4968
Macromolecules 1994,27, 4968-4971
Thermogravitational Thermal Diffusion in Liquid Polymer Solutions 0.Ecenarroti J. A. Madariaga, J. L. Navarro, and C. M. Santamaria Departamento de Fiiica Aplicada 11, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
J. A. Carri6n and J. M. Savirdn Facultad de Ciencias, Universidad de Zaragoza, 50010 Zaragoza, Spain Received February 8, 1994; Revised Manuscript Received June 13, 1994'
ABSTRACT We analyzed the steady state behavior of a thermogravitational columnfor a toluenepolystyrene system at different concentrationsand at molecular weights of polystyrene in the 45 000-540 OOO range. The
results obtained indicate that steady state separation strongly increases with concentration and molecular weight. From steady state separation measurements, the thermal diffusion coefficient of the mixtures considered was determined. It is shown that this quantity is in the dilute region, irrespective of molecular weight and concentration. The close agreement between our values and those reported in the literature obtained using differenttechniquesindicatesthat closed columns can be used to accurately determinethermal diffusion properties of macromolecular solutions. Introduction When a thermal gradient is set up within the bulk of a binary system, its components are partially separated by thermal diffusion. The magnitude of this separation is governed by what is known as the thermal diffusion factor a. Both convective and nonconvective techniques have been used to measure this transport property. In the former, a is determined from steady state separation measurements for a given thermal gradient in the absence of convection. With respect to convective techniques, the best known is that the thermogravitational columns, in which the binary mixture is placed between two vertical walls at different temperatures and the effect of the horizontal separation due to thermal diffusion is combined with vertical convective currents, thus giving rise to a separation between the ends of the column. In general, the thermal diffusion factor is difficult to measure. Indeed, the data obtained, particularly those concerningliquid mixtures, by different authors for a given system differ considerab1y.l In a recent work2we obtained reliable data concerning the thermal diffusion factor in organic binary liquid mixtures from column steady state measurements. There are two reasons for the good results obtained (i) the so-called forgotten effect was included in the theory; (ii) closed columns without reservoirs at their ends were used. As pointed out by Tyrrell' and demonstrated experimentally by Beyerlein and Bearmanx~,~ these reservoirs cause perturbations that lead to large errors upon measuring the thermal diffusion factor. The aim of the present work is to extend the thermogravitational column method to the determination of thermal diffusion in polymer solutions. We studied solutions of polystyrene in toluene at a temperature of 25 O C , analyzing the effect of the molecular weight and the concentration of polystyrene on the thermal diffusion factor. The data in the literature4concerning this mixture present a large scatter and, in particular, the results obtained in thermogravitational columns differ among each other up to 1order of magnitude, probably due to the presence of reservoirs. These reservoirs are nonactive volumes connected at the ends of the columns.
* To whom correspondence should be addressed.
Abstract published in Advance ACS Abstracts, July 15, 1994. 0024-9297/94/2227-4968$04.50/0
From steady state separation measurements in the column one first obtains (see below) the so-called thermal diffusion coefficient D' related to the thermal diffusion factor, a,and the ordinary diffusion coefficient,D, through the expression' D' = aD/T.This quantity which determines the mass flow for a given temperature gradient, seems to be much less sensitive to the nature of the molecular species present in a mixture than a. In fact, we have recently managed to show that even in the neighborhood of a critical point, where a strongly increases, the magnitude of D' is of the same order as that of ordinary organic mixture^.^ A similar result is obtained in the polymer system considered in this work. The thermal diffusion coefficient is of the same order as that of ordinary mixtures, the thermal diffusion factor being 2 orders of magnitude higher. In addition, the results obtained indicate that D' in the dilute region is independent of polymer concentration and molecular weight. Thermal diffusion is an intrinsically interesting transport property, though not well understood, that has recently gained great practical importance because it underlies the high resolution technique of thermal fieldflow fractionation proposed by Giddings and collaborators6 for separation and characterization of macromolecularand colloidalparticles. On the other hand, new optical methods to determine thermal diffusion in liquid mixtures based on the forced and small angle Rayleigh scattering7s8have been recently developed and accurate experimental data are required to check the validity and limitations of these promising methods. Theory According to column theory that takes into account the effect on separation of the dependence of density on concentration (the forgotten effectIgthe separation factor between the ends of the column at steady state is given by
