112
The Journal of Physical Chemistry, Val. 82, No.
1, 1978
Kaatze et al.
Dielectric Relaxation in Aqueous Solutions of Some Oxygen-Containing Linear Hydrocarbon Polymers U. Kaatze," 0. Gottmann,
R. Podbielski, R. Pottel, and U. Terveer
Drittes Physikaiisches Institut, Universitat Gottingen, 0 - 3 4 Gottingen, West Germany (Received Ju/y 27, 1977)
Complex dielectric constants of aqueous solutions of nonionic linear polymers have been measured at 10-18 frequencies in the range 0.4-40 GHz at 25 "C by an interferometric transmission method. The solutes were poly(ethy1ene oxide) and poly(viny1 alcohol) polymers of different molecular weight ( M = 3400-600 000) and one sample of poly(viny1methyl ether). To the observed data were fitted a sum of Debye relaxation functions to yield the hydration number and reorientation time of the hydration water on one hand and the relaxation strength and relaxation time of the solute on the other hand. The values of the former parameters are compared with those for poly(vinylpyrro1idone)samples and for dioxane and other monomers in order to obtain insight into the effect of macromolecular size, shape, and flexibility upon the dynamics of the surrounding water.
Introduction Hydration properties of organic molecules in general reflect in a complex manner the combined effect of different interactions around the solute. In most cases, the solute molecule acts on the surrounding water due to the H-bond accepting ability of its polar groups (the hydrophilic effect) and by the ability of its inert groups to promote the water structure around them (the hydrophobic effect). These two effects interfere with each other leading to destructive or, under certain conditions, to cooperative interaction. It seems likely that the interaction of the hydrophilic and hydrophobic effect strongly depends on the steric arrangement of the respective groups and on the particular size and shape of the solute molecule. If this is accepted, then one may expect the hydration properties of polymers to differ in a characteristic manner from those of small organic molecules of equivalent chemical composition. Moreover, one may expect the properties of the hydration water to depend on the degree of polymerization of the macromolecule. In view of the current wide interest on the hydration properties in aqueous systems, it will be informative to make a comparative study of the solution behavior of different water-soluble polymers. In this communication, we are reporting the results of our complex dielectric constant measurements on aqueous solutions of some nonionic oxygen-containing linear hydrocarbon polymers a t 25 "C. The solutes are the simplest structured water-soluble synthetic polymer, poly(ethy1ene oxide) (PEO),its isomer poly(viny1 alcohol) (PVA),and poly(viny1 methyl ether) (PVME). For one of the polymers (PEO), measurements were performed a t seven degrees of POlymerization (n)between n = 7 5 and 13 600. Three PVA samples were used covering the range n = 1000-4600, whereas PVME solutions were measured a t one n value only ( n 1300). The solute concentration (c) of monomer units was around 1mol/L in most cases. For some solutes measurements were also carried out a t higher concentrations (up to c = 12 mol/L) in order to observe the concentration dependence. From analysis of the complex dielectric constant values, reorientation times of the water molecules surrounding the solutes and the number of disturbed water molecules per monomer unit have been extracted and compared with the corresponding data of poly(vinylpyrro1idone) (PVP) solutions.'*2 The parameters of the oxygen-containing polymers are also discussed in comparison to those of
dioxane obtained by recalculating the complex dielectric constant data of Hasted et al.3 and of this l a b ~ r a t o r y . ~ Experimental Section I. Materials. A survey of the different polymers used is in Table I. Besides its name and average degree of polymerization, n, the tradename of the samples, its supplier, its average molecular weight, M , and the specification of the purity grade are given. The polymers were used without additional purification. The solutions were prepared by addition of a suitable amount of water and simple stirring. The water used for the preparation was doubly distilled and was sterilized by UV radiation during distillation. Concentration data of the solutions are listed in the first columns of Table 11. Here and in the following, the molarity c and the molality m refer to the monomer unit. e, represents the molarity of the solvent water. 11. Viscosities. The viscosity, q, of the solutions has been measured with a falling ball viscosimeter (Haake, Model B) and, in some cases, verified with a Ubbelohde viscosimeter (Schott, Type KPG). The q values as obtained with the two different methods of measurement agree within the limits of experimental error. The viscosity data of the solutions are also shown in Table 11. In Figures 1 and 2 the dependence of the viscosity number, ??red = (q - ro)/(qocg),on the solute concentration, c, and on the degree of polymerization, n, respectively, is displayed for most solutions used. (qodenotes the viscosity of the pure solvent and cg the solute concentration in grams per liter of solution.) A few additional 17 data, not compiled in Table 11, are also presented in the plots. As shown by the two figures, there is a strong dependence of the solution viscosity on both, c and n, as well as on the type of polymer. 111. Complex Dielectric Constants. The dependence of total dielectric constant E.tot(Y)
=
f'(U)
- iEtot"(V)
of the solutions on the frequency v has been determined between 0.4 and 40 GHz. Frequency domain measurements were performed at 10-18 frequencies by use of an interferometric transmission method. A propagating electromagnetic wave was set up in circular waveguides filled with the liquid to be measured. A receiving probe immersed in the liquid was shifted along the direction of wave propagation. The wavelength X and the attenuation exponent aX of the propagating wave were determined by
0022-3654/78/2082-0112$01.00/00 1978 American Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 1, 1978
Solution Behavior of Water-Soluble Polymers
113
TABLE I: Survey of the Polymers Used in the Measurements Comp, formula
Abbr name PEO
Poly(ethy1ene oxide)
75 850
fCH,-CH,Od
Poly(viny1 alcohol)
PVA
Poly(vinyi methyl ether) fCH,-CH
PVME
I
0-CH,] 100
30
t
I
I
PVME
1
v PVA
nc1300 n = 1600
o PEO
n-
7
M 3000-3700
n
I
1
,
35000-40000
2300 4500 6800 9000 13600 1000 1600 4600
100000 200000 300000 400000 600000 45000 72000 200000
1300
76000
I
Manufacturer, supplier SERVA
Purity, wt % 295
SERVA
295
Trade name Polyathy lengl ykol 4000, Polyathylenglykol 40000 Polyox WSR N-10 Polvox WSR N-80 Polyox WSR N-750 Polyox WSR N-3000 Polyox WSR-205 Polyviol M 13/20 Polyviol W 28/20 Mowiol 56-98
Union Carbide Union Carbide Union Carbide Union Carbide Union Carbide Wacker-Chemie Wacker-Chemie Hoechst
298.5 >98.5 2 98.5 298.5 >99.5 295
Lutonal M 40
BASF
295
298.5
2 98.5
30
I
10
75
10
-. -
3
en
. -
r
v
3
?
