4 Results and Discussions

Transcription

1 4 Results nd Discussions 4. Generl Considertion In this chpter, results will be presented nd discussed. Further, if possible, they will be compred with men-field results. As mentioned in previous chpters, the interest is especilly in the interfcil properties of symmetric polymer blends nd their phse behvior. The interfcil properties of polymers with vrious degrees of flexibilities re discussed in section 4.2. The systems consist of flexible nd semiflexible polymers whose flexibility vries fom flexible polymers to stiff rod. Vrious quntities which chrcterize the polymer-polymer interfce hve been studied. The interfcil tension s function of sttisticl segment length of semiflexible component hs been clculted using viril theorem nd cpillry wve spectrum method. Simultion results re compred with the men field results of Helfnd nd Spse [2], nd Liu nd Fredrickson [67]. Similrly, the interfcil width s function of stiffness of semiflexible chins is studied by simultion nd they re compred with the men field results of Helfnd-Spse nd Liu-Fredrickson. The monomer density profiles re lso obtined s function of chin stiffness of semiflexible components. Further, we study the orientions of chins nd bonds. The other interfcil properties which chrcterize the interfce re distribution of chin ends nd center of mss of polymer chins. All of these quntities re studied s function of chin stiffness of semiflexible component. In section 4.3, the interfce properties of polymers with different monomer sizes will be presented. We study nd compre our results for interfcil properties of two different types of systems; () system hving two different types (sy type A nd type B) of polymer chins such tht the dimeter of type B monomer is double thn tht of type A monomer but the number of monomers per chin for both types of polymers is equl i.e, monomer size disprity with equl number of monomers per chin, nd (2) system hving two different types (sy type A nd type B) of polymer chins with lmost equl rdius of gyrtion, however, the dimeter of type B monomers is double thn tht of type A monomers. The results of such symmetric polymer-polymer interfces re compred to the interfcil properties of symmetric system in which the size of monomers of both types of chins s well s number of monomers per chin re equl. The simultion results re compred with men field results of Helfnd nd Spse [2]. 58

2 Other interfcil properties like density profile, chin orienttion, distribution of chin ends ner the interfce nd distribution of center of mss of polymer chins re lso studied. In section 4.4, we estimte the criticl vlue of Flory-Huggins prmeter s function of degree of chin flexibility in system of flexible-semiflexible polymers such tht semiflexible chins re fr from isotropc-nemtic trnsition. In simultion, one cn study phse digrm of polymer mixture by using semi-grndcnonicl techniuqes in which types of chins re fluctuting but totl number of prticles remins constnt. Becuse of high stiffness disprity for our systems of study such techniques will be inefficient for the present study. By clculting interfcil tension for wek segregtion limit, we estimte the vlue of Flory-Huggins prmeter t which the interfcil tension becomes zero, corresponding vlue of is criticl vlue of t which two types of polymers get phse seprted. 4.2 Interfces of Flexible nd Semiflexible Polymers In this section we describe the results bout the interfce properties of flexible nd semiflexible polymers t strong segregtion limit. As mentioned before the present study covers the whole rnge of flexiblity of semiflexible polymers from flexible to stiff rods. We hve studied interfcil tension, interfcil width, density profile, distribution of chin ends ner the interfce, distribution of center of mss of polymers ner the interfce, orienttion of chins nd bonds ner the interfce s function of stiffness of the semiflexible chins. All these results will be presented nd discussed in following subsections Interfcil Tension Fig. 4. shows the obtined results for the interfcil tension of n interfce between chins without dditionl bending restrictions nd semiflexible chins described in previous chpter versus the sttisticl segment length of the stiffer chins estimted ccording to Eq The results obtined by the viril theorem nd by the cpillry wve method gree very well within the error brs estimted by the fluctutions of the single mesurements (see bove). The interfcil tension increses with incresing stiffness of the semiflexible component nd levels of for vlues of stiffness beyond the semiflexible region b L which is visible lso by the violtion of the reltionship b =2l p intble 3.. In Fig. 4. lso simultion results obtined by Mueller nd Werner [38] within the bond-fluctution model for rther limited rnge of stiffness disprity re displyed. The interfcil tensions obtined by using viril theorem re higher thn tht by cpillry wve spectrum method. This systemtic difference cn be ttributed to the fct tht viril theorem gives the difference of free energy per cross sectionl re of the interfce while cpillry method is relted to the interfcil re. Therefore, little bit 59

3 higher interfcil tension obtined by viril theorem is not unexpected. In the simultion results it cn be seen tht there is very strong tendency towrds sturtion of interfcil tension with incresing sttisticl segment length of semiflexible components. Moreover, from these results it cn be seen tht there is no chnge in the sturtion property of interfcil tension even though the sttisticl segment length of semiflexible component crosses the isotropic-nemtic trnsition region nd hence we re deling with isotropic-nemtic (flexible-stiff rod) interfce insted of isotropic-isotropic interfce, provided in the isotropic-nemtic interfce the polymers forming nemtic phse re prllel to the interfce plne (when one considers the isotropic-nemtic interfce the interfcil tension (in fct, interfcil properties) depend on the direction of orienttion of polymers which form the nemtic phse [75]. In the present work, only one cse of the isotropic-nemtic interfce of flexible-stiff poylmers is considered in which the nemtic director is prllel to the interfce. The interfcil tension gets sturted before the stiffer chins form nemtic phse nd this trend of interfcil tension continues. The profile of interfcil tension ginst the sttisticl segment length of semiflexible component is very smooth fter segment length in our model crosses the vlue 6.7. These simultion results re compred with the men-field results of Helfnd nd Spse [2], nd Liu nd Fredrickson [67]. Helfnd nd Spse [2] obtined for the interfcil tension, σ, of plnr interfce between two phses of Gussin chins with different sttisticl segment lengths intercting vi Flory-Huggins-type interction σ k B T = 2 ( (β 3 α A β 3 ) B) (4.) 3 (βa 2 βb) 2 The β i ( i = A,B ) β i = 6 ρ ib i (4.2) re the prmeters which contin the chin sttistics. The stticl segment lengths b i re defined in the sme wy s in Eq. 3.4 nd the ρ i re the number densities of sttisticl segments in both bulk phses respectively. For comprison with simultion dt we will use the mpping ρ i = ρ (4.3) C i which corresponds to the introduction of sttisticl segments by Eq ρ is then the number density of beds which is the sme for both chins. The interction prmeter α of the interction between two sttisticl segments is then given by α = ρ (4.4) 6