In q = where q is the separation factor, which for dilute solutions is given by q = C ~ C T C, B and CT being the mass concentrations of the polymer at the bottom and the top of the 0 1994 American Chemical Society
Thermal Diffusion in Liquid Polymer Solutions 4969
Macromolecules, Vol. 27,No. 18, 1994 column,respectively;\k is a factor related to the geometrical characteristics of the column as \k = 504Llga4, L being the column length g the acceleration of gravity, and Q the annual gap; D’is the thermal diffusion coefficient; t is the viscosity; p is the density; j3 is the thermal expansivity; T is the mean column temperature; andf(S) is a complicated function of the separation parameter, S, defined by
where y is the coefficient of the dependence of density on concentration and a is the thermal diffusion factor related to the thermal diffusion coefficient D‘ since D‘ = &IT. This function was evaluated in a previous work,1° and here we shall only give its expression for small values of S. In that work it was shown that for S I2, f ( S )is
+
f(S) = 1 AS
(3)
where X = 1.33 X 1t2.Thus the influence of the forgotten effect is small for this range of S values. In contrast, for large S values, this influence is quite strong and gives rise to very large separations. This occurs, for example, in mixtures in the vicinity of critical ~ o i n t s .For ~ the mixture of polystyrene and toluene considered in this work, the greatest value of a can be estimated to be (Y lo2, and the maximum working concentration, as approximately 3% ’ . The value of y can be evaluated by considering an ideal mixture and taking values of p1 = 0.87 and p2 = 1.1, respectively, for the densities of toluene and polystyrene. According to eq 2, under these conditions one obtains S = 1.3. Thus, from eq 3 it may be deduced that the forgotten effect can be overlooked for the mixtures analyzed here. As a result, eq 1allows us to extract the value of D‘ from column separation measurements without the previous knowledge of the ordinary diffusion coefficient, in contrast with other alternative methods to determine this quantity.
-
Experimental Results and Discussion The liquid thermal diffusion column used in this work is a conventional stainless-steel concentric-tube type closed a t both ends, which has been described in detail in earlier pub1ications.l’ The total length of the column is L = 0.9 m, and the distance between the sampling ports at the ends is 0.791 m. The annular gap dimension, Q, is 0.95 mm. The temperature difference across the annulus is considered as the temperature difference between the two water baths corrected for the conductivity of the stainlesssteel walls (AT = 4 K). The temperatures of the walls were maintained at constant values using two circulating (15 Llmin) thermostated baths. Figure 1 shows a schematic drawing of the column representing the working space, sampling ports, and auxiliary connecting tubes for thermostated baths. The mean temperature under our experimental conditions is the arithmetic mean between hot and cold temperatures, T = 298 K. Concentrations were determined with a Pulfrich refractometer with a nominal accuracy of 5 X 10-6. Experimental procedures and the preparation and manipulation of the mixtures have been described e1sewhere.l’ We measured the stationary state separation for the polystyrene + toluene system within a concentration range up to 25 g/L and weight average molecular weights of polystyrene of M, = 45 X lo3, 105 X 103,300 X 103, and 540 X lo3. The polymers were synthesized in the laboratory of the Polymer Group of the Faculty of Science of the University of the Basque Country. Preliminary experiments showed that separation is practically independent
B
Figure 1. Cross-sectional view of our TD column, showing the connections for cooling water circulation (A) and heating water circulation (B)and showing the sampling porta (C)connected with the working space. of sample polydispersity. In view of this, samples with a polydispersity of about 2 were used. Figure 2 shows the logarithm of the separation factor as a function of concentration for the four molecular weights considered. As may be seen, the experimental points fit straight lines of increasing slope with the molecular weight and a common intercept at the origin. The value of this oridinate is 0.32 f 0.01 and represents the logarithm of the separation factor within the limit of infinite dilution. Therefore, according to eq 1the thermal diffusion coefficient at this limit should be independent of molecular weight. This result agrees with that obtained by Giddings et al.12J3for several polymer-solvent systems using the technique of thermal field-flow fractionation, based on the different rate of retention of the polymers in a flow between horizontal walls a t different temperatures. Introducing the above limit value for the separation factor in eq 1 and taking for the viscosity that of pure toluene, one obtains for the thermal diffusion coefficient at infinite dilution, D’o, the value
D’, = (1.10 f 0.05) X
cm2s-l K-’
which is in close agreement with that reported by Schimpf cm2 s-l K-l). and Giddingsl2 (D’o = (1.03 f 0.05) X
