Use of Solvent Free-Volume Parameters from13C Relaxation to Study Polymer/Solvent Diffusion Behavior

Polymer International 39 (1996) 243-249

Use of Solvent Free-Volume Parameters from 13C Relaxation to Study

Polymer/Solvent Diffusion Behavior

Seong-Uk Hong,B*Alan J. BenesP & J. L. Dudac*

Department of Materials Science and Engineering, * Department of Chemistry, ' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

(Received 1 November 1994; revised version received 19 April 1995; accepted 24 November 1995)

Abstract: Variable temperature 3C nuclear magnetic resonance (NMR) spin- lattice ( Tl) relaxation studies were carried out using pure ethylbenzene, tetrahydrofuran, and methanol samples. Single correlation times, determined from an isotropic rotational diffusion model, were correlated successfully using the free- volume equation. The solvent free-volume parameters estimated in this study were comparable with those determined from viscosity measurements. The solvent free-volume parameters estimated in this study were then used to correlate and predict diffusion behavior of polymer/solvent systems. The results were comparable with those using the parameters from viscometry when the Vrentas- Duda free-volume diffusion model was used.

Key words: NMR spin-lattice relaxation, polymer-solvent diffusion, solvent free-volume.

I N T R 0 D U CTI 0 N

For many years, theories based on free-volume concepts have been used in correlating and predicting diffusion behavior in polymer-solvent systems. In the theory derived by Vrentas & D ~ d a , ' - ~ the empty space between the molecules that is available for molecular transport, referred to as free-volume, is continuously being redistributed. Molecular transport will occur only when a free-volume of sufficient size appears adjacent to a molecule and the molecule has enough energy to jump into this void. The diffusive jump is considered complete when the void left behind is closed before the molecule returns to its original position.

In order to describe diffusion behavior in concen- trated polymer solutions, the free-volume contribution of both polymer and solvent must be known a priori. Since all transport processes are assumed to be govern- ed by the same free-volume,4 the available free-volume for the diffusion can be extracted from other transport

$ Present address : Department of Chemical Engineering, The Johns Hopkins University, Baltimore, MD 21218-2694, USA. * To whom correspondence should be addressed.

processes. Although polymer free-volume parameters have been determined by various techniques such as viscometry, dynamic mechanical, dielectric, and NMR relaxation only viscosity-temperature data have been used to estimate solvent free-volume parameters. In order to estimate free-volume parameters accu- rately, low-temperature viscosity data are essential. Viscosity data at low temperatures are rare in the liter- ature, however, due to difficult temperature control, lengthy run times, and problems associated with partial crystallization. Recently, solvent free-volume parameters of toluene and chloroform were alternatively obtained from I3C NMR Tl relaxation studies." The purposes of the present work are (1) to estimate solvent free-volume parameters, using this method, for other solvents, and (2) to correlate and predict the diffusion coefficient for polymer/solvent systems using estimated parameters.

EXPERIMENTAL

' 3C NMR spin-lattice relaxation studies were carried out at 7.05 T on a Bruker AM-300 spectrometer. Tl was measured using the inversion recovery method.

243 Polymer International 0959-8103/96/$09.00 0 1996 SCI. Printed in Great Britain

244

- cn Y

kU

10-9

10-'0

lo-"

1 o-=

0 2.6 This s W

3.5 mls study

0 4

2.6

Thts s t w

Brettmaie et. al.

Breitmsier et. ai.

Breitmaier et. ai.

A 3.5

= 4

4 0 6 r

5 2 points

125 175 225 275 325

T (K)

Fig. 1. Ethylbenzene correlation time (z,) determined by 13C NMR spin-lattice relaxation with an isotropic rotational diffusion model, versus absolute temperature (K). The line rep-

resents the theoretical correlation using eqn (8).

k"

1 0 - ' O

10-I'

1 O-'=

2-3

100 150 200 250 300

T (I.0 Fig. 2. Correlation times (7,) of THF and methanol determined by I3C NMR spin-lattice relaxation with an isotropic rotational diffusion model, versus absolute temparature (K).

