The Effect of Neutral Polymer and Nonionic Surfactant Adsorption on the Electroacoustic Signals of Colloidal Silica

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

193, 200–214 (1997)

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The Effect of Neutral Polymer and Nonionic Surfactant Adsorption on the Electroacoustic Signals of Colloidal Silica Melanie L. Carasso, 1 William N. Rowlands, and Richard W. O’Brien Department of Physical and Theoretical Chemistry, University of Sydney, NSW 2006 Australia Received February 7, 1997; accepted July 17, 1997

The effects of adsorbed neutral polymer and nonionic surfactant on the electroacoustic signals of silica particles have been determined. The dynamic mobility of the particles was measured using an AcoustoSizer (Matec Applied Sciences) during the addition of poly(vinyl alcohol) (PVA) and nonyl phenol ethoxylate (C9fEN ) to the system. Large changes in the dynamic mobility of the coated particles were observed. A theoretical treatment was developed to describe the electroacoustic behavior of an elastic gel layer of uncharged adsorbed polymer and provided a reasonable fit to the experimental dynamic mobility data. This analysis provided information about the thickness and elasticity of the adsorbed layers. The adsorbed layer thickness for PVA compared well with results of other techniques reported in the literature and appeared to be influenced by changes in pH. C9fEN adsorbed in a relatively dense layer. For both PVA and C9fEN , the thickness of the inner adsorbed layer remained approximately constant, while the outer layer became thicker with increasing concentration and molecular weight. q 1997 Academic Press Key Words: adsorption; dynamic mobility; electroacoustics; poly(vinyl alcohol); nonyl phenol ethoxylate.

INTRODUCTION

A great deal of information about the adsorption of polymers onto colloidal surfaces has been gained by studying the behavior of model uncharged macromolecules. Two of the most commonly studied examples are the linear, flexible polymers poly(vinyl alcohol), or PVA, and poly(ethylene oxide) (PEO). The interest in the adsorption of these neutral polymers was initiated in the 1970s, when extensive investigations were carried out by several groups of researchers (1– 23). A range of techniques including microelectrophoresis, streaming potential, viscometry, sedimentation, and adsorption isotherms were applied to determine the mechanisms for adsorption and to measure properties of the adsorbed layer such as adsorbed amount, distribution of polymer seg1 To whom correspondence should be addressed at Bell Laboratories, Lucent Technologies, Room 1A-352, 600 Mountain Ave., Murray Hill, NJ 07974-0636.

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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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ments, and thickness. Subsequent to these fundamental studies of neutral polymer adsorption, research has been directed toward the development of new experimental techniques in order to obtain more information about the adsorbed layer. These techniques include small angle neutron scattering (SANS) (24, 25), NMR (24, 26), and surface forces (27). Photon correlation spectroscopy (PCS) has also been used (28–32) to study the thickness and structure of adsorbed layers of polymer. Most studies of nonionic surfactant adsorption involve either alkyl polyoxyethylene, CxEy , or alkyl phenol polyoxyethylene, CxfEy , molecules, as these are commonly used in commercial applications. A review by Clunie and Ingram (33) has highlighted some general characteristics of nonionic surfactant adsorption. The adsorption isotherms are usually of the Langmuir type, with surfactant displacing water from the surface at low concentrations and forming a flat monolayer near the cmc. If the concentration of surfactant is increased further, the adsorbed molecules will interact to induce more adsorption and closer packing. On hydrophilic surfaces such as silica, the hydrophilic part of the surfactant (i.e., the ethoxylate chain) adsorbs to polar sites, with the hydrophobic hydrocarbon chain in the aqueous solution. Another layer then builds up in some sort of reverse orientation as a consequence of hydrophobic interactions. Ongoing investigations into the exact configuration of the surfactant molecules and the thickness of each layer continue to be reported in the literature (34–40). In this paper, the adsorption of PVA and nonyl phenol ethoxylates onto colloidal silica will be investigated by measuring an electroacoustic effect called the electrokinetic sonic amplitude ( ESA ) . The ESA effect results from the application of an alternating electric field across a suspension of charged colloidal particles. The applied field causes the particles to oscillate between the electrodes, generating pressure changes in the dispersion which propagate as sound waves. The ESA signal is related to both the zeta potential and the sizes of colloidal particles ( 41 ) , and these quantities can be experimentally determined in suspensions of arbitrary concentration using

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an electroacoustic device known as the AcoustoSizer 2 ( Matec Applied Sciences, Hopkinton, MA ) ( 42 ) . The first step in the determination of the zeta potential and particle size from ESA measurements is the calculation of the dynamic mobility, mD . This quantity is the electrophoretic mobility in an alternating field. When such a field is applied to a colloid, it causes the particles to move with a sinusoidal velocity. If the frequency v of the applied field is sufficiently high, the inertia forces will cause the particle to lag the applied field. Thus if the applied field is E cos vt, the particle velocity will have the form V cos v(t 0 Dt), where Dt is the time lag in the particle motion. For small applied fields, the velocity amplitude V is proportional to the applied field amplitude E, and the time delay is independent of E. The dynamic mobility is a complex quantity defined by mag( mD ) Å

V , E

Dr » mD …Z, r

[1]

S D

2ez va 2 G (1 / f ( l, v* )), n 3h

The AcoustoSizer is now manufactured by Colloidal Dynamics Inc., Warwick, RI.

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for the case of spherical particles with thin double layers. The function f ( l, v* ) is defined as f ( l, v* ) Å

1 / iv* 0 (2l / iv* ( ep / e )) , 2(1 / iv* ) / (2l / iv* ( ep / e ))

[4]

where ep is the permittivity of the particles, l Å Ks /K `a

[5]

v* Å ve /K ` .

[6]

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K ` is the electrical conductivity of the solvent and Ks is the surface conductance of the double layer. If the particles are coated with an adsorbed layer of polymer or surfactant, the electroacoustic behavior will be altered. This has been demonstrated in the work of Berg et al., where the effect of adsorption of PEO onto silica (44) and of PVA onto titania (45) on the (single-frequency) ESA signal of the particles was monitored using an ESA 8000 device. In both systems, the magnitude of the ESA was lowered by the presence of an adsorbed layer of polymer, and this was interpreted as an outward shift in the location of the plane of shear. In contrast to the ESA 8000, which operates at a single frequency of about 1 MHz, the AcoustoSizer measures the ESA signal at 13 frequencies in the megahertz range. This allows the frequency dependence of the dynamic mobility to be determined, in the form of a dynamic mobility spectrum. In the work presented here, the adsorption of neutral polymers and nonionic surfactants to silica particles is investigated using the AcoustoSizer. The aim is to determine the effects of the adsorbed layers on the dynamic mobility. A further objective is to determine whether, with a suitable theoretical treatment of the system, these effects on the dynamic mobility can be used to provide information about the properties of the adsorbed polymer or surfactant layer.

[2]

2

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[3]

and

where f is the particle volume fraction, r is the density of the solvent, and Dr is the density difference between the particles and the solvent. C is an instrument factor and » mD … is the particle-averaged dynamic mobility. The term Z is related to the acoustic impedances (43) of the suspension and of the glass delay rods in the device. The term ESA in this equation is a complex quantity. The magnitude of the ESA is the amplitude of the voltage from the pressure sensor used to measure the electroacoustic signal, and the argument of the ESA is the phase of the sinusoidal pressure sensor signal relative to the phase of the applied voltage. In the AcoustoSizer the quantity Z is determined by an independent acoustic measurement, and the instrument factor C is determined by a calibration procedure (42). Thus for a suspension in which f and Dr are known, the dynamic mobility can be determined from the measured ESA using Eq. [1]. For the dilute suspensions (i.e., f õ 0.02) of interest here, the dynamic mobility can be expressed by the formula (41) mD Å

q

1 / (1 / i) a /2 q G( a ) Å 1 / (1 / i) a /2 / i( a /9)(3 / 2( Dr / r ))

arg( mD ) Å 0 vDt.

