breeds [element1 element2] ;; two different materials or phases ;; element1 is the main material ;; element2 is the materials which is ;; dispersed inside element1 (second-phase particles) element1-own [ neighbors-6 ;; agentset of 6 neighboring cells n ;; used to store a count of second-phase neighbors ] patches-own [ border? ;; this indicates that a patch is located on the view's border ] turtles-own [ temp ;; atom's temperature neighboring-turtles ;; agentset of surrounding atoms sides-exposed ;; number of sides exposed to walls (between 0 and 4) ] globals [ time ;; keeps track of simulation time logtime ;; log of time colors ;; used both to color turtles, and for histogram xmax ;; max x size ymax ;; max y size intercepts ;; used to calculate grain size (1/intercepts = average grain size) average-grain-size ;; average grain size logaverage-grain-size ;; log of average grain size (for plotting) orientation-for-incercept-count ;; for grain size calculation (normally 90 degrees) MCS ;; Number of Monte Carlo Steps (simulation steps) initial-loggrain-size ;; For grain growth exponent calculation and graphing initial-logtime ;; For grain growth exponent calculation and graphing grain-growth-exponent ;; Grain growth exponent total-atoms ;; Total number of atoms in the system ] to fill-outer-area-with-white ;; fills outer white box set ymax (height + 1) / 2 set xmax (width + 1) / 2 ask patches with [(abs pycor >= ymax) or (abs pxcor >= xmax)] [set pcolor white set border? true ] end to setup-hex-grid ;; setup the hexagonal grid in which atoms will be placed ;; and creates turtles set-default-shape element2 "square" ask patches [ ;; to avoid wrapping around the edge of the screen, ;; we do not create atoms on the screen's edge if border? != true [ sprout 1 [ ifelse (fraction-element2 > random-float 100) ;; if there is a second element, create the corresponding atoms [ ;; element2 is the fixed second-phase particle set breed element2 set color white set heading 360 ] [ ;; element1 is the main material, which grains growth set breed element1 set shape atom-shape set color random 139 set heading color ] ;; shift even columns down if pxcor mod 2 = 0 [ set ycor ycor - 0.5 ] ] ] ] ;; now set up the neighbors6 agentsets ; ask element1 [ set shape3d "sphere" ] ; ask element2 [ set shape3d "cube" ] ; the two above lines are for NetLogo 3D. Uncomment them if you use that version. ask element1 ;; define neighborhood of atoms [ ifelse pxcor mod 2 = 0 [ set neighbors-6 element1-on patches at-points [[0 1] [1 0] [1 -1] [0 -1] [-1 -1] [-1 0]] ] [ set neighbors-6 element1-on patches at-points [[0 1] [1 1] [1 0] [0 -1] [-1 0] [-1 1]] ] ] end ;; makes initial box for image import to makes-initial-box resets-time fill-outer-area-with-white setup-hex-grid end ;; makes initial box for random arrangement to makes-initial-box-random ca resets-time fill-outer-area-with-white setup-hex-grid end to resets-time ;; resets time set time 1 set MCS 0 end ;; import image into turtles to import-image ca let file user-choose-file if file = false [ stop ] ;; imports image into patches import-pcolors file ;; generates a white border around the image to avoid wrapping ;; converts the square grid to an hex grid makes-initial-box ;; transfers the image to the turtles. Rounds the color values to be integers. ask turtles [ set color round (pcolor-of patch-here) set heading color ] ;; erases the patches (sets their color back to black), ;; except the border patches, which will remain white ask patches [if border? != true [ set pcolor black ] ] end to define-neighboring-turtles ;; defines neighboring turtles ask turtles [ set neighboring-turtles (turtles at-points [[-1 1] [ 0 1] [1 1] [-1 0] [ 0 0] [1 0] [-1 -1] [ 0 -1] [1 -1]]) ] end to grain-count ;; count number of grains based on the number of linear intercepts set orientation-for-incercept-count 90 ;; direction of intercepts count set intercepts 0 set total-atoms count turtles ;; ask patches ask turtles [ ;; checks if turtle is before the last 'x' column and 'y' row if ((xcor != (xmax - 1)) and (who < ((width * height) - 1)) and (who < total-atoms)) [ ;; checks if there is a turtle to the right for the intercept calculation ifelse any? turtles-on (patch-at-heading-and-distance orientation-for-incercept-count 1) [ ;; If there is a turtle, checks if the heading is different. let right-neighbor-heading heading-of (random-one-of turtles-on (patch-at-heading-and-distance orientation-for-incercept-count 1)) if (heading != (right-neighbor-heading)) [ ;; If heading is different, add 1 to 'intercepts'. set intercepts (intercepts + 1) ] ] [ ;; if there is no turtle, simply add 1 to 'intercepts'. ;; A turtle/nothing interface is considered as grain boundary. set intercepts (intercepts + 1) ] ] ] ifelse intercepts = 0 [set average-grain-size (total-atoms)] ;; grain size = area of the whole sample (to avoid division by zero) [set average-grain-size ((total-atoms) / intercepts)] ;; grain size = area / grain-grain interface end to do-plots set-current-plot "Grain Size (log-log)" plot average-grain-size end to go ;;initiates grain growth set total-atoms count turtles if average-grain-size >= total-atoms [stop] ;; stops when there is just one grain repeat (total-atoms) [ ;;limits grain growth to element1, element2 represent the stationary second-phase particles ask random-one-of element1 [grain-growth] ] set MCS (MCS + 1) ;; advances Monte Carlo Steps (simulation time) ;; one Monte Carlo Step represents 'n' reorientation attemps, ;; where 'n' is the total number of atoms if remainder MCS measurement-frequency = 0 ;; calculates grain size at a given frequency [ set logtime (log time 10) grain-count if average-grain-size != 0 [ set logaverage-grain-size (log (average-grain-size) 10) ] ;; grain growth is better plotted on log-log scale do-plots if MCS = 20 [ set initial-logtime logtime set initial-loggrain-size logaverage-grain-size ] ;; only initiates grain size calculation after MCS = 20 if MCS > 20 [ ;; calculate the angular coeficient of the grain growth curve ;; since it is a log-log plot, it's the grain growth exponent set grain-growth-exponent (-1 * ((logaverage-grain-size - initial-loggrain-size) / (initial-logtime - logtime))) ] ] end ;; Grain growth procedure - free energy minimization ;; if another random crystalographic heading minimizes energy, switches headings, otherwise keeps the same. to grain-growth ;; increases time - divided by the area, gives the Monte Carlo Steps set time (time + 1) ;; calculates the PRESENT free energy let present-heading (heading) let present-free-energy count neighbors-6 with [heading != present-heading] ;; chooses a random orientation let future-heading (heading-of (random-one-of neighbors-6)) ;; calculates the FUTURE free energy, with the random orientation just chosen let future-free-energy count neighbors-6 with [heading != future-heading] ;; compares PRESENT and FUTURE free-energies; the lower value "wins" ifelse future-free-energy <= present-free-energy [set heading (future-heading)] [if (annealing-temperature > random-float 100) [set heading (future-heading)]] ;; this last line simulates thermal agitation (adds more randomness to the simulation) set color heading ;;update the color of the atoms end ;; drawing procedure to turtle-draw if mouse-down? ;; reports true or false to indicate whether mouse button is down [ ask turtles-at mouse-xcor mouse-ycor [ask turtles in-radius brush-size [set color draw-color set heading color]] ] end ;; in the drawing mode, erases the whole "canvas" with red to erase-all ask turtles [if pcolor != white [set color red set heading color]] end ; *** NetLogo Model Copyright Notice *** ; ; ; Copyright 2005 by Uri Wilensky. All rights reserved. ; ; Permission to use, modify or redistribute this model is hereby granted, ; provided that both of the following requirements are followed: ; a) this copyright notice is included. ; b) this model will not be redistributed for profit without permission ; from Uri Wilensky. ; Contact Uri Wilensky for appropriate licenses for redistribution for ; profit. ; ; To refer to this model in academic publications, please use: ; Blikstein, P. and Wilensky, U. (2005). NetLogo MaterialSim Grain Growth model. ; http://ccl.northwestern.edu/netlogo/models/MaterialSimGrainGrowth. ; Center for Connected Learning and Computer-Based Modeling, ; Northwestern University, Evanston, IL. ; ; In other publications, please use: ; Copyright 2005 by Uri Wilensky. All rights reserved. ; See http://ccl.northwestern.edu/netlogo/models/MaterialSimGrainGrowth ; for terms of use. ; ; We gratefully acknowledge the support of the ; National Science Foundation (REPP, ROLE & ITR programs) ; ; *** End of NetLogo Model Copyright Notice *** @#$#@#$#@ GRAPHICS-WINDOW 476 10 878 433 24 24 8.0 1 10 1 1 1 0 1 1 1 CC-WINDOW 5 580 887 675 Command Center 0 SLIDER 234 37 462 70 width width 3 501 45 2 1 atoms SLIDER 235 73 462 106 height height 3 501 45 1 1 atoms TEXTBOX 12 17 205 35 (1) Simulation starting point SLIDER 235 149 464 182 annealing-temperature annealing-temperature 0 100 0 1 1 % BUTTON 237 308 392 347 measure grains now grain-count NIL 1 T OBSERVER T NIL PLOT 11 378 263 566 Grain Size (log-log) log (time) log (avg grain size) 0.0 200.0 0.0 1.0 true false PENS "average-grain-size" 1.0 0 -16777216 false MONITOR 268 437 376 486 Grain Size average-grain-size 3 1 BUTTON 11 315 111 355 go go T 1 T OBSERVER NIL NIL SLIDER 235 185 463 218 fraction-element2 fraction-element2 0 10 0 1 1 NIL BUTTON 9 36 200 69 start with random arrangement makes-initial-box-random NIL 1 T OBSERVER T NIL BUTTON 9 78 200 111 import image import-image NIL 1 T OBSERVER T NIL MONITOR 380 438 493 487 Log Grain Size logaverage-grain-size 2 1 MONITOR 378 379 472 428 Log time logtime 2 1 MONITOR 267 379 374 428 Simulation time MCS 0 1 MONITOR 380 491 493 540 Growth exponent grain-growth-exponent 2 1 TEXTBOX 11 295 150 313 (4) Run simulation TEXTBOX 236 127 326 145 Special features SLIDER 235 265 423 298 measurement-frequency measurement-frequency 1 100 1 1 1 NIL TEXTBOX 238 18 328 36 Simulation size CHOOSER 10 135 111 180 atom-shape atom-shape "hex" "hexline" "thin-line" "line" "circ" "square" "spikes90" "default" 0 BUTTON 115 135 200 180 apply shape ask turtles [set shape atom-shape] NIL 1 T OBSERVER T NIL BUTTON 10 204 79 237 draw turtle-draw T 1 T OBSERVER T NIL BUTTON 10 243 79 286 erase all erase-all NIL 1 T OBSERVER T NIL SLIDER 82 204 200 237 brush-size brush-size 1 6 4 1 1 NIL TEXTBOX 11 185 101 203 (3) Draw grains CHOOSER 82 242 200 287 draw-color draw-color 45 55 85 105 15 0 TEXTBOX 237 244 367 262 Grain measurement TEXTBOX 13 360 181 378 Grain size plot and calculations TEXTBOX 10 115 186 133 (2) Change the shape of atoms BUTTON 117 315 194 355 go once go\n NIL 1 T OBSERVER T NIL @#$#@#$#@ WHAT IS IT? ----------- Most materials are not continuous arrangements of atoms, but rather composed of thousands or millions of microscopic crystals, known as grains. This model shows how the configuration and sizes of these grains change over time. Grain size is a very important characteristic for evaluating the mechanical properties of materials; it is exhaustively studied in metallurgy and materials science. Usually this kind of study is made by careful analysis and comparison of pictures taken in microscopes, sometimes with the help of image analysis software. Recently, as the processing power of computers has increased, a new and promising approach has been made possible: computer simulation of grain growth. Anderson, Srolovitz et al. proposed the most widely known and employed theory for computer modeling and simulation of grain growth, using the Monte Carlo method. Instead of considering the grains as spheres, and being obliged to make numerous geometrical approximations, Anderson proposed that the computer would simulate the behavior of each individual atom in the system. Each atom would follow a very simple rule: it will always try to have, in its immediate neighborhood, as many atoms as possible with the same orientation as it. This model is part of the MaterialSim (Blikstein & Wilensky, 2004) curricular package. To learn more about MaterialSim, see http://ccl.northwestern.edu/materialsim/. HOW IT WORKS ------------ The basic algorithm of the simulation is simple: atoms are trying to be as stable as possible. Their stability is based on the number of equal neighbors: the more equal neighbors (i.e. atoms with the same orientation) an atom has, the more stable it is. If it has many different neighbors, it is unstable, and not likely to be in that position for long, because during the simulation atoms will try to relocate to more stable positions. Therefore, the steps are: 1) Choose a random atom. 2) Calculate its present energy (which is correlated with the stability). This calculation is done by simply counting the number of different neighbors. 