CN109522675B - Method for simulating microstructure of tin-based binary eutectic alloy and finite element solving and analyzing method - Google Patents
Method for simulating microstructure of tin-based binary eutectic alloy and finite element solving and analyzing method Download PDFInfo
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Abstract
The invention discloses a method for simulating a microscopic structure of a tin-based binary eutectic alloy and solving and analyzing finite elements, and relates to the field of computing material science. The shape of the eutectic structure obtained by simulation of the method is very similar to that of the eutectic structure obtained by acquisition of experimental instruments, and the loading, solving and analysis of the eutectic structure can be realized on mainstream finite element CAE software and platforms (such as ANSYS, MARC, ABAQUS, MSC/PATRAN, COMSOL and the like); the method can make up the defects of the existing tin-based binary eutectic alloy microstructure detection technology and realize the characterization of the physical and mechanical behaviors of the microstructure. The method combines a Monte Carlo method and a finite element method, improves the modeling efficiency of the eutectic structure, realizes the loading solution and analysis of the eutectic structure, provides a new method for the characteristic analysis and the reliability analysis of the eutectic structure, and well solves the problems of the microstructure modeling and the performance characterization of the eutectic structure.
Description
Technical Field
The invention relates to the field of computing material science, in particular to a method for simulating a microscopic structure of a tin-based binary eutectic alloy and solving and analyzing finite elements.
Background
Tin-based binary eutectic alloy solder is widely applied to micro-interconnection welding spots of electronic components and equipment. With the continuous development of high-density packaging technology in the electronic industry, the trend of miniaturization of electronic devices and systems is more and more obvious, so that the size of a welding spot is continuously reduced, and a welding spot solder matrix presents obvious micro-structure (binary eutectic structure, mixture of two solid-phase machines) nonuniformity. Researchers at home and abroad generally consider that solder joints are the weakest part of electronic products and equipment. Researches find that the binary eutectic structure nonuniformity can obviously influence the electromigration, the heat migration, the mechanical reliability and the like of welding points. However, due to the limitation of the test accuracy of the test method and the instrument, the measurement and characterization of the size and distribution of the current density, the temperature gradient, the stress strain and the like in the micro-area are extremely difficult or even impossible to realize, so the research and report on the size and distribution of the current density, the temperature gradient, the stress strain and the like in the micro-area of the binary eutectic structure are very deficient.
Due to the limitations of experimental approaches, finite element methods are used for the evaluation of the magnitude and distribution of micro-zone current density, temperature gradients, or stress strains, among others. The traditional finite element modeling method is based on point, line and surface entity modeling, a user utilizes software interface operation or commands to establish a geometric model by determining the coordinate relation of points, lines, surfaces and bodies, and then the finite element model is obtained by dividing grids. Or establishing a geometric model on computer aided design platforms such as AUTOCAD, UG, SOLIDWORKS, PRO/E and the like, then introducing finite element CAE software, and finally dividing meshes to obtain a finite element model. However, the shape and structure of the binary eutectic structure are extremely complex, and the traditional finite element modeling method takes a lot of time and energy and even cannot realize modeling. Therefore, most previous studies have homogenized the dual eutectic structure of solder joints, i.e., do not consider the difference in physical and mechanical properties of the two eutectic phases (phases in the eutectic structure). It has been reported that the phase field method, the cellular automata method, and the monte carlo method can realize numerical modeling of binary eutectic structures with complex shapes and structures. However, these methods do not allow for solution analysis under their complex physical and mechanical loads.
Disclosure of Invention
The invention aims to provide a method for simulating a microstructure of a tin-based binary eutectic alloy and solving and analyzing a finite element, which combines a Monte Carlo method and a finite element method, improves the modeling efficiency of the eutectic structure, realizes the loading solving and analysis of the eutectic structure, provides a new method for the characteristic analysis and the reliability analysis of the eutectic structure, and well solves the problems of the modeling and the performance characterization of the microstructure of the eutectic structure.
The technical problem to be solved by the invention is realized by adopting the following technical scheme.
The invention provides a method for simulating a microstructure of a tin-based binary eutectic alloy and analyzing a finite element solution, which comprises the following steps of:
s1: randomly generating an initial model according to a phase proportion, considering that the energy of a system is determined by interface energy, and adopting the interface energy within a local 5 multiplied by 5 range during energy calculation;
s2: the energy of the system is reduced through grain boundary migration and long-range diffusion, and whether the grain boundary migration or the long-range diffusion occurs is judged through the orientation value symbols of the sites and the randomly selected adjacent sites.
The probability of occurrence of grain boundary migration is:
the probability of long-range diffusion occurring is:
wherein eta is the interval [0,1]A preset or random value in between; s3: noise points are eliminated; the main phase noise point and the second phase noise point are mutually counteracted, the long-range diffusion continues to occur without the counteracted noise point, and the long-range diffusion is required not to generate new noise points;
s4: according to the unit number and the arrangement mode of the simple finite element model, the pixel number and the arrangement mode are specified;
s5: generating a script file which can be identified by finite element CAE software, wherein the script file contains the serial number of each pixel and corresponding phase information;
s6: reading the script file by using CAE software, giving material attributes and element type information of the elements with the same number in the finite element model according to the object image information of each pixel of the script file, and generating a complex finite element model containing different element types and material attribute information;
s7: and loading and solving analysis are carried out on the complex finite element model containing different unit types and material attribute information, so as to obtain a simulation result.
