Polymer Journal, Vol. 37, No. 2, pp. 65–73 (2005) Mutual- and Self-Diffusion Coefficients of a Semiflexible Polymer in Solution yy y Takayuki KANEMATSU,1; Takahiro SATO,1; Yasuhito IMAI,2 Koichi UTE,2 and Tatsuki KITAYAMA2 1Department of Macromolecular Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka 560-0043, Japan 2Department of Chemistry, Faculty of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan (Received September 1, 2004; Accepted October 27, 2004; Published February 15, 2005) ABSTRACT: Mutual- and self-diffusion coefficients of a semiflexible polymer, cellulose tris(phenyl carbamate) (CTC), in tetrahydrofuran were measured by dynamic light scattering and pulsed field gradient NMR, respectively, as functions of the polymer concentration and molecular weight. The mutual-diffusion coefficient after elimination of the effects of thermodynamic force and solvent back flow agrees with the self-diffusion coefficient for a low molecu- lar weight CTC fraction with the number of Kuhn’s statistical segments N ¼ 1:8 up to high concentrations, but they disagree for a higher molecular weight CTC fraction with N ¼ 4:7 at the highest concentration investigated. The mutual diffusion coefficients for CTC fractions with N ranging from 1.8 to 10.6 after elimination of the above two effects were also compared with the fuzzy cylinder theory for the self-diffusion coefficient. Disagreements start at lower concentra- tion for larger N, which form in a contrast with good agreements for a more stiff polymer, poly(n-hexyl isocyanate), previously studied. [DOI 10.1295/polymj.37.65] KEY WORDS Mutual-Diffusion Coefficient / Self-Diffusion Coefficient / Dynamic Light Scatter- ing / Pulsed Field Gradient NMR / Semiflexible Polymer / Cellulose Derivative / There are two kinds of translational diffusion coef- force and the solvent back flow discriminate Dm from ficients for binary solution systems: the mutual-diffu- Ds, and the two effects may be eliminated by use of 3 sion coefficient Dm and the self-diffusion coefficient the following equation: D .1–3 The conventional concentration gradient meth- s D~ D =½ðM=RTÞð@Å=@cÞð1 À vvc Þ ð1Þ od provides the former diffusion coefficient. Dynamic m light scattering is a more convenient method to deter- where M, c, and v are the molecular weight, the mass mine Dm, where the concentration gradient is generat- concentration, and the partial specific volume of the ed by thermal fluctuation. On the other hand, a certain solute, respectively, RT is the gas constant multiplied solute molecule is labeled, and the Brownian motion by the absolute temperature, and Å is the osmotic of the labeled molecule (the tracer) is monitored by pressure. The effects of the thermodynamic force some method to measure the latter diffusion coeffi- and the solvent back flow are taken into account by cient Ds. Forced Rayleigh scattering, fluorescence re- the factors ðM=RTÞð@Å=@cÞ and 1 À vvc , respectively. covery after photobleaching, pulsed field gradient This new diffusion coefficient D~ exhibits a negative NMR, and so on belong to techniques to measure concentration dependence, and is comparable with 3,4 Ds. These two diffusion coefficients are identical the self-diffusion coefficient Ds. at infinite dilution, but they often exhibit opposite Kitchen et al.5 and more recently Le Bon et al.6 concentration dependencies. showed that D~ agreed well with Ds for aqueous solu- The mutual and self diffusions occur under different tions of globular proteins up to considerably high con- experimental conditions. While the solution is thermo- centrations. Ohshima et al.7 demonstrated that exper- dynamically homogeneous during the measurement of imental results of D~ for semidilute solutions of a stiff Ds, the concentration gradient is necessary to measure polymer poly(n-hexyl isocyanate) (PHIC) were favor- Dm. The concentration gradient provides a thermody- ably compared with a theory for Ds. However, Calla- namic force to make a macroscopic flow of the solute ghan and Pinder8 and also Brown et al.9,10 reported at a finite concentration, and the solute flow is accom- that D~ and Ds were not identical for flexible polymer panied by the solvent back flow to maintain the con- solutions. Moreover, de Gennes1 pointed out that the stant density of the solution. The thermodynamic mutual diffusion in entangled polymer solutions might yTo whom correspondence should be addressed (Tel&Fax: +81-6-6850-5461, E-mail: [email protected]). yyPresent Address: Dainippon Ink and