PHYSICAL REVIEW E VOLUME 54, NUMBER 6 DECEMBER 1996

Depletion interactions in colloid-polymer mixtures

X. Ye, T. Narayanan, and P. Tong* Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078

J. S. Huang and M. Y. Lin Exxon Research and Engineering Company, Route 22 East, Annandale, New Jersey 08801

B. L. Carvalho Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

L. J. Fetters Exxon Research and Engineering Company, Route 22 East, Annandale, New Jersey 08801 ͑Received 10 May 1996͒ We present a neutron-scattering study of depletion interactions in a mixture of a hard-sphere-like colloid and a nonadsorbing polymer. By matching the scattering length density of the solvent with that of the polymer, we measured the partial structure factor Sc(Q) for the colloidal particles. It is found that the measured Sc(Q) for different colloid and polymer concentrations can be well described by an effective interaction potential U(r) for the polymer-induced depletion attraction between the colloidal particles. The magnitude of the attraction is found to increase linearly with the polymer concentration, but it levels off at higher polymer concentrations. This reduction in the depletion attraction presumably arises from the polymer-polymer interactions. The ex- periment demonstrates the effectiveness of using a nonadsorbing polymer to control the magnitude as well as the range of the interaction between the colloidal particles. ͓S1063-651X͑96͒10911-9͔

PACS number͑s͒: 82.70.Dd, 61.12.Ex, 65.50.ϩm, 61.25.Hq

I. INTRODUCTION faces through a loss of conformational entropy. Colloidal surfaces are then maintained at separations large enough to Microscopic interactions between colloidal particles in damp any attractions due to the depletion effect or London– polymer solutions control the phase stability of many van der Waals force and the colloidal suspension is stabilized colloid-polymer mixtures, which are directly of interest to ͓5,6͔. industries. Lubricating oils and paint are examples of the In addition to its important practical applications, the study of the colloid-polymer mixtures is also of fundamental colloid-polymer mixtures in which phase stability is desired. interest in statistical mechanics. The recent theoretical calcu- The interaction between the colloidal particles can be ex- lations ͓7–10͔ for the entropy-driven phase separation in bi- pressed in terms of an effective potential U(r), which is the nary mixtures of hard spheres have stimulated considerable work required to bring two colloidal particles from infinity to experimental efforts to study the phase behavior of various a distance r in a given polymer solution ͓1͔. In the study of binary mixtures, such as liquid emulsions ͓11͔, binary colloi- the interactions in colloid-polymer mixtures, it is important dal mixtures ͓12–15͔, colloid-surfactant mixtures ͓16͔, bi- to distinguish between polymers that are adsorbed on the nary emulsions ͓17͔, and mixtures of colloids with nonad- colloidal surfaces and those that are free in solutions, be- sorbing polymers ͓18͔. While it is successful in explaining cause the two situations usually lead to qualitatively different the phase behavior of these mixtures, the binary hard-sphere effects. In a mixture of a colloid and a nonadsorbing poly- model is nevertheless a highly idealized theoretical model. In mer, the potential U(r) can develop an attractive well be- reality, there are many complicated, non-hard-sphere colloi- cause of the depletion effect ͓1,2͔ in that the polymer chains dal mixtures. Even in the study of the model colloidal mix- are expelled from the region between two colloidal particles tures, the particles used in the experiments are stabilized ei- when their surface separation becomes smaller than the size ther by surface charges or by a layer of surfactant molecules of the polymer chains. The exclusion of polymer molecules and hence the interparticle potential has a weak repulsive from the space between the colloidal particles leads to an tail. The interaction potentials for the surfactant aggregates unbalanced osmotic pressure difference pushing the colloidal and polymer molecules are probably much softer than that of particles together, which results in an effective attraction be- the hard spheres. Measurements of phase behavior and other tween the two colloidal particles. If the attraction is large thermodynamic properties are useful in studies of macro- enough, phase separation occurs in the colloid-polymer mix- scopic properties of the binary mixtures, but they are much ture ͓3,4͔. In the case of adsorption, the adsorbed polymer less sensitive to the details of the molecular interactions in chains, in a good solvent, resist the approach of other sur- the system. Microscopic measurements, such as radiation- scattering experiments, therefore are needed to directly probe the molecular interactions in the mixtures. With this knowl- *Author to whom correspondence should be addressed. edge, one can estimate the phase stability properties of the-

1063-651X/96/54͑6͒/6500͑11͒/$10.0054 6500 © 1996 The American Physical Society 54 DEPLETION INTERACTIONS IN COLLOID-POLYMER... 6501 binary mixtures in a straightforward way. ͑The reverse pro- II. THEORY cess of inferring the intermolecular interactions from the A. Scattering from a mixture of colloidal particles phase behavior is much more problematic and unsure.͒ The and polymer molecules study of microscopic interactions complements the macro- scopic phase measurements. With both microscopic and Consider the scattering medium to be made of a mixture macroscopic measurements, one can verify assumptions and of colloidal particles and nonadsorbing polymer molecules. test predictions of various theoretical models for the deple- The total scattered intensity from the mixture can be written as ͓23,24͔ tion effect ͓3,4,19,20͔. These measurements are also useful for the further development of the depletion theory to a more 2 2 general form, so that non-hard-sphere interactions can be in- I͑Q͒ϭKc ⌺cc͑Q͒ϩ2͑KcKp͒⌺cp͑Q͒ϩKp⌺pp͑Q͒, ͑1͒ cluded. In contrast to many previous experimental studies ͓19– where the subscripts c and p are used, respectively, to iden- 22͔, which mainly focus on the phase behavior of the tify the colloid and polymer and K␣ is the scattering length colloid-polymer mixtures, we have recently carried out a la- density of the ␣ component, which has taken into account the ser light-scattering study of the depletion interaction in a contrast between the ␣ component and the solvent. The scat- mixture of a hard-sphere-like colloid and a nonadsorbing tering wave number Qϭ(4␲/␭0)sin͑␪/2͒, with ␭0 being the polymer ͓23͔. In the experiment, the second virial coefficient wavelength and ␪ the scattering angle. The function ⌺␣␤(Q) of the colloidal particles as a function of the free-polymer in Eq. ͑1͒ has the form concentration was obtained from the measured concentration N␣ N␤ dependence of the scattered light intensity from the mixture. ␣ ␤ ϪiQ•͑r Ϫr ͒ The experiment demonstrated that our light-scattering ⌺␣␤͑Q͒ϭ ͗e j l ͘, ͑2͒ ͚j ͚l method is indeed capable of measuring the depletion effect in the colloid-polymer mixture. However, the interpretation where r ␣ is the position of the jth monomer of the ␣ com- of the measurements was somewhat complicated by the un- j ponent and N is the total number of the monomers in the wanted scattering from the polymer. Assumptions were ␣ sample volume V. The angular brackets indicate an average made in order to deal with the interference effect between the over all possible configurations of the molecules. colloid and the polymer. The function ⌺␣␤(Q) measures the interaction between In this paper we report a small-angle neutron scattering the components ␣ and ␤ in the mixture. In the experiment to ͑SANS͒ study of the depletion interaction in the same be described below, we are interested in the polymer-induced colloid-polymer mixture. The use of SANS with isotopically depletion attraction between the colloidal particles and thus mixed solvents eliminates the undesirable scattering from the the scattering length density of the solvent is chosen to be the polymer. In the experiment, we measure the colloidal ͑par- same as that of the polymer. In this case, the polymer mol- tial͒ structure factor Sc(Q) over a suitable range of the scat- ecules become invisible to neutrons (K pϭ0) and Eq. ͑1͒ tering wave number Q. The measured Sc(Q) is directly re- becomes lated to the interaction potential U(r) and therefore it provides more detailed information about the colloidal inter- 2 I͑Q͒ϭKc ␳cPc͑Q͒Sc͑Q͒. ͑3͒ action in the polymer solution than the second virial coeffi- cient does. The colloidal particle chosen for the study con- In the above, ␳c is the number density of the colloidal par- sisted of a calcium carbonate ͑CaCO3͒ core with an adsorbed ticles and P (Q) is their scattering form factor. The partial monolayer of a randomly branched calcium alkylbenzene c structure factor Sc(Q) measures the interaction between the sulphonate surfactant. The polymer we used was hydroge- colloidal particles and it is proportional to the Fourier trans- nated polyisoprene, a stable straight-chain polymer. Both the form of the radial distribution function gc(r) for the colloidal polymer and the colloidal particles were dispersed in a good particles. Experimentally, Sc(Q) is obtained by solvent, decane. Such a nonaqueous colloid-polymer mixture is ideal for the investigation attempted here since the colloi- dal suspension is approximately a hard-sphere system and I͑Q͒ both the colloid and the polymer have been well character- Sc͑Q͒ϭ 2 , ͑4͒ Kc ␳cPc͑Q͒ ized previously using various experimental techniques. Be- cause the basic molecular interactions are tuned to be simple, 2 where K c Pc(Q) is the scattered intensity per unit concentra- the SANS measurements in the colloid-polymer mixture can tion measured in a dilute pure colloidal suspension, in which be used to critically examine the current theory for the deple- the colloidal interaction is negligible and thus Sc(Q)ϭ1. tion effect ͓3,4,19,20͔. The paper is organized as follows. In Sec. II, we review the scattering theory for a mixture of a colloid and a nonad- sorbing polymer and present the calculation of the colloidal B. Calculation of the form and structure factors for colloidal particles structure factor Sc(Q) using the potential U(r) for the polymer-induced depletion attraction. Experimental details As mentioned in Sec. I, our colloidal particle consists of a appear in Sec. III, and the results are discussed in Sec. IV. calcium carbonate core with an adsorbed monolayer of sur- Finally, the work is summarized in Sec. V. factant molecules. A simple core-shell model, as shown in 6502 X. YE et al. 54

