Investigating the Dynamic Viscoelasticity of Single Polymer Chains using Atomic Force Microscopy

In single‐molecule force spectroscopy (SMFS), many studies have focused on the elasticity and conformation of polymer chains, but little attention has been devoted to the dynamic properties of single polymer chains. In this study, we measured the energy dissipation and elastic properties of single polystyrene (PS) chains in toluene, methanol, and N,N‐dimethylformamide using a homemade piezo‐control and data acquisition system externally coupled to a commercial atomic force microscope (AFM), which provided more accurate information regarding the dynamic properties of the PS chains. We quantitatively measured the chain length‐dependent changes in the stiffness and viscosity of a single chain using a phenomenological model consistent with the theory of viscoelasticity for polymer chains in dilute solution. The effective viscosity of a polymer chain can be determined using the Kirkwood model, which is independent of the intrinsic viscosity of the solvent and dependent on the interaction between the polymer and solvent. The results indicated that the viscosity of a single PS chain is dominated by the interaction between the polymer and solvent. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1736–1743     

Transient Changes in Topology and Energy on Extension of Polybutadiene Networks

The theory of elasticity of polymer networks has been developed along two lines. The phenomenological approach leads to the Mooney-Rivlin relation between stress and extension ratio for uniaxial extension. The statistical theory of elasticity, based on a model for polymer molecules, predicts a similar relation with one of the constants zero. Actual elastic properties of rubbers do not agree fully with either theory.

Experimental results are reported obtained with quantitatively cured polybutadiene and polyisoprene vulcanizates. These data are near-equilibrium results through the use of a cyclic stress sequence which largely eliminates the influence of long-time creep. The dependence of the initial modulus and the parameters of the Mooney-Rivlin relation on the chemical nature and the degree of branching of the polymer, the type of cross-links, and temperature has been investigated. A possible relation between the energy component of the elastic force and one of the parameters is discussed.

These results as well as those in the literature refer to irreversible processes. It is proposed that this irreversibility results from friction accompanying slippage of chain entanglements. This mechanism is compatible with the observed dependences. It is concluded that the variation of elastic properties with elongation is due to changes in network topography.

Some observations are made on the topological changes of vulcanizate networks at very high elongations. Similarities are pointed out between reinforcement by stress crystallization and by addition of carbon black. The effect of blacks is attributed mainly to preferential adsorption on the carbon particles of short network chains which become overstressed at high deformation. On adsorption the kinetic energy of these particular chains will be dissipated in the form of heat of adsorption.

Examples are given of the applicability of F. Bueche's relation between extension of the sample and that of the elastomer matrix in a filled vulcanizate. This equation differs from that based on Einstein's relation for the viscosity of suspensions, which has been shown to be applicable in other filled rubbers. The difference between the two relations may be associated with the absence or presence of chemical bonding of elastomer to filler.

Network topography has an influence on the ultimate properties of vulcanizates. Polybutadiene samples in which the cross-links are single bonds break at lower elongation than those with equal concentrations of cross-links consisting of 18-atom chains.     

Elastic Behavior of Polymer Chains

The elastic behavior of the polymer chain was investigated in a three-dimensional off-lattice model. We sample more than 109 conformations of each kind of polymer chain by using a Monte Carlo algorithm, then analyze them with the non-Gaussian theory of rubberlike elasticity, and end with a statistical study. Through observing the effect of the chain flexibility and the stretching ratio on the mean-square end-to-end distance, the average energy, the average Helmholtz free energy, the elastic force, the contribution of energy to the elastic force, and the entropy contribution to elastic force of the polymer chain, we find that a rigid polymer chain is much easier to stretch than a flexible polymer chain. Also, a rigid polymer chain will become difficult to stretch only at a quite high stretching ratio because of the effect of the entropy contribution. These results of our simulation calculation may explain some of the macroscopic phenomena of polymer and biomacromolecular elasticity.     

