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Taylor dispersion measurement of the diffusivities of polymethylenes in dilute solutions
Chemical Engineering Science Vol. 40, No. 3, pp. 521-526, Printed in Great Britain.
ooo9-2509/85 53.00 + .lhJ o 1985 Pcrgamon Press Ltd.
1985
TAYLOR DISPERSION MEASUREMENT DIFFUSIVITIES OF POLYMETHYLENES DILUTE SOLUTIONS HANG-CHANG Department
CHEN
and SHAW-HORNG
OF THE IN
CHEN*
of Chemical Engineering, University of Rochester, Rochester, NY
14627, U.S.A.
(Receiued 26 September 1983; accepted 16 April 1984) Abstract-Experimental conditions are presented under which the Taylor dispersion technique is capable of generating accurate translational diffusion coefficients for polymers in dilute solutions across a wide temperature range. With the Monte Carlo results on polymethylene conformation, the KirkwoodRiseman theory is found to predict quite well the diffusivities of a series of normal alkane solutes, from n-hexane to n-hexacosane, in carbon tetrachloride,toluene and chlorobenxenefrom 293 to 373 K. ~ODUCLION
coefficients have diffusion implications on the hydrodynamic important behavior of chain-like polymers in dilute solutions. Although the diffusion data have been gathered over the years with the quasi-elastic light scattering (QELS) or interferometry technique, the results are probably questionable according to Vrentas and Duda[l, 21. In addition, the applicability of the Kirkwood-Riseman theory[3] to flexible chain polymers in extremely dilute solutions remains controversial up to the present[4,5]. As a consequence, it is imperative that accurate diffusion data be gathered before meaningful testing of the theory can be accomplished. In an effort to provide a reliable experimental technique, we have adapted the Taylor dispersion (TD) technique[d, 7j for determining diffusivities of polymers in extremely dilute solutions across a wide temperature range. The versatility of the TD technique has recently been demonstrated[& 93, but when it comes to polymer solutions two additional factors must be considered. Firstly, it is well known that even extremely dilute polymer solutions may exhibit non-Newtonian behavior [ lo]. Secondly, dissolved polymers in a non-uniform flow field may also diffuse from a high to low shear stress region. The latter phenomenon is entropic in its origin and is referred to as stress-induced diffusion [ 111. Reliable data can be obtained under experimental conditions where both phenomena peculiar to polymer solutions are suppressed to a tolerable level. Thus, it is important that the flow behavior of a dilute polymer solution be examined and that the extent to which the stress-induced diffusion takes place under experimental conditions be evaluated. In this paper we look into the conditions under which the Taylor dispersion method can be employed to determine The
translational
*Author to whom correspondenceshould be addressed. 521
polymer-diffusion coefficients in extremely dilute solutions. The extension of this technique for an investigation on the concentration dependence is also attempted. We also report the diffusivities of a series of normal alkanes, from n-hexane to n-hexacosane, in carbon tetrachloride, toluene and chlorobenzene from 293-373 K. The results are discussed in terms of the Kirkwood-R&man theory using the Monte Carlo results on polymethylene conformation. EXPERIMENTAL Materials
The solvents carbon tetrachloride (99 + %, J. T. Barker), toluene (99 + %, MCB), chlorobenzene 99 + %, Aldrich) and I-butanol (99%, Aldrich) were filtered before usage on all-glass filtration device with 0.5 pm teflon membrane (Millipore). The solutes n-hexane (99%, Sigma), n-decane (99 + %, Sigma), n-tetradecane (99%, Sigma), n-octadecane (97%, Aldrich), n -docosane (99%, Aldrich) and n -hexacosane (99x, Aldrich) were all used as received without further purification. Viscosity
measurements
The concentric cyclinder viscometer is a modification of Zimm and Crother’s original design[l2]. The test tube like rotor (9 mm ID, 10 mm OD, and 50mm long) is made of aluminum. The clearance between the rotor and the outer tube made of Pyrex glass is 1 mm. The rotor is floated by buoyance and centered by surface forces of the liquid. Constant temperature water (298.15 + 0.05 K) is circulated around the viscometer. Only 3 ml liquid is needed of each measurement. A permanent magnet is driven by a variable speed precision motor (G. K. Heller Corp.), the rotational speed of which measured with a tachometer (Cole-Parmer) is stable to within + 0.3% in the shear rate range 0 - 40 set-’ of interest to diffusion measurements. The rotation of the aluminum rotor driven by the induced eddy
CHEN and
HANG-CHANG
SHAW-HORNG
CHEN
Figure 2 shows the shear stress-shear strain relationship for polymethylene/carbon tetrachloride solutions at concentrations that are injected in diffusion measurements. It is clear that for noctadecane at 1.8 wt% the flow behavior deviates from Newtonian behavior around a shear rate of 40 set-‘. However, for shear rates below 32 set-’ the solution exhibits Newtonian behavior, and the solution viscosity differs from solvent viscosity by less than 5%. All toluene and chlorobenzene solutions show Newtonian behavior with viscosities identical to solvent viscosity to within + 3% (Table 1). The implications of these viscosity measurements on the accuracy of measured diffusivities are discussed in the following section. p=o_904cp I reference I-BuOH
q
jJ-
2.571
i 2.570
cp
(0.868cp F 10
