Computer simulation of solutions of telechelic polymers with associating end-groups

We present the results of numerical Monte Carlo simulations of solutions of telechelic chains with strongly attracting end-groups. Formation of micelles (aggregates), their structure and structural characteristics of the system as a whole are studied in detail. The features revealed in computer experiments are qualitatively similar to the recent theoretical predictions. In particular, we show that micelles formed by telechelic chains attract each other even if the solvent is good for the soluble blocks forming micellar shells. As a result, A “micellar gel” structure with a number of chain “bridges” connecting micelles is formed. The bridging chains turn out to be significantly stretched. Furthermore, we observe a pronounced maximum in the wave-vector dependence of the static structure factor for the associating end-units which is a manifestation of a quasiperiodic pattern of alternating microdomains consisting of dense micellar cores and the swollen soluble chain blocks.     

Linear viscoelastic properties of transient networks formed by associating polymers with multiple stickers

We have developed a single-chain theory that describes dynamics of associating polymer chains carrying multiple associative groups (or stickers) in the transient network formed by themselves and studied linear viscoelastic properties of this network. It is shown that if the average number N of stickers associated with the network junction per chain is large, the terminal relaxation time τ(A) that is proportional to τ(X)N(2) appears. The time τ(X) is the interval during which an associated sticker goes back to its equilibrium position by one or more dissociation steps. In this lower frequency regime ω<1/τ(X), the moduli are well described in terms of the Rouse model with the longest relaxation time τ(A). The large value of N is realized for chains carrying many stickers whose rate of association with the network junction is much larger than the dissociation rate. This associative Rouse behavior stems from the association/dissociation processes of stickers and is different from the ordinary Rouse behavior in the higher frequency regime, which is originated from the thermal segmental motion between stickers. If N is not large, the dynamic shear moduli are well described in terms of the Maxwell model characterized by a single relaxation time τ(X) in the moderate and lower frequency regimes. Thus, the transition occurs in the viscoelastic relaxation behavior from the Maxwell-type to the Rouse-type in ω<1/τ(X) as N increases. All these results are obtained under the affine deformation assumption for junction points. We also studied the effect of the junction fluctuations from the affine motion on the plateau modulus by introducing the virtual spring for bound stickers. It is shown that the plateau modulus is not affected by the junction fluctuations.     

Computer simulation of three‐arm star polymers

We show that Shaffer's version of the bond fluctuation model can be used to simulate three‐arm star polymers. We report a simulation study of both single stars and melts of star polymers with arm lengths up to 90 monomer units (approximately twice the entanglement crossover length for linear chains). Center‐of‐mass self‐diffusion of single stars is Rouse‐like (D ˜ N^–1). Due to a limited range of molecular weights we cannot distinguish between a power‐law and an exponential dependence of the star‐melt self‐diffusion coefficient on arm length.     

Thermoreversible gelation of solutions of vinyl polymers

The thermoreversible gelation of solutions of isotactic poly(methyl methacrylate) is investigated. Amorphous gels can be prepared in l-butanol by a combination of a liquid-liquid demixing with an upper critical demixing temperature and a glass transition. Annealing of the demixed solutions above their glass transition temperature T_G, results in the formation of a crystalline gel. In oxylene, crystalline gels are formed by a liquid-liquid demixing with an lower critical demixing temperature and an annealing at room temperature. Very fast gelation is observed to occur far below room temperature in solvents like 2-butanone.     

Mechanical selfsimilarity of polymers during chemical gelation

Network polymers near their gel point exhibit selfsimilar mechanical behavior, as expressed by power law relaxations. The range of selfsimilarity is defined by two limiting length scales. The upper limit is the correlation length, defined by the linear size of the typical cluster, and a lower limit, roughly given by the size of one preformed linear chain, i. e., the mean distance between crosslinks. The correlation length increases with the approach to the gel point, and diverges at the critical extent of reaction, i. e., the gel point where the infinite cluster is formed. Above the gel point, it decreases again with further crosslinking. Dynamic mechanical measurements of the complex modulus at the gel point show a power law in the frequency dependence over the entire frequency range, monitoring selfsimilarity. Swelling effects reduce the fractal dimension of the percolation cluster form 2.5 to 2. It is shown how the power law G() ^1/2, found by experiment, is connected to the structure of the polymeric cluster.Presented at the Physikertagung 1987 in Berlin.     