Nanoscience of single polymer chains revealed by nanofishing

The invention of atomic force microscopy (AFM) enabled us to study the statistical properties of single polymer chains by a method called "nanofishing," which stretches a single polymer chain adsorbed on a substrate with its one end by picking it at the other end. A force-extension curve obtained for a single polystyrene chain in a Theta solvent (cyclohexane) shows good agreement with a worm-like chain model and, therefore, gives microscopic information about entropic elasticity. Furthermore, the nanofishing technique can be used for dynamic viscoelastic measurement of single polymer chains. An AFM cantilever is mechanically oscillated at its resonant frequency during the stretching process. This technique enables the estimation of quantitative and simultaneous elongation-dependent changes of stiffness and viscosity of a single chain with the use of a phenomenological model. In this study, the effect of solvent on viscosity in low extension regions reveals that the viscosity is attributed to monomer-solvent friction. Thus, static and dynamic nanofishing techniques are shown to give powerful experimental proofs for several basic questions in polymer physics. The techniques are expected to reveal hidden properties of polymer chains or polymer solutions by any types of macroscopic measurements in the future.     

Viscoelastic properties of an isolated polymer chain with arbitrary flexibility,chain length and hydrodynamic interactions from 4‐dimensional Dirac propagator

Kholodenko's theory of semiflexible polymer chains, the conformation and properties of which are obtained from the Dirac propagator, shows applicability to dilute solutions of semiflexible polymers of arbitrary persistence and contour lengths by calculating the static scattering function and the squared end‐to‐end distance of the polymer chain. In the present work, the theory is extended and applied to obtain the intrinsic viscosity with consideration of hydrodynamic interactions. The intrinsic viscosity formula is derived as function of chain length and persistence length. The hydrodynamic interactions are also taken into account following the Kirkwood and Riseman scheme. From this calculation, we obtain the general expression for the intrinsic viscosity and diffusion coefficients covering the whole range of chain flexibilities without confusion with the excluded volume effects. Calculated limiting values of hydrodynamical observables are in complete agreement with those known for random coils and rigid rods.     

The role of intermolecular interactions in the viscoelastic properties of polymer melts

The exact microscopic expression for the stress tensor in polymer liquids contains a tensor product of the the segment position vector with the total, intra- plus inter-chain, force acting on the segment. On the other hand, the widely accepted theory of viscoelasticity of polymer melts¹⁾ is based on the assumption, that contributions from interchain interactions to the viscosity of polymer melts is negligible relative to the effectively intrachain entropic interactions. Starting from the exact Green-Kubo formula for the viscosity, the Rouse dynamic correlation functions, and Newton's second law, we show that the intrachain assumption is inadequate. Rather, the intrachain and interchain forces acting on polymer segments cancel each other largely. The intrachain contribution therefore cannot be dominant as anticipated in the usual treatment¹⁾, or, in other words, the interchain contribution cannot be ignored. The main contribution to viscoelastic properties of polymer melts can only arise from a part of the total stress tensor as already suggested by M. Fixman based on a different argument²⁾. It is concluded that the viscosity is of a purely interchain nature, and is determined by the tensor product of the vector connecting the centers-of-mass of neighboring macromolecules on the one hand, and the total force by which macromolecules interact, on the other, just in the case of simple liquids.     

Stability of a model alkali-soluble associative polymer in the presence of a weak and a strong base

 Hydrophobically modified alkali-soluble emulsion (HASE) polymer is solubilized by the addition of a base. When the pH is increased to greater than 6.5, methacrylic acids on the polymer backbone are neutralized and the carboxylated latex polymer goes into solution causing a large increase in the viscosity due to inter-molecular associations of the hydrophobes. The stability of the viscosity of the polymer solution at pH in the range 9–10 was studied in the presence of a strong (NaOH) and a weak [1-amino-1-methylpropanol (AMP)] base. No change in the viscosity or the moduli was observed for the polymer in AMP. Reduction in the viscous and elastic properties of the polymer solution in NaOH was observed after 4 weeks. Such small changes are detectable using the superposition of oscillation on the steady shear technique. The decrease in the viscoelastic properties is attributed to the hydrolysis reaction of the urethane groups of the macromonomer, which resulted in a decrease in the number of hydrophobes per polymer chain. It is recommended that a weak base be used to neutralise the HASE polymer in order to avoid the possibility of compositional changes in the polymer after neutralisation for more than 6 weeks. Received: 19 May 1998 Accepted in revised form: 26 October 1998     