D@usion measurements The Taylor dispersion apparatus for determining tracer diffusivities (i.e. binary diffusivities at infinite dilution) over extended temperature ranges was as described previously[14]. If Taylor’s original analysis[6, 71 is to apply, two conditions have to be satisfied: (1) that the liquid is Newtonian with constant viscosity; and (2) that the stress-induced diffusion is absent and molecular diffusion follows Fick’s law with constant diffusivity. That the stressinduced diffusion is absent at the flow rates (less than 0.075 ml/min) employed in this work can be substantiated with Janssen’s study on slip phenomena observed for polymer solutions in an inhomogeneous stress field [15]. Under our experimental conditions py,fC,RT is on the order of lo-‘, where 9, is the shear rate at tube wall, C, the polymer concentration in stress-free region, and R the gas constant. From Fig. 2 of Janssen’s paper[ IS] we conclude that the stress-induced diffusion is practically ml. The absence of stress-induced diffusion is further supported by Brunn’s hydrodynamic theory [21] for polymer migration across streamlines. Specifically, the time scale
I CP ;Rdld
)
iR&l3)
cd1
20
30
4
Fig. 1. Calibration and testing of the concentric cylinder viscometer. current is monitored visually with a LCD timer (to 0.01 set). Repeated period measurements show a precision of + O.S”A. In the formulae for computing shear stress and shear strain given by Zimm and Crothers[l2] we have f(oi, Do) = 0.986 for the present design, where Di and D,, are rotor OD and Pyrex tube ID, respectively. The viscosities of lbutanol and n-decane at 298.15 K based on the viscometer calibration using carbon tetrachloride at same temperature are 2.571 and 0.869 CP, respectively (Fig. 1). Both values compare favorably with those available in the literature[ 131.
0.4 -
0.3
-
@a 298.2
'K
cc+ E u 0.2 -:
0.1
2 D
n-cc
3 0
n-C,O/
cc14
4
A
Fig.
2.
10
20
cc14
n-C,4/’
CC14
5 v
bC18/
cc14
6 0
“-
CC14
F OY
/
C26,’
1.5
%wt.
(*-‘I
I 30
40
50
Shear stress-shear strain relationship for dilute solutions of normal alkanes in carbon tetrachloride at 298.2 K
Measurement of polymethylenesin dilute solutions
523
Table 1. Shear viscositiesof dilute solutions in the shear rate range from 0 to 35 see-’ at 298.15 K
tetrachloride w0 =
Carbon b
0.904)
II-C 6 0.65
3
_n-C14
-“+8
0.9
1.3
1 .8
1.5
0.890
0.892
0.904
0.945
0.930
to1 uene
(&
-“-C26
= 0.552)
WC6
5___-__
“-Cl4
“-‘18
“-C22
“-‘26
0.45
0.6
0.8
1 .l
1.45
1 .75
0.548
0.552
0 -555
0.556
0.563
0.570
“-ca6
chlorobenzene
(M,-, = 0.753)
‘l-C6
“-5
“-Cl 4
“-Cl 8
n-C22
0.35
0.5
0.6
0.7
0.9
1 .3
0.745
0.763
0.755
0.760
0.769
0.778
aFor each solvent wt%. the second data .is 2 1%.
0
the first number represents concentration in shear viscosity In CF. Precision of viscosity
for hexacosane migration is found to be on the order of lo6 hr, which is considerably longer than the dispersion time, 6 hr, for diffusion measurement. There should be no question about the Fickian behavior of polymer diffusion in dilute solutions, but the constancy of diffusivity shall be discussed along with the flow behavior in what follows. In our dispersion measurements a dilute solution is injected as a pulse into the flowing solvent. The result of axial dispersion downstream from injection is that the solute concentration diminishes continuously. Hence, one would expect the solute concentration to vary from that of the injected solution to zero (on both ends of the dispersion curve) throughout the whole dispersion process. As a result, the viscosity is also non-uniform as dispersion proceeds, which in turn will affect the accuracy of the measured diffusivity. It is thus important, particularly in dealing with polymer solutions, that the variation of viscosity in the course of dispersion be carefully controlled. In the extreme case of n-octadecane in carbon tetrachloride, 5% difference in viscosity between the injected solution and solvent probably would cause 3% error at most in the observed value; concentration independence is discussed below. From the solution and solvent viscosities listed in Table 1, one can conclude that the diffusivities reported here are generally accurate to within + 2%, which is about
the precision one would expect of the Taylor persion technique. RESULTS
AND
dis-
DISCUSSION
As first attempt at using the Taylor dispersion technique for determining polymer diffusivities in dilute solutions as a function of temperature, we have measured the diffusivities of a series of normal alkanes (from n-hexane to n-hexacosane) in carbon tetrachloride, toluene and chlorobenzene from 293 to 373 K. Measurements were performed under conditions where accuracy is comparable to precision. The observed diffusivities of n-hexacosane in chlorobenzene at 373.2 K are 1.76 f 0.02, 1.81 _t 0.03 and I.75 + 0.02 (in IO-%m2/sec) at injected concentrations 0.5, 1 .O and 1.5 wt”/,, respectively, suggesting that experimental data (Table 2) are virtually tracer diffusivities. Each diffusivity is the average of at least three measurements. In Table 3 a comparison is made between present results and those obtained previously with proton magnetic resonance. (PMR) spinecho[l6] or interferometry[l7] method. Significant disagreement is observed for n-decane/carbon tetrachloride system at 50°C. This discrepancy is not surprising in view of the scattering of PMR spin-echo data at low concentrations (Fig. 1 of Ref. [16]). According to the classical Kirkwood-Riseman theory[3], the translational diffusion coefficient of a
HANG-CHANG
524
CHEN
and SHAW-HORNGCHEN
Table 2. Diffusivitiesof polymethylenesin dilute solutions from 293 to 373 K so1 bK
WC
utesa
1
6
I
“-Cl0
n-C1
Solvent
“-?
