(* 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[ 48855, 1041] NotebookOptionsPosition[ 46386, 998] NotebookOutlinePosition[ 46787, 1014] CellTagsIndexPosition[ 46744, 1011] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Phase diagram of a polymer blend", "Title", CellChangeTimes->{{3.8614252876884317`*^9, 3.861425295541616*^9}},ExpressionUUID->"c83c4dba-47e2-4b87-b246-\ 64c2a50a92df"], Cell[TextData[{ "Mixing free energy in units ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "B"], RowBox[{"T", "/", "v"}]}], TraditionalForm]],ExpressionUUID-> "b812bbe2-91af-42be-940e-1ff05239cc29"], "\nHint: You can write \[OpenCurlyDoubleQuote]\[Phi]\[CloseCurlyDoubleQuote] \ by typing Esc+\[OpenCurlyDoubleQuote]phi\[CloseCurlyDoubleQuote]+Esc.\nDenote \ the chain lengths by \[OpenCurlyDoubleQuote]mA\[CloseCurlyDoubleQuote] and \ \[OpenCurlyDoubleQuote]mB\[CloseCurlyDoubleQuote]." }], "Text", CellChangeTimes->{{3.861425387250803*^9, 3.861425419635903*^9}, 3.861435790894437*^9},ExpressionUUID->"a9708e6d-b9e8-4703-93d3-\ 7ee7ad047e2a"], Cell[BoxData[ RowBox[{ RowBox[{"fMix", "=", RowBox[{ RowBox[{ FractionBox["\[Phi]", "mA"], RowBox[{"Log", "[", "\[Phi]", "]"}]}], "+", RowBox[{ FractionBox[ RowBox[{"1", "-", "\[Phi]"}], "mB"], RowBox[{"Log", "[", RowBox[{"1", "-", "\[Phi]"}], "]"}]}], "+", RowBox[{"\[Chi]", " ", "\[Phi]", RowBox[{"(", RowBox[{"1", "-", "\[Phi]"}], ")"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.861425306630929*^9, 3.861425307912677*^9}, 3.861425368504366*^9, {3.8614254355339117`*^9, 3.8614254811796494`*^9}, { 3.861427389687133*^9, 3.861427391771*^9}}, CellLabel-> "In[443]:=",ExpressionUUID->"76446357-adee-49f3-8368-b55b23f1ffe3"], Cell["Spinodal line", "Text", CellChangeTimes->{{3.861425487973814*^9, 3.8614254923905773`*^9}},ExpressionUUID->"a9926331-13dc-4725-9018-\ 88eee26587b6"], Cell[BoxData[ RowBox[{ RowBox[{"\[Chi]S", "=", RowBox[{ FractionBox["1", "2"], RowBox[{"(", RowBox[{ FractionBox["1", RowBox[{"mA", " ", "\[Phi]"}]], "+", FractionBox["1", RowBox[{"mB", RowBox[{"(", RowBox[{"1", "-", "\[Phi]"}], ")"}]}]]}], ")"}]}]}], ";"}]], "Input",\ CellChangeTimes->{{3.861425496112561*^9, 3.861425497461564*^9}, 3.861425570294504*^9, {3.861425635787285*^9, 3.86142567205492*^9}}, CellLabel-> "In[444]:=",ExpressionUUID->"95e8e024-780e-4cf7-9085-03960fc40880"], Cell["Critical point", "Text", CellChangeTimes->{{3.861428876779009*^9, 3.861428885059338*^9}},ExpressionUUID->"99571618-f209-4c18-9cce-\ fb897b8a32ee"], Cell[BoxData[{ RowBox[{ RowBox[{"\[Phi]C", "=", FractionBox[ SqrtBox["mB"], RowBox[{ SqrtBox["mA"], "+", SqrtBox["mB"]}]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Chi]C", "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{ FractionBox["1", RowBox[{ SqrtBox["mA"], " "}]], "+", FractionBox["1", SqrtBox["mB"]]}], ")"}], "2"]}]}], ";"}]}], "Input", CellChangeTimes->{{3.861428139222185*^9, 3.861428169084404*^9}, { 3.8614282206058817`*^9, 3.861428244558146*^9}, {3.8614288348543453`*^9, 