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Ternary mixture of a homopolymer blend and diblock copolymer studied near the Lifshitz composition by small-angle neutron scattering

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Ternary mixture of a homopolymer blend and diblock copolymer studied near the Lifshitz composition by small-angle neutron scattering
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                                Ternary mixture of homopolymer blendand diblock copolymer studied near theLifshitz composition by small-angleneutron scattering Kell Mortensen, a *  Dietmar Schwahn, b Henrich Frielinghaus a andKristoffer Almdal a a Condensed Matter Physics & Chemistry department, Risø National Laboratory, Roskilde Denmark,  b  Inst. Festk¨ orperforschung,Forschungzentrum J¨ ulich, Germany. E-mail: kell.mortensen@risoe.dk  Aternarymixtureof twohomopolymers and thecorresponding diblock copolymer represents agood model systemtostudycriticalphenomenanear themean fieldpredictedLifshitzpoint. Thestructurefactorof con-centration fluctuations is in this regime of the phase diagram describedby a formula which independently accounts for the peak-position  q    ,the scattering intensity at  q    ,  S     q      , and the forward scattering,  S     0    .The susceptibility, which is given by the maximum value of either  S     0   or  S     q      , shows markedly renormalized critical behavior with criticalexponents significant larger than the 3d-Ising value relevant for simplebinary blends. 1. Introduction The characterization of different states of matter and the phase transi-tions between these states are of fundamental scientific interest. Eachphase transition belongs to a universality class with a set of uniquecritical exponents describing material properties in the vicinity of thephase transition. The critical fluctuations and the associated classifica-tion have been studied experimentally and theoretically in great detailsin both polymer blends and in diblock copolymers.In this paper we investigate the cross over regime from macro-scopic and microscopic phase transitions in a model polymer systemcomposed of a critical blend of two homopolymers mixed with smallamounts of the corresponding symmetrical diblock copolymer. By ap-propriate choice of molar sizes the ternary system has been made tofulfil that the critical temperature of the pure homopolymer blend isclosely matched to the order-disorder temperature of the pure diblock copolymer. With the molar volumes  V   A    V   B  for the homopolymersA and B and with the molar volume  V   AB  for the diblock copolymer,the Flory-Huggins parameters  Γ   must obey the condition  V   AB Γ  ODT     V   A Γ  c    V   B Γ  c ,  Γ  ODT   representing the Flory-Huggins parameter of thediblock copolymer at the order-disorder transition and  Γ  c  that of theblend at the critical point. Within mean field approximation, the molarvolumes must fulfil the ratio  V   AB    V   A    V   AB    V   B    10   495    2     5.If homopolymers are added to a diblock copolymer in the orderedphase, the microdomain size might ideally increases continuously un-til the thermodynamic limit at  q      0 where the domains are of macroscopic size. In mean field theory the critical line of   macrophaseseparation  for homopolymer mixtures and the corresponding line of  microphase separation  in diblock copolymers meet at a multicriti-cal point, the Lifshitz point (Broseta & Fredrickson, 1990; Holyst &Schick, 1992; Kielhorn & Muthhukumar, 1997).In the near vicinity of the critical point, however, fluctuation renor-malized characteristics are different for homopolymer blends and di-block copolymers. The state of homopolymer blends is the 3d-Isingclass (Schwahn  et al. , 1987), while that of diblock copolymers isthe Bradsovskii type (Fredrickson & Helfand, 1987). The cross-overtemperature from mean-field to renormalized states is given by theGinzburg-criterion, which for hopolymer blends is predicted to scalewith the degree of polymerization  N   as  Gi     T      T  c     T      1    N  ,whereas the scaling for diblock copolymers is  Gi     1        N  . The criti-cal behavior near the Lifshitz critical point is therefore expected to behighly complex and strongly influenced by thermal fluctuations.In a recent study on a ternary system of relative high molar masssymmetric polyolefins composed of