I
4
Carbon
8
“-%2
I
“-56
I
Tetrachlorfde
293.7
1 .28
2
13.02~
1 .oo
+ 0.01
0.80
+ 0.01 -
0.67
+O.Ol
0.54
+0.03
308.2
1.65
5
0.01
1.26
50.01
1 .oo
+ 0.01
0.86
50.01
0.69
50.01
323.2
2.02
-+ 0.01
1.53
50.01
1.22
io.01
1.10
zO.01
2.51
20.01
t-66
50.01
1.51
+ 0.02 -
1.39
50.01 +0.03
0.85
337.2
1 .08
+ 0.01
353.2c
2.97
20.02
2.24
LO.01 -
1 .83
+ 0.01 -
1 .54
To.01 -
1.29
-,O.Ol
300.2
2.41
+ 0.01
1 .87
+ 0.01
1.45
+ 0.01
1 .24
+ 0.01
0.99
zo.01
3.25
5
0.02
2.41
5
0.02
1.97
5
1 .66
+ 0.01 + 0.01
1.09
323.2
1.45
1.30
+0.03
346.2
3.99
5 _
0.04
3.21
-:
0.04
2.61
50.04 -
2.26
; -
0.01
1.99
373.2
5.57
-+ 0.09
4.22
-+ 0.03
3.39
-+ 0.06
2.91
+ 0.01 -
2.65
+0.02 + 0.02 + 0.03 -
Tol uene
0.01
1 .79
-+ 0.01
2.43
-+ 0.03
Chl orobenrene 298.2
1.76
+ 0.01
1.85
+ 0.03
1.07
+
0.02
0.92
+ -
0.01
0.74
323.2
2.52
TO.03
1.86
5
0.02
1.53
+
0.02
1 .32
-+
0.03
1.04
0.02
348.2
3.38
:
0.02
2.50
5
373.2
4.32
-+
0.04
3.11
IO.02
a”-C6 = “-‘26 b
Number
‘System
n-Hexane;
“-Cl
0
=
n-Decane;
+ 0.01
0.67
+
0.01
+ 0.04
0.99
-+
0.01
2.03
-T
0.02
1.78
+
0.05
1 .56
+ 0.03
1 .23
+ 0.01
2.51
-+
0.01
2.29
-+
0.02
1 .89
; -
0.01
1.78
Y _
=
n-Docosane;
= n-Tetradecane.
“-Cl4
.-
“-Cl8
=
n-Octadecane;
“-C22
0.04
= n-Hexacosane. give”
pressure:
in
105D,
7
cm2/sec_
psig.
h the solvent viscosity and [R -‘I evaluated from
flexible chain molecule of x segments in a dilute solution can be predicted from the following expression:
I
[R-l]=
of diffusion data (in lo5 D, cr&/sec) between this work and previous results
sys tern
T.‘K
this
previous
work
n-C6/CCl
298.2
n-c,
298.2
“-c18/cc14
323.2
n-c,
o/ cc1 4
1.53
c1 .PO
323.2
n-c,
8/cc1
1 .a0
‘1
aI”terpolated
4
0/Ccl
at
4
al
-39
‘1
.08
results
298.2
type
(r;‘)
ao -73
4
298.2’K
from
equation.
bData
take”
from
Ref.
17.
‘Data
take”
from
Ref.
16.
(2)
where (r; ‘) is the mean reciprocal distance of separation between segments i andj. In most previous applications of the theory[4, 51, the solute chain
in which < is the friction coefficient for each segment,
Table 3. A comparison
1
our
bl
.50
bl
.08
‘0.69
data
at
293-373’K
.80
using
Arrhenius
Measurement of polymethylenes in dilute solutions
I’ I
,”
Gaussian
chain
a’’ /’
525
molecules were assumed to be in random coil, i.e. Gaussian chain. It appears to us that the more realistic molecular conformation should result from a consideration of rotational isomers. To this effect, we believe that Monte Carlo simulations of polymethylenes by Paul and Maze [ 181 shall better represent the [R-l] value. It is made clear in Fig. 3 that there is a considerable discrepancy between the random coil and Monte Carlo results. The lirst term on the r.h.s. of eqn (1) represents the draining of solvent molecules through the solute molecule. When Stokes’ law is applied to the friction coefficient [, c=&r/I&l
(3)
where a is the radius of the friction center. In the present case of polymethylene solutes a = 0.77 di is the van der Waals radius of a methylene group. Equation (1) is then rewritten as
(4) Fig. 3. A comparison of [R -‘I betweenGaussian chain and Monte Carlo results.
