Dynamic Force Spectroscopy of the Silicon Carbon Bond

Transcription

1 Hochschule für angewandte Wissenschaften FH München Fakultät 06 Feinwerk- und Mikrotechnik, Physikalische Technik Mikro- und Nanotechnik Master Thesis Dynamic Force Spectroscopy of the Silicon Carbon Bond von Sebastian W. Schmidt München, den Aufgabensteller: Referent: Korreferent: Labor für Nanoanalytik und Biophysik, Hochschule für angewandte Wissenschaften FH München Prof. Dr. Hauke Clausen-Schaumann Prof. Dr. Attila Vass

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3 Contents III Contents Abstract... V 1 Introduction Materials and Methods Experimental Setup Data Analysis Theory of Force-Induced Bond Rupture Bond Dissociation under Constant Force Bond Dissociation under Constant Force-Loading Rate Potentials with Fixed Barrier Distance Morse Potential Analytical Model Based on the Combination of the Arrhenius Rate Equation and a Morse Potential Analytical Model Connected with High-Level Density Functional Theory Calculations Results and Discussion Experimental Data Modelling of the Experimental Data Modelling Based on a Potential with Fixed Barrier Distance Modelling Based on the Combination of the Arrhenius Rate Equation and a Morse Potential Simulated Rupture Forces Based on an Arrhenius Kinetics Model with a Morse Potential Conclusion References Figures Tables Abbreviations Appendix Appendix A. Sample Preparation Protocol Appendix B. Cantilever Calibration Appendix C. Elasticity of the CM-amylose polymer Appendix D. Analytical Model Based on an x 3 -Binding Potential... 61

4 Contents IV Appendix E. Software and Instruments Acknowledgment... 66

5 Abstract V Abstract In chemical and material sciences, as well as for many practical applications, the strength of covalent bonds is of profound importance. Since bond dissociation under external force is a thermally activated process, the bond strength is not just governed by the structural parameters of the binding potential, like potential depth and width, but is also affected by the timescale of the experiment, i.e., the rate at which the force is applied. In this study the mechanical strength of individual Silicon Carbon bonds was determined as a function of the applied force-loading rate df dt by dynamic single molecule force spectroscopy, using an atomic force microscope (AFM): individual carboxymethylated amylose polymers, which were attached between an amino activated AFM tip and an amino activated glass substrate, were stretched and the loading force was continuously increased until the link between AFM tip and substrate surface was lost. As the weakest bond in this system is the Si C bond, a failure of the connection between AFM tip and substrate can be attributed to the failure of this bond. In an effort to asses the dynamics of this bond rupture process, as well as the structural parameters of the Si C binding potential, force-loading rates were applied in the range of 0.5 to 267 nn/s, spanning three orders of magnitude. As predicted by Arrhenius kinetics models, a logarithmic increase of the bond rupture force with increasing force-loading rate was observed, with average rupture forces ranging from 1.1 nn for 0.5 nn/s to 1.8 nn for 267 nn/s. Three different theoretical models, all based on Arrhenius kinetics and analytic forms of the binding potential, were used to analyze the experimental data and to extract the parameters f max and D e of the binding potential together with the Arrhenius A-factor. All three models well reproduced the experimental data including statistical scattering. However, the three free parameters cover a considerable range and an unambiguously determination of all three free parameters is not possible on the basis of the experimental data. Successful fits with

6 Abstract VI an Arrhenius kinetics model based on a Morse potential, which was linked to density functional theory (DFT) calculations were achieved for the range of the maximum forces f max = nn and the corresponding bond energies D e = kj/mol, with the Arrhenius A-factor covering to /s, respectively.

7 Introduction 1 1 Introduction Besides traditional methods, where chemical reactions are initiated or accelerated through catalysts, heat, light, pressure or electricity, the approach of mechanochemical activation, where the course of chemical reactions is controlled by mechanical stress, is in the focus of investigation since several years [1-3]. The fact that chemical reactions can be activated and controlled by mechanical force was already recognized around 300 B.C. in Greece [4], and today, the mechanical activation of chemical reactions through rubbing, grinding, milling, sonication etc. is familiar to every chemist and chemical engineer [5-9]. One focus in the realm of mechanochemistry is on the investigation of the strength of intermolecular interactions and intramolecular forces with the aim to understand and furthermore control the underlying processes on the molecular level. In addition, by investigating the mechanical properties of materials on the molecular level it is possible to infer to the underlying structure, because structure and mechanical properties are mutually dependent parameters [10-12]. This relation between structure and mechanical parameters is in turn essential for the functionality of materials as has been shown in several studies dealing with biomaterials [13-15]. The macroscopic properties and the mechanical stability of, e.g., polymeric materials are fundamental for many practical applications. These macroscopic properties are governed to a large extent by the intrinsic structures, i.e., the mechanical characteristics of individual polymer chains [16], and it is well understood, that mechanical stress causes failure of polymeric materials, resulting from bond scissions [17-21]. In this context it turns out targetaimed to explore the mechanical properties on the molecular level experimentally in an effort to predict the strength and durability of materials and furthermore develop new materials with enhanced properties. The idea that stress-induced reactions in polymers could be used for the purpose of repairing damage or controlling molecular strengthening was pursued in a study of Hickenboth et al.,

8 Introduction 2 where the possibility to vary the course of a chemical reaction due to mechanical force was demonstrated. Hickenboth et al. were able to alter the shape of activation energy barriers sitespecifically and to obtain chemical reaction pathways that are unattainable through conventional activation with UV light or heat [22]. In the long run, this approach may lead to more efficient chemical reactions as well as the development of stress-sensitive or even selfhealing materials, where the mechanical energy stored in stressed polymers is used to activate desired chemical pathways, e.g., self-healing reactions. Potential applications range from selfhealing coatings to self-healing structural components to self-healing adhesives, to name but a view [23-25]. In order to obtain information about bond energies from bounded atoms or molecules, the traditional approach is to measure these parameters by bulk techniques, e.g., optical spectroscopy [5] or calorimetry [26]. In contrast to energies, however, the extraction of intramolecular forces from ensemble measurements is hardly possible since force is a directional parameter. The division of the acquired overall force by the number of measured bonds turns out to be an inappropriate way to extract the binding force of a single molecule, because the pre-condition would be a synchronisation of all probed molecules in time and space [27]. By probing one molecule at a time this synchronisation is no longer required and with the advancement of single molecule techniques such as the atomic force microscope (AFM) or optical tweezers the force-extension response of single molecules has become experimentally accessible. These instruments are characterized by piconewton force sensitivity and Ångstrom position control and are therefore highly suitable for investigations at the level of individual molecules. In this manner, AFM-based techniques have been applied in the field of life science and material science to study polymer elasticity [28], DNA base pairing [29], protein folding [30], molecular recognition [31], and biomolecular [32] as well as covalent [33] bonds, to name but a view outstanding publications.

9 Introduction 3 Probing the force-extension response of single molecules allows for the extraction of the strength of intermolecular and intramolecular forces. Moreover, an insight into the structural parameters of the binding potential, which is characterized by the reaction energy landscape, is gained [3, 34, 35]. On the basis of theoretical studies, where simplified energy landscapes were modelled [36, 37], it has been suggested that the underlying energy landscapes should be extractable by measuring rupture forces of single bonds as a function of the force-loading rate, which corresponds to an increase in force per time [38]. A widely used term for such experiments, where bonds are probed over many orders of magnitude in force-loading rate, is dynamic force spectroscopy [39, 40]. However, because data about the force-loading rate dependence of rupture forces of covalent bonds is still missing today, in the present study, the dynamic strength of the Silicon Carbon bond was investigated by systematically varying the applied force-loading rate over three orders of magnitude.

10 Materials and Methods 4 2 Materials and Methods 2.1 Experimental Setup As already mentioned, the AFM is a versatile tool to study processes on the single-molecule level and follow intermolecular and intramolecular processes in real time in an effort to obtain information about mechanical, chemical or functional properties of single molecules [16]. One of the main challenges in single molecule force spectroscopy is to find an appropriate coupling chemistry which allows specific binding of individual molecules between, e.g., AFM tip and substrate. To accomplish specific binding, substrate and AFM tip are functionalized chemically offering specific anchor points for the molecule which is subject of investigation [26]. In an effort to minimize non-specific interactions between substrate and AFM tip, i.e., adhesion, electrostatic and van der Waals interactions, force measurements are typically performed by attaching spacer molecules to the interacting molecules. These spacer molecules are stretched during measurement and lead to a bond rupture at tip-substrate distances, where the range of short distance interactions has already been exceeded [41, 42]. In addition, AFM based single molecule measurements are performed in aqueous solution, because it is well known that the electrostatic interactions are screened by the shielding effect of dissolved ions within a few nanometers [5]. If a functionalized AFM tip is moved toward a functionalized surface, where the molecule of interest is already anchored and a spacer molecule is attached (cf. Figure 2-1A), the tip can interact with the spacer molecule and form a link, e.g., a chemical bond (cf. Figure 2-1B). Upon retracting the AFM tip from the surface, the spacer molecule is stretched and a cantilever deflection proportional to the tensile force F tensile can be observed (cf. Figure 2-1C). The cantilever deflection is recorded with an optical lever, where a laser beam is focussed on the backside of the cantilever and reflected to a segmented photodiode. As soon as the tensile

