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Polymer diffusion at intermediate concentrations studied by photon correlation spectroscopy
Chemical Physics 12 (1976) 161-168 Q North-Holland Publishing Company
POLYMER DIFFUSION AT INTERMEDIATE CONCENTRATIONS !XIJDlED BY PHOTON CORRELATION SPECTROSCOPY
D.BAILEY,T.A. KING
and
D.N.PINDER*
Physics Department, Schuster Laboratory, University of Manchester, Mardmter
rCfJ39PL, UK
Received 25 July 1975
Macromolecular dynamics in concentrated systems has been the subject of scvcral thcorctical studies in recent years. Here the technique of photon corrchtion spectroscopy applied in quasiclastic Raylcigh tincwidth laser light scattering investigations, has been eslcnded from dilute solutions to solutions of intermediate concentration. In particular measure mcnts are presented on monodispcrse linear, atactic polystyrene in the good solvent 2-bulanonc over a wide concentraThe lincwidths of the tion range, 5 X low3 to 0.15 g cmm3, and with variation of molecular wcizht and tempcmturc. scattered light in low concentration solutions yield the molecular translational diffusion coefficients, observed with variation of the molecular environrncnt. With incrcasc in concentration B change in diffusion bchaviour occurs at a concentration in the region where chain overlap is present; at higher concentrations the diffusion becomes insensitive to polymer molecular weight. An overall activation energy of about 0.10 cV is found from the temperature dcpcndcncc of the diffusion of which 0.08 eV is attributed to the viscosity activatinn energy and the remainder to cntanglemcnts.
1. Introduction
Macromolecular diffusion in concentrated solutions has excited sufficient interest to engender both recent experimental study [l-4] and theoretical speculation [5-13]_ This paper presents an investigation of the intermediate concentration region, polymer concentration from 1715%; in this region some theoretical progress is being made. By using the ielatively new photon correlation spectroscopy technique experimental data can be provided which can be used to assess and improve the theoretical approaches. Rehage et al. [l] conducted an experimental study of the tlanslational diffusion of monodisperse polystyrene [molecular wei&t,IWn = ISOOOO] in ethyl-benzene; this work employed an interferometiic method and the translational diffusion coefficient,D, of polystyrene was calculated as a function of concentration, c, eve; the range 0.001-0.4 g cmm3 ..A graph consisting of a family of curves of D * On leave from Department Biophysics,
of ~emistry,~Biochemis!ry, Massey University, New +hnd.
as a function of c was presented with temperature as a parameter. T!le most significant findings were: firstly, that the diffusion coefficient increased with concentration until a maximum value was attained at c = 3576, and then the value decreased as the concentration increased further, and secondly that the lower the temperature the smaller the diffusion coefficient, the less it varied with concentration and the lower the concentration at which the diffusion coefficient reached a maximum. Photon correlation spectroscopy has also been used to study the translational diffusion coefficient of polystyrene in solution [3,41. King et al. [ 14,151 have presented an extensive study in the low coneentration range (c < 0.01 g cme3) under both theta and nun-theta conditions. These measurements include studies of monodispcrse polystyrenes in the good solvent 2-butanone +t 25’C. The translational diffusion coefficient, D,wasidentified with the @antity r/X*, where r is the measured linewidth and, K the magnitude of the scattering wave vector. Valu& of D w&e obtained as a function of concentration and molecular weight. The translational dilfusion coefficient was, expanded as a function of concentration.
162
D. B&y et ah/Polymerdi//usiortstirdiedby photon correlationspectroscopy
D“D,,(l
+ kg
+ . ..).
eq. (3) should be rewritten as
(1)
k,=iA,M-k,-I,
where D,-, is the translational diffusion coefficient at infinite dilution, c.the polymer concentration, A2 the second virial coefficient, M the polymer molecular weight, k, the first order frictional coefficient concentration dependence and 1 the specific volume of -the polymer. The values of k, thus obtained were compared with dilute solution theory. Pusey et al. [4] used one highly monodisperse polystyrene, molecular weight 111000, but extended the measurements to higher concentrations, (up to 0.1 g cmW3j in toluene at 21’C. The translational diffusion coefficient was again identified with the quantity r/2KL, although in this case r’ was taken as an average linewidth as calculated by the method of moments [16]. Photon correlation spectroscopy is used here to provide a range of experimental information of the diffusion of linear polystyrene in a reasonably good solvent with variation of polymer molecular weight and concentration and solution kmperature. This provides a range of new information, a confirmation of other studies and a reqkement for new interpretations.
2. Experimental The quantity measured in the light scattering experiment is the intensity autocorrelation function of the light scattered at the angle 8 by the solution, g(2)(T) = 1 +fl$‘(7)12
,
(2)
where g(t)(T)
is the electric field autocorrelation function of the scattered light andfis a constant. ..For light scattered by a large number of identical, “small” macroltiolecules
g(l)(7) = exp (--l;l~); where
(3)
:
‘;, .=Di2,-
(4)
K=.(4j&I
sin+.
(5)
‘.I A sarnp!q qf polydisperse macromolecules has a -distributi&~ of f,,.valtie& p(rh), and for such systems. _. ..
,,
-;
..:.
..
=I P(rh)exp(-I-‘,,7)dI’,,.
lg(‘)(T)I
(6)
0
The fust moment of the distributic-n, rh, is given by m
pr;,
=
s
qv--J dr,.
(7)
0
Using eq. (7), eq. (6) becomes (8)
k”‘(T)1 = exp(-rh;h7) Rearranging eq. (2) and using eq. (8) In[gc2)(T) - 11 = Inf+ 2 =lnft2
c
ln[l&~)ll
lGi;,e2
-~hTt-=--
- @hT)3/3!r$.
