UNPERTURBED DIMENSION OF SODIUMCELLULOSE XANTHATE the system. The question often is settled with stability analyses of these states. The unstable states are usually regarded as physically unobservable. It is clear now that under some situations steady states may exist, be stable, but nevertheless, not prescribe the behavior of the system. It must be noted that in treating the problem of hydrocarbon oxidation the steady-state analysis is important and essential. A portion of the explosion limits and damped oscillations are predicted by such analysis. On the other hand, the problem also extends to the area where the steady-state analysis alone is not sufficient. The observed characteristics of the explosion peninsula and the self-excited oscillations are explained on1,y through the investigations other than those fu rnished by the steady-state analysis.
A . Explosion Peninsula. Figures 6 and 9 show that inside the explosion peninsula (BCD) the stable steady states exist. They exist in terms of stable focuses or nodal points. The explosion characteristics, however, are not characterized by these states. In general, an experiment of this kind is not initiated at the steady-state condition. Examination of the trajec-
3413
tories in the phase plane reveals that the initial values of a system are basically responsible for determining which steady state the trajectory will approach, or it may not go into any of the steady states. In such a case the trajectory is divergent and associated with explosion. When this occurs, the steady-state analysis is not useful. B. Self-Excited Oscillation. The self-excited oscillation region LNKC in Figure 3l is associated with the existence of stable limit cycles. Limit cycles cannot be predicted by the steady-state analysis. The singularity within this area of the Po-Toplane are unstable focuses which predict divergent oscillation in contradiction to the stable oscillation calculated. It is thus clear that the steady-state analysis, while in general very powerful, has its limitations. Elaborate trajectory calculations may often become necessary under special circumstances. The two-stage ignition phenomenon is another example. Acknowledgment. The author is indebted to Professors A. L. Berlad and L. s. Wang of the State University of New York for many stimulating discussions during the course of the work.
Unperturbed Dimension of Sodium Cellulose Xanthate by Bibekananda Dag, Alok K. Ray, and P. K. Choudhury Department of Applied Chemistry, Calcutta University, Calcutta, I n d i a
(Received January 2, 1069)
Sodium cellulose xanthate has been fractionated, and the fractions have been characterized by viscosity, lightscattering, and sedimentation velocity measurements. The data cover the molecular weight range from 0.14 to 1.25 x 106. Unperturbed chain dimeiisioii of sodium cellulose xanthate has been determined from the data of intrinsic viscosity and molecular weight by applying the Stockmayer-Fixman method and also from those of frictioiial coefficient and molecular weight according to the Cowie-Bywater method. I n the former case, the (Ro2/M)'/zvalueobtained was I .38 x 10-8 cm, while in the latter, it was 1.11 X 10-* em.
Introduction Apart from graphical procedures, measurements at the 8 temperature or in 8 solvents are usually necessary for the estimation of unperturbed dimensions of polymer molecules as the excluded volume effect vanishes under these conditions. But unlike the synthetic polymers, the 8 conditions can rarely be attained for cellulose derivatives owing to their semicrystalline nature, and their unperturbed dimensions are, therefore, derived mainly from good solvent data. The estimation of the unperturbed dimension of sodium cellulose xanthate presents additional difficulties owing to its unstable nature and formation of by-product thio salts. Comparatively few studies have, therefore,
been made on the dilute solution properties of this important cellulose derivative. However, detailed investigations on the dilute solution behavior of sodium cellulose xanthate have recently been reported by Elmgren,' who has found a striking similarity of behavior of sodium cellulose xanthate with other cellulose derivatives, when the substituent solvent influence on their hydrodynamic properties is taken into account. Cornell and Swenson2 observed that with increasing degree of substitution coil dimensions of sodium cellulose xanthate show a tendency to decrease and ascribed (1) H. Elmgren, Ark. Kemi,24,237 (1965). (2) R. H. Cornell and H. A. Swenson, J . Appl. PoZyt. Sci., 5, 641 (1961).
Volume 73, Number 10 October 1969
B. DAB,A, K. RAY,AND P. K. CHOUDHURY
3414 this to the decrease in the polymer-solvent interactions. Investigations on the influence of the degree of substitution on the molecular parameters of sodium cellulose xanthate were also made by the present author^.^ Stockmayer and Fixman4 utilized the perturbation theory of excluded volume effect to develop a method for estimating the unperturbed dimensions, and they obtained the very simple equation [7]M-'/2 =
K
+ 0.51&md1/Z
(1)
where K = c#Q(&~/M)~'/", (l?02)1'2 is the unperturbed end-to-end distance, rpois the Flory constant, and B is a polymer-solvent interaction parameter. Perturbation calculations of the frictional coefficient, fo, for flexible polymer coils with excluded volume have been carried out by Kurata and Yamakawa5 and also by Stockmayer and Albrecht6 who derived the relation fo/'~o =
[fl
=
KrM'/'ar
(2)
where
Kr
=
Po[R:02/M]1/z
and at =
1 f 0.6092
+ ...