two homopolymers and the corre-sponding diblock copolymer: PE/PEP/PE-PEP,PE being polyethyleneand PEP poly(ethylene propylene), mean field Lifshitz-like behaviorwas observed near the predicted isotropic Lifshitz critical point (Bates et al. , 1995); the critical exponents of the susceptibility and correlationlength were determined to be    =1 and    =0.25, respectively, and thestructure factor followed the characteristic mean-field Lifshitz behav-ior:  S     q       q    4 . This is opposed to the common  S     q       q    2 charac-teristic known for binary blends. Near the mean field Lifshitz criticalpoint, however, the influence of fluctuations clearly manifested itself by the absence of the Lifshitz critical point, and instead the appear-ance of a one-phase channel of polymeric microemulsion (Bates  et al. ,1997). An equivalent one-phase gap was observed between the micro-and macro-phase separated states in the more low-molar mass systemof poly(ethyl ethylene) and poly(dimethyl siloxane), as shown in Fig 1(see also Schwahn, Mortensen  et al. , 1999a & b). This system showedmoreover, in the near vicinity of the mean field Lifshitz point, criti-cal exponents that are significantly renormalized relative to both meanfield and 3d-Ising behavior. 0.00 0.05 0.10 0.15 0.20050100150   T c,ren Lifshitz Line Two-PhaseOrderedDisordered PEE/PDMS/PEE-PDMS    T  e  m  p  e  r  a   t  u  r  e   [   °   C   ] Composition Diblock  Figure 1 Phase diagram of ternary system of PEE/PDMS/PEE-PDMS, with fixedPEE/PDMS ratio: 0.516/0.484. The present paper presents detailed experimental results on thestructure factor of the poly(ethyl ethylene) and poly(dimethyl silox-ane) system: PEE/PDMS/PEE-PDMS, measured close to the Lifshitzcomposition. 2. Experimental The systems that have been studied are ternary mixtures of PEEand PDMS homopolymers and the corresponding PEE-PDMS diblock                       copolymer, PEE being the acronym for partially deuterated poly(ethylethylene), and PDMS the acronym for poly(dimethylsiloxane).Thesimilarsized PEEand PDMShomopolymers and thesymmetricdiblock copolymer PEE-PDMS were synthesized by anionic polymer-ization followed by catalytic hydrogenation (Almdal  et al.  1996). ThePEE-monomers were partly deuterated during the catalytic saturation,giving the monomer unit: C 4 D 2   8 H 5   2 . The degree of polymerizationof the two homopolymers were close to be equal, while the symmet-ric diblock copolymer had a molar mass approximately 5 times larger,in the attempt to match the microphase separation temperature of thepure PEE-PDMS with the macrophase separation temperature of thepure homopolymer binary blend:  N  PEE     30   5,  N  PDMS     29   2 and  N  PEE     PDMS     168. The ratio of the molar volumes of the homopoly-mers relative to the diblock copolymer,         V   A V   B    V   AB    V  i    V   AB   i    A   B  (1)is for this system,    =0.18. The Lifshitz critical value separating macroand microphase separation is given as  Φ  L    2    2     1    2    2   (Broseta& Fredrickson, 1990; Fredrickson & Bates 1997)The structure factor reflecting the thermal composition fluctuationswas measured by small-angle neutron scattering in a number of mix-tures of varying copolymer content  Φ PEE     PDMS     Φ , keeping the ra-tio of the two homopolymer concentration constant equal the criticalvalue:  Φ PDMS     Φ PEE     0   516    0   484. With the PEE monomers of thehomo- and the block copolymer deuterated to the same degree, themeasured structure factor,  S     q    , reflects thermal composition fluctua-tions with respect to the total PEE/PDMS-fractions.  S     q    is thereforea measure of a scalar ( n =1) order parameter represented by the localcomposition          r     .In the present report we concentrate on the system with block copolymer concentration  Φ PEE     PDMS     10   9%, but extended studieswill be reported elsewhere (Schwahn, Mortensen, Frielinghaus, Alm-dal,1999a &b). The10.9%sampleisclosetotheexperimental Lifshitzcomposition. 