, I
Z!Ff
(9)
where
s=-& j(rh-Fh)~p(rh)drh, h h
r>l
(10)
is the normalized rth moment about the mean. The moments Fh,&/F; etc. can be calculated by applying graphical procedures [ 171. Ten measurements of the correlation function g(z)(T) were made at each scattering angle, correspond$g to ten different values of T,,.,~ suth that 0.4 5 S 4.0, where 7maKis the maximum time delay num er o f correlator channels times the channel ?“‘b time-scale). The principle of the technique is that at low values of 7-m&X the right hand side of eq. (9) is dominated by the term &,r, but also at small values Of ~mrlzzthere is a large random error associated with the measurement of &,. At large values of I-A~ the random error is smaller but the higher order terms are more important so the measured value of i; islikely to have a large systematic error. The preferred vdue of r’ is obtained by fitting the data to eq. (9) by standard least squares procedures and then plotting the .. predicted.values of. r against T,.,,~, for each order of. fit. The preferred value of F is given by the intercepts qn the F gjcis of the straight.lines drawn through the points represefiting each order of fit. If all these
D. Bailey et ai./Polymer diffusion studied by photon correlation spectroscopy
5x
Id’,
I
Y
*
+
-f
+
Fig. 1. Four orders of fitting of lincwidth obtzined for diffcrent esperimcntal time scales and shown as a computer gcnemtcd graphical plot,r= 1 (+). 2 (?I,3 (X) and 4 (x). For sample Ffw = 670 000 zt 0°C observed in 90” scattering. The preferred value of F is given as Ihc intercept on the ordinate.
straight lines are approximately horizontal then the
second moment is very small. The fust four orders of fit were used where the rth order fit assumes all moments of order geater than the rth are zero. Fig. 1 shows a typical graph used to obtain the preferred value bf F/2K2, the intercept on the ordinate. A similar procedure can be used to obtain the notialized second moment. Macromolecular diffusion in solutions of intermediate csncentration is complicated by intermolecular effects which cause the electric field autocorrelation function to be more complex than a simple, single exponential. Consequently these intermolecular effects may be treated in the same fashion as polydispersity and an average linewidth i? may be obtained by the application of the procedures outlines above. The basic experimental arrangement has been described in ref. [14], the only differences being that an ITT FW 130 photomultiplier and a 48.channel clipping digital correlator (Precision Devic;es and-Systems Ltd.) were used. Temperature control and sta-
163
bility were achieved by circulating water from a thermostatically controlled bath, through the massive copper block cell holder. An error of less than G.l% could be assigned to the value of the concentration. The sample cleansing procedure adopted has been described previously [ 14,:5]. All samples were centrifuged at 50 000 g for at least two hours and then 3-5 cm3 were transferred to a 1 cm square spectroscopic cell tith the aid of a clean pipette and sealed, AlI samples were inspected under laser illumination to ensure that they were dust f&e. Very highly concentrated solutions are too viscous to be centrifuged, consequently For highly concentrated samples (> 30% by weight) experiments were carried out in which samples were prepared by adding polymer directly to spectroscopic cells containing previously centrifuged solvent. However, samp!es prepared in this way were found to be too dirty and this procedure was abandoned. This inability to prepare clean highly concentrated samples was a major limitation to the concentration range studied. The samples were left for a few days before measurements were begun and no systematic variations of r/X* with sample age were recorded. Such variations, had they been observed, could possibly have indicated a slow attainment of sample equilibrium. Measurements were taken at the three temperatures O”C, 25°C and SO’C. Each sample was kept in a refrigerator for about two days, and then left in the apparatus at O’C for at least one hour before measurements were made at 0°C. The samples were left in the thermostated apparatus at 50°C for at least two hours before the high temperature observations were recorded. Measurements were made at the four apparent scattering angles 40” through the front face, 60°, 90° and 120’ through the side face. The correlation functions were accumulated for one to two minutes and then punched onto paper tape for computer analysis. The measured quantity, F/X*, did not &ow any systematic variations with scattering angle. Since ten observations were made at each angle, forty correiation functions were recorded for each sample each time it was run at each temperature, and moreover each sample was run two or three times at each temperature. Consequently each point on the graphs.of figs. 2,3 and 4 has been obtained by analysing eighty to a hundred and twenty correlaJion functions in four differect wayi corresponding to the four orders of fit used to
D. Bailey et ab/Pol_vmer difjkiorl
164 ‘Cable 1 Charattcrislics
studied by
pl~~on correlarion spectroscopy
of the pol~styrcnc Qmplcs
Sample
. PC-lc PC-3b PC-133
200000
1.06
392 000 670000
1.1 1.1
obtain the preferred value of F. However it should be
noticed that the third and fourth order fits do not yield any more information then the second, fQr the calculated values of the third and fourth moments are too inaccurate to be useful. Three highly monodispcrse polystyrenes (obtained from the Pressure Chemical Company) were investigated over the solution concentration range 0.005 to 0.133 g cmS3. The characteristics of the polystyrene timpIes are given in table I_
c I 02g cm”) Fig. 3. Diffusion cocfticients for T= 25°C. 200000,0-o~w=392000,s--x~w=670000.
l-ofiw
=
also relates to the results of King et al. [IS] which are found to be in good agreement with the present work. -The behaviour of the lowest and highest molecular weight samples PC-lc and PC-13a at 25°C in fig. 3 is particuliirly interesting. For PC-lc an initial slight decrease in diffusion coefficient with increasing concentration has been explained in terms of thermodynamic and hydrodynamic arguments [14,15] ; however, at higher concentrations the diffusion coefficient attains a minimum value and then increases with concentration. Similarly the slope of the graph for sample PC-13a, for which particular extensive measurements have been made, change! rapidly around the
3. Results The translational diffusion coefficients, D. were derived from the measured quantity F/X2. Over the concentration range 0.01 g cmm3 to 0.13 g cmd3 the measured autocorrelation ftinctions changed from an exponential form to a non-exponential form yet the angular dependence of F follows cIosely a K* dependence. The values of D art drawn in figs. 2,3 and 4 for variation of concentration and temperature. Fig. 3
“I
.c 1
::
-“.