3) Randomly choose a new orientation for the chosen atom, amongst the orientations of its neighbors. We still don't know if that new attempted orientation will be maintained. We have to calculate the energy in this new situation in order to know. 4) Calculate the free energy of the chosen element with the new, tentative orientation. Again, we just count the number of different neighbors. 5) Comparison of the two values for free energy: the lowest value "wins", i.e., the less different neighbors an atom have, more stable it is. 6) Repeat steps 1-6. The ANNEALING-TEMPERATURE slider controls the probability of maintaining an re-orientation which yields more instability. The FRACTION-ELEMENT-2 slider defines the percentage of second-phase particles to be created when the user setups the simulation. Those particles and not movable and are not subject to grain growth. Atoms see those particles as a different neighbor. Note that the actual number of atoms is small compared to a real metal sample. Also, real materials are three-dimensional, while this model is 2D. HOW TO USE IT ------------- (1) Simulation starting point: IMPORT IMAGE: Resets the simulation, and imports an image file in the JPG, BMP, GIF or PNG file formats. The image will be automatically resized to fit into the view, but maintaining its original aspect ratio. Note that the image MUST HAVE THE SAME ASPECT RATIO AS THE VIEW. In other words, if the view is square, the image should be square as well. Prior to importing the image, it is recommended to clean it up using an image editing software (increase contrast, remove noise). Try to experiment various combinations of values for the WIDTH and HEIGHT sliders, the view's size and the patch size to get the best results. START WITH RANDOM ARRANGEMENT: Resets the simulation, and starts it with a random orientation for each atom. GO-ONCE: Runs the simulation, one time step at a time. GO: Runs the simulation continuously until either the GO button is pressed again, or just one grain survives. (2) Change the shape of the atoms The ATOM-SHAPE chooser has many different shapes, such as circle, hexagon, line, circle with spikes, thin line, and square. After choosing the ATOM-SHAPE, click on APPLY to change the shape. This can also be done during the simulation. (3) Draw grains You can draw grains with the mouse, using different brush sizes and colors. The DRAW button activates drawing, the ERASE ALL button erases the screen and sets all the atoms to red, the BRUSH-SIZE slider controls the radius of the brush and the DRAW-COLOR chooser changes the numeric value of the drawing color. (4) Run simulation GO: runs the simulation continuously GO ONCE: runs the simulation, one step at a time. Simulation size WIDTH: x (horizontal) dimension of the sample. HEIGHT: y (vertical) dimension of the sample Special features ANNEALING-TEMP: changes the probability of non-favorable orientation flips to happen. A 10% value, for instance, means that 10% of non-favorable flips will be maintained. This mimics the effect of higher temperatures. FRACTION-ELEMENT2: This slider controls the amount of dispersed second-phase particles throughout the sample. Those particles slow down or stop grain growth. Grain measurement MEASUREMENT-FREQUENCY: to increase the model's speed, the user can choose not to calculate grain size at every time step. If grain size is calculated at every ten time units (20, 30, 40 etc.), the performance is slightly increased. This only affects the plot and the monitors, but not the actual simulation. MEASURE GRAINS NOW: if the MEASUREMENT-FREQUENCY is too large, and the user wants to evaluate grain size at a specific moment, this button can be used. Note that this does not alter the plot. Plots and monitors Grain Size (log-log): Grain size vs. time, in a log-log scale. Under normal conditions (ANNEALING-TEMP = 0 and FRACTION-ELEMENT-2 = 0), this plot should be a straight line with an angular coefficient of approximately 0.5. SIMULATION