The data reported in Table 2 are compared to the prediction using eqn (4) in Fig. 4. The absolute
Carbon
87-
A
298.2’K
0
323.2’
0
348.2’~
0
373.2
Tetrachloride
K o K
6-
Chlorobezene
Fig. 4. Experimental diffusivities compared to Kirkwood-
Riseman theory using Monte Carlo results for [I? -I]. Solid curves represent theoretical prediction.
526
HANG-CHANG
CHEN
and SHAW-HORNG Cnrsr
average deviation between theory and experiment is 4%, and the maximum deviation is 10%. In contrast, the prediction using the Wilke-Chang equation[l9] results in an absolute average deviation of 11% and a maximum deviation of 25%. The present study on polymer diffusion in an extremely dilute solution, where concentration dependence is practically absent, can be readily extended to the investigation on the concentration dependence of polymer diffusivity. Instead of pure solvent a polymer solution shall be delivered continuously through the dispersion coil. A pulse of solution slightly richer in solute polymer shall then be injected into the flowing solution. The accuracy with which the polymer diffusivity can be determined depends on the variation of flow behavior as a result of the dispersion process. For instance, if the flowing solution is a power-law fluid, the diffusivity can be accurately obtained from the dispersion measurement following an analysis[20] similar to Taylor’s as long as the variation of the flow index n in the dispersion process can be carefully controlled. This can be achieved by minimizing the concentration difference between the flowing solution and the injected pulse and by delivering the solution at a lower speed. CONCLUSIONS
In summary, we would like to recapitulate the main points resulting from this work: (1) Under carefully designed experimental conditions the Taylor dispersion technique can be adapted for accurately measuring the diffusivities of polymers in dilute solutions across a large temperature range. (2) The Kirkwood-Riseman theory is applicable to the diffusion of long-chain hydrocarbons from 293 to 373 K if Monte Carlo simulation results for [R-l] are used in the prediction. Acknowledgements-The authors are grateful for the financial support of this work by the National Science Foundation under Grant CPE-8305649. Helpful comments raised by the reviewers are also gratefully acknowledged. NOTATION a C0 D
van der Waals radius of a friction center, A solute concentration in the stress-free region, g-mole/l. translational diffusion coefficient of polymer in dilute solution, cm2/sec.
Di
D0 k
(rii’) [R-i] T X
Greek
inner diameter of the concentric cylinder viscometer, mm. outer diameter of the viscometer, mm. Boltzmann constant, 1.38 x lo-l6 erg/K. mean reciprocal distance of separation between segments i and j, A-‘. as defined by eqn (2), A-‘. absolute temperature, K. number of segments in a chain molecule
symbols solution shear viscosity, CP. p
h fv 6
solvent shear viscosity, CP. shear rate at tube wall, see-‘. friction coefficient, g/set
REFERENCES
HI Vrentas J. S. and Duda J. L., J. Appl. Polymer Sci. 1976 20 1125. PI Vrentas J. S. and Duda J. L.. J. Polymer Sci. Polymer Phys. Edn. 1976 14 101. 131 Kirkwood J. G. and Riseman J., .7. Chem. Phys. 1948 16 565. 141 Edwards C. J. C., Rigby D. and Stepto R. F. T., Macromolecules 198 1 14 1808. 151 Schmidt M. and Burchard W., Macromolecules 1981 14 210. Taylor G. I., Proc. Roy. Sot. London 1953 A219 186. t: Taylor G. I., Proc. Roy. Sot. London 1954 A225 473. 181 Chen S. H.. Davis H. T. and Evans D. F., J. Chem. Phys. 198 1 75 1442. PI Chen S. H., Evans D. F. and Davis H. T., A.Z.Ch.E.J. 1983 29 640. 1101 Bird R. B., Armstrong R. C. and Hassager O., Dynamics of Polvmeric Liauids. Vol. 1. Wiley. New York 1977. Sci. Polymer 1111 Tirrel M: and Malone M. F., J. Po&er Phys. Edn 1977 15 1569. [l21 Zimm 3. H. and Crothers D. M., Proc. Nat. Acad. Sci. (U.S.A.) 1963 48 905. 1131 Raznjevii: K., Handbook of Thermodynamic Tables and Charts. McGraw-Hill, New York 1976. I151 Chen S. H. and Yumet E., Chem. Engng Sci. in press (1983). 1151 Janssen L. P. B. M., Rheol. Acta 1980 19 32. 1161 McCall D. W. and Douglass D. C., J. Chem Phys. 1963 38 2314. t171 Dewan R. K. and Van Holde K. E., J. Chem Phys. 1963 39 1820. Paul E. and Mazo R. M., J. Chem. Phys. 1968 48 1405. 1953 1 264. ;:;; Wilke C. R. and Chana P., A.Z.Ch.E.J. c201 Fan L. T. and Hwang W. s., Proc. Roy. Sot. London 1965 A283 576. 1211 kunn P. O., Znt. J. Mtdtiphase Flow 1983 9 187.
ooo9-2509/85 53.00 + .lhJ o 1985 Pcrgamon Press Ltd.
