Bridging Flocculation in VermiculiteâPEO Mixtures - American

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Bridging Flocculation in Vermiculite-PEO Mixtures M. V. Smalley* 49 Gordon Road, Wealdstone, Harrow HA3 5RY, United Kingdom

H. L. M. Hatharasinghe and I. Osborne Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral L63 3JW, United Kingdom

J. Swenson Department of Applied Physics, Chalmers University of Technology, S-412 96, Go¨ teborg, Sweden

S. M. King ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom Received June 15, 2000. In Final Form: March 27, 2001 The mechanism and strength of polymer bridging flocculation have been investigated by experiments on a four-component clay-polymer-salt-water system consisting of butylammonium vermiculite, poly(ethylene oxide) (PEO), butylammonium chloride, and water. The system was studied by neutron diffraction as functions of the three concentration variables, the volume fraction r of the clay in the condensed matter system, the volume fraction v of the polymer, and the salt concentration c, and the molecular weight M of the PEO molecules. The highly conspicuous result obtained is that the addition of polymer brings about a contraction in the interplate distance between the parallel clay platelets only when the mean end-to-end distance l is greater than or equal to the interplate distance d0 in the corresponding three-component clay-salt-water system. With r, c, and v constant, d is independent of M, ruling out the depletion flocculation mechanism. Instead, the results confirm a new mechanism for bridging flocculation, namely, that equilibrium occurs between a drawing force arising from the segmental motions of the partly stretched bridging polymers and a bonding force to the clay platelets. The PEO concentration inside the clay gel was determined by two methods, gel permeation chromatography with refractive index analysis and chemical analysis. Both gave the result that the ratio of the concentration inside the gel to the added concentration was 0.45. Using this value, the strength of the bridging force was determined by measuring d as a function of v and mapping the contractions onto the previously determined curves of d as a function of uniaxial stress p applied to the clay-salt-water system. The drawing force per polymer bridge was found to be 0.6 pN. This value was constant over 3 orders of magnitude in p, and its calculation represents a new challenge in colloid and polymer science. Another interesting phenomenon observed was that of “crowding”, where the welldefined d-spacing of the gels suddenly disappears at higher volume fractions. At c ) 0.1 M, the crowding transition was found to occur at approximately 10% coverage of the clay surfaces.

Introduction There are two basic mechanisms for the stabilization of colloids, charge stabilization and steric stabilization. The latter is due to the adsorption of polymers onto the surface of colloidal particles, leading to a steric repulsion between them. Under certain circumstances, however, high molecular weight polymers can adsorb on separate particles and draw them together, a phenomenon known as polymer bridging flocculation.1 This phenomenon has many important industrial applications in a large variety of areas, such as in the preparation of paints and papers, in the stabilization of drilling fluids, and in water purification,1 where the flocs are used for the removal of unwanted particles. In view of its widespread occurrence, it is essential to have quantitative results on the forces involved. We recently proposed a new model of polymer bridging flocculation based upon neutron scattering studies of vermiculite-PEO mixtures.2 Here, we test and (1) Everett, D. H. Basic Principles of Colloid Science; Royal Society of Chemistry: London, 1988.

refine the model in an attempt to elucidate the physical origin of bridging flocculation. To investigate the phenomenon quantitatively, it is vital to have a model experimental system. The threecomponent system composed of butylammonium vermiculite, butylammonium chloride, and water, first reported by Walker,3 has proved an ideal system for the investigation of charge stabilization.4 We have chosen to add PEO to this particularly well-defined three-component clay-salt-water system in order to investigate bridging flocculation because the three-component PEO-saltwater system is also well characterized.5-7 The butylammonium vermiculite system is an ideal model system (2) Swenson, J.; Smalley, M. V.; Hatharasinghe, H. L. M. Phys. Rev. Lett. 1998, 81, 5840. (3) Walker, G. F. Nature 1960, 187, 312. (4) Smalley, M. V. Langmuir 1994, 10, 2884. (5) Briscoe, B.; Luckham, P.; Zhu, S. Macromolecules 1996, 29, 6208. (6) Crowther, N. J.; Eagland D. J. Chem. Soc., Faraday Trans. 1996, 92, 1859. (7) Polverari, M.; van den Ven, Th. G. M. J. Phys. Chem. 1996, 100, 13687.

10.1021/la0008232 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/30/2001

Bridging Flocculation in Vermiculite-PEO Mixtures

because it consists of regularly spaced and parallel charged flat two-dimensional colloidal platelets in an electrolyte solution.4 The one-dimensional nature of the system simplifies structural and physical analyses considerably and makes investigations of many important problems in colloidal science possible. For example, we have explored the intermediate range structure and counterion distribution of the interlayer solution8,9 and have obtained detailed information about the effect of uniaxial stress along the swelling axis of the vermiculite gels.9,10 The swelling of vermiculites in aqueous solutions, which occurs perpendicularly to the crystalline silicate layers, is in most cases sufficiently large that the d-spacing of the resulting gels falls in a colloidal range ideal for studying by small-angle neutron scattering. The actual measurement of the d-value is easy. The difficulty in studying the corresponding four-component system with added polymer is that the problem is a many variable one. Even for the unstressed gels, when the vermiculite is allowed to swell freely at atmospheric pressure, we must consider the three concentration variables (the volume fraction r of the clay in the condensed matter system, the volume fraction v of the polymer, and the salt concentration c), temperature T, the molecular weight of the polymer M, and the chemical nature of the polymer x. Preliminary investigations into the cases x ) PEO (poly(ethylene oxide)) and x ) PVME (poly(vinyl methyl ether)) showed that both neutral polymers show similar behavior under identical conditions of the five variables {r, v, c, T, M},11-13 and we here restrict attention to the PEO system. Likewise, we choose a fixed temperature within the colloidally swollen gel phase (T ) 5 °C) in order to study the mechanism and strength of the bridging flocculation by {r, v, c, M} variations. Crucially, in this regime we can use the salt concentration c to control the interlayer spacing (d-spacing) and can thereafter choose polymers with mean end-to-end distances (determined by the molecular weight M) suitable for bridging. There are indeed two temperatureinduced phase transitions in the vermiculite-salt-water system,14-17 and the effect of the addition of PEO on these is a fascinating study12,13 but beyond the scope of the present paper. In the results section, we give detailed descriptions of the {r, v, c, M} variations, including sample-to-sample variability studies at fixed {r, v, c, M} points. In order that the reader not feel too badly the tedium of a long complicated description, we give a brief overview of what has been obtained so far and what we were trying to map out. The main idea was to use the 1991 measurement of the d-spacing as a function of uniaxial pressure p (d vs p curves)10 to correlate any contraction of d with respect to added polymer volume fraction v (d vs v curves) with (8) Swenson, J.; Smalley, M. V.; Thomas, R. K.; Crawford, R. J.; Braganza, L. F. Langmuir 1997, 13, 6654. (9) Swenson, J.; Smalley, M. V.; Thomas, R. K.; Crawford, R. J. J. Phys. Chem. B 1998, 102, 5823. (10) Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Adv. Colloid Interface Sci. 1991, 34, 537. (11) Jinnai, H.; Smalley, M. V.; Hashimoto, T.; Koizumi, S. Langmuir 1996, 12, 1199. (12) Smalley, M. V.; Jinnai, H.; Hashimoto, T.; Koizumi, S. Clays Clay Miner. 1997, 45, 745. (13) Hatharasinghe, H. L. M.; Smalley, M. V.; Swenson, J.; Williams, G. D.; Heenan, R. K.; King, S. M. J. Phys. Chem. B 1998, 102, 6804. (14) Smalley, M. V.; Thomas, R. K.; Braganza, L. F.; Matsuo, T. Clays Clay Miner. 1989, 37, 474. (15) Braganza, L. F.; Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Clays Clay Miner. 1990, 38, 90. (16) Williams, G. D.; Moody, K. R.; Smalley, M. V.; King, S. M. Clays Clay Miner. 1994, 42, 614. (17) Hatharasinghe, H. L. M.; Smalley, M. V.; Swenson, J.; Hannon, A. C.; King, S. M. Langmuir 2000, 16, 5562.

