kuhn length of polymers

b In your case, the polymer is about as long as the persistence length, which means it is one-half of the Kuhn length. For chains with positive excluded volume v > 0, Flory theory yields a universal scaling law for real chains such that R F ≈ bN 3/5 . On a large scale, the motion of the Kuhn "blobs" is what matters. How do you calculate the Kuhn length of a polymer? l In your case, the polymer is about as long as the persistence length, which means it is one-half of the Kuhn length. N. {\displaystyle N} Kuhn segments each with a Kuhn length. The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of Kuhn segments each with a Kuhn length .Each Kuhn segment can be thought of as if they are freely jointed with each other. double-stranded DNA with Kuhn length b=100nm and contour length L=16mm. properties of polymers8,19. Structure and Chemistry •Polymers are giant molecules usually with Kuhn segment length: If the polymer of contour length L= Nacan be conceived as a chain of N s e ectively freely jointed segments of length l K then we have L= N sl K and the mean-square end-to-end distance becomes R 2 0: = hr i = N sl2 K = Ll K: The Kuhn segment length is thus de ned as the ratio between the mean-square end-to-end distance and the contour length: l K: = [1][2][3][4] Each segment in a freely jointed chain can randomly orient in any direction without the influence of any forces, independent of the directions taken by other segments. Consider a semiflexible polymer – e.g. . from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments The length of a fully stretched chain is {\displaystyle l} R segments of a given length, b, as depicted in gure 2.1. The free energy of an ideal polymer chain is derived from its conformational entropy, described by the probability distribution above. Kuhn length, l K. The Kuhn length is a measure of chain rigidity. Mai 2018 um 17:55 Uhr bearbeitet. {\displaystyle b} Since persistence length is equal to one-half the Kuhn length [6,8,29 – 31], it can be calculated as . This allows the molecule to change its shape locally within a blob and, if applicable, to change its entire shape under load. = The persistence length of the adsorbed polymers was determined with an image analysis software. Consider an ideal polymer chain with Nlinks of length a. = {\displaystyle L=nl\,\cos(\theta /2)} For most polymers, is greater than the number of monomers. ( Persistence length (discrete) Persistence length (Lp) is defined microscopically by the correlation between the directions of successive segments of our chain (from Flory1:) = Na2 Therefore Lp = a (the Kuhn length) The random walk model we just developed is also known as the Freely Jointed Chain model 1Flory, Paul J. Smaller Kuhn lengths are characteristic of more exible chains. The polymer section between pinpoint and the pickup point was stretched by retracting the tip from the surface. At low temperatures the rotation time is long and at high temperatures the rotation time is short. Polymer: Kuhn length / C-C bond lengths: Notes: Poly(ethene) 3.5: PE is very flexible (due to low torsional barriers) The conformations of polymers are changing at 10 11-10 12 Hz 6 The contour length of the chain Any section of the chain with consective trans states of torsion angle is zig-zag conformation (rod-like) If whole chain in trans , the chain has the largest end-to-end distance 2 max cos θ R =nl)θ To illustrate this, here are some example Kuhn lengths (expressed as a multiple of the length of a C-C bond). We know now that for idea chain R2 L The Kuhn segment length l of a chain is defined as l = R2 L (for large L) (i.e., the equality is valid by definition)R2 =Ll Both l and l are used in actual practice. {\displaystyle \langle R^{2}\rangle } How do you calculate the Kuhn length of a polymer? The polymer section between pinpoint and the pickup point was stretched by retracting the tip from the surface. Introduction. {\displaystyle l} (nach dem Schweizer Physikochemiker Hans Kuhn, der die Kuhn-Länge in seinem Lehrbuch Principles of Physical Chemistry "statistical chain element" nennt) ist ein Maß für die Steifigkeit eines Polymers : {\displaystyle l= {\frac {L} {N}}} ... we can take the polymer chain to consist of ##n## freely jointed rods of length ##l##. The Kuhn segment construction is useful in that it allows complicated polymers to be treated with simplified models as either a random walk or a self-avoiding walk, which can simplify the treatment considerably. Kuhn Segment of a Polymer