..
I
5
4
IO
c(10-‘gci’)
__. -,
.: ._
I IO g cm-3 1
Fig. 4. Diifusion coefficients for T = 50°C. .* 20Ci000,0-o~w~~92000+----xji&,=670000.
ITk. 2:Diffusi& cocfticients for T= 0°C. ~-0Tf~ = ,200 ooo;,o --$&,, = 392000, x-x& = 670000..
.,
.. ..
_-
I 5 c w1
I IS . l
a,,, =
165
D. Bailey et ai./Polymer diffusion studied hy photon correlationspcctrbscopy
concentration 0.03 g IXII-~. All three curves approach the same dope’at higher concentrations. The transition region changes in a systematic manner with molecular weight and temperature. Figs. 2,3 and 4 indicate that as the concentration is increased the translational diffusion coefficient becomes less dependent on molecular weight. As stated above, the viscosity of highly concentrated solutions was so large that centrifugation of such solutions was not feasible, consequently it was not possible in this study to go to sufficiently high concentrations to investigate the results of Rehage et al. concerning the attainment of a maximum value of the diffusion coefficient at higher concentrations (in the region of
noticed that the variation of the diffusion coeffkient with concentration is significantly reduced at lower temperatures. These results also indicate that the diffusion coefficient is slightly less molecular weight dependent at lower temperatures. The temperature dependence of the trans!ational diffusion coefficient is usually described by the relationship D = D, exp (-ElkT),
where 0, is the translational diffusion coefficient at very high temperature, E is the activation energy, and T the absolute temperature. Fig. 5 is a set of graphs of log10 D against l/T at various concentrations taken from the data shown in figs. 2,3 and 4. At low polymer concentrations the graphs are almost linear, indicating that a relationship similar to eq. (11) is obeyed and that E is approximately 0.10 eV. The curves deviate further from linearity as the concentration is increased, a similar trend has been reported by Rehage et.al. [I] .
0.35 g cmm3). The effect of temperature on the relationship between translational diffusion coefficient and concentration can be gauged by comparison between figs. 2, 3 and 4. The activation energy of viscosity For the solvent 2.butanone has been measured over the temperature range 0-SO’C, this was found to be 0.081 eV. Not surprisingy, the lower the temperature, the smaller the diffusion coefficient. Also it should be
1
(11)
c - 0.12 gd
c i O.Wgcd’
t
i
Fig. 5.33~ wmperature dependence of the diffusion coefficient taken from the data shown in figs. 2,3 ;ind 4. At each conccntration: upper curve & =.200 dO0, middle curve Ew = 392 000, lower curve tiw = 670 000.
.,.
166
4.
D. B&y
et al./Polynwr diffusion studied by photon correlation spectroscopy
Discussion
The quantity F/7X! is fdund to be independent of the scattering angle for all the measurements reported here; consequently intramolecular relaxation analogous to the Rouse modes of isolated molecules may be neglected. However as has been indicated it is not possible to show rigorously that r/2K2 can be identified with a bulk diffusion coefficient in concentrated solutions, although some encouragement may be taken from the fact that the above results compare favourably with the results of Rehage et al. [l] who used a different solvent and Paul et al. [2] for the same solvent. Berne [ 181 has discussed the selfand distinct parts of the space-time autocorrelation function and showed that, if the Vineyard approximation is va!id, the correlation function should be a sintjle exponential even in the presence of intermolecular interactions
which would give rise to a finite dis-
tinct term in the correlation function. These results may be tin il?dication that the Vineyard approximation is valid for extensively overlapping polymer molecuies. The non-exponential nature of the measured autocorrelation function at the highe: concentration carries information on the modification of the diffusion either by modification of the diffusion process itself, e.g. changing to hindered browniun motion or non-fickiar. diffusion, or by modification of the polymer by the onset of entanglements. The nonexponential feature is apparent but the data is insufficiently accurate to permit further analysis. The initial behaviour of all the curves in figs. 2,3 and 4 up WC = 0.01 g cmm3 can be well described by dilute solution theory [ 141, however the behaviour at higher concentrations cannot. This can be readily illustrated by the following simplistic argument. If dilute solution theory were applicable then one would expect the polystyrene molecules to be distinct and-have a radius of gyration similar to that appertaining to infinite dilution conditions. However, it can be calculated that if this were the case then at a concentration of about 0.04 g cm-3 the whole volume of a solution of sample PC-13a in 2-butanone Gould b.e “occupied” by segments of the polystyrene molecules. Clsariy the concentration could not be increased without .increasing intermolecular effects. So at intermediate conc&trations the polystyrene .molecules cannot have the same radius of gyration :.