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an effective uniaxial pressure p (p vs v curves). Because the earlier results for the d versus p curves10 were obtained at r ) 0.01, the main study was restricted to this case. However, we also need to know the distribution of the polymer molecules between the clay gel and its supernatant fluid. It turns out to be much more reliable to measure this distribution at r ) 0.1. To connect the results of the determination of the polymer concentration inside the gel with the microscopic measurements, further neutron diffraction studies were performed at r ) 0.1. In general, the d-value decreases with all three concentration variables, but there are abrupt changes in behavior at fixed molecular weights, corresponding to “bridging” conditions. The exact onset of these transitions was studied in detail as a function of the variables {r, v, c, M}. Broadly speaking, the data confirm that the bridging transition occurs when the mean end-to-end distance of the polymer approximately matches the d-value of the same {r, c} system with no added polymer. It greatly facilitates the understanding of this paper if the reader is familiar with ref 2, where we gave the first determination of the bridging force. There were three essential features of the outcome of the {c, v, M} variations, as follows. (1) The clay keeps its normal d-spacing (dvalue without added polymer) up until a definite molecular weight and thereafter contracts. (2) The contraction depends on the volume fraction of polymer in the same way as the d-value of the vermiculite-salt-water system depends on applied uniaxial stress. (3) The well-defined d-values of the gels disappear above a critical volume fraction, which we name the “crowding” volume fraction. The results suggested that equilibrium occurs between an entropy-driven drawing force, arising from partly stretched bridging polymers, and a bonding force to the clay platelets. In the new mechanism, the force is proportional to the number of bound segments, and the strength per polymer bridge was given as 1.4 pN.2 In the discussion, we describe how this force was calculated from the experimental results and how well it fits with the more rigorous tests applied here. Experimental Section The vermiculite crystals were from Eucatex, Brazil. Crystals about 30 mm2 in area by 1 mm thick were washed and then treated for about a year with 1 M NaCl solution at 50 °C, with regular changes of solution, to produce a pure Na vermiculite, with the chemical formula11

Si6.13Mg5.44Al1.65Fe0.50Ti0.13Ca0.13Cr0.01K0.01O20(OH)4Na1.29 To prepare the n-butylammonium vermiculite, the Na form was soaked in 1 M n-butylammonium chloride solution at 50 °C, with regular changes of solution, for about a month. Chemical analysis of the n-butylammonium vermiculite thus obtained showed that the amount of interlayer sodium remaining was less than 1%. The crystals were stored in a 1 M n-butylammonium chloride solution prior to the swelling experiments. The PEO was purchased from Scientific Polymer Products Ltd (Church Stretton, U.K.) with manufacturer’s specifications Mw/Mn ) 1.02, where Mw and Mn are the weight-average and number-average molecular weights, respectively, and used without further purification. Solutions of the required volume fraction of PEO were prepared by dissolving a known mass of the polymer (F ) 1.13 g/cm3) in a known volume of an 0.1, 0.03, or 0.01 molar n-butylammonium chloride solution, itself prepared by dissolving a known mass of n-butylammonium chloride in D2O. It was necessary to swell the crystals in D2O rather than H2O solutions because of the large incoherent scattering cross section of hydrogen which would otherwise have obscured the scattering of interest.

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Prior to performing an experiment on the clay, the crystals were first washed thoroughly to remove any molar solution that may be trapped in surface imperfections. This was achieved by rinsing the crystals with 500 cm3 of distilled water at 60°-80 °C 15 times before drying on filter paper. The distilled water was heated in order to prevent any swelling occurring during this washing process because, although the absorption of distilled water is rapid, it does not occur above 40 °C.16 After drying, the crystals were cut to dimensions of approximately 0.5 × 0.5 × 0.1 cm. These were individually weighed, and the volume of a crystal in its fully hydrated state was calculated using the density (F ) 1.86 g/cm3). After weighing, a single vermiculite crystal was placed into a quartz sample cell of internal dimensions 1 × 1 × 4.5 cm and an appropriate amount of solution (typically 2.5 cm3) was added to prepare an r ) 0.01 sample. The volume fractions of the clay and salt were fixed such that their numbers were the same in all of the studies and not varied to accommodate the addition of the polymer. That is, addition of the polymer was assumed to have negligible effect on the calculation of the other volume fractions. The volume fraction of polymer was varied between v ) 0.0010 and 0.2000, with accuracy to the fourth decimal place. The cells containing the four components were sealed with Parafilm and allowed to stand at 7 °C for 2 weeks prior to the neutron scattering experiments, to ensure that full equilibrium swelling had been achieved.16 The neutron diffraction experiments were carried out at the ISIS spallation source, using the time-of-flight small-angle scattering instrument LOQ, described in ref 18. A white beam of neutrons with wavelengths in the range between λ ) 2.2 Å and λ ) 10 Å was used, and the incident beam was collimated by passage through an 8 mm wide, 2 mm high rectangular slit. The samples were mounted on a temperature-controlled 20-position sample changer. Neutrons scattered by the gel samples were recorded on a two-dimensional area detector, software coded as 64 × 64 pixels, situated 4.15 m behind the samples, covering the approximate Q-range between 0.01 and 0.2 Å-1. The quartz sample cells used were practically transparent to neutrons at the wavelengths utilized on LOQ, and the small-angle neutron scattering from D2O was of low intensity over the Q-range studied. Subtraction of the background scattering, after making the appropriate transmission corrections, was found to have a negligible effect on the scattering patterns, which were dominated by scattering from the gels. The way in which the patterns recorded on the two-dimensional multidetector were reduced to plots of intensity against the Q-vector perpendicular to the clay layers, expressed as I(Q) versus Q in the following, is described fully in refs 12 and 16. The concentration of polymer in the supernatant fluids was measured by gel permeation chromatography (GPC) with refractive index detection. Measurement of the refractive index of a polymer solution is a widely used method for the determination of polymer concentrations, which we here express as volume fractions. After being in contact with the vermiculite for two weeks, the supernatant fluids containing PEO can be contaminated by other particles, for example, silica particles dissolving from the edges of the clay, that affect the refractive index. Taking advantage of the narrow molecular weight distribution of the PEO molecules (Mw/Mn ) 1.02), we were able to use GPC to obtain a pure solution of the polymer for analysis by differential refractometry. The experiments were carried out at Unilever Research Port Sunlight Laboratory using a Hewlett-Packard HP1100 modular liquid chromatograph connected to a Water 410 differential refractometer. A Tosoh TSK GMPW column and 0.3 M NaNO3 were the stationary and mobile phases used for the separation, respectively.19 The instrument provides automated sample injection and precise high-pressure pumping with computerassisted data treatment. The polymer concentration in each sample was determined by means of an external calibration using the same source of PEO. In each set of measurements under fixed {r, c, v, M} conditions, two samples of the PEO solution (18) Heenan, R. K.; King, S. M. LOQ Instrument Handbook; RAL Report RAL-TR-96-036, 1996. (19) Kato, Y.; Matsuda, T.; Hashimoto, T. J. Chromatogr. 1985, 332, 39.

