Volume 19, number 3 CHEMICAL PHYSICS LETTERS I April 1973

INTERNAL MOTION IN CHAIN POLYMERS

T.A. KING, A. KNOX and J.D.G. McADAM Physics Department, Schuster Laboratory, University of Manchester,

Manchester MI3 9PL. UK

Received 6 February 1973

Internal motion in polystyrene chain polymers with molecular weights greater than IO6 has been masured from the linewidths of the light scattered Rayleigh components. The relaxation time of this motion is around LO4 set for low viscosity solvents and is lower than but in the same region as the fundamental mode of the Rouse-Zimm modsl.

We wish to report the quantitative observation of

internal motion in a chain polymer deduced from the analysis of the lineshape of the Iight scattered Rayleigh component. The observational methods used here have the advantage of not perolrbing the molecular system. Previous investigations using Rayleigh linewidth tech- riques [I] have shown evidence for internal motion but without quantitative analysis.

The dynamics of polymer systems are of fundamen-

tal interest. The early discussions of molecular motion by Kirkwood and Riseman [2], Rouse [3] and Zimm [4] led to the descriptive bead-and-spring Rouse- Zimm model. This model has been extended by many other workers, as described by Fixman and Stock- mayer [5 J , to include excluded voIume and chain flexibility as well as hydrodynamic interaction. Ex- perimental investigations of internal motion, for example by oscillatory flow birefringence 163, has shown that these models can provide a basis for the interpretation of molecular dynamical effects.

The precise nature of the molecular motion, such as the appropriateness of the descriptions along the lines of the bead-and-spring or continuous wire models and the detail of the chain motion, may be capable of test using the methods described here. We present some data on poiystyrene chains under non-theta conditions in a Iow viscosity solvent, butan-2-one. The internal relaxation times are near 1W4 set and are in the region of thedretical estimates. Also some

data on rotational relaxation of tobacco mosaic virus is presented as an indication of the vaIidity of the experimental method.

Light scattered from a fluctuation in dielectric constant can undergo a shift in frequency if the fluc- tuation changes in time. The average Lineshape, which is characteristic of the ensemble average motion of the scattering system, can contain contributions from translational, rotational and internal motions. To de-

termine the linewidth of the scattered light in these studies homodyne (intensity fluctuation spectroscopy) techniques are used and the frequency components in the photomultiplier detected signal determined by time autocorrelation.

An illustration of the instrumentation is shown in fig. 1. Laser light is focused on to the cIean sample contained in a temperature controlled cell. The scattered light is collected at a determined and vari- able angle by a photomultiplier which is,apertured by a pinhole. In the work reported here a 60 nW Spectra- Physics model 125 He-Ne laser operating at 6328 A and a Hewlett-Packard model 3721 A correiator have been used. The stored autocorreIation function can be displayed, X-Y recorded or punched on paper

tape for computer analysis. For a molecule of a few thousand A length showing

both translational and rotationd or conformatiarial internal motion the autocorrelation function of the photocurrent C,(T) is to a good approximation of the

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Volume 19; number 3, CHEMICAL PHYSICS LETI-ERS 1 April 197 3

Sample

Laser

w

Fig. 1. Ray!eigh linewidth spectrometer.

form

C&r) = A 2 exp (-2K2Dr)

(1) +245exp [-(2K2D+$.)r] +B2 exp [-(2K2D+2Pt)r].

A and B are characteristic of the amplitudes of the contributions from translational and rotational or in- ternal motion. K, the scattered wavevector, becomes for incident light having a vacuum wAvelength 1, in a medium of refractive index II at a sccrtering angle O,, K = 4ml sin (46,)/&,. D is the translational diffusion coefficient of the molecular centre-of-mass motion giving rise to an optical linewidth K2D and a homo- dyne linewidth 2K2D. Pi is the linewidth associated with molecuiar motion other than translational dif- fusion such that

translational motion the photocurrent autocorrelation function will show a single exponential form and will

yield a linear plot of linewidth with 2K2. If the addi- tional contribution to the linewidth from rotational or internal motion Pi is present deviations from the linear 2Kz plot wil! appear. Fig. 2 shows the angular dependence of the apparent linewidth for a sample of (rod-like) T.M.V. in a IO-3 M sodium phosphate buffer at a pH of 7.5, a concentration of 1.8 X 10m4 g ml-’ and at 25°C. It is seen that only at small angles is the dependence on 2X2 linear and the depar- ture from linearity is due to rotational diffusion

For a quantitative analysis the angular dependence graphs have not been used. The analytic form of Ci(r) is approximated as

Ci(r) = A2 exp (-2K2Dj + 2AB exp [-(2K’D+fi) r] r;.=6D,, (24

ri=2r;t ) WI

for a rotational diffusion coefficient D, or a relaxa- tion time Ti for one predominant internal mode.

Usually and in particular for the present experimen- tal conditions the amplitudes associated with the rota- tional or internal modes are such [7] that B <A and the form of eq. (1) reduces to the first two terms. We can qualitatively indicate the presence of molecular motion other than translational diffusion. For.purely

= A l exp (-r/r,) + A 2 exp (--T/rb) . (4)

For each measurement at each angle C~(T) is computer fitted to eq. (4). In this analysis r’, can be obtained from the dependence of C,(r) on angle at low angle and mu, A1 and A2 calculated by optimum fitting. Another method has also been used in which the ratio A4 I /A2 is calculated from the data of Pecora [7,8] for a rod or a random coil. This ratio and the value of T, from the low angle linewidth dependence

.352. . . . . ‘. _; :

(3)

Volume 19, number 3 CHEMICAL PHYSICS LETTERS 1 Ap;ilI973

5 10 15

2K2 (10” cm’)

Fig. 2. Angular dependence of the apparent Rayleigh line- width for T.&I .W.

are used to reduce the number of parameters in the fitting program to find rb.

