湘大高分子专业英语翻译3-3聚合物解决方案

1,主讲人:唐浩宇（博士）联系电话mail:cranetang,Charpter3-3,3,3.3.PolymerSolution,3.3.1Introduction,Foragivenpolymer,therearesolventsthatdissolvethepolymerwellandsolventsthatdonotdissolvethepolymer.Theformersolventsarecalled“goodsolvents”andthelatter“nonsolvents”.Tablelistsatypicalgoodsolventandanonsolventforpolystyrene,poly(methylmethacrylate),andpoly(ethyleneglycol).PolymerHandbookhasalonglistofsolventsandnonsolventsformanypolymers.Theconcentrationofthepolymerinthegoodsolventcanbeashighas100%,yetthesolutionremainsclearanduniform.Addinganonsolventtothesolutioncausesthepolymertoprecipitate,ifthenonsolventmixeswiththegoodsolvent.Asolventwithanintermediatequalitydissolvesthepolymertosomeextent.,TableGoodSolventsandNonsolventsforSomePolymers,4,3.3.聚合物解决方案,3.3.1介绍,对于给定的聚合物，有溶剂能够溶解聚合物以及溶剂和不溶解该聚合物。前者的溶剂称为“良溶剂”和后者“非溶剂”。表中列出了典型的良溶剂和不良溶剂的聚苯乙烯，聚（甲基丙烯酸甲酯）和聚（乙二醇）。聚合物手册有溶剂和非溶剂对许多聚合物的一个长长的清单。在良好溶剂中的聚合物的浓度可以是高达100，但该解决方案仍然是清晰和均匀的。加入非溶剂的溶液使聚合物沉淀，如果与良溶剂的非溶剂混合。具有中间质量的溶剂溶解该聚合物在一定程度上。,表良好溶剂和非溶剂的一些聚合物,5,Likelow-molecular-weightsolutes,apolymerdissolvesinasolventwhensolvationlowersthefreeenergy.Agoodsolventlowersthefreeenergysubstantially.Anonsolventincreasesthefreeenergy.Amorphouspolymers(transparentinthesolidstate;tobeprecise,itisnotasolidbutratherasupercooledliquid)areusuallyeasytodissolveinthegoodsolvent.Incontrast,crystallineandsemicrystallinepolymers(opaqueinthesolidstate)aresometimesnoteasytodissolve.Withinacrystallite,polymerchainsarefoldedintoaregular,thermodynamicallystablearrangement.Itisnoteasytounfoldthechainfromtheself-lockedstateintoadisorderedstateinsolutionevenifthelatterstateisthermodynamicallymorestable.Heatingmayhelpthedissolutionbecauseitfacilitatestheunfolding.Oncedissolved,polymerchainstakearandom-coilconformationunlessthechainisrigid.,6,如低分子量溶质中，聚合物溶解于溶剂中时，溶剂化降低的自由能。一个良好的溶剂会降低自由能大增。一个非溶剂增加的自由能。无定形聚合物（透明固体状态;准确地说，它不是固体而是过冷液体）通常容易在良好溶剂中溶解。与此相反，结晶和半结晶聚合物（不透明固体状态），有时不容易溶解。在一个晶粒，聚合物链折叠成有规律的，热力学稳定的安排。它不容易从自锁定状态展开的链成无序状态，在溶液中，即使后者状态在热力学上更加稳定。加热可帮助溶解，因为它有利于展开。一旦溶解后，聚合物链采取无规卷曲构象，除非该链是刚性的。,7,3.3.2Flory-HugginsMean-FieldTheory,LatticeChainModelDissolutionofapolymerintoasolventlowersthefreeenergyofthepolymersolventsystemwhentheenthalpydecreasesbydissolutionor,ifitdoesnot,whentheproductofthetemperatureandtheentropyofmixingisgreaterthantheenthalpyofmixing.Miscibilityismuchlowerinpolymersolventsystemsbecauseaddingsolventmoleculestothepolymerdoesnotincreasetheentropyasmuchasitdoestothelow-molecular-weightsolutes.Solventsthatdissolveagivenpolymerareoftenlimitedtothosethatpreferentiallysurroundthepolymerchain.Wewilllearninthissectionhowsmalltheentropygainisinthepolymersolventmixture.Wewillalsolearnwhatphenomenacharacteristicofpolymersolutionsareexpected.,8,3.3.2弗洛里-哈金斯平均场理论,格子链模型溶解的聚合物在溶剂降低了聚合物-溶剂体系的自由能时的焓通过溶解减小，或者，如果不是这样，当混合的温度和熵的乘积大于混合焓。