Smalley et al. Table 1. Molecular Weights M and Mean End-to-End Distances l of the PEO Molecules Used molecular weight M (au)

mean end-to-end distance, l (Å)

1 000 4 000 18 000 75 000 300 000 1 000 000 2 000 000

28 61 140 330 720 1400 2100

from the supernatant fluid and two samples of the original soaking solution were analyzed. For these experiments, larger samples of total volume typically 5.0 cm3 were prepared at a fixed clay concentration of r ) 0.1 in glass jars sealed with Parafilm. The jars were allowed to stand at 7 °C for 2 weeks prior to the analyses. To make an independent analysis of the PEO concentrations in the supernatant fluids, a chemical method was employed in certain cases, for samples prepared in the same way as those used in the GPC analysis. This method is based on the ability of PEO to form precipitates with large anions.20 The precipitation was performed using phosphomolybdic acid (H3Mo10PO32‚24H2O), barium chloride (BaCl2‚2H2O), and hydrochloric acid, all purchased from Aldrich Chemicals and used without further purification. The reagent was prepared by dissolving 0.5 g of H3Mo10PO32‚24H2O, 0.5 g of BaCl2‚2H2O, and 1.5 mL of concentrated HCl in 250 mL of distilled water. After precipitation by adding 1 mL of the reagent to 1 mL of the PEO sample in a clean, dry centrifuge tube, the tube was shaken gently and stored at 20 °C for 15 min before being centrifuged for 5 min at 10 000 rpm to separate the complex from its supernatant fluid. The supernatant fluid was then analyzed for excess phosphomolybdic acid by diluting 1 mL of the fluid up to 50 mL in a volumetric flask and determining the absorbance at 216 nm.20 As with the GPC analysis, calibrations were performed using solutions containing known concentrations of PEO, and the polymer volume fractions were obtained using the calibration graph.

Results The philosophy of the experiments was to use the salt concentration c to control the interlayer spacing d in the system without added polymer, henceforth referred to as the pure aqueous system. We denote the d-value under fixed {r, c} conditions in the pure aqueous system by d0. Most of the studies were carried out at r ) 0.01, and three salt concentrations were chosen: c ) 0.1 M, for which d0 ) 120 Å; c ) 0.03 M, for which d0 ) 190 Å; c ) 0.01 M, for which d0 ) 330 Å. In giving the d-spacings to two significant figures, we note that there is sample-to-sample variability in the system leading to a 10% fluctuation in the values obtained from different pieces and batches of vermiculite.16 The addition of polymer tends to reduce this variability,11-13 but in each case studied here four separate crystals from within the same batch were swollen in their own cells for the neutron diffraction experiments and examined on LOQ to ensure that the results observed were statistically significant. An example of the sampleto-sample variability obtained in earlier LOQ experiments is shown in Figure 2 of ref 13 for the case c ) 0.1 M, v ) 0.02, M ) 20 000. Initially, seven molecular weights were chosen in the range between 1000 and 2 000 000, as shown in Table 1. To show the data in a form suitable for analyzing the mechanism of the contraction, the molecular weight M has been converted into a mean end-to-end distance of the polymer. Using data given in ref 21, we can derive the approximate empirical relation (20) Nuysink, J.; Koopal, L. K. Talanta 1982, 29, 495. (21) Cohen-Stuart, M. A.; Waajen, F. H. W. H.; Cosgrove, T.; Vincent, B.; Crowley, T. L. Macromolecules 1984, 17, 1825.


l ) M0.57/1.84

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(1)

between the mean end-to-end distance l (in Å) and molecular weight M of PEO dissolved in water. Thus, the chosen molecular weights correspond to mean end-to-end distances in the range 28 Å < l < 2100 Å. Equation 1 is an empirical relationship, which corresponds approximately to that expected for a self-avoiding random walk of the polymer segments in the solution. We have recently shown that PEO segments bond directly to the clay surfaces, displacing water molecules from the interface,22 so when the segment distribution is affected by the surfaces it must be remembered that Table 1 gives only approximate values for the “size” of the polymer molecules. The results are presented in four subsections. First, we give a brief summary of the results previously reported in ref 2 at r ) 0.01. Second, we give new results obtained by neutron scattering at r ) 0.01. Third, neutron scattering results at r ) 0.1 are presented. Fourth, the determination of polymer adsorption isotherms at r ) 0.1 is presented. (1) Summary of Results of {c, v, M} Variations at r ) 0.01 in Reference 2. (a) At c ) 0.1 M, M ) 18 000 (bridging), v was varied as 0, 0.01, 0.02, 0.04, 0.08, 0.12, 0.16. The d-spacing was obtained as a function of v up to 0.08, and the curve was compared with the similar d versus p curve for the pure aqueous system. The well-defined d-spacing was found to disappear for volume fractions of 0.12 and above. We denote the volume fraction at which the Bragg peaks disappear as the crowding volume fraction vc and abbreviate the result to vc ) 0.10 ( 0.02. (b) At c ) 0.03 M, M ) 75 000 (bridging), v was varied as 0, 0.01, 0.02, 0.04. The d-spacing was obtained as a function of v up to 0.02 and also mapped onto the corresponding uniaxial stress case. In this case, no Bragg peaks were observed at v ) 0.04, giving vc ) 0.03 ( 0.01. (c) At c ) 0.1 M, v ) 0.02, M was varied at the seven values shown in Table 1. The well-defined d-spacing measured in each case was found to be equal to d0 for molecular weights up to 4000. For M ) 18 000 and above, the d-value was constant and equal to 80 Å. We denote the molecular weight for which the contraction first occurs as the bridging molecular weight Mb and abbreviate the result to 4000 < Mb < 18 000. (d) At c ) 0.03 M, v ) 0.02, M was varied at the seven values shown in Table 1. In this case, 18 000 < Mb < 75 000 was found for the bridging molecular weight. (2) Neutron Scattering Study of Further {c, v, M} Variations at r ) 0.01. The most urgent object of the new study was to test our value for the strength of bridging flocculation by experiments at c ) 0.01 M. In this case, the d-spacing of 330 Å gives rise to a first-order Bragg peak at Q = 0.02 Å-1, near the limit of the Q-range of LOQ. There is also greater sample-to-sample variability for the more dilute gels,16 so the results are more difficult to obtain, and some contradictions between different data sets were found. (a) At c ) 0.01 M, M ) 2 000 000 (bridging), v was varied between 0 and 0.02. Two experiments were performed 6 months apart on separate clay batches. In the first, N ) 3 where N is the number of samples studied. The volume fraction of polymer v was varied as 0.001, 0.002, 0.004, 0.008, 0.016, and it was confirmed that the batch gave the normal d-spacing of 330 Å for the pure aqueous system. The results are given in the second column of Table 2. In the second experiment, for N ) 4, the results are given (22) Swenson, J.; Smalley, M. V.; Hatharasinghe, H. L. M. J. Chem Phys. 1999, 110, 9750.

Table 2. d-Values Obtained as a Function of v for Two Separate Experiments at r ) 0.01, c ) 0.01 M, T ) 5 °C, M ) 2 000 000a v 0 (pure aqueous) 0.001 0.002 0.004 0.006 0.008 0.01 0.015 0.02 a

d (observed) (Å)

d (observed) (Å)

350, 380, 270 275, 325 350 295, 320, 270 280, 280, 280, 280 260, 260, 260, 260 235, 230, 230 235, 235, 235 220, 220, 220 crowded crowded, 195 crowded

d (average) (Å) 330 330 320 285 260 235 220 195 crowded

The average values are plotted in Figures 1c and 4.

in the third column of Table 2. The final column of Table 2 gives the average d-values obtained. The d-values given in Table 2 were calculated from the simple equation

d ) 2π/Qmax

(2)

