DIFFUSION IN POLYMER SOLUTIONS S. U. L I 1 A N D J . L . G A I N E R University of Virginia, Charlottesuille, V a .
An expression has been developed to account for the variation of the diffusion coefficient, D, in polymer solutions. This expression, with heat of mixing and viscosity data, can be used to predict D in polymer solutions in terms of D for the pure solvent. D was measured for seven polymer systems; the polymer concentration was up to 40% by weight.
HIS study is concerned with the molecular diffusion of T o r d i n a r y , lo\\ molecular weight solutes through polymer solutions. Experimental evidence shows that in some polymer systems (Astarita, 1965a, 1966; Heertjes et al., 1959; Metzner, 1965; Paul, 1967; Saunders, 1959) the diffusion coefficient of a solute diffusing into a solution increases as the concentration of the polymer is increased, but in other systems (Biancheria and Kagles, 1957; Boss et a/., 1967) the diffusion rate is loiver in more concentrated polymer solutions. Since the viscosity of a solution usually increases drastically when a small amount of polymer is dissolved in it, in the first case, the diffusive transport rate of a solute increases as the medium into Ivhich it diffuses becomes more viscous. This type of behavior is not predicted by any existing theoretical equation for diffusion in liquid systems. Classical theories such as the hydrodynamic (Einstein, 1905), kinetic (Arnold, 1930), alxolute rate (Glasstone et al., 1941)> and others (Scheibel, 1954; LYilke and Chang, 1955) predict that diffusivity varies inversely with solution viscosity. Thus, diffusion in polymer solutions is of fundamental interest. \'ery little previous work has been done on this subject, partly because of the lack of a satisfactory liquid-state theory for polymer solutions. Even in the relatively simple case of ordinary liquids, no comprehensive model for the liquid state is available, and added complications are due to the presence of macromolecules in the solution. Clough et al. (1962) and Astarita (1965b) have proposed simple structural models, from which they derived expressions to predict diffusion coefficients in polymer solutions. The resulting correlations of their analyses appear to lack the generality \vhich is required to explain the tivo diffusive phenomena observed in polymer solutions, since the equation of Clough and co\vorkers predicts only that the diffusion coefficient \vi11 decrease, and iistarita's analysis concludes that long-chain molecules increase the rate of diffusion of the solute. 'The purpose of the present investigation \\-as to develop a general expression to explain the phenomena of increasing and decreasing diffusion rates in polymer solutions. Based on the lvork by Eyring (Glasstone et a / . , 1941) and Gainer and Metzner (1965), a simple expression \vas obtained Lvhich relates the diffusion coefficient in a polymer solution to that in the pure solvent. According to this, the cause of increasing (or decreasing) diffusivity appears to be related to the energy of interaction hetiveen the polymer molecule and the solvent, and the derived equation predicts diffusion coefficients in good agreement xvith experimental values.
Background
Polymeric Solutions. Biancheria and Kagles (1 957) studied molecular diffusion in very dilute polymer solutions. 1
Present address, Esso Research and Engineering Co., Florham
Park, N . J .
I n all cases investigated, the diffusion coefficient of the solute decreased when the concentration of polymer molecules in solution increased, but the decrements in diffusivity were never as large as the relative increase in macroscopic viscosity of the medium. Based on the obstruction theory developed by Wang (1954), diffusion data were correlated with the volume fraction of the long-chain organic molecules by the equation
D = Do(1
- CY$)
(1)
where
D = diffusion coefficient of the solute in the polymer solution Do = diffusion coefficient of the solute in the pure solvent = a parameter which describes the geometric shape of CY the polymer molecule; its value always greater than unity (1.5 for spherical shape and polymer molecule) 6 = volume fraction of the polymer in the solution Applying Equation 1 to their data, good correlation was reported. However, this correlation does not apply to concentrated polymer solutions. I n the dilute concentration range, Equation 1 predicts DiDo < 1 ; it does not explain the phenomenon of increasing diffusion coefficient in polymer solutions. Realizing that the diffusivity of a solute in a polymer solution can be higher than in the pure solvent, Saunders (1959) developed a dimensional, empirical correlation which predicts diffusivities in polymer solutions as much as 2.5 times higher than those measured in pure solvents. This magnitude of increase, ho\vever, is higher than any experimental value reported (Metzner. 1965). Clough et ai. (1962) developed a theoretical approach for the prediction of diffusivities in viscous and non-Newtonian fluids. T h e fluid is considered as consisting of tivo regions: a large volume of a continuous phase essentially composed of pure solvent, and a region regarded as highly solvated polymer molecules; each region contributes differently to the overall diffusion process. I n the pure solvent portion, diffusion may take place n i t h a coefiicient of DO,and in the polymerThe concentrated region the diffusion coefficient is D,. over-all diffusivity in the solution is then expressed as a weighted fraction of Do and DB. Assuming that the diffusion rate in the polymer concentrated region is very small-Le., D , + 0-the folloicing expression was obtained for prediction of the diffusivity : €
D = DoXCp-
1 ~
ECP P C P
VOL.
