(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 2886589, 48737] NotebookOptionsPosition[ 2873088, 48497] NotebookOutlinePosition[ 2873631, 48516] CellTagsIndexPosition[ 2873588, 48513] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 18 - 04/06/2022", "Subtitle", CellChangeTimes->{{3.8525512993398438`*^9, 3.8525513206118402`*^9}, { 3.852652054138073*^9, 3.8526520591301193`*^9}, {3.853194369726288*^9, 3.8531943739664793`*^9}, 3.8531971130005827`*^9, {3.853361889945813*^9, 3.853361893353859*^9}, {3.854455749568426*^9, 3.8544557604776297`*^9}, { 3.8545621484661007`*^9, 3.8545621501697083`*^9}, {3.85499650158072*^9, 3.854996504060568*^9}, {3.855176893175762*^9, 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orientational-dependance of \ surface energy and its implications of on the equilibrium shapes of crystals \ using the Wulff construction\ \>", "Item", CellChangeTimes->{{3.85637353525263*^9, 3.856373564586499*^9}, { 3.85637359663395*^9, 3.856373613349485*^9}, {3.856876710077532*^9, 3.856876732987672*^9}, {3.8569466786931887`*^9, 3.856946722662561*^9}, { 3.8574946251575327`*^9, 3.857494626325678*^9}, {3.857998552486617*^9, 3.857998609017055*^9}, {3.858157379071251*^9, 3.858157417827963*^9}},ExpressionUUID->"d4949c4d-60b3-49d3-b0a2-\ 829a872ec2fe"], Cell[CellGroupData[{ Cell["Orientational Dependence", "Section", CellChangeTimes->{{3.8580223639202843`*^9, 3.858022368279603*^9}},ExpressionUUID->"7371501b-69af-4753-bb0d-\ 45b000acdcf3"], Cell[CellGroupData[{ Cell["\<\ We\[CloseCurlyQuote]ve seen in L08 that preferred directions in crystals \ result in anisotropic physical properties.\ \>", "Item", CellChangeTimes->{{3.858024306298699*^9, 3.858024319404174*^9}, { 3.858024355280157*^9, 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TraditionalForm]],ExpressionUUID-> "685c644a-e0d4-4f92-a253-89ed9392ba5b"], " is the unit vector pointing towards the crystal\[CloseCurlyQuote]s \ orientation" }], "Subsubitem", CellChangeTimes->{{3.858268889179755*^9, 3.8582689359160757`*^9}},ExpressionUUID->"d3578b0c-3fd6-4140-86ba-\ ec0feff35b36"], Cell[CellGroupData[{ Cell[TextData[{ "Since we\[CloseCurlyQuote]re in 2D, we can parametrize ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox[ StyleBox["n", "TI"], "^"], TraditionalForm], "errors" -> {}, "input" -> "\\hat{n}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "0bd67e73-f17c-4687-a54f-b615aaaa2423"], " by the angle \[Theta] it makes with the {1,0} orientation" }], "Subsubitem", CellChangeTimes->{{3.858268889179755*^9, 3.858268973666254*^9}},ExpressionUUID->"247c6e0d-35ee-4cf2-a4a8-\ 2fcbb6bc099e"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"squareSurfaceEnergyDensity", "[", "\[Alpha]_", "]"}], "[", 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$CellContext`wulffline[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, Pattern[$CellContext`wulfflength, Blank[]]] := Block[{$CellContext`\[Theta], $CellContext`wulffhalf = \ $CellContext`wulfflength/ 2, $CellContext`x1, $CellContext`x2, $CellContext`y1, \ $CellContext`y2}, $CellContext`\[Theta] = ArcTan[$CellContext`x, $CellContext`y]; $CellContext`x1 = \ $CellContext`x + $CellContext`wulffhalf Cos[$CellContext`\[Theta] + Pi/2]; $CellContext`x2 = $CellContext`x + \ $CellContext`wulffhalf Cos[$CellContext`\[Theta] - Pi/ 2]; $CellContext`y1 = $CellContext`y + $CellContext`wulffhalf Sin[$CellContext`\[Theta] + Pi/2]; $CellContext`y2 = $CellContext`y + \ $CellContext`wulffhalf Sin[$CellContext`\[Theta] - Pi/2]; Line[{{$CellContext`x1, $CellContext`y1}, {$CellContext`x2, \ $CellContext`y2}}]]}; Typeset`initDone$$ = True), SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, 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87a7bfb38c34"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Gamma]", RowBox[{"(", OverscriptBox[ StyleBox["p", "TI"], "\[RightVector]"], ")"}], "\[LongEqual]", StyleBox["r", "TI"], "\[Gamma]", RowBox[{"(", OverscriptBox[ StyleBox["n", "TI"], "^"], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\gamma(\\vec{p}) = r \\gamma(\\hat{n})", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.858268878732563*^9, 3.858268881594473*^9}, 3.858269801028153*^9},ExpressionUUID->"7875d357-d90a-4c7c-84fe-\ d3c6d7ed1afb"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ OverscriptBox[ StyleBox["p", "TI"], "\[RightVector]"], "\[LongEqual]", StyleBox["r", "TI"], OverscriptBox[ StyleBox["n", "TI"], "^"]}], TraditionalForm], "errors" -> {}, "input" -> "\\vec{p} = r \\hat{n}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "6b285c6d-c536-4147-a218-87bc2a37a183"] }], "Subitem", CellChangeTimes->{{3.858269802114736*^9, 3.858269804082041*^9}},ExpressionUUID->"bd35d725-e72f-4389-a4e9-\ 03d9db6f751c"], Cell["\<\ This allows us to introduce a new vector-quantity, known as the capillary \ vector\ \>", "Item", CellChangeTimes->{{3.858269835736891*^9, 3.858269856429121*^9}, 3.858269906359694*^9},ExpressionUUID->"c6aad7aa-d1a0-4e80-8305-\ 434ece0367be"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Xi]", RowBox[{"(", OverscriptBox[ StyleBox["n", "TI"], "^"], ")"}], "\[LongEqual]", "\[Del]", "\[Gamma]", RowBox[{"(", OverscriptBox[ StyleBox["p", "TI"], "\[RightVector]"], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\xi(\\hat{n}) = \\nabla \\gamma(\\vec{p})", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.858268878732563*^9, 3.858268881594473*^9}, 3.858269801028153*^9, 3.858269936618724*^9},ExpressionUUID->"3170c63a-5d8d-4074-acd2-\ a52ce5e42ae0"], Cell["\<\ As defined, the capillary vector achieves the necessary geometric properties \ of the tangent-plane construction\ \>", "Item", CellChangeTimes->{{3.858269990289781*^9, 3.858270036975008*^9}},ExpressionUUID->"120aa566-7a10-4249-a759-\ 27369df56a3e"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ OverscriptBox["\[Xi]", "\[RightVector]"], "\[LongEqual]", "\[Gamma]", OverscriptBox[ StyleBox["r", "TI"], "^"], "+", FractionBox[ RowBox[{"\[PartialD]", "\[Gamma]"}], RowBox[{"\[PartialD]", "\[Theta]"}]], OverscriptBox["\[Theta]", "^"], "+", FractionBox["1", RowBox[{"sin", "\[Theta]"}]], FractionBox[ RowBox[{"\[PartialD]", "\[Gamma]"}], RowBox[{"\[PartialD]", "\[Phi]"}]], OverscriptBox["\[Phi]", "^"]}], TraditionalForm], "errors" -> {}, "input" -> "\\vec{\\xi} = \\gamma \\hat{r} + \\frac{\\partial \\gamma}{\\partial \ \\theta} \\hat{\\theta} + \\frac{1}{\\sin \\theta} \\frac{\\partial \\gamma}{\ \\partial \\phi} \\hat{\\phi}", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.858268878732563*^9, 3.858268881594473*^9}, 3.858269801028153*^9, 3.858269936618724*^9, 3.858270092228195*^9, 3.858270294298286*^9},ExpressionUUID->"6a0e7e78-0eed-4add-8058-\ e66da6d5bc89"], Cell[TextData[{ "Namely that the radial component of the capillary vector ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ OverscriptBox["\[Xi]", "\[RightVector]"], "\[CenterDot]", OverscriptBox[ StyleBox["p", "TI"], "\[RightVector]"], "\[LongEqual]", "\[Gamma]"}], TraditionalForm], "errors" -> {}, "input" -> "\\vec{\\xi} \\cdot \\vec{p} = \\gamma", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "b5de43cb-7051-4f10-a3f8-5e6c6eb9e60a"], " is the surface energy density" }], "Subitem", CellChangeTimes->{{3.858270096928895*^9, 3.8582701480182543`*^9}, { 3.8582702186400747`*^9, 3.858270226326148*^9}},ExpressionUUID->"1a201531-ea88-42f1-8c24-\ 511098064fc6"], Cell[CellGroupData[{ Cell["\<\ and that the remaining components defined a vector in the tangent plane\ \>", "Subitem", CellChangeTimes->{{3.858270096928895*^9, 3.8582701480182543`*^9}, { 3.8582702186400747`*^9, 3.858270259232164*^9}, {3.85827029880869*^9, 