Pois a constant whose limiting value is 5.2, and at is the expansion factor derived from frictional coeficient data. In contrast to the case of viscosity, the difference between L y f and CY is insignificant. Cowie and Bywater7 pointed out the relationship between the frictional coefficient and molecular weight in the form
If] = KtM'l2 + 0 . ~ O ~ P O B A - ~ M (3) where
A
= (202/M)'/~
which can be utilized for the determination of unperturbed dimensions. The present investigation aims to utilize the CowieBywater method for estimating the unperturbed dimension of sodium cellulose xanthate and to compare the data with those obtained from the Stockmayer-Fixman method.
Experimental Section
1'0
30 Sin28/2 t 5 0 0 C
5'0
Figure 1. Zimm plot for sodium cellulose xanthate (S-I, F-2) in 1 M NaOH.
Viscometry. Viscosity measurements were carried out at 30 f 0.1" using a Ubbelohde-type capillary viscometer having a flow time of 120 sec with respect to water. Kinetic energy corrections were negligible, and no correction for shear rate dependence was applied. Light Scatteying. Light-scattering measurements were carried out in a Brice-Phoenix Universal lightscattering photometer, Series 1999-10, at the wavelength of 546 mp. The photometer was calibrated with Cornel1 standard polystyrene. Appropriate corrections for reflection and volume effects were applied. Correction for refraction at the cell wall was accomplished by multiplica,tion by the square of the refractive index of the solution which was assumed to be essentially equal 'to the solvent refractive index. The solvent scattering, after similar treatment, was subtracted from that of the solution. The data were treated according to the method of Zimm, the dn/dc values used in these calculations being 0.20 ml/g.lO A typical Zimm plot is shown in Figure 1.
Preparation of Samples. Sodium cellulose xanthate (3) B. Das and P. K. Choudhury, J . PoEym. Sei., Part A-I, 769 was prepared in the laboratory from filter paper and (1967). cotton cellulose according to the method described (4) W.H.Stockmayer and M. Fixman, ibid., Part C-1, 137 (1963). earlier.3 The two samples were fractionated into three (5) IM. Kurata and H. Yamakawa, J . Chem. Phys., 29, 311 (1958). fractions each by precipitation with methan01.~~~(6) W.H.Stockmayer and A. C. Albrecht, J . Polym. Sci., 32, 215 (1958). The fractionated samples were washed with methanol, (7) J. M.G. Cowie and 8. Bywater, Polymer, 6, 197 (1965). thoroughly solvent-exchanged with ether, and finally (8) B. Das, Dissertation, Calcutta University, 1968. dried under vacuum. Samples were dissolved in 1 (9) K. K. Ghosh and P. K. Choudhury, Makromol. Chem., 102, 217 M NaOH and stored at -5". The degrees of substitu(1967). tion (D. S.) of the samples were determined in the same (10) C. W.Tait, R. J. Vetter, J. M. Swanson and P. Debye, J . Polym. Sci., 7,261 (1951). way as described earlier.3 The Journal of Physical Chemistry
3415
UNPERTURBED DIMENSIOX OF SODIUM CELLULOSE XANTHATE Sedimentation Velocity. Sedimentation velocity mea-
surements were made in a Spinco Model E analytical ultracentrifuge equipped with a rotor temperature control (RTIC) unit, schlieren optics, and a phase plate. A Spinco An-D rotor and Kel-F center piece was used. The sedimentation coefficients were calculated from the distance of the maximum of the refractive index gradient curve to the image of the reference hole measured by a microcomparator. Extrapolation of sedimentation coefficient, X, to infinite dilution was achieved using the method of Newman, Loeb, and Conrad," who proposed the relation (1/S2 - l/Si)(C2
- Ci)
k
+ ~ ( C +I C,)
30-
(4)
where the subscripts 1 and 2 refer to any pair of data. The value of partial specific volume used was 0.603. The relevant data are presented in Tables I and I1 and Figures 2 and 3.
0
M$ Figure 2.
Is],
Rw x
dl/g
10-6
10s, cm
D.S.
F-1 0.78 F-2 F-3
6.6 2.6 1.5
1.25 0.33 0.14
3438 2099 1516
S-I1 D.S.
F-1 0.82 F-2
5.1 3.5 2.9
0.91 0.53 0.39
2271 1798 1216
s-I
x
(R%l/2
K
x
10'
F-3
6'0
-
(RovM)1/2 X 10s. om
1.38
2*ol I
Table I1 : Sedimentation and Intrinsic Frictional Coefficient Data for Sodium Cellulose Xanthate in 1 M NaOH Solution (RO2/M)'/9
So, Sved-
1j1 X
bergs
10'
F-1 F-2 F-3
3.23 2.55 2.43
2.19 0.33
F-1 F-2 F-3
2.86 2.17 2.56
1.80 1.36 0.86
Sample
s-I D.S. = 0.78
Kf X 108
X lW,
1.11
M(1
-
VP)/SON*
I
48
I
1'2
I
~~4 x t d Figure 3. Stockmayer-Fixman plot for sodium cellulose xanthate.