4 .2.5 σ k B T..5 viril theorem cpillry wve spectrum method Helfnd-Spse Helfnd-Spse with finite end correction Liu-Fredrickson Mueller-Werner Figure 4.: Interfcil tension s function of sttisticl segment length of semiflexible polymers. The sttisticl segment lengths re in units of verge bond length. b with the Flory-Huggins-prmeter for the interction of two beds of chins of different kind s defined by Eq For the interfcil tension of chins with the sme segment length the Helfnd Tgmi result [65], σ k B T = ρ b bb 6 (4.5) is reproduced with now ρ b nd b s the number density nd interction prmeter of sttisticl segments. Fig. 4. shows clerly tht the Helfnd-Spse results [2] gree well with the simultion dt in the relly semiflexible rnge of our system but differs incresingly with incresing stiffness. Expected resons re s well effects of finite chin length s lso the formtion of locl order with incresing stiffness. It will be discussed below. Liu nd Fredrickson [67] nlyzed the interfcil tension of binry blends of polymers with different stiffness strting from wormlike chin hmiltonin for both chins nd with n interction hmiltonin qudrtic in both order prmeters, concentrtion nd orienttion. Using Lndu-de Gennes expnsion for the orienttionl prt of the free 6

5 energy, fixing the vlue of the Mier-Supe prmeter nd ssuming wek orienttion only, they obtined; σ k B T = 4 κ 3/2 A κ 3/2 B (4.6) 9 2 κ A κ B where is the monomer length nd κ i ( i = A,B ) is the dimensionless persistence length ( in units of ) of the ith component of the polymer blend. Eq. 4.6 hs the drwbck showing not the expected dependence on monomer density s eg Eqs. 4. nd 4.5 nd lso not greeing with Eq. 4.5 in the limiting cse κ A = κ B =. Using the correction fctor 3 proposed in [67], with the replcement κ = C N + nd 8 2 using our vlues for the verge bond length s monomer length i.e., 2l c for two flexible chins, we get lmost complete numericl greement with the results from Eq With this choice of prmeters the interfcil tension ccording to Eq. 4.6 in Fig. 4. shows less increse with incresing stiffness disprity s the Helfnd-Spse result [2] nd seems to gree better with the simultion result for lrge stiffness. But the bove discussed problems nd the behvior t smll stiffness disprities rules Eq. 4.6 out to be suitble expression for describing the interfcil tension for unsymmetricl polymer blends. Up to now, the simultion results for finite segment numbers re compred with men-field results for long chins. In literture ( see e.g. [87] ) severl corrections for finite segment numbers re discussed. Ermoshkin nd Semenov [87] reconsidered the problem most recently nd proposed corrections for interfces between blends with different moleculr weight nd lso for the cse N. Using the correction ( 4 ln 2 N ) obtined in [87] to Eq. 4. the reduction is too lrge but we get n lmost complete greement for the region of smll stiffness disprity using the correction fctor ( 2 ln 2 N ) obtined in [88] s is visible from Fig. 4.. A detiled discussion of possible physicl resons for this disgreement is beyond the scope of this work but it my be relted to the problem lredy discussed by Binder [] tht minimiztion of free energy functionl in squre grdient pproximtion is not sufficient for the strong segregtion cse N >. As min reson for the differences between men-field results nd simultion t higher stiffness disprities the strong orienttion of bonds nd chins ner the interfce must be considered ( see below ). This is not tken into ccount in the pproches discussed bove. Moreover, when the persistence length ( lp ) of semiflexible chins is beyond 3.6 n isotropic-nemtic trnsition will occur ( see tble 3.). This strong increse in order in bulk is not ccompnied by visible chnge in the stiffness-dependence of the interfcil tension. This is n dditionl hint tht the orienttion ner the interfce is lredy lrge in the cse of isotropic bulk phses nd determines the stiffness dependence of interfcil tension. To derive the formul 4.6, Liu nd Fredrickson ssumed tht the semiflexible polymers, in flexible-semiflexible polymer system, re fr from nemtic phse. By incresing the persistence length of semiflexible component, the system will be closer to isotropic-nemtic trnsition. Therefore, the disgreement with their results 62