The Rheology of Linear and Long-chain Branched Polymer Melts

By generalising the Doi-Edwards tube model to the Molecular Stress Function theory, the non-linear rheology of polymer melts can be described quantitatively. The strain-hardening of linear polymer melts in extensional flows can be accounted for by a strain energy function, which reflects the increase of strain energy due to tube squeeze. In comparison to linear polymer melts, long-chain branched polymer melts show enhanced strain-hardening. This is due to the fact that while the backbone of the branched macromolecule is stretched by deformation, side chains are compressed. It is demonstrated that the experimentally observed slope of the elongational viscosity after inception of strain-hardening depends on the ratio β of total molar mass to backbone molar mass as predicted by the model. The steady-state (plateau) value of the elongational viscosity depends on the maximum relative stretch, , which can be supported by chain segments and which represents the maximum elastic energy storable in the polymeric system.     

The solid-state polymerization and physical properties of bis(ethyl urethane) of 2,4-hexadiyne-1,6-diol. II. Resonance Raman spectroscopy

Resonance Raman spectroscopy has been used to measure the frequencies of the vibrational modes of the polymer backbone of fully and partially polymerized crystals of an ethyl urethane diacetylene. The data have assisted in the assignment of crystallographic structures to the different modifications of the crystals. Effective interatomic force constants have been derived and used to estimate the Young's modulus. For fully polymerized crystals a value of approximately 76 GPa was found. The dependence of the frequencies on elastic tensile strain parallel to the polymer chain direction was determined. By comparing the strain dependence of frequencies of partially polymerized singlecrystal fibers with those expected for model composites, it was concluded that the polymer chains in thermally polymerized crystals are considerably longer than in those polymerized using γ rays.     

Construction of approximate entropic forces for finitely extensible nonlinear elastic (FENE) polymers

When the stress applied to a Rouse-like polymer chain is large enough, one must use anharmonic entropic spring forces in order to keep the chain contour length from increasing to unphysical values. Although one can derive “exact” equations relating the spring extension to the entropic force produced by a finitely extensible non-linear elastic (FENE) random-walk polymer, such expressions are usually of little interest because their complexity would entail large evaluation times in numerical studies by computer. Moreover, these expressions can rarely be used directly in analytical studies. In this article, we describe a systematic method to construct analytically simple yet numerically accurate expressions to relate the entropic force to the extension of an entropic spring for a random-walk polymer chain in arbitrary dimension d ≥ 2. These expressions are modified Pade approximants which yield the correct asymptotic behaviours in both the small and large extension limits. It is shown that the well-known Warner empirical approximation is but a limiting case (for infinite dimensions).     

Structure transfer from a polymeric melt to the solid state Part III: Influence of knots on structure and mechanical properties of semicrystalline polymers

Investigations of the influence of crystallization conditions and molecular weight on mechanical properties of melt spum monofilaments have been carried out in previous studies [1, 2]. The present results emphasize the influence of crystallization upon two classes of molecular entanglements present in the melt:

1. mobile entanglements with short life times;
2. long living knots as a superstructure of the entanglement network.

On the other hand, it is shown that the former number of entanglements controls the thickness of the crystalline lamellae. It is pointed out that knots also have a specific influence on morphology. Knots form the amorphous phase of a semicrystalline polymer and are mechanically active. The present model is comparatively discussed in light of Peterlin's plastic deformation model and Kilian's van der Waals Network Theory.     

Light scattering measurements of anisotropic viscoelastic coefficients of a main-chain polymer nematic liquid crystal

We report the first systematic measurements of elastic coefficients and viscosities that clearly demonstrate the distinction between rigid and semiflexible behaviour of a main-chain polymer nematic liquid crystal, a solution of poly-γ-benzyl glutamate (PBG) in mixed organic solvents. Quasi-elastic Rayleigh scattering studies show a crossover which occurs at a molecular chain length near the persistence length of PBG as the chain length increases. The results are in qualitatively good agreement with recent theoretical predictions for semiflexible chains. It is seen that bending distortions of an individual polymer play an important role in the fundamental nature of nematic elasticity and viscosity.     