Here mag and arg are the magnitude and polar argument (in radians), respectively, of the dynamic mobility. The ESA is related to the dynamic mobility by the formula ESA Å C f

where z is the zeta potential, a is the particle radius, e is the permittivity of the solvent, h is the viscosity of the solvent, and n ( Åh / r ) is its kinematic viscosity. The function G( a ) is given by

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MATERIALS AND METHODS

The silica particles used in the experiments were manufactured by Geltech, Inc. (Florida), and have been characterized

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elsewhere (46). Briefly, the particles were amorphous, nonporous toward nitrogen, and highly spherical and monodisperse. The average particle size was determined by TEM to be 1.2 { 0.2 mm, the specific surface area was measured by BET adsorption to be 2.8 { 0.2 m2 g 01 , and the particle density was 2.15 g cm03 . Silica suspensions were prepared in 5 1 10 03 M solutions of sodium chloride and dispersed thoroughly through a series of stirring, ultrasonicating, and decanting steps (46). The final suspensions had a volume of about 390 ml and a particle volume fraction f of 1.5%. All reagents used were AR grade. Sodium chloride and sodium hydroxide were from Merck, and hydrochloric acid was from Rhoˆne-Poulenc. High purity Millipore water was used for all solutions. Poly(vinyl alcohol) of MW range 30,000–70,000, which will be referred to as PVA 50, and of MW range 70,000–100,000 (PVA 85) were obtained from Sigma. Nonyl phenol ethoxylates, C9fEN , were obtained from ICI Australia Operations Pty. Ltd, with N values of 40 and 100. These will be denoted N40 and N100, respectively. All polymer and surfactant solutions were made up in a background electrolyte solution of 5 1 10 03 M sodium chloride. Adsorption experiments were conducted in the AcoustoSizer measurement cell at 22.07C, with a stirrer speed of 300 rpm. The polymer or surfactant solutions were added to the silica suspensions via an autoburette, and pH titrations were carried out on each PVA/silica suspension at the completion of adsorption, by addition of dilute NaOH and HCl solutions. Background electrolyte corrections were performed for all ESA measurements by adding the required amount of 1 M NaCl and 0.1 M HCl or NaOH to 390 ml of high purity water in the cell to reproduce the conductivity and pH of each colloid ESA measurement. These electrolyte ESA measurements were then subtracted vectorially from the corresponding colloid measurements to eliminate the background salt contribution. RESULTS

The dynamic mobility ( mD ) spectra of silica suspensions at different pH values are shown in Fig. 1 as the magnitude (filled circles) and argument (open circles) components. With increasing pH, the mobility magnitudes shifted to higher values and also decreased more steeply with frequency. The mobility phase angles were constant at low to intermediate pH values, increasing from 0107 to 0437 with frequency. At pH 10, the phase angles leveled out to 0387 at the higher frequencies. These experimental results are in good agreement with electroacoustic theory for model systems, as demonstrated in detail elsewhere (46). The random error involved in the measurement of dynamic mobility is about 1% in the magnitudes and 17 in phase.

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FIG. 1. Dynamic mobility spectra for silica at pH 4 ( h ), pH 7 ( n ), and pH 10 ( , ) in the absence of added polymer. Filled symbols represent magnitudes; open symbols represent arguments.

PVA During the addition of PVA 50 to silica up to a total concentration of 1.1 1 10 06 M, the pH of the suspension increased marginally from 4.70 to 4.85, while the conductivity remained constant at 0.050 S m01 . Estimates of zeta potential and size are obtained by the AcoustoSizer using the dynamic mobility formulae for uncoated particles. The theory does not account for the effect of the polymer layer on the electroosmotic flow, and thus the estimates of size and zeta are invalid. In order to understand the effect of polymer addition it is more meaningful to study the changes in the dynamic mobility. The influence of the adsorbed polymer layer can be examined by calculating the ratio mD (coated)/ mD (uncoated) of the dynamic mobility of the particles in the presence and absence of added polymer. This eliminates the contributions to the mobility from the underlying particles, so that the ratio represents the electroacoustic behavior of the adsorbed layer itself. The accuracy of each ESA measurement is typically 1% in the magnitudes and 17 in the arguments, therefore in ratio the error is 2% and 27, respectively. A ratio of 1.00 { 0.02 in the magnitudes and a phase difference of 0 { 27 in the arguments over all frequencies indicates that the adsorbed layer has no significant effect. The dynamic mobility ratios of the coated and uncoated particles are plotted in Fig. 2 (magnitudes) and Fig. 3 (arguments) at five different PVA 50 concentrations. At the lowest concentration (0.06 1 10 06 M) the magnitude of the mobility ratio was independent of frequency with a value of 0.95, showing only a slight deviation from the mobility of the uncoated particles. As the concentration was increased, the magnitude decreased and showed an increasingly strong dependence on the applied frequency in such a way that at the highest concentration (1.1 1 10 06 M), there was a 30% increase in the magnitudes from the lowest to the highest

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FIG. 2. Magnitude of dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 50.

frequency. The arguments of the mobility ratio were also affected by the increasing concentration of PVA 50. At 0.06 1 10 06 M PVA, the phase angles remained within 17 relative to the uncoated particles, again indicating no significant difference from the uncoated particles. As the concentration was increased, a well-defined trend with frequency developed, consisting of a phase lead which increased progressively with concentration and a peak at about 3 MHz for all concentrations. At 1.1 1 10 06 M PVA, the phase lead was almost 87 with respect to the bare particles. The addition of PVA 85 to the silica generated similar results to those described above for the lower molecular weight PVA experiments. The pH was 5.8 and the conductivity was 0.052 S m01 throughout the experiment, and the magnitudes of the dynamic mobility ratios at each concentration were identical to those for PVA 50. The arguments covered the same range of phase angles as the PVA 50

FIG. 3. Argument of dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 50.

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FIG. 4. Magnitude of dynamic mobility ratio mD (coated)/ mD (uncoated) for 1.1 1 10 06 M PVA 50 at different pH.

results, but the peak occurred at a slightly lower frequency, 2 MHz (see Figs. 8 and 9, symbols). pH Titration for PVA The dynamic mobility ratios mD (coated)/ mD (uncoated) for the pH titration of silica in the presence of 1.1 1 10 06 M PVA 50 were determined using the appropriate mobility spectra from the pH titration of bare silica (Fig. 1) in the denominator. The magnitudes are shown in Fig. 4. At pH 4 and pH 7, there was no significant change from the final result of the adsorption experiment (Fig. 2). At pH 10, the magnitudes of the mobility ratio were higher and showed a stronger increase with frequency. The magnitudes at pH 7 on the reverse titration approached the forward pH 7 results, while at pH 4 the ratio was approximately 1 at all frequencies. The arguments are given in Fig. 5. The phase angles

FIG. 5. Argument of dynamic mobility ratio mD (coated)/ mD (uncoated) for 1.1 1 10 06 M PVA 50 at different pH.

