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 1996! 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. @S1063-651X~96!10911-9# PACS number~s!: 82.70.Dd, 61.12.Ex, 65.50.1m, 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 @5,6#. 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 @7–10# 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 @1#. In the study of binary mixtures, such as liquid emulsions @11#, binary colloi- the interactions in colloid-polymer mixtures, it is important dal mixtures @12–15#, colloid-surfactant mixtures @16#, bi- to distinguish between polymers that are adsorbed on the nary emulsions @17#, and mixtures of colloids with nonad- colloidal surfaces and those that are free in solutions, be- sorbing polymers @18#. 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 @1,2# 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 @3,4#. 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/96/54~6!/6500~11!/$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 @23,24# tion effect @3,4,19,20#. 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- I~Q!5Kc Scc~Q!12~KcKp!Scp~Q!1KpSpp~Q!, ~1! cluded. In contrast to many previous experimental studies @19– where the subscripts c and p are used, respectively, to iden- 22#, which mainly focus on the phase behavior of the tify the colloid and polymer and Ka is the scattering length colloid-polymer mixtures, we have recently carried out a la- density of the a component, which has taken into account the ser light-scattering study of the depletion interaction in a contrast between the a component and the solvent. The scat- mixture of a hard-sphere-like colloid and a nonadsorbing tering wave number Q5(4p/l0)sin~u/2!, with l0 being the polymer @23#. In the experiment, the second virial coefficient wavelength and u the scattering angle. The function Sab(Q) of the colloidal particles as a function of the free-polymer in Eq. ~1! has the form concentration was obtained from the measured concentration Na Nb dependence of the scattered light intensity from the mixture. a b 2iQ•~r 2r ! The experiment demonstrated that our light-scattering Sab~Q!5 ^e j l &, ~2! (j (l method is indeed capable of measuring the depletion effect in the colloid-polymer mixture. However, the interpretation where r a is the position of the jth monomer of the a 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 a 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 Sab(Q) measures the interaction between In this paper we report a small-angle neutron scattering the components a and b in the mixture. In the experiment to ~SANS! 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- tial! structure factor Sc(Q) over a suitable range of the scat- ecules become invisible to neutrons (K p50) and Eq. ~1! 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 I~Q!5Kc rcPc~Q!Sc~Q!. ~3! action in the polymer solution than the second virial coeffi- cient does. The colloidal particle chosen for the study con- In the above, rc is the number density of the colloidal par- sisted of a calcium carbonate ~CaCO3! 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 I~Q! both the colloid and the polymer have been well character- Sc~Q!5 2 , ~4! Kc rcPc~Q! 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)51.