Lines represent theoretical correlations using eqn (8).

cn \ N

E 0 v

n

S-U. Hong, A. J . Benesi, J . L. Duda

10-6

10-

0 T=140"C

A T=160°C

VlSCOSlty ___.

NMR -

0.00 0.05 0.10 0.15 0.20

w , (Solvent Weight Fraction)

Fig. 3. Experimental dataz6 and theoretical correlations for polystyrene/ethylbenzene mutual diffusion using ethylbenzene free-volume parameters obtained by viscometry and 3C

NMR TI relaxation.

lo-'

1 0 - 5

cn \ N

E 0 v

d lod

1 o-'

0 T=28.5'C

A T=50.2'C

0 T=70.2"C

_ _ _ _ v ISCOSI t y

NMR -

0.4 0.6 0.8 1 .o

w , (solvent Weight Fraction)

Fig. 4. Experimental data" and theoretical correlations for THF self-diffusion in polystyrene using THF free-volume parameters obtained by viscometry and 13C NMR Tl relax-

ation.

POLYMER INTERNATIONAL VOL. 39, NO. 3, 1996

Polymerlsolvent difusion behavior 245

Although samples were run unlocked, the field did not drift significantly. The experiments were performed starting from room temperature and proceeding step- wise to approximately the freezing point of the solvents. For all samples the delay time was set longer as the temperature decreased. The sample temperature was regulated by liquid nitrogen, and the temperature reading was calibrated using methanol. After each temperature change, a delay of approximately 30 min was provided for temperature equilibration and both 'H and 3C channels of the probe were tuned.

The methanol, tetrahydrofuran (THF), and ethylbenzene samples used in the NMR studies were of spec- troscopic grade quality and purchased from Aldrich Chemical Company. The samples were used without further purification, and dissolved oxygen was removed by the freeze-pumpthaw cycle. The NMR tube was immersed in a liquid nitrogen bath, and the air space was then evacuated. The frozen tube was then removed from the bath and the solvent allowed to melt. Dis- solved gases then bubbled off into the vacuum in the upper part of the tube. This freeze-pumpthaw process was repeated until no bubbles were observed during the thawing step. After the final evacuation, the tube was flame-sealed, taking care to make the seal symmetrical so that the tube could spin properly.

Since the correlation time, which is equated with the time taken for a molecule to rotate by roughly 1 rad about any axis, is not directly measurable, 7'' values were first calculated from experimental decay curves by the best fit of parameters M , and T,, using a non-linear least-squares program, in the equation

M,(t) = M,[1 - 2e-t'Tl] (1) where M , and M,(t) are the longitudinal magne- tizations at equilibrium and at certain time t , respectively. A Newton-Raphson method was then used to determine z, , using the following expression, from each measured Tl :' '

where z, is the correlation time; p0/4x is a conversion factor for SI units and is equal to 10-7H/m; n is the number of hydrogens attached to the carbon being examined; h is the Planck constant (6.63 x 10-34Js); yc and yH are the gyromagnetic ratios for carbon and hydrogen spins having values of 6.727 x lo3 and 2-675 x 104rad/(sG), respectively; oC and oH are the Larmor frequencies for carbon and hydrogen, respectively; and R,H is the C-H bond length (1.09 x 10-9m).

THEORY

According to the Vrentas-Duda free-volume theory for diffusion,'-3 the solvent self-diffusion coefficient, D,, and the polymer/solvent binary mutual diffusion coefficient, D, are given by

D, = Do exp($)

D = Dl(1 - 4,)2(1 - 2x41) (4)

where the subscripts 1 and 2 refer to the solvent and polymer, respectively. Do is a pre-exponential factor, E is the critical energy that a molecule must obtain to overcome the attraction forces holding it to its neigh- bors, and y i is an overlap factor for the component i that is introduced because the same free-volume is available to more than one jumping unit. Pf is the specific hole free-volume of the component i required for jump, oi is the weight fraction of the component i , and 5 is the ratio of molar volume of the solvent jumping unit to that of the polymer jumping unit. K , , and K,, are free-volume parameters for the solvent, while K,, and K,, are those for the polymer, 4, is the solvent volume fraction, and x is the polymer-solvent interaction parameter.