and remain distinct and moreover the mo!ecules are very probably entangled at these concentrations. From figs. 2,3 and 4 it is clear that there is a transition from the dilute solution behaviour reported previously by King et al. [14,15] and represented by eq. (!.) to a rather different form of behaviour at higher concentrations. To the authors’ knowledge this difference has not been previously observed in diffusional measurements and it seems to confirm the division into dilute and semi-dilute regions as proposed by Edwards [ 191. As anticipated and also as found in other experiments [20] this division occurs around the region of chain overlap. It is therefore clear that explanations of the full concentration range based on an expansion of D in terms of concentration keeping only the first two terms D = D,( 1 + kc), such as have been susested [4] are invalid, since they predict the wrong behaviour at higher dilution. The concentration dependence ofD may be rationalised quantitatively in several possible ways: (a) The sphere model used by King et al. [ 141 may be extended by keeping one or more additional terms in the expansion of the friction coefficient and the osmotic pressure. This approach does not provide significant physical insight. (b) It has been shown that at high concentrations the polymer takes on unperturbed dimensions [21] , if this is so then the molecular radius should decrease at high concentration to give rise to a smaller friction coefficient and thereby a larger diffusion coefficient. Using the intrinsic viscosity relation [Q] = KW = @ m3 M1/213~n-3/2 with m = 52, Q = Flory universal
constantz2.1 X 1023,K=3X 10W2cm3g-’ and 0.60 we obtain the molecular weight dependence of = as a =Z0.77 M”.033. The unperturbed and expanded hydrodynamic radii at 25°C are then: far PClc9.2and 10.6nm,forPC-3b 13.8and 16.3nmand for PC-13a 18.8 and 22.6 nm. The predicted change in
Q=
the djffusion coefficients
are in units of 10e7 cmB2
for PC-lc 5195 to 6.85, for PC-3b 3.95 to 4.66 and for PC-13a 2.80 to 3.37. These changes are in the direction of increasing D with c but they are insufficient to explain in full the observed values of D through the intermediate concentration region. (c) It may be supposed that because of entanglements only the sections of molecules between entanglements are free to diffuse; so that the observed diffusion coefficient at highconcentrations corresponds S-I:
D. Bailey et al.fPolymer diffusion studied by photon correlation
L
=I 5
c
Fig. 6. Illustration of the diffusional behaviour in dilute solutions and intermediate concentration solutions and the transition region for low (a). medium (b) and hiph(c) molecular wciyht polymer in a good solvent. The approximate region of significant chain overlap is shown for low and high molecular weights by the concentrations c2 and cl. For some molccutar weight, shown as (b), there is no abviaus change in the slopes for dilute and intermediate concentration.
to a molecular
weight
much
smaller
than that of the
whole molecule. It is difficuit however to interpret the results of Rehage et al. [l] using this model, since they observed bulk diffusion, without assuming a direct connection between bulk diffusion and segment diffusion. Whether a non-equilibrium distribution of entanglements occurs over the polymer molecules with a greater density near the polymer extremities is still a matter of conjecture. (d) Fig. 3 contains the measurements at 25’C, at this temperature a larger number of samples were observed. The curves suggest that they can be represented by a piecewice-linear behaviour as illustrated in fig. 6. This can be interpreted by assuming that the initial line for each molecular weight is governed by the thermodynamic-hydrodynamic dependences appropriate to high dilution as in eq. (1). The second part for each sample, which is concerned with intermediate concentrations, may be represented by D = DK( 1 + kc) where DK is a constant dependent upon molecular weight but k is a constant only weakly molecular weight dependent. The transition region between the two regions occurs at about the concentration at which molecular overlapping occurs. The temperature dependence of the diffusion coefficient has been interpreted in terms of an activation
167
spccrroscopy
energy of approximately 0.10 eV. Of this 0.08 eV may be aEcounted for as the temperature dependence of the viscosity of the solvent. The remaining 0.02 eV may, in part, represent an activation energy associated with the removal of entanglements. It should be noted that the effective number of entanglements should be less than the number appropriate to dynamical mechanical viscoelastic relaxations performed at higher frequencies. From fig. 2 it is apparent that the molecular weight dependence of D is much reduced at the higher concentrations - indeed at 25°C and 10% concentration D a M-o.1. This may indicate that at high concentrations diffusion tends to occur by solvent flow and the entangled polymer molecules move in a way which is largely independent of the length of the polymer chain.
Acknowledgement The Science Research Council are gratefully acknowledged for providing a Postdoctoral Fellowship to DB and for funding for equipment.
References [1] G. Rehage, 0. Ernst and J. Fuhrmann, Discussions Faraday Sot. 49 (1970) 208. [2] D:R. Paul, V. hfavichak and D.R. Kemp, I. Appl. Pol. Sci. 15 (1971) 1553. [ 3) J.D.G. hIcAdam, T.A. King and A. Knox; Chcm. Phys. Letrers 26 (1974) 64. [41 P.N. Pusey, J.M. Vaughan and G. Williams, J. Chcm. Sot. Faraday Trans. II 70 (1974) 1696. [S] H. Yamakawa, J. Chem. Phys. 43 (1965) 1334. [6 I P.G. de Gcnncs. Physics 3 (1967) 37. [ 7) P.G. dc Gcnncs, J. Chcm. Phys. 55 (197 1) 572. [S] S.F. Edwards and K. Freed, J. Chem. Phys. 61 (1974)
1189. [Y] K. Freed ond 3626.
8-F. Edwards,
[ 10) K. Freed and S.F. Edwards,
J. Chcm.
Phys. 61 (1974)
J.Chem.
Phys. 62 (1975)
4032.
[ 111 M. Doi, Cbem. Phys. Letters 26 (1974) 269. [12] hl. Doi, J. Phys. A 8 (1975) 417. [ 131 hi. Doi, J. Phys. A 8 (1975) 757. [ 141T.A. King, W.I. tee, A. Knox and J.D.G. McAdam, Polymer 14 (1973) 151. [ 1.51T.A. King, A. Knox and J.D.G. McAdam, Polymer 14
(1973) 293. (161 D.E. Koppel, J; Chem. Phys. 57 (1972) 4814.
168
D. Bailey et al./Polymer di/fsion
[1’7j r.N. Pusey, in: Proceeding of NATO Advanced Stud& Institute on Phoibn correlation and light beating spectroscopy (Plenum Press, Newt York, 1974). [181 B J. Berne, in: Physical chemistry, an advanced treatise, .Vol. 8B,Thc liquid sfaic,eds. H. Eyring, D. Henderson and W. Jost (Academic Press, New York, 1971) ch. 9.
studied by photon correlation spectroscopy [191 SF. Edwards, Proc. Phys. Sot. (London) 88 (1966) 265. WI J.P. Cotton, B. Farnoux, G. Jannink and C. Strazielle, J. Pol. Sci. Symp. No. 42 (1973) 981. I211 J.P. Cotton, D. Decker, H. Benoit. B. Farnoux, J. Higgins, C. Jannink, R. Ober, C. Picot and J. des Cloizeaux, M;lc&moleculcs 7 (1974) 863.