1
F
v
E
0.3
F 1
0.3
01
0.1
0.03 I 30
I
I
I
1
,
100
300
1000
3000
10000
30 IO
n -
0
0.3 0.05
0.1
0.2 0.3 0.5
1
2
3
5
10
20
c irnol/ll
Figure 1. The viscosity number, vre,, = (TJ - T J ~ ) / ( T J ~ofC ~aqueous ), solutions of some polymers bilogarithmically plotted vs. the molarity of monomer units. c .
moving the probe and by adjusting a reference signal of calibrated variable amplitude and fixed phase so that the probe signal and the reference signal interfere to zero. From the measured values of A, &A, and u, the dielectric constant was calculated by the following equation
where A. = (speed of light)/u is the wavelength in free space and A, is the cutoff wavelength of the empty waveguide used as the measuring cell. In general, the measured cto/(u) contains two parts but only the part c”(u) due to polarization processes is of interest here. This part has been calculated according to E”(V)
= Etot”(V)
-
2u/v
(2)
by subtracting the contribution 2alu due to ion drift. The ionic conductivity CT has been measured at 10 kHz in the usual manner. The conductivity term 2alu in eq 2 is small (less than 20% of et&), u 2 0.4 GHz) because the con-
Figure 2. Bilogarithmic plot of the viscosity number, T J = , ~(TJ the degree of polymerization, n. The solute concentration is 1 moVL of monomer units. T J ~ ) / ( ~ of ~ Csolutions ~), vs.
ductivity is less than 1.8 X ohm-l cm-l for all solutions. The frequency u was determined and kept constant during the measurement with an uncertainty of less than 0.1% . Imperfections in the apparatus, uncertainty in adjusting zero interference and temperature fluctuations, resulted in an estimated error of less than 2% in the values ) ~ ” ( u ) . A more detailed description of this of ~ ’ ( u and interferometric transmission method and the construction of the microwave bridges as well as a discussion of error sources is given in ref 1. Treatment of Data In Figure 3 the dependence of E’ and E” on the frequency u is displayed for the pure solvent and for three PEO solutions. The dielectric relaxation of pure water can be represented by a Cole-Cole relaxation function5 (3)
The parameter values obtained by fitting the function R&) to the experimental data are cmW= 5.2 f 0.2, esw = 78.5 f 0.2, rw = (8.25 f O.O2)ps, and h, = 0.007 f 0.002 at 25 OC.’ As shown by Figure 3 the relaxation strength ~ ( 0-) c(m) and the frequency of relaxation obviously decrease with
114
Kaatze et al.
The Journal of Physical Chemistry, Vol. 82, No. 1, 1078
TABLE 11: Concentration Data, Viscosities, and Parameter Values of the Relaxation Function R ( v ) for Aqueous Solutions of the Polymers m,
mol(kgof
Abbr name
n
PEO
75 75 850 850 850 2300 4500 6800 9000 13600 PVA 1000 1000 1600 1600 4600 PVME 1300 1300 1300 1300
C, cw, 77, mol/L H,O)-' mol/L cP i0.2% i O . 1 % +0.2% +5% 6.82 9.10 41.65 14.6 11.99 21.03 31.68 79 1,248 1.313 52.78 5.6 3.809 4.445 47.61 68 9.12 41.54 6.82 438 1.001 1.040 53.46 9.4 1.001 1.040 53.45 85 0.992 1.033 53.35 398 1.011 1.059 53.03 706 1.006 1.045 53.49 971 1.908 2.046 51.98 105 3.747 4.309 48.31 0,999 1.037 53.49 30 3.121 3.485 49.76 4627 1.000 1.026 54.11 51 1.742 1.916 50.32 30 3.512 4.310 45.10 397 5.29 7.39 39.65 2954 7.13 11.49 34.35 17970
Th$
E(-)
i0.2 5.2 4.6 5.5 5.3 5.2 5.5 5.9 5.9 5.7 5.8 5.7 5.8 5.5 6.1 5.6 5.9 5.3 4.7 4.1
em
56.5:; 43-21; 71.8:: 6 5.2:;. a 56.5:;., 75.2+;., 75.2;;.* 75.2:E.4 75.21::; 74.01i, 73.5:;*, 66.8:: 76.0:;. 69.8+;., 76.0';:: 68.6 + 0.1 60.1 i 0.1 52.3 i 0.1 44.7 i 0.1
80 1
increasing solute concentration c. In addition, the dispersion (dd/du c O)/absorption (e'' > 0) region of the dielectric relaxation of the solutions extends over a broader frequency band than with the pure solvent. This broadening has to be taken to indicate that the spread of relaxation times is more pronounced in aqueous solutions than in pure water. The increase in the distribution of relaxation times when going from water to a solution, in principle may be due to two effects: (i) the water around the solute molecules will be affected and thus reorientate with a relaxation time 7 h different from that of unaffected water T ~ This . water is called "hydration" water in the following. (ii) Due to its polar groups the solute molecules may contribute to the complex permittivity of the solutions. The relaxation time characteristic of this contribution is denoted by 7,. In view of what we mentioned above we used a sum of three relaxation terms defined by the following equation in order to analytically describe the dependence of E on the frequency Y:
R(v) = E(") e, cw
40) 61.1$,0:: 48.3+;,, 7 5.33.I 68.3;;: 61.5;i,3 76.3+;,, 76.1 i 0.1 76.lf::: 76.4';. * 77.2:2,.2 76.1 i 0.1 72.7 i 0.1 77.4 i 0.1 74.2+;., 77.3:;:; 68.6 + 0.1 60.1 + 0.1 52.3 i 0.1 44.7 + 0.1 I
Zh 5.0?:.,
TU
PS
h