4970 Ecenarro et al.
Macromolecules, Vol. 27, No. 18, 1994
diffusion coefficient in polymer solutions is 2 orders of magnitude smaller than in organic mixtures, one can expect the thermal diffusion factor to be 2 orders higher. Finally, to determine the thermal diffusion factor, a, from D' data, one must know the ordinary diffusion coefficient. In the dilute region, the experimental data14J5 for D in the literature fit the linear approximation corresponding to the expansion in powers of the polymer concentration:
1.5
D = DO(1 + kdc)
(4)
where DOis the diffusion coefficient a t the limit of infinite dilution. The dependence of this quantity with the molecular weight has been calculated by Giddings et al.,16 starting from the theory of Flory17for the frictional factor. These calculations for the system toluene + polystyrene at 25 "C lead to the expression
1.0
c I
( Q I-')
I
I
1
1
I
5
10
15
20
25
Figure 2. Naturallogarithm of the separationfactor as a function of the concentration of polystyrene in toluene for different molecular weights: (e)M Z= 45 OOO; (A)M , = 105 000; (m) M, = 300 000; (V)M , = 540 000.
where A = 2.99 X lo4 cm2/s. The experimental values of different authors quoted in refs 14 and 16 agree with this equation. With respect to the coefficient k d in eq 4, the literature indicates14J5J8that, as with DO,it also follows a power law dependence with molecular weight. Some discrepancies exist concerning the value of the corresponding power law exponent. We took for it the value of 0.99 proposed by Wiltziuset al.,'Bas with this value one can reproduce quite well the extensive diffusion data in ref 15. Using the above equations, we have calculated the diffusion coefficients for the molecular weights and concentrations considered in this work. From these values and those of D'we determined the thermal diffusion factor ff.
1
c
I 5
10
i 5
(g1-1)
10
15
Figure 3. Thermal diffusion coefficient as a function of concentration for polystyrene in toluene: (e)M, = 45 O00; (A) M, = 105 OOO; (m) M, = 300 OOO; (v)M, = 540 000.
Figure 3 plots the values of D as a function of concentration for the four molecular weights of polystyrene considered. These values were obtained from eq 1, using the viscosities of the mixtures measured on a Ubbelohde viscometer. As can be seen, D' is almost independent of the molecular weight and decreases slightly with concentration. The values obtained for D', and thus for the product aD, are of the same order of magnitude as that of the ordinary organic liquid solutions analyzed in a previous work.2 The same also occurs in many other polymerorganic solvent systems.12J3 Therefore, it seems that the thermal diffusion coefficient at least in organic solutions is not very sensitive to the nature of the molecular species present in a mixture. However, this is not the case of the thermal diffusion factor. In fact, since the ordinary
Figure 4 shows the concentration dependence of our experimental thermal diffusion factor a,at 298 K. The continuous lines in this figure represent the experimental results obtained by Meyerhoff and Na~htigall'~ using a thermal diffusion cell at 293 K. This figure also shows the experimental value of Giglio ane Vendraminil@obtained using a laser-beam deflection technique a t 298 K for M , = 110 000 and a concentration of 1 % . The agreement with our experimental results is excellent. As can be seen the dependence of a on the molecular weight is very strong at low concentrations and becomes much smaller a t higher concentrations. In particular, in the limit of infinite dilution, according to eq 5 and taking into account the independence of D' on the molecular weight, cy will depend on M , as M,0.553. Additionally, it can be seen that a is of the order of lo2,i.e. very large in comparison with the current values in ordinary organic mixtures which are of the order of unity. Therefore, a considerable degree of separation can be expected when a polymer solution is exposed to a thermal gradient. The results obtained indicate that the thermogravitational method can be reliably used to determine the thermal diffusion coefficientsof macromolecularsolutions. The contradictory results in earlier thermogravitational experiments performed by other authors are probably due to the presence of reservoirs at the ends of the used columns. These reservoirs cause a temperature difference dependence of the steady separation not described by the thermogravitational theory, as has been shown by Beyerlein and Bearmann.3 On the other hand Tyrrell' has suggestedthat a more adequate hydrodynamical treatment is required to describe the transport from these passive elements. Therefore it seems that the thermogravitational
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Macromolecules, Vol. 27, No. 18, 1994
ties in protein solutions and particle suspensions, a field up to now somewhat unexplored.
