Effect of adsorption on the viscosity of dilute polymer solution

PII:

Eur. Polym. J. Vol. 34, No. 11, pp. 1613±1619, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0014-3057/99 $ - see front matter S0014-3057(98)00009-3

EFFECT OF ADSORPTION ON THE VISCOSITY OF DILUTE POLYMER SOLUTION RONGSHI CHENG,1 YUFANG SHAO,1 MINGZHU LIU2* and RENYUAN QIAN3 1 Department of Chemistry, Nanjing University, Nanjing 210093, P.R. China, 2Department of Chemistry, Lanzhou University, Lanzhou 730000, P.R. China and 3Institute of Chemistry, Academia Sinica, Beijing 100080, P.R. China

(Received 2 June 1997; accepted in ®nal form 15 September 1997) AbstractÐMeasurements of the dilute solution viscosities of polyethylene glycol and polyvinyl alcohol in water were carried out. The reduced viscosities of both polymer solutions show an upward turning of curvature at the extremely dilute concentration region and this is similar to the phenomena observed for many polymer solutions in the early 1950s. Upon the changes of the ¯ow time of pure water and the wall surface wettability of the viscometer after measuring solution viscosity, a judgement was formed that the observed viscosity abnormality at the extremely dilute concentration region is solely due to the e€ect of adsorption of polymer chains on the viscometer wall surface. A theory of the adsorption e€ect based on the Langmuir isotherms was proposed and a mathematical analysis for data treatment was performed. The theory could adequately describe the existing viscosity data. It seems necessary to correct the viscosity of a dilute polymer solution measured by a glass capillary viscometer for e€ect of adsorption in all cases. # 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION

The viscosity of a polymer solution in the extremely dilute concentration region usually reveals some abnormalities such as the curve of reduced viscosity plotted against concentration that shows either an upward or a downward turn as the concentration is below de®nite concentration. These unexpected facts attracted the notice of many investigators since in the early 1950s [1±19]. Most of them attempted to relate the curve turning concentration to the critical concentration at which the physical contact of polymer coil starts. However, the others argued against this judgment as the relative viscosity of an extremely dilute solution is very close to unity, high accuracy would be demanded for the measurement [20, 21]. Furthermore, the reduction in radius of viscometer capillary and actual concentration of solution due to adsorption of the polymer from solution may also in¯uence the result of determination [22±24]. Though the most probable reason for the existence of abnormality is the hypothesis of adsorption, a quantitative explanation is still lacking. All experimentalists who had been involved in measuring the viscosity of a polymer solution certainly su€ered an unforgettable experience that the viscometer should be thoroughly cleaned after use in order to eliminate the in¯uence of adhered polymer residues on the wall of the viscometer. In a recent careful viscometric study on dilute solution of water soluble polymers, we noticed that after use the viscometer should be cleaned by boiling water. If the viscometer was cleaned only by repeated *To whom all correspondence should be addressed.

soaking and rinsing with water at ambient temperature, the ¯ow time of water increased in the vicinity of one percent as compared with that of a thoroughly cleaned viscometer. The wettability of the viscometer wall to water was also changed as realized by visual inspection. These doubtless facts indicate that the polymer chains are rather ®rmly adsorbed on the viscometer wall surface and the ¯ow time of the solvent should be varied in the presence of the adsorbed polymer layer. With this realization in mind, we analyzed the e€ect of adsorption on the viscosity of dilute polymer solutions. The theoretical results obtained could adequately explain the observed abnormalities of viscosity measurements in a quantitative way.

