of Single Polymer Molecules Characterization of Single Polymer Molecules

This chapter offers an overview of the use of atomic force microscopy (AFM) in polymer studies. Soft AFM cantilevers with sharp tips are useful for their relatively high spatial resolution, a few nm, and force resolution, a few tens of pN. AFM imaging is used to characterize conformational properties of single polymer chains at solid-liquid interfaces. AFM force microscopy gives molecular elasticity as well as interaction forces of single polymer chains with solids. Recent technical developments have made possible the characterization of time-resolved mechanical properties of single polymer chains, including the relaxation time and internal friction. AFM force microscopy with biomolecules, supramolecules, and mechanophores reveals the forces required for, and the kinetics of, conformational transitions and chemical reactions in these molecules at the single-chain and single bond levels.


Introduction
From the time of its invention in 1986 [1], atomic force microscopy (AFM) has been influential in polymer studies mainly at the nanoscale. The imaging mode of AFM has been used to visualize polymer chains [2,3], while the force microscopy mode to measure their elasticity, internal friction, and adhesion forces [4][5][6][7]. Moreover, the long-established theories of polymer mechanics and dynamics could be reevaluated and retuned to better interpret the new results obtained from AFM measurements [8,9]. Alongside theories, computational chemistry methods have been adopted to evaluate relevant experimental parameters from ab initio or molecular dynamics calculations, or to model the force response of polymers with conformational transition, for example polysaccharides [10,11].
In imaging application, polymer chains are generally adsorbed from a dilute solution. The dilute condition results in thin polymer films where the chains are isolated. The polymers are deposited on flat solids such as mica, silica (due to roughness, silica is used with thick polymers such as dendronized polymers), gold (for example, gold deposited on mica), or highly oriented pyrolytic graphite (HOPG). The individual chains are then imaged using noncontact or intermittent contact imaging modes [12][13][14][15][16][17]. Analysis of AFM images provides useful information on conformations and sizes of polymer molecules, and conformational transitions because of changing chemical environment [3,[18][19][20][21]. Examples of AFM images of doublestranded DNA [3] and four generations of a dendronized polymer [22] are shown in Figure 1 (a) and (b). Analysis of DNA images shows the effect of chemical environment, solution as well as solid substrate, on DNA conformation and length. Processing of the AFM images of dendronized polymers show that chains thicken with generation of dendronization, while their conformations persist over longer distances.
In a seminal work, Gaub and coworkers showed that AFM can be used to manipulate proteins at single molecule level [4]. This research led to the use of AFM in polymer studies involving the extension and manipulation of single polymer chains. The measurements are realized by adsorbing a polymer film on solid from a dilute to moderately concentrated polymer solution. The tip of the AFM cantilever is then brought into contact with the solid and retracted. This process results in occasional extension of a single chain. The solid substrate and the AFM tip can be functionalized to chemically bind the polymer chains, or to tune between extension and desorption interactions [5,6]. To model the force versus extension profiles, the polymer chain is modeled with a continuous curve, or as a series of discrete segments that are freely jointed or jointed at fixed bond angles with rotational freedom [26]. These models normally incorporate a characteristic length corresponding to entropic elasticity of the polymer and a characteristic elasticity constant corresponding to deformation of bond angles. Examples of AFM force microscopy of poly(ethylene glycol) (PEG) [23] and single-stranded DNA [11] are shown in Figure 1(c) and (d). In both cases, one observes that the force increases monotonically with extension. This is because the polymer chain loses its entropy during elongation causing a restoring force on AFM cantilever. Unlike the response of single-stranded DNA, PEG force response shows conformational transition in electrolyte solution. The transition is absent in non-hydrogen bonding hexadecane.
Among other developments, AFM single molecule force microscopy was combined with electrochemistry to obtain sequential extension-oxidation-relaxation giving a thermodynamic cycle with a single chain of a redox polymer [27]. Using two AFMs in parallel configuration, a correlation force microscope (CFM, or correlation force spectroscopy, CFS) was developed and used to measure the dynamics of single polymer chains, namely elasticity and relaxation time [7,28]. Furthermore, by laterally dragging single polymer chains that are covalently bound to AFM tip and adsorbed onto solid, nanoscale friction mechanisms were investigated using a single polymer chain probe [29,30].
AFM is also used to activate chemical reactions and conformational transitions at single polymer chain level. In this case, the polymers contain force-sensitive units, which are activated by application of mechanical force. Moreover, to measure the strength of chemical bonds, one may incorporate a functional group at free end of polymer and investigate specific interactions between the group and the AFM tip or the solid. Investigation of chemical reactions at singlechain or bond level using AFM has led to insights into forces and kinetics of various chemical reactions and transitions, including complexation and coordination [31,32], receptor-donor type interactions [33], hydrogen bonding [34], and covalent bonding [35,36]. An example of mechanochemistry at single-chain level is shown in Figure 1(e). AFM force microscopy reveals that the force of opening benzocyclobutene ring is about 1400 pN in toluene, but reduces to 920 pN with the help of an alkene lever arm in the structure of the polymer [24]. Below, I have illustrated AFM application in polymer studies with specific examples. Schematics of the AFM applications in imaging, force microscopy, and other modes are shown in Figure 2. The structures of some of the polymers used in the experiments are summarized in Figure 3. The polymers are poly(2-vinyl pyridine) (P2VP), poly(styrene) (PS), poly(ethylene) (PE), poly(ethylene glycol) (PEG), and a triblock copolymer of poly(exo-N-(2-aminoethyl)-5norbornene-2,3-dicarboximide) and poly(exo-N-hexyl-5-norbornene-2,3-dicarboximide) (P1).  . Chemical structure of a triblock copolymer of poly(exo-N-(2-aminoethyl)-5-norbornene-2,3-dicarboximide) and poly(exo-N-hexyl-5-norbornene-2,3-dicarboximide) (P1) [37], poly(2-vinyl pyridine) (P2VP), poly(ethylene glycol) (PEG), poly(styrene) (PS), and poly(ethylene) (PE). The side blocks of P1 are about 11 monomers long (m ≈ 11) and the middle block 544 monomers long (n ≈ 544). The side blocks contain amine, which enhances bonding of the polymer ends to epoxy-functionalized AFM tip and solid. The covalent bonding helps pull the polymer to high forces of about 1 nN. P2VP is positively charged at pH 3.0.

