Correction of systematic errors in single-molecule force spectroscopy with polymeric tethers by atomic force microscopy

Single-molecule force spectroscopy has become a valuable tool for the investigation of intermolecular energy landscapes for a wide range of molecular associations. Atomic force microscopy (AFM) is often used as an experimental technique in these measurements, and the Bell-Evans model is commonly used in the statistical analysis of rupture forces. Most applications of the Bell-Evans model consider a constant loading rate of force applied to the intermolecular bond. The data analysis is often inconsistent because either the probe velocity or the apparent loading rate is being used as an independent parameter. These approaches provide different results when used in AFM-based experiments. Significant variations in results arise from the relative stiffness of the AFM force sensor in comparison with the stiffness of polymeric tethers that link the molecules under study to the solid surfaces. An analytical model presented here accounts for the systematic errors in force-spectroscopy parameters arising from the nonlinear loading induced by polymer tethers. The presented analytical model is based on the Bell-Evans model of the kinetics of forced dissociation and on the asymptotic models of tether stretching. The two most common data reduction procedures are analyzed, and analytical expressions for the systematic errors are provided. The model shows that the barrier width is underestimated and that the dissociation rate is significantly overestimated when force-spectroscopy data are analyzed without taking into account the elasticity of the polymeric tether. Systematic error estimates for asymptotic freely jointed chain and wormlike chain polymer models are given for comparison. The analytical model based on the asymptotic freely jointed chain stretching is employed to analyze and correct the results of the double-tether force-spectroscopy experiments of disjoining "hydrophobic bonds" between individual hexadecane molecules that are covalently tethered via poly(ethylene glycol) linkers of different lengths to the substrates and to the AFM probes. Application of the correction algorithm decreases the spread of the data from the mean value, which is particularly important for measurements of the dissociation rate, and increases the barrier width to 0.43 nm, which might be indicative of the theoretically predicted hydrophobic dewetting.     

A scaling analysis of the fracture mechanisms in glassy polymers

Recent experimental work on craze growth in polymers has shown that the reptation model for polymer motion may be successfully applied, in certain cases, to the behavior of glassy polymers in the solid phase. In this paper, a simple scaling analysis is presented to show how the primitive path model for reptation may be applied to the behavior of glassy polymers before and during slow crack growth. Different dynamic regimes are identified and related to the relaxation times of the polymer chain. The dependence on polymer chain length for each mechanism is described. Estimates for the fracture energy of each process are presented.     

Fracture toughness predictions in polycarbonate

The concepts of fracture mechanics have been used to analyze the fracture toughness of polycarbonate in terms of the polymer relaxation properties. By considering the temperature rise at the tip of a growing crack; a model is proposed which describes the viscoelastic energy loss in the fracture process. The model was used to predict the fracture toughness of polycarbonate as a function of test temperature.     

Molecular and nanoscale factors governing pressure‐sensitive adhesion strength of viscoelastic polymers

Pressure‐sensitive adhesives (PSAs) are finding increasing applications in various areas of industry and medicine. PSAs are a special class of viscoelastic polymers that form strong adhesive joints with substrates of varying chemical nature under application of light external bonding pressures (1–10 Pa) over short periods of time (1–5 s). To be a PSA, a polymer should possess both high fluidity under applied bonding pressure, to form good adhesive contact, and high cohesive strength and elasticity, which are necessary for resistance to debonding stresses and for dissipation of mechanical energy at the stage of adhesive bond failure under detaching force. For rational design of novel PSAs, molecular insight into mechanisms of their adhesive behavior is necessary. As shown in this review, strength of PSA adhesive joints is controlled by a combination of diffusion, viscoelastic, and relaxation mechanisms. At the molecular level, strong adhesion is the result of a narrow balance between two generally conflicting properties: high cohesive strength and large free volume. These conflicting properties are difficult to combine in a single polymer material. Individually, high cohesive interaction energy and large free volume are necessary but insufficient prerequisites for PSA strength. Evident correlations are observed between the adhesive bond strengths of different PSAs, and their relaxation behaviors are described by longer relaxation times. Innovative PSAs with tailored properties can be produced by physical mixing of nonadhesive long‐ and short‐chain linear parent polymers, with groups at the two ends of the short chains complementary to the functional groups in the recurring units of the long chains. Although chemical composition and molecular structure of such innovative adhesives are unrelated to those of conventional PSAs, their mechanical properties and adhesive behaviors obey the same general laws, such as the Dahlquist's criterion of tack. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Interfacial toughness in polymer‐layered laminar composites