THEORETICAL

General consideration In determining the relative viscosity of polymer solutions with a glass capillary viscometer, the ¯ow time of pure solvent to at a given temperature is a constant. This is a matter of common practice. However, if the polymer chains are ®rmly adsorbed on the viscometer capillary and bulb wall surface during the course of determination, the e€ective diameter of the capillary and the wettability of the wall surface to the pure solvent should be changed. Therefore, the ¯ow time of the solvent is no longer a true constant but varies with the change of surface properties of the wall. We may regard the ¯ow time of the solvent as a function of polymer concentration and denote it by to(c). It is plausible to assume that to(c) is directly related to the fractional surface coverage y of the

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Rongshi Cheng et al.

viscometer by the adsorbed polymer chains and the ¯ow time of the solvent changes from to(c) to to(1) as y changes from 0 to 1. to …c† ˆ to …0† ‡ ‰to …1† ÿ to …0†Šy

…1†

If the adsorption process obeys the Langmuir adsorption isotherm, y ˆ bc=…1 ‡ bc†

…2†

in which b is a parameter related to relative adsorption and desorption rates, the function to(c) could be easily deduced. The parameter b may be replaced by its reciprocal ca ca ˆ 1=b

…3†

which means at this particular concentration ca, y = 1/2, i.e., half of the available surface is ®lled with the polymer. Inserting equations (2) and (3) into equation (1) we have to …c† ˆ to …0†
‰1 ‡ kc=…ca ‡ c†Š

…4†

in which the constant k ˆ ‰to …1† ÿ to …0†Š=to …0†

…5†

denotes the maximum fractional change of ¯ow time of pure solvent due to the variation of the surface properties of the viscometer wall in the course of determining the polymer solution viscosity. It is worthy to note that the constant k may be either positive or negative depending on whether the ¯ow time of the solvent is increased or decreased after the adsorption of the polymer. If the ¯ow time of the solvent increases after adsorption of the polymer, thus k>0, the curvature is upwards. The opposite phenomenon may also occur. In other words, it is possible that the ¯ow time of the solvent will decrease after adsorption of the polymer. In this case, k < 0, the curvature is downward. The increase or decrease of the ¯ow time of the solvent depends on the nature of the absorbed polymer and the interaction between the absorbed polymer and the solvent. Both of the above mentioned phenomena were noticed by many investigators as early as the 1950s. Neglecting the corrections such as kinetic energy and drainage for the capillary viscometer [25, 26], the relative viscosity of polymer solutions is conventionally calculated from the ¯ow time of the solutions t(c) and that of pure solvent in the clean viscometer to(0) as Zr ˆ to …c†=to …0†

The above analysis is the starting point of the present discussion. equation (8) re¯ects the e€ect of adsorption on the conventionally measured relative viscosity of polymer solutions and may by regarded as the basis to make the correction for the adsorption e€ect. Figure 1 demonstrates the general trends of this e€ect. Viscosity of extremely dilute solution Theoretically, in the extremely dilute concentration region both hydrodynamic and thermodynamic intermolecular interactions are absent. In this case the Einstein viscosity law is applicable. This is true for solutions of globular solutes with Zr less than 1.1. In this concentration region the relative viscosity varies linearly with concentration and may be expressed as Zr,theo ˆ 1 ‡ ‰ZŠc

…9†

‰ZŠ ˆ Zsp,theo =c ˆ …Zr,theo ÿ 1†=c

…10†

from which It is reasonably to assign Zr,true ˆ Zr,theo

…11†

Therefore, inserting equation (9) into equation (8), we have the expression of the conventional reduced viscosity as Zsp =c ˆ …Zr ÿ 1†=c ˆ ‰ZŠ‰1 ‡ kc=…ca ‡ c†Š ‡ k=…ca ‡ c† …12† equation (12) indicates that the conventional reduced viscosity is no longer a constant equal to [Z], but varies with polymer concentration. The general pattern of such variations is illustrated in Fig. 2 with given values of [Z], k and ca. In the extremely dilute concentration region, from equation (12) we have the extrapolated value of the reduced viscosity to in®nitive dilution I…0† ˆ lim Zsp =c ˆ ‰ZŠ ‡ k=ca cÿ 40

…13†

The limiting value of the reduced viscosity as the concentration approaches to in®nity or the reciprocal of concentration approaches to zero is