11 Materials and Methods 5 force exceeds the strength of the weakest bond in the chain the bond ruptures and the cantilever snaps back into its equilibrium position (cf. Figure 2-1D). Thus one measures the force as a function of the displacement of the base of the AFM cantilever relative to the substrate surface, plotted as a so-called force-distance curve. Figure 2-1. Schematic set of an AFM-based single-molecule force spectroscopy experiment. A functionalized AFM tip and a functionalized substrate with the molecule of interest attached are approaching (A) and a link between the molecule and the AFM tip is formed via a spacer molecule (B). Retracting the AFM tip from the surface is accompanied with a deflection of the cantilever, which is proportional to the tensile force F tensile (C). The deflection is detected via the changing path of the reflected laser beam. The connection is lost when the tensile force exceeds the strength of the link (D). To observe and record rupture events of single covalent bonds, in the present study, individual carboxymethylated amylose (CM-amylose) polymers were covalently coupled between an amino-functionalized glass slide and an amino-functionalized AFM tip and stretched until detachment [33]. The pre-cleaned glass slides and AFM tips were coated with amine functional groups by incubation in a 2% N 1 -[3-(Trimethoxysilyl)-propyl]diethylenetriamine (DETA) solution in ethanol (EtOH) [33, 43, 44]. In order to activate the carboxyl groups of the CM-amylose polymers, CM-amylose, ethyl-dimethylaminopropyl-carbodiimide (EDC), and N-hydroxysuccinimide (NHS) were dissolved in phosphate buffered saline (PBS) and incubated on an amino-silanized glass slide, leading to the formation of covalent anchors between CM-amylose polymers and the surface [33, 45-47]. Prior to force measurements, the

12 Materials and Methods 6 glass slide was thoroughly rinsed with PBS in order to remove non-covalently bound molecules from the surface. The glass slide was subsequently covered with PBS and transferred to the AFM. An amino-functionalized AFM tip was then moved towards the glass slide to couple individual CM-amylose polymers to the AFM tip. As soon as a single CMamylose strand was attached to the tip, the AFM tip was retracted from the surface and the molecular chain was stretched, until the connection ruptured. In order to record rupture events at various force-loading rates, the retraction velocity of the z-piezo was varied between 100 nm/s and nm/s, and force-distance curves were acquired. In order to extract bond rupture forces versus force-loading rates and convert force-distance curves to force-extension curves, 1 the force constants of AFM cantilevers were determined, using the thermal noise method [48]. Since various studies have shown that non-specific attachments of polysaccharides to the AFM tip can be avoided by keeping the maximum indentation force of the AFM tip below 0.3 nn [33, 49, 50], additional negative controls were performed, where the indentation force was kept below 0.3 nn. Negative controls were carried out with CM-amylose, which was not activated by EDC and NHS. No binding events were observed at indentation forces below 0.3 nn [33]. A detailed description of sample and AFM tip preparation and cantilever calibration can be found in Appendix A and Appendix B, respectively. 1 Force-distance curves are given by the overall extension of the probed molecule and the additional deflection of the loaded cantilever. In contrast, force-extension curves exclusively specify the extension of the probed molecule and hence are used to extract information about the stretched polymer, e.g. the polymer length or specific conformational changes (cf. Figure 2-2).

13 Materials and Methods Data Analysis From the obtained force-distance curves the bond rupture forces and force-loading rates were extracted. To determine the bond rupture force f r as well as the force-loading rate df dt at a certain retraction velocity, the deflection signal U photo from the photodiode (recorded in Volts) was converted into force in Nanonewtons [nn]. The cantilever deflection d and the force f, respectively, are then given by d f = U s (1) photo = k d = k U s (2) c c photo where k c is the cantilever spring constant [N/m] and s is the optical lever sensitivity [nm/v], both obtained from the cantilever calibration. The resulting force-distance curve was then used to extract the force-loading rate df dt as well as the rupture force f r. By measuring the slope f / z immediately before the bond rupture and multiplying the slope by the retraction velocity v re, the force-loading rate df dt can be extracted as follows: df dt f = ν re. (3) z The bond rupture force f r is obtained by extracting the maximum force before the connection between AFM tip and surface is finally lost and the cantilever relaxes to its equilibrium position which defines zero force. Here, a constant force-loading rate over the entire time scale of the experiment is assumed, i.e., the applied force is supposed to grow linear with time. Because of the non-linearity of the polymeric spacer this assumption is in fact an approximation [37, 51, 52]. However, for covalent bonds, the probability of a bond failure in the non-linear low force regime (below 0.5 nn) is many orders of magnitude smaller than at higher forces [53], and therefore force dependent force-loading rates may be neglected in the case of covalent bonds (cf. also theoretical section).

14 Materials and Methods 8 In the next step, force-distance curves were converted to force-extension curves. The conversion is of particular importance since the recorded z-piezo distance corresponds to the distance between surface and the base of the AFM cantilever. However, because the stretched molecule is attached to the AFM tip, the cantilever deflection d has to be subtracted from the measured z-piezo distance z p to obtain the tip-surface distance z ts, which corresponds to the actual extension of the stretched molecule (Figure 2-2): z ts f = z p d = z p, (4) k c where f is the force acting on the tip and k c is the cantilever spring constant [5]. Figure 2-2. Correlation of the measured z-piezo distance z p and the actual distance between AFM tip and surface z ts in an AFM-based force spectroscopy experiment (left side). To extract the distance between AFM tip and surface z ts and finally obtain force-extension curves, the cantilever deflection d has to be subtracted from the distance z p (cf. also ref. [5]). On the right side a force-distance curve (black curve) is shown along with the corresponding force-extension curve (grey curve). To confirm that exclusively rupture forces of individual strands were considered for analysis, the shape of all force-extension curves was checked with respect to the characteristic plateau, which is known to be a dominant feature of single-molecule force curves of CM-amylose (shown in Figure 2-3). This plateau, which starts around 0.3 nn and has been attributed to a force-induced chair-to-boat transition of individual glucose rings in the polymer resulting in an enlargement of the CM-amylose monomer length by 0.5 nm, has been observed in various studies [50, 54, 55]. If more than one individual polymer was coupled between surface and

15 Materials and Methods 9 AFM tip, the plateau would appear at significantly higher forces, smear out or even disappear. Thus, the plateau can be seen as a reliable criterion for force-extension curves of single CMamylose strands. Figure 2-3. Typical force-extension curve of a single CM-amylose polymer covalently anchored between substrate and AFM tip along with the detected rupture force f r. The characteristic plateau, which origins from internal transitions of the glucose rings, can be seen as an internal reference for the detection of individual rupture events. The different lengths of the polymer tethers can be scaled in an effort to superimpose individual force-extension curves and to make sure that the rupture force of a single polymer was probed (cf. also ref. [16]). Consequently force-extension curves with significantly higher plateaus than 0.3 nn were discarded. Due to the fact that all the elastic properties of CM-amylose filaments scale linearly with their lengths a superposition of force-extension curves with different contour lengths gives rise to a plateau at the same force [55]. Because of the polydispersity of the CMamylose spacer molecules and the fact that the spacers were picked up at random positions, individual force curves of different polymer tether lengths where scaled with regard to their length and superimposed. A more detailed description of the chair-to-boat transition of single glucose rings due to an external force and the scaling and superposition of different polymer tethers can be found in Appendix C.

16 Theory of Force-Induced Bond Rupture 10 3 Theory of Force-Induced Bond Rupture 3.1 Bond Dissociation under Constant Force A covalent bond can be described by an electron pair shared between two neighbouring atoms [56]. To depict bond separation of diatomic molecules, Bell proposed a potential curve V ( x) with a minimum and a local maximum [36]. The location of the minimum is given by the distance between the two binding partners, the so-called equilibrium bond length (x 0 in Figure 3-1). The local maximum separates the bound from the unbound state and hence can be seen as a barrier against bond dissociation (x b in Figure 3-1). 2 Figure 3-1. Schematic representation of a single-well potential curve V(x) in the equilibrium state, where the internuclear separation between the two atoms of a diatomic molecule corresponds to the potential minimum at x 0. The energetic barrier E a at x b can be seen as the transition state of the bond dissociation event. The distance between minimum and activation barrier is given by x. Generally, bond dissociation is a time-dependent process driven by thermal fluctuations: thermally activated bond scission is a spontaneous process, where the bond lifetime depends on the height of the energy barrier (cf. Figure 3-1, E a ), i.e., the bond lifetime increases with 2 For the sake of completeness it should be mentioned that consecutive rebinding is not considered in the theoretical section.