混溶性低得多的聚合物-溶剂系统，因为加入溶剂分子与聚合物不增加熵一样，因为它不与低分子量溶质。溶剂能够溶解在给定的聚合物往往局限于那些优先包围聚合物链。我们将学习本节中的熵增益是在聚合物-溶剂混合物有多小。我们还将学习什么现象聚合物溶液的特性的预期。,9,Miscibilityofthepolymerwithagivensolventiswellexplainedinthemeanfieldtheory.Thetheoryisanextensionofthelatticefluidtheoryoriginallydevelopedtoexplainthemiscibilityoftwolow-molecular-weightliquids.Florypioneeredtheapplicationofthemean-fieldlatticefluidtheorytopolymersolutions.Thesimplestversionofthislatticechaintheoryisgenerallyreferredtoas,FigureLatticemodelforpolymersolution.Graysitesareoccupiedbypolymerchains,andwhitesitesareoccupiedbysolventmolecules.,10,与给定溶剂中的聚合物的混溶性在meanfield理论很好的解释。理论是格子流体理论的延伸最初开发解释的两种低分子量的液体混溶性。弗洛里开创了平均场格子流体理论应用到聚合物溶液。这个晶格链理论的最简单版本通常被称为,图格子模型聚合物溶液。灰点是由聚合物链中占据，白点是由溶剂分子占据。,11,FloryHugginsmean-fieldtheory.Asimilarmean-fieldtheorysuccessfullydescribesthermodynamicsofpolymerblendsand,withsomemodifications,diblockcopolymersandtheirblendswithhomopolymers.Themean-fieldtheoryforthepolymersolutioncomparesthefreeenergyofthepolymersolventsystembeforemixingandthefreeenergyaftermixing.Weconsiderasimplesituation:thepolymerisamonodispersehomopolymerandisinanamorphousstateorinaliquidstate(melt)beforemixing.,TheFloryHugginstheoryusesthelatticemodeltoarrangethepolymerchainsandsolvents.Thesystemconsistsofnsitesites.Eachsitecanbeoccupiedbyeitheramonomerofthepolymerorasolventmolecule(themonomerandthesolventmoleculeoccupiesthesamevolume).AlinearpolymerchainoccupiesNsitesonastringofN1bonds.,12,弗洛里-哈金斯平均场理论。类似的平均场理论成功地描述了聚合物共混物的热力学，并与一些修改，二嵌段共聚物和均聚物与它们的共混物。平均场理论的聚合物溶液混合后，混合和自由能之前比较对聚合物-溶剂体系的自由能。我们考虑一个简单的情况：该聚合物是单分散的均聚物，并且是无定形状态或处于液体状态（熔融物）混合之前。,在弗洛里-哈金斯理论使用的点阵模型来安排聚合物链和溶剂。该系统由nsite的网站。每个站点可以通过聚合物的任一单体或溶剂分子（被占用的单体和溶剂分子占据相同的体积）。线性聚合物链上占据的N-1个键的字符串N个地点。,13,Thereisnopreferenceinthedirectionthenextbondtakeswhenapolymerchainislaidontothelatticesites(flexible).PolymerchainsconsistingofNmonomersarelaidontoemptysitesonebyoneuntilthereareatotalnPchains.Then,theunoccupiedsitesarefilledwithsolventmolecules.ThevolumefractionofthepolymerisrelatedtonPby,andthenumberofthesolventmoleculesnSisgivenby,Beforemixing,thepolymeroccupiesthevolumeofnPNvsiteandthesolventoccupiesthevolumeofnSvsite,wherevsiteisthevolumepersite.Thetotalvolumensitevsitedoesnotchangeuponmixing(incompressible).Thus,theenthalpyofmixingHmixintheconstant-pressureprocessisequaltothechangeintheinternalenergy,Umix,andtheGibbsfreeenergychangeGmixisequaltotheHelmholtzfreeenergychangeAmix.,(3.12),(3.13),14,还有就是在方向上没有偏好，当聚合物链铺设到格点（灵活的）下一个债券需要。