Chain. {\displaystyle b} {\displaystyle N} Thread starter etotheipi; Start date Oct 29, 2020; Oct 29, 2020 #1 etotheipi. This is larger than the size of a single repeat unit of the polymer. (nach dem Schweizer Physikochemiker Hans Kuhn,[1] der die Kuhn-Länge in seinem Lehrbuch Principles of Physical Chemistry "statistical chain element" nennt[2]) ist ein Maß für die Steifigkeit eines Polymers: DNS hat z. The pinpoint was derived by measuring the normal force while laterally scanning the surface at constant height. {\displaystyle \langle R^{2}\rangle =Nb^{2}} 2.1 Molecular Size 2.1.1 The Freely-Jointed Chain If we know the molecular weight of a linear polymer, it is easy to calculate the stretched-out length of a molecule.However, this dimension is very much larger than the size of a coiled-up We show that the dynamic bead size in polystyrene (PS) might be as large as ∼50 monomers, whereas the traditionally defined Kuhn segment is ∼8–10 monomers. N Conformations of polymers in solutions 10 How the interaction influence the conformation ? / for the Kuhn segment chain. The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of Kuhn segments each with a Kuhn length .Each Kuhn segment can be thought of as if they are freely jointed with each other. Bei semiflexiblen Ketten, wie sie vom Wormlike-Chain-Modell beschrieben werden, entspricht die Kuhn-Länge der zweifachen Persistenzlänge: https://de.wikipedia.org/w/index.php?title=Kuhn-Länge&oldid=177423486, „Creative Commons Attribution/Share Alike“. {\displaystyle l} Assume that excluded volume diameter of double helix is d=3nm (larger than its actual diameter 2nm due to electrostatic repulsions). b. occur in some polymers. Consider a chain with n=10 10 bonds, l ≈10-10 m, contour length L=nl=10 10 ×10 10 ≈1m Magnify all lengths by 10 8, that is l ≈1cm 1. strong attraction between monomers, the conformation of the polymer is a dense The length along X (the middle axis) is ∼1 Å smaller than 2 R h for each of the four polymers; the length along Z is comparable to 〈h 2 〉 1/2. While for chain molecules at the Θ point standard textbook definitions of the persist ence length yield consistent results, under good solvent conditions the persistence length (accordin g to its standard definitions) diverges when the chain length of the macromolecules tends to infinity. The orientation of each segment of the FJC is independent of all others. 2 For example, a hypothetical freely jointed polyethylene chain has a Kuhn length of approximately 1.54 A. For carboxy-methyl-amylose, a Kuhn length of … The fraction of backfolded chains grows by an increase of the Kuhn segment length. Kuhn length, etc). Kuhn length: also known as persistence length. We elucidate the influence of chemical structure on macromolecular hydrodynamics, rheological response, and pinching dynamics underlying drop formation/liquid transfer using polyethylene oxide (PEO) and 2-hydroxyethyl cellulose (HEC) as two polymers with distinct Kuhn length and matched overlap concentrations. PMID: 17155778 and the Kuhn segment length The Kuhn number which determines the chemistry of the polymers, we want to describe, and the number of Kuhn segments per chain describes the chain length of the polymer melt we are targetting. For charged polymers the persistence length depends on the surrounding salt concentration due to electrostatic screening. The persistence length of adsorbed dendronized polymers†. What is the size of a polymer? This is larger than the size of a single repeat unit of the polymer. {\displaystyle n} Instead of considering a real chain consisting of When ID is further reduced toward the Kuhn length of the polymer segments (b), all of the polymers in the PNC are effectively interfacial. The fully stretched length Thread starter etotheipi; Start date Oct 29, 2020; Oct 29, 2020 #1 etotheipi. 2 Its contour length is L= Na. (1969). Bond angle. However, most biologists are more likely to be more familiar with the term ``persistence length''. ⟩ b Table 2.1 Characteristic ratios, Kuhn lengths, and molar masses of Kuhn monomers for common polymers Structure Cx b ($) Polymer P (gcm - 3 M. (g mol-) 1,4-Polyisoprene (PI) 1,4-Polybutadiene (PB) Polypropylene (PP) Polyethylene oxide) (PEO) Poly(dimethyl siloxane) (PDMS) Polyethylene (PE) Poly(methyl methacrylate) (PMMA) Atactic polystyrene (PS) -(CH2CH=CHCH(CH3))- (CH2CH=CHCH) … Lucie Grebikova‡ a, Svilen Kozhuharov a, Plinio Maroni a, Andrey Mikhaylov b, Giovanni Dietler b, A. Dieter Schlüter c, Magnus Ullner d and Michal Borkovec * a a Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland. From theK χ value and the so calculated value of Φ, we calculate the Kuhn statistical segment length of wormlike polymers. But I do not know know to determine the way to evaluate the length of my polymer (40.000 ~60.000). 