Viscosity of aqueous solutions of poly(ethylene glycol)s at 298.15 K

Fluid Phase Equilibria 155 Ž1999. 311–325

Viscosity of aqueous solutions of poly Žethylene glycol .s at 298.15 K Stanislava Kirincic ˇ ˇ a , Cveto Klofutar

b, )

a

b

Institute of Public Health of the Republic of SloÕenia, 1000 Ljubljana, SloÕenia Department of Food Technology, Biotechnical Faculty, UniÕersity of Ljubljana, 1000 Ljubljana, SloÕenia Received 15 September 1998; accepted 30 December 1998

Abstract The viscosities of aqueous solutions of some polyŽethylene glycol.s ŽPEG. with nominal molecular weights ranging from 300 to 35 000 g moly1 were determined up to a concentration of 0.3 g cmy3 at 298.15 K. From these data the intrinsic viscosity and the viscosity average molecular weight of the solute were calculated. The viscosity coefficients B were evaluated and hence the partial molar Gibbs free energy of activation of viscous flow of solute at infinite dilution was calculated and interpreted in terms of the relative effects of solute on the ground and transition state solvent. The hydration numbers were determined and compared with available values in the literature. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Viscosity; Data; Gibbs energy; Hydration numbers; Aqueous solutions; PolyŽethylene glycol.

1. Introduction PolyŽethylene glycol. Ž PEG., an industrially important polymer, has unique solubility properties, dissolving in water in all proportions at moderate temperatures and in a very wide range of degrees of polymerization, contrary to other structurally similar polyethers like polyŽ methylene oxide. , polyŽ acetaldehyde. and polyŽ propylene oxide. , which are all insoluble in water. It is expected that the structure of water w1x would effect the conformation of polyŽethylene glycol. in aqueous solution w2–9x, and play an important role in the physico-chemical properties of the solution. As far as we know, a surprisingly small amount of high quality viscosity data has been published for aqueous solutions of polyŽethylene glycol. s at 298.15 K. Thus for example, Hayduk and Malik w10x measured the viscosity of aqueous solutions of monoethylene glycol and recently Gonzalez-Tello ´ et al. w11x reported results for polyŽ ethylene glycol. 1000, 3350 and 8000 g moly1, but in a higher concentration range than ours. The majority of the other viscosity data for aqueous polyŽ ethylene )

Corresponding author. Tel.: q386-61-123-11-61; fax: q386-61-266-296.

0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 0 0 5 - 9

312

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

glycol. solutions are reported either at other temperatures or in combination with different salts w12,13x. From measurements of the viscosities of aqueous solutions of polyŽ ethylene glycol. s of molecular weights up to 1000 g moly1 at 298.15 K, Bahri and Guveli w14x interpreted the viscosity coefficient B in terms of the Einstein law and determined the average hydration number, which increased with the increasing average molecular weight of solute. A similar study was also made by Moulik and Bisal w15x at a temperature of 303.15 K, and by Herskovits and Kelly w16x, but only for monoethylene glycol. More detailed insight into the relative importance of various solute properties might be obtained by a study of aqueous solutions of a series of oligomers with increasing molecular weight. In this work we investigated systematically the viscosity properties of aqueous solutions of a wide range of polyŽ ethylene glycol. s with the aim of elucidating their dependence on molecular weight. 2. Experimental section 2.1. Materials Samples of polyŽ ethylene glycol. of different average molecular weight, ranging from 300 to 35 000 g moly1, were supplied from Fluka Chemica Ž pract.. , except the sample of 15 000 g moly1, which was from Merck, Suchard Žfor synthesis.. The compounds were used without further purification and stored in a dessicator over P2 O5. Doubly distilled water was used to prepare solutions on a volumetric concentration scale. 2.2. Viscosity measurements The viscosity of the solutions investigated was determined by an Ubbelohde viscometer up to a solute concentration of 0.30 g cmy3 at a temperature of 298.15 " 0.05 K. The absolute viscosity values were calculated by means of the equations w17x Nd h s Ldt y 2 Ž1. t where h is the absolute viscosity of the solution, t is the flow time, d is the density of the solution, L and N are constants characteristic of the viscometer. The viscometer constants L and N were determined by a least-squares fit to Eq. Ž1. of the literature data for the absolute viscosity, ho and density of water, d o w18x, at the respective temperature. The temperature of the water bath was maintained at "0.05 K. The flow time of the investigated solutions was between 250 to 1090 s and was measured with an accuracy better than 0.1 s. For each solution the flow time was measured at least seven times. The maximal error of the measured viscosity, Eh , was 2 P 10y6 kg my1 sy1. The densities and the apparent specific volumes of the investigated solutions were obtained from Ref. w19x. 3. Results and discussion The viscosity data of the systems investigated are presented in Table 1 as a function of concentration, c 2 Žgrams of solute per cm3 of solution., at 298.15 K. It can be seen from Table 1 that