Although there are 14 independent parameters in eqn (4), grouping some of them means that only 10 parameters ultimately need to be evaluated: F:, ^vt, K,,/y,, KZ1 - T,,, K,,/Y,, K , , - T,,, Do, E , t, and x. x is not needed in estimating the solvent self-diffusion coefficient and thermodynamic diffusion coefficient. Six out of 10 parameters are pure component properties and can be easily evaluated from data that are usually available. The methods for determining free-volume parameters to estimate D as a function of temperature and concentration can be summarized as follows:

(1) The two critical volumes, P: and P:, can be estimated as the specific volumes of the solvent and polymer at OK. Molar volumes of the solvent and polymer at OK can be estimated using group contribution methods summarized by Haward," and values of these parameters for a large number of solvents and polymers have been reported.'

(2) The parameters K,,/y, and K,, - q2 are simply related to the WLF constants of the polymers, Cyk" and CY;", as follows : l4


246 S-U. Hong, A. J . Benesi, J. L. Duda

TABLE 1. Free-volume parameters from viscosity and N M R studies

Solvent Wl 1 / Y l ) 1 o3 (cm3/€l lo K,, -Tgl (K)

Viscosity NMR Viscosity N M R

Ethyl benzene 1.49 1.70 -84.40 -83.85 TH F 0.98 1.15 1.67 1.51 Methanol 1.17 2.1 0 -48.41 -64.60

Values of Cy’F, C:tF, and T,, have been tabulated for a large number of polymers,’ whereas values of Kl,/y, and K,, - T g , have been calculated by several researchers.’ 3*1

(3) In 1921, Vogel proposed an empirical equation to describe the viscosity-temperature relation- ship.16 Thirty years later Doolittle postulated that viscosity should be related to the amount of free-volume in a system and derived the Vogel equation from free-volume concepts.” Adopting Doolittle’s expression and using the nomencla- ture of Vrentas & Duda leads to eqn (7) for solvent viscosity, ql, by the equation

( Y l W K l l ) (7) In q, = In A , + ( G 1 - T , A + T

where A , is considered to be effectively constant. Since the correlation time, z, , is determined from inertial and viscosity effects, the model describ- ing the viscosity of a solvent as a function of temperature is assumed to also represent the correlation time-temperature behavior :lo

Hence, the parameters K,,/y, and K,, - T,, can be determined, using viscosity or correlation time data as a function of temperature, from a nonlinear regression analysis of eqn (7) or (8).

(4) It is easy to find polymer/solvent interaction parameters for many polymers and solvents in the l i t e ra t~re . ’~ . ’~ It is also possible, using the Flory-Huggins equation,” to determine x from solubility data where the equilibrium volume fraction of the solvent in the polymer is known as a function of solvent vapor pressure, P , :

(9)

where Py is the solvent saturation vapor pressure. x also can be determined from a semi- empirical equation developed by Bristow & Watson:”

where v, is the solvent molar volume, and 6, and 6, are the solubility parameters of the solvent and polymer, respectively. Since eqn (4) is relatively insensitive to the choice of x , 1 3 3 2 2

either estimation method can be used. ( 5 ) Finally, the parameters D o , . E , and can be

determined, using diffusivity data, by a non- linear regression analysis of eqn (4). In addition, if diffusion data for the particular polymer/ solvent system is not available and the solvent molecule moves as a single unit, the parameter 5 can alternatively be estimated from one of two methods. For the first method, diffusivity data should be available for the diffusion of other solvents that move as single units in the polymer of interest. From these data, assuming the molar volume of polymer jumping unit, PZj, is independent of the solvent, the constant 0: in the following equation can be evaluated for a particular polymer:23