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at pH 7 were similar to those of the final adsorption ratio (Fig. 3). At pH 4, the arguments showed smaller phase leads, and at pH 10 the phase lead had increased to 137 at the lowest frequency. The phase angles on the reverse titration were reasonably similar to the forward titration results. C9fEN During the addition of N40 to silica, there was a small increase in the pH of the suspension from 5.60 to 5.85, accompanied by a slight rise in conductivity from 0.055 to 0.056 S m01 . The magnitude component of the dynamic mobility at 4.5 1 10 04 M N40 increased by about 20% with frequency, in a manner qualitatively similar to the highPVA-concentration mobility ratios described in the previous section. The magnitudes at 6.7, 8.9, and 11 1 10 04 M N40 were all identical, showing the same frequency dependence as the 4.5 1 10 04 M curve, but with slightly lower values (see Fig. 10, symbols). The arguments of the mobility ratios for the N40 experiment displayed a phase lead over all frequencies, as found for PVA, but the nature of the frequency dependence was different in this case. The arguments at all concentrations were identical at low frequencies. The phase angles for the ratio at 4.5 1 10 04 M N40 peaked at a value of 87 at about 5 MHz, a frequency higher than that of the PVA 50 and PVA 85 peaks. The position of the peak shifted to higher frequencies with increasing concentration of N40. At 11 1 10 04 M N40, the phase lead was 107 at frequencies above 7 MHz, with no obvious peak (see Fig. 11, symbols). The effects of addition of N100 to silica were similar to those produced by N40. The pH of the suspension remained constant at 5.3 and the conductivity increased marginally

FIG. 6. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the magnitude of the dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 50. Theoretical parameters: a / h Å 0.023 nm02 , v0 Å 0.8 s 01 ; (a) d Å 2.0 nm, D Å 0.12 nm; (b) d Å 4.5 nm, D Å 0.59 nm; (c) d Å 5.7 nm, D Å 1.04 nm; (d) d Å 6.3 nm, D Å 1.34 nm; (e) d Å 6.9 nm, D Å 1.56 nm.

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FIG. 7. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the argument of the dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 50. Theoretical parameters: a / h Å 0.023 nm02 , v0 Å 0.8 s 01 ; (a) d Å 2.0 nm, D Å 0.12 nm; (b) d Å 4.5 nm, D Å 0.59 nm; (c) d Å 5.7 nm, D Å 1.04 nm; (d) d Å 6.3 nm, D Å 1.34 nm; (e) d Å 6.9 nm, D Å 1.56 nm.

from 0.055 to 0.057 S m01 . The magnitudes and arguments of the mobility ratios for N100 showed a strong resemblance to the N40 results. The magnitudes increased by 40% with increasing frequency and were displaced to lower values with increasing surfactant concentrations. The peak in the phase angles shifted to higher frequencies as the concentration increased. The phase lead increased with concentration, reaching a maximum of 207 in the high frequency region of the 11 1 10 04 M N100 curve (see Figs. 12 and 13, symbols). Comparison of Theory with Experiment In Appendix A we derive a formula for the mobility of a particle coated by a layer of neutral polymer. The dynamic mobility mD is determined by the solution of the equations of motion of the liquid outside the double layer (41), subject to the condition that there is a prescribed jump in the tangential velocity across the double layer. Since these equations of motion are linear, the dynamic mobility will be proportional to this velocity jump. For an uncoated particle the velocity jump is given by Dus . Hence, dividing the dynamic mobility of a particle coated with polymer by that of an uncoated particle gives the ratio ( Du/ Dus ). Thus the ratio of the mobility of a coated particle to that of an uncoated particle is given by the formula [17], and this formula will be used to analyze the measured mobility ratios. The experimental mD (coated)/ mD (uncoated) magnitudes for PVA 50 were replicated theoretically by increasing the outer adsorbed layer thickness d from 2.0 to 6.9 nm, and increasing the inner adsorbed layer thickness D from 0.1 to 1.6 nm, with increasing PVA 50 concentration, as shown in Fig. 6. The corresponding experimental and theoretical phase angles are plotted in Fig. 7. The theory predicts a phase lead

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FIG. 8. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the magnitude of the dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 85. Theoretical parameters: a / h Å 0.018 nm02 , v0 Å 0.7 s 01 ; (a) d Å 2.5 nm, D Å 0 nm; (b) d Å 5.4 nm, D Å 0.72 nm; (c) d Å 6.8 nm, D Å 1.09 nm; (d) d Å 7.4 nm, D Å 1.30 nm; (e) d Å 7.8 nm, D Å 1.44 nm.

at all frequencies, in agreement with experimental observation. A discrepancy of about 37 is evident at high frequency for the 1.1 1 10 06 and 0.28 1 10 06 M curves. The frequency at which the peaks occurred was matched well by a single value of the relaxation frequency v0 (0.8 s 01 ), and reasonable agreement in the peak heights was achieved. In preliminary experiments it was found that the magnitude of the ESA signal began to form a plateau at 1 1 10 05 M PVA 50. At this concentration, an experimental phase lead of 207 in the low frequency arguments was obtained, with magnitudes increasing from 0.25 to 0.5, and the best fit was achieved with d Å 14 nm and D Å 3.0 nm. The PVA 85 results are shown in Fig. 8 (magnitudes) and Fig. 9 (arguments). Compared with PVA 50, a slightly lower value for v0 was required to fit the peaks, and d was larger at each concentration. However, the values of D were similar to the PVA 50 results at each concentration. The best fit to the experimental data for N40 was obtained with d Å 5.4 nm, v0 Å 2.4 s 01 and with d Å 6.0 nm, v0 Å 4.0 s 01 . The magnitudes (Fig. 10) were reasonably well described by the theoretical curves, and the estimate of D was constant at approximately 4.1 nm. A very good fit to the phase angles (Fig. 11) was achieved over the whole frequency spectrum by the increasing values of v0 and d with surfactant concentration, reproducing the respective increases in peak position and height which were found experimentally. For N100, good fits to the experimental magnitudes (Fig. 12) were produced, particularly at low concentrations, by increasing d from 9.1 to 11.5 nm and v0 from 1.5 to 4.8 s 01 with increasing concentration. As found for N40, D for N100 was constant with concentration, at a slightly lower value of about 3.3 nm. Excellent agreement

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FIG. 9. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the argument of the dynamic mobility ratio mD (coated)/ mD (uncoated) for PVA 85. Theoretical parameters: a / h Å 0.018 nm02 , v0 Å 0.7 s 01 ; (a) d Å 2.5 nm, D Å 0 nm; (b) d Å 5.4 nm, D Å 0.72 nm; (c) d Å 6.8 nm, D Å 1.09 nm; (d) d Å 7.4 nm, D Å 1.30 nm; (e) d Å 7.8 nm, D Å 1.44 nm.

between theory and experiment was obtained in the N100 arguments (Fig. 13) with these values of d and v0 . DISCUSSION

PVA At the plateau concentration of 1 1 10 05 M PVA 50, the estimated adsorbed amount Gest was as much as 6 mg/m 2 , based on the assumption that all of the added polymer adsorbed to the surface. This estimate is larger than the reported plateau amounts of about 2.0 { 0.5 mg/m 2 for adsorption

FIG. 10. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the magnitude of the dynamic mobility ratio mD (coated)/ mD (uncoated) for N40. Theoretical parameters: a / h Å 0.05 nm02 ; (a) d Å 5.4 nm, D Å 4.1 nm, v0 Å 2.4 s 01 ; (b) d Å 6.0 nm, D Å 4.0 nm, v0 Å 4.0 s 01 .

