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Triple junction structure and carbide precipitation in 304L stainless steel

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Abstract

Type 304L stainless steel (SS) samples were used to investigate the correlation between carbide precipitation and triple junction structure derived from crystallographic data obtained by the orientation imaging microscopy associated with electron backscattered diffraction. The samples were solution treated and annealed at different sensitization temperatures/time to introduce various degrees of carbide precipitation at the interface region, thus different degrees of selectivity toward triple junctions. Four models were used to characterize triple junction microstructures: (i) the I-line and U-line model, (ii) the coincident axial direction (CAD) model, (iii) the coincident site lattice (CSL)/grain boundary (GB) model and (iv) the plane matching (PM)/GB model. Among them, the I-line and U-line model is the most effective in identifying special triple junctions, i.e., those exhibiting the beneficial property of high resistance to carbide precipitation. The results showed that the percentage of special triple junctions (I-lines) immune to carbide precipitation, increased from 35 to 80%, as the precipitation became more selective toward triple junction structures due to the corresponding sensitization heat treatment conditions, whereas more than 80% of random triple junctions (U-lines) exhibited susceptibility to carbide precipitation regardless of the sensitization conditions.

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Acknowledgments

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Research Fund-Research Excellence (ORF-RE) is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of Toronto, Toronto, Ontario, M5S 3E4, Canada

    Yijian Zhou, Karl T. Aust & Uwe Erb

  2. Integran Technologies Inc., Mississauga, Ontario, L4V 1H7, Canada

    Gino Palumbo

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  1. Yijian Zhou
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  2. Gino Palumbo
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  3. Karl T. Aust
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Corresponding author

Correspondence to Yijian Zhou.

Appendices

Appendix - Triple junction characterization procedure

Here an example is presented to illustrate the calculation procedures of the four triple junction (TJ) models used in this study to determine the triple junction line characters.

First, the measured OIM orientation data are converted into rotation matrices, i.e., R1, R2, and R3 for grains 1, 2, and 3, respectively,

$$ {R1} \;\;\;\; = \;\;\;\; {\left[ { {1.4871}\;\;\;\;{.8178}\;\;\;\;{ -.3065} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { -.8663}\;\;\;\;{.4970}\;\;\;\;{ -.0507} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.1108}\;\;\;\;{.2902}\;\;\;\;{.9505} } \right],}{} \;\;\;\; {} \;\;\;\; {} \\ {R2} \;\;\;\; = \;\;\;\; {\left[ { {.2557}\;\;\;\;{.9362}\;\;\;\;{ -.2410} } \right]} {} \;\;\;\; {} \;\;\;\; {\left[ { { -.8399}\;\;\;\;{.3386}\;\;\;\;{.4242}} \right]} {} \;\;\;\; {} \;\;\;\; {\left[ { {.4787}\;\;\;\;{.09392}\;\;\;\;{.8729} } \right],} \\ $$
FIG. A1
figure A1

A schematic TJ with three grains, 1, 2, and 3, and three GBs, Ra, Rb, and Rc. The arrows show the sequence of misorientation calculation. The three grain orientation data from a sample TJ in terms of three Euler angles with Bunge’s notation are:

Full size image
$$ {{\text{grain}}\,{\text{1:}}\left[ { {159.1}\;\;\;\;{18.1}\;\;\;\;{260.6} } \right]} \\ {{\text{grain}}\,{\text{2:}}\left[ { {101.1}\;\;\;\;{29.2}\;\;\;\;{330.4} } \right].} \\ {{\text{grain}}\,{\text{3:}}\left[ { {43.5}\;\;\;\;{50.0}\;\;\;\;{42.4} } \right]} \\ $$
$$ {R3} \;\;\;\; = \;\;\;\; {\left[ { {.2373}\;\;\;\;{.8227}\;\;\;\;{.5165} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { -.8159}\;\;\;\;{ -.1198}\;\;\;\;{.5657} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.5273}\;\;\;\;{ -.5557}\;\;\;\;{.6428} } \right].} \\ $$