聚合物链包括N个单体铺设到空的网站一个接一个，直到有一个总数为nP链。然后，将未占用的位点都充满了溶剂分子。的体积分数的聚合物是与到nP由,与溶剂分子数nS由下式给出,前混合，聚合物占据nPNvsite溶剂占据nSvsite，其中vsite为每个站点的音量。总体积的nSvsite混合时（不可压缩）不会改变。因此，混合Hmix在恒压过程等于在内部能量，Umix，和吉布斯自由能的变化Gmix是等于亥姆霍兹自由能变化Amix.。,(3.12),(3.13),15,EntropyofMixingFlorycountedthenumberofpossiblearrangementsofnPchainsonnsitesitesandcompareditwiththenumberofarrangementsonnPNsitesbeforemixing,thatis,inthemelt.Thus,heobtainedtheentropyofmixingSmixpersiteas,Thisexpressionissimilartotheentropyofmixingfortwogaseoussubstances.Thedifferenceisthatthevolumefractionappearsintheargumentoflogarithm,inplaceofthemolefraction.Notethatthefirsttermisdividedbythechainlength.ThedivisionbyalargeNmakesSmixsmall,especiallyatlowconcentrations(1;thesecondtermisnearzero)comparedwithSmixforN=1(mixtureoftwosolvents).ThedivisionalsomakesSmixasymmetricwithrespectto=1/2.,(3.14),16,混合熵弗洛里计算的n个可能的排列数nP链，并与n个安排的数量进行了比较nPN地点混合之前，也就是在熔体。因此，他获得Smix每个站点,这个表达式是类似于混合两种气体物质的熵。所不同的是体积分数显示在对数的参数，代替摩尔分数。注意，第一项是由链长度划分。由一个大的部门N使得Smix小，特别是在低浓度下（1，第二项是接近零）与Smix为N=1（的两种溶剂的混合物）。该部门还使S混合不对称就=1/2。,(3.14),17,Asshownintherightfigure,longerchainsdecreaseSmixatlowandshiftthemaximumtotheright.AtN=100,theplotisalreadyclosetotheoneforN=inwhichtheentropyofmixingisdeterminedbysolventmoleculesonly.WecanshowthatSmixgivenbyEq.2.14isgreaterthantheentropyofmixingforanidealsolutionofnPsolutemoleculesandnSsolventmolecules.Thedifferenceisduetoagreaternumberofconformationsapolymerchaincantakewhentherequirementthatallthesitesbeoccupiedbythemonomersislifted.,FigureEntropyofmixingpersite,Smix/(kBnsite),plottedasafunctionofpolymervolumefraction.ThenumberadjacenttoeachcurvedenotesN.,18,如右图所示，长链减少Smix在低和最大转移到右侧。当N=100，剧情已经接近了一个用于N=在其中混合的熵是通过唯一的溶剂分子来确定。我们可以证明，Smix式给出。2.14大于混合的熵的理想解决方案nP溶质分子和nS溶剂分子。该差异是由于较大数目的构象的高分子链可以利用时，所有的位点由单体所占据的规定被提起的。,图中每个站点的混合熵，Smix/(kBnsite)，绘制成的聚合物体积分数的函数。相邻曲线的数字表示N。.,19,ParameterTheentropyofmixingissmallforpolymersolventsystems,especiallyatlowconcentrations.Therefore,thechangeintheinteractionsuponmixing(enthalpyofmixing)governsthemiscibility.Theinteractionsweareconsideringhereareshort-rangedonesonly,typicallyvanderWallsinteractions(alsoknownasdispersions),hydrogen-bonding,anddipoledipoleinteractions.Thelatticefluidmodelconsidersinteractionsbetweennearestneighborsonly.Theinteractionsresideinthecontacts.WedenotebySS,PP,andPStheinteractionsforasolventsolvent(SS)contact,apolymerpolymer(PP)contact,andapolymersolvent(PS)contact,respectively.Mixingthesolventandpolymerchangestheoverallinteractionenergythroughrearrangementofcontacts.Thefollowingfigureillustratesthechangeinthetwo-dimensionalrenderingofthelattice.,FigureChangeinthecontactsbetweennearestneighborswhenapolymerchainmixeswithsolventmolecules.,20,参数混合的熵是小为聚合物-溶剂系统，特别是在低浓度下。