Analytical phase diagrams for colloids and non-adsorbing polymer

We review the free-volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20 (1992) 559] for the phase behavior of colloids in the presence of non-adsorbing polymer and we extend this theory in several aspects: (i) We take the solvent into account as a separate component and show that the natural thermodynamic parameter for the polymer properties is the insertion work Pi(v), where Pi is the osmotic pressure of the (external) polymer solution and v the volume of a colloid particle. (ii) Curvature effects are included along the lines of Aarts et al. [J. Phys.: Condens. Matt. 14 (2002) 7551] but we find accurate simple power laws which simplify the mathematical procedure considerably. (iii) We find analytical forms for the first, second, and third derivatives of the grand potential, needed for the calculation of the colloid chemical potential, the pressure, gas-liquid critical points and the critical endpoint (cep), where the (stable) critical line ends and then coincides with the triple point. This cep determines the boundary condition for a stable liquid. We first apply these modifications to the so-called colloid limit, where the size ratio q(R)=R/a between the radius of gyration R of the polymer and the particle radius a is small. In this limit the binodal polymer concentrations are below overlap: the depletion thickness delta is nearly equal to R, and Pi can be approximated by the ideal (van't Hoff) law Pi=Pi(0)=phi/N, where phi is the polymer volume fraction and N the number of segments per chain. The results are close to those of the original Lekkerkerker theory. However, our analysis enables very simple analytical expressions for the polymer and colloid concentrations in the critical and triple points and along the binodals as a function of q(R). Also the position of the cep is found analytically. In order to make the model applicable to higher size ratio's q(R) (including the so-called protein limit where q(R)>1) further extensions are needed. We introduce the size ratio q=delta/a, where the depletion thickness delta is no longer of order R. In the protein limit the binodal concentrations are above overlap. In such semidilute solutions delta approximately xi, where the De Gennes blob size (correlation length) xi scales as xi approximately phi(-gamma), with gamma=0.77 for good solvents and gamma=1 for a theta solvent. In this limit Pi=Pi(sd) approximately phi(3gamma). We now apply the following additional modifications: With these latter two modifications we obtain again a fully analytical model with simple equations for critical and triple points as a function of q(R). In the protein limit the binodal polymer concentrations scale as q(R)(1/gamma), and phase diagrams phiq(R)(-1/gamma) versus the colloid concentration eta become universal (i.e., independent of the size ratio q(R)). The predictions of this generalized free-volume theory (GFVT) are in excellent agreement with experiment and with computer simulations, not only for the colloid limit but also for the protein limit (and the crossover between these limits). The q(R)(1/gamma) scaling is accurately reproduced by both simulations and other theoretical models. The liquid window is the region between phi(c) (critical point) and phi(t) (triple point). In terms of the ratio phi(t)/phi(c) the liquid window extends from 1 in the cep (here phi(t)-phi(c)=0) to 2.2 in the protein limit. Hence, the liquid window is narrow: it covers at most a factor 2.2 in (external) polymer concentration.     

Microphase separation in two-dimensional athermal polymer solutions on a triangular lattice

The results of Monte Carlo simulations of 2D polymer solutions are presented. The simulations were performed under athermal conditions for long chains (up to 1024 beads) over a full range of polymer concentration phi, explicitly taking into account the solvent molecules. The results obtained for short chains (N < or = 256) are in good agreement with previous simulations whereas for long chains microphase separation is observed below phi = 0.6. This phenomenon is attributed to strong excluded volume interactions in 2D systems. A sort of interpenetration of the coils is also observed.     

Solvent size effect on the depletion layer of a polymer solution near an interface

By varying polymer concentration phi0p and Flory-Huggins parameter chi, the effect of solvent size on the depletion interaction between polymer coils and a hard wall was investigated by the real-space version of self-consistent field theory (SCFT). The depletion profiles and depletion thickness indicated that the depletion effect is strong in less good solvent with large molecular volume. Through the analysis of the respective free energies of polymer coils and solvent molecules, we found that the increment in the translation entropy of the solvent is the key to strengthening the depletion interaction. On the basis of the SCFT results, we define a solvent with volume about one to six times that of the polymer segment as a "middle-sized solvent". The density oscillations previously studied by Van der Gucht et al. and Maassen et al. were also observed in our simulation, and the addition of middle-sized solvent will magnify the amplitude of the oscillations. The solvent-size-dependent depletion interaction may be an explanation for the reduced entanglement and promoted crystallization behavior of polymer coils prepared from the solution with middle-sized solvent.     