To calculate the structure factor Sc(Q), one needs first to solve the Ornstein-Zernike equation for the direct correlation function C(r) using a known interaction potential U(r) ͓30͔. Asakura and Oosawa ͓1͔ have derived an effective interac- tion potential U(r) for the colloidal particles in a nonadsorb- ing polymer solution. It is assumed in the model that the colloidal particles are hard spheres and the polymer mol- ecules behave as hard spheres toward the colloidal particles but can freely penetrate with each other. Under this approxi- mation, U(r) takes the form ͓1,2͔

ϩϱ, rр␴ Ϫ⌸ V ͑r͒, ␴Ͻrр␴ϩ2R FIG. 1. Simple core-shell model for a colloidal particle consist- U͑r͒ϭ p 0 g ͑9͒ ing of a CaCO3 core with an adsorbed monolayer of surfactant ͭ 0, rϾ␴ϩ2R , g molecules. Here Rc is the core radius, ␦ is the shell thickness, and bc , bs , and bm are the coherent scattering length densities of the where ␴ is the particle diameter, ⌸ p is the osmotic pressure core, shell, and solvent, respectively. of the polymer molecules, and Rg is their radius of gyration. The volume of the overlapping depletion zones between the Fig. 1, has been used to characterize the microstructure of the two colloidal particles at a separation r is given by ͓1,2͔ particles ͓25–27͔. The form factor P (Q) for the core-shell c 3 3 model is ␭ 3 r 1 r V ͑r͒ϭv 1Ϫ ϩ , ͑10͒ 0 pͩ ␭Ϫ1ͪ ͫ 2 ͩ ␴␭ͪ 2 ͩ ␴␭ͪ ͬ 3 4␲Rc Pc͑QRc͒ϭ ͑bcϪbs͒ f͑QRc͒ϩ͑bs 3 ͭ 3 where v pϭ(4␲/3)R g is the volume occupied by a polymer chain and ␭ϭ1ϩ2Rg/␴. This potential has been used to 3 2 4␲͑Rcϩ␦͒ calculate the phase diagram of the binary system ͓3,4͔, and Ϫb ͒ f͓Q͑R ϩ␦͔͒ , ͑5͒ m 3 c ͮ recent phase measurements of several colloid-polymer mix- tures have shown qualitative agreement with the calculation with ͓18–20͔. When the potential U(r) is constituted by a hard core plus 3͑sin xϪx cos x͒ a weak attractive tail as in Eq. ͑9͒, the direct correlation f ͑x͒ϭ . ͑6͒ function C(r) can be obtained under the mean spherical ap- x3 proximation ͑MSA͒, which is a perturbative treatment to the Percus-Yevick equation ͓30͔. Under the MSA, we have In the above, Rc is the core radius, ␦ is the shell thickness, and bc , bs , and bm are the coherent scattering length densi- CHS͑r͒, rр␴ ties of the core, shell, and solvent, respectively. C͑r͒ϭ ͑11͒ When the particles are not uniform in size, the average ͭ ϪU͑r͒/kBT, rϾ␴, form factor takes the form where kBT is the thermal energy and CHS(r) is a known ϱ direct correlation function for the simple hard-sphere system ͗Pc͑Q͒͘ϭ ͵ P͑QRc͒g͑Rc͒dRc , ͑7͒ ͓31,32͔. Note that C(r) is constituted by two parts: ͑i͒ a hard 0 core that involves the colloid diameter ␴ and its volume fraction ␾c and ͑ii͒ an attractive tail with the dimensionless where g(Rc) is the probability distribution for the core radius ˜ amplitude Pϭ⌸ pv p/(kBT) and the range parameter ␭. With Rc . To reduce the fitting parameters, we have assumed in Eqs. ͑9͒–͑11͒, we calculate the Fourier transform C(Q)of Eq. ͑7͒ that the shell thickness ␦ is a constant. Previous ex- C(r) ͑see the Appendix͒ and obtain the structure factor periments ͓28,29͔ have shown that for a narrow size distri- Sc(Q) via the well-known relation ͓30͔ bution g(Rc), the measured ͗Pc(Q)͘ is not very sensitive to the detail functional form of g(Rc). For simplicity, we now 1 use the Schultz distribution to model the size distribution of S ͑Q͒ϭ . ͑12͒ c 1Ϫ␳ C͑Q͒ our particles. The Schultz distribution function has the form c

zϩ1 z ͑zϩ1͒ Rc C. Scattering from flexible polymer chains g͑R ͒ϭ eϪ͑zϩ1͒Rc /R0, ͑8͒ c R ⌫͑zϩ1͒ ͩ R ͪ Because of the chain flexibility and interchain penetration, 0 0 the scattered intensity I(Q) from a polymer solution cannot where R0 is the mean core radius and ⌫(x) is the gamma be generally written as a product of the form factor P p(Q) function. The normalized standard deviation ͑or the polydis- and the structure factor S p(Q). However, Zimm has shown persity͒ ⑀ is related to the parameter z via ͓33,34͔ that when two different polymer chains are assumed 2 1/2 Ϫ1/2 ⑀ϵ͗(RcϪR0) ͘ /R0ϭ(zϩ1) . With the Schultz distri- to have only a single contact, P p(Q) can still be factored out bution function, one can calculate ͗Pc(Q)͘ in Eq. ͑7͒ using a from the measured I(Q). Under this single-contact approxi- simple numerical integration method. mation, I(Q) takes the form 54 DEPLETION INTERACTIONS IN COLLOID-POLYMER... 6503