Shear-thickening flow of nanoparticle suspensions flocculated by polymer bridging

The steady-shear viscosity, dynamic viscoelasticity, and stress relaxation behavior were measured for suspensions of silica nanoparticles dispersed in aqueous solutions of poly(ethylene oxide) (PEO). The suspensions of silica with diameters of 8-25 nm show striking shear-thickening profiles in steady shear and highly elastic responses under large strains in oscillatory shear. Since the silica particles are much smaller than the polymer coils, one molecule can extend through several particles by intrachain bridging. Each polymer coil may remain isolated as a floc unit and the silica particles hardly connect two flocs. Therefore, the flow of suspensions is Newtonian with low viscosity at low shear rates. When the polymer coils containing several nanoparticles are subjected to high shear fields, three-dimensional network is developed over the system. The shear-thickening flow may arise from the elastic forces of extended bridges. But, the polymer chain is easily detached from particle surface by thermal energy because of large curvature of particles. As a result, the network structures are reversibly broken down in a quiescent state and the suspensions behaves as viscoelastic fluids with the zero-shear viscosity.     

Elastic behavior of adsorbed polymer chains

Elastic behaviors of single polymer chains adsorbed on the attractive surface are first investigated using Monte Carlo simulation method based on the bond fluctuation model. We investigate the chain size and shape of adsorbed chains, such as mean-square radius of gyration S2, mean-square bond length b2, shape factors sf(i) and delta*, and the orientation of chain segments P2, to illuminate how the shape of polymer chains changes during the process of tensile elongation. There are some special behaviors of the chain size and shape at the beginning of elongation, especially for strong attraction interaction. For example, mean fraction of adsorbed segments decreases abruptly in the region of small elongation ratio and then decreases slowly with increasing elongation ratio. In fact, the chain size and shape also changes abruptly for small elongation ratio with strong attraction interaction. Some thermodynamics properties are also investigated here. Average Helmholtz free energy increases fast for elongation ratio lambda<1.15, especially with strong attraction, and increases slowly for lambda>1.15. Similar behaviors are obtained for average energy per bond. Elastic force (f ) and energy contribution to force (f(U)) are also studied, and we find that elastic force decreases abruptly for lambda<1.15, and there is a minimum of elastic force for strong attraction interaction, then increases very slowly with increasing elongation ratio. However, there are different behaviors for weak attraction interaction. For energy contribution to force (f(U)), there is a maximum value for strong attraction interaction in the region of lambda<1.15. Some comparisons with the atomic force microscopy experiments are also made. These investigations may provide some insights into the elastic behaviors of adsorbed polymer chains.     

Single polymer chains as specific transducers of molecular recognition in scanning probe microscopy

A new approach for the specific detection and mapping of single molecule recognition is presented, based on the nonlinear elastic behavior of a single polymer chain. The process of molecular recognition between a ligand and a receptor is inherently accompanied by a decrease in the translational and rotational degrees of freedom of the two molecules. We show that a polymeric tether linked to the ligand can effectively transduce the configurational constraint imposed by molecular recognition into a measurable force, which is dominated by the entropic elasticity of the polymer. This force is specifically characterized by a strong nonlinearity when the extension of the polymer approaches its contour length. Thus, a polymer chain tethering the ligand to an oscillating cantilevered tip gives rise to a highly anharmonic motion upon ligand-receptor binding. Higher-harmonics atomic force microscopy allows us to detect this phenomenon in real time as a specific signature for the probing and mapping of single-molecule recognition.     

Dynamic viscosity of dilute solutions of poly(L-glutamic acid)

The dynamic viscosity η′ of a dilute solution of poly(L-glutamic acid) (DP = 1370) in a mixed solvent made up of aqueous 0.2M NaCl and dioxane (2:1 by volume) is measured over the pH range 4.2–10 and in the frequency range 2–500 kHz. The frequency dependence of η′ in the helix region (low pH) is interpreted in terms of a model molecule consisting of n rigid helical segments connected by universal joints. The steady-flow viscosity, relaxation time, and high-frequency limiting viscosity at pH 4.75 (helical content 80%) are well explained by this model with n = 5. This value of n is consistent with that estimated from the nucleation parameter σ = 1.4 × 10^?3 obtained from the relation between reduced steady-flow viscosity and helical content. The high-frequency values of η′ in the coil region (high pH) are fitted by Peterlin's theory. The internal viscosity seems to arise in part from the polyelectrolytic character of the molecule. An additional relaxation at low frequencies in the coil region is ascribed to rotation of molecules elongated by the electrostatic interaction. The lower value of reduced steady-flow viscosity in the coil region in the mixed solvent compared with that in water is interpreted in terms of the lower degree of effective ionization and the selective solvation of water by the polypeptide. No anomaly is observed in the helix–coil transition region, indicating that the relaxation time for helix–coil equilibrium is less than 10^?6sec.     