Colloid-polymer mixtures in solution with refractive index matched acrylate colloids

Colloid-polymer (CP) mixtures extend between two limiting cases, the colloid limit with the polymer coil size small compared to the colloid radius Rcol and the protein limit with the colloidal particles much smaller in size than the radius of gyration of the polymer chains Rg. In the present work, model systems are developed for the protein limit. The colloid-solvent pairs are optimized in terms of their isorefractivity in order to facilitate the characterization of large polystyrene chains in suspensions of small colloids. The degree of isorefractivity of colloidal particles was successfully evaluated in terms of a reduced scattering intensity. Two polystyrene samples with radii of gyration of Rg = 96 nm and Rg = 78 nm, respectively, are used. The radii of the colloidal particles are close to Rcol = 12 nm, leading to size ratios of Rg/Rcol = 8 and Rg/Rcol = 6.5. Four colloid solvent systems were found to be suitable for polymer characterization by light scattering, one based on silica particles and three systems with acrylate particles. The present investigation is focused on the three acrylate systems: poly(methyl methacrylate) in ethyl benzoate (ETB) at 7 degrees C, poly(ethyl methacrylate) in toluene at 7 degrees C and poly(ethyl methacrylate) in ETB at 40 degrees C. Characterization of PS chains is for the first time performed in colloid concentrations up to 2.5% by weight. In all cases, the size and shape of the polymer chains remain unchanged. A slight mismatch of the colloid scattering or a limited colloid solubility prevented investigation of PS chains at higher colloid concentration.     

Polymer modifies the critical region of the coexisting liquid phases

Several recent conceptual advances, which take advantage of the polymer conformation in the near critical point of coexisting liquid phases and practical techniques of some unique molecular interactions between polymer chain and the solvent molecules, have been made to allow the investigation of the effect of the well-defined polymer in phase separation of binary mixtures. The behavior of a flexible linear or branched chain polymer (polyethylene oxide, PEO, MW = 9 x 10(5), as an impurity) in the critical binary mixture of isobutyric acid (I) + water (W) was studied by the refractive index (n) measurements using a very accurate and sensitive refractometer. The refractive index in each phase of IW as well as three different PEO concentrations (C = 0.395, 0.796, and 1.605 mg/cm(3)) in the near critical composition of IW have been measured at temperatures below the system's upper critical point. We observed that the polymer was significantly affected in the critical region of IW and these various concentrations of PEO show an important behavior on the critical exponents (beta), the critical temperatures (T(c)), and critical composition (phi(c)), which are depicting the shape of the coexistence curve. The phase-transition region of coexisting phases of IW shifts down with the addition of PEO and T(c) decreases linearly with increasing PEO concentrations. This may indicate that the polymer chain entangles with each phase, thereby the polymer monomers strongly interact with neighbor solvent particles and also intrachain interaction between the polymer segments. At such conditions, the collapse of polymer chain is possible in the vicinity of the critical point. At temperatures T close enough to T(c), the critical exponent beta (defined by the relation (n(1) - n(2)) proportional, variant (T(c) - T)(beta), with n(1) and n(2) being the refractive indices of the coexisting phases) was found to decrease from 0.382 to 0.360 when the PEO concentration changes from 0.395 to 1.605 mg/cm(3). These values are higher than that of 0.326 +/- 0.005 of pure IW, which is compatible with the three-dimensional Ising value beta = 0.325. The observed critical exponents for the PEO in IW are fully renormalized Ising critical exponents. Besides, the phi(c) values decrease with increasing the C values in the mixture of IW. It appears that the shape of the PEO in IW coexistence curves is similar from that of pure IW.     