In detail, in step S1, the system energy is determined by the interface energy, and the interface energy calculation is performed in the following manner:
s11: when the interfacial energy of a local 5 multiplied by 5 area is calculated, the anisotropy is considered, the interfacial energy in the vertical direction is 3 times of that in the horizontal direction, and the interfacial energy in the diagonal direction is 2 times of that in the horizontal direction;
s12: let the boundary interface energy in the horizontal direction be E 0 The interfacial energy in the vertical direction and the interfacial energy in the diagonal direction are 3E, respectively 0 、2E 0 Then according to the formula of the interface energy
And calculating the energy.
In detail, in step S2, grain boundary migration is performed as follows:
s21: when the grain boundary migration occurs, a random selection of a site i and a 5 × 5 area around the site i is started, if the site i falls on the boundary area, the area around the site i is smaller than the 5 × 5 area, and the initial 5 × 5 area or the interfacial energy E smaller than 5 × 5 is calculated b1 ；
S22: comparing the difference and the identity between the orientation value of the i site and the orientation value of a random adjacent site j around the i site; if the sign identity is different, the orientation value of the adjacent site j is assigned to the i site to make the orientation values of the i and j sites the same, and the interfacial energy E of the local area before and after the change (i.e. 5X 5 or smaller array) is calculated b1 And E b2 The amount of change in interfacial energy Δ E ═ E b2 -E b1 If Δ E is less than or equal to zero, the probability P of occurrence of grain boundary migration is 1, otherwise, grain boundary migration does not occur, and the state before grain boundary migration is maintained.
In detail, in step S2, the long-range diffusion is performed as follows:
s23: when long-range diffusion occurs, when the orientation value of a randomly selected site i and the orientation value of a random adjacent site j around the randomly selected site i have opposite signs, the exchange positions of the site i and the adjacent site j, namely the position of the site where the site i is located, move only in different signs, namely the out-of-phase site, until the same sign is touched, namely the same-phase site is touched, and the motion is stopped;
s24: comparing the variation delta E of the front and rear interface energies; if Δ E is less than or equal to zero, the probability of occurrence of long-range diffusion is 1, otherwise the probability of occurrence is η, where η is a preset value or random value between intervals [0,1 ].
In detail, in step S3, noise cancellation is performed as follows:
s31: if there are 5 or more (i.e. more than half) different phase points in 8 adjacent sites of a site, defining the site as a noise point, circularly reading each site in the matrix, and counting two kinds of noise points, so that the noise points in the two kinds of phases form two one-dimensional noise point groups;
s32: and exchanging orientation values of partial elements in the two one-dimensional noise point arrays in a one-to-one correspondence mode, so that negative values in the array elements become positive values, positive values become negative values, phase conservation is kept, and then the remaining partial noise points are continuously eliminated through long-range diffusion.
In detail, step S32 is specifically performed as follows:
in the two kinds of noise points, phase A has a elements, phase B has B elements, when a > B (a is not more than B), the first B (a) elements of the two arrays are taken to be in one-to-one correspondence exchange orientation values, negative values in the array elements are changed into positive values, positive values are changed into negative values, phase conservation is kept, and the rest a-B (B-a) noise points are continuously eliminated through long-range diffusion.
In detail, in step S4, the pixels and arrangement are defined as follows:
and checking the unit number and the arrangement mode of the simple finite element model through CAE software, and numbering and arranging the image pixels according to the unit number and the arrangement rule after the unit number and the arrangement rule are determined, so that the units and the pixels in the finite element model form a one-to-one mapping relation.
In detail, in step S6, generating a complex finite element model containing information of different element types and material properties is performed as follows:
transferring the material attribute and the element type information in the script file to the element corresponding to the character combination through CAE software, enabling the pixel to be related to the physical phase information of the element with the same number, endowing or modifying the element information in the simple finite element model according to the physical phase information, and generating the finite element model containing the complex element type and the material attribute information.
In detail, in step S7, three-dimensional reconstruction and unit node coupling are performed through the position correspondence, and boundary constraint conditions are applied, so that solution analysis of the complex model in finite element CAE software can be realized, and a simulation result is obtained.
The method for simulating the microstructure of the tin-based binary eutectic alloy and analyzing the finite element solution has the advantages that:
(1) the initial model is randomly generated according to the proportion of the real object.
(2) The energy of the system is determined by the interface energy, and the interface energy is calculated by adopting the interface energy which does not exceed the local 5 multiplied by 5 range, but not the interface energy of the whole range, so that the calculation efficiency is improved.
(3) And in the long-range diffusion, the phase environment of the neighbors at the final diffusion position is considered.
