Rouse-Segmental Motion as Probed by Depolarized Photon-Correlation Spectroscopy The usual mode of photon-correlation spectroscopy )self-beating)is based on the

condition that the scattered light field obeys Gaussian statistics.^118,119 This makes it particularly suitable and popular for probing dynamics in systems “populated”by Brownian particles as exemplified by the numerous studies of polymer chain dynamics in solutions.^27,120 Depolarized dynamic light scattering being much affected by the fast fluctuations of polarizability anisotropy, it is expected that depolarized photon-correlation spectroscopy mainly probes the reorientation motion of a correlated region.¹²¹ Since the Rouse segment is the most basic Brownian particle in the Rouse model, which describes very well the polymer viscoelastic behaviour over at least the intermediate- and long-time regions of an entanglement-free concentrated system,^4,14,15the depolarized photon-correlation function may provide the information about the motion of a single Rouse segment. Such an expectation is borne out by recent studies^9,10,21-23as summarised below:

Depolarized photon-correlation spectroscopy was first used to study the chain dynamics in a well-entangled polystyrene melt by Patterson et al.¹³ It was later pointed out by Lin^9,10that the average correlation timecobtained by Patterson follows the same temperature dependence as that of viscosity of nearly monodisperse polystyrene samples obtained by Plazek and

ORourke¹²²from 130 to 110^oC (see Figure 5; the correction for the changes in density and temperature as made in the figure causes only a negligible difference). Thecvalue changing by a factor as large as 356 over this temperature range, the agreement is significant, suggesting strongly that the observed time constant is basicallyv)i.e. of the same order of magnitude)as given by eq 7, which shares the same frictional factor K as that of viscosity (eq 6).

K of polystyrene at 127.5^oC is given by the average value listed in Table 1 of ref. 11 to be 5.2x10^-910%. The structural factors of the relaxation times of the RouseMooney normal modes {Ap

} are independent of molecular weight. At the same time, if the molecular weight is sufficiently high, K/K is at the plateau value 3.3 based on eq 8 of ref. 11. The polystyrene sample studied by Patterson et al. was prepared by thermal polymerization at 90^oC. Under such a condition, its number-average molecular weight is expected to be around 400,000;¹²³ in other words, it is in the highly entangled region where the plateau value of K/K is applicable, even though its molecular-weight distribution is not nearly monodisperse. Thus, we can use the above K value at 127.5^oC and the ratio K/K =3.3 to obtain K. As explained in ref. 11, the mass of a Rouse segment of polystyrene, m, being about 850,6-12, 21-23,124,125

leads to Ne=16. Using the value of K’obtained as described above and Ne=16 or equivalently m=850 we can calculate

vA15

from eq 7 or eq 5 (with K substituted by K; and Nrreplaced by Ne=16) to be 5.1x10^-3 sec, which, clearly as expected, is of the same order of magnitude as thecvalue at 127.5^oC, 3.5x10^-3, obtained from Pattersons results by interpolation.

In the case of polystyrene, it has been shown that the effective optical anisotropy per monomer unit from polystyrene in melt and in solution (cyclohexane as the solvent, whose

depolarized light scattering is negligible) is the same^126,127indicating that the static correlation between segments belonging to different chains is nil. And the dynamic pair correlation is in general much smaller than the static pair correlation.^29,128 On the basis of neglecting both the static and dynamic pair correlation among segments belonging to different chains, and assuming that the size of the polymer coil is much smaller than the scattering wavelength and that the collective reorientation time is much shorter than the time needed for the centre-of-mass of the polymer chain to travel the distance of a scattering wavelength, the time-correlation function for depolarized Rayleigh light scattering can be expressed as.^9,10,21-23

   

C t( )S f t_s( )R P₂ u( )t u( )0 (8)

where P2is the second-order Legendre polynomial; u(t) is the unit vector representing the direction of the symmetry axis of a correlated region)the whole region is regarded as a Kuhn segment or equivalently a Rouse segment¹²⁹)along the polymer chain at time t; fs(t) is a

normalized time-correlation function that reflects the motions associated with the local chemical bonds, which are grossly referred to as the sub-Rouse-segmental motion; the relaxation strength S depends on the details of bond angles and steric interactions among chemical bonds; and R is a constant that is related to how anisotropic the correlated region is.

The depolarized photon-correlation functions of two entanglement-free concentrated solutions (60wt%) of polystyrene with M_w=9100; M_w/M_n=1.02 and M_w=18100; M_w/M_n=1.01 in cyclohexane at thecondition, i.e. at 35^oC)denoted by samples S1 (59.832wt%; 0.552 g/cm³) and S2 (60.287wt%; 0.556 g/cm³), respectively)have been measured and analysed.^21-23 Along with the depolarized photon-correlation measurements, two solution samples with accurately determined concentrations in the close neighbourhood of the concentration of each of the two samples, S1 and S2, are prepared for viscosity measurements by the falling-ball method, which,

with both the ball and solution sealed in a glass tube, is particularly good for studying solution systems as solvent evaporation can be prevented. Then, by interpolation or extrapolation, the viscosity values at the concentrations of samples S1 and S2 can be individually determined;

subsequently, theirvvalues can be calculated (eqs 6 and 7) for comparison with their depolarized photon-correlation results. Furthermore, the obtained information of the concentration dependence of viscosity allows the viscosity results to be compared under the same concentration and can be used to correct for the small concentration difference between samples S1 and S2 when their depolarized photon-correlation results are compared. The

discussions below are all based on the results after the corrections have been made; the details of the corrections can be found in ref. 21.