Molecular conformations obtained from AFM imaging
Conformation of a single polymer chain may be interpreted in terms of average of spatial correlations between unit vectors n tangent to the chain. In the framework of wormlike chain (WLC) model, the average function is of the form: where s is the length, and ℓ p is the characteristic decay length of the correlations, or the persistence length. Image analysis software has been developed that tracks the imaged chains and quantify their persistence lengths using Eq. (1) [38].
The correlations generally decay rapidly for thin and flexible polymers, but persist longer for thick and semiflexible polymers, such as double-stranded DNA, which have inherent bending rigidity [3,18]. For charged polymers such as polyelectrolytes, the persistence length has a contribution from intramolecular electrostatic repulsion, which tends to expand the chain. This contribution may be controlled by pH and the ionic strength of an electrolyte solution. Odijk, Skolnik and Fixman (OSF) theory predicts that the electrostatic contribution decays rapidly with inverse of the ionic strength [39,40]. However, experiments and simulations generally find a slower decay [18,41,42].  3. AFM force microscopy of single polymer chains

Molecular elasticity
From an analysis of the force versus extension response of single polymer chains, one may interpret their elasticity. The elasticity has two contributions: one from the loss of entropy and the other from the deformation of bond angles [23]. Bond angle deformation results in polymer length increasing beyond its contour length (the unperturbed length of polymer chain). The polymer length increases by about 10% at a force of about 2 nN [43].
The crucial step in interpretation of the elasticity of single polymer chains is the identification of single-chain responses, namely that two or more chains are not simultaneously measured.
Oversight of this step would result in force responses that are stiffer than the response of an individual chain. It is equally important to ensure that the ends of the polymer chain are strongly adhered to the solid and the AFM tip; that is, the polymer does not slide over the tip or the solid. Sliding would result in softer response than the pure elastic response of the chain.
The force versus extension response is generally interpreted in terms of freely jointed chain (FJC) model [44]: where L is the contour length, k is the Boltzmann constant, and T is the absolute temperature. The Kuhn length ℓ K and the elasticity constant K represent the mechanical properties of single chains. The FJC model has been successful in the analysis of extension responses of flexible polymers, such as synthetic polymers [44]. Figure 5 shows the force versus extension responses of poly(ethylene) (PE), which were collected in methyl benzoate and on silica [43]. Polymer solution with concentration 100 mg/L dissolved in toluene was used for deposition. After a deposition period of about 40 s, the polymer-coated silica was rinsed multiple times with toluene to remove loosely bound polymer chains. Thereafter, repeated extension-retraction cycles of the AFM tip to and from polymer coated substrate results in the force-extension responses of single polymer chains. After modeling the individual force responses with the FJC, the extension length of each response was normalized to the fitted contour length. The figure displaying the force versus relative extension profiles shows that the responses from different chains agree reasonably well. The overlap of the profiles asserts that the responses were obtained from single chains. An average Kuhn length ℓ K ¼ 0:6 AE 0:1 nm and an elasticity constant K ¼ 24 AE 3 nN were obtained for PE. Figure 6 shows the force versus relative extension responses of P2VP and PS. P2VP responses were collected in 1 mM, pH 3.0 solution and on mica. Sample preparation was like that explained for AFM images in Figure 4. Nanohandling technique was employed to ensure the placement of AFM tip on one end of the adsorbed polymers [5]. An average Kuhn length ℓ K ¼ 0:5 AE 0:1 nm and elasticity constant K

Adhesion force of single polymer chains