因此，在混合（混合焓）的相互作用的变化支配的混溶性。我们正在考虑这里的相互作用是短程那些只，通常范德华相互作用（也称为分散剂），氢键和偶极-偶极相互作用。格子流体模型只考虑最近邻之间的相互作用。的相互作用存在于接触。我们用的SS，PP和PS为溶剂的溶剂（S-S）相接触，聚合物的聚合物（P-P）相接触，并有聚合物的溶剂（P-S）分别接触的相互作用。混合溶剂和聚合物通过触点的重排改变了整体的相互作用能。下面的图说明了在二维格子的呈现的改变。,图变化近邻的交往时，聚合物链与溶剂分子混合。,21,Beforemixing,therearefourPPcontactsandfourSScontactsbetweenatotalofeightsites.MixingreplacestwoPPcontactsandtwoSScontactswithfourPScontacts.Theinteractionenergyontheseeightbondschangesfrom4SS+4PPto4PS+2SS+2PP.Thedifferenceis4PS-2(SS+PP).PernewlycreatedPScontact,thechangeisPS-(SS+PP)/2.The(chi)parameter,alsocalledFlorysparameterorFloryHugginsparameter,isdefinedastheproductofthelatticecoordinateZandtheenergychangereducedbykBT:,(3.15),22,前混合，有四个P-P的接触和共八个站点之间的四个S-S接触。混合取代两个P-P的接触，并与四名P-S接触两个S-S接触。的相互作用能在这八个键从4SS+4PP到4PS+2SS+2PP。所不同的是4PS-2(SS+PP)。每个新创建的p-S接触，改变为的PS-（SS+PP）/2。经（中）参数，也叫弗洛里的参数或弗洛里-哈金斯参数，是定义为晶格坐标Z和能量变化减少kBT：,(3.15),23,Apositivedenotesthatthepolymersolventcontactsarelessfavoredcomparedwiththepolymerpolymerandsolventsolventcontacts.Anegativemeansthatpolymersolventcontactsarepreferred,promotingsolvationofthepolymer.Ingeneral,decreasesitsmagnitudewithanincreasingtemperaturebecauseofkBTinthedenominator,butthepairinteractionsalsodependonthetemperatureinamannercharacteristicofeachpolymersolventsystem.Inahydrogenbondingpair,forinstance,usuallychangesfromnegativetopositivewithincreasingtemperature.,Therearensitesites,eachwithZbonds.TheproductZnsiteistwiceaslargeasthetotalnumberofbondsinthemixturebecauseZnsitecountseachbondtwice.Thus,Z/2isthenumberofbondspersite.Inthetwo-dimensionalsquarelattice,thenumberis2.Inthethree-dimensionalcubiclattice,itis3.,24,正表示该聚合物的溶剂的接触是较不利的聚合物-聚合物和溶剂-溶剂接触进行比较。负意味着聚合物-溶剂接触是首选，促进聚合物的溶剂化。在一般情况下，减小由于其具有随着温度升高幅度kBT，但在一对相互作用也依赖于温度在每个聚合物-溶剂系统的一个方式的特征。在氢键配对，例如，通常从负变化到正随着温度的升高。,有nsite站点，每个站点以Z债券。该产品的Znsite是两倍大，在混合物中键的总数，因为Znsite计数每个键两次。因此，Z/2为每个站点键的数目。在二维正方晶格的数目为2。在该三维立方点阵，它是3。