…6†

Since the e€ective ¯ow time of the solvent in the course of measuring the solution viscosity is to(c) rather than to(0), the relative viscosity calculated by equation (6) is only an apparent value, and the true relative viscosity should be Zr,true ˆ t…c†=to …c†

…7†

Combining equations (4), (6) and (7), we thus get the relationship between the apparent and true relative viscosity as Zr ˆ Zr,true
‰1 ‡ kc=…ca ‡ c†Š

…8†

Fig. 1. E€ect of adsorption on the relative viscosity of polymer solutions. Middle straight line: hypothetical. Zr=1 + 2.5c. Upper dotted line: maximum 1% increase in ¯ow time of solvent due to adsorption of polymer on viscometer wall surface. Lower dotted line: maximum 1% decrease in ¯ow time of solvent due to adsorption of polymer on viscometer wall surface. (ca=5  10ÿ4 g/ml.)

E€ect of adsorption on solution viscosity

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S…0† ˆ …k=c2a †…1 ÿ ca ‰ZŠ†

…20†

and solving simultaneous equations (13), (14) and (20), we have ca ˆ ‰I…0† ÿ I…1†Š=S…0†

…21†

but in practice, the arbitrariness of the drawn initial tangent line limits the accuracy of estimating S(0) and hence ca. Viscosity in dilute solution region For the purpose of determining intrinsic viscosity of polymer solutions, measurements were usually carried out in the concentration region with relative viscosities from 1.1 to 2. In this dilute solution region, intermolecular interactions are operative and thereby the theoretical relative viscosity should include a higher order term such as Fig. 2. E€ect of adsorption on the reduced viscosity of extremely dilute polymer solutions. (ca=5  10ÿ4 g/ml, [Z] = 25 ml/g, k = 0.01, 0.005, 0, ÿ0.005, ÿ0.01 corresponding to curve a, b, c, d and e.)

I…1† ˆ lim Zsp =c ˆ lim Zsp =c ˆ ‰ZŠ…1 ‡ k† cÿ 41

1=cÿ 40

…14†

which is the asymptotic tangent line of the conventional reduced viscosity vs concentration plot. equation (12) also shows that as c = ca, the reduced viscosity is the average of I(0) and I(1) Zsp =ca ˆ ‰I…0† ‡ I…1†Š=2

…15†

equations (13)±(15) provide a simple graphical procedure to treat the experimental viscosity data in the extreme dilute concentration region as: (1) From the Zsp/c vs c and Zsp/c vs 1/c plots to get I(0) and I(1) by graphical extrapolation method. (2) Using equation (15) to calculate Zsp/ca and then using the interpolation method on the curve of Zsp/c vs c plot to get ca. (3) Solving [Z] and k by equations (13) and (14) with known I(0), I(1) and ca subsequently, we have 2

1=2

k ˆ 1=2f‰1 ÿ I…0†ca Š ÿ 4‰I…1† ÿ I…0†Šca g ÿ 1=2‰1 ÿ I…0†ca Š ‰ZŠ ˆ I…1†=…1 ‡ k†

…16†

Zr,theo ˆ 1 ‡ ‰ZŠc ‡ kH ‰ZŠ2 c2

…22†

in which kH is the Huggins slope constant. Inserting equation (22) into equation (8), we have the reduced viscosities of dilute polymer solutions Zsp =c ˆ k=…ca ‡ c† ‡ ‰ZŠ‰1 ‡ kc=…ca ‡ c†Š ‡ kH ‰ZŠ2 c‰1 ‡ kc=…ca ‡ c†Š