applied to the position of the first-order (00l) Bragg reflection from the gel Qmax. Sharp diffraction effects were observed for all values of v, with the best statistics obtained at v ) 0.004 and v ) 0.008. The seven diffraction traces observed in each of these two cases are shown in parts a and b of Figure 1, respectively. The position is approximately constant at Qmax ) 0.022 Å-1 for v ) 0.004 and shifts to Qmax ) 0.027 Å-1 for v ) 0.008. In each case where second and third-order Bragg peaks were observed, these were linear in Q, justifying the simple analysis. The data points for uniaxial stress mapping obtained were v ) 0.004, d ) 285 Å and v ) 0.008, d )235 Å. Figure 1c shows the d-values obtained as a function of volume fraction at c ) 0.01 M. These data are absolutely vital for checking the mechanism of polymer bridging flocculation, as the theory has never been applied to c ) 0.01 M. (b) At c ) 0.01 M, v ) 0.004 (bridging), M was varied between 75 000 and 2 000 000. The sample-to-sample variability shown in Figure 1a,b was not always so good at c ) 0.01 M. In addition to the point at M ) 2 million described above, four samples were prepared at each of four molecular weights in the range between 75 000 and 300 000. In the new cases studied, only 7 of the 16 gels gave clear Bragg peaks, with only one good trace obtained at M ) 75 000 and M ) 300 000. Nevertheless, the d-values observed were all in the range between d ) 270 Å and d ) 300 Å, with an average value of d ) 285 Å. The data point d ) 285 Å for c ) 0.01 M and v ) 0.004 is an important one for testing the mapping of our data onto the uniaxial stress data,10 so it is necessary to prove that it holds for different molecular weights. It is clear that all of the polymers studied were in the bridging regime, so bridging occurs for all molecules with M greater than 75 000. Using Table 1, this corresponds to bridging for l g 330 Å. We denote the end-to-end distance for which the contraction first occurs as lb and write the result as 330 Å > lb. Because d0 ) 330 Å for the pure aqueous system at c ) 0.01 M, the new result substantiates our previous conclusion that bridging occurs for l values greater than or roughly equal to the normal d-value.2 We also attempted to measure the lower limit for the length of PEO molecule required to bridge at c ) 0.01 M. From our previous data (see Table 2), we had anticipated that the crowding volume fraction would be around v ) 0.015, so the experiment was performed at v ) 0.01 for polymers in the M range between 18 000 and 75 000. Unfortunately, we were unable to see any gel peaks for any of the 16 samples

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Figure 1. Parts a and b show sample-to-sample variability studies, the different symbols denoting patterns obtained from different pieces of gel under identical conditions. (a) Diffraction traces obtained from seven samples in the gel phase at T ) 5 °C. The PEO molecular weight was 2 million, and the concentration variables were fixed as r ) 0.01, c ) 0.01 M, v ) 0.004. (b) Diffraction traces obtained at v ) 0.008, with the other variables as defined in part a. (c) Plot of the d-values obtained as a function of v at M ) 2 million, r ) 0.01, c ) 0.01 M, T ) 5 °C.

Smalley et al.

studied and were therefore unable to determine the crossover. The only explanation we could give for the result was that the samples were in the crowded regime. This suggests that the crowding volume fraction is close to 1% in this case but fails to explain why we should have observed clear peaks at v ) 0.010 in the earlier experimental results collected in Table 2. Further experiments are planned to elucidate this point. What we can say is that the fraction is between 0.8% and 1.6% at c ) 0.01 M, or 0.008 < vc < 0.016. (c) Precise Determination of Crowding Volume Fraction at c ) 0.03 M and c ) 0.1 M. The crowding volume fractions vc at c ) 0.03 M and c ) 0.1 M were previously known to lie in the ranges between 0.02 and 0.04 and between 0.08 and 0.12, respectively. At c ) 0.03 M, a bridging polymer with M ) 75 000 was studied with v varied as 0.02, 0.025, 0.03, 0.035, 0.04. Figure 2a shows clearly that the welldefined traces disappear between 0.035 and 0.04, giving us vc ) 0.0375 ( 0.0025. Although this is a very precise result, we should note that in a more recent experiment23 we observed clear gel peaks for three out of four samples at v ) 0.04, with one crowded. This suggests that vc shows some batch-to-batch variability, as noted at c ) 0.01 M above. We are now able to extend our d versus v data at c ) 0.03 M from v ) 0.02 as in ref 2 up to v ) 0.035, providing another test of the bridging model. The results for the average d-values observed are given in Table 3. Figure 2b shows the crossover at c ) 0.1 M between v ) 0.08 and v ) 0.10, giving vc ) 0.09 ( 0.01. We have never seen any exceptions to this result. In this case, a bridging polymer with M ) 20 000 was used and v varied as 0.06, 0.08, 0.10, 0.012. At v ) 0.06, clear peaks were observed for d ) 52 Å, and at v ) 0.08, the d-value was found to be 48 Å, in agreement with the published results.2 (d) Precise Determination of Bridging Molecular Weights at c ) 0.03 M and c ) 0.1 M. The bridging molecular weights at c ) 0.03 M and c ) 0.1 M were previously known to lie in the ranges between 18 000 and 75 000 and between 4000 and 18 000, respectively. At c ) 0.03 M, v ) 0.02, two new M values were studied, M ) 30 000 and M ) 59 000. In both cases, all four samples gave clear gel peaks, with excellent sample-to-sample reproducibility. The bridging molecular weight Mb was therefore found to be in the range between 18 000 and 30 000, corresponding to mean end-to-end polymer distances between 140 and 190 Å. Because the normal d-value is 190 Å in this case, an approximate matching condition of d and l is again suggested. We also note that the d-values obtained were all in the range between 120 and 130 Å, in agreement with the published results at M ) 75 000.2 At c ) 0.1 M, v ) 0.02, M was varied as 7000, 9000, and 12 000. In this case, all 12 samples gave clear diffraction effects with d in the range between 70 and 75 Å, slightly lower than the value of 80 Å obtained at M ) 18 000 but within the errors due to sample variability. The crossover clearly occurs in the range 4000 < Mb < 7000, corresponding to mean endto-end polymer distances between 61 and 85 Å. Because the interlayer spacing in the pure aqueous system is 110 Å in this case, the crossover occurs between l values in the range from 1.3l to 1.8l. Our previous results have given the criterion d0 ≈ l for the crossover. The new result suggests that the polymer can bridge between the colloidal particles at separations considerably greater than the mean end-to-end distance. The mechanism proposed for polymer bridging flocculation was based on the observation that the d-value is (23) Swenson, J.; Smalley, M. V.; Hatharasinghe, H. L. M.; Fragneto, G. Langmuir, in press.


Langmuir, Vol. 17, No. 13, 2001 3805

Figure 3. Molecular weight variations at r ) 0.01, c ) 0.1 M, v ) 0.02 in the gel phase at T ) 5 °C. The open circles, triangles, and closed circles show the patterns for M ) 4000, M ) 20 000, and M ) 2 000 000, respectively.

Figure 2. (a) Example diffraction traces at r ) 0.01, c ) 0.03 M, T ) 5 °C for M ) 75 000 (bridging) with v varied in the range between 0.02 and 0.04. The crowding volume fraction vc is between v ) 0.035 and v ) 0.04. (b) Example diffraction traces at r ) 0.01, c ) 0.1 M, T ) 5 °C for M ) 20 000 (bridging) with v varied in the range between 0.06 and 0.12. In this case, vc ) 0.09 ( 0.01. Table 3. Average d-Values Obtained as a Function of v at r ) 0.01, c ) 0.03 M, T ) 5 °C, M ) 75 000a v 0.02 0.03 a

d (Å) 124 115

v

d (Å)

0.035 0.04

107 crowded

The results are plotted in Figure 4.

constant with respect to M (provided that the polymer is large enough to bridge) for a fixed set of the concentration variables {r, c, v}.2 At c ) 0.1 M, because bridging occurs for M > 7000 or l > 85 Å, we can test the function d versus M over a factor ×300 in molecular weight or a factor ×20 in the mean end-to-end distance l. The volume fraction was fixed at v ) 0.02, and M was varied between 4000 and 2 000 000. Selected results from this study, which gave