Then the GB misorientations between two adjacent grains in reference to the crystal coordinates are calculated as follows (note that R−1 is an inverse matrix of R):

$$ {{R_a}} \;\;\;\; = \;\;\;\; {\left[ { {.9052}\;\;\;\;{.1732}\;\;\;\;{ -.3881} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { -.06935}\;\;\;\;{.9612}\;\;\;\;{.2671} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.41928}\;\;\;\;{ -.2149}\;\;\;\;{.8821} } \right],} \\ {} \;\;\;\; {} \;\;\;\; {} \\ {{R_b}} \;\;\;\; = \;\;\;\; {\left[ { {.9984}\;\;\;\;{04502}\;\;\;\;{ -.3531} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { -.004540}\;\;\;\;{.6775}\;\;\;\;{.7355} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.05704}\;\;\;\;{ -.7342}\;\;\;\;{.6766} } \right],} \\ {} \;\;\;\; {} \;\;\;\; {} \\ {{R_c}} \;\;\;\; = \;\;\;\; {\left[ { {.8808}\;\;\;\;{ -.05836}\;\;\;\;{.4699} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.4430}\;\;\;\;{.4520}\;\;\;\;{ -.7743} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { -.1672}\;\;\;\;{.8901}\;\;\;\;{.4240} } \right].} \\ $$

Determination of TJ character using Bollmann’s model

The main idea in this model is to determine a unimodular matrix that leads to the nearest neighbor relation (NNR) with the adjacent grain. The starting point is the f.c.c. structure matrix S as the crystal unit cell,

$$ S \;\;\;\; = \;\;\;\; {\left[ { {.5}\;\;{ - 5}0 } \right]} \\ {} \;\;\;\; {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left[ { {.5}\;\;\;\;{.5}\;\;\;\;{.5} } \right]} \\ {} \;\;\;\; {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\left[ { 0\;\;\;\;0\;\;\;\;\;{.5} } \right].} \\ $$

Convert the GB matrices with reference to the unit cell, e.g. Ra, and get a new form of GB matrix, e.g. Ra ′

$$ {Ra'} \;\;\;\; { = {{\text{S}}^{ - 1}}{R_a}S = } \;\;\;\; {\left[ { {0.8829}\;\;\;\;{0.4663}\;\;\;\;{0.1731} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { - 0.1955}\;\;\;\;{1.198}\;\;\;\;{0.3880} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {0.2044}\;\;\;\;{ -.6341}\;\;\;\;{0.6672} } \right].} \\ $$

Then Ra ′ is decomposed into external (M_ext) and internal (M_int) matrix. The entries in the external matrix must be integers, whereas those in the internal matrix have values between 0 and 1 so that the column vectors in M_int are all inside the unit cell:

$$ {} \;\;\;\; {{{R'}_a} = M\_{\text{ext + }}M\_{int}} \;\;\;\; {} \\ {\left[ { {.8829}\;\;\;\;{.4663}\;\;\;\;{.1731} } \right]} \;\;\;\; {} \;\;\;\; {\left[ { 000 } \right]} \\ {\left[ { { -.1955}\;\;\;\;{1.198}\;\;\;\;{.3880} } \right]} \;\;\;\; = \;\;\;\; {\left[ { { - 1}10 } \right]} \\ {\left[ { {.2044}\;\;\;\;{ -.6341}\;\;\;\;{.6672} } \right]} \;\;\;\; {} \;\;\;\; {\left[ { 0{ - 1}0 } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.8829}\;\;\;\;{.4663}\;\;\;\;{.1731} } \right]} \\ {} \;\;\;\; {} \;\;\;\; { + \left[ { {.8045}\;\;\;\;{.1984}\;\;\;\;{.3880} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { {.2044}\;\;\;\;{.3659}\;\;\;\;{.6672} } \right]} \\ $$