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FIG. 11. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the argument of the dynamic mobility ratio mD (coated)/ mD (uncoated) for N40. Theoretical parameters: a / h Å 0.05 nm02 ; (a) d Å 5.4 nm, D Å 4.1 nm, v0 Å 2.4 s 01 ; (b) d Å 6.0 nm, D Å 4.0 nm, v0 Å 4.0 s 01 .

of similar-sized PVA onto silver iodide (14, 16), polystyrene (3), and titania (45), but is close to the value of 7 mg/ m 2 obtained by Kavanagh et al. with gibbsite (22). At the highest concentration of PVA in all subsequent experiments (1.1 1 10 06 M), the adsorbed amount was 0.5 mg/m 2 (again assuming all added polymer adsorbed), which is only a fraction of the maximum adsorbed amount. The presence of adsorbed polymer introduces a frequency dependence in the relation between the electroosmotic velocity jump and the tangential field. This is evident in both the magnitudes and arguments of the dynamic mobility ratios, mD (coated)/ mD (uncoated). Thus in the presence of adsorbed polymer, the Smoluchowski equation,

FIG. 12. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the magnitude of the dynamic mobility ratio mD (coated)/ mD (uncoated) for N100. Theoretical parameters: a / h Å 0.02 nm02 ; (a) d Å 9.1 nm, D Å 3.3 nm, v0 Å 1.5 s 01 ; (b) d Å 10.4 nm, D Å 3.3 nm, v0 Å 2.7 s 01 ; (c) d Å 11.0 nm, D Å 3.5 nm, v0 Å 4.0 s 01 ; (d) d Å 11.5 nm, D Å 3.4 nm, v0 Å 4.8 s 01 .

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FIG. 13. Comparison of polymer gel layer theory (lines) with experimental data (symbols) for the argument of the dynamic mobility ratio mD (coated)/ mD (uncoated) for N100. Theoretical parameters: a / h Å 0.02 nm02 ; (a) d Å 9.1 nm, D Å 3.3 nm, v0 Å 1.5 s 01 ; (b) d Å 10.4 nm, D Å 3.3 nm, v0 Å 2.7 s 01 ; (c) d Å 11.0 nm, D Å 3.5 nm, v0 Å 4.0 s 01 ; (d) d Å 11.5 nm, D Å 3.4 nm, v0 Å 4.8 s 01 .

ueo Å 0

ez E, h

[7]

becomes u Å C( v )E.

[8]

Taking the ratio mD (coated)/ mD (uncoated) gives the quantity C( v )/( ez / h ). The polymer gel layer formula (Eq. [17]) for evaluating the quantity C( v ) is based on a relatively simple model, but it can reproduce the experimental results quite well. Although the induced frequency dependence renders the reported zeta potentials unreliable, so that Eq. [30] for the effective electrophoretic layer thickness is not valid, the polymer gel theory allows estimates of the layer thickness to be obtained using the dynamic mobility ratio instead. The thickness of the outer, permeable region of the adsorbed layer is described by the parameter d, and that of the inner dense layer is given by D. These estimates of the layer thickness are derived from dynamic multifrequency electrokinetics. For comparison with the static electrophoretic thicknesses commonly reported in the literature, an effective static mobility thickness Ds can be calculated by setting v Å 0 in Eq. [17], with the e 0 kD correction (Eq. [32]). The formula then reduces to e 0 kDs Å e 0 kD 1

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a/h a/h 0 k2

F S

e 0 kd 1 /

D

k k2 1 tanh kd 0 2 k k cosh kd

G

.

[9]

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TABLE 1 Adsorbed Layer Thicknesses for PVA PVA 50

PVA 85

PVA (1006 M)

Gest (mg m02)

d (nm)

D (nm)

(d / D) (nm)

Ds (nm)

Gest (mg m02)

d (nm)

D (nm)

(d / D) (nm)

Ds (nm)

0.06 0.28 0.56 0.83 1.1 14

0.03 0.1 0.3 0.4 0.5 6

2.0 4.5 5.7 { 1.1 6.3 6.9 { 1.3 14.0

0.1 0.6 1.0 1.3 1.6 3.0

2.1 5.1 6.7 7.6 8.5 17.0

0.2 1.0 1.8 2.3 2.7 7.6

0.05 0.2 0.5 0.7 0.9

2.5 5.4 6.8 { 1.4 7.4 7.8 { 1.7

0.0 0.7 1.1 1.3 1.4

2.5 6.1 7.9 8.7 9.2

0.1 1.3 2.0 2.4 2.7

The total adsorbed layer thickness is equal to the sum of the dynamic outer and inner layer thicknesses, (d / D ). This sum should be comparable with the thicknesses obtained from sedimentation and PCS techniques, in which the outer free-draining layer seems to be included. Using the parameters shown in Figs. 6 to 9 for PVA 50 and PVA 85, Ds and (d / D ) can be determined as functions of the estimated adsorbed amount Gest for comparison with the literature. The results are shown in Table 1. The concentration and molecular weight dependences of d imply that the thickness of the outer, permeable layer increases significantly as more polymer is added, and that this outer layer is slightly thinner for the smaller polymer. If the adsorbed amount Gest is used instead of concentration as the basis for comparison between PVA 50 and PVA 85 and the uncertainty in d is taken into account, the dependence of the outer layer thickness on molecular weight is less apparent. However, the increase in d with concentration is an explicable result and demonstrates that as more polymer is adsorbed, the loops and tails become longer. There was very little difference between PVA 50 and PVA 85 in the effective inner layer thickness D at a given concentration or adsorbed amount. The static effective thicknesses Ds were higher than the corresponding D values, but also showed no significant dependence on molecular weight. This independence in D of PVA molecular weight has been observed by Koopal and Lyklema (14). They also found no molecular weight dependence in the fraction u of the Stern layer occupied by polymer segments, and concluded that the distribution of trains and loops was the same at a given adsorbed amount for all molecular weights. Their D and u data indicated that trains and flat loops were formed for the first 0.15 mg/m 2 , with further amounts of PVA distributed in loops. The D and d values in Table 1 are consistent with this finding. The effective static mobility thicknesses for low adsorbed amounts in Table 1 can be compared with the electrophoretic results of Kavanagh et al. (22) for PVA (MW 70,000) on gibbsite and of Koopal and Lyklema (14) for PVA (MW

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98,000) on silver iodide. They obtained respective values of 1.5 and 1.1 nm at 0.4 mg/m 2 . At about 0.7 mg/m 2 , Koopal and Lyklema determined Ds to be 2.7 nm. The agreement with the values of Ds estimated in this work is good, considering the different approach used here to evaluate Ds and also the differences in the underlying substrates. For the plateau region, the approximate Ds value of 7.6 nm also compares well with the literature. Kavanagh et al. (22) reported a maximum of 11 nm for their system. Fleer et al. (2) obtained a value of 5 nm using electrokinetics for PVA of molecular weight 55,000 adsorbed on AgI, as well as values of 10 nm using viscometry for PVA 105 and 4 nm using their dz /dpAg method (14) for PVA 60. Miller and Berg (45) using an ESA 8000 obtained a thickness of 13 nm for PVA (MW 78,000) adsorbed onto titania. These plateau values are approximately one-third of the adsorbed layer thicknesses of 20–30 nm obtained by the techniques of PCS (28–30) and sedimentation (1, 3, 18, 21). The PCS and sedimentation values represent the total layer thickness, and as such are best compared with the (d / D ) results in Table 1. At the plateau of PVA 50 adsorption, (d / D ) was estimated to be 17 nm, which is almost three times larger than the effective static mobility thickness and only slightly smaller than typical PCS and sedimentation results. Most published data regarding the total layer thickness have been confined to the plateau region of adsorbed polymer, so the layer thicknesses determined here at low adsorbed amounts cannot be directly compared with that found in the literature. However, the (d / D ) results at these low concentrations (except 0.06 1 10 06 M) are 3 to 5 times larger than the corresponding Ds values, in keeping with the apparent relationship between the effective thickness and the total thickness obtained at higher concentration. Although the estimate of the a / h parameter (Appendix B) for use in the polymer gel theory was based on random coil considerations, the theoretical results correlated well with the experimental data and yielded layer thicknesses with relatively small uncertainties, which were in agreement with published values. As a, the drag coefficient, is related