Chemicals, Inc., Kansai R & D Center, 1-3 Takasago, Takaishi 592-0001, Japan 65 T. KANEMATSU et al. Table I. Molecular characteristics of cellulose tris(phenyl carbamate) fractions used12 cÃc D k Fraction M =104 M =M DSa Nb 0 D w w n (g cmÀ3) (10À7 cm2 sÀ1) (cm3 gÀ1) F20 3.94 1.06 1.80 0.028 13d F19 4.06 1.08 2.8 1.86 0.027 8.55 13 F17 6.42 2.94 0.016 6.72 23 F16 10.3 1.08 4.72 0.0091 4.50 35 F14 15.0 1.05 3.0 6.87 0.0065 3.85 39 F12 23.2 1.07 3.1 10.6 0.0041 2.89 51 aDegree of substitution estimated by elemental analysis. bNumber of Kuhn’s statistical segments calculat- À1 ed from Mw with the molar mass per unit contour length ML ¼ 1040 nm and the persistence length c à 2 3=2 q ¼ 10:5 nm. Overlap concentration calculated by c ¼ 3Mw=4NAhS i , where NA is the Avogadro con- stant, and hS2i is the mean-square radius of gyration determined previously by light scattering (fractions F16, F14, and F12) or calculated by the Benoit–Doty equation31 with the above q and N (for fractions F20, F19, and F17). dAssumed value. take place without disentanglement, which is necessa- (THF), calculated from Mw by using the molar mass À1 ry in the case of the self diffusion. Nonequilibrium per unit contour length ML ¼ 1040 nm and the per- statistical mechanics verifies the non-identity of the sistence length q ¼ 10:5 nm determined previously,12 two diffusion coefficients,11 but cannot answer how as well as the overlap concentration cÃ. much the difference is. The present study was undertaken to elucidate the Pulsed Field Gradient NMR (PFG-NMR) relation between the mutual- and self-diffusion coeffi- NMR measurements were performed for fractions cients in entangled polymer solutions. The previous F16 and F20 dissolved in THF-d8 at 25 C on a Varian studies imply that the polymer chain stiffness is a Unity-INOVA 750 spectrometer operating at a proton key parameter in that relation. Thus, we chose as a test NMR frequency of 750 MHz using an 1H{15N–31P} 5- sample a cellulose derivative, cellulose tris(phenyl mm indirect probe (fraction F20) and an 1H{13C/15N} carbamate) (CTC), with a chain stiffness intermediate 5-mm triple resonance probe (fraction F16) with between typical stiff and flexible polymers. This poly- pulsed field gradient coils. Test solutions in precision mer was molecularly well characterized in the previ- co-axial tube inserts of 2 mm o.d. (cf. Figure 1a) were ous study.12 By dynamic light scattering, static light set in the spectrometer, and the DBPPSTE (Dosy scattering, and pulsed field gradient NMR (PFG- NMR), we have determined Dm, @Å=@c, and Ds, re- spectively, as functions of the concentration and mo- lecular weight of CTC, and compared experimental 13,14 D~ and Ds, as well as the fuzzy cylinder theory for Ds which was favorably compared with D~ for PHIC solutions.7 EXPERIMENTAL CTC Fractions Among CTC fractions used in previous studies,12,15 six fractions were chosen for diffusion measurements. The preparation and fractionation methods of the frac- tions were described in a previous paper,12 and molec- ular characteristics previously obtained by light scat- tering and SEC are listed in Table I. The full sub- stitution of CTC fractions was checked previously by elemental analysis.12 Ratios of the weight- to number- average molecular weight Mw= Mn estimated by SEC are less than 1.1, guaranteeing narrow molecular weight distributions of the fractions used. Table I also contains the number of Kuhn’s statisti- Figure 1. (a) Precision co-axial tube set and (b) the DBPPSTE cal segments N of each fraction in tetrahydrofuran pulse sequence used for PFG-NMR. 66 Polym. J., Vol. 37, No. 2, 2005 Mutual- and Self-Diffusion Coefficients in Solution Table II. Experimental conditions in PFG-NMR and results of the self-diffusion coefficient obtained D =10À7 cm2 sÀ1 D =10À7 cm2 sÀ1 Fraction c=10À2 gcmÀ3 Number of g/g cmÀ1 Á/s s,w s,z transientsa DECRA CONTINc MEMc CONTIN MEM F20 7.40 16 14–60 0.80 1.99 2:0 Æ 0:52:1 Æ 0:5 1.85 1.72 4.70 16 14–60 0.50 3.42 3:4 Æ 0:23:5 Æ 0:6 3.40 2.44 2.52 64 20–60 0.20 5.20 5:3 Æ 1:25:2 Æ 1:5 4.92 3.82 0.940 128 20–60 0.20 7.16 7:5 Æ 2:18:0 Æ 2:1 6.67 6.82 0.381 128 20–60 0.20 8.30 9:0 Æ 3:18:4 Æ 1:8 7.45 7.84 0.134 128 20–60 0.20 8.81 10:1 Æ 4:68:7 Æ 1:3 7.42 8.39 F16 4.93 256 20–60 2.30 0.75 0:79 Æ 0:20:75 Æ 0:8 0.750 0.749 2.83 128 20–60 1.00 1.51 1:5 Æ 0:41:5 Æ 0:5 1.45 1.42 1.03 512 20–60 0.60 2.74 2:8 Æ 0:42:8 Æ 0:3 2.72 2.79 0.313 128 20–60 0.55 4.03 4:4 Æ 1:1 NCb 3.97 NCb 0.140 512 20–60 0.50 3.93 5:3 Æ 3:13:5 Æ 0:5 3.31 3.44 aRecycle delay, 14 s.