where Py(0) is the solvent molar volume at OK. Once u is known for a particular polymer, the value of 5 for any solvent in that polymer can be estimated as far as the solvent moves as a single unit. c1 values have been reported for various polymer^.'^^'^ A second method of estimating t has been proposed by Zielinski & Duda” and modified by H ~ n g ~ ~ using more diffusivity data. According to Hong, the molar volume of the polymer jumping unit can be correlated with its glass transition temperature as follows :

V, j(cm3/mol)

= 0*0925Tg,(K) + 69*47(T,, < 295 K) = 0.6224Tgz(K) - 86.95(T,z 2 295 K) (12)

where the glass transition of the polymer, T,, , is in Kelvin. Therefore, if T,, is known for the polymer of interest, the value of t can also be estimated using eqn (12).

RESULTS AND DISCUSSION

13C NMR T, relaxation studies were carried out using


Polymerlsolvent diffusion behavior 247

pure ethylbenzene, THF, and methanol samples at various temperatures. Ethylbenzene has six peaks while THF and methanol possess two and one, respectively. For ethylbenzene only three peaks from hydrogenated carbons in the benzene ring are used in this study, since the relaxation mechanism of other carbons may include contributions from other than the CH heteronuclear dipolar interaction.

The temperature dependence of z, is observed for solvent samples, as shown in Figs 1 and 2. Values of z, from a previous study for ethylbenzeneZ5 are also provided for comparison. The correlation results using eqn

lo-’

- cn .. N

E 0 v

0 T=35“C

A T=45“C

T=55“C

Vlscoslty

0.00 0.05 0.10 0.15


Fig. 5. Experimental data2* and theoretical correlations for poly(viny1 acetate)/methanol thermodynamic diffusion using methanol free-volume parameters obtained by viscometry and

13C NMR Tl relaxation.

1 oo

lo-‘

1 0 -

1 o - ~

0.0 0.2 0.4 0.6 0.8 1 .o


Fig. 6. Experimental datatg and theoretical predictions for ethylbenzene self-diffusion in polystyrene using ethylbenzene free-volume parameters obtained by viscometry and ’ 3C

NMR Tl relaxation.

(8) are shown as solid lines. For all the solvents, the correlations are successful over the temperature ranges studied. The solvent free-volume parameters obtained in this study are compared with values determined from viscometry in Table 1. Viscosity data at temperatures between the melting point of the solvents and room temperature were used for the calculation. The parameters of ethylbenzene and THF were determined from an average of the relaxation results from the three and two ring carbons, respectively. The solvent free-volume parameters from the two techniques are in fair-to-good agreement.

TABLE 2. Parameters used in diffusion coefficient correlations

Parameter PS/Ethyl benzene PSPH F PVAc/Methanol

0.928 0.91 1 0.850 0.850 1.49 (1.70)

5.82 x 1 OW4

0.45 - 0.65 (0.62) 0.54 (0.67) 7.2 x 10’ (4.5 x 10‘) 1.2 x (1.5 x

7.85 (9.93) 0.36 (0.79)

0.98 (1 .I 5) -84.40 (-83.85) 1.67 (1.51)

5.82 x 1 0-4 -327.0 -327.0 -

0.961 0.728 1.17 (2.10)

-48.41 (-64.60)

-258.2 4.33 x 10-4

0.44 (0.37)

-6.08 (2.78) 2.5 x 10-8 (2.1 x 10-3)

Parameters in parentheses were obtained directly from NMR studies (K,,/y, and K2’/Ta,) or by non-linear regression eqns (3) and (4) using the parameters from NMR studies (Do, E, and f).