3. Results and Discussion The basic thermodynamic features of systems near their consolute lineare well described by the Landau expansion of the free energy:  H     12     d  d  r     c 2          2   c 4      2      2      2   u    4   u 6   6 ℄ (2)with the order parameter    (Holyst & Schick, 1992). A principle effectof diblock copolymers solved with a homopolymer blend is a reduc-tion of the surface energy which, according to the Hamitonian (2), isdescribed by a reduction of the parameter  c 2 . The  c 2  parameter is pos-itive at low copolymer content, becomes zero at the Lifshitz criticalcomposition and is negative for large copolymer content.The Hamiltonian (2) predicts composition fluctuations in the ho-mogeneous (disordered) one-phase regime. These fluctuations are de-scribed by the structure factor  S     q    , which within the random phaseapproximation (Leibler, 1980), is giving as: S     q    V   AB      F     x       2 Γ  V   AB ℄  (3)where  F     q    is the inverse form factor. The structure factor  S     q    canbe measured directly in a scattering experiment with  q  given by  q    4          sin    ,    being the wavelength of the used radiation, and    beinghalf of the scattering angle. S     q    can mean field approximation be expanded into powers of   q 2 ,as S     1   q       S     1   0      2 q 2     4 q 4 (4)with the coefficients given by of the parameters of the Hamiltonian (2).The first term of (4) is  S     1   0      , which for compositions less thanthe Lifshitz value represents the susceptibility. The coefficients    2  and   4  are proportional to respectively  c 2  and  c 4  in the Hamiltonian (2), andcan be determined in terms of the polymer parameters and composition(Kielhorn & Muthukumar, 1997) c 2       2     R 2 g    V     4    2   1     Φ  AB       2 Φ  AB ℄      3    2   1     Φ  AB   2 ℄ (5) c 4       4    R 4 g    V     1     Φ  AB   2   4    4   16    2    9      4        1     Φ  AB    16    2    9      8    4 ℄     36    1     Φ  AB   3    3 ℄    1 (6)At the Lifshitz composition ( c 2 =    2 =0) the characteristic meanfield be-havior:  S     1   q       q 4 , clearly appear from (4). 3.1. Blend-like Compositions For positive  c 2 -values the structure factor  S     q    as obtained from theHamiltonian (2) has the basic characteristics of polymer blends, with S     q    maximum at  q =0, and with the susceptibility,      1 , correspond-ingly given by this  S     q    0    value:     S     1   q    2    Γ  c     Γ     (7)where Γ  is the effective Flory-Huggins parameter and Γ  c  represents thevalue at the critical point of phase separation. For dominating  c 2  in (2)(    2  term in (4)),  S     q    approaches the common form of homopolymerblends: S     1   q    S     1   0      2 q 2  (8)At the critical temperature of macrophase separation  T  c  ( Γ  = Γ  c ), thesusceptibility diverges, i.e. the inverse susceptibility      S     1   0    be-comes zero. According tomean fieldtheory,  S     1 scaleswith T       ,    =1.Including fluctuation renormalization, the critical exponent of the sus-ceptibility increases to the 3d-Ising value (    =1.24), but the form of  S     q    as given in (8) remains a good approximation. 3.2. Diblock Copolymer-like Compositions For negative  c 2 -values the structure factor,  S     q    , has the basic char-acteristics of block copolymer melts with the maximum value of   S     q   appearing at a finite  q -value,  q    q    . The formfactor  F     q    in (3) canfor  c 2  dominating in (2) be calculated in terms of the partial struc-ture factors  S   AA ,  S   BB  and  S   AB  describing the correlations between themonomers of type A and B (Leibler, 1980): F     q      V   AB   S   AA    q    S   BB    q    2 S   AB    q   S   AA    q    S   BB    q       S  2  AB    q    (9)For unperturbed Gaussian chains (9) can be written in terms of molec-ular parameters via the generalized Debye-function: g  D    f    x    2    x 2        fx    exp       fx       1 ℄  (10)with  x    q 2  R 2 g    q 2  N     2   6,    being the statistical segment length of the copolymer,  N   the degree of polymerization, and  R g  being the radiusof gyration of the diblock copolymer. Inset into (9) gives: F     x    g  D    1   x        g  D    f    x    g  D    1     f    x      1    4    g  D    1   f        g  D    f    x       g  D    1     f    x ℄  2   (11)                      The susceptibility is in this diblock copolymer case given by the maxi-mumvalueof thestructure factor at  q    , S     q      . Withinmeanfieldtheoryof symmetric copolymers,  S     q      diverges at the critical point  T  c . Below T  c  the polymer system will microphase separate into a mesoscopic or-dered lamellar structure.In the fluctuation renormalized case, the structure factor can be ap-proached a similar form as (3), but with a renormalized Flory-Hugginsparameter given by Γ  ren V   AB    Γ  V   AB    ˜ c    S     q        V   AB  (12)where ˜ c  is given by molecular parameters. 