For the T.M.V. data the computer fitting for C,(r) at low angles provides a value of D = (2.85 f 0.08) X 10e6 cm2 s ec-I corrected to 20°C. From the higher angle data using the derived value of D the value of Dn = 365 + 25 set-1 is obtained. This value is in agreement with the theoretical estimates assuming that T.M.V. is a cylindrical rod 3000 a long by 170 A diameter. A fuller discussion on rotational motion in a range of rod-like molecules will be presented for publication shortly.

For a polystyrene chain sample the angular depen- dence of the linewidth is shown in fig. 3. This sample has a weight average moiecular weight %,v = 8.7 X 10e,&?Wf~fi = 1.32 and is at a concentration of 10M4 g ml-’ in butan-Zone at 25°C. In fig. 3a the linear angular dependence at low angles shows the linewidth due to translational diffusion. In fig. 3b the total apparent linewidth over a wide angular range in-

‘.--::. 2K2(10S cm2:

3

(b)

5 10 15 2 K2(10’%m2)

Fig. 3. Angular dependence of the apparent Rayleigh !in=_ width for a polystyrene chain of nf, = 8.7 X 106. (a) Low angle scattering, (b) full angular dependence.

dicates an additional contribution from motion other than translational diffusion. Using the computer anal- ysis based on eq. (4) a mean value over many measure- ments of ~~ Sas been found of rb = 46.3 c 6 ,usec. Since sample polydispersity can influence the value of ri measurements on polydisperse samples have been carried out to estimate its magnitude. This has been found to be significantiy lower than the internal motion itself.

We can relate the observed motion to long wave internal motion on the.polyrner backbone or to a rotational diffusion. For the internal normal mode treatment of the polymer on the bead-and-spring

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Volume 19, number 3 CHEMICAL PHYSICS LJXTERS 1 April 1973

model of Rouse we content ourselves with the develop. ment of this model as far as the modifications of

D, = 9D/R22).

Zimm; for the case of non-free-draining with segment From eq. (7) we calculate DR = 1020 set-l . In com-

hydrodynamic interaction, as being sufficient and ap- parison with the experimental value we see that the propriate at this stage. Then the relaxation time of the value calculated from Isihara’s equation is slower by a &h mode is factor of about 3.

(6) On alIowing for the different molecular environ-

ments the light scattering linewidths reported here are where .$ characterises the draining effect on the polymer and for non-draining becomes .$ = 2 for k = 1

in the same region as those detemiined by oscillatory

and t = 1 for free draining and (R2) is the mean square flow birefringence [6 ] . The sensitivity. of the experi- mental technique appears such that comparisons with

end-to-end distance. Using eq. (6) we fiid ‘1 = refinements of the Rouse-Zimm model and investi- 600 gsec (non-free-draining) and r1 = 300 I.tsec (free gation of details of po1yme.r dynamics such as coupled draining). From the relationship of eq. (2b) we obtain motion in polymer molecules should be possible. an internal mode relaxation time of pi = 125 + 30 psec. The experimenta! values of Ti are in the ie- We wish to express our thanks to Dr. W.I. Lee for gion of the va.Iue of 7I predicted by the Rouse-Zimm his help during his stay in our laboratory, to Professor model but lower by a factor of about 4. Since these J.B. Bancroft and Dr. J.V. Champion for samples, to observations are on a semiflexible polymer under non- Professor G. Allen and Professor S.F. Edwards for theta conditions discrepancies from the theory ap- discussions and the Science Research Council for propriate to a flexible polymer under random flight support. conditions can be expected. It is notable that the Morite Carlo computer calculations of Verdier and Stockmayer [9] indicate an effective relaxation time References equal to 0.29 TV.

It seems more su!‘table to analyse the motion result- ing in the line broadening Ti in terms of internal motion rather than rotational motion because of the relatively long chains studied here and the degree of flexibility of the polystyrene molecule. It is interest- ing however to compare this with the analysis of the additionai component to the linewidth in terms of rotation. From eq. (2a) we find a rotatio;lal diffusion coefficient of D, = 2700 + 400 set-1. This will be experimentally observable if the molecule in rotation is geometrically or optically aniso:ropic. The Kirk- wood-Riseman theory of the friction coefficient has been developed further by Isihara [IO] to produce an equation for the rotational diffusion coefficient for a

&yible chain polymer of the form

[I] 0. Kramer and J.E. Frederick, Macromolecules 5 (1972) 69.

[2] J.G. Kirlcwood and J. Riseman, J. Chem. Phys. 16 1.1948) 565.

[3] P.E. Rouse, I. Chem. Phys. 21 (1953) 1272. [4] B.A. Zimm, J. Chem. Phys. 24 (1956) 269. [5] 14. Fiuman and W.H. Stockmaycr, Ann. Rev. Phys.

Chem. 21(1970) 407. [6] G.B. Thurston and J.L. Schrag, J. Polymer Sci. A2

(1968) 1331. [ 71 R. Pecora? J. Chem. Phys. 49 (1968) 1032. [S] R. Pecora, J. Chem. Phys. 48 (1968) 4126. 191 P.H. Verdier and W.H. Stockmayer, J. Chem. Phys. 36

(1962) 227. [lo] i\. Isihara, J. Chem. Phys. 47 (1967) 3821.