excellent sample-to-sample reproducibility in the diffraction traces, are shown in Figure 3. As noted previously, there is a dramatic change in behavior between the two lower molecular weights, the M ) 4000 polymer being nonbridging and that at M ) 20 000 being bridging. In all three cases, clear diffraction effects are observed for d-values of 140 Å (M ) 4000) and d ) 75 Å for the bridging polymers. That the d-value remains constant for the two higher molecular weight polymers, for which M differs by a factor of 100, is clear. (e) Summary of Results of {c, v, M} Variations at r ) 0.01. The new data for the d versus v variations at c ) 0.01 M and c ) 0.03 M have been given in Tables 2 and 3, respectively. They have been plotted together with the previous data at c ) 0.03 M and c ) 0.1 M in Figure 4. To check that the new sample batches gave comparable d-values to those obtained before, control experiments were performed at c ) 0.1 M, v ) 0.02 and v ) 0.08. The average values were 75 and 48 Å, respectively. These values are as close to the previously reported 80 and 50 Å as can be expected in clay science. Likewise, control experiments on the pure aqueous samples gave the usual values of d0 ) 120 Å at c ) 0.1 M and d0 ) 330 Å at c ) 0.01 M. At c ) 0.01 M, the occurrence of crowding at the low volume fraction vc ≈ 0.01 means that our test curve for uniaxial stress mapping at the new salt concentration could only be measured over a limited range, but reliable data points were obtained for v ) 0.004, d ) 285 Å and v ) 0.008, d ) 235 Å. The results for the crowding volume fraction at c ) 0.01 M were less satisfactory, with different values of vc ) 0.01 and vc ) 0.015 obtained in different experiments. Similarly, at c ) 0.03 M, different results of vc ) 0.035 and vc ) 0.04 have been obtained. The most reliable data point is vc ) 0.09 ( 0.01 at c ) 0.1 M. The disappearance of the Bragg peaks at volume fractions of around 1%, 4%, and 9% for the cases d0 = 330 Å, d0 = 190 Å, and d0 = 120 Å, respectively, strongly suggests an important role for the proportion of the surface area of the clay platelets occupied by adsorbed polymer segments, as discussed below. The results for vc determined by these experiments are given

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Smalley et al.

Figure 4. Plot of d-values obtained as a function of v at r ) 0.01, T ) 5 °C for the three salt concentrations c ) 0.01 M, c ) 0.03 M, and c ) 0.1 M. The open symbols correspond to the published results,2 and the closed symbols are the new results. The lines between the points are drawn as a guide for the eye. Table 4. Crowding Volume Fractions vc, Crossover Molecular Weights Mb, and Crossover End-to-End Distances lb as Functions of the Salt Concentration c and the Pure Aqueous d-Value d0 c (M) d0 - 10 (Å) 0.01 0.03 0.1

320 180 110

vc

Mb

lb (Å)

0.01 Mb < 75 000 lb < 330 0.04 18 000 < Mb < 30 000 140 < lb < 190 0.09 4 000 < Mb < 7 000 60 < lb < 85

in Table 4. Table 4 also shows the results obtained for Mb and lb. It is instructive to divide lb by the normal aqueous d-value in each case. For c ) 0.01 M, we have determined lb < 1.0d0 (but need further experiments on smaller polymers to define the lower bound), and at c ) 0.03 M, the range was determined as 0.8d0 < lb < 1.0d0. Combining these results with the range 0.6d0< lb < 0.8d0 obtained at c ) 0.1 M, it seems likely that lb = 0.8d0 represents the limiting end-to-end distance for bridging to occur. The results are mutually consistent if the crossover occurs at d0 = 1.3l. (3) Neutron Scattering Experiments at Higher Sol Concentrations. The main aim of these experiments was to connect the results of the neutron scattering experiments at r ) 0.01 with the polymer adsorption isotherm results described below. The bulk of the structural data on {c, v, M} variations had to be taken at r ) 0.01 in order to make accurate comparisons between the systems with and without PEO.13,16 However, in determining the distribution of polymer molecules between the clay gel and the supernatant fluid, greater accuracy can be obtained by performing the analyses at r ) 0.1 because in the main case studied, at r ) 0.01, the gel occupies a small fraction of the condensed matter system and any redistribution of polymer molecules between the two phases is difficult to measure. At r ) 0.1, the gel and supernatant fluid have roughly equal volumes and fractionation of PEO between the two phases was measured by GPC on the supernatant fluids, as described below. The phase space of the variables {r, c, v, M} is too vast to investigate exhaustively, so our comparisons were made

Figure 5. Typical diffraction traces observed at higher volume fractions, with c ) 0.1 M, M ) 18 000, T ) 5 °C: (a) r ) 0.1 and (b) r ) 0.3.

with c ) 0.1 M and M ) 18 000 (bridging). Two sol concentrations were studied (r ) 0.1 and r ) 0.3), and v was varied in the range between 0 and 0.12. The results of the control study on the pure aqueous system at higher sol concentrations are described in detail elsewhere.17 For the batch used in these experiments, the control on the pure aqueous v ) 0 system gave d0 ) 130 Å at r ) 0.01 (slightly above the average value), d0 ) 110 Å at r ) 0.1, and d0 ) 75 Å at r ) 0.3. Typical results of the effect of PEO addition are shown in Figure 5, and the average d-values obtained from the traces are given in Table 5. The contraction of the gel phase at r ) 0.1 is similar to that observed at r ) 0.01, with the overall d-spacings lower. This result is expected because salt trapped in the crystals leaches out to give a higher background salt concentration at the higher sol concentrations9,16 and because of the salt fractionation effect.4,16 It is noteworthy that the Bragg peak disappears at a similar polymer volume fraction, v = 0.10, at r ) 0.1 as it does at r ) 0.01. The crowding volume fraction therefore seems to be fairly


Langmuir, Vol. 17, No. 13, 2001 3807

Table 5. Average d-Values Obtained at Higher Sol Concentrations, with c ) 0.1 M, M ) 18 000, T ) 5 °C v

d (average) (Å) r ) 0.1


0.01 0.02 0.04

77 69 58

70 52

v 0.08 0.12 0.16



46 crowded crowded

insensitive with respect to r in the range of mapping with the isotherm results. At r ) 0.3, our grid of points was too coarse to be able to able to say more than vc ) 0.10 ( 0.06, a compatible result. In ref 9, preliminary data were given on the pure aqueous system at r = 0.4, for which the clay can soak up all of the available salt solution, resulting in a system with no supernatant fluid. Our new results at r ) 0.3 are right on the edge of the one-phase and twophase regions of the colloidal gels, because d ) 70 Å corresponds to an expansion by 31/2 times of the vermiculite crystal. Overall, the results seem to point clearly to the three concentration variables all having a similar effect on the d-value for a PEO molecule with a molecular weight in the bridging range. The additions of all three components bring about a contraction of the clay gel. The important comparison is between the r ) 0.01 and r ) 0.1 results, which are plotted in Figure 6a. This shows that the curves of d versus v are approximately parallel in the two cases, showing that the same mechanism is at work. The magnitude of the contraction is also what is expected from the effect of increasing r at fixed c in the three-component clay-salt-water system.6,17,19 It is also instructive to plot d as a function of r at a fixed polymer volume fraction v, as shown in Figure 6b. The data for Figure 6b have been taken from Table 5 and ref 13. In this case, the curvature of the plot is steeper for the pure aqueous gels. The polymer clearly exerts a drawing force throughout the two-phase region. (4) Analysis of Supernatant Fluids at r ) 0.1. At the salt concentration for which most structural data are available,10-12 c ) 0.1 M, the gel expands to between 2.5 and 6 times the volume of the original vermiculite crystal (d ) 19.4 Å), so at r ) 0.01 the volume of the gel phase at equilibrium is only a few percent of the total volume, with over 90% occupied by the supernatant fluid. This made it difficult to analyze redistribution of polymer molecules between the two phases. However, for r ) 0.1, similar expansions lead to equilibrium conditions in which the gel occupies 25-60% of the total volume, permitting accurate analyses. The main aim was to determine the distribution of PEO molecules between the gel and the supernatant fluid at r ) 0.1, c ) 0.1 M, T ) 5 °C for M ) 18 000 (bridging) and polymer volume fractions in the range between 0 and 0.12. The corresponding neutron diffraction traces are shown in Figure 5a. In comparing these structural analyses with an independent analysis of the concentration of the PEO in the supernatant fluid, we established the following protocol in preparing the samples. (i) The salt was always dissolved first into the heavy water. We use molarity for the salt concentration variable c and note that at the highest concentration studied, c ) 0.1 M, the volume fraction of salt is approximately 1%. (ii) The polymer was then dissolved into the solution such that, for example, in preparing a v ) 0.12 sample 12 cm3 of PEO was added to 100 cm3 of the salt solution. The actual volume fraction of polymer in the standard solution vadd was v/(1 + v), or 0.11 in this case. (iii) The solution containing the polymer and salt was added to the vermiculite clay crystals (d ) 1.94 nm) such

Figure 6. (a) Plot of d-values as a function of v for the two sol concentrations r ) 0.01 (upper curve) and r ) 0.1 (lower curve). (b) Plot of d-values as a function of r for the two volume fractions v ) 0 (upper curve) and v ) 0.01 (lower curve). The lines between the points are drawn as a guide for the eye.

that in preparing an r ) 0.1 sample 100 cm3 of solution was added to 10 cm3 of clay. The global volume fraction of polymer in the condensed matter system vglobal was vadd/ (1 + r), or

vglobal ) v/{(1 + v)(1 + r)}

(3)

with the concentration variables v and r experimentally controlled as above. We note that the global volume includes the volume of the clay platelets (d ≈ 10 Å), which may be regarded as inaccessible to the polymers. In keeping with eq 3, the global volume fraction rglobal of clay in the system is given by

rglobal ) r/(1 + r) or rglobal ) 0.091 for the unswollen crystals.

