(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 154871, 4035]*) (*NotebookOutlinePosition[ 155541, 4058]*) (* CellTagsIndexPosition[ 155497, 4054]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Phase Separations", "Title"], Cell[CellGroupData[{ Cell["Reference", "Section"], Cell[CellGroupData[{ Cell["Title", "Subsection"], Cell["Phase Separations", "Text", FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell["Author", "Subsection"], Cell["Thomas G. Anderson (tga@stanford.edu)", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Summary", "Subsection"], Cell["\<\ This package computes the compositions of coexisting phases in a \ binary mixture.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Copyright", "Subsection"], Cell[TextData[{ "\[Copyright] Copyright 2002 ", StyleBox["Thomas G. Anderson", FontSlant->"Italic"] }], "Text"], Cell["\<\ Permission is granted to distribute this file for any purpose \ except for inclusion in commercial software or program collections. This \ copyright notice must remain intact.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Notebook Version", "Subsection"], Cell["1.0", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Version" }], "Subsection"], Cell["4.1.5", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["History", "Subsection"], Cell["1.0\t21 January 2002\tFirst public version\t", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Keywords", "Subsection"], Cell["phase separation, phase diagram, binary mixture, miscibility", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Sources", "Subsection"], Cell[TextData[{ "Package template adapted from Maeder, R. \"Programming in ", StyleBox["Mathematica,", FontSlant->"Italic"], "\" 3d Ed." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Warnings", "Subsection"], Cell["none", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Limitations", "Subsection"], Cell["\<\ BinaryPhaseSeparations uses a finite sample of points in the free \ energy function to determine the regions of phase separation, so it is \ possible that small two-phase regions will be missed. The points are chosen \ by the adaptive sampling algorithm used in the built-in Plot function, so \ only features resolvable by Plot will be detected by BinaryPhaseSeparations. \ In other words, if you can see a region of phase separation by plotting the \ free energy function, it can be found. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Discussion", "Subsection"], Cell[TextData[{ "In chemical thermodynamics, a mixture of two or more components is \ unstable with respect to phase separation if the second derivative of the \ free energy with respect to composition is negative. At equilibrium, the \ chemical potential of the components is the same between phases. \ Geometrically speaking, this means that the free energy curve has a common \ tangent at the compositions of the coexisting phases. Conversely, the \ composition of coexisting phases may be found by finding a line that is \ tangent to the free energy curve at two points; this is the method of ", StyleBox["double tangent construction.", FontSlant->"Italic"] }], "Text"], Cell["\<\ This package implements the method of double tangent construction \ using an adaptive algorithm to sample the free energy function find the \ approximate composition of coexisting phases, followed by refinement to \ arbitrary precision. Multiple distinct regions of phase separation are found \ automatically by a simple recursive algorithm.