The investigation of the interfacial toughness of polymer layered laminar composites with two different approaches produced results differing by up to an order of magnitude and following opposite trends with respect to the strain rates. The flexural modulus and neutral axis of a constrained epoxy‐adhesive layer bound to a painted metal substrate varied with the thickness of the adhesive layer. The adhesion energy depended on the rate at which the force was transmitted to the adhesion bonds—not just on the strength of the adhesion bonds—and on the concomitant strain hardening at high strain rates. As the strain rate and thickness of the polymer layer increased, the transition from a cohesive mode to an adhesive–cohesive (polymer–polymer interface) mode of debonding led to the observed high adhesion energy. The high adhesion energy and increased strain hardening were attributed to the formation of organic–inorganic composites and nanocomposites within the polymer matrix, which evolved as a result of the interactions between the metal oxide pigments and fillers with the polymer matrix during curing. Scission of the polymer chains at the interface was proposed to be the predominant fracture mechanism; it was based on the high relaxation time (～10¹⁷ s) and the high activation energy (～175 kJ mol^?1). © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3822–3835, 2004     

A Sequential Debonding Fracture Model for Hydrogen-Bonded Hydrogels

Hydrogen bonds are known to play an important role in prescribing the mechanical performance of certain hydrogels such as polyether-based polyurethanes. The quantitative contribution of hydrogen bonds to the toughness of polymer networks, however, has not been elucidated to date. Here, a new physical model is developed to predict the threshold fracture energies of hydrogels physically crosslinked via hydrogen bonds. The model is based on consecutive and sequential dissociation of hydrogen-bonded crosslinks during crack propagation. It is proposed that the scission of hydrogen bonds during crack propagation allows polymer strands in the deformation zone to partially relax and release stored elastic energy. The summation of these partial chain relaxations leads to amplified threshold fracture energies which are 10–45 times larger than those predicted by the classical Lake–Thomas theory. Experiments were performed on a hydrophilic polyurethane hydrogel where urea additions were used to control the density of hydrogen bonds. The measured fracture energies were in good agreement with the calculated values. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 1287–1293     

Statistical mechanical model for diffusion of simple penetrants in polymers. I. Theory

Following Di Benedetto it is proposed that noncrystalline polymer regions possess an approximate semicrystalline order with chain bundles that are locally parallel along distances of several nanometers. Packing with on-average four nearest neighbors is assumed. A spherical molecule may move through such a substrate in two distinct ways: (a) along the axis of a “tube” formed by locally parallel chains or (b) perpendicular to this axis by two polymer chains separating sufficiently to permit passage of the molecule. The first process is relatively fast, generally requires little activation energy, and determines the effective jump length in diffusion. The second is responsible for the activation energy of diffusion, which is taken as the minimum energy necessary to produce a symmetrical chain separation which allows transfer of a molecule. This is calculated as a function of the penetrant diameter d and parameters Γ and β which characterize the interchain cohesion and chain stiffness, respectively. Γ is estimated from the polymer density and cohesive energy density by suitably linearizing a relation given by Di Benedetto for the potential between two polymer chains approximated as infinite strings of Lennard-Jones force centers. β is shown to be approximately obtainable from the polymer chain backbone geometry and bond rotation potentials. An expression for the diffusion coefficient D is developed which contains only one disposable parameter, the effective jump length.     

Energy storage in polymer chains under stress

The local conformation and storage of energy in individual polymer chains during a deformation of a bulk polymer sample are examined by the computer simulation of a relatively simple model. It is shown that as the interaction between the chain atoms and surrounding medium increases, rotational angle motion is suppressed during the deformation, and large amounts of energy are stored in backbone bond angle and bond length distortions. The relationship of this phenomena to T_g and the implications for chain relaxation are discussed.     