(4) The generated script file is in a pure file format, is convenient to program and process, has strong universality, and can be used by CAE software.
(5) The method improves the modeling efficiency of the eutectic structure and realizes the quick loading, solving and analysis of the eutectic structure.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention, and therefore should not be considered as limiting the scope, and for those skilled in the art, other related drawings can be obtained according to the drawings without inventive efforts.
FIG. 1 is a flow chart of a method in an embodiment of the present invention;
FIG. 2 is a random matrix (truncated) generated in an embodiment of the present invention;
FIG. 3 illustrates an initial model prior to microstructural evolution in an embodiment of the present invention;
FIG. 4 is a graph of a Monte Carlo model of a crystal lattice expressed in terms of site orientation values in an embodiment of the present invention;
FIG. 5 illustrates an embodiment of the present invention in which i falls within a local 5 × 5 array of non-bounding regions;
FIG. 6 illustrates a local array area where i falls within the boundary region in an embodiment of the present invention;
FIG. 7 illustrates the migration of grain boundaries in an arbitrary 5X 5 array of non-boundary regions in an embodiment of the present invention;
FIG. 8 is a diagram of a long-range diffusion simulation of an arbitrary 5 × 5 array of non-boundary regions in an embodiment of the present invention;
FIG. 9 is a simulation of grain boundary migration for a 3 × 3 array of boundary regions in an embodiment of the present invention;
FIG. 10 is a simulation of grain boundary migration for a 3 × 4 array of boundary regions in an embodiment of the present invention;
FIG. 11 is a simulation of grain boundary migration for a 3 × 5 array of boundary regions in an embodiment of the present invention;
FIG. 12 is a simulation diagram of grain boundary migration of a 4 × 4 array of boundary regions according to an embodiment of the present invention;
FIG. 13 is an evolutionary diagram of the organizational structure in an embodiment of the present invention;
FIG. 14 is a schematic diagram of an arrangement of cells according to an embodiment of the present invention;
FIG. 15 is a finite element model of a eutectic alloy microstructure established in an embodiment of the present invention;
FIG. 16 is a graph showing simulation results of the magnitude and distribution of current density in the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below. The examples, in which specific conditions are not specified, were carried out according to conventional conditions or conditions recommended by the manufacturer. The reagents or instruments used are not indicated by the manufacturer, and are conventional products available commercially.
The method for simulating the microstructure of the tin-based binary eutectic alloy and the finite element solution analysis according to the embodiment of the present invention will be specifically described below.
A method for simulating a microstructure of a tin-based binary eutectic alloy and analyzing finite element solution comprises the following steps:
s1: the initial model is randomly generated in terms of phase ratios, and consists of an m x n array of sites, where the sites represent domains having a lattice orientation, assigned a random value (i.e., an orientation value, Q being an integer greater than 0) between 1 and Q or-1 and-Q to represent a particular lattice orientation. The orientation value of the site representing the main phase of the binary eutectic structure is randomly distributed as a positive integer in a closed interval from 1 to Q, and the orientation value of the site representing the second phase is randomly distributed as a negative integer in a closed interval from-Q to-1; the phase proportion in the initial model is randomly generated according to the real phase proportion in the microstructure. In the microstructure simulation, a grain in a polycrystalline material may be represented by a set of sites having the same orientation value.
In the embodiment of the invention, the system energy is determined by the interface energy, and the interface energy is calculated by using the interface energy which does not exceed the local 5 x 5 array range; the setting can improve the calculation efficiency and save the calculation time.
S2: the system energy is determined by the interfacial energy, and the system interfacial energy is reduced according to a specified rule, so that the growth of crystal grains is simulated. When the interface energy is judged, the interface energy between two points with the same orientation value is zero, and the interface energy exists between two points with different orientation values. Randomly choose 1 site i and then randomly choose one of its neighbors j. Reading orientation values of i and j, and if the orientation values of i and j have the same sign (namely, are both positive or both negative) and have different values, carrying out grain boundary migration according to a certain probability, as shown in the following a); if the i and j orientation values are different in sign, long-range diffusion occurs with a certain probability, as shown in the following b); in calculating the change in interfacial energy caused by grain boundary migration or long-range diffusion, a local change in interfacial energy is used instead of the overall change in interfacial energy to reduce the amount of calculation, and the calculation of the local change in interfacial energy is divided into several cases as shown in c). The evolution rules in the embodiments of the invention are as follows:
a) the interfacial energy of the system is reduced by grain boundary migration. In the present invention, when the orientation values of i, j have the same sign but different magnitudes, grain boundary migration occurs with a certain probability P, that is, the j site is locatedAnd assigning the orientation value of (b) to the i-site so that the orientation values of the i, j-sites are the same. Calculating the interfacial energy E after the crystal boundary migration 2 Interfacial energy E before grain boundary migration 1 Change in interfacial energy (i.e., Δ E ═ E) 2 -E 1 ) If the interface energy is smaller or unchanged, (namely, the delta E is less than or equal to 0), the probability P of the grain boundary migration is 1, the grain boundary migration can occur, otherwise, the probability P is 0, and the grain boundary migration cannot occur.
