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Confined star polymers
Abstract
The behavior of star polymers confined to a narrow slit in a good solvent is investigated by using scaling ("blob") analysis. In the confined star we distinguish (1) an interior region with spherical symmetry in which the confinement has no effect, (2) an intermediate region where the global spherical symmetry is lost, but the blob structure is still three-dimensional, and (3) an exterior region characterized by cylindrical symmetry. The concentration profile and the radius of the isolated star are obtained as well as its confinement free energy. Concentration effects are discussed. For this system we find six regimes rather than the five found for confined linear chains.
... A polymer chain of size N confined below its gyration radius, say by two parallel walls at a distance h, behaves as an effective chain of blobs. 15,16 The blob size saturates the monomer position fluctuations correlation functions normal to the walls so that the macroscopic properties resulting from the chain entropy are strongly impacted. The chains are laterally stretched. ...
... In this case, the grafted patch resembles a star polymer under confinement. For a dilute solution of star polymers confined in a slit of width h, Halperin and Alexander 16 ...
... Because radial size of all fs is larger than the confinement level ⟨r xy i ⟩ > h (Figure 3a), the fs are located in either the intermediate or exterior domain. The mean position of each fs of the confined star polymer inFigure 3ccan be predicted by integrating the radial concentration of the monomers over radius of that fs and compared with the total volume of the star polymer of length n i leading to16 ...
... The experiments showed that the force acting on the confined polymer chains was repulsive and long-ranged [2]. Recently, such confined systems were frequently studied by means theoretical considerations and computer simulations [3][4][5][6][7][8][9][10]. Differences in the static and dynamic properties as well the phase behavior of chains in the bulk and in the confinement were shown. ...
... However, most of these studies concerned linear homopolymers and block copolymers chains only. Simple scaling theory of confined star-branched chains was done by Halperin and Alexander [5] while Monte Carlo simulations of simple models of confined star polymers were carried out by Romiszowski and Sikorski [9] and Sikorski and Romiszowski [10]. ...
... It can be explained by the fact that near the surfaces polymer segments try to locate parallel to the walls and the perpendicular direction is discriminated. In the middle of the slit more perpendicular orientations can appear in spite of the fact that the segment density is the highest there [5]. ...
We studied the properties of simple models of linear and star-branched polymer chains confined in a slit. The polymer chains were built of united atoms and were restricted to a simple cubic lattice. Two macromolecular architectures of the chain linear and star-branched with three branches (of equal length) were studied. The excluded volume was the only potential introduced into the model and thus, the system was athermal. The chains were put between two parallel and impenetrable surfaces. Monte Carlo simulations with a sampling algorithm based on chain’s local changes of conformation were carried out. The differences and similarities in the global size and the structure and of linear and star-branched chains were shown and discussed.
... irrespective of chain length. The dashed line in Fig. 1b shows that this result is in good semiquantitative agreement with the MD data although it relies on the blob picture of a linear chain with excluded-volume interactions rather than on the geometrically more involved blob analysis of the deformation of a multiarm brush 23 . The agreement suggests that it is worthwhile seeking a coarse-grained interpretation that does not depend on the details of the macromolecular architecture, and here we propose two complementary continuum theories. ...
Softness is an essential mechanical feature of macromolecular particles such as polymer-grafted nanocolloids, polyelectrolyte networks, cross-linked microgels as well as block copolymer and dendrimer micelles. Elasticity of individual particles directly controls their swelling, wetting, and adsorption behaviour, their aggregation and self-assembly as well as structural and rheological properties of suspensions. Here we use numerical simulations and self-consistent field theory to study the deformation behaviour of a single spherical polymer brush upon diametral compression. We observe a universal response, which is rationalised using scaling arguments and interpreted in terms of two coarse-grained models. At small and intermediate compressions the deformation can be accurately reproduced by modelling the brush as a liquid drop, whereas at large compressions the brush behaves as a soft ball. Applicable far beyond the pairwise-additive small-strain regime, the models may be used to describe microelasticity of nanocolloids in severe confinement including dense disordered and crystalline phases.
... However, other combinations of (f, α) can be used to select highervalency particles that can be used to assemble denser, and possibly ordered, phases [19,27]. Additional control could be provided by confining the system, effectively reducing its dimensionality to generate two-dimensional or quasi two-dimensional phases with distinct symmetries and properties [43,44]. ...