Carbon tetrachloride-induced crack healing in polycarbonate

Crack healing induced by carbon tetrachloride in polycarbonate has been studied at temperatures in the range of 40–60°C. The carbon tetrachloride treatment reduces the glass transition temperature of polycarbonate. Crack healing is observed because the effective glass transition temperature in polycarbonate is reduced to below the test temperature by the carbon tetrachloride treatment. Two distinctive stages of crack healing are divided based on the recovery of mechanical strength and fractograph. The first stage corresponds to the progressive healing due to the convolution of wetting and self-diffusion, which has a constant crack closure rate. Immediately following the first stage, the second stage, corresponding to the self-diffusion of polymer chain, enhances the quality of healing behavior. The transport of carbon tetrachloride in polycarbonate consists of case I (concentration gradient controlled) and case II (relaxation controlled) behaviors. The direction of case II is opposite to that of case I. The solubility decreases with increasing temperature, but diffusivity and velocity for mass transfer, crack closure rate, and diffusion coefficient for the diffusion front have the opposite trend. The first stage of crack healing is controlled by case II transport. The transport of carbon tetrachloride changes the fracture behavior of polycarbonate from ductile to brittle. A comparison of crack healing in polycarbonate and poly (methyl methacrylate) is made. © 1994 John Wiley & Sons, Inc.     

Universal properties of a single polymer chain in slit: Scaling versus molecular dynamics simulations

We revisit the classical problem of a polymer confined in a slit in both of its static and dynamic aspects. We confirm a number of well known scaling predictions and analyze their range of validity by means of comprehensive molecular dynamics simulations using a coarse-grained bead-spring model of a flexible polymer chain. The normal and parallel components of the average end-to-end distance, mean radius of gyration and their distributions, the density profile, the force exerted on the slit walls, and the local bond orientation characteristics are obtained in slits of width D=4/10 (in units of the bead diameter) and for chain lengths N=50/300. We demonstrate that a wide range of static chain properties in normal direction can be described quantitatively by analytic model-independent expressions in perfect agreement with computer experiment. In particular, the observed profile of confinement-induced bond orientation is shown to closely match theory predictions. The anisotropy of confinement is found to be manifested most dramatically in the dynamic behavior of the polymer chain. We examine the relation between characteristic times for translational diffusion and lateral relaxation. It is demonstrated that the scaling predictions for lateral and normal relaxation times are in good agreement with our observations. A novel feature is the observed coupling of normal and lateral modes with two vastly different relaxation times. We show that the impact of grafting on lateral relaxation is equivalent to doubling the chain length.     

Molecular basis of healing and fracture at polymer interfaces

The interdiffusion of polymer chains across a polymer–polymer interface, and subsequent fracture to re-create the interface is reviewed. In particular, films formed via latex coalescence provide a very large surface area. Of course, latex film formation is a very important practical problem. Healing of the interface by interdiffusion is treated using the de Gennes reptation theory and the Wool minor chain reptation model. The self-diffusion coefficients of polystyrene and the polymethacrylates obtained by small-angle neutron scattering, SANS, direct non-radiative energy transfer, DET, and other techniques are compared. Reduced to 150,000 g/mol and 135°C, both polystyrene and poly(methyl methacrylate) have diffusion coefficients of the order of 10^?16?10^?17 cm²/sec. Variations in the diffusion coefficient values are attributed to the experimental approaches, theoretical treatments and molecular weight distribution differences. An activation energy of 55 kcal/mol was calculated from an Arrhenius plot of all polystyrene data reduced to a number-average molecular weight of 150,000 g/mol, using an inverse square molecular weight conversion method. Interestingly, this is in between the activation energies for the α and β relaxation processes in polystyrene, 84 and 35 kcal/mol, respectively. Fracture of polystyrene was considered in terms of chain scission and chain pull-out. A dental burr apparatus was used to fracture the films. For low molecular weights, chain pull-out dominates, but for high molecular weights, chain scission dominates. At 150,000 g/mol, the energy to fracture is divided approximately equally between the two mechanisms. Above a certain number average molecular weight (about 400,000 g/mol), the number of chain scissions remains constant at about 10²⁴ scissions/m³. Energy balance calculations for film formation and film fracture processes indicate that the two processes are partly reversible, but have important components of irreversibility. From the interdiffusion SANS data, the diffusion rate is calculated to be about 1 Å/min, which is nine orders of magnitude slower than the dental burr pull-out velocity of about 0.8 cm/sec.     