b) The interfacial energy of the system is reduced by long-range diffusion. If the orientation values of the i and j points are different in sign, long-range diffusion will occur with a certain probability P, that is, the point where i is located moves in position and can only move in points of different signs (out of phase) until a point of the same sign (in phase) is encountered and the motion stops. Calculating the interface energy E after long-range diffusion 2 ' initial interfacial energy E before long-range diffusion 1 ' change in interfacial energy (i.e.. DELTA.E ═ E) 2 ’-E 1 ') if the interfacial energy is small or constant, (i.e. Δ E ≦ 0), then the probability of long-range diffusion P is 1, long-range diffusion may occur, otherwise, long-range diffusion may occur only with a certain probability P of η, where η is the interval [0,1 ≦ 0]A preset value or a random value in between.
c) The change in interfacial energy due to grain boundary migration or long-range diffusion was calculated. When the row and column positions of the randomly selected site i satisfy the conditions that L is more than or equal to 3 and less than or equal to m-2 and C is more than or equal to 3 and less than or equal to n-2, as shown in FIG. 5, whether crystal boundary migration or long-range diffusion occurs is judged by calculating the interface energy change condition of the 5 × 5 local array where the site i is located. When the row and column positions of the randomly selected site i satisfy the conditions 1. ltoreq. L.ltoreq.2, 1. ltoreq. C.ltoreq.2, m-1. ltoreq. L.ltoreq.m, n-1. ltoreq. C.ltoreq.n (i.e., the randomly selected site i falls within the boundary region), referring again to FIG. 5, the local 5X 5 array range of the embodiment of the present invention in which the site i falls within the non-boundary region (i.e., 3. ltoreq. L.ltoreq.148, and 3. ltoreq. C.ltoreq.148). As shown in fig. 6, a 3 × 3 local array when the random bit point i falls on the 1 st row and the 1 st column, a 3 × 4 local array when i falls on the 1 st row and the 2 nd column, a 3 × 5 local array when i falls on the 1 st row and the 3 rd column, a 4 × 4 local array when i falls on the 2 nd row and the 2 nd column, and a 4 × 5 local array when i falls on the 2 nd row and the 3 rd column are selected, and the change of the interface energy before and after the grain boundary migration or the long-range diffusion is calculated to judge whether the grain boundary migration or the long-range diffusion can occur. And similarly, the conditions of grain boundary migration and long-range diffusion energy change when the random site i falls at the lower left corner position, the upper right corner position and the lower right corner position can be calculated.
S3: the model is evolved by N MCSs, according to Step S2, by repeating m × N times (i.e., the total number of sites in the m × N array of sites, defined as one Monte Carlo Step (MCS)).
S4: and (4) eliminating noise points. And determining a noise point and canceling the noise point (the main noise point and the second noise point are cancelled). The un-cancelled noise is long-range diffused as per step S2, and it is required that no new noise is formed after the long-range diffusion.
S5: establishing a simple finite element model corresponding to the microstructure site array (m multiplied by n), numbering the site array and the element array in the same way, and generating finite element CAE software according to the phase information of the site array to identify a script file, wherein the script file contains the number of each site and the phase information corresponding to the site.
S6: reading the script file by using CAE software, endowing or modifying the material attribute and the element type information of the elements with the same number in the simple finite element model according to the phase information of each pixel of the script file, and generating a complex finite element model containing different element types and material attribute information; and automatic loading and solving are realized.
The features and properties of the present invention are described in further detail below with reference to examples.
Example 1
Specifically, as shown in fig. 1, the method for simulating the microstructure of the SnBi binary eutectic alloy and for finite element solution analysis includes the following steps:
and S1, generating an initial model. The initial model consists of a 150 × 150 (i.e., m-150, n-150) array of sites, where a site represents a domain with a lattice orientation that is assigned a random value between 1 and 10 or-1 and-10 (i.e., Q-10) to represent the orientation of the grains. The sites representing the Sn phase were designated as positive orientation (i.e., 1 to 10). The site representing the Bi phase is assigned a negative integer (i.e., -10 to-1) in the interval-10 to-1; then, an initial numerical model is randomly generated according to the proportion of real phases, and the probability of generating Sn phase is 49.26% and the probability of generating Bi phase is 50.74% in the specified program because the volume ratio of two SnBi phases in the eutectic structure is 1: 1.03. The generated random matrix (part) is shown in fig. 2 and the initial model is shown in fig. 3. Thus, the initial model can be randomly generated according to the proportion of the real object.