17 Theory of Force-Induced Bond Rupture 11 increasing height of the activation barrier E a. Therefore, thermally activated bond dissociation is assumed to satisfy a first-order rate equation of the form dp dt () t () t off = k0 P, (5) where P () t is the probability, that the bond is still intact at the time t, and rate constant [57]. Following the reaction rate theory, k off 0 equation off k 0 is the reaction is governed by the Arrhenius k Ea off kbt 0 = A e. (6) Combining Eqs. (5) and (6) leads to dp dt () t Ea () kbt = A e P t. (7) In Eqs. (6) and (7) A is the Arrhenius pre-factor, E a is the activation energy, which is equivalent to the energy difference between the potential minimum and the barrier, and k B T represents the thermal energy with k B being the Boltzmann constant and T the absolute temperature. The pre-exponential factor A is given by q A = κν. (8) q Here, the parameter κ describes the accommodation coefficient, ν the vibrational constant and q q the ratio of the partition functions of the activated complex and the initial state, respectively. The ratio of q q is proportional to exp ( S k T ), where B S is the activation entropy. The vibrational constant ν depends on the curvature of the potential minimum and can be seen as the attempt frequency to break the bond [53, 58-60]. If the bond is subjected to an external force the unperturbed potential V ( x) is deformed because of the additional mechanical energy, which is stored in the deflected cantilever and the stretched polymer spacers (Figure 3-2, dashed inclined line) [61].

18 Theory of Force-Induced Bond Rupture 12 Figure 3-2. Schematic representation of an effective potential curve V eff ( x) when an external force is applied and a fixed barrier distance is assumed. The additional mechanical energy f ( x ) stored in the AFM cantilever and the stretched polymer spacers (dashed inclined line) leads to the deformation of the unperturbed potential V ( x), so that the activation barrier is lowered to E a ( f ) reaction rate constant is increased to k off ( f ). x 0. As a consequence, the force-dependent To obtain the resulting effective binding potential V eff ( x), and to assess the height of the activation barrier, the potential energy stored in the deflected AFM cantilever has to be subtracted from the original potential. Because in single molecule force spectroscopy typically soft effective springs are used, the stored potential energy may be approximated by f x, where x is the direction of the force acting, and the effective potential can be written as V eff ( x) V ( x) f ( x ) x 0 =. (9) Here, f is the applied force and x 0 is the location of the minimum in the undisturbed potential. In the case of a constant distance between potential minimum and activation barrier, the height of the activation barrier is diminished due to the external force to E a ( f ) E f x = (10) a and the reaction rate constant k off ( f ) increases to

19 Theory of Force-Induced Bond Rupture 13 k off ( f ) E f x E ( f ) a a f x k T kbt B off kbt = A e = A e = k0 e. (11) Consequently, with an additional external force the bond lifetime is shortened. However, single molecule force spectroscopy involves a mechanical transient [37] as an instantaneous application of force is usually not possible. Therefore the parameters k off 0 and x, which determine the bond rupture process, are difficult to extract directly. For that reason, in single molecule force spectroscopy, the bond is usually stretched with a constant pulling velocity and the bond rupture force is recorded as a function of the force-loading rate. 3 3 To probe the force-dependent rate of reduction of a disulfide bond, Wiita et al. introduced an alternative technique, the AFM-based single-molecule force-clamp spectroscopy, where the force is the controlled variable and the bond lifetime is probed at constant force [62, 63]. In this study, polyproteins with defined domains and a buried S-S bond were engineered. The multidomain proteins where stretched and unfolded stepwise until a mechanical strong disulfide bond was exposed to a solution of a reducing agent. The extended disulfide bond was then reduced by the reducing agent. Wiita et al. found a linear dependence of the reduction rate on the concentration of the reducing agent and an exponential dependence on the applied force [62].

20 Theory of Force-Induced Bond Rupture Bond Dissociation under Constant Force-Loading Rate Potentials with Fixed Barrier Distance Evans and Ritchie [37] introduced a theoretical model, where the bond is loaded by a force ramp and a constant barrier distance x is assumed. Therefore, a constant pulling velocity is presumed, so that the external force grows linear in time and the slope of the potential energy of soft effective springs is assumed to increase linear in time [64]. With these assumptions the activation barrier E a decreases linearly in time and reads [37, 65, 66] Here, E a () t E ( df dt) t x = (12) a df dt is the force-loading rate and x is the distance between the potential minimum and the activation barrier along the reaction coordinate (cf. Figure 3-2). 4 Combining Eq. (12) with Eq. (6) leads to ( E ( df dt) t x ) off a k0 = A exp (13) k BT Eq. (13) is valid provided that the term ( ) E a df dt t x is larger than a few k B T and rebinding is not considered. Inserting Eq. (13) into Eq. (7) gives an equation for the probability P () t that the bond is still intact at time t dp dt ( t x ) () t E ( df dt) a = A exp P() t. (14) k BT Integrating Eq. (14) yields P () t ( df dt) Ea A k BT k t x BT = exp e ( ) 1 exp df dt x. (15) k BT 4 In a multidimensional coordinate space, the reaction coordinate corresponds to the minimum energy pathway in the tilted energy landscape [59].

21 Theory of Force-Induced Bond Rupture 15 Substituting ( df dt) t by f gives the probability that the molecule is still in the bound state at the applied force f D : P ( f ) Ea A k ( ) BT k f x BT D = exp e 1 exp. (16) df dt x k BT Eq. (16) leads to the distribution of dissociation forces p( f ) D df D ( f ) dp = df D as follows df f D p ( f ) D df D = A e E f x a D kbt Ea A k BT k T f D x B exp e df D ( df dt) x 1 exp k BT.(17) The most probable dissociation force f mp is given by the maximum of the force distribution formulated in Eq. (17). Setting the first derivative ( ) = 0 dp f D df D and using Eq. (6) leads to f mp E x k BT + x ( df dt) x k T ( df dt) B x ln = ln. (18) A x k BT k0 = a off It is apparent from Eq. (18) that the most probable rupture force f mp increases logarithmically with the force-loading rate df dt [37] (cf. also Figure 3-3). Figure 3-3. Following the theoretical model by Evans and Ritchie [37], the most probable rupture force f mp (vertical axis) increases with logarithmically increasing force-loading rate df/dt (horizontal axis). The barrier distance x can be derived from the slope of the graph and the reaction rate k off 0 determining the interception point of the x-coordinate, where the force is zero. can be extracted by

22 Theory of Force-Induced Bond Rupture 16 This result can be understood keeping in mind that, with increasing pulling velocity, the timeframe for thermally activated bond dissociation is reduced. As a consequence, the probability of thermal bond dissociation at lower forces is also reduced and the observed rupture force increases. With this model, the distance between the potential minimum and the activation barrier x can be derived from the slope of force-loading rate-dependent bond rupture force measurements. Furthermore, the reaction rate k off 0 can be determined from the df dt - interception point for which the force vanishes (Figure 3-3). Therefore, x and used as fit parameters in an effort to model experimental results. off k 0 can be Nevertheless one has to be aware of the fact, that the model by Evans and Ritchie is based on the assumption that x does not depend on the applied force, i.e., the location of x 0 and x b is left unchanged. As pointed out by these authors, this approach is only reliable as long as the energy barrier is sufficiently sharp and/or the applied force is sufficiently weak. However, as will be shown in the results and discussion section, for covalent bonds, these assumptions are no longer valid.

23 Theory of Force-Induced Bond Rupture Morse Potential For covalent bonds, a Morse potential, like the one shown in Figure 3-4, is generally chosen as a one-dimensional analytical representation. The Morse potential represents a widely used and convenient empirical approach to describe binding potentials without a sharp activation barrier, so that the force-dependent distance between x 0 and x b is considered [60, 67]. Based on the approach by Kauzmann and Eyring [68] a suitable ansatz to describe the potential curve of a diatomic molecule under tension has been introduced by Beyer [53], who started out with an undisturbed Morse potential V βx ( x) D ( 1 ) 2 e e =, with (19) 2 f β = max, 5 D e where D e is the bond dissociation energy, x is the reaction coordinate, and f max is the maximum binding force which corresponds to the slope at the inflection point of the potential ' curve ( max ( x) f max V = ). 1 β is proportional to the width of the potential 6 Combining Eq. (9) and (19) leads to the effective potential (cf. Figure 3-4) V eff βx 2 ( x) = V ( x) f x = D ( 1 e ) f x e. (20) In order to describe bond dissociation under an applied force the potential given in Eq. (20) can then be combined with the Arrhenius kinetics model, as described in the following chapters. 5 βx The Morse potential reads in its classical notation: V = D ( 1 ) 2 e e, where β π m ( 2D ) ν = r e 2. Here, D e is the depth of the potential, corresponding to the bond dissociation energy, m r is the reduced mass of the diatomic molecule, and ν is the vibrational constant of the harmonic oscillator [60, 67]. 6 x 2 Following Eq. (19), the Morse potentials full width at half minimum can be written as D ( 1 e ) = ( 1 ) De x so that 1 e β = ± 1 2 and β, e 2 ( ) ( ) x β β. 1 ± 1 2 = e βx. From this it follows 1,2 = ( 1 ) ln