…23†

equation (23) indicates that the variation of the reduced viscosity of dilute polymer solutions with concentration is no longer linear down to extremely dilute solutions. The curves of reduced viscosity plotted against concentration will show either an upward or a downward turn depending on whether the constant k is positive or negative as the concentration is below a certain critical value. These features are illustrated in Fig. 3. From equation (23), it is easy to deduce the initial tangent line of reduced viscosity vs concentration plots as c approaches zero in the extremely dilute region and the asymptotic tangent line of the reduced viscosity curve as c approaches in®nity as: Initial tangent line: …Zsp =c†init ˆ …‰ZŠ ‡ k=ca † ÿ …k=c2a ÿ ‰ZŠk=ca ÿ kH ‰ZŠ2 †c …24†

…17†

There are two alternative methods to estimate ca from the initial tangent line …Zsp =c†init ˆ ‰ZŠ ‡ k=ca ÿ …k=c2a †…1 ÿ ca ‰ZŠc†

…18†

It meets with the asymptotic tangent line at the crossover concentration c** c** ˆ ca

…19†

Though equation (19) is the exact solution of equations (14) and (18), the accuracy of the graphical estimation of ca depends on how far the experimental data reach the asymptotic region. Another method is based on the slope of the initial tangent line. Designing it as S(0)

Fig. 3. E€ect of adsorption on reduced viscosity of dilute polymer solutions. ([Z] = 100 ml/g, kH=0.35, others as in Fig. 2.)

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Asymptotic tangent line: 2

…Zsp =c†asymp ˆ …1 ‡ k†‰ZŠ ‡ …1 ‡ k†kH ‰ZŠ c

…25†

equations (23)±(25) indicate that in the extremely dilute concentration region     Zsp Zsp k lim ˆ lim ˆ ‰ZŠ ‡ …26† cÿ 40 cÿ 40 ca c c init and for concentration region with c>>ca   Zsp lim ˆ …1 ‡ k†‰ZŠ cÿ 40 c asymp

…27†

which are identical with equations (13) and (14) for I(0) and I(1) of the extremely dilute solutions. This means that the procedures suggested in the last paragraph for treating experimental data in the extremely dilute concentration region are still a good approximation though the exact solution for ca is slightly changed due to the presence of a higher order term. It is noteworthy that the intercept and slope of the asymptotic tangent line in the region with c>>ca are both increased or decreased by a factor of (1 + k). This factor should be concerned as the correction factor for the e€ect of adsorption of polymers on the viscometer wall surface in the conventional concentration region of intrinsic viscosity determinations. We believe that the adsorption e€ect is nonavoidable in most cases. However, it has unfortunately been overlooked for many years.

EXPERIMENTAL PROCEDURES

Materials Six commercial reagent grade polyethylene glycols (PEG) were used as samples. These samples were characterized by VPO and size exclusion chromatography (SEC). The results obtained are listed in Table 1. The aqueous stock solutions of PEG were prepared by thoroughly weighing dried samples and then the boiled solution. The weight concentration cw(g/g) of PEG was converted to a weight±volume concentration c(g/ml) through its density± concentration relation r ˆ r0 …1 ‡ kcw †

…28†

in which r0 is the density of pure water and constant k = 0.1866 as determined by the pycnometer method. The poly(vinyl alcohol) (PVA) sample used in this work was a commercial product. The degree of polymerization (DP) ranged from 2400 to 2500 and the degree of saponi®cation (DS) from 98.0 to 99.0 mol%. The molecular weight of the sample was 1.14  105 determined by static light scattering. A known amount of PVA was dissolved Table 1. The molecular weights of PEG samples PEG A B C D E F

Nominal Mol. Wt

hMin (VPO)

hMiw/hMin (SEC)

400 1000 4000 6000 10000 20000

ÿ 1060 3795 4733 9150 21090

ÿ 1.58 1.43 1.82 1.34 1.75

in distilled water under re¯ux for 2 h to prepare the stock solution, its concentration was determined by weighing.