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Table 1 Viscosity of aqueous solutions of polyŽethylene glycol.s at 298.15 K c 2 Žg cmy3 .

hP10 3 ŽPL.

c 2 Žg cmy3 .

hP10 3 ŽPL.

c 2 Žg cmy3 .

hP10 3 ŽPL.

PEG 300 0.0188 0.0377 0.0527 0.0753 0.1055 0.1507 0.2109 0.2411 0.3013

0.958 1.030 1.091 1.191 1.338 1.605 2.045 2.323 3.001

PEG 400 0.0169 0.0337 0.0472 0.0674 0.0943 0.1348 0.1887 0.2156 0.2685

0.959 1.031 1.091 1.189 1.333 1.584 2.002 2.263 2.887

PEG 600 0.0195 0.0391 0.0547 0.0781 0.1094 0.1563 0.2188 0.2500 0.3126

0.981 1.076 1.160 1.298 1.515 1.912 2.619 3.067 4.231

PEG 900 0.0138 0.0276 0.0386 0.0551 0.0772 0.1103 0.1544 0.1764 0.2206

0.963 1.042 1.109 1.218 1.380 1.667 2.142 2.425 3.130

PEG 1000 0.0120 0.0241 0.0337 0.0481 0.0674 0.0963 0.1348 0.1540 0.1925

0.963 1.038 1.102 1.204 1.353 1.612 2.037 2.291 2.893

PEG 1500 0.0101 0.0203 0.0284 0.0406 0.0568 0.0811 0.1136 0.1298 0.1623

0.961 1.036 1.098 1.198 1.347 1.600 2.007 2.243 2.806

PEG 2000 0.0087 0.0173 0.0242 0.0346 0.0484 0.0692 0.0969 0.1107 0.1384

0.962 1.036 1.099 1.197 1.345 1.596 1.987 2.214 2.734

PEG 3000 0.0076 0.0153 0.0214 0.0306 0.0428 0.0611 0.0855 0.0978 0.1222

0.970 1.054 1.125 1.239 1.406 1.693 2.145 2.404 3.002

PEG 4000 0.0061 0.0121 0.0170 0.0243 0.0340 0.0486 0.0680 0.0777 0.0971

0.968 1.050 1.119 1.229 1.388 1.660 2.076 2.321 2.866

PEG 6000 0.0038 0.0075 0.0106 0.0151 0.0211 0.0302 0.0422 0.0482 0.0603

0.949 1.010 1.061 1.141 1.255 1.440 1.719 1.869 2.208

PEG 10 000 0.0023 0.0045 0.0063 0.0090 0.0126 0.0180 0.0252 0.0288 0.0360

0.941 0.992 1.034 1.100 1.189 1.333 1.537 1.648 1.881

PEG 12 000 0.0023 0.0046 0.0064 0.0092 0.0129 0.0184 0.0257 0.0294 0.0368

0.962 1.036 1.099 1.196 1.337 1.568 1.914 2.104 2.527

PEG 15 000 0.0027 0.0055 0.0075 0.0110

0.982 1.078 1.151 1.288

PEG 20 000 0.0020 0.0041 0.0057 0.0081

0.968 1.047 1.114 1.218

PEG 35 000 0.0018 0.0036 0.0050 0.0071

0.980 1.074 1.154 1.279

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

314

Table 1 Žcontinued. c 2 Žg cmy3 .

hP10 3 ŽPL.

c 2 Žg cmy3 .

hP10 3 ŽPL.

c 2 Žg cmy3 .

hP10 3 ŽPL.