7 NO. 3
AUGUST 1968
(2)
433
where
D
= diffusivity in the polymer solution
Do
= diffusivity in the pure solvent
X C p= fractional area occupied by the continuous phase = viscosity of the pure solvent p L c p = viscosity of the continuous phase 5 = a parameter which describes the geometrical con,.t
figuration of the diffusing molecule and its nearest neighbors Although this equation predicts D I D O to have an order of magnitude of unity even when the viscosity of the polymer solution increases a hundredfold over that of the pure solvent, this ratio is still less than unity a t all times. I n other words, Equation 2 predicts that for all solutes, the diffusion rate in polymer solutions is very close to, but always smaller than, the rate in the pure solvent. This is a consequence of the assumption that the diffusivity through the region of high polymer concentration, D,, is essentially zero. The result is therefore not general enough to explain the other observed phenomenon of increasing diffusivity in polymer solutions. Astarita (1956b) proposed a simplified structural model for polymer solutions to predict diffusion coefficients, similar to that outlined by Clough and coworkers. The polymer solution is again considered to consist of a region of essentially clear solvent and a region of highly concentrated polymer material, where the long-chain organic molecules are envisaged as a network of channels crisscrossing a continuous medium of clear solvent. However, the diffusion coefficient for this portion of the solution was regarded as having a value even higher than in the clear solvent. This equation does not allow for a value of D I D O less than unity, as is observed in some cases. Recently, in similar research, the diffusion of polymer molecules through mixed solvents has been studied by Cussler and Lightfoot (1965) and Tang and Himmelblau (1965), and the diffusion of a polymer molecule in its solution has received attention (Chalyk and Vasenin, 1965). I n their study of the diffusion of polystyrene through mixed solvents of toluene and cyclohexane, Cussler and Lightfoot employed the generalized Stefan-Maxwell equation to predict ternary diffusion coefficients. Some interdiffusion coeficients reported have negative values, which are dificult to explain with any physical interpretation. I n general, the analyses and information presented in these ivorks, as pointed out by Astarita (1965b), are incomplete and appear to be applicable only for specific systems. Theory
Many theories have been developed for calculation of diffusion coefficients in ordinary liquids, but most available methods lack the generality required for application to the whole spectrum of systems. Difficulties have been encountered where the solute is highly polar (Garner and Marchant, 1961; Kamal and Canjar, 1962) or has a low molecular weight, and where the solvent is highly viscous (Gainer, 1964). Reviexvs are given by Robinson and Stokes (1957), Jost (1952), \Vilke (1961), Gainer (1964), Neinow (1965), and Kamal and Canjar (1966). The familiar equations for the prediction of diffusivities such as those of Stokes-Einstein (Bird et al., 1960), Arnold (1930), Il7ilke and Chang (1955j, and Eyring (Glasstone et al., 1941) have one feature in common: The diffusion coefIicient varies inversely \\-ith viscosity. This is not the observed behavior for polymer systems, nor is i t true for viscous, SeIvtonian systems. This latter case was studied by Gainer and Metzner (1965), 434
l&EC FUNDAMENTALS
who presented an equation based on absolute rate theory which appears to allow for accurate predictions for most Newtonian systems. Their work has been extended here to the case of polymer solutions. The better known absolute rate theory, that due to Eyring, is based on the hole theory of liquid state. Diffusion is described as a process in which molecules rearrange their positions in a liquid structure which contains holes. HOWever, before a molecule can move from a given position to an adjacent hole, it must surmount a potential energy barrier. Using absolute rate theory to predict the frequency of rearrangement, Eyring has suggested the following expression for the prediction of diffusivity :
where
E
= a parameter describing the geometrical configura-
IT
=
V,
=
Ell,
= =
E,,, R
=
tion of the diffusing molecule and its nearest neighbors Avogadro's constant molar volume of solvent energy of activation for viscosity energy of activation for diffusion gas constant
They further assumed that the activation energies for diffusion and viscosity are equal. Hence, Equation 3 becomes:
(s)
1 /3
DAB = &B
(4)
Although this assumption is justified for the case of selfdiffusion where the process involves molecules of a single species, the energies of activation for diffusion and viscosity need not be equal for mutual diffusion. T h e consequence of this assumption is that the diffusivity predicted by Equation 4 is in agreement with the experimentally measured value only when [ is used as an adjustment factor. The value of ( varies over a wide range for diffusion of various solutes into a given solvent (Ree and Eyring, 1958), and no method for estimating the values of (was recommended by Ree and Eyring. Subsequent Lvork to modify Eyring's work \vas aimed at proposing a method for estimating the difference between the energies for diffusion and viscosity, and suggesting an a priori method to calculate the geometric factor, E. Olander (1963) suggested that the activation energy for diffusion may be expressed in terms of the individual viscous activation energies of the solute and solvent and this correlation was reported to predict diffusivity ivith better accuracy than Eyring's expression. Using a similar approach, Gainer and Metzner (1965) developed an equation to estimate the difference in activation energies of diffusion and viscosity, and proposed an a priori method for the calculation of E. They reported that [ has a value of 6 for the 15 compounds they examined, except for methanol and ethanol, where E equals 8. After making several assumptions concerning the liquid models of the diffusion and viscosity processes as defined by absolute rate theory, Gainer and Metzner proposed the follo\ving equation for estimation of the differences in the activation energies:
where = intermolecular distance between two x molecules = (VJLIy3
rrz
= (rrr
TZV
+ rJ/2
where subscript S denotes the medium which is a polymer solution (polymer and solvent B). T h e relationship between the diffusivity of solute A in pure solvent B and in the polymer medium S may be obtained by comparing Equations 3 and 6.
E b l = activation energy for viscosity of compound x due to hydrogen bonding
E,
r-D
= activation energy for viscosity of compound x due
to dispersion forces By utilizing the expression for the activation energy of viscosity derived by Eyring (Glasstone et al., 1941) and the homolog idea of Bondi and Simkin (1957) and Epz-D can be calculated. A detailed procedure for this calculation is outlined by Gainer and Metzner (1965). T h e resulting expression predicts diffusivity values which compare favorably with those predicted by the Wilke-Chang correlation in solvents with viscosity less than 6 centipoises. I n highly viscous solvents such as glycerol and triethylene glycol, the equation of Gainer and Metzner predicts diffusivities with a n error ranging from 10 to 60% compared to a 300 to 1500% error when the Wilke-Chang correlation is used. In dilute and intermediate concentration range polymer solutions, larger regions of the solution are occupied by solvent molecules and hence the properties of the solvent play an important role in the diffusion phenomena. Therefore, a fruitful approach to the problem of diffusion in polymer solutions appears to be to relate this phenomenon to the case of diffusion in an ordinary liquid, and then to take into account the contributions of the long-chain molecules. This approach is followed in this mark. Previous research has indicated that polymer solutions are true solutions and molecular in character, and there is no basic difference from ordinary solutions of lo\ver molecular weights. T h e unique property of polymer solutions is their rheological behavior under flolv : the dependence of the viscosity coefficient on the rate of shear. But, a t extremely lo\v shear rates, solutions of most polymer materials appear to approach Newtonian flow (Eyring et ai.,1964). As pointed out by Volkenstein (1963), there is much similarity between ordinary liquids and polymer solutions. Many physical models originally developed for ordinary liquids have been adapted to polymer systems; among these are the hydrodynamic model for diffusion (Cussler and Lightfoot, 1965) and the lattice model for the liquid state (Flory, 1953). Based on these considerations, it would appear reasonable to extend Eyring’s absolute rate theory of diffusion to polymer systems. This approach was selected because the equations proposed by Gainer and Metzner were based on the work of Eyring and his coworkers, and have been shown to be better for predicting diffusivities