#2009 The Society of Polymer Science, Japan
AWARD ACCOUNTS SPSJ Award Accounts Macromolecular Assemblies in Solution: Characterization by Light Scattering
By Takahiro SATO1;Ã and Yasuhiro MATSUDA2
This review article demonstrates that the light scattering technique provides important structural information of the following macromolecular assemblies, polymer living anions in a non-polar solvent, amphiphilic telechelic polymer and block copolymers in aqueous solution, and a thermally denatured double-helical polysaccharide aggregated during renaturation. To get accurate information from light scattering data, however, we have to analyze the data using proper theoretical tools, which are briefly explained in this article. The structural information of these macromolecular assemblies assists us in understanding, e.g., the kinetics of living anionic polymerization and gelation. KEY WORDS: Macromolecular Assembly / Light Scattering / Micellization / Aggregation / Telechelic Polymer / Block Copolymer / Helical Polymer /
Associating polymers, bearing functional groups of strong have to formulate characteristic parameters using models attractive interactions, hydrophobic, solvophobic, electrostatic suitable for each architectures considering the dispersity effect interactions, or hydrogen-bonding, form macromolecular as- if necessary. semblies in solution.1 There are many types of architecture of The present article reviews recent light scattering studies on macromolecular assemblies (cf. Figure 1). For associating various macromolecular assemblies in dilute solution. Light polymers with a single attractive moiety per chain (e.g., scattering is one of the most suitable techniques to characterize diblock copolymers, polymer living anions, water-soluble macromolecular assemblies in solution. Its measurements are polymers hydrophobically modified at one end), the architec- made on assemblies in solution without any perturbations, and ture may be spherical (star-like), cylindrical, bilayer or nanometer to sub-micrometer structures can be investigated. vesiclar.2–7 On the other hand, if the chain possesses a number However, analyses of light scattering data are accompanied of attractive moieties, its assembly may be a uni-core or multi- with the above mentioned difficulties. In what follows, we core flower micelle.8–13 Multi-stranded helical polymers may will explain the method for each system to cope with the form assemblies of their unique architecture by the hydrogen- difficulties. bonding.14 The macromolecular assemblies formed by associating THEORETICAL polymers in solution can be characterized in terms of the architecture, the size (i.e., the radius of gyration, hydrodynamic Association–Dissociation Equilibrium radius, or intrinsic viscosity), the average and distribution of Let us consider an association–dissociation equilibrium the aggregation number, and inter-assembly interaction. How- between unimer U and m-mer Um in a polymer solution: ever, the determination of these characteristics is not necessa- Km rily an easy task in comparison with molecularly dispersed mU Um ð1:1Þ polymers. The equilibrium constant Km is determined from the condition The difficulty comes from (1) the concentration dependence given by m ¼ m1, where m and 1 are the chemical of the degree of aggregation, (2) the variety and complexity of potentials of m-mer and unimer, respectively. From statistical 16,17 the architecture, and (3) the dispersity in the assembly structure thermodynamics, m can be generally written as in the case of open association systems.15 Because of (1), the w standard procedure of infinite dilution and also size exclusion m ¼ m þ ln m þ mF ð ÞþF ð Þð1:2Þ k T k T m 1 2 chromatography does not necessarily provide right information B B of the assemblies at a given finite concentration. To make where, m , wm, and are the standard chemical potential (or proper characterization, we have to take into account the the internal free energy), the weight fraction of m-mer in the association–dissociation equilibrium and inter-assembly inter- total polymer, and the total volume fraction of the polymer, action of associating polymers in solution of the finite respectively, F1ð Þ and F2ð Þ are functions of , and kBT is the concentration. To overcome the difficulties (2) and (3), we Boltzmann constant multiplied by the absolute temperature.
1Department of Macromolecular Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka 560-0043, Japan 2Department of Materials Science and Chemical Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8561, Japan ÃTo whom correspondence should be addressed (Tel: +81-6-6850-5461, Fax: +81-6-6850-5461, E-mail: [email protected]).
Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 doi:10.1295/polymj.PJ2008235 241 T. SATO and Y. MATSUDA
The expression of Ám depends on the type of association. At the formation of a uni-core star or flower micelle from m single 20,21 chains, Ám may be given by 2 2 Ám ¼ÀðmÞþðmÞ þ 4Rc ð1:6Þ with the number of solvophobic groups per chain in the micelle core. Here, the first term on the right hand side correspond to the free energy gain when m solvophobic groups enter in the micelle core from the solvent, the second term the free energy due to interaction among coronal chains in the micelle, and the third term the interfacial free energy of the micelle core; and are free energy coefficients, Rc the radius of the micellar core, and the interfacial tension. From eqs (1.4)–(1.6), the optimum aggregation number m0 of the 2 1=2 micelle is equal to ð4Rc =Þ , and the micellar size distribution is determined by .If is large enough, the aggregation number can be regarded essentially to be mono- disperse (m ¼ m0). The free energy of the micellar formation Ám=m per monomer at m ¼ m0 is usually denoted as "0, which mainly determines the association constant Km. On the other hand, in the case of the random association of 16 f -functional unimers, Ám may be simply written as
Ám ¼ðm À 1ÞðÁf0=kBTÞÀln Wm ð1:7Þ
where Áf0 is the free energy change at the single physical bond formation and Wm is the number of ways where m unimers form an m-mer. It is noted that m-mer possesses m À 1 physical bonds. Under the tree approximation, Wm is written as22 f mð fm À mÞ! W ¼ ð1:8Þ m ð fm À 2m þ 2Þ!