Monte Carlo investigations of dense copolymer systems. III. Properties of triblock copolymers in good and theta solvent

The present article gives an analysis of XYX triblock copolymers in a good solvent and in a theta solvent, the segments of type X and type Y being repulsive for each other. The results are compared to homopolymers as well as to copolymers in a selective solvent that is a good one for the outer blocks and a theta solvent for the inner one and vice versa, the strength of repulsion between blocks being the same as in the present types of copolymers. A lattice model is used for the investigations and the concentration ranges from a volume fraction phi = 0 up to phi = 0.8. In the limit phi --> 0 the triblocks in good solvent are slightly more expanded than homopolymers and in theta solvent mean square dimensions of triblocks are considerably increased compared to homopolymers due to the repulsion between blocks. With increasing concentration the dimensions decrease but then they increase again and for large concentrations they become similar for all types of copolymers studied, as the effect of the solvent levels off making the repulsive interaction between blocks the dominant interaction. This leads to an orientation effect and as a consequence to microphase separation which is demonstrated by the concentration dependence of various quantities as well as by visualization of snapshots.     

Primary electroviscous effect in a moderately concentrated suspension of charged spherical colloidal particles

On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials zeta at the particle volume fraction phi < or = 0.3 for the case of nonoverlapping double layers, that is, at kappaalpha > 10 (where kappa is the Debye-Hückel parameter). A simple approximate analytic expression for p applicable for particles with large kappaalpha and arbitrary zeta is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction phi < or = 0.3, where the relative error is negligible for kappaalpha > or =100 and even at kappaalpha = 50 the maximum error is approximately 20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.     

Chemically tuned amphiphilic diblock copolymers dispersed in water: from colloids to soluble macromolecules

We investigate by small-angle scattering the structural behavior in water of a family of asymmetric poly(styrene-stat-(acrylic acid))-block-poly(acrylic acid), i.e., P(S-stat-AA)-b-PAA, diblock copolymers. These diblocks are of constant block ratio and increasing molar fraction, phi(AA), ranging from 0 to 1, of acrylic acid in the first P(S-stat-AA) statistical block. We identify three types of structural behavior in water: (i) for phi(AA) /= 0.50, the diblocks dispersions in water are at equilibrium. For high phi(AA), the diblocks are soluble in water, demonstrating that a transition from colloid-like objects to soluble macromolecules is achieved. Close to the transition, (phi(AA) approximately 0.50), the diblocks form objects interpreted as comprising a water-swollen core formed by the P(S-stat-AA) block, surrounded by a swollen brush composed of the majority PAA block, above a apparent critical micelle concentration. However, these diblocks do not behave as macrosurfactants, and their self-association behavior is rather interpreted as a microphase separation which can arise from the incompatibility between two polymer blocks P(S-stat-AA) and PAA placed in a common solvent water.     

Marangoni effects on evaporative lithographic patterning of colloidal films

We investigate the effects of Marangoni stresses on the evaporative lithographic patterning of colloidal films (Harris, D. J.; Hu, H.; Conrad, J. C.; Lewis, J. A. Phys. Rev. Lett. 2007, 98 (14), 148301). Films are dried beneath a mask that induces periodically varying regions of free and hindered evaporation. Direct imaging reveals that silica microspheres suspended within an organic solvent exhibit recirculating flows induced by temperature and surface tension gradients that arise during drying. The films display remarkable pattern formation with a majority of the particles deposited in the masked regions. Above a critical colloid volume fraction, recirculating flows are suppressed, leading to particle deposition in unmasked regions of high evaporative flux.     

Simulation of polymer--polymer interdiffusion using the dynamic lattice liquid model

In this paper, we present computer simulation results concerning interdiffusion of fully compatible components in symmetric binary (AB) polymer mixtures in solutions. The simulation is performed in two dimensions using the algorithm based on the dynamic lattice liquid model. The solvent molecules are taken into account explicitly. The evolution of the concentration profiles in time at an interface is studied for chain lengths N=2,4,8,16 for three polymer concentrations phi=0.1,0.5,0.9. The tracer diffusion coefficients for polymer chains and for the solvent are obtained by monitoring the mean square displacements of their center of mass. The relationships between coefficients of interdiffusion and self-diffusion are tested.     