3.858270308286392*^9}},ExpressionUUID->"c91e96e1-4de4-4724-ada9-\ c7be093b7d20"], Cell["Note: (4) is given in 3D - in 2D the last term is omitted", "Subsubitem", CellChangeTimes->{{3.858270096928895*^9, 3.8582701480182543`*^9}, { 3.8582702186400747`*^9, 3.858270259232164*^9}, {3.85827029880869*^9, 3.858270322062449*^9}},ExpressionUUID->"5c16be62-3de8-4d6b-8c87-\ de95fd09df2f"] }, Open ]], Cell[CellGroupData[{ Cell["Let\[CloseCurlyQuote]s construct the capillary vector analytically", \ "Item", CellChangeTimes->{{3.858270333525448*^9, 3.858270357758053*^9}},ExpressionUUID->"4e78594b-a07d-4b1f-86f2-\ c74d88351fd6"], Cell[BoxData[{ RowBox[{ RowBox[{ 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RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"Grad", "[", RowBox[{"\[Gamma]p", ",", RowBox[{"{", RowBox[{"p1", ",", "p2"}], "}"}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"p1", "->", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", RowBox[{"p2", "->", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], "}"}]}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"0", "<", "\[Theta]", " ", "<", RowBox[{"2", "\[Pi]"}]}]}]}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.858200420889049*^9, 3.858200463265417*^9}, { 3.858200539618195*^9, 3.8582005477946*^9}, {3.8582703774581842`*^9, 3.858270399586103*^9}, {3.858271810047987*^9, 3.858271812801173*^9}}, CellLabel-> "In[195]:=",ExpressionUUID->"e64622a4-516b-47b7-98bb-2d95a669b83c"] }, Open ]], Cell[CellGroupData[{ Cell["And plot it over our surface energy-density plot", "Item", CellChangeTimes->{{3.8582704092542048`*^9, 3.858270423847157*^9}},ExpressionUUID->"6080688e-9dad-46ba-aa50-\ 0be4d1c0bf2b"], 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$CellContext`squareSurfaceEnergyDensityParametric[$CellContext`\[Alpha]$$]}, Show[ Graphics[{ Opacity[0.25], Table[ $CellContext`wulffline[ $CellContext`f$[$CellContext`\[Theta]], 1.5], {$CellContext`\[Theta], 0, 2 Pi - 2 (Pi/250), 2 (Pi/250)}]}], ParametricPlot[ $CellContext`squareXiVector[$CellContext`\[Alpha]$$][$CellContext`\ \[Theta]], {$CellContext`\[Theta], 0, 2 Pi}, PlotStyle -> Directive[Blue, Thick]], PlotRange -> 2 {{-1, 1}, {-1, 1}}, ImageSize -> 500]], "Specifications" :> {{{$CellContext`\[Alpha]$$, Rational[1, 2], "Anistropy"}, 0, 4}}, "Options" :> {Paneled -> False}, "DefaultOptions" :> {}], ImageSizeCache->{650., {352.025390625, 358.974609375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`squareSurfaceEnergyDensityParametric[ Pattern[$CellContext`\[Alpha], Blank[]]][ Pattern[$CellContext`\[Theta], Blank[]]] = { Cos[$CellContext`\[Theta]] ( 1 + ($CellContext`\[Alpha] Cos[$CellContext`\[Theta]]^2) 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Cell["Convexification of 1/\[Gamma]", "Subsection", CellChangeTimes->{{3.858270589394127*^9, 3.858270595140802*^9}},ExpressionUUID->"a805d2ba-f43f-4163-8ea5-\ 98821afb04b2"], Cell["\<\ What is more, the \[Xi]-vector produces these \[OpenCurlyDoubleQuote]ears\ \[CloseCurlyDoubleQuote] of unstable or \[OpenCurlyDoubleQuote]forbidden\ \[CloseCurlyDoubleQuote] crystal orientations\ \>", "Item", CellChangeTimes->{{3.8582705316041517`*^9, 3.858270577987124*^9}},ExpressionUUID->"0ef5f578-5f1d-4fe3-a618-\ 21f20d3de237"], Cell[CellGroupData[{ Cell["We can remove these using a Convex-Hull -based approach", "Item", CellChangeTimes->{{3.8582705316041517`*^9, 3.858270557241267*^9}, { 3.858270599323434*^9, 3.858270636456603*^9}},ExpressionUUID->"8a3d3cef-383d-46c5-871b-\ 00e8fc40dc3c"], Cell[CellGroupData[{ Cell["\<\ However, we have to work with the reciprocal surface energy density\ \>", "Subitem", CellChangeTimes->{{3.8582705316041517`*^9, 3.858270557241267*^9}, { 3.858270599323434*^9, 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