To obtain adhesion interaction forces between single polymer chains and solids, the polymer chains are generally covalently bound to the AFM tip [6,25,45]. The polymer chains are brought in contact with the solid. During contact, a single polymer chain may adsorb onto the solid. Upon retraction of the tip, the polymer chain desorbs resulting in a steplike (constant) force response. This force response is then fitted to a sigmoidal model giving the desorption force and length of the polymer-solid interaction.
An example of these studies is shown in Figure 1(f) [25]. The force versus extension response of poly(isoprene) with 88 kDa PS side chains in water and on hydrogen-terminated diamond shows two force response behaviors. In one case, polymer chains desorb from solid, resulting in steplike response. If two or more polymer chains desorb simultaneously, additional steps are observed in the response. Thereby, the last step is due to the final desorbed polymer chain. The second response behavior involves polymers being extended before detachment from the solid. The desorption force of polymer chains from solid may generally be tuned by the chemical environment of the polymer, polymer chemistry, and the adsorption time on the solid [6,25].

Dynamical mechanical properties of single polymer chains
Elasticity of single polymer chains is only one property that defines their response to force. The other property is the relaxation time, or the time it takes for the polymer chain to respond to the force. Lessons from nature, e.g., wing flapping of hummingbirds, tongue projection of salamanders, or eye retraction of slugs, show that these responses are not infinitely fast but take time. This is especially important for end-tethered polymers [46]. Experiments that measure the elasticity and the relaxation time of single polymer chains generally use the thermal fluctuations of an AFM cantilever [47,48], or externally drive the cantilever by magnetic or acoustic forces [49]. Recently, a correlation force spectroscopy (CFS) is developed that employs two AFM cantilevers in antiparallel configuration as shown in Figure 7(a). The advantage of using two cantilevers in CFS, as compared with one cantilever in AFM, is that in AFM, the proximity of the cantilever to the solid increases the hydrodynamic friction due to thin film lubrication. The increase in the hydrodynamic force (or the hydrodynamic friction coefficient) increases the Brownian forces-a result of fluctuation-dissipation theorem [51]. Brownian forces result in thermal noise that is the major source of noise in AFM force spectroscopy measurements. Because of the thermal noise and the high hydrodynamic force, AFM force resolution is reduced, and polymer chains may only be examined accurately when extended to high forces. To reduce the high force limit, in AFM applications discussed in the above sections 3.1 and 3.2, one applies a low-pass filter to cantilever deflection signal and thereby discards the time-related or dynamical data. Placement of two AFM cantilevers in the configuration shown in Figure 7(a) reduces the hydrodynamic friction and the Brownian forces. Figure 7(b) shows a comparison between the hydrodynamic friction coefficient between AFM and CFS. In all separations (in AFM, tip-solid separation, in CFS, tip-tip separation), CFS has a lower hydrodynamic friction coefficient. Similarly, the Brownian forces or the thermal noise are lower in CFS than in AFM. Thereby, CFS has a higher force resolution. CFS also gives the dynamical mechanical properties of single molecules where no filtering is applied in the data analysis [7,51].
In the measurements, a single polymer chain is tethered between two tips, then extended to a force and clamped. During the clamp period, thermal fluctuations of the top and bottom cantilevers are collected simultaneously. Dynamical mechanical properties of single polymer chains are obtained from an analysis of the time correlations between the two thermal fluctuations. Figure 7(c) and (d) show the stiffness and the relaxation time of end-tethered singlestranded DNA in the force range from 5 to 50 pN, respectively, [28]. One observes that the stiffness of the chain increases with the force, while the relaxation time remains almost constant equal to about 30 μs. Constant relaxation time is consistent with theory [50].