POLYMER DIFFUSION AT INTERMEDIATE CONCENTRATIONS !XIJDlED BY PHOTON CORRELATION SPECTROSCOPY
D.BAILEY,T.A. KING
and
D.N.PINDER*
Physics Department, Schuster Laboratory, University of Manchester, Mardmter
rCfJ39PL, UK
Received 25 July 1975
Macromolecular dynamics in concentrated systems has been the subject of scvcral thcorctical studies in recent years. Here the technique of photon corrchtion spectroscopy applied in quasiclastic Raylcigh tincwidth laser light scattering investigations, has been eslcnded from dilute solutions to solutions of intermediate concentration. In particular measure mcnts are presented on monodispcrse linear, atactic polystyrene in the good solvent 2-bulanonc over a wide concentraThe lincwidths of the tion range, 5 X low3 to 0.15 g cmm3, and with variation of molecular wcizht and tempcmturc. scattered light in low concentration solutions yield the molecular translational diffusion coefficients, observed with variation of the molecular environrncnt. With incrcasc in concentration B change in diffusion bchaviour occurs at a concentration in the region where chain overlap is present; at higher concentrations the diffusion becomes insensitive to polymer molecular weight. An overall activation energy of about 0.10 cV is found from the temperature dcpcndcncc of the diffusion of which 0.08 eV is attributed to the viscosity activatinn energy and the remainder to cntanglemcnts.
1. Introduction
Macromolecular diffusion in concentrated solutions has excited sufficient interest to engender both recent experimental study [l-4] and theoretical speculation [5-13]_ This paper presents an investigation of the intermediate concentration region, polymer concentration from 1715%; in this region some theoretical progress is being made. By using the ielatively new photon correlation spectroscopy technique experimental data can be provided which can be used to assess and improve the theoretical approaches. Rehage et al. [l] conducted an experimental study of the tlanslational diffusion of monodisperse polystyrene [molecular wei&t,IWn = ISOOOO] in ethyl-benzene; this work employed an interferometiic method and the translational diffusion coefficient,D, of polystyrene was calculated as a function of concentration, c, eve; the range 0.001-0.4 g cmm3 ..A graph consisting of a family of curves of D * On leave from Department Biophysics,
of ~emistry,~Biochemis!ry, Massey University, New +hnd.
as a function of c was presented with temperature as a parameter. T!le most significant findings were: firstly, that the diffusion coefficient increased with concentration until a maximum value was attained at c = 3576, and then the value decreased as the concentration increased further, and secondly that the lower the temperature the smaller the diffusion coefficient, the less it varied with concentration and the lower the concentration at which the diffusion coefficient reached a maximum. Photon correlation spectroscopy has also been used to study the translational diffusion coefficient of polystyrene in solution [3,41. King et al. [ 14,151 have presented an extensive study in the low coneentration range (c < 0.01 g cme3) under both theta and nun-theta conditions. These measurements include studies of monodispcrse polystyrenes in the good solvent 2-butanone +t 25’C. The translational diffusion coefficient, D,wasidentified with the @antity r/X*, where r is the measured linewidth and, K the magnitude of the scattering wave vector. Valu& of D w&e obtained as a function of concentration and molecular weight. The translational dilfusion coefficient was, expanded as a function of concentration.
162
D. B&y et ah/Polymerdi//usiortstirdiedby photon correlationspectroscopy
D“D,,(l
+ kg
+ . ..).
eq. (3) should be rewritten as
(1)
k,=iA,M-k,-I,
where D,-, is the translational diffusion coefficient at infinite dilution, c.the polymer concentration, A2 the second virial coefficient, M the polymer molecular weight, k, the first order frictional coefficient concentration dependence and 1 the specific volume of -the polymer. The values of k, thus obtained were compared with dilute solution theory. Pusey et al. [4] used one highly monodisperse polystyrene, molecular weight 111000, but extended the measurements to higher concentrations, (up to 0.1 g cmW3j in toluene at 21’C. The translational diffusion coefficient was again identified with the quantity r/2KL, although in this case r’ was taken as an average linewidth as calculated by the method of moments [16]. Photon correlation spectroscopy is used here to provide a range of experimental information of the diffusion of linear polystyrene in a reasonably good solvent with variation of polymer molecular weight and concentration and solution kmperature. This provides a range of new information, a confirmation of other studies and a reqkement for new interpretations.
2. Experimental The quantity measured in the light scattering experiment is the intensity autocorrelation function of the light scattered at the angle 8 by the solution, g(2)(T) = 1 +fl$‘(7)12
,
(2)
where g(t)(T)
is the electric field autocorrelation function of the scattered light andfis a constant. ..For light scattered by a large number of identical, “small” macroltiolecules
g(l)(7) = exp (--l;l~); where
(3)
:
‘;, .=Di2,-
(4)
K=.(4j&I
sin+.
(5)
‘.I A sarnp!q qf polydisperse macromolecules has a -distributi&~ of f,,.valtie& p(rh), and for such systems. _. ..
,,
-;
..:.
..
=I P(rh)exp(-I-‘,,7)dI’,,.
lg(‘)(T)I
(6)
0
The fust moment of the distributic-n, rh, is given by m
pr;,
=
s
qv--J dr,.
(7)
0
Using eq. (7), eq. (6) becomes (8)
k”‘(T)1 = exp(-rh;h7) Rearranging eq. (2) and using eq. (8) In[gc2)(T) - 11 = Inf+ 2 =lnft2
c
ln[l&~)ll
lGi;,e2
-~hTt-=--
- @hT)3/3!r$.