Polymer fracture—A simple model for chain scission

A simple model for calculating the fracture process for a single extended-chain molecule such as polyethylene is considered. The model consists of a chain of N coupled Morse oscillators. There exists a critical overall extension ΔL_c below which the fracture is energetically unfavorable but above which fracture is favored both energetically and kinetically. This elongation ΔL_c scales as N^1/2. For the critically stretched chain, the activation energy for rupture increases with N. Long chains must be stretched beyond this critical value to fail within experimentally meaningful times. Chains of all lengths subjected to the same force will fail with the same activation energy, provided this force is large enough to stretch each chain to ΔL > ΔL_c. Observed activation energies are less than 1/3D_e, where D_e is the bond energy.     

Energy landscapes of biomolecular adhesion and receptor anchoring at interfaces explored with dynamic force spectroscopy

Beyond covalent connections within protein and lipid molecules, weak noncovalent interactions between large molecules govern properties of cellular structure and interfacial adhesion in biology. These bonds and structures have limited lifetimes and so will fail under any level of force if pulled on for the right length of time. As such, the strength of interaction is the level of force most likely to disrupt a bond on a particular time scale. For instance, strength is zero on time scales longer than the natural lifetime for spontaneous dissociation. On the other hand, if driven to unbind or change structure on time scales shorter than needed for diffusive relaxation, strength will reach an adiabatic limit set by the maximum gradient in a potential of mean force. Over the enormous span of time scales between spontaneous dissociation and adiabatic detachment, theory predicts that bond breakage under steadily rising force occurs most frequently at a force determined by the rate of loading. Moreover, the continuous plot (spectrum) of strength expressed on a scale of loge(loading rate) provides a map of the prominent barriers traversed in the energy landscape along the force-driven pathway and reveals the differences in energy between barriers. Illustrated with results from recent laboratory measurements, dynamic strength spectra provide a new view into the inner complexity of receptor-ligand interactions and receptor lipid anchoring.     

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.     

拉伸分子动力学方法研究聚乙烯单链的力学行为

采用拉伸分子动力学方法(steered molecular dynamics,SMD)研究一端固定的聚乙烯单链(singlepolyethylene chain)在被拉伸过程中的力学性质.在拉伸过程中发现平均拉力〈f〉受拉伸速度v的影响,当v<0.05 nm/ps时,〈f〉在250 pN附近会出现一个拉力平台.聚乙烯单链各部分的形状因子在拉伸过程中表现出一定的规律性,总是头部和尾部的形状因子〈δh〉、〈tδ〉先增加然后才是中间部分〈δm〉增加.如果按顺序再释放被拉开的聚乙烯单链,就会出现力学回滞现象,这与Kellermayer等的力学回滞曲线实验是一致的.力学回滞曲线面积表示耗散能〈Ed〉,与速度v满足方程〈Ed〉=a+b×e-cv,而且在v<0.005 nm/ps和v>0.005 nm/ps两个速度区域有不同的特性.〈Ed〉在不同的分子热运动温度区域,也表现出不同的规律性,当温度T>220 K时,〈Ed〉随着温度T的升高而减小,这与Pegoretti等的实验一致,当T<220 K时〈Ed〉随着温度T的升高而增加.     