Mechanochemistry at the level of single polymer chains
The force versus extension response of biopolymers, such as double-stranded DNA and various proteins, supramolecules, and polymers containing force-sensitive units, namely mechanophores, generally shows a different behavior. In these polymers, specific structural changes or chemical reactions occur, which are triggered by the application of mechanical force [31][32][33][34][52][53][54][55]. The process involves force reducing the energy barrier of transition by an amount FΔx, where Δx is a length scale associated with the transition length [56]. The reduction in the energy barrier facilitates the transition. For example, the rate of transition increases by a factor exp FΔx ð Þ. It has been shown that reactions that do not occur thermally may be triggered by the application of mechanical force [24,57].
Mechanically induced isomerization of cis carbon-carbon double bonds to trans conformation in polymer P1 is shown in Figure 8 [37]. Experiments were realized by adsorbing polymer chains from a solution with concentration 100 mg/L dissolved in dimethyl sulfoxide (DMSO). The deposition period was 2 hr, after which the solid was rinsed multiple times with DMSO to remove loosely bound chains. Finally, DMSO was added to the solid before the measurements. Figure 8 shows the force versus extension responses of polymer P1, which differ from the force behaviors of PE, P2VP, and PS in Figures 5 and 6. In the latter, the force increases with extension until the chain breaks from either the AFM tip or the solid. As shown in Figure 8, a single chain of polymer P1 is tethered between the AFM tip and the solid even after the isomerization. The force response of P1 contains a sudden increase in the extension that is due to the isomerization of some cis monomers in the backbone of P1 to trans conformation. When isomerization occurs, the force shows a sudden reduction that is due to relaxation of stress on the chain because of extension increase. The force where the isomerization occurs is denoted by F ct and has an average value of about 800 pN. The isomerization force is lower than the force of breaking of covalent bonds and rings, 1-2 nN [57,58].

Conclusions
AFM started as a power imaging technique and soon found its way in the diverse field of polymer studies. In this chapter, the focus was placed on those studies that are at the level of single polymer chains, that is nanoscale. AFM imaging in noncontact mode or intermittent contact mode may be used to obtain conformations and sizes of individual polymer chains. The chains ought to be adsorbed from dilute polymer solutions and on atomically flat solids. AFM force microscopy may be used to obtain the elasticity of single polymer chains. The molecular elasticity in this case is interpreted in terms of an entropic elasticity, which can be tuned by the solvent, and an elasticity term that is due to deformation of bond angles. In the case of force-sensitive polymers, AFM may be used to apply force, and thus trigger specific chemical reactions or conformational transitions in the polymer at the level of single chains and even single bonds. Technical development in AFM has resulted in techniques such as correlation force spectroscopy, which is employed to obtain the dynamical mechanical properties of single polymer chains. Finally, one should note that AFM has also been used to characterize the mechanical properties, such as adhesion, friction, and compression support, of dense polymer films and polymer brushes. This level of investigation is not single-molecule level and thereby was not included in this chapter.