\ \>", "Text"], Cell[TextData[{ "Briefly, the algorithm works as follows. First, the free energy function \ is sampled using the built-in Plot function (with the graphic output \ suppressed). These points are used to numerically compute the curvature of \ the function. If any points of negative curvature are found, the algorithm \ proceeds to the next step. The composition having the smallest (most \ negative) curvature is identified as ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], ". The point of minimum free energy is taken as the first approximation of \ one phase, ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ". The first approximation of the second phase, ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], ", is taken to be that point on the other side of ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " for which the slope of the line between it and ", Cell[BoxData[ \(TraditionalForm\`\((x\_1, y\_1)\)\)]], " is closest to zero. Then the set of points is skewed so that ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], " have the same ", StyleBox["y", FontSlant->"Italic"], " coordinate. The process is repeated until the result no longer changes. \ To refine the values of ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], ", the free energy of the two phases present at ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\_sep = \([\((x\_0 - x\_1)\) y\_2 + \((x\_2 - x\_0)\) y\_1]\)/\((x\_2 - x\_1)\)\)]], ", is minimized with respect to ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Requirements", "Subsection"], Cell["\<\ Requires the Standard Packages NumericalMath`NLimit` and \ Utilities`FilterOptions`.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Interface", "Section", InitializationCell->True], Cell["\<\ This part declares the publicly visible functions, options, and \ values.\ \>", "Text", InitializationCell->True], Cell[CellGroupData[{ Cell["Set up the package context, including public imports", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(BeginPackage["\", {"\"}];\ \)\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Usage messages for the exported functions and the context \ itself\ \>", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(PhaseSeparations::usage = "\";\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(BinaryPhaseSeparations::usage = "\";\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(DrawPhaseDiagram::usage = "\";\)\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(\(BinaryPhaseDiagram::usage = "\";\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(SepPoints::usage = "\";\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(SepDivision::usage = "\";\)\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(\(VerticalPoints::usage = "\";\)\ \)], "Input", InitializationCell->True], Cell[BoxData[ \(\(VerticalDivision::usage = "\";\)\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Error messages for the exported objects", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(BinaryPhaseSeparations::noref = "\";\)\)], "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Implementation", "Section", InitializationCell->True], Cell[CellGroupData[{ Cell["Begin the private context (implementation part)", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(Begin["\"];\)\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Read in any hidden imports", "Subsection", InitializationCell->True], Cell[BoxData[ \(Needs["\"]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Unprotect any system functions for which definitions will be made\ \ \>", "Subsection", InitializationCell->True], Cell["n/a", "Text", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Definition of auxiliary functions and local (static) \ variables\ \>", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["skew", "Subsubsection", InitializationCell->True], Cell[TextData[{ "Skews the curve given by the points ", StyleBox["vals", FontSlant->"Italic"], " about the point ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " ", "by addition of a linear function. The slope of the added function is \ determined by finding the minimum point on the curve, ", Cell[BoxData[ \(TraditionalForm\`P\_1\)]], ", and then finding that point ", Cell[BoxData[ \(TraditionalForm\`P\_2\)]], " on the other side of ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " such that the slope of the line ", Cell[BoxData[ \(TraditionalForm\`\(P\_1\) P\_2\)]], " is closest to zero." }], "Text", InitializationCell->True], Cell[TextData[{ "To avoid unnecessary computation, points on the side of ", Cell[BoxData[ \(TraditionalForm\`P\_1\)]], "away from ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " are discarded, as is the point ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "." }], "Text", InitializationCell->True], Cell[BoxData[ \(skew[pts_, x0_] := Module[{pmin, lpts, rpts, slope}, \[IndentingNewLine]pmin = Reverse[First[Sort[Reverse /@ pts]]]; \[IndentingNewLine]If[ pmin\[LeftDoubleBracket]1\[RightDoubleBracket] < x0, \[IndentingNewLine] (*\ minimum\ is\ to\ left\ of\ x0\ *) \[IndentingNewLine]\((\ \[IndentingNewLine]lpts = Select[pts, \((pmin\[LeftDoubleBracket]1\[RightDoubleBracket] \ \[LessEqual] #\[LeftDoubleBracket]1\[RightDoubleBracket] < x0)\) &]; \[IndentingNewLine]rpts = Select[pts, \((x0 < \ #\[LeftDoubleBracket]1\[RightDoubleBracket])\) &]; \[IndentingNewLine]slope = Min[\(\((Divide @@ Reverse[# - pmin])\) &\) /@ rpts];\[IndentingNewLine])\), \[IndentingNewLine] (*\ minimum\ is\ to\ right\ of\ x0\ *) \[IndentingNewLine]\((\ \[IndentingNewLine]lpts = Select[pts, \((#\[LeftDoubleBracket]1\[RightDoubleBracket] < x0)\) &]; \[IndentingNewLine]rpts = Select[pts, \((x0 < #\[LeftDoubleBracket]1\[RightDoubleBracket] \ \[LessEqual] pmin\[LeftDoubleBracket]1\[RightDoubleBracket])\) &]; \ \[IndentingNewLine]slope = Max[\(\((Divide @@ Reverse[# - pmin])\) &\) /@ lpts];\[IndentingNewLine])\)\[IndentingNewLine]]; \ \[IndentingNewLine]\({#\[LeftDoubleBracket]1\[RightDoubleBracket], #\ \[LeftDoubleBracket]2\[RightDoubleBracket] - slope #\[LeftDoubleBracket]1\[RightDoubleBracket]} &\) /@ Join[lpts, rpts]\[IndentingNewLine]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["concavity", "Subsubsection", InitializationCell->True], Cell["\<\ Determines the concavity for a curve running through three points; \ this expression corresponds to the coefficient of the quadratic term of a \ parabola fit to the points.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(concavity[{x1_, y1_}, {x2_, y2_}, {x3_, y3_}] := \((x1 \((y3 - y2)\) + x2 \((y1 - y3)\) + x3 \((y2 - y1)\))\)/\((\((x1 - x2)\) \((x1 - x3)\) \((x2 - x3)\))\)\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["findSeparations", "Subsubsection", InitializationCell->True], Cell["\<\ This function recursively determines the intervals of phase \ separation over the given composition range. First, the point of greatest \ instability (negative concavity) is identified and its phases calculated. \ Then the function calls itself for the composition ranges on either side of \ this separation. If no unstable point is found, returns an empty list.\ \>", \ "Text", InitializationCell->True], Cell[BoxData[ \(\(findSeparations[pts_ /; Length[pts] < 3] = {};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(findSeparations[pts_] := Module[{xc, xcmin, skewpts, x1, x2}, \[IndentingNewLine]xc = Map[{#\[LeftDoubleBracket]2, 1\[RightDoubleBracket], concavity @@ #} &, Partition[pts, 3, 1]]; \[IndentingNewLine]xcmin = Reverse[First[Sort[Reverse /@ xc]]]; \[IndentingNewLine]If[ xcmin\[LeftDoubleBracket]2\[RightDoubleBracket] \[GreaterEqual] 0, Return[{}]]; \[IndentingNewLine]\[IndentingNewLine]skewpts = FixedPoint[ skew[#, xcmin\[LeftDoubleBracket]1\[RightDoubleBracket]] &, pts, 10]; \[IndentingNewLine]{x1, x2} = \(Sort[Reverse /@ skewpts]\)\[LeftDoubleBracket]{1, 2}, 2\[RightDoubleBracket]; \ \[IndentingNewLine]\[IndentingNewLine]{findSeparations[ Select[pts, \((#\[LeftDoubleBracket]1\[RightDoubleBracket] < x1)\) &]], \[IndentingNewLine]Interval[{x1, x2}], \[IndentingNewLine]findSeparations[ Select[pts, \((#\[LeftDoubleBracket]1\[RightDoubleBracket] > x2)\) &]]}\[IndentingNewLine]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["refineSeparation", "Subsubsection", InitializationCell->True], Cell[TextData[{ "Refines a calculated phase separation by minimizing the total free energy \ of the two phases. For a point ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " inside the region of phase separation, ", Cell[BoxData[ \(TraditionalForm\`{x\_1, x\_2}\)]], ", the total free energy of the coexisting phases is, by the lever rule," }], "Text", InitializationCell->True], Cell[BoxData[ \(f \((x\_0)\) = \(\((x\_2 - x\_0)\)\/\((x\_2 - x\_1)\)\) f \((x\_1)\) + \(\((x\_0 - x\_1)\)\/\((x\_2 - x\_1)\)\) f \((x\_2)\)\)], "DisplayFormula", InitializationCell->True], Cell[BoxData[ \(\(Attributes[refineSeparation] = {HoldFirst};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(refineSeparation[f_, x_Symbol, Interval[{x1o_, x2o_}], opts___] := Module[{x0, fm}, \[IndentingNewLine]x0 = \((x1o + x2o)\)/ 2; \[IndentingNewLine]fm = FindMinimum[\[IndentingNewLine]If[\(! \((0 < x1 < x0 < x2 < 1)\)\), $MaxMachineNumber, \ \[IndentingNewLine]ReleaseHold[\((\((x2 - x0)\) \((Hold[f] /. x \[Rule] x1)\) + \((x0 - x1)\) \((Hold[f] /. x \[Rule] x2)\))\)/\((x2 - x1)\)]], \[IndentingNewLine]{x1, {x1o, x1o + \((x2o - x1o)\)/100}}, \[IndentingNewLine]{x2, {x2o, x2o - \((x2o - x1o)\)/ 100}}, \[IndentingNewLine]opts\[IndentingNewLine]]; \ \[IndentingNewLine]If[ Head[fm] === FindMinimum, \((Message[BinaryPhaseSeparations::noref, {x1, x2}]; Interval[{x1o, x2o}])\), \[IndentingNewLine]Interval[{x1, x2} /. fm\[LeftDoubleBracket]2\[RightDoubleBracket]]]\ \[IndentingNewLine]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["connectIntervals", "Subsubsection", InitializationCell->True], Cell["\<\ For a pair of binary phase separation intervals at consecutive \ potential values, connects the endpoints of intervals that overlap in \ composition.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(connectIntervals[{i1_Interval, p1_}, {i2_Interval, p2_}] := Outer[\[IndentingNewLine]If[ IntervalIntersection[#1, #2] === Interval[], {}, {Line[{{#1\[LeftDoubleBracket]1, 1\[RightDoubleBracket], p1}, {#2\[LeftDoubleBracket]1, 1\[RightDoubleBracket], p2}}], Line[{{#1\[LeftDoubleBracket]1, 2\[RightDoubleBracket], p1}, {#2\[LeftDoubleBracket]1, 2\[RightDoubleBracket], p2}}]}] &, \[IndentingNewLine]List @@ Split[i1], List @@ Split[i2]\[IndentingNewLine]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["joinLineSegments", "Subsubsection", InitializationCell->True], Cell["\<\ List of rules for joining together Lines that have common \ endpoints.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(\(joinLineSegments = {{l1___, Line[{p1__, p2_}], l2___, Line[{p2_, p3__}], l3___} \[RuleDelayed] {Line[{p1, p2, p3}], l1, l2, l3}, {l1___, Line[{p2_, p3__}], l2___, Line[{p1__, p2_}], l3___} \[RuleDelayed] {Line[{p1, p2, p3}], l1, l2, l3}};\)\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["criticalPoints", "Subsubsection", InitializationCell->True], Cell["\<\ For a pair of binary phase separation intervals at consecutive \ potential values, identifies separation regions that are present at one \ potential but not the other, and interpolates their coordinates to obtain \ miscibility critical points.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(criticalPoints[{i1_Interval, p1_}, {i2_Interval, p2_}] := Join[Select[ List @@ Split[ i1], \((IntervalIntersection[#, i2] === Interval[])\) &], Select[List @@ Split[i2], \((IntervalIntersection[#, i1] === Interval[])\) &]] /. Interval[{x1_, x2_}] \[RuleDelayed] Point[{\((x1 + x2)\)/2, \((p1 + p2)\)/2}]\)], "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definition of the exported functions", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["BinaryPhaseSeparations", "Subsubsection", InitializationCell->True], Cell["\<\ Calculates the intervals of two-phase coexistence for a given free \ energy function. In the event that a free energy function is not well-behaved \ near the edges of the composition range, we allow explicit limits on the \ composition to be provided as optional arguments.