, I
Z!Ff
(9)
where
s=-& j(rh-Fh)~p(rh)drh, h h
r>l
(10)
is the normalized rth moment about the mean. The moments Fh,&/F; etc. can be calculated by applying graphical procedures [ 171. Ten measurements of the correlation function g(z)(T) were made at each scattering angle, correspond$g to ten different values of T,,.,~ suth that 0.4 5 S 4.0, where 7maKis the maximum time delay num er o f correlator channels times the channel ?“‘b time-scale). The principle of the technique is that at low values of 7-m&X the right hand side of eq. (9) is dominated by the term &,r, but also at small values Of ~mrlzzthere is a large random error associated with the measurement of &,. At large values of I-A~ the random error is smaller but the higher order terms are more important so the measured value of i; islikely to have a large systematic error. The preferred vdue of r’ is obtained by fitting the data to eq. (9) by standard least squares procedures and then plotting the .. predicted.values of. r against T,.,,~, for each order of. fit. The preferred value of F is given by the intercepts qn the F gjcis of the straight.lines drawn through the points represefiting each order of fit. If all these
D. Bailey et ai./Polymer diffusion studied by photon correlation spectroscopy
5x
Id’,
I
Y
*
+
-f
+
Fig. 1. Four orders of fitting of lincwidth obtzined for diffcrent esperimcntal time scales and shown as a computer gcnemtcd graphical plot,r= 1 (+). 2 (?I,3 (X) and 4 (x). For sample Ffw = 670 000 zt 0°C observed in 90” scattering. The preferred value of F is given as Ihc intercept on the ordinate.
straight lines are approximately horizontal then the
second moment is very small. The fust four orders of fit were used where the rth order fit assumes all moments of order geater than the rth are zero. Fig. 1 shows a typical graph used to obtain the preferred value bf F/2K2, the intercept on the ordinate. A similar procedure can be used to obtain the notialized second moment. Macromolecular diffusion in solutions of intermediate csncentration is complicated by intermolecular effects which cause the electric field autocorrelation function to be more complex than a simple, single exponential. Consequently these intermolecular effects may be treated in the same fashion as polydispersity and an average linewidth i? may be obtained by the application of the procedures outlines above. The basic experimental arrangement has been described in ref. [14], the only differences being that an ITT FW 130 photomultiplier and a 48.channel clipping digital correlator (Precision Devic;es and-Systems Ltd.) were used. Temperature control and sta-
163
bility were achieved by circulating water from a thermostatically controlled bath, through the massive copper block cell holder. An error of less than G.l% could be assigned to the value of the concentration. The sample cleansing procedure adopted has been described previously [ 14,:5]. All samples were centrifuged at 50 000 g for at least two hours and then 3-5 cm3 were transferred to a 1 cm square spectroscopic cell tith the aid of a clean pipette and sealed, AlI samples were inspected under laser illumination to ensure that they were dust f&e. Very highly concentrated solutions are too viscous to be centrifuged, consequently For highly concentrated samples (> 30% by weight) experiments were carried out in which samples were prepared by adding polymer directly to spectroscopic cells containing previously centrifuged solvent. However, samp!es prepared in this way were found to be too dirty and this procedure was abandoned. This inability to prepare clean highly concentrated samples was a major limitation to the concentration range studied. The samples were left for a few days before measurements were begun and no systematic variations of r/X* with sample age were recorded. Such variations, had they been observed, could possibly have indicated a slow attainment of sample equilibrium. Measurements were taken at the three temperatures O”C, 25°C and SO’C. Each sample was kept in a refrigerator for about two days, and then left in the apparatus at O’C for at least one hour before measurements were made at 0°C. The samples were left in the thermostated apparatus at 50°C for at least two hours before the high temperature observations were recorded. Measurements were made at the four apparent scattering angles 40” through the front face, 60°, 90° and 120’ through the side face. The correlation functions were accumulated for one to two minutes and then punched onto paper tape for computer analysis. The measured quantity, F/X*, did not &ow any systematic variations with scattering angle. Since ten observations were made at each angle, forty correiation functions were recorded for each sample each time it was run at each temperature, and moreover each sample was run two or three times at each temperature. Consequently each point on the graphs.of figs. 2,3 and 4 has been obtained by analysing eighty to a hundred and twenty correlaJion functions in four differect wayi corresponding to the four orders of fit used to
D. Bailey et ab/Pol_vmer difjkiorl
164 ‘Cable 1 Charattcrislics
studied by
pl~~on correlarion spectroscopy
of the pol~styrcnc Qmplcs
Sample
. PC-lc PC-3b PC-133
200000
1.06
392 000 670000
1.1 1.1
obtain the preferred value of F. However it should be
noticed that the third and fourth order fits do not yield any more information then the second, fQr the calculated values of the third and fourth moments are too inaccurate to be useful. Three highly monodispcrse polystyrenes (obtained from the Pressure Chemical Company) were investigated over the solution concentration range 0.005 to 0.133 g cmS3. The characteristics of the polystyrene timpIes are given in table I_
c I 02g cm”) Fig. 3. Diffusion cocfticients for T= 25°C. 200000,0-o~w=392000,s--x~w=670000.
l-ofiw
=
also relates to the results of King et al. [IS] which are found to be in good agreement with the present work. -The behaviour of the lowest and highest molecular weight samples PC-lc and PC-13a at 25°C in fig. 3 is particuliirly interesting. For PC-lc an initial slight decrease in diffusion coefficient with increasing concentration has been explained in terms of thermodynamic and hydrodynamic arguments [14,15] ; however, at higher concentrations the diffusion coefficient attains a minimum value and then increases with concentration. Similarly the slope of the graph for sample PC-13a, for which particular extensive measurements have been made, change! rapidly around the
3. Results The translational diffusion coefficients, D. were derived from the measured quantity F/X2. Over the concentration range 0.01 g cmm3 to 0.13 g cmd3 the measured autocorrelation ftinctions changed from an exponential form to a non-exponential form yet the angular dependence of F follows cIosely a K* dependence. The values of D art drawn in figs. 2,3 and 4 for variation of concentration and temperature. Fig. 3
“I
.c 1
::
-“.