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FIG. 14. (a) Volume of fluid near a charged surface. (b) Shear displacement caused by adsorbed layer of neutral polymer at a charged interface.

to the porosity of the adsorbed layer, this suggests that the average permeability of the layer is not very different to that of a random coil. This is consistent with the findings of Garvey et al. (3), in which the adsorbed PVA molecules showed no apparent compression with respect to the random coil state, retaining the same volume as in solution. Fitting the peak in the phase angles with the relaxation frequency v0 resulted in values of 0.8 and 0.7 s 01 at all concentrations for PVA 50 and PVA 85, respectively. Using Eq. [27], with h Å 0.95 N sm02 (viscosity of water at 227C), yields a shear modulus g of 0.8 N m02 for an adsorbed layer of PVA 50 and 0.7 N m02 for PVA 85. These results are much lower than the typical values of about 1000 N m02 for the elastic shear modulus of dense polymer melts at similar applied frequencies (47), but in the present case the polymers have a much more open structure and so a low elastic modulus is expected. The accuracy of the results obtained from the polymer gel model rests mainly on the validity of two assumptions: (i) that the adsorbed layer structure resembles the arrangement depicted in Fig. 14b, i.e., a dense layer near the particle surface with an outer, more permeable layer toward the solution, and (ii) that the local polymer geometry is independent of the thickness of the layer and the concentration of polymer. These assumptions lead to the assignment of uniform values to the drag coefficient a and the shear modulus g for the inner layer and for the outer layer, whereas in reality it is likely that these quantities vary with distance from the surface in a smooth fashion. Comparison with literature val-

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ues on the basis of the adsorbed amount Gest involves a further assumption (iii) that all of the added polymer adsorbed to the surface, with none remaining in solution. The uncertainty implicit in these assumptions and the simple nature of the model limit the reliability and scope of the results; however, the values for D and d obtained here are consistent with results from electrophoresis and light scattering measurements. The pH titration of silica with 1.1 1 10 06 M PVA 50 (Figs. 4 and 5) yielded unusual results in the high pH region. The pH 7 curve maintained the 5–87 phase lead caused by the adsorbed layer, while the pH 10 and pH 4 curves were displaced to larger and smaller phase leads, respectively, suggesting expansion and contraction effects in the adsorbed layer. A decrease in adsorbed amount with increasing pH has been reported by Tadros for PVA on silica (48) and by Rubio and Kitchener for PEO on silica (23). Several different explanations were presented in each paper to identify the exact cause of this effect, without resolution, but it was shown clearly that the ionization of silanol groups inhibits the adsorption of neutral polymer. Therefore it is reasonable to expect that at high pH, where adsorption is less favorable, many of those polymer segments adsorbed as trains may become redistributed into loops and tails, thus increasing the thickness of the adsorbed layer. This would account for the increase in phase lead at pH 10, consistent with an increased d. At low pH, more segments adsorb in trains to the surface, and the layer contracts, resulting in a lower d and smaller phase leads. The magnitudes of the mD ratios were in agreement with this trend, with the significant increase with frequency at pH 10 indicative of a large d. The effect at pH 4 was less substantial than that at pH 10, which is reasonable in view of the small change in the surface charge of the silica particles ( õ2 mC/cm2 ) which occurs between pH 4 and pH 7, compared with the increase of approximately 15 mC/cm2 in surface charge which occurs between pH 7 and pH 10 (49). Similar effects were observed in the dynamic mobility spectra for pH titrations of silica suspensions with lower coverages of PVA. The extent of the effect decreased with decreasing coverage; for 0.28 1 10 06 M PVA, the phase lead at pH 10 was less than 37 higher than the pH 4 and pH 7 phase angles, which were unchanged from the small effect induced during the adsorption stage of the experiment. The proportionality between the adsorbed amount, and hence layer thickness, and the magnitude of the mD effect supports the concept of expansion and contraction of the adsorbed layer. C9fEN The development of adsorbed layers of nonyl phenol ethoxylate could not be followed as closely as the PVA

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TABLE 2 Adsorbed Layer Thicknesses for Nonyl Phenol Ethoxylate N40 [C9fEN] (1004 M) 3.3 4.5 6.7 8.9 11.0

N100

d (nm)

D (nm)

(d / D) (nm)

Ds (nm)

d (nm)

D (nm)

(d / D) (nm)

Ds (nm)

{ { { {

4.1 4.3 4.0 4.1

9.5 9.7 10.0 10.1

5.3 5.5 5.5 5.6

9.1 { 1.9 10.4 11.0 11.5 { 2.5 11.5 { 2.5

3.3 3.3 3.5 3.3 3.4

12.4 13.7 14.4 14.8 14.9

5.2 5.7 6.2 6.3 6.4

5.4 5.4 6.0 6.0

1.0 1.0 1.1 1.1

adsorption due to flocculation effects which restricted the measurements to high concentrations. The adsorption mechanism for CmEn and CmfEn surfactants on silica is usually attributed to hydrogen bonding between the oxygen atoms along the ethylene oxide group and the silanol groups on the surface (50–52). Fluorescence decay and neutron reflection studies indicate that nonionic surfactants with ethylene oxide chains which are short compared with the hydrocarbon tail tend to form bilayers (53), while molecules with relatively long ethylene oxide chains seem to form small aggregates on the surface (38, 54). These studies have been confined to n £ 40. The characteristics of the adsorbed layer which are of interest in the present study are its thickness and elasticity. As these properties would presumably be quite similar for bilayers and aggregates, no attempt will be made to distinguish between the two possible structures. However, it should be possible to determine whether N40 and N100 surfactant molecules form some type of bilayer, or whether they adsorb in a monolayer arrangement, similar to neutral polymers. Estimates of the area per molecule involve large uncertainties and cannot be considered as reliable indicators of the adsorbed layer structure. A more accurate assessment of the adsorbed structure is provided by the polymer gel theory analysis. The effective static mobility thickness Ds was calculated for N40 and N100 using Eq. [33] and is given in Table 2. The total adsorbed layer thickness (d / D ) is also shown. The thicknesses for N40 are approximately constant at all concentrations as they all correspond to the plateau region of the ESA magnitudes. The two lowest concentrations of N100 are just below the true plateau, where the layer thickness is still increasing, but at higher concentrations there is little change in d, D, or Ds . If the molecules were adsorbed in a completely flat orientation on the surface via attachment of the ethylene oxide group and in a single layer, there should be no significant difference in d or D for the two surfactants and the quantity (d / D ) would be much less than 5 nm. The thicknesses in Table 2 dismiss the possibility of this configuration. Although there is extensive evidence in the literature for bilayer

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formation for nonionic surfactants, the information supplied by the polymer gel theory is more consistent with the molecules adsorbing in trains, loops, and tails on the surface. The similarity between the experimental dynamic mobility ratios for N40 and N100 and those for PVA suggests that these large surfactants behave like neutral polymers. The successful fitting of the data using the polymer gel theory, which is based on the train/loop/tail concept, does not support the existence of an aggregated adsorbed layer. The total thickness (d / D ) is significantly larger for N100 than for N40, while the thickness D of the dense inner layer is approximately the same for both surfactants. The Ds values of about 5.5 nm for N40 and 6 nm for N100 are comparable with the estimated plateau Ds for PVA 50. These results also favor a train/loop/tail configuration. The observed increase with concentration in the peak position of the mD ratio arguments was described theoretically by increasing the relaxation frequency v0 of the adsorbed layer in the low and intermediate plateau concentrations. This corresponds in physical terms to an increase in the density of the adsorbed layer, an effect which is presumably due to continuing adsorption of surfactant molecules, particularly between 4 1 10 04 and 9 1 10 04 M, so that the adsorbed layer becomes more tightly packed. The exchange of adsorbed molecules for shorter chain homologs in solution may also have an effect in this concentration range. The v0 values result in a maximum shear modulus g of 4 N m02 for N40 and 5 N m02 for N100. These values are much higher than the calculated shear moduli for PVA, implying that nonyl phenol ethoxylate forms more dense adsorbed layers than PVA. Unlike PVA, the surfactant molecules are very small polymers and can form loops and tails only of finite length. Thus further adsorption of N40 and N100 is likely to occur by closer packing on the surface, rather than by the continued extension of the permeable layer which was apparent during the adsorption of PVA. The increase in v0 with concentration implies a corresponding increase in the drag coefficient a, since the adsorbed layer would presumably become less porous as more surfactant molecules pack onto the surface per unit area. In

