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Macromolecular Assemblies in Solution: Characterization by Light Scattering #2009 The Society of Polymer Science, Japan AWARD ACCOUNTS SPSJ Award Accounts Macromolecular Assemblies in Solution: Characterization by Light Scattering By Takahiro SATO1;à 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 ðÞþF ðÞð1:2Þ 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, F1ðÞ and F2ðÞ 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 F1ðÞ and F2ðÞ, 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 F2ðÞ¼1, 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 32 1 þ 2fX À 1 þ 4fX F2ðÞ¼ À lnð1 À Þð1:3Þ ¼ ð1:11Þ 3ð1 À Þ2 2fX Inserting eq (1.1) into the equilibrium condition, we obtain with X exp½F2ðÞÀÁf0=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 ðÞ 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.
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