TABLE I. Scattering length densities of the colloid, polymer, K0␳ЈpM pP p͑Q͒ I͑Q͒ϭ . ͑13͒ and solvents. 1ϩ2͑A2͒pM p␳ЈpP p͑Q͒ Scattering length density In the above, K0 is an instrumental constant, ␳Јp is the poly- ͑1010 cmϪ2͒ 3 mer concentration in g/cm , M p is the molecular weight, and Mass density Material ͑g/cm3͒ x ray neutron (A2) p is the second virial coefficient. Zimm initially derived Eq. ͑13͒ in a low polymer concentration limit and later CaCO core 2 16.8 3.48 Benoit and Benmouna ͓35,36͔ generalized the equation to a 3 surfactant shell 0.9–1.0 8.0–9.5 0.35 high concentration regime using the mean-field theory of PEP polymer 0.856 ϳ6.78 Ϫ0.31 Ornstein and Zernike ͓35͔. For a dilute polymer solution, Eq. Ϫ ͑13͒ can be rearranged to read decane 0.73 ϳ6.78 0.49 decane-d22 0.84 ϳ6.78 6.58

K0␳ЈpM p 1 ϭ ϩ2͑A2͒pM p␳Јp . ͑14͒ I͑Q͒ P͑Q͒ using ␮-Styragel columns and tetrahydrofuran as the elution 2 solvent. To vary the scattering contrast, both hydrogenated For small values of Q, we have P p(Q)Ӎ1Ϫ(QRg) /3 and decane ͑Aldrich, Ͼ99% pure͒ and deuterated decane ͑Cam- thus Eq. ͑14͒ becomes bridge Isotope Laboratories, Ͼ99% deuterated͒ were used as solvents. Decane has been found to be a good solvent for K ␳ЈM 1 0 p p 2 both the colloid and PEP ͓23͔. The scattering length densities ϭ1ϩ ͑QRg͒ ϩ2͑A2͒pMp␳Јp . ͑15͒ I͑Q͒ 3 of the colloid, the PEP polymer, and the solvents used in the

2 experiment are listed in Table I. Some of the values in the With Eq. ͑15͒ one can obtain values of K0M p , R g, and table are taken from Refs. ͓25–27͔. (A2) pM p by linearly extrapolating the scattering data /I(Q) to the limits of 0 and Q2 0, respectively. ␳Јp ␳Јp B. Small-angle neutron- and x-ray-scattering measurements Using Eq. ͑13͒, one can also→ obtain the→ polymer structure The SANS measurements of the pure colloidal samples factor S p(Q)ϭI(Q)/͓K0␳ЈpMpPp(Q)͔, which has the form and the mixture samples were performed at the High Flux 1 Beam Reactor in the Brookhaven National Laboratory. The S p͑Q͒ϭ . ͑16͒ incident neutron wavelength ␭ ϭ7.05Ϯ0.4 Å and the usable 1ϩ2A M ␳ЈP ͑Q͒ 0 2 p p p Q range was 0.007 ÅϪ1рQр0.15 ÅϪ1. The SANS measure- ments of the pure polymer samples were performed on a III. EXPERIMENT SANS instrument ͑NG-7͒ at the National Institute of Stan- dards and Technology. The incident neutron wavelength A. Sample preparation ␭0ϭ5.00Ϯ0.35 Å and the usable Q range was 0.01 As mentioned in Sec. I, our colloidal particle consists of a ÅϪ1рQр0.142 ÅϪ1. The SAXS measurements of dilute CaCO3 core with an adsorbed monolayer of a randomly colloidal samples were performed on a high-resolution spec- branched calcium alkylbenzene sulphonate surfactant. The trometer at the Oak Ridge National Laboratory. The incident synthesis procedures used to prepare the colloid have been x-ray wavelength ␭0ϭ1.54Ϯ0.04 Å and the usable Q range described by Markovic et al. ͓25͔. These particles have been was 0.014 ÅϪ1рQр0.31 ÅϪ1. All the scattering measure- well characterized previously using SANS and small-angle ments were conducted at room temperature. The neutron- x-ray scattering ͑SAXS͒ techniques ͓25–27͔ and they are scattering sample cells were made of quartz and the x-ray- used as an acid-neutralizing aid in lubricating oils. The mo- scattering cells were made of Kapton; both types of cells had lecular weight of the particle is M cϭ300 000Ϯ15%, which a path length of 1 mm. The raw scattered intensity Ir(Q) was obtained from a sedimentation measurement ͓23͔. The ͑counts/h͒ was measured by a two-dimensional detector. The colloidal suspensions were prepared by diluting known corrected intensity I(Q) was obtained by applying the stan- amounts of the concentrated suspension with the solvent, dard corrections due to the background intensity, solvent decane. The suspensions were then centrifuged at an accel- scattering and sample turbility via eration of 108 cm/s2 ͑105 g͒ for 2.5 h to remove any colloidal aggregates and dust. The resulting colloidal suspensions Ir͑Q͒ϪIb͑Q͒ Is͑Q͒ϪIb͑Q͒ I͑Q͒ϭ Ϫ ͑17͒ were found to be relatively monodispersed with ϳ10% stan- Tr Ts dard deviation in the particle radius, as determined by dy- namic light scattering ͓23͔. and subsequently computing the azimuthal average. In the The polymer used in the study was hydrogenated polyiso- above, Ib(Q) is the background scattering when the neutron prene ͓poly-ethylene-propylene ͑PEP͔͒, a straight-chain beam is blocked, Is(Q) is the scattered intensity from the polymer synthesized by an anionic polymerization scheme solvent, and Tr and Ts are the transmission coefficients for ͓37,38͔. The PEP is a model polymer ͑with M w/M nϽ1.1͒, the scattering sample and the solvent, respectively. To elimi- which has been well characterized previously using various nate the inhomogeneity of detector’s sensitivity at different experimental techniques ͓37–39͔. The molecular weight of pixels, we normalized the scattering data ͑both neutron and the PEP was M pϭ26 000 amu. The molecular weight char- x-ray͒ with isotropic scattering standards. The structure fac- acterization was carried out by size exclusion chromatogra- tor Sc(Q) for the pure colloid samples and the mixture phy, which was made with a Waters 150-C SEC instrument samples were obtained by using Eq. ͑4͒. 6504 X. YE et al. 54

TABLE II. Fitting results from dilute colloid and polymer samples.

