Eur. Phys. J. E 5, 275–280 (2001)THE EUROPEANPHYSICAL JOURNAL EcEDP SciencesSociet`a Italiana di FisicaSpringer-Verlag 2001Interaction force between incompatible star-polymers in dilutesolutionM. Benhamouaand F. BenzouineLaboratoire de Physique des Polym`eres et Ph´enom`enes Critiques, Facult´e des Sciences Ben M’sik, B.P. 7955, Casablanca,MoroccoReceived 2 October 2000 and Received in final form 24 January 2001Abstract. We consider a low-density assembly of spherical colloids, such that each is clothed by L end-grafted chemically incompatible polymer chains either of types A or B. These are assumed to be dissolvedin a good common solvent. We assume that colloids are of small size to be considered as star-polymers.Two adjacent star-polymers A and B interact through a force F originating from both excluded-volumeeffects and chemical mismatch between unlike monomers. Using a method developed by Witten and Pin-cus (Macromolecules 19, 2509 (1986)) in the context of star-polymers of the same chemical nature, wedetermine exactly the force F as a function of the center-to-center distance h. We find that this force isthe sum of two contributions Feand Fs. The former, that results from the excluded volume, decays asFe∼ ALh−1,withtheL -dependent universal amplitude AL∼ L3/2. While the second, which comes fromthe chemical mismatch, decays more slowly as Fs∼ χBLh−1−τ,whereτ is a critical exponent whose valueis found to be τ∼=0.40, and χ is the standard Flory interaction parameter. We find that the correspondingL-dependent universal amplitude is BL∼ L(3+τ)/2. Theses forces are comparable near the cores of two ad-jacent star-polymers, i.e. for h ∼ hc∼ aχ1/τ√L (a is the monomer size). Finally, for two star-polymers ofthe same chemical nature (A or B), the force F that simply results from excluded-volume effects coincidesexactly with Fe, and then the known result is recovered.PACS. 82.70.Dd Colloids – 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling –64.75.+g Solubility, segregation, and mixing; phase separation1 IntroductionColloids are particles of mesoscopic size, which are subjectof extensive studies due to their abundant industrial appli-cations. Most colloidal systems are made of grains, whichoften flocculate because of the existence of long-range vander Waals forces. To avoid this flocculation, one may intro-duce adequately soluble polar head polymer chains. Thisis the grafting phenomenon. As a consequence, two adja-cent grains with end-grafted chains repel each other due tothe excluded-volume forces between monomers, which as-sist in stabilizing the colloids. Grafted polymers onto solidsurfaces or fluid interface are of considerable interest, andhave many physicochemical applications [1], such as ad-hesion [2], wetting [3], chromatography [4] and colloidalstabilization [5–7].Consider a set of colloidal particles clothed by L end-grafted polymer chains (or branches) in a dilute solution.We assume that they are of small diameter, so that theycan be regarded as star-polymers. Such an assumptionallows us to take advantage of the results already estab-lished within the star-polymers field. A star-polymer is aae-mail: [email protected] branched macromolecule, which possesses a centralcore from which many end-attached linear chains emerge.The physics of star-polymers is the subject of a great dealof attention from theoretical [8–20] and experimental [21]point of view. Theoretical works pioneered by Daoud andCotton (DC) [9]. Their model is based on a blob picture,according to which, in a dilute solution, each chain orbranch can be viewed as a sequence of growing sphericalblobs.Fourteen years ago, Witten and Pincus (WP) [22] wereinterested in the computation of the interaction force be-tween small spherical colloids with L end-grafted flexiblechains, which is due to the excluded-volume effects. Theyassimilated these clothed particles to star-polymers andused the blob picture of DC. They found that the force de-cays with separation h according to the power law ALh−1,with the L-dependent universal amplitude AL∼ L3/2.Our purpose is precisely an extension of this result to de-termine the interaction force between two chemically in-compatible star-polymers A and B, which are a finite dis-tance h apart. More precisely, the question is: how doesthe chemical mismatch affect the interaction force?276 The European Physical Journal EWe start by considering a low-density assembly of col-loidal particles, such that each is clothed either by end-grafted A or B chains. We assume that the particles aredissolved in a common good solvent. For the sake of sim-plicity, we assume that the particles are small enough tobe considered as star-polymers (A or B). Two adjacentstar-polymers A and B interact via a force F , which re-sults from both the excluded volume and the chemical mis-match between unlike monomers. We are interested in thedetermination of this force as a function of the center-to-center distance h. Since excluded-volume and segregationinteractions are of short range, the effect should be appre-ciable only for separations, h, below the star-polymer size2R ∼ 2aL1/5N3/5, i.e. h<2R, where N is the polymer-ization degree of chains and a is the monomer size.Our findings are the following. Using that method de-veloped by WP [22], we find that the force F is the sumof two contributions Feand Fs. The former results fromthe excluded-volume effects and decays as Fe∼ ALh−1,with the L-dependent universal amplitude AL∼ L3/2.The second, which originates from the chemical segrega-tion between A and B monomers, decays more slowly ac-cording to Fs∼ χBLh−1−τ, with the L-dependent uni-versal amplitude BL∼ L(3+τ)/2. Here, χ is the standardFlory interaction parameter, and τ is a critical exponentwhose value is found to be τ∼=0.40. As we will show be-low, the exponent τ can be related to the cubic anisotropyexponent, as pointed out by Joanny, Leibler and Ball intheir theoretical work [23]. We find that the forces FeandFsbecome comparable near the cores of two adjacent star-polymers, i.e. for h ∼ hc∼ aχ1/τ√L.Finally, for two star-polymers of the same chemicalnature (A or B), the force F simply originates from theexcluded volume, and thus, coincides exactly with Fe.TheWP result is then recovered.The remainder of the presentation proceeds as follows.Section 2 gives a succinct recall of
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