The next step is to find the closest corner of the unit cell for each column vector in M_int. Since the unit cell coordinate is a nonorthonormal coordinate, the metric tensor G is required for the calculation of the (square) distance \({d_i}^2\) between a vector, vi, and a corner, ci,

$$d_i^2 = \Delta x_i^{\text{T}}G{x_i},$$

where G = STS, and Δxi = vi − ci. The superscript T means the transpose of a vector or a matrix. Table A1 shows the results of \({d_i}^2\), whereby the minimum values of \({d_1}^2,{d_2}^2,\), and \({d_3}^2\) (bold numbers) correspond to the corners of [1 1 0], [0 0 1], and [0 0 1], respectively. A matrix consisting of the corners of least distance, M_cor, is then constructed, using the corner coordinates as the column vectors, respectively,

$$ {M\_{\text{cor}}} \;\;\;\; = \;\;\;\; {\left[ { 100 } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { 100 } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { 011 } \right].} \\ $$
Table A1
figure TabA1

Distances between internal vectors and unit cell corners.

Full size image

Adding M_ext to M_cor leads to a reshaped unit cell, M_res. Note that the determinant of M_res has to be +1. Otherwise choose slightly larger values of \({{\text{d}}_{\text{i}}}^2\) until the determinant is + 1. In this case:

$$ {M\_{\text{res}} = M\_{\text{ext}} + M\_cor} \\ {\left[ { 100 } \right] {}\;\;\;\;{\left[ { 000 } \right] {}\;\;\;\;{\left[ { 100 } \right]} } } \\ {\left[ { 010 } \right] ={\left[ { { - 1}10 } \right] +{\left[ { 100 } \right]} } } \\ {\left[ { 001 } \right] {}\;\;\;\;{\left[ { 0{ - 1}0 } \right] {}\;\;\;\;{\left[ { 011 } \right]} } } \\ $$

The unimodular matrix, Ua, is then obtained for Ra, the GB between grains 1 and 2 is given by

$$ {{U_a}} \;\;\;\; = \;\;\;\; {M\_{\text{re}}{{\text{s}}^{ - 1}}} \;\;\;\; = \;\;\;\; {\left[ { 100 } \right]} \\ {} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {\left[ { 010 } \right]} \\ {} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {\left[ { 001 } \right]} \\ $$

Likewise, Ub and Uc can be derived as

$$ {} \;\;\;\; {\left[ { 1{ - 1}\;\;\;\;{ - 1} } \right]} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {\left[ { 111 } \right]} \\ {{U_b}} \;\;\;\; {\left[ { 00{ - 1} } \right]} \;\;\;\; {{U_c}} \;\;\;\; = \;\;\;\; {\left[ { 011 } \right]} \\ {} \;\;\;\; {\left[ { 011 } \right]} \;\;\;\; {} \;\;\;\; {} \;\;\;\; {\left[ { 0{ - 1}0 } \right]} \\ ,$$

and the TJ line character, T, can be readily determined as

$$ {} \;\;\;\; {\left[ { 10{ - 1} } \right]} \\ {{\text{T}}{U_c}{U_b}{U_a}} \;\;\;\; {\left[ { 010 } \right]} \\ {} \;\;\;\; {\left[ { 001 } \right]} \\ .$$

For this particular TJ, it is a U-line because T is not identity.

Determination of TJ character using the CSL/GB model

The OIM software is capable of identifying a CSL GB with Σ value using preset angle deviation allowance. In this study, two criteria were used for the deviation, namely, the Brandon criterion (BR) ΔθBR = 15° Σ−0.5 and the Palumbo—Aust criterion (PA) ΔθPA = 15° Σ−5/6. The results are shown in Table A2 for the same sample triple junction, where Rd stands for random.

Table A2
figure TabA2

Σ values of the three GBs using BR and PA criterion.

Full size image

According to the BR criterion, the TJ has two CSL GBs, while it has no CSL GB using the PA criterion. Therefore, this is a BR2 or PA0 triple junction.

Determination of TJ character using the CAD model and the PM/GB model

For both models, the first step is to construct a list of ∏ values and deviation allowances (ΔθCAD = 20.25 ∏−0.5) as shown in Table A3.

Table A3
figure TabA3

The list of CAD and its ∏ value and deviation allowance.