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(\(Attributes[BinaryPhaseSeparations] = {HoldFirst, ReadProtected};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(Options[BinaryPhaseSeparations] = Join[Options[FindMinimum], {SepPoints \[Rule] 50, SepDivision \[Rule] 30. }];\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(BinaryPhaseSeparations[f_, x_Symbol, opts___?OptionQ] := BinaryPhaseSeparations[f, {x, 0, 1}, opts]\)], "Input", InitializationCell->True], Cell[BoxData[ \(BinaryPhaseSeparations[f_, {x_Symbol, xmin_?NumericQ, xmax_?NumericQ}, opts___?OptionQ] /; \((0 \[LessEqual] xmin < xmax \[LessEqual] 1)\) := Module[{optsfm, sp, sd, pts}, \[IndentingNewLine]optsfm = FilterOptions[FindMinimum, opts, Sequence @@ Options[BinaryPhaseSeparations]]; \[IndentingNewLine]{sp, sd} = {SepPoints, SepDivision} /. Flatten[{opts, Options[ BinaryPhaseSeparations]}]; \[IndentingNewLine]pts = \(Plot[ f, {x, xmin, xmax}, DisplayFunction \[Rule] Identity, PlotPoints \[Rule] sp, PlotDivision \[Rule] sd]\)\[LeftDoubleBracket]1, 1, 1, 1\[RightDoubleBracket]; \[IndentingNewLine]IntervalUnion @@ \ \((\(refineSeparation[f, x, #, Evaluate[optsfm]] &\) /@ Flatten[findSeparations[pts]])\)\[IndentingNewLine]]\)], "Input",\ InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["DrawPhaseDiagram", "Subsubsection", InitializationCell->True], Cell[TextData[{ "Given a list of phase coexistence regions ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox[ StyleBox["int", FontSlant->"Italic"], "1"], ",", \(p\_1\)}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox[ StyleBox["int", FontSlant->"Italic"], "2"], ",", \(p\_2\)}], "}"}], ",", "..."}], "}"}], TraditionalForm]]], " where the ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["int", FontSlant->"Italic"], "i"], TraditionalForm]]], " are the separation Intervals at the potentials ", Cell[BoxData[ \(TraditionalForm\`p\_i\)]], ", \"connects the dots\" to draw the phase diagram over the range of \ potentials given." }], "Text", InitializationCell->True], Cell[BoxData[ \(\(Attributes[DrawPhaseDiagram] = {ReadProtected};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(Options[DrawPhaseDiagram] = Join[{PlotStyle \[Rule] Automatic, CriticalPointStyle \[Rule] PointSize[0.02]}, Options[Graphics] /. {\((Frame \[Rule] _)\) \[Rule] \((Frame \ \[Rule] {True, True, False, False})\)}];\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(DrawPhaseDiagram[seps : {{_Interval, _?NumericQ} .. }, opts___?OptionQ] := Module[{optsgr, ps, cs, splt, lins, cpts}, \[IndentingNewLine]optsgr = FilterOptions[Graphics, opts, Sequence @@ Options[DrawPhaseDiagram]]; \[IndentingNewLine]{ps, cs} = \({PlotStyle, CriticalPointStyle} /. Flatten[{opts, Options[DrawPhaseDiagram]}]\) /. Automatic \[Rule] GrayLevel[0]; \[IndentingNewLine]\[IndentingNewLine]lins = Flatten[Table[ connectIntervals[ seps\[LeftDoubleBracket]i\[RightDoubleBracket], seps\[LeftDoubleBracket]i + 1\[RightDoubleBracket]], {i, 1, Length[seps] - 1}]] //. joinLineSegments; \[IndentingNewLine]splt = Split[seps, \((Length[#1\[LeftDoubleBracket]1\[RightDoubleBracket]] \ \[Equal] Length[#2\[LeftDoubleBracket]1\[RightDoubleBracket]])\) &]; cpts = Table[ criticalPoints[ splt\[LeftDoubleBracket]i, \(-1\)\[RightDoubleBracket], splt\[LeftDoubleBracket]i + 1, 1\[RightDoubleBracket]], {i, 1, Length[splt] - 1}]; \[IndentingNewLine]\[IndentingNewLine]Show[ Graphics[{{ps, lins}, {cs, cpts}}], optsgr]\[IndentingNewLine]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(cpts\)\" is similar to \ existing symbols \!\({opts, pts}\)."