1985
TAYLOR DISPERSION MEASUREMENT DIFFUSIVITIES OF POLYMETHYLENES DILUTE SOLUTIONS HANG-CHANG Department
CHEN
and SHAW-HORNG
OF THE IN
CHEN*
of Chemical Engineering, University of Rochester, Rochester, NY
14627, U.S.A.
(Receiued 26 September 1983; accepted 16 April 1984) Abstract-Experimental conditions are presented under which the Taylor dispersion technique is capable of generating accurate translational diffusion coefficients for polymers in dilute solutions across a wide temperature range. With the Monte Carlo results on polymethylene conformation, the KirkwoodRiseman theory is found to predict quite well the diffusivities of a series of normal alkane solutes, from n-hexane to n-hexacosane, in carbon tetrachloride,toluene and chlorobenxenefrom 293 to 373 K. ~ODUCLION
coefficients have diffusion implications on the hydrodynamic important behavior of chain-like polymers in dilute solutions. Although the diffusion data have been gathered over the years with the quasi-elastic light scattering (QELS) or interferometry technique, the results are probably questionable according to Vrentas and Duda[l, 21. In addition, the applicability of the Kirkwood-Riseman theory[3] to flexible chain polymers in extremely dilute solutions remains controversial up to the present[4,5]. As a consequence, it is imperative that accurate diffusion data be gathered before meaningful testing of the theory can be accomplished. In an effort to provide a reliable experimental technique, we have adapted the Taylor dispersion (TD) technique[d, 7j for determining diffusivities of polymers in extremely dilute solutions across a wide temperature range. The versatility of the TD technique has recently been demonstrated[& 93, but when it comes to polymer solutions two additional factors must be considered. Firstly, it is well known that even extremely dilute polymer solutions may exhibit non-Newtonian behavior [ lo]. Secondly, dissolved polymers in a non-uniform flow field may also diffuse from a high to low shear stress region. The latter phenomenon is entropic in its origin and is referred to as stress-induced diffusion [ 111. Reliable data can be obtained under experimental conditions where both phenomena peculiar to polymer solutions are suppressed to a tolerable level. Thus, it is important that the flow behavior of a dilute polymer solution be examined and that the extent to which the stress-induced diffusion takes place under experimental conditions be evaluated. In this paper we look into the conditions under which the Taylor dispersion method can be employed to determine The
translational
*Author to whom correspondenceshould be addressed. 521
polymer-diffusion coefficients in extremely dilute solutions. The extension of this technique for an investigation on the concentration dependence is also attempted. We also report the diffusivities of a series of normal alkanes, from n-hexane to n-hexacosane, in carbon tetrachloride, toluene and chlorobenzene from 293-373 K. The results are discussed in terms of the Kirkwood-R&man theory using the Monte Carlo results on polymethylene conformation. EXPERIMENTAL Materials
The solvents carbon tetrachloride (99 + %, J. T. Barker), toluene (99 + %, MCB), chlorobenzene 99 + %, Aldrich) and I-butanol (99%, Aldrich) were filtered before usage on all-glass filtration device with 0.5 pm teflon membrane (Millipore). The solutes n-hexane (99%, Sigma), n-decane (99 + %, Sigma), n-tetradecane (99%, Sigma), n-octadecane (97%, Aldrich), n -docosane (99%, Aldrich) and n -hexacosane (99x, Aldrich) were all used as received without further purification. Viscosity
measurements
The concentric cyclinder viscometer is a modification of Zimm and Crother’s original design[l2]. The test tube like rotor (9 mm ID, 10 mm OD, and 50mm long) is made of aluminum. The clearance between the rotor and the outer tube made of Pyrex glass is 1 mm. The rotor is floated by buoyance and centered by surface forces of the liquid. Constant temperature water (298.15 + 0.05 K) is circulated around the viscometer. Only 3 ml liquid is needed of each measurement. A permanent magnet is driven by a variable speed precision motor (G. K. Heller Corp.), the rotational speed of which measured with a tachometer (Cole-Parmer) is stable to within + 0.3% in the shear rate range 0 - 40 set-’ of interest to diffusion measurements. The rotation of the aluminum rotor driven by the induced eddy
CHEN and
HANG-CHANG
SHAW-HORNG
CHEN
Figure 2 shows the shear stress-shear strain relationship for polymethylene/carbon tetrachloride solutions at concentrations that are injected in diffusion measurements. It is clear that for noctadecane at 1.8 wt% the flow behavior deviates from Newtonian behavior around a shear rate of 40 set-‘. However, for shear rates below 32 set-’ the solution exhibits Newtonian behavior, and the solution viscosity differs from solvent viscosity by less than 5%. All toluene and chlorobenzene solutions show Newtonian behavior with viscosities identical to solvent viscosity to within + 3% (Table 1). The implications of these viscosity measurements on the accuracy of measured diffusivities are discussed in the following section. p=o_904cp I reference I-BuOH
q
jJ-
2.571
i 2.570
cp
(0.868cp F 10
