2 2 c′ = (G1 + Fp 1 i ) − A1(C1pi + 2I1pi + J1)

(5) In pi, by computing the intercept lengths from the valid sectional lines intersecting the sectional ellipse SEi2 (of which the computational method has been described in the literature27), the valid area from M valid sectional lines occupying to SEi2 can thus be determined easily. (6) Iterate step 3 to step 5, derive the valid statistical volume from N valid cross-sectional planes occupying to E2, namely, the statistical volume from M × N valid sectional lines occupying to E2. (7) Iterate step 1 to step 6, determine the statistical average volume Vv2 from valid sectional lines intercepting E2 after T iterations. Thus, according to eq 11, the coefficient δ of E1 can be obtained by

Figure 4. Flowchart of the numerical algorithm for deriving Vv2 and δ.

where Q1 and Q2 are coefficient matrixes of the geometrical models of E1 and E2, respectively. XT is the transposed matrix of X. (2) As described above, a family of cross-sectional planes at equal spacing is set normal to the Z-axis, as shown in Figure 2. In the family of sectional planes, it apparently exists at a minimum value (Z = pmin) and maximum value (Z = pmax) tangent to E1 in the direction of Z-axis. In other words, those of cross-sectional planes as {Z = pi, 1 ≤ i ≤ N} normal to the Z-axis located within the interval of [pmin, pmax] are only valid for Vv2, where N is the number of valid cross-sectional planes. By means of the coordinate invariant complete system of a quadratic curve,18,26 pmin and pmax can be obtained by eq 17. A1

D1

Fp 1 i + G1

D1

B1

E1pi + H1

6

δ=

(17)

where A1, B1, ..., and J1 are coefficients of the matrix Q1. (3) Because the cross-sectional morphology from the common intersection between a cross-sectional plane and an ellipsoid must be an ellipse, the geometrical model of the sectional ellipse SEi1 from the intersection between a valid cross-sectional plane pi and E1 is derived,27 as depicted in eq 18. Similarly, the geometrical model of the sectional ellipse SEi2 from the intersection between pi and E2 can also be obtained by eq 19.

(18)

A 2 x 2 + B2 y 2 + 2D2xy + 2(G2 + F2pi )x + 2(H2 + E2pi )y + C 2pi2 + 2I2pi + J2 = 0

(21)

4. TEST To test the numerical model developed above, the analytical solutions for the correction factor and overestimation degree of the interface thickness around the 3D spherical grain from Chen et al.14 are chosen for comparisons. In their solutions, the radius of the spherical particle and the preconfigured interface thickness are supposed to be r and t, the statistical average volume Vv2 from valid sectional lines occupying to the spherical shell can be expressed as14

A1x 2 + B1y 2 + 2D1xy + 2(G1 + Fp 1 i )x + 2(H1 + E1pi )y + C1pi2 + 2I1pi + J1 = 0

Deq2 3 − Deq13

Notably, to guarantee the reliability of the numerical results, the number of valid cross-sectional planes and of valid sectional lines should be selected large enough, and the number T of iterations needs to meet the representative volume element (RVE).28−30 In this work, N and M are both set to 10000, and the coefficient of variation of the statistical result of the coefficient δ used as a criterion for determining RVE is set to 0.01.

=0

Fp 1 i + G1 E1pi + H1 C1

Deq2 3 − π Vv2

Vv2 =

(19)

=

where A2, B2, ..., and J2 are coefficients of the matrix Q2. 17174

∫0

r

4πx (r + t )2 − x 2 dx

4 4 π (r + t )3 − π[(r + t )2 − r 2]3/2 3 3

(22)



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According to eqs 21 and 22, the correction factor for the spherical particle is displayed by α=δ=

[(r + t )2 − r 2]3/2 (r + t )3 − r 3

(23)

On the basis of eqs 14 and 23, the overestimation degree Od of the interface thickness around the spherical grain is written as Od =

2(1 − δ)[(r + t )3 − r 3] 3tr 2

(24)

Figure 5 depicts comparisons of the analytical results from eqs 23 and 24 with the numerical results by the present numerical scheme for δ and Od. From Figure 5, it can be seen