Samples R0 ͑nm͒ ␦ ͑nm͒ zRg͑nm͒ RT ͑nm͒ Colloid-decane 2.0 15.1 ͑SAXS͒

Colloid–decane-d22 2.0 2.0 12.0 ͑SANS͒

PEP–decane-d22 8.28 4.85 ͑SANS͒

lated ͗Pc(Q)͘ for the polydispersed core-shell particles us- ing Eqs. ͑7͒, ͑5͒, and ͑8͒. In the calculation, the neutron- scattering densities for the CaCO3 core, the surfactant shell, and deuterated decane are taken from Table I and R0 is fixed at the value determined from the SAXS data. As a result, there are only three parameters left in the fitting: ␦, z, and I0 . The fitting results for the dilute colloidal suspension are listed in Table II and they agree well with previous SAXS and SANS measurements ͓25–27͔. From the fitted value of z, we find the polydispersity of the core radius to be ⑀Ӎ25%, which is slightly larger than that determined by dynamic light scattering ͑⑀Ӎ10%͓͒23͔. The fact that the value of z obtained from the SANS data is smaller than that from the SAXS data suggests that the surfactant shell is also some- what polydispersed ͓27͔. We now discuss the structure factor Sc(Q) of the concen- trated pure colloidal suspensions in decane measured by SANS. Figure 3 shows the measured Sc(Q) for three colloid FIG. 2. Measured scattering intensity I(Q) as a function of Q concentrations: ͑a͒ ␳cЈϭ26.2 wt. %, ͑b͒ 17.5 wt. %, and ͑c͒ for the dilute pure colloidal suspensions. ͑a͒ SAXS data from the 8.7 wt. %. The solid curves are the fits to the simple hard- sphere structure factor ͓40,41͔, which can be obtained by colloid-decane suspension at the concentration ␳ cЈϭ1 wt. % and ͑b͒ taking U(r)ϭ0 for rϾ␴ in Eq. ͑11͒. There are two fitting SANS data from the colloid–deuterated-decane suspension at ␳Јc ϭ1 wt. %. The solid curve in ͑a͒ is a fit to the form factor of the parameters in the hard-sphere model: the volume fraction ␾c polydispersed spheres and that in ͑b͒ is a fit to the form factor of the and the hard-sphere diameter ␴. The fitted values of ␾c and polydispersed core-shell particles. ␴ for different colloidal concentrations are given in Table III. Because the centrifugation process removed some colloidal IV. RESULTS AND DISCUSSION aggregates and dust from the master colloidal solution, the nominal mass concentration of the colloidal samples has A. Pure colloids some uncertainties. However, the ratios of the concentrations Figure 2͑a͒ shows the SAXS data for the dilute pure col- are accurate and the ratios of the fitted ␾c in Table III agree loidal suspension in decane at the concentration ␳cЈϭ1 well with them. It is seen from Table III that the fitted ␴ wt. %. Because the x-ray-scattering length density for the remains constant for different colloid concentrations and its surfactant shell is about the same as that of decane ͑see Table best fit value is ␴ϭ7.7 nm. This value is very close to the I͒, the scattering is dominated by the CaCO3 core ͓27͔. The measured particle size 2(R0ϩ␦) ͑ϭ8.0 nm͒ as listed in solid curve in Fig. 2͑a͒ is the calculated ͗Pc(Q)͘ for the Table II. Because the surfactant shell of the colloidal particle polydispersed spheres using Eqs. ͑7͒, ͑5͒, and ͑8͒ with is ‘‘soft,’’ we expect its effective hard-shell thickness to be bsϭbm . Since the shell thickness ␦ is absent from the fitting, smaller than ␦. one can determine the core radius more accurately from the Figure 3 clearly shows that the hard-sphere model fits the SAXS data. There are three fitting parameters in the plot: the low concentration data well. As the colloid concentration mean core radius R0 , the polydispersity parameter z, and the increases, the measured Sc(Q) starts to deviate from the scattered intensity I0 at Q 0. Figure 2͑b͒ shows the SANS hard-sphere model in the small-Q region. The deviation can data for the dilute colloidal→ suspension in deuterated decane be attributed to a weak repulsion between the soft surfactant at ␳cЈϭ1 wt. %. Because the neutron-scattering length den- shells, since the measured Sc(0) is smaller than the hard- sity of the surfactant shell is much different from that of sphere value. The colloidal particles ‘‘feel’’ more and more deuterated decane ͑see Table I͒, the scattering from the sur- repulsion when their separation is decreased. Because the factant shell is more pronounced in the SANS data. It is seen deviation in the small-Q region is small, the measured Sc(Q) from Fig. 2͑b͒ that the scattering from the surfactant shell becomes insensitive to the detail functional form of the soft exhibits a sharp minimum, which entails a tight fit for the repulsion in U(r). We have tested several functional forms shell thickness ␦. The solid curve in Fig. 2͑b͒ is the calcu- for the repulsive tail, such as an exponential decaying func- 54 DEPLETION INTERACTIONS IN COLLOID-POLYMER... 6505

TABLE III. Fitting results from concentrated colloid samples.

Hard sphere Hard core plus a square barrier Colloidal samples ͑wt. %͒ ␾c ␴ ͑nm͒ ␾c ␴ ͑nm͒ ␩ (kBT) ␭ 26.2 0.146 7.8 0.146 7.8 Ϫ0.1 2.0 17.5 0.086 7.7 0.086 7.7 Ϫ0.06 2.0 8.7 0.038 7.7 0.038 7.7 Ϫ0.03 2.0

Table III. It is seen that the fitted values of ␴ and ␾c are the same as those from the hard-sphere model. The fitted barrier height ␩ is found to be small compared to the thermal energy kBT ͑the negative sign indicates repulsion͒ and it increases slightly with the colloid concentration as discussed above. It is also found from the fitting that a fixed value of ␭͑ϭ2.0͒ can be used to fit all the data with different concentrations. This value of ␭ suggests that the soft repulsion range is ap- proximately the same as the hard-core diameter ␴. Previous SANS measurements for the same colloid have also shown this relatively large interaction range ͓26͔. The above mea- surements for the pure colloidal samples reveal that the mi- crostructure of the particles can be well described by the simple core-shell model and the colloidal suspension is ap- proximately a hard-sphere system. Our data analysis for the particle form factor is somewhat complicated by the small polydispersity of the samples and the interparticle potential is slightly affected by the weak repulsion between the soft surfactant layers.

B. Pure polymers Figure 4͑a͒ shows the Zimm plot of the SANS data for the pure PEP in deuterated decane with the polymer concentra- 3 tion ␳Јp ranging from 0.0072 to 0.0634 g/cm . The SANS data are found to be well described by the Zimm relation as shown in Eq. ͑15͓͒for clarity we did not draw the linear extrapolation lines in Fig. 4͑a͔͒. From the Zimm analysis, we find that the polymer chains have a radius of gyration Rgϭ8.28 nm and their second virial coefficient A2M pϭ44.4 3 cm /g. With the measured A2 one can define a thermody- namic radius ͑an effective hard-sphere radius͒ RT via 3 2 4(4␲/3)R TϭA2M p. Thus we have RTϭ4.85 nm, which FIG. 3. Measured structure factor S (Q) of the concentrated agrees well with our previous light-scattering measurement c ͓23͔. It is shown in Eq. ͑14͒ that when the scattering data colloid-decane suspensions for three concentrations: ͑a͒ ␳Јc ϭ26.2 wt. %, ͑b͒ 17.5 wt. %, and ͑c͒ 8.7 wt. %. The solid curves are the ␳Јp/I(Q) at a fixed Q is plotted as a function of ␳Јp , a linear Ϫ1 fits to the hard-sphere model and the dashed curves correspond to a extrapolation to ␳Јp 0 gives [K0M pP p(Q)] . The zeroth → Ϫ1 hard-core potential with a repulsive barrier. curve in Fig. 4͑a͒ shows the extrapolated [K0M pP p(Q)] . Using the known value of K0M p ͑ϭ0.0036, determined from tion and a square barrier, and found that all these functions the Zimm plot͒, we obtain the polymer form factor P p(Q)as fit the data equally well. The dashed curves in Fig. 3 show shown in Fig. 4͑b͒. The large-Q portion of P p(Q) is found to 1/␯ the calculated Sc(Q) for the hard spheres with a square re- be well described by the power law 1/[0.64(QRg) ] ͑solid 3 pulsive barrier, which fit the data better than the hard-sphere line͒, with ␯ϭ 5 being the Flory exponent for three- model in the small-Q region. Sharma and Sharma ͓42͔ have dimensional polymer chains ͓43͔. calculated Sc(Q) for a hard-core potential with an attractive With the extrapolated P p(Q), one can obtain the polymer square well. Here we simply change the sign of the attractive structure factor S p(Q)ϭI(Q)/͓K0␳ЈpMpPp(Q)͔ from the well to model the repulsive barrier. concentrated polymer samples. Figure 5 shows the measured There are four parameters in the model: the volume frac- S p(Q) of the PEP polymer in deuterated decane for three tion ␾c , the diameter ␴, the barrier height ␩ ͑in units of polymer concentrations. The solid curves show the calcu- kBT͒, and the barrier width ␥. A dimensionless parameter ␭ lated S p(Q) using Eq. ͑16͒ and the extrapolated P p(Q)in is defined as ␭ϭ1ϩ␥/␴. The fitting results are summarized in Fig. 4͑b͒. Because hydrogenated polymer samples are used 6506 X. YE et al. 54

TABLE IV. Fitting results from concentrated polymer samples.