Comparison of the association process in physical gels and in polymers with small amounts of stickers

The association processes in physical gels and physical networks with smal amount of stickers are compared on the basis of the generalization of literature data and our own results on the gels of gelatine, PVA and olygocarbodiimides as well as on the physical network formation in poly(dimethylcarbosiloxane) with side COOH groups. The measurement of the rheological properties is used as a main method of the association evaluation. The role of the rearrangement of the intra- to intermolecular bonds and of the thermodynamic quality of the solvent in the process of physical network formation is emphasized.     

Growth process of polymers near the gelation threshold

We will review experimental results obtained recently on the determination of the laws governing the growth process of polymer clusters as the gel point is approached. The exponent γ which characterizes the increase of the mean weight-average molecular weight M_w as the gel point is approached and the exponent τ which characterizes the mass distribution were measured on different chemical systems. They were found to be independent of the chemical system (within experimental error) and very close to exponent values calculated by computer simulations following the percolation model. Therefore, the sol-gel transition is a critical phenomenon of connectivity belonging to the same class of universality as percolation.     

Computer simulation of bottle-brush polymers with flexible backbone: good solvent versus theta solvent conditions

By molecular dynamics simulation of a coarse-grained bead-spring-type model for a cylindrical molecular brush with a backbone chain of N(b) effective monomers to which with grafting density σ side chains with N effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range 5 ≤ N ≤ 40, backbone chain lengths are in the range 50 ≤ N(b) ≤ 200, and we perform a comparison to results for the bond fluctuation model on the simple cubic lattice (for which much longer chains are accessible, N(b) ≤ 1027, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of the side chains and the backbone chain and discuss their N-dependence in terms of power laws and the associated effective exponents. We show that even at the theta point the side chains are considerably stretched, their linear dimension depending on the solvent quality only weakly. Effective persistence lengths are extracted both from the orientational correlations and from the backbone end-to-end distance; it is shown that different measures of the persistence length (which would all agree for Gaussian chains) are not mutually consistent with each other and depend distinctly both on N(b) and the solvent quality. A brief discussion of pertinent experiments is given.     

Concentration effects on irreversible colloid cluster aggregation and gelation of silica dispersions

Effects of particle concentration on the irreversible aggregation of colloidal silica are studied using in situ destabilization via the ionic strength increase derived from the enzymatic hydrolysis of urea by urease. Aggregation is monitored by time-resolved optical density and dynamic light scattering measurements. It terminates at a gel boundary, signaled by a prominent increase of the optical density and incipient non-ergodicity. Raman scattering is used to demonstrate that the enzymatic reaction continues, well beyond gelation for the compositions studied here, until the urea is consumed. Calibration of the ionic conductivity permits for constructing stability diagrams in terms of particle and salt concentration. As with reversible gelation, the process exhibits a collective character in that lower ionic strengths are required for gelation of concentrated dispersions and vice versa. However, light scattering demonstrates that the gel boundary is preceded here by a line marking the transition from reversible to irreversible cluster formation, with the two transition boundaries tracking each other. Comparisons are made with dispersions destabilized by direct addition of salt solutions, which gel under very different conditions.     

Spinodal demixing,percolation and gelation of biostructural polymers

We present a variety of new experiments which concern the self-assembly of a polymeric network from homogeneous solutions of Agarose, a representative biostructural polysaccharide used for previous studies at our laboratories. They allow deriving a semi-quantitative phase diagram in the T, C plane. The diagram includes both the spinodal and gelation lines. Below a value of about 2% w/v, concentration is not sufficient for direct gelation; however, quenching of the sol from high temperatures to below the spinodal line initiates the spinodal demixing. The latter generates two sets of regions having respectively, higher- and lower-than-average polymer concentrations. In the higher-concentration regions the functional polymer-polymer interaction (that is, self-assembly) is favoured. In fact, as in the course of demixing the point representative of higher-concentration regions reaches the region below the gelation line, gelation is allowed and indeed observed to occur. This evidences the possibly more general role of spinodal demixing (extended to multi-component systems) as a pathway for specific, local and biofunctional enhancements of concentrations. At very low concentrations (e.g. 10⁻⁴ w/v) the kinetics of demixing is still observed to occur, but the set of higher-concentration regions is no longer percolative. In these conditions, gelation occurs only within each individual region, while the specimen remains free-running. This illustrates a novel aspect of the notion of non-gelling concentrations.     