Figure 6. Comparisons of the numerical results of the volume of E1 from the present scheme with that of the theoretical results for κ1 = 0.8 and κ1 = 3.0. Numerical results of the volume of E1 with the preconfigured thickness of t = 0.05 mm.

in surprise that δ and Od fall in the order n(κ1=1.0) = 1.0 > n(κ1=2.0) = 0.928 > n(κ1=0.4) = 0.852. It suggests that the valid statistical mean apparent interface thickness potentially increase with the increase of sphericity n(κ1). Accordingly, we further investigate the numerical results of δ and Od for various sphericities n(κ1), as shown in Figure 8. As expected, it can be clearly seen that Figure 8 proposed the influence of sphericity on δ and Od is in good agreement with that shown in Figure 7; that is, the valid statistical mean apparent interface thickness is a monotonically increasing function with the increase of sphericity. It is the first time to our knowledge that such a correlation is shown. More interestingly, on the basis of the definition of sphericity, Figures 7 and 8 display that the valid statistical mean apparent interface thickness primarily depends upon the surface area of the grains. It means that the numerical scheme first proposes a quantitative relationship between the surface area of the grains and the valid statistical mean apparent interface thickness, even if characteristics of interfaces related to the surfaces of the grains were mentioned in previous experimental works.2,31 To validate the quantitative scheme proposed above, we examine the numerical results of δ and Od for two types of grains with the same surface areas, namely, with the same sphericity n(κ1). Without loss of generality, the prolate ellipsoidal particle with n(κ1=1.743) = 0.952 and the oblate ellipsoidal particle with n(κ1=0.6) = 0.952 are considered. The numerical results of δ and Od for the two types of grains are derived by the present numerical model, as shown Figure 9. As can be seen from Figure 9, under the same n(κ1), δ and Od for the two types of grains are both in good agreement, respectively. Moreover, the relative errors between them are very small for δ and Od, respectively. It reveals that, for the given other parameters, the coefficient and overestimation degree are only controlled by sphericity, namely, the surface area of the grains, rather than by the aspect ratio of the grains. Therefore, it can be confirmed that the quantitative scheme where the smaller surface area of the grains generates the larger apparent interface thickness is reasonable. This case is theoretically attributable to the rapid increase of the packing density of hard convex particles as particles deviate from spheres developed by Torquato and his co-workers15,32−34 and Yu and his co-workers,17 which leads to the shrinkage of interface zones with the denser packing of convex grains so as to decrease the interface thickness around grains, owing to the fact that the formation of interface zones is originated from the “packing” constraint of grain surface on matrix.2−4,35 On the other hand, experimentally, the required amount of matrix for

Figure 5. Comparisons of the analytical results by Chen et al.14 with the numerical results by the present numerical scheme for (a) δ and (b) Od. Analytical results of δ and Od with the preconfigured thickness of t = 0.05 mm.

that the numerical results of δ and Od are both in good agreement with that of the analytical results. To further test the numerical scheme, we compare the numerical results of the volume V1 of the ellipsoidal particle E1 from valid sectional lines occupying to E1 with that of the theoretical results for two aspect ratios κ1 = 0.8 and κ1 = 3.0, which can reflect the reliability of the numerical scheme for the statistical average volume Vv2 from valid sectional lines occupying to E2, as shown in Figure 6. It can be clearly seen from Figure 6 that the numerical volumes V1 from the present scheme are consistent with the theoretical volumes for different sizes and aspect ratios. Therefore, the validity of the present numerical scheme is favorable.