Equations (1.4), (1.5), and (1.7) provide us wm and the weight Figure 1. Various types of architecture of macromolecular assemblies. average aggregation number mw in the forms 1 w ¼ W ð =f ÞmÀ1ð1 À Þ fmÀ2mþ2 ð1:9Þ m ðm À 1Þ! m (1 can be calculated by eq (1.2) with m ¼ 1.) Various statistical thermodynamic theories of polymer solutions and proposed different functional forms of F1ð Þ and F2ð Þ, 1 þ only the latter of which is important in the equilibrium mw ¼ ð1:10Þ 16,18 1 Àðf À 1Þ condition. The Flory-Huggins theory gives us F2ð Þ¼1, and the scaled particle theory for hard spherocylinders Here is the fraction of physical bonding, which is related to provides17,19 the total polymer volume fraction by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 À 3 2 1 þ 2fX À 1 þ 4fX F2ð Þ¼ À lnð1 À Þð1:3Þ ¼ ð1:11Þ 3ð1 À Þ2 2fX
Inserting eq (1.1) into the equilibrium condition, we obtain with X exp½F2ð ÞÀÁf0=kBT. The distribution wm is broad- m mÀ1 er for larger f and mw diverges at ¼ 1=ð f À 1Þ (the gel point). wm ¼ Kmw1 ð1:4Þ
where w1 is the weight fraction of unimer in the solution, Radius of Gyration 2 1=2 Pwhich can be determined from the normalization condition, The radius of gyration hS i is one of the most important m wm ¼ 1, and the equilibrium constant Km is given by molecular parameters determined by static light scattering and characterizing the structure of macromolecular assemblies. In K m exp½ÀÁ þðm À 1ÞF ð Þ ð1:5Þ m m 2 what follows, we present expressions of hS2i1=2 for (1) the star- where Ám ðm À m1 Þ=kBT is the free energy of the m- like micelle, (2) the core-corona-type micelle, (3) the vesicle, mer formation in units of the thermal energy kBT. and (4) the random aggregate.
242 #2009 The Society of Polymer Science, Japan Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 Macromolecular Assemblies in Solution
The star-like micelle formed by m linear flexible chains, mass Mcorona. The radius Rcore of the core may be calculated each of which possesses an associating group at one chain end, by can be identified by a uniform m-flexible-arm star polymer, if R ¼ð3mM =4N Þ1=3 ð1:13Þ the micellar core is negligibly small. The square radius of core core A core gyration of the star polymer can be written as: where core is the density of the core. The segment density distribution function ðsÞ in the coronal region of the micelle 2 3m À 2 2 hS i ¼ hS i ð1:12aÞ may be approximated by that starðsÞ for a uniform m-flexible- star m2 linear arm star polymer with a molecular weight of Mstar, which may or be approximated by the Gaussian distribution28 hS2i ¼ 1:94mÀ4=5hS2i ð1:12bÞ M 3 3=2 3s2 star linear ðsÞ¼ star exp À ð1:14Þ star 2 2 2 NA 2hS istar 2hS istar where hS ilinear is the square radius of gyration of the linear 2 polymer chain with the same molecular weight as that of the Here, hS istar is the square radius of gyration of the star polymer. star polymer. Although eq (1.12a) was originally derived for Integrating eq (1.14) over the coronal region, we have the Gaussian star polymers by Zimm and Stockmayer,23 its validity relation Z even in good solvents was confirmed by many experiments and 1 2 Monte Carlo simulations (3 m 18).24,25 On the other hand, starðsÞÁ4s ds eq (1.12b) was proposed by Douglas et al.26 to fulfill the Rcore ð1:15Þ 2M 2 mM Daoud–Cotton scaling theory27 with respect to the asymptotic ¼ pffiffiffistar ½erfcð Þþ eÀ ¼ corona behavior, so that it should be applied at large m. NA NA 2 2 2 2 Unfortunately, hS i for flower micelles has not been where R 3Rcore =2hS istar and erfcðxÞ is the error function 1 Àt2 formulated. Instead, a flower micelle with p petals or loops [erfcðxÞ x e dt]. From this equation combined with 2 has been usually approximated by a star-like micelle with 2p eq (1.12), we can determine Mstar and hS istar from mMcorona 2 arms, being neglected a slight increase in hS i by cutting of the and Rcore. loop chains. When refractive index increments ( @n=@c) of the two If the block copolymer forms a star or core-corona micelle block chains are different, the light-scattering radius of in a selective solvent, the micellar core is not necessarily small, gyration hS2iÃ1=2 obtained by static light scattering is not so that we have to take into account the effect of the finite core identical with the conventionally defined radius of gyration. size on hS2i. Consider the core-corona micelle consisting of m From the light scattering theory for copolymers,29 hS2ià of the block copolymer chains each of which comprises a core chain of core-corona micelle is given by the molar mass Mcore and a flexible coronal chain of the molar
Z 1 4 2 NA 4s ðsÞstarðsÞds hS2ià ¼ 0 2 mðMcore þ McoronaÞ ð1:16Þ 3 2 2 4Mstar 3 À 2 2 2 Mcorecore Rcore þ pffiffiffi e þ Mcorona corona hS i 5 3 m star ¼ 2 ðMcore þ McoronaÞ