24 Theory of Force-Induced Bond Rupture 18 Figure 3-4. Undisturbed Morse potential V(x) with the dissociation energy D e together with the effective potential V eff (x) when an additional force is applied to a diatomic molecule. Here, the diatomic molecule Si C was chosen as previously calculated theoretically by Beyer [53]. In the unperturbed potential V(x) the equilibrium bond length is set equal to zero. Used values for D e and f max in the undisturbed potential are kj/mol and 4.78 nn, respectively. The effective potential V eff (x) corresponds to a rupture force of 2.59 nn and an energy barrier height of E a (f) = 67.7 kj/mol Analytical Model Based on the Combination of the Arrhenius Rate Equation and a Morse Potential An analytical approach based on the combination of the Arrhenius rate equation and a Morse potential has been introduced by Hanke and Kreuzer [52, 58, 59]. Here, bond rupture probability and Morse potential are directly related with each other. Starting again from Eq. (7) the probability P ( t) that the bond is intact at the time t satisfies the Arrhenius rate equation dp() t dt A [ E ( k T )] P( t) = exp. By applying an external force, which linearly a B increases in time with a force-loading rate df dt in order that f = f 0 + ( df dt) t, and furthermore eliminating t in favour of f, the probability that the molecule ruptures at the applied force f can be written as

25 Theory of Force-Induced Bond Rupture 19 dp df ( f ) ( df dt) Integrating Eq. (21) gives E ( f ) a A kbt = e P. (21) ( f ) P ( f ) f A = exp ( ) e df dt f0 ( f ') Ea kbt df ' (22) The probability that the bond is already broken at the force f becomes P r f A = = 1 exp ( ) e df dt f0 ( f ) 1 P( f ) ( f ') Ea kbt df ' The rupture force distribution is given by the first derivative dp r ( f ) df (23) of Eq. (23), and the most probable bond rupture f mp, i.e., the maximum of the rupture force distribution, is obtained by setting the second derivative of Eq. (23) to zero. Setting the third derivative of Eq. (23) equal to zero leads to the width of the force distribution f. For a Morse potential, the effective potential in the presence of an external force is V eff βx 2 ( x) = V ( x) f x = D ( 1 e ) f x e as given in Eq. (16). 7 The effective height of the energy barrier E a ( f ) can then be calculated from the energetic difference between the forcedependent local minimum at E a x, eff ( f ) = V ( x ) V ( x ) eff = D e b, eff eff 0 and the maximum at x b, eff (cf. Figure 3-4) as follows: 2 2 ( 1 exp[ βx ]) D ( 1 exp[ βx ]) f ( x x ) b, eff 0, eff The exponential arguments βx b, eff and x 0, eff ( ) and βx ln 2 ln f ( β ) 0, eff D e e 0, eff b, eff 0, eff (24) ( ) β are given by βx = ln 2 ln f ( βd ) b, eff =, respectively. As depicted in Eq. (19) the maximum force at which the barrier vanishes is given by f = βd 2. Accordingly, the exponential arguments may be recast to max e e 7 An alternative notation is used in [69] and reads V eff ( x) = V { exp[ 2 ( x x )] 2 [ β ( x x )]} f ( x x ) β with V 0 being the potential depth. exp

26 Theory of Force-Induced Bond Rupture 20 β x b, eff f = ln 2 ln 1 1 f max (25) β x0, eff f = ln 2 ln 1+ 1 f max (26) With ~ f = f f the activation barrier E a ( f ) can then be written in terms of the maximum max force as follows E a ( f ) ~ ~ ~ = D e 1 f f coth 1 f. (27) This expression can be approximated by (cf. also Figure 3-5) E a ~, (28) ( f ) D ( 1 f ) 2 e if the exact shape of the binding potential is neglected and mainly its depth and width are taken into account. Figure 3-5. Ratio of the activation barrier E a (f) and the dissociation energy D e (vertical axis) as a function of the ratio of the force f and the maximum force f max (horizontal axis). The dotted line represents the analytical result given in Eq. (27) and the solid line shows the parabolic approximation formulated in Eq. (28).

27 Theory of Force-Induced Bond Rupture 21 The parabolic form of E a ( f ) allows the explicit integration of the force distribution given in e x : 2 2 π Eq. (23) by using the error-function = erf ( x) P r ~ ( f ) A 1 π k BT D e D e = 1 exp f max erf erf f ( df dt) 2 D e k BT k BT ~ ( 1 ) (29) According to this model, the most probable bond rupture force f mp at a certain force-loading rate df dt is then given by setting the second derivative of Eq. (29) to zero: f mp 1 2 βd e β A k = 4 B T exp D 1 2 ( df dt) k BT β De e f mp 2, (30) This model again yields an analytical equation for the most probable rupture force f mp. Unlike the model by Evans and Ritchie, which was presented in the previous chapter, this model has three free parameters, which have to be determined by fitting it to experimental data. However, one has to keep in mind that the approximation leading to Eq. (30) neglects the exact shape of the binding potential. Furthermore, the model does not consider the fact that for a Morse potential the Arrhenius A-factor is also a force-dependent parameter Analytical Model Connected with High-Level Density Functional Theory Calculations To model the mechanical strength of covalent bonds Beyer has introduced an Arrhenius kinetics model, which is directly linked to density functional theory (DFT) calculations [33, 53]. Therefore, the mechanical stretching of small model molecules is simulated by relaxed potential energy surface scans [53], where the potential is determined as a function of the distance between the two atoms. Starting from the equilibrium distance, the distance between the two atoms is elongated in steps of 2 Å along the reaction coordinate and kept constant. With this fixed distance, the model molecules geometry is optimized, so that the molecule is relaxed to the most favourable energetic state and a distance-dependent numerical potential is

28 Theory of Force-Induced Bond Rupture 22 obtained. For an analytical representation of the calculated potential, a Morse potential is used as depicted in Eq. (19). To assure, that the analytic potential exhibits the same f max and D e as the one obtained from the DFT calculations, the unperturbed potential V ( x) is connected with ' the DFT calculations by the conditions V max ( x) = f max and V x ( x) V ( ) = De Figure 3-4). 0 (see also The effective potential due to an external force is given in Eq. (20). The difference between the effective minimum and the local maximum of the binding potential may be directly taken to derive the force-dependent activation energy E a ( f ). However, the lowest possible energy level slightly differs from that of the zero point energy of the effective Morse potential and is considered in 1 ( f ) V ( x ) V ( x ) h A( x ) E a eff b, eff eff 0, eff 0, eff = (31) 2 where h is the Planck constant and ( x ) A, eff 0 is the force-dependent pre-exponential factor. As a suitable pre-exponential factor Beyer proposed the maximum frequency of an optical phonon in the one-dimensional polymer in the gas-phase, i.e., influences from the solvent are neglected. This corresponds to the motion of two atoms in a diatomic molecule in opposite directions and to the assumed direction of the bond rupture event along the reaction '' coordinate, as mentioned above. By using the potentials second derivative ( x ) V 0, eff, which corresponds to the curvature of the potential at its minimum, a connection between the Morse potential and the pre-exponential factor A from the Arrhenius kinetics model is achieved: ( ) A x 0, eff ( x ) '' 1 2 V 0, eff =. (32) 2 π M ( A) In Eq. (32) the parameter M ( A) defines the greater of the two molar masses of the diatomic molecule, which is subject of investigation (e.g., for the Si C bond, M ( A) is the molar mass of Si).

29 Theory of Force-Induced Bond Rupture 23 With the two parameters ( f ) E a and ( x ) A, eff 0 given in Eqs. (31) and (32) a numerical integration of Eq. (23) is carried out and the probability P r ( f ) that the bond is broken at the force f is calculated. The results can be converted into a bond rupture probability within a given force interval at a given force-loading rate. This probability can in turn be used to generate simulations of experimental results with a random number generator [53]. With the aim to consider solvent effects and to be able to fit experimental data, the pre-exponential factor may be systematically scaled as the bond dissociation energy and the maximum force may be systematically varied, yielding parameter sets for the values D e, f max and A, which are able to describe the experimental data sufficiently. In contrast to the analytical model outlined in chapter , this model does not yield a straightforward fitting equation, which can be directly applied to the experimental data. On the other hand, the model is connected with DFT calculations and because Eq. (23) is solved numerically, the approximation leading to Eq. (28) is not needed. Furthermore, the force-dependence of the pre-exponential Arrhenius A-factor is accurately incorporated in this model, even if the frequency range of the A-factor is scaled in order to account for solvent effects. An alternative analytical model was recently introduced by Dudko et al., who represented the potential under tension by a cubic polynomial, evoking an x 3 -binding potential [64]. This approach is described in detail in Appendix D.