248

10‘’

1 0 - 2

S-U. Hong, A. J . Benesi, J . L. Duda

6 7’

The diffusion data for three polymer/solvent systems (polystyrene/ethylbenzene, polystyrene/THF, and poly(viny1 acetate)/methanol) were then correlated using the solvent free-volume parameters estimated from both 13C NMR Tl relaxation and viscometry. These correlations are shown in Figs 3-5, while the parameters used for correlations are provided in Table 2. The remaining parameters were determined using the methods described in the previous section. For all the systems investigated, the correlations of diffusion behavior, using both sets of the free-volume parameters, are successful.

In addition, the temperature and concentration dependences of diffusion coefficients were predicted without using any diffusion data. These predictions are

0 1 ‘ - 3

3

v

A ? ,-

Y n

1 o2

10’

100 I I

0.00 0.05 0.10 0.15 0.20

w , (Solvent‘ Weight Fraction)

Fig. 7. Experimental dataz6 and theoretical predictions for polystyrene/ethylbenzene mutual diffusion using ethylbenzene free-volume parameters obtained by viscometry and 13C

NMR TI relaxation.


Fig. 8. Experimental data” and theoretical predictions for THF self-diffusion in polystyrene using THF free-volume parameters obtained by viscometry and I3C NMR 7’’ relax-

ation.

provided in Fig. 6-9; the parameters used for predictions are tabulated in Table 3. For all the parameters except for c, the values are the same as those used pre- viously for correlations. Equation (1 1) was used to estimate the parameter ( in this study. Since the diffusion coefficients are normalized by the values at pure solvent or polymer limit, Do and E in eqn (3) are not required for predictions. In Fig. 6, experimental data at the pure solvent limit were used for the normalization, while correlation results using the parameters from viscometry were used in Figs 7-9 since experimental data at the pure solvent or polymer limit were not available for those systems. The equation for the solvent self- diffusion coefficient was used to generate theoretical curves in Fig. 9 since the thermodynamic diffusion coef-

TABLE 3. Parameters used in diffusion coefficient predictions

Parameter PS/Ethyl benzene PSPH F PVAc/Methanol

0.91 1 0,850 0.98 ( 1 . 1 5) 1.67 (1.51 ) 5.82 x 1 0-4

-327.0 -

0.47

0.961 0.728 1.17 (2.10)

-48.41 (-64.60) 4.33 x 10-4

-258.2 -

0.32

Parameters in parentheses were obtained from NMR studies.


Polymer/solvent diflusion behavior 249

1 O6

1 o5

1 o4 3 ‘I_ 3 + lo3 - 2- 3 1

d 1 o2

1 0 ’

1 oo 1

0.00 0.05 0.10 0.15


Fig. 9. Experimental dataz8 and theoretical predictions for poly(viny1 acetate)/methanol thermodynamic diffusion using methanol free-volume parameters obtained by viscometry and

3C NMR T, relaxation.

ficient has the same meaning as the solvent self-diffusion coefficient.

The prediction results indicate that the solvent free- volume parameters determined from NMR studies predict the diffusion behavior somewhat higher than those from viscometry. For the polystyrene/solvent systems, the predictions using the parameters from viscometry are somewhat better than those using the values from NMR studies, while the results for the poly(viny1 acetate)/methanol system show opposite behavior. In general, the predictions using the solvent free-volume parameters from both viscometry and NMR studies are reasonable.

CONCLUSIONS

The results presented confirm that solvent free-volume parameters can be determined directly from variable temperature 13C NMR TI relaxation studies. For the three polymer/solvent systems investigated in this study, the correlations of diffusion behavior, using the solvent free-volume parameters from NMR studies, are satisfac-

tory when the Vrentas-Duda free-volume diffusion theory is used. In addition, the predictions using the parameters determined in this study are comparable with those using the values from viscometry.

ACKNOWLEDGEMENT

S.U.H. is indebted to the Ministry of Education of Korea for the scholarship it provided.

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Documents

Use of Solvent Free-Volume Parameters from13C Relaxation to Study Polymer/Solvent Diffusion Behavior