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this situation the assumption that the local polymer geometry, and hence a / h, is constant for all concentrations is no longer valid. The lack of a suitable method for evaluating a / h independently is another restriction of the current polymer gel theory. However, it turns out that if both v0 and a / h are increased with concentration, then lower values of d are required to fit the experimental data. It seems likely that the slight increase in d which occurs when a constant value of a / h is used (Table 2) would be eliminated if the change in a / h with concentration could be determined accurately and taken into account. The picture which emerges from the analysis of the experimental and theoretical results is that an adsorbed layer of nonyl phenol ethoxylate has an approximately constant thickness and a density which tends to increase over the low and intermediate concentrations. The high values of v0 are characteristic of a relatively dense elastic gel layer. The layer thickness and density are larger for N100 than for N40, indicating that the longer ethylene oxide chains form thicker layers than the shorter ones, and are more tightly packed. These observations are more indicative of a train/loop/tail arrangement than a bilayered structure of surfactant molecules on the silica surface; however, further studies are necessary to confirm this. CONCLUSIONS

The simple polymer gel theory developed here is capable of describing the effect of an adsorbed neutral polymer or surfactant layer on the dynamic mobility of a particle and provides information about the elasticity and thickness of the adsorbed layer. The adsorbed layer thickness for PVA compared well with results from other techniques reported in the literature and appeared to be influenced by changes in pH. Nonyl phenol ethoxylate adsorbed in a relatively dense layer. For both species, the thickness of the inner adsorbed layer remained constant, while the outer layer became thicker with increasing concentration and molecular weight. The electroacoustic technique, in particular the dynamic mobility spectrum, is an effective method for studying the adsorption of neutral polymers and nonionic surfactants. APPENDIX A: THE POLYMER GEL LAYER THEORY

The derivation of the polymer gel layer formula is similar to that of the Smoluchowski formula for electrophoretic mobility. For a negatively charged surface, the adjacent fluid contains an excess of positive ions. The applied field induces the ions to move, and the surrounding fluid is dragged along with them. The system is illustrated in Fig. 14a, where E(x) represents the electric field. To determine the effect of the polymer layer on the particle mobility we must first derive the differential equations

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which describe the flow in that layer. For a block of fluid moving with velocity u(y), the tangential force on the lower face of the block is given by 0h

Ìu(y 0 Dy/z) D x Dz Ìy

and the tangential force on the upper face is h

Ìu(y / Dy/z) D xDz, Ìy

where h is the viscosity of the fluid, and D x, Dy, and Dz are the lengths of the sides of the block. On subtracting these forces and dividing by the volume of the block, we find that the net tangential force per unit volume is given by h

Ì 2u(y) Ìy 2

in the limit as the block volume approaches zero. In addition to the viscous forces there is an elastic force acting on the material in the block. This force comes about from the deformation of the polymer caused by the viscous drag of the liquid as it flows back and forth through the network. The elastic forces on the polymer strands give rise to a force g ÌV D x Dy iv Ìy

on the top face of the block, where V is the local polymer velocity and g is the effective elastic shear modulus of the polymer. By subtracting the elastic force acting on the bottom of the block and dividing by the block volume, we find that the net elastic force per unit volume on the block is g Ì 2V . iv Ìy 2

In addition to the viscous and elastic forces, which oppose the motion of the block, there is an electric driving force due to the applied field acting on the charges within the block. This field is given by r EE,

where r E is the charge density and E is the local applied electric field. The net force on the block is equal to its mass times

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ELECTROACOUSTIC EFFECTS OF ADSORBED NEUTRAL POLYMERS

acceleration. For the case of an alternating field, the inertia force has the form

without the elastic force term. Solution of Eqs. [11] and [14], with the Debye–Hu¨ckel approximations,

ivurD xDyDz,

Ì 2c Å k 2c Ìy 2

where v is the frequency of the applied field. However, for the frequencies of interest here (1–10 MHz), this term can be neglected. This follows from the order of magnitude estimate,

Z

Z S D S D S D

ivru vru vr v ÅO ÅO ÅO , 2 2 2 2 h(d u/dy ) h uk hk nk 2

[10]

for the ratio of inertia to viscous force on the block, where k is the Debye–Hu¨ckel parameter and n is the kinematic viscosity ( Å h / r ). Using typical values v Å 10 MHz, 1/ k Å 5 nm, and n Å 10 06 m2 s 01 , we obtain a value of 3 1 10 04 m2 s 01 for this ratio. Thus

S D

v O nk 2

Du iva / g 0 k 2 Å0 D us k2 0 k2 1

F S

e 0 kd 1 /

[11]

∑ ni zi e,

iva , gk 2

[17]

ni Å n exp( 0zi e c /kT ).

d 2c , dy 2

[14]

The force balance on the polymer alone is [15]

Here a(V 0 u) is the drag force per unit volume exerted by the liquid on the polymer and a is the drag coefficient. The more densely packed the polymer, the larger the value of a. The fluid motion in the region y ú d is given by Eq. [11]

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S

1/

ivh g

D

[18]

q

k Ç a/h r `,

Du Å e 0 kd . Dus

[13]

From the Poisson–Boltzmann formula (56),

g Ì 2V Å a(V 0 u). iv Ìy 2

a h

[19]

and Eq. [17] reduces to

0 i

/

0

and Dus Å ez / h is the Smoluchowski velocity jump across the double layer without adsorbed polymer. The quantities g and a will depend on the local polymer geometry. The segment density will be highest close to the surface, hence in this region g and a are large. In the limit as g and a approach ` ,

where

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G

[12]

i

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D

k k2 1 tanh kd 0 2 k k cosh kd

k2 Å

in the region 0 õ y õ d. For y ú d the elastic term disappears. The volume charge density is given by

rE Å 0e

[16]

gives the velocity jump across the polymer layer

!1

g Ì 2V Ì 2u(y) / h / r EE Å 0 iv Ìy 2 Ìy 2

r Å

c Å ze 0 k y ,

where

and the inertia force term can be neglected. Thus the equation of motion on the block is (55)

E

and

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[20]

In this case the effect of the polymer layer is simply to displace the shear plane by a distance d. In practice g and a are likely to decrease smoothly with distance from the surface. As a crude model of this distance dependence, we envisage the adsorbed polymer to be made up of two layers: an inner, dense layer near the surface which displaces the shear plane by a distance D and an outer, permeable layer which gives rise to the elastic effects, as illustrated in Fig. 14b. Thus we take g and a to be infinite for the region 0 õ y õ D and assign uniform finite values to them in the region D õ y õ (d / D ). An order-of-magnitude estimate of the drag coefficient a can be obtained for the case of a random polymer coil in solution. Taking s as the cross-sectional radius of the polymer strand and D as the average distance between adjacent

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CARASSO, ROWLANDS, AND O’BRIEN

polymer strands, the drag force per unit length on the polymer chain can be expressed as

S D

F s Å O hu L D

,

[21]

where L is the length of the chain. To estimate the distance D between neighboring strands of the polymer we first note that a plane passing through the center of the coil will cut the polymer at O(N) points, where N is the number of loops in the coil. N is of the order of (L/dc ), where dc is the rms diameter of the coil. Setting n as the number of intersections of the chain with the crosssectional plane per unit area gives

S D

DÅO

1 n

q

.

g , h

[27]

where the shear modulus g of the polymer has units of N m02 and the viscosity h is in N s m02 . The nondimensional quantity a* is defined as a* Å

ad 2 . h

[28]

Hence in terms of random coil dimensions,

r

S D S D

dc DÅO q N

Å O dc

dc L

.