PEG 15 000 0.0150 0.0219 0.0299 0.0342 0.0439

1.456 1.786 2.217 2.469 3.130

PEG 20 000 0.0113 0.0162 0.0227 0.0259 0.0324

1.363 1.601 1.953 2.152 2.593

PEG 35 000 0.0100 0.0142 0.0199 0.0227 0.0284

1.459 1.761 2.208 2.460 3.069

1 PL s1 kg my1 sy1 w51x.

the viscosity h of the investigated solutions depends on the average molecular weight of polyŽ ethylene glycol. and increases in an non-linear manner with its concentration. The viscosity data can be expressed by an empirical relation given by Schulz and Blaschke w20x as

hsp c2

s h Ž 1 q k SBhsp .

Ž2.

where wh x is the intrinsic viscosity, a characteristic function for a single molecule in solution, hsp s Žh y ho .rho is the specific viscosity and k SB is a Schulz–Blaschke constant depending on the solute, solvent and temperature. As an example in Fig. 1 the reduced viscosity, hsprc2 vs. hsp is given for polyŽethylene glycol. of a nominal molecular weight of 2000 g moly1. It can be seen from Fig. 1 that the reduced viscosity increases linearly with specific viscosity, where the regression line is shown

Fig. 1. Dependence of reduced viscosity on specific viscosity of polyŽethylene glycol. 2000 at 298.15 K.

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

315

as a solid line. The calculated values of wh x and k SB from linear regression analysis are given in Table 2. The average standard error of wh x and k SB are 0.05 cm3 gy1 and 0.02, respectively. From these it can be seen that the intrinsic viscosity gradually increases while k SB the constant decreases with increasing molecular weight of the solute. For an ideal solution of spherical solute particles, the value of k SB is 0.4, while for a real solution, taking into account the frictional coefficient, f w21x, this value is w20x 1 k SB s F 0.4 Ž3. 2.5 f 3 At finite polymer concentrations, the relation between the specific viscosity, hsp , and the concentration of solute, c 2 , can be expressed conveniently and most generally by the polynomial w22x 2

3

hsprc2 s h q k 1 h c 2 q k 2 h c22 q . . . ,

Ž4.

Without the third term, this is the familiar Huggins equation w23x and the dimensionless parameter k 1 is called the Huggins slope coefficient or a hydrodynamic interaction constant w24x arising from hydrodynamic interactions between separate polymer chains. In Fig. 2 the dependence of reduced viscosity, hsprc2 , on concentration c 2 of polyŽethylene glycol. with an average molecular weight of 300 g moly1 is represented. It can be seen from Fig. 2 that the ratio of hsprc2 s hred , i.e., the reduced viscosity increases in a non-linear manner with the concentration of solute, which is characteristic of polyŽ ethylene glycol. s. Therefore, we described this dependence by a polynomial of second degree, which is shown as a solid line in Fig. 2. The values of the constants in Eq. Ž4. were calculated from the data given in Table 1 and are collected in Table 2. The average standard error of wh x, k 1 and k 2 Table 2 Intrinsic viscosity, wh x, Schulz–Blaschke constant, k SB , constants k 1 and k 2 , nominal molecular weight, M2,n , and viscosity average molecular weight, M2,h , of polyŽethylene glycol.s at 298.15 K M2,n Žg moly1 .

wh x Eq. Ž2. Žcm3 gy1 .

wh x Eq. Ž4. Žcm3 gy1 .

wh x lit. val. Žcm3 gy1 .

k SB Eq. Ž2.

k1 Eq. Ž4.

k2 Eq. Ž4.

M2,h Eq. Ž6. Žg moly1 .

300 400 600 900 1000 1500 2000 3000 4000 6000 10 000 12 000 15 000 20 000 35 000

3.91 4.41 5.03 5.84 6.54 7.65 9.03 11.50 14.15 17.28 24.86 34.51 36.92 42.02 55.82

3.92 4.51 5.02 5.78 6.61 7.63 8.96 11.30 13.98 17.14 24.92 34.13 36.26 41.95 55.29