6 for the system with flexible nd highly stiff polymer chins is not unexpected. Further in their study, they fixmier-supe prmeter. When we increse the stiffness prmeter of semiflexible chins the Mier-Supe prmeter of semiflexible chins lso increses. These could be the resons why the difference between men-field nd simultion goes on incresing with the stiffness of semiflexible component Density Profile nd Interfcil Width The entnglements in the interfcil zone re of mjor importnce for the mechnicl properties of the blend. Therefore, the monomer density profiles of different components of the polymer blends re lso importnt. Figure 4.2 presents the density profiles of the individul components s well s totl monomer density profile s function of the stiffness prmeter of the semiflexible component. The density profiles for the different stiffness prmeters re not much different until the persistence length ( lp )of semiflexible component is 3.6. The profiles become shrper in the semiflexible side s the stiffness increses. When the persistence length ( lp ) of the semiflexible components is lrger thn 3.6, it forms nemtic phse nd the density profiles lso become quite different nd moleculrly shrp which clerly shows decrese in interfcil width significntly. We cn describe these profiles lso by the tngent hyperbolic function For exmple, figure 4.2 shows the tngent hyperbolic function fitted for the system with flexible-semiflexible polymers in which semiflexible component hs persitence length ( lp ) =2.5. A reduction of the totl monomer density is observed t the center of the interfce nd the effect increses with the increse in the stiffness of semiflexible component. When the persistence length ( lp ) of the semiflexible component of polymers is more thn 3.6 the reduction of the totl monomer density t the interfce is very high s shown in the figure 4.2. The minimum vlue of totl monomer density is.95 in the cse of persistence length ( lp ) of semiflexible component 3.6 wheres it is bout.85 nd.63 for the systems with isotropic-nemtic interfces with persistence length ( lp ) of semiflexible component 28. nd 3.2 see figure 4.3. It should be noted tht in the present work, the nemtic director in isotropic-nemtic interfce is prllel to the interfce plne. Thus it is observed tht s the stiffnes of the semiflexible component increses the density profile becomes shrper in semiflexible side nd the depth of dip in the totl density t the interfce goes on incresing. These results qulittively gree with the previous results of Schmid nd Mueller [66] for symmetric polymer-polymer interfce nd results of Liu nd Fredrickson [67]. However, Mueller nd Werner [38] hve reported tht the reduction of the totl monomer density t the center of the interfce is lmost independent of the stiffness of the semiflexible component nd the density profiles for stiffness disprity re lmost independent of stiffness of semiflexible component. The reson my be tht they considered very smll stiffness disprity (C N =3.3 is highest chrcteristic rtio, estimted from their dt), therefore, in their results the reduction of totl monomer density t the center of the interfce is lmost independent of stiffness. In the present results lso if we just 63

7 consider very smll stiffness disprity s in their cse (e.g. sttisticl segment length of semiflexible component =.5 nd 2.8 only), we cnnot see the smll difference in the reduction of totl monomer density t the center of the interfce nd density profiles re lso lmost independent of sttisticl segement length of semiflexible component which grees very well with their results up to the stiffness disprity they studied. Compring the flexible-flexible polymer interfce with the interfce of flexible polymers nd semiflexible polymers with persistence length ( lp ) 3.6, the mount of reduction of totl monomer density t the interfce is little different nd the density profiles re lso not much different. φ(x).5 l p =.25 l p =2. l p =2.5 l p =4.2 l p =7.2 l p =3.6 l p =28. l p =3.2 tnh x coordintes Figure 4.2: Normlized totl monomer density nd individul component density s function of chin stiffness of the semiflexible polymers. Persistence lengths l p s re in unit of verge bond length. φ(x) is the normlized monomer density. Tngent hyperbolic function describing φ(x) of the system with lp = 2.5 is lso shown. The totl interfce width w (compre to Eq. 3.27) is determined s lredy explined 64

8 in 3.4 by fitting the density profile to the model function The intrinsic width w follows then nlyzing Eq As lredy discussed bove s lower cutoff length the minimum of persistence length l p is used. We get slight decrese of totl interfce width with incresing stiffness disprity within the rnge of n isotropic phse for the semiflexible chins but shrp decrese down to moleculrly shrp interfce s visible in Figure 4.2 for the interfce between flexible chins nd stiff chins in nemtic stte. Figugre 4.4 shows lso the dt for the intrinsic width (considering persistence length s the lower cut off length)..4.3 δφ b Figure 4.3: Depth of the dip in monomer density profile s function of chin stiffness of the semiflexible polymers. Further, the intrinsic width keep on incresing until we consider the flexible-semiflexible interfce such tht semiflexible polymers re isotropic. But the intrinsic width hs smllest vlue for isotropic-nemtic interfce (i.e. the interfce between flexible-stiff rod polymers). Therefore, the intrinsic width lso decreses when we pss from isotropic-isotropic interfce to isotropic-nemtic interfce of polymers i.e. intrinsic width decreses with incresing sttisticl segment length of semiflexible component when the sttisticl segment length is greter thn the vlue t which isotropic-nemtic trnsition tkes plce. In figure 4.4, it cn be seen tht the difference in totl interfcil width nd intrinsic width decreses s function of sttisticl segment length of semiflexible component. This mens the contribution from the cpillry wve to the interfcil width lso de- 65

9 creses s function of sttisticl segment length of semiflexible component. In ll these discussions, we hve considered isotropic-nemtic interfce in which the nemtic director of polymers with nemtic phse is prllel to the interfce plne. 2.5 w.5 totl width (simultion) intrinsic width Helfnd-Spse Helfnd-Spse with finite end correction Liu-Fredrickson Figure 4.4: Interfcil width s function of sttisticl segment length of semiflexible polymers. The width nd sttisticl segment length both re in units of verge bond length. Within the work shortly chrcterized bove Helfnd nd Spse [2] obtined for the intrinsic interfcil width prmeter w, w = b β 2 A + β2 B 2α (4.7) The results for our systems re gin obtined with the replcements ccording to Eqs. 4.4 nd 4.2 nd shown lso in Fig The end effect rises the vlue of interfcil width by fctor of (+ 2ln2 ) [89]. Figure 4.4 lso presents the dt obtined by using N this correction fctor in the men-field expression obtined by Hlfnd-Spse i.e Eq As for the interfcil tension resonble greement between the intrinsic width nd men-field dt with finite end corrections is observed in the semiflexible rnge nd 66