8 where ðsÞ, core, and corona are refractive index increments at the > 0(s Ri) position s, of the core chain, and of the coronal chain, respectively, > 2 0 2 0 <> Ai½Hi ÀðR À sÞ (Ri < s < R ) and is the weight-average refractive index increment of the ðsÞ¼ (R0 s R00) ð1:17Þ copolymer sample. It is noted that eq (1.16) along with eqs (1.12) vesicle > shell > 2 00 2 00 > Ao½Ho Àðs À R Þ (R < s < Ro) and (1.15) has no adjustable parameters, if we know Mcore, Mcorona, : 2 m, core and the molecular weight dependence of hS ilinear for the 0(Ro s) linear coronal chain. where shell is the density in the shell region, and The block copolymer also can form a vesicle in a selective 1 1 R R À a À H ; R0 R À a; solvent under certain conditions.2–6 The vesicle consists of a i i 2 2 ð1:18Þ spherical shell and brushes inside and outside the shell. The 1 1 R00 R þ a; R R þ a þ H thicknesses of the shell and inner and outer brushes are denoted 2 o 2 o 30 as a, Hi, and Ho, and the inner and outer radii of the shell Furthermore, according to the theory of polymer brushes, we as R Æ a=2. According to Milner et al.,30 the concentration have the relations profile of the polymer brush is assume to take a parabolic form. 1=3 1=3 mi Mbrush mo Mbrush Then the segment radial distribution function ðsÞ may be H ¼ ; H ¼ ð1:19Þ vesicle i R02=3 o R002=3 written as
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where is a constant, and mi and mo are the numbers of chains the molar mass Mbrush of the brush chain by the following in the inner and outer brushes, respectively. equations The two constants Ai and Ao in eq (1.17) are related to
8 Z R0 > 2 1 2 miMbrush <> 4s2 ðsÞds ¼ 4A R02 À R0H þ H 2 H 3 ¼ vesicle i 3 2 i 15 i i N ZRi A ð1:20Þ > Ro > 2 2 002 1 00 2 2 3 moMbrush : 4s vesicleðsÞds ¼ 4Ao R þ R Ho þ Ho Ho ¼ R00 3 2 15 NA
On the other hand, the radius R is related to the molar mass Mshell of the block chain forming the shell by the equation 2 1 2 ðmi þ moÞMshell 4shellaRþ a ¼ ð1:21Þ 12 NA The light-scattering radius of gyration hS2iÃ1=2 of the vesicle is given by 2 3 2 3 2 04 03 4 02 2 1 0 3 2 4 6 brush AiHi R À R Hi þ R Hi À R Hi þ Hi 7 6 3 5 3 35 7 6 7 6 1 7 6 2 005 05 7 6 þ shell shellðR À R Þ 7 6 5 7 4 5 2 3 2 004 003 4 002 2 1 00 3 2 4 þbrush AoHo R þ R Ho þ R Ho þ R Ho þ Ho 2 Ã 3 5 3 35 hS i ¼ 2 ð1:22Þ ðmi þ moÞðMshell þ MbrushÞ=4NA
where brush and shell are refractive index increments of the brush and shell chains, respectively. POLYMER LIVING ANIONS IN A NON-POLAR At last, we consider open association systems where SOLVENT f -functional units randomly aggregate according to the formula: Polymer living anions possessing the non-polar main chain and polar active end may form reversed micelles during the U þ U ! U ðm; n ¼ 1; 2; Á Á ÁÞ ð1:23Þ m n mþn living anionic polymerization carried out in non-polar solvents. Similar open association systems have been already treated in Micellization plays an important role in polymerization the previous subsection, but here the systems are in the reaction kinetics, because only dissociated free living anions unidirectional process with functional groups keeping equal are believed to be active species for the propagation reac- reactivity throughout the process. In such cases, the distribution tion.33–37 Thus the precise aggregation number m, the associ- law wm and the weight average aggregation number mw are ation constant (or the free energy of micellar formation), and given by eqs (1.9) and (1.10),22 even if the system is not the structure of the reversed micelle formed by polymer living thermally equilibrium. Furthermore, the cascade theory9,31 anions are prerequisite to the profound understanding of the gives us the following expression of the z-average square anionic polymerization. radius of gyration Matsuda et al.21,38 studied on the reversed micelle of 2 polybutadiene (PB) living anions in cyclohexane at 25 Cby 2 b f 2 hS i ¼ mw þhS i ð1:24Þ light scattering. Light scattering is a convenient method of z ð þ Þ2 u 2 1 investigating moisture-sensitive PB living anions, because the where b is the distance between associating nearest neighbor measurements can be made on test solutions in sealed glass 2 units, is the reacted fraction of functional groups, and hS iu is cells. the square radius of gyration of the associating unit. Figure 2 shows the concentration c dependences of ðKcM1= 1=2 If the above aggregation process is stopped at a certain R0Þ for the major components of three living polybutadiene , and aggregates formed are fractionated according to the (LPB) samples investigated with the molecular weights M1 of aggregation number, the radius of gyration of the fraction with 9,600 (LPB-10k), 35,200 (LPB-35k), and 81,900 (LPB-82k). the aggregation number m can be calculated by32 Here, K is the optical constant, c is the polymer mass