Thermoreversible gelation strongly coupled to coil-to-helix transition of polymers

This paper theoretically studies thermoreversible gelation driven by aggregation of helices formed on the polymer chains. Two fundamentally different cases of (i) multiple association of single helices and (ii) association by multiple helices with multiplicity k (such as double helices (k=2), triple helices (k=3), etc.) are treated on the basis of different equations. The helix length distribution on a polymer chain (or assemble of chains for multiple helices) is derived as a function of polymer concentration and temperature. Theoretical calculation of the total helix content in the solution is compared with experimental data of optical rotation in iota-carrageenan solutions at different polymer concentrations. It is shown that at low temperature there is a sharp transition from network to bundle state (pair, triplet, etc.). To confirm such a network/pairing transition, we carried out Monte Carlo simulation of polymer solution in which hydrogen-bonded zipper-like cross-links are formed.     

Critical association and thermoreversible gelation of some selected polymers

The study of associating polymers is complicated by the fact that at a finite concentration only an apparent molar mass is directly measured. This apparent molar mass deviates from the true one because of thermodynamic interaction. In a good solvent the repulsive interaction can be fully described in terms of the second virial coefficient, which can be measured at very low concentrations, and a correction can be made for the true molar mass. The technique has been successively applied to β-galactosidase in a phosphate buffer and to cellulose 2,5-acetate in acetone. The gel point could be determined, and critical behavior was found in very good agreement with percolation prediction. Two other methods were employed and tested. One is based on the condition that the longest relaxation time must diverge at the gel point, and correspondingly the translational diffusion coefficient should go to zero. The second criterion consists of the prediction that power law behavior should be observed for the time correlation function of dynamic light scattering as well as for the frequency dependent storage and loss moduli. These predictions were indeed found to hold for two polysaccharides which form thermoreversible gels in water at a fairly low salt concentration. The two polysaccharides are the Tamarind seeds polysaccharide in 1M Na₂SO₄ and an exopolysaccharide from Rhizobium leguminosarum, strain 8002 in 0.1M NaCl solutions respectively.     

Computer simulation of globules with microstructure

We present recent data of our Monte Carlo computer simulation study of properties of AB-copolymer globules which depend strongly on the primary sequence of A and B monomeric units. Different primary sequences have been studied: random, random-block, regular and designed ones by using some particular spatial conformation of a homopolymer chain (we have compared here three models: proteinlike copolymers, AB-copolymers modeling membrane proteins and ABC-copolymers modeling proteins with active enzymatic center). We have found several evidences for the fact that an AB-copolymer chain with a primary sequence prepared on the basis of a particular conformation of a homopolymer chain by some “coloring” procedure preserves the “memory” about its “parent” spatial conformation. Analyzing the power spectra of AB-sequences, we find the existence of long-range power-law correlations for the copolymers with specially designed primary sequences.     

Computer simulation of relaxation phenomena in star‐branched polymers. Temperature dependence

Dynamic Monte Carlo simulations of simple models of star‐branched polymers were conducted. A model star macromolecule consisted of f = 3 arms of equal length with a total number of polymer segments up to 800. The chain was confined to a simple cubic lattice with simple nearest neighbor attractive interactions. The relaxation phenomena were studied by means of autocorrelation functions in wide ranges of temperatures. Short‐time‐scale dynamic processes in the entire star‐branched chain were examined. It was found that under good solvent conditions the longest relaxation time of the end‐to‐center vector decreases with decreasing temperature. For low temperatures (below the Θ‐point) where the chain is collapsed, the dependence of the relaxation time on the temperature is opposite.     