18.1!;.8 32.0':,, 18.1:: 5.7:;.1 19.8t_:,, 5.1:;:; 16.9?;., 5.4:::: 17.9:::; 5.6'0,:: 6.0+_;q4 18.7!;., 20.2:;., 5.5:;:: 17.8:Q., 6.1:;:; 19.8?:,, 2.234 18.4 c 0.5 5.210,,, 16.4:::: 5.31;., 4.8tY.' 20.8':. , 5.0 i 0 18.2'0,,, 20.9ti:: 5.4:::: 6.8 i 0.3 20.0 i 0.4 6.3 i 0.2 21.1 i 0 . 3 6.6 i 0.2 19.7 i 0.9 24.7 i: 0.2 I
I
I
l
PS
0.061,0.,3 0.14?;.,, 0.00 0.01 i 0.01 0.061;. 0.00 0.00 0.00 O.OO::.M
0.00 0.021;.,, 0.04; ;. o3 0.04?:., 0.00 0.031;:2 0.00 0.00 0.10 i 0.01 0.16 i: 0.01
l
I
I
I
H20tPE0,n=850 7n .0'
\+'*\
c =
Y\-
Omol/l
= 1 25 moll1
+
c
o
c=3.8lrnol/l
'"1 I 20
+
zh
1+ ( 2 i T V q p h )
+
30
(4) The first relaxation term on the right-hand side of eq 4 represents the dielectric relaxation of the hydration water. In doing so, the number of affected water molecules per monomeric unit is denoted by 2,. The second term is attributed to the uneffected "free" water of molarity c, - Z h C . This water is assumed to reorientate with the reorientation time 7, of pure water. The total contribution of both types of solvent water to the static dielectric constant is characterized by cm, The third relaxation term on the right-hand side of eq 4 represents the contribution of the solute molecules to the complex dielectric constant. It turns out that contribution is rather small as compared to the other contributions. Therefore any relaxation time distribution was ignored with the solute term in order to restrict the number of unknown parameters. In the solutions of very high solute concentration there will be no unaffected water remaining. Thus the free water
:Id
20
10 0 Y
(GHzl
Figure 3. Real part, e', and imaginary part, e", of the dielectric constant plotted vs. the frequency, v, for water and three aqueous poly(ethy1ene oxide) solutions at 25 OC. The points indicate the measured data. Full lines represent the relaxation functions R,(v) and respectively R(u) with the parameter values found by the fitting procedure. Dashed lines indicate the superdivision of the R(u) function into the different terms for the 6.28 moi/L solution.
term has to be omitted in the R(v) function. Therefore we used eq 4 with the restriction Zhc E c, for sufficiently high concentrated solutions (c > 7 mol/L). In order to find the values of the parameters ~ ( m ) ,Th, h, Zh,E,, 7,, and 4 0 ) of the relaxation function defined by
The Journal of Physical Chemistry, Vol. 82,No. 1, 1978
Solution Behavior of Water-Soluble Polymers
00
I
l
T
A
v A
o 0
70
o
e e o o
o 60
PVME PVA PVA PVA PEO PEO PEO PEO PEO PEO PEO PVP PVP PVP
n= n= n= n= n = n = n =
n n n n n
n n
1300 1000 1600 4600 75 850 2300 = 4500 = 6800 9000 =13600 = 100~~ li 230 = 3200~'
25OC
'3
E
W
D
50
40
115
t, = 2...4denotes the dielectric constant of a homogeneous dielectric sphere, representing the solute molecule with respect to its distortion polarizability (electronic and atomic), and tsw= 78.5 the static dielectric constant of the pure solvent. v = 1- c,/c,O is the volume fraction of the solute where c," denotes the molarity of pure water. The parameter w is simply defined in this case by
(7 1
W=V
Equation 6 together with eq 7 is a refined version of the formula of Polder and van Santen6 and is based on a reasonable mean field approximation for a mixture of spherically shaped particles of dielectric constant with continuous matter of dielectric constant E~,. The dotted line in Figure 4 represents an extended version of eq 6 in which w is given by w = 1- 6 ( 1 - v ) (8) 6 denotes an adjustable parameter herein, the value of which was found to be F = 0.728 by fitting the corre-
..... small organic
molecules Equ 16) together with Equ 18)
sponding eq 6 to the measured static dielectric constant values of aqueous solutions of small organic molecules. The third graph in Figure 4 (the broken line) corresponds to the function e, = f,,
\
+
- mixture formula for spherically shaped solutes
Equ. 16) together with Equ. 17)
--- mixture formula
for r o d - l i k e shaped solutes Equ. 19)
(9)
30 c w 1 mol /I 1
Figure 4. The contribution of the solvent water to the static dielectric constant, t,, plotted vs. the water concentration, c,, and graphs of different dielectric mixture formulas.
eq 4 we fitted R(v) to the measured ~ ( v )values. We used a non-linear least-squares fitting procedure to obtain the absolute minimum value ( Vmin)of the variance:
(5) (where I is the number of measurement frequencies). In order to avoid mathematical difficulties the measurement frequency was treated as an exactly known independent variable in these calculations. This simplification is justified because the uncertainty of v is much smaller than that of ~'(v)and ~ " ( v ) as mentioned above. The parameter values found for the different solutions are given in the 7th to 13th column of Table 11. In order to test the significance of the data, uncertainty intervals have been determined for the parameter values by variation of the initial values in the fitting procedure. Within the uncertainty intervals of the parameters given in Table I1 the variance does not markedly differ from Vmin.