10 incresing differences pproching the isotropic-nemtic trnsition. The intrinsic width pproches the totl width for lrge stiffness disprities becuse the incresing interfce stiffness prevents the formtion of cpillry wves within the considered subsystems. Liu nd Fredrickson [67] obtined for their wormlike chin model w = 2 3 κa + κ B (4.8) With the sme tretment s for the interfcil tension the vlues for our systems re lso shown in Fig. 4.4 nd similr reltionship between the nlytic results in [2] nd [67] nd our simultion dt s in Fig. 4. cn be observed for the predicted interfcil width. Moreover, it should be noted tht the results for the interfcil tension nd the interfce width derived in [38] from the Helfnd nd Spse [2] results nd lso the results following from the free-energy-functionl in [87] by minimiztion with the ínterfce profile Eq nd using the sme mpping procedure s in Eqs. 3.4 nd 4.3 gree completely with the results in Figs. 4. nd 4.4 for the cse C A = C B = but show strong incresing devitions t lrger stiffness disprities. Using insted Eqs. 3.4 the forml equivlent mpping procedure b = C N b = (4.9) ρ b = ρ leding lso to Eq. 4.5 but now with prmeters b, b nd ρ b, n lmost complete greement with the nlytic results in Figs. 4. nd 4.4 up to lrge stiffness disprities is obtined. Mueller nd Werner [38] hve lso reported tht the totl interfcil width decreses with increse in the stiffness of semiflexible component in the blend of flexiblesemiflexible polymers which grees with the present results Orienttion of Chins nd Bonds in the Interfce Region The thermodynmic quntities which were discussed in previous subsections re not sufficient to understnd the microscopic structure of the polymer interfces. The width of the interfcil region nd the orienttion of polymers on different length scles influence the mteril properties. They lso ply n importnt role for rections t interfces. The polymers stretch prllel to the interfce. The shpe of polymer, ner the interfce is prolte ellipsoid. The existence of plnr interfce destroys the isotropy in the bulk polymer nd consequently orienttion of bonds s well s chins reltively to the interfce will be observed. 67

11 Quntittively, the orienttion of the bond vectors ner the interfce region hve been studied. We study the following bond orienttionl prmeter (defined in the chpter 3); S (x) = 3 2 x (x) / 2 (4.) 2 where re the bond vectors. The bond orienttion prmeter is positive for perpendiculr nd negtive for the prllel orienttion. The profile of the bond orienttion is shown in the figure 4.5. In figure it is seen tht bond vectors prefer to llign prllel to the interfce. The orienttion effects increses upon incresing the stiffness of the semiflexible component. Further the bond orienttion prmeter re not much different for different systems of studies in the flexible side wheres in the semiflexible side they re different which is the effect of stiffness of the semiflexible component. In the orienttionl profile the oriented region is broder in the comprtment occupied by the stiffer chins. Further, in contrst to the width of the density profile, the sptil rnge over which the orienttion of bonds extends grows with increse in the stiffness of the semiflexible component. Therefore, the orienttionl width nd the width of the compositionl profile re two independent microscopic length scles. Similr results re obtined by Mueller nd Werner [38]. Further, we hve studied the orienttion of the prllel nd the perpendiculr components of the rdius of gyrtion of the polymer chins. We hve studied the following orienttionl prmeter (s defined in the chpter 3) for the polymer chins ner the interfce; Rg = 3 <Rg2 x > <Rg2 > (4.) 2 <Rg 2 > nd Rg = 3(< Rg2 z > + <Rg2 y >)/2 <Rg>2 (4.2) 2 <Rg 2 > where Rg nd Rg re the perpendiculr nd prllel orienttionl prmeters of the rdius of gyrtion of polymer chins (perpendiculr nd prllel with respect to the interfce plne), <Rg 2 > is the men squred rdius of gyrtion of the chins nd <Rgi 2 > (i = x, y, z) is the ith component of men squred rdius of gyrtion of the polymer chins. Therefore, if the chins orient prllel to the interfce, prllel orienttionl prmeter ( Rg ) will be positive (mximum vlue.25) while the perpendiculr orienttionl prmeter ( Rg ) will be negtive (minimum vlue -.5). If the chins orient perpendiculr to the interfce, Rg will be negtive nd Rg will be positive. If there is no preferred orienttion of the chins, Rg nd Rg both the quntities will be equl to zero. The profiles of these quntities re shown in the figure 4.6. As shown in the figure, Rg nd Rg in the flexible side remins lmost constnt (until the persistence length ( lp ) of the semiflexible chins increses upto 3.6) while in the semiflexible side they re different which is not unexpected s semiflexible chins 68

12 -.2 S (x) l p =.25 l p =2. l p =2.5 l p =4.2 l p =7.2 l p = x Figure 4.5: Orienttion prmeters of bonds s function of stiffness of semiflexible component. l p s re in unit of verge bond length. possess different degrees of flexibilities. In the isotropic-nemtic interfce, the chins in the flexible side remins unffected till they re very close to interfce. However, t the vicinity of the interfce, they strongly stretch prllel to the interfce. At the men time, in the semiflexible region the semiflexible chins get stretched more nd more prllel to the interfce s the degree of flexibility decreses. In the isotropic-nemtic interfce the nemtic chins prefer to llign prllel to the interfce nd they do it perfectly which is depicted in the figure 4.6. The polymer chins prefer to llign prllel to the interfce s Rg is positive in ll the cses nd Rg is negtive (t the interfce region) while in both bulk phses no preferentil orienttion tkes plce for the isotropic-isotropic interfce. In the cse of isotropic-nemtic interfce, nemtic chins prefer to llign perfectly prllel to the interfce plne even in the bulk phse. The orienttion effects become stronger with the increse of the stiffness of semiflexible chins. For isotropic -isotropic interfces, the orienttionl effect tends to be smll unless the system is close to the isotropic-nemtic trnsition. Pssing through the interfce region the elliptic chins first ttempt to mximize the homocontcts with their own bulk phse (A-A nd B-B segment contcts, respectively) in order to minimize the energy by rotting their longest xes into the interfce plne s fr s possible. 69