3.861428866787376*^9}}, CellLabel-> "In[445]:=",ExpressionUUID->"9abada32-82db-4cf6-893b-24852e1b140d"], Cell["Specify the polymer lengths:", "Text", CellChangeTimes->{{3.861429661094976*^9, 3.861429671669207*^9}, { 3.8614346234152184`*^9, 3.861434638252256*^9}},ExpressionUUID->"5f337fe4-a41c-4267-9881-\ 1e993a68da18"], Cell[BoxData[ RowBox[{ RowBox[{"parameters", "=", RowBox[{"{", RowBox[{ RowBox[{"mA", "->", "8"}], ",", RowBox[{"mB", "->", "2"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.861426902936556*^9, 3.86142692688238*^9}, { 3.8614273561914873`*^9, 3.861427368682476*^9}, {3.861427407142411*^9, 3.8614274087826777`*^9}, {3.861427982773588*^9, 3.861427983076419*^9}, { 3.861428037124402*^9, 3.861428038805655*^9}, {3.861429494895446*^9, 3.861429496456601*^9}, {3.861429735705369*^9, 3.861429735840904*^9}, { 3.861429804057104*^9, 3.861429806648769*^9}, {3.861429856816042*^9, 3.861429873367073*^9}, {3.8614351528079844`*^9, 3.861435167243968*^9}, { 3.8614407317080193`*^9, 3.861440784987823*^9}, {3.86144100861423*^9, 3.861441030470434*^9}, {3.8614438215878696`*^9, 3.8614438219442177`*^9}, { 3.861443883516361*^9, 3.8614438896729403`*^9}}, CellLabel-> "In[447]:=",ExpressionUUID->"1d0cbbe0-bd47-4252-87f3-9df44067fac3"], Cell["\<\ Explanation: This is a list of replacement rules. Taking any expression, for \ example fMix, and writing fMix/.parameters will replace all occurrences of mA \ by 10 and mB by 3.\ \>", "Text", CellChangeTimes->{{3.861434642890842*^9, 3.861434709888558*^9}},ExpressionUUID->"94a0540b-e9fb-45a8-83cb-\ 8eb8be18a955"], Cell[CellGroupData[{ Cell["Determine binodal line numerically", "Subsection", CellChangeTimes->{{3.861429762191678*^9, 3.861429767495729*^9}},ExpressionUUID->"580821bf-cac6-4515-a12b-\ c001acf409ee"], Cell["Conditions for the binodal line [\[Phi]1(\[Chi]),\[Phi]2(\[Chi])]", \ "Text", CellChangeTimes->{{3.861425709789021*^9, 3.8614257175685463`*^9}, { 3.861425913541346*^9, 3.861425956119673*^9}, {3.861429542797256*^9, 3.8614295440166197`*^9}, {3.8614443201008387`*^9, 3.861444321789201*^9}},ExpressionUUID->"511e8fe5-efdf-4223-9cde-\ f14131fb2f4c"], Cell[BoxData[ RowBox[{ RowBox[{"condB", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{"fMix", ",", "\[Phi]"}], "]"}], "/.", RowBox[{"\[Phi]", "->", "\[Phi]1"}]}], ")"}], "==", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{"fMix", ",", "\[Phi]"}], "]"}], "/.", RowBox[{"\[Phi]", "->", "\[Phi]2"}]}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{"fMix", ",", "\[Phi]"}], "]"}], "/.", RowBox[{"\[Phi]", "->", "\[Phi]1"}]}], ")"}], "==", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"fMix", "/.", RowBox[{"\[Phi]", "->", "\[Phi]2"}]}], ")"}], "-", RowBox[{"(", RowBox[{"fMix", "/.", RowBox[{"\[Phi]", "->", "\[Phi]1"}]}], ")"}]}], RowBox[{"\[Phi]2", "-", "\[Phi]1"}]]}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.86142951985868*^9, 3.861429532438551*^9}, { 3.86144434451295*^9, 3.8614443872769327`*^9}, {3.861444434773614*^9, 