S2: in the microstructure simulation, the regions with the same orientation value in the polycrystalline material can be understood as a crystal grain; interfaces do not exist between sites with the same orientation value, belong to the same crystal grain, and the interface energy is zero; grain boundaries exist among different orientation value sites, and the interface energy is not zero; for example: there is no interfacial energy between the site orientation value 1 at row 4, column 4 and the site orientation value 1 at row 4, column 5 in fig. 4; an interface energy exists between the site orientation value 1 of the 10 th column in the 4 th row and the site orientation value 5 of the 11 th column in the 4 th row; in computing the interfacial energy, we generally use the local 5 × 5 range interfacial energy in a random matrix. As shown in FIG. 5, assuming that point i is located in L rows and C columns of the array, the range of the 5 × 5 array around it is defined as: the array region is located from L-2 to L +2 rows and from C-2 to C +2 columns. If the area around the site i cannot take the local 5 × 5 array, as shown in FIG. 6, the array area ranging from L-2 to L +2 and from C-2 to C +2 is selected according to the actual situation. For example: when i falls in the 1 st row and 1 st column position, selecting a 3X 3 array; when i falls at the position of the 1 st row and the 2 nd column, selecting a 3 multiplied by 4 array; when i falls in the 3 rd column position of the 1 st row, selecting a 3 multiplied by 5 array; when i falls in the 2 nd row and 2 nd column position, selecting a 4 multiplied by 4 array; when i falls in row 2, column 3 position, a 4 x 5 array is selected. In our model, we consider anisotropy, the vertical boundary interfacial energy (interfacial energy between two adjacent columns of the same row) is three times that of the horizontal boundary interfacial energy (interfacial energy between two adjacent rows of the same column), and the diagonal boundary interfacial energy (interfacial energy between two adjacent sites of two rows and adjacent columns) is two times that of the horizontal boundary interfacial energy; for example: in FIG. 5, the interface energy in the horizontal direction is between the L-th row and the C-th column position i and the L-1-th row and the C-th column position i; the interface energy in the vertical direction is between the L-th row and the C-th column position i and the L-th row and the C-1-th column position; and a diagonal interface energy is formed between the L-th row and C-th column site i and the L-1-th row and C-1-th column site. And (3) calculating the interface energy of the local matrix where the position point i is located according to the formula:
wherein E (i, j) represents the interface energy between points of the positions i, j, and Z is the number of neighbors of the position i; n is the number of sites in the local array of sites i.
The evolution in the present invention is as follows: in a 150 x 150 array, one site i is randomly selected, and then one of its neighbors j is randomly selected. Reading orientation values of i and j, and if the orientation values of i and j have the same sign (namely, are both positive or both negative) and have different values, carrying out grain boundary migration according to a certain probability, as shown in the following a); if the i and j orientation values are different in sign, long-range diffusion occurs with a certain probability, as shown in the following b); in calculating the interfacial energy variation caused by grain boundary migration or long-range diffusion, a local interfacial energy variation is used instead of the overall interfacial energy variation to reduce the amount of calculation, and the calculation of the local interfacial energy variation is classified into several cases as shown in c). The evolution rules of the present invention are as follows:
a) the interfacial energy of the system is reduced by grain boundary migration. In the invention, when the orientation values of i and j have the same sign but different values, grain boundary migration occurs with a certain probability P, namely the orientation value of the j site is assigned to the i site, so that the orientation values of the i site and the j site are the same. Calculating the interfacial energy E after grain boundary migration 2 Initial interfacial energy E before migration with grain boundaries 1 (let. DELTA.E be E 2 -E 1 ) If the energy is reduced or unchanged, (namely delta E is less than or equal to 0), the probability P of grain boundary migration is 1, and the grain boundary migration can occur; otherwise, the probability P is 0, and grain boundary migration cannot occur. As shown in FIG. 7, the orientation value of a randomly selected site i in the non-boundary region is-5, and the initial interfacial energy of the 5X 5 array of the sites is calculated as E according to equation (1) 1 ＝137E 0 Then randomly selecting a j orientation value-1 of one adjacent site; since the symbols of-5 and-1 are the same and different, assigning-1 to the site i starting to be randomly selected, as shown in FIG. 7(b), the interfacial energy E after grain boundary migration is calculated 2 ＝132E 0 (ii) a Comparing the energy changes before and after the transition can make the crystal boundary migration process occur. b) The system interfacial energy is reduced by long-range diffusion. In the present invention, when the orientation values of i, j are different in sign, long-range diffusion will occur with a certain probability P, i.e. i is located at a position which moves and can only move in the position of different sign until the position of the same sign (in-phase) is touched and stops moving. Calculating the interfacial energy E after long-range diffusion 2 ' initial interfacial energy E before long-range diffusion 1 ' (let. DELTA.E-E) 2 ’-E 1 ') if the energy is small or constant, (i.e., Δ E ≦ 0), then the probability P of long-range diffusion occurring is 1, and long-range diffusion may occur; otherwise, the probability P of long-range diffusion is 0.1 (i.e., η is made 0.1). In the example shown in FIG. 8, the orientation value of a randomly selected site i in the non-boundary region is-2, and the sum E of interfacial energies of two initial 5X 5 array regions before long-range diffusion of the site i shown in FIG. 8(a) is calculated according to equation (1) 1 ’＝115E 0 Then randomly selecting one adjacent site j 1 The orientation value is 1, because-2 is not signed with 1, the positions i and j 1 Position exchange (orientation value exchange) occurs, and then random diffusion is performed in a region with positive orientation value according to a random path with probability P until a site (-2) with negative orientation value is encountered, wherein the arrows in FIG. 8(a) indicate one of the random diffusion paths, and site i and site j are respectively 1 Position j 2 Position j, position j 3 Position j 4 Position j 5 Position j 6 Position j 7 Swapping positions until a point of the same phase (-2 or-4) is encountered and movement is stopped; the local array after long-range diffusion is shown in FIG. 8(b), and then the sum E of the interfacial energies of the two 5X 5 array regions after long-range diffusion is calculated 2 ’＝107E 0 The energy change before and after comparison can give Δ E<0, the probability P that this process occurs, is 1.