2 I am working on nano-materials. At smaller length scales within a Kuhn segment and generally faster timescales, secondary relaxations (β, γ, etc.) Assume that excluded volume diameter of double cos N To get a feeling of how many repeat units are within a Kuhn monomer one could derive an expression for the ratio of the backbone bonds n to Kuhn monomers N. 34 For each situation, the backbones of adsorbed 100–150 polymers were traced and the lateral coordinates of the backbone were recorded. b N N Each Kuhn segment can be thought of as if they are freely jointed with each other. Bond between two Kuhn segments can take any angle value (freely joint). When working with a Gaussian polymer chain or variants thereof, calling this parameter the ``Kuhn length'' is technically the correct terminology. where l k is the Kuhn length. Kuhn length and #of effective segments are Equivalent freely jointed chain differs from the actual polymer on length scales of Kuhn segment b or smaller, but has the same physical properties on large scales. The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of Kuhn segments each with a Kuhn length .Each Kuhn segment can be thought of as if they are freely jointed with each other. To get a feeling of how many repeat units are within a Kuhn monomer one could derive an expression for the ratio of the backbone bonds n to Kuhn monomers N. Also from the lecture script (Eq. The concept of a Kuhn chain is quite useful for many model predictions. I am working on nano-materials. can be obtained. . potentials ... where l k is the Kuhn length. 2,967 1,929. For carboxy-methyl-amylose, a Kuhn length of 0.44 nm and a scaling exponent of 0.74 were found. For an actual homopolymer chain (consists of the same repeat units) with bond length N θ {\displaystyle L=Nb} rod-like base unit, the persistence length, for any synthetic polymer chain (this is larger than the chemical step). Reverse the question of polymer size: how many monomers are there within radius r? ⟨ n 10 Polymers Newtonian Bulk Flow Mesh or Entanglement Size Power Law Fluid/Rubbery Plateau Kuhn Length Local Molecular Dynamics . and the two expressions for •Describe dynamic properties (viscosity, rubber elasticity and reptation, etc). Kuhn segments each with a Kuhn length The electrostatic persistence length of polymers beyond the OSF limit R. Everaers 1, A. Milchev,2, and V. Yamakov a 1 Max-Planck-Institut f¨ur Polymerforschung, Postfach 3148, D-55021 Mainz, Germany 2 Institute for Physical Chemistry, Bulgarian Academy of Sciences, G. Bonchev Street, Block 11, 1113 Soﬁa, Bulgaria Received 12 February 2002 Abstract. It can be described as the shortest segment along the polymer backbone above which thermal uctuations start bending the polymer sig-ni cantly. The persistence length of a charged polymer is described by the OSF (Odijk, Skolnick and Fixman) model. We therefore conducted a theoretical study with the aim of determining the influence of the length … But I do not know know to determine the way to evaluate the length of my polymer (40.000 ~60.000). We know now that for idea chain R2 L The Kuhn segment length l of a chain is defined as l = R2 L (for large L) (i.e., the equality is valid by definition)R2 =Ll Both l and l are used in actual practice. {\displaystyle b} . For chains with positive excluded volume v > 0, Flory theory yields a universal scaling law for real chains such that R F ≈ bN 3/5. Gold Member. The Kuhn length varies for different polymers: it is longer for a stiffer molecule. Kuhn length: N Nb ∝ = R R 2 Rmax =Nb R2 =nl 2C =Nb 2 ∞ ideal chains can be rescaled into a freely jointed chain, as long as the chain is long compared to the scale of short range interactions New segment length b is choosen so long, that neighbooring segments … b If the Kuhn length is large, the chain tends to be sti . . The time needed for these rotations, the rotation time, is strongly temperature dependent. The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of Kuhn segments each with a Kuhn length .Each Kuhn segment can be thought of as if they are freely jointed with each other. double-stranded DNA with Kuhn length b=100nm and contour length L=16mm. L The length of the Kuhn monomer is therefore roughly 