Plots of [f]M-'/2 us. M'I2 according to Cowie and Bywater are shown in Figure 2. The unperturbed dimension (&2/M)1/z,according to the Stockmayer-Fixman method, was obtained from a [77]M-'/' us. M i l 2 plot (Figure 3). The Kurata-Yamakawa value of +o = 2.87 X 1021was used.5 Heterogeneity corrections were applied by the relation12
4. = [(h Results and Discussion The frict,ional coefficient fo was calculated according to the Svedberg equation fo =
I
44
am
0.74 5.50
s-I1 D.S. = 0.82
Cowie-Bywater plot for sodium cellulose xanthate.
X
2.90 =
xi63
1
Table I : Intrinsic Viscosity and Light-Scattering Data for Sodium Cellulose Xanthate and 1 M XaOH Solution Sample
1'5
0'9
0'3
(5)
N A is the Avogadro number, and the intrinsic frictional
+ I)/@ + 2 ) 1 ' W h + 3/2)(h+ I ) - ' / ~ r ( h + 1)-l
(6)
where is the gamma function and h = 4 for normally fractionated polymers. Considering the experimental errors involved, the (Z@/M) '/' value obtained by the Cowie-Bywater method is found t o be in good agree-
coefficient [f]is
If1 = fo/m T~
is the solvent viscosity in poises
(70 =
0.01104 P).
(11) 5. Newman, L.Loeb, and C. M. Conrad, J. Polym. Sci., 10,463 (1963). (12) M. Kurata and W. H. Stockmayer, Fortschr. Hochpolymer. Forsch., 3 , 196 (1963). volume 76,Number 10 October 1969
B. DAS,A. K. RAY,AND P. K. CHOUDHURY
3416
ment with that obtained from the Stockmayer-Fixman method. Values close to that of sodium cellulose xanthate have been obtained for the unperturbed dimensions of a number of cellulose derivatives, e.g., ethyl hydroxethyl cellulose (1.30 X methyl cellulose (1.50 X lo-*), the data in parentheses being the (R02/M)'/' values.12 Brown and Henley13 noted a very close agreement between the K values of cellulose tricaproate (K = 2.45 X low3) and hydroxyethyl cellulose (K = 2.50 X loms)and also that the configuration of a number of cellulose derivatives in cadoxen is essentially independent of the nature of the substituent. l4 The approximate constancy of the unperturbed coil extensions of a series of cellulose derivatives with increasing weight of the substituents has also been demonstrated.15 It therefore appears to substantiate the view that cellulose backbone is the dominant factor in determining the extension of the chain and consequently its hydrodynamic properties, while the substituents play a relatively minor role. I n this connection, cognizance should also be taken of various shortcomings inherent in both these treatments. The Stockmayer-Fixman treatment assumes that drainage corrections are negligibly small and the relationship5 cy3
= 1
+ 1.552 + . . .
1
+ 0 . 6 0 9 ~+ . . .
(8)
Burchard16 observed that [q]M-'/' us. MI'* plots show curvature at high values of x and attributed this to the
The Journal of Physical Chemistry
+
AcknowZedgment. Thanks are due to the University Grants Commission, New Delhi, for financial support to B. D. and A. K. R.
(7)
could be interrupted after the first power of z without introducing serious errors. The Cowie-Bywater treatment also involves similar tacit assumption regarding the Stockmayer-Albrecht relation6 acf =
neglect of drainage corrections. He proposed an elaborate graphical procedure to rectify this defect, but it has recently been shown by Banks and Greenwood'' that application of this procedure to amylose acetate results in widely different values for K in different solvents. They therefore prefer the original Stockmayer-Fixman plot to this modification. Baumann, l8 on the other hand, suggests that the deviations observed a t high values of z are due to omission of higher powers of x in the Kurata-Yamakawa relation and proposes 2z, be employed that the Fixman equation,lg a3 = 1 instead. Thus, while an adjustment of the numerical constant was made by Stockmayer and Fixman to accommodate the viscosity expansion factor a?,, Baumann's relation follows directly from the Fixman equation. However, this too does not result in any significant improvement as it only extends the range of applicability but does not eliminate the curvature completely. Finally, absence heretofore of an unambiguous value for the Flory constant 40 adds to the uncertainty of the unperturbed dimensions obtained from good solvent data.
(13) W. Brown and D. Henley, Makromol. Chem., 108, 153 (1967). (14) W. Brown, Tappi, 49,367 (1966). (15) W. Brown and D. Henley, Makromol. Chem., 75, 179 (1964). (16) (a) W. Burchard, Makromol. Chem., 50, 20 (1961); (b) paper Bresented at International Symposium on the Solution Behavior of Natural Polymers, Edinburgh, July 1967. (17) W. Banks and C. T. Greenwood, Makromol. Chem., 114, 246 (1968). (18) H. Baumann, J . Polym. Sci., Part B-3, 1069 (1965). (19) M. Fixman, J. Chem. Phys., 40, 1506 (1964).