3 3 ␳Јp ͑g/cm ͒ A2M p ͑cm /g͒ ␶ 0.0038 45.0 0.02 0.0072 47.5 0.005 0.0151 41.0 0.02 0.0223 40.8 0.03 0.0276 43.0 0 0.0352 41.5 0.04 0.0452 42.0 0.05 0.0634 47.0 0.04

duce an additive noise term ␶ to the calculated S p(Q) in Eq. ͑16͒. The fitted values of A2M p and ␶ are listed in Table IV. It is seen from Table IV that the contribution of the incoher- ent scattering to the total intensity is at a few percent level. This is consistent with our calculation of the incoherent scat- tering intensity for the PEP using the known incoherent cross section of the hydrogen atoms. The fitted values of A2M p are very close to that obtained from the Zimm analysis and they do not change very much in our working range of the poly- mer concentration. Figure 5 thus demonstrates that Eqs. ͑13͒ and ͑16͒ can indeed be used to describe the scattering data from the concentrated polymer samples. The measured S p(Q) in Fig. 5 shows a typical correlation hole for the poly- mer chains ͓43͔ and is very different from the structure factor of either a hard-sphere system ͑see Fig. 3͒ or an ideal gas FIG. 4. ͑a͒ Zimm plot of the SANS data for the pure PEP in ͑polymer at ⌰ point͒. These two models have been widely 3 used to describe the polymer molecules in the colloidal sus- deuterated decane with the polymer concentration ␳Јp ͑g/cm ͒ being ͑1͒ 0.0072, ͑2͒ 0.0151, ͑3͒ 0.0223, ͑4͒ 0.0352, ͑5͒ 0.0452, and ͑6͒ pension ͓2,22͔. Ϫ1 0.0634. The zeroth curve is the extrapolated [K0M pP p(Q)] and the scale constant ␬ϭ0.8. ͑b͒ Extrapolated polymer form factor C. Mixtures of colloids and polymers Pp(Q) as a function of Q. The solid line is the fitted function 5/3 We now discuss the SANS data from the colloid-PEP- 1/[0.64(QRg) ] to the circles at large Q. decane mixtures. Some of the results in this section have been briefly reported in Ref. ͓44͔. Because decane and the in the experiment, the hydrogen atoms in the polymer sample PEP are both protonated, the polymer chains in the mixture contribute a small amount of incoherent scattering to the are invisible to neutrons. To reduce the fitting ambiguity and total intensity I(Q) ͓which is proportional to S p(Q)͔.To pinpoint the control parameters for the depletion effect, we describe the Q-independent incoherent scattering, we intro- prepared three series of mixture samples with ␳cЈϭ26.2, 17.5, and 8.7 wt. %, respectively. For each series of the

samples, ␳cЈ was kept the same and the polymer concentra- tion ␳Јp was increased until the mixture became phase sepa- rated ͑except for the series with ␳cЈϭ8.7 wt. %͒ with a vis- ible interface, which separates the dark-brown colloid-rich lower phase from the light-brown colloid-poor upper phase. In this way all the colloidal parameters remain unchanged and one can clearly see the effect of adding polymer to the colloidal suspensions. Our previous light-scattering measure- ments ͓23͔ have revealed that the PEP chains do not adsorb onto the colloidal surfaces and the phase separation in the colloid-PEP mixture samples occurs at the concentrations very close to the depletion prediction. Figure 6 compares the measured Sc(Q) for three values of ␳Јp when ͑a͒ ␳cЈϭ26.2 wt. %, ͑b͒ ␳cЈϭ17.5 wt. %, and ͑c͒ ϭ8.7 wt. %. It is seen that the main effect of adding PEP FIG. 5. Measured structure factor S p(Q) for the PEP polymer in ␳cЈ deuterated decane. The polymer concentrations are ␳Јp is to increase the value of Sc(Q) in the small-Q region, 3 ͑g/cm ͒ϭ0.0072 ͑open circles͒, 0.0276 ͑closed circles͒, and 0.0634 whereas the large-Q behavior of Sc(Q) remains nearly un- ͑open triangles͒. The solid curves are the calculated S p(Q) using changed. As the colloid concentration increases, the effect of Eq. ͑16͒ and the extrapolated Pp(Q) in Fig. 4͑b͒. adding PEP becomes more and more pronounced. Since 54 DEPLETION INTERACTIONS IN COLLOID-POLYMER... 6507

TABLE V. Fitting results from three series of the colloid-PEP

mixtures with three different colloid concentrations: ͑a͒ ␳cЈϭ26.2 wt. %, ͑b͒ ␳cЈϭ17.5 wt. %, and ͑c͒ ␳Јc ϭ8.7 wt. %.

3 ˜ Sample ␳Јp ͑g/cm ͒ ␾c ␴ ͑nm͒ P ␭ ͑a͒ 1 0.0 0.146 7.80 Ϫ0.1 2.9 2 0.004 0.136 7.45 Ϫ0.01 2.9 3 0.009 0.130 7.45 0.095 2.9 4 0.017 0.123 7.30 0.175 2.9 5 0.024 0.123 6.90 0.228 2.9 6 0.031 0.123 6.80 0.265 2.9

͑b͒ 1 0.0 0.086 7.70 Ϫ0.06 2.9 2 0.004 0.081 7.70 Ϫ0.019 2.9 3 0.009 0.081 7.70 0.066 2.9 4 0.016 0.080 7.55 0.143 2.9 5 0.023 0.080 7.20 0.180 2.9 6 0.030 0.080 7.10 0.214 2.9 7 0.049 0.083 6.65 0.260 2.9 8 0.065 0.080 6.85 0.250 2.9

͑c͒ 1 0.0 0.038 7.70 Ϫ0.033 2.9 2 0.004 0.038 7.70 Ϫ0.007 2.9 3 0.008 0.038 7.70 0.054 2.9 4 0.016 0.038 7.50 0.110 2.9 5 0.023 0.038 7.20 0.141 2.9 6 0.029 0.038 7.10 0.172 2.9 7 0.037 0.038 6.90 0.200 2.9 8 0.048 0.038 6.90 0.220 2.9

tude ˜P and the range parameter ␭ describe the attraction tail. The fitting results are summarized in Table V. It is seen from Table V that for a fixed colloid concentra- tion, the fitted values of ␴ and ␾c do not change very much with the polymer concentration and they are close to those obtained from the corresponding pure colloidal suspensions