3.861444470637187*^9}}, CellLabel-> "In[448]:=",ExpressionUUID->"c03a4f6a-224b-447e-a4b7-f06887f57932"], Cell[TextData[{ "We use here [\[Phi]1, \[Phi]2] to denote [", Cell[BoxData[ FormBox[ SubscriptBox["\[Phi]", "1"], TraditionalForm]],ExpressionUUID-> "a78012d0-cb4e-4268-955f-cd0bfda51149"], ", ", Cell[BoxData[ FormBox[ SubscriptBox["\[Phi]", "2"], TraditionalForm]],ExpressionUUID-> "33d48449-83c5-4d52-a21d-e5e3faac2e39"], "] from the exercise sheet. Use replacement rules to replace \[Phi] by \ \[Phi]1, \[Phi]2, respectively." }], "Text", CellChangeTimes->{{3.861434730606482*^9, 3.861434829182969*^9}, { 3.8614443300506287`*^9, 3.861444342798656*^9}},ExpressionUUID->"dd08fe62-df1f-49e9-b0ac-\ 717676754740"], Cell["\<\ The following function solves the conditions for the binodal given a fixed \ density of the B-rich phase: Explanation: The natural thing to do would be to fix \[Chi] and to calculate \ \[Phi]1, \[Phi]2 for this given value of \[Chi]. The binodal \[Chi]B is then \ given by inversion of the this relationship, i.e., by the collection of \ points (\[Phi]1,\[Chi]) and (\[Phi]2,\[Chi]). However, this procedure is numerically unstable: For long polymer lengths, \ \[Phi]1 and \[Phi]2 lie close to 0 and 1. At 0 and 1 fMix diverges and it is \ hard to reliably solutions that lie close to these divergencies. Furthermore, we would need a very dense grid of \[Chi] to resolve the binodal \ line close to the critical point. If we choose w.l.o.g. mA>mB, both difficulties are circumvented by fixing \ \[Phi]1 and solving for the value of \[Chi] at which this density is the \ density of the B-rich phase in coexistence with a A-rich phase of density \ \[Phi]A (which we also solve for). This way, the distance from the \ singularities at 0 and 1 is fixed by our choice of the grid for \[Phi]1. \ Moreover we do not need a particularly dense grid to well determine the \ binodal line around the critical point.\ \>", "Text", CellChangeTimes->{{3.861429616776599*^9, 3.86142964820009*^9}, { 3.86143483613321*^9, 3.861435134204383*^9}, {3.861435173501968*^9, 3.86143522742598*^9}, {3.8614352709821043`*^9, 3.861435487466114*^9}, { 3.8614443956042747`*^9, 3.861444407285408*^9}, {3.8614444551005898`*^9, 3.861444457314754*^9}},ExpressionUUID->"d6214021-3812-4933-af69-\ 875bdf5901c2"], Cell[BoxData[{ RowBox[{ RowBox[{"ClearAll", "@", "binodalDensities"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"binodalDensities", "[", RowBox[{ RowBox[{"\[Phi]1_", "?", "NumericQ"}], ",", RowBox[{"chainLengthA_", "?", "NumericQ"}], ",", RowBox[{"chainLengthB_", "?", "NumericQ"}]}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"\[Chi]", ",", "\[Phi]2"}], "}"}], "/.", RowBox[{"FindRoot", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"condB", "/.", RowBox[{"{", RowBox[{ RowBox[{"mA", "->", "chainLengthA"}], ",", RowBox[{"mB", "->", "chainLengthB"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"\[Chi]", ",", "\[IndentingNewLine]", RowBox[{"\[Chi]C", "/.", RowBox[{"{", RowBox[{ RowBox[{"mA", "->", "chainLengthA"}], ",", RowBox[{"mB", "->", "chainLengthB"}]}], "}"}]}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\[Phi]2", ",", "\[IndentingNewLine]", RowBox[{"1", "-", RowBox[{"10", "^", RowBox[{"-", "5"}]}]}], ",", RowBox[{"\[Phi]C", "/.", RowBox[{"{", RowBox[{ RowBox[{"mA", "->", "chainLengthA"}], ",", RowBox[{"mB", "->", "chainLengthB"}]}], "}"}]}], ",", RowBox[{"1", "-", RowBox[{"10", "^", RowBox[{"-", "10"}]}]}]}], "}"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"PrecisionGoal", "\[Rule]", "20"}], ",", "\[IndentingNewLine]", RowBox[{"WorkingPrecision", "\[Rule]", "100"}]}], "]"}]}]}], ";"}]}], "Input", CellChangeTimes->{{3.861425743251609*^9, 3.861425900663496*^9}, { 3.8614260402017927`*^9, 3.861426137943754*^9}, {3.861426240780078*^9, 3.8614262514157057`*^9}, 3.861426295494813*^9, {3.861426334896339*^9, 3.861426416981773*^9}, {3.861426450867897*^9, 3.861426457005616*^9}, { 3.861426495443101*^9, 3.8614265300542107`*^9}, {3.8614265743833303`*^9, 3.861426579902486*^9}, {3.8614266274538116`*^9, 3.861426799878767*^9}, { 3.8614272032537813`*^9, 3.861427213260881*^9}, {3.8614272899445*^9, 3.861427321891388*^9}, {3.8614275082352037`*^9, 3.8614275358177767`*^9}, { 3.861427836004836*^9, 3.8614278803863993`*^9}, {3.8614287460998983`*^9, 3.8614287491459303`*^9}, {3.861428891530703*^9, 3.8614289212574987`*^9}, { 3.861428956791868*^9, 3.861429015074973*^9}, {3.861429058048188*^9, 3.86142907062656*^9}, {3.861429115917564*^9, 3.86142924210448*^9}, { 3.8614294214320993`*^9, 3.86142942318229*^9}, {3.861429526692397*^9, 3.8614295635880003`*^9}, {3.86143057403699*^9, 3.8614305780784407`*^9}, 3.861444410834589*^9, {3.861444458351955*^9, 3.861444459239562*^9}}, CellLabel-> "In[449]:=",ExpressionUUID->"e3f5ebbc-7ad8-4977-828f-287515bec83c"], Cell["\<\ Iterate over different densities \[Phi]1 and make a list of points [\[Phi], \ \[Chi]B(\[Phi])] by collecting all the pairs (\[Phi]1,\[Chi]) and (\[Phi]2,\ \[Chi]). Afterwards, we sort all these pairs by the density \[Phi].\ \>", "Text", CellChangeTimes->{{3.861429906928069*^9, 3.861429951081214*^9}, { 3.861435491526751*^9, 3.8614355424633627`*^9}, {3.861444414087227*^9, 3.861444420392639*^9}, 3.861444460461351*^9},ExpressionUUID->"669e9234-5d2b-4719-9823-\ 74ff41f8f5f5"], Cell[BoxData[{ RowBox[{ RowBox[{"binodal", "=", RowBox[{"{", "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"res", "=", RowBox[{"binodalDensities", "[", RowBox[{"\[Phi]1", ",", RowBox[{"mA", "/.", "parameters"}], ",", RowBox[{"mB", "/.", "parameters"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"AppendTo", "[", RowBox[{"binodal", ",", RowBox[{"{", RowBox[{"\[Phi]1", ",", RowBox[{"Re", "[", RowBox[{"res", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "}"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"AppendTo", "[", RowBox[{"binodal", ",", RowBox[{"{", RowBox[{ RowBox[{"Re", 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