5. DISCUSSION As described in section 2, key factors that affect the overestimation degree Od and coefficient δ factor include the shape and size of ellipsoidal grains and the preconfigured interface thickness. It is, therefore, of practical significance to evaluate the effects of these factors on δ and Od. Figure 7 shows the effect of the aspect ratio κ1 of the ellipsoidal particle E1 on the coefficient δ and the overestimation degree Od. It can be seen from Figure 7 that, for the same Deq1, δ and Od fall in the order spherical particle κ1 = 1.0 > prolate ellipsoidal particle with κ1 = 2.0 > oblate ellipsoidal particle with κ1 = 0.4. That is to say, for both oblate and prolate spheroidal particles, δ and Od decrease as particles deviate from perfect spheres. It indicates that the assumption of the spherical grain overestimates practically the valid mean apparent interface thickness. Interestingly, when we transform each aspect ratio to the corresponding sphericity according to eq 6, it can be found 17175



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Figure 7. Effect of the aspect ratio κ1 of the ellipsoidal particle E1 on the coefficient δ and the overestimation degree Od. Numerical results of δ and Od with the preconfigured interface thickness of t = 0.05 mm.

Figure 8. Effect of sphericity n(κ1) of E1 on the coefficient δ and the overestimation degree Od. Numerical results of δ and Od with the preconfigured interface thickness of t = 0.05 mm.

Figure 9. Comparisons of the numerical results of the coefficient δ and the overestimation degree Od for the two types of grains with the same sphericity n(κ1). Numerical results of δ and Od with the preconfigured interface thickness of t = 0.05 mm.

Additionally, panels a and b of Figure 10 display the coefficient δ versus the equivalent diameter Deq1 of the ellipsoidal grain E1 and the overestimation degree Od versus Deq1 for various the preconfigured thicknesses t, respectively. From Figure 10a, it can be seen that δ decreases with the increase of Deq1 for a given t but increases with the increase of t for a fixed Deq1. On the other hand, Figure 10b shows that Od increases with the increase of Deq1 for a given t but decreases with the increase of t for a fixed Deq1.

wrapping the surface of grains with large surface areas is normally more than that for wrapping the surface of grains with small surface areas, which induces the interface zones generated in the vicinity of grains with the large surface areas to be lower than those in the vicinity of grains with the small surface areas. Therefore, from the view of the interface thickness, we should select the shape of the grains deviated from the sphere as excellent grains to reduce the interface thickness in experimental works. 17176



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Figure 10. (a) Coefficient δ versus the equivalent diameter Deq1 and (b) the overestimation degree Od versus Deq1 for various the preconfigured interface thicknesses t. Numerical results of δ and Od with sphericity of n(κ1) = 0.952.

and the Ministry of Science and Technology of China “973 Project” (Grant Nos. 2009CB623203 and 2010CB832702).

That is to say, the valid statistical mean apparent interface thickness is an increasing function with the increase of the size of the grains for a constant preconfigured thickness t. Indeed, in experimental works, the required amount of matrix for wrapping the surface of grains with the large size is more than that for wrapping the surface of grains with the small size, so that the interface around the grains with the large size is smaller than that around grains with the small size. Furthermore, according to eqs 2 and 13, it can be found that the valid statistical mean apparent interface thickness is inversely proportional to the preconfigured thickness t for a given size of the grains.

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6. CONCLUSIONS This article proposed a numerical model to address the problem of the determination of the overestimation degree and valid statistical mean apparent interface thickness around an ellipsoidal grain with an arbitrary aspect ratio. The numerical model adopted a theoretical scheme for the valid statistical mean apparent interface thickness and a numerical algorithm for the unknown coefficient, or the correction factor mentioned in the literature. Compared with the theoretical models for spheres and ellipsoids, the numerical model displayed favorable accuracy. Furthermore, a quantitative scheme of the surface area of the grains characterized by sphericity on the valid apparent interface thickness was first proposed; that is, the larger sphericity of the grains generates the higher apparent interface thickness. As a theoretical criterion, it is very important to guide corresponding practical works.

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AUTHOR INFORMATION

Corresponding Author

*W. Xu: tel, +86-25- 83786873; fax, +86-25-83736860; e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors greatly appreciate the reviewers’ helpful comments for the quality of our article. We also acknowledge financial support from the Natural Science Foundation Project for Jiangsu Province (Grant No. BK20130841), National Science Foundation Project for Distinguished Young Scholars (Grant No. 11125208), National Natural Science Foundation Project of China (Grant Nos. 11202065, 51309090, and 20120094120016), 17177



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