Monte Carlo simulation study of the kinetics of brush formation by irreversible adsorption of telechelic polymers onto a solid substrate

The irreversible adsorption of telechelic polymer chains from solution and melts onto solid substrates has been studied using the bond fluctuation Monte Carlo model. Complex brush formation kinetics dominated by diffusion of chains to the substrate at short times (diffusion-limited regime or DLR) and by penetration of chains through the maturing brush at longer times (penetration-limited regime or PLR) were observed. During the entire adsorption process, the rate of chain adsorption decreases monotonically with time. In the DLR, characterized by a maximum in the concentration of singly bound chains and a rapidly increasing fraction of doubly bound chains (loops), this decrease is due primarily to the depletion of free chains near the substrate and the formation of concentration gradients of free (nonadsorbed) chains in the bulk solution. The DLR and PLR are separated by an intermediate regime during which the brush becomes dominated by doubly bound chains and both penetration of the maturing brush and diffusion of chains to the brush surface play a role in determining the kinetics of brush growth. The PLR is characterized by steep gradients of free chains within the growing brush and the disappearance of concentration gradients for free chains in the bulk solution. In the PLR, the concentration of singly bound chains is low and decreases slowly while surface coverage and the fraction of doubly bound chains increase slowly. The rates of adsorption of new chains and the formation of loops in the PLR slow dramatically with increasing surface coverage and increasing chain length and less dramatically with decreasing bulk concentration.     

Computer simulation studies of aggregates of associating polymers: Influence of low-molecular-weight additives solubilizing the aggregates

The structure of aggregates in solutions of chain molecules with associating groups at one of the ends is studied by Monte Carlo computer simulations using the bond fluctuation model. The main attention is paid to the influence of additives of low-molecular-weight solvent solubilizing the aggregates. It is shown that upon the addition of solvent the aggregates adopt a three-layer structure with the ‘lake’ of the solvent molecules in the central region surrounded by the layer of associating end-groups of polymer chains, which in turn is surrounded by the outer corona formed by the chain tails. The equilibrium form of the aggregates becomes close to that of a droplet of low-molecular-weight liquids. The regimes are found when the addition of the low-molecular-weight solvent stabilizes the multiplets and even induces the aggregate formation.     

Aggregation behaviour below the gelation threshold for metal-olefin coordination in diene polymers

This study focuses on aggregation below the gelation threshold in ternary solutions containing diene polymers, atactic 1,2-polybutadiene (PBu) or 3,4-polyisoprene (PI), and an inorganic salt, bis(acetonitrile) dichloropalladium(II). Upon mixing, effective coordination crosslinks are formed because the acetonitrile ligands of the palladium salt are displaced by olefinic pendant groups of the polymers. In all cases, the aggregation kinetics correlate with PdCl₂ concentration because the polymer concentration is 100 times greater than the salt concentration. Aggregation is the process that occurs prior to gelation as the transition-metal salt forms a coordination bridge between two different polymer chains. Differential analysis of the initial aggregation rate on the basis of light scattering data at 45° relative to the incident throughput beam reveals that the ‘true’ order of the coordination reaction is close to unity. This suggests that coordination crosslinking is diffusion-controlled in the early stages of aggregation. Half-life analysis of viscous solutions yields an overall reaction order of 1.6 for aggregation of polybutadiene and palladium chloride in tetrahydorfuran (THF). Scaling analysis of the weight-average molecular weight dependence of the root-mean-square size of isolated scattering particles yields an effective exponent of 0.7 for polymer/metal-salt complexes at infinite dilution. This experimental scaling law agrees with literature values for cross-linked polymer networks. In all cases, the size of the aggregates increases at higher PdCl₂ concentration. For polybutadiene/PdCl₂ mixtures in THF or toluene below the gelation threshold, the light-scattering-detected average aggregation number (AN = M_{w, complex}/M_{w, pure polymer} via Zimm-plot intercepts) for low-viscosity solutions is 2, while AN ∼ 9 for viscous THF solutions. In contrast, low aggregation numbers (AN ∼ 2) were calculated for viscous ternary mixtures of 3,4-polyisoprene and PdCl₂ in THF.