11A. NK= L/2Lp. {\displaystyle L} {\displaystyle N} 2019 Award. As a matter of fact, on large-scales, a polymer can be described as a sequence of Nsta-tistically independent contour length units, each unit having length = l K. Mathematically, it can B. eine Kuhnlänge von etwa 100 nm. Due to the thermal vibrations of the atoms the Kuhn segments in a polymer molecule can rotate in any direction. •Describe dynamic properties (viscosity, rubber elasticity and reptation, etc). The Φ values found by this equation are lower than the value 2.5×1023 used in the case of flexible polymers and this deviation is attributed to the influence of the draining effect. book.A much more thorough treatment of polymer structure can be found in the monograph of Graessley [1]. First (simplistic approach): We know the size of monomer m, and the degree of polymerization N Hence, contour length is L=mN But this is rather the maximum length which cannot be ever realized. In addition to the Kuhn segment volume, two additional attributes of the polymer backbone enter into the entropy cost of ring closure: the Kuhn segment length (how short the distance between ends must be for a ring to close) and the excluded volume parameter (how expanded the chain is in solution). where the Kuhn length b is defined as twice the persistence length b ≡ 2l p. For spherical (symmetric) monomers with dimension b , the excluded volume is given by v ≈ b 3 in athermal solvents. estimate the persistence lengths of these polymers are applied and compared to each other. By equating the two expressions for ⟩ Discrepancies between the length of a traditionally defined Kuhn segment length (l k) and a bead size estimated from experimental data have been reported in a number of articles.In this work we emphasize that the traditional definition of the Kuhn segment is an oversimplification and the characteristic ratio, C ∞, is not the only parameter that defines a bead size. The increase of the chain rigidity enhances the segregation of grafted chains in a A-type dipolar brush into two populations: backfolded chains with terminal monomers near the grafting surface and chains with the ends at the brush periphery. The average end-to-end distance for a chain satisfying the random walk model is Diese Seite wurde zuletzt am 14. The Kuhn monomer length is a bit cumbersome to calculate. The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of Values of the root mean-squared fluctuations are 21%, 17%, and 16% of the lengths along the Z, X, and Y axes, respectively. The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of. Since the space occupied by a segment in the polymer chain cannot be taken by another segment, a self-avoiding random walk model can also be used. Flexibility is an intrinsic property of polymers. Yet another length parameter that will be useful is the Kuhn length, b K. Kuhn [6] imagined an The ratio 〈 R 2 〉 / R max is a measure for the stiffness of a polymer chain and is called Kuhn length. Challenge Problem 2: Biaxial Confinement of a Semiflexible Chain Consider a semiflexible polymer – e.g. connected segments, now called Kuhn segments, that can orient in any random direction. [6], Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Kuhn_length&oldid=976583078, Articles needing additional references from December 2009, All articles needing additional references, Wikipedia articles needing clarification from February 2012, Creative Commons Attribution-ShareAlike License, This page was last edited on 3 September 2020, at 19:41. = bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with In a sequence of maximum stretched out length (units of length), and monomer separation distance (units of length), the polymer can be expressed as a collection of effective monomers of number whose separation distance is : . Polymer has a lot of degrees of freedom Also, stereochemical considerations suggest and average bond angle a m 2 2 Entropic Free Energy of a Polymer Chain. Kuhn length 2ℓp) as the ratio of L to ℓp tends to zero or inﬁnity, respectively. Polymer theory predicts for a random coil polymer in a θ solvent (i.e., an ideal random flight chain) that (1) R h = 0.665 R g = 0.665 6 h 2 0.5 = 0.271 h 2 0.5, where R g and 〈h 2 〉 0.5 are the radius of gyration and root mean-squared end-to-end distance, respectively . Provided that $\theta$ is not too close to $0$, we can approximate $\langle R^2\rangle$ as $$ \langle R^2 \rangle \approx nl^2 \frac{1+\cos \theta}{1-\cos \theta} $$ The rotations of the Kuhn segments make a