Viscosity Two Ubbelohde dilution type viscometers were used, one for PEG, the other for PVA. The ¯ow time of freshly distilled water in the thoroughly cleaned viscometer for PEG was measured in the temperature range of 10 to 608C, and compared with the simultaneously measured ¯ow time of water in a standard viscometer to determine the instrumental constants for the correction of kinetic energy and drainage of the working viscometer [27]. The ¯ow time of water in the thoroughly cleaned working viscometer at 258C, was 86.6 s. The ¯ow time of water increased to 87.7 s or more if the viscometer was only cleaned by repeated soaking and rinsing with water at ambient temperature after measuring solution viscosity and the wettability of the inner wall surface of the viscometer to water was also changed as realized by visual inspection. If the viscometer was cleaned by boiling water, the ¯ow time and wettability reconverted to its original state. Therefore the ¯ow time of a known amount of water in the viscometer was ®rst measured for checking the cleanliness and then weighted amounts of stock solutions were added into the viscometer successively for determining the solution viscosity. The relative viscosities of PEG solutions were calculated according to Zr …c† ˆ

r t…c† A ÿ B=t2 …c† ÿ C=t4 …c† p0 t0 …0† A ÿ B=t20 …0† ÿ C=t40 …0†

…29†

in which A, B and C are predetermined instrumental constants of the working viscometer. The viscosity determination for PVA aqueous solutions was carried out at 308C. The viscometer used had a much longer ¯ow time. The kinetic energy and drainage corrections were neglected. The ¯ow time of water in the thoroughly cleaned viscometer is 310.78 s, while it increased to 311.43 s for a viscometer washed with cold water after contact with a PVA solution. The ¯ow time of water reconverted to the original value as the viscometer was washed with boiling water. The procedure for determining the solution viscosity is similar to that for the PEG solution, but the relative viscosities were calculated simply as the ratio of the ¯ow time of the solution to that of the solvent.

RESULTS AND DISCUSSION

The dependence of the relative viscosities of PEG aqueous solutions on the concentration is shown in Fig. 4. Inspecting it we get an instant feeling about the nearly perfect linearity of the variation in relative viscosity with concentration. However, the results of linear regression analysis listed in Table 2 showed that, though all the linear correlation coecients were greater than 0.999, the most probable straight lines never pass through the origin point (Zr=1 at c = 0) but intersected with the Zr axis at points slightly greater than one. The plots of reduced viscosity against concentration are shown in Fig. 5. The curves show upward turns at extremely dilute concentration region for all the samples as those curves possessed positive k as discussed in Section 2. Treating the experimental data by the graphic procedures outlined above, the extrapolated and interpolated values of I(0), I(1) and ca are listed in Table 2. k and [Z] evaluated by equations (16) and (17) are also listed in the same Table. Though

E€ect of adsorption on solution viscosity

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Fig. 4. Concentration dependence of relative viscosity of PEG aqueous solutions. Experimental points: calculated with the ¯ow time of the solution and pure water in thoroughly cleaned viscometer, corrected for both kinetic energy and drainage.

the accuracies of graphic extrapolation and/or interpolation were limited, yet the evaluated values of ca, k and [Z] could adequately describe all the viscosity data as shown by the calculated lines of relative viscosities and reduced viscosities according to equations (8) and (12) in Figs 4 and 5. The viscosity of dilute PVA aqueous solutions in the concentration range with relative viscosity from 1.1 to 1.6 has been measured previously for estimating intrinsic viscosity and overlap concentration [28]. Linear extrapolation in this concentration range gave [Z] = 97 ml/g. Measurements were extended to the extremely dilute concentration range with relative viscosity less than 1.1. The reduced viscosities obtained in previous experiments were gathered and plotted against concentration as shown in Fig. 6. It can be seen that a rather sharp upward turning appears in the extremely dilute concentration region as that observed by previous investigators in the same concentration region. By the