13 Rg nd Rg Rg l p =.25 l p =2. l p =2.5 l p =4.2 l p =7.2 l p =3.6 l p =3.2 Rg x coordinte Figure 4.6: Orienttion prmeters of chins s function of stiffness of semiflexible polymers. The persistence length re in units of verge bond length Distribution of the Chin Ends nd Density of Center of Mss of the Chins The distribution of the chin ends re importnt for the interdiffusion nd heling properties t interfces between long polymers [5]. Further on the theoreticl side, the behvior of chin ends is relted to corrections to the ground stte pproximtions [9]. Chin end effects give lrge corrections to the interfcil tension nd width (e.g., see the subsections 4.2. nd 4.2.2) nd they lso ply n importnt role for long rnge interctions between interfces. Due to entropic reson polymers orient themselves by putting their ends preferentilly t the center of the interfces. Since A type of chins (in polymer blend of type A nd type B) close to the interfce prefer to put the chin ends into the B phse nd vice vers, the chin ends re more t the interfce thn they re t the bulk side. A chin close to interfce prefers to put its ends to its minority phse becuse of entropic reson. Therefore the density of ends of type A chins is incresed t thesideoftypebchinsnddecreseclosetotheinterfcettheasidendvicevers. When this effect increses the interfce becomes shrper nd we cnnot see such effect in the wek segregtion limit. If we clculte the totl chin end distributions, we find effectively n enrichment of chin ends t the center of the interfce nd depletion in the wings of the profile. The result of polymer chin ends being locted preferentilly 7

14 close to the interfce is tht whole chins tend to orient themselves prllel to the interfce nd hence the shpe of the polymer chins ner the interfce is prolte ellipsoid (see previous subsection). Nρe(x) 2ρ(x).5 l p =.25 l p =4.2 l p = x coodinte Figure 4.7: Distribution of chin ends s function of stiffness of semiflexible components in flexible-semiflexible polymer systems. The persistence lengths l p srein units of verge bond length. Figure 4.7 shows the profiles of the distribution of the chin ends s function of stiffness of semiflexible components. In Fig. 4.7, Nρe(x) versus x coordinte hven been 2ρ(x) plotted where ρ e (x) is the number density of chin ends t x, ρ(x) is the totl monomer density t x ndn is the number of monomer per chin. We hve presented the bove described quntitiy s function of chin stiffnes of semiflexible component. It shows the chin end distributions for flexible chins, semiflexible chins nd sum of them. Chin ends re enriched t the center of the interfce, nd this effect goes long with depletion wy from the interfce. A - polymers like to put their ends into B rich phse nd vice vers. Similr results re obtined theoreticlly for Gussin chins by Schmid nd Mueller [66]. The effect becomes stronger when the chin stiffness increses. In the figure 4.7, it cn be seen tht the mximum vlue of the profile increses with the increse of the stiffness of semiflexible component of the blend. As result the minimum vlue wy from the interfce is smllest for the system hving highest vlue of stiffness for semiflexible component. Further the totl chin end distribution becomes more symmetric s function of stiffness of semiflexible component. In the flexible side the profiles re not much different but in the semiflexible side they re different which is reflected t the totl vlue of end distributions. Anlogous observtions pply to figure 4.8 which presents the normlized density of centers of mss of the chins, ρcm(x) ρ cm (where ρ cm (x) is the number density of center of mss of polymer chins t x nd ρ cm is the verge density of center od mss of polymer 7

15 ρcm(x) ρcm.5 l p =.25 l p =4.2 l p = x coordinte Figure 4.8: Distribution of center of mss of flexible chins, semiflexible chins nd sum of them. The persistence lengths l p s re in units of verge bond length. chins) except for the mxim t the interfce being now exchnged to minim due to the enrichment of centers of mss of chins in the interfce region next to the bulk phse of the component. The density of center of mss hs minim t the center of the interfce. This figure presents the normlized density of center of mss of flexible chins, semiflexible chins nd the sum of them. The profile hs minimum t the interfce nd this effect goes on incresing s the stiffness of the semiflexible component increses while the mxim t the interfce in the profile of the chin ends distribution goes on incresing with the increse in the stiffness of the semiflexible component. As in the cse of chin end distributions, the profiles in the flexible side re not much different but in the semiflexible side they re. Therefore, the totl profiles re not symmetric. The mximum vlue in the profile which contins sum of center of mss of flexible nd semiflexible chins wy from interfce increses s function of stiffness of semiflexible component. 4.3 Interfces of Polymers Hving Different Monomer Sizes In the previous section the results of interfce properties of flexible-semiflexible polymer systems re discussed. This section is devoted to results nd discussions bout the interfce properties of polymers with different monomer sizes. These systems consist of two types of polymer chins, viz; type A nd type B. A type of polymer chins hve monomers with dimeter d A = d min (which is defined lredy in chpter 3) wheres B type of polymer chins hve monomers with dimeter, d B =2 d A. We study nd compre interfcil properties of polymers hving monomers of different sizes monomer size disprity with equl number of monomers per chin nd monomer size disprity with 72