Results and Discussion I. T h e Contribution E , of t h e Solvent W a t e r to t h e Static Dielectric Constant and t h e Extrapolated HighFrequency Dielectric Constant ~ ( m of ) t h e Solutions. In Figure 4 the values of the static dielectric constant E, of the solvent water are plotted vs. the water molarity c, in order to show the concentration dependence. For comparison three graphs are added in this diagram. The full line corresponds to the mixture formula e, =
E,,
-esw)~ + 2Es, 3esw(ee + E , - W ( E , - esw)
with D1 = D 2= 0.5 and D3 = 0. This function represents the extended mixture formula (eq 6 together with eq 7) in the case of solution particles with the shape of very long prolate ellipsoids of revolution. Equation 9 may be taken to approximately describe the dependence of E, on the volume fraction v, respectively on the water molarity c, in the case of rodlike shaped solute molecules. Two characteristics of the E, values found with the polymer solutions attract attention. First, within the limits of experimental error the ,E values are independent of the polymer type (including PVP) and are thus a function of the water concentration c, only. Secondly, as demonstrated by Figure 4 the t, vs. c, relation found with the polymer solutions up to very high solute concentrations nicely agrees with the corresponding relation found with aqueous solutions of nearly spherically shaped small organic molecules. The latter finding is rather striking for reasons of polymer chain conformation, since one may expect the macromolecules to ad-apt nonspherical (mainly elongated) shapes in aqueous solution. The fact that the dependence of the E, values of the polymer solutions on c, appears to be the same as with nearly spherically shaped solutes might be regarded as to be due to a compensation of two effects. A reduction of the ,6 values on grounds of the nonspherical shape of solute particles may be counteracted by an enlargement of the E, values due to an enhancement of the static dielectric constant of the water surrounding the polymer molecules. However, despite of the fact that a distinct enhancement of the dielectric constant of solvent water is found around only one special molecule (triethylenediamine7) up to now, it is rather unlikely that the two opposite effects mentioned above would completely cancel out with all the different types of polymers. So it seems to be more suggestive to explain the t, vs. c, relation found with the polymer solutions by the assumption that the differently orientated segments of a flexible chain molecule
Kaatze et al.
See Figure L for the symbols
.......
6
mixture formula Equ. ( 10)
8
3
5
W
L
I 31 55
....... .............. ......... ....... ............. ............. ....... ...... .... .......... E
I
I
L5
35
-2.'"
e-
1
I
I
I
I
I
I
I
PVA
TX
........ L
I
1
I
1
n =lo00 v n E 1600 A n =4600 A
-4
2 p-A-
I
I
....... ..... I
25
PEO
c w ( m o l l 11
Figure 5. The extrapolated high-frequency permittivity of the solutions, ~ ( m )plotted , vs. the water concentration, c, and the graph of a mixture formula.
act like spherically shaped particles in the dielectric mixture. If this latter explanation is accepted, then no enhancement of the static dielectric constant of the water around the solute particles has to be taken into account. So it is justified to attribute the factor E, - t ( m ) to the hydration water term and to the free water term in the relaxation function R(v), eq 4. We will shortly mention here that the mixture formula (eq 6) for spherically shaped solute particles may be extended to account for different dielectric properties of the solute hydration shells and the uneffected solvent. Inserting the different complex dielectric constants of these two solvent regions the extended mixture formula may be expanded to yield the relaxation function R(v), eq 4, in the case of solute particles not very much larger than the water molecules. The ~ ( m values ) found with the polymer solutions are plotted in Figure 5 showing the dependence on the water molarity e,. For comparison, the mixture formula (eq 6), rewritten to refer to the high frequency dielectric constant is used here.
The value of ernWis 5.2, as mentioned above, and w is defined by eq 8. Again, 6 , denotes the dielectric constant of a homogeneous dielectric sphere, representing the solute molecule with respect to its distortion polarizability. This dielectric constant will be in the range 2-4, but the definite value of E, is unknown. Since (in contrast to trn of solutions with v 5 0.5, c, 2 25 mol/L) the high frequency dielectric constant strongly depends on the value of E,, two limiting graphs of eq 10 are displayed in Figure 5. E, is assumed to be 2 in one case and 4 in the other. These graphs define the range of ~ ( m )values to be expected on the basis of mixture formula 10. With the less concentrated solutions (c 5 3 mol/L, e, 2 47 mol/L) the t ( m ) data exceed the values expected on the basis of our theory of binary mixtures. This finding may be taken to indicate that the high frequency dielectric constant emWof the solvent water is enhanced with respect to the pure water value in those solutions. Two types of polarization mechanisms beside electronic polarization may contribute to E-,: atomic polarization on one hand and orientational polarization of zero-bonded and one-bonded water molecules on the other hand. Both mechanisms may be more pronounced in polymer solutions than in pure water. The atomic polarization may be enhanced by the formation of additional hydrogen bonds with the solute. Zero-bonded and one-bonded water
/ B I-"
$I '
I
1
I
I
I
I
1
2
3
dioxane
0.1
02
0.3
c^ = 1 +
0.5
(Czh-Cw)/Ci
Figure 6. Plot of the hydration waterlpure water reorientation time ratio, 7 h / ~ , , vs. the concentration parameter E for aqueous solutions of polymers and of dioxane.