Dimensions of branched and linear polymer chains in solution: Modifications of flory's theory

It is pointed out that a theory proposed by Vrij for the expansion of a linear polymer chain in solution beyond its unperturbed size is also applicable formally to branched chains. The theory predicts the temperature at which the chain obeys random-flight statistics to be lower for a branched chain than for a linear one (for a polymer–solvent system exhibiting the usual upper critical consolute point). Vrij's derivation follows a well-known procedure of Flory, but refines it to take account of the variation of the time-averaged segment density with distance from the center of mass of the chain. This modification in combination with a somewhat different derivation is shown to lead to a similar result. Yet another scheme for “correcting” Flory's theory is outlined. Some reasons are advanced for questioning the physical significance of these small modifications of Flory's theory, all of which depend on the retention of previously ignored terms in power series expansions.     

Rheological properties of linear and crosslinked polymer blends: Relation between crosslink density and enhancement of elongational viscosity

Strain‐hardening behavior in the elongational viscosity of binary blends composed of a linear polymer and a crosslinked polymer, in which the molecular chains of the linear polymer were incorporated into the network chains of the crosslinked polymer, was studied. Blending the crosslinked polymer characterized as the gel just beyond the sol–gel transition point greatly enhanced the strain‐hardening behavior in the elongational viscosity, even though the amount of the crosslinked polymer was only 0.3 wt %. However, the crosslinked polymer, which was far beyond or below the sol–gel transition point, had little influence on the elongational viscosity as well as the shear viscosity. The stretching of the chain sections between the crosslink points was responsible for the strain‐hardening behavior. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 228–235, 2001     

Failure of elastomeric polymers due to rate dependent bond rupture

A new cohesive zone model is developed in order to study the mechanisms of adhesive and cohesive failures of soft rubbery materials. The fracture energy is estimated here using a strategy similar to that of Lake and Thomas (LT) by considering the dissipation of stored elastic energy followed by the extension and relaxation of polymer chains. The current model, however, departs from that of LT in that the force needed to break an interfacial bond does not have a fixed value; instead, it depends on the thermal state of the system and the rate at which the force is transmitted to the bond. While the force required to rupture a chain is set by the rules of thermomechanically activated bond dissociation kinetics, extension of a polymer chain is modeled within both the linear and nonlinear models of chain elasticity. Closed form asymptotic solutions are obtained for the dependence of crack propagation speed on the energy release rate, which are valid in two regimes: (I) slow crack velocity or short relaxation time for bond dissociation; (II) fast crack velocity or long relaxation time for bond dissociation. The rate independent and the zero temperature limit of this theory correctly reduces to the fracture model of LT. Detailed comparisons are made with a previous work by Chaudhury et al. which carried out an approximate analysis of the same problem.     

Rheology of hydroxypropylcellulose solutions

The rheology of hydroxypropylcellulose (HPC)—acetic acid solutions was investigated by using a cone-and-plate rheometer and a capillary rheometer, for polymer concentrations ranging from 10 to 80%. Isotropic solutions exhibit a Newtonian plateau followed at higher shear rates by a pseudoplastic zone. The apparent viscosity varies as C^5.2 if concentration C is less than 27% and as C¹³ for 27% < C < 30%. A biphasic interval (isotropic and cholesteric phases) exists between 30 and 35%. A maximum in viscosity is observed at C = 30%, the height of the viscosity peak being a decreasing function of shear rate. Anisotropic solutions are strongly viscoelastic. Both isotropic and anisotropic solutions give results (apparent viscosity, first normal-stress difference, relaxation time, etc.) which are not in good agreement with Doi's theory. This is understandable since the HPC chain cannot be modeled by a rigid rod. Upon heating, anisotropic HPC—acetic acid solutions undergo an anisotropic to isotropic phase transition which is easily detected by a maximum in the temperature dependence of the first normal-stress difference and of the apparent viscosity.