(4)

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Smalley et al.

Table 6. Adsorption Isotherm Results at r ) 0.1, c ) 0.1 M, M ) 18 000 (bridging), T ) 5 °Ca v

vex

rgel

vi

f

(a) First Set of GPC Results 0.01 0.02 0.04 0.08 0.12

0.012 0.024 0.044 0.081 0.106

0.36 0.32 0.28 0.21 0.21

0.0046 0.0061 0.014 0.018 0.081

0.46 0.31 0.35 0.22 0.67

v

vex

rgel

vi

f

(b) Second Set of GPC Results 0.06 0.07 0.08 0.09 0.10 0.11 0.12

0.059 0.067 0.077 0.083 0.089 0.099 0.105

0.24 0.22 0.21 0.21 0.21 0.21 0.21

0.034 0.041 0.038 0.056 0.074 0.075 0.086

0.56 0.59 0.47 0.62 0.74 0.68 0.71

a The volume fraction v is the experimentally controlled variable, vex is the measured volume fraction of PEO in the supernatant fluid, rgel is the volume fraction occupied by the gel at equilibrium, vi is the calculated volume fraction of PEO in the fluid inside the gel, and f is the fractionation factor defined by f ) vi/v. The average value of the fractionation factor f is equal to 0.45 in the bridging regime.

In analyzing the supernatant fluid, we measure the volume fraction vex in the fluid external to the clay gel. The results of the GPC analysis at c ) 0.1 M, M ) 18 000 (bridging) are given in Table 6. To determine the volume fraction of polymer vgel inside the gel from vglobal and vex we need to know the relative volume occupied by the two phases. Because of errors due to adhesion of the sticky fluids to the outside of the clay gel, it is better to use the neutron data to determine the gel volume because previous studies have shown the microscopic and macroscopic expansions to match well.8-17 If we let x be the factor by which the crystals have expanded, then x ) dgel (Å)/19.4 and the final volume fraction of clay in the condensed matter system is rgel ) xrglobal ) 0.091x. The results are given in the third column of Table 6. Although it was not possible to obtain the d-values for v g 0.09 from the neutron data, the macroscopic expansions observed in these cases were similar to that at v ) 0.08, so we may estimate that rgel ) 0.21 in these cases also. The volume fraction of polymer in the gel phase is calculated as

vgel ) {vglobal - vex(1 - rgel)}/rgel

(5)

and the volume fraction of polymer vi in the fluid inside the gel is calculated from this quantity by assuming that the clay platelets exclude PEO molecules. The results are given in the fourth column of Table 6. Finally, the fifth column of Table 6 gives the polymer fractionation factor f defined by f ) vi/v. It is clear that f is roughly constant and equal to 0.45 for the lower polymer volume fractions. In the following discussion, this will be written as vi ) 0.45v, relating the volume fraction of polymer inside the gel to the experimentally controlled v. Next, results for a nonbridging polymer at c ) 0.1 M, M ) 4000 provide a comparison with the more widely studied case of polymer adsorption at a single interface.24 The results are given in Table 7. In this case, it is clear that the fractionation factor f is approximately constant, with an average value of 0.95. This means that the polymer is roughly equally divided between the two phases. The usual way of representing polymer adsorption onto clay surfaces is to plot an isotherm showing the amount of polymer adsorbed in grams per gram of clay as a function of the equilibrium concentration of polymer in g cm-3. We know from the density that 1 g of clay occupies 0.54 cm3, so according to our protocol an r ) 0.1 sample is prepared by adding 5.4 cm3 of polymer solution. Because the actual (24) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993.

Table 7. Adsorption Isotherm Results at r ) 0.1, c ) 0.1 M, M ) 4000 (Nonbridging), T ) 5 °Ca v

vex

rgel

vi

f

v

vex

rgel

vi

f

0.01 0.02 0.03 0.04

0.0090 0.018 0.027 0.036

0.56 0.56 0.56 0.56

0.0098 0.019 0.029 0.037

0.98 0.95 0.95 0.93

0.05 0.06 0.07 0.08

0.045 0.055 0.061 0.067

0.56 0.56 0.56 0.56

0.046 0.053 0.064 0.074

0.92 0.89 0.91 0.92

a The volume fraction v is the experimentally controlled variable, vex is the measured volume fraction of PEO in the supernatant fluid, rgel is the volume fraction occupied by the gel at equilibrium, vi is the calculated volume fraction of PEO in the fluid inside the gel, and f is the fractionation factor defined by f ) vi/v.

Table 8. Adsorption Isotherm Results at r ) 0.1, c ) 0.1 M, T ) 5 °Ca mass adsorbed (g) v

mass added (g)

M ) 18000

M ) 4000

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

0.060 0.12 0.18 0.23 0.29 0.35 0.40 0.45 0.50 0.55 0.60 0.66

0.0095 0.011

0.034 0.067 0.099 0.13 0.16 0.19 0.22 0.26

0.021 0.044 0.048 0.030 0.060 0.079 0.081 0.092

a The second column gives the mass of PEO added per gram of clay, and the third and fourth columns give the mass of polymer adsorbed per gram of clay at M ) 18 000 (bridging) and M ) 4000 (nonbridging), respectively. The values at v ) 0.04 and v ) 0.08 for M ) 18 000 are average results from Table 6a,b. The results are plotted in Figure 7.

concentration va of polymer in this fluid is given by va ) v/(1 + v), the total volume of polymer added in such an experiment is 5.4v/(1 + v). Because the density of PEO is 1.13 g cm-3, the mass of polymer added per gram of clay is 6.1v/(1 + v) grams, as shown in column 2 of Table 8. The third column of Table 8 gives the mass of polymer adsorbed (g) per gram of clay for the bridging polymer (M ) 18 000), and the fourth column gives the same quantity for the nonbridging polymer (M ) 4000). The results have been plotted in Figure 7. For both molecular weights, the results seem to be linear to the limit of experimental accuracy, confirming the constant fractionation of polymer in both cases. The much steeper gradient of the M ) 4000 data reflects both the fact that f is approximately twice as great for the nonbridging PEO and the fact that the clay gel occupies a much larger proportion of the condensed matter system when the polymer is unable to bridge. Finally, because the accuracy with which we can determine the strength of the bridging force depends on the accuracy of f, three independent chemical analyses were performed in the bridging regime at r ) 0.1, c ) 0.1 M, M ) 18 000, at the v-values 0.01, 0.04, and 0.08. The results of the phosphomolybdic acid titrations were vex ) 0.012 at v ) 0.01, vex ) 0.044 at v ) 0.04, and vex ) 0.076 at v ) 0.08, in close agreement with the GPC results given in Table 6. The fractionation factor was therefore confirmed to be f ) 0.45 ( 0.10 in the bridging regime. Discussion The main aim of the experiments was to test the newly proposed model of bridging flocculation,2 which we review here. Because Crawford et al. studied the contraction of the interlayer spacing as a function of uniaxial stress for


Langmuir, Vol. 17, No. 13, 2001 3809

Na )

(d - 10)Nv 6.8 × 107(d - 10)vi ) 10 000 M

(9)