,25,InteractionChangeUponMixingForasiteoccupiedbyamonomer,twoofitsZneighborsarealwaysadjacentmonomersonthesamechain,exceptforthechainends.Othermonomersonthesamechainarealsolikelytooccupysomeoftheotherneighbors.Themean-fieldtheoryneglectsthisfactandcalculatesthechangeintheinteraction,Umix,bymixingNnP=nsiteunconnectedmonomersandnS=nsite(1-)solventmoleculesatrandom.TheprobabilityforagivenbondtobeaPPcontactis2,theprobabilityfortheSScontactis(1-)2,andtheprobabilityforthePScontactis2(1-).Thus,thechangeUmixintheinternalenergyis,Thechangepersiteis,(3.16),(3.17),26,相互作用改变混合时对于由单体占据的位点，其中两个是它的Z邻居总是在同一链相邻单体，除链端。在同一个链中的其他单体也可能占据一些其他的邻居。平均场理论忽视了这一事实，并计算在互动的变化，Umix，混合NnP=nsite无关单体和nS=nsite(1-)溶剂分子随机的。对于一个给定键的概率是一个P-P的接触是2，对于S-S接触的概率为(1-)2，和概率为p-S接触是2(1-)。因此，该变化Umix在内部的能量是,每个站点的变化,(3.16),(3.17),27,Asshowninthefollowingfigure,Umixmaximizesat=1/2.ThesignofUmixisthesameasthatof.Umixdependsontheinteractionthrough.AsystemwiththesamehasthesameUmix.Forinstance,amixturewithPP=1andPS=SS=0isthermodynamicallyequivalenttoamixturewithPP=SS=0andPS=-1/2.AnotherwaytoarriveatEq.3.16inthemean-fieldapproximationistotallyallthecontactsbeforeandafterthemixing.TableliststheprobabilityforabondinthepolymersolventsystemtobeaPP,PS,orSScontactbeforeandafterthemixing.TheaveragecontactenergyisPP+(1-)SSbeforethemixing.Afterthemixing,itchangesto2PP+2(1-)PS+(1-)2SS.Thedifferenceis(1-)(2PS-PP-SS).ForatotalZnsite/2bonds,weobtainEq.3.16.,FigureChangeintheinteractionbymixing,Umix、(kBTnsite),plottedasafunctionofpolymervolumefraction.Theplotisshownforapositive.,TableProbabilityofNearest-NeighborContacts.,28,如图所示，在下面的图中，Umix最大化在=1/2。Umix是相同的。Umix依赖于相互作用通过。系统具有相同具有相同的Umix。例如，对于PP=1和PS=SS=0是热力学相当于与的混合物PP=SS=0和PS=-1/2。另一种方法在式到达。在平均场近似3.16的相符所有之前和之后的混合接触。表列出了在聚合物-溶剂体系中的键的概率是P-P，P-S，或之前和之后该混合的S-S接触。平均接触能量PP+(1-)SS混合之前。混合后，它将更改为2PP+2(1-)PS+(1-)2SS。所不同的是(1-)(2PS-PP-SS)。总共Znsite/2键，我们得到方程。3.16。,图中变化的相互作用通过混合，Umix、(kBTnsite)，绘制成的聚合物体积分数的函数。该地块显示为一个积极的。、,表的近邻联系人概率。,29,Asolutionwith=0iscalledanathermalsolution.ThereisnodifferencebetweenthePScontactenergyandtheaverageenergyforthePPandSScontacts.Intheathermalsolution,Umix=Hmix=0regardlessof.Wecanregardthatthepolymerchainisdissolvedinaseaofmonomermolecules.Notehoweverthat,inanactualpolymersolventsystem,themonomerbeforepolymerizationischemicallydifferentfromtherepeatingunitinthepolymer.Forinstance,anoxyethylenerepeatingunit(CH2CH2O)inpoly(ethyleneglycol)isdifferentfromethyleneoxideorethyleneglycol.,30,用=0的溶液称为无热解决方案。有对P-S接触的能量和用于P-P的平均能量和S-S触点之间没有差异。在无热溶液中，Umix=Hmix=0，无论的。我们可以把该聚合物链溶解在单体分子的海洋。注然而，在一个实际的聚合物-溶剂体系，聚合前的单体是从聚合物中的重复单元的化学性质不同。例如，氧化乙烯重复单元(CH2CH2O)的聚（乙二醇）是由环氧乙烷或乙二醇不同。,31,3.3.3FreeEnergy,ChemicalPotentials,andOsmoticPressure,GeneralFormulasFromEqs.3.14and3.17,theHelmholtzfreeenergyofmixing,Amix=Umix-TSmix,persiteisgivenas,Foratotalnsitesites,(3.18),(3.19),(3.20),32,3.3.3自由能，化学势，和渗透压,一般公式从方程。