S D q

F hus L q ÅO L dc dc

.

[24]

Thus the viscous force per unit volume on the polymer is F F Å V L

SD S D L d 3c

Å O hus

L 3/2 q d 4c dc

.

[25]

As a is the force per unit volume per unit velocity, it follows that

S

a Å O hs

L 3/2 d 9c / 2

D

.

[26]

In the following analysis we make the assumption that in the region D õ y õ (d / D ) the polymer geometry is similar to that of a random coil, hence the above estimate of a is applicable in that region. Literature values can be used for the diameter of the polymer coil, dc , and the radius s and length L of the polymer chain can be calculated from bond length and angle considerations. The velocity ratio in Eq. [17] can be written in terms of

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S

a* Å O

[23]

Hence Eq. [21] becomes

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v0 Å

[22]

Since n Å O(N/d 2c ),

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three parameters; the nondimensional drag coefficient a*, the relaxation frequency v0 , and the ratio kd of the gel layer thickness, d, to the electrical double layer thickness, 1/ k. Note that d is the thickness of the gel layer, not the random coil diameter. The relaxation frequency v0 (in units of s 01 ) is given by the expression

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sL 3 / 2 d 2 d 9c / 2

D

.

[29]

In the absence of experimental data for the shear modulus (and hence the relaxation frequency v0 ), an empirical approach was used to set the parameter v0 in the polymer gel theory. The value of v0 determines where the peak in the argument of the dynamic mobility ratio mD (coated)/ mD (uncoated) occurs. A range of v0 values were substituted into the theory, and the value which gave the closest approximation to the experimental phase angle peak was selected. Once v0 has been set there are still two parameters, a* and kd, to adjust in order to fit the measured mobility ratio curves. However, since a* Å ad 2 / h and a / h can be estimated using Eq. [26], there is only one adjustable quantity: d, the thickness of the outer polymer layer. APPENDIX B: ESTIMATING a

The radius s and length L of the PVA chain were calculated assuming a zig-zag conformation of the carbon backbone with a tetrahedral bond angle (109.57 ) and excluding the contributions of bound water molecules. The average value for s was taken to be 0.23 nm. The length L is given by 2nz, where n is the number of segments in the chain and z is half the horizontal distance between alternate carbon atoms. Thus L Å 295 nm for PVA 50 and L Å 502 nm for PVA 85. The diameter of the polymer coil, dc , was determined from the hydrodynamic radius calculations reported by Garvey et al. (3) for free PVA in solution. They obtained a radius of 5.40 nm for PVA of molecular weight 43,000, and a radius of 6.75 nm for MW 67,000. Thus a value of

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ELECTROACOUSTIC EFFECTS OF ADSORBED NEUTRAL POLYMERS

10.8 nm was assigned to the dc parameter for PVA 50, and for PVA 85 dc was set to 13.5 nm. Note that the molecular weights used by Garvey et al. were lower than those used in the present study, hence the diameters determined here will be slightly underestimated. The parameter a / h was then estimated by substituting these values for s , L , and dc into Eq. [ 26 ] . For PVA 50, a / h was approximately 2.3 1 10 02 nm02 , while for PVA 85, a value of 1.8 1 10 02 nm02 was obtained. The silica suspensions were all made up in 5 1 10 03 M sodium chloride background electrolyte. This corresponds to a double layer thickness 1 / k of 4.3 nm ( 56 ) ; hence k is equal to 0.23 nm01 . After setting the relaxation frequency v0 to fit the position of the peak in the experimental mD ratio phase angles, the value of d was varied as required to reproduce the peak heights in the phase angle curves at different PVA concentrations. The theoretical magnitude curves resulting from this procedure were of the same shape as the experimental curves, but the theoretical values were higher; i.e., the theory underestimates the reduction in mobility magnitude caused by the polymer layer. This underestimate could be due to the fact that the polymer is not homogeneous as assumed in the theory, but is more dense near the surface. As mentioned earlier, a dense layer of thickness D has the effect of displacing the shear plane by D (Fig. 14b), and this causes a uniform reduction in the theoretical mobility magnitudes. To estimate this reduction in mobility, we use the Gouy– Chapman theory for the potential distribution in the diffuse double layer (55), from which we find that the electrophoretic hydrodynamic layer thickness D is related to z by the formula tanh

S D zz 4kT

Å tanh

S D

zcd 0 kD e , 4kT

[30]

where z Å z ( D ), i.e., the zeta potential in the presence of adsorbed polymer, and cd É z (0), the zeta potential in the absence of polymer. This formula involves two assumptions: (i) that the shear plane displacement is the only effect of adsorbed polymer on the double layer properties and (ii) that the diffuse layer potential cd is equal to the zeta potential in the absence of polymer. For the low zeta potentials assumed in the gel layer theory, Eq. [30] takes the approximate form z Å cd e 0 kD .

[31]

The effect of this change is to multiply Eq. [17] for Du/ Dus by e 0 kD . Thus the magnitude ratios will be uniformly reduced by this factor. To incorporate this shear plane displacement into the gel

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213

layer theory, the theoretical magnitudes were multiplied by the appropriate e 0 kD factor for each curve. A consistent evaluation of this factor over the frequency range for a particular magnitude ratio was obtained from the experimentally determined magnitude of the mobility ratio and the uncorrected theoretical magnitude of the velocity jump across the gel layer, using the expression e 0 kD Å

ÉmD (coated)/ mD (uncoated)Éexpt , ÉmD (coated)/ mD (uncoated)Étheory

[32]

where the subscript ‘‘theory’’ in the denominator refers to the case D Å 0. This calculation was applied at several frequencies within each curve to obtain an average value for the correction factor. This correction to the theoretical magnitudes resulted in very good agreement with the experimental magnitudes at all concentrations of PVA. An estimate of the thickness D follows directly from the evaluation of e 0 kD . Although the C9fEN molecules are surfactants rather than polymers, the EN group itself is a small polymer and hence the molecules may assume a range of conformations in solution, from fully extended to random coil (33). Here we will treat N40 and N100 as random coils for the estimation of the a / h parameter. The radius and length parameters were determined following the molecular-structure method used for PVA, so that for N40, L Å 15.6 nm, and for N100, L Å 36.6 nm, in agreement with literature values (57). The EN group of the nonyl phenol ethoxylates used here was large compared with the length of the hydrocarbon chain, hence for the estimation of dc the effect of the C9f group was neglected and the molecules were treated as random coils of poly(ethylene oxide). The coil diameter was equated with the unperturbed rms end-to-end distance, r0 (nm), of the PEO chain (56), which was calculated from the molecular weight M using the formula (58) r0 1 10 4 Å 750. M 1/2

[33]

The molecular weight of N40 is 1980, yielding a dc value of 3 nm, while for N100, which has a molecular weight of 4626, dc Å 5 nm. Thus from Eq. [26], a / h was estimated to be 0.05 nm02 for N40 and 0.02 nm02 for N100. The value of k was set to 0.23 nm01 , as determined by the background solution of 5 1 10 03 M NaCl, and the v0 parameter was determined from the position of the experimental argument peak. The e 0 kD factor for the nonyl phenol ethoxylates was determined using Eq. [32]. ACKNOWLEDGMENTS This work was supported by an ARC Postgraduate Scholarship for M.L.C. Dr. Robert Hunter is thanked for helpful discussions.