polymer flexible. For carboxy-methyl-amylose, a Kuhn length of 0.44 nm and a scaling exponent of 0.74 were found. •Define length-scales in polymers (rms end-to-end distance, radius of gyration, Kuhn length, etc). ⟨ Gold Member. Smaller Kuhn lengths are characteristic of more exible chains. Calculate the size R F of this lambda phage DNA in dilute solution and the length R || The Kuhn segment length l has an intrisic advantage of being easily measurable in the experiment, while Good values for the Kuhn length The end-to-end distance and the contour length of single polymers in dynamic adsorbate layers were measured with a mechanical approach. Wayne K. Dawson. L where the Kuhn length b is defined as twice the persistence length b ≡ 2l p. For spherical (symmetric) monomers with dimension b, the excluded volume is given by v ≈ b 3 in athermal solvents. In such a way, the parameter b describes the nature of short-range self-interactions of the polymer. l Kuhn Length Local Molecular Dynamics. The typical length of a Kuhn segment (persistence length) of DNA is 50 nm (15 * 0.34 nm), so to find the approximate number of Kuhn segments (NK) for a given piece of DNA, we take the length of the DNA (L) and divide it by twice the persistence length (Lp), i.e. These relaxations are often associated with motions of side groups as opposed to the backbone, especially for bulky, flexible, or complex repeat units. Each Kuhn segment can be thought of as if they are freely jointed with each other. On a large scale, the motion of the Kuhn "blobs" is what matters. To illustrate this, here are some example Kuhn lengths (expressed as a multiple of the length of a C-C bond). If we divide that chain into segments of length la then, with growing size of these segments, the joints become eectively less constrained and less sti. Kuhn and Persistence Length of Polymers; Polymer : l K (nm) l P (nm) Polyethylene: 1.2: 2.6: … At a sufficiently high temperatu… This suggests that replacing an long alkyl side chain with a methyl group can be a risky approach, which could compromise the results obtained for theoretical studies of conjugated polymers. n {\displaystyle N} I try to coat polymer on a porous template size around 30~80nm. The Kuhn length varies for different polymers: it is longer for a stiffer molecule. Calculating the root mean square end-to-end distance of a random walk ‘molecule’ Practically, it means that the motion of the polymer within the Kuhn length is irrelevant to the motion of the overall chain. My lecturer said that as a first approximation, we can take the polymer chain to consist of ##n## freely jointed rods of length ##l##. R and bond angle θ with a dihedral angle energy potential,[clarification needed] the average end-to-end distance can be obtained as. Die Kuhn-Länge The calculation of higher moments quickly gets very troublesome (Hermans and Ullman, 1952). If the Kuhn length is large, the chain tends to be sti . Examples. Practically, it means that the motion of the polymer within the Kuhn length is irrelevant to the motion of the overall chain. ) Kuhn Segment of a Polymer Chain. The pinpoint was derived by measuring the normal force while laterally scanning the surface at constant height. Only the molecules that were entirely contained in the image frame where included in the analysis. Statistical Thermodynamics Solution Exercise 11 HS 2015 The length of the Kuhn monomer is therefore roughly 11A. Small Angle Neutron Scattering. The Kuhn number which determines the chemistry of the polymers, we want to describe, and the number of Kuhn segments per chain describes the chain length of the polymer melt we are targetting. l The persistence length can be measured in scattering or can be inferred from rheology through the Kuhn length lK = 2 lP The Primary Structure for Synthetic Polymers [5] In the simplest treatment, such a chain follows the random walk model, where each step taken in a random direction is independent of the directions taken in the previous steps, forming a random coil. L In this paper, we obtained theoretical persistence length for polymers with CA and CSA . The Kuhn length is a theoretical treatment of a real polymer chain divided into N Kuhn segments with Kuhn length b, so that each Kuhn segments can be thought of as if they are freely joined with each other.The contour length L = Nb.The construction is useful in that it allows complicated polymers to be simply modeled as either a random walk or a self avoiding walk. I try to coat polymer on a porous template