The fabrication of versatile building blocks that are reliably self-assemble
into desired ordered and disordered phases is amongst the hottest topics in
contemporary material science. To this end, microscopic units of varying
complexity, aimed at assembling the target phases, have been thought, designed,
investigated and built. Such a path usually requires laborious fabrication
techniques, especially when a specific funcionalisation of the building blocks
is required. Telechelic star polymers, i.e., star polymers made of a number $f$
of di-block copolymers consisting of solvophobic and solvophilic monomers
grafted on a central anchoring point, spontaneously self-assemble into soft
patchy particles featuring attractive spots (patches) on the surface. Here we
show that the tunability of such a system can be widely extended by controlling
the physical and chemical parameters of the solution. Indeed, at fixed external
conditions the self-assembly behaviour depends only on the number of arms
and/or on the ratio of solvophobic to solvophilic monomers. However, changes in
temperature and/or solvent quality makes it possible to reliably change the
number and size of the attractive patches. This allows to steer the mesoscopic
self-assembly behaviour without modifying the microscopic constituents.
Interestingly, we also demonstrate that diverse combinations of the parameters
can generate stars with the same number of patches but different radial and
angular stiffness. This mechanism could provide a neat way of further
fine-tuning the elastic properties of the supramolecular network without
changing its topology.
... For high molecular weights (large N), the pc-CP resembles a n-arm star polymer where a ¼ 2. 50 For low N, the system behaves more like n independent chains with no in-plane connement (no interactions between the arms) where a ¼ 1. The free energy expression given in eqn (1) can be seen as a more general form of those derived for cylindrical macromolecular brushes 51 and conned star polymers, 52 which are systems that also exhibit equilibrium conformations derived from a competition between steric repulsions and entropic stretching. By minimizing the free energy with respect to R at constant d, (vDF pc-CP /vR) d ¼ 0, we obtain the scaling behavior of the equilibrium corona width as a function of the degree of polymer conjugation: R $ n (aÀ1)/4 N 3/4 . ...
Peptide self-assembly, ubiquitous in biology, is one of the most promising 'bottom-up' approaches for the generation of synthetic supramolecular architectures. However, directing the self-assembly of functional peptides into predictable ordered structures most often requires precise tuning of weak intermolecular forces. Existing strategies are generally based on specific interactions between molecular mediators that require complex chemical synthesis pathways and elaborated design rules. Here we establish a theoretical framework that delineates a generic route towards directing the self-assembly of small peptides by simply using entropic forces generated by the polymer chains attached to the peptides. We demonstrate the viability of this concept for polymer-conjugated peptide nanotubes using coarse-grained molecular dynamics (CGMD) simulations combined with theoretical calculations. We show that conjugated polymer chains create an entropic penalty due to chain confinement upon assembly, and illustrate that the self-assembly process can be directed by merely varying the degree of polymer conjugation. Specifically, the entropic penalty, and consequently, the binding energy between peptides can be greatly varied by changing the length and the number of conjugated polymers. Extending this concept for peptides with different degrees of conjugation reveals a path towards controlling the stacking sequence of binary mixtures. Remarkably, we find that a large disparity in the conjugation degree of the two peptides results in a preference towards alternating mixed sequences that minimize the entropic penalty of confinement in the thermodynamic limit. Our study explains recent experiments on polymer-peptide conjugates and sets the stage for utilizing entropic forces to guide the stacking sequence of functional macrocycles in tubular assemblies.
... In the present paper, we shall be concerned with the escape transition when the macromolecule is not a simple linear chain but has a star polymer architecture3637383940. In the limit where H is much smaller than the radius of a free star, the configuration of a star polymer with f arms confined into a slit of width H is essentially a quasi-two-dimensional star polymer, where each arm occupies a slice with an angle 2π/f cut from a cylinder of height H and radius R (f ) with41424344 R (f ) = f 1/4 (H/a) −1/4 N 3/4 . ...
The escape transition of a polymer "mushroom" (a flexible chain grafted to a
flat non-adsorbing substrate surface in a good solvent) occurs when the polymer
is compressed by a cylindrical piston of radius $R$, that by far exceeds the
chain gyration radius. At this transition, the chain conformation abruptly
changes from a two-dimensional self-avoiding walk of blobs (of diameter $H$,
the height of the piston above the substrate) to a "flower conformation", i.e.
stretched almost one-dimensional string of blobs (with end-to-end distance
$\approx R$) and an "escaped" part of the chain, the "crown", outside the
piston. The extension of this problem to the case of star polymers with $f$
arms is considered, assuming that the center of the star is grafted to the
substrate. The question is considered whether under compression the arms escape
all together, or whether there occurs an arm by arm escape under increasing
compression. Both self-consistent field calculations and Molecular Dynamics
simulations are found to favor the latter scenario.