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REFERENCES 1. Garvey, M. J., Tadros, Th. F., and Vincent, B., J. Colloid Interface Sci. 55, 440 (1976). 2. Fleer, G. J., Koopal, L. K., and Lyklema, J., Kolloid Z. Z. Polym. 250, 689 (1972). 3. Garvey, M. J., Tadros, Th. F., and Vincent, B., J. Colloid Interface Sci. 49, 57 (1974). 4. Joppien, G. R., J. Phys. Chem. 82, 2210 (1978). 5. Brooks, D. E., and Seaman, G. V. F., J. Colloid Interface Sci. 43, 670 (1973). 6. Brooks, D. E., J. Colloid Interface Sci. 43, 687 (1973). 7. Brooks, D. E., J. Colloid Interface Sci. 43, 700 (1973). 8. Brooks, D. E., J. Colloid Interface Sci. 43, 714 (1973). 9. Eremenko, B. V., Platonov, B. E´., Uskov, I. A., and Lyubchenko, I. N., Kolloidn. Zh. 36, 240 (1974). 10. Eremenko, B. V., Mamchenko, A. V., Platonov, B. E´., and Sergienko, Z. A., Kolloidn. Zh. 37, 853 (1975). 11. Eremenko, B. V., Platonov, B. E´., Baran, A. A., and Mamchenko, A. V., Kolloidn. Zh. 37, 1083 (1975). 12. Eremenko, B. V., Platonov, B. E´., Baran, A. A., and Sergienko, Z. A., Kolloidn. Zh. 38, 680 (1976). 13. Fleer, G. J., and Lyklema, J., J. Colloid Interface Sci. 46, 1 (1974). 14. Koopal, L. K., and Lyklema, J., Faraday Discuss. Chem. Soc. 59, 230 (1975). 15. Fleer, G. J., and Lyklema, J., J. Colloid Interface Sci. 55, 228 (1976). 16. Koopal, L. K., and Lyklema, J., J. Electroanal. Chem. 100, 895 (1979). 17. Tadros, Th. F., J. Colloid Interface Sci. 46, 528 (1974). 18. van den Boomgaard, Th., King, T. A., Tadros, Th. F., Tang, H., and Vincent, B., J. Colloid Interface Sci. 66, 68 (1978). 19. Lambe, R., Tadros, Th. F., and Vincent, B., J. Colloid Interface Sci. 66, 77 (1978). 20. Tadros, Th. F., and Vincent, B., J. Colloid Interface Sci. 72, 505 (1979). 21. Barker, M. C., and Garvey, M. J., J. Colloid Interface Sci. 74, 331 (1980). 22. Kavanagh, B. V., Posner, A. M., and Quirk, J. P., Disc. Faraday Soc. 59, 242 (1975). 23. Rubio, J., and Kitchener, J. A., J. Colloid Interface Sci. 57, 132 (1976). 24. Barnett, K. G., Cosgrove, T., Vincent, B., Burgess, A. N., Crowley, T. L., King, T., Turner, J. D., and Tadros, Th. F., Polymer 22, 283 (1981). 25. Cosgrove, T., Heath, T. G., Ryan, K., and van Lent, B., Polym. Commun. 28, 64 (1987). 26. Barnett, K. G., Cosgrove, T., Vincent, B., Sissons, D. S., and CohenStuart, M., Macromolecules 14, 1018 (1981). 27. Klein, J., J. Colloid Interface Sci. 111, 305 (1986). 28. Couture, L., and van de Ven, T. G. M., Colloids Surf. 54, 245 (1991). 29. de Witt, J. A., and van de Ven, T. G. M., Adv. Coll. Int. Sci. 42, 41 (1992).

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30. Polverari, M., and van de Ven, T. G. M., Colloids Surf. A 86, 209 (1994). 31. Killmann, E., Maier, H., Kaniut, P., and Gu¨tling, N., Colloids Surf. 15, 261 (1985). 32. Killmann, E., and Sapuntzjis, P., Colloids Surf. A 86, 229 (1994). 33. Clunie, J. S., and Ingram, B. T., in ‘‘Adsorption from Solution at the Solid/Liquid Interface’’ (G. D. Parfitt and C. H. Rochester, Eds.). Academic Press, New York, 1983. 34. Sidorova, M., Golub, T., and Musabekov, K., Adv. Coll. Int. Sci. 43, 1 (1993). 35. Lindheimer, M., Keh, E., Zaini, S., and Partyka, S., J. Colloid Interface Sci. 138, 83 (1990). 36. McDermott, D. C., Lu, J. R., Lee, E. M., Thomas, R. K., and Rennie, A. R., Langmuir 8, 1204 (1992). 37. Cummins, P. G., Staples, E., and Penfold, J., J. Phys. Chem. 94, 3740 (1990). 38. Levitz, P., and van Damme, H., J. Phys. Chem. 90, 1302 (1986). 39. Gu, T., and Zhu, B., Colloids Surf. 44, 81 (1990). 40. Tiberg, F., Jo¨nsson, B., Tang, J., and Lindman, B., Langmuir 10, 2294 (1994). 41. O’Brien, R. W., J. Fluid Mech. 190, 71 (1988). 42. O’Brien, R. W., Cannon, D. W., and Rowlands, W. N., J. Colloid Interface Sci. 173, 406 (1995). 43. Ford, R. D., ‘‘Introduction to Acoustics,’’ p. 83. Elsevier, London, 1970. 44. Maier, H., Baker, J. A., and Berg, J. C., J. Colloid Interface Sci. 119, 512 (1987). 45. Miller, N. P., and Berg, J. C., Colloids Surf. 59, 119 (1991). 46. Carasso, M. L., and Rowlands, W. N., to be submitted, J. Colloid Interface Sci. 47. Nielsen, L. E., ‘‘Polymer Rheology,’’ p. 55. Dekker, New York, 1977. 48. Tadros, Th. F., J. Colloid Interface Sci. 64, 36 (1978). 49. Carasso, M. L., Ph.D. thesis, Univ. of Sydney, 1996. 50. Rupprecht, H., and Liebl, H., Kolloid Z. Z. Polym. 250, 719 (1972). 51. Rupprecht, H., Progr. Colloid Polymer Sci. 65, 29 (1978). 52. Partyka, S., Zaini, S., Lindheimer, M., and Brun, B., Colloids Surf. 12, 255 (1984). 53. Lee, E. M., Thomas, R. K., Cummins, P. G., Staples, E. J., Penfold, J., and Rennie, A. R., Chem. Phys. Lett. 162, 196 (1989). 54. Bo¨hmer, M. R., Koopal, L. K., Janssen, R., Lee, E. M., Thomas, R. K., and Rennie, A. R., Langmuir 8, 2228 (1992). 55. Hunter, R. J., ‘‘The Zeta Potential in Colloid Science.’’ Academic Press, London, 1981. 56. Hunter, R. J., ‘‘Foundations of Colloid Science,’’ Vol. 1. Oxford Univ. Press, New York, 1987. 57. Urbina-Villalba, G., Reif, I., Ma´rquez, M. L., and Rogel, E., Colloids Surf. 99, 207 (1995). 58. Kurata, M., Tsunashima, Y., Iwama, M., and Kamada, K., in ‘‘Polymer Handbook’’ (J. Brandrup and E. H. Immergut, Eds.), p. IV-45. WileyInterscience, New York, 1975.

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