3.4 a 4.1a 5.2 a – 6.86 a – – – – – – – – 37.7 b –

0.43 0.40 0.38 0.40 0.35 0.35 0.32 0.30 0.28 0.29 0.22 0.25 0.23 0.21 0.22

0.35 0.22 0.28 0.38 0.25 0.32 0.34 0.36 0.33 0.33 0.20 0.30 0.29 0.21 0.25

0.42 0.40 0.39 0.30 0.28 0.22 0.17 0.12 0.11 0.08 0.09 0.06 0.05 0.06 0.06

404 584 751 1025 1344 1767 2363 3512 4969 6849 12 083 19 194 20 961 25 868 38 386

a

Bahri and Guveli w14x. Ataman w25x.

b

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

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Fig. 2. Concentration dependence of reduced viscosity for aqueous solutions of polyŽethylene glycol. 300 at 298.15 K.

are 0.05 cm3 gy1, 0.02 and 0.005, respectively. From these data it can be seen that the values of intrinsic viscosity calculated on the basis of Eq. Ž4. are, within experimental error, equal to those values calculated by the Schulz–Blaschke Eq. Ž 2. . It can be also seen from Table 2 that our data for intrinsic viscosity wh x are in a good agreement with the works of Bahri and Guveli w14x and Ataman w25x, and that the intrinsic viscosity increases with increasing molecular weight of the solute. The coefficient k 1 of Eq. Ž4. shows almost no dependence, while coefficient k 2 gradually decreases with increasing average molecular weight of the solute. The theoretical interpretation of the parameters in Eq. Ž 4. is not clear at present. It is well known that the intrinsic viscosity depends on the size and shape of the macromolecule w24x. The dependence of the intrinsic viscosity on the molecular weight of solute is most frequently described by the Mark–Houwink Kuhn–Sakurade equation w26x. Sadron and Rempp w27x found that the dependence of intrinsic viscosity on the molecular weight of polyŽ ethylene glycol. s in cyclohexane, dioxane, methanol and water could be expressed as

h s A q KM a

Ž5.

where A depends on the solvent properties in the same manner as a and K depend on the statistical configuration of the polymer chains. Ring et al. w28x correlated their data and data from the literature for aqueous solutions of polyŽ ethylene glycol. s in the manner suggested by Sadron and Rempp w27x, and found that the relationship

h s 0.02 q 2.4 P 10y4 M2,0.73 h

Ž6.

satisfactorily describes the dependence of the intrinsic viscosity of polyŽ ethylene glycol. on its average molecular weight. By assuming that the polydispersity of the samples of polyŽ ethylene

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

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glycol.s investigated does not differ much from those of w28x, we calculated from the intrinsic viscosity data the average viscosity molecular weights, M2,h . The calculated values of M2,h are given in Table 2. From Table 2 it can be seen that these values are systematically higher than the nominal average molecular weights, M2,h . A relatively high difference between nominal average molecular weight and viscosity average molecular weight was also observed by Hey et al. w29x for a polyŽethylene glycol. with a number average molecular weight of about 8000 g moly1. The weight average molecular weight obtained from the intrinsic viscosity for this sample at 298.15 K was 9700 g moly1. The observed difference in the molecular weight can be attributed to polyŽ ethylene glycol. , which is a polydisperse polymer w30x, and to the different experimental methods used in the determination of molecular weight. Viscosity data on dilute to moderate concentrations of a non-electrolyte can also be interpreted by the relation w16x

hr s 1 q BX w q CX w 2 q DX w 3 Ž7. X X X where hr s hrho is the relative viscosity, B , C and D are adjustable parameters and w is the weight molality Ž w s c 2rd y c 2 . w31x. Thus the viscosity coefficients defined by Herskovits and Kelly w16x are: B s BX M2 P 10y3 Ž kg moly1 . which is related to the size and shape of the solute molecule and to the solvation effect, and C s CX Ž M2 P 10y3 . 2 Žkg 2 moly2 . and D s DX Ž M2 P 10y3 . 3 Žkg 3 moly3 . to the solute–solute interactions which are not accounted for by the B term. M2 is the average molecular weight of solute which was substituted by the viscosity average molecular weight, M2,h calculated from Eq. Ž6.. The coefficients BX , CX and DX , calculated from the experimental data Ž Table 1. and density data w19x are given in Table 3. Thereby, the average relative standard error of viscosity coefficients BX , CX and DX are 0.40, 5.0 and 18.0%, respectively. From Table 3 it can be seen that coefficient BX increases non-linearly with increasing average molecular weight of solute. This dependence can be

Table 3 Values of coefficients BX , CX and DX of the investigated polyŽethylene glycol.s in aqueous solution as a function of viscosity average molecular weight of solute, M2,h at 298.15 K M2,h Žg moly1 .