3.3. Structure Factor and Susceptibility near the Lifshitz Com-position With  c 2    0 in the Hamiltonian (2) near the Lifshitz point, the forthorder term of the gradient energy,  c 4 , becomes a leading factor in thefree energy, giving rise to the mean field characteristic Lifshitz behav-ior of the structure factor,  S     q      q    4 . Such behavior has been ob-served in a relative high molar mass ternary system of poly(ethylenepropylene) and poly(ethyl ethylene) (Bates  et al. , 1995).The expression for the structure factor of a three component mix-ture of a polymer blend and the corresponding diblock copolymer canwithin the random phase approximation be described by the same  S     q   as given for pure block copolymers, (3) and (9), (Kielhorn & Muth-hukumar, 1997). For a ternary system composed of a critical mixtureof A and B homopolymers of equal volume,  V   A    V   B , and conforma-tion (and thereby also of equal composition Φ  A    Φ  B  and equal partialstructure factors  S   AA  and  S   BB ,  S   AA    S   BB .) and an AB diblock withvolume V   AB ,  F     q    can be reduced to F     q      V   AB    2      S   AA    q      S   AB    q ℄  (13)which in analogy with (11) can be written in terms of the generalizedDebye function (10) (Kielhorn & Muthukar) F     x    4      1    Φ  AB      g  D    1     x     Φ  AB g  D    1   x    4 Φ  AB g  D    0   5   x ℄  (14) 0.0 0.5 1.0 1.5 2.005101520 PEE/PDMS @ 10.9% Diblock 21.2°C38.8°C48.3°C59.3°C66.4°C77.7°C86.7°C95.6°C    S   (   Q   )   [   1   0    5   c  m    3    /  m  o   l   ] Q [10 -2 Å -1 ] Figure 2 Structure factor of PEE-PDMS/PEE-PDMS with composition Φ PEE     PDMS  =10.9%. In this composition transitions from diblock to blend andfrom blend to diblock character is observed by increasing the temperature. 3.4. Effect of thermal fluctuations in blend/copolymer mix-tures. The structure factor of blend/diblock mixtures has been derived be-yond the mean field approximation (Kielhorn & Muthukumar, 1997),using the Hartree approximation in the Bradzovskii formalism, equiv-alent to the procedure developed for pure diblock copolymer melts(Fredrickson & Helfand, 1987). The structure factor (3) & (14) wasthereby parameterized into the simple form S     1   q    a      b    q 2   c    dq 2 (15)where the parameters,  a ,  b ,  c , and  d   have been calculated assuming thatthe general shape of   S     q    is unaltered compared to the mean field re-sult (Kielhorn &Muthukumar, 1997). The parametrization of   S     q    (15)with four parameters reflects the four characteristics: S     1   0    a    b    c  q        a    d     b ℄  1    2  S     q        d     a      da    c    db   (16)and the width of the structure factor peak.The structure factor of the 10.9% sample, as measured at tempera-tures in the range from 21-95 Æ   C is shown in Fig.2. The structure fac-tor shows at low temperatures the characteristic behavior of diblockcopolymers, i.e.  S     q    has maximum at finite  q    -value, while at highertemperatures,  S     q    shows the characteristics of homopolymer blends: S     q    maximum at  q    0. At even higher temperatures, the struc-ture factor is again block copolymer like (for details, see Schwahn,Mortensen  et al.  1999b). The curved Lifshitz line shown in the phasediagram in Fig.1 reflects this behavior. The experimental Lifshitz lineis, in opposition to theoretical predictions, not constant in diblockcopolymer content.The structure factor  S     q    is very well described by the Kielhorn-Muthukumar expression (15), as demonstrated by the solid lines giv-ing the best fits. From these fits we obtain the parameters  a ,  b ,  c  and  d  ,and thereby the values of   q    ,  S     0    and  S     q      .  S     0    and  S     q      are bothshown in Fig.3, plotted versus inverse temperature. 2.4 2.6 2.8 3.0 3.2 3.40.1110 PEE/PDMS @ 10.9% Diblock T LL S -1 (0) S -1 (Q max ) Fit S(0) Fit S(Q max )    S   -   1    (   Q    *   ;   0   )   [   1   0   -   5   m  o   l   /  c  m    3    ] 1/T [10 -3  /K] 2.4 2.5 2.6 2.7 2.803691215T LL Figure 3 Susceptibilities  S     0    and  S     q      from fits to the experimental structure factor S     q    shown in Fig.2. The lines represents fits using the fluctuation renormalizedscaling ansatz.                          