If we combine eqs 7 and 9, the total number of polymer bridges N per unit clay area (µm2) is given by

N ) NaNp )

Figure 7. Adsorption isotherm data at r ) 0.1, c ) 0.1 M. The closed circles show the amount of PEO taken up by the gel for M ) 18 000 (bridging), and the open circles are for M ) 4000 (nonbridging).

the same system without any added polymer,10 we are able to convert the observed d-spacings to effective uniaxial pressures caused by the bridging polymers. If we assume that we have one polymer bridge when the end-to-end polymer distance l (calculated according to eq 1) exactly matches the d-spacing with the thickness (∼10 Å) of the clay platelets subtracted, we are able to calculate the total number of bridges per unit area of the clay platelets from the observed d-spacings and polymer volume fractions, as follows. One polymer bridge is obtained when

M0.57 min d - 10 ≈ l ≈ 1.84

(6)

where Mmin stands for the lowest molecular weight polymer that is able to bridge. For larger polymer molecules, the number of bridges is proportional to the molecular weight of the polymer, as indicated by the molecular weight independent d-spacing. If the molecular weight of the polymer is M, then the number Np of bridges per polymer chain is given by

Np )

M M ) Mmin [1.84(d - 10)]1/0.57

(7)

The average number of polymer chains per unit volume solution Nv (µm3) can be obtained from the density F ) 1.13 g/cm3 of PEO and the polymer volume fraction vi inside the clay gel through the relation

6.8 × 1011vi Nv ) 12 ) M 10 M FNAvi

(8)

where NA is Avogadro’s number (6.02 × 1023). If the solution inside the gel initially occupied one unit volume (µm3), then the total available area (µm2) of the clay platelets is given by 20 000/(d0 - 10), where d0 stands for the initial d-spacing, for adsorption onto a single platelet. However, because the polymers bridge between two clay platelets, the available area of each platelet is 10 000/(d0 - 10). The average number of polymer chains per unit clay area Na (µm2) can therefore be expressed as

2.3 × 107(d0 - 10)vi (d0 - 10)1/0.57

(10)

In Figure 8, we show the relation between the calculated number of bridges/µm2 and the measured uniaxial pressures,10 corresponding to the observed d-spacings. It can be seen that the effective uniaxial pressure, caused by the bridging polymers, increases linearly (within the experimental errors) with the number of polymer bridges. This means that the average drawing force per polymer bridge must be effectively the same for all the samples studied, independent of the salt concentration, polymer volume fraction, and molecular weight. Using the data points given in Figure 8, we can calculate the average drawing force per polymer bridge to be approximately 0.6 pN, which corresponds to the straight line shown in Figure 8. The presence of a constant force is consistent with the assumptions made in deriving eqs 6-10. In Figure 8a, we have plotted the data on a log-log scale to emphasize that the new data points at c ) 0.01 M and c ) 0.03 M, shown by the open symbols, fall on exactly the same line as the previously obtained data points at c ) 0.03 M and c ) 0.1 M, shown by the closed symbols. The linearity of the plot can now be seen to hold over 3 orders of magnitude in the effective drawing force exerted by the polymer chains. This confirms the essential feature of the bridging mechanism proposed in ref 2. To demonstrate the linear relation between the effective uniaxial pressure p and N more convincingly, the data have been replotted on a linear scale in Figure 8b. This has the demerit of crushing the new data points at c ) 0.01 M together close to the origin but the advantage of showing that the linearity is not an artifact of using a logarithmic scale. The qualitative features of the proposed mechanism are therefore confirmed. However, our new value of 0.6 pN for the drawing force per polymer bridge differs very significantly from the value of 1.4 pN given in ref 2. There are three reasons for this. First, when we originally discovered the mechanism, we expressed the effective size of the polymer molecules as 2Rg, where Rg is the radius of gyration of the PEO in solution. Our idea was to work with an effective “diameter of gyration” of the molecules, related directly to the physically observable Rg. However, we amended our expression for the size to the mean end-to-end distance l, as shown by eqs 6-10, which are identical to eqs 3-7 in ref 2. This in itself was unexceptionable, but when we changed from 2Rg to l ) (x6)Rg we neglected to change the value given by the calculation using 2Rg. This means that our estimate of the number of bridges was too low by a factor of (x6/2)1.75 ) 1.42, as is clear from inspection of eqs 6 and 7. This error alone would have meant decreasing the value of the bridging force from 1.4 to 1.0 pN. This serves to emphasize that the absolute value of the force is very sensitive to the exact nature of the condition d0 ≈ l, which we have retained in the present calculation. If the condition really turns out to be d0 ≈ 1.3l, as suggested by the new results at c ) 0.1 M described here, the estimated force would be lower again, by a further factor of 1.31.75 ) 1.58. This would imply a value of 0.4 pN per polymer bridge. It is obvious from these considerations

3810


Figure 8. Effective uniaxial pressure as a function of the calculated number of bridges. The circles, triangles, and diamonds correspond to c ) 0.01, 0.03, and 0.1 M, respectively, with open symbols showing the new data and closed symbols showing data given in ref 2. The log-log plot in (a) shows that the linear relation between p and N extends over 3 orders of magnitude, and the linear plot in (b) emphasizes the direct proportionality of p and N. The straight lines correspond to an effective uniaxial force of 0.6 pN per bridge.

that we cannot give the force to better than one significant figure. It is certainly of the order of 1 pN. The second reason our new value for the strength of the bridging force is lower than that given in ref 2 is more subtle and demands an intimate knowledge of ref 10. In determining the effect of uniaxial stress on these fragile gels by compressing them with a quartz plate in the small cell appropriate for neutron diffraction experiments, substantial sample-to-sample variability was observed but clear global trends emerged. The global outcome of many experiments was that the intercepts of the pressuredistance curves pointed decisively to a constant surface potential of ψo ) 70 mV and that the slopes of the curves κin were described by the relation κind0 ) 9, where d0 is the zero stress d-spacing. These relations have been used in the present mapping of the contractions, whereas in ref 2 we used the global relation ψo ) 70 mV together with the values for κin determined from the individual ln p versus d curves obtained at c ) 0.03 M and c ) 0.1 M. This

Smalley et al.

caused us to overestimate the magnitude of the bridging force by a factor of approximately 1.8, this time by overestimating the pressure necessary to bring about a given contraction. This error in compound with the first would have meant decreasing the value of the bridging force from 1.4 to 0.55 pN. This serves to emphasize that the absolute value of the force is very sensitive to the exact form of the pressure-distance curves, which are difficult to measure. The third reason our new value for the strength of the bridging force is different from that given in ref 2 is a trivial one. Our new value for the polymer fractionation factor f is 10% lower than that of f ) 0.5 obtained from our preliminary isotherm data. Because we have 10% fewer bridges than previously estimated, our value for the force per bridge is correspondingly 10% higher, giving us a final value of 0.6 pN. This serves to emphasize that the absolute value of the force is also sensitive to the exact form of the isotherms and to the uncertainties of mapping f between different sol and salt concentrations. This all may seem to make our final estimate of the strength of the bridges rather vague, but in a multicomponent system with the sample-to-sample variability inherent in colloid science we believe it may not be possible to do better than limit the range to 0.6 ( 0.2 pN. Although our value for the force lies necessarily within a fairly wide range, the resolution of the measurement is very high. Going back over twenty years, there has been considerable advancement in the understanding of the effect of polymers in modifying surface interactions in colloidal dispersions. In particular, the forces between surfaces bearing adsorbed PEO layers have been investigated using the mica surface force apparatus.25-27 It is difficult to compare the results of our experiments with these for three reasons. First, the weakest forces measured in these experiments were of the order of 10 nN, a force resolution 4 orders of magnitude lower than ours. Second, the mica force balance measures the complete forcedistance curve, whereas we are sensitive only to the force at the equilibrium separation of the surfaces. Third, and probably most important, the PEO is introduced into the system in different ways in the two types of experiment. In the force balance, the mica surfaces are incubated in the presence of polymer for about a day at a macroscopic separation, so that adsorbed layers are formed separately on each surface prior to their approach. In the work described here, the polymer penetrates the clay gel as the interlayer spacing expands from 1 nm, so the bridging configuration is achieved before the surfaces become crowded with polymer chains. There are also some uncertainties in our interpretation of the crowding volume fraction vc. To understand the abrupt disappearance of the well-defined Bragg peaks as v is increased, it is necessary to have a model of the polymer segments. The PEO chain is built up by covalently bonded CH2-CH2-O monomers. Using the known bond lengths and bond angles,28,29 the length of each monomer in a zigzag trans conformation can be estimated to be about 3.8 Å. Because the density of PEO is 1.13 g/cm3, the polymer chain with its excluded volume can be approximated by a coil with a diameter of approximately 4.7 Å. We therefore anticipate that the area of clay surface (25) Klein, J.; Luckham, P. F. Nature 1982, 300, 429. (26) Klein, J.; Luckham, P. F. Nature 1984, 308, 836. (27) Luckham, P. F.; Klein, J. J. Chem. Soc., Faraday Trans. 1990, 86, 1363. (28) Rosi-Schwartz, B.; Mitchell, G. R. Polymer 1994, 35, 5398. (29) Carlsson, P.; Swenson, J.; Bo¨rjesson, L.; Torell, L. M.; McGreevy, R. L.; Howells, W. S. J. Chem. Phys. 1998, 109, 8719.