(␳Јpϭ0). Furthermore, the fitted ␭ also remains constant for different ␳cЈ and ␳Јp and its best fit value is ␭ϭ2.9. This value is close to the calculated ␭ϭ1ϩRg/(R0ϩ␦)ϭ3.07. In the literature, Rg is commonly used as the effective colloid- FIG. 6. Measured colloidal structure factor Sc(Q) of the colloid- polymer interaction range ͓18͔ and we have adopted the PEP mixtures for different polymer concentration ␳Јp when ͑a͒ ␳cЈ same convention in Eq. ͑9͒. Because polymer chains are pen- ϭ26.2 wt. %, ͑b͒ ␳Јc ϭ17.5 wt. %, and ͑c͒ ␳Јc ϭ8.7 wt. %. The val- etrable, we expect that the actual colloid-polymer interaction ues of Ј g/cm3 in a are 0.0039 ᭺ , 0.0165 ᭞ , and 0.0308 ␳ p ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ range is in between the radius of gyration R and the ther- ᮀ ; those in b are 0.0038 ᭺ , 0.0233 ᭞ , and 0.0652 ᮀ ; and g ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ modynamic radius R , which measures the polymer-polymer those in ͑c͒ are 0.0081 ͑᭺͒, 0.0226 ͑᭞͒, and 0.048 ͑ᮀ͒. The solid T interaction range. From the fitted value of ␭, we find the curves are the fits to Eq. ͑12͒ using the potential U(r) in Eq. ͑9͒. effective colloid-polymer interaction range to be 7.6 nm, which is about 10% smaller than Rg but 60% larger than RT . Sc(0) is proportional to the isothermal ͑osmotic͒ compress- To the authors’ knowledge, this is the first time that a micro- ibility of the colloidal solution, an increasing Sc(0) with ␳Јp scopic measurement is performed to provide quantitative in- indicates an increase in the attraction between the colloidal formation about the interaction range of the depletion attrac- particles. Figure 6 thus reveals a notable feature for the tion. With the above three fitting parameters fixed, we were polymer-induced depletion attraction between the colloidal able to fit all the scattering data from different mixture particles. The solid curves in Fig. 6 are the fits to Eq. ͑12͒ samples ͑19 samples in total͒ with only one free parameter: using the potential U(r) in Eq. ͑9͒. There are four param- the interaction amplitude ˜P. eters in the fitting: The volume fraction ␾c and the diameter It should be pointed out that in the above data analysis, ␴ are for the hard-core potential; and the interaction ampli- we have assumed that the colloidal particles are identical and 6508 X. YE et al. 54

effect of adding polymer, as shown in Fig. 6, is to increase the value of Sc(Q) in the small-Q range. Figure 7͑a͒ shows the fitted ˜P as a function of the effec- tive polymer volume fraction ␾ pϭ␳Јp/␳*, where ␳* 3 ϭM p/[(4␲/3)R g] is the polymer overlap concentration. It is ˜ seen that P first increases linearly with ␾ p up to ␾ pӍ1 and ˜ then it levels off. For a given ␾ p , P also depends upon ␳cЈ . If the polymer molecules in the mixture are treated as an ideal gas, their osmotic pressure ⌸ pϭn pkBT and hence ˜ Pϭ⌸ pv p/(kBT)ϭ␾p . Recently, Lekkerkerker et al. ͓4,18͔ have pointed out that the polymer number density n p should be defined as n pϭN p/V f , where N p is the total number of the polymer molecules in the mixture and V f ϭ␣(␾c)V is the free volume not occupied by the colloidal particles and their surrounding depletion zones. The free volume V f is propor- tional to the sample volume V and the proportionality con- stant ␣(␾c) ͑р1͒ has the form ͓4͔