... With progress in microtechniques, efforts have been directed toward separation of DNA molecules with the help of well-defined electrophoretic matrices that lend themselves to mass fabrication of reproducible matrices [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Novel and efficient matrices based on the new concept of entropic barrier transport have been developed which allow better resolutions in electrophoretic separation of DNA molecules [28][29][30][31][32]. These studies have had two main purposes: (i) to understand the mechanism of separation of DNA molecules, especially after the development of fluorescence microscopy with sensitive cameras, which allows tracing a single DNA molecule inside such microstructures [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]; and (ii) to develop an easier, faster, and more dependable method for genomic analyses [33]. ...
The dependence of the mobility of DNA molecules through an hexagonal array of micropillars on their length and the applied electric field was investigated and it was found that mobility is a nonmonotonic function of their length. Results also revealed that the size dependence of the DNA mobility depends on the applied electric field and there is a crossover around E approximately 25 V/cm for the mobility of lambda-DNA and T4-DNA. These observations are explained in terms of the diffusion process inside the structure affected by the solvent and are modeled using the Langevin and its corresponding Fokker-Planck equations. The phenomenon is generalized under three regimes in a phase diagram relating the electric field and the DNA lengths. The model and the associated phase diagram described here provide an explanation for the conflicting results reported by previous authors (Han et al. on the one hand, and Duong et al. and Inatomi et al. on the other) about the dependence of mobility on the DNA size in lattices near or below the radius of gyration.
Using molecular dynamics and Monte Carlo simulations as well as analytical considerations, we study the arm-retraction dynamics of a three-arm star polymer in a narrow nanotube as a function of arm length, N, and tube diameter, D. The system dynamics is analyzed and compared to the bistable collective behavior of a pair of polymer chains tethered in a nanopore. The bistability arises from alternate flipping of one arm of the star into the pore section occupied to that moment by a single arm only. We derive analytical expressions for the free-energy landscape of an arm flip and determine the barrier height as a function of N and D. In the related case of two chains in a narrow tube, we demonstrate that correlations lead to a bimodal distribution of the chain-end positions whereas in a polymer brush, made of equivalent chains at the same grafting density, one observes a single peak only. The residence time distribution between consecutive arm flips is shown to follow a power-exponential relationship, demonstrating good agreement between theory and simulation.
抄録
ポリエチレングリコール–ブロック–ポリ部分ベンジルエステル化アスパラギン酸が水中で形成するミセルの内部構造に関して，化学組成との関係を検討した．ベンジル化率(FBzl)とアスパラギン酸残基数(DPAsp)の異なる9試料を合成し，光散乱とX線散乱を用いてミセルの会合数(Nagg,w)，コアの半径(RC)，およびその他のパラメータを決定した．Nagg,wとRCを決定する主な因子は，コアの疎水性を反映するFBzlであった．これまで高分子ミセルの光散乱は光学精製が難しく会合数などの測定は困難だったが，フィールドフローフラクショネーション(FFF)で光学精製後，これに接続された光散乱検出器で各分画の分子量を測定することで解決される．また，疎水性薬剤としてレチノイドのひとつであるLE540をコアに内包させたときのミセル構造の変化についても紹介する．
The free energy cost of confining a star polymer where $f$ flexible polymer
chains containing $N$ monomeric units are tethered to a central unit in a slit
with two parallel repulsive walls a distance $D$ apart is considered, for good
solvent conditions. Also the parallel and perpendicular components of the
gyration radius of the star polymer, and the monomer density profile across the
slit are obtained. Theoretical descriptions via Flory theory and scaling
treatments are outlined, and compared to numerical self-consistent field
calculations (applying the Scheutjens-Fleer lattice theory) and to Molecular
Dynamics results for a bead-spring model. It is shown that Flory theory and
self-consistent field (SCF) theory yield the correct scaling of the parallel
linear dimension of the star with $N$, $f$ and $D$, but cannot be used for
estimating the free energy cost reliably. We demonstrate that the same problem
occurs already for the confinement of chains in cylindrical tubes. We also
briefly discuss the problem of a free or grafted star polymer interacting with
a single wall, and show that the dependence of confining force on the
functionality of the star is different for a star confined in a nanoslit and a
star interacting with a single wall, which is due to the absence of a symmetry
plane in the latter case.