16 lmost equl rdius of gyrtion (see chpter 3). The interfce properties of monomer size disprity systems re compred to tht of symmetricl system. The interfcil properties which hve been studied, re s those of stiffness disprity presented in section 4.2. The interfcil properties nmely, interfcil tension, interfcil width, density profile, orienttion of chins, distribution of chin ends nd center of mss of chins for the systems described bove hve been studied Interfcil Tension The interfcil tension for unsymmetric systems of two types of polymers which differ in the sizes of monomers re clculted s described bove by viril theorem nd these results re compred with the nlytic expression given by Helfnd nd Spse [2]; σ k B T = 2 ( (β 3 α A β 3 ) B) 3 (βa 2 β2 B ) (4.3) which is sme s in Eq. 4.. The β i ( i = A,B) re sme s in eqution 4.2 nd ρ i re lso sme s in eqution 4.3 in the section 4.2. However, the interction prmeter, α, between two sttisticl segments is now given by, α = ρ A ρ B (4.4) with the Flory-Huggins prmeter for the interction of two beds of chins of different types s defined in eqution 3.3. The mpping procedures to compre simultion results with men field re sme s in the subsection From eqution 4. it is seen tht the interfcil tension depends minly on number density of different components, the interction prmeter nd sttisticl segment length of the two components. Tble 4. shows simultion results nd the dt obtined from men-field expression of Helfnd nd Spse eqution (4.3). From tble 4., it cn be seen tht the interfcil tension decreses with the increse in monomer size disprity. It is seen tht the interfcil tension decreses in both cses, nmely in monomer size disprity with equl number of monomers per chin nd monomer size disprity with lmost equl rdius of gyrtion, in comprison to symmetricl system contining smller size of beds. Moreover, for the symmetric systems the interfcil tension is higher for the monomer size disprity with lmost equl rdius of gyrtion thn tht for the monomer size disprity with equl number of monomers per chin. We hve compred lso the rtio of interfcil tension to the Flory-Huggins prmeter,, for different systems of studies. Tble σ 4. presents the rtio of k B nd for the systems of study. The rtio of interfcil T tension ( σ k B ) to Flory-Huggins prmeter, lso hs lrger vlue for the monomer size T disprity with lmost equl size of polymer chins thn tht of monomer size disprity with equl number of monomers per chin. It is cler tht simultion dt grees very well with the men-field dt by tking into ccount the finite chin length effects. Following the description in subsection

17 type of system σ k B T σ/k B T σ H-S w k B T symmetricl.65 ± N A = N B.26 ± Rg A Rg B.33 ± w (H-S) Tble 4.: Interfcil tension nd width for size disprity systems nd symmetricl system. The totl interfcil width re divided by respective bond lengths tht is,.998 in symmetricl system nd in the unsymmetricl systems. we tke into ccount of finite length effects to describe the smll difference between simultion nd men field results. Helfnd et l. [88] hve obtined the corrections in the interfcil tension due to the finite length of the chins. They hve obtined tht the interfcil tension is reduced by fctor of ( ( ln 2 N A + ln 2 N B ) becuse of finite chin length of the polymer chins. This fctor becomes.8 for unsymmetric systems nd describes very well the very smll difference between simultion results nd men-field dt obtined from the nlytic expression of Helfnd nd Spse. When the sttisticl segment length of one of the components increses i.e., when the symmetry in two types of chins increses, the interfcil tension increses but in our system of study it decreses becuse the Flory-Huggins prmeter nd number density of the type B chins lso decrese. From tble 4., it is seen tht the interfcil tension for the symmetric system is lower thn tht for the symmetric system. As seen in the tble 4. for the size disprity cse simultion dt gree very well with the men field results which implies tht the men field results describe the systems of lower vlues of nd number density. The min resons for the lower vlue of interfcil tension for symmetric systems re thelowervlueofnd monomer density. As seen from the Eq. 4.5 the interfcil tension depends up on both of these quntities nd increse with the increse with ny of these quntities. It is seen tht the smll increse in interfcil tension due to increse in sttisticl segment length of the symmetric system cnnot compenste the huge reduction due to number density nd Density Profile nd Interfcil Width Figure 4.9 shows the normlized monomer density profile for the type A monomers type B monomers nd their sum. The number density of type B monomers is multiplied by eight s one type B bed hs volume 8 times lrger thn tht of the type A in monomer size disprity systems. As in the cse of stiffness disprity, one cn describe these profiles by the tngent hyperbolic function, Eq A reduction of the totl monomer density is observed t the center of the interfce. It cn be seen from figure tht the monomer density profile for symmetric polymer-polymer interfces re not significntly different thn tht of the symmetric interfces. In the cse of symmetric system the smller beds penetrte more deep inside the bulk phse of lrger beds thn in the symmetricl system incresing the size of the interfcil region. The dip in the center of the interfce increses slightly with monomer size disprity. But this increse is not 74

18 symmetricl system equl N lmost equl Rg φ(x) x coordinte Figure 4.9: Monomer density profile of individul components nd sum of them in the symmetric system nd monomer size disprity systems. significnt s in the cse of isotropic-nemtic interfce (see subsection 4.2.2). The dip t the center of the interfce is highest for the system of monomer size disprity with equl number of monomers per chin for both types of chins wheres it hs lowest vlue for the symmetric system. Tble 4. shows totl interfcil width s function of dimeters of different types of beds. It depicted both the dt one by simultion, nd nother by men field expression using Eq The sme mpping s in the stiffness disprity cse hs been used. These dt show tht totl interfcil width increses with the symmetry in the monomer volume, i.e., the rdius of monomers. The interfcil width is lrger for the system in which two types of monomers hve differnt size but there re equl number of monomers per chin thn tht the system in which two different types of monomers hve lmost equl rdius of gyrtion Orienttions of Chins In polymer blends the orienttion of the polymers on different length scles influence the mterils properties. The thermodynmic propterties like interfcil tensions re not enough to understnd the microscopic structure of the polymer interfces. The shpe of polymer chin is prolte ellipsoid ner the interfce. Due to entropic resons polymers orient themselves by putting their ends preferentilly t the center of the 75