c) In the two steps 2(a) and 2(b), whether grain boundary migration or long-range diffusion occurs is judged by calculating the energy change of the local 5X 5 array interface where the site i is located. If the 5 × 5 array is not sufficient, we discuss several cases where the calculation site i is located in the upper left corner region: as shown in FIG. 9, when the random site i falls on the firstRow 1, column 1, forming a 3 x 3 partial array, the initial interfacial energy of the partial array (a) calculated from equation (1) is: 40E 0 The interfacial energy of the local array (b) after the grain boundary migration occurs is: 39E 0 (ii) a As shown in fig. 10, when the random site i falls on row 1, column 2, a 3 × 4 local array is formed, and the initial interface energy of the local array (a) is calculated by equation (1): 59E 0 The interface energy of the local array (b) after the grain boundary migration is as follows: 57E 0 (ii) a As shown in fig. 11, when the random site i falls on row 1, column 3, a 3 × 5 local array is formed, and the initial interfacial energy of the local array (a) is calculated by equation (1): 78E 0 The interface energy of the local array (b) after the crystal boundary migration occurs is: 76E 0 (ii) a As shown in FIG. 12, when a random bit point i falls on row 2, column 2, a 4 × 4 partial array is formed, and the initial interfacial energy of the partial array (a) is calculated by equation (1): 84E 0 The interfacial energy of the local array (b) after the grain boundary migration occurs is: 81E 0 . If the 5X 5 local array (i falls in the boundary area) cannot be taken near the point i where the long-range diffusion occurs, the 3X 3 array, the 3X 4 array, the 3X 5 array, the 4X 4 array and the 4X 5 array are selected according to the actual situation as the condition of the grain boundary migration, and then the condition of the interface energy change before and after calculation is carried out. Similarly, the situation that the random position point i falls in the lower left corner boundary area, the upper right corner boundary area and the lower right corner boundary area can be calculated.
To facilitate visual comparison, the initial interfacial energy E is determined b1 Interfacial energy E after grain boundary migration b2 The results are shown in Table 1:
TABLE 1 variation of boundary migration interfacial energy in local array (part) of boundary region
S3: repeating 150 × 150 times (i.e., the total number of sites in the 150 × 150 array of sites, defined as one monte carlo step size (MCS)) according to step S2, 700, 4900, 9800, 14700, 19600, and 2450000 MCSs were evolved for the model as shown in fig. 13.
S4: and (4) eliminating noise points. Reading each site in the matrix in a cyclic manner, when more than half of adjacent sites of a site are out-of-phase points (for example, 5 or more out of 8 adjacent sites are out-of-phase points), determining that the site is a noise point, then counting the noise points of two phases, each site has its own orientation value, and can form two one-dimensional arrays, for example, phase a has a elements, phase B has B elements, when a > B (a ≦ B), then taking the first B (a) elements of the two arrays to exchange the orientation values one by one, and changing the negative value in the array elements to a positive value and the positive value to a negative value, so that we can ensure the phase to be conserved, and the remaining B-a (a-B) noise points are according to step S2: (b) long range diffusion occurs and is required to be unable to form new noise points.
S5: specifying the unit number and the arrangement mode of the model pixel; checking the number and the arrangement mode of the elements of the simple finite element model in CAE software, wherein FIG. 14 shows the arrangement mode of the elements of the simple finite element model in the embodiment, and the numerical values in the figure represent the number of the elements; after the number and the arrangement rule of the units are determined, the pixels of the tin-based binary eutectic alloy microstructure model image are numbered and arranged in the same way according to the number and the arrangement rule of the units, so that the units in the finite element model and the pixels in the image form a one-to-one mapping relation.
S6: generating finite element CAE software capable of identifying script files, wherein the script files contain the serial number of each pixel and corresponding phase information thereof; the script file is composed of a series of character combinations, each character combination corresponds to one pixel in the image and comprises the number of the pixel, the attribute number of the material and the material type number information; in the embodiment, the expression numbered i character combination is an expression numbered "character combination, where A, B, C is a code, and is determined by an operation command of finite element CAE software, i is a unit number, j is a material attribute number, and k is a unit type number, for example, in the embodiment, a pixel numbered 5 represents a tin phase, the material attribute number is 2, and the unit type number of the tin phase is 2, and thus, the element type number corresponds to information that" the middle number is an attribute number and 2 "in a script file in a" piece "in a script file applicable to ANSYS software, that is, the middle number is an attribute number and a material type number in a script file in which a ═ is an attribute number and a material type number; carrying out; by identifying this array, ANSYS software numbers the material attribute of the unit defined with the number 5 to 2, the unit type number to 2, and the unit type with the unit type number 2 specified in the embodiment to be PLANE 67; the language or code used for each character combination can be identified by ANSYS software, and the material property parameters in the example are shown in table 2.