Intramolecular structures in amphiphilic graft copolymers with hydrophilic side chains and hydrophobic backbone have been studied by molecular dynamics simulations. In accordance with earlier theoretical predictions [Borisov, O.; Zhulina, E. Macromolecules 2005, 38, 2506−2514], we have found that balance of repulsive and attractive intramolecular interactions may result in the pearl-necklace-type conformations. The globular “pearls” are formed by collapsed segments of the main chain comprising multiple spacers; these pearls are stabilized against aggregation by repulsive interactions of the hydrophilic grafts. The size of the pearls is controlled by the intramolecular hydrophilic−hydrophobic balance, whereas the total number of pearls in the graft copolymer depends on the length of the main chain. For graft copolymer with relatively long spacers we have observed intramolecular conformational transition from pearl-necklace structure to unimolecular cylindrical micelle upon a decrease in the solvent quality for the main chain. The results of simulations, in particular behavior of polymers of finite chain length, are rationalized on the basis of revised and extended scaling theory.
Transport of star polymers under pressure-driven flow in a pipe with pipe radius being at least twice the size of polymers has been examined with standard dissipative particle dynamics (DPD) simulations. Equilibrium dynamics of star polymers in bulk solution were found to obey the Zimm model very well, indicating that DPD simulation correctly incorporates the hydrodynamic interaction in the stars. Under pressure-driven flow, star polymers with more arms were found to migrate toward the center of pipe more, leading to a net faster velocity and hence a shorter retention time in the pipe. The stretching of star polymers along the flow was found to follow similar scaling behavior as the linear polymer chains, except that the Weissenberg number Wi for the stars should be reduced by arm number f. After rescaling of the Weissenberg number, the stretch ratio Sx, defined as the ratio of square of radius gyration of the chains along the flow, Rgx2, over its corresponding value in dilute bulk solution, was found to scale with Wi linearly when Wi 1. The compression of the chains in the dimension perpendicular to the flow Sy were found to scale with Wi−0.5 when Wi 1.0.
The influence of polymer concentration on the partitioning of flexible macromolecules into adsorptive pores was examined by simulations in an open system under good solvent conditions. In dilute solutions, the partition coefficient K(ε,φ) is sensitive to small variations of both polymer−pore attraction strength ε and concentration φ, whereas in semidilute solutions both influences level off. At weak attraction below critical adsorption energy εc, the coefficient K increases with φ in a fashion that resembles the weak-to-strong penetration transition found for purely steric exclusion (ε = 0) but modified as if the width of pores effectively increased. At moderate attraction above the critical condition the increasing concentration brings about a dramatic drop in the value of K. Additionally, the K(φ) dependences in attractive pores were calculated by an approximate method based on expressions for chemical potentials, and the results were in good agreement with the simulation data. The critical adsorption energy εc for excluded volume chains was found to depend on concentration; the compensation point K = 1 was located at about φ = 0.012 irrespective of the pore width. The energy and entropy contributions to the free energy of confinement were calculated and the compensation of their values at the critical condition was demonstrated. Furthermore, it was shown that the effect of concentration in polymer partitioning with adsorptive pores can alternatively be regarded as a confined adsorption and two alternatives of the adsorbed amount Γ were calculated. The adsorption and depletion concentration profiles φI(x) in the pore in equilibrium with the bulk solution were presented and their variation with concentration φ and attraction ε was analyzed.
The flux of linear and star-branched polyisoprene through micropprous membranes was measured in order to determine the role of molecular structure in hindered diffusion. As the radius of the pore approached the hydrodynamic radius of the polymer, the effective diffusion coefficient decreased. The reduction for the star-branched polymers was much larger than for the linear polymers. The magnitude of the decrease did not agree with a theoretical model that combined the pore partition coefficient of random flight chains with the hydrodynamic resistance of hard spheres. However, the data correspond qualitatively with recent results on size exclusion chromatography of branched polymers.
The structure of polyelectrolyte stars and spherical brushes is discussed using a local force balance argument that balances the osmotic pressure and the tension in the chains locally for a given radial position. In general, there are four distinct regions where the segment density profile Phi(r); is dominated by one of the interactions: (i) in the outer region, electrostatic interactions dominate, (ii) in the intermediate region, binary interactions dominate, (iii) in the inner region ternary interactions dominate, and finally (iv) at the center of the star (of the order of a few segment lengths), the density is unity. For poor solvents we predict a collapse induced by the attractive binary interactions that leads to a two-phase star or brush. The inner dense layer is stabilized by ternary interactions whereas the outer dilute layer is dominated by electrostatic repulsions-the segment density changes discontinuously from one to the other (and the intermediate density region disappears). The magnitude of the collapse transition is found to increase with the chain length for stars and spherical brushes, unlike planar brushes. The star is expected to collapse continuously with the reduction in the solvent quality-unlike a first-order transition for planar brushes. Planar polyelectrolyte brushes can also display two-phase regions; however, the full SCF approach is warranged in this case. Collapse induced by n-clusters in uncharged polymers is also shown to lead to two-phase brushes.