3.14和3.17，混合，Amix=Umix-TSmix,，每个站点可表示为,对于一个总共nsite的网站，,(3.18),(3.19),(3.20),33,Thefollowingidentitiesareuseful:.,ThechemicalpotentialdifferencePofthepolymerchainbetweenthesolutionandthepolymermeltiscalculatedfromEqs.3.20and3.21as,whereGmix=Amixwasused.Likewise,thechemicalpotentialdifferencesofthesolventmoleculebetweenthesolutionandthepuresolventiscalculatedfromEqs.3.19and3.22as,(3.21),(3.22),(3.23),(3.24),34,下面的标识是有用的：,该化学电位差P溶液和聚合物熔体之间的聚合物链的被从方程计算。3.20和3.21,其中Gmix=Amix使用。同样，化学电位差s的溶液与纯溶剂之间的溶剂分子从方程计算。3.19和3.22为,(3.21),(3.22),(3.23),(3.24),35,Theosmoticpressureisgivenas,whereV=vsitensiteisthevolumeofthesolution.Membraneosmometryandvaporpressureosmometrymeasuretheosmoticpressure,ChemicalPotentialofaPolymerChaininSolutionWhenN1,Eq.3.23isrearrangedto,(3.25),(3.26),36,渗透压被给定为,其中V=vsitensite是溶液的体积。膜渗透压测定法和蒸气压渗透压测定法测量的渗透压,在溶液中高分子链的化学势当N1，方程。3.23被重新安排，以,(3.25),(3.26),37,Thefirsttermisthechemicalpotentialoftheidealsolution.N(-1)isjustaconstantandthereforeirrelevanttothefurtherdiscussion.N(1-2)+2representsthenonidealpart.ThefollowingfigureillustrateshowPchangeswith.Atlowconcentrations,P/kBTln,andthesolutionisnearlyideal.Asincreases,thenonidealtermincreasesitsmagnitude.Theshapeoftheplotdependsprimarilyonwhether1/2ontheotherhand,theleadingterminthenonidealpartisnegativeandthereforetheplotdeviatesdownward.Athigherconcentrations,thepositivesecond-ordertermletstheploteventuallycrossthelinefortheidealsolution.Thenonidealityminimizesat=1/2.,FigureChemicalpotentialPofthepolymerchainplottedasafunctionoflnfortheidealsolution(dashedline)andfornonidealsolutionsof1/2.AnincreaseinNinflatesthedeviationofthesolidlinesfromthedashedline.,38,第一项是理想的解决方案的化学势。N(-1)仅仅是一个常数，因此无关的进一步讨论。N(1-2)+2表示非理想的部分。下图说明P与的变化。在低浓度时，P/kBTln，并将该溶液几乎是理想的。由于增大，长期不理想增大它的大小。情节的形状主要取决于是否1/2，在非理想部分的首项是负的，因此情节向下偏离。在较高浓度，正二阶项让剧情最终交行的理想解决方案。非理想最小化在=1/2。,图化学势P聚合物链的绘制为ln为理想溶液（虚线）和1/2。在增加N膨胀的虚线，实线的偏差。,39,ThedeviationfromtheidealsolutionismagnifiedbyN.Asmalldifferenceoffrom1/2showsupasalargenonidealitywhenNislarge.Thusthepolymersolutions,especiallythoseofhigh-molecular-weightpolymers,canbeeasilynonideal.When1/2,inparticular,N(1-2)canbeeasilyaslargeastocauseadipintheplotofP.TheFloryHugginstheoryneglectsthechainconnectivity,thechainrigidity,andtheshapeofthemonomer.ModificationstotheFloryHugginstheoryarepossiblebytakingintoaccounttheseeffects.Thechainrigiditycanbeincorporatedbygivingpreferencetostraightbondsoverangledones,forinstance.,40,从理想的解决方案的偏差是由放大N。从1/2小的差异表现为当大非理想N大。因此，聚合物溶液，尤其是那些高分子量的聚合物，可以很容易地不理想。