TABLE 2 Material Properties of tin-based binary eutectic alloys
S7: and establishing a complex finite element model containing object out-of-phase information. Reading the script file by using CAE software, modifying the material attribute and the element type information of the elements with the same number in the simple finite element model according to the physical phase information of each pixel of the script file, and generating a complex finite element model containing different element types and material attribute information as shown in FIG. 15; and automatic loading and solution solving are realized to obtain a simulation result.
Generating a complex finite element model containing information on different element types and material properties is performed as follows: and transmitting the material attribute and element type information in the script file to the element corresponding to the character combination through CAE software, associating the pixel with the phase information of the element with the same number, endowing the element information in the simple finite element model according to the phase information, and generating the complex finite element model containing different element types and material attribute information.
S8: solving and analyzing to obtain a model result: the finite element models shown in fig. 14 obtained from (1) to (7) can be subjected to solution analysis by applying boundary conditions and loads in the CAE software, and can be subjected to solution analysis in the finite element CAE software by applying boundary constraint conditions through node coupling, so as to obtain simulation results. In the embodiment, when the voltage applied between two ends of the solder joint of the two-dimensional SnBi alloy finite element model is 0.001V, the current density conditions of different tissue forms (fine, medium and coarse) can be studied to obtain the current density condition shown in fig. 16A simulated view is shown; when the two ends of the welding spot of the two-dimensional SnBi alloy finite element model are 1 multiplied by 10 8 A/m 2 The effect of different relative pad electromigration behavior can be studied with applied pressure at current densities of (a) to (b) to obtain a simulated plot as shown in fig. 16, with the data results collated as shown in tables 3 and 4 below.
TABLE 3 Current Density (0.001V) for three pads at the same applied Voltage
TABLE 4 Current Density at the same application (1X 10) 8 A/m 2 ) Current density of three welding spots
In summary, the present invention provides a new method for binary eutectic structure characteristic analysis and reliability analysis, which is specifically represented as follows:
(1) the initial model is randomly generated according to the proportion of real objects.
(2) The energy of the system is determined by the interface energy, and the interface energy is calculated by adopting the interface energy which does not exceed the local 5 multiplied by 5 range, but not the interface energy of the whole range, so that the calculation efficiency is improved.
(3) And in the long-range diffusion, the phase environment of the neighbors at the final diffusion position is considered.
(4) The generated script file is in a pure file format, is convenient to program and process, has strong universality, and can be used by CAE software.
(5) The method improves the modeling efficiency of the eutectic structure and realizes the quick loading, solving and analysis of the eutectic structure.
The embodiments described above are some, but not all embodiments of the invention. The detailed description of the embodiments of the present invention is not intended to limit the scope of the invention as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Claims (7)
1. A method for simulating a microstructure of a tin-based binary eutectic alloy and solving and analyzing finite elements is characterized by comprising the following steps of:
s1: randomly generating an initial model according to a phase proportion, considering that system energy is determined by interface energy, and adopting the interface energy within a local 5 multiplied by 5 range during energy calculation;
s2: reducing system energy through grain boundary migration and long-range diffusion, and judging whether grain boundary migration or long-range diffusion occurs through orientation value symbols of the sites and adjacent sites randomly selected by the sites; the probability of occurrence of grain boundary migration is:
the probability of long-range diffusion occurring is:
wherein eta is the interval [0,1]A preset or random value therebetween; Δ E is the amount of change in interfacial energy;
and in step S2, the grain boundary migration is performed as follows:
s21: when the grain boundary migration occurs, a random selection of a site i and a 5 × 5 area around the site i is started, if the site i falls on the boundary area, the area around the site i is smaller than the 5 × 5 area, and the interfacial energy E of the initial 5 × 5 area or the area smaller than 5 × 5 is calculated b1 ；
S22: comparing the difference and the identity between the orientation value of the i site and the orientation value of a random adjacent site j around the i site; if the same sign value is different, assigning the orientation value of the adjacent position j to the i position to ensure that the orientation values of the i position and the j position are the same, and then calculating the local area before and after the change, the interface energy E b1 And E b2 Change in interfacial energy Δ E ═ E b2 -E b1 If Δ E is less than or equal to zero, the probability P of occurrence of grain boundary migration is 1, otherwiseThen, the grain boundary migration does not occur, and the state before the grain boundary migration is maintained;
the long-range diffusion is performed as follows:
s23: when long-range diffusion occurs, when the orientation value of a randomly selected site i and the orientation value of a random adjacent site j around the randomly selected site i have opposite signs, the exchange positions of the site i and the adjacent site j, namely the position of the site where the site i is located, move only in different signs, namely the out-of-phase site, until the same sign is touched, namely the same-phase site is touched, and the motion is stopped;
s24: comparing the variation delta E of the front and rear interface energies; if the delta E is less than or equal to zero, the probability of occurrence of long-range diffusion is 1, otherwise the probability of occurrence is eta, wherein eta is a preset value or random value between intervals [0,1 ]; s3: noise points are eliminated; the main phase noise point and the second phase noise point are mutually counteracted, the long-range diffusion continues to occur without the counteracted noise point, and the long-range diffusion is required not to generate new noise points;
s4: according to the unit number and the arrangement mode of the simple finite element model, the pixel number and the arrangement mode are specified;
s5: generating a script file which can be identified by finite element CAE software, wherein the script file contains the serial number of each pixel and corresponding phase information;
s6: reading the script file by using CAE software, giving material attributes and element type information of the elements with the same number in the finite element model according to the object image information of each pixel of the script file, and generating a complex finite element model containing different element types and material attribute information;
s7: and loading, solving and analyzing the complex finite element model containing different unit types and material attribute information to obtain a simulation result.