2 XmÀ2 concentration, M1 is the molecular weight of PB sample after 2 b ðm À 2Þ! n hS i ¼ ð f À 1Þ quench, and R0 is the excess Rayleigh ratio at the zero scattering m 2 ½ð f À 1Þm! n¼0 angle. For all three LPB samples, data points obey slightly ½ð f À 1Þm À n À 1! ð1:25Þ concave curves, and the intercepts of the curves seem to be less  ½nð f À 2Þþ2ð f À 1Þ ðm À n À 2Þ! than unity, indicating that LPB aggregates in cyclohexane. 2 The strong c dependences however arise from the virial Âðn þ 1ÞþhS iu terms because cyclohexane is a good solvent for PB, and we
244 #2009 The Society of Polymer Science, Japan Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 Macromolecular Assemblies in Solution
Figure 2. Static light scattering results for three polybutadiene living anion samples in cyclohexane; solid curves, theoretical values calcu- Figure 3. Dynamic light scattering results for three polybutadiene living À1=2 lated by eqs (2.1)–(2.3) along with eq (1.4); dotted curves, mw anion samples in cyclohexane; solid curve, theoretical values calculated by the same equations with the zero virial coefficients. calculated by eqs (2.4) and (2.5). Reproduced with permission Reproduced with permission from ref 21. Copyright (2007) Amer- from ref 21. Copyright (2007) American Chemical Society. ican Chemical Society. have to analyze the light scattering data considering the from eqs (1.4) and (2.3) as a function of or c (see contribution of the virial terms. Using the third virial THEORETICAL SECTION). approximation, we have28 Eqs (2.1)–(2.3) contain two adjustable parameters, the optimum aggregation number m and the free energy " of KcM1 1 0 0 ¼ þ 2A2,zðM1cÞ the micellar formation per monomer and in units of the thermal R0 mw ð2:1Þ energy kBT. (Here, in eq (1.6) is assumed to be so large that 2A3,z þ A3,w 2 2 2 þ À mwðA2z À A2,z Þ ðM1cÞ the aggregation number of the micelle is monodisperse.) A best M1 1=2 fit to the experimental ðKcM1=R0Þ for LPB micellar solutions whereP mw is the weight average aggregation number was obtained when m0 was chosen to be 4 for all three LPB 2 (¼ m1 mwm), and A2,z, A2z, A3,z, and A3,w are average samples. Solid curves in Figure 2 indicate theoretical values second and third virial coefficients defined by calculated from eqs (2.1) and (2.2) using m0 ¼ 4 and 8 X X "0 ¼À14, demonstrating good fittings for all three LPB > A m À1 mw A ; A2 m À1 mw A 2 <> 2,z w m mm 2z w m mm samples. On the other hand, dashed curves in the same figure m1 m1 X X ð2:2Þ indicate ðKcM =R Þ1=2 after subtracting the virial terms, i.e., > À1 1 0 :> A3,z mw mwmAmmm; A3,w wmAmmm 1=2 1=mw . From the results, mw is so close to 4 in the c range m 1 m 1 investigated that most of LPB anions form reversed micelles with the second virial coefficient Amm and the third virial and the dissociation little takes place. In such a c range, 1=2 coefficient Ammm of m-mer. theoretical ðKcM1=R0Þ little changes even if we choose a As demonstrated below, the LPB aggregate is a reversed star value of "0 less than À14, so that the value À14 should be micelle, and Amm and Ammm of the star micelle (or the star taken as the upper limit of "0. It was difficult to make light polymer) can be calculated from the corresponding coefficients scattering measurements at lower c because of low light A2;m(linear) and A3;m(linear) for the linear chain with the molar scattering intensities. mass mM1 using relations Amm ¼ gA2;mA2;m(linear) and To investigate the micellar structure of LPB, we have 26,39,40 Ammm ¼ gA3;mA3;m(linear). From literature data, we have estimated the apparent hydrodynamic radius RH,app at finite the following empirical equations of the g-factors for flexible concentrations divided by the weight-average molar mass star polymers in good solvents: gA2;m ¼ 1 À 0:04ðm À 2Þ Mw for the major component, from the first cumulant À and gA3;m ¼ 1 À 0:08ðm À 2Þ. Furthermore, we can write obtained by dynamic light scattering and Kc=R0 using the 2 43,44 A3;mðlinearÞ¼ A2;mðlinearÞ Mm where is empirically known equation to be 0.3 for linear flexible polymers in good solvents.41 RH,app kBTð1 À cÞ Kc=R0 42 ¼ ð2:4Þ Literature data are available for A2;m(linear). Under the same 2 Mw 60 lim À=k third virial approximation, eq (1.5) is rewritten in the form k!0 where is the partial specific volume, 0 is the solvent Km ¼ m exp ÀÁm þ mM1ðA11 À AmmÞc viscosity coefficient, and k is the magnitude of the scattering ð2:3Þ vector. Figure 3 shows RH,app=Mw for cyclohexane solutions of 1 2 þ mM1ðA111 À AmmmÞc the three LPB samples plotted against cM1. The strong c 2 dependences come from the hydrodynamic interaction among The weight fraction wm of m-mer can be calculated polymers.
Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 #2009 The Society of Polymer Science, Japan 245 T. SATO and Y. MATSUDA
For micellar solutions of finite concentrations, RH,app=Mw may be calculated by21 RH,app 0 à w1M1 wm0 Mm0 ¼ð1 þ kHc=c Þ þ ð2:5Þ Mw RH,1 RH;m0 0 where k H is the strength of the intermolecular hydrodynamic à interaction, c is the overlap concentration, and RH,1 and RH;m0 are hydrodynamic radii of unimer and m0-mer, respectively. The overlap concentration may be calculated by cà ¼ 3 3Mw=4NARH where NA is the Avogadro constant and À1 À1 RH ¼ Mw=ðw1M1RH,1 þ wm0Mm0RH;m0 Þ. Using literature data of the hydrodynamic radius for linear PB in cyclohexane42 and of its g-factor for 4-arm star polymers in a good solvent 26 0 (0.93) as well as reasonable values for k H, we can draw the lines shown in Figure 2. The favorable fitting of RH,app=Mw demonstrates that LPB exists as a four-arm star micelle in cyclohexane. On the assumption that only free LPB (unimer) is the active species for the propagation reaction, the propagation rate is calculated by d lnð½M=½M Þ À 0 ¼ k w C ð2:6Þ dt p 1 0 Figure 4. (a) Propagation reaction rate of the polybutadiene living anion where [M] and [M]0 are molar concentrations of the butadiene in cyclohexane determined by Johnson and Worsfold46 and Spirin monomer at time t and t ¼ 0, respectively, C is the total molar et al.;45 solid curve, theoretical values calculated by eq (2.6) along 0 with eqs (1.4) and (2.3); (b) Non-single-exponential decay curves concentration of LPB (i.e., the molar concentration of the of butadiene monomer concentration [M] divided by the initial concentration [M] during propagation reaction of the living initiator), kp is the rate constant of the propagation reaction for 0 anionic polymerization, calculated with w1 from eq (2.3) with À5 3 À2 free LPB, and w1 is the weight fraction of the free LPB "0 ¼À14, A11 ¼ 3:5  10 cm mol g , A44 ¼ 0, A111 ¼ 5  À5 6 À3 À5 6 À3 calculated from the free energy " of the micellar formation. 10 cm mol g , and A444 ¼ 4  10 cm mol g under condi- 0 tions of Batch codes 9(0) and 10(16,0.12) in ref 37; the ideal Figure 4a compares literature propagation reaction rates of solution curve, obtained with all the virial coefficients to be zero. LPB in cyclohexane45,46 with eq (2.6) along with eqs (2.3) and (1.4) (and the normalization condition), using "0 ¼À14, and À1 À1 kp ¼ 133 M s . The reaction rate slightly depends on M1 due to the non-ideality of the solution at higher LPB AMPHIPHILIC TELECHELIC POLYMER concentration or C0. In Figure 4a, we have calculated the 4 theoretical reaction rate choosing M1 ¼ 10 . As mentioned Amphiphilic telechelic polymers, i.e., hydrophilic polymer above, the value À14 was the upper limit of "0, and thus the kP chains bearing hydrophobic groups on both ends, may be À1 À1 value 133 M s chosen is its lower limit. It is noted that kp regarded as model polymers forming flower micelles in of the free LPB is larger than that at the radical polymerization aqueous solution, where terminal hydrophobic groups are ( 20 MÀ1 sÀ1),47 and thus that the LPB anion is more reactive associated into a hydrophobic core and each hydrophilic than the polybutadiene radical. polymer chain constructs a loop.10,13 With increasing the Recently, Oishi et al.37 investigated the kinetics of living polymer concentration, however, such flower micelles may be anionic polymerization for butadiene in deuterated benzene interconnected by bridging chains with termini entrapped in the (a poor solvent for PB) by 1H NMR. They observed non- core of two distinct micelles, and exhibit the characteristic single-exponential decays of the monomer concentration [M] viscoelastic behavior.8,50–53 The association behavior of flower during the propagation reaction, and the deviation from the micelles is a delicate problem, because repulsive forces single exponential decay was enhanced by addition of inert between loops tend to prevent micellar association whereas deuterated PB. This kinetic behavior may be explained by non- the loop-to-bridge transition by the entropic penalty associated ideality of the polymerization solution, as demonstrated in with loop formation favors association.54,55 Figure 4b, where non-single-exponential curves were obtained Recently, Nojima et al.9 investigated the association behav- from eqs (2.6), (1.4), and (2.3) with reasonable values of virial ior of flower micelles formed by telechelic hydrophobically- coefficients given in the caption of Figure 4b; the chain-end modified poly(N-isopropylacrylamide) (HM-PNIPAM) in effect of the free LPB accounts for the positive A11 in a poor 25 C water by static light scattering. The sample used was solvent.48,49 The curves are favorably compared with Oishi PNIPAM with octadecyl groups at both ends and the weight- et al.’s experimental results shown in Figures 4 and 9 in average molecular weight of 47,500.52,56,57 Water is a good ref 37. solvent for PNIPAM at 25 C investigated. The weight-average