The confinement of liquid crystalline polymers dissolved in nematic solvents is analyzed using a mean field theory. For homeotropic boundary conditions the confinement is associated with a tilting phase transition reminescent of the Frederiks transition. When a confined layer is subjected to a suitable magnetic field, reentrant phase behavior is predicted. A partial repression of the tilting transition is expected when the orientation of the field is appropriately changed.
The confinement of liquid crystalline polymers dissolved in a nematic solvent is analysed using a mean-field theory. For homeotropic anchoring the confinement triggers a tilting phase transition. The transition is reminiscent of the Frederiks transition, however it may be either first or second order.
In this article, we investigate the glass transition in polystyrene melts and free-standing ultra-thin films by means of large-scale computer simulations. The transition temperatures are obtained from static (density) and dynamic (diffusion and orientational relaxation) measurements. As it turns out, the glass transition temperature of a 3 nm thin film is ∼60 °K lower than that of the bulk. Local orientational mobility of the phenyl bonds is studied with the help of Legendre polynomials of the second-order P2(t). The α and β relaxation times are obtained from the spectral density of P2(t). Our simulations reveal that interfaces affect α and β-relaxation processes differently. The β relaxation rate is faster in the center of the film than near a free surface; for the α relaxation rate, an opposite trend is observed. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1160–1167, 2010
Using the change in conductance of a single nanometer-wide protein pore of the R-hemolysin channel to detect pore occupancy by polymers, we measure the equilibrium partitioning of differently sized poly(ethylene glycol)s as a function of polymer concentration in the bulk solution. In the semidilute regime, increased polymer concentration results in a sharp increase in polymer partitioning. Quantifying solution nonideality by osmotic pressure and taking the free energy of polymer confinement by the pore at infinite dilution as an adjustable parameter allows us to describe polymer partitioning only at low polymer concentrations. At larger concentrations the increase in partitioning is much sharper than the model predictions. The nature of this sharp transition between strong exclusion and strong partitioning might be rationalized within the concepts of scaling theory predicting this kind of behavior whenever the correlation length of the monomer density in the semidilute bulk solution becomes smaller than the pore radius. Specific attractive interactions between the protein pore and the polymer that exist in addition to the entropic repulsion accounted for in the present study may also play a role.
Probing the thermodynamics of polymer confinement in small pores, single DNA molecules were imaged by fluorescence microscopy as they partitioned within a pair of adjacent interconnected spherical cavities prepared by the colloidal templating method. This method produces. pores of precisely known geometry and controlled level of confinement. As expected.. a polymer weakly or moderately confined by cavities of unequal diameter was observed to maximize its configurational entropy by partitioning toward the larger cavity. The polymer's cavity-to-cavity partition coefficient could be derived from -the visualized bias in cavity occupation, and segmental excluded volume notably influences how this coefficient depends on chain length and two cavity sizes. A DNA molecule strongly confined within a pair of equalsized cavities predominately adopts "bridging" configurations, splitting its segments between the two cavities. Segmental excluded-volume stabilizes such configurations; the stability, reveak that a chain must overcome a doubly peaked energy barrier to move between cavities.
As a step towards understanding the thermodynamics of multi-branched polymer systems, we look at a lattice model of a uniform branched polymer with fixed topology interacting with a surface and ask for the free energy of the polymer as the number of monomers which compose the polymer goes to infinity. The conformations of a uniform branched polymer with fixed topology are modelled by embeddings of a graph in the simple cubic lattice. Rigorous results about this model are reviewed. The results suggest that large branched polymers in three dimensions interacting with a plane have the same free energy as large linear polymers interacting with a plane; the same is not true, however, for the corresponding two-dimensional problem where the polymer interacts with a line.
Poly(ethylene glycol)-block-poly(partially benzyl-esterified aspartic acid), denoted by PEG-P(Asp(Bzl)), is one of the most examined blockcopolymers for drug carriers. However, little is known about fundamental physical properties. Nine samples of PEG-P(Asp(Bzl)) with different benzylation fractions (F(Bzl)) and aspartic chain lengths (DP(Asp)) were synthesized, and the aggregation number (N(agg)), core radius (R(C)), and other structural parameters were determined with combination of light scattering and synchrotron X-ray small-angle scattering. The major factor to determine N(agg) and R(C) was found to be F(Bzl), i.e., the hydrophobic nature of the core, even though F(Bzl) was changed in the relatively small composition range from 66 to 89 mol %. When we compared the data for the same F(Bzl), the scaling theory was consistent with the core chain length dependence of both core and micelle sizes. The overcrowding nature of the tethered PEG chains on the micelles was increased about 1.3-2.9 times with increasing N(agg) compared with the unperturbed state in solutions.