molecules may increase in number by the disturbing action of the solute particles. It should be noticed that the errors given in Table I1 and in Figure 5 for the extrapolated high frequency dielectric constant reflect the uncertainty of the t ( m ) values if the relaxation function R(v), eq 4, is taken for the extrapolation. However, an additional uncertainty for the ~ ( m ) values comes into play by the fact that we do not know the true R(v) term and that several relaxation functions are appropriate to describe the measured t(v) values in the limited frequency range of measurements. So we restrict our discussion on the ~ ( m )data to the general remarks given above instead of discussing these parameter values in a more definite manner. 11. The Reorientation Time Th, the Number Zh, and the Relaxation Time Distribution Parameter h of the Hydration Water. In Figure 6 the reorientation time of the hydration molecules Th normalized to the dielectric relaxation time of pure water T , is plotted against a certain concentration parameter c^ of the solutions expressed by
c^ = 1 .f
( C Z h - C,)/C,O
(11)
This parameter c^ denotes the sum of the volume fractions of the solute u and of that of the hydration water if c^ is less than 1. For comparison the results of the PVP solutions and of aqueous solutions of dioxane are added in the 7 h / T , vs. c^ plot. The latter results were obtained by fitting to former data the relaxation function R(v) defined by eq 4 with the restriction ~ ( 0E) E,. We used complex dielectric constant data as obtained at 3.3 and 23.8 GHz by Hasted et al.3 and static dielectric constant values of Akerlof et alU8 in these calculations. One aqueous dioxane solution (6 mol/L) had been measured in this laboratory
Solution Behavior of Water-Soluble Polymers
The Journal of Physical Chemistry, Vol. 82, No. 1, 1978 117
at nine frequencies between 1.8 and 40 G H z . ~For the conversion of the proportion of dioxane by weight into the solute concentration c and the water molarity c, density data were taken from the l i t e r a t ~ r e A . ~comparison ~~~ of our results for the PEO solutions with the corresponding dioxane data seems to be of interest because this molecule may be regarded as the cyclic ethylene oxide dimer. With all solutions shown in Figure 6, the same characteristics of the T h / T w vs. t relation emerge. The T h is independent of t for the less concentrated solutions and shows an increase with increasing c^ if t exceeds a value near 1. It seems to be noteworthy that the same properties of the ?h/Tw vs. t relation have been found with other solutes which have hydrophobic parts as well as groups capable of forming hydrogen bonds with water. With each solute the number of perturbed water molecules per solute molecule, respectively per monomer unit, are also nearly independent of the solute concentration with less concentrated solutions (t 5 1). This finding together with the fact that the T h values are constant for solutions of sufficiently low solute concentration can be taken as an indication that our solution model as expressed by the R(v) function is self-consistent if applied to the dielectric relaxation of aqueous solutions of organic molecules. We are aware that there will be overlaps of hydration regions even if t is smaller than about 1and that the finding of constant Th and Z h values may be simply an empirical fact. However in view of the self-consistency mentioned above it seems to be possible to compare the hydration behavior of the different series of aqueous solutions using our model. As mentioned above, the strong increase of Th/T, with increasing t at c^ values larger than about 1 is also found with other aqueous solutions of organic molecules. Several factors may be important in influencing the hydration behavior at very high solute concentrations. These factors may include (a) strong overlap of hydration regions which might lead to enhancement of water structure rather than to destruction, (b) more tightly hydrogen bonding between solute and solvent molecules if less water molecules are available to surround the unpolar groups of the solute particles, (c) the formation of water bridges connecting two polar groups of the solute via hydrogen bonds, and (d) enhancement of the hydrogen bonds between those water molecules which are not connected to the polar groups of the solute via hydrogen bonds and which may form small islands of isolated solvent. This water may be considered to be in the hydrophobic hydration state. It turns out as one important result of our complex dielectric constant measurements that there exists no dependence of the Z h and Th values on the degree of polymerization (n)exceeding the limits of experimental error. In view of this finding it is justified to average the z h and ?h values as found with the less concentrated solutions (c^ 5 1) for each polymer type regardless of the fact that samples of different n values have been used in the measurements. The mean values of Zh and of the ratio Th/?w are displayed in Figure 7 showing the dependence for the polymers (including PVP) and for on (c#I/c#Iw)2'3 dioxane. 4 and 4, denote the solute and solvent apparent molar volume, respectively. With the macromolecules 4 refers to the monomer unit. Very surprisingly the Z h as well as the Th/ T , values of PEO are nearly identical with the corresponding values of the isomeric PVA. This result is worthy to note since a different hydration behavior for the two polymers would be expected from the following reasons: (a) In contrast to the ether group of the PEO molecule the hydroxyl group
20
I
v A
l5
N"
-
0
PEO
0
PVP dioxane
x
10-
I
I
PVME PVA
Z act to Equ. 12
25OC &-
_&e
Z act to Equ 13
/-4I
I
I
I
,., . f. . -.j
heterocyclic hydrocarb0n5'~~.''''"
1
1
I
I
I
1.5
2
2.5
IO IO,)
3
2/3
Flgure 7. The number of affected water molecules per monomer unit, Zh,and the hydration waterlpure water reorientation time ratio, Th/T,, plotted vs. the ratio of apparent molar volumes raised to the power 2/3, ($ /$ w)2'3, for aqueous solutions of polymers and of dioxane.