in highly viscous mediums (Gainer and Metzner, 1965). I n addition, the equation predicts the change of diffusivity with variation of molecular structure. T h e general form of the Eyring equation for diffusion is Equation 3, where the value of 4 has been assumed by Gainer (1964) to depend solely on the diffusing solute A, and subscript B denotes the medium, which is a low molecular lveight solvent. Since diffusion is a process of molecular mass transport, and the system is not under any external force to flo~v, the rheological property of the polymer solution may be assumed to approach Newtonian behavior in the diffusion process. With this in mind, the Eyring equation should apply equally well to polymer solutions:
(7) Assuming that diffusion is through the solvent phase only in the polymer solution, and that the polymer molecules do not interfere with the diffusion process, the activation energy for diffusion of solute A in pure solvent B might be expected to have a value close to that in the polymer solution-i.e., E,,, = EDAs-then Equation 7 simplifies LO:
Equation 8 is expected to apply well to dilute solutions, where the polymer molecules are located in separate regions. and large regions of the solution are occupied by pure solvent, As the concentration of the polymer increases, the presence of the polymer molecules cannot be neglected and Equation 8 is no longer valid; one must then use the more general form, Equation 7. T o estimate the activation energy for viscosity in Equation 8, Eyring’s rate theory for prediction of the liquid viscosity coefficient is utilized: (9) where K = a constant whose value depends on the type of molecular packing in the liquid. At very low shear rates, as in the diffusion process, polymer solutions exhibit Newtonian behavior, so Equation 9 can be applied to these systems. By introducing the expressions for the activation energy of viscosity into Equation 8 and assuming that in dilute solution K B = K,, one obtains:
With the further assumption that
Equation 10 becomes
bvhere
AH,,, AH,,, tion S
= molar heat of vaporization of solvent B = molar heat of vaporization of polymer solu-
Since polymers do not exist in the gas phase, AH,,,, can be expressed in terms of the heat of mixing of the polymer and solvent molecules, and the heat of vaporization of the pure solvent molecules in the solution, using the following expression :
AHba,s = (-AHPA
+ AH,.,, )
(13)
Combining Equations 13 and 12, one obtains:
VOL. 7
NO. 3
AUGUST 1968
435
Diffusion Coefficients of Various Solutes in Aqueous CMC Solutionsa
Table 1.
Solutes and Volume Fraction of Solute Allyl Alcohol 0 . 0 1 2
Ethanol 0.025 w t . 7c of Aqueous C M C Solutions 0 0.5 1. o 2.0 a
Temp.,
c.
D
x
23 23 23 23
sq. em. 105 set.
0.98 1.23 1.55 1.50
(E) 1 .o 1.3 1.6 1.5
sq. em. D X 105- see. 0.84
Glycerol 0.06 sq. em.
1. o 1.1 1.2
0.94 0.97
...
1 .o 1, I 1.2 1.30
0.93 1.06 1. I 4 1.21
, . .
Diffusivity data of Astarita et al., 7966.
According to Equation 14, DA,/DAB is closely related to the enthalpy of mixing between the solvent and polymer molecules. Equation 14 is, of course, valid only for very dilute polymer solutions because of the inherent assumptions that K B = K,, and ED,, = E,,,. Thus, when the concentration of the polymers is increased, the following equation should be employed :
of 1.6, about 30% higher than the 1.2 calculated for the diffusion of allyl alcohol and glycerol in the same solution. Unfortunately, no other data are available to test this proposal further. Diffusivity. QUALITATIVE RESULTS. Equation 14 might be rewritten as:
where
MB A comparison of Equations 15 and 14 sho\ss that the exponential term ( E D A B - EDA,)/RTand the ratio KB/Ks are a measure of the effects of the long-chain molecules on the diffusion process. Based on the work of Gainer and Metzner (1965), the value of the activation energy for diffusion can be correlated to that for viscosity. Thus, the two terms mentioned above may be closely related to the specific viscosity, Ivhich is a measure of the contributions of the polymer molecules to the viscosity of the solution.
.‘=(.Lis)
l/Z
V,
113
(E)
K1 will always be positive and will decrease from unity as the concentration of the polymer in the solution is increased. The heat of mixing, AH,, will be negative (exothermic) for systems in which the polymer is dissolved in a “good” solvent and positive (endothermic) for systems in which the polymer is dissolved in a “poor” solvent. The ratio AH,/AH,,, 1. will always be small-Le., Summarizing these points, we see : POORSOLVENTS