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Then next, assuming that the flower micelle is regarded as an f -functional monomer with the molar mass Mu and the 2 1=2 2 1=2 radius of gyration hS iu , we calculate Mw and hS iz by using eqs (1.10), (1.11), and (1.22) as functions of c.In eq (1.11), we can calculate F2ð Þ by eq (1.3) with ¼ c and the experimental partial specific volume (¼ 0:88 cm3/g) of 2 1=2 PNIPAM in water. Extrapolating Mw and hS iz to the zero c 5 in Figure 5a, we can estimate Mu ¼ 8:5  10 g/mol and 2 1=2 hS iu ¼ 12:8 nm for the unimer flower micelle, and the bond length b between the nearest neighboring flower micelles 1=2 2 1=2 approximately by the equation b 2 Âð5=3Þ hS iu (¼ 33 nm). Then the remaining unknown parameters in eqs (1.10), (1.11), and (1.22) are f and Áf0=kBT. The solid curve in Figure 5 indicates theoretical values calculated by these equations with f ¼ 3 and Áf0=kBT ¼À3:5, which are in a fairly good agreement with experimental results. If we choose 9 f ¼ 4, the fitting to the c dependence of Mw becomes slightly 2 1=2 worse. The theory seems to slightly underestimate hS iz at high c and Mw, which may come from the intra-assembly excluded volume effect. Equation (1.11) predicts the threshold concentration of gelation, corresponding to ¼ 1=ð f À 1Þ, to be 0.024 g/cm3. This prediction is consistent with Kujawa et al.’s report52 that aqueous solutions of a telechelic HM-PNIPAM with Mw ¼ 37;000 formed gels above c 0:02 g/cm3. (One may view each telechelic HM-PNIPAM chain as a bi-functional mono- mer and the hydrophobic core as multiple junction or crosslink Figure 5. (a) Concentration dependences of the molar mass and the radius of gyration and (b) the molar mass dependence of the radius of point, instead of the unit flower micelle as the f -functional gyration at different concentration for a telechelic HM-PNIPAM in monomer. In such a view, however, the theory explained in the 25 C water. Panel (b), reproduced with permission from ref 9. 59,60 Copyright (2008) American Chemical Society. above THEORETICAL SECTION cannot be applied. )
AMPHIPHILIC DIBLOCK COPOLYMER 2 molar mass Mw and z-average square radius of gyration hS iz were determined as functions of the polymer concentration c by The micellization behavior of amphiphilic diblock copoly- static light scattering, where virial term corrections were made mers in selective solvents has been investigated most exten- using theoretical second and third virial coefficients.9 Figure 5 sively among associating polymers for many years, and a large displays their concentration dependences as well as the plot of amount of experimental and theoretical work has been 2 1=2 2 1=2 4,6,7,61–66 hS iz against Mw. Both Mw and hS iz sharply increase accumulated so far. However, to our best knowledge, 2 1=2 with c, and hS iz exhibits a strong Mw dependence. quantitative arguments have not thoroughly carried out yet on If telechelic HM-PNIPAM chains are assumed to form uni- the detailed micellar structure using the radius of gyration and core flower micelles with an aggregation number m, the radius hydrodynamic radius obtained by light scattering. of gyration may be approximated to that of a star polymer Antonietti et al.67 and subsequently Fo¨rster et al.68 exten- which consists of 2m arms of half PNIPAM chains. The radius sively studied micelles of diblock copolymer consisting of 2 1=2 of gyration hS istar of this star polymer may be calculated by polystyrene (PS) and poly(4-vinylpyridine) (P4VP) blocks in 2 eq (1.12b) where hS ilinear for PNIPAM in 25 C water is organic solvents. They prepared a number of PS-b-P4VP 58 empirically expressed by samples with different degrees of polymerization of PS (N0,core) 2 1=2 0:54 and P4VP (N0,corona) blocks, and measured their molar masses hS ilinear ¼ 0:0188 nm  Mw ð3:1Þ 2 Ã1=2 Mw and light-scattering radii of gyration hS i of micelles and m is calculated from Mw/47,500. The dotted line in formed in selective solvents by light scattering. (Since PS 2 1=2 2 Ã1=2 Figure 5b indicates the result for hS istar . The experimental and P4VP are almost iso-refractive, hS i of their copoly- 2 1=2 2 1=2 data points remarkably deviate from hS istar for all solutions mers can be identified with true hS i .) They focused on and exhibit a much stronger molar mass dependence. This N0,core and N0,corona dependences of the aggregation number mw observation is a strong indication that the flower micelle calculated from Mw, and obtained the semi-empirical scaling association occurs in telechelic HM-PNIPAM solutions of c as relation68 low as 1  10À3 g/cm3 by the bridging attraction due to the m / N 2N À0:8 ð4:1Þ telechelic chain. w 0,core 0,corona
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eq (1.22) provides hS2iÃ1=2 in agreement with the experimental result. As expected, the brush heights Hi and Ho are larger than the average end-to-end distance hR2i1=2 of the PNIPAM block chain in water (¼ 8:8 nm)58 and smaller than hR2i1=2 at the full extension (¼ 44 nm). To increase the distance between neighboring brush chains, the PS block chain in the shell have to be compressed from the unperturbed state, so that the a value considerably smaller than hR2i1=2 of the PS chain in the state (¼ 10:7 nm),73 is not unreasonable. Eisenberg et al.6,7 found various morphologies including vesicles, tubules, and lamellae for crew-cut aggregates of asymmetric block copolymers PS-b-poly(acrylic acid) with 74 N0,corona=N0,core < 0:2 by TEM. Iyama and