Microwave spectroscopy measurements of rotational spectra and DFT calculations for two distinct structural isomers of 1,1'-dimethylferrocene

Microwave spectra were obtained for two distinct structural isomers of 1,1'-dimethylferrocene, an eclipsed synperiplanar isomer (phi = 0 degrees, the E0 isomer), with A = 1176.9003(2) MHz, B = 898.3343(2) MHz, C = 668.7469(2) MHz, and an eclipsed synclinal isomer (phi = 72 degrees, the E72 isomer) with A = 1208.7117(14) MHz, B = 806.4101(12) MHz, and C = 718.7179(8) MHz. The b-dipole, asymmetric-top spectra of both structural isomers were measured in the frequency range of 5-12 GHz using a Flygare-Balle type of spectrometer. A very good fit to observed transitions, with small distortion constants, was obtained for the E0 conformer, indicating that this conformer is nearly rigid. The deviations obtained in a similar least-squares fit for the E72 confomer are significantly larger, indicating possible fluxional behavior for this conformer. In addition, 7 out of the 26 transitions observed for the E72 isomer conformer clearly exhibit very small splittings, giving further evidence for internal motion. DFT calculations for the different possible conformations of 1,1'- dimethylferrocene arising from rotation of one methyl cyclopentadienyl ligand relative to the other about the nominal C5 axis by an angle phi (dihedral angle) were performed using the B3PW91 functional. The calculations converged and were optimized for five structures on this torsional potential energy surface corresponding to different dihedral angles phi; three yielded energy minima, and two gave energy maxima, corresponding to transition states. The experimental results are in very good agreement with the results of the DFT calculations.     

Small-angle neutron scattering study of microemulsion-polymer mixtures in the protein limit

Contrast variation small-angle neutron scattering (SANS) has been employed to study complex fluids comprising model microemulsions and polymers. The systems are water-in-oil microemulsions with added non-adsorbing polymer, under good polymer solvency conditions and semidilute polymer concentrations. The polymer/colloid size ratio was q approximately 11, which is well within the "protein limit". Four scattering contrasts were produced by selective deuteration of the dispersed and continuous phases and also the surfactant. In this way, the separate partial structure factors (PSF) for colloid-colloid (c-c), polymer-polymer (p-p), and colloid-polymer (c-p) have been obtained. The c-c PSF has been compared with theoretical predictions, allowing determination of a polymer correlation length. This is compared with a similar correlation length obtained from the p-p PSF, which is shown to increase with colloid concentration. In this sense, adding microemulsion has a similar effect on the dissolved polymer as reducing the solvent quality, and an effective Flory-Huggins chi parameter has been calculated. The cross-term PSF shows a distinct anti-correlation. This is the first time such structure factors have been determined experimentally for colloid-polymer systems in the protein limit and these allow a more detailed understanding of the structural interactions in these systems.     

Characterization of the pore-surface gel phase in functionalized macroporous polymeric materials

Covalently immobilized pore-surface gel phases were prepared in a functionalized macroporous ultra-high-molecular-weight polyethylene by covalent coupling of lightly cross-linked polymer colloid particles [50% styrene, 49.8% (chloromethyl)stryrene, 0.2% divinylbenzene] to the interstitial pore surfaces. Swelling the covalently coupled colloid particles in a good solvent followed by chemical derivitization resulted in an immobilized pore-surface gel phase rich in primary amine groups. The macromolecular reactivity and molecular size-exclusion characteristics of the aminated pore-surface gel phase were then determined using monofunctional, amine-reactive, poly (ethylene glycol)s (PEG). Pegylated pore-surface gel phases that ranged from 71% (10,000 molecular weight PEG) to 56% (40,000 molecular weight PEG) PEG by weight resulted from reaction of the aminated gel phase with the PEG probe molecules. The number of PEG molecules reacting with the aminated pore-surface gel phase depends only on the Flory radius (or radius of gyration) of the PEG molecule to the negative 2.49th power i.e., 1/R _f ^2.49, corresponding to a M^−1.48 dependence. The immobilized and pegylated polymer colloid particles swell by a factor of 16–25 times the diameter of the original polymer colloid particles in water, thereby demonstrating that pegylation occurred though a substantial fraction of the volume of the immobilized colloid particles. Received: 18 January 1999 Accepted in revised form: 8 June 1999     

Modeling of drying in films of colloidal particles

The process of film formation on a solid substrate from polymer colloid dispersion during solvent evaporation has been investigated by means of the Monte Carlo simulation method. Colloid particles are modeled as hard spheres. Time evolution of the colloid density distribution and coverage of the solid substrate are studied. Both density and structure of colloid film is shown to depend strongly on the evaporation rate. At a low evaporation rate, the coexistence of hexagonal and tetragonal domains of dried colloid monolayer has been observed. The results of monolayer structure are in good agreement with the confocal scanning laser microscopy observations of Dullens et al. (2004).     