of PVA is capable of both accepting and donating binding protons. Thus there are more possibilities of solute-solvent interactions with PVA than with PEO. (b) In the general class of polyethers [-(CH,),-0-1, neither molecules with m = 1 nor those with m 2 3 are water soluble. Thus in the series of polyethers the water solubility of PEO ( m = 2 ) has to be considered rather anomalous. This particular anomaly has led to suggestions that it results from an ability of the PEO molecule to specially correspond to the water structure.11J2 The slight increase of the Z h value when going from PVA (2, = 5 . l Z f ) to PVME (2, = 6.6 f 0.2) seems mostly due to the fact that the repeat unit of the latter is more voluminous than that of the former. The enhancement of the T ~ / T ,value of the PVME solutions ( T h / T , = 2.46 f 0.06) with respect to the corresponding value found with the PVA solutions ( T h / T W = 2.302fj) may be expected on the grounds that former results of dielectric relaxation studies generally indicate an increase in the reorientation time of hydration water if an inert group is added to a solute particle. This common feature may be illustrated by the T h / T , ratios found in the series pyrazine ( T h / T , = 1.45 f ().I), 2-methylpyrazine ( T h / T w = 1.65 f 0.15), and 2,6-dimethylpyrazine ( T h / T w = 1.9 f 0.1) and by comparison of the data found with quinoxaline ( T h / T , = 1.9 f 0.2) and 2-methylquinoxaline ( T h / T w = 2.25 & 0.i).13 More examples may be taken from the n-alkylammonium ion and tetraalkylammonium ion series presented in Figure 8. However with the molecules mentioned above the additional inert group is added to another inert group. With PVME, however, the methyl group is bound to the hydroxyl group removing the proton-donating power for hydrogen bonds from this hydrophilic part of the repeat unit. So, since the transition from PVA to PVME is accompanied with a rather "normal" change in T h / T~ and since there is no difference in the Th/Tw values of PVA and
118
The Journal of Physical Chemistry, Vol. 82, No. 1, 1978
1
/ 25
3
35
4
45
5
55
6
[ O I O ~ P ~
Figure 8. Plot of the reorientation time ratio, rh/rw,vs. the ratio of apparent molar volumes raised to the power 2/3,(q5/4)*’3, for the series of n-alkylammonium ions, tetraalkylammonium ions, and azoniaspiroalkane ions.”
PEO, one may conclude that the H-bond proton-donating ability of the hydroxyl group is of minor importance for the solution behavior of PVA. Some characteristic hydration properties of our polymer solutions emerge if the Z h and T h / T , values are compared with those of the PVP and dioxane solutions. The Z h values of PEO, PVA, and PVME (2, = 5.5, 5.1, 6.6, respectively) are distinctly smaller than those of PVP (2, = 16.4) and of dioxane (2,= 14.3). It turns out that this difference in the number of affected water molecules per solute molecule or per monomer unit, respectively, is a mainly matter of solute particle surface area. To illustrate this statement, rough estimates of number (2)of water molecules adjacent to the solute particles are given in the 2, vs. (4/4,)2/3plot in Figure 7. These estimates were obained by division of surface areas assuming the solvent molecules to be cube shaped and the solute particles to be spherically shaped if they are small and cylindrically shaped if they are polymers. In the former case, 2 is given by
2 = (36n)”3(@/$,)2’3
(12)
This equation is represented by the solid line in the 2, vs. (4/1$,)~/~ plot. The number of water molecules in the first layer around a cylinder jacket is calculated with 2= (167~)”~(4/$~)”~
(13)
if the diameter of the cylinder is assumed equal to its height. This ratio of the diameter to its height seems reasonable for the monomer unit of oxygen-containing polymers under consideration. The dashed line in the upper diagram of Figure 7 is the graph of this formula. As can be seen from Figure 7 the Z h values of PVP, in which the nearly spherically shaped pyrrolidone ring dominates, and those of the likewise nearly spherically shaped dioxane, are a little greater than the corresponding number of adjacent water molecules as estimated with formula 12. The number of affected water molecules per repeat unit Z h of the oxygen-containing linear polymers PEO, PVA, and PVME is very close to the respective 2 value as found according to eq 13. Since the 2 values are rough estimates only, the result of this comparison of the 2, data with the 2 values should not be overemphasized. However it may be taken as an indication that the hydration region (Th # )7 , essentially is restricted to water molecules adjacent to a solute particle and that the difference in the values
Kaatze et al.
of different solutes is mainly due to size and shape effects here. A considerable difference in the reorientation time ratio of PEO and PVA ( T h / T w = 2.25 and 2.30, respectively) on one hand and of dioxane (Th/Tw = 1.59) on the other attracts attention. This difference clearly indicates that there are factors other than the chemical composition which are important in influencing the reorientational motion of the hydration water. These factors may include (a) the totalsize of the solute particle, (b) the overall shape of the solute molecule with respect to the water arrangement around it, and (c) the flexibility of the linear polymer chains vs. the inflexibility of the tightly packed cyclic groups. To extract the size effect by comparison of the reorientation time ratio of the solutions of macromolecular PEO and PVA with that of dioxane solutions is difficult owing to the considerable difference in the total molar volume of these two types of solutes. Nothing is known up to now on the dependence of Th/T, on the total molar volume in the large gap between monomers and polymers. So let us first ignore the size effect and comment on points b and c. To substantiate the statements made in these two points, some additional information is given in the Th/Tw vs. (4/4w)2/3 plot in Figure 7 . The dotted line represents the data for nitrogen-containing heterocyclic hydrocarbons (pyrazine, quinoxaline, and their deri~ativesl~). Despite of the other chemical composition of solute the Th/Tw value of the dioxane solution nicely fits these data. The dashed line is derived from deuteron magnetic relaxation rates in D 2 0 solutions of aliphatic noncyclic small organic molec u l e ~ .This ~ ~ line indicates an enhancement of the reorientation time ratio T h / T , when going from cyclic particles to more elongated molecules of equal molar volume. Due to this finding with small molecules, it seems likely that the pronounced enhancement of the T h / T w values of PEO and PVA with respect to that of dioxane may, at least in part, reflect the noncyclic arrangement of groups of these polymers and/or the marked flexibility of the chains. Owing to the different chemical composition of the solute, especially since the hydrophilic component is an imide group instead of an ether or hydroxyl group, the reorientation time ratio of PVP solutions is less comparable to the other. So it may be an empirical fact only that the T h / T w value of PVP in a way nicely fits the conception outlined above. Since PVP is built up by inflexible rings which are strung together on an elongated flexible chain, it may be regarded to be in the ambiguous position between small cyclic molecules on one hand and linear polymers on the other. Thus a reorientation time ratio beyond the value expected for 1-ethyl-2-pyrrolidone on the basis of the data of heterocyclic hydrocarbons (the dotted line in Figure 7 ) seems quite reasonable in this connection, if again the effect of total size is ignored. Our assumption that the overall shape and/or the flexibility of solute particles may be important factors in influencing the hydration behavior is supported by the results of dielectric constant measurements on aqueous solutions of organic ion^.'^,^^ These results are summerized in Figure 8, where a T h / T , vs. (4/4w)2/3 plot is given for n-alkylammonium ions, tetraalkylammonium ions, and azoniaspiroalkane ions. An appreciable decrease of Th/Tw values emerges when going from the elongated flexible n-alkylammonium cations to the nearly spherically shaped tetraalkylammonium ions with decreased chain flexibility. Forming closed loops from the alkyl chains of tetraalkylammonium ions additionally decreases the reorientation time of the hydration water.