30 Results and Discussion 24 4 Results and Discussion 4.1 Experimental Data A typical force-extension curve of a single CM-amylose polymer covalently bound between substrate and AFM tip is shown in Figure 4-1A. Figure 4-1. Force-extension curve of a covalently anchored CM-amylose polymer (A) and a control experiment (B) recorded with an AFM. In (A) the CM-amylose polymer uncoils up to an extension of approximately 800 nm followed by a pronounced plateau starting at around 0.3 nn. The plateau may be attributed to the chair-boat transition of the sugar rings leading to an increase of CM-amylose monomer length by 0.5 nm. With increasing distance between substrate and AFM tip, the molecule is stretched further, thus leading to a strong force increase until the connection is lost. At a force of 1.69 nn the Si C bond in the surface linker of the molecule ruptures and the force drops to zero. (B): A control experiment, where no activation of CM-amylose with EDC and NHS was carried out, is shown. It is clearly visible that no molecule was picked up by the AFM tip. The force curve exhibits a characteristic plateau starting at around 300 pn, followed by a rapid increase of the force, until the connection is lost at 1.69 nn and the AFM cantilever snaps back to its equilibrium position at zero force. The force curve was recorded at a z-piezo velocity of 10 µm/s. If the indentation force between AFM tip and substrate surface was kept below 0.3 nn, approximately 10-20% of all tip-substrate contacts lead to a recordable forcedistance curve, like the one depicted in Figure 4-1A.

31 Results and Discussion 25 In control experiments, where the CM-amylose polymer was not activated with NHS and EDC (Figure 4-1B), no molecules were picked up by the AFM tip under similar conditions. In Figure 4-1B it is clearly visible that the measured force remains zero over the whole range and no molecule was bound to the AFM tip. Because force-induced bond rupture is a thermally activated statistical process, for a fixed force-loading rate, the rupture forces are distributed over a wide range. By varying the forceloading rate over some orders of magnitude, many different distributions are obtained, so that an adequate characterization of a certain chemical bond is neither attainable by a single rupture event nor by the average rupture force at a given force-loading rate. Therefore, 165 rupture events are displayed in a scattered data plot as shown in Figure 4-2. In this plot, the bond rupture force is plotted versus the force-loading rate in a semi-logarithmic representation. As pointed out above, for all displayed data points, the contact force between AFM tip and substrate surface was below 0.3 nn and the characteristic plateau of CMamylose was starting at around 0.3 nn. In Figure 4-2, the force-loading rates span a range from 0.5 to 267 nn/s and the forces range from 0.7 to 2.36 nn, with average rupture forces of 1.1 nn at 0.5 nn/s and 1.8 nn at 267 nn/s. Figure 4-2. Semi-logarithmic scatter plot of 165 bond rupture events. The rupture force (vertical axis) is plotted versus the force-loading rate (horizontal axis).

32 Results and Discussion 26 An alternative and widely used way of data representation is shown in Figure 4-3, where the rupture events are displayed as rupture force distributions for three different loading rate intervals. The three loading rate intervals range from 0.5 to 5 nn/s (Figure 4-3A), from 5 to 50 nn/s (Figure 4-3B), and from 50 to 267 nn/s (Figure 4-3C), respectively. The upper limit of the latter is given by the highest measured loading rate value. Figure 4-3. Rupture force distributions for the force-loading rate intervals from 0.5 to 5 nn/s (A), from 5 to 50 nn/s (B), and from 50 to 267 nn/s (highest value) (C). The total number of rupture events in the three histograms is 165. In the range from 0.5 to 5 nn/s (Figure 4-3A), the mean rupture force f r is 1.14 ± 0.25 nn. The maximum of the rupture force distribution, which corresponds to the most probable bond rupture force f mp, is between 1.1 nn and 1.2 nn. In the loading rate interval between 5 and 50 nn/s (Figure 4-3B) the mean rupture force is 1.49 ± 0.23 nn/s and the most probable bond rupture force lies between 1.5 nn and 1.6 nn. Finally, in the range between 50 and 267 nn/s, the mean rupture force is 1.67 ± 0.27 nn/s and the most probable rupture force lies between 1.7 and 1.8 nn (Figure 4-3C). Comparing the two different ways of data representation in Figure 4-2 and Figure 4-3, the main disadvantage of the histograms arises from the arbitrarily chosen data intervals and the arbitrary binning, because these parameters may have a tremendous effect on the shape of the rupture force distributions and the position of its maximum. Choosing a different range for the loading rate intervals and/or another bin width may result in a different rupture force

33 Results and Discussion 27 distribution and yield a different value for the most probable rupture force. A scattered data plot as shown in Figure 4-2, on the other hand, is not affected by any arbitrary data binning. Therefore, scattered data plots will be used consecutively.

34 Results and Discussion Modelling of the Experimental Data In order to provide insight into structural information as well as the dissociation pathway, all three theoretical models described in the theoretical section (cf. chapter 3) have been used to analyse the experimental data. In the following section the results obtained with these models are presented, compared and analyzed with regard to their significance Modelling Based on a Potential with Fixed Barrier Distance Figure 4-4 represents a scattered data plot of the experimental data along with a fit (dotted line) of Eq. (18), which is based on the widely used Arrhenius kinetics model introduced by Evans and Ritchie [37]. The authors predict a logarithmic increase of the most probable bond rupture force f mp with the applied force-loading rate. 8 As described in the theoretical section, this model assumes that the distance between the binding potential minimum and the transition state is a force-independent parameter. 8 It should be mentioned in this context, that the most probable bond rupture force f mp is slightly below the mean bond rupture force f r due to the asymmetry of the bond rupture probability distributions [53]. This effect should be of minor influence, however. On the other hand, by fitting this analytical model to the experimental scatter plot instead of fitting it to the f mp values, which are extracted from the rupture probability distributions depicted in Figure 4-3, errors caused by arbitrary binning of data in the experimental rupture force distributions can be avoided.

35 Results and Discussion 29 Figure 4-4. Scattered data plot of bond rupture forces versus force-loading rates together with a fit of Eq. (18) to the experimental data. Here, the free parameters are x, which represents the distance between the binding potential minimum and the transition state, and the reaction rate constant k off 0 at zero force. The fit yields values of 0.35 Å for x and /s for off k 0, respectively. The logarithmic correlation between rupture forces and force-loading rates, which was predicted by Evans and Ritchie, seems to be confirmed in Figure 4-4. However, comparing the fit parameters x and off k 0 to typical bond lengths and lifetimes of covalent bonds suggests that especially the bond lifetime is significantly underestimated by this model: the reaction rate constant off k 0 = /s corresponds to a lifetime at zero force of off τ 0 = 1 k s or 45 minutes, which is considerably below the lifetime of a covalent bond [53]. 9 As mentioned in the theoretical section (3.2.1), this model assumes that x is not forcedependent as it is the case e.g. for sharp energy barriers and/or weak applied forces. For covalent bonds, however, which are typically represented by a Morse potential, this is no longer valid. Here, the potential minimum is shifted to larger x-values while the position of 9 As a rough approximation of the lifetime of a covalent bond one may take the parameter values obtained from DFT calculations in the gas-phase and use the Arrhenius equation (cf. Eq. (6) in the theoretical section). For A opt = /s, E a = D e = kj/mol [53], and k B T = kj at room temperature, the reaction rate constant is off k /s, so that the bond lifetime is τ s.

36 Results and Discussion 30 the activation barrier moves, with increasing force, from infinity to smaller values along the reaction coordinate. Therefore, a Morse potential based Arrhenius kinetics model is used for data analysis in the next section Modelling Based on the Combination of the Arrhenius Rate Equation and a Morse Potential Here, an analytical model based on a Morse potential under tension, which was recently introduced by Hanke and Kreuzer [59] and which is described in detail in section , was used to fit the experimental data (cf. also Eq. 30). 10 It turned out that fitting this analytical model to the experimental data was not possible with the three free parameters A, D e, and f max. Therefore, the parameter f max was set to a reasonable predetermined value and A, and D e were treated as free parameters in the fitting procedure. Figure 4-4 represents the experimental data together with a fit function obtained with the fit parameters A = /s, D e = 73.1 kj/mol, and f max = 4.8 nn, which is one of several possible parameter sets that can be used to fit the experimental data. The agreement with the experimental data appears to be rather good. 10 ). Again, fitting was carried out considering the mean bond rupture force f r instead of the most probable bond rupture force f mp (cf. also section 4.2.1).

37 Results and Discussion 31 Figure 4-5. Scattered data plot of bond rupture forces versus force-loading rates together with a fit of Eq. (32) to the experimental data. The dotted line shows a fit to the experimental data generated with the fit parameters A = /s, D e = 73.1 kj/mol, and f max = 4.8 nn using the model by Hanke and Kreuzer [59]. This parameter set is one of several possible parameter sets that can be used to fit the experimental data. Table 4-1 summarizes all parameter sets, which were obtained by fitting this model to the experimental data. Table 4-1. Free parameters based on the analytical model introduced by Hanke and Kreuzer [59]. Fitting this model to the experimental data was carried out by varying the free parameters A and D e, whereas f max was set to a reasonable predetermined value. Setting one of the values to a predetermined value was required, because fitting with three free parameters proved to be inapplicable. scale factor A opt = /s D e [kj/mol] f max [nn] 1/β [Å] As can be seen from Table 4-1, the possible sets of free parameters span a wide range. Only the parameter set to f max = 1.7 nn given in the last line of Table 4-1 can be ruled out, because the fit is not able to reproduce the experimental scatter plot very well (Figure 4-6).