Intramolecular self-assembly in polysoaps qualitatively modifies their deformation behavior. The modified elasticity reflects the coupling of the chain deformation to the internal degrees of freedom associated with the existence of intrachain micelles. The equilibration of this secondary structure tends to lower the restoring force associated with the deformation in accordance with the LeChatelier principle. The force laws for extension and for confinement into a slit are derived for the case of a linear string of spherical intrachain micelles. In both situations the chain behaves as a string of stars for weak deformations. Force laws characteristic of simple linear chains are obtained for strong deformations. Novel force laws, reflecting strong coupling to the secondary structure, are obtained at the intermediate regime. For intermediate extensions, the tension in the chain, f, is independent of R, the end to end distance, up to logarithmic corrections, i.e., f Ro ln g(R), where g(R) is a slowly increasing function of R. For intermediate confinement f D-28/23, where D is the spacing between the two nonadsorbing walls.
Movement of macromolecules through low concentration agarose gels was investigated with linear poly(styrenesulfonate), linear
DNA, star-shaped poly(styrenesulfonate), and circular DNA. Mobilities of weakly entangled flexible macromolecules were independent
of molecular radius; within a homologous chemical sequence, electrophoretic separation at low field strengths depended solely
on the degree of polymerization. These observations cannot be explained either by sieving or by reptation mechanisms; transport
was apparently controlled by spatial variations of chain configurational entropy. Only when the chain was highly entangled
did chain topology affect mobility. Evidence for entropically regulated transport clarifies how gel electrophoresis separates
flexible macromolecules.
We consider solid surfaces, partly covered with flexible, neutral, linear polymers (by adsorption or by grafting), wetted by a liquid which is a good solvent of the poymer. We give formulae for the spreading coefficient S as a function of chain length, solvent quality and adsorption strength. We also discuss the wetting films obtained in spreading a droplet of (non volatile) solvent : the equilibrium thickness e of the film is a compromise between S (favouring thin films) and the coil entropies (favouring thick films) : in most cases e is rather small. Finally we analyse the spreading on certain « patchy » coatings where the film thicknesses can change dramatically with time. Nous discutons le comportement de surfaces greffées, ou couvertes de polymères adsorbés, et exposées à un liquide qui est un bon solvant du polymère. Notre premier objectif est le paramètre d'étalement S, que nous calculons en fonction de la longueur des chaînes, de la qualité du solvant, et aussi de la force de l'adsorption. Par ailleurs, nous analysons les films de mouillage, obtenus à S > 0 par étalement d'un solvant non volatil. L'épaisseur e du film est un compromis entre S (qui favorise des films minces) et l'entropie des chaînes confinées dans le film (qui favorise des films épais). Dans la plupart des cas, l'épaisseur d'équilibre e s'avère plutôt petite. Enfin, nous analysons l'étalement sur une surface bigarrée : par exemple, une zone circulaire couverte de polymère, entourée par du solide nu, ou l'inverse. Dans ces situations, l'épaisseur du film sur la zone centrale doit changer brutalement quand la goutte atteint le bord de cette zone.
The dimensions of star-branched macromolecules in dilute solutions were investigated using scaling concepts. The quality of the solvent, the rigidity of star branches, their number and degree of polymerization are taken into account. The relationship between the approaches based on scaling approximations and mean-field approximations is shown. The results obtained are compared to the data of Monte-Carlo simulations and other theoretical results.
The molecular weight (Mw) and temperature (T) dependence of the radius of gyration (RG) and hydrodynamic radius (RH) of a polymer in a dilute solution are investigated. The theoretical predictions are compared with experimental results on polystyrene in various solvents as functions of Mw and T. It is found that the existing data fall in a region of values where RH cannot be represented by a simple power law RH ∼ Nν′, whereas most of the data on RG satisfy RG ∼ Nν. It is concluded that a power law fit to data would yield a ν′ < ν in this region, even though the theory predicts ν′ = ν in the asymptotic region. The quantitative aspects of the blob theory are also discussed and compared in some cases to the modified Flory theory.
The chain conformational renormalization group method is applied to the calculation of the distribution functions for intersegment distance vectors, the mean square intersegment distances, the mean square radius of gyration, the osmotic second virial coefficient, and the interpenetration function for f-branched star polymers. The calculations provide the full crossover dependence of all quantities on the strength of the excluded volume interaction and the chain length. The results provide a picture of excluded volume effects on the conformations of star polymers which is rather different from that assumed in recent scaling models. Some of the important information concerning the structure of the star polymers emerges from the prefactors since many power law exponents are independent of the number of branches, thereby emphasizing the importance of evaluating these prefactors.