Structure-defined c60/polymer colloids supramolecular nanocomposites in water

We report a new supramolecular method for the synthesis of well-defined pristine C 60/polymer colloid nanocomposites in water. The colloids include polymer micelles and emulsion particles. To a polymer colloid solution in water or alcohol, we introduced C 60 solution in a solvent that is miscible with water or alcohol. After the two solutions mixed, polymer colloids and C 60 spontaneously assembled into stable colloidal nanocomposites. After a dialysis process, a nanocomposite dispersion in pure water was obtained. As characterized by DLS and (Cryo-)TEM, the nanocomposites have a core-shell structure with C 60 aggregated on the surface of emulsion particles or micellar cores. The resulting nanocomposites have many potential applications such as biomedicals and photovoltaics.     

Adsorption behavior of oxyethylated surfactants at the air/water interface

The low-shear viscosity eta(0) of colloidal suspensions of acrylic latex or silica in aqueous gelatin has been measured at a temperature above the sol-gel transition. Measurements were made on dilution of a concentrated suspension with water or a gelatin solution. Thus, either the gelatin : colloid ratio was maintained or it was varied at constant aqueous gelatin concentration. Systems were studied with four lime-processed gelatins of different molecular weights at two concentrations of added salt. In addition, the latex particle size and the thickness of the adsorbed gelatin layer were measured by photon correlation spectroscopy (PCS) under dilute conditions. The dependence of the low-shear viscosity eta(0) on particle concentration was exponential and did not follow the well-established Krieger-Dougherty model for simple hard-sphere suspensions over the concentration range studied. A simple phenomenological model, eta(0)=eta(o)10(phi(e)/phi(s)), was found to predict the behavior well. Here, eta(o) is the viscosity of a gelatin solution of the corresponding solution concentration, phi(e) is proportional to the volume fraction of the particles, and phi(s) is a scaling factor, which was determined to have a value of 0.85. With this value of phi(s), the dimensions determined from PCS could be used to predict the viscosity values.     

Re-entrant melting and freezing in a model system of charged colloids

We studied the phase behavior of charged and sterically stabilized colloids using confocal microscopy in a low polarity solvent (dielectric constant 5.4). Upon increasing the colloid volume fraction we found a transition from a fluid to a body centered cubic crystal at 0.0415+/-0.0005, followed by reentrant melting at 0.1165+/-0.0015. A second crystal of different symmetry, random hexagonal close packed, was formed at a volume fraction around 0.5, similar to that of hard spheres. We attribute the intriguing phase behavior to the particle interactions that depend strongly on volume fraction, mainly due to the changes in the colloid charge. In this low polarity system the colloids acquire charge through ion adsorption. The low ionic strength leads to fewer ions per colloid at elevated volume fractions and consequently a density-dependent colloid charge.     

A dynamic density functional theory for particles in a flowing solvent

We present a dynamic density functional theory (dDFT) which takes into account the advection of the particles by a flowing solvent. For potential flows, we can use the same closure as in the absence of solvent flow. The structure of the resulting advected dDFT suggests that it could be used for nonpotential flows as well. We apply this dDFT to Brownian particles (e.g., polymer coils) in a solvent flowing around a spherical obstacle (e.g., a colloid) and compare the results with direct simulations of the underlying Brownian dynamics. Although numerical limitations do not allow for an accurate quantitative check of the advected dDFT both show the same qualitative features. In contrast to previous works which neglected the deformation of the flow by the obstacle, we find that the bow wave in the density distribution of particles in front of the obstacle as well as the wake behind it are reduced dramatically. As a consequence, the friction force exerted by the (polymer) particles on the colloid can be reduced drastically.