BX

CX

DX

404 584 751 1025 1344 1767 2363 3512 4969 6849 12 083 19 194 20 961 25 868 38 386

3.84 4.42 4.86 5.71 6.52 7.56 8.90 11.23 13.91 17.08 24.80 34.01 36.13 41.81 55.10

4.16 3.43 7.44 10.65 9.00 15.38 22.13 39.02 55.08 84.86 109.98 323.26 357.30 345.96 737.01

1.97 6.95 4.59 9.09 22.58 27.54 34.22 44.82 107.58 186.49 721.45 1410.90 1252.10 3464.90 7987.20

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Fig. 3. Viscosity coefficients B of polyŽethylene glycol.s in aqueous solution plotted against partial molar volumes, V2` at 298.15 K: Ž`. this work; Žv . values from Bahri and Guveli w14x; Ž- - - - -. log B s log 2.5qlog V2`.

ascribed to the change of the configuration of polyŽ ethylene glycol. molecule, which is hydrated through hydrogen bonding w32,33x. PolyŽethylene glycol. molecules with relatively long chains are presumably randomly coiled, as reported for polyoxyethylene polymers w26,34x and for polyoxyethylene surfactants w35x. Owing to the entropy effect, the polyŽ ethylene glycol. s chain persists in a random coil configuration w36x. In Fig. 3 the dependence of log B on log V2` is presented, where V2` Ždm3 moly1 . is the partial molar volume of solute at infinite dilution Ž V2` s Õ`2 M2,h ., calculated from Õ`2 , Žcm3 gy1 . w19x. As can be seen, this dependence is not linear as is usually the case for aqueous solutions of electrolytes and non-electrolytes w16x. It can be seen from Fig. 3 that the data for the viscosity coefficient B determined by Bahri and Guveli w14x are in a good agreement with ours. The dotted line in Fig. 3 represents the function log B s log 2.5 q log V2`, which is valid for a spherically shaped macromolecule w37x. Considering Eyring’s theory of rate processes applied to viscous flow, the relation of viscosity coefficient B Ž kg moly1 ., which is derived from Eq. Ž7., is w38x B s d1o ž V1o y V2` / q

bG M 1 RT

Ž8.

where V1o is the molar volume of the solvent, M1 is the molecular weight of the solvent, R is the gas constant, T is the absolute temperature, d1o is the density of the solvent and bG is a coefficient defined as

bG s DaG`2 y Da G 1o

Ž9.

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319

where DaG`2 is the partial molar Gibbs free energy of activation for viscous flow of solute at infinite dilution and DaG 1o for pure solvent. From Eq. Ž 9. it follows that the parameter bG may be considered as the difference between the partial molar function of activation for viscous flow of a solute at infinite dilution and the respective molar quantity of the pure solvent. From Eq. Ž 8. it is evident that the solvent structure can influence the viscosity coefficient B in two terms: the volumetric term, which gives a negative contribution if V1o - V2`, and the term for changes of partial molar Gibbs free energy of activation for viscous flow of solute at infinite dilution, DaG`2 and pure solvent, DaG 1o . For a system in which Da G 1o - Da G`2 this term gives a positive contribution to the B value. Thus Eq. Ž8. allows the comparison of viscosity coefficients B in different solvents at a definite temperature. For the investigated systems the viscosity coefficients B are positive Žsee Table 3., which means that the second term of the right-hand side of Eq. Ž 8. prevails over the first one which is negative. The partial molar Gibbs free energy of activation for viscous flow of the solute at infinite dilution, DaG 1o , and parameter bG at 298.15 K were calculated via Eqs. Ž8. and Ž9., and are presented in Table 4, using the value of the molar functions of activation for viscous flow for pure solvent, DaG 10 s 9.164 kJ moly1 w38x. The values of DaG`2 can be interpreted in term of the relative effects of solute on the ground and transition state of the solvent w39,40x. If a solute is completely coordinated in the ground state solvent, formation of the transition state involves solute–solvent bond breaking and a reduction in the coordination number of the solute. However, if the ground state solvent is highly structured and thus resists complete coordination of the solute, then the additional solute–solvent bonds can be formed in the less rigid transition state solvent with an increase in the coordination number of the solute.