Two colloidal particles, each with f long polymers grafted to it, experience an effective mutual repulsion in a good solvent. The effective interaction potential depends logarithmically on the separation between the particles, with a universal coefficient depending on the functionality f. We deduce this coefficient for f = 1 and 2 using results of des Cloizeaux for interior correlations within a self-avoiding walk. For f ≫ 1 we deduce that this coefficient varies as f3/2, using the semidilute scaling properties of the polymeric "corona" around a colloidal particle. We show that the repulsive interaction arising from the polymers can be sufficient to stabilize a dispersion of colloidal particles, despite their inherent tendency to precipitate owing to van der Waals attractions. The repulsions also give rise to a peak in the scattering structure factor S(q) for concentrations near the overlap concentration c*. We show that the height of this peak scales as f3/2 and predict a macrocrystalline phase for c near c* and for sufficiently large f.
The conformational and dynamic behavior of polystyrene star molecules with 3, 12, and 18 arms has been studied by means of dynamic and static light scattering. Measurements have been carried out in cyclohexane at 34.5°C and in toluene at 20°C. The g (≡〈S2〉b/〈S2〉 1),) and h (≡Rhb/Rhl) factors were determined in both solvents by comparison of the properties of branched and linear chains. In addition the ρ (≡〈S2〉1/2/Rh) parameters were obtained from the combined measurements of the radius of gyration 〈S2〉1/2 and the hydrodynamic radius Rh. The g and h factors have nearly the same values in both solvents, with the exception of the 18-arm star, where ge > g and hθ > h. Furthermore, the experimental data are larger than predicted by theory, and the deviations appear to increase with increasing number of arms. This effect is taken as evidence for a perturbation of Gaussian statistics of a single chain by the presence of the others. The ρ values were found to be 14-25% lower than expected from theory. In the good solvent this ρ parameter is about 20% larger than in the θ-solvent. The three parameters g, h, and ρ also show a chain length dependence. The increase of g and h with decreasing M̄w is explained by a chain expansion due to the high segment density, which becomes more effective for short arms. This result is in qualitative agreement with a recent theory by Daoud and Cotton. The increase of ρ at lower M̄w indicates an increasing influence of the free-draining term. The effect of excluded volume is represented by the factor g(α) = αSb2/αSl2> where αSb and αSl are the chain expansion factors of the branched and linear chains, respectively. The curve g(α) as a function of M̄w shows a minimum at about M̄w = 1.5 × 105. In this region αSbt51, ≃ 1; the star molecule is already in the θ-solvent considerably stretched and the excluded volume has no further effect. For the linear chains αSl is still appreciably larger than those of the linear chains. The second virial coefficients were found to be smaller than those of the linear chains. The effect is, however, less pronounced than predicted by theory.
The authors study the total numbers of lattice trees with specified topologies. For strongly embeddable (or site) clusters with ni branching points of degree i, they show how to prove rigorously that the growth constants exist and are all equal to the neighbour-avoiding walk limit nu . They derive some exact upper bounds for the critical exponents associated with the 'star', 'comb' and 'brush' topologies. Exact enumeration data are derived and analysed for both weak and strong embeddings of some stars, combs and brushes on the square, triangular, simple cubic and d=4 simple hypercubic lattices. Using the exact enumeration data for the general d-dimensional simple hypercubic lattice and the exact results for the interior of a Bethe lattice, they derive expansions for the growth constants in inverse powers of the dimensionality. These results are consistent with the growth constants being equal to the appropriate walk limits ( mu or nu ).
It is shown that the dynamical properties of long polymers made of N links, in solutions, converge much more slowly to the asymptotic limit (N → oo) than the static properties. This effect, which has a simple origin, explains why dynamical measurements of critical indices lead to results which seem to disagree with static measurements and recent theories. On montre que les propriétés dynamiques de longs polymères de N maillons en solution convergent beaucoup plus lentement vers leur limite asymptotique (N → oo) que les propriétés statiques. Cet effet dont la raison est simple explique pourquoi les mesures dynamiques des indices critiques conduisent à des résultats qui semblent en désaccord avec les mesures statiques et les théories récentes.