38 Results and Discussion 32 Figure 4-6. Fit to the experimental data based on the model by Hanke and Kreuzer[59] with the predetermined force f max = 1.7 nn. This parameter set can be ruled out since the experimental data is visibly not described appropriately Simulated Rupture Forces Based on an Arrhenius Kinetics Model with a Morse Potential 11 The simulated rupture force data in this chapter was generated using a Morse potential Arrhenius kinetics model as described in chapter Figure 4-7 shows the experimental data (black dots) along with two sets of simulated data (open squares and triangles), with f max = 4.8 nn, D e = 337 kj/mol and a load-dependent pre-exponential factor covering the range A = /s to /s. The parameters f max and D e as well as the pre-exponential factor A, which was chosen as the maximum optical phonon frequency of the polymer, were derived from previously published DFT calculations in the gas-phase [53], as outlined in chapter The rupture force probability distributions were obtained by integrating Eq. (23). Based on these probability distributions, scattered data plots were generated using a random number generator. 11 The simulations presented in this chapter were carried out by Prof. Dr. M. K. Beyer, Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, Olshausenstraße 40, Kiel, Germany.

39 Results and Discussion 33 Figure 4-7. Comparison of the experimental rupture force distribution (black dots and dotted line) and simulated data sets taken from DFT calculations. The open squares and triangles show two simulated data sets, which were generated with a Morse potential based Arrhenius kinetics model. The parameters A = /s, D e = 337 kj/mol and f max = 4.8 nn were extracted from gas-phase DFT calculations of a stretched model molecule as previously published [53]. The straight line represents a logarithmic fit to the simulated data sets. Obviously, the simulated data sets do not describe the measured rupture force distribution satisfactory since the simulated data lie well above the experimentally observed rupture forces. In addition, the loading rate dependence of the rupture force seems to be less pronounced in the simulation, as can be seen from the slope of the two logarithmic fits (dotted line and black line in Figure 4-7). The pronounced difference in terms of rupture forces has been attributed to the fact that the modelled rupture forces are not subjected to solvent effects, because the simulated data sets were extracted from DFT calculations in the gas-phase. Hence solvent effects associated with the solvation energy of fragment radicals, which lower the bond dissociation energy and weaken the maximum force, may be considered as an explanation [33, 52, 58, 59, 70]. Another influence may emanate from protons or other ions from the solvent, which may attack the strained bond [71-73]. In this context even a mechanically activated chemical reaction [5, 22, 74] of the mechanoradicals with the linker chemicals EDC and NHS, which might still be present in the reaction medium in small quantities, seems imaginable. Granted that one of those molecules may attack the mechanically activated bond the chemical reaction

40 Results and Discussion 34 could precede the homolytic bond scission. In this case, the bond dissociation energy derived from the fit corresponds to the activation energy of a mechanically activated polymerization; the attack of the diffusing reactant molecule would widen the potential thus lowering the maximum force. If the diffusion of a second reactant into the correct position is involved, a significant decrease of the pre-exponential factor relative to the gas-phase values may be caused. Furthermore, by choosing the pre-exponential factor A as the maximum optical phonon frequency the experimental conditions are not described accurately. In contrast to the theoretical approach in vacuo, where the value of the pre-exponential factor reaches an upper limit, the experiments were performed in liquid environment. As described in chapter 3.1, the pre-exponential factor is given by A ( q / q) = κν. For reactions in liquid environment, the accommodation coefficient κ is usually smaller than one and the attempt frequency ν may be drastically reduced by solvation and viscous damping [37, 59]. The ratio of the partition functions q / q, which is proportional to exp ( S k T ), is usually smaller than one, due to conformational constraints of the activated complex [60]. Altogether this gives rise to the assumption that the pre-exponential factor is much smaller for reactions in liquid environment, than the optical phonon frequency in vacuum. In order to reproduce experimentally observed rupture forces including the load dependence and the statistical scattering with the simulated data sets, the two parameters f max and D e from the Morse potential were systematically varied and the pre-exponential factor A was scaled until a close resemblance between simulated and experimental data was obtained. Table 4-2 summarizes the fit parameters which were obtained in this way. B

41 Results and Discussion 35 Table 4-2. Free parameters of simulated scattered data plots. The maximum optical phonon frequency in vacuo A opt was systematically scaled, because the pre-exponential factor A of the Arrhenius kinetics model is assumed to be much lower in liquid environment. D e is the bond dissociation energy and f max is the maximum rupture force of the Morse potential. 1/β is proportional to the width of the Morse potential (cf. also Eq. (19)). scale factor A opt /s D e [kj/mol] f max [nn] 1/β [Å] If one compares the free parameters in Table 4-1 and Table 4-2 a close resemblance of the results can be noticed. Again, the possible parameter sets cover a considerable range and an unambiguous determination of all three free parameters is not possible. Nevertheless, the lines printed in bold represent an upper and a lower limit for f max. The maximum force f max = 4.8 nn corresponds to the result from the DFT calculations in the gas-phase. Higher values of f max could only be attributed to the rupture of a different bond in the polymer, which seems unlikely, because all other bonds should be significantly stronger than the Si C bond [53]. Moreover, micro-ruptures in the high-force regime of the force extension curve (cf. Figure 4-8) point to ruptures of Si C bonds in the surface anchor of the molecule, revealing a stepwise break-up of the polymer from the substrate.

42 Results and Discussion 36 Figure 4-8. Micro-rupture in the high-force regime of a force-extension curve. Prior to the final detachment of the polymer the bonds in the surface anchor of the molecule break up stepwise, while the polymer backbone remains unaffected (B shows a schematic representation). The enlarged curve section in figure A shows a microrupture indicating that the force drops instantly after secession, leading to a length increase of the polymer about 2.8 nm [33]. Considering that the backbone of the CM-amylose may form up multiple connections with the surface via the COO - -moieties of the sugar monomers, the polymer backbone remains intact, while the bonds between the surface break one by one, until the AFM tip and the substrate surface are completely detached from each other [33]. The lower limit of f max is approximately 2 nn, because f max can not lie below the highest experimentally observed rupture forces. 12 Furthermore, the parameter sets in Table 4-2 exhibit a maximum of the relative potential width In the gas-phase the value for 1 β = 0.31 Å at a maximum force of f max = 2 nn. 1 β is 0.58 Å. This might be an indication that the upper limit of f max is already reached at 2 nn, presuming that the potentials width is less influenced by the solvent than the other parameters. However, to clarify this speculation the experimentally covered force-loading range must be expanded to higher rates. Altogether it should be noted, that for the relatively wide range of the maximum forces f max = 2.0 nn to 4.8 nn, the 12 The two data points around 2.3 nn are considered as statistical scattering.

43 Results and Discussion 37 corresponding bond energies amount to D e = kj/mol, spanning a relative narrow range, while the pre-exponential factor A varies over five orders of magnitude. Figure 4-9 shows the experimental data (black dots) together with two simulated data sets (open squares and triangles) revealing an example of one possible parameter set. Here, the optical phonon frequency was scaled by corresponding to A /s. The dissociation energy and the maximum force are D e = 86.5 kj/mol and f max = 4.8 nn. Figure 4-9. Experimental data (black dots) along with two simulated data sets (open squares and triangles). In order to obtain simulated data, which is closely resembles the experimental results, the parameters D e, f max and A from the Morse potential were varied systematically. Here, the parameter set with A /s, D e = 86.5 kj/mol, and f max = 4.8 nn from Table 4-2 was used. This is one of several possible parameter sets that show good agreement with the experimental data. The solid line represents a logarithmic fit to the experimental data. Figure 4-10 shows the experimental data along with simulated values over an experimentally inaccessible range of force-loading rates of 21 decades. The solid line in Figure 4-10 represents the analytical model by Hanke and Kreuzer [59] (cf. also Eq. 30).

44 Results and Discussion 38 Figure Experimental data (black dots) together with simulated data over an experimentally inaccessible range of force-loading rates of 21 decades (red dots). The solid line represents the analytical model by Hanke and Kreuzer [59]. As predicted by Hanke and Kreuzer, the rupture force distribution reaches an upper limit as soon as the rupture force reaches f max, in this particular case at 4.8 nn. Concerning the lower loading rate regime the model by Hanke and Kreuzer [59] predicts zero bond rupture force at a force-loading rate around 10-6 nn/s, whereas the simulated data approaches zero at forceloading rates below 10-8 nn/s. This may be attributed to the fact that in the model by Hanke and Kreuzer the pre-exponential factor A is treated as a force-independent parameter compared to the simulation based on the approach by Beyer (cf and in the theoretical section).