2. A method for simulating a tin-based binary eutectic alloy microstructure and finite element solution analysis according to claim 1, wherein in step S1, the interfacial energy calculation is performed as follows:
s11: when the interface energy of a local 5 multiplied by 5 area is calculated, the anisotropy is considered, the interface energy in the vertical direction is 3 times of the interface energy in the horizontal direction, and the interface energy in the diagonal direction is 2 times of the interface energy in the horizontal direction;
s12: let the boundary interface energy in the horizontal direction be E 0 The interfacial energy in the vertical direction and the interfacial energy in the diagonal direction are 3E respectively 0 、2E 0 Then according to the formula of the interface energy
Calculating the interface energy; wherein E (i, j) represents the interface energy between points of the position i, j, and Z is the number of neighbors of the position i; n is the number of sites in the local array of sites i.
3. A method for simulating a tin-based binary eutectic alloy microstructure and finite element solution analysis according to claim 1, wherein noise cancellation is performed in step S3 as follows:
s31: if 5 or more different phase points exist in 8 adjacent points of one point, the point is defined as a noise point, each point in the matrix is read circularly, two kinds of noise points can be counted, and finally the noise points in the two kinds of phases form two one-dimensional noise point arrays;
s32: and exchanging orientation values of partial elements in the two one-dimensional noise point arrays in a one-to-one correspondence mode, so that negative values in the array elements become positive values, positive values become negative values, phase conservation is kept, and then the remaining partial noise points are continuously eliminated through long-range diffusion.
4. The method for simulating a microstructure of a tin-based binary eutectic alloy and finite element solution analysis according to claim 1, wherein step S32 is performed in the following manner:
in the two kinds of noise points, phase A has a elements, phase B has B elements, when a > B, the first B elements of the two arrays are taken to be in one-to-one correspondence exchange orientation values, so that the negative value in the array elements becomes a positive value, the positive value becomes a negative value, the phase conservation is kept, and the remaining a-B noise points are continuously eliminated through long-range diffusion;
and when a is less than or equal to b, the first a elements of the two arrays are taken to be in one-to-one correspondence exchange orientation values, negative values in the elements of the arrays are changed into positive values, positive values are changed into negative values, phase conservation is kept, and the remaining b-a noise points are continuously eliminated through long-range diffusion.
5. A method for simulating a tin-based binary eutectic alloy microstructure and finite element solution analysis according to claim 1, wherein in step S4, the pixel and arrangement are defined as follows:
and checking the unit number and the arrangement mode of the simple finite element model through CAE software, and numbering and arranging the image pixels according to the unit number and the arrangement rule after the unit number and the arrangement rule are determined, so that the units and the pixels in the finite element model form a one-to-one mapping relation.
6. A method for simulating a tin-based binary eutectic alloy microstructure and finite element solution analysis according to claim 1, wherein in step S6, generating complex finite element models containing different element types and material property information is performed as follows:
transferring the material attribute and element type information in the script file to the element corresponding to the character combination through CAE software, associating the pixel with the phase information of the element with the same number, endowing or modifying the element information in the simple finite element model according to the phase information, and generating the finite element model containing the complex element type and the material attribute information.
7. The method of claim 1 for simulating a tin-based binary eutectic alloy microstructure and finite element solution analysis, wherein:
in step S7, three-dimensional reconstruction and unit node coupling are performed through the position correspondence, and boundary constraint conditions are applied, so that solution analysis of the complex model in finite element CAE software can be realized, and a simulation result is obtained.
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Application publication date: 20190326 Assignee: Guilin Qunmei Technology Co.,Ltd. Assignor: GUILIN University OF ELECTRONIC TECHNOLOGY Contract record no.: X2023980044664 Denomination of invention: Simulation and finite element analysis method for microstructure of tin based binary eutectic alloy Granted publication date: 20220816 License type: Common License Record date: 20231031 |