Acknowledgment. This work was supported by grants from the Basque Country Government (PGV9023). We gratefully acknowledge Prof. I. Katime and Prof. A. Santamaria, who supplied the polymer samples. References and Notes
..
I
'
c
O
5
10
15
(0) 9
20
25
Figure 4. Thermal diffusion factor as a function of concentration for polystyrene in toluene: ( 0 )M, = 45 OOO; (A)M, = 105 OOO; (m) M y= 300 OOO; (v)M, = 540 OOO, (-) after Meyerlioff and Nachtigall15 at 298 K and weights of 540 OOO and 44 OOO, ( 0 ) Giglio and Vendramini.19
theory in the actual form describes correctly the separation process in columns without reservoirs. Our aim is now focused on the application of the proposed method to determine thermal diffusion proper-
(1) Tyrrell, H. J. V. Diffusionand Heat Flows in Liquids;Butterw o r t h London, 1961. (2) Ecenarro, 0.;Madariaga, J. A.; Navarro, J.; Santamarla, C. M.; Carribn, J. A,; Savirbn, J. M. J. Phys.: Condens. Matter 1990, 2, 2289. (3) Beyerlein, A.; Bearman, R. J. J. Chem. Phys. 1968, 49, 5022. (4) Emery, A. H. In Polymer Fractionation; Cantow, M. J. R., Ed.; Academic: New York, 1967. (5) Ecenarro, 0.;Madariaga, J. A. Navarro, J.; Santamarla, C.M.; Carribn, J. A.; Savirbn, J. M. J. Phys.: Condens. Matter 1993, 5, 2289. (6) Giddings, J. C. Science 1993,260, 1456. (7) Kohler, W. J. Chem. Phys. 1993,98,660. (8) SBgre, P. N.; Gammon, R. W.; Sengers, J. V. Phys. Rev. E 1993, 47, 1026. (9) Navarro, J. L.; Madariaga, J. A.; Savirbn, J. M. J. Phys. A.: Math. Gen. 1982,15, 1683. (10) Ecenarro, 0.;Madariaga, J. A.; Navarro, J.; Santamarla, C. M.; Carribn, J. A.; Savirbn, J. M. J. Phys.: Condens. Matter 1989, 1,9741. (11) Ecenarro, 0.;Madariaga, J. A.; Navarro, J.; Santamarla, C. M.; Carribn, J. A.; Savirbn, J. M. Sep. Sci. Technol. 1989,24,555. (12) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci. 1989,27, 1317. (13) Schimpf, M. E.; Giddings, J. C. Macromolecules 1987,20,1564. (14) Freeman, B. D.; Soane, D. S.; Denn, M. M. Macromolecules 1990, 23, 245. (15) Meyerhoff, G.; Machtigall, K. J. Polym. Sci. 1962,57, 227. (16) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976, 9, 106.
(17) Flory, P. J.Principles ofPolymerChemistry;CornellUniversity Press: Ithaca, NY 1953. (18) Wiltzius, P.; Haller, H. R.; Cannell, D. S. Phys. Rev. Lett. 1984, 53, 834. (19) Giglio, M.; Vandramini, V. Phys. Rev. Lett. 1977,38, 26.