..
I
5
4
IO
c(10-‘gci’)
__. -,
.: ._
I IO g cm-3 1
Fig. 4. Diifusion coefficients for T = 50°C. .* 20Ci000,0-o~w~~92000+----xji&,=670000.
ITk. 2:Diffusi& cocfticients for T= 0°C. ~-0Tf~ = ,200 ooo;,o --$&,, = 392000, x-x& = 670000..
.,
.. ..
_-
I 5 c w1
I IS . l
a,,, =
165
D. Bailey et ai./Polymer diffusion studied hy photon correlationspcctrbscopy
concentration 0.03 g IXII-~. All three curves approach the same dope’at higher concentrations. The transition region changes in a systematic manner with molecular weight and temperature. Figs. 2,3 and 4 indicate that as the concentration is increased the translational diffusion coefficient becomes less dependent on molecular weight. As stated above, the viscosity of highly concentrated solutions was so large that centrifugation of such solutions was not feasible, consequently it was not possible in this study to go to sufficiently high concentrations to investigate the results of Rehage et al. concerning the attainment of a maximum value of the diffusion coefficient at higher concentrations (in the region of
noticed that the variation of the diffusion coeffkient with concentration is significantly reduced at lower temperatures. These results also indicate that the diffusion coefficient is slightly less molecular weight dependent at lower temperatures. The temperature dependence of the trans!ational diffusion coefficient is usually described by the relationship D = D, exp (-ElkT),
where 0, is the translational diffusion coefficient at very high temperature, E is the activation energy, and T the absolute temperature. Fig. 5 is a set of graphs of log10 D against l/T at various concentrations taken from the data shown in figs. 2,3 and 4. At low polymer concentrations the graphs are almost linear, indicating that a relationship similar to eq. (11) is obeyed and that E is approximately 0.10 eV. The curves deviate further from linearity as the concentration is increased, a similar trend has been reported by Rehage et.al. [I] .
0.35 g cmm3). The effect of temperature on the relationship between translational diffusion coefficient and concentration can be gauged by comparison between figs. 2, 3 and 4. The activation energy of viscosity For the solvent 2.butanone has been measured over the temperature range 0-SO’C, this was found to be 0.081 eV. Not surprisingy, the lower the temperature, the smaller the diffusion coefficient. Also it should be
1
(11)
c - 0.12 gd
c i O.Wgcd’
t
i
Fig. 5.33~ wmperature dependence of the diffusion coefficient taken from the data shown in figs. 2,3 ;ind 4. At each conccntration: upper curve & =.200 dO0, middle curve Ew = 392 000, lower curve tiw = 670 000.
.,.
166
4.
D. B&y
et al./Polynwr diffusion studied by photon correlation spectroscopy
Discussion
The quantity F/7X! is fdund to be independent of the scattering angle for all the measurements reported here; consequently intramolecular relaxation analogous to the Rouse modes of isolated molecules may be neglected. However as has been indicated it is not possible to show rigorously that r/2K2 can be identified with a bulk diffusion coefficient in concentrated solutions, although some encouragement may be taken from the fact that the above results compare favourably with the results of Rehage et al. [l] who used a different solvent and Paul et al. [2] for the same solvent. Berne [ 181 has discussed the selfand distinct parts of the space-time autocorrelation function and showed that, if the Vineyard approximation is va!id, the correlation function should be a sintjle exponential even in the presence of intermolecular interactions
which would give rise to a finite dis-
tinct term in the correlation function. These results may be tin il?dication that the Vineyard approximation is valid for extensively overlapping polymer molecuies. The non-exponential nature of the measured autocorrelation function at the highe: concentration carries information on the modification of the diffusion either by modification of the diffusion process itself, e.g. changing to hindered browniun motion or non-fickiar. diffusion, or by modification of the polymer by the onset of entanglements. The nonexponential feature is apparent but the data is insufficiently accurate to permit further analysis. The initial behaviour of all the curves in figs. 2,3 and 4 up WC = 0.01 g cmm3 can be well described by dilute solution theory [ 141, however the behaviour at higher concentrations cannot. This can be readily illustrated by the following simplistic argument. If dilute solution theory were applicable then one would expect the polystyrene molecules to be distinct and-have a radius of gyration similar to that appertaining to infinite dilution conditions. However, it can be calculated that if this were the case then at a concentration of about 0.04 g cm-3 the whole volume of a solution of sample PC-13a in 2-butanone Gould b.e “occupied” by segments of the polystyrene molecules. Clsariy the concentration could not be increased without .increasing intermolecular effects. So at intermediate conc&trations the polystyrene .molecules cannot have the same radius of gyration :.