The obtained molecular-weight dependence of viscosity at the same concentration (60 wt%) indicates that the Rouse theory is applicable; in other words, the concentrations of the studied polystyrene solutions are high enough to screen out the hydrodynamic interactions.²This conclusion is further confirmed by analyses in terms of the Rouse theory in other aspects of experiments as will be described below. Through the multi-exponential singular-value decomposition (MSVD) analysis,²⁷ a bimodal relaxation-time distribution can clearly be obtained from the depolarized photon-correlation functions of both S1 and S2, as corresponding to the two modes of motion in eq 8. Because of the limitation of the time window of photon-correlation spectroscopy, only the tail region of the fast mode fs(t) can be observed. Thus, as far as the fast mode is concerned, one can only show its existence from the MSVD analysis.

However, much information about the slow mode P2[u(t)u(0)]has been obtained from the analysis of the experimental results.^21-23 It has been shown that the slow mode, with a rather narrow relaxation-time distribution)extending over slightly less than two decades, is

independent of scattering angle and molecular weight in accordance with eq 8. In the polystyrene melt case, the depolarized photon-correlation function is well described by the stretched exponential form with the stretching exponentnear 0.4. This corresponds to a

unimodal broad relaxation-time distribution, covering more than five decades.¹³⁰ The fact that the two modes of motion as contained in eq 8 cannot be separated in melt as in the concentrated-solution case is explained as due to the stronger interactions among segments causing the two modes to overlap extensively.

Assuming u(t)=b(t)/b(t), the time-correlation functionP2[u(t)u(0)]can be calculated by the Monte-Carlo simulation based on the Langevin equation of the Rouse model.^4,22 Also, from the simulation, the ratio betweenv (corresponding to eq 7) and the average correlation timerobtained from integrating the simulatedP2[u(t)u(0)]curve can be calculated for comparison with the experimental results2/v (2denotes the average correlation time of the slow mode obtained from resolving the measured photon-correlation function, whilevis calculated from the viscosity data through eqs 6 and 7). In comparing the analyses of the depolarized photon-correlation function, viscosity and Monte-Carlo simulation results, we have found that m=1130 gives a good overall agreement: Corresponding to m=1130, Nr=8 and 16 for samples S1 and S2, respectively. From the results of the depolarized photon-correlation function and viscosity, we obtained2/v=2.4 and 2.6 for samples S1 and S2, respectively; from the simulation, we obtainedr/v=2.2 and 2.5 for Nr=8 and 16, respectively.

Furthermore, as shown in Figure 1, the line shapes of the time-correlation functions of the slow mode of both samples S1 and S2 (denoted byC2(t)) are in close agreement with the simulation results of P₂[u(t)u(0)]for N_r=8 and 16. Thus, in spite of the crudeness of the Rouse segment, the effect of chain connectivity as contained in the Rouse model can quite fully account for the detailed aspect of the dynamics as showing up in the depolarized photon-correlation function and its relation with viscosity, supporting the physical picture that the dynamic process probed by depolarized photon-correlation spectroscopy is the reorientation motion of a Rouse segment.

The mass of a Rouse segment obtained for the studied concentrated polystyrene solutions, m=1130, is about 25% larger than that in the melt. This small difference should be due to the presence of solvent; indeed, the small solvent-enhancement effect is about that expected from the

concentration-dependence of the Rouse segment size obtained by Inoue et al.¹²from analyzing the dynamic mechanical and birefringence results)the expected m value at the studied

concentration is about 1100, versus 850 in the melt (see Figure 10 of ref. 12). The agreement between the two independent studies based on very different premises¹³¹reconfirms that the Rouse segment size can be defined and that the motion associated with a single Rouse segment can indeed be studied; in other words, the study of the Rouse-segmental motion as presented above is well supported.

In summing up the above studies of polystyrene melt and concentrated solutions, we can notice differences and common points: The differences between the melt case and the

concentrated-solution case are mainly two: (1) The relaxation-time distribution is much broader in the former than in the latter, and (2) thecvratio is smaller in the former than in the latter (denoted by2vin the latter case). These two differences can be accounted for by the stronger interactions among segments in the melt)in the concentrated-solution case, the

interactions among segments can be much reduced by the “lubrication”of the solvent molecules.

Due to the stronger interactions in the melt case, the fast and slow modes as contained in eq 8 overlap extensively; the photon-correlation function cannot be resolved into the two modes.

While the effect leads to a broad unimodal relaxation-time distribution,^13,38the fast component in the distribution also causes the observed average relaxation timecto be smaller than when only the slow component contributes to it as in the concentrated-solution case. The main shared common point is the applicability of the Rouse model)either asR(t/R) or asA(t/A), which is a part of ERT)in relating the viscoelasticity results to the dynamics observed by depolarized photon-correlation spectroscopy. As the melt system and the concentrated-solution system at the

point are very similar dynamically and thermodynamically)both free of the hydrodynamic interactions and excluded-volume effect,²the precise analysis achieved in the concentrated-solution case lends additional support to the analysis of the melt results, in which some of the details are prevented by the much broader relaxation-time distribution in C(t) from being

revealed.

In summary, the recent studies as briefly described above confirm the initial expectation that the motion of a single Rouse segment can be studied by depolarized photon-correlation spectroscopy. This conclusion has a bearing on the comparison of therelaxation with the highest RouseMooney normal mode, both extracted from the creep compliance J(t) as reported in ref. 11.