Ϫ͑A␥ϩB␥2ϩC␥3͒ ␣͑␾c͒ϭ͑1Ϫ␾c͒e , ͑18͒

2 3 2 3 where ␥ϭ␾c/(1Ϫ␾c), Aϭ3␰ϩ3␰ ϩ␰ , Bϭ9␰ /2ϩ3␰ , 3 and Cϭ3␰ . In the above, ␰ϭ2Rg/␴ is the size ratio of the polymer chain to the colloidal particle. At low ␾c , we have 3 3 ␣(␾c)ϭ1Ϫ␾c(1ϩ␰) , where ␾c(1ϩ␰) is the volume fraction occupied by the colloidal particles and their sur- rounding depletion zones. It should be mentioned that Eq. ͑18͒ holds only when ␰р1. For our mixture samples, how- ever, the polymer chains are larger than the colloidal par- ticles. In this case, Eq. ͑18͒ overestimates the volume frac- tion of the depletion zones and the resulting ␣(␾c) becomes ˜ underestimated. For example, when ␾cϭ0.14 and FIG. 7. Fitted interaction amplitude P as a function of ͑a͒ the 8.28 3 ␰ϭ 4.0 ϭ2.07, we have ␣(␾c)Ӎ1Ϫ␾c(1ϩ␰) ϭϪ3.05. effective polymer volume fraction ␾ pϭ␳Ј/␳* and ͑b͒ the scaled p Clearly, this is an unphysical value for ␣(␾ ). Gast, Hall, polymer volume fraction ␾ p/␣. The colloid concentrations of the c and Russel ͓3͔ have shown that this failure in getting the mixture samples are ␳Јc ϭ26.2 wt. % ͑᭺͒, ␳Јc ϭ17.5 wt. % ͑᭞͒, and correct volume for the overlapping depletion zones results ␳cЈϭ8.7 wt. % ͑ᮀ͒. The solid curve in ͑b͒ is the fitted function ˜ 2 from the many-body effect on the mutual overlap of the ex- PϭϪ0.054ϩ0.178(␾ p/␣)Ϫ0.0245(␾ p/␣) . cluded shells of three or more colloidal particles. In analyz- ˜ have considered only the polymer-induced depletion interac- ing the fitting results for P, we find that if ␴/(2Rg), instead tion between the colloidal particles. In principle, the polydis- of 2Rg/␴, is defined as the size ratio ␰, Eq. ͑18͒ can still be persity in particle size and a small amount of surfactant ag- used to calculate ␣(␾c). As shown in Fig. 7͑b͒, once ␾ p is gregates in the mixture could also introduce depletion scaled by the calculated ␣(␾c) using Eq. ͑18͒ with attractions, which may cause some degree of fractionation or ␰ϭ␴/(2Rg), the three curves in Fig. 7͑a͒ collapse into a association in the pure colloidal suspension ͓11,45͔. How- single master curve. The solid curve in Fig. 7͑b͒ is the fitted ˜ 2 ever, because the size difference between the colloidal par- function PϭϪ0.054ϩ0.178(␾ p/␣)Ϫ0.0245(␾ p/␣) . ticles is small and the polymer chains are much larger than The fitted ˜P consists of three terms. The small negative the surfactant aggregates, the depletion effect due to the sur- intercept indicates that there is a weak repulsive interaction factant aggregates and the polydispersity in particle size between the soft surfactant shells of the colloidal particles should be very weak compared to the polymer-induced ͑see the discussion in Sec. IV A͒. To have a meaningful com- depletion effect and therefore it is not likely to affect our parison with the fitted ˜P for the mixture samples, here we data analysis. This argument is further supported by the fol- have used the same U(r)asinEq.͑9͒, but changed the sign lowing facts. ͑i͒ We did not observe any substantial amount of U(r) for rϾ␴ to fit the measured Sc(Q) of the pure of attraction in the measured Sc(Q) for the pure colloidal colloidal samples. The three data points at ␾ pϭ0 in Fig. 7͑a͒ samples. As shown in Fig. 3, the measured Sc(Q) of the pure were obtained using the potential U(r) in Eq. ͑9͒ with colloidal suspensions can be well described by a hard-core ␭ϭ2.9. The linear coefficient of ˜P should be unity for non- potential with a weak repulsive tail. ͑ii͒ Because the colloidal interacting polymer chains ͑an ideal gas͒, but our fitted value parameters are the same for each series of the mixture is 0.178. One plausible reason for the deviation is that with samples with same ␾c , the effect seen in Fig. 6 is solely due the effective potential approach, the polymer molecules are to the addition of PEP into the colloidal suspension. ͑iii͒ assumed to be smaller than the colloidal particles and their Furthermore, the main effect of the small polydispersity in number density should be much higher than that of the col- particle size is to average out the oscillations of Sc(Q) in the loidal particles. In our experiment, however, these two as- large-Q range beyond the first peak position, whereas the sumptions are not strictly satisfied and thereby the overlap 54 DEPLETION INTERACTIONS IN COLLOID-POLYMER... 6509 volume V0(r) in Eq. ͑10͒ is overestimated. As a result, the that of the polymer, we measured the ͑partial͒ colloidal ˜ fitted P is smaller than its actual magnitude because U(r)in structure factor Sc(Q) of the mixture samples as well as the ˜ Eq. ͑9͒ is proportional to the product of V0(r) and P. An- form and the structure factors of the pure colloidal suspen- other possibility is that in calculating ␾ pϭ␳Јp/␳*, a charac- sion and the pure polymer solution. Under the mean spheri- teristic length smaller than Rg should be used for the poly- cal approximation, we calculated Sc(Q) using an effective mer chains. For example, if the thermodynamic radius RT is interaction potential U(r) for the polymer-induced depletion used to estimate ␾ p , the linear coefficient will be increased attraction between the colloidal particles. It is found that the 3 3 by a factor of (Rg/RT) ϭ(8.28/4.85) Ӎ5. measured Sc(Q) for different colloid and polymer concen- The polymer-polymer interaction, which gives rise to the trations can be well described by the depletion potential quadratic term in the fitted ˜P, can have two competing ef- U(r). The fitted amplitude ˜P of the depletion attraction is fects on the depletion attraction. It may either increase ˜P found to increase linearly with the polymer volume fraction because the osmotic pressure of the bulk polymer solution is ␾ p up to ␾ pӍ1. At higher volume fractions, the polymer- increased or reduce the depletion attraction because the os- polymer interaction becomes important and results in a motic pressure inside the narrow gap between two colloidal gradual reduction in the depletion attraction. The experiment particles is substantially increased when the distance of the reveals that the interaction amplitude ˜P becomes indepen- two colloidal surfaces becomes large enough such that a dent of the colloid concentration, once the free volume not monolayer of polymer molecules can stay in the gap ͓46͔. occupied by the colloidal particles and their surrounding Figure 7͑b͒ clearly shows that the polymer-polymer interac- depletion zones is used to calculate ␾ p . The experiment also tion tends to reduce the depletion attraction. Recent theoreti- shows that the interaction range for the depletion attraction is cal calculations ͓46–48͔ of the depletion attraction between about 10% smaller than the radius of gyration Rg of the two parallel plates immersed in an interacting polymer ͑or polymer chains. Because the polymer molecules can be used particle͒ solution have shown that the enhanced osmotic to continuously change the range and the amplitude of the pressure within the gap between the two colloidal particles depletion interaction, our experiment depicts the effective- can overcome the osmotic pressure increase in the bulk poly- ness of using polymer to study interaction-related phenom- mer solution and thus produce a repulsive barrier in addition ena in colloidal suspensions. to the usual attractive well shown in Eq. ͑9͒. In these calcu- lations, the effective colloid-polymer interaction range is as- ACKNOWLEDGMENTS sumed to be a constant ͑say, Rg͒ independent of the polymer concentration. As a result, the calculations are applicable We have benefited from useful discussions with T. A. only to the binary hard-sphere systems, in which the deplet- Witten, B. J. Ackerson, J. H. H. Perk, W. B. Russel, and J. B. ants are rigid spheres ͓46͔. For flexible polymer chains in the Hayter. We thank D. Schneider for his assistance with the SANS measurements at the Brookhaven National Labora- semidilute regime (␾ pϾ1), the polymer chain-chain interac- tions are screened over a distance of the order of the corre- tory. We also acknowledge the Brookhaven National Labo- Ϫ3/4 ratory, the National Institute of Standards and Technology, lation length ͑or mesh size͒ ␰bӍRg␾ p , which decreases with increasing polymer concentration ͓43͔. In this regime, and the Oak Ridge National Laboratory for granting neutron the self-consistent mean-field calculation by Joanny, Leibler, and x-ray beam times. This work was supported by the Na- and de Gennes ͓49͔ indicates that the polymer segments are tional Aeronautics and Space Administration under Grants Nos. NAG3-1613 and NAG3-1852. depleted from a shell of thickness ␰b around each colloidal particle. The fact that the depletion layer thickness ␰b shrinks with increasing ␾ p in the semidilute regime can be used to APPENDIX explain the leveling-off effect of the fitted ˜P in Fig. 7͑b͒.We The Fourier transform of the direct correlation function note from Fig. 7͑b͒ that at the highest polymer concentration ˜ C(r)is ␾ pӍ4, the fitted P even starts to decrease. Clearly, if this trend for ˜P continues, the colloid-polymer mixtures can be ␴ ϱ ϪiQ•r ϪiQ•r C͑Q͒ϭ CHS͑r͒e drϩ ͓ϪU͑r͒/͑kBT͔͒e dr restabilized at higher polymer concentrations. Gast and co- ͵0 ͵␴ workers ͓50,51͔ have used the shrinking effect of ␰b to pre- dict the equilibrium restabilization of electrically and steri- ϭCHS͑Q͒ϩCdep͑Q͒. ͑A1͒ cally stabilized colloidal particles in free polymer solutions. Further experimental and theoretical studies are needed in In the above, U(r) is the interaction potential given by Eq. order to have a quantitative comparison for the polymer- ͑9͒ and 1 3 induced depletion potential U(r) in the semidilute regime. CHS͑r͒ϭϪ␣Ϫ␤rϪ 2 ␾c␣r ͑A2͒

is the direct correlation function for a hard-sphere system V. CONCLUSION ͓31,32͔. The coefficients ␣ and ␤ in Eq. ͑A2͒ are given by Small-angle neutron- and x-ray-scattering techniques ͓31,32͔ 2 3 have been used to study the depletion interaction in a mix- ͑1ϩ2␾c͒ ϩ␾c͑␾cϪ4͒ ␣ϭ , ͑A3͒ ture of a hard-sphere-like colloid and a nonadsorbing poly- ͑1Ϫ␾ ͒4 mer. The colloidal particles used in the study were sterically c stabilized CaCO nanoparticles, the polymer used was hy- 3 18ϩ20␾ Ϫ12␾2ϩ␾4 drogenated polyisoprene, and the solvent was decane. By c c c ␤ϭϪ␾c 4 , ͑A4͒ matching the scattering length density of the solvent with 3͑1Ϫ␾c͒ 6510 X. YE et al. 54 where ␾ ϭ␲␳ ␴3/6 is the volume fraction of the colloidal ˜ c c 24␾c P particles. The final results for C(Q) in Eq. ͑A1͒ are ␳ C ͑Q͒ϭ ͑3␭2Q2␴2Ϫ12͒cos͑␭Q␴͒ c dep ͑Q␴͒6 ͑␭Ϫ1͒3 ͭ 24␾c 1 3 3 4 4 4 ␳cCHS͑Q͒ϭϪ 6 ͕␣͑Q␴͒ ͓sin͑Q␴͒ϪQ␴ cos͑Q␴͔͒ Ϫ12␭Q␴ sin͑␭Q␴͒ϩ ␭ Q ␴ Ϫ ͑Q␴͒ ͫ ͩ ␭ 2␭2 ϩ␤͑Q␴͒2͓2Q␴ sin͑Q␴͒Ϫ͑Q2␴2Ϫ2͒ 1 6 ϩ ϩ␭2Q2␴2 3Ϫ ϩ12 cos͑Q␴͒ 2␭4 ͪ ͩ ␭2 ͪ ͬ 1 3 3 ϫcos͑Q␴͒Ϫ2͔ϩ 2 ␾c␣͓͑4Q ␴ Ϫ24Q␴͒ 3 2 4 4 2 2 ϩ 12Q␴Ϫ␭3Q3␴3 1Ϫ ϩ sin͑Q␴͒ . ϫsin͑Q␴͒Ϫ͑Q ␴ Ϫ12Q ␴ ϩ24͒ ͫ ͩ ␭ ␭3 ͪͬ ͮ ϫcos͑Q␴͒ϩ24͔͖, ͑A5͒ ͑A6͒