Nose also reported Figure 6. Comparison of experimental light-scattering radius of gyration vesicle and tubule formations by a PS-b-poly(dimethylsilox- h 2iÃ1=2 S for amphiphilic block copolymers in selective solvents ane) sample with N0,corona=N0,core ¼ 0:58. Therefore, the pos- with the theoretical hS2iÃ1=2 for the core-corona micelle; solid curve, eye guide. sibility of bilayer morphologies should be checked for asymmetric block copolymers with smaller N0,corona=N0,core. Here, we focus on their hS2iÃ1=2 data. Assuming that all PS- Equation (1.16) is useful for this check. b-P4VP samples they studied form core-corona micelles in toluene and THF (good solvents for PS), we calculate radii of THERMALLY-DENATURED DOUBLE-HELICAL gyration of their micelles using eq (1.16). Since we know the POLYSACCHARIDE 3 density core of P4VP (¼ 1:05 g/cm ) and the molecular weight 2 69,70 dependence of hS ilinear for PS in toluene and THF, we can Helical biopolymers (polysaccharides and proteins) are calculate hS2iÃ1=2 for the core-corona micelle of each PS-b- industrially important materials used as additives to foods, P4VP sample with no adjustable parameters (core ¼ corona ¼ detergents, cosmetics, etc. to control their rheological proper- for PS-b-P4VP). ties.75–78 A microbial polysaccharide, xanthan, is one of such Figure 6 compares the theoretical hS2iÃ1=2 calculated by biopolymers with versatile properties in aqueous media and eq (1.16) with the experimental data for PS-b-P4VP in toluene used in many areas of technology. and THF. Since eq (1.16) was derived on the basis of the The main chain of xanthan is the same as cellulose and every segment density distribution for star polymers, it should be other glucose residue possesses an ionic trisaccharide side more accurate for core-corona micelles of smaller core sizes, chain. The native xanthan is known to take a double-helical or at larger N0,corona=N0,core. In fact, ratios of the experimental structure and the double helix exhibits an order-disorder to theoretical hS2iÃ1=2 seem to approach to unity at large transition in aqueous solution by changing temperature and 79–82 N0,corona=N0,core in Figure 6 as shown by the eye guide curve. ionic strength. The disordered or denatured xanthan However, there are several points being appreciably deviated recovers its native circular dichroism (CD) by quenching and from unity in an irregular fashion, of which reasons are under adding salt to the solution, and in what follows we call this consideration. phenomenon as the renaturation of xanthan. However, the Recently, Zhang et al.71 investigated micelles of PNIPAM- viscosity of the renatured xanthan solution does not usually b-PS formed in water by transmission electron microscopy recover the native one,83 so that the local conformation of (TEM) and light scattering. From the TEM image, the authors renatured xanthan is similar to that of the native but its global judged that a PNIPAM-b-PS sample they investigated forms conformation is not necessarily identical with that of native the vesicle in water. On the other hand, using eq (1.16), we can xanthan. When the concentration of xanthan is high enough, the calculate hS2iÃ1=2 for the core-corona micelle formed by this renatured xanthan solution even forms a gel, which is another 3 72 84 PNIPAM-b-PS sample (core ¼ 0:25 cm /g and corona ¼ important property of xanthan in industrial applications. 0:162 cm3/g58). As shown by the filled circle in Figure 6, the Liu et al.85–87 carried out static light scattering to show that deviation from unity of the ratio of the experimental hS2iÃ1=2 the double helix of xanthan was unwound from the ends, but for the vesicle to the theoretical hS2iÃ1=2 for the core-corona did not dissociate into two dissociated single chains. Kawakami micelle is comparable to the data of PS-b-P4VP micelles with et al.88 demonstrated by sedimentation equilibrium and similar N0,corona=N0,core. viscometry that the denatured xanthan formed aggregates after We can also calculate hS2iÃ1=2 for the vesicle of Zhang renaturation, so that we can expect the mismatched winding of et al.’s PNIPAM-b-PS sample (N0,shell ¼ 207; N0,brush ¼ 176) partially unwound double helices as illustrated in Figure 1h. by eq (1.22) along with eqs (1.18)–(1.21). These equations Recently, Matsuda et al.14 characterized aggregates of the contain three adjustable parameters, a, , and mi=mo. Although renatured xanthan by size exclusion chromatography equipped it was difficult to determine those parameters uniquely, by with an online multi-angle light scattering detector (SEC- choosing a ¼ 1:9 nm, ¼ 0:0015, and mi=mo ¼ 0:3 (i.e., MALS). They prepared aqueous solutions of three xanthan À2 3 R ¼ 60 nm, Hi ¼ 18 nm, and Ho ¼ 23 nm) for example, samples listed in Table I with c ¼ 1 Â 10 g/cm , and heated
248 #2009 The Society of Polymer Science, Japan Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 Macromolecular Assemblies in Solution
Table I. Molecular weights and polydispersities of native xanthan samples used for the renaturation experiments
5 Sample Mw/10 Mw=Mn S14 14 1.6 S5 5.1 1.1 S1 1.2 <1:1
Figure 8. Molar mass dependences of hS2i1=2 for three renatured xanthan samples, S14, S5, and S1, with different molecular weights.