当1/2，尤其是，N(1-2)可以很容易地大，容易引起浸在P的情节。的FloryHuggins理论忽略了链连接，该链的刚性，并且该单体的形状。修改到的Flory-Huggins理论是可能考虑到这些影响。环刚度可通过直债券优先考虑了角度的，比如被纳入。,41,3.3.4DiluteSolution,Mean-FieldTheoryInthissubsection,weconsiderdilutesolutions.When1,ln(1)=(1/2)2(1/3)3.Then,Eq.3.25isrewrittento,(3.27),Inthelowconcentrationlimit,Eq.3.27givestheosmoticpressureidealoftheidealsolution:,Thustheratiooftoidealcomparedatthesameconcentrationis,Theratioisoftencalledthe(osmotic)compressibility.,(3.28),(3.29),42,3.3.4稀溶液,平均场理论在本小节中，我们考虑稀溶液。当When1,ln(1)=(1/2)2(1/3)3。然后，方程3.25被改写为,(3.27),在低浓度限值，公式。3.27使渗透压ideal的理想解决方案：,从而到ideal相比，在相同的浓度是,的比例通常被称为（渗透压）的可压缩性。,(3.28),(3.29),43,Therightfigureshows/idealasafunctionof.Threecurveswith=0.4,0.5,and0.55areplottedforchainsofN=100.Theupwardordownwarddeviationoffromidealdependsonwhether0;PScontactsaredisfavored)ofmixingiscompensatedbytheentropyofmixing,whichgivesrisetothecoefficient1/2inthelineartermofEq.3.29.When1/2,theentropyofmixingisnotsufficienttooffsettheincreaseintheinteractionduetounfavorablepolymersolventcontacts.Then,thepolymerpolymercontactsarepromoted,effectivelyloweringtheosmoticpressure.,Athigherconcentrations,thepositive(N/3)2drivesaboveideal.Whenidealintheentirerangeof.Asdecreasesandturnsnegative,thepolymersolventcontactsarefavoredalsointermsoftheinteraction.Then,isevengreater.ItisapparentinEq.3.29thatthedeviationfromidealismagnifiedbyN.Thenonidealityislargeinpolymersolutions.,FigureOsmoticcompressibility(/ideal)plottedasafunctionoffortheidealsolution(dashedline)andnonidealsolutionswithN=100and=0.4,0.5,and0.55(solidlines).,44,右图显示/ideal为的函数。三条曲线与=0.4，0.5和0.55绘制对于N=100的链。从向上或向下偏离理想取决于是否0，P-S接触是混合的不受欢迎)是由混合的熵，这就会引起该系数的1/2式中的线性项补偿。3.29。当1/2，混合熵不足以增加的相互作用抵消由于不利的聚合物-溶剂接触。接着，聚合物-聚合物接触被促进，从而有效地降低了渗透压。,在较高浓度，正(N/3)2驱动器以上ideal。当ideal在的整个范围。如减少，而变为负时，聚合物-溶剂接触也赞成在相互作用方面。然后，就更大了。很明显在式。3.29，该偏差从ideal是由N.放大的非理想性较大的聚合物解决方案。,图渗透可压缩性(/ideal)绘制为的理想的解决方案（虚线）和非理想的解决方案，具有N=100的功能以及=0.4，0.5和0.55（实线）。,45,VirialExpansionTocomparethetheorywithexperiments,needstobeexpressedintermsofmassconcentrationc,typicallying/Lormg/L.Usingtheidentity,(3.30),whereM(NAN)isthemassofthemonomer(notmolarmass),(NAkBT)is,ingeneral,expandedinapowerseriesofc:,(3.31),Inthisvirialexpansion,A2isthe(osmotic)secondvirialcoefficient,andA3isthethirdvirialcoefficient.ApositiveA2deviatesupwardcomparedwiththatoftheidealsolution(ideal/(NAkBT)=c/M).WhenA2=0,thesolutionisclosetotheidealsolutioninawiderangeofconcentrations.,46,维里展开要比较的理论与实验，需要被表达的质量浓度c，通常以g/L或mg/L条件使用身份,(3.30),其中M(NAN)是单体（未摩尔质量），(NAkBT)的，在一般情况下，扩大了幂级数的c：,(3.31),