D@usion measurements The Taylor dispersion apparatus for determining tracer diffusivities (i.e. binary diffusivities at infinite dilution) over extended temperature ranges was as described previously[14]. If Taylor’s original analysis[6, 71 is to apply, two conditions have to be satisfied: (1) that the liquid is Newtonian with constant viscosity; and (2) that the stress-induced diffusion is absent and molecular diffusion follows Fick’s law with constant diffusivity. That the stressinduced diffusion is absent at the flow rates (less than 0.075 ml/min) employed in this work can be substantiated with Janssen’s study on slip phenomena observed for polymer solutions in an inhomogeneous stress field [15]. Under our experimental conditions py,fC,RT is on the order of lo-‘, where 9, is the shear rate at tube wall, C, the polymer concentration in stress-free region, and R the gas constant. From Fig. 2 of Janssen’s paper[ IS] we conclude that the stress-induced diffusion is practically ml. The absence of stress-induced diffusion is further supported by Brunn’s hydrodynamic theory [21] for polymer migration across streamlines. Specifically, the time scale
I CP ;Rdld
)
iR&l3)
cd1
20
30
4
Fig. 1. Calibration and testing of the concentric cylinder viscometer. current is monitored visually with a LCD timer (to 0.01 set). Repeated period measurements show a precision of + O.S”A. In the formulae for computing shear stress and shear strain given by Zimm and Crothers[l2] we have f(oi, Do) = 0.986 for the present design, where Di and D,, are rotor OD and Pyrex tube ID, respectively. The viscosities of lbutanol and n-decane at 298.15 K based on the viscometer calibration using carbon tetrachloride at same temperature are 2.571 and 0.869 CP, respectively (Fig. 1). Both values compare favorably with those available in the literature[ 131.
0.4 -
0.3
-
@a 298.2
'K
cc+ E u 0.2 -:
0.1
2 D
n-cc
3 0
n-C,O/
cc14
4
A
Fig.
2.
10
20
cc14
n-C,4/’
CC14
5 v
bC18/
cc14
6 0
“-
CC14
F OY
/
C26,’
1.5
%wt.
(*-‘I
I 30
40
50
Shear stress-shear strain relationship for dilute solutions of normal alkanes in carbon tetrachloride at 298.2 K
Measurement of polymethylenesin dilute solutions
523
Table 1. Shear viscositiesof dilute solutions in the shear rate range from 0 to 35 see-’ at 298.15 K
tetrachloride w0 =
Carbon b
0.904)
II-C 6 0.65
3
_n-C14
-“+8
0.9
1.3
1 .8
1.5
0.890
0.892
0.904
0.945
0.930
to1 uene
(&
-“-C26
= 0.552)
WC6
5___-__
“-Cl4
“-‘18
“-C22
“-‘26
0.45
0.6
0.8
1 .l
1.45
1 .75
0.548
0.552
0 -555
0.556
0.563
0.570
“-ca6
chlorobenzene
(M,-, = 0.753)
‘l-C6
“-5
“-Cl 4
“-Cl 8
n-C22
0.35
0.5
0.6
0.7
0.9
1 .3
0.745
0.763
0.755
0.760
0.769
0.778
aFor each solvent wt%. the second data .is 2 1%.
0
the first number represents concentration in shear viscosity In CF. Precision of viscosity
for hexacosane migration is found to be on the order of lo6 hr, which is considerably longer than the dispersion time, 6 hr, for diffusion measurement. There should be no question about the Fickian behavior of polymer diffusion in dilute solutions, but the constancy of diffusivity shall be discussed along with the flow behavior in what follows. In our dispersion measurements a dilute solution is injected as a pulse into the flowing solvent. The result of axial dispersion downstream from injection is that the solute concentration diminishes continuously. Hence, one would expect the solute concentration to vary from that of the injected solution to zero (on both ends of the dispersion curve) throughout the whole dispersion process. As a result, the viscosity is also non-uniform as dispersion proceeds, which in turn will affect the accuracy of the measured diffusivity. It is thus important, particularly in dealing with polymer solutions, that the variation of viscosity in the course of dispersion be carefully controlled. In the extreme case of n-octadecane in carbon tetrachloride, 5% difference in viscosity between the injected solution and solvent probably would cause 3% error at most in the observed value; concentration independence is discussed below. From the solution and solvent viscosities listed in Table 1, one can conclude that the diffusivities reported here are generally accurate to within + 2%, which is about
the precision one would expect of the Taylor persion technique. RESULTS
AND
dis-
DISCUSSION
As first attempt at using the Taylor dispersion technique for determining polymer diffusivities in dilute solutions as a function of temperature, we have measured the diffusivities of a series of normal alkanes (from n-hexane to n-hexacosane) in carbon tetrachloride, toluene and chlorobenzene from 293 to 373 K. Measurements were performed under conditions where accuracy is comparable to precision. The observed diffusivities of n-hexacosane in chlorobenzene at 373.2 K are 1.76 f 0.02, 1.81 _t 0.03 and I.75 + 0.02 (in IO-%m2/sec) at injected concentrations 0.5, 1 .O and 1.5 wt”/,, respectively, suggesting that experimental data (Table 2) are virtually tracer diffusivities. Each diffusivity is the average of at least three measurements. In Table 3 a comparison is made between present results and those obtained previously with proton magnetic resonance. (PMR) spinecho[l6] or interferometry[l7] method. Significant disagreement is observed for n-decane/carbon tetrachloride system at 50°C. This discrepancy is not surprising in view of the scattering of PMR spin-echo data at low concentrations (Fig. 1 of Ref. [16]). According to the classical Kirkwood-Riseman theory[3], the translational diffusion coefficient of a
HANG-CHANG
524
CHEN
and SHAW-HORNGCHEN
Table 2. Diffusivitiesof polymethylenesin dilute solutions from 293 to 373 K so1 bK
WC
utesa
1
6
I
“-Cl0
n-C1
Solvent
“-?