data treatment procedures outlined in Section 2, for the whole concentration range of aqueous PVA solution studied we found k = 0.001, ca=4.1  105, [Z] = 95.6 and kH=0.51. Using these evaluated parameters, the reduced viscosity curve calculated according to equation (23) is also shown in Fig. 6. As can be seen, it ®ts the experimental points well. The ®tness of the calculated curve to experimental data is only phenomenological. The most encouraging fact is that the evaluated parameter k is in agreement with the direct measured value from the changes of ¯ow time of water in the viscometer for PEG aqueous solution, the latter value being 0.012 while the average value of k is 0.014 (Table 2). The fractional increasing of ¯ow time of water in the viscometer for PVA aqueous solution is 0.002 while k is 0.001. Therefore, we can decide that the mathematical analysis in the present article is correct and the abnormality of reduced viscosity appearing in the extremely dilute concentration

Table 2. The derived viscosity data of PEG aqueous solution The results of linear regression for Zr±c lines PEG samples A B C D E F

intercept

slope

corr. coef

1.0075 1.0050 1.0058 1.0011 1.0024 1.0038

4.26 6.81 15.51 22.85 27.07 42.51

0.9992 0.9987 0.9999 0.9996 1.0000 0.9989

Graphic evaluation

I(0)

I(1)

ca

k

[Z]

6.5 13 22 26 31 58

4 6 15 22 26 40

0.008 0.0025 0.0025 0.0025 0.0025 0.001

0.0206 0.0178 0.0182 0.0106 0.0134 0.0187

3.90 5.90 14.73 21.77 25.66 39.26

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Rongshi Cheng et al.

Fig. 5. Concentration dependence of reduced viscosity of PEG aqueous solutions. Experimental points: reduced viscosities calculated with to(0) corrected for kinetic energy and drainage. Dotted lines: calculated by equation (12) with evaluated k, [Z] and ca. Others as in Fig. 4.

region is solely due to the e€ect of adsorption of polymer chains on the viscometer inner surface. Another interesting problem has to do with the critical concentration ca. Its values are of the same order of magnitude as that of the critical concentrations for coil shrinking cs [29, 30]. The values of ca for PEG aqueous solution in Table 2 decrease with increasing molecular weight which also agrees with the cs of PEG in tetrahydrofurane determined by SEC as shown in Fig. 7. The resemblance

Fig. 6. Concentration dependence of reduced viscosity of PVA aqueous solution. Box points: in dilute concentration region, Zr>1.1. Cross points: in extremely dilute concentration region, Zr<1.1; Dotted lines: calculated according to equation (23) with k = 0.001, [Z] = 95.6, kH=0.51, ca=4.1  10ÿ5 g/ml.

between ca and cs stimulates investigators to make further intensive e€orts in seeking the natures of them. The e€ects of adsorption are usually ignored in traditional viscometry for polymer solutions. As analyzed in Section 2 and illustrated by experimental results, in the conventional concentration region for determining intrinsic viscosity, an error of one to two percent for both intrinsic viscosity and slope constant may result from it. Since adsorption is non-avoidable, we may conclude that it seems necessary to correct the viscosity of dilute polymer solution measured by glass capillary viscometer for adsorption in all cases.

Fig. 7. Molecular weight dependence of ca and cs of PEG solutions. Box points: ca determined by viscometry for the aqueous solution. Cross points: cs measured by SEC for the THF solution.

E€ect of adsorption on solution viscosity

The purpose of this work is to examine the behavior of viscosity of the polymer solutions from extremely dilute to concentrated solution and try to explain the abnormalities of the curve of the reduced viscosity plotted against the concentration in the extremely dilute concentration region rather than for the simple measurement of viscosity for concrete system, such as PVA or PEG. As mentioned above, though the phenomena of upward or downward turn of the curvature were observed by early investigators, a quantitative explanation for this is still lacking. Thus, a theory of adsorption e€ect was proposed and a mathematical analysis for the adsorption e€ect was performed on the basis of our experimental results. AcknowledgementÐThe ®nancial support of this work by the Chinese National Basic Research Project ``Macromolecular Condensed State'', is gratefully acknowledged. REFERENCES

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