Shear and extensional rheology of entangled polymer melts: Similarities and differences

This work extends our previous understanding concerning the nonlinear responses of entangled polymer solutions and melts to large external deformation in both simple shear and uniaxial extension. Many similarities have recently been identified for both step strain and startup continuous deformation, including elastic yielding, i.e., chain disentanglement after cessation of shear or extension, and emergence of a yield point during startup deformation that involves a deformation rate in excess of the dominant molecular relaxation rate. At a sufficiently high constant Hencky rate, uniaxial extension of an entangled melt is known to produce window-glass-like rupture. The present study provides evidence against the speculation that chain entanglements tie up into "dead knots" in constant-rate extension because of the exponentially growing chain stretching with time. In particular, it is shown that even Instron-style tensile stretching, i.e., extending a specimen by applying a constant velocity on both ends, results in rupture. Yet, in the same rate range, the same entangled melt only yields in simple shear, and the resulting shear banding is clearly not a characteristic of rupture. Thus, we conclude that chain entanglements respond to simple shear in the manner of yielding whereas uniaxial extension is rather effective in causing some entanglements to lock up, making it impossible for the entanglement network to yield at high rates.     

拉伸分子动力学方法研究高分子单链的吸附现象

采用拉伸分子动力学方法研究聚乙烯单链从一个强吸附表面被拉伸的动力学过程.当聚乙烯单链被拉伸时,从理想弹簧测量到的平均力与形变的曲线上出现了一个力的平台,这与实验结果非常一致.在研究这个平台力与拉伸速度的关系时发现,当速度v<0.001 nm/ps时,不随速度的变化而变化;当速度v>0.001 nm/ps时,随着拉伸速度的增加而迅速上升,这表明拉伸速度0.001 nm/ps是平衡态拉伸和非平衡态拉伸的临界点,并与链长无关.按顺序拉伸和释放这条链,观察到有趣的力回滞现象,并与实验结果进行了比较.     

Determination of fracture energy of high strength double network hydrogels

The fracture energy G of double network (DN) gels, consisting of poly(2-acrylamido-2-methylpropanesulfonic acid) (PAMPS) as the first network and poly(acrylamide) (PAAm) as the second network, was measured by the tearing test as a function of the crack velocity V. The following results were obtained: (i) The fracture energy G ranges from 10(2) to approximately 10(3) J/m2, which is 100-1000 times larger than that of normal PAAm gels (10(0) J/m2) or PAMPS gels (10(-1) J/m2) with similar polymer concentrations to the DN gels. (ii) G shows weak dependence on the crack velocity V. (iii) G at a given value of V increases with decreasing of cross-linking density of the 2nd network. The measured values of G were compared with three theories that describe different mechanisms enhancing the fracture energy of soft polymeric systems. A mechanism relating to a heterogeneous structure of the DN gel is convincing for the remarkable large values of G.     

Experimental studies on dynamic fracture behavior of thin plates with parallel single edge cracks

Dynamic fracture behavior of polymer PMMA thin plates with three- and four-parallel edge cracks was studied by means of the method of caustics in combination with a high-speed Schardin camera. A series of dynamic caustic patterns surrounding the crack tip and fracture path of the specimen were recorded simultaneously by two types of focused images. Some dynamic fracture parameters such as the dynamic stress intensity factor, crack velocity and acceleration were determined. The evolution of dynamic stress intensity factors on the parallel edge cracks, due to the dynamic unloading effect, was analyzed from the viewpoint of the release of elastic strain energy.     

Fracture behavior in nylon 6 fibers

A reaction rate model of fracture in polymer fibers is described. This model assumes that bond rupture is governed by absolute reaction rate theory with a stress-aided activation energy. It is demonstrated that the key in obtaining good agreement between the model and experiment lies in taking proper account of the variation of stress on the tie-chain molecules. The more taut chains rupture first, and the load is redistributed among the remaining unruptured tie chains. The effect of varying the temperature both in the model and in experiments on fracture in fibers is explored. Good agreement between predictions of the model and experiment is possible only with an undeterstanding of the distribution in stress on the tie chains. The distribution in stress on the chains was experimentally determined by monitoring the kinetics of bond rupture with electron paramagnetic resonance (EPR) spectroscopy. Temperature is found to have two effects on macroscopic strength. (1) The thermal energy aids the atomic stress in breaking the atomic bonds; as a consequence the rate of bond rupture of a family of bonds under a given molecular stress is increased. In this respect temperature might be viewed as decreasing the “strength” of a bond. (2) Temperature also serves to “loosen” the molecular structure and in this way modify the distribution in stress on the tie chains. To explain bond rupture and macroscopic fracture behavior quantitatively, account must be taken of both effects.