Table 4 Values of coefficient bG , partial molar Gibbs free energy of activation for viscous flow of polyŽethylene glycol. at infinite dilution, DaG`2 , and per mol of monomeric unit, Da G`2,m at 298.15 K as a function of viscosity average molecular weight, M2,h M2,h Žg moly1 .

bG ŽkJ moly1 .

Da G`2 ŽkJ moly1 .

Da G`2 r P s Da G`2,m ŽkJ moly1 .

404 584 751 1025 1344 1767 2363 3512 4969 6849 12 083 19 194 20 961 25 868 38 386

258 420 586 921 1360 2040 3164 5830 10 078 16 882 42 624 92 036 106 623 151 771 295 413

267 430 596 930 1369 2049 3173 5839 10 088 16 892 42 633 92 045 106 632 151 780 295 422

29 32 35 40 45 51 59 73 89 109 155 211 224 259 339

320

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

From Table 4 it follows that the values Ž DaG`2 y Da G 1o . are positive. As Ž Da G`2 ) Da G 1o ., it may be anticipated that in the systems studied the solute–solvent bonds are stronger than the solvent–solvent bonds in the transition state. For the investigated systems the values of Ž DaG`2 y Da G 1o ., like the coefficients B, increase with the molar volumes or average molecular weight of the solute Ž see Figs. 3 and 4.. According to the transition state theory of viscosity w39x, the values of DaG`2 do not contain any contributions from the changes of solvent–solvent interactions caused by the molecule of solute. Thus the value of DaG`2 and the viscosity coefficient B depend only on the difference in the solute–solvent interactions in the ground and transition state. In contrast, the changes of the corresponding enthalpy, DaH2`, and entropy of activation for viscous flow of solute at infinite dilution, DaS`2 are affected not only by the changes of solute–solvent interactions but also by the changes of solvent–solvent interactions. In Fig. 4 DaG`2,m , i.e., the Gibbs free energy of activation for viscous flow of the solute at infinite dilution per monomeric unit, is given as a function of degree of polymerization. As can be seen from Fig. 4 and Table 4, DaG`2,m values gradually but non-linearly increase with degree of polymerization. On the other hand, the coefficients CX and DX , which are higher than coefficient BX , show a similar trend as observed for the coefficients BX , i.e., they depend on the average molecular weight or degree of polymerization of the investigated solutes. From these it may be assumed that with increasing solute concentration and increasing chain length when polyŽ ethylene glycol. molecules approach each other, effects of aggregation occur. Information on the hydration of solutes can be obtained by a procedure given by Linow and Philipp w20x. In a relatively simple geometric model, based on Einstein’s viscosity law w37x, a calculation of

Fig. 4. The average numbers of moles of water bound per mole of monomeric unit of poly Žethylene glycol., hX Žv . and Gibbs free energy of activation for viscous flow of the solute at infinite dilution per monomeric unit of polyŽethylene glycol., DaG`2,m Ž^. as a function of degree of polymerization, P at 298.15 K.

S. Kirincic, ˇ ˇ C. Klofutarr Fluid Phase Equilibria 155 (1999) 311–325

321

solvation numbers from the change of solution viscosity with solute concentration can be achieved. From Eq. Ž6. in Ref. w20x

hsp c2

2.5 s d

X

ž

1q

1 2.5

hsp

/

Ž 10.

which is formally analogous to the Schultz and Blaschke relation w20x, the value 1rdX s Õh , i.e., the specific volume of hydrated solute can be obtained. It follows from Eq. Ž 10. that the value of specific volume Õh does not depend on the numerical factor 2.5, which means that for the model used, the shape of the solute particles is not involved in the calculation of the hydration number. On the other hand, from the dependence of density on solute concentration, the apparent specific volume of the solute, Õ 2f , can be obtained. So the difference of Õh y Õ 2 f is the volume of solvent bound per gram of solute. Thus all volume changes connected with the solvation process are included in the apparent specific volume of solute. Hence, the hydration number, h, i.e., the number of moles of bonded water molecules in the inner hydration sphere per mole of solute, is given by hs

d1o ž Õh y Õ 2 f / M2 M1

Ž 11.