Solution Behavior of Water-Soluble Polymers
To summarize, it seems rather likely that the shape and/or the flexibility of solute particles are important factors in influencing its hydration properties in aqueous solutions. However it cannot be excluded on the strength of our data that the difference (or part of it) in the rh/r, value of PEO and PVA on one hand and dioxane on the other is a result of the overall size of the solute particle. Certainly, no dependence of 731 on the degree of polymerization has been found with our solutions. However it may be that the size effect in particular acts if the solutes are not too voluminous and that it is imperceptible if the solute size exceeds a certain value. As can be seen from the different series of small molecules or organic ions presented as an example in Figures 7 and 8 an increase of rh/rw with increasing q5 emerges within each series. However this enhancement of the reorientation time ratio may mostly reflect the increasing amount of inert groups within such a series and may be to due in a minor way to a size effect only. For the different reasons mentioned above it seems to be impossible to draw a clear-cut conclusion on the relative influence of the polymer size, shape, and flexibility on the reorientational motion of the hydration water until data on oligomer solutions are available. The discussion of the hydration water properties is based on the assumption that (as expressed by eq 4) the ColeCole relaxation function is appropriate to describe the dielectric relaxation of the affected solvent. However as can be seen from the next to last column of Table I1 the relaxation time distribution parameter h is small if not zero with most solutions and achieves values distinctly different from zero if the concentration parameter c^ is close to or above 1. This fact may be taken to indicate that there is a rather uniform reorientational motion of hydration water in less concentrated solutions (e 5 1) of PEO, PVA, and PVME. In addition the finding that the h values are close to zero implies that the Z h and Th values extracted from the analysis of the complex dielectric constant data are nearly independent of the type of relaxation time distribution function attributed to the dielectric relaxation of the hydration water. 111. Dielectric Relaxation of the Solute. Due to the polar heteroatom in the chain, the PEO molecule has an electric dipole moment perpendicular to the chain direction. The relaxation of this dipole moment by some segmental mode of motion is represented by the solute relaxation term of the R(v) function (eq 4). The relaxation strength of this process 4 0 ) - ,6 is small as compared to that of the water process E , - t ( m ) . For that reason the values of the relaxation time of the solute r, extracted from the complex dielectric constant data are too uncertain to allow for a detailed discussion. However we can state at least that the ru values are strikingly small (13 ps 5 r, 5 270). This finding suggests that the polymer chain of PEO molecules is very flexible though forming hydrogen bonds with the solvent water. A very short relaxation time r, PEO in solution (10 ps at 20 " C ) has already been observed in the case of a nonassociating solvent by Davies et al.,zO who examined a series of PEO polymers in benzene solutions. The reorientational motion of the hydroxyl side group of PVA leads to a dielectric relaxation with a relaxation time r, somewhere in the range 36-110 ps. With the PVME molecule, however, the reorientational motion of the polar side group seems to be hindered by the additional methyl group so that no relaxation is found in the range of frequencies of measurements (7, 2 300 ps). As shown in Figure 9, the plot of the relaxation strength 4 0 ) - t, of the PVA and PEO solutions vs. the solute
n 2
0
4
6
8
1 0 1 2 1 4
c Irnol/ll
Flgure 9. The strength of the solute relaxation, e(0) - e,,, plotted vs. the molarity of monomer units, c, for poly(viny1alcohol) and poly(ethy1ene oxide) solutions.
TABLE IKK: Observed Dipole Moment and Calculated Group Moment of the Polar Unit of Poly(viny1 alcohol) and Poly(ethy1ene oxide) Obsd dipole moment/ monomeric unita Solute PVA PEO a
~ G / D 1.6 c 0.15 1.1f 0.15
From eq 14.
Calcd group momentb Group C-OH C-0-CH,
p0/D (MG/Y~Y 1.7 0.89 i 0.2 1.3 0.77 c 0.15
From ref 24.
concentration c is (within the limits of experimental errors) a straight line which can be extrapolated back to pass through zero for pure water. The slope of this line ~ ( 0-) E J / C is (1.6 f 0.3) (mol/L)-l and (0.8 f 0.15) (mol/L)-' in the case of PVA and PEO solutions, respectively. These values can be used to estimate the rms dipole moment pG, which the polar parts of the polymers would have in the gaseous state. We use a relation for this estimation which was originally presented by Brownz1 and which leads to reasonable results in the case of aqueous solutions of moderately and strongly dipolar solutes.22 With (~(0) t,,)2/t(0)z and t ( m ) / 2 t B Wneglected in comparison to 1 this relation becomes
(14) Here NL denotes Avogadro's number, h is Boltzmann's constant, T represents the temperature on the Kelvin scale, and c is the solute concentration in mol/L, as mentioned above. The value of E , has been empirically found to be about 4.3 in the case of pure water23 and may be approximately used for the solute polar components consisting of ether oxygen or a hydroxyl group. The values of the rms dipole moment obtained for the monomeric unit of PVA and PEO by application of eq 14 are given in Table 111. The effective dipole moment of a polymer repeat unit is usually expressed as WG = g1/2po, where po is the group dipole moment of the polar unit and g a factor depending on the degree of flexibility of the chain, and on the nature of the monomeric unit. Comparison of the observed ,uG values with values of group moments from the l i t e r a t ~ r e , 'which ~ are also given in
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The Journal of Physical Chemistry, Vol. 82, No. 1, 1978
Table 111, indicate that pG po holds. So one may conclude that with both PVA and PEO solutions, the value of the ratio ( p ~ / p $ is close to but a little lower than the predicted value 0.92 of the g factor24 which has been calculated for a (-CH&HR-) type polymer and for poly(ethy1ene oxide), assuming free rotation within the chain backbone.