Crossing the Lifshitz temperature becomes clearly visible from thetemperature behavior of the resulting  q    -values of the  S     q    -peak, asshown in Fig.4. At the Lifshitz temperature  q    becomes zero. The be-havior of   q    near the Lifshitz line can approximately be described bya scaling law  q     T     T   L       with an exponent    between 0.3and 0.4 when approaching  T   L  from both low and high temperatures.  q   becomes relatively constant at low temperatures far from the Lifshitzline. 10 30 50 70 90 110 130 1500246810121416 ∆ T 0.33 T LL =(81.2±0.3)°C = 113.2°CQ * =2.48 10 -3 (T-81.2) 0.33 T LL PEE/PDMS @ 10.9% Diblock     Q    *  [   1   0   -   3  Å   -   1  ] T [°C] Figure 4 q    value versus temperature for the 10.9% diblock composition of thePEE/PDMS/PEE-PDMSternary mixture. Near the Lifshitz temperatures q    canbe fitted by a scaling ansatz. Both  S     0    and the  S     q      decrease continuously with temperature,but it clearly appears that at  T     80 Æ   C ( T     1   2.83   10    3 K    1 ),the susceptibility crosses over from being determined by  S     q     to  S     0    , and again, somewhat less pronounced, at  T     115 Æ   C( T     1   2.58   10    3 K    1 ), the susceptibility gets again determined by S     q      . This reflects the crossing of the Lifshitz line separating blendand copolymer like character.Thefluctuation renormalized susceptibility S     q      isassumed tohavethe usual form (7): S     1   q      2    Γ  c    Γ  ren ℄  (17)but with the renormalized Flory-Huggins parameter  Γ  ren  that includesthe effect of thermal fluctuations. The detailed form of   Γ  ren  is givenseparately for the two cases,  Φ    Φ  LL  and  Φ    Φ  LL  correspondingto the susceptibility represented by respectively  S     q      at finite  q    and q      0. Both  S     q    0    and  S     q    q      have been fitted with the theo-retical expressions according to (Kielhorn & Muthukumar, 1997). Thecorresponding fits are depicted as respectively solid and dashed linesin the figure. The  S     0    -susceptibility is well described only above thelower Lifshitz temperature and the  S     q    q      -value above and belowthe Lifshitz temperature can only be fitted with different sets of param-eters. 4. Conclusions Ternary mixtures of two homopolymers and the corresponding diblock copolymer represent a very good model system to study critical phe-nomena near the mean field predicted Lifshitz point. The structure fac-tor asderived by Kielhornand Mathukumar based on therandom phaseapproximation on unperturbed chains, describes the experimental scat-tering function very well, and makes it possibly to derive the relevantparameters describing the systemthermodynamics, e.g.thesusceptibil-ity      1 , given by either  S     0    or  S     q      , ( q      0) depending on the com-position relative to the Lifshitz value, Φ  L . The temperature dependenceof the susceptibilities are well described by the fluctuation renormal-ized parametrization, derived by Kielhorn and Mathukumar based onthe Bradsovskii formalism. References Almdal, K., Mortensen, K., Ryan, A.J., & Bates, F.S. (1996).  Macromolecules 29 , 5940-5947.Bates, F.S., Maurer, W., Lodge, T.P., Schulz, M.F., Matsen, M.W., Almdal, K.& Mortensen, K. (1995).  Phys. Rev. Lett.  75 , 4429-4432.Bates, F.S., Maurer, W.W.,Lipic, P.M., Hillmyer, M.A., Almdal, K., Mortensen,K., Fredrickson, G.H., & Lodge, T.P. (1997).  Phys. Rev. Lett.  79 , 849-852.Broseta, D. & Fredrickson, G.H. (1990).  J. Chem. Phys.  93 , 2927-2938.Fredrickson, G.H. & Helfand, E. (1987).  J.Chem.Phys.  87  697-705.Fredrickson, G.H. & Bates, F.S. (1997).  J. Polym. Science B, Polym. Phys.  35 ,2775-2786.Fredrickson, G.H. & Helfand, E. (1987)  J. Chem. Phys.  87 , 697-705.Holyst, R. & Schick, M. (1992).  J. Chem. Phys.  96 , 7728-7737.Kielhorn, L. & Muthukumar, M. (1997).  J. Chem. Phys.  107 , 4079–4089.Leibler, L. (1980).  Macromolecules  13 , 1602-1617.Schwahn, D., Mortensen, K. & Yee-Madeira, H. (1987).  Phys. Rev. Lett.  58 ,1544-1548.Schwahn, D., Mortensen, K., Frielinghaus, H. & Almdal, K. (1999).  Phys. Rev. Lett.  82  1756-1759.Schwahn, D., Mortensen, K., Frielinghaus, H. & Almdal, K. (1999).  J. Chem.Phys.  submitted.
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