19 interfce. In the present work, the orienttion of chins hve been studied by defining orienttion prmeter in terms of rdius of gyrtion of polymer chins. The orienttionl profiles of monomer size disprity systems re compred with tht of symmetric system. Orienttions of the perpendiculr nd prllel (perpendiculr nd prllel with respect to interfce) components of rdius of gyrtion re studied. The orienttionl prmeters which re studied re sme s in the cse of stiffness disprity Eq. 4. nd 4.2. The orienttionl profiles re shown in the figure 4.. The figure shows tht the polymers in symmetric systems with lrger rdius of beds orient more prllel to the interfce plne compred to symmetricl systems. Further the effect is less for the system of the monomer size disprity with lmost equl rdius of gyrtion of chins thn the system of the monomer size disprity with equl number of monomers per chin. Mostnotbly, the orienttion prmeter in the side of smll beds re not much different. If we see only A types of chins (chins with smll beds, i.e. left side in the figure 4.), they stretch more in the symmetric system thn tht in the symmetric system. It is seen tht A types of chins re not much influenced by the interfce nd B types of chins get more ffected by the interfce. Generlly, there is wek nemtic ordering ner the interfce but it is seen from our results tht the effect is stronger in symmetric system with smller nd lrger beds thn the symmetric system with the smller beds. Pssing through the interfce region the elliptic chins first ttempt to mximize the homocontcts with their own bulk phse (A-A nd B-B segment contcts, respectively) in order to minimize the energy by rotting their longest xes into the interfce plne s fr s possible Distribution of the Chin Ends nd Density of Center of Mss of the Chins Since A types of chins (in polymer blend of type A nd type B chins) close to the interfce prefer to put the chin ends into the B phse nd vice vers, the chin ends re more t the interfce thn they re t the bulk side. A chin close to interfce prefers to put its ends to its minority phse becuse of entropic reson. Therefore the density of ends of type A chins is incresed t the side of type B chins nd decrese close to the interfce t the A side nd vice vers. When this effect increses the interfce becomes shrper nd we cnnot see such effect in the wek segregtion limit. If we clculte the totl chin end distributions, we find effectively n enrichment of chin ends t the center of the interfce nd depletion in the wings of the profile. As result whole chins tend to orient themselves prllel to the interfce nd hence the shpe of the polymer chins ner the interfce is prolte ellipsoid (see bove). Figure 4. shows the distribution of the chin ends. Here we hve plotted Nρe(x) 2ρ(x) versus x coordinte where ρ e (x) is the number density of chin ends, ρ(x) is the totl monomer density nd N is the number of monomers per chin. Here we hve presented described quntity s function of monomer size disprity in our systems of study. Figure shows the chin end distributions for type A chins (i. e. chins hving smll size of beds), for type B chins ( i.e. beds with lrge size of beds) nd sum of them. Fur- 76

20 Rg nd Rg.5 Equl N Almost equl Rg Symmetric system x coordintes Figure 4.: Orienttion of chins in the symmetric system nd monomer size disprity systems. 77

21 .2.9 Nρe(x) 2ρ(x).6 Equl N Almost equl Rg Symmetricl system x coordintes Figure 4.: Distribution of chin ends in symmetric systems nd in systems with different monomer sizes. 78

22 ρcm(x) ρcm.3 Symmetric system Equl N Almost equl Rg x coordintes Figure 4.2: Distribution of center of mss of chins in symmetric system nd systems with different monomer sizes. 79

23 ther, it depicted chin end distribution for symmetricl system. It is seen tht the chin ends re enriched t the center of the interfce. The effect increses with the monomer size disprity. The pek hs highest vlue for symmetric system such tht lrger bed size component hs equl number of monomers per chin s tht of chins with smller beds wheres the pek hs lowest vlue for the symmetricl system. Totl chin end distribution becomes symmetric for the symmetric systems which we studied. The distribution of the center of mss of chins re lso importnt to understnd the interfcil properties s they give us insight of the loction of chins. The profile for the distribution of chin center of mss, viz; ρcm(x) ρ cm, is shown in figure 4.2. ρ cm (x) isthe density of center of mss of polymer chins t x nd ρ cm is the verge density of center of mss of chins. Therefore, this figure presents the normlized density of centers of mss of the chins. In the chin end distributions, there is mxim t the center of interfce (in totl chin end distribution) wheres the totl density of center of mss hs minim t the center of interfce. In figure 4.2 we hve presented the normlized density of center of mss for type A chins (i.e. chins hving smller beds), for type B chins (i.e. chins hving lrger beds) nd sum of them. Further, to compre the results of symmetric systems to tht of symmetricl system, we hve presented distribution of center of mss for the symmetricl system lso. The minimum vlue t the center of interfce is lowest for symmetricl system in which both types of chins hve equl number of monomers per chin nd it is highest for the symmetricl system. 4.4 Phse Behvior in Flexible-Semiflexible Polymer Blend We study the dependence of criticl vlue of Flory-Hugins prmeter,, nd hence the criticl temperture T c on stiffness of semiflexible component. In this section, the criticl vlue of ( c ) hs been estimted s function of stiffness of semiflexible components in flexible-semiflexible polymer blend. As described in chpter 2, one cn study phse digrm in polymeric systems by using semi-grndcnonicl ensemble. In this method one types of chins re converted into other type nd vice vers, to tke into ccount density fluctution. The totl number of prticles in the system remins constnt. However, we consider very high stiffness disprity, so this method becomes inefficient s it violtes excluded volume effect. The behvior of interfcil tension nd interfcil width in wek segregtion limit hs been studied nd from these dt criticl vlue of hs been estimeted. In the present work, we estimte vlue of t which interfcil tension vnishes by studying behvior of interfcil tension t wek segregtion limit. The behvior of interfcil tension t strong segregtion limit will be lso discussed. To know whether the system hs ttined equilibrium in the wek segregtion limit, 8