\)], "Message", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["BinaryPhaseDiagram", "Subsubsection", InitializationCell->True], Cell[BoxData[ \(\(Attributes[BinaryPhaseDiagram] = {HoldFirst, ReadProtected};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(Options[BinaryPhaseDiagram] = Join[Options[BinaryPhaseSeparations], {VerticalPoints \[Rule] 25, VerticalDivision \[Rule] 50. }, Options[DrawPhaseDiagram]];\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(BinaryPhaseDiagram[f_, x_Symbol, p_List, opts___Rule] := BinaryPhaseDiagram[f, {x, 0, 1}, p, opts]\)], "Input", InitializationCell->True], Cell[BoxData[ \(BinaryPhaseDiagram[ f_, {x_Symbol, xmin_?NumericQ, xmax_?NumericQ} /; \((0 \[LessEqual] xmin < xmax \[LessEqual] 1)\), {p_Symbol, pmin_?NumericQ, pmax_?NumericQ} /; \((pmin < pmax)\), opts___Rule] := Module[{optsbps, optsdpd, pr, vp, vd, ps, seps = {}, srt}, \[IndentingNewLine]optsbps = FilterOptions[BinaryPhaseSeparations, opts, Sequence @@ Options[BinaryPhaseDiagram]]; \[IndentingNewLine]optsdpd = FilterOptions[DrawPhaseDiagram, opts, Sequence @@ Options[BinaryPhaseDiagram]]; \[IndentingNewLine]pr = \ \(\(\(PlotRange /. {optsdpd}\) /. \(({Automatic, pr_} \[RuleDelayed] {{xmin - \((xmax - xmin)\)/40. , xmax + \((xmax - xmin)\)/40}, pr})\)\) /. \(({xr_, Automatic} \[RuleDelayed] {xr, {pmin - \((pmax - pmin)\)/ 40. , pmax + \((pmax - pmin)\)/ 40. }})\)\) /. \((Automatic \[Rule] {{xmin - \ \((xmax - xmin)\)/40. , xmax + \((xmax - xmin)\)/40. }, {pmin - \((pmax - pmin)\)/ 40. , pmax + \((pmax - pmin)\)/ 40. }})\); \[IndentingNewLine]{vp, vd} = {VerticalPoints, VerticalDivision} /. Flatten[{opts, Options[ BinaryPhaseDiagram]}]; \[IndentingNewLine]\ \[IndentingNewLine]Plot[\[IndentingNewLine]\((\[IndentingNewLine]ps = BinaryPhaseSeparations[With[{p = y}, f], x, optsbps]; \[IndentingNewLine]seps = Join[seps, {{ps, y}}]; \[IndentingNewLine]If[ps === Interval[], 0, Min[\(\((#\[LeftDoubleBracket]2\[RightDoubleBracket] - #\ \[LeftDoubleBracket]1\[RightDoubleBracket])\) &\) /@ \((List @@ ps)\)]]\[IndentingNewLine])\), \[IndentingNewLine]{y, pmin, pmax}, DisplayFunction \[Rule] Identity, PlotPoints \[Rule] vp, PlotDivision \[Rule] vd]; \[IndentingNewLine]\[IndentingNewLine]srt = Sort[Union[ seps], \((#1\[LeftDoubleBracket]2\[RightDoubleBracket] < #2\ \[LeftDoubleBracket]2\[RightDoubleBracket])\) &]; \[IndentingNewLine]\ \[IndentingNewLine]{srt, DrawPhaseDiagram[srt, PlotRange \[Rule] pr, optsdpd]}\[IndentingNewLine]]\)], "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definitions for system functions", "Subsection", InitializationCell->True], Cell["n/a", "Text", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Restore protection of system symbols", "Subsection", InitializationCell->True], Cell["n/a", "Text", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["End the private context", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(End[];\)\)], "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Epilog", "Section", InitializationCell->True], Cell["This section protects exported symbols and ends the package.", "Text", InitializationCell->True], Cell[CellGroupData[{ Cell["Protect exported symbols", "Subsection", InitializationCell->True], Cell[BoxData[ \(Protect[Evaluate[$Context <> "\<*\>"]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["End the package context", "Subsection", InitializationCell->True], Cell[BoxData[ \(EndPackage[]\)], "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Examples, Tests", "Section"], Cell["Disable spelling error messages and set options for Plot.", "Text"], Cell[BoxData[ \(Off[General::spell, General::spell1]\)], "Input"], Cell[BoxData[ \(\(SetOptions[Plot, Axes \[Rule] False, Frame \[Rule] {True, True, False, False}];\)\)], "Input"], Cell["This reads in the package.", "Text"], Cell[BoxData[ \(<< PhaseSeparations`\)], "Input"], Cell[CellGroupData[{ Cell["1. 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*******************************************************************)