where in place of M2 we used the viscosity average molecular weight, M2,h calculated from Eq. Ž6.. As an example, in Fig. 1, hsprc2 s hred vs. hsp for polyŽ ethylene glycol. of nominal molecular weight of 2000 g moly1 is given. From Fig. 1 it can be seen that Eq. Ž 10. satisfactorily represents the experimental results. Since the apparent specific volume of polyŽ ethylene glycol. s are slightly dependent on solute concentration w19x, the apparent specific volume of solute at infinite dilution, Õ`2f from Ref. w19x, was used in calculating the hydration numbers, h in Eq. Ž11.. The calculated values of hydration numbers hX Žthe number of moles of bonded water molecules per monomer unit of polymer. are given in Table 5 and are shown in Fig. 4 as a function of the degree of polymerization, P. It can be seen from Table 5 and Fig. 4 that hX values increase with the average molecular weight of the solute. The same was also observed by other authors w14,20,41x, whose hydration numbers, hX , determined by viscometry are compiled in Table 5. It follows from Table 5 that our hydration numbers are in a good agreement with the literature ones and that they increase remarkably with increasing molecular weight of polyŽ ethylene glycol. , especially above a nominal molecular weight of 3000 g moly1. It can also be seen from Table 5 that the hydration values determined by Bisal et al. w41x at 303.15 K are lower than the other ones, which were determined at 298.15 K. The nature and the extent of hydration of polyŽethylene glycol. in aqueous mixtures remains uncertain despite numerous investigations by different experimental methods used, including conductometry w41,42x, neutron scattering w7,9,43x, differential scanning calorimetry w3,44–47x, nuclear magnetic resonance w48,49x, a molecular dynamics study w6x and Raman spectroscopy w50x. It seems that the basic hydration of polyŽ ethylene glycol. is not satisfied until two water molecules have been added to each –CH 2 CH 2 O– group w2x. However, estimates of the literatures hydration numbers, except by viscometry, vary from less than one to more than five. Hydration numbers determined by viscometry, which are presented in Table 5, are unusually large for higher molecular weight polyŽ ethylene glycol. samples, compared with those of the literature determined by other experimental methods. Antonsen and Hoffman w3x attempted an explanation and proposed that at low molecular

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Table 5 Comparison of hydration numbers Žmoles of hydrated water molecules per monomer unit., hX determined by viscometry at specific temperatures for polyŽethylene glycol.s of nominal molecular weights, M 2,n , from 200 to 35 000 g moly1 , found in this work, and literature values Hydration numbers, hX determined by viscometry M2,n Žg moly1 .

This work at 298.15 K

Linow and Philipp w20x at 298.15 K

Bahri and Guveli w14x at 298.15 K

Bisal et al. w41x at 303.15 K

200 300 400 600 900 1000 1500 2000 3000 4000 6000 10 000 12 000 15 000 20 000 35 000

– 2.06 2.23 2.58 3.44 3.54 4.40 5.08 6.31 7.73 10.14 11.23 18.98 18.30 19.79 28.44

1.8 2.3 2.9 3.2 – 3.8 – – – 8.8 11.8 – – – 22.9 –

0.16 1.41 2.00 2.90 – 3.20 – – – – – – – – – –

0.90 1.03 1.12 1.39 – 2.49 – – – – – – – – – –

weights of polyŽ ethylene glycol. , only tightly bound water is associated with the polyŽ ethylene glycol. chain. As the molecular weight increase the chain begins to fold in on itself, forming segment–segment interactions as it traps additional, more loosely bound water between the segments. Thus chain coiling and water hydration interactions provide the common link connecting a number of molecular weight dependent properties of polyŽ ethylene glycol. . The latter hypothesis can be tested by investigations on the structure of polyŽ ethylene glycol. in aqueous solution.