Full size image

Because of the symmetry, different misorientation matrices can be used to describe a given GB. It is crucial to use the matrix that gives the least rotation angle with the axis pointing into the standard stereographic triangle ([100], [110], [111]), the so-called disorientation matrix.

To do this, the trace (sum of the diagonal elements) of a GB matrix, e.g. Rb, is maximized by interchanging rows and columns of the matrix. It results in Rb _max:

$$ {{R_b}\_\max } \;\;\;\; = \;\;\;\; {\left[ { {0.9984}\;\;\;\;{ -.03531}\;\;\;\;{0.04502} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { - 0.004540}\;\;\;\;{0.7355}\;\;\;\;{0.66775} } \right]} \\ {} \;\;\;\; {} \;\;\;\; {\left[ { { - 0.05704}\;\;\;\;{ -.6466}\;\;\;\;{0.7342} } \right].} \\ $$

Then Rb– max is converted into angle-axis pair format,

$${{\theta }}\left[ { {{\text{u,}}}\;\;\;\;{{\text{v,}}}\;\;\;\;{\text{w}} } \right] = 42.79^\circ \left[ { { - 43.97}\;\;\;\;{3314}\;\;\;\;{1.000} } \right].$$

The entries of the axis is re-arranged so that u ≥ v ≥ w ≥ 0 to obtain the axis pointing correctly. This leads to a new angle-axis pair:

$$42.79^\circ \left[ { {43.97}\;\;\;\;{3314}\;\;\;\;{1.000} } \right].$$

Converting the pair back into the matrix gives the disorientation matrix, Rb– dis, between grains 2 and 3:

$$ {{R_{b\_{\text{dis}}}}}={\left[ { {0.9984}\;\;\;\;{0.004524}\;\;\;\;{0.05704} } \right]} \\ {}\;\;\;\;{}\;\;\;\;{\left[ { {0.03532}\;\;\;\;{0.7355}\;\;\;\;{ -.6766} } \right]} \\ {}\;\;\;\;{}\;\;\;\;{\left[ { { - 0.04501}\;\;\;\;{0.6775}\;\;\;\;{0.7342} } \right].} \\ $$

Beginning with CAD = [1, 1, 1] or ∏ = 3, grain 2 is taken as reference grain. So the CAD vector, V1, in grain 2, is V1 = [1, 1, 1], and the CAD vector, V2, in the adjacent grain (grain 3) is

$$V2 = {R_b}\_{\text{dis}}V1 = \left[ {1.060,0.09424,1.367} \right].$$

The angle, θb, between V1 and V2, is obtained using the vector scalar product:

$$V{\text{1}} \cdot V{\text{2 = }}\left| {V{\text{1}}} \right|\left| {V{\text{2}}} \right|\cos {{{\theta }}_b}.$$

So θb = 32.82°, which is larger than the deviation allowance, i.e., 11.7° (see Table A3). Moving on to ∏ = 4 or CAD = [2, 0, 0], and repeating the procedure gives V1 = [2, 0, 0], and V2:

$$V2 = {R_b}\_{\text{dis}}V1 = \left[ {1.997,0.07064, -.09002} \right].$$

Since θb = 3.28°, smaller than 10.1° (see Table A3), this is a ∏4 GB. Likewise, Rc is found to be a ∏8 GB with the deviation angle of 3.41°, while Ra is a random GB because all its deviation angles up to ∏ = 20 are greater than the allowances. The results are summarized in Table A4.

Table A4
figure TabA4

TJ characters of the CAD and PM/GB models.

Full size image

Therefore, according to the CAD model, the triple junction is a random TJ since no common ∏ ≤ 20 value is found among three GBs.

The same TJ is defined as a PM2 triple junction because it has two special GBs, ∏8 and ∏4 according to the PM/GB model (Table A4).

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Zhou, Y., Palumbo, G., Aust, K.T. et al. Triple junction structure and carbide precipitation in 304L stainless steel. Journal of Materials Research 28, 1589–1600 (2013). https://doi.org/10.1557/jmr.2013.148

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