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{"title":"交联聚合物的 z 平均值","authors":"Rolf Bachmann","doi":"10.1002/mats.202300065","DOIUrl":null,"url":null,"abstract":"<p>Stockmayer's formula for the weight average of cross-linked primary chains is extended to the <i>z</i>-average degree of polymerization <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <msub>\n <mi>P</mi>\n <mi>z</mi>\n </msub>\n </mrow>\n <annotation>$DP_z$</annotation>\n </semantics></math>. This average is a function of the weight- and <i>z</i>-average degree of polymerization λ<sub><i>w</i></sub> and λ<sub><i>z</i></sub> of the primary chain distribution and the branching density α: <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <msub>\n <mi>P</mi>\n <mi>z</mi>\n </msub>\n <mo>=</mo>\n <mfrac>\n <mrow>\n <msub>\n <mi>λ</mi>\n <mi>z</mi>\n </msub>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>+</mo>\n <mi>α</mi>\n <mo>)</mo>\n </mrow>\n <mn>3</mn>\n </msup>\n <mo>−</mo>\n <msubsup>\n <mi>λ</mi>\n <mi>w</mi>\n <mn>2</mn>\n </msubsup>\n <msup>\n <mi>α</mi>\n <mn>2</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <mn>3</mn>\n <mo>+</mo>\n <mi>α</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>−</mo>\n <mi>α</mi>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>λ</mi>\n <mi>w</mi>\n </msub>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <mn>2</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>+</mo>\n <mi>α</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n </mrow>\n <annotation>$DP_z= \\frac{\\lambda _z(1+\\alpha)^3-\\lambda _w^2\\alpha ^2(3+\\alpha)}{(1-\\alpha (\\lambda _w-1))^2(1+\\alpha)}$</annotation>\n </semantics></math>. Higher averages of branched polymers from monomers of type <span></span><math>\n <semantics>\n <msub>\n <mi>A</mi>\n <mi>f</mi>\n </msub>\n <annotation>$A_f$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>B</mi>\n <msub>\n <mi>A</mi>\n <mrow>\n <mi>f</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation>$BA_{f-1}$</annotation>\n </semantics></math> with functionalities <i>f</i> ⩾ 2 are discussed.</p>","PeriodicalId":18157,"journal":{"name":"Macromolecular Theory and Simulations","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The z-Average of Cross-Linked Polymers\",\"authors\":\"Rolf Bachmann\",\"doi\":\"10.1002/mats.202300065\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Stockmayer's formula for the weight average of cross-linked primary chains is extended to the <i>z</i>-average degree of polymerization <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n <msub>\\n <mi>P</mi>\\n <mi>z</mi>\\n </msub>\\n </mrow>\\n <annotation>$DP_z$</annotation>\\n </semantics></math>. This average is a function of the weight- and <i>z</i>-average degree of polymerization λ<sub><i>w</i></sub> and λ<sub><i>z</i></sub> of the primary chain distribution and the branching density α: <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n <msub>\\n <mi>P</mi>\\n <mi>z</mi>\\n </msub>\\n <mo>=</mo>\\n <mfrac>\\n <mrow>\\n <msub>\\n <mi>λ</mi>\\n <mi>z</mi>\\n </msub>\\n <msup>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>+</mo>\\n <mi>α</mi>\\n <mo>)</mo>\\n </mrow>\\n <mn>3</mn>\\n </msup>\\n <mo>−</mo>\\n <msubsup>\\n <mi>λ</mi>\\n <mi>w</mi>\\n <mn>2</mn>\\n </msubsup>\\n <msup>\\n <mi>α</mi>\\n <mn>2</mn>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mn>3</mn>\\n <mo>+</mo>\\n <mi>α</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>α</mi>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>λ</mi>\\n <mi>w</mi>\\n </msub>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <mn>2</mn>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>+</mo>\\n <mi>α</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mfrac>\\n </mrow>\\n <annotation>$DP_z= \\\\frac{\\\\lambda _z(1+\\\\alpha)^3-\\\\lambda _w^2\\\\alpha ^2(3+\\\\alpha)}{(1-\\\\alpha (\\\\lambda _w-1))^2(1+\\\\alpha)}$</annotation>\\n </semantics></math>. Higher averages of branched polymers from monomers of type <span></span><math>\\n <semantics>\\n <msub>\\n <mi>A</mi>\\n <mi>f</mi>\\n </msub>\\n <annotation>$A_f$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>B</mi>\\n <msub>\\n <mi>A</mi>\\n <mrow>\\n <mi>f</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation>$BA_{f-1}$</annotation>\\n </semantics></math> with functionalities <i>f</i> ⩾ 2 are discussed.</p>\",\"PeriodicalId\":18157,\"journal\":{\"name\":\"Macromolecular Theory and Simulations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-02-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Macromolecular Theory and Simulations\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mats.202300065\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"POLYMER SCIENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Macromolecular Theory and Simulations","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mats.202300065","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"POLYMER SCIENCE","Score":null,"Total":0}
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