45 Conclusion 39 5 Conclusion The dynamic strength of the Silicon Carbon bond was investigated using AFM-based single molecule force spectroscopy and varying the force-loading rate over 3 orders of magnitude. As predicted by Arrhenius kinetics models, a logarithmic increase in bond rupture force as function of the applied force-loading rate was observed over the experimentally accessible force-loading range. This clearly demonstrates that the force induced dissociation of covalent bonds follows Arrhenius kinetics and shows that the bond strength is directly related to the bond lifetime. However, the data also show that if an Arrhenius model with structural parameters extracted from gas-phase quantum chemical calculations is used, the model overestimates the bond rupture forces and underestimates the force-loading rate dependence. Therefore the parameters for the Arrhenius kinetics model have to be treated as free parameters and their values have to be extracted from the experimental data. Nevertheless, because with all three Arrhenius kinetics models used, it is impossible to unambiguously extract all three free parameters from the presented experimental data, more experimental data spanning an even wider force-loading range and modifying additional experimental parameters, like temperature, are required. Hanke and Kreuzer [59] pointed out, that if it was possible to determine f max independently from experiments with high force-loading rates, it was much easier to derive the remaining two parameters from experimental data. Especially fast experiments with stiffer cantilever springs might allow to actually reach f max experimentally, but presumably the effect of hydrodynamic drag, which acts on the cantilever, may be of increasing influence at very high piezo velocities [75]. Following the strategy of probing the bond lifetime at constant force [62, 63], rather than measuring bond rupture forces as a function of force-loading rate, might be an alternative to further spanning the force-loading range. Nevertheless, as pointed out above, force induced bond rupture is a thermally activated

46 Conclusion 40 process, and its kinetics is controlled to a large extent by the ratio of activation energy E a to thermal energy k B T, which enters the exponent of the Arrhenius equation. For this reason, gathering temperature dependent data should allow for the determination of the free parameters of the Arrhenius model more accurately. More specifically, because the exponent of the Arrhenius equation is temperature dependent and the entropic term which enters the Arrhenius pre-exponential factor does not depend on temperature, temperature dependent measurements might be a versatile strategy to separate the activation energy E a and the activation entropy S, and determine D e and A independently. The finding that the bond lifetime of covalent bonds is drastically reduced by external forces indicates that bond ruptures in polymeric materials already appear at tensile forces well below the maximum binding force f max. As a consequence, the assumption that repeated loading is the main reason for material fatigue and failure has to be reconsidered to some extend, because the timescale of the applied force also seems to play a key role in bond rupture processes. The effect of dynamic loading on the durability of materials is therefore of outstanding importance for the determination of material properties and has to be further investigated in an effort to obtain a deeper insight into the fundamental mechanisms of material fatigue and failure on the molecular level and further contribute to the improvement of polymeric materials in the long run.

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53 Figures 47 Figures Figure 2-1. Schematic set of an AFM-based single-molecule force spectroscopy experiment.5 Figure 2-2. Correlation of the measured z-piezo distance and the actual distance between AFM tip and surface in an AFM-based force spectroscopy experiment... 8 Figure 2-3. Typical force-extension curve of a single CM-amylose polymer covalently bound between substrate and AFM tip along with the detected rupture force... 9 Figure 3-1. Schematic representation of a single-well potential curve in the equilibrium state, where the internuclear separation between the two atoms of a diatomic molecule corresponds to the potential minimum Figure 3-2. Schematic representation of an effective potential curve when an external force is applied and a fixed barrier distance is assumed Figure 3-3. Following the theoretical model by Evans and Ritchie [37], the most probable rupture force fmp increases with logarithmically increasing force-loading rate df/dt Figure 3-4. Undisturbed Morse potential with the dissociation energy D e together with the effective potential when an additional force is applied to a diatomic molecule.. 18 Figure 3-5. Ratio of the activation barrier and the dissociation energy as a function of the ratio of the force f and the maximum force Figure 4-1. Force-extension curve of a covalently anchored CM-amylose polymer and a control experiment recorded with an AFM Figure 4-2. Semi-logarithmic scatter plot of 165 bond rupture events Figure 4-3. Rupture force distributions for the force-loading rate intervals from 0.5 to 5 nn/s, from 5 to 50 nn/s, and from 50 to 267 nn/s Figure 4-4. Scattered data plot of bond rupture forces versus force-loading rates together with a fit of Eq. (18) to the experimental data Figure 4-5. Scattered data plot of bond rupture forces versus force-loading rates together with a fit of Eq. (32) to the experimental data Figure 4-6. Analytical fit to the experimental data based on the model by Hanke and Kreuzer[59] with the predetermined force f max = 1.7 nn Figure 4-7. Comparison of the experimental rupture force distribution and simulated data sets taken from DFT calculations Figure 4-8. Micro-rupture in the high-force regime of a force-extension curve Figure 4-9. Experimental data along with two simulated data sets... 37

54 Figures 48 Figure Experimental data together with simulated data over an experimentally inaccessible range of force-loading rates of 21 decades... 38

55 Tables 49 Tables Table 4-1. Free parameters based on the analytical model by Hanke and Kreuzer Table 4-2. Free parameters of simulated scattered data plots... 35

56 Abbreviations 50 Abbreviations AFM CM-amylose ddh 2 O DETA DFT EDC EtOH NHS PBS Atomic force microscopy / atomic force microscope Carboxymethylated amylose Double distilled water N 1 -[3-(Trimethoxysilyl)-propyl]diethylenetriamine Density functional theory Ethyl-dimethylaminopropyl-carbodiimide Ethanol N-hydroxysuccinimide Phosphate buffered saline

57 Appendix 51 Appendix

58 Appendix A. Sample Preparation Protocol 52 Appendix A. Sample Preparation Protocol Initially, cleaning of glass microscope slides (Roth, Karlsruhe, Germany) was carried out through sonication in a 1:1-solution of double distilled water (ddh 2 O) and ethanol (EtOH, p.a., Merck, Darmstadt, Germany) for 15 minutes. To avoid damage or destruction, silicon nitride AFM cantilevers (MLCT-AU, Veeco Instruments GmbH, Mannheim, Germany) were submerged and rinsed carefully in a 1:1-solution of ddh 2 O and EtOH without sonication. Subsequently, glass slides and AFM cantilevers were incubated in a 2 % N 1 -[3-(Trimethoxysilyl)-propyl]diethylenetriamine (DETA, Sigma-Aldrich, Deisenhofen, Germany) solution in EtOH at 90 C for 15 minutes, rinsed in a 1:1 solution of ddh 2 O and EtOH, and cured at 120 C for 30 minutes. DETA is a widely used agent to promote adhesion of polymer films on glass surfaces since all three methyl groups ( CH 3 ) may be replaced by siloxane bonds at the glass surface and a coverage of amine functional groups ( NH 2 ) is obtained as shown in Scheme A-1 [43, 44].

59 Appendix A. Sample Preparation Protocol 53 Scheme A-1. DETA and its hydrolysis followed by condensation at a hydrated glass surface and the resulting formation of siloxane crosslinks (adapted from [43, 44]). Pre-treated glass slides and cantilevers with amine functional groups were transferred to an exsiccator and preserved for five days maximum. In order to covalently couple CM-amylose to the amine functional groups of the pre-treated glass slides, 10 mg/ml CM-amylose (Sigma-Aldrich, Deisenhofen, Germany), 20 mg/ml 1-(3- Dimethylaminopropyl)-3-ethylcarbodiimide (EDC, Sigma-Aldrich, Deisenhofen, Germany), and 10 mg/ml N-Hydroxy-Succinimide (NHS, Sigma-Aldrich, Deisenhofen, Germany) were dissolved in 1 ml phosphate buffered saline (PBS, ph 7.4, Sigma-Aldrich, Deisenhofen, Germany). The entire solution was then transferred to an amino-silanized glass slide and incubated on it for 10 minutes. Amylose is a linear polymer of n glucose residues, which are linked by α-d-1,4-glucosidic bonds. CM-amylose is an amylose which is carboxymethylated in order to increase the hydrophilicity of the polysaccharide and achieve a better solubility in aqueous solutions. In

60 Appendix A. Sample Preparation Protocol 54 CM-amylose some of the hydroxyl groups ( OH) of the glucose monomers are substituted by carboxyl groups ( COOH) as illustrated in Scheme A-2. Scheme A-2. Basic repeat unit of the molecular structure of CM-amylose. CM-amylose is a linear poly-dglucose in which the monosaccharides are connected by α-d-1,4 linkages and some hydroxyl groups of the glucose monomers are replaced by (ionized) carboxymethyl groups (adapted from [55]). EDC (cf. Scheme A-3) was used to promote the formation of amide bonds between carboxyl groups and amino groups and couple DETA to CM-amylose (shown in Scheme A-4). 13 Scheme A-3. Molecular structure of the coupling agent EDC. 13 EDC catalyzes the formation of amide bonds between carboxyl-containing compounds and amines by activating the carboxyl groups to form highly reactive O-acylurea (cf. Scheme A-4 I). This intermediate, however, is subjected to a considerable degree of hydrolysis when the reaction is carried out in aqueous solution, and hence has an extremely short half-life. In order to obtain a longer time-frame for the formation of amide bonds and thus increase the overall reaction yield, NHS is added to the solution. The reactive O-acylurea species is stabilized by NHS through the formation of O-acylurea ester, which hydrolyzes slowly in aqueous media. As a result, an increased stability to aqueous hydrolysis is achieved (cf. Scheme A-4 II) [45, 46, 47, 76, 77].