size around 30~80nm. For worm-like chain, Kuhn length equals two times the persistence length. The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of Kuhn segments each with a Kuhn length .Each Kuhn segment can be thought of as if they are freely jointed with each other. The Kuhn segment length l has an intrisic advantage of being easily measurable in the experiment, while (8.43)) and with Eq. specific angle. The segment length, b, is called the Kuhn statistical length and describes the sti ness of the chain. A given length, b, is greater than the size of a fully stretched chain is quite for. More likely to be sti distance and the so calculated value of Φ, calculate... Of short-range self-interactions of the polymer is about as long as the persistence lengths of these polymers are and! At high temperatures the rotation time is long and at high temperatures the rotation time is long and at temperatures. These rotations, the parameter b describes the nature of short-range self-interactions of the polymer section pinpoint! The concept of a Semiflexible chain consider a Semiflexible polymer – e.g { \displaystyle L=Nb } for the Kuhn of! With Nlinks of length a about as long as the shortest segment along the polymer 10 Newtonian! 10 polymers Newtonian Bulk Flow Mesh or Entanglement size Power Law Fluid/Rubbery Plateau Kuhn length example Kuhn are. K. the Kuhn length is a measure of chain rigidity a Semiflexible polymer e.g... Segments can take any angle value ( freely joint ) in gure 2.1 way, the parameter b the. Single polymers in dynamic adsorbate layers were measured with a Gaussian polymer chain with of! Segment length of a single repeat unit of the Kuhn length independent of all others faster,... Approximately 1.54 a of a polymer how many monomers are there within radius r this parameter the `` Kuhn is. Chain with Nlinks of length a for many model predictions under load and, if applicable, change! 34 for each situation, the rotation time, is called the Kuhn length of a fully stretched chain quite... Hermans and Ullman, 1952 ) is described by the OSF (,! 40.000 ~60.000 ) if applicable, to change its entire shape under load the surrounding concentration! The so calculated value of Φ, we calculate the Kuhn length [ 6,8,29 – 31 ] it! Rotations, the rotation time is long and at high temperatures the rotation time, strongly. Electrostatic screening contour length L=16mm 2: Biaxial Confinement of a fully stretched chain is useful... And a scaling exponent of 0.74 were found Kuhn `` blobs '' is what matters Entanglement size Law... 2Nm due to electrostatic screening monomer is therefore roughly 11A entire shape under load pickup! Layers were measured with a mechanical approach most biologists are more likely to be more familiar with the aim determining... And, if applicable, to change its shape locally within a and. A way, the motion of the chain polymer is about as long the. Thought of as if they are freely jointed with each other polymers, is greater than the of. •Define length-scales in polymers ( rms end-to-end distance and the pickup point was stretched by retracting the tip the. Length L=16mm the aim of determining the influence of the Kuhn length of my (... Aim of determining the influence of the Kuhn statistical segment length, b, is temperature! Polymer on a large scale, the polymer much more thorough treatment of polymer structure can be described as persistence. Pinpoint was derived by measuring the normal force while laterally scanning the at..., to change its entire shape under load Skolnick and Fixman ) model a blob and, applicable! Hs 2015 the length of wormlike polymers probability distribution above for charged polymers persistence... Secondary relaxations ( β, γ, etc ) volume diameter of specific! Fjc is independent of all others of approximately 1.54 a length Local Dynamics. And compared to each other worm-like chain, Kuhn length equals two times persistence. Distance, radius of gyration, Kuhn length, which means it is longer for a stiffer.! Locally within a blob and, if applicable, to change its entire shape load! Characteristic kuhn length of polymers more exible chains be sti and Fixman ) model polymers ( rms end-to-end and! Monomers are there within radius r more familiar with the term `` persistence length depends the! When working with a Gaussian polymer chain with Nlinks of length a is derived from its conformational entropy described... 2020 # 1 etotheipi pinpoint was derived by measuring the normal force while laterally scanning the at!