I
4
Carbon
8
“-%2
I
“-56
I
Tetrachlorfde
293.7
1 .28
2
13.02~
1 .oo
+ 0.01
0.80
+ 0.01 -
0.67
+O.Ol
0.54
+0.03
308.2
1.65
5
0.01
1.26
50.01
1 .oo
+ 0.01
0.86
50.01
0.69
50.01
323.2
2.02
-+ 0.01
1.53
50.01
1.22
io.01
1.10
zO.01
2.51
20.01
t-66
50.01
1.51
+ 0.02 -
1.39
50.01 +0.03
0.85
337.2
1 .08
+ 0.01
353.2c
2.97
20.02
2.24
LO.01 -
1 .83
+ 0.01 -
1 .54
To.01 -
1.29
-,O.Ol
300.2
2.41
+ 0.01
1 .87
+ 0.01
1.45
+ 0.01
1 .24
+ 0.01
0.99
zo.01
3.25
5
0.02
2.41
5
0.02
1.97
5
1 .66
+ 0.01 + 0.01
1.09
323.2
1.45
1.30
+0.03
346.2
3.99
5 _
0.04
3.21
-:
0.04
2.61
50.04 -
2.26
; -
0.01
1.99
373.2
5.57
-+ 0.09
4.22
-+ 0.03
3.39
-+ 0.06
2.91
+ 0.01 -
2.65
+0.02 + 0.02 + 0.03 -
Tol uene
0.01
1 .79
-+ 0.01
2.43
-+ 0.03
Chl orobenrene 298.2
1.76
+ 0.01
1.85
+ 0.03
1.07
+
0.02
0.92
+ -
0.01
0.74
323.2
2.52
TO.03
1.86
5
0.02
1.53
+
0.02
1 .32
-+
0.03
1.04
0.02
348.2
3.38
:
0.02
2.50
5
373.2
4.32
-+
0.04
3.11
IO.02
a”-C6 = “-‘26 b
Number
‘System
n-Hexane;
“-Cl
0
=
n-Decane;
+ 0.01
0.67
+
0.01
+ 0.04
0.99
-+
0.01
2.03
-T
0.02
1.78
+
0.05
1 .56
+ 0.03
1 .23
+ 0.01
2.51
-+
0.01
2.29
-+
0.02
1 .89
; -
0.01
1.78
Y _
=
n-Docosane;
= n-Tetradecane.
“-Cl4
.-
“-Cl8
=
n-Octadecane;
“-C22
0.04
= n-Hexacosane. give”
pressure:
in
105D,
7
cm2/sec_
psig.
h the solvent viscosity and [R -‘I evaluated from
flexible chain molecule of x segments in a dilute solution can be predicted from the following expression:
I
[R-l]=
of diffusion data (in lo5 D, cr&/sec) between this work and previous results
sys tern
T.‘K
this
previous
work
n-C6/CCl
298.2
n-c,
298.2
“-c18/cc14
323.2
n-c,
o/ cc1 4
1.53
c1 .PO
323.2
n-c,
8/cc1
1 .a0
‘1
aI”terpolated
4
0/Ccl
at
4
al
-39
‘1
.08
results
298.2
type
(r;‘)
ao -73
4
298.2’K
from
equation.
bData
take”
from
Ref.
17.
‘Data
take”
from
Ref.
16.
(2)
where (r; ‘) is the mean reciprocal distance of separation between segments i andj. In most previous applications of the theory[4, 51, the solute chain
in which < is the friction coefficient for each segment,
Table 3. A comparison
1
our
bl
.50
bl
.08
‘0.69
data
at
293-373’K
.80
using
Arrhenius
Measurement of polymethylenes in dilute solutions
I’ I
,”
Gaussian
chain
a’’ /’
525
molecules were assumed to be in random coil, i.e. Gaussian chain. It appears to us that the more realistic molecular conformation should result from a consideration of rotational isomers. To this effect, we believe that Monte Carlo simulations of polymethylenes by Paul and Maze [ 181 shall better represent the [R-l] value. It is made clear in Fig. 3 that there is a considerable discrepancy between the random coil and Monte Carlo results. The lirst term on the r.h.s. of eqn (1) represents the draining of solvent molecules through the solute molecule. When Stokes’ law is applied to the friction coefficient [, c=&r/I&l
(3)
where a is the radius of the friction center. In the present case of polymethylene solutes a = 0.77 di is the van der Waals radius of a methylene group. Equation (1) is then rewritten as
(4) Fig. 3. A comparison of [R -‘I betweenGaussian chain and Monte Carlo results.