61 Appendix A. Sample Preparation Protocol 55 Scheme A-4. (I) EDC-mediated amidation reaction without the assistance of NHS. The reaction of EDC with CM-amylose results in reactive O-acylurea, an intermediate with an extremely short half-life in aqueous solution, because it is subjected to an extensive degree of hydrolysis. As a result, a reduced rate of yield is obtained. (II) By adding NHS to the solution, the intermediate is stabilized by the formation of O-acylurea ester. O-acylurea ester hydrolyzes slowly in aqueous media, and an increased stability along with an increased overall reaction yield is obtained (adapted from [46]). Immediately after incubation of activated CM-amylose on the silanized glass slide, the slide was thoroughly rinsed with ddh 2 O using a 20ml-syringe and a 0.7mm-injection needle, to remove excess CM-amylose which was non-covalently bound to the amino-silanized glass slide. Finally, the glass slide was transferred to the AFM and covered with several 100 µl PBS. The AFM cantilever was then brought into contact with the CM-amylose covered glass slide, so that amylose polymers could attach to the amino-silanized AFM tip. The resulting

62 Appendix A. Sample Preparation Protocol 56 coupling chemistry is schematically shown in Scheme A-5, where the probed Si C bond can be found in the surface anchor. Scheme A-5. Schematic representation of the resulting coupling chemistry between glass surface and AFM tip.

63 Appendix B. Cantilever Calibration 57 Appendix B. Cantilever Calibration AFM cantilevers were individually calibrated using the thermal noise method [48, 78]. In the first step, the optical lever sensitivity was determined and the deflection signal was converted from Volts to Nanometers. This was achieved by recording a force-distance curve on the glass substrate and extracting the slope in the indentation range as shown in Figure B-1. Figure B-1. Determination of the optical lever sensitivity from the slope of the indentation range of a forcedistance curve. To determine the optical lever sensitivity and to convert the deflection signal from Volts into force units, the slope in the indentation range of the force-distance curve was extracted. After that, the cantilever spring constant was calibrated on the basis of its thermal motion: The cantilever was displaced minimum 500 nm away from the surface and a power spectrum was recorded. A Lorentz curve was fitted to the maximum representing the first resonance frequency of the cantilever (shown in Figure B-2). The fit then yields the cantilever spring constant k c.

64 Appendix B. Cantilever Calibration 58 Figure B-2. Calibration of the cantilever spring constant from its power spectrum. In order to calibrate the cantilever spring constant the power spectrum is displayed (grey curve) and a Lorentz fit (black dotted curve) is fitted to the first harmonic resonance frequency. Three independent measurements of the optical lever sensitivity as well as the cantilever force constant were performed for every cantilever spring. The three determined values were averaged in each case and used for data analysis.

65 Appendix C. Elasticity of the CM-amylose polymer 59 Appendix C. Elasticity of the CM-amylose polymer As described in chapter 2.2 the elastic properties of the CM-amylose polymer can be used to ensure that force-extension curves of single CM-amylose strands are recorded. Figure C-1 shows a schematic representation of a force-induced chair-to-boat transition of an individual glucose ring in the CM-amylose polymer. Due to the transition the distance between the two glycosidic oxygens in every monomer is enlarged by 0.5 nm, leading to an observable plateau as described in chapter 2.2 and shown in Figure C-2. Figure C-1. Schematic representation of a chair-to-boat transition of a single glucose ring in the CM-amylose polymer due to the application of an external force F. The transition leads to an increased separation of the glycosidic oxygen atoms resulting in an enlargement of the CM-amylose monomer length by 0.5 nm (adapted from [50] and [79]). Figure C-2A shows three force-extension curves with different contour lengths and Figure C- 2B shows the same data scaled with regard to the length and superimposed. As can be seen in Figure C-2B, the CM-amylose elasticity scales linearly with their lengths, so that a characteristic plateau is observable.

66 Appendix C. Elasticity of the CM-amylose polymer 60 Figure C-2. (A): three force-extension curves with different contour lengths. (B): because the CM-amylose polymers scale linear with their length, scaling with respect to the tether length and superimposing the forceextension curves reveals the a characteristic plateau at around 300 pn (cf. also chapter 2.2).

67 Appendix D. Analytical Model Based on an x3-binding Potential 61 Appendix D. Analytical Model Based on an x 3 -Binding Potential Theory Recently, an analytical model, which represents the potential of a diatomic molecule under tension by a cubic polynomial, evoking an x 3 -binding potential, has been introduced by Dudko et al.[64] These authors took advantage of the fact that the potential energy of a diatomic molecule can be expanded into a Taylor-series around the inflection point x * of the potential when an external force is applied: V eff 1 1! 1 2! * * * * 2 * * 3 * ( x) V ( x ) + ( x x ) V '( x ) + ( x x ) V ''( x ) + ( x x ) V '''( x ) fx '' * Taking into account that the curvature at the inflection point is zero ( ( ) 0 leads to (see also Figure D-1): V eff * * * * 3 * ( x) V x ) + ( x x ) V '( x ) + ( x x ) V '''( x ) fx 1 3! V eff x = (D-01) ), Eq. (D-01) 1 (. (D-02) 6. Figure D-1. Schematic representation of the x 3 -binding potential as introduced by Dudko et al. The height of the force-dependent activation barrier E a ( f ) is given by the distance between ( ) at x, in the effective potential ( x) b eff V eff. V eff x 0 at x, eff 0 and barrier V eff ( x b )

68 Appendix D. Analytical Model Based on an x3-binding Potential 62 ' The extrema of this truncated potential may be extracted by setting ( x) V eff to zero, resulting in a distance from the minimum to the barrier of activation barrier given by ( f ) V ( x ) V ( x ) E a eff b, eff eff 0, eff and a height of the x eff = xb, eff x0, eff = (cf. Figure D-1). By taking the formal limit f 0, the parameters x and E a, which are actually the parameters from the undisturbed potential, may be derived from the first and third derivative of the linear-cubic potential at the inflection point as follows: * ( x ) * ( x ) x = 2 ' 2 V (D-03) V ''' E a ( x ) * ( x ) 3 ' * 2 2 V = (D-04) 3 V ''' The asymptotic expression for the mean rupture force f r as a function of the force-loading rate df dt is then given by E 1 1 k ln off a 0 f r a E ( 2 3) x E x ( df dt) e a ( 2 3). (D-05) The three free parameters which can be used in order to fit experimental data are the height of the energy barrier E a, the distance between the potential minimum and the local maximum x, and the reaction rate constant k off 0. Since fitting this model to the experimental data turned out to be impossible with the three free parameters, the parameter E a was set to a reasonable predetermined value and the two remaining factors were treated as free parameters, as it was done with f max in the model by Hanke and Kreuzer (cf. chapter 4.2.2). Results In Figure D-2 the fit of one possible parameter set to the experimental data is shown (dotted line). In this case, the fit was generated with the fit parameters E a = kj/mol, x = 0.09 Å, and off k 0 = /s.

69 Appendix D. Analytical Model Based on an x3-binding Potential 63 Figure D-2. Scattered data plot of bond rupture forces versus force-loading rates together with a fit of Eq. (D- 05) to the experimental data. The dotted line reveals a fit to the experimental data following the analytical model by Dudko et al. [64]. The fit was generated with the fit parameters E a = kj/mol, x = 0.09 Å, and off k 0 = /s Table D-1 summarizes all fit parameter sets which have been obtained using this model along ( x ) with the maximum force f max, which is given by the pre-factor E ( 3) a 2 in Eq. (D-05). Table D-1. Free parameters based on the analytical model by Dudko et al. using an x 3 -binding potential. Fitting this model to the experimental data was carried out by varying the free parameters x and off k 0. The parameter E a was set to a reasonable predetermined value. This predetermination was required, because fitting with three free parameters proved to be inappropriate. The maximum force f max is given by the pre-factor in Eq. (D-05). E a [kj/mol] x [Å] k 0 off [1/s] f max [nn] As can be seen from Table D-1, this model does also not furnish unique fit-parameters, when fitted to the experimental data. Moreover, as pointed out by Dudko et al., the high force

70 Appendix D. Analytical Model Based on an x3-binding Potential 64 regime, where f r is expected to approach f max, the experimental data is not accurately represented by this model.

71 Appendix E. Software and Instruments 65 Appendix E. Software and Instruments Experimental data was acquired with an atomic force microscope (NanoWizard, JPK Instruments, Berlin, Germany) using the implemented software (SPM software Version 3.17, JPK Instruments, Berlin, Germany) in the force spectroscopy mode. Figure E-1. Lateral view (A) and bottom view (B) from the used AFM NanoWizard by JPK Instruments. In (B) the red arrow shows the cantilever chip, where the AFM tip is located. Data analysis, fitting and display were carried out by customized functions and procedures using Igor Pro Version (WaveMetrics Inc., Lake Oswego, Orgeon, USA). Figures were made using Igor Pro or Adobe Illustrator CS (Adobe Systems Inc., San Jose, California, USA). Schemes were made using ChemDraw Version 11.0 (CambridgeSoft Corporation, Cambridge, Massachusetts, USA).