Conclusion The dependence of the complex dielectric constant of aqueous solutions of PEO, PVA, and PVME on the frequency can easily be regarded as due to a superposition of three relaxation processes: (i) that of unaffected solvent water, (ii) that of disturbed water (hydration water), and (iii) that of the solute molecules. If the measured data are analyzed corresponding to this superdivision into three relaxation terms, no dependence of the parameter values on the degree of polymerization n is found in the range 75 5 n 5 13600. However the values of the hydration water reorientation times T h differ in a characteristic manner from what one expects on the basis of the data for aqueous solutions of small organic molecules. This difference in the hydration properties of polymers on one hand and of monomers on the other indicates that there are factors other than the chemical composition of the solute, the steric arrangement of its groups, and the size of the repeat unit or of the monomer molecule which are important in influencing the reorientational motion of the hydration water. These factors may include (a) an additional size parameter of the total molecule, (b) the overall shape of the solute particle, and (c) the flexibility of the solute. On the basis of our data no clear-out conclusion can be drawn on the relative influence of these factors. However more detailed insight into the relative importance of the different solute properties with respect to the hydration behavior might be obtainable by a study of aqueous solutions of a series of oligomers. It can be seen from the dielectric relaxation time of the solute relaxation that there is some rapid motion of the polar components in the PEO and PVA molecules. The strength of the solute relaxation can be analyzed to show that there is some hindrance of this motion as compared to a free rotation within the chain backbone.
Acknowledgment. It is a pleasure for us to acknowledge the donation of polymer samples by the BASF AG,
Kaatze et al.
Ludwigshafen, by the Farbwerke Hoechst AG, Frankfurt, by the Union Carbide Deutschland GmbH, Dusseldorf, and by the Wacker-Chemie GmbH, Munchen. We also thank the Deutsche Forschungsgemeinschaft for financial assistance. The numerical calculations have been done on a computer financed by the Stiftung Volkswagenwerk.
References and Notes U. Kaatze, Adv. Mol. Relaxation Processes, 7, 71 (1975). U. Kaatze, unpublished results. J. B. Hasted, G. H. Haggis, and P. Hutton, Trans. Faraday SOC.,47, 577 (1951). U. Kaatze, Diplom-Thesis, Gottingen, 1967. K. S.Cole and R. H. Cole, J. Chem. Phys., 9,341 (1941). D. Polder and J. H. van Santen, Physica, 12, 257 (1946). U. Kaatze and W.-Y. Wen, J. Phys. Chem., precedlng paper in this issue. G. Akerlof and A. 0. Short, J . Am. Chem. SOC.,58, 1241 (1936). H. S.Harned and B. B. Owen, "The Physical Chemistry of Electrolyte Solutions", Reinhold, New York, N.Y., 1950, pp 545-546. J. Timmermans, "The Physico-Chemical Constants of Binary Systems", Interscience, New York, N.Y., 1960, Vol. 4. D. T. Warner, Ann. N . Y . Acad. Sci., 125, 605 (1965). M. J. Blandamer, M. F. Fox, E. Powell, and J. W. Stafford, Makromol. Chem., 124, 222 (1969). R. Pottel and U. Kaatze, Ber. Bunsenges. Phys. Chem., 73, 437 (1969). G. Engel and H. G. Hertz, Ber. Bunsenges. Phys. Chem., 72, 808 (1968) (dashed curve in Figure 19). U. Kaatze, C. H. Limberg, and R. Pottel, Ber. Bunsenges. Phys. Chem., 78, 555 (1974). W.-Y. Wen and U. Kaatze, J. Phys. Chem., 81, 177 (1977). The reorientation time values have been derived from the results on solutions of n-alkylammonium chlorides,15 tetraalkylammonium bromides," and azoniaspiroalkane bromides." The solute concentration c was at around 1 mol/L except with n-heptylammonium and n-octylammonium chloride solutions. To prevent formation of micelles, solutions with cvalues smaller than 0.4 and 0.175 mol/L, respectively, were used in these cases. C#I denotes the apparent molar cationic volume at infinite dilution, C#I in this diagram. The 4 ,"' vaiues for tetraalkylammoniumand azoniaspiroalkane ions were taken from the literature," those for the n-alkylammonium ions have been calculated from the respective values of the bromides" with the assumption of C#I:(Br-) = 30.21 mL/mol.'* W.-Y. Wen, A. LoSurdo, C. Jolicoeur, and J. Boileau, J. phys. Chem., 80,486 (1978). J. E. Desnoyers and M. Arel, Can. J. Chem., 45, 359 (1967). M. Davies, G. Williams, and G. D. Loveluck, Z . Hektrochem., 84, 575 (1960). W. F. Brown, "Dielectrics" in "Handbuch der Physik", S.Flugge, Ed., Springer, Berlin, 1956, Section 46, eq 46.9. R. Pottel, D. Adolph, and U. Kaatze, Ber. Bunsenges. Phys. Chem., 79, 278 (1975). N. E. Hill, J . Phys. C , 3, 238 (1970). A. H. Price, "The Permittivity of Liquids" in "Dielectric Properties and Molecular Behavior", N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davies, Ed., Van Nostrand, London, 1969, Chapter 4, pp 232-279.
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