and remain distinct and moreover the mo!ecules are very probably entangled at these concentrations. From figs. 2,3 and 4 it is clear that there is a transition from the dilute solution behaviour reported previously by King et al. [14,15] and represented by eq. (!.) to a rather different form of behaviour at higher concentrations. To the authors’ knowledge this difference has not been previously observed in diffusional measurements and it seems to confirm the division into dilute and semi-dilute regions as proposed by Edwards [ 191. As anticipated and also as found in other experiments [20] this division occurs around the region of chain overlap. It is therefore clear that explanations of the full concentration range based on an expansion of D in terms of concentration keeping only the first two terms D = D,( 1 + kc), such as have been susested [4] are invalid, since they predict the wrong behaviour at higher dilution. The concentration dependence ofD may be rationalised quantitatively in several possible ways: (a) The sphere model used by King et al. [ 141 may be extended by keeping one or more additional terms in the expansion of the friction coefficient and the osmotic pressure. This approach does not provide significant physical insight. (b) It has been shown that at high concentrations the polymer takes on unperturbed dimensions [21] , if this is so then the molecular radius should decrease at high concentration to give rise to a smaller friction coefficient and thereby a larger diffusion coefficient. Using the intrinsic viscosity relation [Q] = KW = @ m3 M1/213~n-3/2 with m = 52, Q = Flory universal
constantz2.1 X 1023,K=3X 10W2cm3g-’ and 0.60 we obtain the molecular weight dependence of = as a =Z0.77 M”.033. The unperturbed and expanded hydrodynamic radii at 25°C are then: far PClc9.2and 10.6nm,forPC-3b 13.8and 16.3nmand for PC-13a 18.8 and 22.6 nm. The predicted change in
Q=
the djffusion coefficients
are in units of 10e7 cmB2
for PC-lc 5195 to 6.85, for PC-3b 3.95 to 4.66 and for PC-13a 2.80 to 3.37. These changes are in the direction of increasing D with c but they are insufficient to explain in full the observed values of D through the intermediate concentration region. (c) It may be supposed that because of entanglements only the sections of molecules between entanglements are free to diffuse; so that the observed diffusion coefficient at highconcentrations corresponds S-I:
D. Bailey et al.fPolymer diffusion studied by photon correlation
L
=I 5
c
Fig. 6. Illustration of the diffusional behaviour in dilute solutions and intermediate concentration solutions and the transition region for low (a). medium (b) and hiph(c) molecular wciyht polymer in a good solvent. The approximate region of significant chain overlap is shown for low and high molecular weights by the concentrations c2 and cl. For some molccutar weight, shown as (b), there is no abviaus change in the slopes for dilute and intermediate concentration.
to a molecular
weight
much
smaller
than that of the
whole molecule. It is difficuit however to interpret the results of Rehage et al. [l] using this model, since they observed bulk diffusion, without assuming a direct connection between bulk diffusion and segment diffusion. Whether a non-equilibrium distribution of entanglements occurs over the polymer molecules with a greater density near the polymer extremities is still a matter of conjecture. (d) Fig. 3 contains the measurements at 25’C, at this temperature a larger number of samples were observed. The curves suggest that they can be represented by a piecewice-linear behaviour as illustrated in fig. 6. This can be interpreted by assuming that the initial line for each molecular weight is governed by the thermodynamic-hydrodynamic dependences appropriate to high dilution as in eq. (1). The second part for each sample, which is concerned with intermediate concentrations, may be represented by D = DK( 1 + kc) where DK is a constant dependent upon molecular weight but k is a constant only weakly molecular weight dependent. The transition region between the two regions occurs at about the concentration at which molecular overlapping occurs. The temperature dependence of the diffusion coefficient has been interpreted in terms of an activation
167
spccrroscopy
energy of approximately 0.10 eV. Of this 0.08 eV may be aEcounted for as the temperature dependence of the viscosity of the solvent. The remaining 0.02 eV may, in part, represent an activation energy associated with the removal of entanglements. It should be noted that the effective number of entanglements should be less than the number appropriate to dynamical mechanical viscoelastic relaxations performed at higher frequencies. From fig. 2 it is apparent that the molecular weight dependence of D is much reduced at the higher concentrations - indeed at 25°C and 10% concentration D a M-o.1. This may indicate that at high concentrations diffusion tends to occur by solvent flow and the entangled polymer molecules move in a way which is largely independent of the length of the polymer chain.
Acknowledgement The Science Research Council are gratefully acknowledged for providing a Postdoctoral Fellowship to DB and for funding for equipment.
References [1] G. Rehage, 0. Ernst and J. Fuhrmann, Discussions Faraday Sot. 49 (1970) 208. [2] D:R. Paul, V. hfavichak and D.R. Kemp, I. Appl. Pol. Sci. 15 (1971) 1553. [ 3) J.D.G. hIcAdam, T.A. King and A. Knox; Chcm. Phys. Letrers 26 (1974) 64. [41 P.N. Pusey, J.M. Vaughan and G. Williams, J. Chcm. Sot. Faraday Trans. II 70 (1974) 1696. [S] H. Yamakawa, J. Chem. Phys. 43 (1965) 1334. [6 I P.G. de Gcnncs. Physics 3 (1967) 37. [ 7) P.G. dc Gcnncs, J. Chcm. Phys. 55 (197 1) 572. [S] S.F. Edwards and K. Freed, J. Chem. Phys. 61 (1974)
1189. [Y] K. Freed ond 3626.
8-F. Edwards,
[ 10) K. Freed and S.F. Edwards,
J. Chcm.
Phys. 61 (1974)
J.Chem.
Phys. 62 (1975)
4032.
[ 111 M. Doi, Cbem. Phys. Letters 26 (1974) 269. [12] hl. Doi, J. Phys. A 8 (1975) 417. [ 131 hi. Doi, J. Phys. A 8 (1975) 757. [ 141T.A. King, W.I. tee, A. Knox and J.D.G. McAdam, Polymer 14 (1973) 151. [ 1.51T.A. King, A. Knox and J.D.G. McAdam, Polymer 14
(1973) 293. (161 D.E. Koppel, J; Chem. Phys. 57 (1972) 4814.
168
D. Bailey et al./Polymer di/fsion
[1’7j r.N. Pusey, in: Proceeding of NATO Advanced Stud& Institute on Phoibn correlation and light beating spectroscopy (Plenum Press, Newt York, 1974). [181 B J. Berne, in: Physical chemistry, an advanced treatise, .Vol. 8B,Thc liquid sfaic,eds. H. Eyring, D. Henderson and W. Jost (Academic Press, New York, 1971) ch. 9.
studied by photon correlation spectroscopy [191 SF. Edwards, Proc. Phys. Sot. (London) 88 (1966) 265. WI J.P. Cotton, B. Farnoux, G. Jannink and C. Strazielle, J. Pol. Sci. Symp. No. 42 (1973) 981. I211 J.P. Cotton, D. Decker, H. Benoit. B. Farnoux, J. Higgins, C. Jannink, R. Ober, C. Picot and J. des Cloizeaux, M;lc&moleculcs 7 (1974) 863.