The conformation of macromolecular chains in dilute or semi-dilute solution in a good solvent, confined into a finite slit with thickness D1 and width D2 is examined The conformational evolution (sphere, pancake, cigar with 3d or 2d local correlations) is studied as a function of D1/D2 in the (x, z)-plane where x = R3/D 1, z = (C/C*)3/4, R3 is the radius of gyration and C* the semi-dilution limit of unconfined chains. On examine la conformation de chaînes macromoléculaires en solution diluec ou semi-diluée en bon solvant, confinées dans une fente d'épaisseur D 1 et de largeur D2. L'évolution des différents régimes (sphère, gateau, cigare avec corrélations locales tri- ou bidimensionnelles) est étudiée en fonction du rapport D1/D2 dans le plan (x, z) où x = R3/D1, z = (C/C*)3/4, R3 étant le rayon de giration et C* la limite de semi-dilution pour une chaîne non confinée.
The statistics of crosslinked polymer chains, produced by condensation of polyfunctional units, is described by constrained equilibrium ensembles with fugacities controlling dimer, trimer, endpoint and polymer number. It is shown that this description can be reproduced identically by a statistical field theory from which polymer size distribution functions can be calculated. The field theory, in the mean field approximation in the absence of repulsive interactions, gives critical probabilities and polymer size distribution functions identical to those of Flory and Stockmayer. Modifications to the Flory-Stockmayer theory resulting from repulsive interactions are studied. Within the context of the constrained equilibrium ensembles studied here, the critical properties of gelation and percolation are identical. On donne une description de la statistique des chaînes polymériques basée sur des ensembles à l'équilibre dont les contraintes sont le nombre de dimères, de trimères et de polymères et celui des extrémités. On montre que cette description correspond exactement à une théorie des champs avec laquelle la distribution des tailles des polyméres peut être calculée. La théorie des champs, dans l'approximation de champ moyen en l'absence d'interactions répulsives, donne le seuil de gélation et la distribution des tailles des polymères en accord avec ceux calculés par Flory et Stockmayer. On étudie les modifications de la théorie de Flory-Stockmayer dues à l'addition d'interactions répulsives. Pour les ensembles à l'equilibre avec contraintes étudiés ici, les propriétés critiques de gélation et de percolation sont identiques.
We analyse the theoretical behaviour of macromolecular chains dissolved in a good solvent, and confined into tubes (or slits) of diameter D comparable to the coil radius. The repulsive interactions between monomers are taken into account by a scaling method which goes beyond the usual Flory Huggins approach. For the slit problem, we find five different regimes (depending on the concentration C and on the diameter D) with smooth cross-overs at all boundaries. For the tube problem, one of these regimes disappears and two cross-over lines merge, giving rise to a line of stronger discontinuity. This is related to the fact that, when D decreases down to the monomer size, different coils cannot overlap each other at all. For all regimes that thermodynamic properties, the local correlations and the overall chain size are estimated. The scaling arguments, however, predict only the power laws (in C and D) for all these quantities, and do not give precise numerical coefficients.
A model giving the conformation of a star shaped polymer is proposed by taking into account the radial variation of the monomer concentration phi (r). For an isolated star when increasing r (at the center of the star r equals 0), the variation of phi (r) is first given by a constant value (r less than f**1**/**2 l) then has a (r/1)** minus **1 variation (for f**1**/**2 l less than r less than f**1**/**2 upsilon ** minus **1 l) and finally a(r/l)** minus **4**/**3 variation (for r greater than f**1**/**2 upsilon ** minus **1 l); where f is the number of branches, N the number of monomers in a branch and upsilon and l are the excluded volume and the length associated to a monomer. For all these cases, it is shown that the size of a branch is always larger than that of a linear polymer made of N monomers. Beyond the overlapping concentration the star conformation is obtained from two characteristic lengths essentially: chi (c) a radius inside which the branches of the other stars do not penetrate, this radius defines a domain where the conformation of a star is similar to that of an isolated one. Beyond chi (c) the interpenetration of branches is characterized by a screening length xi (c) very similar to that for semi-dilute solutions of linear polymers. For all these regimes the variation of the size of a star is predicted as a function of N, f, upsilon and c.
Direct measurements of the forces acting between polymer surface phases adsorbed at interfaces have only recently been reported1-3. These were carried out in good solvents and showed that repulsive forces were acting between the adsorbed layers; these forces increased monotonically as the surfaces approached each other. In certain conditions, however, for example, those leading to the flocculation of sterically stabilized colloidal systems, one expects the surfaces bearing the adsorbed phases to attract before repelling each other4,5. I have measured the forces acting between two curved mica surfaces immersed in cyclohexane at 24 °C (a worse than theta solvent at this temperature), each bearing a surface layer of adsorbed polystyrene (M w = 6 × 105). No forces are observed at surface separations larger than about three radii of gyration of the polymer; on closer approach a strong attraction develops between the surfaces, changing to an ultimate repulsion as the surfaces approach closer than about one radius of gyration. Between times of a few minutes and several hours the forces are stable, well behaved and reproducible.