The data reported in Table 2 are compared to the prediction using eqn (4) in Fig. 4. The absolute
Carbon
87-
A
298.2’K
0
323.2’
0
348.2’~
0
373.2
Tetrachloride
K o K
6-
Chlorobezene
Fig. 4. Experimental diffusivities compared to Kirkwood-
Riseman theory using Monte Carlo results for [I? -I]. Solid curves represent theoretical prediction.
526
HANG-CHANG
CHEN
and SHAW-HORNG Cnrsr
average deviation between theory and experiment is 4%, and the maximum deviation is 10%. In contrast, the prediction using the Wilke-Chang equation[l9] results in an absolute average deviation of 11% and a maximum deviation of 25%. The present study on polymer diffusion in an extremely dilute solution, where concentration dependence is practically absent, can be readily extended to the investigation on the concentration dependence of polymer diffusivity. Instead of pure solvent a polymer solution shall be delivered continuously through the dispersion coil. A pulse of solution slightly richer in solute polymer shall then be injected into the flowing solution. The accuracy with which the polymer diffusivity can be determined depends on the variation of flow behavior as a result of the dispersion process. For instance, if the flowing solution is a power-law fluid, the diffusivity can be accurately obtained from the dispersion measurement following an analysis[20] similar to Taylor’s as long as the variation of the flow index n in the dispersion process can be carefully controlled. This can be achieved by minimizing the concentration difference between the flowing solution and the injected pulse and by delivering the solution at a lower speed. CONCLUSIONS
In summary, we would like to recapitulate the main points resulting from this work: (1) Under carefully designed experimental conditions the Taylor dispersion technique can be adapted for accurately measuring the diffusivities of polymers in dilute solutions across a large temperature range. (2) The Kirkwood-Riseman theory is applicable to the diffusion of long-chain hydrocarbons from 293 to 373 K if Monte Carlo simulation results for [R-l] are used in the prediction. Acknowledgements-The authors are grateful for the financial support of this work by the National Science Foundation under Grant CPE-8305649. Helpful comments raised by the reviewers are also gratefully acknowledged. NOTATION a C0 D
van der Waals radius of a friction center, A solute concentration in the stress-free region, g-mole/l. translational diffusion coefficient of polymer in dilute solution, cm2/sec.
Di
D0 k
(rii’) [R-i] T X
Greek
inner diameter of the concentric cylinder viscometer, mm. outer diameter of the viscometer, mm. Boltzmann constant, 1.38 x lo-l6 erg/K. mean reciprocal distance of separation between segments i and j, A-‘. as defined by eqn (2), A-‘. absolute temperature, K. number of segments in a chain molecule
symbols solution shear viscosity, CP. p
h fv 6
solvent shear viscosity, CP. shear rate at tube wall, see-‘. friction coefficient, g/set
REFERENCES
HI Vrentas J. S. and Duda J. L., J. Appl. Polymer Sci. 1976 20 1125. PI Vrentas J. S. and Duda J. L.. J. Polymer Sci. Polymer Phys. Edn. 1976 14 101. 131 Kirkwood J. G. and Riseman J., .7. Chem. Phys. 1948 16 565. 141 Edwards C. J. C., Rigby D. and Stepto R. F. T., Macromolecules 198 1 14 1808. 151 Schmidt M. and Burchard W., Macromolecules 1981 14 210. Taylor G. I., Proc. Roy. Sot. London 1953 A219 186. t: Taylor G. I., Proc. Roy. Sot. London 1954 A225 473. 181 Chen S. H.. Davis H. T. and Evans D. F., J. Chem. Phys. 198 1 75 1442. PI Chen S. H., Evans D. F. and Davis H. T., A.Z.Ch.E.J. 1983 29 640. 1101 Bird R. B., Armstrong R. C. and Hassager O., Dynamics of Polvmeric Liauids. Vol. 1. Wiley. New York 1977. Sci. Polymer 1111 Tirrel M: and Malone M. F., J. Po&er Phys. Edn 1977 15 1569. [l21 Zimm 3. H. and Crothers D. M., Proc. Nat. Acad. Sci. (U.S.A.) 1963 48 905. 1131 Raznjevii: K., Handbook of Thermodynamic Tables and Charts. McGraw-Hill, New York 1976. I151 Chen S. H. and Yumet E., Chem. Engng Sci. in press (1983). 1151 Janssen L. P. B. M., Rheol. Acta 1980 19 32. 1161 McCall D. W. and Douglass D. C., J. Chem Phys. 1963 38 2314. t171 Dewan R. K. and Van Holde K. E., J. Chem Phys. 1963 39 1820. Paul E. and Mazo R. M., J. Chem. Phys. 1968 48 1405. 1953 1 264. ;:;; Wilke C. R. and Chana P., A.Z.Ch.E.J. c201 Fan L. T. and Hwang W. s., Proc. Roy. Sot. London 1965 A283 576. 1211 kunn P. O., Znt. J. Mtdtiphase Flow 1983 9 187.