Figure 7. Molar mass distributions wðMÞ of native and renatured sample S14, obtained by SEC-MALS. the solutions at 80 C for 30 min. Aqueous NaCl was added to the denatured xanthan solutions to adjust the concentrations of xanthan and NaCl to 5  10À3 g/cm3 and 0.1 M, respectively. Those solutions were left at the room temperature for 24 h after the addition of NaCl, and then further diluted by 0.1 M aqueous NaCl to be used for SEC-MALS measurements. The aggrega- tion was stopped by this dilution. Circular dichroism demon- strated that the renatured xanthan recovered locally the original double helical structure. Figure 7 compares the molar mass distribution of a xanthan Figure 9. Schematic illustrate of a xanthan aggregate formed from m sample S14 after renaturation with that of the native sample, partially unwound double helices. where wðMÞ is the weight fraction of the xanthan component with the molar mass M, obtained by SEC-MALS. The helix and mismatched double helix portion as L1 and Lmis, distribution curve of the renatured sample shifts to higher respectively. The square radius of gyration of the m-mer can be 2 M region, and becomes broader than that of native one, given by eq (1.25). In the equation, b and hS iu may be demonstrating the aggregation of the denatured xanthan dimer. calculated by Figure 8 shows the molar mass M dependences of the radius b2 ¼hR2iðð1 À f ÞN Þ; hS2i ¼hS2ið f N Þð5:1Þ of gyration hS2i1=2 obtained by SEC-MALS for fractions of the mis 1 u mis 1 renatured xanthan samples S14, S5, and S1. Data points for all where hR2iðNÞ and hS2iðNÞ are the mean square end-to-end the fractions do not agree with that for the native xanthan distance and the mean square radius of gyration of the double helix indicated by the dot-dashed curve, demonstrating wormlike chain, respectively, as a function of the Kuhn that the renatured samples do not recover the original double statistical segment number N, fmis is the mismatch fraction helix. This is in a contrast with the CD result almost recovering ( 2Lmis=L1), and N1 is the Kuhn statistical segment number of the native spectrum after renaturation. the native xanthan sample ( L1=2q; q: the persistence length). From the above SEC-MALS and CD results, we can Kratky and Porod89 and Benoit and Doty90 presented hR2iðNÞ imagine that denatured xanthan dimers form aggregates by the and hS2iðNÞ, respectively, for the wormlike chain model in the mismatched winding as illustrated in Figure 1h. This aggrega- forms tion can be viewed as the polycondensation of monomers 1 indicated by circles in Figure 9, i.e., m partially unwound hR2iðNÞ¼ð2qÞ2 N À ð1 À eÀ2N Þ ð5:2Þ 2 xanthan double helices form (m þ 1)-mer. The monomers 1 1 1 1 shown in Figure 9 are bi-functional ( f ¼ 2) or tri-functional hS2iðNÞ¼ð2qÞ2 N À þ 1 À ð1 À eÀ2N Þ ð5:3Þ ( f ¼ 3). We denote the contour lengths of the intact double 6 4 4N 2N
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Solid and dotted curves in Figure 8 indicate theoretical 7. D. E. Discher and A. Eisenberg, Science, 297, 967 (2002). values calculated by the polycondensate model for f ¼ 2 and 8. B. Xu, L. Li, A. Yekta, Z. Masoumi, S. Kanagalingam, M. A. Winnik, K. Zhang, P. M. Macdonald, and S. Menchen, Langmuir, 13, 2447 3, respectively. Here, N1 was calculated from Mw for each (1997). À1 sample (cf. Table I) with ML ¼ 1940 nm and q ¼ 100 nm, by 9. R. Nojima, T. Sato, X. Qiu, and F. M. Winnik, Macromolecules, 41, neglecting polydispersity in the molecular weight. For all 292 (2008). xanthan samples, experimental data points are almost fitted to 10. A. N. Semenov, J.-F. Joanny, and A. R. Khokhlov, Macromolecules, 28, 1066 (1995). the solid curve with fmis indicated in the figure. The mismatch 11. O. V. Borisov and A. Halperin, Langmuir, 11, 2911 (1995). fraction fmis chosen increases with Mw of the sample. The 12. O. V. Borisov and A. Halperin, Macromolecules, 29, 2612 (1996). fitting is slightly worse if one chooses f ¼ 3. The junction 13. A. Halperin, ‘‘Supramolecular Polymers,’’ A. Ciferri, Ed., Marcel for f ¼ 3 must be formed only at simultaneous gathering of Dekker, New York and Basel, 2000, p. 93. three denatured xanthan dimers which may seldom occur in 14. Y. Matsuda, Y. Biyajima, and T. Sato, Polym. J., submitted (2008). relatively dilute solution. 15. T. Kimura, A. Hashidzume, and T. Sato, Prog. Theor. Phys., in press (2008). 16. F. Tanaka, Macromolecules, 22, 1988 (1989). SUMMARY 17. T. Sato, Langmuir, 20, 1095 (2004). 18. P. J. Flory, ‘‘Principles of Polymer Chemistry,’’ Cornell Univ. Press, Light scattering studies on the following macromolecular Ithaca, New York, 1953. assemblies were overviewed: 19. T. Sato, Y. Jinbo, and A. Teramoto, Macromolecules, 30, 590 (1997). 20. J. N. Israelachvili, ‘‘Intermolecular and Surface Forces,’’ 2nd ed, (1) Polybutadiene living anions in cyclohexane. 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Takahiro SATO was born in Osaka, Japan, in 1957, graduated from the Department of Macromolecular Science, Osaka University in 1980, and received his PhD from Osaka University in 1985. He started working in 1985 as a guest scientist at National Bureau of Standards (now National Institute of Standards and Technology) in USA, and then joined the Department of Macromolecular Science, Osaka University as an assistant professor in 1987. He promoted to an associate professor in 1996 and a full professor in 2002 at Osaka University. He received the Award of the Society of Polymer Science, Japan in 2008. His major research field is polymer solution properties, and he is interested in interrelations among the conformation, interaction, higher-order structure, and various solution properties of stiff-chain polymers, helical polymers, and associating polymers.
Yasuhiro MATSUDA was born in Shizuoka, Japan in 1979, graduated from Department of Chemistry, Osaka University in 2002, and received his PhD from Osaka University in 2006. He started working in 2006 as a postdoctoral fellow at Institute for Materials Chemistry and Engineering, Kyushu University, and then joined the Department of Materials Science and Chemical Engineering, Shizuoka University as an assistant professor in 2007. His present research interest is relations between solution properties and surface and interfacial properties of various polymers.
Polymer Journal, Vol. 41, No. 4, pp. 241–251, 2009 #2009 The Society of Polymer Science, Japan 251