TIME and LOG TIME: time steps of the simulation so far (and its log) GRAIN SIZE and LOG GRAIN SIZE: grain size (is atoms) and its log. GROWTH EXPONENT: the angular coefficient of the GRAIN SIZE plot. This number should approach 0.5 with ANNEALING-TEMP = 0 and FRACTION-ELEMENT2 = 0. THINGS TO NOTICE ---------------- When you setup with a random orientation and run the simulation, notice that the speed of growth decreases with time. Toward the end of the simulation, you might see just two or three grains that fight with each other for along time. One will eventually prevail, but this logarithmic decrease of speed is an important characteristic of grain growth. That is why the GRAIN SIZE plot is a straight line in a "log-log" scale. Notice also that if you draw two grains, one concave and one convex, their boundary will tend to be a straight line, if you let the simulation run long enough. Every curved boundary is unstable because many atoms at its interface will have more different than equal neighbors. THINGS TO TRY ------------- Increase the value of the ANNEALING-TEMP slider. What happens to the GRAIN SIZE plot, and to the boundaries' shapes? Try to increase the FRACTION-ELEMENT2 slider to 5%. Then press START WITH RANDOM ARRANGEMENT and GO. What happens to grain growth? Now try several values (1, 3, 5, 7, 9%), for instance. What happens with the final grain size? What about the GRAIN SIZE plot and the GROWTH EXPONENT? One advanced use of this model would be to get a digital picture of a real metallic sample, reduce noice and increase contrast with image editing programs, and load into this model using the IMPORT IMAGE button. Don't forget to update the WIDTH and HEIGHT sliders and the view's size to accommodate the picture, and also to change the patch size in order to be able to see the whole sample. EXTENDING THE MODEL ------------------- This models assumes that the misorientation between two grains has no effect on their growth rates. Two grains with a very similar crystallographic orientation have the same growth rate as grains which orientations differ by a lot. Try to take the angular misorientation into consideration. When we insert second-phase particles, all of them have the same size. Try to create a slider that changes the size of the particles. NETLOGO FEATURES ---------------- This model uses some special features: It uses a hexagonal grid (as opposed to a square one) it uses different shapes for different visualization purposes and it uses the import-pcolors primitive to enable the image import capability. RELATED MODELS -------------- Crystallization Basic Crystallization Directed CREDITS AND REFERENCES ---------------------- ?This model is part of the MaterialSim (Blikstein & Wilensky, 2004) curricular package. To learn more about MaterialSim, see http://ccl.northwestern.edu/materialsim/. To refer to this model in academic publications, please use: Blikstein, P. and Wilensky, U. (2005). NetLogo MaterialSim Grain Growth model. http://ccl.northwestern.edu/netlogo/models/MaterialSimGrainGrowth. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. In other publications, please use: Copyright 2005 by Uri Wilensky. All rights reserved. See http://ccl.northwestern.edu/netlogo/models/MaterialSimGrainGrowth for terms of use. Two papers describing the use of this model in education are: Blikstein, P. & Wilensky, U. (2005) Less is More: Agent-Based Simulation as a Powerful Learning Tool in Materials Science. The IV International Conference on Autonomous Agents and Multiagent Systems. Utrecht, Netherlands. Blikstein, P. & Wilensky, U. (2004) MaterialSim: An agent-based simulation toolkit for Materials Science learning. (PDF, 1.5 MB) Proceedings of the International Conference on Engineering Education. Gainesville, Florida. The core algorithm of the model was developed at the University of Sao Paulo and published in: Blikstein, P. and Tschiptschin, A. P. Monte Carlo simulation of grain growth (II). Materials Research, Sao Carlos, 2 (3), p. 133-138, jul. 1999. Available for download at: http://www.blikstein.com/paulo/documents/papers/BliksteinTschiptschin-MonteCarlo-MaterialsResearch1999.pdf. 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