4. Conclusion In this work the viscosity of aqueous solutions of polyŽethylene glycol. s with nominal molecular weights ranging from 300 to 35 000 g moly1 were determined. The viscosity of the investigated solutions can be satisfactorily described by empirical Schulz–Blaschke, Huggins and Herskovits equations. The Huggin’s slope coefficient for the investigated systems shows almost no dependence on the molecular weight of the solute, though it is expected that the slope increase with molecular weight of the solute. The available data in the literature show that it can either increase, decrease or remain constant, depending on the polymer–solvent system. Attempts have been made to clarify the theoretical significance of this coefficient, but a completely quantitative interpretation has not yet been formulated. From the partial molar Gibbs free energy of activation for viscous flow of solute at infinite dilution it follows that the solute–solvent bonds are stronger than the solvent–solvent bonds in the transition state. The relatively high value of this thermodynamic function is a consequence of

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strong hydrogen bonding between water molecules and ether oxygens of polyŽ ethylene glycol. molecules, which also follows from the hydration numbers of the solute which are unusually large, especially for high molecular-weight samples. This can be explained by the formation of segment– segment interactions and consequently by more loose of water molecules between the segments.

5. List of symbols a and A B BX c2 C CX d do d1o 1rdX D DX f DaG 1o DaG`2 DaG`2,m h hX DaH2` k 1 and k 2 k SB L M M1 M2 M2,n M2,h N P R DaS2 ` t

Constant in Eq. Ž5. Viscosity coefficient Ž B s BX M2 10y3 . Žkg moly1 . Constant in Eq. Ž7. Concentration of solute Žg cmy3 . Viscosity coefficient Ž C s CX ŽM 2 P 10y3 . 2 . Ž kg 2 moly2 . Constant in Eq. Ž7. Density of solution Žg cmy3 . Density of water Ž g cmy3 . Density of solvent Žg cmy3 . Specific volume of hydrated solute Ž1rdX s Õh . Žcm3 gy1 . Viscosity coefficient Ž D s DX Ž M2 P 10y3 . 3 . Žkg 3 moly3 . Constant in Eq. Ž7. Frictional coefficient in Eq. Ž3. Molar Gibbs free energy of activation for viscous flow of pure solvent Ž kJ moly1 . Partial molar Gibbs free energy of activation for viscous flow of solute at infinite dilution ŽkJ moly1 . Partial molar Gibbs free energy of activation for viscous flow of solute at infinite dilution per monomeric unit of solute ŽkJ moly1 . Hydration number Ž number of bonded water molecules per mole of solute. hydration number Žnumber of bonded water molecules per monomeric unit of solute. Partial molar enthalpy of activation for viscous flow of solute at infinite dilution Ž kJ moly1 . Constants in Eq. Ž4. Schulz–Blaschke constant in Eq. Ž2. Empirical constant in Eq. Ž1. Ž m2 sy2 . Molecular weight Žg moly1 . Molecular weight of solvent Žg moly1 . Average molecular weight of solute Ž g moly1 . Nominal average molecular weight of solute Ž g moly1 . Viscosity average molecular weight of solute Ž g moly1 . Empirical constant in Eq. Ž1. Ž m2 s. Degree of polymerization Gas constant Partial molar entropy of activation for viscous flow of solute at infinite dilution Ž kJ moly1 . Flow time of solution Žs.

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T Õ`2 Õ`2f Õ`2f Õh V1o V2` w

Absolute temperature ŽK. Partial specific volume of solute at infinite dilution Žcm3 gy1 . Apparent specific volume of solute Žcm3 gy1 . Apparent specific volume of solute at infinite dilution Žcm3 gy1 . Specific volume of hydrated solute Ž1rdX s Õh . Žcm3 gy1 . Molar volume of solvent Ždm3 moly1 . Partial molar volume of solute at infinite dilution Ždm3 moly1 . Weight molality Ž w s c 2rŽ d y c 2 ..

Greek letters bG h ho hsp hr hred wh x Eh

Coefficient from Eq. Ž8. Absolute viscosity of solution Žkg my1 sy1, PL. Absolute viscosity of water Ž kg my1 sy1 , PL. Specific viscosity Relative viscosity Žhr s hrho . Reduced viscosity Žhred s hsprc2 . Žcm3 gy1 . Intrinsic viscosity Žcm3 gy1 . Maximal error of measured viscosity Žkg my1 sy1 .

Superscripts o `

Pure component Infinite dilution

Subscripts 1 2 1,2 m n f h

Solvent Solute Solution Monomeric unit Nominal Apparent quantity Viscosity

Acknowledgements We thank the Ministry of Science and Technology of the Republic of Slovenia, Ljubljana for financial support.

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