͓1͔ S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 ͑1954͒. ͓26͔ I. Markovic and R. H. Ottewill, Colloid Polym. Sci. 264,65 ͓2͔ A. Vrij, Pure Appl. Chem. 48, 471 ͑1976͒. ͑1986͒. ͓3͔ A. P. Gast, C. K. Hall, and W. B. Russel, J. Colloid Interface ͓27͔ T. P. O’Sullivan and M. E. Vickers, J. Appl. Cryst. 24, 732 Sci. 96, 251 ͑1983͒. ͑1991͒. ͓4͔ H. N. W. Lekkerkerker et al., Europhys. Lett. 20, 559 ͑1992͒. ͓28͔ M. Kotlarchyk, S.-H. Chen, J. S. Huang, and M. W. Kim, ͓5͔ D. H. Napper, Polymeric Stabilization of Colloidal Disper- Phys. Rev. A 29, 2054 ͑1984͒. sions ͑Academic, New York, 1983͒. ͓29͔ X.-L. Wu, P. Tong, and J. S. Huang, J. Colloid Interface Sci. ͓6͔ B. L. Carvalho et al., Macromolecules 26, 4632 ͑1993͒. 148, 104 ͑1992͒. ͓7͔ T. Biben and J.-P. Hansen, Phys. Rev. Lett. 66, 2215 ͑1991͒. ͓30͔ J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, ͓8͔ D. Frenkel and A. A. Louis, Phys. Rev. Lett. 68, 3363 ͑1992͒. 2nd ed. ͑Academic, London, 1991͒. ͓9͔ M. Dijkstra and D. Frenkel, Phys. Rev. Lett. 72, 298 ͑1994͒. ͓31͔ M. Wertheim, Phys. Rev. Lett. 8, 321 ͑1963͒. ͓10͔ Y. Rosenfeld, Phys. Rev. Lett. 72, 3831 ͑1994͒. ͓32͔ E. Thiele, J. Chem. Phys. 38, 1959 ͑1963͒. ͓11͔ J. Bibette, D. Roux, and F. Nallet, Phys. Rev. Lett. 65, 2470 ͓33͔ B. H. Zimm, J. Chem. Phys. 14, 164 ͑1946͒. ͑1990͒. ͓34͔ B. H. Zimm, J. Chem. Phys. 16, 1093 ͑1948͒. ͓12͔ S. Sanyal, N. Easwar, S. Ramaswamy, and A. K. Sood, Euro- ͓35͔ H. Benoit and M. Benmouna, Polymer 25, 1059 ͑1984͒. phys. Lett. 18, 107 ͑1992͒. ͓36͔ B. Hammouda, Adv. Polym. Sci. 106,87͑1993͒. ͓13͔ J. S. van Duijneveldt, A. W. Heinen, and H. N. W. Lek- ͓37͔ J. Mays, N. Hadjichristidis, and L. J. Fetters, Macromolecules kerkerker, Europhys. Lett. 21, 369 ͑1993͒. 17, 2723 ͑1984͒. ͓14͔ P. Kaplan, J. L. Rouke, A. G. Yodh, and D. J. Pine, Phys. Rev. ͓38͔ J. Mays and L. J. Fetters, Macromolecules 22, 921 ͑1989͒. Lett. 72, 582 ͑1994͒. ͓39͔ N. S. Davidson et al., Macromolecules 20, 2614 ͑1987͒. ͓15͔ A. Imhof and J. K. G. Dhont, Phys. Rev. Lett. 75, 1662 ͑1995͒. ͓40͔ N. W. Ashcroft and J. Lekner, Phys. Rev. 154,83͑1966͒. ͓16͔ R. Piazza and G. D. Pietro, Europhys. Lett. 28, 445 ͑1994͒. ͓41͔ R. V. Sharma and K. C. Sharma, Phys. Lett. 56A, 107 ͑1976͒. ͓17͔ U. Steiner, A. Meller, and J. Stavans, Phys. Rev. Lett. 74, 7450 ͓42͔ R. V. Sharma and K. C. Sharma, Physica A 89, 213 ͑1977͒. ͑1995͒. ͓43͔ P.-G. de Gennes, Scaling Concepts in Polymer Physics ͑Cor- ͓18͔ S. M. Ilett, A. Orrock, W. Poon, and P. Pusey, Phys. Rev. E nell University Press, Ithaca, 1979͒. 51, 1344 ͑1995͒. ͓44͔ X. Ye, T. Narayanan, P. Tong, and J. Huang, Phys. Rev. Lett. ͓19͔ W. B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal 76, 4640 ͑1996͒. Dispersions ͑Cambridge University Press, Cambridge, 1989͒. ͓45͔ J. Bibette, J. Colloid Interface Sci. 147, 474 ͑1991͒. ͓20͔ Colloid-Polymer Interactions, ACS Symposium Series 532, ͓46͔ Y. Mao, M. E. Gates, and H. N. W. Lekkerkerker, Physica A edited by P. Dubin and P. Tong ͑American Chemical Society, 222,10͑1995͒. Washington, DC, 1993͒. ͓47͔ P. B. Warren, S. M. Ilett, and W. C. K. Poon, Phys. Rev. E 52, ͓21͔ B. Vincent, P. F. Luckham, and F. A. Waite, J. Colloid Inter- 5205 ͑1995͒. face Sci. 73, 508 ͑1980͒. ͓48͔ J. Y. Walz and A. Sharma, J. Colloid Interface Sci. 168, 485 ͓22͔ H. D. Hek and A. Vrij, J. Colloid Interface Sci. 84, 409 ͑1981͒. ͑1994͒. ͓23͔ P. Tong, T. A. Witten, J. S. Huang, and L. J. Fetters, J. Phys. ͓49͔ J. F. Joanny, L. Leibler, and P.-G de Gennes, J. Polym. Sci. ͑Paris͒ 51, 2813 ͑1990͒. Polym. Phys. Ed. 17, 1073 ͑1979͒. ͓24͔ M. Nierlich, F. Boue, A. Lapp, and R. Oberthur, Colloid ͓50͔ A. P. Gast, C. K. Hall, and W. B. Russel, Faraday Discuss. Polym. Sci. 263, 955 ͑1985͒. Chem. Soc. 76, 189 ͑1983͒. ͓25͔ I. Markovic et al., Colloid Polym. Sci. 262, 648 ͑1984͒. ͓51͔ A. P. Gast and L. Leibler, Macromolecules 19, 686 ͑1986͒.