occupied by a monomer unit will be approximately 3.8 × 4.7 Å2 ≈ 18 Å2. Now let us consider the 1 µm3 box inside the clay gel introduced in the calculation of the bridging force, taking as an example the case r ) 0.1, c ) 0.1 M, v ) 0.08, at the edge of the crowding regime. Using the experimentally determined results d ) 44 Å and f ) 0.45, the number of monomer units inside 1 µm3 of gel is 5.6 × 108 and the available area of clay surfaces is 450 µm2. To calculate the surface area occupied by the polymer segments, we need independent data on the polymer segment density profile. This has been obtained from wideangle neutron diffraction experiments,22 which gave an approximate picture of the PEO chain inside the gel that has one-quarter of the segments stuck to each of the bridged layers, with the remaining one-half of the segments in the fluid in the interlayer region. We therefore have 2.8 × 108 segments stuck to the surfaces, occupying 2.8 × 108 × 1.8 × 10-7 µm2 ≈ 50 µm2. At crowding, approximately 10% of the vermiculite surfaces are therefore covered by PEO segments. We can obtain an independent estimate of the surface coverage at crowding from the isotherm data displayed in Figure 7, which show that for the bridged polymer at v ) 0.08, approximately 0.05 g of PEO is adsorbed per gram of clay. It is known that the butylammonium vermiculite used in the present studies has a surface area of approximately 500 m2 per gram,12 so we have 0.1 mg of PEO adsorbed per square meter of surface. A general rule of thumb for monolayer adsorption of homopolymers is 1 mg per square meter,24 so the macroscopic property also corresponds to 10% coverage. Although the figure of 10% coverage is a rough one, it is clear that the surface is still very open when the crowding transition occurs. One possible interpretation of this result is that we have found evidence for the hypothesis proposed by de Gennes that polymers cannot entangle in two dimensions.30 De Gennes based this idea on analytical calculations, and it has been substantiated by recent numerical simulations.31 In refs 30 and 31, “twodimensional” really refers to a confined geometry in which the confinement distance d is shorter than the mean endto-end distance l. This is clearly the case studied here. In these circumstances, the large part of each polymer chain that remains in a Gaussian-like distribution in the middle between the clay layers must exclude both other parts of the same chain between different bridges and other polymer molecules, leading to a wide exclusion zone about each bridge. The idea is illustrated schematically in Figure 9. Although Figure 9 is only a sketch, we draw attention to several features that are consistent with all the results of the present paper and refs 2 and 22. First, the PEO segments do zigzag across the surface, in the layer directly adjacent to the surface oxygens of the vermiculite layers. Second, there is some kind of random coil, comprising approximately half the segments, in the middle between the vermiculite layers.22 Third, the region between the surface and the coil is relatively depleted of polymer segments, as represented by the stringy bits connecting these two parts. Fourth, the only difference between bridging polymers of (a) molecular weights just sufficient to bridge and (b) much higher molecular weights, under identical {c, v, M} conditions, is that in the former case there is one bridge per molecule and in the latter case there are many bridges per molecule, with the number of (30) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (31) Shannon, S. R.; Choy, T. C. Phys. Rev. Lett. 1997, 79, 1455.

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Figure 9. Schematic illustration of the model for the bridging polymers: (a) shows the case for a polymer that is just large enough to form a single bridge, and (b) is for a much larger polymer, with multiple bridges. The drawing force between the vermiculite plates is the same in both cases.

bridges per unit amount of polymer the same in both cases. Fifth, our final speculative point about crowding is represented by showing that the “trains” on the surface are well separated from each other because of the nonentanglement of the chains. We cannot substantiate the final point in advance of isotherm data and segment density profile determinations at lower salt concentrations. The mechanism of polymer bridging flocculation can therefore be outlined as follows. The polymer chains diffuse between the clay platelets and each chain adsorbs at both surfaces, provided that the mean end-to-end distance of the polymer is approximately equal to or larger than the d-spacing. At this point, it is likely that each chain is like a random coil with many bends and that the distance along a chain between the two adsorbed points is much larger than the actual d-spacing. Thereafter, larger and larger parts of the polymer chains will adsorb at the initial clay-polymer contact points, so the length of the adsorbed trains increases at the same time as the chains stretch out between the two clay surfaces. This will continue until equilibrium occurs between the drawing force, arising from the segmental motions of a “stretched” polymer chain, and the bonding force to the clay surfaces. Thus, the bonding force determines the strength of each polymer bridge, which then explains why we obtain a constant effective drawing force per bridge. This conclusion is in good agreement with a recent lattice Monte Carlo simulation of a bridging polymer.32 The widely invoked model to explain why interparticle separations between colloidal particles decrease when a large polymer is introduced into the system is known as depletion flocculation.33,34 This is basically an equilibrium osmotic exclusion model, in which the reduction in d-spacing is driven by an osmotic pressure of polymer molecules excluded from the interparticle region. We have (32) Jiminez, J.; de Joannis, J.; Bitsainis, I.; Rajagopalan, R. Macromolecules 2000, 33, 7157.

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indeed discovered that the PEO molecules are partially excluded from the gel, but the constancy of the d-spacing with respect to molecular weight at a fixed volume fraction rules out the depletion flocculation mechanism. This is because osmotic pressure is a colligative property, and the change in the number of polymer molecules at fixed v and variable M would give rise to a varying contraction. Furthermore, there is no obvious way in which the crowding transition could come about via depletion flocculation. Instead, the bridging mechanism gives a coherent explanation of all the available data.

in which equilibrium occurs between an entropy-driven drawing force and a bonding force to the clay platelets. The force is proportional to the number of bound segments, and the strength per polymer bridge is constant over 3 orders of magnitude in the drawing force. In ref 2, an erroneously high value of 1.4 pN was given for the strength of the polymer bridges. The actual value of the strength of the polymer bridges in the vermiculite-PEO system has been found to be 0.6 pN. The calculation of the absolute value of the drawing force constitutes a new challenge in colloid and polymer science.

Conclusion The main outcome of the study has been the validation of the new mechanism for polymer bridging flocculation,

Acknowledgment. J.S. thanks the Swedish Natural Science Research Council for their support, and H.L.M.H. thanks Unilever PLC for a studentship in support of the work. This work was financially supported by Unilever PLC. We thank Dr. Gordon Welch of Unilever PLC for his help in coordinating the project.

(33) Dickinson, E. J. Chem. Soc., Faraday Trans. 1995, 91, 4413. (34) Jenkins, P.; Snowden, M. Adv. Colloid Interface Sci. 1996, 68, 57.

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Bridging Flocculation in VermiculiteâPEO Mixtures - American

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