24 42 w 2 nd MSD, MSD MSD w AMM/ 5 Figure 4.3: MSD of center of mss of polymers nd squre of interfcil width versus the AMM for wek segregtion limit for flexible-semiflexible system in which semiflexible polymers hve persistence length ( lp )=4.2when =.36. the interfcil width nd the men squred displcement ( MSD) of center of mss re monitored. For the wek segregtion limit the interfcil width increses (see below) nd hence the ide which ws used for the strong segrgtion limit (see section 3.2.2) my not be enough for the equilibrtion. If the squre of interfcil width nd MSD re comprble, the clcultions of interfcil tensions nd interfcil widths re strted. Figure 4.3 shows the grph of MSD nd squre of width for persistence length ( lp )= 4.2. In ll the systems of wek segregtion limit, such grphs re produced before strting ny clcultions to ensure the system hs ttined equilibrium condition Interfcil Tension in Strong nd Wek Segregtion Limit In the strong segregtion limit, men-field theory predicts tht the interfcil tension vries s the squre root of Flory-Huggins prmeter, (Eq. 4.). However, in the wek segregtion limit the behvior is different. In wek segregtion limit Flory-Huggins-de Geenes formul for interfcil tension nd interfcil width re given by (from chpter 2); σ k B T = 9 b 2 N ( c ) 3/2 (4.5) 8

25 .5 ( σ ) 2 3 kbt c Figure 4.4: ( σ ) 2 3 k B T versus in flexible-flexible polymer system. The solid line is the stright line fitted to the wek segregtion dt nd dshed line is the curve from the formul of strong segregtion limit (Eq. 4.)..5 ( σ ) 2 3 kbt c Figure 4.5: ( σ ) 2 3 k B T versus in flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =2.). The solid line is the stright line fitted to the wek segregtion dt nd dshed line is the curve from the formul of strong segregtion limit (Eq. 4.). 82

26 .2.5 ( σ ) 2 3 kbt c Figure 4.6: ( σ ) 2 3 k B T versus in flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =2.5). The solid line is the stright line fitted to the wek segregtion dt nd dshed line is the curve from the formul of strong segregtion limit (Eq. 4.)..2.6 ( σ ) 2 3 kbt c Figure 4.7: ( σ ) 2 3 k B T versus in flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =4.2). The solid line is the stright line fitted to the wek segregtion dt nd dshed line is the curve from the formul of strong segregtion limit (Eq. 4.). 83

27 σ kbt Figure 4.8: Interfcil tension versus for flexible-flexible polymer system. The solid line is the stright line from the Eq. 4. nd the dshed line is the curve from Eq σ kbt Figure 4.9: Interfcil tension versus for flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =2.). The solid line is the stright line from the Eq. 4. nd the dshed line is the curve from Eq

28 .8.6 σ kbt Figure 4.2: Interfcil tension versus for flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =2.5). The solid line is the stright line from the Eq. 4. nd the dshed line is the curve from Eq σ kbt Figure 4.2: Interfcil tension versus for flexible-semiflexible polymer system in which semiflexible component hs persistence length ( lp =4.2). The solid line is the stright line from the Eq. 4. nd the dshed line is the curve from Eq

29 systems Eq. 4.5 Eq. 4. Eq. 4.5 M-F Eq. 4. M-F l p = l p = l p = l p = Tble 4.2: Tble of the prefctors in equtions 4.5 nd 4. obtined fter compring our dt with men filed. Here l p s re in units of verge bond length. In the column of system mens, the system contins flexible polymers nd semiflexible polymers of given persistence length. nd w = b N 3 ( ) ( /2) (4.6) c respectively. In these equtions, b is sttisticl segment length, N is the number of monomers per chin nd c is the criticl vlue of. To derive these formuls, it is ssumed tht the system is symmetricl, i.e. the number of monomers per chin in type A nd type B polymers is equl nd further the sttisticl segment length for both types of chins re equl. Therefore, when one considers symmetricl systems the prefctors in equtions 4.5 nd 4.6 will be different. However, exponents in right hnd side will be the sme. From the eqution 4.5, it is seen tht interfcil tension decreses with decrsing nd finlly vnishes when = c. Similrly, from 4.6 the interfcil width increses with decrsing nd finlly becomes infinite when = c. As described bove (see Eq. 4.5), ( σ ) 2 3 k B T linerly vries with nd the interfcil tension vnishes when = c. Figure 4.4 shows the dependence of ( σ ) 2 3 k B T on for the flexible-flexible polymer system. We cn describe the behvior by stright line s shown. Further, the figure shows interfcil tension t strong segregtion limit nd described curve (see Eq. 4. nd 4.5). It cn be seen in the figure tht the dt of strong segregtion limit (SSL) does not follow the stright line fitted for the dt of wek segregtion limit (WSL). From the fitted stright line for the dt of interfcil tension in wek segregtion limit, we hve estimted the criticl vlue of. Further, the prefctors in eqution 4.5 lso hs been clculted. Tble 4.2 shows the prefctors in the eqution 4.5, from men field nd present work. The prefctors from men-field nd present work re not much different. The difference is less thn 5%. Figure 4.8 presents interfcil tension s function of squre root of for ll the rnge of study in the present work for the flexible-flexible polymer system. From figure 4.8 it cn be seen tht the dt of strong segregtion limit follow liner behvior wheres tht of wek segregtion limit shows different behvior. We hve clculted the slope of the stright line described to the dt t SSL which is not fr from the menfiled vlue. Tble 4.2 shows the slope of the fitted line from present work nd men filed. 86